Syntonic–chromatic equivalence continuum
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
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–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 | k | m | Temperament | Comma |
|---|---|---|---|---|
| 7/3 = 2.3 | 1/3 = 0.3 | 7/4 = 1.75 | Seville | [-5 -7 7⟩ |
| 5/2 = 2.5 | 1/2 = 0.5 | 5/3 = 1.6 | Sixix | [-2 -6 5⟩ |
| 7/2 = 3.5 | 3/2 = 1.5 | 7/5 = 1.4 | Sevond | [6 -14 7⟩ |
| 9/2 = 4.5 | 5/2 = 2.5 | 9/7 = 1.285714 | Artoneutral | [14 -22 9⟩ |
| 21/4 = 5.25 | 13/4 = 3.25 | 21/17 = 1.235… | Brahmagupta | [40 -56 21⟩ |
| 37/7 = 5.285714 | 37/7 = 3+2/7 | 37/30 = 1.23 | Raider | [71 -99 37⟩ |
| 16/3 = 5.3 | 10/3 = 3.3 | 16/13 = 1.230769 | Geb | [-31 43 -16⟩ |
| 11/2 = 5.5 | 7/2 = 3.5 | 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
- WE: ~2 = 1203.7664 ¢, ~3/2 = 691.1525 ¢
- error map: ⟨+3.766 -7.036 +1.298]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 689.3122 ¢
- error map: ⟨0.000 -12.643 -3.879]
Optimal ET sequence: 7, 33, 40, 47, 54b
Badness (Sintel): 9.44
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
- WE: ~2 = 1206.3211 ¢, ~3/2 = 686.6308 ¢
- error map: ⟨+6.321 -9.003 -2.053]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 682.6617 ¢
- error map: ⟨0.000 -19.293 -12.931]
Optimal ET sequence: 7, 30b, 37b, 44b, 51b, 58bc, 65bbc
Badness (Sintel): 15.6
Nethertone
Subgroup: 2.3.5
Comma list: 14348907/13107200
Mapping: [⟨1 1 -1], ⟨0 2 15]]
- mapping generators: ~2, ~2560/2187
- WE: ~2 = 1203.1196 ¢, ~2560/2187 = 346.2857 ¢
- error map: ⟨+3.120 -6.264 +1.733]
- CWE: ~2 = 1200.0000 ¢, ~2560/2187 = 345.5992 ¢
- error map: ⟨0.000 -10.757 -2.326]
Optimal ET sequence: 7, 38c, 45c, 52, 59b, 66b
Badness (Sintel): 19.4
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
- WE: ~2 = 1210.1294 ¢, ~16/15 = 174.5613 ¢
- error map: ⟨+3.120 -6.264 +1.733]
- CWE: ~2 = 1200.0000 ¢, ~16/15 = 172.4620 ¢
- error map: ⟨0.000 -10.757 -2.326]
Badness (Sintel): 3.38
Nadir
Subgroup: 2.3.5
Comma list: 1162261467/1048576000
Mapping: [⟨1 2 5], ⟨0 -3 -19]]
- mapping generators: ~2, ~729/640
- WE: ~2 = 1202.743 ¢, ~729/640 = 169.7633 ¢
- error map: ⟨+2.743 -5.758 +1.900]
- CWE: ~2 = 1200.000 ¢, ~729/640 = 169.2234 ¢
- error map: ⟨0.000 -9.625 -1.559]
Optimal ET sequence: 7, 57c, 64, 71b, 78b, 85b
Badness (Sintel): 34.6
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
- WE: ~2 = 1200.9934 ¢, ~6/5 = 338.6456 ¢
- error map: ⟨+0.993 +7.797 -14.214]
- CWE: ~2 = 1200.0000 ¢, ~6/5 = 338.1959 ¢
- error map: ⟨0.000 +7.065 -15.489]
Optimal ET sequence: 7, 25, 32, 39c
Badness (Sintel): 3.59
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). It tempers out the absurditon, 10460353203/10240000000.
Subgroup: 2.3.5
Comma list: 10460353203/10240000000
Mapping: [⟨7 0 -17], ⟨0 1 3]]
- mapping generators: ~800/729, ~3
- WE: ~800/729 = 171.4824 ¢, ~3/2 = 700.4067 ¢ (~81/80 = 14.4772 ¢)
- error map: ⟨+0.376 -1.172 +0.836]
- CWE: ~800/729 = 171.4286 ¢, ~3/2 = 700.3453 ¢ (~81/80 = 14.6310 ¢)
- error map: ⟨0.000 +7.065 -15.489]
Optimal ET sequence: 7, …, 70, 77, 84, 329, 413b, 497b
Badness (Sintel): 8.00
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]]
- mapping generators: ~10/9, ~3
- WE: ~10/9 = 171.3446 ¢, ~3/2 = 705.9421 ¢ (~250/243 = 20.5637 ¢)
- error map: ⟨-0.588 +3.399 -3.673]
- CWE: ~10/9 = 171.4286 ¢, ~3/2 = 705.9119 ¢ (~250/243 = 20.1977 ¢)
- error map: ⟨0.000 +3.957 -3.061]
Optimal ET sequence: 7, …, 42, 49, 56, 119
Badness (Sintel): 7.96
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]]
- mapping generators: ~125/108, ~3
- WE: ~125/108 = 171.7481 ¢, ~3/2 = 707.7272 ¢ (~25/24 = 20.7346 ¢)
- error map: ⟨+2.237 +8.009 -17.609]
- CWE: ~125/108 = 171.4286 ¢, ~3/2 = 708.3739 ¢ (~25/24 = 22.6596 ¢)
- error map: ⟨0.000 +6.419 -20.797]
Optimal ET sequence: 7, 35b, 42c
Badness (Sintel): 10.3
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 -1 -4], ⟨0 9 22]]
- mapping generators: ~2, ~243/200
- WE: ~2 = 1199.7434 ¢, ~243/200 = 344.7454 ¢
- error map: ⟨-0.257 +1.010 -0.889]
- CWE: ~2 = 1200.0000 ¢, ~243/200 = 344.8041 ¢
- error map: ⟨0.000 +1.282 -0.624]
Optimal ET sequence: 7, … 73, 80, 87
Badness (Sintel): 8.17
Geb (5-limit)
- For extensions, see Metric microtemperaments #Geb and Breedsmic temperaments #Osiris.
Subgroup: 2.3.5
Comma list: [-31 43 -16⟩
Mapping: [⟨1 -3 -10], ⟨0 16 43]]
- mapping generators: ~2, ~8000/6561
- WE: ~2 = 1200.0135 ¢, ~8000/6561 = 343.8718 ¢
- error map: ⟨+0.014 -0.047 +0.038]
- CWE: ~2 = 1200.0000 ¢, ~8000/6561 = 343.8683 ¢
- error map: ⟨0.000 -0.062 +0.024]
Optimal ET sequence: 7, …, 157, 164, 171, 506, 677, 848
Badness (Sintel): 2.78