Holdrian comma: Difference between revisions

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One of these intervals was first described by [[Jing Fang]] in 45 BCE.<ref name=Touma/> Mercator applied logarithms to determine that <math>\ \sqrt[55]{2\;}\ </math> (≈ 21.8182 cents), exactly one step of [[55edo]], was nearly equivalent to a syntonic comma of ≈ 21.5063 cents (a feature of the [[historical temperaments|prevalent]] [[meantone]] temperament of the time). He also considered that an "artificial comma" of <math>\ \sqrt[53]{2\;}\ </math> might be useful, because 31 octaves could be practically approximated by a cycle of 53 [[just fifth]]s.

William Holder, for whom the ''Holdrian'' comma is named, favored this latter unit because the intervals of [[53edo]] are closer to [[just intonation]] than to [[55edo]]. Thus Mercator's old comma and the Holdrian comma are two distinct but nearly equal intervals.

William Holder, for whom the ''Holdrian'' comma is named, favored this latter unit because the intervals of 53edo are closer to [[just intonation]] than to [[55edo]]. Thus Mercator's old comma and the Holdrian comma are two distinct but nearly equal intervals.

There is another comma named ‘[[Mercator's comma]]’ which receives much more usage in modern musical tuning. It a small comma of 3.615 cents which is the amount by which 53 [[perfect fifth]]s exceed 31 [[octave]]s, in other words (3/2)<sup>53</sup>/2<sup>31</sup>. It has its own dedicated article.

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 One of these intervals was first described by [[Jing Fang]] in 45 BCE.<ref name=Touma/> Mercator applied logarithms to determine that <math>\ \sqrt[55]{2\;}\ </math> (≈ 21.8182 cents), exactly one step of [[55edo]], was nearly equivalent to a syntonic comma of ≈ 21.5063 cents (a feature of the [[historical temperaments|prevalent]] [[meantone]] temperament of the time). He also considered that an "artificial comma" of <math>\ \sqrt[53]{2\;}\ </math> might be useful, because 31 octaves could be practically approximated by a cycle of 53 [[just fifth]]s.
-William Holder, for whom the ''Holdrian'' comma is named, favored this latter unit because the intervals of [[53edo]] are closer to [[just intonation]] than to [[55edo]]. Thus Mercator's old comma and the Holdrian comma are two distinct but nearly equal intervals.
+William Holder, for whom the ''Holdrian'' comma is named, favored this latter unit because the intervals of 53edo are closer to [[just intonation]] than to [[55edo]]. Thus Mercator's old comma and the Holdrian comma are two distinct but nearly equal intervals.
 There is another comma named ‘[[Mercator's comma]]’ which receives much more usage in modern musical tuning. It a small comma of 3.615 cents which is the amount by which 53 [[perfect fifth]]s exceed 31 [[octave]]s, in other words (3/2)<sup>53</sup>/2<sup>31</sup>. It has its own dedicated article.