**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 ( ) 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. ( ) 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:^{[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: 21/20, 7/4, 8/7, 7/6, 9/7, 12/7, 7/5, and 10/7.^{[4]}
Archytas of Tarantum is the oldest recorded musicologist to calculate 7-limit tuning systems. 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, Arthur von Oettingen, Hugo Riemann, Colin Brown, and Paul Hindemith ("chaos"^{[6]}).^{[4]}

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*.

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

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.^{[7]} One, called by Ptolemy the "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 "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.