Pythagorean tuning: Difference between revisions

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.

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

: {{dash| G♭, D♭, A♭, E♭, B♭, F, C, G, D, A, E, B }}.

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.

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

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

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

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

* [[Pythagorean17]] – improper [[12L&nbsp;5s]], also known as p-enharmonic, the ''pythagorean enharmonic scale''.

* [[Pythagorean29]] – improper [[12L&nbsp;17s]], sometimes known as ''pythagotonic''.

* [[Pythagorean41]] – proper [[12L&nbsp;29s]], sometimes known as ''pythamystonic.''

* [[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, 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.

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