Diaschismic–gothmic equivalence continuum: Difference between revisions
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m clean up beginning, avoid "gothic-" in title as diaschisma and tetracot both play key roles; diaschisma for the reasons FloraC stated and tetracot for being the point at which inverting conveniently yields the kleisma^n-based continuum, as well as for other reasons such as diaschismic and tetracot tempering being the most strongly characteristic of 34et while being the simplest and most efficient |
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The '''diaschismic- | The '''diaschismic-tetracot equivalence continuum''' (which is the '''diaschismic-gothmic equivalence continuum''' with an offset of 2) 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 }}, equating a number of [[2048/2025|diaschismas (2048/2025)]] with the [[gothic comma|gothic comma (134217728/129140163)]]. 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|gothic comma (134217728/129140163)]]. At ''n'' = 2 (which we align with ''r'' = 0) we get '''tetracot''', which is an important offset for a number of reasons discussed later. 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 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]]. | 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]]. | ||
Revision as of 00:15, 24 July 2024
The diaschismic-tetracot equivalence continuum (which is the diaschismic-gothmic equivalence continuum with an offset of 2) 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 (134217728/129140163). At n = 2 (which we align with r = 0) we get tetracot, which is an important offset for a number of reasons discussed later. 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 tetracot comma (20000/19683), so that (2048/2025)r ~ 20000/19683. As a result, r = n - 2, and this labeling may also be called the diaschismic-tetracot equivalence continuum. The just value of r is 1.4146…, and temperaments near this tend to be the most accurate.
| r | n | Temperament | Comma | |
|---|---|---|---|---|
| Ratio | Monzo | |||
| -2 | 0 | Gothic | 134217728/129140163 | [27 -17⟩ |
| -1 | 1 | Immunity | 1638400/1594323 | [16 -13 2⟩ |
| 0 | 2 | Tetracot | 20000/19683 | [5 -9 4⟩ |
| 1 | 3 | Hanson | 15625/15552 | [-6 -5 6⟩ |
| 2 | 4 | Würschmidt | 393216/390625 | [17 1 -8⟩ |
| 3 | 5 | Mabila | 268435456/263671875 | [28 -3 -10⟩ |
| 4 | 6 | Goldis | 549755813888/533935546875 | [39 -7 -12⟩ |
| … | … | … | … | … |
| ∞* | ∞ | Srutal | 2048/2025 | [11 -4 -2⟩ |
- * in projective tuning space, ∞ = -∞.
All temperaments in the continuum also satisfy (15625/15552)s ~ 20000/19683, for a value of s defined such that 1/r + 1/s = 1; equivalently, we can offset s by 1, and equate a number of kleismas (15625/15552) with the diaschisma, giving rise to the name diaschismic-kleismic equivalence continuum. Varying s results in different temperaments listed in the table below. It converges to hanson as s approaches infinity, and is motivated by the fact that many important temperaments of 34edo follow a chain of commas connected by kleismas. The just value of s is 3.4117…, and temperaments near this tend to be the most accurate.
| s | n | Temperament | Comma | |
|---|---|---|---|---|
| Ratio | Monzo | |||
| -2 | 8/3 | 34 & 113 | 152587890625/148769467776 | [-7 -19 16⟩ |
| -1 | 5/2 | Fifive | 9765625/9565938 | [-1 -14 10⟩ |
| 0 | 2 | Tetracot | 20000/19683 | [5 -9 4⟩ |
| 1 | ∞ | Srutal | 2048/2025 | [11 -4 -2⟩ |
| 2 | 4 | Würschmidt | 393216/390625 | [17 1 -8⟩ |
| 3 | 7/2 | Vishnu | 6115295232/6103515625 | [23 6 -14⟩ |
| 4 | 10/3 | Gammic | (28 digits) | [-29 -11 20⟩ |
| 5 | 13/4 | Quatracot | (38 digits) | [-35 -16 26⟩ |
| … | … | … | … | |
| ∞ | 3 | Hanson | 15625/15552 | [-6 -5 6⟩ |
| s | r | n | Temperament | Comma |
|---|---|---|---|---|
| 5/2 = 2.5 | 5/3 = 1.6 | 11/3 = 3.6 | Majvam | [40 7 -22⟩ |
| 7/2 = 3.5 | 7/5 = 1.4 | 17/5 = 3.4 | Chlorine | [-52 -17 34⟩ |
| 5/3 = 1.6 | 5/2 = 2.5 | 9/2 = 4.5 | 34 & 142 | [45 -2 -18⟩ |