Septimal comma

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3-limit 9:8 major tone
7-limit 8:7 septimal whole tone

A septimal comma is a small

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 (27.26 cents).[1][2]

Use of septimal commas introduces new intervals that extend tuning beyond common-practice, extending music to the

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 "7 upside-down") to indicate a note is raised 49 cents (36/35).[3]

Specific commas

Septimal comma, 64/63

The 64/63 septimal comma, also known as Archytas' Comma,[1] 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.[4] Its size is 27.264 cents, slightly larger than the Pythagorean comma.

The composition of the septimal comma and the

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.

Septimal diesis

Other septimal commas include 49/48 (occasionally called the slendro diesis[1]), 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,[1] 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 C7 upside-down- Septimal comma, Archytas' comma
50/49 34.98 B7 upside-down7 upside-down♯-
Septimal sixth-tone
, tritonic diesis, Erlich's decatonic comma
49/48 35.7 D77♭ Slendro diesis
36/35 48.77 C7 upside-down Septimal quarter tone

References

  1. ^ a b c d e Manuel Op de Coul. "List of intervals". Huygens-Fokker Foundation. Retrieved 2006-07-29.
  2. JSTOR 09588442
    .
  3. ^ 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.
  4. ISBN 0-521-85387-7
    .
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