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. 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.
Scales
Because Pythagorean tuning is a rank-2 system, 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