Holdrian comma: Difference between revisions

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{{Infobox Interval

| Ratio = ~~53root(~~2)

| Ratio = 2^{1/53}

| Cents = 22.6415

| Name = Holdrian comma, Holder’s comma, Arabian comma, 1 step of 53edo

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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) 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 comma and the Holdrian comma are two distinct but nearly equal intervals.

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== See also ==

* [[Mercator's comma]]

* [[Syntonic comma]]

* [[53edo]]

* [[Historical temperaments]]

== Notes ==

[[Category:53edo]]

[[Category:Small commas]]

[[Category:Commas named after individuals]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Ratio = 53root(2)
+| Ratio = 2^{1/53}
 | Cents = 22.6415
 | Name = Holdrian comma, Holder’s comma, Arabian comma, 1 step of 53edo
@@ 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) 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 comma and the Holdrian comma are two distinct but nearly equal intervals.
@@ Line 30: / Line 30: @@
 == See also ==
 * [[Mercator's comma]]
-* [[Syntonic comma]]
-* [[53edo]]
 * [[Historical temperaments]]
 == Notes ==
+[[Category:53edo]]
 [[Category:Small commas]]
 [[Category:Commas named after individuals]]