|Just interval||18:11 or 13:8|
|Equal temperament||800 or 900|
|24 equal temperament||850|
|Just intonation||853 or 841|
- 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. 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.
13:8 (840.53 cents), the ratio of the thirteenth harmonic is notated in Ben Johnston's system as A13♭.
- 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.