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 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 5.2861…, and temperaments near this tend to be the most accurate ones.

2187/2048 is the characteristic 3-limit comma tempered out in 7edo. In each case, we notice that n equals the order of harmonic 5 in the corresponding comma, and equals the number of steps to obtain a harmonic 3 in the generator chain.

However, if we 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. Some prefer this way of conceptualising it because:

• 25/24 is the classic chromatic semitone, notable in the 5-limit as the difference between 5/4 and 6/5, so this shifted continuum could also logically be termed the "syntonic-chromatic equivalence continuum". This means that at k = 0, 5/4 and 6/5 are mapped to the same interval while 81/80 becomes independent of 25/24 (meaning 81/80 may or may not be tempered) because the relation becomes (81/80)⁰ ~ 1/1 ~ 25/24.
• k = 1 and upwards (up to a point) represent temperaments with (the potential for) 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), with the only exception being meantone (n = k = (unsigned) infinity). (Temperaments corresponding to k = 0, -1, -2 are comparatively low-accuracy to the point of developing various intriguing structures and consequences.)
• 25/24 is the simplest ratio to be tempered in the continuum, the second simplest being 81/80 at (unsigned) infinity, which together are the two smallest 5-limit superparticular intervals and the only superparticular intervals in the continuum.

Temperaments in the continuum

k = n − 2	n = k + 2	Temperament	Comma
			Ratio	Monzo
-5	-3	7 & 64 "nadir?"	1162261467/1048576000	[-23 19 -3⟩
-4	-2	7 & 52	14348907/13107200	[-19 15 -2⟩
-3	-1	7 & 33 (unofficially named "tragicomical", "deeptone")	177147/163840	[-15 11 -1⟩
-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⟩
…	…	…	…
∞	∞	Meantone	81/80	[-4 4 -1⟩

We may also invert the continuum by setting m such that 1/m + 1/n = 1. The just value of m is 1.2333…

Temperaments in the continuum

m	Temperament	Comma
		Ratio	Monzo
0	Whitewood	2187/2048	[-11 7⟩
1	Meantone	81/80	[-4 4 -1⟩
2	Dicot	25/24	[-3 -1 2⟩
3	Enipucrop	1125/1024	[-10 2 3⟩
…	…	…	…
∞	Mavila	135/128	[-7 3 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)
• Geb (n = 16/3 = 5.3)
• Undetrita (n = 5.5)

Enipucrop

The 5-limit 6b&7 temperament. Its name is "porcupine" spelled backwards, because that's what this temperament is – it is porcupine, with the generator sharp of 2\7 such that the major and minor thirds switch places. The fifths are very flat, meaning that this is more of a melodic temperament than a harmonic one.

Subgroup: 2.3.5

Comma list: 1125/1024

Mapping: [⟨1 2 2], ⟨0 -3 2]]

POTE generator: ~16/15 = 173.101

Badness: 0.1439

Absurdity

See also: Porwell temperaments § Absurdity

The 5-limit 7&84 temperament, so named because it truly is 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.

Subgroup: 2.3.5

Comma list: 10460353203/10240000000

Mapping: [⟨7 0 -17], ⟨0 1 3]]

Mapping generators: ~800/729, ~3

POTE generator: ~3/2 = 700.1870 (or ~81/80 = 14.4727)

Badness: 0.341202

Sevond

See also: Keemic temperaments § 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.

Subgroup: 2.3.5

Comma list: 5000000/4782969

Mapping: [⟨7 0 -6], ⟨0 1 2]]

POTE generator: ~3/2 = 706.288

Badness: 0.339335

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.

Subgroup: 2.3.5

Comma list: 78125/69984

Mapping: [⟨7 0 5], ⟨0 1 1]]

POTE generator: ~3/2 = 706.410

Badness: 0.4377