Holdrian comma - Revision history

FloraC: Misc. cleanup

2025-10-27T12:53:43Z

Misc. cleanup

← Older revision		Revision as of 12:53, 27 October 2025
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	\| Ratio = 2^{1/53}		\| Ratio = 2^{1/53}
	\| Cents = 22.6415		\| 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)		\| Calc = 2^(1/53)
	\| Comma = yes		\| 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 '''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 ==		== 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 Pythagorean [[diatonic semitone]]s and a comma (the Pythagorean 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 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.

	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, 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.		'''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.		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 ==		== See also ==
	* [[Historical temperaments]]		* [[Historical temperaments]]
	* [[Interval size measure]]: both the Holdrian comma and ~~Mercator’s~~ old comma are examples of this		* [[Interval size measure]]: both the Holdrian comma and Mercator's old comma are examples of this

	== Notes ==		== Notes ==
			<references group="note"/>

			== References ==
			<references/>

	[[Category:53edo]]		[[Category:53edo]]
	[[Category:Small commas]]		[[Category:Small commas]]
	[[Category:Commas named after individuals]]		[[Category:Commas named after individuals]]

BudjarnLambeth: /* See also */ add link to interval size measure

2024-11-06T07:50:18Z

See also: add link to interval size measure

← Older revision		Revision as of 07:50, 6 November 2024
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	== See also ==		== See also ==
	* [[Historical temperaments]]		* [[Historical temperaments]]
			* [[Interval size measure]]: both the Holdrian comma and Mercator’s old comma are examples of this

	== Notes ==		== Notes ==

BudjarnLambeth: /* Historical origin */ Clarify specific type of diatonic semitone & chromatic semitone

2024-11-06T07:12:32Z

Historical origin: Clarify specific type of diatonic semitone & chromatic semitone

← Older revision		Revision as of 07:12, 6 November 2024
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	== Historical origin ==		== 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 [[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.		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)]]<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 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.

	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".

BudjarnLambeth: /* Historical origin */

2024-11-06T07:09:28Z

Historical origin

← Older revision		Revision as of 07:09, 6 November 2024
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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)]]<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.		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".		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".

BudjarnLambeth: /* Mercator's comma, Mercator’s old comma, and the Holdrian comma */

2024-11-06T06:58:32Z

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

← Older revision		Revision as of 06:58, 6 November 2024
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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.		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.		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.

BudjarnLambeth: /* Mercator's comma, Mercator’s old comma, and the Holdrian comma */

2024-11-06T06:58:05Z

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

← Older revision		Revision as of 06:58, 6 November 2024
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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/>		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.		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.

BudjarnLambeth: /* Historical origin */

2024-11-06T06:56:31Z

Historical origin

← Older revision		Revision as of 06:56, 6 November 2024
Line 12:		Line 12:
	== 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)]]<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>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.		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 [[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".		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".

BudjarnLambeth at 06:54, 6 November 2024

2024-11-06T06:54:05Z

← Older revision		Revision as of 06:54, 6 November 2024
Line 12:		Line 12:
	== 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)]]<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.		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 [[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>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".

BudjarnLambeth: /* Mercator's comma, Mercator’s old comma, and the Holdrian comma */

2024-11-06T06:49:50Z

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

← Older revision		Revision as of 06:49, 6 November 2024
Line 17:		Line 17:

	== 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.		'''Mercator's old comma''' is a name sometimes used for a closely related interval because of its association with Nicholas Mercator.

	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/>

BudjarnLambeth: Rewrite Mercator comma section to be about old vs new Mercator comma, explicitly state the diaschisma mentioned is different to the modern one

2024-11-06T06:49:26Z

Rewrite Mercator comma section to be about old vs new Mercator comma, explicitly state the diaschisma mentioned is different to the modern one

← Older revision		Revision as of 06:49, 6 November 2024
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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)\|diaschismata]], 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.		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 ~~often~~ used for a closely related interval because of its association with Nicholas Mercator.

	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 ==
	* [[Mercator's comma]]
	* [[Historical temperaments]]		* [[Historical temperaments]]