Diaschismic–gothmic equivalence continuum: Difference between revisions

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All temperaments in the continuum satisfy (15625/15552)''n'' ~ 393216/390625. Equivalently, we can offset ''n'' by 1, and equate a number of kleismas with the [[2048/2025|diaschisma (2048/2025)]], hence the name. Varying ''n'' results in different temperaments listed in the table below. It converges to [[Hanson_and_cata|hanson]] 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 both commas and thus tempers all combinations of them. The just value of ''n'' is 1.4117…, and temperaments near this tend to be the most accurate ones.

An equally reasonable way of defining this continuum equates a number of diaschismas with the Würschmidt comma, so that (2048/2025)''m'' ~ 393216/390625. The value of ''m'' is defined such that 1/''m'' - 1/''n'' = 1, and its just value is 0.5853…. The [[gothic comma]] (134217728/129140163) is the characteristic [[3-limit]] comma tempered out in 34edo, and it has a value of ''m'' = 4. Therefore, one can additionally define ''k'' = 4 - ''m'', 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, the value of ''k'' represents the number of generator steps required to reach the 3rd harmonic, even as ''n'' comes naturally from examining a chain of commas connected by kleismas.

An equally reasonable way of defining this continuum equates a number of diaschismas with the Würschmidt comma, so that (2048/2025)''m'' ~ 393216/390625. The value of ''m'' is defined such that 1/''m'' - 1/''n'' = 1, and its just value is 0.5853…. The [[17-comma|gothic comma]] (134217728/129140163) is the characteristic [[3-limit]] comma tempered out in 34edo, and it has a value of ''m'' = 4. Therefore, one can additionally define ''k'' = 4 - ''m'', 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, the value of ''k'' represents the number of generator steps required to reach the 3rd harmonic, even as ''n'' comes naturally from examining a chain of commas connected by kleismas.

{| class="wikitable center-1 center-2"

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 All temperaments in the continuum satisfy (15625/15552)<sup>''n''</sup> ~ 393216/390625. Equivalently, we can offset ''n'' by 1, and equate a number of kleismas with the [[2048/2025|diaschisma (2048/2025)]], hence the name. Varying ''n'' results in different temperaments listed in the table below. It converges to [[Hanson_and_cata|hanson]] 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 both commas and thus tempers all combinations of them. The just value of ''n'' is 1.4117…, and temperaments near this tend to be the most accurate ones.
-An equally reasonable way of defining this continuum equates a number of diaschismas with the Würschmidt comma, so that (2048/2025)<sup>''m''</sup> ~ 393216/390625. The value of ''m'' is defined such that 1/''m'' - 1/''n'' = 1, and its just value is 0.5853…. The [[gothic comma]] (134217728/129140163) is the characteristic [[3-limit]] comma tempered out in 34edo, and it has a value of ''m'' = 4. Therefore, one can additionally define ''k'' = 4 - ''m'', 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, the value of ''k'' represents the number of generator steps required to reach the 3rd harmonic, even as ''n'' comes naturally from examining a chain of commas connected by kleismas.
+An equally reasonable way of defining this continuum equates a number of diaschismas with the Würschmidt comma, so that (2048/2025)<sup>''m''</sup> ~ 393216/390625. The value of ''m'' is defined such that 1/''m'' - 1/''n'' = 1, and its just value is 0.5853…. The [[17-comma|gothic comma]] (134217728/129140163) is the characteristic [[3-limit]] comma tempered out in 34edo, and it has a value of ''m'' = 4. Therefore, one can additionally define ''k'' = 4 - ''m'', 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, the value of ''k'' represents the number of generator steps required to reach the 3rd harmonic, even as ''n'' comes naturally from examining a chain of commas connected by kleismas.
 {| class="wikitable center-1 center-2"