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Inmusical tuning theory, aPythagorean interval is amusical interval with afrequency ratio equal to apower of two divided by a power of three, orvice versa.^[1] For instance, theperfect fifth with ratio 3/2 (equivalent to 3¹/ 2¹) and theperfect fourth with ratio 4/3 (equivalent to 2²/ 3¹) are Pythagorean intervals.

All the intervals between the notes of a scale are Pythagorean if they are tuned using thePythagorean tuning system. However, some Pythagorean intervals are also used in other tuning systems. For instance, the above-mentioned Pythagorean perfect fifth and fourth are also used injust intonation.

Notice that the termsditone andsemiditone are specific for Pythagorean tuning, whiletone andtritone are used generically for all tuning systems. 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).

The table shows from which notes some of the above listed intervals can be played on an instrument using a repeated-octave 12-tone scale (such as a piano) tuned with D-based symmetric Pythagorean tuning. Further details about this table can be found inSize of Pythagorean intervals.

The fundamental intervals are thesuperparticular ratios 2/1, 3/2, and 4/3. 2/1 is theoctave ordiapason (Greek for "across all"). 3/2 is theperfect fifth,diapente ("across five"), orsesquialterum. 4/3 is theperfect fourth,diatessaron ("across four"), orsesquitertium. These three intervals and their octave equivalents, such as the perfect eleventh and twelfth, are the only absoluteconsonances 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 thetone ormajor second. This has the ratio 9/8, also known asepogdoon and it is the only other superparticular ratio of Pythagorean tuning, as shown byStørmer's theorem.

Two tones make aditone, a dissonantly widemajor third, ratio 81/64. The ditone differs from the just major third (5/4) by thesyntonic comma (81/80). Likewise, the difference between the tone and the perfect fourth is thesemiditone, a narrowminor third, 32/27, which differs from 6/5 by the syntonic comma. These differences are "tempered out" or eliminated by using compromises inmeantone temperament.

The difference between the minor third and the tone is theminor semitone orlimma of 256/243. The difference between the tone and the limma is themajor semitone orapotome ("part cut off") of 2187/2048. Although the limma and the apotome are both represented by one step of 12-pitchequal temperament, they are not equal in Pythagorean tuning, and their difference, 531441/524288, is known as thePythagorean 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).

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• Just intonation
• List of meantone intervals
• List of intervals in 5-limit just intonation
• Shí-èr-lǜ
• Whole-tone scale

• Neo-Gothic usage by Margo Schulter

• 3-limit tuning and intervals

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