Chromatic-diatonic equivalence continuum
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.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| -1 | Yellow | 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:
- Alteraugment (n = -0.5)
- Smate (n = 0.5)
- Würschmidt (n = 1.5)
- Isnes (n = 1.6)
- Magus (n = 5/3 = 1.6)
3 & 33c
Comma list: [16 3 -9⟩
POTE generator: 34.0971 cents
Mapping: [⟨3 5 7], ⟨0 -3 -1]]
Optimal ET sequence: 3, 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]]