Holdrian comma: Difference between revisions

Line 21:

Holder 1731 writes that [[Marin Mersenne]] had calculated 58<sup>1</sup>/<sub>4</sub>s in the octave; Mercator "working by the logarithms, finds out but 55, and a little more."<ref name=Holder-1731/>

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) 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.

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.

@@ Line 21: / Line 21: @@
 Holder 1731 writes that [[Marin Mersenne]] had calculated 58<sup>1</sup>/<sub>4</sub>s in the octave; Mercator "working by the logarithms, finds out but 55, and a little more."<ref name=Holder-1731/>
-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) 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.
+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.