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.
| 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 | 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:
| 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 sequence: 8c, 11, 19
Badness: 1.061630
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 sequence: 19, 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 sequence: 19, 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 sequence: 19, 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 sequence: 19, …, 1600, 3219, 4819
Badness: 32.0