Diaschismic–gothmic equivalence continuum: Difference between revisions
The mixed table looked pretty awful as it mixed temps of utterly different complexity levels. I'm making a separate table for fractional-numbered temps. |
Again just normalize this to the right name in order to avoid further confusion |
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The '''diaschismic- | The '''diaschismic-gothmic equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] that describes the set of all [[5-limit]] temperaments supported by [[34edo]]. | ||
All temperaments in the continuum satisfy (2048/2025)<sup>''n''</sup> ~ {{monzo| 27 -17 }}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[diaschismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 34edo due to it being the unique equal temperament that [[tempering out|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 3.41464…, and temperaments having ''n'' near this value tend to be the most accurate ones | All temperaments in the continuum satisfy (2048/2025)<sup>''n''</sup> ~ {{monzo| 27 -17 }}, equating a number of [[2048/2025|diaschismas (2048/2025)]] with the [[gothic comma]]. Varying ''n'' results in different temperaments listed in the table below. It converges to [[diaschismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 34edo due to it being the unique equal temperament that [[tempering out|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 3.41464…, and temperaments having ''n'' near this value tend to be the most accurate ones. | ||
The | The gothic comma is the characteristic [[3-limit]] comma tempered out in 34edo. Describing the continuum this way 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 ''n'' represents the number of generator steps required to reach the interval class of [[3/1|3]]. | ||
Another reasonable way of defining this continuum equates a number of | Another reasonable way of defining this continuum equates a number of diaschismas with the [[393216/390625|würschmidt comma (393216/390625)]], so that (2048/2025)<sup>''k''</sup> ~ 393216/390625. As a result, ''k'' = 4 - ''n'', and this may also be called the ''diaschismic-würschmidt equivalence continuum'', which is more or less the same thing. The just value of ''k'' is 0.5853…, and temperaments near this tend to be the most accurate. | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
|+ Temperaments | |+ Temperaments with integer ''n'' | ||
|- | |- | ||
! rowspan="2" | ''k'' | ! rowspan="2" | ''k'' | ||
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All temperaments in the continuum also satisfy (15625/15552)<sup>''l''</sup> ~ 393216/390625, for a value of ''l'' defined such that 1/''k'' - 1/''l'' = 1; equivalently, we can offset ''l'' by 1, and equate a number of [[15625/15552|kleismas (15625/15552)]] with the diaschisma. Varying ''l'' results in different temperaments listed in the table below. It converges to [[hanson]] as ''l'' approaches infinity, and is motivated by the fact that many important temperaments of 34edo follow a chain of commas connected by kleismas. | All temperaments in the continuum also satisfy (15625/15552)<sup>''l''</sup> ~ 393216/390625, for a value of ''l'' defined such that 1/''k'' - 1/''l'' = 1; equivalently, we can offset ''l'' by 1, and equate a number of [[15625/15552|kleismas (15625/15552)]] with the diaschisma, giving rise to the name ''diaschismic-kleismic equivalence continuum''. Varying ''l'' results in different temperaments listed in the table below. It converges to [[hanson]] as ''l'' approaches infinity, and is motivated by the fact that many important temperaments of 34edo follow a chain of commas connected by kleismas. | ||
{| class="wikitable center-1" | {| class="wikitable center-1" | ||
Revision as of 13:59, 23 July 2024
The diaschismic-gothmic equivalence continuum is a continuum of 5-limit temperaments that describes the set of all 5-limit temperaments supported by 34edo.
All temperaments in the continuum satisfy (2048/2025)n ~ [27 -17⟩, equating a number of diaschismas (2048/2025) with the gothic comma. Varying n results in different temperaments listed in the table below. It converges to diaschismic as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 34edo due to it being the unique equal temperament that tempers out both commas and thus tempers out all combinations of them. The just value of n is approximately 3.41464…, and temperaments having n near this value tend to be the most accurate ones.
The gothic comma is the characteristic 3-limit comma tempered out in 34edo. Describing the continuum this way 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 n represents the number of generator steps required to reach the interval class of 3.
Another reasonable way of defining this continuum equates a number of diaschismas with the würschmidt comma (393216/390625), so that (2048/2025)k ~ 393216/390625. As a result, k = 4 - n, and this may also be called the diaschismic-würschmidt equivalence continuum, which is more or less the same thing. The just value of k is 0.5853…, and temperaments near this tend to be the most accurate.
| k | n | Temperament | Comma | |
|---|---|---|---|---|
| Ratio | Monzo | |||
| 4 | 0 | Gothic | 134217728/129140163 | [27 -17⟩ |
| 3 | 1 | Immunity | 1638400/1594323 | [16 -13 2⟩ |
| 2 | 2 | Tetracot | 20000/19683 | [5 -9 4⟩ |
| 1 | 3 | Hanson | 15625/15552 | [-6 -5 6⟩ |
| 0 | 4 | Würschmidt | 393216/390625 | [17 1 -8⟩ |
| -1 | 5 | Mabila | 268435456/263671875 | [28 -3 -10⟩ |
| -2 | 6 | Goldis | 549755813888/533935546875 | [39 -7 -12⟩ |
| … | … | … | … | … |
| ∞* | ∞ | Srutal | 2048/2025 | [11 -4 -2⟩ |
- * in projective tuning space, ∞ = -∞.
We may invert the continuum by setting m such that 1/n + 1/m = 1. The just value of m is 1.41414…, and temperaments near this tend to be the most accurate ones.
| m | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | Gothic | 134217728/129140163 | [27 -17⟩ |
| 1 | Srutal | 2048/2025 | [11 -4 -2⟩ |
| 2 | Tetracot | 20000/19683 | [5 -9 4⟩ |
| … | … | … | … |
| ∞ | Immunity | 1638400/1594323 | [16 -13 2⟩ |
| n | m | Temperament | Comma |
|---|---|---|---|
| 5/2 = 2.5 | 5/3 = 1.6 | Fifive | [-1 -14 10⟩ |
| 13/4 = 3.25 | 13/9 = 1.4 | Quatracot | [-33 -16 26⟩ |
| 10/3 = 3.3 | 10/7 = 1.428571 | Gammic | [-29 -11 20⟩ |
| 17/5 = 3.4 | 17/12 = 1.416 | Chlorine | [-52 -17 34⟩ |
| 7/2 = 3.5 | 7/5 = 1.4 | Vishnu | [23 6 -14⟩ |
| 11/3 = 3.6 | 11/8 = 1.375 | Majvam | [40 7 -22⟩ |
| 9/2 = 4.5 | 9/7 = 1.285714 | 34 & 142 | [45 -2 -18⟩ |
All temperaments in the continuum also satisfy (15625/15552)l ~ 393216/390625, for a value of l defined such that 1/k - 1/l = 1; equivalently, we can offset l by 1, and equate a number of kleismas (15625/15552) with the diaschisma, giving rise to the name diaschismic-kleismic equivalence continuum. Varying l results in different temperaments listed in the table below. It converges to hanson as l approaches infinity, and is motivated by the fact that many important temperaments of 34edo follow a chain of commas connected by kleismas.
| l | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| -4 | 34 & 113 | 152587890625/148769467776 | [-7 -19 16⟩ |
| -3 | Fifive | 9765625/9565938 | [-1 -14 10⟩ |
| -2 | Tetracot | 20000/19683 | [5 -9 4⟩ |
| -1 | Srutal | 2048/2025 | [11 -4 -2⟩ |
| 0 | Würschmidt | 393216/390625 | [17 1 -8⟩ |
| 1 | Vishnu | 6115295232/6103515625 | [23 6 -14⟩ |
| 2 | Gammic | (28 digits) | [-29 -11 20⟩ |
| 3 | Quatracot | (38 digits) | [-35 -16 26⟩ |
| … | … | … | … |
| ∞ | Hanson | 15625/15552 | [-6 -5 6⟩ |
| l | k | Temperament | Comma |
|---|---|---|---|
| 1/2 = 0.5 | 1/3 = 0.3 | Majvam | [40 7 -22⟩ |
| 3/2 = 1.5 | 3/5 = 0.6 | Chlorine | [-52 -17 34⟩ |
| -1/3 = -0.3 | -1/2 = -0.5 | 34 & 142 | [45 -2 -18⟩ |