A NON-PYTHAGOREAN MUSICAL SCALE, INCLUDING A TWELVE-TONE OCTAVE THAT MAY BE ASSOCIATED WITH TRADITIONAL MUSICAL NOTATION AND WITH THE KEYS OF A RE-TUNED PIANO

By Robert Schneider

A new musical scale may be defined by letting successive pitches have the ratios to one another of the natural logarithms of successive whole numbers. Most of these pitches bear no correspondence to the twelve tones of the tuning system attributed to Pythagoras, based on ratios of whole numbers, or to the twelve chromatic tones of equal temperament tuning.

Successive pitches in the logarithmic scale grow closer and closer to one another, and the number of distinct tones in each octave increases nearly exponentially, with each successive octave. When listening to the lower octaves of the scale, the ear strains to bend the tones to traditional chromatic pitches lying near them. However, most of the intervals of the scale are irrational, and do not correspond to any of the traditional twelve tones. As successive tones grow closer together they become almost indistinguishable, and the resemblance to a traditional scale disappears.

To begin the scale on middle C we define a fundamental pitch F by

F = c / (2 log 2) = 190.44 Hz

where c = 264 Hz is a typical frequency for middle C and 2 log 2 = 1.3863 approximately, with the notation log M referring to the natural logarithm of a whole number M. We note that any frequency may be substituted for c = 264 Hz in this expression to establish the scale in another key.

Then the Mth pitch in the logarithmic scale will be defined and notated as

M* = F log M

We indicate a logarithmic tone with an asterisk. If we let M = 1 we find the first tone 1* is silence, as F log 1 = 0. The second tone 2* is C one octave below middle C, the fourth tone 4* is middle C, and the sixteenth tone 16* is one octave above middle C. The eighth tone 8* is a perfect fifth above tone 4*, corresponding to the pitch G in the traditional scale, but all other intervals of pitches having M < 16 are irrational, aside from the octaves of C; the tones do not fall into the chromatic scale or correspond to any pitch on a piano keyboard.

As noted above, successive pitches grow closer together as M increases, and each octave contains a greater number of tones than the preceding octave. The C two octaves above middle C is tone 64*, so there are 48 tones in the octave beginning with tone 16*. The next C is tone 256*, so there are 192 tones in the octave beginning with 64*. The next octave of C is 1024*, and so on. The construction of a piano with such a great number of keys in the higher octaves seems impossible. It is likely that the electronic medium is better suited to composing with the logarithmic scale.

The beat frequency is a kind of overtone defined as the difference between two pitches sounded simultaneously. Due to the subtraction property of logarithms (log x - log y = log (x/y)), there is potential in the logarithmic scale to compose with not only the tones themselves, but with the beat frequencies within chords. If whole-numbered tones X* and Y* are chosen such that X/Y is a whole number, the beat frequency created by sounding the two pitches together will itself be a tone (X/Y)* of the logarithmic scale, allowing an additional layer of manipulation of tones within the scale-- that is, purposeful melodies and harmonies may be constructed in the realm of the overtones. The author wishes to explore such avenues for composition in the future.

To generate a twelve-tone scale suitable for playing upon a re-tuned traditional piano keyboard, and for composing using traditional notation, we associate tone 4* = F log 4 with both the pitch and keyboard position middle C on the keyboard; and associate subsequent logarithmic pitches with the subsequent keys in the octave, such that the formula for the Nth tone in the octave above middle C has the pitch

F log(3 + N),

for N ranging from 1 to 12. Higher and lower octaves of the resulting sequence of tones may be associated with the other octaves of the keyboard by multiplying or dividing the following frequencies (approximated to two decimal places) by powers of 2:

C* = tone 4* = F log 4 = 264 Hz
C#* = tone 5* = F log 5 = 306.49 Hz
D* = tone 6* = F log 6 = 341.22 Hz
D#* = tone 7* = F log 7 = 370.57 Hz
E* = tone 8* = F log 8 = 396 Hz
F* = tone 9* = F log 9 = 418.43 Hz
F#* = tone 10* = F log 10 = 438.49 Hz
G* = tone 11* = F log 11 = 456.64 Hz
G#* = tone 12* = F log 12 = 473.22 Hz
A* = tone 13* = F log 13 = 488.46 Hz
A#* = tone 14* = F log 14 = 502.57 Hz
B* = tone 15* = F log 15 = 515.71 Hz,

where we employ the convention of adjoining an asterisk to the traditional name of the keyboard position of a tone, to indicate that it is a logarithmic note, e.g. C*, D*, E*, F#*, etc.

It would be desirable to play an acoustic piano tuned to these pitches. The author has composed a number of short pieces in the twelve-tone logarithmic scale using a tone generator and computer, using the H-Pi Tuning Box for MIDI synthesizers, and also using a re-tuned kalimba, or African thumb piano; and intends to attempt longer works exploiting the special properties of the scale.

The melodies and harmonies possible in the logarithmic scale have a different nature from those produced by the chromatic scale, as logarithms obey a special arithmetic compared to whole and rational numbers, allowing for novel avenues of musical expression. For example, logarithmic chords have distinctive "textures," at times resembling crickets chirping, the ringing of bells and other environmental sounds, differing in feeling from chords in the chromatic scale.

There is an alien beauty to this non-Pythagorean musical scale, stemming from the mathematics of logarithms, when the listener becomes accustomed to the strange intervals.