Pythagorean interval
Encyclopedia
In musical tuning
Musical tuning
In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...

 theory, a Pythagorean interval is a musical interval with frequency ratio
Interval ratio
In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth is 3:2 , 1.5, and may be approximated by an equal tempered perfect fifth which is 27/12, 1.498...

 equal to a power
Exponentiation
Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...

 of two divided by a power of three, or vice versa
Vice Versa
Vice Versa: A Lesson to Fathers is a novel by F. Anstey, first published in 1882.The title originates from the Latin phrase, "vice versa", meaning "the other way around".-Plot summary:...

. For instance, the perfect fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...

 with ratio 3/2 (equivalent to 31/21) and the perfect fourth
Perfect fourth
In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there...

 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
Pythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...

 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
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

.

Interval table

Name Short Other name(s) Ratio
Interval ratio
In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth is 3:2 , 1.5, and may be approximated by an equal tempered perfect fifth which is 27/12, 1.498...

 
Factors Derivation Cents
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...

 
ET
Cents
MIDI file Fifths
diminished second
Diminished second
In modern Western tonal music theory a diminished second is the interval between notes on two adjacent staff positions, or having adjacent note letters, whose alterations cause them, in ordinary equal temperament, to have no pitch difference, such as B and C or B and C...

d2 524288/531441 219/312 -23.460 0 -12
(perfect) unison
Unison
In music, the word unison can be applied in more than one way. In general terms, it may refer to two notes sounding the same pitch, often but not always at the same time; or to the same musical voice being sounded by several voices or instruments together, either at the same pitch or at a distance...

P1 1/1 1/1 0.000 0 0
Pythagorean comma
Pythagorean comma
In musical tuning, the Pythagorean comma , named after the ancient mathematician and philosopher Pythagoras, is the small interval existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B , or D and C...

 
531441/524288 23.460 0 12
minor second
Minor second
In modern Western tonal music theory a minor second is the interval between two notes on adjacent staff positions, or having adjacent note letters, whose alterations cause them to be one semitone or half-step apart, such as B and C or C and D....

m2 limma,
diatonic semitone,
minor semitone
256/243 28/35 90.225 100 -5
augmented unison
Augmented unison
In modern Western tonal music theory an augmented unison is the interval between two notes on the same staff position, or having the same note letter, whose alterations cause them, in ordinary equal temperament, to be one semitone apart. In other words, it is a unison where one note has been raised...

 
A1 apotome,
chromatic semitone,
major semitone
2187/2048 37/211 113.685 100 7
diminished third
Diminished third
In classical music from Western culture, a diminished third is the musical interval produced by narrowing a minor third by a chromatic semitone. For instance, the interval from A to C is a minor third, three semitones wide, and both the intervals from A to C, and from A to C are diminished thirds,...

 
d3 tone,
whole tone,
whole step
65536/59049 216/310 180.450 200 -10
major second
Major second
In Western music theory, a major second is a musical interval spanning two semitones, and encompassing two adjacent staff positions . For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff postions...

 
M2 9/8 32/23 3·3/2·2 203.910 200 2
semiditone  m3 (Pythagorean minor third
Minor third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...

)
32/27 25/33 294.135 300 -3
augmented second
Augmented second
In classical music from Western culture, an augmented second is an interval produced by widening a major second by a chromatic semitone. For instance, the interval from C to D is a major second, two semitones wide, and both the intervals from C to D, and from C to D are augmented seconds, spanning...

A2 19683/16384 39/214 317.595 300 9
diminished fourth
Diminished fourth
In classical music from Western culture, a diminished fourth is an interval produced by narrowing a perfect fourth by a chromatic semitone. For example, the interval from C to F is a perfect fourth, five semitones wide, and both the intervals from C to F, and from C to F are diminished fourths,...

d4 8192/6561 213/38 384.360 400 -8
ditone
Ditone
In music, a ditone is the interval 81:64, which is 407.82 cents. It is the major third in Pythagorean tuning. The ditone is evenly divisible by two major tones and is wider than a just major third by a syntonic comma...

 
M3 (Pythagorean major third
Major third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...

)
81/64 34/26 27·3/16·2 407.820 400 4
perfect fourth
Perfect fourth
In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there...

 
P4 diatessaron,
sesquitertium 
4/3 22/3 2/3 498.045 500 -1
augmented third
Augmented third
In classical music from Western culture, an augmented third is an interval produced by widening a major third by a chromatic semitone. For instance, the interval from C to E is a major third, four semitones wide, and both the intervals from C to E, and from C to E are augmented thirds, spanning...

 
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
Diminished sixth
In classical music from Western culture, a diminished sixth is an interval produced by narrowing a minor sixth by a chromatic semitone. For example, the interval from A to F is a minor sixth, eight semitones wide, and both the intervals from A to F, and from A to F are diminished sixths, spanning...

d6 262144/177147 218/311 678.495 700 -11
perfect fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...

P5 diapente,
sesquialterum 
3/2 3/2 701.955 700 1
minor sixth
Minor sixth
-Subminor sixth:In music, a subminor sixth or septimal sixth is an interval that is noticeably narrower than a minor sixth but noticeably wider than a diminished sixth.The sub-minor sixth is an interval of a 14:9 ratio or alternately 11:7....

m6 128/81 27/34 792.180 800 -4
augmented fifth
Augmented fifth
In classical music from Western culture, an augmented fifth is an interval produced by widening a perfect fifth by a chromatic semitone. For instance, the interval from C to G is a perfect fifth, seven semitones wide, and both the intervals from C to G, and from C to G are augmented fifths,...

A5 6561/4096 38/212 815.640 800 8
diminished seventh
Diminished seventh
In classical music from Western culture, a diminished seventh is an interval produced by narrowing a minor seventh by a chromatic semitone. For instance, the interval from A to G is a minor seventh, ten semitones wide, and both the intervals from A to G, and from A to G are diminished sevenths,...

d7 32768/19683 215/39 882.405 900 -9
major sixth
Major sixth
In classical music from Western culture, a sixth is a musical interval encompassing six staff positions , and the major sixth is one of two commonly occurring sixths. It is qualified as major because it is the largest of the two...

 
M6 27/16 33/24 9·3/8·2 905.865 900 3
minor seventh
Minor seventh
In classical music from Western culture, a seventh is a musical interval encompassing seven staff positions , and the minor seventh is one of two commonly occurring sevenths. The minor quality specification identifies it as being the smallest of the two: the minor seventh spans ten semitones, the...

 
m7 16/9 24/32 996.090 1000 -2
augmented sixth
Augmented sixth
In classical music from Western culture, an augmented sixth is an interval produced by widening a major sixth by a chromatic semitone. For instance, the interval from C to A is a major sixth, nine semitones wide, and both the intervals from C to A, and from C to A are augmented sixths, spanning...

 
A6 59049/32768 310/215 1019.550 1000 10
diminished octave
Diminished octave
In classical music from Western culture, a diminished octave is an interval produced by narrowing a perfect octave by a chromatic semitone. As such, the two notes are denoted by the same letter but have different accidentals...

 
d8 4096/2187 212/37 1086.315 1100 -7
major seventh
Major seventh
In classical music from Western culture, a seventh is a musical interval encompassing seven staff positions , and the major seventh is one of two commonly occurring sevenths. It is qualified as major because it is the larger of the two...

 
M7 243/128 35/27 81·3/64·2 1109.775 1100 5
augmented seventh
Augmented seventh
In classical music from Western culture, an augmented seventh is an interval produced by widening a major seventh by a chromatic semitone. For instance, the interval from C to B is a major seventh, eleven semitones wide, and both the intervals from C to B, and from C to B are augmented sevenths,...

 
A7 (octave − comma) 1048576/531441 1176.540 1200 -12
(perfect) octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

 
P8 diapason 2/1 2/1 1200.000 1200 0
diminished ninth d9 (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. 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).

12-tone Pythagorean scale

The table shows from which notes some of the above listed intervals can be played on a 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
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

 or diapason (Greek
Ancient Greek
Ancient Greek is the stage of the Greek language in the periods spanning the times c. 9th–6th centuries BC, , c. 5th–4th centuries BC , and the c. 3rd century BC – 6th century AD of ancient Greece and the ancient world; being predated in the 2nd millennium BC by Mycenaean Greek...

 for "across all"). 3/2 is the perfect fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...

, diapente ("across five"), or sesquialterum. 4/3 is the perfect fourth
Perfect fourth
In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there...

, diatessaron ("across four"), or sesquitertium. These three intervals and their octave equivalents, such as the perfect eleventh and twelfth, are the only absolute consonances
Consonance and dissonance
In music, a consonance is a harmony, chord, or interval considered stable, as opposed to a dissonance , which is considered to be unstable...

 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
Major second
In Western music theory, a major second is a musical interval spanning two semitones, and encompassing two adjacent staff positions . For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff postions...

. This has the ratio 9/8, and it is the only other superparticular ratio of Pythagorean tuning, as shown by Størmer's theorem
Størmer's theorem
In number theory, Størmer's theorem, named after Carl Størmer, gives a finite bound on the number of consecutive pairs of smooth numbers that exist, for a given degree of smoothness, and provides a method for finding all such pairs using Pell equations...

.

Two tones make a ditone, a dissonantly wide major third
Major third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...

, ratio 81/64. The ditone differs from the just major third (5/4) by the syntonic comma
Syntonic comma
In music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...

 (81/80). Likewise, the difference between the tone and the perfect fourth is the semiditone, a narrow minor third
Minor third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...

, 32/27, which differs from 6/5 by the syntonic comma. These differences are "tempered out" or eliminated by using compromises in meantone temperament
Meantone temperament
Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...

.

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
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...

, they are not equal in Pythagorean tuning, and their difference, 531441/524288, is known as the Pythagorean comma
Pythagorean comma
In musical tuning, the Pythagorean comma , named after the ancient mathematician and philosopher Pythagoras, is the small interval existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B , or D and C...

.

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

  • Just intonation
    Just intonation
    In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

  • List of meantone intervals
  • List of intervals in 5-limit just intonation
  • Shi Er Lü
    Shi Er Lü
    Shí-èr-lǜ |pitches]]) is a standardized gamut of twelve notes. Also known, rather misleadingly, as the Chinese chromatic scale it is one kind of chromatic scale used in ancient Chinese music. The Chinese scale uses the same intervals as the Pythagorean scale, based on 2:3 ratios...

  • Whole-tone scale

External links

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