Holdrian comma: Difference between revisions

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| Ratio = 2^{1/53}

| Cents = 22.6415

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

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

| Calc = 2^(1/53)

| Comma = yes

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The '''Holdrian comma''', also called '''Holder's comma''', rarely the '''Arabian comma''',<ref name=Touma>Habib Hassan Touma & Laurie Schwartz - ''The Music of the Arabs'' - p23 (1993) - ISBN=0-931340-88-8</ref> is a small [[interval]] of approximately 22.6415 [[cents]],<ref name=Touma/> equal to exactly one step of [[53edo]], or <math>\ \sqrt[53]{2\;}\ </math>.

The name "[[comma]]~~", however, is technically misleading, since this interval is an irrational number~~ and it does not describe a compromise between intervals of any tuning system. The interval gets the name "comma" because it is a close approximation of several commas, most notably the [[syntonic comma]] (21.51 [[cents]]), which was widely used as a unit of tonal measurement during [[William Holder]]’s time.

The name ''[[comma]]'' describes its size and does not describe a compromise between intervals of any tuning system, since this interval is an irrational number. The interval gets the name ''comma'' because it is a close approximation of several commas, most notably the [[syntonic comma]] (21.51 [[cents]]), which was widely used as a unit of tonal measurement during [[William Holder]]'s time.

== Historical origin ==

The origin of Holder's comma resides in 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 Pythagorean [[diatonic semitone]]s and a comma (the Pythagorean diatonic semitone consisted in two [[diaschisma (Ancient Greek music)|diaschismata]]<ref group="note">different to modern-day [[diaschisma|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>) and believed that in the [[Pythagorean tuning]] the tone could be divided in nine commas, four of which forming the Pythagorean diatonic semitone and five the Pythagorean [[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.

The origin of Holder's comma resides in 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 [[diaschisma|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>) and 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".

== Mercator's comma, ~~Mercator’s~~ old comma, and the Holdrian comma ==

== Mercator's comma, Mercator's old 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.

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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)53/231. It has its own dedicated article.

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)53/231. It has its own dedicated article.

== See also ==

* [[Historical temperaments]]

* [[Interval size measure]]: both the Holdrian comma and Mercator's old comma are examples of this

== Notes ==

== References ==

[[Category:53edo]]

[[Category:Small commas]]

[[Category:Commas named after individuals]]

@@ Line 2: / Line 2: @@
 | Ratio = 2^{1/53}
 | Cents = 22.6415
-| Name = Holdrian comma, Holder’s comma, Arabian comma, 1 step of 53edo
+| Name = Holdrian comma, Holder's comma, Arabian comma, 1 step of 53edo
 | Calc = 2^(1/53)
 | Comma = yes
@@ Line 8: / Line 8: @@
 The '''Holdrian comma''', also called '''Holder's comma''', rarely the '''Arabian comma''',<ref name=Touma>Habib Hassan Touma & Laurie Schwartz - ''The Music of the Arabs'' - p23 (1993) - ISBN=0-931340-88-8</ref> is a small [[interval]] of approximately 22.6415 [[cents]],<ref name=Touma/> equal to exactly one step of [[53edo]], or <math>\ \sqrt[53]{2\;}\ </math>.
-The name "[[comma]]", however, is technically misleading, since this interval is an irrational number and it does not describe a compromise between intervals of any tuning system. The interval gets the name "comma" because it is a close approximation of several commas, most notably the [[syntonic comma]] (21.51 [[cents]]), which was widely used as a unit of tonal measurement during [[William Holder]]’s time.
+The name ''[[comma]]'' describes its size and does not describe a compromise between intervals of any tuning system, since this interval is an irrational number. The interval gets the name ''comma'' because it is a close approximation of several commas, most notably the [[syntonic comma]] (21.51 [[cents]]), which was widely used as a unit of tonal measurement during [[William Holder]]'s time.
 == Historical origin ==
+The origin of Holder's comma resides in 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 Pythagorean [[diatonic semitone]]s and a comma (the Pythagorean diatonic semitone consisted in two [[diaschisma (Ancient Greek music)|diaschismata]]<ref group="note">different to modern-day [[diaschisma|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>) and believed that in the [[Pythagorean tuning]] the tone could be divided in nine commas, four of which forming the Pythagorean diatonic semitone and five the Pythagorean [[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.
-The origin of Holder's comma resides in 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 [[diaschisma|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>) and 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".
-== Mercator's comma, Mercator’s old comma, and the Holdrian comma ==
+== Mercator's comma, Mercator's old 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.
@@ Line 25: / Line 24: @@
 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.
+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 ==
 * [[Historical temperaments]]
+* [[Interval size measure]]: both the Holdrian comma and Mercator's old comma are examples of this
 == Notes ==
+<references group="note"/>
+== References ==
+<references/>
 [[Category:53edo]]
 [[Category:Small commas]]
 [[Category:Commas named after individuals]]