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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. 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 | -12 | |||
| (perfect) unison | P1 | 1/1 | 30/20 | 1/1 | 0.000 | 0 | 0 | ||
| Pythagorean comma | 531441/524288 | 312/219 | 23.460 | 0 | 12 | ||||
| minor second | m2 | limma, diatonic semitone, minor semitone | 256/243 | 28/35 | 90.225 | 100 | -5 | ||
| augmented unison | A1 | apotome, chromatic semitone, major semitone | 2187/2048 | 37/211 | 113.685 | 100 | 7 | ||
| diminished third | d3 | tone, whole tone, whole step | 65536/59049 | 216/310 | 180.450 | 200 | -10 | ||
| major second | M2 | 9/8 | 32/23 | 3·3/2·2 | 203.910 | 200 | 2 | ||
| semiditone | m3 | (Pythagorean minor third) | 32/27 | 25/33 | 294.135 | 300 | -3 | ||
| augmented second | A2 | 19683/16384 | 39/214 | 317.595 | 300 | 9 | |||
| diminished fourth | d4 | 8192/6561 | 213/38 | 384.360 | 400 | -8 | |||
| ditone | M3 | (Pythagorean major third) | 81/64 | 34/26 | 27·3/32·2 | 407.820 | 400 | 4 | |
| perfect fourth | P4 | diatessaron, sesquitertium | 4/3 | 22/3 | 2·2/3 | 498.045 | 500 | -1 | |
| augmented third | A3 | 177147/131072 | 311/217 | 521.505 | 500 | 11 | |||
| diminished fifth | d5 | tritone | 1024/729 | 210/36 | 588.270 | 600 | -6 | ||
| augmented fourth | A4 | 729/512 | 36/29 | 611.730 | 600 | 6 | |||
| diminished sixth | d6 | 262144/177147 | 218/311 | 678.495 | 700 | -11 | |||
| perfect fifth | P5 | diapente, sesquialterum | 3/2 | 31/21 | 3/2 | 701.955 | 700 | 1 | |
| minor sixth | m6 | 128/81 | 27/34 | 792.180 | 800 | -4 | |||
| augmented fifth | A5 | 6561/4096 | 38/212 | 815.640 | 800 | 8 | |||
| diminished seventh | d7 | 32768/19683 | 215/39 | 882.405 | 900 | -9 | |||
| major sixth | M6 | 27/16 | 33/24 | 9·3/8·2 | 905.865 | 900 | 3 | ||
| minor seventh | m7 | 16/9 | 24/32 | 996.090 | 1000 | -2 | |||
| augmented sixth | A6 | 59049/32768 | 310/215 | 1019.550 | 1000 | 10 | |||
| diminished octave | d8 | 4096/2187 | 212/37 | 1086.315 | 1100 | -7 | |||
| major seventh | M7 | 243/128 | 35/27 | 81·3/64·2 | 1109.775 | 1100 | 5 | ||
| diminished ninth | d9 | (octave − comma) | 1048576/531441 | 220/312 | 1176.540 | 1200 | -12 | ||
| (perfect) octave | P8 | diapason | 2/1 | 2/1 | 1200.000 | 1200 | 0 | ||
| augmented seventh | A7 | (octave + comma) | 531441/262144 | 312/218 | 1223.460 | 1200 | 12 |
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).
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.
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).
See also
- Generated collection
- Just intonation
- List of meantone intervals
- List of intervals in 5-limit just intonation
- Shí-èr-lǜ
- Whole-tone scale
References
- 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."
External links
- Neo-Gothic usage by Margo Schulter
The article s lead section may need to be rewritten Please help improve the lead and read the lead layout guide November 2010 Learn how and when to remove this message 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 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 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 tableName Short Other name s Ratio Factors Derivation Cents ET Cents MIDI file Fifthsdiminished second d2 524288 531441 219 312 23 460 0 12 perfect unison P1 1 1 30 20 1 1 0 000 0 0Pythagorean comma 531441 524288 312 219 23 460 0 12minor second m2 limma diatonic semitone minor semitone 256 243 28 35 90 225 100 5augmented unison A1 apotome chromatic semitone major semitone 2187 2048 37 211 113 685 100 7diminished third d3 tone whole tone whole step 65536 59049 216 310 180 450 200 10major second M2 9 8 32 23 3 3 2 2 203 910 200 2semiditone m3 Pythagorean minor third 32 27 25 33 294 135 300 3augmented second A2 19683 16384 39 214 317 595 300 9diminished fourth d4 8192 6561 213 38 384 360 400 8ditone M3 Pythagorean major third 81 64 34 26 27 3 32 2 407 820 400 4perfect fourth P4 diatessaron sesquitertium 4 3 22 3 2 2 3 498 045 500 1augmented third A3 177147 131072 311 217 521 505 500 11diminished fifth d5 tritone 1024 729 210 36 588 270 600 6augmented fourth A4 729 512 36 29 611 730 600 6diminished sixth d6 262144 177147 218 311 678 495 700 11perfect fifth P5 diapente sesquialterum 3 2 31 21 3 2 701 955 700 1minor sixth m6 128 81 27 34 792 180 800 4augmented fifth A5 6561 4096 38 212 815 640 800 8diminished seventh d7 32768 19683 215 39 882 405 900 9major sixth M6 27 16 33 24 9 3 8 2 905 865 900 3minor seventh m7 16 9 24 32 996 090 1000 2augmented sixth A6 59049 32768 310 215 1019 550 1000 10diminished octave d8 4096 2187 212 37 1086 315 1100 7major seventh M7 243 128 35 27 81 3 64 2 1109 775 1100 5diminished ninth d9 octave comma 1048576 531441 220 312 1176 540 1200 12 perfect octave P8 diapason 2 1 2 1 1200 000 1200 0augmented seventh A7 octave comma 531441 262144 312 218 1223 460 1200 12 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 D A 27 16 9 8 3 2 Just perfect fourth one perfect fifth inverted 4 3 1 1 4 3 Major tone on C C D 9 8 3 2 3 2 two Pythagorean perfect fifths Pythagorean small minor seventh 1 1 16 9 two perfect fifths inverted Pythagorean major sixth on C 1 1 27 16 three Pythagorean perfect fifths Semiditone on C 1 1 32 27 three Pythagorean perfect fifths inverted Ditone on C 1 1 81 64 four Pythagorean perfect fifths Pythagorean minor sixth on C 1 1 128 81 four Pythagorean perfect fifths inverted Pythagorean major seventh on C 1 1 243 128 five Pythagorean perfect fifths Pythagorean augmented fourth tritone on C 1 1 729 512 six Pythagorean perfect fifths Pythagorean diminished fifth tritone on C 1 1 1024 729 six Pythagorean perfect fifths inverted Fundamental intervalsThe 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 Stormer 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 nomenclatureThere 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 See alsoGenerated collection Just intonation List of meantone intervals List of intervals in 5 limit just intonation Shi er lǜ Whole tone scaleReferencesBenson 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 External linksNeo Gothic usage by Margo Schulter
