Subminor and supermajor

File:Origin of seconds and thirds in harmonic series.png.{{cite book|editor-last=Miller|editor-first=Leta E.|year=1988|title=Lou Harrison: Selected keyboard and chamber music, 1937-1994|page=XLIII|isbn=978-0-89579-414-7}}.]]

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.

class="wikitable"

|diminished

!subminor

|minor

|neutral

|major

!supermajor

|augmented

seconds

|D{{music|bb}}

|≊ D{{music|db}}

|D{{music|b}}

|D{{music|d}}

|≊ D{{music|t}}

|D{{music|#}}

thirds

|E{{music|bb}}

|≊ E{{music|db}}

|E{{music|b}}

|E{{music|d}}

|≊ E{{music|t}}

|E{{music|#}}

sixths

|A{{music|bb}}

|≊ A{{music|db}}

|A{{music|b}}

|A{{music|d}}

|≊ A{{music|t}}

|A{{music|#}}

sevenths

|B{{music|bb}}

|≊ B{{music|db}}

|B{{music|b}}

|B{{music|d}}

|≊ B{{music|t}}

|B{{music|#}}

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."Brabner, John H. F. (1884). [https://books.google.com/books?id=cy2P8q3RRI0C&pg=PA134 The National Encyclopaedia], vol. 13, p. 182. London. {{pre-ISBN}}

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{{music|13}}{{music|bb}}{{music|minus}} {{audio|Tridecimal third tone on C.mid|Play}}). Another example is the ratio 28:27, or 62.96 cents (C{{music|7}}{{music|#}}{{music|minus}} {{audio|Septimal minor second on C.mid|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{{music|13 upside down}}{{music|#}}); the ratio 27:14, or 1137.04 cents (B{{music|L}} {{audio|Septimal major seventh on C.mid|Play}}); and 35:18, or 1151.23 cents (C{{music|7}} {{audio|Septimal supermajor seventh on C.mid|Play}}).

Subminor third and supermajor sixth

File:Septimal minor third on C.png

{{multiple image|caption_align=center|header_align=center

| width = 150

| image1 = Subminor third on G.png

| alt1 =

| image2 = Supermajor sixth on B7b.png

| alt2 =

| footer = Subminor third on G {{audio|Subminor third on G.mid|Play}} and its inverse, the supermajor sixth on B{{music|7}}{{music|b}} {{audio|Supermajor sixth on B7b.mid|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{{music|7}}{{music|b}}), or 266.87 cents,{{cite book|last=Helmholtz|first=Hermann L. F. von|author-link=Hermann von Helmholtz|year=2007|title=On the Sensations of Tone|pages=195, 212|isbn=978-1-60206-639-7}}{{sfn|Miller|1988|p=XLII}} the septimal minor third, the inverse of the supermajor sixth. Another example is the ratio 13:11, or 289.21 cents (E{{music|13}}{{music|down}}{{music|b}}).

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{{music|L}}).Andrew Horner, Lydia Ayres (2002). Cooking with Csound: Woodwind and Brass Recipes, p. 131. {{ISBN|0-89579-507-8}}.Royal Society (Great Britain) (1880, digitized February 26, 2008). Proceedings of the Royal Society of London, vol. 30, p. 531. Harvard University.Society of Arts (Great Britain) (1877, digitized November 19, 2009). Journal of the Society of Arts, vol. 25, p. 670. In 24 equal temperament A{{music|t}} = {{nowrap|B{{music|db}}}}. The septimal major sixth is an interval of 12:7 ratio (A{{music|L}} {{audio|Septimal major sixth on C.mid|Play}}),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}}. or about 933 cents.{{sfn|Helmholtz|2007|p=456}} It is the inversion of the 7:6 subminor third.

Subminor sixth and supermajor third

{{anchor|1=Major fifth|2=Subminor sixth|3=Supermajor third|4=Minor fourth}}

File:Septimal minor sixth on C.png minor sixth (14/9) on C.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. {{audio|Septimal minor sixth on C.mid|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 (A{{music|7}}{{music|b}}) or alternately 11:7. (G{{music|up}}{{music|minus}} {{audio|Undecimal minor sixth on C.mid|Play}}) The 21st subharmonic (see subharmonic) is 729.22 cents. {{audio|21st subharmonic on C.mid|Play}}

File:Septimal major third on C.png

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{{music|L}}). Another example is the ratio 50:39, or 430.14 cents (E{{music|13U}}{{music|sharp}}).

Subminor seventh and supermajor second

{{multiple image|caption_align=center|header_align=center

| width = 150

| image1 = Harmonic seventh on C.png

| alt1 =

| image2 = Septimal major second on B7b.png

| alt2 =

| footer = Harmonic seventh {{audio|Harmonic seventh on C.mid|Play}} and its inverse, the septimal whole tone {{audio|Septimal major second on B7b.mid|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{{music|7}}{{music|b}}).

A supermajor second (or supersecond) 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, also known as the septimal whole tone (D{{music|L}}{{music|minus}} {{audio|Septimal major second on C.mid|Play}}) and the inverse of the subminor seventh. Another example is the ratio 15:13, or 247.74 cents (D{{music|13U}}{{music|sharp}}).

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).Miller and Lieberman (2006), p. 72.{{incomplete short citation|date=October 2021}} Together the two produce the 4:3 just perfect fourth.Miller & Lieberman (2006), p. 74. "The subminor third and supermajor second combine to create a pure fourth ({{math|{{frac|8|7}} x {{frac|7|6}} {{=}} {{frac|4|3}}}})."{{incomplete short citation|date=October 2021}}

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

Category:Intervals (music)