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7-limit Tuning

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Title: 7-limit Tuning  
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Subject: Musical tuning, Cent (music), List of intervals in 5-limit just intonation, Interval ratio, Ben Johnston (composer)
Collection: 7-Limit Tuning and Intervals
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7-limit Tuning

Harmonic seventh About this sound   , septimal seventh.
Septimal chromatic semitone on C About this sound   .
9/7 major third from C to E7 upside-down [1] About this sound   . This, "extremely large third," may resemble a neutral third or blue note.[2]
Septimal minor third on C About this sound   .

7-limit or septimal tunings and intervals are musical instrument tunings that have a limit of seven: the largest number contained in the interval ratios between pitches is a multiple of seven.

For example, the greater just minor seventh, 9:5 About this sound    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. (About this sound   ) 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]

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.

See also

Sources

  1. ^ 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.
  2. ^ Fonville (1991), p.128.
  3. ^ Benson, Dave (2007). Music: A Mathematical Offering, p.212. ISBN 9780521853873.
  4. ^ 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.
  5. ^ Shirlaw, Matthew (1900). Theory of Harmony, p.32. ISBN 978-1-4510-1534-8.
  6. ^ Hindemith, Paul (1942). Craft of Musical Composition, v.1, p.38. ISBN 0901938300.
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