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%I #13 Jan 05 2023 18:28:57
%S 1,2,3,5,7,12,19,31,34,53,118,171,289,323,441,612,730,1171,1783,2513,
%T 4296,12276,16572,20868,25164,46032,48545,52841,73709,78005,151714,
%U 229719,537443,714321,792326,944040,1022045,1251764,3755292,3985011
%N A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the six simple ratios of musical harmony: 6/5, 5/4, 4/3, 3/2, 8/5 and 5/3.
%C The sequence was found by a computer search of all of the equal divisions of the octave from 1 to over 3985011. There seems to be a hidden aspect or mystery here: what is it about the more and more harmonious equal temperaments that causes them to express themselves collectively as a perfect, self-accumulating recurrent sequence?
%C From _Eliora Ben-Gurion_, Dec 15 2022: (Start)
%C The answer is because temperament mappings can be added. If harmonic correspondences are written in a bra, that is <N2 N3 N5], where Nx is the step corresponding to the x-th harmonic, then these types of one-row matrices can be added and the resulting temperament will represent them as well. In case of temperaments with high precision, this also leads to another high-precision temperament. Such a bra notation is referred to as "val" by the microtonal music community, and in simple words, vals can be added together to produce another val.
%C Example: a tuning with 118 equal steps to the octave has a second harmonic on the 118th step by definition, the third harmonic is approximated with 187 steps, and the fifth is with 274 steps, which leads to <118 187 274]. A 171 equal division system will have a corresponding bra <171 271 397]. When these two are added, we obtain <289 458 671], which is exactly how the 2nd, 3rd, and 5th harmonics are represented in 289 equal divisions of the octave. (End)
%H Tonalsoft - Encyclopedia of Microtonal Music Theory, <a href="http://tonalsoft.com/enc/v/val.aspx">Val - a linear functional on the vectors in the tonespace lattice</a>.
%H Xenharmonic Wiki, <a href="https://en.xen.wiki/w/Val">Val</a>
%F Stochastic recurrence rule - the next term equals the current term plus one or more previous terms: a(n+1) = a(n) + a(n-x) + ... + a(n-y) + ... + a(n-z), etc.
%e 34 = 31 + the earlier term 3. Again, 118 = 53 + the earlier terms 34 and 31.
%Y Cf. A001149, A018065, A001856, A002858, A007335, A060525, A060526, A060527.
%K nonn
%O 0,2
%A Mark William Rankin (MarkRankin95511(AT)Yahoo.com), Apr 09 2000; Dec 17 2000
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