Neutral sixth

Neutral sixth
Inverse	neutral third
Name
Other names	-
Abbreviation	n6
Size
Semitones	~8½
Interval class	~3½
Just interval	18:11 or 13:8
Cents
12-Tone equal temperament	N/A
24-Tone 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 halfway 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 (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

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♭. In 24-ET is approximated by A. 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]

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.
^ [1] 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.