Holdrian comma: Difference between revisions
That "diaschisma" isn't the diaschisma in our modern sense |
Rewrite Mercator comma section to be about old vs new Mercator comma, explicitly state the diaschisma mentioned is different to the modern one |
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== Historical origin == | == Historical origin == | ||
The origin of Holder's comma resides in the fact that the [[Ancient Greek]]s (or at least to the Roman [[Anicius Manlius Severinus Boethius]]). According to Boethius, [[Pythagoras of Samos |Pythagoras]]' disciple [[Philolaus of Croton]] would have said that the tone consisted in two [[diatonic semitone]]s and a comma; the diatonic semitone consisted in two [[diaschisma (Ancient Greek music) | The origin of Holder's comma resides in the fact that the [[Ancient Greek]]s (or at least to the Roman [[Anicius Manlius Severinus Boethius]]). According to Boethius, [[Pythagoras of Samos |Pythagoras]]' disciple [[Philolaus of Croton]] would have said that the tone consisted in two [[diatonic semitone]]s and a comma; the diatonic semitone consisted in two [[diaschisma (Ancient Greek music)]]<ref>different to modern-day [[diaschismata]].</ref>, each formed of two commas.<ref>Anicius Manlius Severinus Boethius - ''De institutione musica'' - book 3 ch8</ref><ref>J. Murray Barbour - ''Tuning and Temperament: A historical survey'' (1951) - p123</ref>believed that in the [[Pythagorean tuning]] the tone could be divided in nine commas, four of which forming the diatonic semitone and five the chromatic semitone. If all these commas are exactly of the same size, there results an octave of ''5 tones + 2 diatonic'' semitones, ''5 × 9 + 2 × 4 = 53 equal'' commas. | ||
Holder<ref name=Holder-1731>William Holder - ''A Treatise of the Natural Grounds, and Principles of Harmony'' (1731) - ed3 p79</ref> attributes the division of the octave in 53 equal parts to [[Nicholas Mercator]]: "The late ''Nicholas Mercator'', a Modest Person, and a Learned and Judicious Mathematician, in a Manuscript of his, of which I have had a Sight."<ref name=Holder-1731/> who himself had proposed that 1/53 of the octave be named the "artificial comma". | Holder<ref name=Holder-1731>William Holder - ''A Treatise of the Natural Grounds, and Principles of Harmony'' (1731) - ed3 p79</ref> attributes the division of the octave in 53 equal parts to [[Nicholas Mercator]]: "The late ''Nicholas Mercator'', a Modest Person, and a Learned and Judicious Mathematician, in a Manuscript of his, of which I have had a Sight."<ref name=Holder-1731/> who himself had proposed that 1/53 of the octave be named the "artificial comma". | ||
== Mercator's comma, Mercator’s old comma, and the Holdrian comma == | |||
== Mercator's comma and the Holdrian comma == | [[Mercator's old comma]] is a name sometimes used for a closely related interval because of its association with Nicholas Mercator. | ||
[[Mercator's comma]] is a name | |||
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/> | 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/> | ||
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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) 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. | 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. | |||
== See also == | == See also == | ||
* [[Historical temperaments]] | * [[Historical temperaments]] | ||