Pythagorean Intervals
Four of the above mentioned intervals take a specific name in Pythagorean tuning. In the following table, these specific names are provided, together with alternative names used generically for some other intervals. Notice that the Pythagorean comma does not coincide with the diminished second, as its size (524288:531441) is the reciprocal of the Pythagorean diminished second (531441:524288). Also ditone and semiditone are specific for Pythagorean tuning, while tone and tritone are used generically for all tuning systems. Interestingly, despite its name, a semiditone (3 semitones, or about 300 cents) can hardly be viewed as half of a ditone (4 semitones, or about 400 cents). All the intervals with prefix sesqui- are justly tuned, and their frequency ratio, shown in the table, is a superparticular number (or epimoric ratio). The same is true for the octave.
| Number of semitones |
Generic names | Specific names | |||||
|---|---|---|---|---|---|---|---|
| Quality and number | Other naming conventions | Pythagorean tuning | 5-limit tuning | 1/4-comma meantone |
|||
| Full | Short | ||||||
| 0 | comma | Pythagorean comma (524288:531441) |
diesis (128:125) | ||||
| 0 | diminished second | d2 | (531441:524288) | ||||
| 1 | minor second | m2 | semitone, half tone, half step |
diatonic semitone, minor semitone |
limma (256:243) | ||
| 1 | augmented unison | A1 | chromatic semitone, major semitone |
apotome (2187:2048) | |||
| 2 | diminished third | d3 | tone, whole tone, whole step | ||||
| 2 | major second | M2 | sesquioctavum (9:8) | ||||
| 3 | minor third | m3 | semiditone (32:27) | sesquiquintum (6:5) | |||
| 4 | major third | M3 | ditone (81:64) | sesquiquartum (5:4) | |||
| 5 | perfect fourth | P4 | diatessaron | sesquitertium (4:3) | |||
| 6 | diminished fifth | d5 | tritone | ||||
| 6 | augmented fourth | A4 | |||||
| 7 | perfect fifth | P5 | diapente | sesquialterum (3:2) | |||
| 12 | (perfect) octave | P8 | diapason | duplex (2:1) | |||
Read more about this topic: Pythagorean Tuning
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