Subminor and supermajor

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 major supermajor augmented
seconds D ≊ D D♭ D ≊ D D♯
thirds E ≊ E E♭ E ≊ E E♯
sixths A ≊ A A♭ A ≊ A A♯
sevenths B ≊ B B♭ B ≊ B 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 (D-  Play ). Another example is the ratio 28:27, or 62.96 cents (C♯-  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 (B♯); the ratio 27:14, or 1137.04 cents (B  Play ); and 35:18, or 1151.23 cents (C  Play ).

Subminor third and supermajor sixth

Septimal minor third on C  Play 
Subminor third on G  Play  and its inverse, the supermajor sixth on B♭  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 (E♭), 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 (E↓♭).

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 (A).[5][6][7] In 24 equal temperament A = B. The septimal major sixth is an interval of 12:7 ratio (A  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[12][13] (A♭) or alternately 11:7.[14] (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 (E). Another example is the ratio 50:39, or 430.14 cents (E♯).

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 (B♭).

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] the septimal whole tone (D-  Play ) and the inverse of the subminor seventh. Another example is the ratio 15:13, or 247.74 cents (D♯).

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).[15] Together the two produce the 4:3 just perfect fourth.[16]

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

Sources

  1. 1 2 Leta E. Miller, 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. ↑ Von Helmholtz, Hermann L. F (2007). On the Sensations of Tone, p.195&212. ISBN 978-1-60206-639-7.
  4. ↑ Miller (1988), p.xlii.
  5. ↑ Andrew Horner, Lydia Ayres (2002). Cooking with Csound: Woodwind and Brass Recipes, p.131. ISBN 0-89579-507-8.
  6. ↑ Royal Society (Great Britain) (1880, digitized Feb 26, 2008). Proceedings of the Royal Society of London, Volume 30, p.531. Harvard University.
  7. ↑ Society of Arts (Great Britain) (1877, digitized Nov 19, 2009). Journal of the Society of Arts, Volume 25, p.670. The Society.
  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. ↑ Hermann L. F Von Helmholtz (2007). On the Sensations of Tone, p.456. ISBN 978-1-60206-639-7.
  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. ↑ Royal Society (Great Britain) (1880, digitized Feb 26, 2008). Proceedings of the Royal Society of London, Volume 30, p.531. Harvard University.
  13. ↑ Society of Arts (Great Britain) (1877, digitized Nov 19, 2009). Journal of the Society of Arts, Volume 25, p.670. The Society.
  14. ↑ Andrew Horner, Lydia Ayres (2002). Cooking with Csound: Woodwind and Brass Recipes, p.131. ISBN 0-89579-507-8.
  15. ↑ Miller and Lieberman (2006), p.72.
  16. ↑ 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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