Minor sixth: Difference between revisions

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{{Infobox interval region|Cents lower=~~750~~|Cents upper=~~825~~|JI intervals=8/5

{{Infobox interval region|Cents lower=760|Cents upper=830|JI intervals=8/5

128/81}}

128/81|Cents upper wide=840|Cents lower wide=740}}

~~The~~ minor sixth is ~~an interval region reasonably classified as~~ intervals ~~of 750 to 825 ¢~~. It is found in the ~~diatonic scale as~~ the minor ~~6-diastep~~, ~~or as~~ the sixth ~~degree of the Aeolian mode~~.

A '''minor sixth (m6)''' is the smaller of the two "sixths" - intervals spanning 6 degrees or 5 scale steps in the diatonic scale. It is found between the 1st and 6th notes of the minor scale, hence its name. Another diatonic interval around the same size is the '''augmented fifth.''' More generally, an interval close to 800 cents in size can be called a minor sixth.

== ~~Just intervals~~ ==

== As an interval region ==

The minor sixth, as a concrete interval region, is typically near 800{{c}} in size, distinct from the [[major sixth]] of roughly 900{{c}} and the [[neutral sixth]] of roughly 850{{c}}. A rough tuning range for the minor sixth is about 760 to 828{{c}} according to [[Margo Schulter]]'s theory of interval regions. ''Minor sixth'' in this sense refers both to the ~740–840{{c}} range as a whole, and to a specific subdivision within it (~785–840{{c}}) as opposed to subminor sixths; minor sixths flat of this are often called "subminor sixths".

=== ~~3-limit~~ ===

=== In MOS scales ===

~~The 3-limit minor 6th, or~~ the ~~Pythagorean minor 6th, has a ratio of~~ [[~~128/81~~]]~~. There is another 3-limit ratio in the range of the minor 6th~~: ~~the Pythagorean augmented 5th, at about 816¢.~~

Intervals between 720 and 840 cents generate the following [[mos]] scales:

~~=== 5-limit ===~~

These tables start from the last monolarge mos generated by the interval range.

~~5-limit minor 6ths include:~~

[[~~8/5~~]], [[~~25/16~~]]

Scales with more than 12 notes are not included.

{| class="wikitable"

!Range

! colspan="5" |Mos

|-

|800–840{{c}}

|[[1L 2s]]

|[[3L 1s]]

|[[3L 4s]]

| colspan="2" |[[3L 7s]]

|-

|764–800{{c}}

| rowspan="3" |[[1L 1s]]

| rowspan="3" |[[2L 1s]]

| rowspan="3" |[[3L 2s]]

| rowspan="2" |[[3L 5s]]

|[[3L 8s]]

|-

|750–764{{c}}

|[[8L 3s]]

|-

|720–750{{c}}

| colspan="2" |[[5L 3s]]

|}

=== 7-~~limit~~ ===

== As a diatonic interval category ==

7~~-limit~~ minor ~~6ths include:~~

{{Infobox|Title=Diatonic minor sixth|Header 1=MOS|Data 1=[[5L 2s]]|Header 2=Other names|Data 2=Minor 5-diastep|Header 3=Generator span|Data 3=-4 generators|Header 4=Tuning range|Data 4=720–857{{c}}|Header 5=Basic tuning|Data 5=800{{c}}|Header 6=Function on root|Data 6=Submediant|Header 7=Interval regions|Data 7=[[Minor sixth (interval region)|Minor sixth]], [[neutral sixth (interval region)|neutral sixth]]|Header 8=Associated just intervals|Data 8=[[8/5]], [[128/81]]|Header 9=Octave complement|Data 9=[[Major third (interval region)|Major third]]}}As a diatonic interval category, a minor sixth is an interval that spans five scale steps in the [[5L 2s|diatonic]] scale with the minor (narrower) quality. It is generated by stacking 4 fourths [[Octave reduction|octave reduced]], and depending on the specific tuning, it ranges from 720 to 857{{cent}} ([[5edo|3\5]] to [[7edo|5\7]]).

~~54/35~~, ~~14/9~~

In [[just intonation]], an interval may be classified as a minor sixth if it is reasonably mapped to five steps of the diatonic scale and eight steps of the chromatic scale.

~~=== 11-limit~~ and ~~higher ===~~

The minor sixth is often the bounding interval of a [[Triad|tertian triad]] chord in inversion, and as such is often involved in chord structures in diatonic harmony.

~~11-limit-plus minor 6ths include:~~

~~11/7~~, ~~31/20, 17/11, 36/23, 37/24, 39/25, 41/26, 43/27, 45/29, 47/30, 48/31, 53/34, 58/37, 59/38, 61/39, 64/41, 65/42, 67/43, 71/46, 73/47, 76/49, 79/50~~

In [[TAMNAMS]], this interval is called the '''minor 5-diastep'''.

~~== In EDOs ==~~

The augmented fifth is enharmonic with the minor sixth, ranging from 686 to 960{{c}} (4\7 to 4\5). It is arguably the most common of the enharmonic intervals besides the chromatic semitone itself, appearing in the augmented triad. It is generated by stacking 8 fifths octave reduced, and is as such not found in the diatonic scale. Regardless, in TAMNAMS, it may be called the '''augmented 4-diastep'''.

~~EDOs~~ with minor ~~6ths include:~~

3n, 8n, ~~11n~~, ~~14n~~, ~~17n~~, ~~19n~~, ~~20n~~, ~~23n~~, ~~25n~~, ~~26n~~

In [[just intonation]], an interval may be classified as an augmented fifth if it is reasonably mapped to '''four''' steps of the diatonic scale and eight steps of the chromatic scale.

=== Scale info ===

The diatonic scale contains three minor sixths. In the Ionian mode, minor sixths are found on the third, sixth, and seventh degrees of the scale; the other four degrees have major sixths. This roughly equal distribution is analogous to that of the [[Third|thirds]].

=== Tunings ===

Being an abstract mos degree, and not a specific interval, the diatonic minor sixth does not have a fixed tuning, but instead has a range of ways it can be tuned, based on the tuning of the generator used in making the scale. This is similar for the augmented fifth.

The tuning range of the diatonic minor sixth ranges from 720 to 857.2{{c}}. The generator for a given tuning in cents, ''n'', for the diatonic minor sixth can be found by {{nowrap|(3600-''n'')/4}}. For example, the sixth 816{{c}} gives us {{nowrap|(3600-816)/4 {{=}} 2784/4 {{=}} 696{{c}}}}, corresponding to 50edo.

The tuning range of the diatonic augmented fifth ranges from 686 to 960{{c}}. The generator for a given tuning in cents, n, for the augmented fifth can be found by (4800-n)/8. For example, the augmented fifth 816{{c}} gives us (4800-816)/8 = 3984/8 = 498{{c}}, corresponding to 200edo.

== In just intonation ==

=== By prime limit ===

The simplest 3-limit minor sixth is the Pythagorean minor sixth of [[128/81]], 792{{c}} in size, which is generated by [[stacking]] four just perfect fourths of [[4/3]]. There is also a Pythagorean augmented fifth of about 816{{c}}.

Much [[Odd limit|simpler]] minor sixths and augmented fifths exist in higher [[Prime limit|limits]], however, for example:

* The 5-limit '''classical minor sixth''' is a ratio of [[8/5]] and is about 814{{c}}.

* The 7-limit '''(septimal) subminor sixth''' is a ratio of [[14/9]] and is almost exactly 765{{c}}.

* The 11-limit '''neogothic minor sixth''' is a ratio of [[11/7]], and is about 782{{c}}. (Note that this is often considered an imperfect or augmented fifth.)

* The 13-limit '''(tridecimal) inframinor sixth''' is a ratio of [[20/13]], and is about 746{{c}}.

** There is also a 13-limit '''(tridecimal) supraminor sixth''', which is a ratio of [[21/13]], and is about 830{{c}}.

* The 17-limit '''(septendecimal) supraminor sixth''' is a ratio of [[34/21]], and is about 834{{c}}.

Note that the ratios of higher-limit supraminor sixths approximate the golden ratio - the [[golden ratio]] itself as a musical interval is a supraminor sixth of about 833 cents.

== In regular temperaments ==

See [[Major third (interval region)#In regular temperaments]]

@@ Line 1: / Line 1: @@
-{{Infobox interval region|Cents lower=750|Cents upper=825|JI intervals=8/5
+{{Infobox interval region|Cents lower=760|Cents upper=830|JI intervals=8/5
-/81}}
+/81|Cents upper wide=840|Cents lower wide=740}}
-The minor sixth is an interval region reasonably classified as intervals of 750 to 825 ¢. It is found in the diatonic scale as the minor 6-diastep, or as the sixth degree of the Aeolian mode.
+A '''minor sixth (m6)''' is the smaller of the two "sixths" - intervals spanning 6 degrees or 5 scale steps in the diatonic scale. It is found between the 1st and 6th notes of the minor scale, hence its name. Another diatonic interval around the same size is the '''augmented fifth.''' More generally, an interval close to 800 cents in size can be called a minor sixth.
-== Just intervals ==
+== As an interval region ==
+The minor sixth, as a concrete interval region, is typically near 800{{c}} in size, distinct from the [[major sixth]] of roughly 900{{c}} and the [[neutral sixth]] of roughly 850{{c}}. A rough tuning range for the minor sixth is about 760 to 828{{c}} according to [[Margo Schulter]]'s theory of interval regions. ''Minor sixth'' in this sense refers both to the ~740–840{{c}} range as a whole, and to a specific subdivision within it (~785–840{{c}}) as opposed to subminor sixths; minor sixths flat of this are often called "subminor sixths".
-=== 3-limit ===
+=== In MOS scales ===
-The 3-limit minor 6th, or the Pythagorean minor 6th, has a ratio of [[128/81]]. There is another 3-limit ratio in the range of the minor 6th: the Pythagorean augmented 5th, at about 816¢.
+Intervals between 720 and 840 cents generate the following [[mos]] scales:
-=== 5-limit ===
+These tables start from the last monolarge mos generated by the interval range.
--limit minor 6ths include:
-[[8/5]], [[25/16]]
+Scales with more than 12 notes are not included.
+{| class="wikitable"
+!Range
+! colspan="5" |Mos
+|-
+|800–840{{c}}
+|[[1L 2s]]
+|[[3L 1s]]
+|[[3L 4s]]
+| colspan="2" |[[3L 7s]]
+|-
+|764–800{{c}}
+| rowspan="3" |[[1L 1s]]
+| rowspan="3" |[[2L 1s]]
+| rowspan="3" |[[3L 2s]]
+| rowspan="2" |[[3L 5s]]
+|[[3L 8s]]
+|-
+|750–764{{c}}
+|[[8L 3s]]
+|-
+|720–750{{c}}
+| colspan="2" |[[5L 3s]]
+|}
-=== 7-limit ===
+== As a diatonic interval category ==
--limit minor 6ths include:
+{{Infobox|Title=Diatonic minor sixth|Header 1=MOS|Data 1=[[5L&nbsp;2s]]|Header 2=Other names|Data 2=Minor 5-diastep|Header 3=Generator span|Data 3=-4 generators|Header 4=Tuning range|Data 4=720–857{{c}}|Header 5=Basic tuning|Data 5=800{{c}}|Header 6=Function on root|Data 6=Submediant|Header 7=Interval regions|Data 7=[[Minor sixth (interval region)|Minor sixth]], [[neutral sixth (interval region)|neutral sixth]]|Header 8=Associated just intervals|Data 8=[[8/5]], [[128/81]]|Header 9=Octave complement|Data 9=[[Major third (interval region)|Major third]]}}As a diatonic interval category, a minor sixth is an interval that spans five scale steps in the [[5L 2s|diatonic]] scale with the minor (narrower) quality. It is generated by stacking 4 fourths [[Octave reduction|octave reduced]], and depending on the specific tuning, it ranges from 720 to 857{{cent}} ([[5edo|3\5]] to [[7edo|5\7]]).
-/35, 14/9
+In [[just intonation]], an interval may be classified as a minor sixth if it is reasonably mapped to five steps of the diatonic scale and eight steps of the chromatic scale.
-=== 11-limit and higher ===
+The minor sixth is often the bounding interval of a [[Triad|tertian triad]] chord in inversion, and as such is often involved in chord structures in diatonic harmony.
--limit-plus minor 6ths include:
-/7, 31/20, 17/11, 36/23, 37/24, 39/25, 41/26, 43/27, 45/29, 47/30, 48/31, 53/34, 58/37, 59/38, 61/39, 64/41, 65/42, 67/43, 71/46, 73/47, 76/49, 79/50
+In [[TAMNAMS]], this interval is called the '''minor 5-diastep'''.
-== In EDOs ==
+The augmented fifth is enharmonic with the minor sixth, ranging from 686 to 960{{c}} (4\7 to 4\5). It is arguably the most common of the enharmonic intervals besides the chromatic semitone itself, appearing in the augmented triad. It is generated by stacking 8 fifths octave reduced, and is as such not found in the diatonic scale. Regardless, in TAMNAMS, it may be called the '''augmented 4-diastep'''.
-EDOs with minor 6ths include:
-n, 8n, 11n, 14n, 17n, 19n, 20n, 23n, 25n, 26n
+In [[just intonation]], an interval may be classified as an augmented fifth if it is reasonably mapped to '''four''' steps of the diatonic scale and eight steps of the chromatic scale.
+=== Scale info ===
+The diatonic scale contains three minor sixths. In the Ionian mode, minor sixths are found on the third, sixth, and seventh degrees of the scale; the other four degrees have major sixths. This roughly equal distribution is analogous to that of the [[Third|thirds]].
+=== Tunings ===
+Being an abstract mos degree, and not a specific interval, the diatonic minor sixth does not have a fixed tuning, but instead has a range of ways it can be tuned, based on the tuning of the generator used in making the scale. This is similar for the augmented fifth.
+The tuning range of the diatonic minor sixth ranges from 720 to 857.2{{c}}. The generator for a given tuning in cents, ''n'', for the diatonic minor sixth can be found by {{nowrap|(3600-''n'')/4}}. For example, the sixth 816{{c}} gives us {{nowrap|(3600-816)/4 {{=}} 2784/4 {{=}} 696{{c}}}}, corresponding to 50edo.
+The tuning range of the diatonic augmented fifth ranges from 686 to 960{{c}}. The generator for a given tuning in cents, n, for the augmented fifth can be found by (4800-n)/8. For example, the augmented fifth 816{{c}} gives us (4800-816)/8 = 3984/8 = 498{{c}}, corresponding to 200edo.
+== In just intonation ==
+=== By prime limit ===
+The simplest 3-limit minor sixth is the Pythagorean minor sixth of [[128/81]], 792{{c}} in size, which is generated by [[stacking]] four just perfect fourths of [[4/3]]. There is also a Pythagorean augmented fifth of about 816{{c}}.
+Much [[Odd limit|simpler]] minor sixths and augmented fifths exist in higher [[Prime limit|limits]], however, for example:
+* The 5-limit '''classical minor sixth''' is a ratio of [[8/5]] and is about 814{{c}}.
+* The 7-limit '''(septimal) subminor sixth''' is a ratio of [[14/9]] and is almost exactly 765{{c}}.
+* The 11-limit '''neogothic minor sixth''' is a ratio of [[11/7]], and is about 782{{c}}. (Note that this is often considered an imperfect or augmented fifth.)
+* The 13-limit '''(tridecimal) inframinor sixth''' is a ratio of [[20/13]], and is about 746{{c}}.
+** There is also a 13-limit '''(tridecimal) supraminor sixth''', which is a ratio of [[21/13]], and is about 830{{c}}.
+* The 17-limit '''(septendecimal) supraminor sixth''' is a ratio of [[34/21]], and is about 834{{c}}.
+Note that the ratios of higher-limit supraminor sixths approximate the golden ratio - the [[golden ratio]] itself as a musical interval is a supraminor sixth of about 833 cents.
+== In regular temperaments ==
+See [[Major third (interval region)#In regular temperaments]]