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2^(y/x) = 3/2 (either y or x must not be a whole number for this to be true)
→Inconsistencies between 12 Tones and Pure Intervals
n * (3/2)^x = n * 2^y
(3/2)^x = 2^y (but no power of 2 can have 3 in its prime factorization, which a power of 3/2 must have)
x * ln(3/2) = y * ln(2)
x * ln(3/2) / ln(2) = y
ln(3/2) / ln(2) = y/x
log2(3/2) = y/x
2^(y/x + 1) = 3
Older tuning methods typically started at a certain note and derived other notes by moving up and down from it using some small whole number ratio. For example, moving up and down from A 440 HZ in fifths defined as 3:2...