Major second: Difference between revisions

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A '''major second''' ('''M2''') is the larger of two seconds – intervals spanning 2 degrees or 1 scale step in the diatonic scale. It is found on the 1st note of the major scale, hence its name. Because it is one large step, it is also called a '''tone''' or '''whole tone'''. Another diatonic interval around the same size is the '''diminished third''' ('''d3'''). More generally, an interval close to 200 cents can be called a major second.

Many JI intervals are called major seconds, but the term usually refers to one of these three intervals:

* [[9/8]], the major tone of about 204 cents.

* [[10/9]], the minor tone of about 182 cents.

* [[8/7]], the septimal major second of about 231 cents.

== As an interval region ==

{{Infobox interval region

| Name = Major second, whole tone

Line 6:

Line 14:

| Cents upper wide = 260

| JI intervals = 8/7, 9/8, 10/9

| MOSes = 1L 6s, 7L 1s, 6L 1s, 1L 5s, 5L 1s, 5L 4s

| MOSes = 1L 6s, 7L 1s, 6L 1s, 1L 5s, 5L 1s, 5L 4s

| Complement = [[Minor seventh]]

| Lower region = [[Neutral second]]

| Higher region = [[Minor ~~third~~]]

| Higher region = [[Minor third]]

}}

As a concrete [[interval region]], a major second is typically near 200{{c}} in size, distinct from the [[Semitone (interval region)|semitone]] of roughly 100{{c}} and the [[neutral second]] of roughly 150{{c}}. A rough tuning range for the major second is about 180 to 240{{c}} according to [[Margo Schulter]]'s theory of interval regions.

This article covers intervals between 160 and 260{{c}}. The outer range of this might be too extreme to call "major seconds", but this is done so that one can find what they are looking for easily.

=== In mos scales ===

Being a small interval, major seconds generate a number of monosmall and monolarge [[mos scale]]s.

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="3" | Mos

|-

| 150–171{{c}}

| [[1L 6s]]

| colspan="2" | [[7L 1s]]

|-

| 171–200{{c}}

| [[1L 5s]]

| colspan="2" | [[6L 1s]]

|-

| 200–218{{c}}

| rowspan="2" | [[1L 4s]]

| rowspan="2" | [[5L 1s]]

| [[6L 5s]]

|-

| 218–240{{c}}

| [[5L 6s]]

|-

| 240–267{{c}}

| [[1L 3s]]

| [[4L 1s]]

| [[5L 4s]]

|}

== As a diatonic interval category ==

{{Infobox

| Title = Diatonic major second

| Header 1 = MOS| Data 1 = [[5L 2s]]

| Header 2 = Other names| Data 2 = Major 1-diastep

| Header 3 = Generator span| Data 3 = +2 generators

| Header 4 = Tuning range| Data 4 = 171–240{{c}}

| Header 5 = Basic tuning| Data 5 = 200{{c}}

| Header 6 = Function on root| Data 6 = Supertonic

| Header 7 = Interval regions| Data 7 = Major second

| Header 8=Associated just intervals| Data 8 = [[10/9]], [[9/8]]

| Header 9=Octave complement| Data 9 = [[Minor seventh]]

}}

~~A '''~~major second~~''' ('''M2''')~~ is an interval that spans one scale step in the [[5L 2s|diatonic]] scale with the major (wider) quality. It is generated by stacking 2 fifths [[octave reduction|octave reduced]], and depending on the specific tuning, it ranges from 171 to 240{{~~cent~~}} ([[7edo|1\7]] to [[5edo|1\5]]). It can be considered the large step of the diatonic scale.

As a diatonic interval category, a major second is an interval that spans one scale step in the [[5L 2s|diatonic]] scale with the major (wider) quality. It is generated by stacking 2 fifths [[octave reduction|octave reduced]], and depending on the specific tuning, it ranges from 171 to 240{{c}} ([[7edo|1\7]] to [[5edo|1\5]]). It can be considered the large step of the diatonic scale.

In [[just intonation]], an interval may be classified as a major second if it is reasonably mapped to 2 steps of the chromatic scale - formally, this is 4\24, which is used as opposed to [[12edo]]'s 2\12 to better capture the characteristics of many intervals in the [[11-limit|11-]] and [[13-limit]] – and 1 step of the diatonic scale. Diminished thirds are mapped to 2 steps of the chromatic scale and 2 steps of the diatonic scale.

In TAMNAMS, the major second is called the ''major 1-diastep''.

Given its role as the large step, it can be used to construct other diatonic intervals, along with the [[minor second]]: two major seconds make a [[major third]], a major second and a minor second make a [[minor third]], and three major seconds result in an [[augmented fourth]], also called a tritone for that reason.

=== Scale info ===

The diatonic scale contains five major seconds. In the Ionian mode, major seconds are found on the 1st, 2nd, 4th, 5th, and 6th degrees of the scale; the other two degrees have minor seconds. The large number of major seconds compared to minor seconds ensures that thirds that include minor seconds (that is, minor thirds) are roughly evenly distributed with major thirds; in a scale with three small steps and four large steps, for example, six out of the seven thirds are minor.

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

=== Tunings ===

Being an abstract mos degree, and not a specific interval, the diatonic major second 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.

~~As a concrete [[interval region]], it is typically near 200~~{{c}} in ~~size~~, ~~distinct from~~ the ~~[[Semitone~~ (~~interval region~~)~~|semitone]] of roughly 100 ¢ and~~ the ~~[[neutral~~ second~~]] of roughly 150~~{{c}}~~. A rough tuning range for the major second is about 180 to 240~~{{c}} ~~according~~ to ~~[[Margo Schulter]]'s theory of interval regions~~.

The tuning range of the diatonic major second ranges from 171.4 to 240{{c}}. The generator for a given tuning in cents, ''n'', for the diatonic major second can be found by {{nowrap| (''n'' + 1200)/2. For example, the second 192{{c}} gives us {{nowrap| (192 + 1200)/2 {{=}} 1392/2 {{=}} 696{{c}} }}, corresponding to 50edo.

~~This article covers intervals between 160 and 260~~{{c}}~~. The outer range of this might be too extreme to call "major seconds", but this is done so that one can find what they are looking for easily.~~

Several example tunings are provided below:

{| class="wikitable center-all left-1"

! Tuning

! Step ratio

! Edo

! Cents

|-

| Equalized

| 1:1

| 7

| 171{{c}}

|-

| Supersoft

| 4:3

| 26

| 184{{c}}

|-

| Soft

| 3:2

| 19

| 189{{c}}

|-

| Semisoft

| 5:3

| 31

| 194{{c}}

|-

| Basic

| 2:1

| 12

| 200{{c}}

|-

| Semihard

| 5:2

| 29

| 207{{c}}

|-

| Hard

| 3:1

| 17

| 212{{c}}

|-

| Superhard

| 4:1

| 22

| 218{{c}}

|-

| Collapsed

| 1:0

| 5

| 240{{c}}

|}

== In just intonation ==

Line 24:

Line 143:

Other major seconds exist in higher [[prime limit|limits]], however, for example:

* The 5-limit ptolemaic major second is a ratio of [[10/9]], however in 5-limit harmony it is used alongside 9/8. It is about 182{{c}}.

* The 5-limit ~~'''~~ptolemaic major second~~'''~~ is a ratio of [[10/9]], however in 5-limit harmony it is used alongside 9/8. It is about 182{{c}}.

* The 7-limit (septimal) supermajor second is a ratio of [[8/7]], and is about 231{{c}}.

* The 7-limit ~~'''~~(septimal) supermajor second~~'''~~ is a ratio of [[8/7]], and is about 231{{c}}.

* The 11-limit (undecimal) submajor second is a ratio of [[11/10]], and is about 165{{c}}, though it can also be analyzed as a [[neutral second]]. Despite that, it is also here for completeness.

* The 11-limit ~~'''~~(undecimal) submajor second~~'''~~ is a ratio of [[11/10]], and is about 165{{c}}, though it can also be analyzed as a [[neutral second]]. Despite that, it is also here for completeness.

* The 13-limit (tridecimal) ultramajor second is a ratio of [[15/13]], and is about 248{{c}}, though it can also be analyzed as an [[minor third|inframinor third]]. Despite that, it is also here for completeness.

* The 13-limit ~~'''~~(tridecimal) ultramajor second~~'''~~ is a ratio of [[15/13]], and is about 248{{c}}, though it can also be analyzed as an [[minor third|inframinor third]]. Despite that, it is also here for completeness.

=== By delta ===

Line 99:

Line 217:

The following table lists the best tuning of 10/9, 9/8, and 8/7, as well as other major seconds if present, in various significant [[edo]]s.

{| class="wikitable"

{| class="wikitable" style="text-align: center;"

|-

! Edo

Line 193:

Line 311:

|}

== In ~~mos scales~~ ==

== In regular temperaments ==

~~Being~~ a ~~small interval~~, major seconds ~~generate~~ a ~~number of monosmall~~ and ~~monolarge~~ [[~~mos~~]].

The three simplest major second ratios are 10/9, 9/8, and 8/7, and these along with other more complex interpretations serve as [[generator]]s for a variety of [[regular temperament]]s.

* The generator of the 7L 1s scale can be interpreted as a [[10/9]] major second, that is equated to [[11/10]] and [[12/11]] [[neutral second]]s by [[porcupine]], so that three generators reach [[4/3]]. Its tuning range is therefore somewhat ambiguous between major and neutral second.

* The generator of the 6L 1s and 7L 6s scales can be interpreted as a 10/9 major second in [[tetracot]] (which is sometimes equated to 11/10), where four of these seconds reach [[3/2]].

* The generator of the 6L 1s and 6L 7s scales can be interpreted in terms of [[2.5.7 subgroup|2.5.7]] [[didacus]], whose generator represents [[28/25]] and which splits the septimal tritone [[7/5]] in three, with one step making the generator 28/25 and two making [[5/4]]. This generator can also stand in for 10/9 and [[9/8]] in the 2.9.5.7 [[subgroup]], if it is treated as an index-2 restriction of [[septimal meantone]].

* The generator of the 5L 6s scale can be interpreted as [[8/7]] in [[2.3.7 subgroup|2.3.7]] [[slendric]], where three of them are equated to 3/2.

* The generator of the 5L 4s scale can be interpreted in terms of 2.3.7 [[semaphore]], where 8/7 is equated to the subminor third [[7/6]] so that two generators reach 4/3, or more accurately as 2.3.13/5 [[barbados]] if 8/7 is eschewed in favor of [[15/13]]. Either way, it is tuned as an [[interseptimal interval|interseptimal]] ambiguous between a major second and [[minor third]].

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

~~Scales~~ with ~~more than 12 notes are not included~~.

Due to the [[9/8]] major second being closely related to the perfect fifth, it is often useful to detune the fifth to approach other intervals with the diatonic major second. If the diatonic perfect fifth is treated as [[3/2]], approximating various intervals with the diatonic major second leads to the following temperaments:

{| class="wikitable"

{| class="wikitable center-2 center-5"

|+

! Just<br>interval

! Cents

! Temperament

! Vanishing<br>comma

! Generator<br>(eigenmonzo tuning)

|-

~~! Range~~

| 21/19

~~! colspan="3"~~ | ~~Mos~~

| 173{{c}}

| [[Surprise]]

| [[57/56]]

| 687{{c}}

|-

| ~~150–171~~{{c}}

| 10/9

| [[~~1L 6s~~]]

| 182{{c}}

~~| colspan="2"~~ | [[~~7L 1s~~]]

| [[Meantone]]

| [[81/80]]

| 691{{c}}

|-

| ~~171–200~~{{c}}

| 19/17

| [[~~1L 5s~~]]

| 193{{c}}

~~| colspan="2"~~ | [[~~6L 1s~~]]

| [[Little ganassi]]

| [[153/152]]

| 696{{c}}

|-

| ~~200–218~~{{c}}

| 9/8

~~| rowspan="2"~~ | [[~~1L 4s~~]]

| 204{{c}}

~~| rowspan="2"~~ | [[~~5L 1s~~]]

| [[Pythagorean]]

| ~~[[6L 5s]]~~

| [[1/1]]

| 702{{c}}

|-

| ~~218–240~~{{c}}

| 17/15

| [[~~5L 6s~~]]

| 217{{c}}

| [[Fiventeen]]

| [[136/135]]

| 708{{c}}

|-

| ~~240–267~~{{c}}

| 8/7

| [[~~1L 3s~~]]

| 231{{c}}

| [[~~4L 1s~~]]

| [[Archy]]

| ~~[[5L 4s]]~~

| [[64/63]]

| 716{{c}}

|}

~~=== Temperament interpretations ===~~

~~The three simplest major second ratios are 10/9, 9/8, and 8/7, and these along with other more complex interpretations serve as [[generator]]s for a variety of [[regular temperament]]s.~~

* The generator of the 7L 1s scale can be interpreted as a [[10/9]] major second, that is equated to [[11/10]] and [[12/11]] [[neutral second]]s by [[porcupine]], so that three generators reach [[4/3]]. Its tuning range is therefore somewhat ambiguous between major and neutral second.

* The generator of the 6L 1s and 7L 6s scales can be interpreted as a 10/9 major second in [[tetracot]] (which is sometimes equated to 11/10), where four of these seconds reach [[3/2]].

* The generator of the 6L 1s and 6L 7s scales can be interpreted in terms of [[2.5.7 subgroup|2.5.7]] [[didacus]], whose generator represents [[28/25]] and which splits the septimal tritone [[7/5]] in three, with one step making the generator 28/25 and two making [[5/4]]. This generator can also stand in for 10/9 and [[9/8]] in the 2.9.5.7 [[subgroup]], if it is treated as an index-2 restriction of [[septimal meantone]].

* The generator of the 5L 6s scale can be interpreted as [[8/7]] in [[2.3.7 subgroup|2.3.7]] [[slendric]], where three of them are equated to 3/2.

* The generator of the 5L 4s scale can be interpreted in terms of 2.3.7 [[semaphore]], where 8/7 is equated to the subminor third [[7/6]] so that two generators reach 4/3, or more accurately as 2.3.13/5 [[barbados]] if 8/7 is eschewed in favor of [[15/13]]. Either way, it is tuned as an [[interseptimal interval|interseptimal]] ambiguous between a major second and [[minor third]].

~~{{Todo|complete list|inline=1}}~~

@@ Line 1: / Line 1: @@
+A '''major second''' ('''M2''') is the larger of two seconds – intervals spanning 2 degrees or 1 scale step in the diatonic scale. It is found on the 1st note of the major scale, hence its name. Because it is one large step, it is also called a '''tone''' or '''whole tone'''. Another diatonic interval around the same size is the '''diminished third''' ('''d3'''). More generally, an interval close to 200 cents can be called a major second.
+Many JI intervals are called major seconds, but the term usually refers to one of these three intervals:
+* [[9/8]], the major tone of about 204 cents.
+* [[10/9]], the minor tone of about 182 cents.
+* [[8/7]], the septimal major second of about 231 cents.
+== As an interval region ==
 {{Infobox interval region
 | Name = Major second, whole tone
@@ Line 6: / Line 14: @@
 | Cents upper wide = 260
 | JI intervals = 8/7, 9/8, 10/9
-| MOSes = 1L 6s, 7L 1s, 6L 1s, 1L 5s, 5L 1s, 5L 4s
+| MOSes = 1L&nbsp;6s, 7L&nbsp;1s, 6L&nbsp;1s, 1L&nbsp;5s, 5L&nbsp;1s, 5L&nbsp;4s
 | Complement = [[Minor seventh]]
 | Lower region = [[Neutral&nbsp;second]]
-| Higher region = [[Minor&nbsp;third]]
+| Higher region = [[Minor third]]
+}}
+As a concrete [[interval region]], a major second is typically near 200{{c}} in size, distinct from the [[Semitone (interval region)|semitone]] of roughly 100{{c}} and the [[neutral second]] of roughly 150{{c}}. A rough tuning range for the major second is about 180 to 240{{c}} according to [[Margo Schulter]]'s theory of interval regions.
+This article covers intervals between 160 and 260{{c}}. The outer range of this might be too extreme to call "major seconds", but this is done so that one can find what they are looking for easily.
+=== In mos scales ===
+Being a small interval, major seconds generate a number of monosmall and monolarge [[mos scale]]s.
+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="3" | Mos
+|-
+| 150–171{{c}}
+| [[1L 6s]]
+| colspan="2" | [[7L 1s]]
+|-
+| 171–200{{c}}
+| [[1L 5s]]
+| colspan="2" | [[6L 1s]]
+|-
+| 200–218{{c}}
+| rowspan="2" | [[1L 4s]]
+| rowspan="2" | [[5L 1s]]
+| [[6L 5s]]
+|-
+| 218–240{{c}}
+| [[5L 6s]]
+|-
+| 240–267{{c}}
+| [[1L 3s]]
+| [[4L 1s]]
+| [[5L 4s]]
+|}
+== As a diatonic interval category ==
+{{Infobox
+| Title = Diatonic major second
+| Header 1 = MOS| Data 1 = [[5L&nbsp;2s]]
+| Header 2 = Other names| Data 2 = Major 1-diastep
+| Header 3 = Generator span| Data 3 = +2 generators
+| Header 4 = Tuning range| Data 4 = 171–240{{c}}
+| Header 5 = Basic tuning| Data 5 = 200{{c}}
+| Header 6 = Function on root| Data 6 = Supertonic
+| Header 7 = Interval regions| Data 7 = Major second
+| Header 8=Associated just intervals| Data 8 = [[10/9]], [[9/8]]
+| Header 9=Octave complement| Data 9 = [[Minor seventh]]
 }}
-A '''major second''' ('''M2''') is an interval that spans one scale step in the [[5L 2s|diatonic]] scale with the major (wider) quality. It is generated by stacking 2 fifths [[octave reduction|octave reduced]], and depending on the specific tuning, it ranges from 171 to 240{{cent}} ([[7edo|1\7]] to [[5edo|1\5]]). It can be considered the large step of the diatonic scale.
+As a diatonic interval category, a major second is an interval that spans one scale step in the [[5L 2s|diatonic]] scale with the major (wider) quality. It is generated by stacking 2 fifths [[octave reduction|octave reduced]], and depending on the specific tuning, it ranges from 171 to 240{{c}} ([[7edo|1\7]] to [[5edo|1\5]]). It can be considered the large step of the diatonic scale.
+In [[just intonation]], an interval may be classified as a major second if it is reasonably mapped to 2 steps of the chromatic scale - formally, this is 4\24, which is used as opposed to [[12edo]]'s 2\12 to better capture the characteristics of many intervals in the [[11-limit|11-]] and [[13-limit]] – and 1 step of the diatonic scale. Diminished thirds are mapped to 2 steps of the chromatic scale and 2 steps of the diatonic scale.
+In TAMNAMS, the major second is called the ''major 1-diastep''.
+Given its role as the large step, it can be used to construct other diatonic intervals, along with the [[minor second]]: two major seconds make a [[major third]], a major second and a minor second make a [[minor third]], and three major seconds result in an [[augmented fourth]], also called a tritone for that reason.
+=== Scale info ===
+The diatonic scale contains five major seconds. In the Ionian mode, major seconds are found on the 1st, 2nd, 4th, 5th, and 6th degrees of the scale; the other two degrees have minor seconds. The large number of major seconds compared to minor seconds ensures that thirds that include minor seconds (that is, minor thirds) are roughly evenly distributed with major thirds; in a scale with three small steps and four large steps, for example, six out of the seven thirds are minor.
-In [[just intonation]], an interval may be classified as a major second if it is reasonably mapped to one step of the diatonic scale and two steps of the chromatic scale.
+=== Tunings ===
+Being an abstract mos degree, and not a specific interval, the diatonic major second 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.
-As a concrete [[interval region]], it is typically near 200{{c}} in size, distinct from the [[Semitone (interval region)|semitone]] of roughly 100 ¢ and the [[neutral second]] of roughly 150{{c}}. A rough tuning range for the major second is about 180 to 240{{c}} according to [[Margo Schulter]]'s theory of interval regions.
+The tuning range of the diatonic major second ranges from 171.4 to 240{{c}}. The generator for a given tuning in cents, ''n'', for the diatonic major second can be found by {{nowrap| (''n'' + 1200)/2. For example, the second 192{{c}} gives us {{nowrap| (192 + 1200)/2 {{=}} 1392/2 {{=}} 696{{c}} }}, corresponding to 50edo.
-This article covers intervals between 160 and 260{{c}}. The outer range of this might be too extreme to call "major seconds", but this is done so that one can find what they are looking for easily.
+Several example tunings are provided below:
+{| class="wikitable center-all left-1"
+! Tuning
+! Step ratio
+! Edo
+! Cents
+|-
+| Equalized
+| 1:1
+| 7
+| 171{{c}}
+|-
+| Supersoft
+| 4:3
+| 26
+| 184{{c}}
+|-
+| Soft
+| 3:2
+| 19
+| 189{{c}}
+|-
+| Semisoft
+| 5:3
+| 31
+| 194{{c}}
+|-
+| Basic
+| 2:1
+| 12
+| 200{{c}}
+|-
+| Semihard
+| 5:2
+| 29
+| 207{{c}}
+|-
+| Hard
+| 3:1
+| 17
+| 212{{c}}
+|-
+| Superhard
+| 4:1
+| 22
+| 218{{c}}
+|-
+| Collapsed
+| 1:0
+| 5
+| 240{{c}}
+|}
 == In just intonation ==
@@ Line 24: / Line 143: @@
 Other major seconds exist in higher [[prime limit|limits]], however, for example:
+* The 5-limit ptolemaic major second is a ratio of [[10/9]], however in 5-limit harmony it is used alongside 9/8. It is about 182{{c}}.
-* The 5-limit '''ptolemaic major second''' is a ratio of [[10/9]], however in 5-limit harmony it is used alongside 9/8. It is about 182{{c}}.
+* The 7-limit (septimal) supermajor second is a ratio of [[8/7]], and is about 231{{c}}.
-* The 7-limit '''(septimal) supermajor second''' is a ratio of [[8/7]], and is about 231{{c}}.
+* The 11-limit (undecimal) submajor second is a ratio of [[11/10]], and is about 165{{c}}, though it can also be analyzed as a [[neutral second]]. Despite that, it is also here for completeness.
-* The 11-limit '''(undecimal) submajor second''' is a ratio of [[11/10]], and is about 165{{c}}, though it can also be analyzed as a [[neutral second]]. Despite that, it is also here for completeness.
+* The 13-limit (tridecimal) ultramajor second is a ratio of [[15/13]], and is about 248{{c}}, though it can also be analyzed as an [[minor third|inframinor third]]. Despite that, it is also here for completeness.
-* The 13-limit '''(tridecimal) ultramajor second''' is a ratio of [[15/13]], and is about 248{{c}}, though it can also be analyzed as an [[minor third|inframinor third]]. Despite that, it is also here for completeness.
 === By delta ===
@@ Line 99: / Line 217: @@
 The following table lists the best tuning of 10/9, 9/8, and 8/7, as well as other major seconds if present, in various significant [[edo]]s.
-{| class="wikitable"
+{| class="wikitable" style="text-align: center;"
 |-
 ! Edo
@@ Line 193: / Line 311: @@
 |}
-== In mos scales ==
+== In regular temperaments ==
-Being a small interval, major seconds generate a number of monosmall and monolarge [[mos]].
+The three simplest major second ratios are 10/9, 9/8, and 8/7, and these along with other more complex interpretations serve as [[generator]]s for a variety of [[regular temperament]]s.
+* The generator of the 7L 1s scale can be interpreted as a [[10/9]] major second, that is equated to [[11/10]] and [[12/11]] [[neutral second]]s by [[porcupine]], so that three generators reach [[4/3]]. Its tuning range is therefore somewhat ambiguous between major and neutral second.
+* The generator of the 6L 1s and 7L 6s scales can be interpreted as a 10/9 major second in [[tetracot]] (which is sometimes equated to 11/10), where four of these seconds reach [[3/2]].
+* The generator of the 6L 1s and 6L 7s scales can be interpreted in terms of [[2.5.7 subgroup|2.5.7]] [[didacus]], whose generator represents [[28/25]] and which splits the septimal tritone [[7/5]] in three, with one step making the generator 28/25 and two making [[5/4]]. This generator can also stand in for 10/9 and [[9/8]] in the 2.9.5.7 [[subgroup]], if it is treated as an index-2 restriction of [[septimal meantone]].
+* The generator of the 5L 6s scale can be interpreted as [[8/7]] in [[2.3.7 subgroup|2.3.7]] [[slendric]], where three of them are equated to 3/2.
+* The generator of the 5L 4s scale can be interpreted in terms of 2.3.7 [[semaphore]], where 8/7 is equated to the subminor third [[7/6]] so that two generators reach 4/3, or more accurately as 2.3.13/5 [[barbados]] if 8/7 is eschewed in favor of [[15/13]]. Either way, it is tuned as an [[interseptimal interval|interseptimal]] ambiguous between a major second and [[minor third]].
-These tables start from the last monolarge mos generated by the interval range.
+{{Todo|complete list|inline=1}}
-Scales with more than 12 notes are not included.
+Due to the [[9/8]] major second being closely related to the perfect fifth, it is often useful to detune the fifth to approach other intervals with the diatonic major second. If the diatonic perfect fifth is treated as [[3/2]], approximating various intervals with the diatonic major second leads to the following temperaments:
-{| class="wikitable"
+{| class="wikitable center-2 center-5"
+|+
+! Just<br>interval
+! Cents
+! Temperament
+! Vanishing<br>comma
+! Generator<br>(eigenmonzo tuning)
 |-
-! Range
+| 21/19
-! colspan="3" | Mos
+| 173{{c}}
+| [[Surprise]]
+| [[57/56]]
+| 687{{c}}
 |-
-| 150–171{{c}}
+| 10/9
-| [[1L&nbsp;6s]]
+| 182{{c}}
-| colspan="2" | [[7L&nbsp;1s]]
+| [[Meantone]]
+| [[81/80]]
+| 691{{c}}
 |-
-| 171–200{{c}}
+| 19/17
-| [[1L&nbsp;5s]]
+| 193{{c}}
-| colspan="2" | [[6L&nbsp;1s]]
+| [[Little ganassi]]
+| [[153/152]]
+| 696{{c}}
 |-
-| 200–218{{c}}
+| 9/8
-| rowspan="2" | [[1L&nbsp;4s]]
+| 204{{c}}
-| rowspan="2" | [[5L&nbsp;1s]]
+| [[Pythagorean]]
-| [[6L&nbsp;5s]]
+| [[1/1]]
+| 702{{c}}
 |-
-| 218–240{{c}}
+| 17/15
-| [[5L&nbsp;6s]]
+| 217{{c}}
+| [[Fiventeen]]
+| [[136/135]]
+| 708{{c}}
 |-
-| 240–267{{c}}
+| 8/7
-| [[1L&nbsp;3s]]
+| 231{{c}}
-| [[4L&nbsp;1s]]
+| [[Archy]]
-| [[5L&nbsp;4s]]
+| [[64/63]]
+| 716{{c}}
 |}
-=== Temperament interpretations ===
-The three simplest major second ratios are 10/9, 9/8, and 8/7, and these along with other more complex interpretations serve as [[generator]]s for a variety of [[regular temperament]]s.
-* The generator of the 7L 1s scale can be interpreted as a [[10/9]] major second, that is equated to [[11/10]] and [[12/11]] [[neutral second]]s by [[porcupine]], so that three generators reach [[4/3]]. Its tuning range is therefore somewhat ambiguous between major and neutral second.
-* The generator of the 6L 1s and 7L 6s scales can be interpreted as a 10/9 major second in [[tetracot]] (which is sometimes equated to 11/10), where four of these seconds reach [[3/2]].
-* The generator of the 6L 1s and 6L 7s scales can be interpreted in terms of [[2.5.7 subgroup|2.5.7]] [[didacus]], whose generator represents [[28/25]] and which splits the septimal tritone [[7/5]] in three, with one step making the generator 28/25 and two making [[5/4]]. This generator can also stand in for 10/9 and [[9/8]] in the 2.9.5.7 [[subgroup]], if it is treated as an index-2 restriction of [[septimal meantone]].
-* The generator of the 5L 6s scale can be interpreted as [[8/7]] in [[2.3.7 subgroup|2.3.7]] [[slendric]], where three of them are equated to 3/2.
-* The generator of the 5L 4s scale can be interpreted in terms of 2.3.7 [[semaphore]], where 8/7 is equated to the subminor third [[7/6]] so that two generators reach 4/3, or more accurately as 2.3.13/5 [[barbados]] if 8/7 is eschewed in favor of [[15/13]]. Either way, it is tuned as an [[interseptimal interval|interseptimal]] ambiguous between a major second and [[minor third]].
-{{Todo|complete list|inline=1}}
 {{Navbox intervals}}