7-limit or septimal tunings and intervals are musical instrument tunings that have a limit of seven: 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 temperament, equal tempered perfect ...

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 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 ...

, 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 not ...

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 cent (music), cents). This is about 32 cents narrower, with a more stable and consonant sound, than a ...

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 Cent (music), cents. The septimal chromatic semitone may be derived ...

, 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 chord, major triad plus the harmonic seventh interval (ratio of 7:4, about 968.826 cent (music), centsBosanquet, Robert Holford Macdowall (1876). ''An elementary treatise on musical intervals and temperamen ...

and

music Music is the arrangement of sound to create some combination of Musical form, form, harmony, melody, rhythm, or otherwise Musical expression, expressive content. Music is generally agreed to be a cultural universal that is present in all hum ...

. () 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 k ...

's ''

The Well-Tuned Piano ''The Well-Tuned Piano'' is an ongoing improvisatory solo piano work begun in 1964 by La Monte Young. Young has never considered the composition or performance of this piece finished, and he has performed it differently several times since its ...

'', Ben Johnston's String Quartet No. 4,

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 ...

's ''Incidental Music for Corneille's Cinna'', and Michael Harrison's ''Revelation: Music in Pure Intonation''. The

Great Highland bagpipe The great Highland bagpipe ( '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 Armed Forces, British mili ...

is tuned to a ten-note seven-limit scale: 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 (; , ; ; – 160s/170s AD) was a Greco-Roman mathematician, astronomer, astrologer, geographer, and music theorist who wrote about a dozen scientific treatises, three of which were important to later Byzantine science, Byzant ...

described the septimal intervals: 21/20, 7/4, 8/7, 7/6, 9/7, 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 ...

: An Account of a Creative Work, Its Roots, and Its Fulfillments'', pp. 90–91. .

Archytas Archytas (; ; 435/410–360/350 BC) was an Ancient Greek mathematician, music theorist, statesman, and strategist from the ancient city of Taras (Tarentum) in Southern Italy. He was a scientist and philosopher affiliated with the Pythagorean ...

of Tarantum is the oldest recorded musicologist to calculate 7-limit tuning systems. 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, except for the h sound, which is pronounced without any stricture in the vocal tract. Examples are and pronou ...

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 Pirano in the Republic of Venice (now Piran, Slovenia). Tartini was a prolific composer, composing over a hundred pieces for the ...

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

François-Joseph Fétis François-Joseph Fétis (; 25 March 1784 – 26 March 1871) was a Belgian musicologist, critic, teacher and composer. He was among the most influential music intellectuals in continental Europe. His enormous compilation of biographical data in the ...

, J. A. Serre, Moritz Hauptmann,

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 nam ...

, Wilfred Perrett, Max Friedrich Meyer. Those considering 7 to be dissonant include

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

René Descartes René Descartes ( , ; ; 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 Modern science, science. Mathematics was paramou ...

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 ...

Hermann von Helmholtz Hermann Ludwig Ferdinand von Helmholtz (; ; 31 August 1821 – 8 September 1894; "von" since 1883) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The ...

Arthur von Oettingen Arthur Joachim von Oettingen ( – 5 September 1920) was a Baltic German physicist and music theorist. He was the brother of theologian Alexander von Oettingen (1827–1905) and ophthalmologist Georg von Oettingen (1824–1916). Biography ...

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 mus ...

, Colin Brown, and

Paul Hindemith Paul Hindemith ( ; ; 16 November 189528 December 1963) was a German and American 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 advo ...

("chaos").

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 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.

Ptolemy's ''Harmonikon''

Claudius Ptolemy Claudius Ptolemy (; , ; ; – 160s/170s AD) was a Greco-Roman mathematician, astronomer, astrologer, geographer, and music theorist who wrote about a dozen scientific treatises, three of which were important to later Byzantine, Islamic, and ...

of Alexandria described several 7-limit tuning systems for the

diatonic Diatonic and chromatic are terms in music theory that are used to characterize scales. The terms are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a pair ...

and

chromatic Diatonic and chromatic are terms in music theory that are used to characterize scales. The terms are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a pair, es ...

genera. He describes several "soft" ( μαλακός) diatonic tunings which all use 7-limit intervals. One, called by Ptolemy the
tonic diatonic
" is ascribed to the

Pythagorean Pythagorean, meaning of or pertaining to the ancient Ionian mathematician, philosopher, and music theorist Pythagoras, may refer to: Philosophy * Pythagoreanism, the esoteric and metaphysical beliefs purported to have been held by Pythagoras * Ne ...

philosopher and statesman

Archytas of Tarentum Archytas (; ; 435/410–360/350 BC) was an Ancient Greek mathematician, music theorist, statesman, and strategist from the ancient city of Taras (Tarentum) in Southern Italy. He was a scientist and philosopher affiliated with the Pythagorean sch ...

. It used the following tetrachord: 28:27, 8:7, 9:8. Ptolemy also shares the
soft diatonic
according to peripatetic philosopher

Aristoxenus Aristoxenus of Tarentum (; born 375, fl. 335 BC) was a Ancient Greece, Greek Peripatetic school, Peripatetic philosopher, and a pupil of Aristotle. Most of his writings, which dealt with philosophy, ethics and music, have been lost, but one musi ...

of Tarentum: 20:19, 38:35, 7:6. Ptolemy offers his own
soft diatonic
as the best alternative to Archytas and Aristoxenus, with a tetrachord of: 21:20, 10:9, 8:7. Ptolemy also describes a "tense chromatic" tuning that utilizes the following tetrachord: 22:21, 12:11, 7:6.

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

Ptolemy's ''Harmonikon''

See also

References

External links