A054540 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.

1, 2, 3, 5, 7, 12, 19, 31, 34, 53, 118, 171, 289, 323, 441, 612, 730, 1171, 1783, 2513, 4296, 12276, 16572, 20868, 25164, 46032, 48545, 52841, 73709, 78005, 151714, 229719, 537443, 714321, 792326, 944040, 1022045, 1251764, 3755292, 3985011

Offset: 0

Author

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), Apr 09 2000; Dec 17 2000

Keywords

• nonn

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

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)

Examples

			34 = 31 + the earlier term 3. Again, 118 = 53 + the earlier terms 34 and 31.
		

Links

• Tonalsoft - Encyclopedia of Microtonal Music Theory, Val - a linear functional on the vectors in the tonespace lattice.
• Xenharmonic Wiki, Val

Crossrefs

Cf. A001149, A018065, A001856, A002858, A007335, A060525, A060526, A060527.

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.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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