Syntonic–kleismic equivalence continuum

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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–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 7c & 12c 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 Sensipent 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:

Notable temperaments of fractional n
Temperament n Comma
Unsmate 9/2 = 4.5 [-24 2 9
Sycamore 11/2 = 5.5 [-16 -6 11
Counterhanson 25/4 = 6.25 [-20 -24 25
Enneadecal 19/3 = 6.3 [-14 -19 19
Egads 51/8 = 6.375 [-36 -52 51
Acrokleismic 32/5 = 6.4 [22 33 -32
Parakleismic 13/2 = 6.5 [8 14 -13
Countermeantone 20/3 = 6.6 [10 23 -20
Mowgli 15/2 = 7.5 [0 22 -15

Lalasepyo (8c & 11)

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

Optimal ET sequence8c, 11, 19

Badness: 1.061630

The temperament finder - 5-limit 19 & 8c

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

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

Badness: 0.317551

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

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

Badness: 0.373477

Mowgli

Subgroup: 2.3.5

Comma list: [0 22 -15

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

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

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

Badness: 0.653871

Oviminor

Oviminor is named after the facts that it takes 184 minor thirds of 6/5 to reach 4/3, the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past egads, though it is less accurate.

Subgroup: 2.3.5

Comma list: [-134 -185 184

Mapping: [1 50 51], 0 -184 -185]]

Optimal tuning (CTE): ~6/5 = 315.7501

Optimal ET sequence19, …, 1600, 3219, 4819

Badness: 32.0