Origin of seconds and thirds in harmonic series

music Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspect ...

, a subminor interval is an interval that is noticeably wider than a diminished interval but noticeably narrower than a

minor interval In music theory, an interval is a difference in pitch (music), pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and ...

. 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 In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or h ...

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 Inversion or inversions may refer to: Arts * , a French gay magazine (1924/1925) * ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas * Inversion (music), a term with various meanings in music theory and musical set theory * ...

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. Traditionally, "supermajor and superminor, rethe names given to certain thirds :7 and 17:14found in the justly intoned scale with a natural or subminor seventh."Brabner, John H. F. (1884).
The National Encyclopaedia
', vol. 13, p. 182. London.

Subminor second and supermajor seventh

Thus, a subminor second is intermediate between a

minor second A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent no ...

and a

diminished second In modern Western tonal music theory, a diminished second is the interval produced by narrowing a minor second by one chromatic semitone.Bruce Benward and Marilyn Saker (2003). ''Music: In Theory and Practice, Vol. I'', p. 54. . Specific example ...

(

enharmonic In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. The enharmonic spelling of a written n ...

unison In music, unison is two or more musical parts that sound either the same pitch or pitches separated by intervals of one or more octaves, usually at the same time. ''Rhythmic unison'' is another term for homorhythm. Definition Unison or per ...

). An example of such an interval is the ratio 26:25, or 67.90 cents (D ). Another example is the ratio 28:27, or 62.96 cents (C ). A supermajor seventh is an interval intermediate between a

major seventh In music from Western culture, a seventh is a musical interval encompassing seven staff positions (see Interval number for more details), and the major seventh is one of two commonly occurring sevenths. It is qualified as ''major'' because it i ...

and an

augmented seventh In classical music from Western culture, an augmented seventh is an interval produced by widening a major seventh by a chromatic semitone. For instance, the interval from C to B is a major seventh, eleven semitones wide, and both the interv ...

. 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 ); and 35:18, or 1151.23 cents (C ).

Subminor third and supermajor sixth

A subminor third is in between a

minor third In music theory, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions (see: interval number). The minor third is one of two com ...

and a

diminished third In classical music from Western culture, a diminished third () is the musical interval produced by narrowing a minor third by a chromatic semitone.Benward & Saker (2003). ''Music: In Theory and Practice, Vol. I'', p.54. . For instance, the inte ...

. An example of such an interval is the ratio 7:6 (E), or 266.87 cents, the

septimal minor third In music, the septimal minor third, also called the subminor third (e.g., by Ellis), is the musical interval exactly or approximately equal to a 7/6 ratio of frequencies. In terms of cents, it is 267 cents, a quartertone of size 36/35 flatter ...

, 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 In music from Western culture, a sixth is a musical interval encompassing six note letter names or staff positions (see Interval number for more details), and the major sixth is one of two commonly occurring sixths. It is qualified as ''major ...

but noticeably narrower than an

augmented sixth In classical music from Western culture, an augmented sixth () is an interval produced by widening a major sixth by a chromatic semitone.Benward & Saker (2003). ''Music: In Theory and Practice, Vol. I'', p.54. . Specific example of an A6 not g ...

, and may be a just interval of 12:7 (A).Andrew Horner, Lydia Ayres (2002). ''Cooking with Csound: Woodwind and Brass Recipes'', p. 131. .Royal Society (Great Britain) (1880, digitized February 26, 2008). ''

Proceedings of the Royal Society of London ''Proceedings of the Royal Society'' is the main research journal of the Royal Society. The journal began in 1831 and was split into two series in 1905: * Series A: for papers in physical sciences and mathematics. * Series B: for papers in life s ...

'', vol. 30, p. 531. Harvard University.Society of Arts (Great Britain) (1877, digitized November 19, 2009). ''

Journal of the Society of Arts The Royal Society for the Encouragement of Arts, Manufactures and Commerce (RSA), also known as the Royal Society of Arts, is a London-based organisation committed to finding practical solutions to social challenges. The RSA acronym is used m ...

'', vol. 25, p. 670. In 24 equal temperament A = . The septimal major sixth is an interval of 12:7 ratio (A ), or about 933 cents. It is the

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 In classical music from Western culture, a diminished sixth () is an interval produced by narrowing a minor sixth by a chromatic semitone.Benward & Saker (2003). ''Music: In Theory and Practice, Vol. I'', p.54. . Specific example of an d6 not ...

, enharmonically equivalent to the major fifth. The sub-minor sixth is an interval of a 14:9 ratio (A) or alternately 11:7. (G ) The 21st subharmonic (see subharmonic) is 729.22 cents. Septimal major third on C

A supermajor third is in between a

major third In classical music, a third is a musical interval encompassing three staff positions (see Interval number for more details), and the major third () is a third spanning four semitones. Forte, Allen (1979). ''Tonal Harmony in Concept and P ...

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 In music, the septimal major third , also called the supermajor third (by Hermann von Helmholtz among others Hermann L. F. von Helmholtz (2007). ''Sensations of Tone'', p. 187. .) and sometimes '' Bohlen–Pierce third'' is the musical interval ...

(E). Another example is the ratio 50:39, or 430.14 cents (E).

Subminor seventh and supermajor second

A subminor seventh is an interval between a

minor seventh In music theory, a minor seventh is one of two musical intervals that span seven staff positions. It is ''minor'' because it is the smaller of the two sevenths, spanning ten semitones. The major seventh spans eleven. For example, the interval f ...

and a

diminished seventh In classical music from Western culture, a diminished seventh () is an interval produced by narrowing a minor seventh by a chromatic semitone.Benward & Saker (2003). ''Music: In Theory and Practice, Vol. I'', p.54. . Specific example of an d7 ...

. An example of such an interval is the 7:4 ratio, the

harmonic seventh The harmonic seventh interval, also known as the septimal minor seventh, or subminor seventh, is one with an exact 7:4 ratio (about 969 cents). This is somewhat narrower than and is, "particularly sweet", "sweeter in quality" than an "ordinar ...

(B). A supermajor second (or supersecond) is intermediate to a

major second In Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (). A second is a musical interval encompassing two adjacent staff positions (see Interval number for more deta ...

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 In music, the septimal whole tone, septimal major second, or supermajor second is the musical interval exactly or approximately equal to an 8/7 ratio of frequencies.Andrew Horner, Lydia Ayres (2002). ''Cooking with Csound: Woodwind and Brass R ...

(D ) and the inverse of the subminor seventh. Another example is the ratio 15:13, or 247.74 cents (D).

Use

Composer

Lou Harrison Lou Silver Harrison (May 14, 1917 – February 2, 2003) was an American composer, music critic, music theorist, painter, and creator of unique musical instruments. Harrison initially wrote in a dissonant, ultramodernist style similar to his for ...

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 Gamelan () ( jv, ꦒꦩꦼꦭꦤ꧀, su, ᮌᮙᮨᮜᮔ᮪, ban, ᬕᬫᭂᬮᬦ᭄) is the traditional ensemble music of the Javanese, Sundanese, and Balinese peoples of Indonesia, made up predominantly of percussive instruments. T ...

'', ''Cinna'' for tack-piano, and ''Strict Songs'' (for voices and orchestra). Together the two produce the 4:3 just

perfect fourth A fourth is a musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth () is the fourth spanning five semitones (half steps, or half tones). For example, the ascending interval from C to ...

.Miller & Lieberman (2006), p. 74. "The subminor third and supermajor second combine to create a pure fourth ()."

19 equal temperament In music, 19 Tone Equal Temperament, called 19 TET, 19 EDO ("Equal Division of the Octave"), or 19 ET, is the tempered scale derived by dividing the octave into 19 equal steps (equal frequency ratios). Each step represent ...

has several intervals which are simultaneously subminor, supermajor, augmented, and diminished, due to tempering and

enharmonic equivalence In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. The enharmonic spelling of a written no ...

(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

{{Intervals Intervals (music)

Subminor second and supermajor seventh

Subminor third and supermajor sixth

Subminor sixth and supermajor third

Subminor seventh and supermajor second

Use

See also

References