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).

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 the interval class of 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 with integer n

k	n	Temperament	Comma
			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…

Temperaments with integer m

m	Temperament	Comma
		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

Temperament	n	m
Seville	7/3 = 2.3	7/4 = 1.75
Sixix	5/2 = 2.5	5/3 = 1.6
Sevond	7/2 = 3.5	7/5 = 1.4
Artoneutral	9/2 = 4.5	9/7 = 1.285714
Brahmagupta	21/4 = 5.25	21/17 = 1.235…
Raider	37/7 = 5.285714	37/30 = 1.23
Geb	16/3 = 5.3	16/13 = 1.230769
Undetrita	11/2 = 5.5	11/9 = 1.2

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 87 & 94 temperament.

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

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, 54b

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-Ex) in melodic antidiatonic notation and a diminished third (C-Ebb) 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:

• CTE: ~2 = 1\1, ~3/2 = 681.8012
• CWE: ~2 = 1\1, ~3/2 = 682.6617

Optimal ET sequence: 7, 30b, 37b, 44b, 51b, 58bc, 65bbc

Badness: 0.666

Nethertone

Subgroup: 2.3.5

Comma list: 14348907/13107200‎‎

Mapping: [⟨1 1 -1], ⟨0 2 15]]

mapping generators: ~2, ~2560/2187

Optimal tunings:

• CTE: ~2 = 1\1, ~2560/2187 = 345.9462
• CWE: ~2 = 1\1, ~2560/2187 = 345.5992

Optimal ET sequence: 7, 38c, 45c, 52, 59b, 66b

Badness: 0.828

Nadir

Subgroup: 2.3.5

Comma list: 1162261467/1048576000

Mapping: [⟨1 2 5], ⟨0 -3 -19]]

mapping generators: ~2, ~729/640

Optimal tunings:

• CTE: ~2 = 1\1, ~729/640 = 168.9826
• CWE: ~2 = 1\1, ~729/640 = 169.2234

Optimal ET sequence: 7, 57c, 64, 71b, 78b, 85b

Badness: 1.47