OFFSET
0,2
COMMENTS
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?
From Eliora Ben-Gurion, Dec 15 2022: (Start)
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.
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)
LINKS
Tonalsoft - Encyclopedia of Microtonal Music Theory, Val - a linear functional on the vectors in the tonespace lattice.
Xenharmonic Wiki, Val
FORMULA
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.
EXAMPLE
34 = 31 + the earlier term 3. Again, 118 = 53 + the earlier terms 34 and 31.
CROSSREFS
KEYWORD
nonn
AUTHOR
Mark William Rankin (MarkRankin95511(AT)Yahoo.com), Apr 09 2000; Dec 17 2000
STATUS
approved