A neutral second or medium second is an interval wider than a minor second and narrower than a major second. Three distinct intervals may be termed neutral seconds:
The intermediate neutral second, called the lesser undecimal neutral second, has a ratio between the higher-frequency tone to the lower-frequency tone of 12:11 and is about 150.64 cents wide, while the larger one,
the greater undecimal neutral second, has a ratio of 11:10 between the two tones and is about 165.00 cents wide. The lesser undecimal neutral second may be derived from the harmonic series as the interval between the eleventh and twelfth harmonics. The greater undecimal neutral second may be derived from the harmonic series as the interval between the tenth and eleventh harmonics.
An equal-tempered neutral second is characterized by a difference in 150 cents between the two tones, a hairsmaller than a ratio of frequencies between the two tones of 12:11, and exactly half of an equal-tempered minor third.
The equal-tempered neutral is found in some traditional Arabic music. Because the equal tempered neutral second is essentially a semitone plus a quarter-tone, they may be considered three-quarter tones in the quarter tone scale.
In equal temperament
Approximations to the 12:11 and 11:10 neutral seconds can be found in a number of equally temperedtuning systems. 11:10 is very closely matched by 22-ET, whereas 12:11 is matched by 24-ET, 31-ET and 41-ET. 72-ET matches both intervals closely and is also the smallest widely used equal temperament that uniquely matches both intervals. Tuning systems that temper out the comma of 121:120 do not distinguish between the two intervals. 17-ET has a neutral second between 12:11 and 13:12, and a neutral third between 16:13 and 11:9.
A septimal neutral seventh has a ratio of 64:35 or about 1045 cents.
The just undecimal neutral seventh has a ratio of 11:6 between the frequencies of the two tones, or about 1049 cents. Alternately, 13:7 or about 1071.7 cents.
A tridecimal neutral seventh has a ratio of 24:13 between the frequencies of the two tones, or about 1061 cents. This is the largest neutral seventh, and occurs infrequently in music, as little music utilizes the 13th harmonic.
An equal-tempered neutral seventh is characterized by a difference in 1050 cents between the two tones, a hair larger than the 11:6 ratio, and exactly half of an equal-tempered major thirteenth.
These intervals are all within about 12 cents of each other and are difficult for most people to distinguish. A neutral seventh can be formed by stacking a neutral third together with a perfect fifth. Based on its positioning in the harmonic series, the undecimal neutral seventh implies a root one perfect fifth below the lower of the two notes.
