Pythagorean comma

The Pythagorean comma or ditonic comma is the interval with the ratio 531441/524288 (monzo: [-19 12⟩). It is the amount by which twelve fifths exceed seven octaves, or in other words (3/2)¹²/2⁷. It also can be written as the ratio between the apotome and limma, (2187/2048)/(256/243), and as the ratio between the Pythagorean augmented fourth and the Pythagorean diminished fifth, (729/512)/(1024/729). In addition, it is also the difference between six 9/8 major seconds and an octave.

Interval information

Ratio	531441/524288
Factorization	2-19 × 312
Monzo	[-19 12⟩
Size in cents	23.46001¢
Names	Pythagorean comma, ditonic comma
Color name	LLw-2, Lalawa comma
FJS name	[math]\displaystyle{ \text{d}{-2} }[/math]
Special properties	reduced, reduced harmonic
Tenney norm (log2 nd)	38.0196
Weil norm (log2 max(n, d))	38.0391
Wilson norm (sopfr(nd))	74
Comma size	small
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Pythagorean comma

In pythagorean tuning or tunings close to it, this interval is a negative diesis. This is because adding adding pythagorean commas makes the interval go up in pitch, down the scale.

This counterintuitive notion is a result of just fifths naturally producing a hard diatonic MOS, which means that the chroma (chromatic semitone) is wider, not narrower, than the small step (diatonic semitone).

Temperaments

If the pythagorean comma is tempered out, then the circle of fifths closes at 12 notes. This circle of fifths covers the entirety of 12edo, while larger multiples of 12edo such as 24edo and 72edo contain multiple such circles. If one takes this circle of fifths and adds an independent generator for prime 5, this leads to the 5-limit rank-2 compton temperament. See Compton family for the family of rank-2 temperaments where it is tempered out.

Edos with a fifth sharper than the 12edo fifth of 700 ¢, such as 41edo and 53edo, map the pythagorean comma to a positive small number of steps rather than tempering it out. The pythagorean comma is quite close to the syntonic comma, only exceeding it by a schisma. It is also fairly close to the septimal comma, with the septimal comma exceeding the pythagorean comma by the garischisma. Tempering out both the schisma and the garischisma leads to garibaldi temperament, which is the simplest 7-limit interpretation of the pythagorean chain of fifths.

Edos with a fifth flatter than the 12edo fifth, such as 19edo and 31edo, map the pythagorean comma negatively, and thus have a positive diminished second (also known as a diesis). The majority of these edos support meantone, which equates the pythagorean major third 81/64 to the 5-limit major third 5/4.

Since it is reached by 12 fifths, a highly composite number, there are many temperaments that split this comma whilst keeping fifths unsplit. Notably:

• Kalismic, splitting it into 2 fwiwismas.
• Landscape, splitting it into 3 marvel commas.
• Nexus, splitting it into 3 rastmas.
• Atomic, splitting it into 12 schismas.

See also

• Mercator's comma, the difference between 53 perfect fifths and 31 octaves
• Gallery of just intervals
• Small comma