User:Eliora/Schismic-parakleismic equivalence continuum
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
The schismic–parakleismic equivalence continuum is a continuum of 5-limit temperaments which equate a number of schismas (32805/32768) with parakleisma (. This continuum is theoretically interesting in that the temperaments associated with its various commas are all 5-limit microtemperaments.
All temperaments in the continuum satisfy (32805/32768)n = [8 14 -13⟩. Varying n results in different temperaments listed in the table below. It converges to parakleismic as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 118edo (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 2.70854..., and temperaments having n near this value tend to be the most accurate ones.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | Mercator | (52 digits) | [-84 53⟩ |
| 1 | Counterschismic | (44 digits) | [-69 45 -1⟩ |
| 2 | Monzismic | (36 digits) | [54 -37 2⟩ |
| 3 | Alphatricot | (28 digits) | [39 -29 3⟩ |
| 4 | Vulture | (22 digits) | [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 | Demibuzzard | (22 digits) | [-36 11 8⟩ |
| 9 | Untriton | (32 digits) | [-51 19 9⟩ |
| … | … | … | … |
| ∞ | Schismic | 32805/32768 | [-15 8 1⟩ |