# 7-limit tuning

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 (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. (Play) 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: 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. Archytas of Tarantum is the oldest recorded musicologist to calculate 7-limit tuning systems. Those considering 7 to be consonant include Marin Mersenne, 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").

## 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.

### Approximation using equal temperament

It is possible to approximate 7-limit music using equal temperament, for example 31-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's Harmonikon

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. 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.