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Neutral sixth

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Title: Neutral sixth  
Author: World Heritage Encyclopedia
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Subject: Neutral interval, Orwell comma, Breedsma, Ragisma, Major limma
Collection: Neutral Intervals, Quarter Tones, Sixths (Music)
Publisher: World Heritage Encyclopedia

Neutral sixth

Neutral Sixth
Inverse neutral third
Other names -
Abbreviation n6
Semitones ~8½
Interval class ~3½
Just interval 18:11[1] or 13:8[2]
Equal temperament 800 or 900
Just intonation 853 or 841
Neutral sixth on C About this sound   .

A neutral sixth is a musical interval wider than a minor sixth About this sound    but narrower than a major sixth About this sound   . 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. About this sound   
  • 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. About this sound   
  • 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). About this sound   

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

Thirteenth harmonic on C. About this sound   

13:8 (840.53 cents), the ratio of the thirteenth harmonic is notated in Ben Johnston's system as A13.

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]

Use of the thirteenth harmonic in the prologue to Britten's Serenade for tenor, horn and strings.

See also


  1. ^ Haluska, Jan (2003). The Mathematical Theory of Tone Systems, p.xxiv. ISBN 0-8247-4714-3. Undecimal neutral sixth.
  2. ^ Haluska (2003), p.xxiii. Tridecimal neutral sixth.
  3. ^ [1] Jan Haluska, The Mathematical Theory of Tone Systems, CRC (2004).
  4. ^ Fauvel, John; Flood, Raymond; and Wilson, Robin J. (2006). Music And Mathematics, p.21-22. ISBN 9780199298938.
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