Subminor and supermajor

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	D	≊ D	D♭	D	D	≊ D	D♯
thirds	E	≊ E	E♭	E	E	≊ E	E♯
sixths	A	≊ A	A♭	A	A	≊ A	A♯
sevenths	B	≊ B	B♭	B	B	≊ B	B♯

Traditionally, "supermajor and subminor, [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

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

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] (A♭) or alternately 11:7.^[5] (G↑- Play^ⓘ) The 21st subharmonic (see subharmonic) is 729.22 cents. 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] also known as 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).^[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.

References

^ ^a ^b Miller, Leta E., ed. (1988). Lou Harrison: Selected keyboard and chamber music, 1937-1994. p. XLIII. ISBN 978-0-89579-414-7..
^ ^a ^b Brabner, John H. F. (1884). The National Encyclopaedia, vol. 13, p. 182. London. [ISBN unspecified]
^ Helmholtz, Hermann L. F. von (2007). On the Sensations of Tone. pp. 195, 212. ISBN 978-1-60206-639-7.
^ Miller 1988, p. XLII.
^ ^a ^b Andrew Horner, Lydia Ayres (2002). Cooking with Csound: Woodwind and Brass Recipes, p. 131. ISBN 0-89579-507-8.
^ ^a ^b Royal Society (Great Britain) (1880, digitized February 26, 2008). Proceedings of the Royal Society of London, vol. 30, p. 531. Harvard University.
^ ^a ^b Society of Arts (Great Britain) (1877, digitized November 19, 2009). Journal of the Society of Arts, vol. 25, p. 670.
^ Partch, Harry (1979). Genesis of a Music, p. 68. ISBN 0-306-80106-X.
^ Haluska, Jan (2003). The Mathematical Theory of Tone Systems, p. xxiii. ISBN 0-8247-4714-3.
^ Helmholtz 2007, p. 456.
^ 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.
^ Miller and Lieberman (2006), p. 72.^{[incomplete short citation]}
^ Miller & Lieberman (2006), p. 74. "The subminor third and supermajor second combine to create a pure fourth ( $.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}8⁄7 x 7⁄6 = 4⁄3$ )."^{[incomplete short citation]}