# Pythagorean comma

Diatonic intervals
Prime
second
third
fourth
fifth
sixth
seventh
octave
none
decime
undezime
duodecime
tredezime
semitone / whole tone
Special intervals
Microinterval
Comma
Diësis
Limma
Apotome
Ditone Tritone
Wolf
fifth
Natural septime
units
Cent
Millioctave
Octave
Savart

In music, the Pythagorean comma is an interval of about an eighth tone (23.46 cents ), which is not used as an independent musical step. While seven (pure) octaves correspond to exactly twelve (equal) fifths in the more common equal tuning today , there is a difference between seven (pure) octaves and twelve (pure) fifths in the early Pythagorean tuning (or also with the pure tuning ).

By definition: Pythagorean comma = 12 fifths - 7 octaves.

This difference is evenly distributed over the twelve fifths in the equal tuning. A tempering is obtained in which these equal fifths (700 cents) differ only insignificantly from the pure fifths (702 cents). However, the equal thirds (300 or 400 cents) - and this is often overlooked - differ from the pure thirds (315.5 and 386.5 cents). The syntonic comma , the difference between the Pythagorean and the pure third (408 - 386.5 = 21.5 cents) is almost the same as the Pythagorean comma.

The comma is of practical relevance when tuning instruments with fixed pitches. This includes, for example, keyboard instruments and string instruments with frets .

## Size and frequency ratio

The size of the Pythagorean comma is calculated from the definition equation:

Pythagorean comma = 12 fifths - 7 octaves 23.46 cents .${\ displaystyle \ approx}$

Since the frequency ratios are multiplied or divided when adding or subtracting intervals, the frequency ratio of the Pythagorean comma is calculated as:

${\ displaystyle {\ frac {\ left ({\ frac {3} {2}} \ right) ^ {12}} {2 ^ {7}}} = {\ frac {3 ^ {12}} {2 ^ {19}}} \ approx {1 {,} 01364}.}$

## The Pythagorean comma as a problem when tuning keyboard instruments

An instrument (like modern keyboard instruments) that produces only twelve different tones per octave cannot be tuned so that it can be played in all keys with absolutely pure intervals.

Twelve perfect fifths (frequency ratio 3: 2) result in 8423.46 cents , while seven octaves are only 8400 cents. The difference of 23.46 cents is called the Pythagorean comma. Four perfect fifths give the Pythagorean major third with 407.82 cents, while the pure major third is only 386.31 cents. The difference of 21.51 cents is called the syntonic comma .

The Pythagorean tuning was used in Gregorian chant and music up until the late Middle Ages . The Pythagorean major third resulting from the Pythagorean tuning played no role in one- or two-part (fifths, fourths) music. With the advent of the chord connections formed in polyphony, the pure major third with the frequency ratio of 5: 4 was soon recognized as a consonance . This made the Pythagorean mood unusable. For a long time, mid-tone tunings were used , which reproduced the pure major third exactly at the expense of the fifths, but excluded many keys. During JS Bach's time, the need to be able to play in all keys grew. Countless attempts with well-tempered tunings , which tried to make the major thirds sound as pure as possible in keys close to C major, or with keyboard instruments whose octaves comprised more than twelve tones (e.g. through divided keys), has now become a reality the equal mood prevailed almost throughout .

The fifths of equal tuning differ from those of the pure or Pythagorean tuning by only 2 cents; the major third, 14 cents too high compared to the pure major third, is inevitably accepted as "sharpened".

Perfect fifth:, Equal fifth: 700 cents. ${\ displaystyle 1200 \ cdot \ log _ {2} \ left ({3 \ over 2} \ right) \; \ mathrm {Cent} \ approx 701 {,} 96 \; \ mathrm {Cent}}$

Pure major third:, Major third of equal order: 400 cents. ${\ displaystyle 1200 \ cdot \ log _ {2} \ left ({5 \ over 4} \ right) \; \ mathrm {Cent} \ approx 386 {,} 31 \; \ mathrm {Cent}}$

## history

The Pythagorean Philolaos was the first to define the Pythagorean comma. He based himself on the tuning of a lyre and assigned ratios of string lengths to quotients:

${\ displaystyle {\ frac {2} {1}}}$for the octave, for the fifth and for the fourth${\ displaystyle {\ frac {3} {2}}}$${\ displaystyle {\ frac {4} {3}}}$

He explains the whole tone as the difference between a fourth and a fifth. Since the addition of intervals corresponds to the multiplication and the subtraction corresponds to the division of the associated ratios, the following calculation results:

The frequency ratio = corresponds to the whole tone = fifth - fourth .${\ displaystyle {\ frac {3} {2}}: {\ frac {4} {3}} = {\ frac {9} {8}}}$

Philolaos now defines the (small) semitone as the difference between a fourth and two whole tones.

The (small) semitone = fourth - 2 whole tone corresponds to the frequency ratio .${\ displaystyle {\ frac {4} {3}}: \ left ({\ frac {9} {8}} \ right) ^ {2} = {\ frac {256} {243}}}$

However, two Pythagorean semitones do not add up to a whole tone. Philolaos defines the difference as a (Pythagorean) comma.

The frequency ratio corresponds to the Pythagorean comma = whole tone - 2 · (smaller) semitone .${\ displaystyle {\ frac {9} {8}}: \ left ({\ frac {256} {243}} \ right) ^ {2} = {\ frac {531441} {524288}} = \ left ({ \ frac {3} {2}} \ right) ^ {12}: 2 ^ {7}}$

Philolaos defines the whole tone and the small semitone (he called Diesis , later called Limma ), but does not calculate the corresponding ratios. The first mention of the comma proportion 531441: 524288 can be found in Euclid . He notes that 6 whole tones form a larger interval than an octave. The difference is again the Pythagorean comma.

According to this definition, the frequency ratio also corresponds to the Pythagorean comma = 6 whole tone octave .${\ displaystyle \ left ({\ frac {9} {8}} \ right) ^ {6}: 2 = {\ frac {531441} {524288}}}$

## literature

• Euclid: Katatome kanonos (Latin Sectio canonis ). Engl. Transl. In: Andrew Barker (Ed.): Greek Musical Writings. Vol. 2: Harmonic and Acoustic Theory , Cambridge Mass .: Cambridge University Press, 2004, pp. 190–208, here: p. 199.
• Hermann Diels: The fragments of the pre-Socratics , 1st volume. 2nd Edition. Weidmannsche Buchhandlung, Berlin 1906