View source for Neutral third

{{Infobox musical interval
| main_interval_name = Neutral third
| inverse = [[neutral sixth]]
| complement = [[neutral sixth]]
| other_names = –
| abbreviation = n3
| semitones = ~3½
| interval_class = ~3½
| just_interval = 11:9,<ref name="Haluska">Haluska, Jan (2003). ''The Mathematical Theory of Tone Systems'', p. xxiii. {{ISBN|0-8247-4714-3}}. Undecimal neutral third and Zalzal's wosta.</ref> 27:22,<ref name="Haluska" /><ref>{{Xenharmonic wiki|Neutral_third_scales|Neutral third scales}}</ref> or 16:13<ref name="Haluska-xxiv">Haluska (2003), [https://books.google.com/books?id=w7yZ1Nzb5awC&pg=PR24 p. xxiv]. Tridecimal neutral third.</ref>
| cents_equal_temperament =
| cents_24T_equal_temperament = 350
| cents_just_intonation = 347, 355, or 359
}}

[[File:Neutral third on C.png|thumb|[[File:Neutral third on C.mid|thumb|Neutral third on C]][[File:Zalzal's wosta on C.mid|thumb|27/22]]]]
[[File:Neutral interval.png|thumb|Neutral third]]

A '''neutral third''' is a [[interval (music)|musical interval]] wider than a [[minor third]] {{Audio|Minor third on C.mid|play}} but narrower than a [[major third]] {{Audio|Major third on C.mid|play}}, named by Jan Pieter Land in 1880.<ref>J. P. Land, ''Over de toonladders der Arabische muziek'', 1880; ''Recherches sur l'histoire de la gamme arabe'', 1884. See [[Hermann von Helmholtz]], ''On the Sensations of Tone,'' ([[Alexander John Ellis]], trans.) (3rd ed., 1895), p. 281, note † (addition by Ellis).</ref> Land makes reference to the neutral third attributed to [[Mansour Zalzal|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).<ref>Liberty Manik (1969). ''Das Arabische Tonsystem im Mittelalter'' (Leiden: Brill, 1969), 46–49.</ref> The Zalzalian third may have been a mobile interval.

Three distinct intervals may be termed neutral thirds:<ref>Alois Hába, ''Neue Harmonielehre des diatonischen, chromatischen, Viertel-, Drittel-, Sechstel- und Zwölftel-Tonsystems'' (Leipzig: Fr. Kistner & C.F.W. Sigel, 1927), 143. {{pre-ISBN}}. Cited in Skinner, Miles Leigh (2007). ''Toward a Quarter-tone Syntax: Analyses of Selected Works by Blackwood, Haba, Ives, and Wyschnegradsky'', p. 25. ProQuest. {{ISBN|9780542998478}}.</ref>

* The ''undecimal neutral third'' has a ratio of 11:9<ref>Andrew Horner, Lydia Ayres (2002). ''Cooking with [[Csound]]: Woodwind and Brass Recipes'', p. 131. {{ISBN|0-89579-507-8}}. "Super-Major Second".</ref> between the [[frequencies]] of the two tones, or about 347.41 cents {{Audio|Undecimal neutral third on C.mid|play}}. This ratio is the mathematical [[mediant (mathematics) | 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. <math>|5 f - 4 (11/9) f| = (1/9) f</math>, is the same as the beat frequency of the 6th harmonic of the lower pitch against the 5th of the upper, i.e. <math>|6 f - 5 (11/9) f| = |-(1/9) f| = (1/9) f</math>. In this sense, it is the unique ratio which is equally well-tuned as a major and minor third.
* A ''tridecimal neutral third'' {{Audio|Tridecimal neutral third on C.mid|play}} has a ratio of 16:13 between the [[frequencies]] of the two tones, or about 359.47 cents.<ref  name="Haluska-xxiv" /> This is the largest neutral third, and occurs infrequently in music, as little music utilizes the 13th [[harmonic]]. It is the [[mediant (mathematics) |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, <math>|7 f - 6 (16/13) f| = |9 f - 7 (16/13) f| = (5/13) f</math>.
* An ''[[equal temperament|equal-tempered]] neutral third'' {{Audio|Neutral third on C.mid|play}} is characterized by a difference of 350 [[Cent (music)|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 equal temperament|31-ET]], [[41 equal temperament|41-ET]], or [[72 equal temperament|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 <math>\sqrt{3/2}</math> 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 {{Audio|Neutral thirds triad on C.mid|Play}} and 31-tet {{Audio|Neutral thirds triad in 31-tet.mid|Play}}.
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== Occurrence in human music ==

=== In infants' song ===

[[Infants]] experiment with singing, and a few studies of individual infants' singing found that neutral thirds regularly arise in their improvisations. In two separate case studies of the progression and development of these improvisations, neutral thirds were found to arise in infants' songs after major and minor seconds and thirds, but before intervals smaller than a [[semitone]] and also before intervals as large as a [[perfect fourth]] or larger.<ref>[[Bruno Nettl|Nettl, Bruno]]. "Infant Musical Development and Primitive Music", ''[[Southwestern Journal of Anthropology]]'', vol. 12, no. 1, pp. 87–91. (Spring 1956) {{JSTOR|3628859}}</ref>

=== In modern classical Western music ===

The neutral third has been used by a number of modern composers, including [[Charles Ives]], [[James Tenney]], and [[Gayle Young]].<ref>Young, Gayle. "The Pitch Organization of Harmonium for James Tenney", ''[[Perspectives of New Music]]'', vol. 26, no. 2. pp. 204–212 (Summer 1988) {{JSTOR|833190}}</ref>

=== In traditional music ===

[[Ptolemy|Claudius Ptolemy]] describes an "even diatonic" tuning which uses two justly tuned neutral thirds in [[iarchive:solomon-1999-ptolemy-harmonics|''Harmonikon'']] built off the 12:11 and 11:10 neutral seconds in compound intervals with 9:8 and 10:9 whole tones, forming the intervals: (12/11)*(9/8) = 27/22, (11/10)*(10/9) = 11/9. The latter of these is an interval found in the [[Harmonic series (music)|harmonic series]] as the interval between partials 9 and 11. 

The equal-tempered neutral third may be found in the [[quarter tone scale]] and in some traditional [[Arab music]] (see also [[Arab tone system]]). Undecimal neutral thirds appear in traditional [[Music of Georgia (country)|Georgian music]].<ref name="Caucasia1994">{{cite book|editor=Aronson, Howard I.|title=The Annual of the Society for the Study of Caucasia|url=https://books.google.com/books?id=YbYWAQAAMAAJ|access-date=14 April 2011|year=1994|publisher=Society for the Study of Caucasia|page=93}}</ref> Neutral thirds are also found in American folk music.<ref>Boswell, George W. "The Neutral Tone as a Function of Folk-Song Text", ''Yearbook of the International Folk Music Council'', vol. 2, 1970, pp. 127–132 (1970) {{JSTOR|767430}}</ref>

=== In contemporary popular music ===

[[Blue note]]s (a note found in [[country music]], [[blues]], and some [[rock music]]) on the third note of a scale can be seen as a variant of a neutral third with the tonic, as they fall in between a major third and a minor third. Similarly the blue note on the seventh note of the scale can be seen as a neutral third with the dominant.

== In equal temperaments ==

Two steps of [[Equal temperament#Five-, seven-, and nine-tone temperaments in ethnomusicology|seven-tone equal temperament]] form an interval of 342.8571 cents, which is within 5 cents of 347.4079 for the undecimal (11:9) neutral third.<ref>{{Xenharmonic wiki|7edo}}</ref> This is an equal temperament in reasonably common use, at least in the form of "near seven equal", as it is a tuning used for Thai music as well as the Ugandan Chopi tradition of music.<ref>Morton, David (1980). "The Music of Thailand", ''Musics of Many Cultures'', p. 70. May, Elizabeth, ed. {{ISBN|0-520-04778-8}}.</ref>

The neutral third also has good approximations in other commonly used equal temperaments including [[quarter tone|24-ET]] (7 steps, 350 cents<ref>{{Xenharmonic wiki|24edo}}</ref>) and similarly by all multiples of 24 equal steps such as 48-ET<ref name=48edo>{{Xenharmonic wiki|48edo}}</ref> and 72-ET,<ref name=72edo>{{Xenharmonic wiki|72edo}}</ref> [[31 equal temperament|31-ET]] (9 steps, 348.39),<ref>{{Xenharmonic wiki|31edo}}</ref> [[34 equal temperament|34-ET]] (10 steps, 352.941 cents<ref>{{Xenharmonic wiki|34edo}}</ref>), [[41 equal temperament|41-ET]] (12 steps, 351.22 cents<ref>{{Xenharmonic wiki|41edo}}</ref>), and slightly less closely by [[53 equal temperament|53-ET]] (15 steps, 339.62 cents<ref name=53edo>{{Xenharmonic wiki|53edo}}</ref>).

Close approximations to the tridecimal neutral third (16:13) appear in [[53 equal temperament|53-ET]]<ref name=53edo /> and [[72 equal temperament|72-ET]].<ref name=72edo /> Both of these temperaments distinguish between the tridecimal (16:13) and undecimal (11:9) neutral thirds. All the other tuning systems mentioned above fail to distinguish between these intervals; they temper out the comma 144:143.

== See also ==
* [[List of pitch intervals]]
* [[Microtonal music]]

== References ==
{{reflist}}

{{Intervals}}

[[Category:Neutral intervals]]
[[Category:Quarter tones]]
[[Category:Thirds (music)]]