Derivation
As described in the introduction, the Pythagorean comma may be derived in multiple ways:
- Difference between two enharmonically equivalent notes in a Pythagorean scale, such as C and B♯ ( Play), or D♭ and C♯ (see below).
- Difference between Pythagorean apotome and Pythagorean limma.
- Difference between twelve just perfect fifths and seven octaves.
- Difference between three Pythagorean ditones (major thirds) and one octave.
A just perfect fifth has a 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 ratio of any other note.
Apotome and limma are the two kinds of semitones defined in Pythagorean tuning. Namely, the apotome (about 113.69 cents, e.g. from C to C♯) is the chromatic semitone, or augmented unison (A1), while the limma (about 90.23 cents, e.g. from C to D♭) is the diatonic semitone, or minor second (m2).
A ditone (or major third) is an interval formed by two major tones. 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.
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