Non-Pythagorean Music Scale
The Non-Pythagorean Music Scale
By Robert Schneider
A Non-Pythagorean Musical Scale (Including A Twelve Tone Octave Which May Be Associated With Traditional Musical Notation And With The Keys Of A Re-Tuned Piano)
A new musical scale is created 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 generated by the circle of fifths, attributed to Pythagoras, or to the twelve tones of equal temperament tuning.
Successive notes in this scale grow closer and closer together, and the number of discrete tones in each octave increases nearly exponentially, with each successive octave.
Listening to lower octaves the ear strains to bend the tones to traditional pitches lying near them. However most of the intervals of this 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 K = C / 2 ln 2 = 264 Hz / 1.3863 = 190.44 Hz, where 264 Hz is a typical frequency for middle C and 1.3863 = 2 ln 2 (the notation ln M refers to the natural logarithm of M). Of course, any frequency may be substituted for C = 264 Hz in equation K to establish the scale in another key.
Then each pitch in the scale is M = K ln M (pitch M is the Mth tone).
For M=1 the first tone is silence, as ln 1 = 0. The second tone is middle C, the fourth tone is C one octave above middle C, and the sixteenth tone is two octaves above middle C.
The eighth tone is G when the scale begins on C.
The rest of the intervals M < 16 are not rational tones. Their ratios are irrational numbers, and they do not fall into the traditional scale or correspond to any pitches on a piano keyboard.
Successive tones grow closer together as M increases. The C an octave above M = 16 is M = 64 so there are 48 tones in that octave. The next C is M = 256 so there are 192 tones in that octave. The next octave of C is M = 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 this scale.
To create a 12-note scale suitable for playing upon a re-tuned traditional piano keyboard, and suitable for composing using traditional notation, we associate Tone 1 = K ln 4 with middle C on the keyboard.
The equation for the Nth tone in the octave above middle C is Tone N = K ln (3 + N), for N = 1 through 13.
We then associate the successive tones with successive piano keys in the octave above middle C, to Tone 12 = K ln 15. Tone 13 falls on C one octave above Tone 1 (again, K = 264 / 2 ln 2 Hz = 190.44 Hz, so that Tone 1 = 264 Hz or middle C). Higher and lower octaves of these pitches may be associated with the other octaves of the keyboard by multiplying and dividing the pitches by powers of 2.
Tone 1 = K ln 4 = 264 Hz (middle C)
Tone 2 = K ln 5 = 306.24 Hz
Tone 3 = K ln 6 = 340.56 Hz
Tone 4 = K ln 7 = 369.6 Hz
Tone 5 = K ln 8 = 396 Hz
Tone 6 = K ln 9 = 417.12 Hz
Tone 7 = K ln 10 = 438.24 Hz
Tone 8 = K ln 11 = 456.72 Hz
Tone 9 = K ln 12 = 472.56 Hz
Tone 10= K ln 13 = 488.4 Hz
Tone 11= K ln 14 = 501.6 Hz
Tone 12= K ln 15 = 514.8 Hz
Tone 13= K ln 16 = 528 Hz (C one octave above middle C)
It would be desirable to play a piano tuned to these pitches, and to hear compositions based on this sequence of tones.
The melodies and chords possible in this twelve-tone scale have a completely different nature from those produced by the traditional scale, as logarithms add according to a different algebra from whole and rational numbers, allowing for novel avenues of musical expression.
Non-Pythagorean chords have distinct “textures” resulting from beats produced by the superposition of frequencies in logarithmic intervals, which differ in feeling from chords in the traditional scale.
There is an alien beauty to the non-Pythagorean musical scale when the listener becomes accustomed to the strange intervals.