Pythagorean tuning

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Pythagorean tuning

Pythagorean tuning is a system where all intervals are determined by perfect fifths tuned to 3/2 and 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

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Music of Mesopotamia#Music theory

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Shi'er lü

Pythagorean tuning was not actually invented by Pythagoras. The earliest records are from Ancient Mesopotamia, and it was later inherited by the Ancient Greeks.

The 12-tone form of Pythagorean tuning was (probably independently) invented in Ancient China between 600 BCE and 240 CE, where it was called shi'er lü (十二律).

Relation to temperaments

Pythagorean tuning can be considered a 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 tuning.

A series of just fifths can also be considered a reasonable tuning of the schismatic temperament, where the 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, with keyboards tuned to nearly pure fifths as

G♭ – D♭ – A♭ – E♭ – B♭ – F – C – G – D – A – E – B.

This makes triads such as D – G♭ – A (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.

Scales

Pythagorean tuning generates the following 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 hardnesses 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 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.