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.
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…
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⟩ |
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
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 sequence: 7, 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 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 tuning (POTE): ~10/9 = 1\7, ~3/2 = 706.288
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 tuning (POTE): ~125/108 = 1\7, ~3/2 = 706.410
Optimal ET sequence: 7, 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 sequence: 7, 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 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 tuning (CTE): 2/1 = 1\1, ~2560/2187 = 345.9462
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 tuning (CTE): 2/1 = 1\1, ~729/640 = 168.9826
Optimal ET sequence: 7, 57c, 64, 71b, 78b, 85b
Badness: 1.47