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''' 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 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 ==

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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''' 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 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 ==