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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. 20
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; graph; refs; listen; history; text; internal format)
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?

LINKS

Table of n, a(n) for n=0..39.

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

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

Sequence in context: A143642 A192685 A060986 * A216231 A117537 A137713

Adjacent sequences:  A054537 A054538 A054539 * A054541 A054542 A054543

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified August 27 13:20 EDT 2014. Contains 246135 sequences.