Pythagorean tuning is based on a stack of perfect fifths, each tuned in the ratio 3:2, the next simplest ratio after 2:1, which is the ratio of an octave. The two notes A and D, for example, are tuned so that their frequencies are in the ratio 3:2 - if D is tuned to 200Hz, then the A is tuned to 300Hz. The E a fifth above that A is also tuned in the ratio 3:2 - with the A at 300Hz, this puts the E at 450Hz, a 9:4 above the original D. When describing tunings, it is usual to speak of all notes as being within an octave of each other, and as this E is over an octave above the original D, it is usual to halve its frequency to move it down an octave. Therefore, the E is tuned to 225Hz, a 9:8 above the D. The B a 3:2 above that E is tuned to the ratio 27:16 and so on, until the starting note, D, is arrived at again.
In applying this tuning to the chromatic scale, however, a problem arises: no number of 3:2s will fit exactly into an octave. Because of this, the D arrived at after twelve fifths have been tuned up is about a quarter of a semitone sharper than the D used to begin the process. The below table (starting at E flat rather than D) illustrates this, showing the note name, the ratio above D, and the value in centss above the D for each note in the chromatic scale. The cent values of the same notes in equal temperament are also given for comparison (marked in the table below as "et-Cents").
In order to keep the ratios in this table relatively simple, fifths are tuned down from D as well as up. The first note in the circle of fifths given here is E flat (equivalent to D#), from which five perfect fifths are tuned before arriving at D, the nominal unison note.
Note Ratio Cents et-Cents Interval Eb 256:243 90.22 100 minor second Bb 128:81 792.18 800 minor sixth F 32:27 294.13 300 minor third C 16:9 996.09 1000 minor seventh G 4:3 498.04 500 perfect fourth D 1:1 0 0 unison A 3:2 701.96 700 perfect fifth E 9:8 203.91 200 major second B 27:16 905.87 900 major sixth F# 81:64 407.82 400 major third C# 243:128 1109.78 1100 major seventh G# 729:512 611.73 600 augmented fourth [D#][2187:2048] [113.69] [100] [augmented unison]In equal temperament, and most other modern tunings of the chromatic scale, E flat and D sharp are thought of as the same note - however, as the above table indicates, in Pythagorean tuning, they theoretically have different ratios, and are at a different frequency. This discrepency, of about 23.5 cents, or one quarter of a semitone, is known as a Pythagorean comma.
To get around this problem, Pythagorean tuning uses the above 12 notes from E flat to G sharp shown above, and then places above the G sharp another E flat, starting the sequence again. This leaves the interval G#-Eb sounding badly out of tune, meaning that any music which combines those two notes is unplayable in this tuning. A very out of tune interval such as this one is known as a wolf interval. In the case of Pythagorean tuning, all the fifths are 701.96 cents wide, in the exact ratio 3:2, except the wolf fifth, which is only 678.49 cents wide, nearly a quarter of a semitone flatter.
(33.1KB) is a sound file demonstrating this out of tune fifth. The first two fifths are perfectly tuned in the ratio 3:2, the third is the G#-Eb wolf fifth. It may be useful to compare this to (38.2KB), which is the same three fifths tuned in equal temperament, each of them tolerably well in tune.
If the notes G# and Eb need to be sounded together, the position of the wolf fifth can be changed (for example, the above table could run from A to E, making that the wolf interval instead of Eb to G#). However, there will always be one wolf fifth in Pythagorean tuning, making it impossible to play in all keys in tune.
Because of the wolf interval, this tuning is rarely used nowadays, although it is thought it was once 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.
Because fifths in Pythagorean tuning are in the simple ratio of 3:2, they sound very "smooth" and consonant. The thirds, by contrast, which are in the relatively complex ratios of 81:64 (for major thirds) and 32:27 (for minor thirds), sound less smooth. For this reason, Pythagorean tuning is particularly well suited to music which treats fifths as consonances, and thirds as dissonances. In classical music, this usually means music written prior to the 16th century. As thirds became to be treated as consonances, so meantone temperament, and particularly quarter comma meantone, which tunes thirds to the relatively simple ratio of 5:4, became more popular. However, meantone still has a wolf interval, so is not suitable for all music.
From around the 18th century, the need for instruments to change key widely, and therefore to avoid a wolf interval, led to the widespread use of well temperaments and eventually equal temperament.