Pythagorean tuning: Difference between revisions

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~~The~~ '''Pythagorean tuning''' is ~~the 3-limit version of~~ [[~~just intonation~~]]~~. '''Pythagorean''' can be considered a~~ [[~~trivial temperament~~|~~trivial~~]] ~~rank-2 temperament in the 2~~.~~3 subgroup, as~~ 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) are extensions of pythagorean~~.

'''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]].

~~The Pythagorean temperament consists of all intervals generated by a just 3/2 and 2/1. Musically, the 2/1 is most often interpreted as an equave~~, ~~and as such~~ 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]].

~~See [[3-limit]] for more information~~.

== History ==

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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'''.

== 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.

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]].

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♭, C♭, 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.

== Scales ==

~~Because~~ Pythagorean tuning ~~is a rank-2 system,~~ the ~~moment-of-symmetry~~ scales ~~generated by its fifth can be named the same way scales corresponding to other rank-2 temperaments are, as follows~~:

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

* [[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.

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== See also ==

* [[3-limit]], the JI subgroup ~~which pythagorean is the trivial temperament of~~

* [[3-limit]], the corresponding JI subgroup.

* [[Chain of fifths]], a harmonic structure based on the concepts of Pythagorean tuning

* [[Chain of fifths]], a harmonic structure based on the concepts of Pythagorean tuning.

== References ==

[[Category:3-limit| ]]

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 }}
 {{Wikipedia|Pythagorean tuning}}
-The '''Pythagorean tuning''' is the 3-limit version of [[just intonation]]. '''Pythagorean''' can be considered a [[trivial temperament|trivial]] rank-2 temperament in the 2.3 subgroup, as 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) are extensions of pythagorean.
+'''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]].
-The Pythagorean temperament consists of all intervals generated by a just 3/2 and 2/1. Musically, the 2/1 is most often interpreted as an equave, and as such 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]].
-See [[3-limit]] for more information.
 == History ==
@@ Line 18: / Line 16: @@
 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'''.
+== 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.
+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]].
+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♭, C♭, 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.
 == Scales ==
-Because Pythagorean tuning is a rank-2 system, the moment-of-symmetry scales generated by its fifth can be named the same way scales corresponding to other rank-2 temperaments are, as follows:
+Pythagorean tuning generates the following [[MOS]] scales:
 * [[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.
@@ Line 35: / Line 43: @@
 == See also ==
-* [[3-limit]], the JI subgroup which pythagorean is the trivial temperament of
+* [[3-limit]], the corresponding JI subgroup.
-* [[Chain of fifths]], a harmonic structure based on the concepts of Pythagorean tuning
+* [[Chain of fifths]], a harmonic structure based on the concepts of Pythagorean tuning.
+== References ==
+<references />
 [[Category:3-limit| ]] <!-- main article -->