# Syntonic-kleismic equivalence continuum

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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

Badness: 0.803397

## 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

Badness: 1.061630

## Counterhanson

See also: Ragismic microtemperaments #Counterkleismic

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

Badness: 0.317551

## 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

Badness: 2.105450

## 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

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

Badness: 0.653871

## Oviminor

See also: Ragismic microtemperaments #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

Badness: 32.0