The Harmony Problem

by Steven H. Cullinane on October 23, 1996

What subdivisions of the octave allow us to play, starting with any note, a rising sequence of three notes whose frequencies are in the ratios 3:4:5 or 4:5:6 or 5:6:8?

(Because any frequency and its double sound like the same note, these three ratios are different forms of the same ratio, with terms in varying order.)

In other words, the harmony problem is to find a frequency for each (vertical) column in the matrix below, whose (horizontal) rows indicate the desired frequency ratios

The reader may verify that the harmony problem has no exact solution.

1. "Just intonation," which uses whole-number ratios to make most of the entries exactly right, and
2. "Equal temperament," which uses irrational numbers to make all of the entries in the matrix nearly right.

For a discussion of these solutions, see the article on "Interval" in The New Harvard Dictionary of Music, Harvard University Press, 1986.

The near-harmonies in equal temperament can be summarized by noting that in this system, all the exact (but impossible) ratios become approximate (and attainable) as follows:

3:4:5 becomes 3.0000: 4.0045: 5.0454,
5:6:8 becomes 5.0454: 6.0000: 8.0090,
4:5:6 becomes 4.0045: 5.0454: 6.0000.

Another approach is to subdivide the octave into fewer than 12 notes (see "Pentatonic" in the Harvard dictionary) or more than 12 notes (see "Interval" in The New Grove Dictionary of Music and "South Asia" and "Near and Middle East" in the Harvard dictionary.)

Hindu music uses a 22-note octave, Turkish music a 24-note octave with unequal intervals.

Since the 12-note equal-temperament system is based on powers of the root-note 2^1/12, it is natural to ask whether powers of any other irrational numbers yield satisfactory scales. The author discovered such a system, perhaps new [but see postscript], on October 17, 1996. Using as the root-tone the square root of 25/24, which approximately equals 2^1/34, we have

JUST                 34-TONE EQUAL             TONAL
INTONATION           TEMPERAMENT               STEPS
C  =  1/1  = 1.0000  C  = 1.0000 = (25/24)0
C# = 16/15 = 1.0667  C# = 1.0631 = (25/24)3/2   3 steps
D  =  9/8  = 1.1250  D  = 1.1303 = (25/24)6/2   6 steps
D# =  6/5  = 1.2000  D# = 1.2017 = (25/24)9/2   9 steps
E  =  5/4  = 1.2500  E  = 1.2517 = (25/24)11/2 11 steps
F  =  4/3  = 1.3333  F  = 1.3308 = (25/24)14/2 14 steps
F# =  21/2  = 1.4142  F# = 1.4148 = (25/24)17/2 17 steps
G  =  3/2  = 1.5000  G  = 1.5041 = (25/24)20/2 20 steps
G# =  8/5  = 1.6000  G# = 1.5991 = (25/24)23/2 23 steps
A  =  5/3  = 1.6667  A  = 1.6657 = (25/24)25/2 25 steps
A# = 16/9  = 1.7778  A# = 1.7709 = (25/24)28/2 28 steps
B  = 15/8  = 1.8750  B  = 1.8828 = (25/24)31/2 31 steps
C' =  2/1  = 2.0000  C' = 2.0017 = (25/24)34/2 34 steps

Thus the standard 12 notes are embedded in the 34-note equal-temperament octave rather neatly.

Therefore the 34-note octave may be of interest to music theorists -- or even to some working musicians....

See the November 1996 issue of Keyboard magazine (p. 78), where the alternate tunings of Robert Rich (expanded just intonation) and of Harry Partch (a 43-note octave) are discussed.

Copyright © 1996 S. H. Cullinane. All rights reserved.

Postscript of March 10, 2001:

For more information on actual use of the 34-tone scale, see Neil Haverstick's Virtual Chautauqua page.

For more information on how well the 34-tone scale approximates the principal natural harmonies discussed by Helmholtz (On the Sensation of Tone), see Natural Temperament.

Page created March 10, 2001. Return to Journal.