A septimal comma is a small musical interval

Interval (music)

In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

 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...

 that contains the number seven in its prime factorization. There is more than one such interval, so the term septimal comma is ambiguous, but it most commonly refers to the interval 64/63.

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

Septimal minor third

In music, the septimal minor third , also called the subminor third, is the musical interval exactly or approximately equal to a 7/6 ratio of frequencies. In terms of cents, it is 267 cents, a quartertone of size 36/35 flatter than a just minor third of 6/5...

, the 7/5 septimal tritone and the 8/7 septimal major second. Composers who made extensive use of these intervals include Harry Partch

Harry Partch

Harry Partch was an American composer and instrument creator. He was one of the first twentieth-century composers to work extensively and systematically with microtonal scales, writing much of his music for custom-made instruments that he built himself, tuned in 11-limit just intonation.-Early...

 and Ben Johnston. Johnston uses a "7" as an accidental to indicate a note is lowered 49 cents, or an upside down seven to indicate a note is raised 49 cents.

Specific commas

The 64/63 septimal comma , also known as Archytas

Archytas

Archytas was an Ancient Greek philosopher, mathematician, astronomer, statesman, and strategist. He was a scientist of the Pythagorean school and famous for being the reputed founder of mathematical mechanics, as well as a good friend of Plato....

' 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 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...

 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...

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

Harmonic seventh

The harmonic seventh interval , also known as the septimal minor seventh, or subminor seventh, is one with an exact 7:4 ratio . This is somewhat narrower than and is "sweeter in quality" than an "ordinary" minor seventh, which has a just-intonation ratio of 9:5 , or an equal-temperament ratio of...

. Its size is 27.264 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...

, slightly larger than 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...

.

The composition of the septimal comma and 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...

 is 36/35, known as the septimal diesis

Septimal diesis

In music, septimal 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 wider than the septimal comma...

. Its size is 48.8 cents, making it practically a quarter tone

Quarter tone

A quarter tone , is a pitch halfway between the usual notes of a chromatic scale, an interval about half as wide as a semitone, which is half a whole tone....

. The septimal diesis appears as the difference between many septimal intervals and their 5-limit

Limit (music)

In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name...

 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

Septimal minor third

In music, the septimal minor third , also called the subminor third, is the musical interval exactly or approximately equal to a 7/6 ratio of frequencies. In terms of cents, it is 267 cents, a quartertone of size 36/35 flatter than a just minor third of 6/5...

 and the 6/5 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...

, the 9/7 septimal major third

Septimal major third

In music, the septimal major third , also called the supermajor third and sometimes Bohlen–Pierce third is the musical interval exactly or approximately equal to a just 9:7 ratio of frequencies, or alternately 14:11. It is equal to 435 cents, sharper than a just major third by the septimal...

 and the 5/4 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...

, 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

Tritone

In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones...

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

Paul Erlich 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 discover pajara temperament and its decatonic scales in 22-ET. He holds a Bachelor of Science degree in...

 (the intervals are tempered so that 50/49 vanishes).

The septimal kleisma

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 2 . That says that it is the amount that two major thirds of 5/4 and a septimal...

 and the septimal semicomma

Septimal semicomma

In music, the ratio 126/125 is called the septimal semicomma . It is also called the starling comma after its use in starling temperament, as well as the small septimal comma...

 are smaller septimal commas.

Summary

Ratio	Size in cents	Names
64/63	27.3	Septimal comma, Archytas' comma
50/49	35.0	Septimal sixth-tone Septimal sixth-tone In music, septimal sixth-tone is a septimal sixth-tone, an interval with the ratio of 50:49 , about 34.98 cents, which in just intonation is the difference between the lesser septimal tritone, and its inversion, the greater septimal tritone .... , tritonic diesis, Erlich's decatonic comma
49/48	35.7	Slendro diesis Septimal diesis In music, septimal 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 wider than the septimal comma...
36/35	48.4	Septimal quarter tone Septimal quarter tone A septimal quarter-tone is an interval with the ratio of 36:35 , which is the difference between the septimal minor third and the Just minor third , or about 48.77 cents wide. The name derives from the interval being the 7-limit approximation of a quarter tone...