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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 Tun ...
theory, a Pythagorean interval is a
musical interval In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or ha ...
with frequency ratio equal to a
power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may ...
of two divided by a power of three, or
vice versa References Additional references * * {{Latin phrases V ca:Locució llatina#V da:Latinske ord og vendinger#V fr:Liste de locutions latines#V id:Daftar frasa Latin#V it:Locuzioni latine#V nl:Lijst van Latijnse spreekwoorden en ui ...
.Benson, Donald C. (2003). ''A Smoother Pebble: Mathematical Explorations'', p.56. . "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." For instance, the
perfect fifth In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval from the first to the last of five ...
with ratio 3/2 (equivalent to 31/ 21) and the
perfect fourth A fourth is a musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth () is the fourth spanning five semitones (half steps, or half tones). For example, the ascending interval from C to t ...
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 is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2.Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: McG ...
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 In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios (such as 3:2 or 4:3) of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals (and c ...
.


Interval table

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 In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval from the first to the last of five ...
, ''diapente'' ("across five"), or ''sesquialterum''. 4/3 is the
perfect fourth A fourth is a musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth () is the fourth spanning five semitones (half steps, or half tones). For example, the ascending interval from C to t ...
, ''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 In Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (). A second is a musical interval encompassing two adjacent staff positions (see Interval number for more detai ...
and it is the only other superparticular ratio of Pythagorean tuning, as shown by
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 equ ...
. 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 In music theory, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions (see: interval number). The minor third is one of two common ...
, 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 In musical tuning, the Pythagorean comma (or ditonic comma), named after the ancient mathematician and philosopher Pythagoras, is the small interval (or comma) existing in Pythagorean tuning between two enharmonically equivalent notes such as ...
.


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 In diatonic set theory, a generated collection is a collection or scale formed by repeatedly adding a constant interval in integer notation, the generator, also known as an interval cycle, around the chromatic circle until a complete collect ...
*
Just intonation In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios (such as 3:2 or 4:3) of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals (and c ...
* List of meantone intervals *
List of intervals in 5-limit just intonation The intervals of 5- limit just intonation (prime limit, not odd limit) are ratios involving only the powers of 2, 3, and 5. The fundamental intervals are the superparticular ratios 2/1 (the octave), 3/2 (the perfect fifth) and 5/4 (the major thi ...
*
Shí-èr-lǜ ''Shí-èr-lǜ'' (, , ''12 pitches'') (twelve-pitch scale) was a standardized gamut of twelve notes. Also known, rather misleadingly, as the Chinese chromatic scale, it was one kind of chromatic scale used in ancient Chinese music. The Chinese sc ...
*
Whole-tone scale In music, a whole-tone scale is a scale in which each note is separated from its neighbors by the interval of a whole tone. In twelve-tone equal temperament, there are only two complementary whole-tone scales, both six-note or ''hexatonic'' sc ...


References


External links


Neo-Gothic usage by Margo Schulter
{{DEFAULTSORT:Pythagorean Interval 3-limit tuning and intervals cs:Ditón de:Ditonus es:Ditono eo:Ditono nl:Ditonus ru:Дитон