Syntonic-31 equivalence continuum
The syntonic-31 equivalence continuum is a continuum of 5-limit temperaments which equate a number of syntonic commas (81/80) with a 31-comma ([-49 31⟩). This continuum is theoretically interesting in that these are all 5-limit temperaments supported by 31edo.
All temperaments in the continuum satisfy (81/80)n ~ [-49 31⟩. 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 31edo 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 7.46781…, and temperaments having n near this value tend to be the most accurate ones.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | 31 & 31c | [-49 31⟩ | |
| 1 | 31 & 12c | [-45 27 1⟩ | |
| 2 | Quasimoha | 2353579470675/2199023255552 | [-41 23 2⟩ |
| 3 | Oncle | 145282683375/137438953472 | [-37 19 3⟩ |
| 4 | Sentinel | 8968066875/8589934592 | [-33 15 4⟩ |
| 5 | Tritonic | 553584375/536870912 | [-29 11 5⟩ |
| 6 | Ampersand | 34171875/33554432 | [-25 7 6⟩ |
| 7 | Orson | 2109375/2097152 | [-21 3 7⟩ |
| 8 | Würschmidt | 393216/390625 | [17 1 -8⟩ |
| 9 | Valentine | 1990656/1953125 | [13 5 -9⟩ |
| 10 | Mynic | 10077696/9765625 | [9 9 -10⟩ |
| 11 | Nusecond | 51018336/48828125 | [5 13 -11⟩ |
| 12 | Cypress | 258280326/244140625 | [1 17 -12⟩ |
| 13 | Diesic | 10460353203/9765625000 | [-3 21 -13⟩ |
| 14 | 31 & 13c | 847288609443/781250000000 | [-7 25 -14⟩ |
| … | … | … | … |
| ∞ | Meantone | 81/80 | [-4 4 -1⟩ |
Examples of temperaments with fractional values of n:
| Temperament | n | Comma |
|---|---|---|
| Slender | 13/2 = 6.5 | [-46 10 13⟩ |
| Eris | 29/4 = 7.25 | [-80 8 29⟩ |
| Tertiaseptal | 22/3 = 7.3 | [-59 5 22⟩ |
| Luna | 15/2 = 7.5 | [38 -2 -15⟩ |
| Quasiorwell | 38/5 = 7.6 | [93 -3 -38⟩ |
| Counterwürschmidt | 23/3 = 7.6 | [55 -1 -23⟩ |
| Birds | 31/4 = 7.75 | [72 0 -31⟩ |
| Countermiracle | 25/3 = 8.3 | [47 7 -25⟩ |
| Casablanca | 19/2 = 9.5 | [22 14 -19⟩ |
Quadlayo (31 & 12c)
In the circle-of-fifths notation, 5/4 is mapped to the quadruple-diminished fifth (C-Gbbbb).
Subgroup: 2.3.5
Comma list: [-45 27 1⟩ = 38127987424935/35184372088832
Mapping: [⟨1 0 45], ⟨0 1 -27]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.950
Optimal ET sequence: 12c, 19c, 31, 43c, 50c
Badness: 2.993628
The temperament finder - 5-limit 31 & 12c
Quadlaleyo (31 & 70c)
Subgroup: 2.3.5
Comma list: [-54 18 11⟩ = 18917016064453125/18014398509481984
Mapping: [⟨1 3 0], ⟨0 -11 18]]
Optimal tuning (POTE): ~2 = 1\1, ~32768/30375 = 154.597
Optimal ET sequence: 8c, 23c, 31, 39c, 132, 163
Badness: 2.067160
The temperament finder - 5-limit 31 & 70c
Ampersand
Subgroup: 2.3.5
Comma list: [-25 7 6⟩ = 34171875/33554432
Mapping: [⟨1 1 3], ⟨0 6 -7]]
Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 116.673
Optimal ET sequence: 10, 21, 31, 41, 72
Badness: 0.165755
Counterwürschmidt
Subgroup: 2.3.5
Comma list: [55 -1 -23⟩
Mapping: [⟨1 9 2], ⟨0 -23 1]]
- mapping generators: ~2, ~5/4
Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 386.8710
Optimal ET sequence: 28b, 31, 90, 121, 152, 335, 822, 1157c, 1492c
Badness: 0.420
Lalasepbigu (31 & 13c)
Subgroup: 2.3.5
Comma list: [-7 25 -14⟩ = 847288609443/781250000000
Mapping: [⟨1 7 12], ⟨0 -14 -25]]
Optimal tuning (POTE): ~2 = 1\1, ~25000/19683 = 464.423
Optimal ET sequence: 13c, 18bc, 31, 44c, 49bc, 75c, 80bc
Badness: 2.094918