7-limit orseptimaltunings andintervals are musical instrument tunings that have alimit of seven: the largestprime factor contained in theinterval ratios betweenpitches is seven. Thus, for example, 50:49 is a 7-limit interval, but 14:11 is not.
.[1] This, "extremely large third", may resemble aneutral third orblue note.[2]
For example, the greater justminor seventh, 9:5 (Playⓘ) is a5-limit ratio, theharmonic seventh has the ratio 7:4 and is thus a septimal interval. Similarly, theseptimal chromatic semitone, 21:20, is a septimal interval as 21÷7=3. The harmonic seventh is used in thebarbershop seventh chord andmusic. (Playⓘ) Compositions with septimal tunings includeLa Monte Young'sThe Well-Tuned Piano,Ben Johnston's String Quartet No. 4,Lou Harrison'sIncidental Music for Corneille's Cinna, andMichael Harrison'sRevelation: Music in Pure Intonation.
TheGreat Highland bagpipe is tuned to a ten-note seven-limitscale:[3]1:1,9:8,5:4,4:3,27:20,3:2,5:3,7:4,16:9,9:5.
In the 2nd centuryPtolemy described the septimal intervals: 21/20, 7/4, 8/7, 7/6, 9/7, 12/7, 7/5, and 10/7.[4]Archytas ofTarantum is the oldest recorded musicologist to calculate 7-limit tuning systems. Those considering 7 to beconsonant includeMarin 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 includeGioseffo Zarlino,René Descartes,Jean-Philippe Rameau,Hermann von Helmholtz,Arthur von Oettingen,Hugo Riemann, Colin Brown, andPaul Hindemith ("chaos"[6]).[4]
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 fouridentities (1, 3, 5, 7 [P8, P5, M3, H7]). Similarly, the 2,3,5,7pitch 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, forThe Well-Tuned Piano.
Approximation using equal temperament
editIt is possible to approximate 7-limit music using equal temperament, for example31-ET.
| Fraction | Cents | Degree (31-ET) | Name (31-ET) |
|---|---|---|---|
| 1/1 | 0 | 0.0 | C |
| 8/7 | 231 | 6.0 | D or E |
| 7/6 | 267 | 6.9 | D♯ |
| 6/5 | 316 | 8.2 | E♭ |
| 5/4 | 386 | 10.0 | E |
| 4/3 | 498 | 12.9 | F |
| 7/5 | 583 | 15.0 | F♯ |
| 10/7 | 617 | 16.0 | G♭ |
| 3/2 | 702 | 18.1 | G |
| 8/5 | 814 | 21.0 | A♭ |
| 5/3 | 884 | 22.8 | A |
| 12/7 | 933 | 24.1 | A or B |
| 7/4 | 969 | 25.0 | A♯ |
| 2/1 | 1200 | 31.0 | C |
Ptolemy'sHarmonikon
editClaudius Ptolemy of Alexandria described several 7-limit tuning systems for thediatonic andchromatic 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 thePythagorean philosopher and statesmanArchytas of Tarentum. It used the followingtetrachord: 28:27, 8:7, 9:8. Ptolemy also shares the "soft diatonic" according toperipatetic philosopherAristoxenus 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.
See also
editReferences
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.
- ^abcPartch, Harry (2009).Genesis of a Music: An Account of a Creative Work, Its Roots, and Its Fulfillments, pp. 90–91.ISBN 9780786751006.
- ^Shirlaw, Matthew (1900).Theory of Harmony, p. 32.ISBN 978-1-4510-1534-8.
- ^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
edit- Centaur a 7 limit tuning shows Centaur tuning plus other related 7 tone tunings by others