Pythagorean tuning
"[3] Alternatively, it can be described as the tuning of the syntonic temperament[1] in which the generator is the ratio 3:2 (i.e., the untempered perfect fifth), which is ≈ 702 cents wide.
It is named, and has been widely misattributed, to Ancient Greeks, notably Pythagoras (sixth century BC) by modern authors of music theory.
Ptolemy, and later Boethius, ascribed the division of the tetrachord by only two intervals, called "semitonium" and "tonus" in Latin (256:243 × 9:8 × 9:8), to Eratosthenes.
Extended Pythagorean tuning corresponds 1-on-1 with western music notation and there is no limit to the number of fifths.
Starting from D for example (D-based tuning), six other notes are produced by moving six times a ratio 3:2 up, and the remaining ones by moving the same ratio down: This succession of eleven 3:2 intervals spans across a wide range of frequency (on a piano keyboard, it encompasses 77 keys).
This, as shown above, implies that only eleven just fifths are used to build the entire chromatic scale.
The remaining interval (the diminished sixth from G♯ to E♭) is left badly out-of-tune, meaning that any music which combines those two notes is unplayable in this tuning.
For example, a C-based Pythagorean tuning would produce a stack of fifths running from D♭ to F♯, making F♯-D♭ the wolf interval.
As shown in the table, the latter interval, although enharmonically equivalent to a fifth, is more properly called a diminished sixth (d6).
As an obvious consequence, each augmented or diminished interval is exactly 12ε (≈ 23.460) cents narrower or wider than its enharmonic equivalent.
[2][9] Because of the wolf interval when using a 12-tone Pythagorean temperament, this tuning is rarely used today, although it is thought to have been widespread.
In music which does not change key very often, or which is not very harmonically adventurous, the wolf interval is unlikely to be a problem, as not all the possible fifths will be heard in such pieces.
At the same time, syntonic-diatonic just intonation was posited first by Ramos and then by Zarlino as the normal tuning for singers.
Pythagorean temperament can still be heard in some parts of modern classical music from singers and from instruments with no fixed tuning such as the violin family.
Where a performer has an unaccompanied passage based on scales, they will tend towards using Pythagorean intonation as that will make the scale sound best in tune, then reverting to other temperaments for other passages (just intonation for chordal or arpeggiated figures, and equal temperament when accompanied with piano or orchestra).
