Chromatic-diatonic equivalence continuum

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The chromatic-diatonic equivalence continuum is a continuum of 5-limit temperaments which equate a number of chromatic semitones (25/24) with diatonic semitones (16/15).

All temperaments in the continuum satisfy (25/24)n ~ 16/15. Varying n results in different temperaments listed in the table below. It converges to dicot as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 3edo (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them). The just value of n is approximately 1.58097..., and temperaments having n near this value tend to be the most accurate ones.

Temperaments in the continuum
n Temperament Comma
Ratio Monzo
-1 Yo 10/9 [1 -2 1
0 Father 16/15 [4 -1 -1
1 Augmented 128/125 [7 0 -3
2 Magic 3125/3072 [10 1 -5
3 Wesley 78125/73728 [13 2 -7
4 3 & 33c 1953125/1769472 [16 3 -9
Dicot 25/24 [-3 -1 2

Examples of temperaments with fractional values of n:

3 & 33c

Comma list: [16 3 -9

POTE generator: 34.0971 cents

Mapping: [3 5 7], 0 -3 -1]]

Optimal GPV sequence3, 6, 9b, 33c

The temperament finder - 5-limit 3 & 33c

Isnes

So called because the generator is half of a 8/5 minor sixth, in a similar way that sensi has a generator of half a 5/3.

Comma list: [41 2 -19

POTE generator: 12582912/9765625 ~ 1953125/1572864 = 405.1047 cents

Mapping: [1 8 3], 0 -19 -2]]

Optimal GPV sequence3, 74, 77, 80, 83, 154, 157, 160

The temperament finder - 5-limit 3 & 77