Neutral second: Difference between revisions

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A '''neutral second (n2)''' is an interval that ~~spans~~ one ~~step~~ 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~~ [[minor ~~third~~]].

{{Infobox interval region

| Name = Neutral second

| Cents lower = 130

| Cents lower wide = 120

| Cents upper = 160

| Cents upper wide = 170

| JI intervals = 11/10, 12/11, 13/12

| MOSes = [[1L 8s]], [[1L 7s]], [[1L 6s]], [[9L 1s]], [[8L 1s]], [[7L 1s]]

| Complement = [[Neutral seventh]]

| Lower region = [[Semitone (interval region)|Semitone]]

| Higher region = [[Major second]]

}}

A '''neutral second''' ('''n2''') is an interval that exists as exactly one half of a [[minor third]] in a variant of [[5L 2s|diatonic]] with its original [[perfect fifth|perfect-fifth]] generator halved. Like the [[major second]] and [[minor second]], it is considered a second, so it spans one step in diatonic-based notation, but has a quality between major and minor.

In [[just intonation]], an interval may be classified as a neutral second if it is reasonably mapped to ~~1\7 and 3[[24edo|\24]] (precisely~~ one step of the diatonic scale and one and a half steps of the chromatic scale).

In [[just intonation]], an interval may be classified as a neutral second if it is reasonably mapped to one step of the diatonic scale and one and a half steps of the chromatic scale.

As a concrete [[interval region]], it is typically near 150 ~~[[cents]]~~ in size, distinct from the [[Semitone (interval region)|semitone]] of roughly 100 ~~[[Cent|cents]]~~ and the [[major second]] of roughly ~~200 ¢~~. A rough tuning range for the neutral ~~third~~ is 130 to ~~170 ¢~~ according to [[Margo Schulter]]'s theory of interval regions.

As a concrete [[interval region]], it is typically near 150{{cent}} in size, distinct from the [[Semitone (interval region)|semitone]] of roughly 100{{c}} and the [[major second]] of roughly 200{{c}}. A rough tuning range for the neutral second is 130 to 170{{c}} according to [[Margo Schulter]]'s theory of interval regions. This page will consider intervals between about 120 and 170{{c}}. The outer range of this might be too extreme to call neutral seconds, but this is done so that one can find what they're looking for easily.

== In just intonation ==

=== By prime limit ===

The [[3-limit]] does not have a simple neutral second, so we start with the 5-limit:

* The 5-limit acute minor second or large limma is a ratio of [[27/25]], and is about 133{{c}}.

* The 7-limit septimal neutral second is a ratio of [[35/32]], and is about 155{{c}}.

** There is also a 7-limit swetismic neutral second, which is a ratio of [[49/45]], and is about 147{{c}}.

* The 11-limit (undecimal) neutral/submajor seconds are the ratios of [[12/11]] and [[11/10]], which are about 151{{c}} and 165{{c}}, respectively; 11/10 in particular can also be analyzed as a [[major second]]. Despite that, it is also here for completeness.

* The 13-limit (tridecimal) neutral/supraminor seconds are the ratios of [[14/13]] and [[13/12]], which are about 128{{c}} and 139{{c}}, respectively; 14/13 in particular can also be analyzed as a [[semitone]]. Despite that, it is also here for completeness.

=== ~~By prime limit~~ ===

== In mos scales ==

~~The~~ [[3-~~limit~~]] ~~and 5~~-~~limit do not have simple neutral seconds, so we start with the 7-limit:~~

Intervals between 120 and 171{{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="2" | Mos

|-

| 120–133{{c}}

| [[1L 8s]]

| [[9L 1s]]

|-

| 133–150{{c}}

| [[1L 7s]]

| [[8L 1s]]

|-

| 150–171{{c}}

| [[1L 6s]]

| [[7L 1s]]

|}

* The 7-limit '''septimal neutral second'''

~~{{Todo|complete page|inline=1}}~~{{Navbox intervals}}

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-A '''neutral second (n2)''' is an interval that spans one step 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 [[minor third]].
+{{Infobox interval region
+| Name = Neutral second
+| Cents lower = 130
+| Cents lower wide = 120
+| Cents upper = 160
+| Cents upper wide = 170
+| JI intervals = 11/10, 12/11, 13/12
+| MOSes = [[1L 8s]], [[1L 7s]], [[1L 6s]], [[9L 1s]], [[8L 1s]], [[7L 1s]]
+| Complement = [[Neutral seventh]]
+| Lower region = [[Semitone (interval region)|Semitone]]
+| Higher region = [[Major second]]
+}}
+A '''neutral second''' ('''n2''') is an interval that exists as exactly one half of a [[minor third]] in a variant of [[5L 2s|diatonic]] with its original [[perfect fifth|perfect-fifth]] generator halved. Like the [[major second]] and [[minor second]], it is considered a second, so it spans one step in diatonic-based notation, but has a quality between major and minor.
-In [[just intonation]], an interval may be classified as a neutral second if it is reasonably mapped to 1\7 and 3[[24edo|\24]] (precisely one step of the diatonic scale and one and a half steps of the chromatic scale).
+In [[just intonation]], an interval may be classified as a neutral second if it is reasonably mapped to one step of the diatonic scale and one and a half steps of the chromatic scale.
-As a concrete [[interval region]], it is typically near 150 [[cents]] in size, distinct from the [[Semitone (interval region)|semitone]] of roughly 100 [[Cent|cents]] and the [[major second]] of roughly 200 ¢. A rough tuning range for the neutral third is 130 to 170 ¢ according to [[Margo Schulter]]'s theory of interval regions.
+As a concrete [[interval region]], it is typically near 150{{cent}} in size, distinct from the [[Semitone (interval region)|semitone]] of roughly 100{{c}} and the [[major second]] of roughly 200{{c}}. A rough tuning range for the neutral second is 130 to 170{{c}} according to [[Margo Schulter]]'s theory of interval regions. This page will consider intervals between about 120 and 170{{c}}. The outer range of this might be too extreme to call neutral seconds, but this is done so that one can find what they're looking for easily.
 == In just intonation ==
+=== By prime limit ===
+The [[3-limit]] does not have a simple neutral second, so we start with the 5-limit:
+* The 5-limit acute minor second or large limma is a ratio of [[27/25]], and is about 133{{c}}.
+* The 7-limit septimal neutral second is a ratio of [[35/32]], and is about 155{{c}}.
+** There is also a 7-limit swetismic neutral second, which is a ratio of [[49/45]], and is about 147{{c}}.
+* The 11-limit (undecimal) neutral/submajor seconds are the ratios of [[12/11]] and [[11/10]], which are about 151{{c}} and 165{{c}}, respectively; 11/10 in particular can also be analyzed as a [[major second]]. Despite that, it is also here for completeness.
+* The 13-limit (tridecimal) neutral/supraminor seconds are the ratios of [[14/13]] and [[13/12]], which are about 128{{c}} and 139{{c}}, respectively; 14/13 in particular can also be analyzed as a [[semitone]]. Despite that, it is also here for completeness.
-=== By prime limit ===
+== In mos scales ==
-The [[3-limit]] and 5-limit do not have simple neutral seconds, so we start with the 7-limit:
+Intervals between 120 and 171{{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="2" | Mos
+|-
+| 120–133{{c}}
+| [[1L&nbsp;8s]]
+| [[9L&nbsp;1s]]
+|-
+| 133–150{{c}}
+| [[1L&nbsp;7s]]
+| [[8L&nbsp;1s]]
+|-
+| 150–171{{c}}
+| [[1L&nbsp;6s]]
+| [[7L&nbsp;1s]]
+|}
-* The 7-limit '''septimal neutral second'''
+{{Navbox intervals}}
-{{Todo|complete page|inline=1}}{{Navbox intervals}}