Clavier tuning optimization

Optimized temperaments for clavier instruments
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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.

(More bla to be added)

Bla about finding a better compromise between just harmonic ratios and equal temperament, after which I present to you:

Note	Cents
C	0.000000
C#Db	118.927630
D	192.428948
D#Eb	311.356578
E	384.857896
F	503.785526
F#Gb	577.286843
G	696.214474
G#Ab	815.142104
A	888.643422
A#Bb	1007.571052
B	1081.072370
C	1200.000000

The distances between the notes are related by the golden ratio. Every x-step interval thus gets two possible lengths, in contrast to equal temperament, where all x-step intervals have the same length. In practice, this results in very consonant intervals, e.g. minor thirds now have either near perfect 6/5 or 7/6 ratios, rather than all having the same harsh sounding inbetween value.

A complete list of intervals per key is to be added.

Interval name	Harmonic ratio
Minor second	short: 24:23 (5x)
	long: 15:14 (7x)
Major second	short: 19:17 (10x)
	long: 8:7 (2x)
Minor third	short: 7:6 (3x)
	long: 6:5 (9x)
Major third	short: 5:4 (8x)
	long: 9:7 (4x)
Fourth	short: 13:10 (1x, C#-F#)
	long: 4:3 (11x)
Tritonus	short: 7:5 (6x)
	long: 10:7 (6x)
Fifth	short: 3:2 (11x)
	long: 20:13 (1x, F#-C#)
Minor sixth	short: 14:9 (4x)
	long: 8:5 (8x)
Major sixth	short: 5:3 (9x)
	long: 12:7 (3x)
Minor seventh	short: 7:4 (2x)
	long: 9:5 (10x)
Major seventh	short: ~13:7 ~15:8 (7x) (28:15 precise)
	long: ~21:11 (5x) (23:12 precise)
Octave	2:1 (12x)