Schismic–Mercator equivalence continuum
The syntonic-chromatic equivalence continuum is a continuum of temperaments which equate a number of schismas (32805/32768) with Mercator's comma ([-84 53⟩).
All temperaments in the continuum satisfy (32805/32768)n ~ [-84 53⟩. Varying n results in different temperaments listed in the table below. It converges to schismic as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 53edo (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 1.8503390493..., and temperaments having n near this value tend to be the most accurate ones.
For a similar but perhaps more intuitive and practical concept, see Syntonic-chromatic equivalence continuum.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | Mercator | [-84 53⟩ | |
| 1 | Counterschismic | [-69 45 -1⟩ | |
| 2 | Monzismic | [54 -37 2⟩ | |
| 3 | Tricot | [39 -29 3⟩ | |
| 4 | Vulture | [24 -21 4⟩ | |
| 5 | Amity | 1600000/1594323 | [9 -13 5⟩ |
| 6 | Kleismic | 15625/15552 | [-6 -5 6⟩ |
| 7 | Orson | 2109375/2097152 | [-21 3 7⟩ |
| … | … | … | … |
| ∞ | Schismic | 32805/32768 | [-15 8 1⟩ |
Examples of temperaments with fractional values of n:
- 3684 & 11105 (n = 11/6 = 1.83)
Counterschismic
Comma: [-69 45 -1⟩
Map: [<1 2 21|, <0 -1 -45|]
Wedgie: <<1 45 69||
POTE generator: ~3/2 = 701.9175
EDOs: 53, 412, 465, 518, 571, 624, 677, 730, 2973, 3703, 4433, 5163, 11056
Badness: 0.09123
3684 & 11105
Comma: [-339 230 -11⟩
Map: ⟨1 2 11], ⟨0 -11 -230]
Wedgie: ⟨⟨11 230 339]]
POTE generator: 45.2769
EDOs: Template:Val list
Unnamed temperament
Comma: [393 -267 13⟩