Syntonic-chromatic equivalence continuum

From Xenharmonic Wiki
Jump to navigation Jump to search

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.
Temperaments in the continuum
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

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

Optimal GPV sequence6b, 7

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

Badness: 0.4377