Origin of large and small seconds and thirds (including 7:6) in harmonic series.[1]

In music, a subminor interval is an interval that is noticeably wider than a diminished interval but noticeably narrower than a minor interval. It is found in between a minor and diminished interval, thus making it below, or subminor to, the minor interval. A supermajor interval is a musical interval that is noticeably wider than a major interval but noticeably narrower than an augmented interval. It is found in between a major and augmented interval, thus making it above, or supermajor to, the major interval. The inversion of a supermajor interval is a subminor interval, and there are four major and four minor intervals, allowing for eight supermajor and subminor intervals, each with variants.

diminished subminor minor neutral major supermajor augmented
seconds Ddouble flat ≊ Dthree quarter flat D♭ Dhalf flat D ≊ Dhalf sharp D♯
thirds Edouble flat ≊ Ethree quarter flat E♭ Ehalf flat E ≊ Ehalf sharp E♯
sixths Adouble flat ≊ Athree quarter flat A♭ Ahalf flat A ≊ Ahalf sharp A♯
sevenths Bdouble flat ≊ Bthree quarter flat B♭ Bhalf flat B ≊ Bhalf sharp B♯

Traditionally, "supermajor and superminor, [are] the names given to certain thirds [9:7 and 17:14] found in the justly intoned scale with a natural or subminor seventh."[2]

Subminor second and supermajor seventh

Thus, a subminor second is intermediate between a minor second and a diminished second (enharmonic to unison). An example of such an interval is the ratio 26:25, or 67.90 cents (D13double flat- Playⓘ). Another example is the ratio 28:27, or 62.96 cents (C7♯- Playⓘ).

A supermajor seventh is an interval intermediate between a major seventh and an augmented seventh. It is the inverse of a subminor second. Examples of such an interval is the ratio 25:13, or 1132.10 cents (B13 upside down♯); the ratio 27:14, or 1137.04 cents (B7 upside-down Playⓘ); and 35:18, or 1151.23 cents (C7 Playⓘ).

Subminor third and supermajor sixth

Septimal minor third on C Playⓘ
Subminor third on G Playⓘ and its inverse, the supermajor sixth on B7♭ Playⓘ

A subminor third is in between a minor third and a diminished third. An example of such an interval is the ratio 7:6 (E7♭), or 266.87 cents,[3][4] the septimal minor third, the inverse of the supermajor sixth. Another example is the ratio 13:11, or 289.21 cents (E13↓♭).

A supermajor sixth is noticeably wider than a major sixth but noticeably narrower than an augmented sixth, and may be a just interval of 12:7 (A7 upside-down).[5][6][7] In 24 equal temperament Ahalf sharp = Bthree quarter flat. The septimal major sixth is an interval of 12:7 ratio (A7 upside-down Playⓘ),[8][9] or about 933 cents.[10] It is the inversion of the 7:6 subminor third.

Subminor sixth and supermajor third

Septimal minor sixth (14/9) on C.[11] Playⓘ

A subminor sixth or septimal sixth is noticeably narrower than a minor sixth but noticeably wider than a diminished sixth, enharmonically equivalent to the major fifth. The sub-minor sixth is an interval of a 14:9 ratio[6][7] (A7♭) or alternately 11:7.[5] (G↑- Playⓘ) The 21st subharmonic (see subharmonic) is 729.22 cents. Playⓘ

Septimal major third on C Playⓘ

A supermajor third is in between a major third and an augmented third, enharmonically equivalent to the minor fourth. An example of such an interval is the ratio 9:7, or 435.08 cents, the septimal major third (E7 upside-down). Another example is the ratio 50:39, or 430.14 cents (E13 upside down♯).

Subminor seventh and supermajor second

Harmonic seventh Playⓘ and its inverse, the septimal whole tone Playⓘ

A subminor seventh is an interval between a minor seventh and a diminished seventh. An example of such an interval is the 7:4 ratio, the harmonic seventh (B7♭).

A supermajor second (or supersecond[2]) is intermediate to a major second and an augmented second. An example of such an interval is the ratio 8:7, or 231.17 cents,[1] also known as the septimal whole tone (D7 upside-down- Playⓘ) and the inverse of the subminor seventh. Another example is the ratio 15:13, or 247.74 cents (D13 upside down♯).

Use

Composer Lou Harrison was fascinated with the 7:6 subminor third and 8:7 supermajor second, using them in pieces such as Concerto for Piano with Javanese Gamelan, Cinna for tack-piano, and Strict Songs (for voices and orchestra).[12] Together the two produce the 4:3 just perfect fourth.[13]

19 equal temperament has several intervals which are simultaneously subminor, supermajor, augmented, and diminished, due to tempering and enharmonic equivalence (both of which work differently in 19-ET than standard tuning). For example, four steps of 19-ET (an interval of roughly 253 cents) is all of the following: subminor third, supermajor second, augmented second, and diminished third.

See also

References

  1. 1 2 Miller, Leta E., ed. (1988). Lou Harrison: Selected keyboard and chamber music, 1937-1994. p. XLIII. ISBN 978-0-89579-414-7..
  2. 1 2 Brabner, John H. F. (1884). The National Encyclopaedia, vol. 13, p. 182. London. [ISBN unspecified]
  3. ↑ Helmholtz, Hermann L. F. von (2007). On the Sensations of Tone. pp. 195, 212. ISBN 978-1-60206-639-7.
  4. ↑ Miller 1988, p. XLII.
  5. 1 2 Andrew Horner, Lydia Ayres (2002). Cooking with Csound: Woodwind and Brass Recipes, p. 131. ISBN 0-89579-507-8.
  6. 1 2 Royal Society (Great Britain) (1880, digitized February 26, 2008). Proceedings of the Royal Society of London, vol. 30, p. 531. Harvard University.
  7. 1 2 Society of Arts (Great Britain) (1877, digitized November 19, 2009). Journal of the Society of Arts, vol. 25, p. 670.
  8. ↑ Partch, Harry (1979). Genesis of a Music, p. 68. ISBN 0-306-80106-X.
  9. ↑ Haluska, Jan (2003). The Mathematical Theory of Tone Systems, p. xxiii. ISBN 0-8247-4714-3.
  10. ↑ Helmholtz 2007, p. 456.
  11. ↑ John Fonville. "Ben Johnston's Extended Just Intonation- A Guide for Interpreters", p. 122, Perspectives of New Music, vol. 29, no. 2 (Summer 1991), pp. 106–137.
  12. ↑ Miller and Lieberman (2006), p. 72.
  13. ↑ Miller & Lieberman (2006), p. 74. "The subminor third and supermajor second combine to create a pure fourth (8⁄7 x 7⁄6 = 4⁄3)."
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