Music is all about patterns. From rhythmic patterns to melodies built from harmonic intervals, mathematics are everywhere. This article illustrates a possible tuning scale which is, like all tuning scales, built around prime numbers.
The goal is to better tune any clavier to create better sounding intervals. Current standard tuning methods revolve around the 81/80 ratio, also known as the syntonic comma. The origin lies with Pythagorean scales, which contained only the 2nd and 3rd harmonics. Musicians found that slightly adjusting some intervals created a much more consonant sound with more usable intervals.
The minor adjusments of the syntonic comma resulted in a slightly stretched octave to accentuate the 3rd and 5th harmonics and to avoid the 7th and up. The 7th can be clearly heard in less mainstream music like jazz and blues and is often referred to as "the blue note". Wind instruments easily accomodate higher harmonics because without tuning they play notes from the harmonic series. Other instruments like claviers are not as accomodating, e.g. most piano's are not tuned to play the blue note.
This article presents a tuning scale based on the microtonal 28/27 ratio, originating from a 19-keyed piano. The scale uses this ratio to create distances of two (E-F, B-C) or three (C-D, D-E, F-G, G-A, A-B) quarter tones, rather than the usual semitones. The 3 quartertone intervals leave room for the black keys to be used for the 18/17 interval relative to the key before it. C#/Db is a pure 18/17 tone, D#/Eb becomes (28/27)^3 * (18/17). Note how the factor 9 is present in both 27 and 18, creating optimal consonance despite using the high prime number 17.
Octaves are very slightly narrowed using this scale, but are closer to an actual 1:2 ratio. All other just ratio's like C:G = 2:3 and C:F = 3:4 are very well maintained. The C and following A are closer together, so the scale as a whole is tuned to A-438.
The calculated frequencies for a five octave keyboard:
C2 66.09554941 69.98352291 73.71485550 78.05102347 82.21249341 88.41508208 93.61596926 98.60732012 104.40775071 109.97449023 116.44357789 122.65203523 C3 131.90561814 139.66477215 147.11132090 155.76492802 164.06989364 176.44828068 186.82759130 196.78873431 208.36454221 219.47397732 232.38421128 244.77431032 C4 263.24150795 278.72630253 293.58723683 310.85707429 327.43113462 352.13444382 372.84823463 392.72749636 415.82911379 A4 438.00000000 463.76470588 488.49138851 C5 525.34602001 556.24872707 585.90640814 620.37149097 653.44802476 702.74794433 744.08605870 783.75871940 829.86217348 874.10818514 925.52631368 974.87287915 C6 1048.42295919 1110.09489797 1169.28216228 1238.06346595 1304.07366897 1402.46057130 1484.95825196 1564.13221872 1656.13999629 1744.44091172 1847.05508300 1945.53507565 C7 2092.31755736
Note that this is a work in progress. Samples for a 61-key clavier are available from cu64 and mathematical equations for each key are to be published.