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)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 - an augmented seventh - and an octave.

The '''Pythagorean comma''' or '''ditonic comma'''{{cn}} is the interval with the ratio '''531441/524288''' ([[monzo]]: {{monzo| -19 12 }}). It is the amount by which twelve [[3/2|perfect fifths]] exceed seven [[octave]]s, or in other words (3/2)12/27. It also can be written as the ratio between the Pythagorean 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 (i.e. an augmented seventh) and an octave.

~~== Importance in diatonic ==~~

In [[Pythagorean tuning]], this interval is an ''inverse'' diminished second. This is because adding Pythagorean commas makes the interval go up in pitch, down in [[5L 2s|diatonic]] [[degree]]s. This apparently counterintuitive notion is a result of just fifths naturally producing a [[TAMNAMS #Step ratio spectrum|hard-of-basic]] diatonic scale, which means that the [[chromatic semitone]] is wider, not narrower, than the [[diatonic semitone]].

In [[Pythagorean tuning]] ~~or tunings close to it~~, this interval is a ''~~negative~~'' 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 ==

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~~.

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.

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.

See [[Compton family]] for the [[family of 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 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 [[32805/32768|schisma]]. It is also fairly close to the [[64/63|septimal comma]], with the septimal comma exceeding the Pythagorean comma by the [[septimal schisma]]. Tempering out both the classical and septimal schismas leads to the [[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 (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]].

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 {{Wikipedia| Pythagorean comma }}
-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 - an augmented seventh - and an octave.
+The '''Pythagorean comma''' or '''ditonic comma'''{{cn}} is the interval with the ratio '''531441/524288''' ([[monzo]]: {{monzo| -19 12 }}). It is the amount by which twelve [[3/2|perfect fifths]] exceed seven [[octave]]s, or in other words (3/2)<sup>12</sup>/2<sup>7</sup>. It also can be written as the ratio between the Pythagorean 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 (i.e. an augmented seventh) and an octave.
-== Importance in diatonic ==
+In [[Pythagorean tuning]], this interval is an ''inverse'' diminished second. This is because adding Pythagorean commas makes the interval go up in pitch, down in [[5L 2s|diatonic]] [[degree]]s. This apparently counterintuitive notion is a result of just fifths naturally producing a [[TAMNAMS #Step ratio spectrum|hard-of-basic]] diatonic scale, which means that the [[chromatic semitone]] is wider, not narrower, than the [[diatonic semitone]].
-In [[Pythagorean tuning]] or tunings close to it, this interval is a ''negative'' 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 ==
-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.
+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.
-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.
+See [[Compton family]] for the [[family of 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 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 [[32805/32768|schisma]]. It is also fairly close to the [[64/63|septimal comma]], with the septimal comma exceeding the Pythagorean comma by the [[septimal schisma]]. Tempering out both the classical and septimal schismas leads to the [[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 (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]].

Pythagorean comma: Difference between revisions

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