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       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) 1993 Society for Music Theory
| Volume 0, Number 3      June, 1993      ISSN:  1067-3040    |
  All queries to: mto-editor@husc.harvard.edu
AUTHOR: Lindley, Mark and Turner-Smith, Ronald 
TITLE: An Algebraic Approach to Mathematical Models of Scales 
KEYWORDS: group, halfgroup, harmony, dodecaphony, semitones, temperament, 
Landini, Matteo da Perugia, Dufay, Schutz, Louis Couperin 
Mark Lindley
P.O. Box 1125
Cambridge MA 02238-1125
Ronald Turner-Smith
19 Risingholme Road
Harrow, Middx HA3 7EP 
ABSTRACT: Although mathematical models of the scale have always been 
characteristic of Western music theory, in the last 200 years they have 
not been very much improved (although some interesting properties of 
scales have been defined in recent years). This article describes our 
effort in a new book to contribute to this part of music theory by using 
some appropriate concepts of modern algebra. 
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[1] Our sequence of mathematical constructions is listed in Fig 1.(1) 
Our point of departure is the pitch continuum, which is not a 
mathematical construction but an intuition, since pitch is subjective.  
(Pitches as we hear them cannot be measured, but only judged.)  Our first 
construction is a positive-number line for the sound-wave frequencies, 
which can be measured and which almost wholly determine the pitches. 
1. All the Figs are in the file mto.93.0.3.lindley.fig, which is
available through the MTO FileServer, mto-serv.  Consult the MTO
Guide (information.txt) on how to retrieve files with mto-serv.
[2] It is the differences between pitches that interest us musically.  
Experience teaches us that there is a logarithmic relation between 
these subjectively judged differences and the differences among the 
corresponding pitch frequencies. So our second construction is a number 
line for the logarithms of the frequencies.  We use logarithms to the 
base 2; this choice is due to another aspect of musical experience: the 
musical interval between two notes whose frequencies are in the ratio 2:1 
is an octave, and notes one or more octaves apart from each other are 
intuitively heard as manifestations of the same note on different levels.  
Normally a scale repeats itself in each octave (there are exceptions, but 
they can be treated as special cases), so musicians today speak of pitch 
classes (see Fig 2) - that is, of equivalence classes of notes that are 
one or more octaves apart from each other - and of "pitch-class 
intervals" or "pitch-class relations", which are the analogous 
equivalence classes of musical intervals.  With logarithms to the base 2 
it is very easy to define the addition of logarithms-mod-1 (which we call 
"flogs") in such a way that the sum will be musically valid when the 
flogs for two pitch-class relations are added.  
[3] For a simple example, let us see how the flog for a 5th with a 
frequency ratio of 3:2, when added to itself, yields a flog for a whole-
tone.  It is readily reckoned that, to three decimal places, log 3 to the 
base 2 = 1.585, and hence log (3/2) to the base 2 = 0.585.  Now instead 
of adding 0.585 + 0.585 and getting 1.170 for a major 9th (which differs 
by 1 from the log for a whole-tone), we reckon in terms of flogs and 
write .585 + .585 = .170.  This is illustrated in GIF 1.(2) 
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command: "send mto.93.0.3.lindley*.gif" will retrieve all GIFs for the 
present essay from mto-serv.
[4] According to Max Weber,(3) there are two rational ways to construct a 
system of tones: by means of harmonic relations or else by dividing the 
octave into equal parts.  From this hypothesis can be derived two types 
of generators for our pitch-class relations (see Fig 1e): either equal-
division (whereby 1/n-octave is taken as a flog), or else harmonic (which 
will be described below). We are content to consider these two types in 
our book, but would admit any other valid type that could be adequately 
3. Max Weber, *Die rationalen und soziologischen Grundlagen der Musik*, 
ed. Theodor Kroyer (Munich 1921).  (The English translation published in 
1958 is, alas, so inadequate that it quite misrepresents Weber's 
[5] Our next construction is the algebraic pair: (set, Abelian group), 
to which such generators give rise.  The groups operate on the sets; the 
operation is addition.  A positive number in a group means "add so much," 
but in a set means an amount that is *per se* so much.  (There is of 
course no such thing as a "negative" or an "absolute-zero" pitch.) 
[6] Although music theorists have always represented notes as points, we 
provide, for the sake of greater validity, that each note in the scale 
will occupy a small neighborhood on the number line, in order to give 
every note some leeway for things like vibrato, inexact intonation etc.  
(We define the musical intervals as between the centres of these 
neighborhoods.)  Thus the elements of our pitch-class sets are 
equivalence classes of neighborhoods around points-mod-1 (see Fig 3).  
To reach the highest degree of validity, one ought to allow that the 
neighborhoods for different notes in the same system may differ in size, 
and that in certain cases (for example, when a violinist has a wide 
vibrato) the neighborhoods for notes adjacent in the scale may overlap, 
obliging one to treat the neighborhoods as fuzzy sets.  For the sake of 
simplicity, however, our book uses a kind of a modelling in which the 
neighborhoods have definite borders, do not overlap, and are uniform in 
any one system.  We also postulate that in every system the leeway is at 
least a couple of ten-thousandths of an octave.  Thus we reach (at Fig 
1h) our next construction: "ideal systems".  Each ideal system has a set 
of non-overlapping neighborhoods (and is thus finite) and a subset of one 
of our groups.  This subset, which we call an "embedded halfgroup", is an 
unusual, indeed novel, algebraic structure: it is associative (when the 
sums are defined) and has an identity element and inverses, but is not 
closed with regard to group multiplication (compounds of the operation).  
The place of halfgroups vis a vis semigroups, quasigroups etc. in 
halfalgebra is indicated in GIF 2.(4) 
4. All the terms in GIF 2 except "halfgroup" are defined in Richard 
Herbert Bruck, *A Survey of Binary Systems* (Springer, Hamburg 1958).  
On the basis of our cordial correspondence with Eytan Agmon we think 
that a 12-oriented theory of diatonicism (as described in his "A 
mathematical model of the diatonic system" in *Journal of Music Theory*, 
33:1) can be improved upon, insofar as validity in regard to certain music 
is concerned (e.g. Gregorian chant), by an acceptance of the concept of 
halfgroups as applicable to music. 
[7] We wonder if in some cases the size of the neighborhood might 
guarantee a set of pitch classes small enough to be musically useful.  
Music is not only an art of sonorities but also (among other things) a 
cognitive game, and most composers wish to "juggle" with a set of pitch 
classes that is big enough to enable them to sustain an interesting 5-, 
10- or 20-minute-long game of this kind - for which only three or four 
pitch classes would be insufficient - but not so big that the listener 
cannot grasp the cognitive game intuitively - for which, say, 40 pitch 
classes would be too many.  One needs an intermediate number, something 
like 7, 8, 10, 12, 15 or 20.  Gregorian chant normally has 7 or 8; 
Giovanni Gabrieli and Schutz had 14; some other well-known Renaissance 
composers (Costeley, Bull) composed music for 19 pitch classes; Bach, 
Debussy and the Beatles had 12. In non-Western music, the sizes of the 
sets are comparable.  Now we have noticed that in outdoor genres - for 
instance, marching-band music - the intonation is not very exact (that 
is, the neighborhoods for the notes are quite wide) and the number of 
pitch classes in a phrase is normally closer to 7 than to 12.  Is there 
a cause-and-effect relation here, in that such bands are usually unable 
to project chromatic harmony because their intonation is so inexact?  The 
question has not been investigated (as the concept of pitch-class-leeways 
is new); we raise it in order to show that just where we come to our 
unusual algebraic structure (the halfgroup) we find a music-theory 
question which calls for empirical treatment - namely, the possible 
relation between *(a)* the limiting of the set which is due to the pitch-
class leeways and *(b)* the limiting which one would want in any case for 
the sake of a cognitively viable juggling of the pitch classes. 
[8] It was by means of a natural mapping that we went over from notes and 
musical intervals to pitch classes and pitch-class relations; so now we 
return by reversing the natural mapping and thereby pulling back the 
system to an "unbounded" scale (repeating itself indefinitely, octave 
after octave) from which limited scales, each with a highest and a lowest 
note, can readily be derived (Fig 1i-j).  One could then go farther, to 
scales in which certain pitch classes are omitted in certain octaves, or 
to scales in which every interval has a little something extra added to 
it (and thus the frequency-ratio for the octave, for instance, is a 
little bigger than 2:1, as on the piano) and so on.  We prefer, however, 
to concentrate our attention on systems. 
[9] Most systems of Western music have had harmonic generators.  There is 
a series of such generators, which - so experience teaches us - can be 
derived mathematically from the following series of primes: 2, 3, 5, 7.  
Our adaptation of the traditional Roman numerals of music theory for 
these generators is shown in Fig  4.  With the first generator alone 
(which we write with the Roman numeral "I"), one can make a kind of 
minimal music in which all the notes belong to the same pitch class; with 
the first two generators (I and V), one obtains the most familiar kind of 
Medieval harmony, in which the 5ths (and their compounds and inversions), 
but not any 3rds, are used as consonant intervals; with the first three 
generators (I, V and III) one obtains the triadic harmony of the 
Renaissance; and with all four (I, V, III and VII), one has certain 
aspects of later harmony.  
[10] The extra "smaller flogs" (positive or negative) referred to in 
Fig 4 are used to obtain one or more equations between the generators, 
and thereby more pitch-class relations with a pitch-class set of a given 
size.  These small amounts, which we designate with the letter "t", are 
necessary for this, because no multiple of the flog of one prime number 
can be equal to another such flog.  (It is well known that no power of 
one prime number can have another prime as a factor.)  The use of such 
small extra amounts is traditionally called "tempering", and in practice, 
temperaments - systems with tempered consonances - have been normal since 
the Renaissance. 
[11] How small should these small amounts be?  To answer this question 
we have to classify empirically certain intervallic magnitudes, or 
rather, certain ranges of magnitude.  Some amounts that are too small 
to be used for melodic intervals between notes (or for pitch-class 
relations) are nonetheless big enough - in the form of deviations from, 
say, log 3 or log 5 - to disturb a generating harmonic pitch-class 
relation by making the resulting intervals sound out of tune.  A rough 
classification is shown in Fig 5.  This is only a first approximation; 
empirically there is no particular validity in a tenfold relation 
between the different ranges.  For example, while the semitones in 
equal temperament are each 1/12 octave, a semitone between an untempered 
major 3rd (i.e. where t^III^ = 0) and an untempered minor 3rd would be 
only some 6% of an octave (this is reckoned in GIF 3), so it would be 
better to say that the range of magnitudes for melodic steps is "20ths" 
rather than "10ths" of an octave.  Also, an amount which would very 
likely render a 5th sour can in certain cases serve for the tempering of 
a major 3rd.  (Such is the case in equal temperament, where the major 
3rds are tempered by a little more than 1% of an octave; whereas a 5th 
tempered by such an amount would be melodically and harmonically too ugly 
to use in many kinds of music.)  Thus the concept of ranges or orders of 
intervallic magnitude needs to be refined empirically by means of psycho-
acoustical probings and a reading of the old music treatises (which often 
discuss temperaments).  In general, however, tempering is taken to mean 
the dividing up of such inconvenient magnitudes as are labeled "out-of-
tune-ness" in Fig 5 into smaller, less noxious amounts to be distributed 
amongst a suitable chain of generating pitch-class relations.  
[12] In order to find the most feasible possibilities for tempering "two-
dimensional" systems (systems with V and III as generators, but not VII), 
we ask the following question: If one multiplies flog 3 by 1, -1, 2, -2, 
3, -3 and so on, then which are the multiples that approach successively 
closer to flog 5 or its inversion?  In the last (full) column of Fig 6 we 
see the smallest flogs by which it is possible to temper V and III at 
once by distributing the various such differences evenly amongst the 
group of generators.(5)  The first two such flogs - T^1^ and T^2^ - are 
too big (they would mar the pitch-class relations); T^3^ is good; T^4^ is 
nearly insignificant, hence very good; to make use of T^5^ would involve 
more than 45 pitch classes - too many for traditional composition.  
Thus the equations at the far right in Fig 6 represent the most likely 
possibilities.  They are the most feasible equations between harmonic 
generators for a two-dimensional system.  
5. When we go from one n to the next we get a rather lower T^n^ because 
s^n^ is smaller while at the same time m^n^ is bigger.  (However, since 
each T is an average of some t^V^s and a t^III^; one of those t's may be 
less than T if the other is more.)  Some intermediate multiple might 
yield a *slightly* lower average t (if the quasi-s is not as much bigger 
as the multiple is smaller), but it is reasonable, once the multiple 
becomes bigger than, say, 15 or 20, to demand a distinctly lower average 
t in return for involving more pitch classes. 
[13] Now we are ready to discuss our system tree (see Fig 7).  We include 
those possibilities that have been significant in the history of Western 
composition and theory.  There are harmonic and equal-division systems, 
according to the type of generator.  Among harmonic systems we have those 
of one, two or three dimensions, according to the number of generators 
(apart from the identity element).  Also among harmonic systems, we 
distinguish between coherent sytems (in which all the pitch classes make 
one chain of 5ths) and non-coherent systems - which have proven musically 
so awkward that no well-known composer has ever written music for such a 
system, even though many theorists since the 16th century have described 
non-coherent, two-dimensional systems without any tempered intervals.  
(Mostly they were theorists who did not understand the problem to which 
tempering is the solution.) 
[14] Among coherent systems, we have temperaments and (one-dimensional) 
untempered systems.  Then we have regular temperaments (in which each 
kind of consonant interval is of uniform size), semi-regular temperaments 
(discussed below), and irregular temperaments, in which some 5ths are 
tempered a little more than others and hence the 3rds etc. also vary.  
Certain irregular temperaments have been quite important historically. 
[15] Among regular, two-dimensional temperaments, there are two main 
types (as we have seen in connection with Fig 6); and when both of their 
defining equations (4V = III; -8V = III) are true for the same system, 
then we have an intersection of the two types, which is so important 
historically that it has its own name, "equal temperament"; and this name 
can refer as well to an equivalent one-dimensional type (i.e. with a 
"circle" of twelve 5ths but no consonant 3rds) which may conceivably have 
played a role in the history of lute music in the 15th century, and also 
to a three-dimensional type which practically everyone would agree is to 
be found in the music of, say, Villa-Lobos (and which we believe is to be 
found in some earlier music as well: think of how Wagner will resolve an 
appoggiatura to a 7th-chord from which the harmony is then just as free 
to move as it would be, in 17th-century harmony, from a triad).  
[16] Apart from equal temperament, there is a spectrum of musically good 
possibilities for each of the two types MT and QP (see Fig 7h).  Meantone 
temperaments were very important for Renaissance and early Baroque music.  
They usually put at the composer's disposition two or three flats and 
three or four sharps: if three flats and four sharps, then in all the 14 
pitch classes (7 chromatic and 7 diatonic) mentioned above in connection 
with Gabrieli and Schutz. (See Fig 8. It is well known that some keyboard 
instruments had 14 keys per octave, i.e. with "split keys" for Eb/D# and 
for Ab/G#.)  The chain of 5ths had a beginning and end, and this was very 
important for the scheme of Renaissance modes, and often important also 
for the planning of compositions. In GIF 4a, for example, we see how 
Schutz in one of his pieces timed the successive steps toward the edges 
of his chain of 5ths.(6)  In an 18th-century composition one would 
normally find the richest harmony in the middle of the movement, not just 
before the end.  And why?  Because in the 18th century, the chain of 5ths 
was closed to make a circle; to modulate "far away" did not mean to 
approach a border; so towards the end of the piece one would merely 
return to the freely-chosen central pitch class, with no opportunity to 
draw upon the structural discipline of an impending fence.  In many late 
19th-century compositions, on the other hand, all the pitch classes are 
introduced already in the first few bars. 
6. GIF 4a shows a diagram from page 144 of our book and GIFs 4b, 4c,
and 4d show some relevant excerpts from the piece to which the diagram 
refers, "Die so ihr den Herren fuerchtet" (SWV 364).  Twenty diagrams of 
this kind (with relevant musical examples) are included in Lindley, 
"Heinrich Schutz: intonazione della scala e struttura tonale" (with a 
long abstract in English), in *Recercare*, vol. i (1990). 
[17] A quasi-Pythagorean temperament with twelve pitch classes was of 
some importance in the first half of the 15th century.  The 5ths were 
either untempered or else so little tempered that no one at all was 
aware of it.  The five chromatic pitch classes were linked to Bb in 
the chain of 5ths - we know this from contemporary treatises(7) - and 
so there was a "wolf 5th" between B and F#, as F# was tuned so low that 
it made a sour 5th with B.  By chance, however, all these rather low 
chromatic notes made remarkably euphonious 3rds (hardly tempered at all) 
with the diatonic notes, as indicated by the slanted lines in the 
diagram at the beginning of GIF 5a.(8)  Now in this transitional period 
between one-dimensional and two-dimensional harmony, certain composers 
would sometimes use such a 3rd at the end of a section of a piece (as 
can be seen in the musical examples in GIFs 5b-5g),(9) but no one would 
make such use of a 3rd without a sharp.  To understand this interesting 
moment in the history of harmony, one must appreciate properly the 
significance of the system; we will say more about this below. 
7. Lindley, "Pythagorean Intonation and the Rise of the Triad", Royal 
Musical Association *Research Chronicle* 16 (1980). 
8. The diagram is on page 55 of our book. 
9. These examples (GIFs 5b-5g) show the conclusions of sections from 
the following pieces:  Landini, "O fanciula giulia;" Matteo da Perugia, 
"A qui fortune" and "Le grant desir;" a Kyrie for organ from the Faenza 
Codex;  Dufay, "Mon chier amy;" and a prelude from the Buxheim Organ 
Book (no. 242). 
[18] For some of the regular temperaments in our system tree, there is 
an *equivalent* equal-division system - that is, physically the same 
though differently conceived (see Fig 7i-j).  In a harmonic system there 
exists consonance (since the harmonic generators are the most consonant 
pitch-class relations) and hence nearly always also its counterpart, 
dissonance; and there is an important distinction between diatonic 
semitones (between two notes with different letter-names) and chromatic 
semitones (between two notes with the same letter-name).  In an equal-
division system there is no consonance, and therefore no dissonance.  
If the generator is 1/12-octave and if all twelve of the ensuing pitch 
classes are in the system, then one composes "mit zwoelf nur auf 
einander bezogenen Toenen" (in Arnold Schoenberg's words) and there 
is no distinction between diatonic and chromatic semitones: they are 
all qualitatively as well as quantitatively alike.  This aspect of 
Schoenberg's dodecaphonic music is just as important as his atonality 
(the fact that each movement or piece is not somehow centered on one 
privileged pitch class), which is often said to be its most basic 
technical characteristic. 
[19] With such a perspective on how various systems have affected the art 
of composition, one can appreciate better the technical significance of 
enharmonic modulations in Romantic music.  In most enharmonic modulations 
a given semitone is used first as a chromatic semitone and then as a 
diatonic one, or *vice versa*.  More and more in the course of the 19th 
century, the significance of enharmonic modulations lay not so much in 
their momentary effect as in the way they enabled composers to exploit 
the same physical scale in terms of two systems at once: harmonic and 
equal-division.  Thus David Lewin's analytical sketch (reproduced in 
GIF 6) of a well-known phrase in the prelude to Wagner's *Parsifal*(10)  
includes not only Roman numerals for a traditional harmonic analysis, but 
also Arabic numerals to show how 3 + 3 + 1 = 7 semitones (adumbrating the 
salient "Zauber-motif" in *Parsifal*) lead from Ab to a cadence on Eb. 
10. David Lewin, *Generalized Musical Intervals and Transformations* 
(Yale, 1987), p. 161.  The 3 + 3 + 1 diagram ends with a high Eb, but 
the tune really goes to the Eb an octave lower after gliding down, step 
by step, from high Ebb (a minor 3rd above Cb) to middle D (a diatonic 
semitone below Eb).  One hears an implicit equation between the Ebb and 
the D, inferring that they are an octave apart, and this gives the 
passage its enharmonic character.  
[20] It is possible to distinguish certain "families" of equal-division 
systems (see Fig 7j) equivalent to the various kinds of temperaments.  
Fig 9 includes formulas (derived in one of the appendices of our book) 
for their generators.  GIF 7 reproduces a diagram by Isaac Newton(11) 
showing how an equal-division system with 1/53-octave as generator and 
with 15 pitch classes is equivalent to the harmonic system represented 
in Fig 10.  The diatonic semitones, labeled "mi-fa" in GIF 7, amount to 
5/53-octave; the chromatic semitones amount to 4/53.  Newton's harmonic 
system is not coherent, but if he had provided for an additional pitch 
class at "4" in the diagram, it would have made at once a good Ab to his 
Eb (at "35") and a good G# to his C# (at "26"), and thus he would have 
had a coherent, quasi-Pythagorean system.(12) 
11. *GB-Cu* add. 4000, fol. 105*v*. (Reproduced on page 57 of our book.) 
12. Helmholtz for his "Harmonium mit natuerlicher Stimmung" used a system 
of this latter kind with 24 pitch classes.  See Hermann von Helmholtz, 
*On the Sensations of Tone*, tr. Alexander J. Ellis, 2nd ed. (London, 
1885), 316-19, or for a more succinct account, the entry on "Just 
intonation" in *The new Grove Dictionary of Musical Instruments*. 
[21] Among the irregular temperaments, the most important historically 
were those used in the late 17th and 18th centuries.  For most composers 
of that time, the various keys had much more individual character than 
they do today, and many contemporary music theorists said that it was 
due to the irregular temperaments of the day.  An irregular temperament 
based on a circle of twelve Vs can be described as a variant of equal 
temperament, so in GIF 8 the sizes of each semitone in three such 18th-
century schemes (by J. G. Neidhardt, J. H. Lambert and Vallotti) are 
described as some percent of 1/12-octave.  The numbers at the outer 
edges of those diagrams show the differences between semitones that 
are adjacent in the circle of 5ths (in the sense that E-F and F#-G are 
adjacent to B-C) and thereby show that in each of these competently 
designed schemes, the semitones vary quite gradually, with B-C and E-F 
being the largest and F-Gb and A#-B the smallest.  There is an analogous 
pattern of gradually varied nuances among the 3rds and 6ths, with C-E-G 
being tempered least and Gb-Bb-Db-F most. 
[22] On the silent screen we cannot demonstrate the acoustical 
differences amongst the different keys in such a system.  But we can 
describe how, in the first section of Louis Couperin's famous F#-minor 
Pavane (see GIF 9), the composer used the pitch class F = E# in a special 
way.  E#, which is essential to the key of F#-minor, was in the French-
style irregular temperament tuned so high in relation to C# that the 
resulting interval was acoustically rather harsh.  In bar 2 (at the first 
asterisk in GIF 9) E# is avoided: contrapuntally, our little ancillary 
example in GIF 9 would sound so much more natural that Couperin's 
alternative resolution of the chord F#-C#-G#-A is obviously an artful 
evasion.  A similar avoidance of E# at the end of bar 5 (at the second 
asterisk) precipitates a modulation to A-major in the next two bars.  In 
bars 10, 11 and 16 (at the next three asterisks) E# does appear, but each 
time in so dissonant a context (notice the A's and B's) that the acoustic 
sourness of E# with C# merely gilds the lily, as it were.  C#-E# is at 
last heard in a straightforward triad at the end of the section; but then 
in the next bar (not included in GIF 9) the composer reverts immediately 
to a C#-minor chord, as if to say, "Alas! E# is *too* sharp for a 
straightfoward triad; let us revert to E-natural."  This is an extreme 
case in that the consonant status of the major 3rd (or 10th) was actually 
compromised by its heavy tempering.(13)  A wealth of subtler nuances 
involving some of the other 3rds in this piece are just as telling when 
the music is heard in a stylistically appropriate tuning (matching the 
conceptual system). It would be far more intelligent, however, to 
demonstrate such nuances than to try to describe them *in absentia*. 
13. It may be worth repeating that this is a French style.  According to 
Bach's concept of the chromatic scale, which is reflected in what we know 
about his tuning (Lindley, "Bach's Harpsichord Tuning", *The Musical 
Times*, vol. 126, December) as well as his music, harmony in "extreme" 
keys is less constrained. 
[23] In early 15th-century music such as represented in GIF 6, the 
distinctly euphonious quality of a 3rd or 6th with a sharp in the quasi-
Pythagorean temperament is sometimes especially salient because it occurs 
right after (or right before) a prominent harmonic 3rd or 6th that is 
tempered by an entire comma (that is, by nearly as much as C#-E# in 
Louis Couperin's pavane).  We may therefore speak of a "semi-regular" 
temperament (Fig 7m), because while the 5ths are uniform, the composer 
has evidently found two sizes of consonant or virtually consonant major 
3rd, major 6th etc. in the scale.  This is a queer kind of system, 
destined to play only a brief (though important), transitional role in 
the history of harmony even though it is physically the same as a regular 
[24] To measure the difference between any two systems that are 
physically almost but not quite the same, we have devised a "margin of 
equivalence", and with it the concept of "quasi-systems" which have no 
generators (and thus no subset of a group of pitch-class relations) but 
only a set of pitch classes, whose neighborhoods are, however, unequal.  
To put it very briefly: if two systems have the same number of pitch 
classes, then the margin of equivalence is the smallest overlap - i.e. 
where the notes differ most when the two systems are aligned as well 
as possible (as illustrated in GIF 10).  
[25] We hope that our book in which these and some related ideas are 
elaborated upon(14) will prove of value to music theory and to the study 
of music history.  Renaissance and Baroque theorists took only some 
limited steps away from Medieval models of scales (by accepting ratios 
involving 5 and 7 as prime factors, and then by accepting irrational 
ratios), and even today many music theorists more or less vaguely favor 
the ancient Pythagorean idea that "Music is sonorous number."  Here our 
algebraic approach could be of value, not only with regard to irregular 
temperaments (where the pitch-class relations have to be represented as 
functions of the pitch classes and not as numbers in their own right), 
but also for the designing of experiments to investigate the various 
musical and psycho-acoustical phenomena that give rise to pitch-frequency 
leeways for the notes. (Instead of a general leeway u, one could 
distinguish u^1^, u^2^, u^3^....)  Some refinement of concepts pertinent 
to music history may also be derived from our work, as music historians 
have generally either neglected most of the various kinds of system which 
we describe or else have mistakenly treated them as a negligible aspect 
of performance practice - that is, as unconscious and inconsequential 
variants of equal temperament insofar as composition is concerned.  
14. *Mathematical Models of Musical Scales* (Verlag fur Systematische 
Musikwissenschaft, Postfach 9026, DW-5300 Bonn, Germany). 

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