7-limit or septimal tunings and

intervals Interval may refer to: Mathematics and physics * Interval (mathematics), a range of numbers ** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets * A statistical level of measurement * Interval est ...

are musical instrument tunings that have a

limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

seven 7 is a number, numeral, and glyph. 7 or seven may also refer to: * AD 7, the seventh year of the AD era * 7 BC, the seventh year before the AD era * The month of July Music Artists * Seven (Swiss singer) (born 1978), a Swiss recording artist ...

: the largest

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

factor contained in the

interval ratio In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth (for example C to G) is 3:2 (), 1.5, and may be approximated by an equal tempered perfect fifth () which is 27 ...

s between pitches is seven. Thus, for example, 50:49 is a 7-limit interval, but 14:11 is not. For example, the greater just minor seventh, 9:5 () is a

5-limit Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note ...

ratio, the harmonic seventh has the ratio 7:4 and is thus a septimal interval. Similarly, the

septimal chromatic semitone In music, a septimal chromatic semitone or minor semitoneHaluska, Jan (2003). ''The Mathematical Theory of Tone Systems'', p.xxiv. . is the interval 21:20 (). It is about 84.47 cents. The septimal chromatic semitone may be derived from the har ...

, 21:20, is a septimal interval as 21÷7=3. The harmonic seventh is used in the

barbershop seventh chord The harmonic seventh chord is a major triad plus the harmonic seventh interval (ratio of 7:4, about 968.826 centsBosanquet, Robert Holford Macdowall (1876). ''An elementary treatise on musical intervals and temperament'', pp. 41-42. Diapason P ...

and

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 ...

. () Compositions with septimal tunings include

La Monte Young La Monte Thornton Young (born October 14, 1935) is an American composer, musician, and performance artist recognized as one of the first American minimalist composers and a central figure in Fluxus and post-war avant-garde music. He is best kno ...

's '' The Well-Tuned Piano'', Ben Johnston's String Quartet No. 4, Lou Harrison's ''Incidental Music for Corneille's Cinna'', and Michael Harrison's ''Revelation: Music in Pure Intonation''. The

Great Highland bagpipe The Great Highland bagpipe ( gd, a' phìob mhòr "the great pipe") is a type of bagpipe native to Scotland, and the Scottish analogue to the Great Irish Warpipes. It has acquired widespread recognition through its usage in the British milit ...

is tuned to a ten-note seven-limit

scale Scale or scales may refer to: Mathematics * Scale (descriptive set theory), an object defined on a set of points * Scale (ratio), the ratio of a linear dimension of a model to the corresponding dimension of the original * Scale factor, a number ...

: 1:1, 9:8, 5:4, 4:3, 27:20, 3:2, 5:3, 7:4, 16:9, 9:5. In the 2nd century

Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...

described the septimal intervals: 7/4, 8/7, 7/6, 12/7, 7/5, and 10/7. Partch, Harry (2009). ''

Genesis of a Music ''Genesis of a Music'' is a book first published in 1949 by microtonal composer Harry Partch (1901–1974). Partch first presents a polemic against both equal temperament and the long history of stagnation in the teaching of music; according t ...

: An Account of a Creative Work, Its Roots, and Its Fulfillments'', pp. 90–91. . Those considering 7 to be

consonant In articulatory phonetics, a consonant is a speech sound that is articulated with complete or partial closure of the vocal tract. Examples are and pronounced with the lips; and pronounced with the front of the tongue; and pronounced wit ...

include

Marin Mersenne Marin Mersenne, OM (also known as Marinus Mersennus or ''le Père'' Mersenne; ; 8 September 1588 – 1 September 1648) was a French polymath whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for ...

Giuseppe Tartini Giuseppe Tartini (8 April 1692 – 26 February 1770) was an Italian composer and violinist of the Baroque era born in the Republic of Venice. Tartini was a prolific composer, composing over a hundred of pieces for the violin with the majority of ...

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

, François-Joseph Fétis, J. A. Serre,

Moritz Hauptmann Moritz Hauptmann (13 October 1792, Dresden – 3 January 1868, Leipzig), was a German music theorist, teacher and composer. His principal theoretical work is the 1853 ''Die Natur der Harmonie und der Metrik'' explores numerous topics, particular ...

Alexander John Ellis Alexander John Ellis, (14 June 1814 – 28 October 1890), was an English mathematician, philologist and early phonetician who also influenced the field of musicology. He changed his name from his father's name, Sharpe, to his mother's maiden na ...

, Wilfred Perrett,

Max Friedrich Meyer Max Friedrich Meyer (June 14, 1873 – March 14, 1967) was the first psychology professor who worked on psychoacoustics and taught at the University of Missouri. He was the founder of the theory of cochlear function, and was also an advocate ...

. Those considering 7 to be dissonant include

Gioseffo Zarlino Gioseffo Zarlino (31 January or 22 March 1517 – 4 February 1590) was an Italian music theorist and composer of the Renaissance. He made a large contribution to the theory of counterpoint as well as to musical tuning. Life and career Zarlino w ...

René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...

Jean-Philippe Rameau Jean-Philippe Rameau (; – ) was a French composer and music theory, music theorist. Regarded as one of the most important French composers and music theorists of the 18th century, he replaced Jean-Baptiste Lully as the dominant composer of Fr ...

Hermann von Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association, ...

, Arthur von Oettingen,

Hugo Riemann Karl Wilhelm Julius Hugo Riemann (18 July 1849 – 10 July 1919) was a German musicologist and composer who was among the founders of modern musicology. The leading European music scholar of his time, he was active and influential as both a musi ...

, Colin Brown, and

Paul Hindemith Paul Hindemith (; 16 November 189528 December 1963) was a German composer, music theorist, teacher, violist and conductor. He founded the Amar Quartet in 1921, touring extensively in Europe. As a composer, he became a major advocate of the ''Ne ...

("chaos" Hindemith, Paul (1942). ''Craft of Musical Composition'', vol. 1, p. 38. .).

Lattice and tonality diamond

The 7-limit tonality diamond: This diamond contains four identities (1, 3, 5, 7 8, P5, M3, H7. Similarly, the 2,3,5,7

pitch lattice In musical tuning, a lattice "is a way of modeling the tuning relationships of a just intonation system. It is an array of points in a periodic multidimensional pattern. Each point on the lattice corresponds to a ratio (i.e., a pitch, or an i ...

contains four identities and thus 3-4 axes, but a potentially infinite number of pitches. LaMonte Young created a lattice containing only identities 3 and 7, thus requiring only two axes, for ''The Well-Tuned Piano''.

Approximation using equal temperament

It is possible to approximate 7-limit music using equal temperament, for example 31-ET.

References

External links

Centaur a 7 limit tuning
shows Centaur tuning plus other related 7 tone tunings by others {{Musical tuning

Lattice and tonality diamond

Approximation using equal temperament

See also

References

External links