Major second: Difference between revisions

Diatonic major second
MOS	5L 2s
Other names	Major 1-diastep
Generator span	+2 generators
Tuning range	171–240 ¢
Basic tuning	200 ¢
Function on root	Supertonic
Interval regions	Major second
Associated just intervals	10/9, 9/8
Octave complement	Minor seventh

← Neutral second Major second, whole tone Minor third →
Name	Major second, whole tone
Lower bound	160¢ – 180¢
Upper bound	240¢ – 260¢
Generated MOSes	1L 6s, 7L 1s, 6L 1s, 1L 5s, 5L 1s, 5L 4s
	Example JI intervals
Intervals	8/7 (231.2¢); 9/8 (203.9¢); 10/9 (182.4¢)
	Related regions
Complement	Minor seventh

← Older edit

VisualWikitext

Latest revision as of 12:56, 31 March 2026

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

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

This article covers intervals between 160 and 260 ¢. 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 scales.

These tables start from the last monolarge mos generated by the interval range. Scales with more than 12 notes are not included.

Range	Mos
150–171 ¢	1L 6s	7L 1s
171–200 ¢	1L 5s	6L 1s
200–218 ¢	1L 4s	5L 1s	6L 5s
218–240 ¢	1L 4s	5L 1s	5L 6s
240–267 ¢	1L 3s	4L 1s	5L 4s

As a diatonic interval category

As a diatonic interval category, a major second is an interval that spans one scale step in the diatonic scale with the major (wider) quality. It is generated by stacking 2 fifths octave reduced, and depending on the specific tuning, it ranges from 171 to 240 ¢ (1\7 to 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- 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.

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.

The tuning range of the diatonic major second ranges from 171.4 to 240 ¢. 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 ¢ gives us (192 + 1200)/2 = 1392/2 = 696 ¢, corresponding to 50edo.

Several example tunings are provided below:

Tuning	Step ratio	Edo	Cents
Equalized	1:1	7	171 ¢
Supersoft	4:3	26	184 ¢
Soft	3:2	19	189 ¢
Semisoft	5:3	31	194 ¢
Basic	2:1	12	200 ¢
Semihard	5:2	29	207 ¢
Hard	3:1	17	212 ¢
Superhard	4:1	22	218 ¢
Collapsed	1:0	5	240 ¢

In just intonation

By prime limit

The Pythagorean (3-limit) major second is 9/8, which is 204 ¢ in size and corresponds to the mos-based interval category of the diatonic major second. It is generated by stacking two just perfect fifths of 3/2. There is also a Pythagorean diminished third of 65536/59049, which is about 180 cents in size. While called a "third", it is within the range of major seconds.

Other major seconds exist in higher 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 ¢.
The 7-limit (septimal) supermajor second is a ratio of 8/7, and is about 231 ¢.
The 11-limit (undecimal) submajor second is a ratio of 11/10, and is about 165 ¢, 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 ¢, though it can also be analyzed as an inframinor third. Despite that, it is also here for completeness.

By delta

See Delta-N ratio. Ratios that are marginal within the interval category and ambiguous with an adjoining one are marked with an asterisk.

Delta-1		Delta-2		Delta-3
8/7	231 ¢	15/13*	248 ¢	22/19*	253 ¢
9/8	204 ¢	17/15	217 ¢	23/20*	242 ¢
10/9	182 ¢	19/17	193 ¢	25/22	221 ¢
11/10*	165 ¢	21/19	173 ¢	26/23	212 ¢
				28/25	196 ¢
				29/26	189 ¢
				31/28	176 ¢
				32/29*	170 ¢

In edos

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 edos.

Edo	10/9	9/8	8/7	Other major seconds
5	240 ¢
7	171 ¢
12	200 ¢
15	160 ¢	240 ¢
16	*	225 ¢
17	212 ¢
19	189 ¢		253 ¢
22	164 ¢	218 ¢
24	200 ¢		250 ¢
25	192 ¢		240 ¢
26	185 ¢		231c
27	178 ¢	222 ¢
29	166 ¢	207 ¢	248 ¢
31	194 ¢		232 ¢
34	176 ¢	212 ¢	247 ¢
41	176 ¢	205 ¢	234 ¢
53	181 ¢	204 ¢	226 ¢	249 ¢ ≈ 15/13

In regular temperaments

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

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 seconds 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 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 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 ambiguous between a major second and minor third.

Todo: complete list

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:


Just interval	Cents	Temperament	Vanishing comma	Generator (eigenmonzo tuning)
21/19	173 ¢	Surprise	57/56	687 ¢
10/9	182 ¢	Meantone	81/80	691 ¢
19/17	193 ¢	Little ganassi	153/152	696 ¢
9/8	204 ¢	Pythagorean	1/1	702 ¢
17/15	217 ¢	Fiventeen	136/135	708 ¢
8/7	231 ¢	Archy	64/63	716 ¢

View • Talk • EditInterval classification
Interval regions
Unison and octave	Unison • Comma and diesis • Octave
Seconds	Minor second • Neutral second • Major second
Thirds	Minor third • Neutral third • Major third
Fourths and fifths	Perfect fourth • Superfourth • Tritone • Subfifth • Perfect fifth
Sixths	Minor sixth • Neutral sixth • Major sixth
Sevenths	Minor seventh • Neutral seventh • Major seventh
Interseptimal intervals	Interseptimal 2nd-3rd • Interseptimal 3rd-4th • Interseptimal 5th-6th • Interseptimal 6th-7th
Interval qualities
Diatonic qualities	Diminished • Minor • Perfect • Major • Augmented
Tuning ranges	Neutral (interval quality) • Submajor and supraminor • Pental major and minor • Novamajor and novaminor • Neogothic major and minor • Supermajor and subminor • Ultramajor and inframinor

Major second: Difference between revisions

Latest revision as of 12:56, 31 March 2026

Contents

As an interval region

In mos scales

As a diatonic interval category

Scale info

Tunings

In just intonation

By prime limit

By delta

In edos

In regular temperaments

Navigation menu

@@ Line 1: / Line 1: @@
-A '''major second (M2)''' in the [[5L 2s|diatonic scale]] is an interval that spans one scale step 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.
+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.
-In [[just intonation]], an interval may be classified as a major second if it is reasonably mapped to 1\7 and [[24edo|4\24]] (precisely one step of the diatonic scale and two steps of the chromatic scale). The use of 24edo's 4\24 as the mapping criteria here rather than [[12edo]]'s 2\12 better captures the characteristics of many intervals in the [[11-limit|11-]] and [[13-limit]].
+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 a concrete [[interval region]], it is typically near 200 ¢ in size, distinct from the [[Semitone (interval region)|semitone]] of roughly 100 ¢ and the [[neutral second]] of roughly 150 ¢. A rough tuning range for the major second is about 180 to 240 ¢ according to [[Margo Schulter]]'s theory of interval regions.
+== As an interval region ==
+{{Infobox interval region
+| Name = Major second, whole tone
+| Cents lower = 180
+| Cents lower wide = 160
+| Cents upper = 240
+| Cents upper wide = 260
+| JI intervals = 8/7, 9/8, 10/9
+| 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 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 ¢. The outer range of this might be too extreme to call "major seconds", but this is done so that one can find what they're looking for easily.
+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 just intonation ==
+=== 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]]
+}}
+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.
-=== By prime limit ===
+=== Tunings ===
-The '''Pythagorean ([[3-limit]]) major second''' is [[9/8]], which is 204 cents in size and corresponds to the MOS-based interval category of the diatonic major second. It is generated by [[stacking]] two just perfect fifths of [[3/2]]. There is also a '''Pythagorean diminished third''' of 65536/59049, which is about 180 cents in size. While called a "third", it is within the range of major seconds.
+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.
-Other major seconds exist in higher [[Prime limit|limits]], however, for example:
+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.
-* 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 cents.
+Several example tunings are provided below:
-* The 7-limit '''(septimal) supermajor second''' is a ratio of [[8/7]], and is about 231 cents.
-* The 11-limit '''(undecimal) submajor second''' is a ratio of [[11/10]], and is about 165 cents, 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 cents, though it can also be analyzed as an [[minor third|inframinor third]]. Despite that, it is also here for completeness.
-=== By delta ===
+{| class="wikitable center-all left-1"
-See [[Delta-N ratio]]. Ratios that are marginal within the interval category and ambiguous with an adjoining one are marked with an asterisk.
+! Tuning
-{| class="wikitable"
+! Step ratio
-! colspan="2" |Delta-1
+! Edo
-! colspan="2" |Delta-2
+! Cents
-! colspan="2" |Delta-3
+|-
+| Equalized
+| 1:1
+| 7
+| 171{{c}}
 |-
-|8/7
+| Supersoft
-|231 ¢
+| 4:3
-|15/13*
+| 26
-|248 ¢
+| 184{{c}}
-|22/19*
-|253 ¢
 |-
-|9/8
+| Soft
-|204 ¢
+| 3:2
-|17/15
+| 19
-|217 ¢
+| 189{{c}}
-|23/20*
-|242 ¢
 |-
-|10/9
+| Semisoft
-|182 ¢
+| 5:3
-|19/17
+| 31
-|193 ¢
+| 194{{c}}
-|25/22
-|221 ¢
 |-
-|11/10*
+| Basic
-|165 ¢
+| 2:1
-|21/19
+| 12
-|173 ¢
+| 200{{c}}
-|26/23
-|212 ¢
 |-
-|
+| Semihard
-|
+| 5:2
-|
+| 29
-|
+| 207{{c}}
-|28/25
-|196 ¢
 |-
-|
+| Hard
-|
+| 3:1
-|
+| 17
-|
+| 212{{c}}
-|29/26
-|189 ¢
 |-
-|
+| Superhard
-|
+| 4:1
-|
+| 22
-|
+| 218{{c}}
-|31/28
-|176 ¢
 |-
-|
+| Collapsed
-|
+| 1:0
-|
+| 5
-|
+| 240{{c}}
-|32/29*
-|170 ¢
 |}
-== In EDOs ==
+== In just intonation ==
-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|EDOs]].
+=== By prime limit ===
+The Pythagorean ([[3-limit]]) major second is [[9/8]], which is 204{{c}} in size and corresponds to the mos-based interval category of the diatonic major second. It is generated by [[stacking]] two just perfect fifths of [[3/2]]. There is also a Pythagorean diminished third of [[65536/59049]], which is about 180 cents in size. While called a "third", it is within the range of major seconds.
+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 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 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 ===
+See [[Delta-N ratio]]. Ratios that are marginal within the interval category and ambiguous with an adjoining one are marked with an asterisk.
 {| class="wikitable"
-!EDO
-!10/9
-!9/8
-!8/7
-!Other major seconds
 |-
-|5
+! colspan="2" | Delta-1
-| colspan="3" |240c
+! colspan="2" | Delta-2
-|
+! colspan="2" | Delta-3
+|-
+| 8/7
+| 231{{c}}
+| 15/13*
+| 248{{c}}
+| 22/19*
+| 253{{c}}
+|-
+| 9/8
+| 204{{c}}
+| 17/15
+| 217{{c}}
+| 23/20*
+| 242{{c}}
+|-
+| 10/9
+| 182{{c}}
+| 19/17
+| 193{{c}}
+| 25/22
+| 221{{c}}
+|-
+| 11/10*
+| 165{{c}}
+| 21/19
+| 173{{c}}
+| 26/23
+| 212{{c}}
+|-
+|
+|
+|
+|
+| 28/25
+| 196{{c}}
 |-
-|7
+|
-| colspan="3" |171c
+|
 |
+|
+| 29/26
+| 189{{c}}
 |-
-|12
+|
-| colspan="3" |200c
+|
 |
+|
+| 31/28
+| 176{{c}}
 |-
-|15
+|
-|160c
+|
-| colspan="2" |240c
+|
 |
+| 32/29*
+| 170{{c}}
+|}
+== In edos ==
+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" style="text-align: center;"
 |-
-|16
+! Edo
-|*
+! 10/9
-| colspan="2" |225c
+! 9/8
-|
+! 8/7
+! Other major seconds
 |-
-|17
+| 5
-| colspan="3" |212c
+| colspan="3" | 240{{c}}
 |
 |-
-|19
+| 7
-| colspan="2" |189c
+| colspan="3" | 171{{c}}
-|253c
+|
-|
 |-
-|22
+| 12
-|164c
+| colspan="3" | 200{{c}}
-| colspan="2" |218c
+|
-|
 |-
-|24
+| 15
-| colspan="2" |200c
+| 160{{c}}
-|250c
+| colspan="2" | 240{{c}}
 |
 |-
-|25
+| 16
-| colspan="2" |192c
+| *
-|240c
+| colspan="2" | 225{{c}}
 |
 |-
-|26
+| 17
-| colspan="2" |185c
+| colspan="3" | 212{{c}}
-|231c
+|
-|
 |-
-|27
+| 19
-|178c
+| colspan="2" | 189{{c}}
-| colspan="2" |222c
+| 253{{c}}
 |
 |-
-|29
+| 22
-|166c
+| 164{{c}}
-|207c
+| colspan="2" | 218{{c}}
-|248c
+|
-|
 |-
-|31
+| 24
-| colspan="2" |194c
+| colspan="2" | 200{{c}}
-|232c
+| 250{{c}}
 |
 |-
-|34
+| 25
-|176c
+| colspan="2" | 192{{c}}
-|212c
+| 240{{c}}
-|247c
 |
 |-
-|41
+| 26
-|176c
+| colspan="2" | 185{{c}}
-|205c
+| 231c
-|234c
 |
 |-
-|53
+| 27
-|181c
+| 178{{c}}
-|204c
+| colspan="2" | 222{{c}}
-|226c
+|
-|249c ≈ 15/13
-|}
-== In moment-of-symmetry scales ==
-Being a small interval, major seconds generate a number of monosmall and monolarge MOSes.
-These tables start from the last monolarge [[MOS]] generated by the interval range.
-MOSes with more than 12 notes are not included.
-{| class="wikitable"
 |-
-! Range
+| 29
-! colspan="3" | MOS
+| 166{{c}}
+| 207{{c}}
+| 248{{c}}
+|
 |-
-| 150–171{{c}}
+| 31
-| [[1L&nbsp;6s]]
+| colspan="2" | 194{{c}}
-| colspan="2" | [[7L&nbsp;1s]]
+| 232{{c}}
-|-
+|
-| 171–200{{c}}
-| [[1L&nbsp;5s]]
-| colspan="2" | [[6L&nbsp;1s]]
 |-
-| 200–218{{c}}
+| 34
-| rowspan="2" | [[1L&nbsp;4s]]
+| 176{{c}}
-| rowspan="2" | [[5L&nbsp;1s]]
+| 212{{c}}
-| [[6L&nbsp;5s]]
+| 247{{c}}
+|
 |-
-| 218–240{{c}}
+| 41
-| [[5L&nbsp;6s]]
+| 176{{c}}
+| 205{{c}}
+| 234{{c}}
+|
 |-
-| 240–267{{c}}
+| 53
-| [[1L&nbsp;3s]]
+| 181{{c}}
-| [[4L&nbsp;1s]]
+| 204{{c}}
-| [[5L&nbsp;4s]]
+| 226{{c}}
+| {{nowrap|249{{c}} ≈ 15/13}}
 |}
-=== Temperament interpretations ===
+== In regular temperaments ==
 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 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}}
+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 center-2 center-5"
+|+
+! Just<br>interval
+! Cents
+! Temperament
+! Vanishing<br>comma
+! Generator<br>(eigenmonzo tuning)
+|-
+| 21/19
+| 173{{c}}
+| [[Surprise]]
+| [[57/56]]
+| 687{{c}}
+|-
+| 10/9
+| 182{{c}}
+| [[Meantone]]
+| [[81/80]]
+| 691{{c}}
+|-
+| 19/17
+| 193{{c}}
+| [[Little ganassi]]
+| [[153/152]]
+| 696{{c}}
+|-
+| 9/8
+| 204{{c}}
+| [[Pythagorean]]
+| [[1/1]]
+| 702{{c}}
+|-
+| 17/15
+| 217{{c}}
+| [[Fiventeen]]
+| [[136/135]]
+| 708{{c}}
+|-
+| 8/7
+| 231{{c}}
+| [[Archy]]
+| [[64/63]]
+| 716{{c}}
+|}
 {{Navbox intervals}}

Major second: Difference between revisions

Latest revision as of 12:56, 31 March 2026

As an interval region

In mos scales

As a diatonic interval category

Scale info

Tunings

In just intonation

By prime limit

By delta

In edos

In regular temperaments

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