# The Harmony Problem

### by Steven H. Cullinane on October 23, 1996

The 12-note scale of Western music is an attempt to solve the following problem:

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 Cullinane Frequency Matrix
C . D . E F . G . A . B
3 .... 4 ... 5 ..
. 3 .... 4 ... 5 .
.. 3 .... 4 ... 5
5 .. 6 .... 8 ...
. 5 .. 6 .... 8 ..
.. 5 .. 6 .... 8 .
... 5 .. 6 .... 8
4 ... 5 .. 6 ....
. 4 ... 5 .. 6 ...
.. 4 ... 5 .. 6 ..
... 4 ... 5 .. 6 .
.... 4 ... 5 .. 6

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

There are two approximate solutions:
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 21/12, 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 21/34, 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
```
These values are all within plus or minus half a percent of the corresponding values in just intonation, if we consider the square root of two (approximately 1.4142) to be the value of F# in just intonation.

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.