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A '''neutral third (n3)''' is an interval that is near 350 [[cents]] in size, distinct from the [[minor third]] of roughly 300 [[Cent|cents]] and the [[major third]] of roughly 400 cents. A rough tuning range for the neutral third is 330 to 370 cents, as in Schulter's theory of [[Interval region|interval regions]].
{{Interwiki
| en = Neutral third
| zh = 中三度
}}
{{Infobox interval region
| Name = Neutral third
| Cents lower = 340
| Cents lower wide = 330
| Cents upper = 360
| Cents upper wide = 370
| JI intervals = 11/9, 16/13
| MOSes = [[4L 3s]], [[3L 4s]], [[7L 3s]], [[3L 7s]]
| Complement = [[Neutral sixth]]
| Lower region = [[Minor Third]]
| Higher region = [[Major third]]
}}
{{Wikipedia}}
A '''neutral third''' ('''n3''') is an interval that generates a variant of [[5L 2s|diatonic]] with its original [[perfect fifth|perfect-fifth]] generator halved. Like the [[major third]] and [[minor third]], it is considered a third, so it spans two steps in diatonic-based notation, but has a quality between major and minor.  


The neutral third range is generally divided at roughly 350 cents into '''artoneutral''' (flat) and '''tendoneutral''' (sharp) thirds. An artoneutral third has a tendoneutral third as a fifth complement, and so neutral thirds will often be provided in pairs.
In [[just intonation]], an interval may be classified as a neutral third if it is reasonably mapped to two steps of the diatonic scale and three and a half steps of the chromatic scale.
 
As a concrete [[interval region]], it is typically near 350 [[cents]] in size, distinct from the minor third of roughly 300 [[cent]]s and the major third of roughly 400{{c}}. A rough tuning range for the neutral third is 330 to 370{{c}} according to [[Margo Schulter]]'s theory of interval regions; intervals in this range may be also called ''Zalzalian thirds''.
 
The neutral third range is generally divided at roughly 350{{c}} into [[neutral (interval quality)|artoneutral]] (flatter) and [[neutral (interval quality)|tendoneutral]] (sharper) thirds. As such, neutral thirds tend to exist in pairs.


== In just intonation ==
== In just intonation ==
The 3-[[Prime limit|limit]] and 5-limit do not have simple neutral thirds, so we start with the 7-limit:
=== By prime limit ===
The [[3-limit]] and 5-limit do not have simple neutral thirds (though hemipythagorean has an irrational [[sqrt(3/2)]] interval that might be considered the "canonical" neutral third), so we start with the 7-limit:
* The 7-limit artoneutral and tendoneutral thirds are the ratios of [[60/49]] and [[49/40]] respectively, and they are slightly flat of and slightly sharp of 351{{c}} respectively.
* The 11-limit alpharabian artoneutral and tendoneutral thirds are the ratios of [[11/9]] and [[27/22]] respectively, and they are about 347 and 355{{c}} respectively.
* The 13-limit artoneutral and tendoneutral thirds are the ratios of [[39/32]] and [[16/13]] respectively, and they are about 342 and 359{{c}} respectively.
* The 17-limit supraminor and submajor thirds are the ratios of [[17/14]] and [[21/17]] respectively, and they are about 336 and 366{{c}} respectively.
 
=== By delta ===
See [[Delta-N ratio]].
 
{| class="wikitable"
|-
! colspan="2" | Delta-2
! colspan="2" | Delta-3
! colspan="2" | Delta-4
! colspan="2" | Delta-5
|-
| [[11/9]]
| 347{{c}}
| [[16/13]]
| 359{{c}}
| [[21/17]]
| 365{{c}}
| [[26/21]]
| 370{{c}}
|-
|
|
| [[17/14]]
| 336{{c}}
| [[23/19]]
| 330{{c}}
| [[27/22]]
| 355{{c}}
|-
|
|
|
|
|
|
| [[28/23]]
| 341{{c}}
|}


* The 7-limit '''artoneutral''' and '''tendoneutral thirds''' are the ratios of [[60/49]] and [[49/40]] respectively, and they are slightly flat of and slightly sharp of 351 cents respectively.
== In edos ==
* The 11-limit '''alpharabian artoneutral''' and '''tendoneutral thirds''' are the ratios of [[11/9]] and [[27/22]] respectively, and they are about 347 and 355 cents respectively.
The following table lists the best tuning of 39/32 and 16/13 in various significant [[edo]]s. For applicable edos, it also lists one half of the edo's perfect fifth, approximating [[sqrt(3/2)]], which, while not a just interval, is the "canonical" neutral third tuning, as stacking two of them gives [[3/2]].
* The 13-limit '''artoneutral''' and '''tendoneutral thirds''' are the ratios of [[39/32]] and [[16/13]] respectively, and they are about 342 and 359 cents respectively.
* The 17-limit '''supraminor''' and '''submajor thirds''' are the ratios of [[17/14]] and [[21/17]] respectively, and they are about 336 and 366 cents respectively.


== In EDOs ==
The following table lists the best tuning of 39/32 and 16/13 in various significant [[EDOs]]. For applicable EDOs, it also lists one half of the EDO's perfect fifth, approximating 1\[[2edf]], which, while not a just interval, is the "canonical" neutral third tuning, as stacking two of them gives [[3/2]].
{| class="wikitable"
{| class="wikitable"
!EDO
!1\2edf
!39/32
!16/13
|-
|-
|7
! Edo
| colspan="3" |343c
! 1\2edf
! 39/32
! 16/13
|-
|-
|17
| 7
| colspan="3" |353c
| colspan="3" | 343{{c}}
|-
|-
|24
| 17
| colspan="3" |350c
| colspan="3" | 353{{c}}
|-
|-
|25
| 24
| -
| colspan="3" | 350{{c}}
| colspan="2" |336c
|-
|-
|26
| 25
| -
| —
|*
| colspan="2" | 336{{c}}
|369c
|-
|-
|27
| 26
| colspan="3" |356c
| —
| *
| 369{{c}}
|-
|-
|29
| 27
| -
| colspan="3" | 356{{c}}
|331c
|*
|-
|-
|31
| 29
| colspan="3" |348c
| —
| 331{{c}}
| *
|-
|-
|34
| 31
| colspan="3" |353c
| colspan="3" | 348{{c}}
|-
|-
|41
| 34
| colspan="3" |351c
| colspan="3" | 353{{c}}
|-
|-
|53
| 41
| -
| colspan="3" | 351{{c}}
|340c
|-
|362c
| 53
| —
| 340{{c}}
| 362{{c}}
|}
|}


== In regular temperaments ==
== In regular temperaments ==
Temperaments generated by neutral thirds often involve tempering a pair of neutral thirds together. As such, each pair of neutral thirds has a corresponding temperament, which equates both neutral thirds to half of a perfect fifth:
Temperaments generated by neutral thirds often involve tempering a pair of neutral thirds together. As such, each pair of neutral thirds has a corresponding temperament, which equates both neutral thirds to half of a perfect fifth:
{| class="wikitable"
|-
! Pair of neutral thirds
! Temperament
|-
| 60/49, 49/40
| [[Breed (temperament)|Breed]] retraction*
|-
| 11/9, 27/22
| [[Neutral (temperament)|Neutral]]
|-
| 39/32, 16/13
| Temperament of [[512/507]]
|-
| 17/14, 21/17
| Temperament of 294/289
|}
<nowiki/>* Breed is a rank-3 temperament, the other generator being ~7/5
== In moment-of-symmetry scales ==
Intervals between 327 and 400{{c}} generate the following [[mos]] scales:
These tables start from the last monolarge mos generated by the interval range.
Scales with more than 12 notes are not included.
{| class="wikitable"
{| class="wikitable"
|+
!Pair of neutral thirds
!Temperament
|-
|-
|60/49, 49/40
! Range
|[[Breedsmic temperaments|Breedsmic]]
! colspan="4" | Mos
|-
|-
|11/9, 27/22
| 327–343{{c}}
|[[Rastmic clan|Rastmic]]
| colspan="1" rowspan="3" | [[1L&nbsp;2s]]
| colspan="1" rowspan="3" | [[3L&nbsp;1s]]
| rowspan="1" | [[4L&nbsp;3s]]
| [[7L&nbsp;4s]]
|-
|-
|39/32, 16/13
| 343–360{{c}}
|Temperament of [[512/507]]
| rowspan="2" | [[3L&nbsp;4s]]
| [[7L&nbsp;3s]]
|-
|-
|17/14, 21/17
| 360–400{{c}}
|Temperament of 294/289
| [[3L&nbsp;7s]]
|}
|}
{{Navbox intervals}}
{{Navbox intervals}}
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