Diaschismic–gothmic equivalence continuum: Difference between revisions

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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 in [[#~~Significance~~ of tetracot]]. 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 in [[#significance of tetracot]]. 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 [[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:

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 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 in [[#Significance of tetracot]]. 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 in [[#significance of tetracot]]. 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 [[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: