7-limit tuning

7-limit is a musical tuning where the largest prime number factor of the interval ratios between pitches is seven. The only primes available in septimal tuning are 2, 3, 5, and 7. Limit is a term devised by Harry Partch.Wolf, Daniel James. "Alternative Tunings, Alternative Tonalities", Contemporary Music Review, vol. 22, nos. 1–2. March 2003. 13.

History In the 2nd century, Ptolemy 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: An Account of a Creative Work, Its Roots, and Its Fulfillments, pp. 90–91. . Archytas of Tarantum is the oldest recorded musicologist to calculate 7-limit tuning systems. Those considering 7 to be consonant include Marin Mersenne,Shirlaw, Matthew. Theory of Harmony. Da Capo Press, 1969. 32. Giuseppe Tartini, Leonhard Euler, François-Joseph Fétis, J. A. Serre, Moritz Hauptmann, Alexander John Ellis, Wilfred Perrett, Max Friedrich Meyer. Those considering 7 to be dissonant include Gioseffo Zarlino, René Descartes, Jean-Philippe Rameau, Hermann von Helmholtz, Arthur von Oettingen, Hugo Riemann, Colin Brown, and Paul Hindemith ("chaos"Hindemith, Paul (1942). Craft of Musical Composition, vol. 1, p. 38. .).

Claudius Ptolemy of Alexandria described several 7-limit tuning systems for the diatonic and chromatic 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 philosopher and statesman Archytas of Tarentum. It used the following tetrachord: 28:27, 8:7, 9:8. Ptolemy also shares the "soft diatonic" according to peripatetic philosopher Aristoxenus 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. Usage The lesser just minor seventh, 16:9 () is a 3-limit ratio, the harmonic seventh has the ratio 7:4 and is thus a septimal interval. Similarly, the septimal chromatic semitone, 21:20, is a septimal interval as 21÷7=3. The harmonic seventh is used in the barbershop seventh chord and music. () Compositions with septimal tunings include La Monte Young'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.

Great Highland bagpipe tuning can be described as a seven tone 7-limit scale. The instrument's drone is a slightly sharper A than standard. The scale ratios are (7:8), 1:1(A), 9:8, 5:4, 4:3, 3:2, 5:3, 7:4, (2:1).Benson, Dave. Music: A Mathematical Offering. University of Aberdeen, 2008.

Lattice and tonality diamond

The 7-limit tonality diamond: 7/4 3/2 7/5 5/4 6/5 7/6 1/1 1/1 1/1 1/1 8/5 5/3 12/7 4/3 10/7 8/7

This diamond contains four identities (1, 3, 5, 7 [P8, 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.

Fraction Cents Degree (31-ET) Name (31-ET) 1/1 0 0 C 8/7 231.174 6 D 7/6 266.871 7 D 6/5 315.641 8 E 5/4 386.314 10 E 4/3 498.045 13 F 7/5 582.512 15 F 10/7 617.488 16 G 3/2 701.955 18 G 8/5 814.686 21 A 5/3 884.359 23 A 12/7 933.129 24 A 7/4 968.826 25 A 2/1 1200 31 C

7-limit tuning

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