Neutral third: Difference between revisions

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{{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}}
{{Wikipedia}}
A '''neutral third (n3)''' is an interval that spans two steps of the [[5L 2s|diatonic]] scale with a quality between major and minor. It exists in [[neutralization|neutralized]] diatonic scales as exactly one half of a [[perfect fifth]].  
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.  


In [[just intonation]], an interval may be classified as a neutral third if it is reasonably mapped to 2\7 and [[24edo|7\24]] (precisely two steps of the diatonic scale and three and a half steps of the chromatic scale).  
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.
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''.


Two neutral thirds stack to a perfect fifth, and as such the neutral third range is generally divided at roughly 350{{c}} into '''artoneutral''' (flatter) and '''tendoneutral''' (sharper) thirds. As such, neutral thirds tend to exist in pairs.
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-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{{c}} 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{{c}} 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{{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 ===
{| class="wikitable"
{| class="wikitable"
|-
|-
! Delta 2
! colspan="2" | Delta-2
! Cents
! colspan="2" | Delta-3
! Delta 3
! colspan="2" | Delta-4
! Cents
! colspan="2" | Delta-5
! Delta 4
! Cents
! Delta 5
! Cents
|-
|-
| [[11/9]]
| [[11/9]]
Line 58: Line 71:


== 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"
|-
|-
Line 109: Line 123:
== 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
! Pair of neutral thirds
! Temperament
! Temperament
|-
|-
| 60/49, 49/40
| 60/49, 49/40
| [[Breedsmic temperaments|Breedsmic]]
| [[Breed (temperament)|Breed]] retraction*
|-
|-
| 11/9, 27/22
| 11/9, 27/22
| [[Rastmic clan|Rastmic]]
| [[Neutral (temperament)|Neutral]]
|-
|-
| 39/32, 16/13
| 39/32, 16/13
Line 125: Line 140:
| 17/14, 21/17
| 17/14, 21/17
| Temperament of 294/289
| 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 09:35, 29 July 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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