Syntonic-chromatic equivalence continuum

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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 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 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 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
Notable temperaments of fractional n
Temperament n m
Shallowtone 1/2 = 0.5 -1
Enipucrop 3/2 = 1.5 3
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]]

Optimal tuning (POTE): ~2 = 1/1, ~16/15 = 173.101

Optimal ET sequence6b, 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 tuning (POTE): ~800/729 = 1\7, ~3/2 = 700.1870 (or ~81/80 = 14.4727)

Optimal ET sequence7, 70, 77, 84, 329

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 tuning (POTE): ~2 = 1\1, ~400/243 = 855.2127

Optimal ET sequence7, … 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 tuning (POTE): ~10/9 = 1\7, ~3/2 = 706.288

Optimal ET sequence7, 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 tuning (POTE): ~125/108 = 1\7, ~3/2 = 706.410

Optimal ET sequence7, 35b, 42c, 49c, 56cc, 119cccc

Badness: 0.4377

Deeptone a.k.a. tragicomical

Deeptone is generated by a fifth. 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 tuning (CTE): ~2/1 = 1\1, ~3/2 = 689.8791

Optimal ET sequence7, 33, 40, 47, 54b

Badness: 0.403

Shallowtone

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

Shallowtone is generated by a fifth. The ~5/4 is reached by minus ten fifths octave-reduced, which is a diminished third (C-Ebb).

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 sequence7, 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 tuning (CTE): 2/1 = 1\1, ~2560/2187 = 345.9462

Optimal ET sequence7, 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 tuning (CTE): 2/1 = 1\1, ~729/640 = 168.9826

Optimal ET sequence7, 57c, 64, 71b, 78b, 85b

Badness: 1.47