Schismic–Mercator equivalence continuum
The schismic-Mercator equivalence continuum is a continuum of 5-limit temperaments which equate a number of schismas (32805/32768) with Mercator's comma ([-84 53⟩). This continuum is theoretically interesting in that these are all 5-limit microtemperaments.
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⟩ |
| 8 | Submajor | [-36 11 8⟩ | |
| 9 | Untriton | [-51 19 9⟩ | |
| … | … | … | … |
| ∞ | Schismic | 32805/32768 | [-15 8 1⟩ |
Examples of temperaments with fractional values of n:
Mercator
- and Mercator family
Comma list: [-84 53⟩
POTE generator: ~5/4 = 386.264
Mapping: [⟨53 84 123], ⟨0 0 1]]
⟨⟨0 53 84]]
Badness: 0.2843
Counterschismic
Counterschismic is much like schismic, but the harmonic 5 is located at +45 fifths instead of schismic's -8. They unite in 53edo, of course.
Comma list: [-69 45 -1⟩
POTE generator: ~3/2 = 701.9175
Mapping: [⟨1 2 21], ⟨0 -1 -45]]
⟨⟨1 45 69]]
Badness: 0.09123
53 & 3684
Comma list: [-339 230 -11⟩
POTE generator: ~10737418240/10460353203 = 45.2769
Mapping: [⟨1 2 11], ⟨0 -11 -230]]
Wedgie: ⟨⟨11 230 339]]
Badness: 0.276036
53 & 4190
Comma list: [393 -267 13⟩
POTE generator: ~[-60 41 -2⟩ = 407.5419
Mapping: [⟨1 6 93], ⟨0 -13 -267]]
Wedgie: ⟨⟨13 267 393]]
Badness: 0.173433