Pythagorean tuning: Difference between revisions

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: {{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 ==

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 ==

@@ Line 24: / Line 24: @@
 : {{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–&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&nbsp;3s]], also known as the ''pythagorean pentic scale''.
+* [[Pythagorean5]] – proper [[2L&nbsp;3s]], also known as pentic, the ''pythagorean pentatonic scale''.
-* [[Pythagorean7]] – improper  [[5L&nbsp;2s]], also known as the ''pythagorean diatonic scale''.
+* [[Pythagorean7]] – improper [[5L&nbsp;2s]], also known as diatonic,the ''pythagorean diatonic scale''.
-* [[Pythagorean12]] – proper [[5L&nbsp;7s]], also known as the ''pythagorean chromatic scale''.
+* [[Pythagorean12]] – proper [[5L&nbsp;7s]], also known as p-chromatic, the ''pythagorean chromatic scale''.
-* [[Pythagorean17]] – improper [[12L&nbsp;5s]], also known as the ''pythagorean enharmonic scale''.
+* [[Pythagorean17]] – improper [[12L&nbsp;5s]], also known as p-enharmonic, the ''pythagorean enharmonic scale''.
-* [[Pythagorean29]] – improper [[12L&nbsp;17s]].
+* [[Pythagorean29]] – improper [[12L&nbsp;17s]], sometimes known as ''pythagotonic''.
-* [[Pythagorean41]] – proper [[12L&nbsp;29s]].
+* [[Pythagorean41]] – proper [[12L&nbsp;29s]], sometimes known as ''pythamystonic.''
-* [[Pythagorean53]] – proper [[41L&nbsp;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, 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 ==