Neutral third

A neutral third is a <a href="/facts/Interval_(music)/aFwkkgpY">musical interval</a> wider than a <a href="/facts/Minor_third/4y8zBxwY">minor third</a> playⓘ but narrower than a <a href="/facts/Major_third/7CPRURg4">major third</a> playⓘ, named by Jan Pieter Land in 1880. Land makes reference to the neutral third attributed to <a href="/facts/Mansour_Zalzal/Lb5SQoyM">Zalzal</a> (8th c.), described by <a href="/facts/Al-Farabi/8tnl33pR">Al-Farabi</a> (10th c.) as corresponding to a ratio of 27:22 (354.5 cents) and by <a href="/facts/Avicenna/EYKG4Vez">Avicenna</a> (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:

<ul><li>The undecimal neutral third has a ratio of 11:9 between the <a href="/facts/Frequencies/QUSbPaW8">frequencies</a> of the two tones, or about 347.41 cents playⓘ. This ratio is the mathematical <a href="/facts/Mediant_(mathematics)/AXo8Alrr"> mediant</a> of the <a href="/facts/Major_third/7CPRURg4">major third</a> 5/4 and the <a href="/facts/Minor_third/4y8zBxwY">minor third</a> 6/5, and as such, has the property that if harmonic notes of frequency f and (11/9) f are played together, the <a href="/facts/Beat_frequency/zKhaKnMy">beat frequency</a> of the 5th harmonic of the lower pitch against the 4th of the upper, i.e. 
 
 
 
 
 |
 
 5
 f
 −
 4
 (
 11
 
 /
 
 9
 )
 f
 
 |
 
 =
 (
 1
 
 /
 
 9
 )
 f
 
 
 {\displaystyle |5f-4(11/9)f|=(1/9)f}
 
, is the same as the beat frequency of the 6th harmonic of the lower pitch against the 5th of the upper, i.e. 
 
 
 
 
 |
 
 6
 f
 −
 5
 (
 11
 
 /
 
 9
 )
 f
 
 |
 
 =
 
 |
 
 −
 (
 1
 
 /
 
 9
 )
 f
 
 |
 
 =
 (
 1
 
 /
 
 9
 )
 f
 
 
 {\displaystyle |6f-5(11/9)f|=|-(1/9)f|=(1/9)f}
 
. In this sense, it is the unique ratio which is equally well-tuned as a major and minor third.</li>
<li>A tridecimal neutral third playⓘ has a ratio of 16:13 between the <a href="/facts/Frequencies/QUSbPaW8">frequencies</a> 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 <a href="/facts/Harmonic/9U9REzeR">harmonic</a>. It is the <a href="/facts/Mediant_(mathematics)/AXo8Alrr">mediant</a> of the <a href="/facts/Septimal_major_third/ygaqB6TF">septimal major third</a> 9/7 and <a href="/facts/Septimal_minor_third/eNuNrSXe">septimal minor third</a> 7/6, and as such, enjoys an analogous property with regard to the beating of the corresponding harmonics as above. That is, 
 
 
 
 
 |
 
 7
 f
 −
 6
 (
 16
 
 /
 
 13
 )
 f
 
 |
 
 =
 
 |
 
 9
 f
 −
 7
 (
 16
 
 /
 
 13
 )
 f
 
 |
 
 =
 (
 5
 
 /
 
 13
 )
 f
 
 
 {\displaystyle |7f-6(16/13)f|=|9f-7(16/13)f|=(5/13)f}
 
.</li>
<li>An <a href="/facts/Equal_temperament/8AwSblW3">equal-tempered</a> neutral third playⓘ is characterized by a difference of 350 <a href="/facts/Cent_(music)/lsm8TJgX">cents</a> between the two tones, slightly wider than the 11:9 ratio, and exactly half of an equal-tempered perfect fifth.</li></ul>
These intervals are all within about 12 cents and are difficult for most people to distinguish by ear. Neutral thirds are roughly a <a href="/facts/Quarter_tone/9QwftMsC">quarter tone</a> sharp from <a href="/facts/12_equal_temperament/89BK2lpj">12 equal temperament</a> minor thirds and a quarter tone flat from 12-ET major thirds. In just intonation, as well as in tunings such as <a href="/facts/31_equal_temperament/XfXmqNrR">31-ET</a>, <a href="/facts/41_equal_temperament/jJPTDWjo">41-ET</a>, or <a href="/facts/72_equal_temperament/4X4TylBA">72-ET</a>, 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 
 
 
 
 
 
 3
 
 /
 
 2
 
 
 
 
 {\displaystyle {\sqrt {3/2}}}
 
 between two frequencies, being exactly half of a just <a href="/facts/Perfect_fifth/PTIaHPuI">perfect fifth</a> 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 Playⓘ and 31-tet Playⓘ.

Neutral third open-in-new

Neutral third