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.^[2]^: 232 Limit is a term devised by Harry Partch.^[3]

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.^[4] Archytas of Tarantum is the oldest recorded musicologist to calculate 7-limit tuning systems. Those considering 7 to be consonant include Marin Mersenne,^[5] Giuseppe Tartini, Leonhard Euler, François-Joseph Fétis, J. A. Serre, Moritz Hauptmann, Alexander John Ellis, Wilfred Perrett, Max Friedrich Meyer.^[4] 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"^[6]).^[4]

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.^[7] 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 (Play^ⓘ) 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. (Play^ⓘ) 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).^[2]^: 201

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

References

↑ Fonville, John. "Ben Johnston's Extended Just Intonation – A Guide for Interpreters", Perspectives of New Music, vol. 29, no. 2. Summer, 1991. pp. 106–137.
1 2 Benson, Dave. Music: A Mathematical Offering. University of Aberdeen, 2008.
↑ Wolf, Daniel James. "Alternative Tunings, Alternative Tonalities", Contemporary Music Review, vol. 22, nos. 1–2. March 2003. 13.
1 2 3 Partch, Harry (2009). Genesis of a Music: An Account of a Creative Work, Its Roots, and Its Fulfillments, pp. 90–91. ISBN 9780786751006.
↑ Shirlaw, Matthew. Theory of Harmony. Da Capo Press, 1969. 32.
↑ Hindemith, Paul (1942). Craft of Musical Composition, vol. 1, p. 38. ISBN 0901938300.
↑ Barker, Andrew (1989). Greek Musical Writings: II Harmonic and Acoustic Theory. Cambridge: Cambridge University Press. ISBN 0521616972.

External links

Centaur a 7 limit tuning shows Centaur tuning plus other related 7 tone tunings by others
Soft diatonic scale at Sevish.com

[JF-1] Fonville, John. "Ben Johnston's Extended Just Intonation – A Guide for Interpreters", Perspectives of New Music, vol. 29, no. 2. Summer, 1991. pp. 106–137.

[DB-2] 1 2 Benson, Dave. Music: A Mathematical Offering. University of Aberdeen, 2008.

[3] Wolf, Daniel James. "Alternative Tunings, Alternative Tonalities", Contemporary Music Review, vol. 22, nos. 1–2. March 2003. 13.

[Partch-4] 1 2 3 Partch, Harry (2009). Genesis of a Music: An Account of a Creative Work, Its Roots, and Its Fulfillments, pp. 90–91. ISBN 9780786751006.

[5] Shirlaw, Matthew. Theory of Harmony. Da Capo Press, 1969. 32.

[6] Hindemith, Paul (1942). Craft of Musical Composition, vol. 1, p. 38. ISBN 0901938300.

[7] Barker, Andrew (1989). Greek Musical Writings: II Harmonic and Acoustic Theory. Cambridge: Cambridge University Press. ISBN 0521616972.

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