Pythagorean tuning - Revision history

Eufalesio: Added more things and changed some others

2026-02-13T23:46:28Z

Added more things and changed some others

FloraC: Recategorize

2026-02-06T08:19:19Z

Recategorize

← Older revision		Revision as of 08:19, 6 February 2026
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	== References ==		== References ==
	<references />		<references/>

	[[Category:3-limit\| ]] <!-- main article -->		[[Category:3-limit\| ]] <!-- main article -->
			[[Category:Rank-2 temperaments]]
	[[Category:Historical]]		[[Category:Historical]]
	[[Category:Tuning]]		[[Category:Tuning]]
	[[Category:Listen]]		[[Category:Listen]]
	~~[[Category:Rank 2]]~~

FloraC: /* Relation to temperaments */ don't say "pythagorean temperament" even if it's treated as a temperament. Misc. cleanup & style improvements

2026-02-06T08:18:39Z

Relation to temperaments: don't say "pythagorean temperament" even if it's treated as a temperament. Misc. cleanup & style improvements

← Older revision		Revision as of 08:18, 6 February 2026
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	== Relation to temperaments ==		== Relation to temperaments ==
	Pythagorean tuning can be considered a [[trivial temperament\|trivial]] rank-2 temperament in the 2.3 subgroup, where it tempers out no commas (providing no additional mappings for intervals other than the pure just structure). As such, all rank-2 temperaments generated by 3/2 and 2/1 in the 5-limit or higher (e.g. meantone) can be seen as ~~extensions~~ of Pythagorean ~~temperament~~.		Pythagorean tuning can be considered a [[trivial temperament\|trivial]] [[rank-2 temperament\|rank-2]] [[regular temperament\|temperament]] in the [[3-limit\|2.3 subgroup]], where it tempers out no commas (providing no additional mappings for intervals other than the pure just structure). As such, all rank-2 temperaments generated by 3/2 and 2/1 in the [[5-limit]] or higher (e.g. [[meantone]]) can be seen as [[extension]]s of Pythagorean tuning.

	A series of just fifths can also be considered a reasonable tuning of the [[schismatic]] temperament, where the [[Pythagorean diminished fourth\|diminished fourth]] (e.g. ~~{{dash\|C, F♭}}~~) approximates [[5/4]], since the [[schisma]] is so small.		A series of just fifths can also be considered a reasonable tuning of the [[schismatic]] temperament, where the [[Pythagorean diminished fourth\|diminished fourth]] (e.g. C–F♭) approximates [[5/4]], since the [[schisma]] is so small. Mark Lindley argues such a system was used in Europe during the 15th century<ref>Mark Lindley, ''Pythagorean Intonation and the Rise of the Triad'', Royal Musical Association Research Chronicle, 1980</ref>, with keyboards tuned to nearly pure fifths as
	Mark Lindley<ref>Mark Lindley, ''Pythagorean Intonation and the Rise of the Triad'', Royal Musical Association Research Chronicle, 1980</ref> ~~argues such a system was used in Europe during the 15th century~~, with keyboards tuned to nearly pure fifths as

	: {{dash\|G♭, D♭, A♭, E♭, B♭, F, C, G, D, A, E, B}}.		: {{dash\| G♭, D♭, A♭, E♭, B♭, F, C, G, D, A, E, B }}.

	When respelled enharmonically, triads such as ~~{{dash\|D, F♯, A}}~~ are very close to [[4:5:6]] in this tuning.		When respelled enharmonically, triads such as D–F♯–A are very close to [[4:5:6]] in this tuning.

	It can also be used to generate a more xenharmonic [[2.3.5.13 subgroup]] [[~~Marveltwin~~]] temperament, as the triple-augmented fourth ~~{{dash\|C, F♯♯♯}}~~ is incredibly close to [[13/8]], differing by the [[~~Tridecapyth~~ comma]] which is even smaller than the schisma.		It can also be used to generate a more xenharmonic [[2.3.5.13 subgroup]] [[marveltwin]] temperament, as the triple-augmented fourth C–F♯♯♯ is incredibly close to [[13/8]], differing by the [[tridecapyth comma]] which is even smaller than the schisma.

	== Scales ==		== Scales ==

FloraC: /* Scales */ de-bold nonlemma

2026-02-06T08:14:08Z

Scales: de-bold nonlemma

← Older revision		Revision as of 08:14, 6 February 2026
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	== Scales ==		== Scales ==
	Pythagorean tuning generates the following [[~~MOS~~]] scales:		Pythagorean tuning generates the following [[mos]] scales:
	* [[Pythagorean5]] – proper [[2L 3s]]~~. Also~~ known as the '''pythagorean pentic scale.'''		* [[Pythagorean5]] – proper [[2L 3s]], also known as the ''pythagorean pentic scale''.
	* [[Pythagorean7]] – improper [[5L 2s]]~~. Also~~ known as the '''pythagorean diatonic scale.'''		* [[Pythagorean7]] – improper [[5L 2s]], also known as the ''pythagorean diatonic scale''.
	* [[Pythagorean12]] – proper [[5L 7s]]~~. Also~~ known as the '''pythagorean chromatic scale.'''		* [[Pythagorean12]] – proper [[5L 7s]], also known as the ''pythagorean chromatic scale''.
	* [[Pythagorean17]] – improper [[12L 5s]]~~. Also~~ known as the '''pythagorean enharmonic scale.'''		* [[Pythagorean17]] – improper [[12L 5s]], also known as the ''pythagorean enharmonic scale''.
	* [[Pythagorean29]] – improper [[12L 17s]].		* [[Pythagorean29]] – improper [[12L 17s]].
	* [[Pythagorean41]] – proper [[12L 29s]].		* [[Pythagorean41]] – proper [[12L 29s]].

ArrowHead294: /* History */

2025-12-11T22:54:42Z

History

← Older revision		Revision as of 22:54, 11 December 2025
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	Pythagorean tuning was not actually invented by [[Pythagoras of Samos\|Pythagoras]]. The earliest records are from [[Mesopotamian music\|Ancient Mesopotamia]], and it was later inherited by the [[Ancient Greek]]s.		Pythagorean tuning was not actually invented by [[Pythagoras of Samos\|Pythagoras]]. The earliest records are from [[Mesopotamian music\|Ancient Mesopotamia]], and it was later inherited by the [[Ancient Greek]]s.

	The 12-tone form of Pythagorean tuning was (probably independently) invented in [[Chinese music\|Ancient China]] between 600 BCE and 240 CE, where it was called '''shi'er lü'''.{{clear}}		The 12-tone form of Pythagorean tuning was (probably independently) invented in [[Chinese music\|Ancient China]] between 600 BCE and 240 CE, where it was called '''shi'er lü''' (十二律).{{clear}}

	== Relation to temperaments ==		== Relation to temperaments ==

Eufalesio: Added some things and made articles for the missing MOS scales

2025-12-08T23:15:06Z

Added some things and made articles for the missing MOS scales

ArrowHead294 at 14:01, 8 December 2025

2025-12-08T14:01:00Z

VectorGraphics at 08:32, 19 October 2025

2025-10-19T08:32:59Z

← Older revision		Revision as of 08:32, 19 October 2025
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	}}		}}
	{{Wikipedia\|Pythagorean tuning}}		{{Wikipedia\|Pythagorean tuning}}
	'''Pythagorean tuning''' is a system where all intervals are determined by [[3/2~~\|pure fifths~~]] and [[2/1\|octaves]]. ~~This makes it essentially~~ the same as [[3-limit]] [[just intonation]].		'''Pythagorean tuning''' is a system where all intervals are determined by perfect fifths tuned to [[3/2]] and [[2/1\|octaves]]. As such, Pythagorean tuning contains the same intervals as [[3-limit]] [[just intonation]],

	When accounting for octave equivalence, Pythagorean tuning mirrors the structure of the [[chain of fifths]].		When accounting for octave equivalence, Pythagorean tuning mirrors the structure of the [[chain of fifths]].

FloraC: Undo revision 211144 by M1k1 H2tsun2 (talk)

2025-09-27T11:53:50Z

Undo revision 211144 by M1k1 H2tsun2 (talk)

← Older revision		Revision as of 11:53, 27 September 2025
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	}}		}}
	{{Wikipedia\|Pythagorean tuning}}		{{Wikipedia\|Pythagorean tuning}}
	'''Pythagorean ~~tuning''' or '''Threerish~~ tuning''' is a system where all intervals are determined by [[3/2\|pure fifths]] and [[2/1\|octaves]]. This makes it essentially the same as [[3-limit]] [[just intonation]].		'''Pythagorean tuning''' is a system where all intervals are determined by [[3/2\|pure fifths]] and [[2/1\|octaves]]. This makes it essentially the same as [[3-limit]] [[just intonation]].

	When accounting for octave equivalence, Pythagorean tuning mirrors the structure of the [[chain of fifths]].		When accounting for octave equivalence, Pythagorean tuning mirrors the structure of the [[chain of fifths]].

M1k1 H2tsun2: add

2025-09-27T11:04:48Z

add

← Older revision		Revision as of 11:04, 27 September 2025
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	}}		}}
	{{Wikipedia\|Pythagorean tuning}}		{{Wikipedia\|Pythagorean tuning}}
	'''Pythagorean tuning''' is a system where all intervals are determined by [[3/2\|pure fifths]] and [[2/1\|octaves]]. This makes it essentially the same as [[3-limit]] [[just intonation]].		'''Pythagorean tuning''' or '''Threerish tuning''' is a system where all intervals are determined by [[3/2\|pure fifths]] and [[2/1\|octaves]]. This makes it essentially the same as [[3-limit]] [[just intonation]].

	When accounting for octave equivalence, Pythagorean tuning mirrors the structure of the [[chain of fifths]].		When accounting for octave equivalence, Pythagorean tuning mirrors the structure of the [[chain of fifths]].

← Older revision		Revision as of 23:46, 13 February 2026
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	: {{dash\| G♭, D♭, A♭, E♭, B♭, F, C, G, D, A, E, B }}.		: {{dash\| G♭, D♭, A♭, E♭, B♭, F, C, G, D, A, E, B }}.

	~~When respelled enharmonically,~~ triads such as ~~D–F♯–A are~~ very close to [[4:5:6]] in this tuning.		This makes triads such as {{dash\|D, G♭, A}} ({{dash\|D, vF#, A}} more intuitively) very close to [[4:5:6]] in this tuning.

	It can also be used to generate a more xenharmonic [[2.3.5.13 subgroup]] [[marveltwin]] temperament, as the triple-augmented fourth ~~C–F♯♯♯~~ is incredibly close to [[13/8]], differing by the [[tridecapyth comma]] which is even smaller than the schisma.		It can also be used to generate a more xenharmonic [[2.3.5.13 subgroup]] [[marveltwin]] [[Schismatic family#Tridecaschismic (2.3.5.13)\|temperament]], as the triple-augmented fourth C– F♯♯♯ is incredibly close to [[13/8]], differing by the [[tridecapyth comma]] which is even smaller than the schisma.

	== Scales ==		== Scales ==
	Pythagorean tuning generates the following [[mos]] scales:		Pythagorean tuning generates the following [[mos\|MOS]] scales:
	* [[Pythagorean5]] – proper [[2L 3s]], also known as the ''pythagorean ~~pentic~~ scale''.		* [[Pythagorean5]] – proper [[2L 3s]], also known as pentic, the ''pythagorean pentatonic scale''.
	* [[Pythagorean7]] – improper [[5L 2s]], also known as the ''pythagorean diatonic scale''.		* [[Pythagorean7]] – improper [[5L 2s]], also known as diatonic,the ''pythagorean diatonic scale''.
	* [[Pythagorean12]] – proper [[5L 7s]], also known as the ''pythagorean chromatic scale''.		* [[Pythagorean12]] – proper [[5L 7s]], also known as p-chromatic, the ''pythagorean chromatic scale''.
	* [[Pythagorean17]] – improper [[12L 5s]], also known as the ''pythagorean enharmonic scale''.		* [[Pythagorean17]] – improper [[12L 5s]], also known as p-enharmonic, the ''pythagorean enharmonic scale''.
	* [[Pythagorean29]] – improper [[12L 17s]].		* [[Pythagorean29]] – improper [[12L 17s]], sometimes known as ''pythagotonic''.
	* [[Pythagorean41]] – proper [[12L 29s]].		* [[Pythagorean41]] – proper [[12L 29s]], sometimes known as ''pythamystonic.''
	* [[Pythagorean53]] – proper [[41L 12s]].		* [[Pythagorean53]] – proper [[41L 12s]], sometimes known as ''pythomerc''.

	The [[hardness]]es of the Pythagorean scales are about 1.442 for pentic, 2.260 for diatonic, 1.260 for chromatic, 3.846 for enharmonic, ~~and~~ 2.8459, 1.8459, 1.1822 for the ~~other three~~.		The [[hardness]]es of the Pythagorean scales are about 1.442 for pentic, 2.260 for diatonic, 1.260 for chromatic, 3.846 for enharmonic, 2.8459 for pythagotonic, 1.8459 for pythamystonic, and 1.1822 for pythomerc. Pythamystonic, pentic, p-chromatic and pythomerc generate the softest MOS scales generated by a just fifth as their equalized tunings represent edos with convergent fifths.

	== Approaches ==		== Approaches ==

← Older revision		Revision as of 23:15, 8 December 2025
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	Pythagorean tuning can be considered a [[trivial temperament\|trivial]] rank-2 temperament in the 2.3 subgroup, where it tempers out no commas (providing no additional mappings for intervals other than the pure just structure). As such, all rank-2 temperaments generated by 3/2 and 2/1 in the 5-limit or higher (e.g. meantone) can be seen as extensions of Pythagorean temperament.		Pythagorean tuning can be considered a [[trivial temperament\|trivial]] rank-2 temperament in the 2.3 subgroup, where it tempers out no commas (providing no additional mappings for intervals other than the pure just structure). As such, all rank-2 temperaments generated by 3/2 and 2/1 in the 5-limit or higher (e.g. meantone) can be seen as extensions of Pythagorean temperament.

	~~Because the [[schisma]] is so small, a~~ series of just fifths can also be considered a reasonable tuning of the [[schismatic]] temperament, where the [[Pythagorean diminished fourth\|diminished fourth]] (e.g. {{dash\|C, F♭}}) approximates [[5/4]].		A series of just fifths can also be considered a reasonable tuning of the [[schismatic]] temperament, where the [[Pythagorean diminished fourth\|diminished fourth]] (e.g. {{dash\|C, F♭}}) approximates [[5/4]], since the [[schisma]] is so small.
	Mark Lindley<ref>Mark Lindley, ''Pythagorean Intonation and the Rise of the Triad'', Royal Musical Association Research Chronicle, 1980</ref> argues such a system was used in Europe during the 15th century, with keyboards tuned to nearly pure fifths as		Mark Lindley<ref>Mark Lindley, ''Pythagorean Intonation and the Rise of the Triad'', Royal Musical Association Research Chronicle, 1980</ref> argues such a system was used in Europe during the 15th century, with keyboards tuned to nearly pure fifths as

	: {{dash\|G♭, D♭, A♭, E♭, B♭, F, C, G, D, A, E, B}}.		: {{dash\|G♭, D♭, A♭, E♭, B♭, F, C, G, D, A, E, B}}.

	When respelled enharmonically, triads such as {{dash\|D, F♯, A}} are close to 4:5:6 in this tuning.		When respelled enharmonically, triads such as {{dash\|D, F♯, A}} are very close to [[4:5:6]] in this tuning.

			It can also be used to generate a more xenharmonic [[2.3.5.13 subgroup]] [[Marveltwin]] temperament, as the triple-augmented fourth {{dash\|C, F♯♯♯}} is incredibly close to [[13/8]], differing by the [[Tridecapyth comma]] which is even smaller than the schisma.

	== Scales ==		== Scales ==
	Pythagorean tuning generates the following [[MOS]] scales:		Pythagorean tuning generates the following [[MOS]] scales:
	* [[Pythagorean5]] – proper [[2L 3s]]. Also known as pythagorean pentic scale		* [[Pythagorean5]] – proper [[2L 3s]]. Also known as the '''pythagorean pentic scale.'''
	* [[Pythagorean7]] – improper [[5L 2s]]. Also known as pythagorean diatonic scale		* [[Pythagorean7]] – improper [[5L 2s]]. Also known as the '''pythagorean diatonic scale.'''
	* [[Pythagorean12]] – proper [[5L 7s]]. Also known as pythagorean chromatic scale		* [[Pythagorean12]] – proper [[5L 7s]]. Also known as the '''pythagorean chromatic scale.'''
	* [[Pythagorean17]] – improper [[12L 5s]]. Also known as pythagorean enharmonic scale		* [[Pythagorean17]] – improper [[12L 5s]]. Also known as the '''pythagorean enharmonic scale.'''
	* [[Pythagorean29]] – improper [[12L 17s]]		* [[Pythagorean29]] – improper [[12L 17s]].
	* [[Pythagorean41]] – proper [[12L 29s]]		* [[Pythagorean41]] – proper [[12L 29s]].
	* [[Pythagorean53]] – proper [[41L 12s]]		* [[Pythagorean53]] – proper [[41L 12s]].

	The [[hardness]]es of the Pythagorean scales are about 1.442 for pentic, 2.260 for diatonic, 1.260 for chromatic, ~~and~~ 3.846 for enharmonic.		The [[hardness]]es of the Pythagorean scales are about 1.442 for pentic, 2.260 for diatonic, 1.260 for chromatic, 3.846 for enharmonic, and 2.8459, 1.8459, 1.1822 for the other three.

	== Approaches ==		== Approaches ==
	There are many possible approaches to Pythagorean tuning, and each approach is associated with a different Pythagorean equave. The two most widely-used are [[octave]]-based and [[tritave]]-based Pythagorean.		There are many possible approaches to Pythagorean tuning, and each approach is associated with a different Pythagorean equave. The two most widely-used are [[octave]]-based and [[tritave]]-based Pythagorean.

	[[Octave]]-based Pythagorean tuning is essentially how it is used in the common-practice music of the West. This gives MOS sizes of 2, 3, 5 ([[2L 3s]] pentic), 7 ([[5L 2s]] diatonic), 12 ([[5L 7s]] chromatic), 17 ([[12L 5s]] enharmonic), 29, 41, and 53.		[[Octave]]-based Pythagorean tuning is essentially how it is used in the common-practice music of the West. This gives MOS sizes of 2, 3, 5 ([[2L 3s]] pentic), 7 ([[5L 2s]] diatonic), 12 ([[5L 7s]] chromatic), 17 ([[12L 5s]] enharmonic), 29, 41, and 53.

	[[Tritave]]-based Pythagorean tuning is an approach described in [https://arxiv.org/abs/1709.00375 this paper] by M. Schmidmeier. This gives MOS sizes of 2, 3, 5, 8 (3L 5s), 11 (8L 3s), 19 (8L 11s), 27 (19L 8s), 46, and 65. The 11-note scale can be regarded as the diatonic-like scale of tritave-equivalent Pythagorean, and the 19-note scale can be regarded as its respective chromatic-like scale.		[[Tritave]]-based Pythagorean tuning is an approach described in [https://arxiv.org/abs/1709.00375 this paper] by M. Schmidmeier. This gives MOS sizes of 2, 3, 5, 8 (3L 5s), 11 (8L 3s), 19 (8L 11s), 27 (19L 8s), 46, and 65. The 11-note scale can be regarded as the diatonic-like scale of tritave-equivalent Pythagorean, and the 19-note scale can be regarded as its respective chromatic-like scale.

← Older revision		Revision as of 14:01, 8 December 2025
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	Pythagorean tuning was not actually invented by [[Pythagoras of Samos\|Pythagoras]]. The earliest records are from [[Mesopotamian music\|Ancient Mesopotamia]], and it was later inherited by the [[Ancient Greek]]s.		Pythagorean tuning was not actually invented by [[Pythagoras of Samos\|Pythagoras]]. The earliest records are from [[Mesopotamian music\|Ancient Mesopotamia]], and it was later inherited by the [[Ancient Greek]]s.

	The 12-tone form of Pythagorean tuning was (probably independently) invented in [[Chinese music\|Ancient China]] between 600 BCE and 240 CE, where it was called '''shi'er lü'''.		The 12-tone form of Pythagorean tuning was (probably independently) invented in [[Chinese music\|Ancient China]] between 600 BCE and 240 CE, where it was called '''shi'er lü'''.{{clear}}

	== Relation to temperaments ==		== Relation to temperaments ==
Line 23:		Line 23:
	Mark Lindley<ref>Mark Lindley, ''Pythagorean Intonation and the Rise of the Triad'', Royal Musical Association Research Chronicle, 1980</ref> argues such a system was used in Europe during the 15th century, with keyboards tuned to nearly pure fifths as		Mark Lindley<ref>Mark Lindley, ''Pythagorean Intonation and the Rise of the Triad'', Royal Musical Association Research Chronicle, 1980</ref> argues such a system was used in Europe during the 15th century, with keyboards tuned to nearly pure fifths as

	:{{dash\|G♭, D♭, A♭, E♭, B♭, F, C, G, D, A, E, B}}.		: {{dash\|G♭, D♭, A♭, E♭, B♭, F, C, G, D, A, E, B}}.

	When respelled enharmonically, triads such as {{dash\|D, F♯, A}} are close to 4:5:6 in this tuning.		When respelled enharmonically, triads such as {{dash\|D, F♯, A}} are close to 4:5:6 in this tuning.
Line 29:		Line 29:
	== Scales ==		== Scales ==
	Pythagorean tuning generates the following [[MOS]] scales:		Pythagorean tuning generates the following [[MOS]] scales:
	* [[Pythagorean5]] – proper [[2L 3s]]. Also known as pythagorean pentic scale		* [[Pythagorean5]] – proper [[2L 3s]]. Also known as pythagorean pentic scale
	* [[Pythagorean7]] – improper [[5L 2s]]. Also known as pythagorean diatonic scale		* [[Pythagorean7]] – improper [[5L 2s]]. Also known as pythagorean diatonic scale
	* [[Pythagorean12]] – proper [[5L 7s]]. Also known as pythagorean chromatic scale		* [[Pythagorean12]] – proper [[5L 7s]]. Also known as pythagorean chromatic scale
	* [[Pythagorean17]] – improper [[12L 5s]]. Also known as pythagorean enharmonic scale		* [[Pythagorean17]] – improper [[12L 5s]]. Also known as pythagorean enharmonic scale
	* [[Pythagorean29]] – improper [[12L 17s]]		* [[Pythagorean29]] – improper [[12L 17s]]
	* [[Pythagorean41]] – proper [[12L 29s]]		* [[Pythagorean41]] – proper [[12L 29s]]
	* [[Pythagorean53]] – proper [[41L 12s]]		* [[Pythagorean53]] – proper [[41L 12s]]

	The [[hardness]]es of the Pythagorean scales are about 1.442 for pentic, 2.260 for diatonic, 1.260 for chromatic, and 3.846 for enharmonic.		The [[hardness]]es of the Pythagorean scales are about 1.442 for pentic, 2.260 for diatonic, 1.260 for chromatic, and 3.846 for enharmonic.

	== Approaches ==		== Approaches ==

	There are many possible approaches to Pythagorean tuning, and each approach is associated with a different Pythagorean equave. The two most widely-used are [[octave]]-based and [[tritave]]-based Pythagorean.		There are many possible approaches to Pythagorean tuning, and each approach is associated with a different Pythagorean equave. The two most widely-used are [[octave]]-based and [[tritave]]-based Pythagorean.

	[[Octave]]-based Pythagorean tuning is essentially how it is used in the common-practice music of the West. This gives MOS sizes of 2, 3, 5 ([[2L 3s]] pentic), 7 ([[5L 2s]] diatonic), 12 ([[5L 7s]] chromatic), 17 ([[12L 5s]] enharmonic), 29, 41, and 53.		[[Octave]]-based Pythagorean tuning is essentially how it is used in the common-practice music of the West. This gives MOS sizes of 2, 3, 5 ([[2L 3s]] pentic), 7 ([[5L 2s]] diatonic), 12 ([[5L 7s]] chromatic), 17 ([[12L 5s]] enharmonic), 29, 41, and 53.

	[[Tritave]]-based Pythagorean tuning is an approach described in [https://arxiv.org/abs/1709.00375 this paper] by M. Schmidmeier. This gives MOS sizes of 2, 3, 5, 8 (3L 5s), 11 (8L 3s), 19 (8L 11s), 27 (19L 8s), 46, and 65. The 11-note scale can be regarded as the diatonic-like scale of tritave-equivalent Pythagorean, and the 19-note scale can be regarded as its respective chromatic-like scale.		[[Tritave]]-based Pythagorean tuning is an approach described in [https://arxiv.org/abs/1709.00375 this paper] by M. Schmidmeier. This gives MOS sizes of 2, 3, 5, 8 (3L 5s), 11 (8L 3s), 19 (8L 11s), 27 (19L 8s), 46, and 65. The 11-note scale can be regarded as the diatonic-like scale of tritave-equivalent Pythagorean, and the 19-note scale can be regarded as its respective chromatic-like scale.

	== Music ==		== Music ==