Pythagorean tuning: Difference between revisions

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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.

~~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

: {{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 ==

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 ==

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.

[[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.

@@ Line 20: / Line 20: @@
 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
 : {{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 ==
 Pythagorean tuning generates the following [[MOS]] scales:
-* [[Pythagorean5]] – proper [[2L&nbsp;3s]]. Also known as pythagorean pentic scale
+* [[Pythagorean5]] – proper [[2L&nbsp;3s]]. Also known as the '''pythagorean pentic scale.'''
-* [[Pythagorean7]] – improper [[5L&nbsp;2s]]. Also known as pythagorean diatonic scale
+* [[Pythagorean7]] – improper  [[5L&nbsp;2s]]. Also known as the '''pythagorean diatonic scale.'''
-* [[Pythagorean12]] – proper [[5L&nbsp;7s]]. Also known as pythagorean chromatic scale
+* [[Pythagorean12]] – proper [[5L&nbsp;7s]]. Also known as the '''pythagorean chromatic scale.'''
-* [[Pythagorean17]] – improper [[12L&nbsp;5s]]. Also known as pythagorean enharmonic scale
+* [[Pythagorean17]] – improper [[12L&nbsp;5s]]. Also known as the '''pythagorean enharmonic scale.'''
-* [[Pythagorean29]] – improper [[12L&nbsp;17s]]
+* [[Pythagorean29]] – improper [[12L&nbsp;17s]].
-* [[Pythagorean41]] – proper [[12L&nbsp;29s]]
+* [[Pythagorean41]] – proper [[12L&nbsp;29s]].
-* [[Pythagorean53]] – proper [[41L&nbsp;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.
+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 ==
 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&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&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.