Pythagorean tuning

Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on on the ratio 3:2. Its discovery is generally credited to Pythagoras. It is the oldest way of tuning the 12-note chromatic scale and, as such, it is the basis for many other methods of tuning.

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 cents 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.

Wolf_fifth.ogg (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 Et_fifths.ogg (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.