Pythagorean comma

The Pythagorean comma is in the music, an interval of about one eighth-note, which is not used as an independent musical tonal step. While in today's more common equal temperament tuning seven octaves correspond exactly twelve fifths, there are in theory wohlklingendsten, so-called pure mood a difference between seven octaves and twelve fifths.

This difference is distributed in the same stage atmosphere on the twelve fifths. This gives a temperature at which the fifths (700 cents) differ only slightly from the perfect fifth (702 cents). However, the equal- thirds differ - and that is often overlooked - clearly audible from the pure thirds. This difference in thirds - the syntonic comma - is almost equal to the Pythagorean comma.

Practical Relevance receives the comma when tuning instruments with fixed pitches. These include, for instance keyboards and stringed instruments with frets.

Frequency ratio and size in cents

See: Structure of the interval space.

Since the addition or subtraction of intervals multiplied by the frequency ratios or be divided, the frequency ratio of the Pythagorean comma is calculated as:

The size in cents is calculated directly from the defining equation:

The Pythagorean comma as a problem when tuning of keyboard instruments

An instrument (such as the modern keyboard instruments ) that can generate per octave twelve different tones, can not agree so that it can be played in any key with absolutely pure intervals. On the one hand, there are different sized whole tones that differ by a syntonic comma, on the other hand, twelve fifths of seven octaves differ by the Pythagorean comma. In practice, tried to be as useful to distribute the syntonic and Pythagorean comma when tuning of keyboard instruments on all tones. According to various theories then provide the various musical moods. There have also been attempts to keyboard instruments, the octave over twelve tones comprises (e.g. divided black keys).

Twelve perfect fifths (2:3) yield 8423.46 cents, seven octaves, however, only 8400 cents. Thus, the fifth race to seven octaves includes the circle of fifths in equally- tempered tuning, which Pythagorean comma must be distributed when tuning to the twelve fifths. Thus, mere fifth of 701.9550 cents is reduced only slightly by 1.9550 cents to 700 cents.

Perfect fifth: Gleichstufige Quinte: 700 cents.

Pure major third: Gleichstufige major third: 400 cents.

History

As the first of the Pythagoreans Philolaus defined the Pythagorean comma. It was based on the mood of a lyre, and ordered ratios of string lengths quotient to:

The whole tone he explains the difference between fourth and fifth. Since the addition of intervals, the multiplication and subtraction corresponds to the division of the associated conditions, the following calculation:

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

Two Pythagorean semitones arise together but still no whole tone. The difference is defined as Philolaus ( Pythagorean ) comma.

Although Philolaos defines the whole tone and the little sharp ( of him as diesis called, later called Limma ), but does not calculate the associated relations. The first mention of the comma proportion 531441:524288 found in Euclid. He notes that six whole tones form a larger interval than an octave. The difference is the Pythagorean comma again.