Pythagorean interval.html |
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| Names | Ratio | Cents | ET Cents | Midi |
|---|---|---|---|---|
| (perfect) unison | 1/1 | 0.00 | 0 | play |
| comma | 531441/524288 | 23.46 | 0 | play |
| minor second limma minor semitone |
256/243 | 90.22 | 100 | play |
| augmented unison apotome major semitone |
2187/2048 | 113.69 | 100 | play |
| diminished third | 65536/59049 | 180.45 | 200 | |
| major second tone |
9/8 | 203.91 | 200 | play |
| minor third semiditone |
32/27 | 294.13 | 300 | play |
| augmented second | 19683/16384 | 317.60 | 300 | |
| diminished fourth | 8192/6561 | 384.36 | 400 | |
| major third ditone |
81/64 | 407.82 | 400 | play |
| perfect fourth diatessaron sesquitertium |
4/3 | 498.04 | 500 | play |
| augmented third | 177147/131072 | 521.51 | 500 | |
| diminished fifth | 1024/729 | 588.27 | 600 | |
| augmented fourth tritone |
729/512 | 611.73 | 600 | |
| diminished sixth | 262144/177147 | 678.49 | 700 | |
| perfect fifth diapente sesquialterum |
3/2 | 701.96 | 700 | play |
| minor sixth | 128/81 | 792.18 | 800 | play |
| augmented fifth | 6561/4096 | 815.64 | 800 | |
| diminished seventh | 32768/19683 | 882.40 | 900 | |
| major sixth | 27/16 | 905.87 | 900 | play |
| minor seventh | 16/9 | 996.09 | 1000 | play |
| augmented sixth | 59049/32768 | 1019.55 | 1000 | |
| diminished octave | 4096/2187 | 1086.31 | 1100 | |
| major seventh | 243/128 | 1109.78 | 1100 | play |
| (perfect) octave diapason |
2/1 | 1200.00 | 1200 | play |
| augmented seventh octave + comma |
531441/262144 | 1223.46 | 1200 | play |
The intervals of Pythagorean tuning are just intervals involving only powers of two and three.
The fundamental intervals are the superparticular ratios 2/1, 3/2, and 4/3. 2/1 is the octave or diapason (Greek for "across all"). 3/2 is the perfect fifth, diapente ("across five"), or sesquialterum. 4/3 is the perfect fourth, diatessaron ("across four"), or sesquitertium. These three intervals and their octave equivalents, such as the perfect eleventh and twelfth, are the only absolute consonances of the Pythagorean system. All other intervals have varying degrees of dissonance, ranging from smooth to rough.
The difference between the perfect fourth and the perfect fifth is the tone or major second. This has the ratio 9/8, and it is the only other superparticular ratio of Pythagorean tuning, as shown by Størmer's theorem.
Two tones make a ditone, a dissonantly wide major third, ratio 81/64. The ditone differs from the just major third (5/4) by the syntonic comma (81/80). Likewise, the difference between the tone and the perfect fourth is the semiditone, a narrow minor third, 32/27, which differs from 6/5 by the syntonic comma. These differences are "tempered out" or eliminated by using compromises in meantone temperament.
The difference between the minor third and the tone is the minor semitone or limma of 256/243. The difference between the tone and the limma is the major semitone or apotome ("part cut off") of 2187/2048. Although the limma and the apotome are both represented by one step of 12-pitch equal temperament, they are not equal in Pythagorean tuning, and their difference, 531441/524288, is known as the Pythagorean comma.
There is a one-to-one correspondence between interval names (number of scale steps + quality) and frequency ratios. This contrasts with equal temperament, in which intervals with the same frequency ratio can have different names (e.g., the diminished fifth and the augmented fourth); and with other forms of just intonation, in which intervals with the same name can have different frequency ratios (e.g., 9/8 for the major second from C to D, but 10/9 for the major second from D to E).
play diatonic scale in Pythagorean tuning