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**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 (help·info)) 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:^{[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: 7/4, 8/7, 7/6, 12/7, 7/5, and 10/7.^{[4]}
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.

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 |

**^**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.- ^
^{a}^{b}^{c}Partch, 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.

- Centaur a 7 limit tuning shows Centaur tuning plus other related 7 tone tunings by others