Diaschismic–gothmic equivalence continuum: Difference between revisions

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The '''diaschismic-~~tetracot~~ equivalence continuum''' (~~which is the~~ '''diaschismic-~~gothmic~~ equivalence continuum''' ~~with offset 2~~) is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] describing the set of all [[5-limit]] temperaments ~~supported~~ by [[34edo]].

The '''diaschismic-gothmic equivalence continuum''' (or '''diaschismic-tetracot equivalence continuum''') is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] describing the set of all [[5-limit]] temperaments [[support]]ed 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)]]. 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 [[17-comma|Pythagorean gothma]] a.k.a. 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|harmonic 3]]. For example:

* [[Immunity]] (''n'' = 1) splits its twelfth in two;

* [[Tetracot]] (''n'' = 2) splits its fifth in four;

* [[Hanson]] (''n'' = 3) splits its twelfth in six;

* Etc.

The factor of 2 between ''n'' and the split of the interval class of 3 has to do with the fact that 34et has two [[ring number|rings]] of 17et's.

Another reasonable way of defining this continuum equates a number of diaschismas with the [[20000/19683|tetracot comma (20000/19683)]], so that (2048/2025)<sup>''r''</sup> ~ 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.

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-The '''diaschismic-tetracot equivalence continuum''' (which is the '''diaschismic-gothmic equivalence continuum''' with offset 2) is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] describing the set of all [[5-limit]] temperaments supported by [[34edo]].
+The '''diaschismic-gothmic equivalence continuum''' (or '''diaschismic-tetracot equivalence continuum''') is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] describing the set of all [[5-limit]] temperaments [[support]]ed 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)]]. 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 [[17-comma|Pythagorean gothma]] a.k.a. 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|harmonic 3]]. For example:
+* [[Immunity]] (''n'' = 1) splits its twelfth in two;
+* [[Tetracot]] (''n'' = 2) splits its fifth in four;
+* [[Hanson]] (''n'' = 3) splits its twelfth in six;
+* Etc.
+The factor of 2 between ''n'' and the split of the interval class of 3 has to do with the fact that 34et has two [[ring number|rings]] of 17et's.
 Another reasonable way of defining this continuum equates a number of diaschismas with the [[20000/19683|tetracot comma (20000/19683)]], so that (2048/2025)<sup>''r''</sup> ~ 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.