Clavier tuning optimization

An equally tempered 12-key scale for clavier instruments that is optimized for more usable intervals and an overall groovy sound.
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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.

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 66,0955494142999 70,4678514975879 73,7148554967589 78,5911840712824 82,2124934138525 88,4150820801927 94,2638487671923 98,6073201150427 105,130315914109 109,974490228391 117,249438345096 122,652035233127 C3 131,905618138233 140,63133740517 147,111320904866 156,842916157004 164,069893639365 176,448280676629 188,120551987352 196,788734309473 209,806551705855 219,473977318577 233,99245150876 244,774310323497 C4 263,241507947355 280,655258249839 293,58723682672 313,008394507975 327,431134624812 352,134443821473 375,428571428571 392,727496355684 418,706904435299 A4 438 466,97424001238 488,491388507849 C5 525,346020013928 560,098306939574 585,906408136244 624,66483940139 653,448024762836 702,747944326561 749,235587265204 783,758719395246 835,605324983272 874,10818514274 931,931519282263 974,872879147151 C6 1048,42295919255 1117,77743054902 1169,2821622819 1246,63161893066 1304,07366897386 1402,46057129699 1495,23506718891 1564,13221872232 1667,60149341721 1744,44091172039 1859,83782876058 1945,53507565341 C7 2092,31755735565

Note that this is a work in progress. Samples for a 61-key clavier are available at [1] (Last updated to v1.0 as of 18-3-2019) Mathematical equations for each key will be made available.