Syntonic-enneadecal equivalence continuum

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The 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)k ~ [-30 19. Varying k results in different temperaments listed in the table below. It converges to meantone as k approaches infinity. If we allow non-integer and infinite k, 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 k is approximately 6.376..., and temperaments having k near this value tend to be the most accurate ones.

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

Temperaments in the continuum
k 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

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

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

POTE generator: ~4/3 = 505.348 cents

Optimal GPV sequence7c, 12c, 19

Badness: 0.803397

Lalasepyo (8c & 19)

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

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

POTE generator: ~675/512 = 442.2674 cents

Optimal GPV sequence8c, 11, 19

Badness: 1.061630

The temperament finder - 5-limit 19 & 8c

Counterhanson

See also: Ragismic microtemperaments #Counterkleismic

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

Mapping: [1 -5 -4], 0 25 24]]

POTE generator: ~6/5 = 316.081 cents

Optimal GPV sequence19, 148, 167, 186, 205, 224, 429, 653, 1082, 1735c

Badness: 0.317551

19 & 506

Comma list: [38 61 -58

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

POTE generator: ~[-12 -20 19 = 505.1394 cents

Optimal GPV sequence19, 468, 487, 506, 1031

Badness: 2.105450

The temperament finder - 5-limit 19 & 506

Countermeantone

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

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

POTE generator: ~104976/78125 = 504.913 cents

Optimal GPV sequence19, 126, 145, 164, 183, 713, 896c, 1079c, 1262c

Badness: 0.373477

Mowgli

Comma list: [0 22 -15

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

POTE generator: ~27/25 = 126.7237 cents

Optimal GPV sequence19, 85c, 104c, 123, 142, 161

Badness: 0.653871