 Pythagorean interval

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. For instance, the perfect fifth with ratio 3/2 (equivalent to 3^{1}/2^{1}) and the perfect fourth with ratio 4/3 (equivalent to 2^{2}/3^{1}) 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.
Contents
Interval table
Name Short Other name(s) Ratio Factors Derivation Cents ET
CentsMIDI file Fifths diminished second d2 524288/531441 2^{19}/3^{12} 23.460 0 play (help·info) 12 (perfect) unison P1 1/1 1/1 0.000 0 play (help·info) 0 Pythagorean comma 531441/524288 23.460 0 play (help·info) 12 minor second m2 limma,
diatonic semitone,
minor semitone256/243 2^{8}/3^{5} 90.225 100 play (help·info) 5 augmented unison A1 apotome,
chromatic semitone,
major semitone2187/2048 3^{7}/2^{11} 113.685 100 play (help·info) 7 diminished third d3 tone,
whole tone,
whole step65536/59049 2^{16}/3^{10} 180.450 200 play (help·info) 10 major second M2 9/8 3^{2}/2^{3} 3·3/2·2 203.910 200 play (help·info) 2 semiditone m3 (Pythagorean minor third) 32/27 2^{5}/3^{3} 294.135 300 play (help·info) 3 augmented second A2 19683/16384 3^{9}/2^{14} 317.595 300 play (help·info) 9 diminished fourth d4 8192/6561 2^{13}/3^{8} 384.360 400 play (help·info) 8 ditone M3 (Pythagorean major third) 81/64 3^{4}/2^{6} 27·3/16·2 407.820 400 play (help·info) 4 perfect fourth P4 diatessaron,
sesquitertium4/3 2^{2}/3 2/3 498.045 500 play (help·info) 1 augmented third A3 177147/131072 3^{11}/2^{17} 521.505 500 play (help·info) 11 diminished fifth d5 tritone 1024/729 2^{10}/3^{6} 588.270 600 play (help·info) 6 augmented fourth A4 729/512 3^{6}/2^{9} 611.730 600 play (help·info) 6 diminished sixth d6 262144/177147 2^{18}/3^{11} 678.495 700 play (help·info) 11 perfect fifth P5 diapente,
sesquialterum3/2 3/2 701.955 700 play (help·info) 1 minor sixth m6 128/81 2^{7}/3^{4} 792.180 800 play (help·info) 4 augmented fifth A5 6561/4096 3^{8}/2^{12} 815.640 800 play (help·info) 8 diminished seventh d7 32768/19683 2^{15}/3^{9} 882.405 900 play (help·info) 9 major sixth M6 27/16 3^{3}/2^{4} 9·3/8·2 905.865 900 play (help·info) 3 minor seventh m7 16/9 2^{4}/3^{2} 996.090 1000 play (help·info) 2 augmented sixth A6 59049/32768 3^{10}/2^{15} 1019.550 1000 play (help·info) 10 diminished octave d8 4096/2187 2^{12}/3^{7} 1086.315 1100 play (help·info) 7 major seventh M7 243/128 3^{5}/2^{7} 81·3/64·2 1109.775 1100 play (help·info) 5 augmented seventh A7 (octave − comma) 1048576/531441 1176.540 1200 play (help·info) 12 (perfect) octave P8 diapason 2/1 2/1 1200.000 1200 play (help·info) 0 diminished ninth d9 (octave + comma) 531441/262144 3^{12}/2^{18} 1223.460 1200 play (help·info) 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).
12tone Pythagorean scale
The table shows from which notes some of the above listed intervals can be played on a instrument using a repeatedoctave 12tone scale (such as a piano) tuned with Dbased 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, 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 12pitch 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 onetoone 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
 Just intonation
 List of meantone intervals
 List of intervals in 5limit just intonation
 Shi Er Lü
 Wholetone scale
External links
Intervals (list) Numbers in brackets are the number of semitones in the interval.
Fractional semitones are approximate.Twelvesemitone
(Western)PerfectMajorMinorAugmentedDiminishedCompoundOther systems SupermajorNeutralSubminor7limitchromatic semitone (⅔) · diatonic semitone (1⅙) · whole tone (2⅓) · subminor third (2⅔) · supermajor third (4⅓) · harmonic (subminor) seventh (9⅔)Other intervals GroupsPythagorean comma · Pythagorean apotome · Pythagorean limma · Diesis · Septimal diesis · Septimal comma · Syntonic comma · Schisma · Diaschisma · Major limma · Ragisma · Breedsma · Kleisma · Septimal kleisma · Septimal semicomma · Orwell comma · Semicomma · Septimal sixthtone · Septimal quarter tone · Septimal thirdtone
MeasurementOthersCategories: Mathematics of music
 Intervals
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