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.

Temperaments 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|]

EDOs: 7, 70, 77, 84, 329

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|]

EDOs: 7, 42, 49, 56, 119

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|]

EDOs: 7, 56, 63, 119

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 sake 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

The temperament finder - 5-limit Seville