neutral sixth#Thirteenth harmonic

{{Infobox Interval|

main_interval_name = Neutral sixth|

inverse = neutral third|

complement = neutral third|

other_names = - |

abbreviation = n6 |

semitones = ~8½ |

interval_class = ~3½ |

just_interval = 18:11Haluska, Jan (2003). The Mathematical Theory of Tone Systems, p.xxiv. {{ISBN|0-8247-4714-3}}. Undecimal neutral sixth. or 13:8Haluska (2003), p.xxiii. Tridecimal neutral sixth.|

cents_equal_temperament = N/A|

cents_24T_equal_temperament = 850|

cents_just_intonation = 853 or 841

}}

File:Neutral sixth on C.png

A neutral sixth is a musical interval wider than a minor sixth {{Audio|Minor sixth on C.mid|play}} but narrower than a major sixth {{Audio|Major sixth on C.mid|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. {{Audio|Undecimal neutral sixth on C.mid|play}}
• A tridecimal neutral sixth has a ratio of 13:8 between the frequencies of the two tones, or about 840.53 cents.[https://books.google.com/books?id=w7yZ1Nzb5awC&dq=%22tridecimal+neutral+third%22&pg=PR24] Jan Haluska, The Mathematical Theory of Tone Systems, CRC (2004). This is the smallest neutral sixth, and occurs infrequently in music, as little music utilizes the 13th harmonic. {{Audio|Tridecimal neutral sixth on C.mid|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 halfway between the equal-tempered minor and major sixths, and half of an equal-tempered perfect eleventh (octave plus fourth). {{Audio|Neutral sixth on C.mid|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 (12-ET) 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==

Image:Thirteenth harmonic on C.png

The pitch ratio 13:8 (840.53 cents) is the ratio of the thirteenth harmonic and is notated in Ben Johnston's system as A¹³{{music|b}}. In 24-ET is approximated by A{{music|d}}. 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.Fauvel, John; Flood, Raymond; and Wilson, Robin J. (2006). Music And Mathematics, p.21-22. {{ISBN|9780199298938}}.