Pythagorean interval

Names	Ratio	Cents	ET Cents
(perfect) unison	1/1	0.00	0
comma	531441/524288	23.46	0
minor second limma minor semitone	256/243	90.22	100
augmented unison apotome major semitone	2187/2048	113.69	100
diminished third	65536/59049	180.45	200
major second tone	9/8	203.91	200
minor third semiditone	32/27	294.13	300
augmented second	19683/16384	317.60	300
diminished fourth	8192/6561	384.36	400
major third ditone	81/64	407.82	400
perfect fourth diatessaron sesquitertium	4/3	498.04	500
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
minor sixth	128/81	792.18	800
augmented fifth	6561/4096	815.64	800
diminished seventh	32768/19683	882.40	900
major sixth	27/16	905.87	900
minor seventh	16/9	996.09	1000
augmented sixth	59049/32768	1019.55	1000
diminished octave	4096/2187	1086.31	1100
major seventh	243/128	1109.78	1100
(perfect) octave diapason	2/1	1200.00	1200
augmented seventh octave + comma	531441/262144	1223.46	1200

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).

Pythagorean interval

See also

External links