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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.  


Two neutral thirds stack to a [[perfect fifth]], and as such the neutral third range is generally divided at roughly 350 cents into '''artoneutral''' (flat) and '''tendoneutral''' (sharp) thirds. As such, 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 ==
=== By prime limit ===
=== By prime limit ===
The 3-[[Prime limit|limit]] and 5-limit do not have simple neutral thirds, so we start with the 7-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.


* 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.
=== By delta ===
* 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.
See [[Delta-N ratio]].
* 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.


=== By delta ===
{| class="wikitable"
{| class="wikitable"
|+
!Delta 2
!Cents
!Delta 3
!Cents
!Delta 4
!Cents
!Delta 5
!Cents
|-
|-
|[[11/9]]
! colspan="2" | Delta-2
|347c
! colspan="2" | Delta-3
|[[16/13]]
! colspan="2" | Delta-4
|359c
! colspan="2" | Delta-5
|[[21/17]]
|365c
|[[26/21]]
|370c
|-
|-
|
| [[11/9]]
|
| 347{{c}}
|[[17/14]]
| [[16/13]]
|336c
| 359{{c}}
|[[23/19]]
| [[21/17]]
|330c
| 365{{c}}
|[[27/22]]
| [[26/21]]
|355c
| 370{{c}}
|-
|-
|
|
|
|
|
| [[17/14]]
|
| 336{{c}}
|
| [[23/19]]
|
| 330{{c}}
|[[28/23]]
| [[27/22]]
|341c
| 355{{c}}
|-
|  
|  
|  
|  
|  
|  
| [[28/23]]
| 341{{c}}
|}
|}


== In EDOs ==
== 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]].
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]].
 
{| 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"
{| class="wikitable"
|+
!Pair of neutral thirds
!Temperament
|-
|-
|60/49, 49/40
! Pair of neutral thirds
|[[Breedsmic temperaments|Breedsmic]]
! Temperament
|-
|-
|11/9, 27/22
| 60/49, 49/40
|[[Rastmic clan|Rastmic]]
| [[Breed (temperament)|Breed]] retraction*
|-
|-
|39/32, 16/13
| 11/9, 27/22
|Temperament of [[512/507]]
| [[Neutral (temperament)|Neutral]]
|-
|-
|17/14, 21/17
| 39/32, 16/13
|Temperament of 294/289
| 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"
|-
! Range
! colspan="4" | Mos
|-
| 327–343{{c}}
| colspan="1" rowspan="3" | [[1L&nbsp;2s]]
| colspan="1" rowspan="3" | [[3L&nbsp;1s]]
| rowspan="1" | [[4L&nbsp;3s]]
| [[7L&nbsp;4s]]
|-
| 343–360{{c}}
| rowspan="2" | [[3L&nbsp;4s]]
| [[7L&nbsp;3s]]
|-
| 360–400{{c}}
| [[3L&nbsp;7s]]
|}
{{Navbox intervals}}
{{Navbox intervals}}

Latest revision as of 14:43, 30 March 2026

← Minor Third Neutral third Major third →
Name Neutral third
Lower bound 330¢ – 340¢
Upper bound 360¢ – 370¢
Generated MOSes 4L 3s, 3L 4s, 7L 3s, 3L 7s
Example JI intervals
Intervals 11/9 (347.4¢)
16/13 (359.5¢)
Related regions
Complement Neutral sixth
English Wikipedia has an article on:

A neutral third (n3) is an interval that generates a variant of diatonic with its original 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.

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 cents and the major third of roughly 400 ¢. A rough tuning range for the neutral third is 330 to 370 ¢ 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 ¢ into artoneutral (flatter) and tendoneutral (sharper) thirds. As such, neutral thirds tend to exist in pairs.

In just intonation

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 ¢ 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 ¢ 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 ¢ 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 ¢ respectively.

By delta

See Delta-N ratio.

Delta-2 Delta-3 Delta-4 Delta-5
11/9 347 ¢ 16/13 359 ¢ 21/17 365 ¢ 26/21 370 ¢
17/14 336 ¢ 23/19 330 ¢ 27/22 355 ¢
28/23 341 ¢

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 sqrt(3/2), which, while not a just interval, is the "canonical" neutral third tuning, as stacking two of them gives 3/2.

Edo 1\2edf 39/32 16/13
7 343 ¢
17 353 ¢
24 350 ¢
25 — 336 ¢
26 — * 369 ¢
27 356 ¢
29 — 331 ¢ *
31 348 ¢
34 353 ¢
41 351 ¢
53 — 340 ¢ 362 ¢

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:

Pair of neutral thirds Temperament
60/49, 49/40 Breed retraction*
11/9, 27/22 Neutral
39/32, 16/13 Temperament of 512/507
17/14, 21/17 Temperament of 294/289

* Breed is a rank-3 temperament, the other generator being ~7/5

In moment-of-symmetry scales

Intervals between 327 and 400 ¢ 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.

Range Mos
327–343 ¢ 1L 2s 3L 1s 4L 3s 7L 4s
343–360 ¢ 3L 4s 7L 3s
360–400 ¢ 3L 7s


View • Talk • EditInterval classification
Interval regions
Unison and octave Unison • Comma and diesis • Octave
Seconds Minor second • Neutral second • Major second
Thirds Minor third • Neutral third • Major third
Fourths and fifths Perfect fourth • Superfourth • Tritone • Subfifth • Perfect fifth
Sixths Minor sixth • Neutral sixth • Major sixth
Sevenths Minor seventh • Neutral seventh • Major seventh
Interseptimal intervals Interseptimal 2nd-3rd • Interseptimal 3rd-4th • Interseptimal 5th-6th • Interseptimal 6th-7th
Interval qualities
Diatonic qualities Diminished • Minor • Perfect • Major • Augmented
Tuning ranges Neutral (interval quality) • Submajor and supraminor • Pental major and minor • Novamajor and novaminor • Neogothic major and minor • Supermajor and subminor • Ultramajor and inframinor
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