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 has the advantage of being the characteristic 3-limit comma tempered out in 7edo. For each case, we notice that n equals the order of harmonic 5 in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the MOS scale. 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, which might be a preferred way of conceptualising it because:
- 25/24 is the 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 if we don't count non-integer k.
k = n − 2 | n = k + 2 | Temperament | Comma | |
---|---|---|---|---|
Ratio | Monzo | |||
-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⟩ |
Examples of temperaments with fractional values of n:
- Enipucrop (n = 1.5)
- Seville (n = 7/3 = 2.3)
- Sixix (n = 2.5)
- Sevond (n = 3.5)
- Brahmagupta (n = 21/4 = 5.25)
- Geb (n = 16/3 = 5.3)
- Raider (n = 37/7 = 5.285714)
Enipucrop
The 5-limit 6b&7 temperament. Its name is "porcupine" spelled backwards, because that's what this temperament is - it's 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]]
POTE generator: ~16/15 = 173.101
Badness: 0.1439
Absurdity
The 5-limit 7&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). This is also part of the syntonic-chromatic equivalence continuum, in this case where (81/80)^{5} = 25/24.
Subgroup: 2.3.5
Comma list: 10460353203/10240000000
Mapping: [⟨7 0 -17], ⟨0 1 3]]
POTE generator: ~10/9 = 185.901 cents
Optimal GPV sequence: 7, 70, 77, 84, 329
Badness: 0.3412
Sevond
This is a fairly obvious temperament; it just equates 7 10/9's with a 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.
Subgroup: 2.3.5
Comma list: 5000000/4782969
Mapping: [⟨7 0 -6], ⟨0 1 2]]
POTE generator: ~3/2 = 706.288 cents
Optimal GPV sequence: 7, 42, 49, 56, 119
Badness: 0.339335
7-limit
Adding 875/864 to the commas extends this to the 7-limit:
Subgroup: 2.3.5.7
Comma list: 875/864, 327680/321489
Mapping: [⟨7 0 -6 53], ⟨0 1 2 -3]]
POTE generator: ~3/2 = 705.613
Optimal GPV sequence: 7, 56, 63, 119
Badness: 0.206592
Seville
This is similar to the above, 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.
Subgroup: 2.3.5
Comma list: 78125/69984
Mapping: [⟨7 0 5], ⟨0 1 1]]
POTE generator: ~3/2 = 706.410
Optimal GPV sequence: 7, 35b, 42c, 49c, 56cc, 119cccc
Badness: 0.4377