Neutral third: Difference between revisions

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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 2\7 and [[24edo|7\24]] (precisely 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 cents. A rough tuning range for the neutral third is 330 to 370 cents 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.


Two neutral thirds stack to a perfect fifth, and as such the neutral third range is generally divided at roughly 350 cents into '''artoneutral''' (flatter) and '''tendoneutral''' (sharper) thirds. As such, neutral thirds tend to exist in pairs.
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.


== In just intonation ==
== In just intonation ==
Line 11: Line 11:
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, 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 cents 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.
* 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 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 cents 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 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{{c}} respectively.


=== By delta ===
=== By delta ===
{| class="wikitable"
{| class="wikitable"
|+
!Delta 2
!Cents
!Delta 3
!Cents
!Delta 4
!Cents
!Delta 5
!Cents
|-
|-
|[[11/9]]
! Delta 2
|347c
! Cents
|[[16/13]]
! Delta 3
|359c
! Cents
|[[21/17]]
! Delta 4
|365c
! Cents
|[[26/21]]
! Delta 5
|370c
! Cents
|-
|-
|
| [[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}}
|}
|}


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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 [[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}}
|}
|}


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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]]
| [[Breedsmic temperaments|Breedsmic]]
|-
|-
|11/9, 27/22
| 11/9, 27/22
|[[Rastmic clan|Rastmic]]
| [[Rastmic clan|Rastmic]]
|-
|-
|39/32, 16/13
| 39/32, 16/13
|Temperament of [[512/507]]
| Temperament of [[512/507]]
|-
|-
|17/14, 21/17
| 17/14, 21/17
|Temperament of 294/289
| Temperament of 294/289
|}
|}
{{Navbox intervals}}
{{Navbox intervals}}

Revision as of 19:37, 26 February 2025

A neutral third (n3) is an interval that spans two steps of the diatonic scale with a quality between major and minor. It exists in neutralized diatonic scales as exactly one half of a perfect fifth.

In just intonation, an interval may be classified as a neutral third if it is reasonably mapped to 2\7 and 7\24 (precisely 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.

Two neutral thirds stack to a perfect fifth, and as such 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, 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

Delta 2 Cents Delta 3 Cents Delta 4 Cents Delta 5 Cents
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 1\2edf, 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 Breedsmic
11/9, 27/22 Rastmic
39/32, 16/13 Temperament of 512/507
17/14, 21/17 Temperament of 294/289


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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