User:Godtone/diaschismic-tetracot equivalence continuum
The diaschismic-tetracot equivalence continuum is a continuum of 5-limit temperaments that describes the set of all 5-limit temperaments supported by 34edo. It is equivalent to the diaschismic-gothic equivalence continuum, where the gothic comma is found as (2048/2025)2 * 20000/19683, at n = -2.
Each n on the continuum is defined by equating (2048/2025)n with 20000/19683. The just value of n is 1.41464… = log2(20000/19683)/log2(2048/2025), and temperaments near this tend to be the most accurate. However, due to this continuum being defined through two reasonably-accurate temperaments and due to the strength of 34edo as a 5-limit temperament (supporting many notable tempered equivalences), simple fractional values of n in the general proximity of the just value are also often notable.
- A reasonable way of defining this continuum equates a number of diaschismas (2048/2025) with the Würschmidt comma (393216/390625), so that (2048/2025)n ~ 393216/390625. As a result, this may also be called the wurschmidt-diaschismic equivalence continuum, or the diaschismic-gothic equivalence continuum, which is more or less the same thing. The just value of n is 0.5853…, and temperaments near this tend to be the most accurate. The gothic comma (134217728/129140163) is the characteristic 3-limit comma tempered out in 34edo, and it has a value of n = 4. Therefore, one can additionally define k = 4 - n, which has notable advantages - in particular, due to being determined in terms of the 3-limit comma and the comma with the next lowest power of 5, (twice the numerator of) the value of k represents the number of generator steps required to reach the 3rd harmonic.
- All temperaments in the continuum also satisfy (15625/15552)m ~ 393216/390625, for a value of m defined such that 1/n - 1/m = 1; equivalently, we can offset m by 1, and equate a number of kleismas (15625/15552) with the diaschisma, hence the name. Varying m results in different temperaments listed in the second table below. It converges to hanson as m approaches infinity, and is motivated by the fact that many important temperaments of 34edo follow a chain of commas connected by kleismas.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| -2 | Gothic | 134217728/129140163 | [27 -17⟩ |
| -1 | Immunity | 1638400/1594323 | [16 -13 2⟩ |
| 0 | Tetracot | 20000/19683 | [5 -9 4⟩ |
| 1/2 | Fifive | 9765625/9565938 | [-1 -14 10⟩ |
| 1 | Hanson/kleismic | 15625/15552 | [-6 -5 6⟩ |
| 4/3 | Gammic | 95367431640625/95105071448064 | [-29 -11 20⟩ |
| 7/5 | Chlorine | [very long but equal to (25/24)17 / 2] | [-52 -17 34⟩ |
| 3/2 | Vishnu | 6115295232/6103515625 | [23 6 -14⟩ |
| 5/3 | Majvam | 2404631929946112/2384185791015625 | [40 7 -22⟩ |
| 2 | Würschmidt | 393216/390625 | [17 1 -8⟩ |
| 5/2 | 34&142 | 35184372088832/34332275390625 | [45 -2 18⟩ |
| 3 | Mabila | 268435456/263671875 | [28 -3 -10⟩ |
| 4 | Goldis | 549755813888/533935546875 | [39 -7 -12⟩ |
| … | … | … | … |
| ∞ | Srutal | 2048/2025 | [11 -4 -2⟩ |
- * in projective tuning space, ∞ = -∞.
We may invert the continuum by setting k such that 1/n - 1/k = 1, resulting in a continuum for (15625/15552)k ~ 20000/19683, motivated by many important temperaments of 34edo being defined by commas connected by kleismas.
The just value of k is 3.41173… = log2(20000/19683)/log2(15625/15552), and temperaments near this tend to be the most accurate ones.
| m | n | Temperament | Comma | |
|---|---|---|---|---|
| Ratio | Monzo | |||
| -2 | ? | 34 & 113 | 152587890625/148769467776 | [-7 -19 16⟩ |
| -1 | ? | Fifive | 9765625/9565938 | [-1 -14 10⟩ |
| 0 | ? | Tetracot | 20000/19683 | [5 -9 4⟩ |
| 1 | ? | Srutal | 2048/2025 | [11 -4 -2⟩ |
| 2 | ? | Würschmidt | 393216/390625 | [17 1 -8⟩ |
| 3 | ? | Vishnu | 6115295232/6103515625 | [23 6 -14⟩ |
| 4 | ? | Gammic | 95367431640625/95105071448064 | [-29 -11 20⟩ |
| 5 | ? | Quatracot | 1490116119384765625/1479074071160291328 | [-35 -16 26⟩ |
| … | … | … | … | … |
| ∞ | ? | Hanson/Kleismic | 15625/15552 | [-6 -5 6⟩ |