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# Septimal comma

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### Septimal comma

Octave minus septimal diesis (36:35) on C  Play  .
3-limit 9:8 major tone  Play  .
7-limit 8:7 septimal whole tone  Play  .

A septimal comma is a small musical interval in Just intonation that contains the number seven in its prime factorization. It is based on the seventh partial of the harmonic series. This partial is a flat + a septimal comma (50 + 41 cents). The comma is approximately 41 cents. It is used to tune an interval justly so that is sounds more "in tune." A pythagorean seventh would be slightly too big.

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).[1]

## Specific commas

The 64/63 septimal comma ( play  ), also known as Archytas' Comma,[2] 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.[3] 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.[2] Its size is 48.8 cents, making it practically a quarter tone. The septimal diesis appears as the difference between many septimal intervals and their 5-limit counterparts: the minor seventh (9/5) and the seventh harmonic (7/4),[1] 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 and the 5/4 major third, and many more.

Other septimal commas include 49/48 (occasionally called the slendro diesis[2]) ( play  ), 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,[2] 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 (the intervals are tempered so that 50/49 vanishes).

The septimal kleisma and the septimal semicomma are smaller septimal commas.

## Summary

Ratio Size in cents Ben Johnston's
notation
Names
64/63 27.26 C- Septimal comma, Archytas' comma
50/49 34.98 B- Septimal sixth-tone, tritonic diesis, Erlich's decatonic comma
49/48 35.7 D Slendro diesis
36/35 48.77 C Septimal quarter tone

## References

1. ^ a b John Fonville. "Ben Johnston's Extended Just Intonation- A Guide for Interpreters", p.113, Perspectives of New Music, Vol. 29, No. 2 (Summer, 1991), pp. 106-137.
2. ^ a b c d Manuel Op de Coul. "List of intervals".
3. ^ Benson, Dave (2006). Music: A Mathematical Offering, p.171. ISBN 0-521-85387-7.
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