Pythagorean tuning
The Pythagorean tuning is the 3-limit version of just intonation. Pythagorean can be considered a trivial rank-2 temperament in the 2.3 subgroup, as 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) are extensions of pythagorean.
The Pythagorean temperament consists of all intervals generated by a just 3/2 and 2/1. Musically, the 2/1 is most often interpreted as an equave, and as such Pythagorean tuning mirrors the structure of the chain of fifths.
See 3-limit for more information.
History
Pythagorean tuning was not actually invented by Pythagoras. It was invented in Ancient Mesopotamia and 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.
Scales
Because Pythagorean tuning is a rank-2 temperament, the moment-of-symmetry scales generated by its fifth can be named the same way scales corresponding to other rank-2 temperaments are, as follows:
- Pythagorean5 – proper 2L 3s. Also known as pythagorean pentic scale
- Pythagorean7 – improper 5L 2s. Also known as pythagorean diatonic scale
- Pythagorean12 – proper 5L 7s. Also known as pythagorean chromatic scale
- Pythagorean17 – improper 12L 5s. Also known as pythagorean enharmonic scale
- Pythagorean29 – improper 12L 17s
- Pythagorean41 – proper 12L 29s
- Pythagorean53 – proper 41L 12s
The hardnesses of the Pythagorean scales are about 1.442 for pentic, 2.260 for diatonic, 1.260 for chromatic, and 3.846 for enharmonic.
Music
See 3-limit #Music.
See also
- 3-limit, the JI subgroup which pythagorean is the trivial temperament of
- Chain of fifths, a harmonic structure based on the concepts of Pythagorean tuning