=== === ============= ==== === === == == == == == ==== == == = == ==== === == == == == == == == = == == == == == == == == == ==== 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 4 July, 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 Articles AUTHOR: Scholtz, Kenneth P. TITLE: Algorithms for Mapping Diatonic Keyboard Tunings and Temperaments KEYWORDS: Algorithm, chain of fifths, diatonic scale, equal temperament, enharmonic, just intonation, meantone, Pythagorean tuning, schisma, syntonic comma Kenneth P. Scholtz 2821 Anchor Ave. Los Angeles, CA 90064-4605 kscholtz@earthlink.net ABSTRACT: Diatonic keyboard tunings in equal temperament, just intonation, meantone and well tempered scales are derived from Pythagorean tuning using algorithms whose operations involve combinations of pure fifths and syntonic commas. Graphic diagrams of the line of fifths are used to show the harmonic and mathematical relationships between common tunings and temperaments. Four modes of just intonation are derived from Pythagorean tuning by an algorithm that narrows each major third by one syntonic comma. Equal temperament is approximated with imperceptible error by algorithms that narrow Pythagorean and justly tuned enharmonic intervals by one or more syntonic commas. [1] Introduction [1.1] This article describes algorithms that map the traditional harmonic tunings and temperaments for the keyboard: Pythagorean tuning, equal temperament, just intonation, and historical variations on meantone temperaments. Keyboard tuning is used because it has been worked out in detail over hundreds of years and is understood to give a reasonable approximation of the pitch choices made by performers on instruments without fixed pitches. I have excluded analysis of multiple divisions of the octave into more than twelve notes because they were not commonly used temperaments. Working from the chain of fifths, we will demonstrate how the intervals in any tuning or temperament can be mapped from Pythagorean tuning by algorithms that combine pure fifths and syntonic commas.(1) The algorithms that map the tunings and temperaments into one another are not purely mathematical, but are derived from the harmonic structure of the chain of fifths and enharmonic relationships between notes. ================ 1. The syntonic comma is defined as the difference between the Pythagorean tuning and just tuning of the major third. The difference between the just third (5/4) and the Pythagorean third (81/64) is 81/80, calculated as follows: 81/64 x 4/5 = 81/16 x 1/5 = 81/80. The syntonic comma is also the difference between the Pythagorean and just tunings for all diatonic intervals other than the fourth and fifth, which are the same in both tunings. The reason for the repeated appearance of the syntonic comma will be apparent from the discussion of the four modes of just intonation in section 5. ================ [1.2] This article is not concerned with generating diatonic scale tuning from prime numbers or otherwise deriving its form from physical or mathematical principles. We will use the historic definition of a diatonic scale as two tetrachords plus one additional tone that completes the octave. The diatonic octave is divided into five whole tones and two semitones in all tunings and temperaments. The convention of assigning letter names to notes in a 12-note chromatic keyboard for diatonic scales will be observed. The chain of fifths will be introduced and used as the defining feature common to all scales without initially assuming any particular tuning. [1.3] Several modern theorists, including Mark Lindley(2) and Easley Blackwood(3) have applied algebraic techniques in describing the tonal relationships which form traditional diatonic scales. Others have developed diagrams to compare the differences in pitch for a note in various tunings.(4) Although such techniques have their uses, I did not personally find them particularly helpful in visualizing the overall difference between one tuning system or temperament and another. For my own analysis, I prefer to diagram the chain of fifths and use rational fractions to express the size of the intervals.(5) Differences between tuning systems are indicated by placing commas or fractional commas between adjacent fifths. This provides a visual comparison between the separate diatonic tunings and temperaments of a complete scale and makes it easier to demonstrate the mathematical and musical relationships between them. ============== 2. Mark Lindley and Ronald Turner-Smith, "An Algebraic Approach to Mathematical Models of Scales," *Music Theory Online* 0.3 (1993), which is a commentary based upon their book, *Mathematical Models of Musical Scales* (Bonn: Verlag fuer Systematische Musikwissenschaft, 1993). 3. Blackwood, *The Structure of Recognizable Diatonic Tunings*, (Princeton: Princeton University Press, 1985). 4. See, for example, L. L. Lloyd's diagrams in his articles in the 1954 *Grove's Dictionary of Music and Musicians* on "Just Intonation" and "Temperaments." 5. Intervals measured by rational fractions can be converted into cents using the following approximate values: octave = 1200 cents, perfect fifth = 702 cents; Pythagorean comma = 24 cents; syntonic comma = 22 cents. =============== [1.4] Theorists classify tuning systems as either cyclic, generated by a reiterative sequence of fifths, or divisive, tunings that subdivide the octave.(6) Historically, this division is somewhat arbitrary. Euclid and Boethius derived the Pythagorean tuning that we now associate with a sequence of fifths by dividing a monochord into two octaves.(7) Methods based upon the overtone series or other systems which attempt to derive just intonation from acoustic phenomena are often considered to be divisive in nature. The typographical diagrams utilized in this article interpret tunings and temperaments in terms of the line of fifths. =============== 6. See Lindley and Turner-Smith, "An Algebraic Approach," paragraph 4. 7. In modern terminology, dividing the string in half gives two octaves, in thirds gives a fifth and an eleventh, and in fourths gives a fourth, octave and double octave. The other diatonic notes are then determined by calculating intervals of a fifth from these intervals. This produces two diatonic octaves in Pythagorean tuning. =============== [2] The Chain of Fifths [2.1] It is well known that every diatonic scale can be reordered into a sequence of fifths. For the C-major scale, the sequence of fifths is F-C-G-D-A-E-B, renumbering the notes of the scale in the order 4-1-5-2-6-3-7. This is the sequence of notes produced when one starts with C and then tunes a keyboard alternately down a fourth and up a fifth until the last note of the scale is reached. For consistency of metaphor, I call the sequence of fifths a "chain" in which the fifths are "links," without assuming any specific tuning system. [2.2] The chain of fifths has its origin at C and extends to the sharp notes on the right and the flat notes on the left. The chain is theoretically endless, neither closing nor repeating, so that each note in the chain is musically unique.(8) Figure 1 shows the central 21 links in the chain, with C at the origin. All the notes in the chain are considered to be within a single octave bounded by two C's. Since fifths and fourths are inversions, each note is a fifth above the note to its left and a fourth above the note to its right. As a matter of convention, it is useful to consider the links to the right of C to be ascending fifths and the those to the left to be ascending fourths. Bbb-Fb-Cb-Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#-D#-A#-E#-B# Fig. 1. A Section of the Chain of Fifths The same seven letter names repeat themselves over and over in the same invariant sequence of the C-major scale-- F-C-G-D-A-E-B--with each repetition augmented by sharps to the right and flats to the left. Any group of six adjacent links defines a major diatonic scale whose tonic note is one link from the left end of the group, and a natural minor scale which starts two links from the right end. Horizontal movement of any six-link group is the equivalent of transposition of the scale into a new key as exemplified by Figure 2. A twelve-note chromatic keyboard commonly contained the notes from Eb to G#, before the acceptance of equal temperament. When all the fifths were pure, the keyboard would be suitable for the diatonic major and minor scales in the keys depicted in Figure 2. Other keys would be unavailable because the required sharp or flat notes would be outside the range of the chain of fifths. Bb major, G-minor Eb-Bb-F-C-G-D-A F major, D-minor Bb-F-C-G-D-A-E C major, A-minor F-C-G-D-A-E-B G major, E-minor C-G-D-A-E-B-F# D major, B-minor G-D-A-E-B-F#-C# A major, F#-minor D-A-E-B-F#-C#-G# Fig. 2. Six Diatonic Scales in a Chromatic Chain From Eb to G# ============== 8. Perfect fifths do not combine to produce a perfect octave. The nth fifth in a sequence of n fifths is defined mathematically by the expression (3/2)^(n), which can never be an exact multiple of 2 since every power of 3 is an odd number. The mathematical proof that no sequence of fourths can ever produce an octave is less obvious since every even number can be expressed as a fraction with an odd denominator, and every power of 3 can be associated with an infinite sequence of even numbers, such as 18/9, 36/9, ...; 162/81,..., etc. However, an intuitive musical proof can be deduced from the fact that fourths and fifths are inversions of one another. No sequence of fourths can generate an octave because the inverse sequence of fifths can never do so. ================ [2.3] The chain of fifths can be used to rationally order the harmonic structure of the diatonic scale. Figure 3 lists the harmonic intervals available on a twelve-note chromatic keyboard for each pairing of links in the chain of fifths. The pairs of notes listed are examples that start with C. Any interval can be horizontally transposed along the chain of fifths. Ascending Harmonic Intervals Links Notes Left to Right Right to Left 1 C-G Perfect fifth Perfect fourth 2 C-D Major second Minor seventh 3 C-A Major sixth Minor third 4 C-E Major third Minor sixth 5 C-B Major seventh Diatonic semitone or minor second 6 C-F# Augmented Fourth Diminished Fifth or tritone 7 C-C# Chromatic semitone Diminished octave 8 C-G# Augmented fifth Diminished fourth 9 C-D# Augmented second Diminished minor seventh 10 C-A# Augmented sixth Diminished minor third 11 C-E# Augmented third Diminished minor sixth Fig. 3. Harmonic Intervals Ordered From the Chain of Fifths Figure 3 classifies harmonic intervals by the number of links and their direction on the chain of fifths. Each pair of intervals are inversions of one another. Chromatic intervals have more than six links, diatonic intervals have six or fewer. Generally speaking, major intervals and augmented intervals ascend from left to right, while minor intervals and diminished intervals ascend from right to left. The perfect fourth is the only consonant interval that ascends from right to left, which may reflect its harmonic ambiguity. Intervals that ascend to the right from a flat to a natural are always major, such as Bb-D, a four link ascending major third transposed from C-E. In the same manner, intervals that ascend to the right from a flat to a sharp are considered "augmented," as in Bb-D#, an 11-link augmented third transposed from C-E#. In the opposite direction, intervals that ascend to the left from a sharp to a natural are always minor, as in C#-E, a three link minor third transposed from C-Eb. Intervals that ascend to the left from a sharp to a flat are diminished, as in C#-Eb, a ten-link diminished third transposed from C-Ebb. Each pair of enharmonically equivalent notes is separated by twelve links. It is also well known that the progressions of diatonic harmony correspond to the chain of fifths, but this is a topic which will not be further detailed. [3] Pythagorean Tuning [3.1] Tuning the diatonic scale with pure fifths, now known as Pythagorean tuning, was the norm for nearly 2,000 years. The Pythagorean tuning of twelve notes in a standard Eb-G# keyboard is shown in Figure 4. The links are a sequence of powers of 3/2 to the right of C and powers of 4/3 to the left of C. Whenever necessary, the fraction is divided by two to keep the pitch within the compass of a single 2/1 octave. Thus, 3/2 x 3/2 = 9/4. 9/4 is divided by 2, making D = 9/8. Logarithms allow the sequence to be calculated by addition instead of multiplication. In logarithmic measure, each perfect fifth is approximately 702 cents. The tuning of D is equal to 204 cents, obtained by subtracting a an octave of 1200 cents from 702 + 702 = 1404. Since all the links are the same size, the harmonic intervals can be transposed freely. All major seconds are 9/8, all major thirds, 81/64, and etc. 16/9 27/16 243/128 2187/2048 | | | | Ab-w--Eb--Bb---F---C---G---D----A---E----B---F#----C#---G# | | | | | | | | 32/27 4/3 1 3/2 9/8 81/64 729/512 6561/4096 Fig. 4 Pythagorean Tuning The dashes between the letter names of the notes in Figure 4 indicate that the fifths are pure. [3.2] Although D# (19683/16384) and Eb(32/27) are enharmonically identical in equal temperament, their pitches in Pythagorean tuning are different; D# is higher than Eb by 531441/524288 -- the Pythagorean comma(P). The more common method for deriving the Pythagorean comma is to calculate it as the difference between 12 consecutive fifths and seven octaves.(9) As Euclid knew, it is also the difference between six whole tones and one octave.(10) =============== 9. In logarithmic measure, a perfect fifth is 702 cents. Therefore twelve perfect fifths equals 8424 cents while seven octaves is 8400 cents. The difference, the Pythagorean comma, is therefore equal to 24 cents. 10. Euclid (attrib.), *Section of the Canon*, in Barker, *Greek Musical Writings*, vol. 2 (Cambridge: Cambridge University Press, 1989), 199. Twelve links on the chain of fifths can be interpreted as 12 fifths, 6 whole tones, 4 minor thirds and 3 major thirds. Thus, the Pythagorean comma is also equal to the difference between three major thirds or four minor thirds and an octave. In Pythagorean tuning, a Pythagorean comma is the interval between any two notes separated by 12 links on the chain of fifths. =============== [3.3] A wolf fifth, "w," occurs at the end of the line when Eb has to be used enharmonically because D# is unavailable; the "fifth," which is more precisely a diminished sixth, from G# to Eb(D#) will be too narrow by a Pythagorean comma. Blackwood describes such a narrow fifth as discordant and sounding badly out of place in a scale whose other fifths are pure.(11) It was known as the wolf fifth, because of the howling sound that it made on an organ. ============== 11. Blackwood, *Recognizable Diatonic Tunings*, 58. ============== [4] Equal Temperament [4.1] Equal temperament has been commonly used for the past 150 years to tune pianos and organs and, before that, was used for fretted instruments such as guitars and lutes. Equal temperament flattens the fifths (and sharpens the fourths) by 1/12 of a Pythagorean comma, "p," as shown in Figure 5. This is just enough to give a true octave from a chain of 12 links. In musical terms, the 12 fifths in the chromatic line are made exactly equal to 7 octaves. As can be seen from Figure 5, the term "equal temperament" only applies to the fifths. The other intervals are not tempered equally with respect to Pythagorean tuning. |<----------------octave--------------(-1P)--------->| | |------seventh----->| (-5/12P) | | | -maj. Third-->| | (-1/3P) | | fifth ----}|-->| |-->| (-1/12P) | | | | | | Eb-p-Bb-p-F-p-C-p-G-p-D-p-A-p-E-p-B-p-F#-p-C#-p-G#-p-D# | | | | fourth --}|<--|<--| | (+1/12P) maj. snd. |------>| | (-1/6P) |-mj. sixth>| (-1/4P) Fig. 5. Schematic of Equal Temperament In Figure 5, the Pythagorean comma is indicated by the upper-case "P" and 1/12 of the Pythagorean comma by the lower-case "p." The direction of the arrow indicates the ascending harmonic interval. Reversing the arrow would invert the interval and change the sign of the fraction, as illustrated by the fifth (-1/4p) and the fourth (+1/4p). [4.2] Since equal temperament makes every thirteenth note the same, it makes all enharmonic pairs equal; for example, G# becomes identical with Ab and D# becomes identical with Eb. This superposition is quite different from a Pythagorean keyboard in which substitution of enharmonically equivalent notes gives a fifth that is narrowed by a Pythagorean comma. Compressing the chromatic line of twelve pure fifths by a Pythagorean comma makes the tuning of Ab equal to the tuning of G#. Instead of an infinite series of links extending along the chain of fifths in both directions, equal temperament reduces the chromatic scale to exactly 12 notes which end on an octave and then repeat. In equal temperament, one does not have to choose between an available G# and an unavailable Ab; both are available and the tuning of the two notes is now the same. Equal temperament made possible modern keyboard music with full modulation between all keys. It has become common to consider equal temperament as being the division of an octave into twelve equal semitones, since the ratio of an octave is 2/1 and an equally tempered semitone multiplied by itself 12 times produces an exact octave.(12) ============ 12. The development of the 12-tone school of composition was a logical consequence of accepting the 12-semitone model of equal temperament in place of a chain of harmonic fifths. ============ [4.3] We can easily demonstrate from the chain of fifths that equal temperament makes both the tuning and the musical function of each pair of enharmonically equivalent notes equal, using the 12 links from C to B#. After subtracting the Pythagorean comma, B# becomes the same musical note as C. C can be substituted for B#, which is indicated by placing it vertically below B# in Figure 6. E# is a fifth below B#, but since B# has been replaced by C, F must be placed below E# since F is a fifth below C. The substitutions continue to the left along the top row from B# to C until the bottom row is complete from C on the right to Dbb on the left. Figure 6 shows that the two rows created by these substitutions are the same as if the chain of fifths was split into two rows, with the sharp notes above the flat notes. The substitutions derived from equal temperament have made the tuning of the top row identical to that of the bottom row. Equal temperament combines each vertical pair of enharmonically equivalent notes into one single note.(13) C G D A E B F# C# G# D# A# E# B# | | | | | | | | | | | | | Dbb Abb Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C Fig. 6. Enharmonic Pairs in Equal Temperament This process can be extended out to infinity by vertically stacking each group of 12 notes in rows below Dbb-C and above C-B#. Each vertical stack of enharmonic notes will be identical. The profound consequence of equal temperament is that the chain of twelve tempered links absorbs the infinite chain into a single row of 12 chromatic notes that corresponds exactly to the 12 notes-to-the- octave (7 diatonic and 5 chromatic) equally-tempered keyboard. =============== 13. These stacked rows appear in Helmholtz's *On the Sensations of Tone* (London: Longman, 1885; New York: Dover, 1954; orig., 4th ed., Braunshweig: Verlag von Fr. Vieweg u. Sohn, 1877), 312, where it is used to illustrate the enharmonic relation between notes in Pythagorean intonation. =============== [4.4] Calculating the tuning of an equally tempered scale generates irrational numbers,(14) because the rational fraction that measures the Pythagorean comma must be divided into twelve roots. One equally tempered semitone equals the twelfth root of 2, an irrational number whose value is normally given in a decimal approximation as 1.05946 (100 cents.) The tuning of the chain of equal tempered fifths in six place decimals is given in Figure 7. Note Tuning Cents G#/Ab 1.587401 800 C#/Db 1.059463 100 (1300-1200) F#/Gb 1.414214 600 (1800-1200) B 1.887749 1100 E 1.259921 400 (1600-1200) A 1.6817793 900 D 1.122462 200 (1400-1200) G 1.498307 700 C 1.0 0/1200 F 1.334840 500 (1700-1200) Bb/A# 1.781797 1000 Eb/D# 1.189207 300 Fig. 7. Equal Temperament Using the diagram in Figure 5, the 700-cent logarithmic tuning of an equally tempered fifth is narrower than a 702 cent perfect fifth by 2 cents, 1/12 of the 24-cent Pythagorean comma. The last column in Figure 7 displays this pattern; the pitch of each note in the vertical chain is 700 cents above the note below it before subtracting out the octave. ================== 14. An irrational number, such as pi or the square root of two, is one that cannot be expressed as the ratio of two integers. An irrational number includes a nonterminating decimal written to the number of decimals needed for practical accuracy. ================== [5] The Four Modes of Just Intonation [5.1] In Pythagorean tuning, only the fifths and fourths are tuned as consonant intervals. The thirds and sixths generated by a sequence of perfect fifths are not generally considered consonant by comparison to the most consonant major third which is 5/4, a minor third of 5/6 and their inversions, a major sixth of 5/3 and a minor sixth of 8/5. The diatonic tuning that includes both consonant thirds and pure fifths is known as just intonation. One traditional method of forming a C-major scale in just intonation is to combine consonant tonic, dominant and subdominant triads, F-A-C, C-E-G and G-B-D. This tunes the just diatonic scale shown in Figure 8. The D-A link is narrowed by a syntonic comma indicated by the "k." Since the other links are pure, the pitch of A, E and B will also be lower than Pythagorean by a syntonic comma. Even though each note has its optimum tuning relative to C, just intonation is a theoretical scale which is unsatisfactory for tuning an instrument with fixed pitch because the narrow D-A results in the unjust intervals shown in Figure 8. The narrow D-A sounds much the same as a Pythagorean wolf fifth. The Pythagorean minor third and its inversion, the major sixth between D and F also sound badly out of place in an otherwise just scale. F----C----G----D--k--A----E-----B 4/3 1 3/2 9/8 5/3 5/4 15/8 D-A = 5/3 x 8/9 = 40/27, A-D = 27/20 D-F = 4/3 x 8/9 = 32/27, F-D = 27/16 Fig. 8. Dissonant Internal Intervals in Just Intonation Scale [5.2] Tuning a chromatic octave in just intonation creates further difficulties since alternate tunings are required for each note. A theoretical scale of just intonation can be calculated for four diatonic major scales in the chromatic octave by replicating the procedure used to form the scale of C-major. A scale of Bb-major would have triads in which Eb-G, Bb-D and F-A were pure major thirds. The pure thirds would be Bb-D, F-A, C-E in F-major and C-E, G-B and D-F# in G-major. Figure 9 shows the two alternative tunings for G, D and A that are required for these four scales. The multiplicity of alternate tunings makes just intonation even more impractical for the normal keyboard with twelve keys to the octave. Eb----Bb----F----C--k--G-----D---A 32/27 16/9 4/3 1 27/20 10/9 5/3 Bb----F----C----G--k--D----A----E 16/9 4/3 1 3/2 10/9 5/3 5/4 F----C----G----D--k--A----E----B 4/3 1 3/2 9/8 5/3 5/4 15/8 C----G----D----A--k--E----B----F# 1 3/2 9/8 27/16 5/4 15/8 45/32 Fig. 9. Just scales in Bb, F, C and G It can also be seen from Figure 9 that the absolute location of the syntonic comma moves but that its relative location is always on the fourth link of the scale. Each just scale could be considered to be virtually a separate entity whose tuning is fixed by the tuning of the triads that it contains. [5.3] It is helpful at this point to introduce the concept of an algorithm, which is a set of instructions for making a series of calculations. Each of the tunings we have discussed so far can be derived from an algorithm for tuning the chain of fifths into one 12-note chromatic octave on a keyboard. The Pythagorean tuning algorithm has four steps: 1. C is tuned to an arbitrary unit pitch of 1. 2. Tune the pitch of each note to be a perfect fifth (3/2) from the notes immediately adjacent to it. 3. If the pitch of any note is greater than 2, divide the pitch by 2 to keep it within the same octave. 4. Stop after the 12 notes from Eb to G# have been tuned. The equal temperament algorithm is the same as the Pythagorean algorithm except that the interval in step 2 is 1/12 of a Pythagorean comma narrower than a pure fifth. Although justly tuned scales often have been considered to represent a different species from the Pythagorean or the equally tempered scale, we will show in Figure 10 how an algorithm can be derived that maps Pythagorean tuning into just chromatic scales. [5.4] The scale in Figure 10 has a syntonic comma at every fourth link. This produces a chromatic scale in which the diatonic scale of C-major is just and the maximum number of major thirds are a consonant 5/4. This scale is generated by an algorithm in which (1) the fourth link of a Pythagorean C-major scale is narrowed by a syntonic comma and (2) every fourth link above and below that link is also narrowed by a syntonic comma. The tuning of the just chromatic scale from Eb to G# depicted in Figure 10 corresponds to the tuning given by Rossing for these same 12 chromatic notes within a larger chromatic scale of just intonation from Fb to B#.(12) Just Intonation 6/5 9/5 4/3 1 3/2 9/8 5/3 5/4 15/8 45/32 25/24 25/16 | | | | | | | | | | | | Eb---Bb--k--F---C---G---D--k--A----E----B----F#--k--C#----G# Eb---Bb----F--C--G---D----A-----E------B-------F#------C#--------G# | | | | | | | | | | | | 32/27 16/9 4/3 1 3/2 9/8 27/16 81/64 243/128 729/512 2187/2048 6561/4096 Pythagorean Tuning Fig. 10. Just Intonation Compared to Pythagorean Tuning We can use the graphic diagram in Figure 10 to evaluate the difference between the just and Pythagorean pitch of each note without having to multiply out the 81/80 syntonic comma. In this just chromatic scale, Bb and Eb are each a syntonic comma higher in pitch while justly tuned A, E, B and F# are each a syntonic comma lower in pitch. Just C# and G# are lower than Pythagorean by two syntonic commas. Examining Figure 10 demonstrates still more reasons why a chromatic scale of just intonation is unusable on a standard keyboard. Narrowing every fourth link by a comma does not leave all the thirds and fifths consonant. The fifths that are narrowed (which means that the fourths are inversely broadened) by a full syntonic comma are obviously dissonant. Thus, the just chromatic scale has two more dissonant links than a Pythagorean chromatic scale. The wolf fifth from enharmonic D#(Eb) to G# will be narrowed by three syntonic commas. Any group of three adjacent links that does not include a comma will form a Pythagorean minor third and a Pythagorean major sixth. Whole tones in this scale that include a syntonic comma are 10/9, while the other whole tones are in the just and Pythagorean proportion of 9/8. To assess the severe impact of these tuning discrepancies we summarize the most significant consequences when a keyboard tuned in just intonation as in Figure 10 is used to play the six diatonic major scales listed in Figure 2. Bb-major: fifth of tonic triad narrowed by a syntonic comma. F-major: major fourth broadened by a syntonic comma. C-major: fifth of the II chord narrowed by a syntonic comma. G-major: fifth of the dominant triad narrowed by a comma. D-major: fifth of tonic triad narrowed by a syntonic comma. A-major: major fourth broadened by a syntonic comma. It is, thus, amply clear that just scales with consonant thirds cannot be meaningfully used on a chromatic keyboard. =============== 15. Rossing, *The Science of Sound* (Reading, Mass: Addison Wesley, 1982), 161. This is a general textbook on acoustics. Rossing's chart lists the equally-tempered, just and Pythagorean tunings for a 22-note chromatic scale in cents and decimals. The just scale corresponds to the "C" scale in Figure 10. =============== [5.5] Since the scale in Figure 10 was created by an algorithm that narrowed every fourth link by one syntonic comma, one can generate four chromatic modes of just intonation, one for each of the scales shown in Figure 9. These are the only just chromatic scales that are possible when C has the relative tuning of 1, since the pattern of the algorithm repeats after the fourth link.(16) Figure 11 presents the tuning of the four modes of just intonation along a chromatic scale from Dbb to B# for a vertical chain of fifths in Pythagorean tuning and the four just chromatic modes in Bb, F, C, and G. The number of syntonic commas by which each note has been altered from Pythagorean tuning is indicated in parentheses, the pitch of the notes above C being lowered by the number of syntonic commas indicated and the ones below C being raised. The remaining notes are Pythagorean. Nt Pythagorean Just (G) Just (C) Just (F) Just (Bb) B# 531441/524288 125/64 (-3k) 125/64 (-3k) 125/64 (-3k) 125/64 (-3k) E# 177147/131072 675/512(-2k) 125/96 (-3k) 125/96 (-3k) 125/96 (-3k) A# 59049/32768 225/128(-2k) 225/128(-2k) 125/72 (-3k) 125/72 (-3k) D# 19683/16384 75/64 (-2k) 75/64 (-2k) 75/64 (-2k) 125/108(-3k) G# 6561/4096 25/16 (-2k) 25/16 (-2k) 25/16 (-2k) 25/16 (-2k) C# 2187/2048 135/128(-1k) 25/24 (-2k) 25/24 (-2k) 25/24 (-2k) F# 729/512 45/32 (-1k) 45/32 (-1k) 25/18 (-2k) 25/18 (-2k) B 243/128 15/8 (-1k) 15/8 (-1k) 15/8 (-1k) 50/27 (-2k) E 81/64 5/4 (-1k) 5/4 (-1k) 5/4 (-1k) 5/4 (-1k) A 27/16 27/16 5/3 (-1k) 5/3 (-1k) 5/3 (-1k) D 9/8 9/8 9/8 10/9 (-1k) 10/9 (-1k) G 3/2 3/2 3/2 3/2 40/27 (-1k) C 1 1 1 1 1 F 4/3 27/20 (+1k) 4/3 4/3 4/3 Bb 16/9 9/5 (+1k) 9/5 (+1k) 16/9 16/9 Eb 32/27 6/5 (+1k) 6/5 (+1k) 6/5 (+1k) 32/27 Ab 128/81 8/5 (+1k) 8/5 (+1k) 8/5 (+1k) 8/5 (+1k) Db 256/243 27/25 (+2k) 16/15 (+1k) 16/15 (+1k) 16/15 (+1k) Gb 1024/729 36/25 (+2k) 36/25 (+2k) 64/45 (+1k) 64/45 (+1k) Cb 2048/2187 48/25 (+2k) 48/25 (+2k) 48/25 (+2k) 256/135(+1k) Fb 8192/6561 32/25 (+2k) 32/25 (+2k) 32/25 (+2k) 32/25 (+2k) Bbb 32768/19683 216/125(+3k) 128/75 (+2k) 128/75 (+2k) 128/75 (+2k) Ebb 65536/59049 144/125(+3k) 144/125(+3k) 256/125(+2k) 256/125(+2k) Abb 262144/177147 192/125(+3k) 192/125(+3k) 192/125(+3k) 1024/675(+2k) Dbb 524288/531441 128/125(+3k) 128/125(+3k) 128/125(+3k) 128/125(+3k) Fig. 11. The Four Modes of Just Intonation Figure 11 provides an insight on the relationship between Pythagorean and just tuning.(17) Instead of presenting just intonation as a series of pitches that are independently derived, Figure 11 shows that the pitch of the sharp and flat notes are modified in a regular progression as syntonic commas are sequentially added to or subtracted from Pythagorean tuning. The logarithmic tuning of each note can be easily calculated using the fact that a pure fifth is 702 cents, a pure fourth, its inversion, is 498 cents, and a syntonic comma is 22 cents. As an example, 27/25, the tuning of Db in the just G-major mode is 5 links plus 2 syntonic commas left of C: 5 x 498 = 2490 + 44 = 2534. After subtracting 2400, Db + 2k = 134 cents. ============== 16. It is possible to generate an infinite number of chromatic just scales by allowing the four adjacent links in Pythagorean tuning to freely slide along the chain of fifths. Whenever C is not part of the Pythagorean notes, it will necessarily be raised or lowered by one or more syntonic commas and no note in the chromatic scale will be tuned to 1. However, changing the absolute tuning of the notes in this manner does not change their relative tuning. Therefore, all of the just chromatic scales generated in this manner can be transposed into one of the four modes listed in Figure 11. The major and minor scales available in each of the four modes are the following: G-mode: Cb, Eb, G, B, D# E-minor mode: Fb, Ab, C, E, G# C-mode: Fb, Ab, C, E, G# A-minor mode: Bbb, Db, F, A, C# F-mode: Bbb, Db, F, A C# D-minor mode: Gb, Bb, D, F#, A# Bb-mode: Ebb, Gb, Bb, D, F# G-minor mode: Cb, Eb, G, B, D# 17. The algorithm used to construct just chromatic scales can be used to illustrate why intervals generated by the prime 7 cannot be systematically included in diatonic keyboard tunings. Pleasant sounding intervals can include sevens. Examples which are commonly cited are a 7/4 minor seventh, a 7/5 diminished fifth, both of which are part of a diminished seventh chord in which the notes C, E, G, Bb are in the proportion of 4:5:6:7, or 1:5/4:3/2:7/4. A single scale in which these intervals are used can be contrived. Eb- Bb-s-F--C--G-s-D----A 7/6 7/4 4/3 1 3/2 7/4 21/16 However, temperaments based on 7's or factors of 7 do not produce a usable family of chromatic keyboard scales from the chain of fifths, which would have the desired ratios for the diminished seventh chord. The "septimal" comma which produces a 7/4 minor seventh and an 8/7 major second on the second link of the chain of fifths is 64/63 (27 cents). Therefore, reiteration of the septimal comma, "s", in an algorithm which tunes every minor seventh to 7/4 produces two modes of septimal chromatic scales in which E is 64/49, not the desired 5/4. Mode 1 (F) Mode 2 (G) G# 4096/2041 (+4s) 4096/2041 (+4s) C# 8192/7203 (+4s) 348/343 (+3s) F# 512/343 (+3s) 512/343 (+3s) B 2048/1029 (+3s) 96/49 (+2s) E 64/49 (+2s) 64/49 (+2s) A 246/147 (+2s) 12/7 (+1s) D 8/7 (+1s) 8/7 (+1s) G 32/21 (+1s) 3/2 C 1 1 F 4/3 21/16 (-1s) Bb 7/4 (-1s) 7/4 (-1s) Eb 7/6 (-1s) 147/128 (-2s) Ab 49/32 (-2s) 49/32 (-2s) Db 49/48 (-2s) 1029/1024 (-3s) Gb 343/256 (-3s) 343/256 (-3s) Cb 343/192 (-3s) 7203/4096 (-4s) Fb 2401/2048 (-4s) 2401/2048 (-4s) These septimal modes are worse than the just modes because the septimal comma is broader than the syntonic comma by five cents. The deviation from just intonation is further aggravated because septimal commas increase the pitch of notes to the right of C and decrease the pitch of notes to the left of C, contrary to the action of the syntonic comma. Thus, compared to just intonation, the pitch of notes to the right of C will be painfully sharp and notes to the left will be dismally flat. For example, the major third necessary for the diminished seventh chord will be 2s+1k (76 cents) broader than in a just scale while the minor sixth will be 2s + 1k (49 cents) narrower. ============== [5.6] F. Murray Barbour listed 22 historical scales of just intonation dating from 1482 to 1776 in his historical survey of tuning and temperament.(18) Barbour defined just intonation more broadly than we have done, including within the concept any 12-note chromatic scale that possessed some pure fifths and at least one pure third. Barbour described a late 18th-century tuning by Marpurg which is the same as Figure 10 as "the model form of just intonation."(19) Two scales which correspond to the F-major mode shown in Figure 11 were described by Barbour as "the most symmetric arrangement of all." The first was an early 17th century scale by Salomon de Caus which started with Bb; the second was by Mersenne and started on Gb.(20) An even earlier tuning by Fogliano in the F-major mode starting with Eb was also included.(21) None of the other tunings are precisely in the form of Figure 11. A complete listing of these historical just tunings is given in Appendix I. =============== 18. Barbour, *Tuning and Temperament: A Historical Survey* (East Lansing, Michigan: Michigan State College Press, 1953), 90-102. 19. Ibid., 100. 20. Ibid., 97-98. 21. Ibid., 94. =============== [5.7] It is also interesting to compare the actual tuning of notes in Figure 11 with Ellis's "Table of Intervals not exceeding One Octave" in his appendix to Helmholtz.(22) Figure 11 gives new significance to intervals whose names as given by Ellis imply a diatonic origin. Thus Ellis's list includes the following varieties of fourths: "acute," 27/20, a "superfluous," 25/18 and another "superfluous," 125/96. Figure 11 shows that the "acute" fourth is the just fourth in the G-major mode, the first "superfluous" fourth is an augmented fourth, F# in the F-major and Bb-major modes, and the second "superfluous" fourth is an augmented third, E#, in the C, F and Bb modes. In all, Ellis lists 33 intervals obtained by combining one or more perfect fifths with one or more just thirds, only 3 less than the 36 tunings in Figure 11 that are altered by one or more Pythagorean commas. All but two of Ellis's 33 intervals are included in Figure 11; one is a well-tempered interval and the other is not part of a chromatic scale in which C is tuned to 1. The complete list of Ellis's intervals and their notation in terms of Pythagorean tuning and syntonic commas is contained in Appendix II. ============== 22. Helmholtz, *On the Sensations of Tone*, 453. A similar list of "Extended Just Tuning" is found in Blackwood, *Recognizable Diatonic Tunings*, 116-119. =============== [5.8] We can now appreciate why the chain of fifths is useful for evaluating the harmonic consequences of alternate tunings of a diatonic scale embedded in a chromatic keyboard. It is not possible to arbitrarily change the pitch of one note without altering its relationship with all other notes in the chromatic space. The interdependence of tuning is not limited to instruments with fixed pitch. For example, a string quartet could not play passages containing a sequence of triads in just intonation without altering the melodic intervals and, possibly the overall level of pitch.(23) The problems of tuning harmonic intervals can only be solved by dynamic ad hoc adjustments of pitch to obtain optimum consonance while preserving the melodic line and stability of pitch.(24) This is one of the reasons why it is so difficult for inexperienced musicians to achieve good ensemble intonation. =============== 23. Blackwood, *Recognizable Diatonic Tunings*, 74, demonstrates that in a common progression of C-major triads from II to V, just intonation would require that D as the root of the II chord be one syntonic comma lower in pitch than D as the fifth of the V chord. In *The Science of Musical Sound* (*Scientific American Books*, 1983, 67), John Pierce shows a five chord progression, I, IV, II, V, I, in which the just tuning of C drops by a syntonic comma from the first chord to the last. 24. It is for this reason that Lloyd in his article on Just Intonation in the 1954 Grove Dictionary adopted the position that instruments without fixed pitch and vocalists use a flexible scale in which the size of the intervals vary according to the context and part of the reason that Lindley and Turner-Smith introduced the concept of "leeway" into their algebraic tuning theory. =============== [5.9] The syntonic comma is not an independent variable; the independent variables are the tuning ratios for the consonant fifth and major third as determined by psychoacoustic measurements, which are 3/2 and 5/4. The syntonic comma, even though it has been known and separately named for two thousand years is merely derived from the difference between the consonant or just major third and the Pythagorean major third generated by four consecutive fifths. However, one would not intuitively expect that this single dependent variable could be used to measure all the differences in tuning between Pythagorean and just scales, including intervals that are not generated by major thirds.(25) The just intonation algorithms provide a coherent framework for this ancient and well respected dependent variable known as the syntonic comma. As was noted above, just intonation and Pythagorean tuning are commonly thought of as different species of tuning. The just intonation algorithms demonstrate that just intonation can more completely be understood in terms of the chain of fifths and syntonic commas than if considered independently. ============= 25. Inspection of the chain of fifths tells us that a major third can only generate one-quarter of the infinite series of chromatic notes that are generated by the fifth. This is why the tuning of the major third is a subsidiary factor in the generation of diatonic scales. ============= [6] Meantone Temperament [6.1] Meantone temperament is a keyboard tuning that makes a chromatic scale with consonant major thirds playable in diatonic scales in a few closely related keys. It does so by equally dividing the syntonic comma over the four links of each major third. This results in a whole tone that is exactly half- way between the alternate 9/8 and 10/9 found in just scales. The name "meantone" is derived from the resulting mid-size interval for the whole tone even though the division of the whole tone was only a by-product of distributing the syntonic comma. Dividing the comma into four "quarter commas" and distributing the quarter-commas so that every fifth was tempered by one quarter-comma also smooths out the three narrow fifths created by the undivided comma in the just scale, replacing them with a sequence of tempered fifths that are far more tolerable to the ear. The earliest generally accepted meantone scale was described by Pietro Aron in the early sixteenth century for a chromatic scale from Eb to G#.(26) ============== 26. Barbour, *Tuning and Temperament*, 26. ============== [6.2] The quarter-comma (4q = 1k) narrowing of each link in the chain of fifths is shown in Figure 12, which also depicts the resulting alteration of other intervals from Pythagorean tuning. Compared to Pythagorean tuning, it is evident that the fifths to the right of C are all tuned a quarter-comma flat, while the fourths to the left of C become a quarter-comma sharp. |---mj. seventh -5/4k--->| | |---mj. third -1k-->| |--->|{-fifth -1/4k | | | | | Eb--q-Bb-q--F--q-C--q-G--q-D--q-A--q-E--q-B--q-F#-q-C#-q-G# | | | | fourth +1/4k -}|<---| |-second->|{---(-1/2k) |-sixth -3/4k->| Fig. 12. Schematic of Quarter-Comma Meantone Temperament As before, Figures 10 and 12 allow us to measure the difference between the just and meantone scales without using arithmetic. As an example, a mean-tone chromatic semitone is narrowed by 7/4 syntonic commas compared to Pythagorean tuning and 3/4 of a syntonic comma compared to just intonation. [6.3] Meantone temperament and its variations was the established mode for tuning keyboards for nearly three centuries. English pianos and organs were tuned this way until the middle of the nineteenth century.(27) As Thurston Dart explained in his treatise on early music: Thus mean-tone provides the player with a group of about a dozen 'central' keys in which all the important chords are more in tune than they are in the modern piano. ... Mean-tone is admittedly imperfect as a tuning for chromatic music; for diatonic music, however, it cannot be bettered, as the musicians of earlier times knew very well.(28) For a time, organs were built with more than twelve keys per octave in order to utilize more of the extended chromatic scale. The keys could be split or other devices used to enable the performer to play either note of an enharmonic pair such as Ab/G# or Db/C#.(29) There was a practice, apparently, of raising or lowering the location of the wolf in the tuning of harpsichords to fit the composition being played at the moment.(30) However, manufacturers refrained from constructing instruments with more than 12 keys per octave and performers refrained from learning how to play such instruments, probably because the standard keyboard was suitable for most uses. As musical development led to the use of more keys outside its central compass, meantone temperament became increasingly impractical. =============== 27. *Grove's Dictionary of Music and Musicians* (1954), "Temperaments" (380). 28. Dart, *The Interpretation of Music* (New York: Harper & Row, 1963), 47. 29. Barbour, *Tuning and Temperament*, 108-9. Since the first reference to split keys found by Barbour goes back to 1484, this device must have been used for both Pythagorean and meantone tunings. 30. *Grove's Dictionary of Music and Musicians*, "Temperaments" (379). =============== [6.4] Since a syntonic comma is 81/80, a quarter-comma, its fourth root, is an irrational number. However, rational fractions that closely approximate the size of a quarter-comma and which cumulatively equal a full syntonic comma can be derived by arithmetically dividing a syntonic comma of 324/320 into four parts as follows: 324/320 = 321/320 x 322/321 x 323/322 x 324/323 Therefore, a quarter-comma fifth may calculated as being: 3/2 x 320/321 = 160/107 Rational fractions approximating each of the other meantone intervals may be calculated in a similar manner, using the other fractions in the expansion. A chromatic meantone tuning for a keyboard is shown in Figure 13 for a chain of fifths tempered by a quarter-comma (q).(31) 323/270 107/80 180/161 540/323 225/161 25/16 | | | | | | Eb-q-Bb--q-F--q-C--q-G--q-D--q-A--q-E--q-B--q-F#--q-C#--q-G# | | | | | 161/90 1 160/107 200/107 675/626 Fig. 13. Quarter-Comma Meantone Temperament Logarithmic values for a meantone scale can be easily calculated by subtracting one-fourth of a syntonic comma (22/4 = 5.5 cents) from every pure fifth. Thus each meantone fifth will be 696.5 cents. =============== 31. Barbour's table 24 gives a monochord mean-tone tuning derived by Gibelius in 1666 by arithmetic division of the comma which is the same tuning shown in Figure 12. Gibelius's monochord is divided into an octave between 216000 and 108000, in which G = 144450, D = 193200 and A = 129200. The equivalence of these monochord tunings to Figure 12 is calculated as follows: G = 216000/144450 = 4320/2889 = 480/321 = 160/107 D = 216000/193200 = 540/483 = 180/161 A = 216000/129200 = 540/323 Barbour states that these approximations "check closely with numbers obtained by taking roots," with the G being off by 0.000003. Barbour, *Tuning and Temperament*, 29. =============== [7] Well Temperament [7.1] The term well temperament includes a family of temperaments that modified meantone temperament to eliminate wolves and to expand the range of playable keys by taking advantage of the small difference between the Pythagorean comma and the syntonic comma. The difference between the two commas is an interval of 32805/32786 (2 cents), which is called the schisma. If only four links of a chromatic keyboard scale are tempered by a quarter-comma, with the remainder being tuned pure, the chromatic scale will exceed an acoustic octave by only a schisma and the wolf fifth will be thereby minimized to the point of nonexistence. |------ C-E ------>| (-1k) |<--- G-B ------>| (-3/4k) |-------F-A------>| (-3/4k) |-----Bb-D----->|---- D-F# ---->| (-1/2k) |-----Eb-G---->| | |------A-C#-- >| (-1/4k) | | | | | Ab--Eb--Bb--F--C--q-G--q-D--q-A--q-E--B--F#--C#--G# | | |<----------(P-k)--------Chromatic Octave------>| Figure 14. Schematic of Well Temperament [7.2] Figure 14 illustrates a simplified form of well temperament in which all but the central four links are Pythagorean and the major thirds vary from just (C-E) to fully Pythagorean (Ab-C and E-G#). No major triad will be exactly consonant. The wolf fifth between Ab(G#) and Eb has been essentially eliminated, and notes which are enharmonic to Eb, Bb, F#, C# and G# will differ in pitch only by a schisma. Therefore, use of these notes enharmonically increases the number of available scales from 6 to 11. The tuning for this version of well temperament are given in Figure 15. 32/37 180/161 5/4 45/32 405/256 | | | | \ Ab---Eb---Bb---F---C--q--G--q--D--q--A--q--E---B---F#---C#---G# \ | | | | | | | 128/81 16/9 4/3 1 160/107 540/323 15/8 135/128 Figure 15. Well Temperament A number of historical well temperaments were devised that separated the quarter commas by one or more links to improve playability in desired keys. Other temperaments subdivided the syntonic even further into 2/7 and 1/6 commas. The more that the commas were divided and dispersed, the more that well temperament approached equal temperament. [8] Algorithms For Schismatic and Syntonic Equal Temperament [8.1] We will develop and expand upon the method advocated by Kirnberger in the eighteenth century for deriving a scale of equal temperament from Pythagorean intonation.(32) A scale obtained by the first procedure is schismatic equal temperament and a scale obtained by the second procedure is syntonic equal temperament. The tuning of schismatic ET is exactly equal to the tuning for syntonic ET, the only difference being in their algorithms. Both match the tuning of equal temperament to several decimal places as shown in Figure 16. ============== 32. Barbour, *Tuning and Temperament*, 64. ============== [8.2] Schismatic and syntonic equal temperament is derived by extending a well-tempered scale eleven links to the left and right of C. When the eleventh note to the left of C is raised by one syntonic comma and the eleventh note to the right of C is lowered by one syntonic comma the results are tunings for E# and for Abb that are almost exactly the same as the equally tempered tunings for F and G respectively. E#-1k = 10935/8192 = 1.334839 F(ET) = 1.334840 Abb+1k = 16384/10935 = 1.498308 G(ET) = 1.498307 This strategy works because a syntonic comma is very nearly 11/12 of a Pythagorean comma, expressed as k = 11/12P. Therefore a schisma, defined as sk = P-k = 32805/32786, is very nearly 1/12 of a Pythagorean comma "p," the amount by which each link is narrowed in equal temperament. The pitch of E# in Pythagorean tuning is higher than its enharmonic note, F, by a Pythagorean comma(P). Therefore, E#-k ~ F+P-k ~ F+sk ~ F+p. As shown in Figure 5, F+p is the pitch of F in equal temperament. Since the precise value of a schisma is 1.954 cents and 1/12 of a Pythagorean comma is 1.955 cents, the error is only 1/1000 of a cent. One can then obtain decimal tunings for equal temperament by reiterating the proportions for E#-1k to a full chromatic chain of 12 links that would exceed an exact octave by a ratio of only 2.000018/2 or 0.012 cents. [8.3] We will now derive and apply two algorithms which map Pythagorean tuning into equal temperament. The algorithm based upon the schisma produces schismatic ET. As with ordinary equal temperament, this procedure equates the tuning of enharmonic pairs, substituting G for Abb + k, D for Ebb + k + 1sk, and so on. Every 10th link to the left of Abb + k and to the right of E#-1k is tempered by one schisma. The five schismatic notes tuned to the right of Abb+k are enharmonically equivalent to the notes normally located to the right of G, while the five schismatic notes tuned to the left of E#-k are enharmonically equivalent to the notes normally located to the left of F. The steps of the algorithm are as follows. 1. Starting with C, move 10 links to the left to Abb+1k. Substitute G enharmonically for Abb+1k. 2. Move 1 link to the right 5 times, adding 1 schisma to the width of each interval. Rename each note enharmonically. 3. Returning to C, move 10 links to the right to E#-1k. Substitute F enharmonically for E#-1k. 4. Move 1 link to the left 4 times, subtracting 1 schisma from the width from each interval. Rename each note enharmonically. [8.4] The algorithm that produces syntonic equal temperament also starts with the intervals C-Abb, ten links to the left of C, and C-E#, ten links to the right of C. Substituting G, which is enharmonic to Abb+1k in syntonic ET, and reiterating the process, D in Syntonic ET would be 10 more links to the left of Abb plus one more syntonic comma, giving D enharmonically equivalent to Fbbb +2k. At the other end of the chain, enharmonic Bb in syntonic ET would be 10 links minus one syntonic comma to the right of E#-1k(F), written as Bb = G###-2k. Syntonic ET is based upon substitution of enharmonic pairs just as is equal temperament. The tuning of each note is exactly the same in both syntonic and schismatic ET. The notation and tuning for schismatic and syntonic ET are given in Figure 16. The algorithm for syntonic ET is as follows. 1. Starting with C, move 10 links to the left to Abb + k. Substitute G enharmonically for Abb+1k. 2. Reiterate the action 5 more times, adding 1 syntonic comma to the width of each 10-link interval. Rename each note enharmonically. 3. Returning to C, move 10 links to the right of C to E#-1k. Substitute F enharmonically for E#-1k. 4. Reiterate the action 4 more times, subtracting 1 syntonic comma for each 10-link interval. Rename each note enharmonically. Tuning Note Schismatic ET Syntonic ET 1.000009 Dbb = C - k + 11sk 1.498295 Abb = G - k + 10sk 1.22454 Ebb = D - k + 9sk 1.683681 Bbb = A - k + 8sk 1.261336 Fb = E - k + 7sk 1.887739 Cb = B - k + 6sk 1.414207 Gb = F# - k + 5sk ---------------------------------------------------------- 1.059459 Db = C# - k + 4sk = F######## - 5k 1.587396 Ab = G# - k + 3sk = D###### - 4k 1.189205 Eb = D# - k + 2sk = B#### - 3k 1.781795 Bb = A# - k + 1sk = G### - 2k 1.334839 F = E# - k + 0sk = E# - 1k 1.0 C C 1.498308 G = Abb + k = Abb + 1k 1.22464 D = Ebb + k - 1sk = Fbbb + 2k 1.681797 A = Bbb + k - 2sk = Dbbbbb + 3k 1.259925 E = Fb + k - 3sk = Bbbbbbbb + 4k 1.887756 B = Cb + k - 4sk = Gbbbbbbbb + 5k 1.41422 F# = Gb + k - 5sk = Ebbbbbbbbbb + 6k ---------------------------------------------------------- 1.059468 C# = Db + k - 6sk 1.58741 G# = Ab + k - 7sk 1.189215 D# = Eb + k - 8sk 1.781811 A# = Bb + k - 9sk 1.334851 E# = F + k - 10sk 1.000009 B# = C + k - 11sk Fig. 16. Schismatic and Syntonic Equal Temperament [8.5] We can now go a step further and derive the algorithm that maps just intonation into syntonic ET from the algorithms previously used to map Pythagorean tuning into just intonation. For example, E# in the just mode of C-major is equal to Pythagorean E#-3k. Adding back the three syntonic commas, E#-1k in schismatic and syntonic ET is enharmonically equivalent to E# + 2k in the just C-major mode. The algorithm for just schismatic ET parallels the algorithm given in section 8.3, adding back to each interval the number of commas listed in Figure 11. In syntonic ET, Bb is enharmonically equivalent to G###-2k. If the C-mode just scale were extended to G###, the Pythagorean tuning would be narrowed by 5 syntonic commas. Replacing the 5 commas, G### + 3k maps just G### into Bb in just syntonic ET. We have not set out the full just chromatic scale for the 121 links necessary in Figure 11, but it can be done simply by adding 1 syntonic comma for every four links to the left and subtracting 1 syntonic comma for every four links to the right as many times as is necessary. Applying these algorithms in Figure 17 gives scales of just schismatic and just syntonic ET for the C-major scale. Tuning Note Just Schis. ET Just Synt. ET 1.059459 Db = C# + 1k + 4sk = F######## + 9k 1.587396 Ab = G# + 1k + 3sk = D###### + 7k 1.189205 Eb = D# + 1k + 2sk = B#### + 5k 1.781795 Bb = A# + 1k + 1sk = G### + 3k 1.334839 F = E# + 2k = E# + 2k 1.0 C C 1.498308 G = Abb - 2k = Abb - 2k 1.22464 D = Ebb - 2k - 1sk = Fbbb - 3k 1.681797 A = Bbb - 1k - 2sk = Dbbbbb - 5k 1.259925 E = Fb - 1k - 3sk = Bbbbbbbb - 7k 1.887756 B = Cb - 1k - 4sk = Gbbbbbbbb - 9k 1.41422 F# = Gb - 1k - 5sk = Ebbbbbbbbbb - 10k Fig. 17. Just (C-Major mode) Schismatic and Syntonic Equal Temperaments [8.6] Of course, neither syntonic nor schismatic ET are precisely equal. They do not merge all enharmonically equivalent notes, as shown by the outer sections of Figure 16 which contains notes from Gb to F# in schismatic ET. The enharmonic value for Db is not exactly equal to the value of C# as it is in equal temperament and C is not exactly equal to Dbb and B#. Therefore, the number of possible notes is still endless and does not collapse to twelve. The divergences between the tunings produced by the two equal temperaments are of no audible or acoustic significance either to the tuning of individual notes or to the overall sound of the temperament. A keyboard instrument tuned to syntonic ET would not sound distinguishable from one tuned with unlikely mathematical perfection to traditional equal temperament. The twelve chromatic links tuned to syntonic ET would have to be replicated nearly 163 times before the divergence in tuning C would exceed even one schisma. [9] Concluding Thoughts [9.1] This article has explored the relations between keyboard tunings based upon the chain of fifths. We began by accepting the historical definition of a diatonic scale embedded in a 12-note chromatic keyboard. We demonstrated that the terms used to name diatonic and chromatic intervals can be rationally ordered according to the number of links between the intervals on the chain of fifths. We then derived algorithms that map Pythagorean tuning into just intonation and equal temperament and map just intonation into equal temperament. Pythagorean tuning and just intonation have traditionally been classified as distinct tuning strategies, while Pythagorean tuning and equal temperament have been related only by the Pythagorean comma. This article has shown that the three tunings can be related using algorithms that utilize combinations of pure fifths and syntonic commas. ====================== APPENDIX I Historical Scales of Just Intonation Listed by Barbour. The numbers in the left hand margin correspond to the number used by Barbour. The table lists the author of the temperament, the date of its publication, the chromatic compass of the tuning on the chain of fifths, and the number of intervals which are pure or are altered by one or more commas, starting at the left end of the chain. 81. Ramis, 1482. Ab-F# Ab-G, 8=0, 4=-1k. 82. Erlangen, 15c. Ebb, Bbb, Gb-B, 2=+1k, 8=0, 2=-1. 83. Erlangen revised, Eb-G#, 7=0, 3=-1k. 84. Fogliano 1529 Eb-G#, 1=+1k, 4=0, 4=-1k, 3=-2k. 85. Fogliano 1529 Eb-G#, 2=+1k, 4=0, 3=-1k, 3=-2k. 86. Fogliano 1529 Eb-G#, 1=+1k, 1=+1/2k, 3=0, 1=-1/2k, 3=-1k, 3=-2k. 88. Agricola 1539. Bb-D#, 8=0, 4=-1k 89. De Caus 17c. Bb-D#, 4=0, 4=-1k, 4=-2k 90. Kepler 1619 Eb-G#, 2=+2, 5=0, 5=-1 91. Kepler 17c. Ab-C#, 3=+1, 5=0, 4=-1 92. Mersenne 1637 Gb-B, 4=+1, 4=0, 4=-1 93. Mersenne 1637 Bb-D#, 4=0, 4=-1, 4=-2 94. Mersenne 1637 Gb-B, 5=+1, 3=0, 4=-1 95. Mersenne 1637 Gb-B, 5=+1, 4=0, 3=-1 96. Marpurg 1776 Eb-G#, 2=+1, 4=0, 4=-1, 2=-2 97. Marpurg 1776 Eb-G#, 1=+1, 6=0, 2=-1, 2=-2 98. Marpurg 1776 Eb-G#, 2=+1, 3=0, 4=-1, 3=-2 99. Malcolm 1721 Db-F#, 3=+1, 5=0, 4=-1 100. Rousseau 1768 Ab-C#, 3=+1, 4=0, 3=-1, 2=-2 101. Euler 1739 F-A#, 4=0, 3=-1, 5=-2 102. Montvallon 1742 Eb-G#, 1=+1, 5=0, 6=-1 103. Romieu 1758 Eb-G#, 1=+1, 5=0, 4=-1, 2=-2 APPENDIX II. Ellis's Table of Intervals not Exceeding an Octave This chart lists the intervals identified by Ellis as being formed from pure fifths and major thirds. It provides the inverted proportions used by Ellis and his name for the interval along with the Pythagorean equivalent of the interval from Figure 11. F#-1k was not part of Figure 11 because it is not included in a scale in which C = 0. G#-1k is a well-tempered interval from Figure 14. The remainder of the intervals are from Figure 11. 24:25 Small semitone C#-2k 128:135 Larger limma C#-1k 15:16 Diatonic or just semitone Db+1k 25:27 Great limma Db+2k 9:10 The minor tone of just intonation D-1k 125:144 Acute diminished third Ebb+3k 108:125 Grave augmented tone D#-3k 64:75 Augmented tone D#-2k 5:6 Just minor third Eb+1k 4:5 Just major third E-1k 25:32 Diminished fourth Fb+2k 96:125 Superfluous fourth E#-3k 243:320 Grave fourth (F-1k) 20:27 Acute fourth F+1k 18:25 Superfluous fourth F#-2k 32:45 Tritone, augmented fourth F#-1k 45:64 Diminished fifth Gb+1k 25:36 Acute diminished fifth Gb+2k 27:40 Grave fifth G-1k 16:25 Grave superfluous fifth G#-2k 256:405 Extreme sharp fifth (G#-1k) 5:8 Just minor sixth Ab+1k 3:5 Just major sixth A-1k 75:128 Just diminished seventh Bbb+2k 125:216 Acute diminished seventh Bbb+3k 72:125 Just superfluous sixth A#-3k 128:225 Extreme sharp sixth A#-1k 5:9 Acute minor seventh Bb+1k 27:50 Grave major seventh B-2k 8:15 Just major seventh B-1k 25:48 Diminished octave Cb+2k 64:125 Superfluous seventh B#-3k *-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-* AUTHOR: Lewin, David, B. TITLE: The D Major Fugue Subject from WTC II: Spatial Saturation? KEYWORDS: Bach, fugue, subject, saturation, tonality, hexachord David B. Lewin Harvard University Music Department Cambridge, MA 02138 lewin@fas.harvard.edu ABSTRACT: In what ways does the Subject of Bach's fugue in D major from Book II of the WTC define a structure "in D"? To what degree is there something "missing" in the Subject (e.g. a C-or-C-sharp)? To what degree does the Subject "saturate" a hexachordal space? In what ways is the Answer required, to define a tonality? ACCOMPANYING FILE: mto.98.4.4.lewin.gif FIGURE 1 [A4 A4 A4 D4 ... D4 D4 D4 G3 B3 E3 A3 G3 F#3 (D3) [1] Figure 1 schematically represents the Subject; on the line above it, the beginning of the Answer appears after a square bracket. [2] For many years I wanted to hear the Subject in G major, to the extent that I would as often as not think of and even refer to the piece as "the G major fugue from Book II," and have to correct myself. The following brief essay pursues several lines of thought that have radiated from my misapprehension. [3] Before undertaking this exercise, I had believed that a Bach fugue subject, among other things, exposes the tonality of its piece in some way. The Subject of Example 1 demonstrates that the belief is not exact. For this fugue, the Subject itself is ambiguous as regards a common-practice D major or G major key. Here the key is determined by subject *plus answer*. The key of D major is specifically determined when we hear the beginning of a real answer on the note A. Were the fugue to proceed from its Subject in the key of G major, a tonal answer would be normative: G G G D E A D C B (G).(1) =========================== 1. Of course this tonal answer could not enter during the F# of the Subject. I suspect the point is relevant to the stretto character of the fugue. Christoph Wolff points out that Friedrich Wilhelm Marpurg, soon after Bach's death, had already observed the features of this fugue so far discussed. In an essay entitled *Analysen von Bachschen Fugenthemen, Fugen und Kanons* (Berlin, 1753-54), Marpurg points out that "the leap down a fifth, which the octave above the tonic note makes here, was prohibited by older authorities, on the grounds that it renders the tonality uncertain" ("Der Sprung, den hier die Oktave der Hauptnote in die Unterquint thut, wurde bey den Alten verboten, weil er die Tonart ungewiss macht."). Some seven years later, in an essay entitled *Themenbeantwortung und Durchfuehrung in einigen Fugen des Wohltemperierten Klaviers* (Berlin, 1760), Marpurg repeats the above sentence and elaborates it with the following continuation: "Indeed, the Subject at issue here does not proclaim the key of D major, but much more G major; and one does not know where one is at home tonally, until the entrance of the Answer" ("In der That zeigt der hier vorhandene Fuehrer nicht die Tonart d dur, sondern vielmehr g dur an; und man merkt es erst bey dem Eintritt des Gefaehrten, wo man zu Hause ist."). Marpurg recognizes the problematic character of this phenomenon, but does not pursue it farther from a theoretical point of view, contenting himself with pedagogical advice: "Such exceptions to the rules can be ventured only by Masters, and beginners will do well to cleave to the rule that requires a fugue theme to indicate the key unambiguously" ("Dergleichen Ausnahmen von der Regel koennen nur von Meistern vorgenommen werden, und Anfaenger thun wohl bey der Regel zu bleiben, welche einen deutlichen und die Tonart gehoerig anzeigenden Fugensatz erfordert."). The Marpurg passages are reproduced in *Bach-Dokumente*, ed. Bach-Archiv Leipzig, Volume III: *Dokumente zum Nachwirken Johann Sebastian Bachs 1750-1800*, ed. Hans-Joachim Schulze (Leipzig and Kassel, 1972). The quote from 1753 appears on page 28, and the quote from 1760 on page 156. =================================== [4] Now as regards the Subject in its own context, one reason that it appears tonally ambiguous to our ears is that it contains neither a C-natural nor a C-sharp. When I first thought about that, I expressed the idea by thinking that a C-or-C# was "missing."(2) Behind my thought lay a covert assumption, that traditional "Tonality" is expressed by exposing a complete diatonic gamut.(3) In my provisional thoughts I regarded the real Answer as "solving" the issue "raised" by the Subject, when the Answer provides the "missing" C-or-C#. By providing C#, the Answer (finally) determines the tonality as D major. ========================== 2. Since the Subject contains neither a C nor a C#, it contains no tritone-dyad. Related issues in musical cognition are discussed by Richmond Browne in "The Tonal Implications of the Diatonic Set," *In Theory Only* 5.6-7 (1981), 3-21; also by Helen Brown and David Butler in "Diatonic Trichords as Minimal Tonal Cue Cells," *In Theory Only* 5.6-7 (1981), 37-55. 3. "A tonality is expressed by the exclusive use of all its tones." Arnold Schoenberg, *Structural Functions of Harmony* (New York: W.W. Norton & Company, Inc., 1954), page 11. The sentence leads off a section entitled "Establishment of Tonality." ============================== [5] The covert assumption above caused me to hear "something missing" in the Subject's pitch material, and that had a decided influence upon the sort of character I attributed to the theme. But I am no longer so satisfied with this way of listening. For one thing--as pointed out in [3] above--our psychological recognition of D major occurs with the *first* note of the Answer, its incipit A; we do not have to wait for the C# that is its penultimate note, to ascertain a tonality of D. The incipit of the Answer determines our perception as a matter of *rhetoric*, not a matter of pitch-class saturation (or the lack thereof). [6] The issues can be pointed even more sharply by observing that Bach has the total chromatic available to him in the WTC. Why not say that this fugue subject--like most others in the WTC--is "missing" quite a few notes of the total chromatic? Why not presume that we are "waiting" to hear those notes? [7] Well, for some fugues that might be an interesting trail to pursue--for example the E-minor fugue in Book I, where F-natural (or E-sharp), G-sharp, and A, and only those pitch-classes, are "missing" from the subject.(4) However, for the fugue of Figure 1, and for the more diatonic fugues in general, the issue of chromatic saturation seems pretty well beside the point. Should we really focus our ears on the fact that the Subject in Figure 1 is "missing" A-flat, B-flat, C, D-flat, E-flat, and F? Who cares? =========================== 4. The phenomenon is suggestive as regards the end of the answer. Since the end of the subject in this fugue modulates from E minor to B minor, the end of the answer should normatively modulate back from B minor to E minor, requiring adjustment of the melody accordingly. Instead Bach gives a real answer, modulating from B minor to F# minor. One hears how the tones missing from the subject--E#, G#, and A--play a characteristic role over the second half of the answer, during the modulation to F# minor. No doubt the chromatic closure is subordinate to other aspects of the real answer. Since the fugue has only two voices, the exposition is complete at the end of the answer, and Bach seems eager to move on tonally at once, rather than returning to E minor. Still, the chromatic closure has a certain effect, not least in that very connection. =========================== [8] Nevertheless, one can be struck (as I was) when one hears how the missing notes of the total chromatic form a diatonic hexachord. Returning from that outer space to the real world of the fugue at hand, one will then hear (as I did) how the pitch content of the fugue's Subject in fact *does* sound "complete" if one regards it as projecting a diatonic *hexachord*, rather than as projecting six notes from an incomplete diatonic scale, the seventh tone being "missing." In the hexachordal context the Subject *does* saturate its pitch-space, and that gives it a very different character, from the major-scale subject that "fails" to provide a C-or-C#. [9] Since the diatonic hexachord at issue has the tone D as its "UT," the hexachordal hearing provides a rationale for perceiving the Subject as some sort of structure "in D." Namely, the Subject projects the (complete) D hexachord. The point is aurally clear if one could sing the subject as UT UT UT FA - LA -- RE SOL FA MI - (UT). No mutation would be required. [10] In what sense might it be legitimate for us to sing the subject as above? This, as it turns out, is an interesting and complicated historical question. I am fortunate to have had a perspicacious reader for this article who alerted me to pertinent issues, and I am fortunate to have Christoph Wolff as a colleague with whom I can confer. [11] First of all, we should ask: was Bach accustomed to Guidonian solmization in the context of his work, and of the WTC in particular? Yes, responds Professor Wolff, without doubt.(5) ======================= 5. Lewis Lockwood recalls an analysis course taught by Edward Lowinsky at Queens College, in which Lowinsky approached Book I as a whole by stressing the completely hexachordal nature of the subject for the first fugue in the Book, and the completely chromatic nature of the subject for the last fugue. ======================= [12] Next: would Bach have solmized the Subject as in [9] above? Or would he not, rather, have mutated, taking his cue from the *high* D with which the Subject begins, thus: SOL SOL SOL UT/FA - LA - - RE SOL FA MI - (UT)? Or thus: SOL SOL SOL UT - MI/LA - - RE SOL FA MI - (UT)? Let us call these productions the versions of the Subject "with mutation," as opposed to the "unmutated" version of [9] above: UT UT UT FA - LA -- RE SOL FA MI - (UT). The question we are currently considering can then be put: could Bach have heard (or sung) the unmutated version of [9]? Or would he have heard (or sung) one of the mutated versions above? [13] My reader brought up this salient issue, without professing a definitive answer for the question. When I consulted with Professor Wolff, he also could not provide a definitive answer, and was much engaged by the question. As of this publication, he is still researching the issue. [14] One could put the crux of the matter as follows. Was "UT" a pitch *class* for Bach, in the way that "DO" is for us in modern solfege? Or would Bach have heard his hexachords positioned in register along an extended gamut--in which case there could be no such thing as a *high* "UT"? [15] All this taken into account, the fact remains that even the mutated versions of the Subject do reference the entire natural D hexachord--taking the final low "(UT)" into account--and they do not reference the *entire* soft G hexachord, no matter where one mutates. In that sense, the Subject, regarded as a hexachordal structure, *does* saturate a tonal space. The weight of the Subject as a whole definitively goes on the D hexachord, as opposed to the G hexachord. This is in decided contrast to the ambivalent structure of the Subject as a production of common-practice tonality, where we can equally well hear the theme proceeding I - V in G major, or IV - I in D major. [16] To sum up: I am (at present) happy to hear the Subject by itself "in D," when I understand the "D" at issue to be the UT of a pertinent hexachord, rather than the tonic note of a common-practice tonality. And then, so far as such tonality is concerned, I hear the fugue's continuation "in the key of D" as a matter of rhetoric--not pitch content--when I hear the Answer come in on the note A. (The rhetoric of the real Answer in this connection was discussed toward the end of [5] above.) The idea that the Subject is "missing" some C-or-C# now seems to me relatively tangential. *-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-* 2. Review AUTHOR: Roeder, John TITLE: Review of Christopher F. Hasty, *Meter as Rhythm* (New York and Oxford: Oxford University Press, 1997). KEYWORDS: Meter, rhythm, time, analysis, Hasty, perception, projection John Roeder University of British Columbia School of Music 6361 Memorial Road Vancouver, B.C. V6T 1Z2 CANADA jroeder@unixg.ubc.ca ABSTRACT: Christopher Hasty's new book presents an evocative and analytically expressive view of meter as the "projection" of duration. This review summarizes Hasty's principal points, demonstrates the concept of projection with an interactive Java applet, and evaluates the theory. ACCOMPANYING FILES: Java applet (Example 1) mto.98.4.4.roeder2.gif mto.98.4.4.roeder3.gif 1. Introduction [1.1] Rhythm and meter are difficult to theorize--not only to formalize but also simply to discuss--because of their elusive and variegated nature. For twenty years Christopher Hasty has explored fundamental questions about musical time. In a series of important articles in the 1980's,(1) he developed some conceptually and analytically productive ways to conceive of temporal processes in music. Although the topics of his articles were not explicitly linked, one would naturally expect a book-length study to follow. ================================== 1. Hasty's articles include: "Rhythm in Post-tonal Music: Preliminary Questions of Duration and Motion," *Journal of Music Theory* 25.2 (1981): 183-216; "Segmentation and Process in Post-tonal Music," *Music Theory Spectrum* 3 (1981): 54-73; "Phrase Formation in Post-tonal music," *Journal of Music Theory* 28.2 (1984): 167-90; and "On the Problem of Succession and Continuity in Twentieth-century Music," *Music Theory Spectrum* 8 (1986): 58-74. ================================== [1.2] After a decade of suspenseful silence, the anticipated book has indeed appeared. Its processive perspective is familiar from the earlier articles, but its topic, the music it discusses, and its grounding in philosophy, psychology, and music theory are fresh. For the importance of its subject, for the quality, consistency, and depth of its theoretical insight, and for the wisdom and clarity of its exposition, *Meter as Rhythm* should be read and appreciated. It requires a willingness to engage with seemingly small musical details, with diverse musics, and with a speculative, philosophical mode of exposition, but the reader's persistent efforts will be well rewarded. 2. A New View of Meter as Projection [2.1] Many modern conceptions of rhythm and meter place them in opposition. Rhythm is often defined to consist of the actually sounding durations of music, while meter is the alternation of strong and weak beats, or the interaction of pulse strata, that are inferred from the rhythm. Rhythm is thus conceived as emerging and active--a "concrete" patterning that is measured by, and heard to work with or against the "abstract," deterministic, rigid metrical grid. These contrasting conceptions are ingrained deeply into much theorizing and into most practical communication about rhythm. [2.2] Hasty problematizes this opposition on both perceptual and logical grounds. As an alternative, he defines meter simply, without reference to accent or pulse, as the potential of a duration to be immediately reproduced. This potential arises in connection with sensations that we gather about events as they "become," or unfold in time. To describe meter from the perspective of an active listener, Hasty appropriates concepts and terminology from Alfred North Whitehead's *Process and Reality*. From this perspective, he says, meter is "a process in which the determinacy of the past is molded to the demands of the emerging novelty of the present." (168). It is thus "rhythmic" in the sense that it, too, is emerging, active, creative and perpetually novel. [2.3] The essence of meter, in Hasty's theory, is the process of "projection." Specifically, as an event becomes, its duration accumulates "projective potential," or the potential to condition our expectations about the duration of future events. In the simplest case, the duration's projective potential is "realized" when "there is a new beginning whose durational potential is determined by the now past first event." We perceive this realization when we project that the second event will replicate the duration that it makes past. The second event's duration may or may not actually realize this projection, and the parity or disparity that is manifest at its end (at the beginning of a third event) corresponds to musical perceptions of "too short" or "too long" that may be expressively and analytically significant. [2.4] Hasty illustrates metrical projection with graphical diagrams that (he ruefully notes) necessarily rely on the very spatial metaphors that he is striving to avoid. To gain a better understanding, and a more truly temporal representation of the nature of projection, please examine the Java applet at http://smt.ucsb.edu/ mto/issues/mto.98.4.4/mto.98.4.4.roeder_ex1.html.(2) It allows the user to determine, in sequence, the beginnings and ends of three line segments, representing events, and to examine the resulting projections and other metrical effects, such as speeding up, slowing down, and hiatus. Click the mouse to define the beginning of the event, then move the mouse to the right to define the duration of the event, then click to end the event. Up to three events of any length may be so determined; however, to keep the demonstration focused on Hasty' theory, the applet will not permit the user to define events that do not involve metrical projections. The scale of time in the example is calibrated by a constant "Lim" signifying the maximum duration that can be projective, which Hasty speculates to be about two seconds in this highly simplified monophonic texture. Try defining events of various lengths, and observe how the projections accumulate, disappear, and changes accordingly, as explained in the running commentary. (I'm sorry that the current version of the Java programming language does not offer a practical method for playing the sound sequences.) ================================== 2. I gratefully acknowledge Tom Roeder for realizing my design of this example in the Java programming language. ================================== 3. Consequences for Metrical Theory [3.1] Although the concept of projection is simple, Hasty uses it to develop detailed and enlightening readings of durational patterns. His readings address many fundamental musical issues, for example: the metrical nature of beginning, ending, and continuation; the differences among various patterns ostensibly having the "same" meter; and metrical effects that have been observed by many other analysts. [3.2] Chapter 9 reworks some traditional conceptions of meter in terms of projection. Metrical accent is defined as "the accent of beginning"; this resembles Jonathan Kramer's definition (*The Time of Music*, Schirmer, 1988, p. 86), but has substantive differences. "Beginning" itself is a function that is determined processively--that is, the function of beginning is only perceived as a duration that is begun becomes. The significance of a beginning depends on whether the duration of interest is that of a single event, or of a harmony, or of a series of associated successive durations. In the last case, the later durations in the series contribute to the continuation of the duration of the whole group. [3.3] Accordingly the distinction between strong and weak (metrical) accent is replaced with a functional distinction between beginning and continuation, as shown in Example 2 (Hasty's 9.2). Each small vertical stroke above a note head symbolizes not a beat but to the "becoming" of the entire mensurally determinate duration that is begun by the attack of the note. The larger vertical stroke over the first notehead refers to the becoming of entire duration of two quarter notes; because the half note duration is continued through a series of briefer durations, its onset is called a "dominant beginning." The slanted stroke over the second notehead indicates that, even while its duration begins, the earlier beginning (of the two-note group) is still actively becoming, so the second event's becoming is a "continuation" of that duration. [3.4] The functions of beginning and continuation are not immutable. They may be reassessed as later events modify the listener's sense of what has begun. For instance, anacrusis is understood in this context as a continuation that becomes "detached" from a beginning. [3.5] Hasty first employs the concepts of projection, beginning and continuation to analyze the meter of common brief durational patterns. The analyses show how patterns with the same meter signature are actually quite different in projective action. They also permit some original and interesting distinctions, such as between acceleration and increased activity of division. Other effects, such as "silent beginning," are also explained as results of projection. [3.6] Since meter entails the projected reproduction of a duration, duple meter is privileged in the theory. (Indeed the duple bias is reflected frankly in the two-column layout of the book's text.) Triple meter can nevertheless be explained, however, in the manner shown in Example 3 (Hasty's 9.18f). This shows a half-note projection Q, corresponding to a potential duple meter, that is denied by events starting at the asterisk that favor (say, by motivic repetition) a dotted-half projection R. The third quarter note in this situation is then regarded as a "deferral"--it functions to delay the next dominant beginning. More complex meters are also analyzed in similar terms. [3.7] The traditional terms "beat" and "measure" take on novel connotations in this context. A "beat" is a beginning--not a time point, but a function that is evaluated as the event becomes. "Beats" need not recur at equal time intervals: "the flexibility of mensural determinacy frees projective activity from the narrow confines of 'precise' equality" (277). "Measure" refers not to the notated bar, but to any duration that is mensurally determinate. Any projection is therefore a "measure." Measures "are not given--they are created under the pressure of antecedent events and are creative for present and future events." A given passage of music may present a polyphony of "measures," some reinforcing regular meter, and others possibly acting against it. 4. Analytical Applications [4.1] Chapter 10 initiates a move from the general to the particular by applying the theory in analysis. Meter, as projection, is always specific to the given musical context. For example, Hasty demonstrates, identical durational patterns may have different metrical interpretations depending on their pitches and tempi. [4.2] Analyzing the meter of a musical passage in these terms entails identifying its projections, and distinguishing beginnings, continuations, and anacruses at the beat and measure level. The theory allows one to explain how, in many cases, the basic projection--functions are modified subtly and richly by denials, overlappings, and contractions. A complete analysis thus explains the continuity or discontinuity in a passage, locates certain common types of projections that execute such formal functions as closure, and identifies certain common metrical processes such as the "contraction of projective focus." [4.3] The emphasis on beat and measure does not preclude a consideration of longer excerpts. A whole chapter is devoted to examining how projection--which is perceived only in relatively brief time spans up to a few seconds--may be "simulated" over larger time spans by means of grouping parallelisms and mutually reinforcing briefer projections. Indeed, in the analyses of common-practice tonal music, the focus is not so much the smaller patterns (which generally accord with the notated meter signature) but on the varieties of larger-scale "hypermetrical" contractions and elisions, and their effect on musical continuity. To further clarify how he regards this "simulated" meter, Hasty compares his readings to Schenkerian-oriented analyses by Schachter and Rothstein. [4.4] One of the most impressive aspects of the book is the variety of music that is analyzed. The tonal works analyzed in depth include the first moveme nt of Beethoven's First Symphony and the Trio of Mozart's Symphony No. 35. But there are also quite convincing accounts of meter in non-tonal modernist music by Wolpe and Carter, showing what thwarts larger-scale projections, and explaining how various denials and deferrals of dominant beginnings facilitate continuity and the sustaining of energy. With reference to the Webern Saxophone Quartet Hasty explains how hiatus contributes to the creation of a "phrase" built of successive projective "consituents," each of which possesses varying degrees of openness or closure. He also provides detailed and exciting readings of vocal polyphony by Monteverdi and Schuetz, readings that convincingly connect the observed metrical effects to the syntax and semantics of the poetry. No one has succeeded better in expressing verbally the rhythmic nature of these musics. 5. Strengths, evidences, and a few weaknesses [5.1] Hasty's theory verges on the systematic, even on the programmable, as Example 1 begins to demonstrate, but he constantly eschews the temptation and dangers of system building. He is frank about the speculative nature of his concepts, and very clear about the intensely personal nature of hearing rhythm and meter. It seems that he is less interested in providing a basis for "verifiable" analyses (however that quality might be defined) than he is in provided a clear language for discussing musicians' possibly disparate metrical intuitions. He acknowledges, even celebrates, the interaction of composer, performer, listener in creating meter, and shows how tempo, the listener's attention, and various performance decisions affect the perception of projection--for instance comparing Boulez's and Craft's recordings of Boulez's *le marteau sans maitre*. [5.2] It is also remarkable that this carefully elaborated theory--in sharp contrast to some contemporary theories of pitch--emphasizes sensation over structure and feeling over logic. The analyses treat not quantities but qualities of (to cite a few examples) "breadth and spaciousness," "suspense and elasticity," "urgency," "mobility," "relaxation," "focus," and getting "lost in a moment of ending." Style, in this view, "is above all an environmentally (culturally and personally) specific manner of feeling duration." Such statements may appear foreign to some theorists, but Hasty makes them meaningful through a carefully reasoned and clearly expressed theory of how such feelings arise from a single, intuitively plausible principle. [5.3] Although the content of his projective theory is primarily speculative, Hasty grounds it with telling references to a variety of psychological studies. Further important precedents for his ideas are revealed in the original glosses and interpretations he provides of earlier music theory. Heinrich Koch, for example, is cited to describe meter "as a creative act of attention, not bound by law, but arising from the exercise of our cognitive or imaginative powers. Mattheson's conception of Bewegung--movement through time--is also cited as an early recognition of the essentially processes aspects of rhythm. The 19th-century theorist who most anticipates Hasty's theory is Hauptmann, who clearly discusses meter as a process whereby duration is created, a process grounded in an innate perceptual disposition for measure. [5.4] Hasty also gives due consideration to recent rhythmic theories--Cooper and Meyer, Lerdahl and Jackendoff, as well as Schachter and Rothstein--finding a basis for dialogue even with those that maintain an opposition of meter and rhythm. Other supporting references range widely through twentieth-century philosophy, science, German music theory, and musicology. [5.5] One important antecedent is inexplicably omitted. There is no mention of Wallace Berry's *Structural Functions in Music* (Prentice-Hall, 1976), which is perhaps the most explicit previous treatment of music in terms of process, as it is notably attuned to "the musical work transpiring in time" (p. 25). Berry anticipates Hasty's terminology: "metric units are initiated by ... impulses of relatively strong projection" (p. 317). He hears impulses to have different functions, and represents impulse functions with vertical and slanting arrows and lines on scores, as Hasty does. He regards these impulses as functional at different "levels of metric structure," and also allows for functional "dualities," "as when the final impulse of a metric unit is both conclusive and anticipative at the same level" (p. 328). Since Hasty spends so much effort at acknowledging tangential theories, it would have been enlightening to read his comments on a theory that is a perhaps distant but nevertheless distinctive precursor of his own. [Perhaps he or others will now consider how his work relates to this and to more recent treatments of meter as process.(3)] ================================== 3. For instance: Justin M. London, "Loud Rests and Other Strange Metric Phenomena (or, Meter as Heard)," *Music Theory Online* 0.2 (1993): 0-16, and John Roeder, "Formal functions of hypermeter in the 'Dies irae' of Verdi's *Messa da Requiem*," *Theory and Practice* 19 (1994): 83-104. ================================== [5.6] Some other minor negligences also bother me. The book suffers from a few production problems: a smattering of typographical mistakes in the text, and, more seriously, some rhythmic errors in the musical examples (Ex. 15.2, an excerpt from *le marteau*, is an especially egregious and by no means solitary instance). More substantively, many analytical decisions about interpreting the onset of an event as a dominant beginning or a continuation hinge on questions of segmentation that are not addressed directly by the theory. Hasty gives the impression that these questions are impossible to address systematically, but a more explicit discussion of segmentation would help justify some of his analytical decisions. [5.7] This book is not an easy read. The concepts are original and their author is thorough, so their exposition is long, detailed, and philosophical. Since the theory does not assume any of the common ways of thinking about meter and rhythm, it may not fall into most reader's comfort zones. Indeed it demands sustained concentration--and a good memory, since the brief index provides little help. But I found it absolutely compelling. It is a vital contribution that has broad potential to affect how we listen, value, and teach music of all styles--potential that will be realized to the extent that we attend to the insights it contains. *-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-* 3. Announcements a. 3rd Triennial British Musicological Societies' Conference 1999 b. Music Theory Society of New York State: Annual Meeting c. SMT Publication Subvention Grants ------------ UNIVERSITY OF SURREY 3rd Triennial British Musicological Societies' Conference 1999 CALL FOR PAPERS The Department of Music at the University of Surrey will host the 3rd Triennial British Musicological Societies' Conference at the University's Guildford campus from 15-18 July 1999. The University of Surrey is 35 minutes from London by train, and is easily accessible from London's two main airports, Heathrow and Gatwick. (Details of the University's location can be found on the University's web page: .) As with the previous conferences at Southampton ('93) and King's College, London ('96), the Critical Musicology Forum, the Royal Musical Association, and the Society for Music Analysis will be represented. Joining them this time will be the British Forum for Ethnomusicology and the inaugural Conference on Twentieth-century Music. Proposals are invited for: * Individual presentations of 20 minutes' duration * Themed sessions of one-and-a-half hours' duration comprising three or four papers * Round-table sessions of one-and-a-half hours' duration * Poster presentations Proposals addressing issues of music education, sociology, psychology, therapy, and cognate areas are encouraged as well as papers on historical musicology, analysis and theory, critical theory, etc. Membership of the participating societies is not a requirement for either the submission of proposals or attendance. Conference and session themes will be determined by the proposals. Individuals may make ONE proposal only. Abstracts of approx. 250 words should be submitted by 30 October 1998 if sent by mail or fax, or 13 November 1998 if sent by email. Include the following details: NAME, INSTITUTION, PHONE, FAX and EMAIL ADDRESS. Proposals should be sent to the Chair of the Programme Committee: Christopher Mark Department of Music University of Surrey Guildford GU2 5XH England Tel: +44 (0)1483 259317 Fax: +44 (0)1483 259386 Email: c.mark@surrey.ac.uk The Programme Committee is as follows: * Christopher Mark (University of Surrey; Royal Musical Association, Conference on Twentieth-century Music) * Nicholas Marston (University of Oxford; Society for Music Analysis) * Allan Moore (Thames Valley University; Critical Musicology Forum) * Jonathan Stock (University of Sheffield; British Forum for Ethnomusicology) Dr Christopher Mark Department of Music University of Surrey Guildford Surrey GU2 5XH ENGLAND Tel: (01483) 259317 International: +44 1483 259317 Fax: (01483) 259386 +44 1483 259386 WWW: ============== EVENT: MTSNYS 1999 Meeting HOST: School of Music Ithaca College Ithaca, NY 14850-7240 DATE: 23 June 1998 DESCRIPTION: Music Theory Society of New York State Annual Meeting Ithaca College Ithaca, New York 10-11 April 1999 CALL FOR PAPERS The Program Committee invites proposals for papers and presentations on any topic. Areas of particular interest include: Analysis symposium on Bergs *Vier Stuecke*, Op. 5 and/or both settings of *Schliesse mir die Augen beide* (1900 and 1925) Analysis of jazz or popular music Technology in music theory pedagogy or analysis Pedagogy of twentieth-century theory Papers given at national conferences or previously published will not be considered. Any number of proposals may be submitted by an individual, but no more than one will be accepted. Most papers will be placed in 45-minute slots, with about 30 minutes for reading and 15 minutes for possible response or discussion. Paper submission should include: 1. Six copies of a proposal of at least three but no more than five double-spaced pages of text. Each copy should include the title of the paper and its duration as read aloud, but not the authors name. 2. An abstract of 200-250 words, suitable for publication. 3. A cover letter listing the title of the paper and the name, address, telephone number, and e-mail address (if applicable) of the author. Proposals should be sent to: Craig Cummings, MTSNYS Program Chair School of Music Ithaca College 208 Ford Hall Ithaca, NY 14850-7240 POSTMARK DEADLINE IS 1 OCTOBER 1998 Members of the MTSNYS 1999 Program Committee are Craig Cummings, Chair (Ithaca College); Cynthia Folio (Temple University); Deborah Kessler (Hunter College, CUNY); Jocelyn Neal (Eastman School of Music); Edward Murray (Cornell University) and Timothy Nord (Ithaca College) PAPER/PROPOSAL DEADLINE: 1 October 1998 CONTACT: Craig Cummings, MTSNYS Program Chair School of Music Ithaca College 208 Ford Hall Ithaca, NY 14850-7240 Submitted by Mary Arlin (arlin@ithaca.edu) =============== SMT Publication Subvention Grants The Society for Music Theory is pleased to announce the establishment of a publication subvention fund. The grants from this fund will be awarded on a competitive basis to any member in good standing of the Society, and are intended to help authors offset out-of-pocket costs associated with the preparation and publishing of any article or book in the field of music theory that has been accepted for publication. Among the possible expenses to which the fund may be applied are the copying and setting of musical examples, the payment of copyright or permission fees, the production of unusually complex graphic and illustrative material, and the development of any relevant computer software, audio material, or other multi-media components essential to the text's production. Grants awarded may be up to $2000. Interested applications should prepare: 1. A short abstract (approx. 1000 words) describing the work to be published and its contribution to the field of music theory. 2. A copy of the article in question, or in the case of a book, one or two representative chapters. 3. A letter from the publisher or journal editor indicating acceptance of the publication. 4. A detailed explanation of the expenses to which the grant would be applied. Where possible, documentation itemizing these expenses should be included. Applicants may request funding up to $2000, although given the limited funds available and the desire to support as many deserving requests as possible, most grants will probably be made at significantly lower amounts. Applicants are particularly encouraged to seek out matching funding from their home institutions. Grants will be evaluated on a bi-annual basis beginning on March 15 and October 15 of each year. The evaluating subcommittee will be co-chaired by the SMT Vice-president and the chair of the Publications Committee. Additional members will be made up of the two At-Large Members of the Publications Committee and one additional member to be appointed by the President from the Executive Board of the Society. Decisions will be announced within three weeks of the submission deadline. Applications for the first review, to begin on October 15, 1998, should be sent in five copies to Professor Thomas Christensen, Vice-president of the Society for Music Theory: School of Music, University of Iowa, Iowa City IA, 52242. For the second round of reviews (to begin March 15, 1999), applications should be sent to Professor Thomas Christensen, Department of Music, University of Chicago, 1010 E 59th St., Chicago IL 60637 Any questions may be directed to the Vice-president at the addresses given above, or by email . *-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-* 4. Employment a. University of Nottingham: Chair of Music b. Eastman School of Music: Asst. Prof. of Music Theory ------------- UNIVERSITY OF NOTTINGHAM CHAIR OF MUSIC Applications are invited for a Chair of Music. Preference may be given to a candidate with strengths in the 19th century or in music analysis. The successful applicant, who will have an international reputation in research, will take up the appointment on a date to be agreed. Closing date: 24 July 1998. Interviews are provisionally scheduled to take place between 18 and 25 September. Informal enquiries may be addressed in confidence to Professor John Morehen, tel: +44 (0)115 951 4761, E-mail: John.Morehen@Nottingham.ac.uk. Further details may be found on the musicology-all list home page at: http://www.mailbase.ac.uk/lists/musicology-all/files/nottingham ============= POSITION/RANK: Asst. Prof., Music Theory INSTITUTION: Eastman School of Music QUALIFICATIONS: Preference will be given to candidates with a completed doctorate and evidence of excellence in teaching. JOB DESCRIPTION/RESPONSIBILITIES: The Eastman School of Music of the University of Rochester seeks an Assistant Professor of Music Theory (tenure-track), with a specialty in the theory and analysis of tonal music. We are looking for an individual with strong musical skills, who will be actively and creatively involved in teaching and coordinating courses in our undergraduate theory "core" as we launch a new curriculum. The successful candidate will also teach in our graduate programs and will be engaged in on-going music research. Position available: July 1, 1999. The Eastman School of Music of the University of Rochester is an equal opportunity employer (M/F). SALARY RANGE: Dependent upon experience and qualifications. ITEMS TO SEND: Applicants should send cover letter, curriculum vitae, references, and supporting documentation. DEADLINE: November 16, 1998 CONTACT: Robert Wason, Chair, Department of Music Theory Eastman School of Music 26 Gibbs Street Rochester, NY 14604 rwsn@theory.esm.rochester.edu phone: 716/274-1552 fax: 716/274-1088 *-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-* 5. New Dissertations AUTHOR: Linklater, Mary L. TITLE: From Procedure to Technique: Canonic Works of the Fifteenth Century INSTITUTION: University of Rochester BEGUN: May, 1998 ABSTRACT: This dissertation examines the canonic works of the fifteenth century, traces their compositional development from ad-hoc measure-by-measure procedures to a systematic large-scale technique, and demonstrates how the use of underlying structural frameworks contributes to accomplishing this shift. Background study of fourteenth-century literature identifies the origin of canon as a compositional device, while Renaissance theoretical treatises provide contemporary descriptions. By employing reductive graphs in the analysis of the canonic voices, the study demonstrates common underlying structural frameworks. The use of structural frameworks first appears in the second half of the fifteenth century in works by Ockeghem, and later develops into a sophisticated manner of composition with Josquin des Prez. The construction of these frameworks greatly facilitated the creation of canonic compositions which, once decorated, concealed the underlying patterns. The dissertation provides the link between early canonic procedures in the fourteenth century and Robert Gauldin's demonstration of canonic paradigms in Palestrina and Lassus, tracing the progression of canonic composition throughout the century from a measure-by-measure procedure to a systematic technique of structural frameworks. KEYWORDS: canon, Josquin, fuga, fifteenth-century, early music analysis CONTACT: Mary Linklater 16201 Comus Road Comus, MD 20871 301-407-0024 popinfo@juno.com ===================== AUTHOR: Diane Luchese TITLE: Olivier Messiaen's Slow Music: Glimpses of Eternity in Time INSTITUTION: Northwestern University BEGUN: Sept, 1996 COMPLETED: May, 1998 ABSTRACT: Messiaen frequently indicated his intention to express the sacred mysteries of the Catholic faith through his music. This study, based on an examination of twenty-five different slow compositions of a similar aesthetic, points out some of the characteristics of Messiaen's slow music that seem to contribute to its distinctive, yet elusive, sound quality. Although this music has been described by others as "static," such a description neglects the music's many progressive elements. These compositions seem to manifest a paradox; the music gives an impression of changelessness while it also allows its listeners to experience progressive change. It appears that this tension between stillness and forward motion is the result of the composer's attempt to express the unchangeableness of eternity through a means that also reflects the Western notion of linear time. Chapter I points out how scholars have described the sound quality of Messiaen's music and suggests a new description of the sound quality as paradoxical. Chapter II shows how the music evokes a transcendental atmosphere through means such as harmonic and instrumental color, unchanging elements, avoidance of conflict, neutralized dissonances, and often relatively stable levels of tension. Chapter III shows how the music reflects a linear conception of time, experienced as varying degrees of thematic and tonal closure that give rise to expectations of the attainment of musical goals. Chapter IV points out musical characteristics that divert our attention from progressive change to other aspects, further contributing to the illusion of changelessness. These characteristics include extreme slowness, deceleration, thematic expansion, and coloristic harmonies. The final chapter contains the analyses of four compositions, illustrating the interaction of the characteristics discussed in earlier chapters, showing how they combine to create the music' paradoxical effect. KEYWORDS: Messiaen, stasis, color, eternity,time, slowness, musical atmosphere TOC: I. Introduction II. A Reflection of Eternity III. A Reflection of Time IV. Diverting Attention From Progressive Change V. Analyses: "Desseins eternels" of La Nativite "Je dors, mais mon coeur veille" of Vingt Regards "Les ressuscites et le chant d'etoile Aldebaran" of Des canyons "Adoro te" of Livre du Saint Sacrement VI. Conclusion Bibliography CONTACT: Diane Luchese 2529 Jackson Ave., Apt. 2W Evanston, IL 60201 (847) 492-9949 dlu316@nwu.edu ================ AUTHOR: MacKay, James S. TITLE: Motivic Structure and Tonal Organization in Selected Motets of William Byrd INSTITUTION: McGill University BEGUN: November 1997 COMPLETED: August 1999 ABSTRACT: In this dissertation, I will examine the 16 motets in William Byrd's 1589 collection of CANTIONES SACRAE from two analytical perspectives: motivic and linear-reductive. Through motivic analysis, I will identify and categorize contrapuntal combinations (labelled according to Peter Schubert's three presentation types: non-imitative module, imitative duo, and canon), and note their patterns of recurrence within each motet. I will focus on how these presentation types are loosened and reformulated to create distinctions between beginning and middle sections, as an extension of Joseph Kerman's "cell technique". Through linear-reductive analysis, I will identify a wide variety of background voice-leading models for Byrd's modally-organized motets, showing their distinctiveness in comparison with later, tonally-conceived works. Following the example of Felix Salzer, Saul Novack, Lori Burns, Cristle Collins Judd and David Stern, I will propose modifications to, and extensions of Heinrich Schenker's analytic method to account for Byrd's modally-based, long-range structural procedures, in particular his emphasis of the subdominant at the background level. By combining linear-reductive and motivic analysis, I will locate links between foreground pattern and background structure in Byrd's CANTIONES SACRAE of 1589, and demon- strate his use of enlargement to project a melodic or contrapuntal motive into the deeper structural levels of a composition. This combination of analytic techniques will provide a flexible model which can be applied to Byrd's works in all genres, as well as to Renaissance music in general. KEYWORDS: motive, counterpoint, Byrd, Schenker, analysis, form, mode, motet, Renaissance CONTACT: 3616 Durocher #501, Montreal, Quebec (Canada), H2X 2E8; (514)-288-1442; bhym@musicb.mcgill.ca ====================== AUTHOR: Palombini, C. V. TITLE: Pierre Schaeffer's Typo-Morphology of Sonic Objects INSTITUTION: University of Durham BEGUN: October 1989 COMPLETED: September 1992 ABSTRACT: "Pierre Schaeffer's Typo-Morphology of Sonic Objects" proposes to present to the English-speaking reader the two achieved stages of Schaeffer's 1966 solfge, namely typology and morphology, as expounded in "Trait des objets musicaux", situating them in the larger context of Schaeffer's musicological work and in the specific context of the solfge. This is done through translation of and commentary on Schaeffer's writing. Chapter I surveys the years 1948-57, exposing the shifts of priorities that define three phases: research into noises, concrete music and experimental music; particular attention is paid to Schaeffer's conception of experimental music and, through the analysis of "Vers une musique exprimentale", what has generally been seen as an antagonism between the Paris and Cologne studios emerges as the conflict between two opposing approaches to technology and tradition. Chapter II delineates three notions that underpin the fourth phase of Schaeffer's musicological work, musical research, of which the 1966 solfge is the programme: acousmatic listening, four functions of listening and sonic object. Chapter III elaborates on the premisses of typology and morphology. Chapter IV expounds typology proper while Chapter V presents morphology and the sketch of the subsequent operations of solfge: characterology and analysis. From this study, it emerges that "Trait des objets musicaux" is first and foremost an inexhaustible repository of insights into sound perception. Typology, the first stage of the solfge, is doubtless a successfully accomplished project. However, as a method for discovering a universal musicality, the solfge enterprise needs to be viewed with caution. It suffers from the almost open-ended nature of its metaphorical vocabulary, the emphasis the text lays on reactive rhetoric, its reliance on "methods of approximation", and a gradual distancing from perceptual reality itself. This notwithstanding, "Trait des objets musicaux" appears as a fundamental text of twentieth century musicology. It brings to the fore two crucial issues: technology and the ways it alters our manner of perceiving and expressing reality, and reality itself thereby; the friction between sounds and musical structures, transparent in the text as the friction between isolated words and the discourse, transparent in Schaeffer's life as the f riction between the man and the social structures he has neede to fit in. KEYWORDS: music technology, musique concrte, sonic object, musical object, experimental music, elektronische Musik, electroacoustic music, listening, virtual instrument, sound analysis TOC: Introduction: By Writing, p. vi. Chapter I: From Research into Noises to Musical Research: 1. Research into Noises (1948-49), p. 3; 2. Concrete Music (1948-58), p. 6; 3. Towards an Experimental Music (1953), p. 9. Chapter II: Three Fundamental Notions: 4. Acousmatic Listening, p. 30; 5. Four Functions of Listening, p. 31; 6. Hearing, Listening to, Listening Out for, Comprehending, p. 34; 7. The Phenomenological Status of the Sonic Object, p. 46. Chapter III: The Premisses of Typology and Morphology: 8. The Method of Research after Concrete Music, p. 60; 9. The Four Operations of Solfge, p. 62; 10. Typo-Morphology and the Prose/Translation Metaphor, p. 66; 11. The Timbre of the Instrument that Does Not Exist, p. 73; 12. First Morphology, Identificatory Typology, Second Morphology, p. 84. Chapter IV: Classificatory Typology: "When the Piping Starts to Sing": 13. The Three Pairs of Criteria of Classificatory Typology, p. 93; 14. Well Balanced Objects, Redundant Objects, Eccentric Sounds, p. 111. Chapter V: Third Typology, Morphology Proper, Characterology, Analysis: 15. Morphology Proper, p. 121; 16. Solfge of Homogeneous Sounds: Criteria of Mass and Harmonic Timbre, p. 123; Solfge of Fixed Masses: Dynamic Criterion, p. 141; Solfge of Maintenance: Grain and Allure, p. 151; 19. Solfge of Variations: Melodic Profile and Mass Profile, p. 163. Conclusion: By Reading, p. 178. Addendum: "Musicology and Linguistics" (an English translation of Roman Jakobson's 1932 "Musikwissenschaft und Linguistik"), p. 178. Notes, p. 188. Bibliography, p. 194. Figures (tables of "Trait des objets musicaux" translated into English), p. 211. CONTACT: Departamento de Música Universidade Federal de Pernambuco CAC Av. Acadêmico Hélio Ramos s/n Cidade Universitária Recife PE 50740-530 Brazil Fax: +55 81 2718300 (c/o Departamento de Música) Tel.: +55 81 2718308, +55 81 2718318 Palombini@altavista.net *-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-* 6. New Books University of Nebraska Press *Brahms Studies 2* Edited by David Brodbeck The eight essays in *Brahms Studies 2* provide a rich sampling of contemporary Brahms research. In his examination of editions of Brahms's music, George Bozarth questions the popular notion that most of the composer's music already exists in reliable critical editions. Daniel Beller-McKenna reconsiders the younger Brahms's involvement in musical politics at midcentury. The cantata "Rinaldo" is the centerpiece of Carol Hess's consideration of Brahms's music as autobiographical statement. Heather Platt's exploration of the twentieth-century reception of Brahms's Lieder reveals that advoactes of Hugo Wolf's aesthetics have shaped the discourse concerning the composer's songs and calls for an approach more clearly based on Brahms's aesthetics. In his examination of the rise of the "great symphony" as a critical category that carried with it a nearly impossible standard to meet, Walter Frisch provides a rich context in which to understand Brahms's well-known early struggle with the genre. Kenneth Hull suggests that Brahms used ironic allusions to Bach and Beethoven in the tragic Fourth Symphony in order to subvert the enduring assumption that a minor-key symphony will end triumphantly in the major mode. Peter H. Smith examines Brahms's late style by concentrating on Neapolitan tonal relations in the Clarinet Sonata in F Minor. Finally, David Brodbeck delineates the complex evolution of Brahms's reception of Mendelssohn's music. David Brodbeck is an associate professor at the University of Pittsburgh, where he chairs the Department of Music. He is a former president of the American Brahms Society and the author of *Brahms: Symphony No. 1*. December Music 208 pp. 7 x 11 48 musical examples, 6 plates, 6 tables, 2 indexes $65.00s cloth 0-8032-1287-9 BRAST2 http://nebraskapress.unl.edu ============================ Princeton University Press http://pup.princeton.edu/order_info Leonora's Last Act Essays in Verdian Discourse Roger Parker "Parker's wit and high irony pervade every page. In a sense, his central aim (though never self-identified as such) is to brandish a new `tone' or attitude toward scholarship-and toward Verdi-in the postmodern 1990s. The book is immensely engaging throughout: verbal surprises and astonishing suggestions lurk around every corner. Leonora's Last Act is a joy to read even as it seeks mischievously to unsettle our views of this composer."--James Hepokoski, University of Minnesota In these essays, Roger Parker brings a series of valuable insights to bear on Verdian analysis and criticism, and does so in a way that responds both to an opera-goer's love of musical drama and to a scholar's concern for recent critical trends. As he writes at one point: "opera challenges us by means of its brash impurity, its loose ends and excess of meaning, its superfluity of narrative secrets." Verdi's works, many of which underwent drastic revisions over the years and which sometimes bore marks of an unusual collaboration between composer and librettist, illustrate in particular why it can sometimes be misleading to assign fixed meanings to an opera. Parker instead explores works like Rigoletto, Il trovatore, La forza del destino, and Falstaff from a variety of angles, and addresses such contentious topics as the composer's involvement with Italian politics, the possibilities of an "authentic" staging of his work, and the advantages and pitfalls of analyzing his operas according to terms that his contemporaries might have understood. Parker takes into account many of the interdisciplinary influences currently engaging musicologists, in particular narrative and feminist theory. But he also demonstrates that close attention to the documentary evidence-especially that offered by autograph scores-can stimulate equal interpretive activity. This book serves as a model of research and critical thinking about opera, while nevertheless retaining a deep respect for opera's continuing power to touch generations of listeners. Roger Parker is Professor of Music and Fellow of St. Hugh's College, Oxford. He is the founding coeditor of the Cambridge Opera Journal and the Donizetti Critical Edition, and the author of several books and articles on nineteenth-century Italian opera. He is the coeditor, with Arthur Groos, of Reading Opera (Princeton). Princeton Studies in Opera Carolyn Abbate and Roger Parker, Editors 208 pages. 4 halftones 2 line illus. 46 music exs. 6 x 9 0-691-01557-0 Cloth $32.50 US and L20.95 UK and Europe Contact: Edith Gimm fax: 609-258-1335 e-mail: edith@pupress.princeton.edu PUP Web site: http://pup.princeton.edu ====================== University of California Press *A Question of Balance: Charles Seeger's Philosophy of Music* Taylor Aitken Greer One of this century's most influential musical intellects takes center stage in Taylor Greer's meticulously wrought study of Charles Seeger (1886-1979). Seeger left an indelible mark in the fields of musicology, music criticism, ethnomusicology, and avant-garde musical composition, but until now there has been no extended appreciation and critique of Seeger's work as a whole, nor has an accessible guide to his texts been available. Exploring the entire corpus of Charles Seeger's writing, *A Question of Balance* highlights the work of those persons who most influenced him, especially Henri Bergson, Bertrand Russell, and Ralph Perry. Invited to inaugurate the music department at the University of California's Berkeley campus in 1912, Seeger became keenly aware of his deficiencies in general education and put himself on a rigorous regimen of intellectual development that included studying history, anthropology, political theory, and philosophy. For the remainder of his life his ideas about music heavily influenced the development of ethnomusicology and systematic musicology. Charles Seeger is perhaps best known as the father of the folksingers Pete, Mike, and Peggy Seeger and as the husband of the innovative American composer Ruth Crawford. This book makes clear that Seeger was an extremely important thinker and educator in his own right. Seeger's intellectual curiosity was as eclectic as it was enthusiastic, and Greer skillfully weaves together the connections Seeger made between music, the humanities, and the sciences. The result is a luminous tapestry depicting Seeger's ideal schemes of musicology. At the same time it reflects the turbulence and vitality in American musical life during the early decades of the century. "Greer offers the first extended appreciation and critique of Seeger's work taken as a whole. A superior work [that] will profoundly shape our understanding of a major figure in American music."--Joseph Straus, author of *The Music of Ruth Crawford Seeger* "Taylor Greer has taken Charles Seeger's fragmented philosophy and theory of music and made it whole, in beautifully concise and lucid prose."--Severine Neff, University of North Carolina, Chapel Hill Taylor Aiken Greer is Associate Professor of Music at Pennsylvania State University December (122) 0-520-21152-9 $55.00x cloth 278 pages, 6 x 9", 22 line figures, 13 musical examples Music World -------------- *The Quest for Voice: Music, Politics, and the Limits of Philosophy* Lydia Goehr Concentrating on the music, politics, and philosophy of Richard Wagner, Lydia Goehr addresses classic questions of German Romanticism. What is musical meaning? How different in meaning is music composed as a symphony or as a song? What is musical autonomy? And what is the relation between music's meaning and its purported political power? Goehr examines the peculiar relationship established between philosophy and music in the nineteenth century and offers a philosophical and political reading of Wagner's "Die Meistersinger," an account of the Wagner-Hanslick debate on musical formalism, an examination of the competing performance ideals embodied in Wagner's "Bayreuth," and an interpretation of Wagner's legacy as experienced by composers exiled from Nazi Germany. Goehr's inquiries are unified by her background as a distinguished philosopher and a finely trained musician with a sophisticated sense of history. Music means something, she observes, not because it is a well-formed symbolic language, but because human beings mean something when they engage as composers, performers, and listeners with music. Within that engagement is found the unending philosophical quest for the cultivation of the soul, or in modern terms, the political quest for agency and freedom. Lydia Goehr is Professor of Philosophy at Columbia University. Her previous publications include *The Imaginary Museum of Musical Works: An Essay in the Philosophy of Music*. Ernest Bloch Lectures September (124) 0-520-21412-9 $45.00x cloth 240 pages, 5-1/2 x 8-1/2" Music/Philosophy/German Studies North America Copub: Oxford University Press +=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+ Copyright Statement [1] *Music Theory Online* (MTO) as a whole is Copyright (c) 1998, all rights reserved, by the Society for Music Theory, which is the owner of the journal. Copyrights for individual items published in MTO are held by their authors. Items appearing in MTO may be saved and stored in electronic or paper form, and may be shared among individuals for purposes of scholarly research or discussion, but may *not* be republished in any form, electronic or print, without prior, written permission from the author(s), and advance notification of the editors of MTO. 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