Pythagorean tuning: Difference between revisions

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'''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 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]].

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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.

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''' (十二律).{{clear}}

== 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.

~~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. 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~~ 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]] [[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 ==

Pythagorean tuning generates the following [[MOS]] scales:

Pythagorean tuning generates the following [[mos|MOS]] scales:

* [[Pythagorean5]] – proper [[2L 3s]]~~. Also~~ known as pythagorean ~~pentic~~ scale

* [[Pythagorean5]] – proper [[2L 3s]], also known as pentic, the ''pythagorean pentatonic scale''.

* [[Pythagorean7]] – improper [[5L 2s]]~~. Also~~ known as pythagorean diatonic scale

* [[Pythagorean7]] – improper [[5L 2s]], also known as diatonic,the ''pythagorean diatonic scale''.

* [[Pythagorean12]] – proper [[5L 7s]]~~. Also~~ known as pythagorean chromatic scale

* [[Pythagorean12]] – proper [[5L 7s]], also known as p-chromatic, the ''pythagorean chromatic scale''.

* [[Pythagorean17]] – improper [[12L 5s]]~~. Also~~ known as 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, ~~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, 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 ==

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 ==

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== References ==

[[Category:3-limit| ]]

[[Category:Rank-2 temperaments]]

[[Category:Historical]]

[[Category:Tuning]]

[[Category:Listen]]

~~[[Category:Rank 2]]~~

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 }}
 {{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 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]].
@@ Line 15: / Line 15: @@
 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&nbsp;BCE and 240&nbsp;CE, where it was called '''shi'er lü''' (十二律).{{clear}}
 == 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.
-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. 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 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]] [[Schismatic family#Tridecaschismic (2.3.5.13)|temperament]], as the triple-augmented fourth C–&nbsp;F♯♯♯ is incredibly close to [[13/8]], differing by the [[tridecapyth comma]] which is even smaller than the schisma.
 == Scales ==
-Pythagorean tuning generates the following [[MOS]] scales:
+Pythagorean tuning generates the following [[mos|MOS]] scales:
-* [[Pythagorean5]] – proper [[2L 3s]]. Also known as pythagorean pentic scale
+* [[Pythagorean5]] – proper [[2L&nbsp;3s]], also known as pentic, the ''pythagorean pentatonic scale''.
-* [[Pythagorean7]] – improper [[5L 2s]]. Also known as pythagorean diatonic scale
+* [[Pythagorean7]] – improper [[5L&nbsp;2s]], also known as diatonic,the ''pythagorean diatonic scale''.
-* [[Pythagorean12]] – proper [[5L 7s]]. Also known as pythagorean chromatic scale
+* [[Pythagorean12]] – proper [[5L&nbsp;7s]], also known as p-chromatic, the ''pythagorean chromatic scale''.
-* [[Pythagorean17]] – improper [[12L 5s]]. Also known as pythagorean enharmonic scale
+* [[Pythagorean17]] – improper [[12L&nbsp;5s]], also known as p-enharmonic, the ''pythagorean enharmonic scale''.
-* [[Pythagorean29]] – improper [[12L 17s]]
+* [[Pythagorean29]] – improper [[12L&nbsp;17s]], sometimes known as ''pythagotonic''.
-* [[Pythagorean41]] – proper [[12L 29s]]
+* [[Pythagorean41]] – proper [[12L&nbsp;29s]], sometimes known as ''pythamystonic.''
-* [[Pythagorean53]] – proper [[41L 12s]]
+* [[Pythagorean53]] – proper [[41L&nbsp;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, 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, 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 ==
 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&nbsp;3s]] pentic), 7 ([[5L&nbsp;2s]] diatonic), 12 ([[5L&nbsp;7s]] chromatic), 17 ([[12L&nbsp;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&nbsp;5s), 11 (8L&nbsp;3s), 19 (8L&nbsp;11s), 27 (19L&nbsp;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 ==
@@ Line 55: / Line 55: @@
 == References ==
-<references />
+<references/>
 [[Category:3-limit| ]] <!-- main article -->
+[[Category:Rank-2 temperaments]]
 [[Category:Historical]]
 [[Category:Tuning]]
 [[Category:Listen]]
-[[Category:Rank 2]]