Pythagorean tuning
Pythagorean tuning is a system where all intervals are determined by pure fifths and octaves. This makes it essentially the same as 3-limit just intonation.
When accounting for octave equivalence, Pythagorean tuning mirrors the structure of the chain of fifths.
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.
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 temperament.
Because the schisma is so small, 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. Mark Lindley[1] 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.
When respelled enharmonically, triads such as D – F♯ – A are close to 4:5:6 in this tuning.
Scales
Pythagorean tuning generates the following MOS scales:
- 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.
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.
References
- ↑ Mark Lindley, Pythagorean Intonation and the Rise of the Triad, Royal Musical Association Research Chronicle, 1980