=== === ============= ==== === === == == == == == ==== == == = == ==== === == == == == == == == = == == == == == == == == == ==== M U S I C T H E O R Y O N L I N E A Publication of the Society for Music Theory Copyright (c) 1998 Society for Music Theory +-------------------------------------------------------------+ | Volume 4, Number 6 November, 1998 ISSN: 1067-3040 | +-------------------------------------------------------------+ General Editor Lee Rothfarb Co-Editors Henry Klumpenhouwer Justin London Catherine Nolan Reviews Editor Robert Gjerdingen Manager Ichiro Fujinaga mto-talk Manager Jay Rahn Consulting Editors Bo Alphonce Richard Littlefield Jonathan Bernard Thomas Mathiesen John Clough Benito Rivera Nicholas Cook John Rothgeb Allen Forte Arvid Vollsnes Stephen Hinton Robert Wason Marianne Kielian-Gilbert Gary Wittlich MTO Correspondents Per F. Broman, Sweden Nicolas Meeus, Belgium, France Peter Castine, Germany Ken-ichi Sakakibara, Japan Wai-ling Cheong, Hong Kong Roberto Saltini, Brazil Geoffrey Chew, England Michiel Schuijer, Holland Gerold W. Gruber, Austria Uwe Seifert, Germany Henry Klumpenhouwer, Canada Arvid Vollsnes, Norway Marco Renoldi, Italy Tess James, England Editorial Assistants Martin Steffen Cindy Nicholson Nicholas Blanchard Jon Koriagin Music Example Designer William Loewe Midi Consultant David Patrick Watts HTML and Java Consultant Bruce Petherick All queries to: mto-editor@smt.ucsb.edu or to mto-manager@smt.ucsb.edu +=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+ 1. Target Article AUTHOR: Jay Rahn TITLE: Practical Aspects of Marchetto's Tuning KEYWORDS: semitone, monochord, geometry, perception, beats, counterpoint, chant, pedagogy, scale Jay Rahn York University Fine Arts Dept., Atkinson College 4700 Keele St. Toronto, Ontario M3J 1P3 jayrahn@yorku.ca ABSTRACT: In his historically momentous account of tuning, Marchetto of Padua (fl. 1305-19) proposed dividing the whole tone of received theory (9/8=204 cents) into 5 parts. This report develops determinate arithmetic and geometric realizations of Marchetto's formulation--directly applicable to the medieval monochord, and sonically illustrated by digital files. The resulting intervals' feasibility is compared with current findings in the psychology of interval perception. Conjectures are offered as to how and to whom Marchetto's tuning was taught. A dual formulation of "nuanced," heterogenous tonal systems is advanced to assess structural effects of Marchetto's tuning on pieces and to suggest ways one could learn to perceive and sing his intervals nowadays. ACCOMPANYING FILES: GIF files: mto.98.4.6.rahn1.gif mto.98.4.6.rahn2.gif mto.98.4.6.rahn3.gif mto.98.4.6.rahn4.gif MIDI files: mto.98.4.6.1305.mid mto.98.4.6.1330-38.mid mto.98.4.6.a2.mid mto.98.4.6.end.mid RA files (all with .ra filename extensions): 0pc 10pc 100pc 15pc 180-185 20pc 25pc 291 292 294 30pc 316 35pc 386 40pc 402 404 408 45pc 5pc 50pc 55pc 60pc 65pc 70pc 75pc 79pc 80pc 85pc 88pc 90pc 95pc a a1 aa ac74 af74 b b1 b3711 b761 b771 bb1 bb3961 bb77 bb771 bd c c1 c74 c74d CCf74 CCg d d41 d74 d741 dd DDa DDc74 DDd DDda1 DDf74 e e1 EEb EEc74 EEcg# EEe f f74 FF74a FFc GG74b GGbb77 GGd GGg g 0. INTRODUCTION [0.0] Marchetto of Padua (fl. 1305-19) was arguably the most important European music theorist between Guido of Arezzo (fl. ~991/2-1033+) and Gioseffo Zarlino (1517- 90). A leading experimental composer of his era, Marchetto taught at Padua's Cathedral, a job that would involve training choirboys, leading the chorus in liturgical chant, performing his own part in written and/or improvised discant (counterpoint), and both performing and rehearsing/directing others in cutting- edge music of his own creation (Gallo 1974: 42-43; Vecchi 1954: 166-68). [0.1] Under aristocratic patronage, Marchetto published pathbreaking contributions to all the main areas of music theory, then or now: in his Lucidarium (1317-18), musical philosophy, semiotics, numerology, and applied mathematics, as well as more narrowly technical matters of tuning, discant, pitch notation, melodic analysis, and modality; in its sequel, the Pomerium (1318-19), rhythmic theory and notation (Herlinger 1985: 3-21; Gallo 1977/1985: 113-16). [0.2] These innovations influenced pre-eminent theorists and musicians of Europe for almost 300 years (cf., e.g., Niemoeller 1956; Herlinger 1981a, 1981b, 1990). Nonetheless, the novelty for which Marchetto was, and remains, best known--namely, his proposal that whole tones be divided into 5 parts--has been interpreted somewhat indeterminately. E.g., Jan Herlinger's important account concludes (1985: 17): Just as Marchetto's enharmonic and diatonic semitones must be approximations, so must be his diesis and chromatic semitone. [The latter] differ from each other in size to a greater degree than [Marchetto's] enharmonic and diatonic semitones, but *just how much* we cannot say [my emphasis]. [0.3] The present report advances determinate interpretations of Marchetto's often difficult account of tuning. Briefly, Marchetto's account favours 2 main readings of his whole-tone division: into 9 or 5 parts. From |Luc. 2.5.15| onward, Marchetto writes uniformly of 5 dieses per whole tone.(1) However, to regard the whole tone as divided into 5 parts *tout court* is to discount Marchetto's extended preliminary account of how "the nature of the whole tone, its essence, would consist in the 9-fold number compared to the 8-fold number [i.e., the ratio 9/8]" (quod natura toni et essentia eius consistat in novenario numero ad octo- narium comparato: |Luc. 2.4.1-42|) and "in the per- fection of the 9-fold number" (in perfectione numeri novenarii: |Luc. 2.5.8|), as well as his theses that: a) "the 9-fold number [i.e., as such, in contrast to, e.g., 9*2=18 or 3*3=9] can never be divided into equal parts" (novenarius numerus numquam potest dividi in partes equales: |Luc. 2.5.9|); b) "its parts must be unequal" (partes ipsius debeant esse inequales: |Luc. 2.5.13|); c) "1 would be its 1st part; from 1 to 3, its 2d [part]; from 3 to 5, its 3d [part]; from 5 to 7, its 4th [part]; from 7 to 9, its 5th [part]; and such a 5th part is the 5th odd number of the 9-fold totality" (unus sit prima pars; de uno ad tres, secunda; de tribus ad quinque, tertia; de quinque ad septem, quarta; de septem ad novem, quinta; et talis quinta pars est quintus numerus impar totius novenarii: |Luc.2.5.14|)--in other words, that Marchetto construes the 5 unequal parts of the whole tone as comprising 1+(4*2) 9th-parts, a point clarified by his having emphasized that the 9-fold number can never be divided into equal parts, because: d) "a unit is in it that resists being divided" (est ... ibi unitas que resistit dividi: |Luc. 2.5.10|), an idea Herlinger astutely connects with Marchetto's later, more complete statement of Remigius s doctrine that: e) "an even number is mutable and divisible [i.e., into equal segments], whereas an odd number is indivisible, containing a unit *in its middle* that resists division" (numerus par mutabilis et divisibilis est; numerus vero impar indivisibilis est continens unitatem *in medio sui* que divisioni resistit: |Luc. 6.3.14|: [my emphasis]). ======================================================= 1. References to the Lucidarium follow Herlinger's numbering of its tractata, chapters, and clauses (1985). The | sign indicates links to the edition at the TML website |http://www.music.indiana.edu/tml/14th|. [EDITORIAL NOTE: The URL leads to fourteenth-century texts at the TML site., which has both the Gerbert version (MARLU) and the Herlinger edition (MARLUC). The *Ludicarium* tractata are in separate HTML files, in the form MARLUC#_TEXT.html, where # is the number of an individual tractate in the treatise (e.g. MARLUC8_TEXT.html).] ======================================================= [0.4] Although a 5-fold division partitioned into 2+1+2 would satisfy this last point as well as a 9-fold division partitioned into (2*2)+1+(2*2), it would miss the plausibility that Marchetto regarded as an important aspect of his original formulation of the 9/8 ratio, "not yet discovered demonstrated by writers [on music theory]" (nondum invenitur ... ab auctoribus demonstratum: |Luc.2.4.3|), the following idea: each of the 9 parts of the 9th part of any whole tone division is the same size as each of the immediately preceding 8 parts of the 8th part (cf. 81/72 and 72/64, and [0.5], [2.3-5], below). Like other aspects of his tuning, this is something "[he could] display to perception on sonorous bodies, i.e., the monochord etc." (cf.: ostendimus ad sensum in corporibus sonoribus, puta in monocordo et aliis: |Luc. 2.5.8|). It jibes with the earlier medieval privileging of supernumerary ratios, (x+1)/x, of which the smaller number, x, is a power of 2, i.e., 2^n, as in 2/1=(1+(2^0))/(2^0) (trivially), or more importantly, 3/2=(1+(2^1))/(2^1). [0.5] Such an approach facilitated geometric construction of ratios by reducing all steps to bisection or addition/transposition of parts resulting from bisection and could be confirmed on the monochord by eye and ear. For Marchetto's novel division, all 9 parts of a whole-tone ratio: 9/8(=((1+8)/8)=((1+(2^3))/(2^3))) would be evident in this way, as would all 9 parts within a whole tone: 81/72(=((9*(1+8))/((8*(8))=((9*(1+(2^3))/((8*(2^3)))). That each of the 9 (=81-72) parts in the 9th part of a 9/8 ratio would be of the same length as each of the preceding 8 (=72-64) parts of the preceding, 8th part, which also is a whole tone, since 72/64=9/8, would necessarily result from the general inequality: (x^2)-x(x-1)=x(x-1)-((x-1)^2)+1 (e.g., (2^2)-2(1)=2(1)-(1^2)+1 or 4-2=2-1+1 or 2=2; (3^2)-3(2)=3(2)-(2^2)+1 or 9-6=6-4+1) or 3=3). [0.6] The 9 spaces marked off within a whole tone's space would be partitioned into 5 spaces, each of which would be a diesis, the middlemost of which (77/76) would comprise the "unit that cannot be divided": 81 80 79 78 77 76 75 74 73 72 81 79 77 76 74 72 <-diesis-><-diesis-><-diesis-><-diesis-> Arithmetically, this would involve dividing the frequency-ratio for the whole tone (e.g., C/D), which for almost 2000 years had been formulated as 9/8: C D 9 8 into 9 parts (cf. Gurlitt and Eggebrecht 1967: v.3, 225): C D 9 8 7 6 5 4 3 2 1 0 so that the entire whole tone could be understood as 81/72: C D 81 80 79 78 77 76 75 74 73 72 Within this newly construed whole tone, Marchetto's most important diesis (e.g., C74/D) would be 74/72 (=37/36), whereas its complement within the whole tone (e.g., C/C74), would be Marchetto's "chromatic semitone," 81/74:(2) |c| |c74| |d| 81 74 72 <--------chromatic semitone--------><--diesis--> 37 36 ======================================================= 2. Below, a tone designated by letter-name within | signs, e.g., |c|, can be heard, in the html version, by clicking on the letter-name, or getting the corresponding RealAudio (ra) file, e.g., http://mto/audio/4.6/c.ra. For sound files, bass-clef staff letter-names are capitalized and tripled (GGG for 1st-line), or doubled (AA, BB, ... GG for 1st space to top space). Upper registers are designated by single letters (from a, just below middle c to g--with a numeral 1 for an 8ve higher: a1, b1, ...). By contrast, single capitals (A, B, ...G) designate any member of a pitch class. Flats are designated by b (e.g., Bb for B-flat; Marchetto's high sharps, by their characteristic numbers (e.g., C74); non-Marchettan sharps, by # (e.g., C#). In the sonic exx., c is middle c (256Hz); unless otherwise indicated, tuning is Pythagorean (e.g., g=(3/2)*256=384Hz). ======================================================= [0.7] Marchetto's other semitones (i.e., parts of a whole tone; incomplete, imperfect tones--not 1/2-tones: cf. |Luc.2.5.18|) would be, to use further terms he appropriated from Ancient Greek theory (via Boethius),(3) the "enharmonic" semitone, whose 2 dieses would comprise 81/77 (e.g., for A/Bb77) and its whole-tone complement, the 3-diesis, "diatonic semitone" (e.g., Bb77/B), whose ratio would be 77/72 (cf., however, below): |a| |bb77| |b| 81 77 72 <-enharmonic semitone-><---diatonic semitone---> ======================================================= 3. Oliver B. Ellsworth (1987: 340) gives a clear account of how Marchetto adapted earlier semitone terms by shifting each "up 1 notch." ======================================================= 1. OVERVIEW [1.0] Although Marchetto's numerical formulation is highly interesting in its own right, as is its place in the stylistic, intellectual, cultural, ethnic, and gendered ideological history of Western European music, I focus here on immediate practical aspects of his tuning. Most important, Marchetto's diesis was a very narrow interval, ~48%, i.e., ca. 48 cents, much closer to a tempered quartertone (50%) than to a tempered semitone (100%). With this in mind, the following questions arise: a) How could Marchetto geometrically draft marks on a monochord to realize his tuning? b) What physical constraints would a monochord place on conveying Marchetto's tuning: i) visually? ii) sonically? c) How readily could musicians identify Marchettan intervals? d) How might Marchetto have taught singers to produce reliably and fluently his new intervals? e) How well would compositions (and discant improvisation) using Marchettan intervals survive the almost inevitable misunderstandings and re-formulations of later centuries? f) What new conceptual/perceptual understanding would be involved in learning to hear and sing Marchetto's dieses? [1.1] Briefly, my answers are as follow (extended discussions are in paragraphs indicated below): a) Monochord marks for Marchetto's new intervals could be drawn using the same kinds of geometric constructions as had been needed to realize the proportions for earlier medieval (Pythagorean) tuning. Either of 2 propositions from Euclid would suffice--even to realize alternative Renaissance versions of Marchetto's tuning.[2;6] b.i) Visually, the marks for Marchetto's truly novel intervals (involving F74, C74, G74, D74) would be quite distinct from those of previous tunings. The intervals in his tuning that involved B or Bb were alternatives to, substitutes for, or re- "ratio"-nalizations of, intervals in previous medieval tuning. These novelties would be hard to distinguish by eye from those they would have replaced, especially in upper registers.[3.0-8] b.ii) Sonically, the medieval monochord would be much more accurate than its post-1500 successor. All the same, any sources of measurement error would help persuade listeners of Marchetto's time that differences between his enharmonic/diatonic semitones and their earlier, Pythagorean versions were negligible.[3.9-10] c) Melodically or in discant, musicians would have little difficulty distinguishing aurally intervals produced by Marchetto's novel F74, C74, G74, and D74 from intervals of previous medieval practice. Conversely, Marchetto's substitutes for standard Pythagorean notes (B, Bb) generally would pass unnoticed.[4] d) Marchetto's musical examples would serve not only as an excellent lab demonstration for theorists but also as a superb, step-by-step curriculum for novice singers. His schematic fragments easily could be memorized as exercises and used as a basis for group- or self-instruction.[5] e) Even if simplistically realized (e.g., as "1/5- tones"), characteristically Marchettan pitch- structures would survive--as they would if the most important mathematical flaw in Marchetto's formulation were removed in a straightforward manner [3.6-7;6] f) Even if Marchetto's intervals were performed in Pythagorean, equally-tempered, etc. versions, Marchettan structures would persist--albeit to varying degrees--in pieces closest in provenance to his original formulation. Each such version can be understood in its own right--or in fully "de- centred" fashion, as a variant of the others. In principle, fluency in each could be acquired by refining or "de-refining" skills learned for any of the others.[7] 2. GEOMETRIC CONSTRUCTIONS FOR THE MONOCHORD [2.0] Geometric constructions required for a fastidious, "ideal" realization of Marchetto's tuning on the monochord had been known for about 2 millenia. E.g., Euclid VI,9 (Heath 1926/1956: v.2, 211-12) gave a formulation for dividing any line segment into any number of subsegments having equal lengths. This powerful construction would more than suffice for both Marchetto's dieses and the previously standard medieval tuning. However, both this earlier, Pythagorean tuning and Marchetto's innovative dieses could be constructed entirely by applying Euclid's well known construction (I,10) for bisecting any given line segment (Heath 1926/1956: v.1, 267-68: cf. Adkins 1980). [2.1] Because the location of the mark for GGG (gamma- ut), the monochord's lowest note, was largely arbitrary (cf., however, [3.0-3.2], below), GGG's sounding length could be established indirectly at the outset by setting BB (a M3 above GGG) at 3 times any feasible length, x, where x=~1/4 the length available: (bridge) BB (bridge) <----x----><---------------3x---------------> Merely by cutting off three consecutive segments of length x, BB's effective, sounding string-length (i.e., its distance from the rightmost bridge), would be 3x: (bridge) BB (bridge) <----x----><----x----><----x----><----x----> Bisecting BB's length once, b, a p8 above BB, would be at (1/2)*3x=(3/2)x: (bridge) BB b (bridge) <--(2/2)x-><----(3/2)x----><----(3/2)x----> <-----------(5/2)x--------><----(3/2)x----> Bisecting BB's length a second time, b1, a p15 above BB, would be at (1/4)*(3x)= (1/2)*(3/2)x=(3/4)x, whereas EE, a p4 above BB (or a p5 below b, or a p12 below b1) would be at (3/4)*3x = (3/2)*(3/2)x=(3/1)*(3/4)x=(9/4)x: (bridge) BB EE b b1 (bridge) <--(4/4)x-><(3/4)x><(3/4)x><(3/4)x><(3/4)x> <----(7/4)x-------><--------(9/4)x--------> And so forth, downward through the cycle of p5s, for A, D, G, C, F, and Bb. [2.2] To add Marchetto's new, sharpened notes (e.g., C74), one need only bisect the whole tone above (D/E) twice, and cut off 1 of these 1/4s *below* the lower note (D): |c| |c74| |d| |e| 9 8 36 32 36 35 34 33 32 37 36 32 81 74 72 70 68 66 64 9 8 Because the whole tone (D/E) would form the ratio 9/8=72/64, its 1/4s would be formed by marks for 70, 68, 66; the 1/4 below the lower note (D=72) would be at C74--and C, a whole tone below D, would be at 81. [2.3] To construct *all* Marchetto's sharpened notes (F74, C74, G74, D74), one could begin the original tuning at f# ~1/2-way along the available string, rather than at BB (~1/4 from the bottom) and construct all other marks relative to this f#. BB's length would be 3/2 times f#'s; d74's would (37/36)*(9/8) greater; etc.: |d| |d74| |e| (f#) 9 8 36 32 36 35 34 33 32 37 36 32 81 74 72 70 68 66 64 9 8 The Pythagorean F# (parenthesized) would be much (~42%) lower than Marchetto's new F74: 81 74 72 |f| (f#) |f74| |g| <--42--><--48--> <------90------> [2.4] To divide a whole tone (e.g., A/B) into Marchetto's enharmonic semitone (A/Bb77) and diatonic semitone (Bb77/B), one would only have to bisect a whole tone above (B/C#, where C# would be a Pythagorean note not actually used by Marchetto) and cut off one of these 1/2s above the enharmonic semitone's lower note (A): |a| |bb77| |b| (c#) 9 (8) 18 17 (16) 72 68 (64) 81 77 72 Because the whole tone above (B/C#) would be 9/8=72/64, its precise 1/2 (i.e., arithmetic mean) would be at 68. Since 72-68=4, the Bb77 could be marked readily at 77=81-4. 3. FEASIBILITY OF DRAFTING MARCHETTO'S 9-FOLD DIVISION [3.0] In Marchetto's period, fundamental frequencies corresponding to notated pitches were not standardized as they are for today's concert musicians (for whom the fundamental frequency of a1 above middle c is ~440-446 Hz). Nonetheless, there were constraints, as always, on vocal ranges of the adults and children who might sing such notes as Marchetto prescribed. [3.1] Expressed in modern notation, Marchetto's gamut ranged from GGG (gamma-ut) on the lowest line of the bass-clef staff to e1 in the top space of the treble- clef staff. In the late 19th century, Alexander Ellis gave absolute values between 403.9 and 425.2 Hz for various versions of a1 above middle c produced by tuning forks and pitch pipes used in Padua to tune bells and other fixed-frequency instruments (1885/1954: 510). Although his measurements are based on local musical practices during the period 1730-80, i.e., 4 centuries after Marchetto's time, there is little to suggest pitch standards in the Middle Ages diverged much more from modern norms than Ellis's measurements suggest--especially in leading churches, where such fixed-frequency instruments as organs might be played with other instruments or voices, and vocal music called for increasingly large ensemble ranges (cf., e.g., Mendel 1948/1968, esp. 167). [3.2] Additionally, manuscript illustrations, though unreliable for certain details, indicate that monochords of the time were ~3-4 ft. long (cf. the general estimate of 90-122 cm. in Adkins 1980:495). E.g., well-known medieval illustration shows a monochord held by |Guido| and Bishop Theobaldus, whose adult heights provide rough estimates of the instrument's absolute dimensions (as do the plates in Adkins (1992: v.2, 500-10) and the ~1150 drawing of |Boethius| (at the smt homepage). [3.3] Presuming, at least for the sake of illustration, a monochord whose sounding string was about a yard long, one can estimate quite closely the absolute distances between various marks for the notes it would produce. GGG gamma-ut, corresponding to the lowest line of the modern bass-clef staff could result from sounding a string-length of 36 inches. The highest notes in Marchetto's system that would produce his narrow diesis are e1 and d741 at the modern treble-clef staff's top. Relative to a GGG gamma-ut of 36", their string-lengths would be: for |e1|, 36*(1/2)*(1/2)*(2/3)*(8/9) =5.33", and for |d741|, 5.33*(74/72) =5.48". [3.4] Such marks would be about 1/6" apart, i.e., readily distinguishable from each other by instruments used at the time for geometric diagrams. Even if "concert pitch" for such church musicians were fully a p4 higher, one would still be dealing with smallest distances of about 1/8", as one would if a monochord's open string-length were only ~27". An 8ve below, this distance would be twice as great; a 15th below, 4 times as great, i.e., ~.5"--for this, the very smallest interval of Marchetto's formulation. [3.5] The notes b' and c' (in the middle of the modern treble-clef staff) were the highest for which Marchetto would use his revised, "enharmonic" version of the minor semitone. According to earlier medieval tuning, these notes would be marked off at the following points on a 36" monochord: for |c1|, 36*(1/2)*(1/2)*(4/3) =6.75", and for |b1|, 6.75*(256/243) =7.11". Relative to c1 at 6.75", Marchetto's version might be located at: |b771|, (81/77)*6.75" =7.101", or |b761|, (76/72)*6.75" =7.125", i.e., ~1/100" to ~1/70" from the Pythagorean mark, from which neither would be easily distinguished by eye (nor from the other). [3.6] As a precise calculation, Marchetto's tuning disregards the incommensuracy between 9-fold subdivisions of the spaces from A to B, Bb to C, and B to (Pythagorean) C#. Overlooked (or ignored) is the mathematical difficulty that 81/77 (in modern decimals, ~1.052) is a smaller ratio than 76/72 (~1.056), but each would be 2 dieses: |a1| |bb771| |b1| (c1) (c#1) 81 77 72 68? 64 81 76 72 68? (a1) |bb1| |b761| |c1| (c#1) Marchetto should have known that for (positive) numbers generally, a/b > (a+x)/(b+x), an important inequality he invoked in discussing the ratios 17/16 and 18/17 (|Luc.4.11.4|: cf. 72/68 and 68/64, here and [2.6], above; cf. also Euclid V.8 in Heath 1926/1956: v. 2, 149-53). [3.7] A plausible way out of this difficulty would be to divide the m3 a1/c1 space into 4+5+4=13 dieses: |a1| |bb3961| |b3711| |c1| 32 27 <--4*5---><------5*5------><--4*5---> 416 396 371 351 The 32/27 m3 ratio could be expanded to 416/351 (by multiplying both numbers by 13, a mathematical "trick" reported by Boethius, e.g., in 5.16.366), so that the 4/9-whole-tone, 2-diesis minor semitones would be 20=4*5 of the intervening 65=13*5=13*(32-27) parts. Graphically the resulting differences would be only ~1/25" in the highest register, where they could hardly be distinguished by drawing tools of the time--or eyes of any time. [3.8] In sum, Marchetto's new version of the minor semitone (and hence, its whole-tone complement, the major semitone, e.g., from Bb396 or Bb77 to B371 or B76) produced differences from the earlier medieval values so slight that they could be ignored or exploited persuasively in a visual demonstration, whereas his most striking innovation, greatly raised scale degrees, would be clearly, visibly distinct on the monochord. [3.9] In addition to problems of drawing and discerning geometric figures, one can assess how vulnerable the sounds of Marchetto's tuning might have been when realized on a necessarily fallible mechanism like the monochord. Although Ellis reported substantial errors in the fundamental frequencies produced on well regarded monochords of his day (1885/1954: 441-42), it should be emphasized that the 3-bridge monochord of Marchetto's period (2 fixed at the ends, 1 movable between--touching the string from below) greatly excelled in accuracy the post-1500 instrument, with fixed bridge and nut with movable tangent between--to press the string to the belly from above. For the latter, Cecil Adkins (1963: 4) reported accuracy of ~0.5 mm.--at e1, ~6.5 cents. [3.10] Nonetheless, any span of tolerance resulting from changes of tension, friction between bridge and string, aligning bridge and marks by eye, matching by ear pitches on another string (e.g., for Marchetto's dyad exx.), variations in temperature and humidity, etc. would add persuasive auditory force to the notion that divergence of Marchetto's enharmonic semitone from the earlier minor semitone was negligible. 4. INTERVAL PERCEPTION AND MARCHETTO'S DIESES [4.0] In recent music perception experiments I had undertaken quite independently of my Marchetto studies--or so I thought!--I found that for *melodic* (i.e., successive) intervals, my subjects (8 undergraduate music majors) were quite uncertain whether to label a particular melodic interval as, e.g., M3 or m3 if its frequency-ratio was close to 350% (i.e., midway between the ideal values of 300% and 400% for modern, equally tempered, 3- and 4-st intervals). Such uncertainty generally extended from ~330% to ~370%. Moreover, much the same held for other intervals involving notes "in the cracks": the students generally displayed uncertainty in labeling frequency-ratios within the ranges 130%-170%, 230%-270%, 430%-470%, 530%-570%, ... 1130%-1170% (see |1-12st|). [4.1] These results agreed with other recent studies of the so-called "categorical perception" of intervals (Harnad 1987; Butler 1992: 55; Krumhansl 1991: 281-83). Of greater moment here was an apparent anomaly that arose consistently for each subject. The region of uncertainty in deciding whether to label a melodic interval "unison" or "semitone" appeared not between ~30% and ~70%; instead, from ~15% to ~35%. In other words, a much smaller difference was required to distinguish a semitone from a unison. This is not to say that the students heard all tones differing by less than ~15% as "the same." Post- experiment de-briefing indicated that, if perceived as differing, such tones were heard as not differing enough to constitute a semitone. [4.2] All students applied with full certainty the label "semitone" to differences on the order of Marchetto's 48%. For differences smaller than ~25%, some offered such responses as "sharp unison." The distinction here seems to have been between 2 distinct pitches, e.g., C# and C, and 2 versions of the same pitch, e.g., C# and sharp C#--in other words, between a functional difference and a nuanced sameness. In converse fashion, some offered the idea that smaller versions of intervals heard clearly as semitones were "flat semitones." The following tones illustrate intervals they heard, where 0%=c and 100%=(equally tempered) c#: |0%| |5%| |10%| |15%| |20%| |25%| |30%| |35%| |40%| |45%| |50%| |55%| |60%| |65%| |70%| |75%| |80%| |85%| |90%| |95%| |100%| [4.3] 3 students volunteered the adjectives "sharp" and "flat" for intervals close to each maximally uncertain interval. E.g., at ~340% they volunteered "sharp m3;" ~360%, "flat M3." This tendency to label intervals of ((2n+1)*50) +/-10)% sharp or flat recurred in all mid-regions--except for the unison/semitone, where ~20% intervals were labeled "sharp unisons," and ~30% intervals "flat semitones." [4.4] Marchetto emphasized his new dieses were not for use in chant, rather, in discant, i.e., 2-part *polyphony* and closely related idioms. In the Lucidarium, Marchetto's 27 mus. exx. of the diesis interval are all cast in a2, note-against-note, "first-species" counterpoint, each illustration presenting a sharpened tone in the 2d of its 3 sonorities. [4.5] 3 of his mus. exx. are counter-examples; regular in all other respects, they counter-indicate use of his sharps in oblique motion, specifically against repeated notes that would form with them the following sonority successions (|Luc.5.6.8-12|): |DDd|_|DDc74|_|DDd| |CCg|_|CCf74|_|CCg| p8 M7 p8 p12 x11 p12 |bd|_|c74d|_|dd| m3 m2 p1 [4.6] The wide M7 and x11, and especially the narrow m2 he proscribed would produce beating (smooth, continuous fluctuation in loudness: ~4-18Hz) or roughness (~18-100+Hz), particularly in lower registers (Stevens and Davis 1938/1983: 242-45). Just below top-space GG in the modern bass-clef staff, a narrow Marchettan m2 (e.g., a dyad, |180-185|, with fundamental frequencies of 180Hz and 185Hz: cf. 74/72= 37/36) would produce beats of 5Hz (i.e., 185-180 = 5 pulses/second: this 'dyad' can help confirm the accuracy of transferred ra files). [4.7] Well within the range where beating is most prominent (~4 to 18 Hz), such effects could be clearly heard and/or felt whenever narrow m2s were produced by a 2-course monochord (or "dichord": Adkins 1980 and 1991: 33-40, 500-10), or by voices--from top-space GG of the bass-clef staff to e1 at the top of Marchetto's pitch system, i.e., throughout the range where they would occur in Marchettan works if he had not proscribed them. [4.8] Secondary, "subjective" beating would be especially prominent between the 2d partial (i.e., 1st "overtone") and 1st partial ("fundamental") of tones close to an 8ve apart (i.e., within 4-18 Hz of the 2/1 ratio), if sounded at relatively high intensities (as they would be in vocal music or on monochords/dichords with resonators, post-1200: Stevens and Davis 1938/83: 184-87, 244; Adkins 1980). Throughout Marchetto's pitch system such effects could be heard in the wide M7s he warns against, esp. in lower registers: from GGGFF74 cf. 96Hz vs 2*96*72/74 = ~186.8Hz to ed741 cf. 324Hz vs 2*324*72/74 = ~630.5Hz which yield secondary beat-rates of (resp.): (2*96)-186.8 = 5.2Hz (cf. [4.6], above) and (2*324)-630.5 = 17.5Hz. [4.9] Divergences greater than ~18Hz give rise to pronounced "roughness." Peaking in intensity at ~50 Hz, such roughness would be especially audible in wide M3s of the lowest register, where DDFF74 would produce a roughness rate of 144-186.8 = ~42.8Hz. Such effects also would be prominent in the lowest p5s (e.g., around AAEE in the bass-clef staff, where 108Hz and 162Hz produce roughness of 162-108=54Hz) and in narrow m3s above middle c (e.g., around f74a1 in the treble-clef staff, where 432*(8/9)*(72/74)=~373.6Hz and 432Hz result in roughness of 432-373.6=~58.4Hz). By contrast, Marchetto's musical examples tend to locate such potentially problematic intervals well outside their respective regions of maximal roughness: both illustrations of his narrow m3 about an 8ve lower (at FF74a and GG74b, where roughness rates would be ~1/2), and all 7 examples of p5s half an 8ve or more above (1 at EEb, 4 at FFc, and 1 each at ae and bb-f). The only exceptions are 2 of his 3 illustrations of the wide M3 (at EEGG74)--the other appearing at bd74 (i.e., half an 8ve above the interval's roughness peak). [4.10] In short, one can understand Marchetto's prohibition of coloured m2s and M7s, as well as his tendency to illustrate p5s and coloured M3s (or m3s) in particular registers as a way of preventing--if only in lab demonstrations--salient beats and roughness in discant. Nonetheless, Marchetto (citing Isidore of Seville) tried to realize in his dissonances harsh, unpleasant effects, and his sharpened degrees served as means to such closely controlled ends: specifically, "extreme mixtures of two sounds thoroughly mixed with one another, coming harsh and unpleasant to the ear" (duorum sonorum sibimet permixtorum ad aurem veniens aspera atque iniocunda permixtio: |Luc.5.2.2|). [4.11] Except his counter-examples (of wide M7s and x11s, and narrow m2s), all Marchetto's diesis illustrations exemplify sharpened degrees participating in narrow m3s, and wide M3s, M6s, and M10s. Each diverges from its ideal, medieval (or, for that matter, modern, equally tempered) frequency-ratio by about half a semitone (all sizes in cents): interval Marchettan Pythagorean equal-temp't m3 252 294 300 M3 450 408 400 M6 948 906 900 M10 1650 1608 1600 [4.12] Although one might expect musicians to respond erratically to Marchetto's m3, M3, M6, and M10, important studies by E.M. Burns and W.D. Ward (1978) and Donald E. Hall and Joan Taylor Hess (1984) argue against such a prediction. Burns and Ward emphasized their subjects were most uncertain (i.e., closest to a 50-50, flip-a-coin response) when asked to label (melodic) frequency-ratios close to 350% as "m3" or "M3." However, several of their formally trained subjects were very certain if asked to isolate intervals "in the cracks" as such--i.e., in contrast to intervals closer to the ideal sizes of 12-st equal temperament. Indeed, the step-like curves for their responses merely shifted ~50%. [4.13] Hall and Hess asked similar subjects not merely to label simultaneous intervals (with spectra ranging from partials 1-5 to 1-10) but also to characterize each on a 7-point scale of "acceptability." Their results confirm for simultaneous intervals earlier findings concerning category boundaries between melodic intervals a semitone or larger: in general, these appear at (2n-1)*50%, where n=1, 2, 3,... (i.e., even for the m2 sonority Marchetto proscribed and that had a ~25% boundary in my study of melodic interval perception: the single outstanding exception in Hall and Hess's study of simultaneous intervals was their high M6/m7 boundary, ~970%). [4.14] Hall and Hess also emphasized the importance of beats and roughness in describing sonorities, esp. if such effects can be traced to the first 5 partials (e.g., C, c, g, c1, e1). Their subjects tended to label "acceptable" intervals close to the following values: 2/1, 3/2, and 4/3 for p8s, p5s, and p4s (as in Marchetto's, and earlier medieval tunings, as well as subsequent formulations of just intonation): 5/4, 5/3, 5/2, and 6/5, 8/5, and 12/5 for, resp., M3, M6, M10, and m3, m6, and m10, (i.e., the ideal values sought in just intonation). Subjects displayed small ranges of acceptance for perfects, and esp. M and m imperfects, in contrast to x4, d5, and all 2s and 7s. [4.15] Though appearing to support, in particular, just intonation, these data jibe well with Marchetto's diesis tuning. Marchetto's ideal values for perfect intervals would yield no beats or roughness; his coloured intervals (esp. his wide M3, M6 and M10, but also his narrow m3, m6, and m10) would fall well within the "unacceptable" characterization of the modern-day musicians Hall and Hess studied. Also important, all intervals in Hall and Hess's study lasted 3 secs.; similarly long durations were notated for simultaneous rough coloured imperfects in music closest in provenance to Marchetto s formulation (and in the dyad exx. here). [4.16] In Marchetto's tuning, "non-coloured" 3, 6 and 10 (e.g., c/e, e/g, e/c , g/e, CC/e, EE/g) diverged quite far from the ideal values Hall and Hess discerned. In Marchettan works, these non-coloured imperfects had rather short durations, as did 2 and 7--especially the (prominently rough or beating) coloured m2, M7, and x11 Marchetto emphatically proscribed. The non-coloured imperfect intervals Marchetto retained from earlier Pythagorean tuning necessarily diverged as much as their precursors from the focal, beat-less, ideal values Hall and Hess found. Moreover, Marchettan substitutes for Bb and/or B would produce similar divergences (all sizes in cents): tuning G/Bb G/B 1/13s |291| |402| =(416/396)*(9/8) =(371/351)*(81/64) 1/9s |292| |404| =(81/77)*(9/8) =(76/72)*(81/64) Pythagorean |294| |408| =32/27 =81/64 just |316| |386| =6/5 =5/4 [4.17] In Hall and Hess's study, the relatively narrow m3s (291-300%) and wide M3s (402-411%) would all fall in the "unacceptable" half of their subjects' scaling. The pronounced sensitivity to rather slight divergences from 5/4 etc. Hall and Hess found can be explained by beating, which is especially salient for sonorities of long duration. In actual Marchettan works, the sensitivity Hall and Hess tapped would arise only in such long-held cadence sonorities as |EEcg#|. Moreover, EEg# would be rough irrespective of the beating in cg# or EEc, whereas other non-coloured, non-cadential imperfect intervals were much shorter. [4.18] Recent research on interval perception uniformly shows responses are learned and learnable. Formally trained musicians display much less uncertainty than non-trained (e.g., as measured by better fits to steeper ogives: cf. |1-12st|, [4.0], above);(4) some also sub-categorize reliably in distinguishing among m3, wide m3, narrow M3, etc.; and adapt readily to novel intervals "in the cracks." Cross-cutting these achievements are acoustically and physiologically based, non-cognitive phenomena: beating and roughness. Though these effects can be tapped experimentally by requiring such polar, arguably ethnocentric (or "hodiecentric") responses as "acceptable" vs "unacceptable," they can be channeled stylistically in many ways within particular cultural settings. ======================================================== 4. In my study of melodic interval perception, all ogives (cf. Guilford 1954) were significantly close to (i.e., diverged non-significantly from) the students' responses at the p<=.05 level in a standard chi-squared test for grouped data (on which see Smith 1985: 319-414). For intervals of 1 to 12 semitones, responses were grouped into 5-cent increments within the medial, (100n +/-30)%) range; for the unison/semitone pair, within the (5-45)% range. ======================================================== [4.19] E.g., in great contrast to Western European ideals of beat-less perfect intervals are the precisely patterned, "shimmering" beat-rates for p1, p8, etc. among the bronze keys and gongs of gamelan; through such carefully crafted timbral structures, professional Indonesian tuners have shaped the individual personalities of entire ensembles--in principle, for centuries (cf. Hood 1960; Susilo 1975; Rahn 1996). By comparison, Marchetto's tuning intensified an earlier medieval opposition between beat-less perfect intervals and all others--esp. M3, M6, their inversions, and 8ve- compounds--and provided for vividly sharpened leading tones. 5. PEDAGOGICAL ASPECTS OF MARCHETTO'S FORMULATION [5.0] As in the Lucidarium, pieces closest in provenance to Marchetto's original account of dieses tend to locate his sharpened notes in the highest voice(s): specifically, from FF74 to d741, in parts designated for boys--in particular, pairs of boys (duo pueri: Vecchi 1954). That boys originally were the main performers of Marchetto's sharps illuminates the reception of his challenging account. [5.1] Because Marchetto complained his new sharp-sign had been drawn wrongly and, as Karol Berger rightly stresses, his sharps "commonly" (a vulgo: |Luc. 8.1.4, 17|; |Pom. p. [-40-]|) had been called "falsa musica" (lit. false music, in contrast to "color fictitius," lit. fictitious colour: imaginative, in one's head, by ear-- |Luc. 2.8.9|; |5.6.27|: cf. Berger 1987: 16), one can conclude his diesis chromaticism had circulated outside his direct purview before 1317-18. Beyond some scores that clearly specify Marchetto's sharp notes (e.g., by a natural, square-B sign with upward stem to the right) and the many Lucidarium copies made, transmission of his tuning must have been largely oral. [5.2] Marchetto's tuning was absorbed into elementary music instruction (Herlinger 1990). E.g., a rudimentary digest of Marchetto's modal theory, seemingly used to teach neophytes at St. Mark's in Venice, the Ars magistri marchetti (Monterosso 1966), presumed knowledge of his dieses in order to determine whether problematically narrow melodies were authentic or plagal. Plausibly, too, the Hebrew translation of an originally vernacular, Italian digest of Marchetto's modal theory, brought to light by Israel Adler (1971), also referred to dieses after the point where the only surviving copy breaks off. If so, this remarkable work would testify to an unusually wide readership for Marchetto's diesis doctrine.(5) [5.3] Significantly, too, the extensive compilation of selections from the Lucidarium in ms Vatican, BAV Capp. lat. 206 (ca. 1500: ff. 138-67'; cf. Herlinger 1990: 239- 40), which seems aimed at more advanced practising musicians, e.g., composers or choral directors (rather than their charges), retains the complex argument Marchetto adduced to support his division of the whole tone--an indication that this apparently speculative material, not readily available outside the Lucidarium, formed the basis of lab demos for the actual mathematics underlying Marchetto's tuning for 200 years (as in the Ancient tradition of Euclid's canon: Mathiesen 1975; cf. Szabo 1978 on the centrality of the canon = monochord = qanun? = qun? for mathematics instruction generally). [5.4] As well as providing further lab demos for speculative aspects of Marchetto's tuning, the Lucidarium's 25 mus. exx. of sharps could have introduced novices to his chromatic practice. Among pieces using these sharps, Marchetto's a3 motet Ave, Regina Celorum/ Mater innocencie/[Ite, missa est] (Sanders 1973: 571-73; Gallo 1974; Fischer and Gallo 1985: #37) stands out for the microtonal fluency it presumes of its upper voices at a very early date (|1305|). Conversely, the much later anon. a4 motet Ave, Corpus Sanctum/Gloriosi Stefani/ Protomartiris (|1330-38|) evidences, in its frequent doublings at the lower 8ve, enduring concern for accuracy in the highest voices (Gallo 1968; Fischer and Gallo 1985: #38). [5.5] Most of Marchetto's semitone exx. are schematic: 17 present the diesis sharp as a chromatic passing tone, e.g.: ascending: |FFc|_|EEc74|_|DDd| or descending: |DDd|_|EEc74|_|FFc| thereby illustrating directly Marchetto's chromatic division and providing rudimentary exercises for learning to sing the new intervals. 4 more exemplify the diesis sharp as a chromatic lower neighbour. Of these 2 are censured (cf. [4.6] above): |DDd|_|DDc74|_|DDd| |CCg|_|CCf74|_|CCg| in contrast to a repeated ex. of approved usage: |DDd|_|EEc74|__|DDd| [5.6] Of the rest, 6 provide parallel realizations of Marchetto's substitutes for minor and major semitones: |aa|_|GGbb77|_|EEb| |EEb|_|GGbb77|_|aa| Although Marchetto said direct chromatic progressions between Bb and B could occur in any kind of music (chant, discant, etc.), he emphasized in the Pomerium (|Pom. [-69]-[-72]|) that they were not properly used to form leading tones in cadences. [5.7] Because Marchettan pieces do not employ such progressions, one can understand his prominent examples of them merely as showing how his tuning would replace earlier versions of B and Bb. Setting such B-Bb progressions in discant thus concretized his distinction between diatonic and chromatic semitones, especially as he tightly juxtaposed their contrasting exx. (|Luc. 8.1.3|). His idea that enharmonic and diatonic semitones were not to be used in cadences clarified greatly his concept of cadence and emphasized further that the earlier, Pythagorean versions of imperfect intervals he retained would not draw attention to themselves. [5.8] Generally, Marchetto presented the simplest, most schematic exx. both earlier in a group of 2 to 4, and repeatedly throughout his entire discussion. Such distinctions as between using sharp dieses more or less "properly" (proprie) or "naturally" (naturaliter) are exemplified by changing only a single variable--as in inductive ascent and incremental pedagogy, from Francis Bacon and Johann Heinrich Pestalozzi onward). [5.9] Later introduced in a few groups of examples are illustrations that presume basic knowledge of his novel division, but set in contexts that extend melodically beyond a whole tone: first in a lower range, plausibly to be sung by older instructors, illustrating the proper, natural, rising resolution: |DDa|_|GG74b|_|aa| thereupon, the less proper, less natural, falling resolution: |EEb|_|FF74a|_|FFc| followed by the proscribed, oblique resolution (cf. [4.6], above: |bd|_|c74d|_|dd| culminating in a non-schematic, but thoroughly idiomatic approach and resolution, entirely within an upper voice: |EEe|_|DDf74|_|CCg| Thus, the principles of Marchetto's tuning could be understood by following his words, observing his monochord marks, listening to, and eventually singing, the accompanying exx.: in sum, a cumulative process facilitated by his own cross-referencing of relevant passages in the Lucidarium and Pomerium. [5.10] Emphasized throughout Marchetto's exposition of diesis tuning were new possibilities for sharp chromaticism, even in monophony. Among Paduan dramatic offices of the time, the Lamentum Beate Marie Virginis (Vecchi 1954: 56-63) realized this possibility amply, gradually unfolding (like other Marchettan works) increasing chromatic complication, before opening into a thorough mixture of unisons and constantly crossing 3ds and 5ths in its binatim-like close. [5.11] That Marchetto's sharps "properly" participated in one of 2 basic progressions: 3-5 cf. |ac74|_|GGd| or 6-8 cf. |af74|_|GGg| (and their inversions) suggests an incipient "chromatic discant modality" that might comprise not only originally modal melodies adapted to polyphony as tenors but also, at least intermittently, unaccompanied melody. E.g., at the |end| of the Lamentum's main, monophonic section, a concluding discant cadence to D (or G), |EEc74|_|DDd| (or |ac74|_|GGd|), is strongly implied by the melodic progression |c74|_|d|, and intensifies the previous centrality of D, just before the work presses to its |a2| conclusion on G (tuned to equal temperament in its MIDI file). [5.12] Marchetto carefully delineated genre-based differences in semitone usage. Because he provided the same kinds of mus. exx. for his initial demonstrations of tuning and dissonance (|Luc. 2.6.4-2.8.9|; |5.6.8- 27|) as for his innovative account of chromatic permutation, i.e., sharp and flat/natural solmization (|Luc. 8.8.3|), one can surmize he intended all his mus. exx. to be sung by his readers/pupils only when understood, conceptually and aurally. [5.13] Since his first groups of exx. appeared well before his treatment of sight singing, they must have served first as sounding illustrations on the monochord --or more precisely and plausibly, as all are a2--on 2 monochord or dichord courses tuned in unison. Such an instrumental realization would also make available a constant check on initial attempts to sing Marchetto's sharps against a (generally lower) non-chromatic part. A "tenor" of this sort could be performed on a single course with movable bridge as support for, or challenge to, an upper voice, which could be checked readily by a 2d course with (independently) movable bridge. In this way, Marchetto's pupils could proceed from initial stages of comprehension to fluent vocal application in his more demanding works. 6. OTHER REALIZATIONS OF MARCHETTO'S DIESES [6.0] Such later writers as Tinctoris (Berger 1987: 22- 29) mention Marchetto's 5-diesis whole tone without mentioning its basis in 1/9-tone division. The space for a whole tone could be divided into 5 equal segments by construing its 9/8 ratio as 45/40. The leading-tone diesis would be 41/40: 42% as compared with 48% for the 1/9-tone ratio 74/72 (=37/36), and similarly perceived as a melodic semitone rather than as a wide unison. In comparison with a string-length of 5.33" for e1 (see [3.3], above), this 1/5-tone diesis's d411 string-length would be ~5.4635": readily visible ~1/7" away from its resolution, but less than 1/50" from its 1/9-tone counterpart (at 5.48"). Respective rates of beating and roughness also would be similar. [6.1] A 1/5-tone, enharmonic-semitone ratio would be 45/43: ~|79%| as compared with ~|88%| for the 1/9-tone version, and ~|90%| for the earlier, Pythagorean value it would replace. For c1 with string-length 6.75", the 1/5-tone version of b1 would be at 7.06": ~1/20" from the 1/9-tone b1 (at 7.11"), and slightly further from the Pythagorean value (7.125"). Although well within the central range for stepwise melodic semitones (a usual context for mi-fa progressions), B or Bb tuned this way would produce beats if combined with Pythagorean E or F: above a 162Hz EE, a 1/5-tone b would be ~249.8Hz, in comparison with a 243Hz Pythagorean b, producing 13.6Hz beating an 8ve higher (cf. 486Hz and 499.6Hz). [6.2] To construct such a 1/5-tone division of, e.g., d/e at d41, one could "back up" even further than for the 1/9-tone tuning: to a Pythagorean g# at a little less than 1/4 GGG's length. Regarding this as 4x, bisecting its length twice (i.e., into 4 x-units) would provide e at 5x. Bisecting e's length thrice (into 8 divisions of .625x) would result in d at 5.625x (=(9/8)*5x). Bisecting the space between the g# and e thrice, into 8 divisions of .125x, would give d41 at 5.125x (=(41/40)*5x), etc.: |d| |d41| |e| (g#) 45 41 40 32 4x--> 5x--> 5.625x--> 5.125x--> [6.3] As a plausible solution to incommensuracy problems for B and Bb in any such tuning, the space for the Pythagorean m3 A/C (32/27) could be construed as 7(=2+3+2) "1/5-tone" dieses. A/C divided into 7 equal segments would produce 224/214 (=112/107=80%) for A/Bb and 199/189 (=89%) for B/C: cf. 90% for Pythagorean B/C. Combined with Pythagorean values, this 1/7-m3 tuning would produce virtually beat-less realizations of E/B, but Bb/F would beat as in 1/9-tone tuning. Because none of these p5s would appear in a Marchettan cadence (since no sharp leading tone, i.e., sharp E or A, was available for them to resolve--though 1/7-m3 B would support 1/5-tone D41 in a cadence to A/E: BD41_AE)-- such effects would pass unnoticed, like those of their uncoloured imperfect counterparts. [6.4] Among copies of the Lucidarium there was great inconsistency in notating Marchetto's dieses. Even more difficult to assess are pieces that originally might have been composed and/or performed with Marchetto's chromaticisms but that survive only in copies lacking his explicit signs. Determining intonation for such pieces is all the more difficult because of the continuing controversy and confusion his doctrine provoked and wide variation in successors' usage of Boethian semitone terms he adopted.(5) ======================================================= 5. Karol Berger (1987: 16-29) comprehensively surveys successors' responses to Marchetto's terms and notational signs. As Berger (26) and Ellsworth (1987: 337-38) emphasize, the Lucidarium survives in 15 full and 3 incomplete copies (~1317-1500), and the Marchettan orthodoxy of Bonaventura da Brescia's Breviloquium enjoyed 19 early editions (1497-1570). ======================================================= [6.5] Among later tunings, the recurrent 4-dieses-plus- comma formulation seems quite parallel to Marchetto's 5-fold partition of 1/9-tones, esp. as his odd-number doctrine would identify the comma (or diacisma) with his middlemost diesis, 77/76 (=~23%--cf. the Pythagorean comma: 24%). Moreover, terse references to such seemingly non-Marchettan whole-tone divisions as into 4 or 8 parts (e.g., in Tinctoris 1475/1963) might merely record widespread tuning mnemonics for 9- or 5-fold division via adjacent whole tones: 72/64 or 40/36. Nonetheless, the possibility remains that works originally conceived in 1/9- or 1/5-tone tuning were actually sung with other, e.g., Pythagorean, intervals. [6.6] If all Marchetto's sharps were rendered in Pythagorean tuning--or as today, in equal temperament-- only Marchetto's leading tones would be affected greatly. The extremely wide, coloured intervals could assume the Pythagorean sizes and qualities of other early idioms; p8s, p5s etc. would sound and be performed much the same as in 1/9- and 1/5-tone tunings; not highlighted in cadences, non-coloured imperfect intervals and dissonances would pass as little noticed as in a 1/9- or 1/5-tone performance: in metrically weak positions, for short durations, or as additions to more salient structures. [6.7] Prominent in Marchettan a3 and a4 works were such progressions as |EEcg#|_|DDda1|. Arguably, 1/9- or 1/5-tone tuning alone would highlight physiologically and cognitively the core, contrary-motion, leading-tone, discant structure of such arresting cadences: EEg#_DDa1. Relegated to a subsidiary structural role would be the (much) augmented 5's similar motion to p5: cg#_da1, c being construed also as forming a structurally less salient, but beating, m6 with EE, with which it would proceed stepwise in non-cadential contrary motion to d: EEc_DDd. [6.8] In listening to a modern, equal-temperament rendering of Ave, Regina Celorum/Mater innocencie (|1305|), a work in which this a3 progression forms the initial cadence, one can merely imagine, "fictitiously," as it were, the effect produced if the already prominent cadential sharps were realized only ~48% (or ~42%) from their resolutions. 7. MARCHETTO'S CHROMATICISMS AND RECENT SCALE THEORY [7.0] Reversing such an exercise, one can consider the effects Marchetto's tuning would have on the diatonic collection as understood of late. In such a view, one construes sharpened and flattened forms of degrees as replacing, at least temporarily, their natural counterparts. As well, one specifies the structural changes that take place when one or more degrees are altered in various ways (cf. Rahn 1991: 35-44; Clough and Douthett 1991: 125-44). [7.1] Of particular concern here are contradictions and ambiguities. E.g., if the "white-key" collection is understood as 12 equally tempered semitones, it is remarkable for comprising no contradictions and only 1 ambiguity, namely, between its x4 (FB) and its d5 (BF), where intervals of differing degree-sizes (4th, 5th) have the same sizes in cents or semitones: 600% or 6 st. In a Pythagorean construal, there are no ambiguities, but the FB/BF pair forms a contradiction: an interval of smaller degree-size (4th) has a larger frequency-ratio (612%) than an interval of larger degree-size (5th: 588%). [7.2] In Marchettan works, sharps generally appear 1 at a time: e.g., C-sharp returns to, is "re-replaced" by, C before F-sharp, G-sharp, D-sharp, or Bb replaces F, G, D, or B--in contrast to later use of 2 sharpened degrees in "double-leading-tone" cadences: e.g., EEGG#c#_DDad. The following fig. compares consequences of replacing B by Bb, F by F-sharp, C by C-sharp, G by G-sharp, and D by D-sharp in 2 frameworks: a) equally tempered 12-semitone (cf. modern, e.g., keyboard, realizations); b) Pythagorean tuning (cf. historical reconstructions of early music): altered intervals ideal tuning (in cents): degree: affected: a) equal temp't b) Pythag'n Bb BbE/EBb 600/600 612/588 F# CF#/F#C 600/600 612/588 C# GC#,FB/C#G,BF 600/600 612/588 C#F/FA,GB,AC# 400/400 384/408 G# DG#,FB/G#D,BF 600/600 612/588 G#C/CE,EG#,FA 400/400 384/408 FG#/DF,G#B,AC,BD 300/300 318/294 D# AD#,FB/D#A,BF 600/600 612/588 D#G/CE,FA,GB,BD# 400/400 384/408 *CD#*/EG,AC,BD/*D#F* *300*/300/*200* *318*/294/*180* D#F/FG,GA,AB 200/200 180/204 [7.3] As the above fig. shows: i) replacing C by C# (cf. "ascending melodic minor") yields 12-st ambiguities or Pythagorean contradictions between the d4 C#F and the M3s AC#, FA, and GB; ii) a corresponding result obtains for the d4 G#C if G is replaced by G# (cf. "harmonic minor"); also the x2 FG# produces ambiguities/contradictions with the m3s DF, G#B, AC, and BD. iii) substituting D# for D results not only in further ambiguities/contradictions (between the d3 D#F and the M2s FG, GA, and AB), but also a 12-st contradiction between the D#F d3 (200%) and the CD# x2 (300%), and for a Pythagorean construal, a "doubly contradictory" relation: not only is the CD# x2 (318%) larger than the m3s EG, AC, and BD (294%), but also both sorts are larger than the M2s FG, GA, AB (204%), which are larger than the d3 D#F (180%)! [7.4] Whether Marchetto's chromaticisms are realized by modern equal temperament or historical Pythagorean tuning, four gradations can be acknowledged: a) substituting F# for F (or Bb for B) complicates an originally diatonic collection merely by virtue of adding a new pitch class to the piece or passage as a whole; otherwise, the new collection created by the substitution has the same profile of ambiguities or contradictions as the diatonic original--the tritone merely moves to another pair of scale degrees; b) replacing C by C# adds a further tritone pair as well as ambiguities/contradictions around the d4 C#F; c) whereas all such complications appear if G# replaces G, ambiguities/contradictions around the x2 FG# are also introduced; d) finally, if D# replaces D, the same kinds of ambiguities/contradictions arise: additionally, there are ambiguities/contradictions around the D#F d3, and most important, whether the framework be equally tempered or Pythagorean, this interval not only contradicts the larger x2 CD#, but does so doubly. In short, even without adopting Marchetto's tuning, chromaticisms of these kinds produce a coherent, gradated increase in complication: from a simple, diatonic starting-point to the complications of D#. [7.5] Taken at face value or understood as an approximation to quartertone equal temperament, Marchetto's intervals intensify complications found already in simpler tunings. As the following fig. shows, from F# onward all the ambiguities in an equally tempered, 12-st construal become contradictions, whether the framework comprises 24 quartertones or Marchetto's reconfiguration of Pythagorean values. Of these possible construals, Marchetto's tuning produces greater contrasts between contradictory intervals, reaching a climax (or crisis) at the d3 D#F (138%), which is fully 222% smaller than the x2 CD# (360%), which is, to re-work a once-popular song title, "its own scale-degree construal's frequency-ratio grandparent." If Marchetto's diesis-based versions of B and Bb are incorporated, these contrasts increase, but only slightly (on the order of ~2-6%): altered intervals ideal tuning (in cents): degree: affected: equal temperament Marchettan (quartertone) (1/9-tone) Bb BbE/EBb 600/600 612/588 F#+ CF#+ 650 654 F#+C 550 546 C#+ GC#+;FB 650;600 654;612 BF;C#+G 600;550 588;546 AC#+;FA,GB 450;400 450;408 C#+F 350 342 G#+ DG#+;FB 650;600 654;612 BF;G#+D 600;550 588;546 EG#+;CE,FA 450;400 450;408 G#+C 350 342 FG#+ 350 360 DF,AC,BD;G#+B 300;250 294;252 D#+ AD#+;FB 650;600 654;612 BF;D#+A 600;550 588;546 BD#+;CE,FA,GB 450;400 450;408 D#+G 350 342 *CD#+* *350* *360* EG,AC,BD;*D#+F* 300;*150* 294;*138* FG,GA,AB 200 204 D#+F 150 138 [7.6] All differences effected by adopting narrow semitones merely intensify complications that would arise in earlier Pythagorean, or modern equally tempered, 12-semitone or 24-quartertone construals. Moreover, in each of these, the divergent tones form a coherent grouping of their own, parallel to the diatonic originals--a "displaced cycle," as it were, shadowing the cycle of 5ths. F#, C#, G#, and D# form, among themselves, a secondary cycle of 702% or 700%, at distances of 100(=600-500)%, 114(=612-498)%, 150(=650- 500)%, or 156(=654-498)% above F, C, G, and D. For 2/9- tone dieses, these cycles are as follow:(6) F74 <--702%--> C74 <--702%--> G74 <--702%--> D74 / / / / 156% 156% 156% 156% / / / / F <--702%--> C <--702%--> G <--702%--> D ======================================================= 6. Mieczyslaw Kolinski's Pythagorean formulation of the 22 srutis of Ancient South Asian tuning (1961) would also result in a displaced cycle and satisfy the scale analysis of Clough et al. (1993), the latter re-framed as comprising 2 kinds of intervals: a=90% and b=24%. In such a construal, a sruti could be understood, like Marchetto's diesis, as the difference between consecutive strings, marks, or frets on a tuning instrument (e.g., of the vina variety). ======================================================= [7.7] If Marchetto's single-diesis semitones are heard or performed as nuanced versions of more usual semitones, i.e., not merely as "semitones" nor as full-fledged "quartertone" intervals, but as "narrow, small, or sharp semitones," the nuances that result can be construed as forming similarity relations among themselves, e.g., narrow, small, or sharp "to the same extent" or "by the same amount." In this way, putatively quantitative divergences can be understood as proportionally qualitative or qualified--as it were, "adverbially" (e.g., DF74 is smaller than DG "by as much as" F74A exceeds GA), rather than "adjectivally" or as "nouns" in their own right (e.g., DF74 and F74A are a "large" M3 and a "small" m3, or "a p4 minus a diesis" and "a M2 plus a diesis"): put another way, not as separate, distinct "kinds" of intervals nor merely as "marked" intervals, but as intervals aletered or varied in a shared, common way and forming a cycle of their own. [7.8] That a wide M3 would be understood as a version "of" a diatonic M3, rather than vice versa--and rather than each being construed as "allophonic" or "in free variation" with the other (cf. allophones or phonetic variations within a single phoneme)--is assured by the consequences: CE and GB match each other within a passage where F#+ is the only chromatic note; FA and GB match each other within a passage where C#+ is the only chromatic note, whereas DF#+ and AC#+ match each other only between such passages; however, CE and FA match each other across such passages also; instances of GB match within and between such passages; etc. [7.9] Because Marchettan chromatic intervals idiomatically are always "out-numbered" by their diatonic counterparts, they are sites of complexity. Disadvantaged by their opposition to the many matching relations among other intervals of the same scale-degree size, each chromatic interval, on its own, would be rather difficult to learn (as are the similarly rare tritones within the diatonic collection). However, that they share extents by which they diverge from their majorities (e.g., AC#+ and C#+E vs FA, GB, and DF, EG, BF, etc.) provides a starting-point for a "progressive" Marchettan pedagogy, with direct extensions available through matching across time-spans (e.g., AC#+, EG#+). [7.10] Cross-cutting the simple-to-complex ordering of chromatic effects from F# to D# are the potential reinforcers served up by the observation that each sharp forms with several of its diatonic passage-mates similar intervals as the others (cf. F#+ vs G, C, D, and D#+ vs E, A, B). Rather than shaping each sharp's intonation merely with its resolution (e.g., by carefully tuning F#+ relative to G, as in chromatic neighbour-tone figures), or construing its intonation merely as diverging from a referential value (e.g., by tuning F#+ relative to F, as in a chromatic passing- tone figure), or attending only to such connections within an initial stage (e.g., F#+ along with C#+, G#+, and D#+, and/or F#+/G along with C#+/D, G#+/A, and D#+/E), Marchettan sharp-structures suggest a richer curriculum that would introduce non-leading-tone/ altered-tone successions early on: D/F#+, A/C#+, C/F#+, etc.), and amplify such melodic tasks with discant--much as the Lucidarium's mus. exx. imply and Marchettan works require. REFERENCES CITED Adkins, Cecil. (1963). *The Theory and Practice of the Monochord*. unpub. Ph.D. diss. (Iowa, Music), repr. Ann Arbor: UMI Dissertation Services (64-3344). Adkins, Cecil. (1980). Monochord. *New Grove Dictionary of Music and Musicians*. London: Macmillan. v.12, 495-96. Adkins, Cecil. (1991). *A Trumpet by Any Other Name: A History of the Trumpet Marine*. Buren: Frits Knuf. 2 vv. Adler, Israel. (1971). Fragment h'ebraieque d'un trait'e attribu'e a' Marchetto de Padoue. *Yuval* 2: 1-10 (cf. RISM Bix2, 490). Berger, Karol. (1987). *Musica Ficta: Theories of Accidental Inflection in Vocal Polyphony from Marchetto da Padova to Gioseffo Zarlino*. Cambridge, Eng.: Cambridge Univ. Press. Bernhard, Michael, and Calvin Bower. (1996). *Glossa Maior in Institutionem Musicam Boethii*. Munich: Bayerischen Akademie der Wissenschaften. Billanovich, G. (1940). Uffici Drammatici della Chiesa Padovana. *Rivisita Italiana del Dramma* 4: 72-100. Boethius, Anicius Manlius Severinus. (~520/1989). *Fundamentals of Music*, trans. Calvin M. Bower, ed. Claude V. Palisca. New Haven: Yale Univ. Press (cf. Bernhard and Bower 1996). Burns, E.M., and W.D. Ward. (1978). Categorical Perception of Musical Intervals--Phenomenon or Epiphenomenon? Evidence and Experiments in the Perception of Melodic Intervals. *Journal of the Acoustical Society of America* 63: 456-68. Butler, David. (1992). *The Musician's Guide to Perception and Cognition*. New York: Schirmer. Clough, John, and Jack Douthett. (1991). Maximally Even Sets. *Journal of Music Theory* 35/1-2: 93-173. Clough, John, Jack Douthett, N. Ramanathan, and Lewis Rowell. (1993). Early Indian Heptatonic Scales and Recent Diatonic Theory. *Music Theory Spectrum* 15/1: 36-58. Ellis, Alexander J. (1885/1954). Additions by the Translator. in Hermann L.F. Helmholtz. *On the Sensations of Tone as a Physiological Basis for the Theory of Music* 2d Eng. ed. repr. New York: Dover. 430-556. Ellsworth, Oliver B. (1987). rev. of Herlinger (1985) in *Journal of Music Theory* 31/2: 337-45. Fischer, Kurt von, and F. Alberto Gallo, eds.(1985-87). *Italian Sacred Music*. Monaco: Oiseau-Lyre. v.12-13. Gallo, F. Alberto. (1966). Cantus Planus Binatim: Polifonica Primitiva in Fonti Tardive. *Quadrivium* 7: 79-89. Gallo, F. Alberto. (1968). Da un Codice Italiano di Motetti del Primo Trecento. *Quadrivium* 9: 25-44. Gallo, F. Alberto. (1974). Marchetus in Padua und die franco-venetische Musik des fruehen Trecento. *Archiv fuer Musikwissenschaft* 31: 42-56. Gallo, F. Alberto. (1977/1985). *Il Medioevo II* Torino: Edizioni di Torino (trans. Karen Eales, as *Music of the Middle Ages II*. Cambridge, Eng.: Cambridge Univ. Press). Guilford, J.P. (1954). *Psychometric Methods*, 2d. ed. New York: McGraw Hill. Gurlitt, Willibald, and Hans Heinrich Eggebrecht, comp., ed. (1959-67). Diesis. *Riemann Musik Lexicon*, 12th ed. Mainz: Schott. 3 vv., *Sachteil*: 225. Hall, Donald E., and Joan Taylor Hess. (1984). Perception of Musical Interval Tuning. *Music Perception* 2/2: 166-95. Harnad, Stevan, ed. (1987). *Categorical Perception*. Cambridge, Eng.: Cambridge Univ. Press. Heath, (Sir) Thomas L. (1925/1956). trans., ed. *Euclid: The Thirteen Books of The Elements*, 2d ed. repr. New York: Dover. 3 vv. Herlinger, Jan W. (1978). The *Lucidarium* of Marchetto of Padua: A Critical Edition, Translation, and Commentary. Ph.D. diss., Univ. of Chicago. Herlinger, Jan W. (1981a). Fractional Divisions of the Whole Tone. *Music Theory Spectrum* 3: 74-83. Herlinger, Jan W. (1981b). Marchetto's Division of the Whole Tone. *Journal of the American Musicological Society* 34/2: 193-216. Herlinger, Jan W., trans., ed. (1985). *The Lucidarium of Marchetto of Padua* Chicago: Univ. of Chicago Press (cf. also |http://www.music.indiana.edu/tml/ 14th/MARLUC#_TEXT|: # = the Arabic numeral for each of the Lucidarium's 16 tractata). Herlinger, Jan. W. (1990). Marchetto's Influence: The Manuscript Evidence. in Andr'e Barbera, ed. *Music Theory and Its Sources: Antiquity and the Middle Ages* Notre Dame: Notre Dame University Press, 235-58. Kolinski, Mieczyslaw. (1961). The Origin of the Indian 22-Tone System. (1961). *Studies in Ethnomusicology* 1: 3-18. Krumhansl, Carol. (1991). Music Psychology: Tonal Structures in Perception and Memory. *Annual Review of Psychology* 42: 277-303. Marchetto of Padua. ([1318-19]/1961). *Pomerium* ed. Giuseppe Vecchi. Florence: American Institute of Musicology (cf. also |http://www.music.indiana.edu/ tml/14th/MARPOME_TEXT|). Martinez-[Goellner], Marie Louise. (1963). *Die Musik des fruehen Trecento*. Tutzing: Schneider. Mathiesen, Thomas J. (1975). An Annotated Translation of Euclid's Division of a Monochord. *Journal of Music Theory* 19: 236-57. Mendel, Arthur. (1948/1968). Pitch in the 16th and Early 17th Centuries. *Musical Quarterly* 48/1-4, repr. in Arthur Mendel, ed. *Studies in the History of Musical Pitch* Amsterdam: Frits Knuf, 88-169. Monterosso, Raffaello. (1966). Un Compendio Inedito del 'Lucidarium' di Marchetto da Padova. *Studi Medievali* 7/2: 914-31 (cf. also MSS Pavia, Aldini 450: 7'-10, and Seville, Biblioteca Capitular y Colombina 5.2.25: 66'- 68'). Niemoeller, Klaus Wolfgang.(1956). Zur Tonus-Lehre der italienischen Musiktheorie des ausgehenden Mittelalter. *Kirchenmusikalisches Jahrbuch* 60: 23-32. Pirrotta, Nino.(1955). Marchettus de Padua and the Italian Ars Nova. *Musica Disciplina* 9: 55-71. Rahn, Jay.(1991). Coordination of Interval Sizes in Seven-Tone Collections. *Journal of Music Theory* 35/1-2: 33-60. Rahn, Jay. (1996). Perceptual Aspects of Tuning in a Balinese Gamelan Angklung for North American Students. *Canadian University Music Review* 16/2: 1-44. Sanders, Ernest. (1973). The Medieval Motet. in Wolf Arlt et al., eds. *Gattungen der Musik in Einzeldar- stellungen*. Bern: Francke, 497-573. Smith, Gary. (1985). *Statistical Reasoning*. Boston: Allyn and Bacon. Stevens, Stanley Smith, and Hallowell Davis. (1938/1983). *Hearing: Its Psychology and Physiology*. New York: American Institute of Physics. Susilo, Hardja. (1975). rev. of Jaap Kunst. *Music in Java*, 3d ed., Ernst Heins in *Asian Music* 7/1: 58-68. Szabo, Arpad. (1978). *The Beginnings of Greek Mathematics*, trans. A. M. Ungar. Hingham, Mass.: D. Reidel. Tinctoris, Johannes. (1475/1963). *Dictionary of Musical Terms*, trans. Carl Parrish. Glencoe, Ill.: Free Press. Vecchi, Giuseppe. (1954). *Uffici Drammatici Padovani*. Florence: Olschki (cf. Billanovich 1940). ====================== 2. Review AUTHOR: Grave, Floyd K. TITLE: Review of William E. Caplin, *Classical Form: A Theory of Formal Functions for the Instrumental Music of Haydn, Mozart, and Beethoven* (New York and Oxford: Oxford University Press, 1998). KEYWORDS: analysis, classical, style, Schoenberg, Ratz, Formenlehre Floyd K. Grave Rutgers University Department of Music New Brunswick, NJ 08903 grave@rci.rutgers.edu ABSTRACT: Reviving the *Formenlehre* tradition as taught by Arnold Schoenberg and Erwin Ratz, Caplin attempts to accommodate the method to a specific but stylistically complex repertory: instrumental works of Haydn, Mozart, and Beethoven within the approximate time-span 1780-1810. He strives not only to retain the unambiguous formal distinctions that constitute both the strength and limitation of Schoenberg's *Formenlehre*, but also to fasten them securely to the music of his protagonists: a challenging project that Caplin handles with disquieting aplomb. His demonstration of terminological reform and analytical precision constitutes a noble effort, and despite the Procrustean edge to some of the analyses, there is much in this book that merits notice by scholars concerned with style and compositional technique of the late 18th and early 19th centuries. [1] William Caplin's new book startles by declaring that "the time is ripe for a new theory of classical music" (p. 3), and surprises even more by its choice of models on which to build: rather than starting from scratch or adapting more recent critical, historical, and analytical research by Charles Rosen, Leonard Ratner, Jan LaRue, and others, Caplin looks back to the *Formenlehre* tradition promulgated by Arnold Schoenberg(1) and his pupil Erwin Ratz(2). ============================= 1. Arnold Schoenberg, *Fundamentals of Musical Composition*, edited by Gerald Strang and Leonard Stein (London: Faber and Faber, 1967); *Structural Functions of Harmony*, rev. ed., edited by Leonard Stein (New York: W. W. Norton, 1969). 2. Erwin Ratz, *Einfuehrung in die musikalische Formenlehre*, 3rd ed. (Vienna: Universal, 1973). ============================= [2] Progressing from small dimensions to large, and from simple structures to complex, the older discipline attempts to specify the structure and deployment of formal functions in standard instrumental movement-types. Elements examined include the motive, the phrase, and the relationships among phrases within archetypal sentences and periods. Small forms are categorized, and theoretical models are provided for the allocation of theme-types, including main theme, transition, and subordinate group, for rondo, sonata, and other large-scale designs. Fully aware of current scholarly disdain for the ennui of pigeonholing, Caplin nevertheless sees the potential of *Formenlehre*, when suitably modified, for addressing present-day theoretical and analytical concerns. [3] He adapts the method first of all by limiting the chronological span to be encompassed. In contrast to Schoenberg, whose rules and functional categories applied to a broad historical spectrum, extending at least from Bach to Brahms, Caplin restricts his study to the high-classical repertory: sonatas, trios, quartets, quintets, overtures, concertos, and symphonies composed by Haydn, Mozart, and Beethoven in the years 1780-1810. He then proposes a vast elaboration of Schoenberg's categories in an effort to accommodate compositional diversity without losing the theoretical appeal of plausible generalization. Embarking on this route, he aims to show how instrumental music by the great classical masters does indeed exemplify certain formal archetypes, and that a methodical dissection of the archetypes can furnish suitable tools both for analyzing specific compositions and for drawing historically useful conclusions about style. He thus invites us to ponder afresh the paradox by which the classical masters seemed bent on undermining and transforming formal conventions, even while celebrating their enduring currency. [4] As Caplin leads us through a bewildering array of hybrids, fusions, transformations, framing and medial functions, failed consequents, post-cadentials, interpolations, expansions, and extensions, an odd disparity becomes apparent between the homely (but functionally precise) terminology and the musical elegance it describes. His anatomy of development sections, for example, isolates such curiosities as transitional introductions, initiating regions, sequenced models, cores, pre-cores, core substitutes, pseudo-cores, and false recapitulations. In trying to comprehend all these subcategories, alternatives, and deviations, the reader is assisted by the clear organization of the text, the detailed explanations and commentaries on illustrative examples, and a helpful glossary that gives thumbnail definitions for many terms with their special, form-functional connotations. Additional help comes from the splendid bibliography, a selective but wide-ranging list of books and articles dealing with theory, analysis, and stylistic criticism pertinent to late 18th- and early 19th-century music. [5] An ingeniously compact format permits inclusion of an extraordinary wealth of music examples: each occupies a single staff, with novel and copious use of divided stems and octave transposition signs to maximize the representation of separate parts and minimize the use of leger lines. (Readers who wish to study the examples in detail will nevertheless want to have full-size scores at hand: in excerpts that involve textural complexity, much is inevitably lost, and while most of the notation signs are easily legible, the bar numbers are so small and fine that readers with even mild visual impairment may find themselves groping for a magnifying glass.) [6] Since Caplin aims to direct our attention to functional categories, and not to the inspirations and idiosyncrasies of his protagonists *per se*, it would appear that we are dealing with norms, customs, and rates of occurrence. But perhaps sensing that a frankly quantitative approach might lean too far in the direction of style description rather than theory, he proposes that his categories "represent abstractions based on generalized compositional tendencies in the classical repertory. A category is not necessarily meant to reflect frequency of occurrence in a statistical sense: it is often the case that relatively few instances in the repertory correspond identically to the complete definition of a given category" (p. 4). But by choosing a specific repertory and probing it in close detail, his inquiry inevitably becomes enmeshed with style criticism--and with style criticism's shadowy companion, statistics. At one point, he actually does resort to numbers, telling us that about 10 percent of classical minuets resemble small binary (p. 220). Elsewhere, he relies on an abundance of adverbs, including "mostly," "rarely," "commonly," and "frequently," to give an idea of how often certain functions, patterns, and relationships occur. Given the author's comprehensive knowledge of the music in question, readers may sometimes wish for more use of numbers. For example, we read that "frequently...the composer adds a *coda*" (p. 179), that "sometimes the notation indicates that the coda starts after the double bar lines that instruct the performer to repeat the development and recapitulation together" (p. 181), and that "on occasion, a genuine coda is included in the repeat of the development and recapitulation" (p. 278, n. 8). "Frequently" tells us that codas are common, while "sometimes" and "on occasion" indicate the modest size of specified subcategories within the total number; but this leaves us with the queasy sense that a majority of instances have been left unaccounted for. True, we can infer a relatively large number of codas in movements where there are no repeat signs for the latter part of the movement, but this subcategory is never identified as such. [7] Coaxing "abstractions based on generalized compositional tendencies" from a collection of masterworks famously packed with novelty is clearly an endeavor fraught with perils, and Caplin usually succeeds in facing them with chilling equanimity. Following his predecessors' example, he places heavy emphasis on harmony as primary determinant of form. Lest there be any doubt, the very opening words of this book proclaim loudly, in 16-point type, Schoenberg's triad of admonitions from *Fundamentals*: "Watch the harmony; watch the root progressions; watch the bass line" (p. 2). [8] The preoccupation with harmony promotes theoretical stability, but it can also limit our analytical purview if it downplays the often prominent role of other elements, including register, dynamics, timbre, surface rhythm, and melodic profile. Thus Caplin's broad distinction between "main theme" and "subordinate theme" specifies a tonal hierarchy unequivocally--a subordinate theme is that which occurs in the subordinate key--but it implies exclusion of the possibility that a movement might assign more than subordinate emphasis (on grounds other than tonality) to an exposition theme stated in the dominant or relative major key. [9] More specifically troublesome is Caplin's elaboration of distinctions drawn by Schoenberg and Ratz between tight-knit and loose organization(3). Caplin observes that "in the classical repertory, subordinate themes are, with rare exceptions, more loosely organized than their preceding main themes" (p. 97). This makes sense from the vantage point of Caplin's method, which typically labels as "subordinate theme" not only a (possibly tight-knit) contrasting period with its own special consistencies of timbre, dynamics, surface rhythm, and register, but also a diversity of more loosely organized, anticipatory or summarizing functions preceding and following such a theme. Caplin's allowances for exceptions and alternatives notwithstanding, the broad "subordinate theme" category tends to swallow up potentially important distinctions, for example the not unfamiliar option of contrasting an open-ended, expansive primary theme with a stabilizing, closed period in the second key (exemplified in the first movements of Mozart's Piano Concerto in D minor, K. 466, and Haydn's "Reiter" Quartet, Op. 74/3). ============================= 3. Caplin defines "tight knit" as "formal organization characterized by the use of conventional theme-types, harmonic-tonal stability, a symmetrical grouping structure, form-functional efficiency, and a unity of melodic-motivic material" (p. 257), whereas "loose" describes "formal organization characterized by the use of non-conventional thematic structures, harmonic-tonal instability (modulation, chromaticism), an asymmetrical grouping structure, phrase-structural extension and expansion, form-functional redundancy, and a diversity of melodic-motivic material" (p. 255). ============================= [10] Also likely to be swallowed up by Caplin's rather strictly defined categories are the processes of gradual transition and secondary-theme solidification that often characterize classical expositions. In Mozart's Piano Sonata in D, K. 576, first movement, a half cadence in bar 27 places us on a dominant plateau, lifting the music from its tonic underpinnings but scarcely confirming arrival in the new key. The open-ended, sequential figuration that follows, featuring a virtually unbroken stream of sixteenth notes (bars 33-40), climaxes in a trilled cadential flourish, followed by a marked contrast of register that emphasizes the sense of a fresh, contrasting theme in the new key at bar 42. But this is not where Caplin begins the subordinate theme group. Rather, he places it at bar 28. Because the thematic statement of bars 28-41 closes with a full cadence in the dominant, Caplin's theory requires him to designate it as a subordinate theme. Thus the emphatic punctuation in bar 41, no matter how salient rhetorically, merely marks an interior divide within a larger subordinate group. [11] Not surprisingly, Mozart deletes the material of bars 28-41 from the recapitulation, where establishment of a new key is no longer an issue. Now, with simple and impeccable logic, the recurrence of the half cadence from bar 27 serves to prepare the contrasting theme first heard at bar 42. But since Caplin's theory commits him to designating bars 28-41 as a first subordinate theme, not merely a dispensable transition, he finds himself explaining the omission in relatively complicated terms: "Mozart begins the subordinate theme of the recapitulation with material from the *second* subordinate theme of the exposition....He does so presumably to avoid a redundant appearance of the main theme's basic idea in the home key, which would arise from using the first subordinate theme" (p. 169). [12] We confront a similar dilemma in an analysis of Mozart's Violin Concerto in A, K. 219, first movement. Here, the problematical passage spans measures 74-80 in the solo exposition. Appearing after a half cadence in the tonic key, and leading to a half cadence in the dominant, this lightly scored material can readily be heard as an introductory gesture, preparing the featured subordinate-theme entry on the upbeat to bar 81. But Caplin's theory obliges him to disagree: "Following traditional notions of form, some analysts might see the true 'second subject' as beginning in measure 81 because of the catchier tune and because that idea was also found in the opening ritornello (m. 20). But this view ignores the fact that measures 74- 80 reside entirely in the new key. Thus for tonal reasons, as well as phrase-structural ones, this passage is consistent with the definition of a subordinate theme (first part) and should not be regarded as belonging to the transition" (p. 117). [13] While rejecting the idea of beginning the second-theme function at bar 81, he acknowledges that this material was indeed announced in the opening ritornello. What he omits noting is how that initial appearance of the theme was marked for emphasis: it designated a point of decisive contrast in register, dynamics, texture, and surface rhythm, where the punctuating rest--actually the only strong-beat rest in the entire movement--occurred at virtually the exact midpoint of the ritornello; thus there seems little doubt as to the role of this event in signaling a major landmark in the design. To be sure, bars 74-80 soften the abrupt contrast experienced in the ritornello, but nonetheless it seems easier to hear bar 81 as a major structural articulation than as merely the start of "a second part to this subordinate theme" begun in bar 74. [14] In contrast to the two examples cited above, where clearly marked cadences guided Caplin's choice (perhaps too unequivocally) of where to place the "subordinate theme" label, expositions whose cadences are less well defined may lead to uncertainty about thematic punctuation. In describing the notoriously ambiguous exposition of Haydn's String Quartet in B minor, Op. 64/2, first movement, Caplin states that "the beginning of this subordinate theme is especially difficult to determine because little in the way of any rhythmic, textural, or dynamic change helps articulate the boundary between the transition and the subordinate theme. After the transition arrives on the dominant of the subordinate key (downbeat of m. 15), a new melodic idea, featuring a chromatic stepwise descent, prolongs the half cadence by means of another half-cadential progression" (p. 114). In a note to the passage, he observes that "an additional difficulty arises from the question of whether each real measure equals a notated measure or one-half a notated measure" (p. 272, n. 53). [15] To begin with, the movement in question exemplifies the so-called compound 4/4 measure, frequently encountered in later 18th-century chamber music, in which the metrical impulse is virtually that of 2/4 with every other bar line removed. Thus the fact that a theme begins in the middle of a notated measure should not, in and of itself, cause confusion(4). Moreover, Haydn marks the entry of the new theme with a change of register in the cello, which leaps up a tenth on the upbeat to beat 3 (i.e., the midbar downbeat). Arguably, in the context of an exceptionally subtle and tonally unsettled exposition, the signals are outstandingly clear at this point: a contrasting melodic figure in the first violin and a register shift in the cello confirm the thematic articulation, while other elements lend continuity and reinforce persisting harmonic tension. The harmony may be unsettled, but the location of the new theme's beginning seems nonetheless unambiguous. ============================= 4. See Floyd K. Grave, "Metrical Displacement and the Compound Measure in Eighteenth-Century Theory and Practice," *Theoria* 1 (1985): 25-60, and "Common-Time Displacement in Mozart," *Journal of Musicology* 3 (1984): 423- 442. ============================= [16] The somewhat disgruntled response of Caplin's theory to the Haydn movement cited above seems at least partially a reaction to the work's undermining an element on which the method depends: a point of unambiguous functional contrast between transition and subordinate theme. But it may also be symptomatic of a tendency to substitute the archetype for the music in question as the point of reference for analytical discussion: the music, not fitting the archetype very well, is judged to be more problematic than it might if viewed in terms of its own environment of expectations and functional nuances. [17] Absorption with the archetype is most clearly evident in Caplin's treatment of the form he labels *sonata without development*. "If a development is eliminated, then the section following the exposition will seem to function more as a *repetition* than a return. Indeed, the listener hearing the movement for the 'first time' would not necessarily know that the appearance of the main theme following the exposition marks the beginning of a recapitulation (of a sonata without development) and could just as likely believe that the exposition is simply being repeated according to the norms of sonata form" (p. 216). [18] This statement seems reasonable in the abstract: the archetypal exposition closes with a full cadence, and the ensuing recurrence of the main theme in tonic could therefore signal the start of either a recapitulation or a repeated exposition. But the list of examples that Caplin cites includes few movements that resemble the archetype closely enough to illustrate the ambivalence. For example, in the slow movement of Mozart's String Quartet in C, K. 465, the cadence marking the end of the exposition (bar 39) elides with an intense, six-measure transition whose rising melodic line, motivic repetitions, sustained crescendo, and prolonged harmonic tension leave no doubt as to its function in anticipating a major landmark--the recapitulation--much as we would expect from a retransition after a development section. Two of Caplin's examples from Haydn's quartets stand even further removed from the model: the second movement of Op. 76/4 prohibits any confusion by turning unexpectedly from major to minor in the first measure of the recapitulation; and the second movement of Op. 50/2 appropriates a design familiar from 18th-century operatic practice: ritornello-like passages at the beginning, middle, and end of the movement frame the two main sections (analogous to exposition and recapitulation), each conceived as an embellished violin solo that culminates in a cadential trill. At no time does the possibility of a repeated exposition come into question. [19] Attesting to the scholarly energy that Caplin has bestowed on this project, an almost overwhelming extravagance of notes accompanies the text--not footnotes, alas, placed in easy viewing range at the bottom of the page, but endnotes, stuffed in the back of the book. Many consist of bibliographical citations, others offer supplementary commentary, and still others list compositions that exemplify functional categories described in the text. Since there are close to 750 of these notes, for less than 250 pages of text, encountering them is a common event on virtually every page, and thus a constant interruption. Fearful of missing something essential, the reader must stop, reach for a bookmark, then begin clawing through the back of the book. Given the irritating necessity of endnote format, it might have been good to incorporate the lists of supplementary examples, and perhaps much of the illuminating commentary, in the text proper. [20] But ferreting out the notes is a minor annoyance, measured against the value of this book as a clear and uniquely detailed presentation of standard forms and their constituent parts, a compilation of intriguing examples illustrating classical design and function, an up-to-date bibliographical guide, and a source of fresh insight into the accomplishments of the classical masters. Caplin's approach, buttressed by methodological rigor and theoretical detail, makes a persuasive case for the revival of *Formenlehre* as a pedagogical tool and analytical discipline; but it also underscores the limitations of a method that sometimes enforces Procrustean choices on music that may use convention as much for nuance and ambiguity as for conformity to functional norms. ====================== 3. Announcements A conference "Musical Borrowing from the Middle Ages to the Present" will be held at the Crane School of Music at the State University College at Potsdam, New York, on February 20-21, 1999. Papers dealing with all historical periods and genres (including vernacular) will be presented. For more information, contact Dr. Stephen Johnson, "Musical Borrowings Conference" c/o Crane School of Music, State University College at Potsdam, Potsdam, New York 13676. E-mail: johnsoss@potsdam.edu or phone: (315) 267-2427. --------------- Call for Papers Music Theory Southeast Eighth Annual Meeting Davidson College, Davidson, NC March 12 and 13, 1999 Proposals are solicited on any theory-related topic. These may include papers (approximately 30 minutes in length), panel discussions, or special interest sessions. Proposals for panel discussions should include a list of participants. Submissions must include: 1) seven copies of a proposal approximately 3-4 pages in length, with the author's name omitted; 2) an abstract of approximately 250-300 words, suitable for publication, with the author's name omitted; and 3) a cover letter giving the title of the proposal, the author's name, address (including e-mail address, if available), telephone number, and specification of technical requirements. Submissions should be postmarked no later than December 15, 1998 and should be sent to: Mark Parker, Program Chair Box 34441 Bob Jones University Greenville, SC 29614 work phone: 864 242-5100, ext. 2793 email (work):mparker@bju.edu home phone: 864 271-3460 e-mail(home):markmparker@juno.com The Program Committee for the 1999 meeting includes Ellen Archambault (student, Florida State University), Mauro Botelho (Davidson College), Amy Carr-Richardson (East Carolina University), Renee McCachren (Catawba College), Jairo Moreno (Duke University), John Nelson (Georgia State University), Mark Parker (Bob Jones University) and Paul Wilson (University of Miami). Submitted by, J. Kent Williams, MTSE secretary --------------- International Conference on "Meter, Rhythm, and Performance - Metrum, Rhythmus, Performanz" at the Hochschule Vechta, 26-28 May 1999 Call for Papers The International Conference on Meter, Rhythm, and Performance will be held from 26 to 28 May 1999 at the Hochschule Vechta, Germany. It will be organized by Dr. Christoph Kueper, Professor of English Linguistics, Fachbereich Geistes-, Kultur- und Sozialwissenschaften, Driverstr. 22, D-49377 Vechta. Tel. ++49-4441-15301, Fax ++49-4441 15444. E-mail: Christoph.Kueper@uni-vechta.de. Please address all conference correspondence to Christoph Kueper. * A special feature of this conference will be a recital of Goethe's "Reineke Fuchs" by Lutz Goerner. A Registration Form can be downloaded from the Internet: ----------------- CHAMP D'ACTION - a meta-composition for computer-controlled ensemble (utilizing MAX) was recently performed at the RadioKulturHaus in Vienna. A RealAudio recording of this concert is available at: http://www.essl.at/works/champ.html Dr. Karlheinz Essl - Composer Vienna / Austria Studio for Advanced Music & Media Technology http://www.essl.at/ ------------------ ANNOUNCEMENT AND CALL FOR PAPERS POLISH MUSIC JOURNAL http://www.usc.edu/go/polish_music/PMJ We are pleased to announce the creation of a new electronic journal. This an academic, peer-reviewed publication devoted to musicological studies of Polish music and music in Poland. The Journal's purpose is to provide a convenient, modern forum for publication of studies of the music that is not well known in the West. Its first issue has just been published on the web page of the Polish Music Reference Center. The content includes three articles (by Tyrone Greive, Jill Timmons and Sylvain Fremaux, and student winner, Timothy Cooley) that have been awarded the 1997 Wilk Prizes for Research in Polish Music. One more issue will be published this year. In 1999 we hope to be able to convert the journal into a quarterly. This publication is to fill in the gap between the Polish researchers, publishing in their native language, and the English-speaking world. Therefore, one or more issues of the PMJ will consists of translations of selected articles originally published in Polish, in the Polish Musicological Quarterly *Muzyka*. In order to make use of the capabilities of the electronic media, the Journal includes scanned musical examples (score excerpts) and samples of sound illustrations (recordings) for some, or all, of the articles published. The Journal's ID number is: ISSN 1521-6039. Maria Anna Harley serves as the Editor. The Editorial Board includes: Prof. Maciej Golab (Associate Professor of musicology, Institute of Musicology, University of Warsaw; also General Editor of *Muzyka*, Warsaw, Poland), Dr. Martina Homma (General Editor, Bela Verlag Music Publisher, Cologne, Germany), Prof. Jeffrey Kallberg (Professor of Music at the University of Pennsylvania, Philadelphia, U.S.), Prof. Zygmunt Szweykowski (Professor of Musicology, Institute of Musicology, Jagiellonian University, Cracow, Poland; also Member of the Editorial Board, Musica Iagellonica), Prof. Adrian Thomas (Professor of Music, Cardiff University of Wales, U.K), Dr. Elzbieta Witkowska Zaremba (Associate Professor, Institute of Fine Arts, Polish Academy of Sciences; also Member of the Editorial Board of *Muzyka*, Warsaw, Poland). For more information and guidelines for contributors visit the Journal's site. http://www.usc.edu/go/polish_music/PMJ The Editor may be reached at: Polish Music Reference Center. School of Music, University of Southern California 840 West 34th St. Los Angeles, CA 90089-0851 USA. Tel: 213-740-9369. Fax: 213-740-3217. E-mail: polmusic@usc.edu ------------------- Rotterdams Conservatorium and the Dutch Society for Music Theory Fourth European Music Analysis Conference ANALYSIS IN EUROPE TODAY 21-24 October 1999 CALL FOR PAPERS The Fourth European Music Analysis Conference will be hosted by Rotterdams Conservatorium in conjunction with the Dutch Society for Music Theory, and will be held in Rotterdam from 21 to 24 October 1999. The Conference's theme - "Analysis in Europe Today" - will be explored in a number of analytical symposia, round-table discussions and other sessions. In addition, there will be an all-day plenary session "Analysis in Europe Today: The Different Traditions"; among other things, this will present the results of a European-wide survey on teaching practices and research activity in theory and analysis. Conference delegates will also be able to attend a number of concerts, including a performance by Irvin Arditti of Ligeti's Violin Concerto with the Rotterdam Philharmonic Orchestra conducted by Reinbert de Leeuw. During the conference the International Gaudeamus Competition for young composers and performers of contemporary music will take place. Proposals for papers are invited for the following sessions: * Fragmentation and Integration in Beethoven's Bagatelles Op. 126 * The Trois Poeme Mallarme Debussy and Ravel * Ligeti's Violin Concerto: Historical Reflections through Modern Music * Nicolas Gombert and the Principle of Parody * Interactions between Acoustic and Electro-Acoustic Music * Analysing Structure in Improvised Music The Programme Committee also welcomes proposals in any area of music theory and analysis (Free Papers). Individual papers, which should last no more than 20 minutes, should be delivered in English if possible, but French and German are also acceptable. Abstracts (maximum 500 words) should be submitted on diskette or by e-mail to Patrick van Deurzen, Conference Director, at the address below, by 1 February 1999. Abstracts in languages other than English should be accompanied by an English translation if possible. The programme will be announced in July 1999. Further information can be obtained from: Patrick van Deurzen, Analysis in Europe Today Rotterdams Conservatorium Pieter de Hoochweg 222 3024 BJ ROTTERDAM HOLLAND Tel: +31 (0)10 213 3197 fax: +31 (0)10 413 1222 e-mail: pdeurzen@xs4all.nl ----------------- New England Conference of Music Theorists Call for Papers Fourteenth Annual Meeting March 27-28, 1999 (Sat.-Sun.) Harvard University Cambridge, Massachusetts Gerald Zaritzky (New England Conservatory), President Janet Hander-Powers (Topsfield, Mass.), Secretary David Cohen (Harvard University), Treasurer Web site: Peter Westergaard, Princeton University Keynote Speaker Featuring a recital, Saturday, March 27, at 8:30 p.m., by Richard Lalli, Yale School of Music, baritone Janet Schmalfeldt, Tufts University, pianist performing Schubert Sechs Moments musicaux, a group of Schubert songs, and Schumann Dichterliebe. Program Committee: Michael Schiano (Hartt School of Music), Chair Deborah Burton (Harvard University) Hali Fieldman (University of Massachusetts at Amherst) Gerald Zaritzky (New England Conservatory), ex officio Proposal deadline: January 15, 1999 Proposals on all topics are invited from interested theorists, both members and non-members of the Conference, whether or not they have presented papers at NECMT in the past. Particularly, papers on the music of Schubert and Schumann will be considered for a session to which performers Lalli and Schmalfeldt might informally respond. Sessions provide 30 minutes for presentation and 10 minutes for discussion of each paper.This year, a prize, consisting of a set of Music Theory Spectrum issues, will be awarded for an outstanding student presentation. All who wish to propose papers should send four copies of a three-to-five-page proposal and an abstract suitable for publication, by January 15, 1999, to the following address (a floppy disk of the abstract also should be provided in the mailing): Janet Hander-Powers NECMT Secretary 37 Mansion Drive Topsfield, Mass. 01983-1109 Proposals are read blind; they should contain no identification of the author. With your proposal and abstract copies, please include a cover letter giving your name, address, phone, email, affiliation, the title of your proposal, and any special equipment or arrangements required. Proposals of session topics may be sent, signed, directly to the Program Chair. NECMT 1998-99 MEMBERSHIP APPLICATION / RENEWAL: The New England Conference of Music Theorists is an organization serving the music theory community in New England and the surrounding areas. Your academic-year membership fee -- $15 professional / $7.50 student -- places you on our mailing list and admits you to our annual meeting. To join the conference or to renew your membership, please print out, complete, and mail this form. Member name: (please print) Mailing address: (please print) ZIP+4 code: Telephone: work ( ) home ( ) E-mail address: Academic affiliation: Membership status: Renewal ( ) New member ( ) Dues payment $15 professional ( ) $7.50 student/emeritus ( ) Please make checks payable to NECMT Mail this completed form with dues to: Janet Hander-Powers NECMT Secretary 37 Mansion Drive Topsfield, Mass. 01983-1109 Fall, 1998 Dear Fellow Music Theorist: Please join us in this fourteenth year of the New England Conference of Music Theorists! With happy memories of the stimulating presentations, congenial interactions, and gracious accommodations of our thirteenth annual meeting last spring at the University of Connecticut, we are looking forward to gathering at our next annual meeting, March 27-28, 1999, at Harvard University, where we will be honored by the presence of Peter Westergaard, Princeton University, as keynote speaker, and by SMT President Janet Schmalfeldt, Tufts University, pianist, and her colleague, Richard Lalli, Yale School of Music, baritone, performing music of Schubert and Schumann (and responding to related papers). Meeting arrangements chair David Cohen and his colleagues are organizing our visit, and program committee chair Michael Schiano and his committee are organizing the program. We hope many of you will propose papers for the meeting. (See our Call for Papers, above, and take note of its January 15th deadline.) Especially, we encourage student theorists to join and participate. We continue our practice of charging no separate fee for the meeting and again will offer arrangements for ride-sharing and home hospitality. So, except for lunch and a modest fee for our Saturday evening banquet--do plan to attend it!--your expenses should be low. Also, this year, through a generous gift from SMT (with special thanks to Cynthia Folio), we are able to offer a virtually-complete set of Music Theory Spectrum as a prize for an outstanding student presentation. We are pleased to welcome to the executive committee our newly-elected secretary, Janet Hander-Powers. We thank the nominating committee for its services and particularly thank Lisa Cleveland, of St. Anselm College and U.Mass.-Lowell, for accepting co-nomination to the ballot. Our outgoing secretary, Hali Fieldman, has accepted an appointment as Assistant Professor of Music Theory at the University of Missouri-Kansas City, starting in January. I know I speak for the membership when I express my special thanks to Hali for her many contributions to NECMT throughout her tenure here. We will miss her, and we wish her well! The terms of president and treasurer expire this spring; nominations for successors are welcome and may be directed to the secretary or to the nominating committee chair, Allan Keiler (keiler@binah.cc. brandeis.edu). Please note also the new email addresses for the secretary and president, above. As you may by now know, our regional website is up and running, available by link from the SMT Regional Societies webpage, or directly at http://mario.harvard.edu/necmt/index.html. NECMT is most grateful to website development chair Edward Gollin, at Harvard University, for constructing and administering the site and to Harvard University for hosting it. The program, abstracts, and minutes from our March meeting can be viewed there, as well as current news and selected documents from our long and illustrious history. Please log on often, read the latest news and comments (maybe your own!), and let us know how else our website can serve you. As always, we invite you to be in touch with us--especially those of you from areas currently under-represented in the conference. We hope still this year to contact many of you individually. Volunteers and suggestions for this and other activities are most welcome. We look forward to seeing many of you this fall in Chapel Hill--and all of you this spring in Cambridge! Sincerely yours, Gerald Zaritzky, President For the Executive Committee P.O. Box 390082 Cambridge, Massachusetts 02139 (617-492-5493) ------------------- West Coast Conference of Music Theory Annual Meeting ANNOUNCING a Joint Meeting of The West Coast Conference of Music Theory and Analysis (the 8th annual meeting of WCCMTA) and the Rocky Mountain Society for Music Theory 16-18 April 1999 at Stanford University The program committee invites proposals for posters, short talks (15 minutes), or long talks (30 minutes) concerning any aspect of music theory. Proposals should be between one and two pages long and should indicate whether they are for a poster, 15-minute, or 30-minute presentation. Since proposals are to be reviewed blind, please list your title, name, and contact information separately; do not reveal your identity within the proposal itself. Because this meeting is sponsored by The Center for Computer Assisted Research in the Humanities (CCARH), The Center for Computer Research in Music and Acoustics (CCRMA), and the Music Department of Stanford University, the Program Committee especially encourages proposals on the topic of computer applications. Proposals should be mailed (postmarked by 15 Dec 1998) to WCCMTA-RMSMT 99 c/o Leigh VanHandel CCARH Braun Music Center Stanford University Stanford, CA 94305-3076 emailed (on or before 15 Dec 1998) to leigh@ccrma.stanford.edu or faxed (by 15 Dec 1998, ATTN Leigh VanHandel) to (650) 725-9290 Program Committee Jonathan Berger (Stanford University) Jonathan Bernard (University of Washington) Jack Boss (University of Oregon) Lisa Derry (Albertson College of Idaho) Steve Larson, chair (University of Oregon) Steve Lindeman (Brigham Young University) ----------------- University of Silesia Institute of British and American Culture and Literature 41-205 Sosnowiec, Poland tel./fax: +48 32 2917417 Organs, Organisms, Organisations: Organic Form in 19th-Century Discourse. Call for papers Conference: Cieszyn, Poland, May 27- 28 1999 The aim of the conference is to re-think the role and place of the notion of "organicity" in nineteenth-century discourse. The idea of evolution and the shift from the mechanical view of nature and the world to the organic one were also projected upon areas other than natural sciences. The view of organisms as beginning and ceasing to exist in time underwent a qualification in numerous social and political theories as well as in other discursive practices of the epoch. For this reason we expect conference presentations to address the question of organicity in the 19th century from various methodological perspectives and approaches representing possibly the widest range of academic disciplines. Please send titles of prospective presentations along with short abstracts (about 200 words) to: Professor Tadeusz Sawek University of Silesia Bankowa 12, Katowice, Poland (tel./fax: +48 32 2917417) or by e-mail to professor Tadeusz Rachwa (rachwal@us.edu.pl) Deadline for the submission of conference presentations is January 31, 1999. ------------- SOUTHWESTERN COLLEGE ANNOUNCES THE SECOND ANNUAL CONFERENCE OF BRIDGES: Mathematical Connections in Art, Music, and Science JULY 30 - AUGUST 1, 1999 Suggested Topics: Fractals, Math and Music, Tessellations, Geometry in Quilting M.C. Escher Work, Math and 3-Dimensional Art, Origami Mathematics and Architecture, Computer-Generated Art Math and Art in Culture, Art in Hyperbolic Geometry The Conference publishes a refereed proceedings of presented papers. Papers accepted for publication should follow the proceedings format and be camera ready; however, for review they don't need to follow a set format. Interested authors must submit their papers by 1/15/99 for review. The authors will be notified of their papers' status and in case of acceptance will receive papers for revision by 3/15/99. Authors need to resubmit the papers in formatted, revised form by 5/2/99. In this step, the original clear figures and graphs should be submitted with the papers. If a presenter is not able to submit a paper for presentation, he or she may send an abstract (not more than 1 page) to be published in the Conference Proceedings . There is no reviewing process for abstracts. The deadline for abstracts is 4/23/99. There is a registration fee of $40.00 for each day or $100.00 for the entire conference plus $25.00 for a Proceedings. The 1998 Bridges Proceedings is available for purchase (Barnes & Noble, Phone: (316) 685-3600, Fax: (316) 685-7729). Besides area motels, Southwestern College offers lodging and meals on campus. For participants who fly to Wichita (the closest airport to Winfield) and report their arrival and departure times in advance, there will be transportation available for the evening of July 29 and for the morning of August 2. For more information (or if you want to add your e-mail to the mailing list) you may contact: Professor Reza Sarhangi, Bridges, Southwestern College, 100 College Street, Winfield, KS, 67156. E-mail: sarhangi@jinx.sckans.edu, (316) 221-8373, Home Page: http://www.sckans.edu/~bridges/ You may also contact the following Bridges Advisory Board members regarding the conference: East: Professor Nat Friedman, Department of Mathematics and Statistics, University At Albany, State University of New York, Albany, NY 12222, E-mail: artmath@math.albany.edu, (518) 442-4621 West: Professor Carlo Sequin, Computer Science Division, EECS Department, University of California, Berkeley, CA 94720, E-mail: sequin@cs.berkeley.edu, (510) 642-5103 Reza Sarhangi, Ph.D. Chair, Mathematics Department Southwestern College 100 College Street Winfield, KS 67156 E-mail: sarhangi@jinx.sckans.edu Tel: (316) 221-8373 Fax: (316) 221-8224 http: //www.sckans.edu/~math/ http://www.sckans.edu/~bridges/ ============= 4. Employment Eastman School of Music VACANCY Position: Musicologist, full-time, tenure-track. Responsibilities: Teach undergraduate and graduate courses, advise dissertations, contribute to the discipline through scholarship and research, participate in the musical and intellectual life of a comprehensive music school within the University of Rochester, a private research institution. Rank: Assistant Professor. Salary: Commensurate with experience. Qualifications: Demonstrable excellence in teaching. Primary scholarly specialization in Western music since 1880; a secondary interest is desirable in one or more of the following areas and repertories: American music; analysis, criticism, and cultural theory/studies; folk and non-Western musics; jazz and popular music. Evidence of, or potential for, scholarly achievement on an international level; Ph.D. completed by September 1999. Review of applications begins 15 November 1998 and will continue until position is filled. Applications materials: Letter of application, curriculum vitae, and dossier (or transcript and three letters of recommendation). Please send no other materials at this time. Please send applications to: Professor Ralph P. Locke Chair, Musicology Search Committee Department of Musicology Eastman School of Music Rochester, NY 14604-2599 rlph@uhura.cc.rochester.edu The Eastman School of Music of the University of Rochester is an EO/AA employer. ------------- University in Indianapolis, IN: BUTLER UNIVERSITY - Associate Professor of Music Theory, appointment effective Auguest 1, 1999. The position is tenure track and the salary competitive. The Jordan College of Fine Arts, Department of Music, is seeking a colleague with a distinctive record of teaching at the undergraduate and preferably graduate levels and of scholarship in music analysis or theory pedagogy. A completed Ph.D. in music theory or the equivalent is required. Responsibilities include primary attention to the undergraduate theory core and master's teaching, advising master's theses in music theory and history, and contributing scholarship regularly. Serving on appropriate committees and occasionally in interdisciplinary teaching is expected, as Butler is a private, comprehensive university with a strong liberal arts component. Additional teaching might respond to personal interests and departmental needs, especially in world music and jazz. Initial interviews will be held at 1998 SMT-Chapel Hill and AMS-Boston but are not required. Women and other minority candidates are invited to apply. Applications should be received by 1/15/99. Send a letter of application, three letters of recommendation, curriculum vitae, and supportive credentials to Dr. James Briscoe, Chair of Theory Search, Jordan College of Fine Arts, Butler University, 4600 Sunset, Indianapolis, IN 46208. EOE/AA. If more information is desired regarding the position or the university, interested persons are more than welcome to contact me (Jeff) via email or phone! Thank you. Jeff Gillespie Jeffrey L. Gillespie Jordan College of Fine Arts Butler University Indianapolis, IN 317-940-6416 jgillesp@thomas.butler.edu -------------- POSITION/RANK: Assistant Professor of Music Theory, full-time. Contract: one-year, renewable. Tenure track. salary: $30,829-$53,779. Appointment date: September 1, 1999. INSTITUTION: Hunter College, CUNY, Department of Music QUALIFICATIONS: Doctoral degree in music theory and evidence of success as a college teacher and emerging scholar. Highly desirable: advanced piano skills and ability to read scores fluently at the keyboard; expertise with MIDI and music-publishing programs (such as Finale); experience in composition. DUTIES: Teach undergraduate and graduate courses in music theory and analysis. Administrative duties include committee and placement exam assignments, and may also include, in time, direction of the music theory program or direction of undergraduate or graduate studies. SEND: Application letter, detailed resume, and the names, addresses, and telephone numbers of three references. Please do not send any other materials at this time. DEADLINE: December 15, 1998. CONTACT: Professor George B. Stauffer, Chair, Department of Music, Hunter College of CUNY, 695 Park Avenue, New York, NY 10021. Email: george.stauffer@hunter.cuny.edu. Phone: 212-772-5020. Fax: 212-772-5022. ---------------- University of Southampton (Full) Professorship in Music The Department of Music, University of Southampton, anticipates advertising a Professorship in Music in approximately two months' time. We are giving advance notice of the position so that prospective applicants may discuss it, if they so wish, with Professor Mark Everist (the Head of Department) at the upcoming AMS meeting at Boston. We shall be seeking an outstanding scholar with an international reputation in any musicological or music-theoretical area. (We are unlikely to appoint a composer to this position, since we are currently appointing to a separate professorship in that area.) The appointment will be a permanent one and will be at the equivalent grade to an American full professorship; salary will be negotiable but not less than $60,000 (equivalent). Appointment will be from 1 October 1999 or as soon as possible thereafter. The Southampton Department of Music has an outstanding research profile with a wide range of strengths ranging from medieval to contemporary music, including popular and film music. Theory and the study of performance are also stongly represented. For further details, visit the Department's web site at http://www.soton.ac.uk/~musicbox/index.html If you would like to receive details of the post when available, please email either Mark Everist (me@soton.ac.uk) or Nicholas Cook (ncook@soton.ac.uk). If you would like to speak with Mark Everist at the Boston meeting, please email him in advance to arrange a time, failing which he may be contacted via the message board. -------------- POSITION/RANK: Assistant Professor of Jazz Studies INSTITUTION: University of Connecticut QUALIFICATIONS: Masters degree or equivalent professional experience; demonstrated success and a record of achievement in performance, teaching, and composition or arranging. JOB DESCRIPTION/RESPONSIBILITIES: Nine-month, tenure-track position to begin August 23, 1999. Conduct and administer university jazz ensembles and coordinate the jazz program. Teach courses in improvisation, arranging, jazz history for music majors and non-majors, and/or other subjects in the jazz area. Teach in an appropriate applied jazz area and supervise adjunct jazz faculty. Organize, coordinate and coach small ensembles as needed. Participate actively in departmental recruitment and outreach efforts. Assist with student advising, serve on department, school and university committees and perform other service as appropriate to the position. SALARY RANGE: negotiable ITEMS TO SEND: letter of application, resume, three current letters of recommendation and a list of references. Do not send recordings, video tapes or scores until requested. DEADLINE: til position is filled CONTACT: Dr. Robert Stephens University of Connecticut Department of Music U-12 876 Coventry Road Storrs, CT 06269-1012 860-486-3731 phone 860-486-3796 fax rstephens@finearts.sfa.uconn.edu --------------------- POSITION ANNOUNCEMENT Indiana University Libraries, Bloomington Head, Music Library Located fifty miles south of Indianapolis, the Bloomington campus of Indiana University supports 26,000 undergraduates and 9,000 graduate, professional, and non-degree students. Indiana University's School of Music is widely respected as one of the world's most comprehensive institutions for musical studies. Central to this program is a faculty of 140 teachers and scholars and a select student body. The facilities of the School of Music include six buildings housing offices and studios, practice rooms, choral and instrumental rehearsal rooms, three recital halls, the Musical Arts Center, and the Music Library which is part of the IU Libraries system. The recently completed William and Gayle Cook Music Library is recognized nationally as one of the finest music libraries in the United States. It occupies a four floor 55,000 square foot facility and featuring state of the art technology. The collections number more than 537,000 items. The staff includes six librarians, two professional staff, seven clerical staff, and approximately seventy student assistants. Further information about the Music Library is available at http://www.music.indiana.edu/muslib.html. The Music Library is one of nineteen Bloomington campus libraries. Librarians have tenure-track academic appointments; support staff are represented by the Communications Workers of America. Responsibilities: The head of the William and Gayle Cook Music Library is responsible for leadership and overall management of the library, including supervision of staff, service to the School of Music in respect of performance, research, and teaching, service to the campus and growth and management of the collections. The head reports operationally to the Executive Associate Dean of the University Libraries and works with the Libraries Director for Information Technology on technology matters and with the Dean of the Libraries and Dean of the School of Music in all development matters. The head supervises the development of the Variations Project with the technological support of the University's Digital Library Program. (The Variations Project is conceived as a digital library of music information, including text, images, sound, and music notation, accessible through a graphic user interface.) The head directs the Specialization in Music Librarianship (offered through the School of Library and Information Science) and has the capability of teaching in the School of Music. Although the Music Library is administered by the University Libraries, the head is responsible for maintaining a close relationship with the music faculty and administration through the Music Library Advisory Committee and other committees within the School. The head is an ex officio member of the Music Library Advisory Committee which is chaired by a member of the music faculty. Other significant duties include setting standards for and evaluating staff; overseeing all functions of the Music Library, including public and technical services and collection development; preparing and overseeing the Music Library budget; developing Music Library policies; and interpreting and applying general library policies. The head of the Music Library oversees assistance to faculty, students and off-campus users, and guides the direction of technology development in the Music Library. The head also maintains a dialogue with users of the Music Library regarding programs and issues. The Head of the Music Library is expected to make contributions in the areas of professional development, research/creative activity and service. Qualifications: MLS degree and graduate degree in music or a combination of equivalent education and experience; substantial experience in music librarianship; Demonstrated ability to work both creatively and pragmatically in a changing environment; a record of accomplishment in both music and music information technology using digital library concepts; solid experience in the leadership and supervision of a diverse staff; outstanding oral and written communication skills and interpersonal skills. Salary and Benefits: Salary negotiable and competitive, dependent upon experience, qualifications, and rank. Rank will be either Associate Librarian or Librarian. This is a tenure track academic appointment in the Libraries, which includes eligibility for sabbatical leaves. Benefits include a university health care plan, TIAA/CREF retirement/annuity plan, group life insurance, and liberal vacation and sick leave. To apply, send letter of application, professional vita, and names, addresses, and phone numbers of four references to: Lila Fredenburg, Libraries Human Resources Officer Indiana University Libraries Main Library C-201 Bloomington, In 47405 812-855-8196; fax: 812-855-2576; e-mail: lfredenb@indiana.edu Review of applications will begin no later than November 23, 1998. The search will remain open until the position is filled. For further information concerning Indiana University: http://www.indiana.edu/iub. Indiana University is an affirmative action/equal opportunity employer. ---------------- Announcement of Position Opening Dean, College of Fine Arts Western Michigan University Western Michigan University seeks nominations and applications for the position of Dean of the College of Fine Arts. Located in Kalamazoo, WMU is a Carnegie Doctoral I university with an enrollment of 26,500 students, 25% at the graduate level. Six colleges employ approximately 775 faculty. The College of Fine Arts consists of the School of Music and the Departments of Art, Dance, and Theatre. Music and Art offer graduate as well as undergraduate programs. The College has 110 faculty and staff, more than 1800 majors and minors, and 4000 other students enrolled each year. The College has state-of-the-art facilities for rehearsal, production, and performance. As the College's chief academic and executive officer, the dean reports to the provost and is responsible for instructional programs, maintaining accreditation, promoting the creative and scholarly work of the faculty, leading fund-raising efforts, and extending the college's outreach locally, in the state, in the nation, and abroad. The successful candidate will have an appropriate terminal degree (or equivalent background and experience), a record of scholarly and/or artistic achievement suitable for tenure and a full professorship in one of the units of the College, a strong grasp of the instructional, scholarly, artistic and performance components of the mission of a college of fine arts, and demonstrated administrative and leadership ability. Send applications and nominations to: Auerbach Associates/Western Michigan 65 Franklin St., Suite 400 Boston, MA 02110 email: jaa@auerbach-assc.com Review of applications will begin immediately and continue until the position is filled. WMU is an equal opportunity employer and encourages qualified women and members of under-represented groups to apply. For more information about Western Michigan University, please visit WMU's Web site at: http://www.wmich.edu ============== 5. New Dissertations AUTHOR: Carey, Norman, A. TITLE: Distribution Modulo 1 and Musical Scales INSTITUTION: University of Rochester BEGUN: July 1996 COMPLETED: February 1998 ABSTRACT: This dissertation examines the relationships between the mathematics of distribution modulo 1 and the theory of well-formed scales. Distribution modulo 1 concerns the distribution of real numbers between 0 and 1. In particular, finite sets of real numbers have been studied with respect to the Steinhaus Conjecture, proven by Sós and others. Well-formed scales, first introduced in Carey and Clampitt 1989, are generated by iterations of a given musical interval modulo the octave, the standard musical interval of periodicity. An introductory survey of ten scale theorists provides a context in which to understand the properties of the well-formed scale. A scale is well-formed if each generic interval comes in two specific sizes, or if it consists of equal step intervals. The structure of the well-formed scale is a function of the continued fraction representing the log ratio of the generator ("fifth") and the interval of periodicity ("octave"). The diatonic scale in Pythagorean tuning serves as the prototype: the generator is the overtone fifth (3:2) and the interval of periodicity is the octave (2:1). The diatonic is a member of an infinite hierarchy of well-formed scales, recursively generated by the continued fraction of Log 2 (3/2). This hierarchy also includes the pentatonic and chromatic collections. In general, the well-formed scale belongs to a hierarchy determined by the continued fraction of, Log I (G), where I is the frequency ratio of the interval of periodicity and G is the frequency ratio of the generator. Five theorems are presented that characterize well-formed scales, their hierarchies, and the patterns of step intervals they exhibit. The step patterns themselves form the basis for a secondary system of well-formed scale classification. The conditions on "coherence" for well-formed scales are fully characterized. Also discussed are applications and extensions of the theory, including tuning theory, rhythmic analysis, and composition. KEYWORDS: scale theory, well-formed, maximally even, Myhill's Property, diatonic, coherence, microtonal, rhythm, distribution modulo 1, continued fractions TOC: I Diatonic Theory A Introduction B Foundational and Structural Properties C Definitions D Diatonic Theory - Antecedents E Three Diatonic Theories II Well-formed Scales A Diatonic Theory and Well-formed Scales B The Theory of Well-formed Scales III Five Theorems Concerning Well-formed Scales A Introduction B Distribution Modulo 1: The Three-Gap Theorem C Well-formed Scales and the Multiplicative Permutation D Symmetry and Closure E Well-formed Scales and Myhill's Property F The Well-formed Scale Sequence G Generic Ordering ("Coherence") IV Applications and Extensions A Microtonalism and Well-formed Scales B Rhythm and Well-formedness C Diatonic Theory and Compostion D Conclusion and Prospects CONTACT: Eastman School of Music nac@theory.esm.rochester.edu ------------------ AUTHOR: Djordjevic, Michael, L. TITLE: Discrete Tone Relations Determined by the Hearing Phenomenon within Five-Dimensional Sound-Musical Continuum INSTITUTION: Radio Belgrade, Hilandarska 2, 11000 Belgrade, Serbia, YU BEGUN: January, 1990. COMPLETITION: June, 1995. ABSTRACT: I. Theory Five-Dimensional Sound-Musical Continuum materializes itself into objective Sound Space and subjective Musical Space defined as MOS-Musical Organisation of Sound. Sound-Musical Space is defined as union of the sets: A the set of parameters related to the sound signal at its source, B the set of parameters of the sound signal with a listener. A and B interactions with binary relations of ordered pairs represents the new set C. MOS defined as set X implicates Y as set of aural, final psychological perception of music. II. Experiments Quantum of Hearing Discrimination was elaborated in experiments MATEST 1 and 2 with values in cents ranged from 3-7, extreme 2, being constant along the greater part of the frequency spectrum, 4 cents. Unit called Unitary Distance was defined. Phase Space Cell signifies distribution of DTR within the Sound-Musical Space as matrix. MATEST 2 included several different tonal systems for comparison and DTR system justification. III. Practice DTR-Discrete Tone Relations: 1. Hearing discrimination 2. Temperament 3. Intonation 4. Musical abilities 5. Compositional organization of DTR 6.Ortography and notation 7. Reproductive medium with its limits 8. Constructing new musical instruments 9. Aesthetics and axiology KEYWORDS: discrete, tone, relations, DTR, UD, unitary, distance, STTR, PSC, phase, space, cells, QHd, quantum, hearing, discrimination, phenomenon, sound, musical, continuum, fivedimensional, matest, system TOC: Introduction I. Theoretical aspects: 1. General conception of universal sound-musical manifestation 2. Materialization of sound-musical continuum into union of sets of all musical parameters, 3. Defining of tone pitches and their relations on the base of quantum theory i.e. discrete atributs of sound-musical space, 4. Presentation of discrete tone relations system and its implications in regard to hearing phenomenon II. Experimental aspects: 5. Experimental determination of hearing threshold value and quantum of hearing discrimination within the range of discrete tone pitches and their relations 6. Projection of discrete tone relation systems and selection of its optimal variants 7. Testing of discrete tone relations system in relation to already existing tone systems and proofs of its universality, validity and practical benefits III. Practical aspects: 8. Practical application of discrete tone relations from the aspect of contemporary musical science 9. Discrete tone relations application from the aspect of reproductive musical media, 10. Systematization of theoretical and practical results of discrete tone relations phenomenon research, Summary, Literature, Application of references towards fundamental theses of dissertation. CONTACT: P. Lekovica 44/4 11000 Belgrade Serbia, YU phone: 381-11-3546262, e-mail: mihajlod@EUnet.yu URL:http://SOLAIR.EUnet.yu/~mihajlod -------------------- AUTHOR: Lemieux, Glenn C. TITLE: "Music in Twelve Parts" by Philip Glass: Reconstruction, Construction and Deconstruction" INSTITUTION: University of Iowa BEGUN: January 1998 COMPLETED: June 1999, projected completion ABSTRACT: "Music in Twelve Parts" is a major work by Philip Glass written between 1971 and 1974. In essense, it summarizes the compositional techniques Glass was using which have come under the rubric of minimalism. Glass says: "All of my works which predate 1976 fall within the highly reductive style known as minimalism. I feel that minimalism can be traced to a fairly specific timeframe, from 1965 through 1975, and nearly all my compositions during this period may be placed in this general category. All such categories are arbitrary, however, and can be misleading. For example, although "Music in 12 Parts" would most likely be classified as a minimal work, it was a breakthrough for me and contains many of the structural and harmonic ideas that would be fleshed out in my later works. It is a modular work, one of the first such compositions, with twelve distinct parts which can be performed separately in one long sequence, or in any combination or variation. Each part concentrated on several of these techniques (cyclic, additive and repetitive structure), and by the completion of Part 10, the cataloguing was fairly intact. Thus Part 11 concentrated on the joining places of the other parts, which, to the listener, appeared as modulations. Part 12 turned to cadence--the formal closing phrases we are accustomed to hearing in western music--as a fitting end to such an extended piece. The major problem in examing the music is that a full score of "Music in Twelve Parts" does not exist. In fact, it is only available as a partially-orchestrated sketch and a set of hand-written instrumental parts. To solve this problem, Dunvagen Music Publishers, Glass's own company, has approved the engraving of this piece in FINALE as part of this project. In addition, an in-depth analysis of whole piece does not exist, although various general descriptions and single movement analyses can be found. To date, three recordings of the piece have been made: in 1974, parts one and two were recorded on the Caroline label; in 1989, a recording of the entire piece was made by Virgin Records; and in 1996, Nonesuch released another recording of the entire piece. KEYWORDS: minimalism, cyclicism, additive, repetitive, rhythm, modular TOC: I. Introduction II. Reconstruction: background to the music 1. Minimalism 2. Previous works by Glass 3. Theoretical and philosophical basis III. Construction: making the score 1. Full score 2. Commentary IV. Deconstruction: Analysis of the piece 1. Macrostructure 2. Proportions and microstructures 3. Harmonic analysis 4. Orchestration V. Conclusion CONTACT: Glenn Lemieux P.O. Box 2771 Iowa City, IA 52244 319-339-1651 (h) 319-335-5877 (w) glenn-lemieux@uiowa.edu --------------------- AUTHOR: Quaglia, Bruce, W. TITLE: Compositional Practice and Analytic Technique; Schoenberg's Atonal Works: Reconciling Approaches to Sets, Lines and Developing Variation. INSTITUTION: University of Utah BEGUN: 9/97 COMPLETED: 10/98 ABSTRACT: This thesis examines the application of linear reductive analytic techniques as applied to pieces from Arnold Schoenberg's atonal period. Basic nomenclature and concepts from pitch-class set theory are invoked as well. Analyses of Schoenberg's Op. 11 no. 1 and Op. 15 song X are presented within the context of Schoenberg's own compositional theories as suggested by his pedagogical and critical writings. A brief discussion of the relevant theoretical and analytical literature is also presented in order to place the present analyses within the relevant context. The remaining dissertation requirement is an original composition: "In A Mirror Dimly" for Soprano, Violin, Picc./Alto Flute, Percussion and Computer Synthesized Tape. KEYWORDS: Schoenberg, Forte, Developing Variation, 20th C. Analysis, Linear Analysis, Set Theory. CONTACT: Bruce Quaglia c/o Dept. of Music 204 Gardner Hall University of Utah SLC, UT 84 bruce.quaglia@m.cc.utah.edu --------------------- AUTHOR: Van Colle, Sue, J. TITLE: "Music therapy process with cerebral palsied children: connections with psychoanalytic models, particularly that of Winnicott." INSTITUTION: University of Reading, Department of Music BEGUN: 4, 1988 COMPLETION: 10, 1999 ABSTRACT: This research aims to make a detailed description of processes that occur in interactive music therapy with cerebral palsied children. The writer has made a video-tape collection of her clinical work which she undertook, over one academic year, with two groups each of four severely and multiply handicapped cerebral palsied children. Data analysis involves both manual and computerised systems, and draws on ethological methods. There are two major aims: (1) to generate the hypothesis that the role of the music therapist is like that of the good-enough mother as described by Donald W Winnicott, and (2) to generate some broad rules of music therapy. KEYWORDS: music therapy, child, handicap, cerebral palsy, piano, process, interaction, psycoanalytic, Winnicott, ethological TOC: 1. Music Therapy: Introduction and Broad Historical Overview 2. Music Therapy Research in Great Britain 3. The Use and Significance of the Piano in Music Therapy 4. The Writer's Clinical Work 5. Research Method 6. Measures of Behaviour: Description of Observables 7. Descriptive Analysis 8. Results of Teacher's Ratings 9. Examining the Process of Music Therapy 10. Conclusion CONTACT: University of Reading 35 Upper Redlands Road, Reading, Berkshire RG1 5JE, UK Email: svc@clive.jenkins.clara.net ------------------- AUTHOR: Vives, Thomas E. TITLE: The Effect of Timbre on the Chord Identification Accuracy of Sophomore-Level College Music Theory Students INSTITUTION: University of Florida BEGUN: September, 1997 COMPLETED: August, 1998 ABSTRACT: This study examined electronic keyboards in use at college and university music departments, the available sounds these keyboards have in common, and the effects of these sounds on students' identification accuracy in harmonic aural skills exercises. Correlations were made between the different timbres and the students' levels of accuracy. The primary question that this study attempted to address was as follows: Does any one specific timbre facilitate greater student accuracy? This study comprised a single experimental trial--a prepared treatment that contained forty-five random examples of triads and seventh chords--that tested several timbral conditions under a single test condition--specifically, the identification of triad or seventh chord quality. The dependent variable for this trial was the subjects' accuracy in identifying both triads and seventh chords. The independent variables were (1) the nine different types of triads and seventh chords (5 seventh chord types and 4 triads) and (2) the five different timbres. The five timbres (electric piano, harpsichord, organ, acoustic piano, and vibraphone) were selected for this study based on communication with keyboard manufacturers. All timbres were generated via digital synthesizers. Two intact first-semester sophomore-level music theory classes served as the sample for this study. Multiple analysis of variance showed that timbre by itself had no significant overall effect on the accuracy of subjects’ responses, although the electric piano and acoustic piano timbres produced slightly, but not significantly, more accurate responses. Closer analysis of the data, including the interaction of gender and principle performance instrument, indicated that neither gender nor principle performance instrument significantly affected subjects' response accuracy, although due to the small size of the subject pool, the results for principle performance instrument were considered unreliable. Suggestions for further research and future studies are included. KEYWORDS: Timbre, Perception, Pedagogy, Aural Skills, CAI, CBI. TOC: DEDICATION ACKNOWLEDGMENTS ABSTRACT CHAPTERS 1 INTRODUCTION Statement of Purpose Purpose of the Study Need for the Study Definitions Limitations of the Study Delimitations of the Study 2 REVIEW OF LITERATURE General Timbre Studies Computer-Assisted and Computer-Based Instruction Timbre Perception and Timbre Preference in Children Timbre Perception in College Students Discussion of the Literature 3 REVIEWING THE PRODUCTS Keyboard Manufacturers Models and Specifications Keyboards in use at University Music Departments 4 METHODOLOGY Introduction The Timbres Design and Apparatus Preparation of the Stimuli Validity and Reliability Pilot Study Subjects Administration of the Test Instrument Hypotheses Analysis 5 RESULTS Timbre Analysis Gender Analysis Principal Performance Instrument Analysis Interactions Between Gender, PPI, Timbre, and Triad and Seventh Chord Quality 6 SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS FOR FURTHER RESEARCH Summary Conclusions Recommendations for Further Research APPENDICES A TABLES B FIGURES C TEST INSTRUMENT MATERIALS AND CORRESPONDENCE D SUBJECTS' WRITTEN RESPONSES TO TEST INSTRUMENT BIBLIOGRAPHY BIOGRAPHICAL SKETCH CONTACT: Ted Vives 5724 NW 25th Terrace Gainesville, Fl 32653 (352) 338-2785 tvives@atlantic.net ----------------- AUTHOR: Weisser, Benedict J. TITLE: Notational Practice in Contemporary Music: A Critique of Three Compositional Models (Luciano Berio, John Cage, and Brian Ferneyhough) INSTITUTION: The City University of New York BEGUN: July, 1995 COMPLETED: July, 1998 ABSTRACT: The purpose of this dissertation is to to examine the integration of notation and content in contemporary music. In particular, it is to show that for the three composers I have chosen, Luciano Berio, John Cage, and Brian Ferneyhough, the notation of a work is not just a traditional 'encoding' but is inextricably linked to its form and content. Their compositional agendas are in many respects defined by their notation. Following an introductory chapter, in which the breadth of twentieth-century notational innovation and experimentation is presented, chapter two deals with the music of Luciano Berio. I compare the 1958 version of his Sequenza I with the 1992 version in metered notation. The title of chapter two, 'notation-as-play within a predefined system,' is the basis of what I see as the success of Berio's works both from a compositional as well as a performance standpoint. In chapter three I study notational aspects of the late music of John Cage, the works known as the 'time-bracket' or 'number' pieces. In these late works, Cage uses notation to reconcile and accommodate himself to certain elements of musical expression, most notably harmony and the very notion of vertical relationships. Purely notational considerations produce harmonic situations that Cage could accept, a flexible, 'anarchic harmony' which is also highly determinate and 'coherent.' In the case of Brian Ferneyhough, the subject of chapter four, notation is approached as a kind of 'inventory of processes,' where various pre-compositional generations of multi-metric structures and compositional transformations of material are presented in an ostensibly unfiltered manner. One now encounters a situation where the composer has no discernible interest in compromising his material to the predispositions of the performer. Instead, Ferneyhough is interested in using notation as a 'behavior-altering agent,' a new notion of 'communication' radically different from both Berio and Cage. Finally, in a concluding chapter I put Berio, Cage, and Ferneyhough in a deeper context, comparing them to each other and reflecting on their importance. I also venture my own opinions as to the future influence of the kind of notational thought they each embody. KEYWORDS: graphic notation, proportional notation, Eco, time-brackets, number pieces, experimental music, McLuhan, new complexity TOC: Abstract Preface Acknowledgements List of Examples Chapter 1 - An Introduction to Notational Practice since 1945 Chapter 2 - Luciano Berio: Notation-as-play within a predefined systemChapter 3 - John Cage: '...the whole paper would potentially be sound': Time-Brackets and the Number Pieces (1981-92) Chapter 4 - Brian Ferneyhough: Notation-as-Inventory Chapter 5 - Conclusions Appendix A - Interview with Luciano Berio Appendix B - Berio, Sequenza I (1958 version) Appendix C - Berio, Sequenza I (revised version, 1992; marked up by B.W.) Bibliography Autobiographical Statement CONTACT: Benedict Weisser Visiting Assistant Professor of Composition, Oberlin Conservatory of Music 77 West College Street Oberlin, OH 44074 phone: 440 775 8254 e-mail: BenWeisser@aol.com Ben_Weisser@qmgate.cc.oberlin.edu home address: 140 Elm Street apt. 2 Oberlin, OH 44074 phone: 440 775 0248 FAX: 440 775 8942 ============== 6. New Books Oxford University Press *Encyclopedia of Aesthetics* 4-Volume Set Edited by Michael Kelly, *Journal of Philosophy*, Columbia University Now available for the first time--a comprehensive, interdisciplinary survey of critical reflections on art and culture, society, and nature. In four volumes and 600 original articles, this landmark reference source surveys an extraordinary range of ideas about the arts in human life. Hundreds of leading scholars--philosophers, art critics and historians, literary theorists, musicologists, anthropologists, sociologists, and others--offer writings that link current critical thought about the contexts of art to long-standing philosophical questions on the nature, meaning, and experience of art. From Theodor Adorno to Arnold Schoenberg, improvisation to interpretation, and African aesthetics to video, the articles in this encyclopedia examine a remarkable range of ideas and issues: * the full breadth of critical thought on the arts, from classical philosophy to current literary theory * an in-depth survey of Western and non-Western traditions * the political, social, economic, and ideological contexts of creativity * the critical vocabularies of many different forms, genres, and periods, from classicism to jazz and folk art to digital media * the work of dozens of leading thinkers, from Bakhtin and Barthes to Schapiro and Stravinsky Other features include: * 600 original articles * 450 distinguished contributors * 100 illustrations * Bibliographies, cross-references, and an exhaustive index * 2,240 pages Four volumes 1998 2224pp.; 100 photos 140. 511307-1 $495.00a/$396.00 * Special shipping charge of $25.00 *The Sound of Medieval Song: Ornamentation and Vocal Style According to the Treatises* Timothy J. McGee, University of Toronto Latin translations by Randall A. Rosenfeld This study of how sacred and secular music were actually sung during the Middle Ages draws upon manuscript notations and statements found in approximately 50 theoretical treatises written between 600 and 1500. The writings describe various singing practices and illustrate desirable and undesirable vocal techniques, thus showing how singers approached the music of this period. (Oxford Monographs on Music) 1998 232 pp.; 11 b/w plates, 3 maps, music examples 377. 816619-2 $65.00w/$52.00 *Modal Counterpoint, Renaissance Style* Peter Schubert, McGill University This book introduces the rules of writing and analyzing 16th-century music through a wide variety of carefully graded exercises. Unlike similar works, which bear little stylistic relation to the music of the period, this is the only species counterpoint book that uses examples and concepts taken directly from 16th-century treatises and contemporaneous theoretical sources. The authors make a clear distinction between technical requirements ("hard rules") and general stylistic guidelines ("soft rules"). Their selection of Renaissance repertoire examples comprises many genres, including the French chanson, the Spanish organ hymn, the Italian falsobordone, the British keyboard in nomine, and the Lutheran chorale, illustrating the range of possibilities within a given technical formula. February 1999 304 pp.; 582 linecuts 66. 510912-0 spiral bound paper $45.00w*/$36.00 *Analyzing Bach Cantatas* Eric Chafe, Brandeis University The Bach cantatas are among the highest achievements of Western musical art, yet studies of individual Bach cantatas that are both illuminating and detailed are few in number. Here, Eric Chafe combines theological, historical, analytical, and interpretive approaches to the cantatas to offer the reader and listener the richest possible experience in light of the composer's intentions as well as the enduring and universal qualities of the works. Concentrating on a small number of representative cantatas, mostly from the Leipzig cycles of 1723-24 and 1724-25, and in particular on Cantata 77, Chafe illustrates how Bach strove to mirror both the dogma and mystery of religious experience in musical allegory. January 1999 336 pp.; 39 music examples 215. 512099-X $55.00w/$44.00 *Fantasy Pieces: Metrical Dissonance in the Music of Robert Schumann* Harald Krebs, University of Victoria, Canada This book presents a theory of metrical conflict and applies it to the music of Robert Schumann, thereby placing that compser's distinctive metrical style in focus. Krebs describes various categories of metrical conflict that characterize the music of Schumann, investigates how states of conflict arise and are manipulated and resolved in the course of compositions, and studies the interaction of metrical conflict with form, pitch structure, and text. The theoretical material is fancifully interwoven with commentary by Florestan and Eusebius, fictional characters based on Schumann's names for contrasting aspects of his own personality, who provide numerous illustrations from "their" compositions. Rich in allusions to Schumann's titles, his writings, and his life, this work will appeal to all students and fans of this composer's music, as well as to theorists and scholars. February 1999 304 pp.; 287 music examples 133. 511623-2 $70.00w/$56.00 *The Gershwin Style: New Looks at the Music of George Gershwin* Edited by Wayne Schneider, University of Vermont George Gershwin is at once one of America's most popular and least appreciated composers. He is loved and revered for his wonderful songs, a few instrumental works, and for the opera Porgy and Bess. But most of his music is virtually unknown--hundreds of songs, show music, and even several large and important instrumental works are gradually fading with the generations that first heard them. For an essay collection that promises to make a key contribution to American music research, Schneider has corralled some of the leading authorities in the area of Gershwin research alongside a few contributors who are approaching Gershwin for the first time (from the perspective of American music or popular music, generally). Essayists include Wayne Shirley, Charles Hamm, Edward Jablonski, and Artis Wodehouse, who has transcribed most of Gershwin's piano performances. October 1998 336 pp.; 3 halftones, 2 linecuts 135. 509020-9 $35.00w*/$28.00 *Rethinking Music* Edited by Nicholas Cook and Mark Everist, both at University of Southampton Offering a comprehensive re-evaluation of current thinking about music, this book collects the work of 24 distinguished musicologists, music theorists, and ethnomusicologists. The contributors review different dimensions of musical study, revealing a range of concerns that are shared across the discipline: the nature of musicological practice, its social and ethical dimensions, issues of canon and value, and the relationship between academic study and musical experience. October 1998 752 pp.; 33 music examples 205. 879003-1 $98.00w/$78.40 *Unfoldings: Essays in Schenkerian Theory and Analysis* Carl Schachter Edited by Joseph N. Straus, both at Queens College and Graduate School, City University of New York Schachter is, by common consent, one of the most important music theorists currently at work in North America. He is the preeminent practioner in the world of the Schenkerian approach to the music of the 18th and 19th centuries, which focuses on the linear organization of music and now and again dominates discussions of the standard repertoire in both university classrooms and professional journals. His articles have appeared in a variety of journals, including some that are obscure or hard to obtain. This volume, then, gathers several of his finest essays, including discussions of rhythm in tonal music, Schenkerian theory, and text setting, as well as a pair of analytical monographs. November 1998 352 pp.; 147 music examples 252. 512590-8 paper $24.95w*/$20.00 163. 512013-2 cloth $65.00w/$52.00 *Philosophical Perspectives on Music* Wayne D. Bowman, Brandon University, Manitoba, Canada This challenging introduction to the issues and problems of music philosophy explores diverse accounts of the nature and value of music. It offers an accessible, even-handed consideration of philosophical orientations without advocating any single one, demonstrating that there are a number of ways in which music may reasonably be understood. Bowman thus examines the strengths and advantages of each perspective--as well as its inevitable shortcomings. From the pre-Socratic Greeks to idealism, through phenomenology, and on to contemporary socio-cultural critiques, this survey examines the views of selected influential thinkers. Examining what music is, how it works, and what it is good for, the book encourages music students and musicians to join in important conversations that shape both how they practice their art and how they and others understand it. 1998 496 pp. 217. 511296-2 $45.00s/$36.00 *Art and Emotion* Derek Matravers, The Open University Here, Matravers examines how emotions form the bridge between our experience of art and life. We often find that a particular poem, painting, or piece of music carries an emotional charge; and we may experience emotions toward, or on behalf of, a particular fictional character. Matravers shows that what these experiences have in common, and what links them to the expression of emotion in non-artistic cases, is the role played by feeling. He carries out a critical survey of various accounts of the nature of fiction, attacks contemporary cognitivist accounts of expression, and offers an uncompromising defense of a controversial view about musical expression: that music works by expressing the emotions it causes its listeners to feel. 1998 248 pp. 175. 823638-7 $65.00w/$52.00 ------------------- Princeton University Press *The Art and Science of Renaissance Music* James Haar Harking back to a time when the inability to sing was met with consternation and disdain, when dancing offered a method of assessing a potential spouse's health, when people played virginals and serpents and shawms and racketts, *The Science and Art of Renaissance Music* (Princeton University Press; November 5, 1998; $45.00 US) presents noteworthy essays of analysis. James Haar, a distinguished scholar of Renaissance music, has had an abiding influence on how musicology is undertaken owing in great measure to a substantial body of articles published over the past three decades, and here provides representative pieces from those years. He discusses in turn the accomplished Antonfrancesco Doni; the Italian madrigal; various problems of theory; and the role of music in Renaissance culture, from advice given by sixteenth-century critics to the nineteenth century's attitudes to early music. In *The Art and Science of Renaissance Music*, Haar explores the same subject from several angles, giving a rich context for further exploration. He was one of the first to recognize the value of cultural study in general, and his work serves to remind us of the merit in examining music specifically. The articles contained in *The Art and Science of Renaissance Music*, published together for the first time, show the author's conviction that a good way to address large problems is to begin by focussing on small ones. James Haar is William Rand Kenan, Jr., Emeritus Professor of Music at the University of North Carolina, Chapel Hill. Among his books are *Essays on Italian Poetry and Music in the Renaissance* and, with Iain Fenlon, *The Italian Madrigal in the Early Sixteenth Century*. 389 pages Cloth $45.00 ISBN 0-691-02874-5 U.S. Publication date: November 5, 1998 Foreign Publication date: December 7, 1998 Contact: Sara Lerner fax: 609/258-1335 e-mail: Sara_Lerner@pupress.princeton.edu PUP Web site: http://pup.princeton.edu ------------------ University of Rochester Press *Theories of Fugue from the Age of Josquin to the Age of Bach* Paul Mark Walker Few bodies of Western music are as widely respected, studied, and emulated as the fugues of Johann Sebastian Bach. Despite the esteem which Bach's contributions brought to this genre, however, the origian and early history of the fugue remain surprisingly poorly understood. *Theories of Fugue* addresses both the history and methodology of the pre-Bach fugue (from roughly 1500-1700), and, of greatest significance, to the literature, it seeks to present a way out of the methodological dilemma of uncertainty which has plagued previous scholarly attempts by considering what musicians of the time had to say about the fugue: what it was, what it was not, how important it was, and where and how a composer should (or shouldn't) use it. Paul Mark Walker is professor of musicology at the University of Virginia. 4 b/w illustrations 352 pp, 6 x 9 March 1999 Eastman Studies in Music Series, vol. 13 ISSN 1071-9989 Distributed by Boyd & Brewer, Ltd. *Elliot Carter: Colleged Essays and Lectures, 1937-1995* Jonathan Bernard, editor "Carter is a very articulate and entertaining writer...This collection is a treasure and will be enjoyed by anyone interested in the serious music of the 20th-century. Very highly recommended." (Choice) 27 b/w illustrations 380 pages, 228 x 152 $24.95 (paperback) 1 580460 25 9 ------------------- Routledge *Musical Performance: A Philosophical Study* Stan Godlovitch, Lincoln University, Canterbury, New Zealand Most music we hear comes to us via a recording medium on which sound has been stored. Such remoteness of music heard from music made has become so commonplace it is rarely considered. *Musical Performance: A Philosophical Study* considers the implications of this separation for live musical performance and music-making. Rather than examining the composition or perception of music as most philosophical accounts of music do, Stan Godlovitch takes up the problem of how the tradition of active music playing and performing has been challenged by technology and what problems this poses for philosophical aesthetics. Where does does the value of musical performance lie? Is human performance of music a mere transfer medium? Is the performance of music more expressive than recorded music? Musical Performance poses questions such as these to develop a fascinating account of music today. musicians - but via some recording medium on which sound has been stored. 'This is an impressive piece of work, which represents an original and imaginative contribution to aesthetics and the philosophy of music. - Alex Neill, University of St Andrews 'Godlovitch brings to bear not only a through knowledge philsophical aesthetics, but also a considerable understanding of musicianship. His book represents an original and imaginative contribution to aesthetics and the philosphy of music.' - Alex Neill, University of St. Andrews 'Stan Godlovitch is a distinctive voice in musical aesthetics. Musical Performancen offers a compelling picture of the performance of music and is essential reading for anyone in the field.' - Jerrold Levinson, University of Maryland Published in the EU: September 1998 Published in North America: November 1998 234x156mm 6.25x9.25 inches: 184pp Hb: 0-415-19128-9 EU List Price: #45.00 US List Price: $75.00 Canadian List Price: $105.00 Pb: 0-415-19129-7 EU List Price: #14.99 US List Price: $24.99 Canadian List Price: $34.99 Within the Americas, please direct your order to: Routledge 7625 Empire Drive Florence, KY 41042 Telephone: 800 634-7064 FAX: 800 248-4724 email: cserve@routledge-ny.com Rest of World and for all journal subscriptions, please direct your order/subscription to: thomson.com Cheriton House North Way Andover SP10 5BE England UK Telephone: 44 1264 342922 FAX: 44 1264 342761 email: row-info@list.thomson.com Please note that prices of products in locations outside the country of origin may vary. Orders may be subject to local taxes. 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