| 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 harmony and equal temperament, after which I present to you:
5-limit meantone approximation
C 1 2^(3/31) D 2^(5/31) 2^(8/31) E 2^(10/31) F 2^(13/31) 2^(16/31) (OR 2^(15/31)) G 2^(18/31) 2^(21/31) A 2^(23/31) 2^(26/31) B 2^(28/31) C 2
7-limit just approximation
C 1 2^(3/31) D 2^(6/31) 2^(8/31) E 2^(10/31) F 2^(13/31) 2^(15/31) G 2^(18/31) 2^(21/31) A 2^(23/31) 2^(25/31) B 2^(28/31) C 2
Note that this is a work in progress. Samples for a 61-key clavier are available here:
5-limit meantone approximation: meantone approx.zip (Last updated 29-3-2019)
7-limit just approximation: just approx.zip (Last updated 29-3-2019)