Syntonic-kleismic equivalence continuum

The syntonic-kleismic equivalence continuum (or syntonic-enneadecal equivalence continuum) is a continuum of 5-limit temperaments which equate a number of syntonic commas (81/80) with the 19-comma ([-30 19).

All temperaments in the continuum satisfy (81/80)n ~ [-30 19. 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 19edo (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 approximately 6.376…, and temperaments having n near this value tend to be the most accurate ones.

This continuum can also be expressed as the relationship between 81/80 and the enneadeca ([-14 -19 19). That is, (81/80)k ~ [-14 -19 19. In this case, k = 3n - 19.

Temperaments in the continuum
n Temperament Comma
Ratio Monzo
0 19 & 19c 1162261467/1073741824 [-30 19
1 Lalayo 71744535/67108864 [-26 15 1
2 Hogzilla 4428675/4194304 [-22 11 2
3 Stump 273375/262144 [-18 7 3
4 Negri 16875/16384 [-14 3 4
5 Magic 3125/3072 [-10 -1 5
6 Hanson 15625/15552 [-6 -5 6
7 Sensi 78732/78125 [2 9 -7
8 Unicorn 1594323/1562500 [-2 13 -8
9 19 & 51c 129140163/125000000 [-6 17 -9
Meantone 81/80 [-4 4 -1

Examples of temperaments with fractional values of k:

Lalayo

Subgroup: 2.3.5

Comma list: [-26 15 1 = 71744535/67108864

Mapping: [1 2 -4], 0 -1 15]]

POTE generator: ~4/3 = 505.348 cents

Lalasepyo (8c & 19)

Subgroup: 2.3.5

Comma list: [-32 10 7 = 4613203125/4294967296

Mapping: [1 -1 6], 0 7 -10]]

POTE generator: ~675/512 = 442.2674 cents

Counterhanson

Subgroup: 2.3.5

Comma list: [-20 -24 25 = 298023223876953125/296148833645101056

Mapping: [1 -5 -4], 0 25 2 4]]

Optimal tuning (POTE): ~6/5 = 316.081

19 & 506

Subgroup: 2.3.5

Comma list: [38 61 -58

Mapping: [1 26 28], 0 -58 -61]]

Optimal tuning (POTE): ~[-12 -20 19 = 505.1394

Countermeantone

Subgroup: 2.3.5

Comma list: [10 23 -20 = 96402615118848/95367431640625

Mapping: [1 10 12], 0 -20 -23]]

Optimal tuning (POTE): ~104976/78125 = 504.913

Mowgli

Subgroup: 2.3.5

Comma list: [0 22 -15

Mapping: [1 0 0], 0 15 22]]

Optimal tuning (POTE): ~27/25 = 126.7237