Syntonic-enneadecal equivalence continuum
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.
| 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:
- 19 & 8c (k = 3.5)
- Unsmate (k = 4.5)
- Sycamore (k = 5.5)
- Counterhanson (k = 25/4 = 6.25)
- Enneadecal (k = 19/3 = 6.3)
- Egads (k = 51/8 = 6.375)
- Acrokleismic (k = 32/5 = 6.4)
- 19 & 506 (k = 58/9 = 6.4)
- Parakleismic (k = 6.5)
- Countermeantone (k = 20/3 = 6.6)
- Mowgli (k = 7.5)
Lalayo
Comma list: [-26 15 1⟩ = 71744535/67108864
Mapping: [⟨1 2 -4], ⟨0 -1 15]]
POTE generator: ~4/3 = 505.348 cents
Optimal GPV sequence: 7c, 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 sequence: 8c, 11, 19
Badness: 1.061630
The temperament finder - 5-limit 19 & 8c
Counterhanson
Comma list: [-20 -24 25⟩ = 298023223876953125/296148833645101056
Mapping: [⟨1 -5 -4], ⟨0 25 24]]
POTE generator: ~6/5 = 316.081 cents
Optimal GPV sequence: 19, 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 sequence: 19, 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 sequence: 19, 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 sequence: 19, 85c, 104c, 123, 142, 161
Badness: 0.653871