2187/2048 - Revision history

FloraC: Cleanup

2025-11-27T16:33:30Z

Cleanup

← Older revision		Revision as of 16:33, 27 November 2025
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	{{Wikipedia\|Semitone #Pythagorean tuning}}		{{Wikipedia\|Semitone #Pythagorean tuning}}

	'''2187/2048''', the '''Pythagorean apotome''' (pronounced /əˈpɒtəmi/, like "a-POT-o'-me"), also known as the '''Pythagorean chromatic semitone''' or the '''Pythagorean chroma~~''' or the '''3-limit augmented unison~~''', is the [[chromatic semitone]] in the [[Pythagorean tuning]]. It is the [[3-limit]] interval between seven perfect ~~just~~ fifths ([[3/2]]) and four octaves ([[2/1]]): 3<sup>7</sup>/2<sup>11</sup> = 2187/2048~~, and~~ measures about 113.7¢. Unlike the situation in [[meantone]] tunings, it is larger, not smaller, than the corresponding diatonic semitone, which is the Pythagorean minor second of [[256/243]].		'''2187/2048''', the '''Pythagorean apotome''' (pronounced /əˈpɒtəmi/, like "a-POT-o'-me"), also known as the '''Pythagorean chromatic semitone''' or the '''Pythagorean chroma''', is the [[chromatic semitone]] in the [[Pythagorean tuning]]. In other words, it is the [[3-limit]] augmented unison, that is, the interval between seven just perfect fifths ([[3/2]]) and four octaves ([[2/1]]): 3<sup>7</sup>/2<sup>11</sup> = 2187/2048. It measures about 113.7{{c}}. Unlike the situation in [[meantone]] tunings, it is larger, not smaller, than the corresponding [[diatonic semitone]], which is the Pythagorean minor second of [[256/243]].

	== Approximation ==		== Approximation ==
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	== Temperaments ==		== Temperaments ==
	When this ratio is taken as a comma to be tempered in the 5-limit, it produces the [[whitewood]] temperament, and it may be called the '''whitewood comma'''. See [[~~apotome~~ family]] for extensions thereof.		When this ratio is taken as a comma to be tempered in the 5-limit, it produces the [[whitewood]] temperament, and it may be called the '''whitewood comma'''. See [[Whitewood family]] for extensions thereof.

	== Notation ==		== Notation ==
	The apotome is the interval by which a sharp (#) or flat (b) modifies a note in the [[5L 2s\|diatonic]] [[chain-of-fifths notation]]. For example, in Pythagorean tuning, C and C# in the same octave are exactly an apotome apart. In tempered tuning systems, the mapping of the apotome dictates the size of sharps and flats. For instance, if the apotome is [[tempered out]], then sharps and flats have no effect on pitch in these systems.		The apotome is the interval by which a sharp (#) or flat (b) modifies a note in the [[5L 2s\|diatonic]] [[chain-of-fifths notation]]. For example, in Pythagorean tuning, C and C# in the same octave are exactly an apotome apart. In tempered tuning systems, the mapping of the apotome dictates the size of sharps and flats. For instance, if the apotome is [[tempered out]], then sharps and flats have no effect on pitch in these systems.

	The number of steps an apotome is mapped to in an ~~EDO~~ is referred to as its [[sharpness]].		The number of steps an apotome is mapped to in an edo is referred to as its [[sharpness]].

	== Etymology ==		== Etymology ==

FloraC: Pythagorean apotome seems more standard according to Wikipedia

2025-11-27T16:27:18Z

Pythagorean apotome seems more standard according to Wikipedia

← Older revision		Revision as of 16:27, 27 November 2025
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	{{Infobox Interval		{{Infobox Interval
	\| Name = apotome, Pythagorean chroma, Pythagorean chromatic semitone, whitewood comma		\| Name = Pythagorean apotome, Pythagorean chroma, Pythagorean chromatic semitone, whitewood comma
	\| Color name = Lw1, lawa unison		\| Color name = Lw1, lawa unison
	\| Sound = jid_2187_2048_pluck_adu_dr220.mp3		\| Sound = jid_2187_2048_pluck_adu_dr220.mp3
	\| Comma = yes		\| Comma = yes
	}}		}}
	{{Wikipedia\|Semitone#Pythagorean tuning}}		{{Wikipedia\|Semitone #Pythagorean tuning}}

	'''2187/2048''', the '''apotome''' (pronounced /əˈpɒtəmi/, like "a-POT-o'-me"), also known as the '''Pythagorean chromatic semitone''' or the '''Pythagorean chroma''' or the '''3-limit augmented unison''', is the [[chromatic semitone]] in the [[Pythagorean tuning]]. It is the [[3-limit]] interval between seven perfect just fifths ([[3/2]]) and four octaves ([[2/1]]): 3<sup>7</sup>/2<sup>11</sup> = 2187/2048, and measures about 113.7¢. Unlike the situation in [[meantone]] tunings, it is larger, not smaller, than the corresponding diatonic semitone, which is the Pythagorean minor second of [[256/243]].		'''2187/2048''', the '''Pythagorean apotome''' (pronounced /əˈpɒtəmi/, like "a-POT-o'-me"), also known as the '''Pythagorean chromatic semitone''' or the '''Pythagorean chroma''' or the '''3-limit augmented unison''', is the [[chromatic semitone]] in the [[Pythagorean tuning]]. It is the [[3-limit]] interval between seven perfect just fifths ([[3/2]]) and four octaves ([[2/1]]): 3<sup>7</sup>/2<sup>11</sup> = 2187/2048, and measures about 113.7¢. Unlike the situation in [[meantone]] tunings, it is larger, not smaller, than the corresponding diatonic semitone, which is the Pythagorean minor second of [[256/243]].

	== Approximation ==		== Approximation ==
Line 22:		Line 22:
	== Etymology ==		== Etymology ==
	According to the OED, the earliest English use of [[limma]] and apotome (alt. spelling "apotomy") with its musical as opposed to mathematical<ref>https://www.scientificlib.com/en/Mathematics/LX/Apotome.html</ref> meaning, is in 1694 in ''A Treatise of the Natural Grounds and Principles of Harmony''<ref>https://ota.bodleian.ox.ac.uk/repository/xmlui/bitstream/handle/20.500.12024/A44132/A44132.html</ref> by Church of England clergyman and natural philosopher William Holder. A relevant quote is "Difference between ... Tone Maj. and Limma. Apotome 2187 to 2048". The words are formed from the Greek, with "apo" meaning "away", "tome" meaning "cut" and limma meaning "remnant". So we begin with a major whole tone; the part cut away is the apotome (chromatic semitone) and the remnant is the limma (diatonic semitone).		According to the OED, the earliest English use of [[limma]] and apotome (alt. spelling "apotomy") with its musical as opposed to mathematical<ref>https://www.scientificlib.com/en/Mathematics/LX/Apotome.html</ref> meaning, is in 1694 in ''A Treatise of the Natural Grounds and Principles of Harmony''<ref>https://ota.bodleian.ox.ac.uk/repository/xmlui/bitstream/handle/20.500.12024/A44132/A44132.html</ref> by Church of England clergyman and natural philosopher William Holder. A relevant quote is "Difference between ... Tone Maj. and Limma. Apotome 2187 to 2048". The words are formed from the Greek, with "apo" meaning "away", "tome" meaning "cut" and limma meaning "remnant". So we begin with a major whole tone; the part cut away is the apotome (chromatic semitone) and the remnant is the limma (diatonic semitone).

			While ''apotome'' alone refers to this interval, the standard term is ''Pythagorean apotome'', which forms a pair with ''Pythagorean limma''.

	== See also ==		== See also ==

FloraC: Undo revision 211134 by M1k1 H2tsun2 (talk)

2025-09-27T11:51:56Z

Undo revision 211134 by M1k1 H2tsun2 (talk)

← Older revision		Revision as of 11:51, 27 September 2025
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	{{Infobox Interval		{{Infobox Interval
	\| Name = ~~Threerish augmented onet,~~ apotome, Pythagorean chroma, Pythagorean chromatic semitone, whitewood comma		\| Name = apotome, Pythagorean chroma, Pythagorean chromatic semitone, whitewood comma
	\| Color name = Lw1, lawa unison		\| Color name = Lw1, lawa unison
	\| Sound = jid_2187_2048_pluck_adu_dr220.mp3		\| Sound = jid_2187_2048_pluck_adu_dr220.mp3

M1k1 H2tsun2: add it them

2025-09-27T10:49:16Z

add it them

← Older revision		Revision as of 10:49, 27 September 2025
Line 1:		Line 1:
	{{Infobox Interval		{{Infobox Interval
	\| Name = apotome, Pythagorean chroma, Pythagorean chromatic semitone, whitewood comma		\| Name = Threerish augmented onet, apotome, Pythagorean chroma, Pythagorean chromatic semitone, whitewood comma
	\| Color name = Lw1, lawa unison		\| Color name = Lw1, lawa unison
	\| Sound = jid_2187_2048_pluck_adu_dr220.mp3		\| Sound = jid_2187_2048_pluck_adu_dr220.mp3

TallKite: added a less jargon-y and more musical name for 2187/2048

2025-02-08T21:36:47Z

added a less jargon-y and more musical name for 2187/2048

← Older revision		Revision as of 21:36, 8 February 2025
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	{{Wikipedia\|Semitone#Pythagorean tuning}}		{{Wikipedia\|Semitone#Pythagorean tuning}}

	'''2187/2048''', the '''apotome''' (pronounced /əˈpɒtəmi/, like "a-POT-o'-me"), also known as the '''Pythagorean chromatic semitone''' or the '''Pythagorean chroma''', is the [[chromatic semitone]] in the [[Pythagorean tuning]]. It is the [[3-limit]] interval between seven perfect just fifths ([[3/2]]) and four octaves ([[2/1]]): 3<sup>7</sup>/2<sup>11</sup> = 2187/2048, and measures about 113.7¢. Unlike the situation in [[meantone]] tunings, it is larger, not smaller, than the corresponding diatonic semitone, which is the Pythagorean minor second of [[256/243]].		'''2187/2048''', the '''apotome''' (pronounced /əˈpɒtəmi/, like "a-POT-o'-me"), also known as the '''Pythagorean chromatic semitone''' or the '''Pythagorean chroma''' or the '''3-limit augmented unison''', is the [[chromatic semitone]] in the [[Pythagorean tuning]]. It is the [[3-limit]] interval between seven perfect just fifths ([[3/2]]) and four octaves ([[2/1]]): 3<sup>7</sup>/2<sup>11</sup> = 2187/2048, and measures about 113.7¢. Unlike the situation in [[meantone]] tunings, it is larger, not smaller, than the corresponding diatonic semitone, which is the Pythagorean minor second of [[256/243]].

	== Approximation ==		== Approximation ==

Cmloegcmluin: /* Notation */

2024-10-21T15:02:00Z

Notation

← Older revision		Revision as of 15:02, 21 October 2024
Line 18:		Line 18:
	The apotome is the interval by which a sharp (#) or flat (b) modifies a note in the [[5L 2s\|diatonic]] [[chain-of-fifths notation]]. For example, in Pythagorean tuning, C and C# in the same octave are exactly an apotome apart. In tempered tuning systems, the mapping of the apotome dictates the size of sharps and flats. For instance, if the apotome is [[tempered out]], then sharps and flats have no effect on pitch in these systems.		The apotome is the interval by which a sharp (#) or flat (b) modifies a note in the [[5L 2s\|diatonic]] [[chain-of-fifths notation]]. For example, in Pythagorean tuning, C and C# in the same octave are exactly an apotome apart. In tempered tuning systems, the mapping of the apotome dictates the size of sharps and flats. For instance, if the apotome is [[tempered out]], then sharps and flats have no effect on pitch in these systems.

	The number of steps an apotome is mapped to in an ~~edo~~ is referred to as ~~the~~ [[sharpness]].		The number of steps an apotome is mapped to in an EDO is referred to as its [[sharpness]].

	== Etymology ==		== Etymology ==

Cmloegcmluin: separate etymological notes

2024-10-21T14:59:40Z

separate etymological notes

← Older revision		Revision as of 14:59, 21 October 2024
Line 8:		Line 8:

	'''2187/2048''', the '''apotome''' (pronounced /əˈpɒtəmi/, like "a-POT-o'-me"), also known as the '''Pythagorean chromatic semitone''' or the '''Pythagorean chroma''', is the [[chromatic semitone]] in the [[Pythagorean tuning]]. It is the [[3-limit]] interval between seven perfect just fifths ([[3/2]]) and four octaves ([[2/1]]): 3<sup>7</sup>/2<sup>11</sup> = 2187/2048, and measures about 113.7¢. Unlike the situation in [[meantone]] tunings, it is larger, not smaller, than the corresponding diatonic semitone, which is the Pythagorean minor second of [[256/243]].		'''2187/2048''', the '''apotome''' (pronounced /əˈpɒtəmi/, like "a-POT-o'-me"), also known as the '''Pythagorean chromatic semitone''' or the '''Pythagorean chroma''', is the [[chromatic semitone]] in the [[Pythagorean tuning]]. It is the [[3-limit]] interval between seven perfect just fifths ([[3/2]]) and four octaves ([[2/1]]): 3<sup>7</sup>/2<sup>11</sup> = 2187/2048, and measures about 113.7¢. Unlike the situation in [[meantone]] tunings, it is larger, not smaller, than the corresponding diatonic semitone, which is the Pythagorean minor second of [[256/243]].

	According to the OED, the earliest English use of [[limma]] and apotome (alt. spelling "apotomy") with its musical as opposed to mathematical<ref>https://www.scientificlib.com/en/Mathematics/LX/Apotome.html</ref> meaning, is in 1694 in ''A Treatise of the Natural Grounds and Principles of Harmony''<ref>https://ota.bodleian.ox.ac.uk/repository/xmlui/bitstream/handle/20.500.12024/A44132/A44132.html</ref> by Church of England clergyman and natural philosopher William Holder. A relevant quote is "Difference between ... Tone Maj. and Limma. Apotome 2187 to 2048". The words are formed from the Greek, with "apo" meaning "away", "tome" meaning "cut" and limma meaning "remnant". So we begin with a major whole tone; the part cut away is the apotome (chromatic semitone) and the remnant is the limma (diatonic semitone).

	== Approximation ==		== Approximation ==
Line 21:		Line 19:

	The number of steps an apotome is mapped to in an edo is referred to as the [[sharpness]].		The number of steps an apotome is mapped to in an edo is referred to as the [[sharpness]].

			== Etymology ==
			According to the OED, the earliest English use of [[limma]] and apotome (alt. spelling "apotomy") with its musical as opposed to mathematical<ref>https://www.scientificlib.com/en/Mathematics/LX/Apotome.html</ref> meaning, is in 1694 in ''A Treatise of the Natural Grounds and Principles of Harmony''<ref>https://ota.bodleian.ox.ac.uk/repository/xmlui/bitstream/handle/20.500.12024/A44132/A44132.html</ref> by Church of England clergyman and natural philosopher William Holder. A relevant quote is "Difference between ... Tone Maj. and Limma. Apotome 2187 to 2048". The words are formed from the Greek, with "apo" meaning "away", "tome" meaning "cut" and limma meaning "remnant". So we begin with a major whole tone; the part cut away is the apotome (chromatic semitone) and the remnant is the limma (diatonic semitone).

	== See also ==		== See also ==

Cmloegcmluin: add references section

2024-10-21T14:59:13Z

add references section

← Older revision		Revision as of 14:59, 21 October 2024
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	* [[Large comma]]		* [[Large comma]]
	* [[25/24]] – classic chromatic semitone		* [[25/24]] – classic chromatic semitone

			== References ==

	[[Category:Second]]		[[Category:Second]]

Cmloegcmluin: unbold terms

2024-10-21T14:58:44Z

unbold terms

← Older revision		Revision as of 14:58, 21 October 2024
Line 9:		Line 9:
	'''2187/2048''', the '''apotome''' (pronounced /əˈpɒtəmi/, like "a-POT-o'-me"), also known as the '''Pythagorean chromatic semitone''' or the '''Pythagorean chroma''', is the [[chromatic semitone]] in the [[Pythagorean tuning]]. It is the [[3-limit]] interval between seven perfect just fifths ([[3/2]]) and four octaves ([[2/1]]): 3<sup>7</sup>/2<sup>11</sup> = 2187/2048, and measures about 113.7¢. Unlike the situation in [[meantone]] tunings, it is larger, not smaller, than the corresponding diatonic semitone, which is the Pythagorean minor second of [[256/243]].		'''2187/2048''', the '''apotome''' (pronounced /əˈpɒtəmi/, like "a-POT-o'-me"), also known as the '''Pythagorean chromatic semitone''' or the '''Pythagorean chroma''', is the [[chromatic semitone]] in the [[Pythagorean tuning]]. It is the [[3-limit]] interval between seven perfect just fifths ([[3/2]]) and four octaves ([[2/1]]): 3<sup>7</sup>/2<sup>11</sup> = 2187/2048, and measures about 113.7¢. Unlike the situation in [[meantone]] tunings, it is larger, not smaller, than the corresponding diatonic semitone, which is the Pythagorean minor second of [[256/243]].

	According to the OED, the earliest English use of ~~'''~~[[limma]]~~'''~~ and ~~'''~~apotome~~'''~~ (alt. spelling "apotomy") with its musical as opposed to mathematical<ref>https://www.scientificlib.com/en/Mathematics/LX/Apotome.html</ref> meaning, is in 1694 in ''A Treatise of the Natural Grounds and Principles of Harmony''<ref>https://ota.bodleian.ox.ac.uk/repository/xmlui/bitstream/handle/20.500.12024/A44132/A44132.html</ref> by Church of England clergyman and natural philosopher William Holder. A relevant quote is "Difference between ... Tone Maj. and Limma. Apotome 2187 to 2048". The words are formed from the Greek, with "apo" meaning "away", "tome" meaning "cut" and limma meaning "remnant". So we begin with a major whole tone; the part cut away is the apotome (chromatic semitone) and the remnant is the limma (diatonic semitone).		According to the OED, the earliest English use of [[limma]] and apotome (alt. spelling "apotomy") with its musical as opposed to mathematical<ref>https://www.scientificlib.com/en/Mathematics/LX/Apotome.html</ref> meaning, is in 1694 in ''A Treatise of the Natural Grounds and Principles of Harmony''<ref>https://ota.bodleian.ox.ac.uk/repository/xmlui/bitstream/handle/20.500.12024/A44132/A44132.html</ref> by Church of England clergyman and natural philosopher William Holder. A relevant quote is "Difference between ... Tone Maj. and Limma. Apotome 2187 to 2048". The words are formed from the Greek, with "apo" meaning "away", "tome" meaning "cut" and limma meaning "remnant". So we begin with a major whole tone; the part cut away is the apotome (chromatic semitone) and the remnant is the limma (diatonic semitone).

	== Approximation ==		== Approximation ==

Dave Keenan: Made "limma" into a link

2024-10-21T07:23:32Z

Made "limma" into a link

← Older revision		Revision as of 07:23, 21 October 2024
Line 9:		Line 9:
	'''2187/2048''', the '''apotome''' (pronounced /əˈpɒtəmi/, like "a-POT-o'-me"), also known as the '''Pythagorean chromatic semitone''' or the '''Pythagorean chroma''', is the [[chromatic semitone]] in the [[Pythagorean tuning]]. It is the [[3-limit]] interval between seven perfect just fifths ([[3/2]]) and four octaves ([[2/1]]): 3<sup>7</sup>/2<sup>11</sup> = 2187/2048, and measures about 113.7¢. Unlike the situation in [[meantone]] tunings, it is larger, not smaller, than the corresponding diatonic semitone, which is the Pythagorean minor second of [[256/243]].		'''2187/2048''', the '''apotome''' (pronounced /əˈpɒtəmi/, like "a-POT-o'-me"), also known as the '''Pythagorean chromatic semitone''' or the '''Pythagorean chroma''', is the [[chromatic semitone]] in the [[Pythagorean tuning]]. It is the [[3-limit]] interval between seven perfect just fifths ([[3/2]]) and four octaves ([[2/1]]): 3<sup>7</sup>/2<sup>11</sup> = 2187/2048, and measures about 113.7¢. Unlike the situation in [[meantone]] tunings, it is larger, not smaller, than the corresponding diatonic semitone, which is the Pythagorean minor second of [[256/243]].

	According to the OED, the earliest English use of '''limma''' and '''apotome''' (alt. spelling "apotomy") with its musical as opposed to mathematical<ref>https://www.scientificlib.com/en/Mathematics/LX/Apotome.html</ref> meaning, is in 1694 in ''A Treatise of the Natural Grounds and Principles of Harmony''<ref>https://ota.bodleian.ox.ac.uk/repository/xmlui/bitstream/handle/20.500.12024/A44132/A44132.html</ref> by Church of England clergyman and natural philosopher William Holder. A relevant quote is "Difference between ... Tone Maj. and Limma. Apotome 2187 to 2048". The words are formed from the Greek, with "apo" meaning "away", "tome" meaning "cut" and limma meaning "remnant". So we begin with a major whole tone; the part cut away is the apotome (chromatic semitone) and the remnant is the limma (diatonic semitone).		According to the OED, the earliest English use of '''[[limma]]''' and '''apotome''' (alt. spelling "apotomy") with its musical as opposed to mathematical<ref>https://www.scientificlib.com/en/Mathematics/LX/Apotome.html</ref> meaning, is in 1694 in ''A Treatise of the Natural Grounds and Principles of Harmony''<ref>https://ota.bodleian.ox.ac.uk/repository/xmlui/bitstream/handle/20.500.12024/A44132/A44132.html</ref> by Church of England clergyman and natural philosopher William Holder. A relevant quote is "Difference between ... Tone Maj. and Limma. Apotome 2187 to 2048". The words are formed from the Greek, with "apo" meaning "away", "tome" meaning "cut" and limma meaning "remnant". So we begin with a major whole tone; the part cut away is the apotome (chromatic semitone) and the remnant is the limma (diatonic semitone).

	== Approximation ==		== Approximation ==