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Pythagorean interval

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

In musical tuning theory, a Pythagorean interval is a musical interval with 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.

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

Name Short Other name(s) Ratio Factors Derivation Cents ET
Cents		 MIDI file Fifths
diminished second d2 524288/531441 219/312 -23.460 0 playⓘ -12
(perfect) unison P1 1/1 1/1 0.000 0 playⓘ 0
Pythagorean comma 531441/524288 312/219 23.460 0 playⓘ 12
minor second m2 limma,
diatonic semitone,
minor semitone		 256/243 28/35 90.225 100 playⓘ -5
augmented unison A1 apotome,
chromatic semitone,
major semitone		 2187/2048 37/211 113.685 100 playⓘ 7
diminished third d3 tone,
whole tone,
whole step		 65536/59049 216/310 180.450 200 playⓘ -10
major second M2 9/8 32/23 3·3/2·2 203.910 200 playⓘ 2
semiditone m3 (Pythagorean minor third) 32/27 25/33 294.135 300 playⓘ -3
augmented second A2 19683/16384 39/214 317.595 300 playⓘ 9
diminished fourth d4 8192/6561 213/38 384.360 400 playⓘ -8
ditone M3 (Pythagorean major third) 81/64 34/26 27·3/16·2 407.820 400 playⓘ 4
perfect fourth P4 diatessaron,
sesquitertium		 4/3 22/3 2/3 498.045 500 playⓘ -1
augmented third A3 177147/131072 311/217 521.505 500 playⓘ 11
diminished fifth d5 tritone 1024/729 210/36 588.270 600 playⓘ -6
augmented fourth A4 729/512 36/29 611.730 600 playⓘ 6
diminished sixth d6 262144/177147 218/311 678.495 700 playⓘ -11
perfect fifth P5 diapente,
sesquialterum		 3/2 3/2 701.955 700 playⓘ 1
minor sixth m6 128/81 27/34 792.180 800 playⓘ -4
augmented fifth A5 6561/4096 38/212 815.640 800 playⓘ 8
diminished seventh d7 32768/19683 215/39 882.405 900 playⓘ -9
major sixth M6 27/16 33/24 9·3/8·2 905.865 900 playⓘ 3
minor seventh m7 16/9 24/32 996.090 1000 playⓘ -2
augmented sixth A6 59049/32768 310/215 1019.550 1000 playⓘ 10
diminished octave d8 4096/2187 212/37 1086.315 1100 playⓘ -7
major seventh M7 243/128 35/27 81·3/64·2 1109.775 1100 playⓘ 5
diminished ninth d9 (octave − comma) 1048576/531441 220/312 1176.540 1200 playⓘ -12
(perfect) octave P8 diapason 2/1 2/1 1200.000 1200 playⓘ 0
augmented seventh A7 (octave + comma) 531441/262144 312/218 1223.460 1200 playⓘ 12

Notice that the terms 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).

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

Sources

  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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