7-limit tuning

{{Short description|Musical instrument tuning with a limit of seven}}

File:Harmonic seventh on C.png]]

File:Septimal chromatic semitone on C.png]]

File:Septimal major third on C.png from C to E{{music|L}}.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. This, "extremely large third", may resemble a neutral third or blue note.Fonville (1991), p. 128.File:Septimal major third on C.mid]]

File:Septimal minor third on C.png on CFile:Septimal minor third on C.mid]]

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 ({{audio|Greater just minor seventh on C.mid|Play}}) 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. ({{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.

The Great Highland bagpipe is tuned to a ten-note seven-limit scale:Benson, Dave (2007). Music: A Mathematical Offering, p. 212. {{ISBN|9780521853873}}. 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: 21/20, 7/4, 8/7, 7/6, 9/7, 12/7, 7/5, and 10/7.Partch, Harry (2009). Genesis of a Music: An Account of a Creative Work, Its Roots, and Its Fulfillments, pp. 90–91. {{ISBN|9780786751006}}.

Archytas of Tarantum is the oldest recorded musicologist to calculate 7-limit tuning systems. Those considering 7 to be consonant include Marin Mersenne,Shirlaw, Matthew (1900). Theory of Harmony, p. 32. {{ISBN|978-1-4510-1534-8}}. Giuseppe Tartini, Leonhard Euler, François-Joseph Fétis, J. A. Serre, Moritz Hauptmann, Alexander John Ellis, Wilfred Perrett, Max Friedrich Meyer. 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"Hindemith, Paul (1942). Craft of Musical Composition, vol. 1, p. 38. {{ISBN|0901938300}}.).