Pythagorean comma

You can arrive at this ratio as follows: starting with 1, get the next term by multiplying the previous term by 3/2 and then again by 1/2 if the result is greater than 1 (except at the very end.) This gives you:

                 c   1
                 g   3/2
'g  3/4
                 d   9/8
'd  9/16
                 a   27/32
                 e   81/64
'e  81/128
                 b   243/256
                 f#  729/512
'f# 729/1024
                 c#  2187/2048
'c# 2187/4096
                 g#  6561/8192
                 d#  19683/16384
'd# 19683/32768
                 a#  59049/65536
                 f   177147/131072
'f  177147/262144
                 c   531441/524288

Essentially, the value represents the amount of disharmony to be distributed around the 12-tone musical scale. It amounts to about 1/55 of an octave, or, in the modern nomenclature, about 23 cents (0.23 of a semitone in equal temperament).

Different tuning systems or intonations may be characterised by where they put this extra 23 cents: the pythagorean tuning (or temperament) hides it all in one fifth (the so-called "wolf fifth" or "wolf tone"), which has the effect of making keys harmonically distant from that particular fifth sound very concordant and harmonically close ones discordant.

Other systems, like just intonation and mean temperament distribute the disharmony using an uneven sequence of rational intervals (ie ratios composed of two integers) between successive semitones, while the modern equal temperament distributes it evenly over all 12 notes, so that each successive note (going up in semitones) has the same ratio to its successor as its predecessor has to it, namely 1::2^1/12, making all keys equally discordant.

The Pythagorean Comma
(The enigma of tuning keyboard instruments)

On a keyboard instrument, an octave is the interval created by any two notes with 11 successive notes (semi-tones) between them. The frequencies, in Hertz, of the two notes in an octave constitute a ratio of 2:1, and there are 12 notes in an octave. A fifth is the interval created by any two notes with 6 successive notes between them. The frequencies, in Hertz, of the two notes in a fifth constitute a ratio of 3:2 (or 1.5:1), and there are 7 notes in a fifth.

A fifth is arguably the most musically pleasing interval, and its importance in musical composition cannot be over-stated. The sharing of overtones produced by the two notes in a pure fifth (with a true ratio of 3:2) is a main reason but not the only one. An octave must maintain its purity (with a true ratio of 2:1) so that a piece of music can move seamlessly among octaves and can also change key signatures. The octave and the fifth are both vital to Western music, but Pythagoras said we could not have both at the same time.

For illustration, a keyboard with 85 keys is appropriate because the Lowest Common Multiple of 7 (the number of notes in a fifth) and 12 (the number of notes in an octave) is 84. The 85th key is necessary to complete the last interval. Let us start with an A of 55 Hertz in frequency. Using the octave's ratio of 2:1, we come up with an A of 7040 Hertz as the 85th note. Using the fifth's ratio of 3:2, we come up with an A of 7136 Hertz as the same 85th note. It is thus apparent that octaves and fifths progress at different mathematical progressions, and that it is physically impossible to accommodate both pure octaves and pure fifths on the same keyboard. The discrepancy between the two mathematical progressions is expressed in a constant ratio of 1.0136:1, and is called the Pythagorean Comma because Pythagoras had devoted much attention to the phenomenon.

Nearly all Western music utilizes more than one octave and often also employs more than one key signature, so the reconciliation of the Pythagorean Comma obviously lies in maintaining the purity of the octaves but compromising the purity of the fifths and that of the other intervals. Different methods have been used in tuning keyboard instruments throughout the history of Western music; the principal ones were the Pythagorean Scale, the Diatonic Scale, the Meantone temperament, the genre of Well-tempered tunings, the Salinas 1/3 Comma, and the Werckmeister III (Circular) Temperament. The Pythagorean Scale gave priority to the fifths. The other methods favored the important key signatures and intervals at the expense of the less important ones, thus rendering music written in different key signatures with distinctly different moods and characteristics, and afforded dramatic effects when a piece of music modulates from one key signature to another.

Modern keyboard instruments are tuned to the Equal Temperament, which dictates the frequency ratio between any two notes to be1.0595:1, the result of dividing the interval of an octave into twelve equal intervals (the 12th root of 2). This mathematical and mechanical process ensures the equality of all key signatures: the octaves are pure but other intervals are all tempered with to the same extent.

Without the opportunity to compare, we have now come to accept Equal Temperament as the standard. Seldom do we stop to ponder how much more effectively a Beethoven Piano Sonata, for example, could touch our hearts when performed on a piano tuned to a temperament of his era.