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A358029
Decimal expansion of the ratio between step sizes of the diatonic and chromatic semitones produced by a circle of 12 perfect fifths in Pythagorean tuning.
0
1, 2, 6, 0, 0, 1, 6, 7, 5, 2, 6, 7, 0, 8, 2, 4, 5, 3, 5, 9, 3, 1, 2, 7, 6, 1, 2, 2, 6, 0, 3, 9, 2, 4, 2, 3, 3, 7, 1, 8, 1, 1, 5, 5, 7, 9, 2, 3, 2, 7, 6, 7, 8, 3, 3, 4, 1, 0, 6, 5, 2, 0, 1, 6, 1, 6, 2, 0, 8, 7, 4, 8, 0, 0, 8, 3, 1, 2, 2, 7, 8, 4, 6, 8, 8, 1, 4
OFFSET
1,2
COMMENTS
Pythagorean tuning is a form of tuning produced by repeated stacking of the perfect fifth, which has the frequency ratio of 3:2. A circle of 12 perfect fifths is approximately equal to the tuning system predominantly in use in the world today. If the perfect fifth is stacked 12 times and the resulting sequence is octave-reduced, then this divides the octave into 5 chromatic semitones which are equal to 2187/2048 (A229948), and 7 diatonic semitones which are equal to 256/243 (A229943). Diatonic semitones are those which are derived from a circle of 7 perfect fifths, the diatonic scale, and 5 chromatic semitones are a byproduct of an addition of 5 more perfect fifths, that is, another rotation, to the scale.
FORMULA
Equals log(2187/2048) / log(256/243).
Equals log(A229948) / log(A229943).
Equals (7*log(3) - 11*log(2))/(8*log(2) - 5*log(3)).
EXAMPLE
1.2600167526708245359312761226039242337181155792327678334106520161...
MATHEMATICA
RealDigits[(7*Log[3] - 11*Log[2])/(8*Log[2] - 5*Log[3]), 10, 120][[1]] (* Amiram Eldar, Jun 21 2023 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Eliora Ben-Gurion, Oct 25 2022
STATUS
approved