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.
| 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… 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.
| 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⟩ |
| 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⟩ |
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 ET sequence: 7, 33, 40, 47, 87b
Badness (Smith): 0.403
Shallowtone (5-limit)
- For 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 ET sequence: 7, 30b, 37b, 44b, 51b, 58bc, 65bbc
Badness (Smith): 0.666
Nethertone
Subgroup: 2.3.5
Comma list: 14348907/13107200
Mapping: [⟨1 1 -1], ⟨0 2 15]]
- mapping generators: ~2, ~2560/2187
Optimal ET sequence: 7, 38c, 45c, 52, 59b, 66b
Badness (Smith): 0.828
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
Badness (Smith): 0.1439
Nadir
Subgroup: 2.3.5
Comma list: 1162261467/1048576000
Mapping: [⟨1 2 5], ⟨0 -3 -19]]
- mapping generators: ~2, ~729/640
Optimal ET sequence: 7, 57c, 64, 71b, 78b, 85b
Badness (Smith): 1.47
Sixix (5-limit)
- For extensions, see Archytas clan #Sixix.
Subgroup: 2.3.5
Comma list: 3125/2916
Mapping: [⟨1 3 4], ⟨0 -5 -6]]
- mapping generators: ~2, ~6/5
- CTE: ~2 = 1200.000, ~6/5 = 338.005
- error map: ⟨0.000 +8.020 -14.344]
- POTE: ~2 = 1200.000, ~6/5 = 338.365
- error map: ⟨0.000 +6.217 -16.507]
Optimal ET sequence: 7, 25, 32, 39c
Badness (Smith): 0.153088
Absurdity (5-limit)
- For extensions, see 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
- CTE: ~800/729 = 171.429, ~3/2 = 700.538 (~81/80 = 14.824)
- POTE: ~800/729 = 171.429, ~3/2 = 700.187 (~81/80 = 14.473)
Optimal ET sequence: 7, …, 70, 77, 84, 329, 413b, 497b
Badness (Smith): 0.341202
Sevond (5-limit)
- For extensions, see 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]]
- CTE: ~10/9 = 171.429, ~3/2 = 705.526 (~250/243 = 19.812)
- POTE: ~10/9 = 171.429, ~3/2 = 706.288 (~250/243 = 20.574)
Optimal ET sequence: 7, 42, 49, 56, 119
Badness (Smith): 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]]
- CTE: ~125/108 = 171.429, ~3/2 = 710.606 (~25/24 = 24.891)
- POTE: ~125/108 = 171.429, ~3/2 = 706.410 (~25/24 = 20.696)
Optimal ET sequence: 7, 35b, 42c
Badness (Smith): 0.4377
Artoneutral (5-limit)
- For extensions, see 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 ET sequence: 7, … 73, 80, 87
Badness (Smith): 0.348