{{Main|Limit (music)}} {{Short description|Musical instrument tuning with a limit of seven}} [[File:Harmonic seventh on C.png|thumb|Harmonic seventh, septimal seventhFile:Harmonic seventh on C.mid]] [[File:Septimal chromatic semitone on C.png|thumb|Septimal chromatic semitone on CFile:Septimal chromatic semitone on C.mid]] [[File:Septimal major third on C.png|thumb|A 9/7 major third from C to E{{music|L}} resembles a supermajor third or blue note.<ref name=JF>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.</ref>{{rp|112, 128}}File:Septimal major third on C.mid]] [[File:Septimal minor third on C.png|thumb|Septimal minor third on CFile:Septimal minor third on C.mid]]
'''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.<ref name=DB/>{{rp|232}} Limit is a term devised by Harry Partch.<ref>Wolf, Daniel James. "[https://doi.org/10.1080/0749446032000134715 Alternative Tunings, Alternative Tonalities]", ''Contemporary Music Review'', vol. 22, nos. 1–2. March 2003. 13.</ref>
==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.<ref name="Partch">Partch, Harry (2009). ''Genesis of a Music: An Account of a Creative Work, Its Roots, and Its Fulfillments'', pp. 90–91. {{ISBN|9780786751006}}.</ref> Archytas of Tarantum is the oldest recorded musicologist to calculate 7-limit tuning systems. Those considering 7 to be consonant include Marin Mersenne,<ref>Shirlaw, Matthew. ''[https://archive.org/details/theoryofharmonyi0000shir/page/32/mode/1up Theory of Harmony]''. Da Capo Press, 1969. 32.</ref> Giuseppe Tartini, Leonhard Euler, François-Joseph Fétis, J. A. Serre, Moritz Hauptmann, Alexander John Ellis, Wilfred Perrett, Max Friedrich Meyer.<ref name="Partch" /> 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"<ref>Hindemith, Paul (1942). ''Craft of Musical Composition'', vol. 1, p. 38. {{ISBN|0901938300}}.</ref>).<ref name="Partch" />
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.<ref>{{Cite book |last=Barker |first=Andrew |title=Greek Musical Writings: II Harmonic and Acoustic Theory |publisher=Cambridge University Press |year=1989 |isbn=0521616972 |location=Cambridge}}</ref> One, called by Ptolemy the "[https://sevish.com/scaleworkshop/?n=Tonic%20Diatonic&l=sFr_wFr_4F3_3F2_eF9_gF9_2F1&version=2.1.0 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 "[https://sevish.com/scaleworkshop/?n=Aristoxenus%20soft%20diatonic&l=kFj_8F7_4F3_3F2_uFj_cF7_2F1__&version=2.1.0 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 ({{audio|Lesser just minor seventh on C.mid|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. ({{audio|Barbershop secondary dominant.mid|Play|help=no}}) 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).<ref name=DB>Benson, Dave. ''[https://logosfoundation.org/kursus/music_math.pdf Music: A Mathematical Offering]''. University of Aberdeen, 2008.</ref>{{rp|201}}
==Lattice and tonality diamond==
The 7-limit tonality diamond: {| border="0" cellspacing="0" cols="7" frame="void" rules="none" |- | || || || 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.
{| class="wikitable" |- ! Fraction !! Cents !! Degree (31-ET) !! Name (31-ET) |- | 420/420 = 1/1 || 0 || 0 || C |- | 480/420 = 8/7 || 231.174 || 6 || D{{music|t}} |- | 490/420 = 7/6 || 266.871 || 7 || D{{music|sharp}} |- | 504/420 = 6/5 || 315.641 || 8 || E{{music|flat}} |- | 525/420 = 5/4 || 386.314 || 10 || E |- | 560/420 = 4/3 || 498.045 || 13 || F |- | 588/420 = 7/5 || 582.512 || 15 || F{{music|sharp}} |- | 600/420 = 10/7 || 617.488 || 16 || G{{music|flat}} |- | 630/420 = 3/2 || 701.955 || 18 || G |- | 672/420 = 8/5 || 814.686 || 21 || A{{music|flat}} |- | 700/420 = 5/3 || 884.359 || 23 || A |- | 720/420 = 12/7 || 933.129 || 24 || A{{music|t}} |- | 735/420 = 7/4 || 968.826 || 25 || A{{music|sharp}} |- | 840/420 = 2/1 || 1200 || 31 || C |}
==See also== *Đàn bầu
==References== {{reflist}}
==External links== *[http://anaphoria.com/centaur.html Centaur a 7 limit tuning] shows Centaur tuning plus other related 7 tone tunings by others *[https://sevish.com/scaleworkshop/?n=soft%20diatonic&l=lFk_7F6_4F3_3F2_1rF14_7F4_2F1&version=2.1.0 Soft diatonic scale] at Sevish.com
{{Musical tuning}}
Category:7-limit tuning and intervals