Pythagorean tuning: Difference between revisions
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{{Wikipedia|Pythagorean tuning}} | |||
'''Pythagorean tuning''' is a system where all intervals are determined by perfect fifths tuned to [[3/2]] and [[2/1|octaves]]. As such, Pythagorean tuning contains the same intervals as [[3-limit]] [[just intonation]], | |||
[[ | When accounting for octave equivalence, Pythagorean tuning mirrors the structure of the [[chain of fifths]]. | ||
== History == | |||
& | {{wikipedia|Music of Mesopotamia#Music theory}} | ||
& | {{wikipedia|Shi'er lü}} | ||
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''' (十二律).{{clear}} | |||
== Relation to temperaments == | |||
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– 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|MOS]] scales: | |||
* [[Pythagorean5]] – proper [[2L 3s]], also known as pentic, the ''pythagorean pentatonic scale''. | |||
* [[Pythagorean7]] – improper [[5L 2s]], also known as diatonic,the ''pythagorean diatonic scale''. | |||
* [[Pythagorean12]] – proper [[5L 7s]], also known as p-chromatic, the ''pythagorean chromatic scale''. | |||
* [[Pythagorean17]] – improper [[12L 5s]], also known as p-enharmonic, the ''pythagorean enharmonic scale''. | |||
* [[Pythagorean29]] – improper [[12L 17s]], sometimes known as ''pythagotonic''. | |||
* [[Pythagorean41]] – proper [[12L 29s]], sometimes known as ''pythamystonic.'' | |||
* [[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, 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 == | |||
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. | |||
== Music == | |||
See [[3-limit #Music]]. | |||
== See also == | |||
* [[3-limit]], the corresponding JI subgroup. | |||
* [[Chain of fifths]], a harmonic structure based on the concepts of Pythagorean tuning. | |||
== References == | |||
<references/> | |||
[[Category:3-limit| ]] <!-- main article --> | |||
[[Category:Rank-2 temperaments]] | |||
[[Category:Historical]] | |||
[[Category:Tuning]] | |||
[[Category:Listen]] | |||