Pythagorean comma: Difference between revisions
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The '''Pythagorean comma''' or '''ditonic comma''' is the interval with the ratio '''531441/524288''' ([[monzo]]: {{monzo| -19 12 }}). It is the amount by which twelve [[3/2|fifths]] exceed seven [[2/1|octaves]], or in other words (3/2)<sup>12</sup>/2<sup>7</sup>. 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. | The '''Pythagorean comma''' or '''ditonic comma''' is the interval with the ratio '''531441/524288''' ([[monzo]]: {{monzo| -19 12 }}). It is the amount by which twelve [[3/2|fifths]] exceed seven [[2/1|octaves]], or in other words (3/2)<sup>12</sup>/2<sup>7</sup>. 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. | ||
In [[ | In [[Pythagorean tuning]] or tunings close to it, this interval is an ''inverse'' diminished second. This is because adding Pythagorean commas makes the interval go up in pitch, down the scale. This apparently counterintuitive notion is a result of just fifths naturally producing a [[TAMNAMS #Step ratio spectrum|hard-of-basic]] [[5L 2s|diatonic]] scale, which means that the [[chromatic semitone]] is wider, not narrower, than the [[diatonic semitone]]. | ||
== Temperaments == | == Temperaments == | ||
If the | If the Pythagorean comma is [[tempering out|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 [[5/1|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{{c}}, such as [[41edo]] and [[53edo]], map the | Edos with a fifth sharper than the 12edo fifth of 700{{c}}, 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 [[81/80|syntonic comma]], only exceeding it by a [[schisma]]. It is also fairly close to the [[64/63|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 one of the most intuitive [[7-limit]] interpretations of the Pythagorean chain of fifths. | ||
Edos with a fifth flatter than the 12edo fifth, such as [[19edo]] and [[31edo]], map the | 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 (scale theory)|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: | Since it is reached by 12 fifths, a highly composite number, there are many temperaments that split this comma whilst keeping fifths unsplit. Notably: | ||
Revision as of 15:14, 9 April 2026
| Interval information |
ditonic comma
reduced harmonic
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.
In Pythagorean tuning or tunings close to it, this interval is an inverse diminished second. This is because adding Pythagorean commas makes the interval go up in pitch, down the scale. This apparently counterintuitive notion is a result of just fifths naturally producing a hard-of-basic diatonic scale, which means that the chromatic semitone is wider, not narrower, than the 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 one of the most intuitive 7-limit interpretations 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