Neutral third
A neutral third is a musical interval wider than a minor third but narrower than a major third , named by Jan Pieter Land in 1880. Land makes reference to the neutral third attributed to Zalzal (8th c.), described by Al-Farabi (10th c.) as corresponding to a ratio of 27:22 (354.5 cents) and by Avicenna (Ibn Sina, 11th c.) as 39:32 (342.5 cents). The Zalzalian third may have been a mobile interval. Three distinct intervals may be termed neutral thirds: * The undecimal neutral third has a ratio of 11:9 between the frequencies of the two tones, or about 347.41 cents . This ratio is the mathematical mediant of the major third 5/4 and the minor third 6/5, and as such, has the property that if harmonic notes of frequency f and (11/9) f are played together, the beat frequency of the 5th harmonic of the lower pitch against the 4th of the upper, i.e. , is the same as the beat frequency of the 6th harmonic of the lower pitch against the 5th of the upper, i.e. . In this sense, it is the unique ratio which is equally well-tuned as a major and minor third. * A tridecimal neutral third has a ratio of 16:13 between the frequencies of the two tones, or about 359.47 cents. This is the largest neutral third, and occurs infrequently in music, as little music utilizes the 13th harmonic. It is the mediant of the septimal major third 9/7 and septimal minor third 7/6, and as such, enjoys an analogous property with regard to the beating of the corresponding harmonics as above. That is, . * An equal-tempered neutral third is characterized by a difference of 350 cents between the two tones, slightly wider than the 11:9 ratio, and exactly half of an equal-tempered perfect fifth. These intervals are all within about 12 cents and are difficult for most people to distinguish by ear. Neutral thirds are roughly a quarter tone sharp from 12 equal temperament minor thirds and a quarter tone flat from 12-ET major thirds. 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. In addition to the above examples, a square root neutral third can be characterized by a ratio of between two frequencies, being exactly half of a just perfect fifth of 3/2 and measuring about 350.98 cents. Such a definition stems from the two thirds traditionally making a fifth-based triad. A triad formed by two neutral thirds is neither major nor minor, thus the neutral thirds triad is ambiguous. While it is not found in twelve tone equal temperament it is found in others such as the quarter tone scale and 31-tet . (Wikipedia).
