Subminor and supermajor

[[File:Origin of seconds and thirds in harmonic series.png|thumb|upright=1.4|Origin of large and small seconds and thirds (including 7:6) in harmonic series.<ref name="Harrison">{{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}}.</ref>]]

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 |≊ D{{music|t}} |D{{music|#}} |- |thirds |E{{music|bb}} |≊ E{{music|db}} |E{{music|b}} |E{{music|d}} |E |≊ E{{music|t}} |E{{music|#}} |- |sixths |A{{music|bb}} |≊ A{{music|db}} |A{{music|b}} |A{{music|d}} |A |≊ A{{music|t}} |A{{music|#}} |- |sevenths |B{{music|bb}} |≊ B{{music|db}} |B{{music|b}} |B{{music|d}} |B |≊ B{{music|t}} |B{{music|#}} |}

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

==Subminor second and supermajor seventh== {{anchor|Subminor second|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== {{anchor|Subminor third|Supermajor sixth}} thumb|Septimal minor third on C {{audio|Septimal minor third on C.mid|Play}} {{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,<ref>{{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}}</ref>{{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}}).<ref name="Andrew Horner 2002 p.131">Andrew Horner, Lydia Ayres (2002). ''Cooking with Csound: Woodwind and Brass Recipes'', p. 131. {{ISBN|0-89579-507-8}}.</ref><ref name="Royal Society 1880, p.531">Royal Society (Great Britain) (1880, digitized February 26, 2008). ''Proceedings of the Royal Society of London'', vol. 30, p. 531. Harvard University.</ref><ref name="Arts 1877, p.670">Society of Arts (Great Britain) (1877, digitized November 19, 2009). ''Journal of the Society of Arts'', vol. 25, p. 670.</ref> 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}}),<ref>Partch, Harry (1979). ''Genesis of a Music'', p. 68. {{ISBN|0-306-80106-X}}.</ref><ref>Haluska, Jan (2003). ''The Mathematical Theory of Tone Systems'', p. xxiii. {{ISBN|0-8247-4714-3}}.</ref> 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|thumb|Septimal minor sixth (14/9) on C.<ref>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.</ref> {{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<ref name="Royal Society 1880, p.531"/><ref name="Arts 1877, p.670"/> (A{{music|7}}{{music|b}}) or alternately 11:7.<ref name="Andrew Horner 2002 p.131"/> (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}}

thumb|Septimal major third on C {{audio|Septimal major third on C.mid|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{{music|L}}). Another example is the ratio 50:39, or 430.14 cents (E{{music|13U}}{{music|sharp}}).

==Subminor seventh and supermajor second== {{main|Harmonic seventh|Septimal whole tone}} {{anchor|Subminor seventh|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<ref name="National"/>) 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,<ref name="Harrison"/> 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).<ref>Miller and Lieberman (2006), p. 72.{{incomplete short citation|date=October 2021}}</ref> Together the two produce the 4:3 just perfect fourth.<ref>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}}</ref>

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== {{reflist}}

Category:Intervals (music)