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A054540
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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.
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21
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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
(list;
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refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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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?
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)
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LINKS
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FORMULA
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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.
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EXAMPLE
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34 = 31 + the earlier term 3. Again, 118 = 53 + the earlier terms 34 and 31.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Mark William Rankin (MarkRankin95511(AT)Yahoo.com), Apr 09 2000; Dec 17 2000
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STATUS
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approved
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