Pythagorean tuning: Difference between revisions

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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'''.{{clear}}

== Relation to temperaments ==

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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.

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== 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.

== 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 ==

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