# Neutral sixth

Inverse | neutral third |
---|---|

Name | |

Other names | - |

Abbreviation | n6 |

Size | |

Semitones | ~8½ |

Interval class | ~3½ |

Just interval |
18:11^{[1]} or 13:8^{[2]} |

Cents | |

Equal temperament | 800 or 900 |

24 equal temperament | 850 |

Just intonation | 853 or 841 |

A **neutral sixth** is a musical interval wider than a minor sixth play but narrower than a major sixth play . Three distinct intervals may be termed neutral sixths:

- The
*undecimal neutral sixth*has a ratio of 18:11 between the frequencies of the two tones, or about 852.59 cents. play - A
*tridecimal neutral sixth*has a ratio of 13:8 between the frequencies of the two tones, or about 840.53 cents.^{[3]}This is the smallest neutral sixth, and occurs infrequently in music, as little music utilizes the 13th harmonic. play - An
*equal-tempered neutral sixth*is 850 cents, a hair narrower than the 18:11 ratio. It is an equal-tempered quarter tone exactly half way between the equal-tempered minor and major sixths, and half of an equal-tempered perfect eleventh (octave plus fourth). play

These intervals are all within about 12 cents of each other and are difficult for most people to distinguish. Neutral sixths are roughly a quarter tone sharp from 12 equal temperament minor sixths and a quarter tone flat from 12-ET major sixths. In just intonation, as well as in tunings such as 31-ET, 41-ET, or 72-ET, which more closely approximate just intonation, the intervals are closer together.

A neutral sixth can be formed by subtracting a neutral second from a minor seventh. Based on its positioning in the harmonic series, the undecimal neutral sixth implies a root one minor seventh above the higher of the two notes.

## Thirteenth harmonic

13:8 (840.53 cents), the ratio of the thirteenth harmonic is notated in Ben Johnston's system as A^{13}♭.

This note is often corrected to a just or pythagorean ratio on the natural horn, but the pure thirteenth harmonic was used in pieces including Britten's *Serenade for tenor, horn and strings*.^{[4]}

## See also

## References

- ↑ Haluska, Jan (2003). The Mathematical Theory of Tone Systems, p.xxiv. ISBN 0-8247-4714-3. Undecimal neutral sixth.
- ↑ Haluska (2003), p.xxiii. Tridecimal neutral sixth.
- ↑ Jan Haluska,
*The Mathematical Theory of Tone Systems*, CRC (2004). - ↑ Fauvel, John; Flood, Raymond; and Wilson, Robin J. (2006).
*Music And Mathematics*, p.21-22. ISBN 9780199298938.