Editing Pythagorean comma (section)

==Derivation==
As described in the introduction, the Pythagorean comma may be derived in multiple ways:

* Difference between two [[enharmonic|enharmonically equivalent]] notes in a Pythagorean scale, such as C and B{{Music|#}}, or D{{Music|b}} and C{{Music|#}} (see [[Pythagorean comma#Circle of fifths and enharmonic change|below]]).
* Difference between [[Pythagorean apotome]] and [[Pythagorean limma]].
* Difference between 12 just [[perfect fifth]]s and seven [[Perfect octave|octaves]].
* Difference between three Pythagorean [[ditone]]s ([[major third]]s) and one octave.

A just perfect fifth has a [[Interval ratio|frequency ratio]] of 3:2. It is used in Pythagorean tuning, together with the octave, as a yardstick to define, with respect to a given initial note, the frequency of any other note.

Apotome and limma are the two kinds of [[semitone]]s defined in Pythagorean tuning. Namely, the apotome (about 113.69 cents, e.g. from C to C{{Music|#}}) is the chromatic semitone, or augmented unison (A1), while the limma (about 90.23 cents, e.g. from C to D{{Music|b}}) is the diatonic semitone, or minor second (m2).

A ditone (or [[major third]]) is an interval formed by two [[major tone]]s. In Pythagorean tuning, a major tone has a size of about 203.9 cents (frequency ratio 9:8), thus a Pythagorean ditone is about 407.8 cents.
{| style="margin-left: auto; margin-right: auto; border: none;"
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|{{Wide image|Octaves versus fifths Cuisenaire rods Pythagorean.png|1600px|Octaves (7 × 1200 {{=}} 8400) versus fifths (12 × 701.96 {{=}} 8423.52), depicted as with [[Cuisenaire rods]] (red (2) is used for 1200, black (7) is used for 701.96).|500px|center}}
|[[File:Octaves versus major thirds Cuisenaire rods Pythagorean.png|thumb|center|Octaves (1 × 1200 {{=}} 1200) versus ditones (3 × 407.82 {{=}} 1223.46), depicted as with Cuisenaire rods (red (2) is used for 1200, magenta (4) is used for 407.82).]]
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