# 7-limit tuning

**7-limit** or **septimal** tunings and intervals are musical instrument tunings that have a limit of seven: the largest prime factor contained in the interval ratios 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 Play (help·info) is a 5-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 (help·info)) Compositions with septimal tunings include La Monte Young's *The Well-Tuned Piano*, Ben Johnston's String Quartet No. 4, and Lou Harrison's *Incidental Music for Corneille's Cinna*.

The Great Highland Bagpipe is tuned to a ten-note seven-limit scale:^{[3]} 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 described the septimal intervals: 7/4, 8/7, 7/6, 12/7, 7/5, and 10/7,^{[4]} 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, A. J. von Öttingen, Hugo Riemann, Colin Brown, and Paul Hindemith ("chaos"^{[6]}).^{[4]}

## Contents

## Lattice and tonality diamond[edit]

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[edit]

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

## See also[edit]

## Sources[edit]

**^**Fonville, John. "Ben Johnston's Extended Just Intonation- A Guide for Interpreters", p.112,*Perspectives of New Music*, Vol. 29, No. 2 (Summer, 1991), pp. 106-137.**^**Fonville (1991), p.128.**^**Benson, Dave (2007).*Music: A Mathematical Offering*, p.212. ISBN 9780521853873.- ^
^{a}^{b}^{c}Partch, Harry (2009).*Genesis of a Music: An Account of a Creative Work, Its Roots, and Its Fulfillments*, p.90-1. ISBN 9780786751006. **^**Shirlaw, Matthew (1900).*Theory of Harmony*, p.32. ISBN 978-1-4510-1534-8.**^**Hindemith, Paul (1942).*Craft of Musical Composition*, v.1, p.38. ISBN 0901938300.