Pythagorean interval

In musical tuning theory, a Pythagorean interval is a musical interval with a frequency ratio equal to a power of two divided by a power of three, or vice versa.[1] For instance, the perfect fifth with ratio 3/2 (equivalent to 31/ 21) and the perfect fourth with ratio 4/3 (equivalent to 22/ 31) are Pythagorean intervals.

Pythagorean perfect fifth on C Play: C-G (3/2 ÷ 1/1 = 3/2).

All the intervals between the notes of a scale are Pythagorean if they are tuned using the Pythagorean 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 in just intonation.

Interval table

NameShortOther name(s)RatioFactorsDerivationCentsET
Cents
MIDI fileFifths
diminished secondd2524288/531441219/312-23.4600play-12
(perfect) unisonP11/130/201/10.0000play0
Pythagorean comma531441/524288312/21923.4600play12
minor secondm2limma,
diatonic semitone,
minor semitone
256/24328/3590.225100play-5
augmented unisonA1apotome,
chromatic semitone,
major semitone
2187/204837/211113.685100play7
diminished thirdd3tone,
whole tone,
whole step
65536/59049216/310180.450200play-10
major secondM29/832/233·3/2·2203.910200play2
semiditonem3(Pythagorean minor third)32/2725/33294.135300play-3
augmented secondA219683/1638439/214317.595300play9
diminished fourthd48192/6561213/38384.360400play-8
ditoneM3(Pythagorean major third)81/6434/2627·3/32·2407.820400play4
perfect fourthP4diatessaron,
sesquitertium
4/322/32·2/3498.045500play-1
augmented thirdA3177147/131072311/217521.505500play11
diminished fifthd5tritone1024/729210/36588.270600play-6
augmented fourthA4729/51236/29611.730600play6
diminished sixthd6262144/177147218/311678.495700play-11
perfect fifthP5diapente,
sesquialterum
3/231/213/2701.955700play1
minor sixthm6128/8127/34792.180800play-4
augmented fifthA56561/409638/212815.640800play8
diminished seventhd732768/19683215/39882.405900play-9
major sixthM627/1633/249·3/8·2905.865900play3
minor seventhm716/924/32996.0901000play-2
augmented sixthA659049/32768310/2151019.5501000play10
diminished octaved84096/2187212/371086.3151100play-7
major seventhM7243/12835/2781·3/64·21109.7751100play5
diminished ninthd9(octave comma)1048576/531441220/3121176.5401200play-12
(perfect) octaveP8diapason2/12/11200.0001200play0
augmented seventhA7(octave + comma)531441/262144312/2181223.4601200play12

Notice that the terms ditone and semiditone are specific for Pythagorean tuning, while tone and tritone 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).

Frequency ratio of the 144 intervals in D-based Pythagorean tuning. Interval names are given in their shortened form. Pure intervals are shown in bold font. Wolf intervals are highlighted in red. Numbers larger than 999 are shown as powers of 2 or 3. Other versions of this table are provided here and here.

12-tone Pythagorean scale

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 in Size of Pythagorean intervals.

Pythagorean perfect fifth on D Play: D-A+ (27/16 ÷ 9/8 = 3/2).
Just perfect fourth Play, one perfect fifth inverted (4/3 ÷ 1/1 = 4/3).
Major tone on C Play: C-D (9/8 ÷ 3/2 = 3/2), two Pythagorean perfect fifths.
Pythagorean small minor seventh (1/1 - 16/9) Play, two perfect fifths inverted.
Pythagorean major sixth on C (1/1 - 27/16) Play, three Pythagorean perfect fifths.
Semiditone on C (1/1 - 32/27) Play, three Pythagorean perfect fifths inverted.
Ditone on C (1/1 - 81/64) Play, four Pythagorean perfect fifths.
Pythagorean minor sixth on C (1/1 - 128/81) Play, four Pythagorean perfect fifths inverted.
Pythagorean major seventh on C (1/1 - 243/128) Play, five Pythagorean perfect fifths.
Pythagorean augmented fourth tritone on C (1/1 - 729/512) Play, six Pythagorean perfect fifths.
Pythagorean diminished fifth tritone on C (1/1 - 1024/729) Play, six Pythagorean perfect fifths inverted.

Fundamental intervals

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, also known as epogdoon 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.

Contrast with modern nomenclature

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 diatonic scale on C Play.

See also

References

  1. Benson, Donald C. (2003). A Smoother Pebble: Mathematical Explorations, p.56. ISBN 978-0-19-514436-9. "The frequency ratio of every Pythagorean interval is a ratio between a power of two and a power of three...confirming the Pythagorean requirements that all intervals be associated with ratios of whole numbers."
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