Musical Mathematics

on the art and
science of acoustic instruments

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* * *

Despite
Harry Partch’s attempt to recognize Max F. Meyer’s “. . . salutary effect
. . . [on the] . . . presentation of material . . .” in |

* * *

CHAPTER 10:* WESTERN TUNING THEORY AND PRACTICE*

Part VI: Just Intonation

Section 10.65

In 1929, one hundred and seventy-nine
years after the appearance of Rameau’s *Démonstration*,
*The Musician’s Arithmetic*.[1]
Meyer wrote this book as a primer in mathematics for music students.

* * ^{↓} * *

The great practical problem for the
student of “a musician’s arithmetic” consists in
*learning to talk of ratios* whose terms
contain the prime numbers 1, 3, 5, and 7 . . . as factors, in the numerators and
denominators of the fractions expressing the ratios.
(“2” is omitted from the list of prime numbers
because . . . in “discovering the octave” we have dispossessed ourselves of all
even numbers; “11” is omitted because a prime number as large as that is
unlikely to be needed, but whoever wants an intellectual chastisement may put it
back.)

The student cannot talk intelligently of ratios
unless they mean something to his ear.
Unfortunately he has never met a teacher (have you met one?) capable of training
him to connect melodic phrases with ratios of numbers.[2]

* * ^{↑} * *

Throughout his text, Meyer is extremely critical
of Rameau’s theories. He finds Rameau’s conclusion “. . . that there are
only two modes, the major and the minor . . .”[3]
particularly offensive and damaging to the future development of music:

* * ^{↓} * *

Rameau’s mistake in substituting two modes for
the ancient seven modes was an error of over-emphasis . . . His mistake
has resulted in denying to the composer the freedom of further inventions, and
has thus hampered musical progress.[4]

* * ^{↑} * *

Despite such criticisms, there are many
indications that Rameau had a considerable influence on Meyer’s understanding of
musical mathematics. For example, Meyer’s highly original illustration on p. 22 of his
book shows a mathematical and musical structure that would have been unthinkable
without knowledge of Rameau’s *Génération
harmonique*. Figure 10.57(a) is an exact copy of Meyer’s illustration.
Notice immediately that this figure is remarkably similar to Al-Jurjani’s
triangular table in *La Musique Arabe,
Volume 3*, p. 230, and to Salinas’ engraving in Figure 10.46. All three
figures consist of diamond-shaped tiles organized into a two-dimensional pattern
of just intoned ratios. In Meyer’s figure, evidence of just ratios is not immediately
apparent because, with the exception of integers 1, 3, 5, 7, all other numbers
represent cent values. However, the ratio
analysis in Figure 10.57(b) reveals that these cent values indicate four
ascending major tonalities, and four descending minor tonalities.

**
**

*dash* between two numbers will always
mean an interval — or ‘span,’
as we call it, — between the two tones represented by the two numbers.”[5]
Although such an ordered sequence of integers is numerically appealing,
it does
not express a graduated progression of intervals: here a large
“fifth,” ratio 3/2, is followed by a small “major third,” ratio 5/4, which in
turn is followed by an even larger “minor seventh,” ratio 7/4. Therefore,
I decided to reverse the order of prime numbers 3 and 5, resulting in the
sequence: 1–5–3–7. Figure 10.58(a)
shows that without changing the contents of the diagonals, every tetrad now
expresses the graduated sequence: “tonic,” “major third,” “fifth,” “minor
seventh.” To help clarify these ascending and descending relations,
Figure 10.58(b) gives the approximate note names of Meyer’s tonality diamond
based on a “tonic,” ratio 1/1, tuned to middle C, or C4.

**
**

We may arrange the thirteen unique frequency ratios of Meyer’s 7-limit Tonality Diamond — plus the “octave,” ratio 2/1 — into the following scale, which contains three kinds of “thirds,” two kinds of “tritones,” and three kinds of “sixths”:

The musical organization of Meyer’s tonality diamond is based on Rameau’s theory
of a *dual-generator*. In other
words, in Figure 10.58, pitch C4
generates the first ascending major tonality C4–E4–G4–Bb4, expressed as ratios
1/1–5/4–3/2–7/4, and pitch C4 generates the first descending minor tonality
C4–Ab3–F3–D3,
expressed as ratios
1/1–8/5–4/3–8/7.
Turn now to Table 10.34, which
lists the exact ratios and approximate note names of Meyer’s 7-limit Tonality
Diamond. Here the left column lists four ascending diagonals that
constitute four ascending major tonalitites, and the right column, four
descending diagonals that constitute four descending minor tonalitites.
Observe that since the first interval of the first ascending diagonal consists
of an ascending “major third,” 1/1 ×
5/4 = 5/4, or interval C4–E4, the first interval of the first descending
diagonal sounds a descending “major third,” 2/1 ÷ 5/4 = 8/5, or interval
C4–Ab3.
Consequently, the latter pitch Ab3, ratio 8/5, acts as the first tone of the
second ascending major tonality Ab3–C4–Eb4–Gb4.
Similarly, since the second interval of the first ascending diagonal
consists of an ascending “fifth,” 1/1 ×
3/2 = 3/2, or interval C4–G4, the second interval of the first descending
diagonal sounds a descending “fifth,” 2/1 ÷ 3/2 = 4/3, or interval
C4–F3.
Consequently, the latter pitch F3, ratio 4/3, acts as the first tone of the
third ascending major tonality F3–A3–C4–Eb4. Finally, since the third
interval of the first ascending diagonal consists of an ascending “minor
seventh,” 1/1 × 7/4 =
7/4, or interval C4–Bb4, the third interval of the first descending diagonal
sounds a descending “minor seventh,” 2/1 ÷ 7/4 = 8/7, or interval
C4–D3.
Consequently, the latter pitch D3, ratio 8/7, acts as the first tone of the
fourth ascending major tonality D3–F#3–A3–C4.

With respect to the second column in Table 10.34, observe that E4, or the “major third” of the first ascending diagonal, acts as the first tone of the second descending minor tonality E4–C4–A3–F#3; similarly, G4, or the “fifth” of the first ascending diagonal acts as the first tone of the third descending minor tonality G4–Eb4–C4–A3; and finally, Bb4, or the “minor seventh” of the first ascending diagonal acts as the first tone of the fourth descending minor tonality Bb4–Gb4–Eb4–C4.

Finally, and most importantly to composers and musicians, the tonality diamond
furthers the study of just intonation because its unique lattice design of
crisscrossed diagonals reveals that every ratio may be taken in two different
senses. In this respect, the diamond pattern sheds new light on the
Western practice of modulation. Students of traditional harmony learn that modulation from one key
to another key — say, from C-major to Bb-major — requires a transitional chord
that both keys have in common; in this
case, an F-major triad qualifies because it represents the chord of the
subdominant in C-major, and the chord of the dominant in Bb-major.
Similarly, every just ratio in Meyer’s tonality diamond serves a double
function. For example, Eb4, ratio 6/5, functions as a “major third,”
ratio 5/4, in the descending minor tonality that begins on G4, ratio 3/2;
and Eb4 also functions as a “fifth,” ratio 3/2, in the ascending major tonality
that begins on Ab3, ratio 8/5.

On many occasions, Meyer refers to his tonality diamond as a “table of spans.”
It is his single most important teaching tool. Meyer uses the diamond not only for convenient interval
calculations, but also as a musical mandala designed to symbolize myriad
mathematical possibilities of just intoned harmonies and scales. Toward
the end of his book, Meyer acknowledges Rameau’s contributions, and offers his
own thoughts on the importance of just ratios in the study of music:

* * ^{↓} * *

**Rameau
and beyond Rameau**. Rameau brought a certain clearness into the theory
of chords by giving each tone *an absolute
name* (fundamental, third, fifth, etc.) in the chord without any reference to
the actual intervals. This reference to
the actual intervals, he discovered, could be avoided by using the concept of
the inversion of chords . . .

The limits of this
clarifying influence of Rameau we recognized when we studied his “theorem.”
It is only by substituting *number
symbols* for such terms as Rameau’s “fundamental, third, fifth, seventh,
etc., natural, diminished, augmented” that we can free the theory from
artificial fetters . . .

The number
symbol has the advantage over all other terms that it is both absolute and
relative . . . Only number symbols can
simply and directly and without modifying epithets fulfill this double
condition. All other names force us to use queer modifiers like
“augmented, sharpened,” or what not, of little definiteness.

A number is
always absolute, individual, in being distinct from all other numbers.
And it is always relative because it permits and invites the formation of
a ratio. **
And there is no lower nor upper limit to
the quantity of terms which may enter a ratio, — two, three, four, five, any
multitude.** The crazy concept of a “triad” as the only legitimate
tone family, outlawing all smaller and larger families as being of illegal size,
is safely avoided. For examples compare in the body of our text our numerous “scales”
varying in size from two [tones and] up.

Ratios, when
*reduced* to their lowest terms or
*translated* into “spans” are quickly comparable with other ratios
without any possibility of ambiguity. No other terms are safe from
ambiguity.[6]
(Bold italics, and text in brackets mine.
Text in parentheses in Meyer’s original text.)

* * ^{↑} * *

We conclude, therefore, that Meyer considered the four diagonal lines that
enclose his tonality diamond as moveable boundaries, and that he
did not
rigidly limit the musical possibilities of just intoned frequency ratios to
prime numbers 2, 3, 5, and 7.

*
*
Meyer, M.F. (1929).
*The Musician’s Arithmetic*. Oliver Ditson Company, Boston,
Massachusetts.

Under the
mentorship of renowned physicist Max Planck (1858–1947), and distinguished
acoustician Carl Stumpf (1848–1936), Max F. Meyer received his Ph.D. from the
University of Berlin in 1896 at the age of twenty-three. With the approval of
Planck and Stumpf, Meyer’s dissertation, *Über Kombinationstöne und einige
hierzu in Beziehung stehende akustische Erscheinungen*, which describes a
mathematically based theory of hearing, was published in the same year. In
1900, Meyer founded the Psychology Department at the University of Missouri,
and held the position as Professor of Experimental Psychology until 1929.

*
*
*Ibid*.,
p. 20.

[3]
Hayes, D., Translator
(1968). Rameau’s*
Theory of Harmonic Generation; *
*An Annotated Translation and Commentary
of “Génération harmonique” by Jean-Philippe Rameau*, pp. 163–164.
Ph.D. dissertation printed and distributed by University Microfilms,
Inc., Ann Arbor, Michigan.

[4]*
*
*The Musician’s Arithmetic*,
p. 51.

[5]*
*
*Ibid*.,
p. 6.

[6]*
*
*Ibid*.,
pp. 103–105.