Pythagorean tuning
The Pythagorean tuning is the 3-limit version of just intonation.
See 3-limit for more information.
History
Pythagorean tuning is a system of musical tuning based on the mathematical ratios of pitches. It is named after the ancient Greek philosopher Pythagoras, who, according to legend, discovered the foundational principles of this tuning system through an experiment with hammers of different weights. Pythagoras' fascination with numerical ratios and their relation to the cosmos, particularly his concept of the 'music of the spheres', significantly influenced this tuning method.
The Greeks used two systems of tuning based on ideal integer ratios: Pythagorean and Ptolemaic. The major difference is, Ptolemaic tuning uses simpler ratios, where as Pythagorean tuning uses a chain of fifths and fourths. For example, a major third in Pythagorean would be 81/64 where as in Ptolemaic it is 5/4. Later music theorists, such as Gioseffo Zarlino[1]during the Renaissance, would prefer the Ptolemaic tuning. Tuning systems based on those ratios are called just intonation.
Pythagorean tuning was developed using method called the 'chain of fifths', where you multiply the pitch/frequency by a fifth (3/2) until you pass an octave. When you pass an octave, you take that same note, and move it down an octave by multiplying it by another ratio. Every ratio can be generated by a combination of 3/2 and 4/3. One old account of this method is ascribed to an anonymous source in a book by Iacobus de Ispania in the 13th century)[2]
Scales
- Pythagorean5 - proper 2L 3s. Also known as pythagorean pentatonic 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
Music
See Also
- ↑ Chisholm, Hugh (1911). The Encyclopædia Britannica, Vol. 28, p. 961. The Encyclopædia Britannica Company.
- ↑ Schulter, Margo “Pythagorean Tuning and Medieval Polyphony"