A septimal comma is a small

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 ...

just intonation In music, just intonation or pure intonation is the tuning of musical intervals Interval may refer to: Mathematics and physics * Interval (mathematics), a range of numbers ** Partially ordered set#Intervals, its generalization from numbers to ...

that contains the number seven in its

prime factorization In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are suf ...

. There is more than one such interval, so the term ''septimal comma'' is ambiguous, but it most commonly refers to the interval 64/63 (27.26 cents). Use of septimal commas introduces new intervals that extend tuning beyond common-practice, extending music to the 7-limit, including the 7/6 septimal minor third, the 7/5 septimal tritone and the 8/7 septimal major second. Composers who made extensive use of these intervals include Harry Partch and Ben Johnston. Johnston uses a "7" as an accidental to indicate a note is lowered 49 cents, or an upside down seven ("ㄥ" or "") to indicate a note is raised 49 cents (36/35). John Fonville. "Ben Johnston's Extended Just Intonation – A Guide for Interpreters", p. 113, ''

Perspectives of New Music ''Perspectives of New Music'' (PNM) is a peer-reviewed academic journal specializing in music theory and analysis. It was established in 1962 by Arthur Berger and Benjamin Boretz (who were its initial editors-in-chief). ''Perspectives'' was first ...

'', vol. 29, no. 2 (Summer 1991), pp. 106–137.

Specific commas

The 64/63 septimal comma, also known as ''

Archytas Archytas (; el, Ἀρχύτας; 435/410–360/350 BC) was an Ancient Greek philosopher, mathematician, music theorist, astronomer, statesman, and strategist. He was a scientist of the Pythagorean school and famous for being the reputed founder ...

' Comma'', is the interval equal to the difference between a major and septimal whole tone (with 9/8 and 8/7 ratios, respectively). Alternatively, it can be viewed as the difference between the 16/9 Pythagorean minor seventh (the composition of two 4/3 perfect fourths) and the 7/4 harmonic seventh.Benson, Dave (2006). ''Music: A Mathematical Offering'', p. 171. . Its size is 27.264 cents, slightly larger than the Pythagorean comma. The composition of the septimal comma and the syntonic comma is 36/35, known as the ''

septimal diesis In music, septimal diesis (or slendro diesis) is an interval with the ratio of 49:48 , which is the difference between the septimal whole tone and the septimal minor third. It is about 35.7 cents wide, which is narrower than a quarter-tone but ...

''. Its size is 48.8 cents, making it practically a

quarter tone A quarter tone is a pitch halfway between the usual notes of a chromatic scale or an interval about half as wide (aurally, or logarithmically) as a semitone, which itself is half a whole tone. Quarter tones divide the octave by 50 cents each, a ...

. The septimal diesis appears as the difference between many septimal intervals and their

5-limit Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note ...

counterparts: the minor seventh (9/5) and the seventh harmonic (7/4), the 8/7 septimal whole tone and the 10/9 minor whole tone, the 7/6 septimal minor third and the 6/5 minor third, the 9/7

septimal major third In music, the septimal major third , also called the supermajor third (by Hermann von Helmholtz among others Hermann L. F. von Helmholtz (2007). ''Sensations of Tone'', p. 187. .) and sometimes '' Bohlen–Pierce third'' is the musical interval ...

and the 5/4 major third, and many more. Other septimal commas include 49/48 (occasionally called the ''slendro diesis''), which commonly appears as the difference between a ratio with 7 in the denominator and another with 7 in the numerator, like 8/7 and 7/6; and 50/49, called the ''tritonic diesis'', because it is the difference between the two septimal tritones, 7/5 and 10/7, or ''Erlich's decatonic comma'', because it plays an important role in the ten-tone scales of

Paul Erlich Paul Erlich (born 1972) is a guitarist and music theorist living near Boston, Massachusetts. He is known for his seminal role in developing the theory of regular temperaments, including being the first to define pajara temperament Accessed 2013 ...

(the intervals are tempered so that 50/49 vanishes). The septimal kleisma and the septimal semicomma are smaller septimal commas.

Summary

References

{{Intervals, state=expanded Commas (music) 0064:0063 7-limit tuning and intervals