Syntonic–chromatic equivalence continuum
The syntonic-chromatic equivalence continuum is a continuum of temperaments which equate a number of syntonic commas (81/80) with the apotome (2187/2048).
All temperaments in the continuum satisfy (81/80)n ~ 2187/2048. Varying n results in different temperaments listed in the table below. It converges to meantone as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 7edo. The just value of n is 5.2861…, and temperaments near this tend to be the most accurate ones.
Note that if you let k=n-2 (meaning n=k+2) so that k=0 means n=2, k=-1 means n=1, etc. then the continuum corresponds to (81/80)^k = 25/24, which might be a preferred way of conceptualising it because:
- 25/24 is the chromatic semitone, highly notable in the 5-limit as the difference between 5/4 and 6/5, so this shifted continuum could also be termed the "syntonic-chromatic equivalence continuum".
- k=1 and upwards (up to a point) represent temperaments of reasonably good accuracy as equating at least one 81/80 with 25/24 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be Gravity (k=4) or at most Absurdity (k=5), with the only exception being Meantone (n = k = (unsigned) infinity) which is in a sense a simplicity that is the reverse of Dicot's as in Meantone, 81/80 is tempered and 25/24 is no longer dependent on 81/80.
- 25/24 is the simplest ratio to be tempered in the continuum.
| k = n - 2 | n = k + 2 | Temperament | Comma | |
|---|---|---|---|---|
| Ratio | Monzo | |||
| -2 | 0 | Whitewood | 2187/2048 | [-11 7⟩ |
| -1 | 1 | Mavila | 135/128 | [-7 3 1⟩ |
| 0 | 2 | Dicot | 25/24 | [-3 -1 2⟩ |
| 1 | 3 | Porcupine | 250/243 | [1 -5 3⟩ |
| 2 | 4 | Tetracot | 20000/19683 | [5 -9 4⟩ |
| 3 | 5 | Amity | 1600000/1594323 | [9 -13 5⟩ |
| 4 | 6 | Gravity | 129140163/128000000 | [-13 17 -6⟩ |
| 5 | 7 | Absurdity | 10460353203/10240000000 | [-17 21 -7⟩ |
| … | … | … | … | |
| Inf | Inf | Meantone | 81/80 | [-4 4 -1⟩ |
Examples of temperaments with fractional values of n:
- Enipucrop (n = 1.5)
- Seville (n = 7/3 = 2.3)
- Sixix (n = 2.5)
- Sevond (n = 3.5)
- Brahmagupta (n = 21/4 = 5.25)
- Raider (n = 37/7 = 5.285714)
Absurdity
The 5-limit 7&84 temperament. So named because it is truly an absurd temperament. The generator is 81/80 and the period is 800/729, which is (10/9) / (81/80). This is also part of the syntonic-chromatic equivalence continuum, in this case where (81/80)5 = 25/24.
Commas: 10460353203/10240000000
POTE generator: ~10/9 = 185.901 cents
Map: [<7 0 -17|, <0 1 3|]
Badness: 0.3412
The temperament finder - 5-limit Absurdity
Sevond
This is a fairly obvious temperament; it just equates 7 10/9's with a 2/1, hence the period is 10/9. One generator from 5\7 puts you at 3/2, two generators from 2\7 puts you at 5/4.
Comma: 5000000/4782969
POTE generator: ~3/2 = 706.288 cents
Map: [<7 0 -6|, <0 1 2|]
Badness: 0.3393
7-limit
Adding 875/864 to the commas extends this to the 7-limit:
Commas: 875/864, 327680/321489
POTE generator: ~3/2 = 705.613 cents
Map: [<7 0 -6 53|, <0 1 2 -3|]
The temperament finder - 5-limit Sevond
Seville
This is similar to the above, but provides a less complex avenue to 5, but this time at the cost of accuracy. One generator from 5\7 puts you at 3/2, and one generator from 2\7 puts you at 5/4.
Comma: 78125/69984
POTE generator: ~3/2 = 706.410 cents
Map: [<7 0 5|, <0 1 1|]
EDOs: 7, 35b, 42c, 49c, 56cc, 119cccc
Badness: 0.4377