A septimal comma is a small

musical interval In music theory, an interval is a difference in pitch (music), 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 v ...

just intonation In music, just intonation or pure intonation is a musical tuning, tuning system in which the space between notes' frequency, frequencies (called interval (music), intervals) is a natural number, whole number ratio, ratio. Intervals spaced in thi ...

that contains the number

seven 7 (seven) is the natural number following 6 and preceding 8. It is the only prime number preceding a cube. As an early prime number in the series of positive integers, the number seven has symbolic associations in religion, mythology, supers ...

in its

prime factorization In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a comp ...

. 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 7-limit or septimal tunings and intervals are musical instrument tunings that have a limit of seven: the largest prime factor contained in the interval ratios between pitches is seven. Thus, for example, 50:49 is a 7-limit interval, but 14 ...

, including the 7/6

septimal minor third Septimal may refer to: *Septimal chromatic semitone, the interval 21:20, about 84.47 cents *Septimal comma, a small musical interval in just intonation divisible by 7 *Septimal diatonic semitone, the interval 15:14, about 119.44 cents *S ...

, the 7/5 septimal tritone and the 8/7

septimal major second In music, the septimal whole tone, septimal major second, supermajor second, or septimal supermajor second is the interval (music), musical interval exactly or approximately equal to an 8/7 ratio of frequencies.Andrew Horner, Lydia Ayres (2002 ...

. Composers who made extensive use of these intervals include

Harry Partch Harry Partch (June 24, 1901 – September 3, 1974) was an American composer, music theorist, and creator of unique musical instruments. He composed using scales of unequal intervals in just intonation, and was one of the first 20th-century com ...

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 John Fonville is a flutist and composer. Fonville specializes in extended techniques on the flute, especially microtonality, and performs on instruments including a complete set of quarter tone ( Kingma system) flutes.Perspectives of New Music ''Perspectives of New Music'' (PNM) is a peer-reviewed academic journal specializing in music theory Music theory is the study of theoretical frameworks for understanding the practices and possibilities of music. ''The Oxford Companion to Musi ...

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

Specific commas

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

Archytas Archytas (; ; 435/410–360/350 BC) was an Ancient Greek mathematician, music theorist, statesman, and strategist from the ancient city of Taras (Tarentum) in Southern Italy. He was a scientist and philosopher affiliated with the Pythagorean ...

' 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 Pythagorean, meaning of or pertaining to the ancient Ionian mathematician, philosopher, and music theorist Pythagoras, may refer to: Philosophy * Pythagoreanism, the esoteric and metaphysical beliefs purported to have been held by Pythagoras * Ne ...

minor seventh In music theory, a minor seventh is one of two musical intervals that span seven staff positions. It is ''minor'' because it is the smaller of the two sevenths, spanning ten semitones. The major seventh spans eleven. For example, the interval ...

(the composition of two 4/3 perfect fourths) and the 7/4

harmonic seventh The harmonic seventh interval, also known as the septimal minor seventh, or subminor seventh, is one with an exact 7:4 ratio (about 969 cent (music), cents). This is about 32 cents narrower, with a more stable and consonant sound, than a ...

.Benson, Dave (2006). ''Music: A Mathematical Offering'', p. 171. . Its size is 27.264 cents, slightly larger than 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 ...

. The composition of the septimal comma and the

syntonic comma In music theory Music theory is the study of theoretical frameworks for understanding the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory": The first i ...

is 36/35, known as the ''

septimal diesis In music, septimal diesis (or slendro diesis) is an Interval (music), 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 q ...

''. 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 (orally, 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 not ...

counterparts: the minor seventh (9/5) and the seventh harmonic (7/4), the 8/7 septimal

whole tone 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 deta ...

and the 10/9 minor whole tone, the 7/6

and the 6/5

minor third In music theory, a minor third is a interval (music), musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions (see: interval (music)#Number, interval numb ...

, 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. .), septimal supermajor third, and sometimes '' Bohlen–Pierce thir ...

and the 5/4

major third In music theory, a third is a Interval (music), musical interval encompassing three staff positions (see Interval (music)#Number, Interval number for more details), and the major third () is a third spanning four Semitone, half steps or two ...

, 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

tritone In music theory, the tritone is defined as a interval (music), musical interval spanning three adjacent Major second, whole tones (six semitones). For instance, the interval from F up to the B above it (in short, F–B) is a tritone as it can be ...

s, 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 In music, the ratio 225/224 is called the septimal kleisma (). It is a minute comma type interval of approximately 7.7 cents. Factoring it into primes gives 2−5 32 52 7−1, which can be rewritten 2−1 (5/4)2 (9/7). That says th ...

and the

septimal semicomma In music, the septimal semicomma, a seven- limit semicomma, is the ratio 126/125 and is equal to approximately 13.79 cents (). It is also called the ''small septimal comma''Haluska, Jan (2003). ''The Mathematical Theory of Tone Systems'', p.xxvi. ...

are smaller septimal commas.

Summary

References

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