Syntonic-chromatic equivalence continuum

The syntonic-chromatic equivalence continuum is a continuum of 5-limit temperaments which equate a number of syntonic commas (81/80) with the apotome (2187/2048). This continuum is theoretically interesting in that these are all 5-limit temperaments supported by 7edo.

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 out both commas and thus tempers out 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, and has many advantages as a target. In each case, n equals the order of harmonic 5 in the corresponding comma, and equals the number of steps to obtain the interval class of harmonic 3 in the generator chain. For example:

Mavila (n = 1) is generated by a fifth;
Dicot (n = 2) splits its fifth in two;
Porcupine (n = 3) splits its fourth in three;
Etc.

At n = 7, the corresponding temperament splits the octave into seven instead, as after a stack of seven syntonic commas, both the orders of 3 and 5 are multiples of 7 again.

If we let k = n − 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 = ∞). 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 with integer n
k	n	Temperament	Comma
k	n	Temperament	Ratio	Monzo
−5	−3	Nadir	1162261467/1048576000	[-23 19 -3⟩
−4	−2	Nethertone	14348907/13107200	[-19 15 -2⟩
−3	−1	Deeptone a.k.a. tragicomical	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 invert the continuum by setting m such that 1⁄m + 1⁄n = 1. This may be called the mavila/pelogic-chromatic equivalence continuum, which is essentially the same thing. The just value of m is 1.2333… The mavila comma is both larger and more complex than the syntonic comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless.

Temperaments with integer m
m	Temperament	Comma
m	Temperament	Ratio	Monzo
−1	Shallowtone	295245/262144	[-18 10 1⟩
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⟩

Temperaments with fractional n and m
n	m	Temperament	Comma
7/3 = 2.3	7/4 = 1.75	Seville	[-5 -7 7⟩
5/2 = 2.5	5/3 = 1.6	Sixix	[-2 -6 5⟩
7/2 = 3.5	7/5 = 1.4	Sevond	[6 -14 7⟩
9/2 = 4.5	9/7 = 1.285714	Artoneutral	[14 -22 9⟩
21/4 = 5.25	21/17 = 1.235…	Brahmagupta	[40 -56 21⟩
37/7 = 5.285714	37/30 = 1.23	Raider	[71 -99 37⟩
16/3 = 5.3	16/13 = 1.230769	Geb	[-31 43 -16⟩
11/2 = 5.5	11/9 = 1.2	Undetrita	[-22 30 -11⟩

Enipucrop

Enipucrop corresponds to n = 3/2 and m = 3, and can be described as the 6b & 7 temperament. Its name is porcupine spelled backwards, because that is 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]]

mapping generators: ~2, ~16/15

Optimal tunings:

CTE: ~2 = 1\1, ~16/15 = 170.6715
POTE: ~2 = 1/1, ~16/15 = 173.101

Optimal ET sequence: 6b, 7

Badness: 0.1439

Absurdity

See also: Porwell temperaments § Absurdity

Absurdity corresponds to n = 7, and can be described as the 77 & 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).

Subgroup: 2.3.5

Comma list: 10460353203/10240000000

Mapping: [⟨7 0 -17], ⟨0 1 3]]

mapping generators: ~800/729, ~3

Optimal tunings:

CTE: ~800/729 = 1\7, ~3/2 = 700.5378 (~81/80 = 14.8235)
POTE: ~800/729 = 1\7, ~3/2 = 700.1870 (~81/80 = 14.4727)

Optimal ET sequence: 7, …, 70, 77, 84, 329, 413b, 497b

Badness: 0.341202

Artoneutral

See also: Hemifamity temperaments § Artoneutral

5-limit artoneutral is generated by ~243/200 (or ~400/243), same as that of amity but sharper. This corresponds to n = 9/2 and m = 9/7 and can be described as the 80 & 87 temperament, though 94edo is a notable tuning not appearing in the optimal ET sequence.

Subgroup: 2.3.5

Comma list: [14 -22 9⟩

Mapping: [⟨1 8 18], ⟨0 -9 -22]]

mapping generators: ~2, ~400/243

Optimal tunings:

CTE: ~2 = 1\1, ~400/243 = 855.2127
CWE: ~2 = 1\1, ~400/243 = 855.1959

Optimal ET sequence: 7, … 73, 80, 87

Badness: 0.348

Sevond

See also: Keemic temperaments § Sevond

Sevond is a fairly obvious temperament; it just equates a stack of seven ~10/9's with ~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. This corresponds to n = 7/2 and m = 7/5 and can be described as the 56 & 63 temperament.

Subgroup: 2.3.5

Comma list: 5000000/4782969

Mapping: [⟨7 0 -6], ⟨0 1 2]]

Optimal tunings:

CTE: ~10/9 = 1\7, ~3/2 = 705.5264 (~250/243 = 19.8121)
POTE: ~10/9 = 1\7, ~3/2 = 706.288 (~250/243 = 20.574)

Optimal ET sequence: 7, 42, 49, 56, 119

Badness: 0.339335

Seville

Seville is similar to sevond, 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. This corresponds to n = 7/3 and m = 7/4.

Subgroup: 2.3.5

Comma list: 78125/69984

Mapping: [⟨7 0 5], ⟨0 1 1]]

Optimal tunings:

CTE: ~125/108 = 1\7, ~3/2 = 710.6056 (~25/24 = 24.8913)
POTE: ~125/108 = 1\7, ~3/2 = 706.410 (~25/24 = 20.696)

Optimal ET sequence: 7, 35b, 42c

Badness: 0.4377

Deeptone a.k.a. tragicomical

Main article: Deeptone

Deeptone is generated by a fifth, which is typically sharper than in 7edo but flatter than in flattone. The ~5/4 is reached by eleven fifths octave-reduced, which is an augmented third (C–E♯).

Subgroup: 2.3.5

Comma list: 177147/163840

Mapping: [⟨1 0 -15], ⟨0 1 11]]

mapping generators: ~2, ~3

Optimal tunings:

CTE: ~2/1 = 1\1, ~3/2 = 689.8791
CWE: ~2/1 = 1\1, ~3/2 = 689.3122

Optimal ET sequence: 7, 33, 40, 47, 87b

Badness: 0.403

Shallowtone

For 7-limit extensions, see Mint temperaments #Shallowtone.

Shallowtone is generated by a fifth, which is typically sharper than in mavila but flatter than in 7edo. The ~5/4 is reached by minus ten fifths octave-reduced, which is an augmented third (C-E𝄪) in melodic antidiatonic notation and a diminished third (C-E𝄫) in harmonic antidiatonic notation.

Subgroup: 2.3.5

Comma list: 295245/262144

Mapping: [⟨1 0 18], ⟨0 1 -10]]

mapping generators: ~2, ~3

Optimal tunings: