Neutral third: Difference between revisions

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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~~]].

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]].

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 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.

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.

== In just intonation ==

=== 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, 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.

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|}

== 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 [[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"

!~~EDO~~

!Edo

!1\2edf

!39/32

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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]].
+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]].
-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 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.
+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.
 == In just intonation ==
 === 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, 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.
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 |}
-== 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 [[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"
-!EDO
+!Edo
 !1\2edf
 !39/32