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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 ==
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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}}
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