Perfect fifth: Difference between revisions

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A '''perfect fifth (P5)''' is an interval ~~that is near~~ 700 [[~~cent~~]]s in size, distinct from ~~diminished~~ fifths ~~(a type~~ of ~~[[tritone]], about 600 cents)~~. A rough tuning range for the perfect fifth is about ~~650~~ to ~~750~~ [[cents]]~~, though this is extremely wide; some might prefer~~ to ~~restrict it to around 670-730 cents~~. Another common range is the ~~even~~ stricter ~~[[diatonic~~ range~~]],~~ from ~686 to 720 ~~cents~~, which ~~corresponds~~ to [[diatonic perfect fifth]]s that ~~can be used to generate a~~ [[5L 2s|diatonic scale]].

A '''perfect fifth''' ('''P5'''), is the large and most common of the two fifths – intervals spanning 5 degrees or 4 scale steps in the diatonic scale. It is found between the 1st and 5th degrees of all diatonic modes except Locrian. Another diatonic interval around the same size is the '''diminished sixth''' ('''d6'''). More generally, an interval close to 700 cents can be called a perfect fifth.

== As an interval region ==

{{Infobox interval region

| Name = Perfect fifth

| Cents lower = 686

| Cents lower wide = 650

| Cents upper = 720

| Cents upper wide = 750

| JI intervals = 3/2

| MOSes = [[5L 2s]], [[2L 5s]], [[5L 3s]], [[7L 2s]], [[2L 7s]]

| Complement = [[Perfect fourth]]

| Lower region = [[Semidiminished fifth]] <br> [[Tritone]]

| Higher region = [[Minor sixth]]

}}

As a concrete [[interval region]], a perfect fifth is typically near 700{{c}} in size, distinct from semidiminished fifths of rougly 650{{c}}. A rough tuning range for the perfect fifth is about 670 to 730 [[cents]] according to [[Margo Schulter]]'s theory of interval regions. Another common range is the stricter range from 686 to 720{{c}}, which generates a diatonic scale.

This article covers intervals from 660 to 750{{c}}, but intervals between 650 and 660 cents have been "grandfathered in" due to the fact that superfourths and subfifths were not originally given their own articles.

=== In mos scales ===

Intervals between 654 and 750{{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="6" | Mos

|-

| 720–750{{c}}

| rowspan="5" | [[1L 1s]]

| rowspan="5" | [[2L 1s]]

| [[3L 2s]]

| colspan="3" | [[5L 3s]]

|-

| 700–720{{c}}

| rowspan="4" | [[2L 3s]]

| rowspan="2" | [[5L 2s]]

| colspan="2" | [[5L 7s]]

|-

| 686–700{{c}}

| colspan="2" | [[7L 5s]]

|-

| 667–686{{c}}

| rowspan="2" | [[2L 5s]]

| colspan="2" | [[7L 2s]]

|-

| 654–667{{c}}

| [[2L 7s]]

| [[9L 2s]]

|}

== As a diatonic interval category ==

{{Infobox

| Title = Diatonic perfect fifth

| Header 1 = MOS | Data 1 = [[5L 2s]]

| Header 2 = Other names | Data 2 = Perfect 4-diastep

| Header 3 = Generator span | Data 3 = +1 generator

| Header 4 = Tuning range | Data 4 = 686–720{{c}}

| Header 5 = Basic tuning | Data 5 = 700{{c}}

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

| Header 7 = Interval regions | Data 7 = Perfect fifth

| Header 8 = Associated just intervals | Data 8 = [[3/2]]

| Header 9 = Octave complement | Data 9 = [[Perfect fourth]]

}}

A perfect fourth is an interval that spans four steps of the [[5L 2s|diatonic]] scale with a perfect quality, i.e. the quality that exists in all but one modes. It is a [[generator]] of the diatonic scale. Depending on the specific tuning, it ranges from 686 to 720{{cent}} ([[7edo|4\7]] to [[5edo|3\5]]).

In [[just intonation]], the just perfect fifth is [[3/2]]. Other intervals are also classified as perfect fifths, sometimes called ''wolf fifths'' or ''imperfect fifths'', if they are reasonably mapped to four steps of the diatonic scale and seven steps of the chromatic scale, or formally 4\7 and [[24edo|14\24]]. The use of 24edo's 14\24 as the mapping criteria here rather than [[12edo]]'s 7\12 better captures the characteristics of many intervals in the [[11-limit|11-]] and [[13-limit]].

In [[TAMNAMS]], this interval is called the '''perfect 4-diastep'''.

== In just intonation ==

The only "perfect" ~~fourth~~ in JI is the ~~'''~~Pythagorean perfect fifth~~'''~~ of [[3/2]], about 702 ~~cents~~ in size, which corresponds to the ~~MOS~~-based interval category of the diatonic perfect fifth, and is the generator for Pythagorean tuning and the diatonic scale. However, various "out of tune" fifths exist, such as the ~~'''~~Pythagorean wolf fifth [[262144/177147]]~~'''~~, which is flat of 3/2 by one [[Pythagorean comma]], and is about 678 ~~cents~~ in size.

The only "perfect" fifth in JI is the Pythagorean perfect fifth of [[3/2]], about 702{{c}} in size, which corresponds to the mos-based interval category of the diatonic perfect fifth, and is the generator for Pythagorean tuning and the diatonic scale. However, various "out of tune" fifths exist, such as the Pythagorean wolf fifth [[262144/177147]], which is flat of 3/2 by one [[Pythagorean comma]], and is about 678{{c}} in size.

Other "out of tune" fifths in higher [[~~Prime~~ limit|limits]] include:

Other "out of tune" fifths in higher [[prime limit|limits]] include:

*The 5-limit '''grave fifth''' is a ratio of 40/27, and is about 680 ~~cents~~

* The 5-limit '''grave fifth''' is a ratio of 40/27, and is about 680{{c}}

*The 7-limit '''superfifth''' is a ratio of 32/21, and is about 729 ~~cents~~.

* The 7-limit '''superfifth''' is a ratio of 32/21, and is about 729{{c}}.

*The 11-limit '''diminished fifth''' is a ratio of 22/15, and is about 663 ~~cents~~.

* The 11-limit '''diminished fifth''' is a ratio of 22/15, and is about 663{{c}}.

**There is also an 11-limit '''acute fifth,''' which is a ratio of 50/33, and is about 720 ~~cents~~.

** There is also an 11-limit '''acute fifth,''' which is a ratio of 50/33, and is about 720{{c}}.

*The 13-limit '''ultrafifth''' is a ratio of 20/13, and is about 746 ~~cents~~, but it might be better analyzed as an [[~~Minor~~ sixth|inframinor sixth]]. Despite that, it is also here for completeness.

* The 13-limit '''ultrafifth''' is a ratio of 20/13, and is about 746{{c}}, but it might be better analyzed as an [[minor sixth|inframinor sixth]]. Despite that, it is also here for completeness.

== In edos ==

The following table lists the best tuning of 3/2, as well as other fifths if present, in various significant [[edo]]s.

~~== In EDOs ==~~

~~The following table lists the best tuning of 3/2, as well as other fifths if present, in various significant [[EDOs]].~~

{| class="wikitable"

|+

~~!EDO~~

~~!3/2~~

~~!Other fifths~~

|-

|5

! Edo

~~|720c~~

! 3/2

|

! Other fifths

|-

|7

| 5

|~~686c~~

| 720{{c}}

|

|-

|12

| 7

|~~700c~~

| 686{{c}}

|

|-

|15

| 12

|~~720c~~

| 700{{c}}

|

|-

|16

| 15

|~~675c~~

| 720{{c}}

|~~750c ≈ 20/13~~

|

|-

|17

| 16

|~~706c~~

| 675{{c}}

|

| {{nowrap|750{{c}} ≈ 20/13}}

|-

|19

| 17

|~~694c~~

| 706{{c}}

|

|-

|22

| 19

|~~709c~~

| 694{{c}}

|~~654c ≈ 22/15~~

|

|-

|24

| 22

|~~700c~~

| 709{{c}}

|~~750c ≈ 20/13, 650c~~ ≈ 22/15

| {{nowrap|654{{c}} ≈ 22/15}}

|-

|25

| 24

|~~720c~~

| 700{{c}}

|~~672c~~ ≈ 40/27

| {{nowrap|750{{c}} ≈ 20/13|650{{c}} ≈ 22/15}}

|-

|26

| 25

|~~692c~~

| 720{{c}}

|~~738c~~ ≈ ~~32/21, 20~~/13

| {{nowrap|672{{c}} ≈ 40/27}}

|-

|27

| 26

|~~711c~~

| 692{{c}}

|~~666c~~ ≈ 22/15

| {{nowrap|738{{c}} ≈ 32/21, 20/13}}

|-

|29

| 27

|~~704c~~

| 711{{c}}

|~~745c ≈ 20/13, 663c~~ ≈ 22/15

| {{nowrap|666{{c}} ≈ 22/15}}

|-

|31

| 29

|~~697c~~

| 704{{c}}

|~~736c~~ ≈ 32/~~21, 659c~~ ≈ 22/15

| {{nowrap|745{{c}} ≈ 20/13|663{{c}} ≈ 22/15}}

|-

|34

| 31

|~~706c~~

| 697{{c}}

|~~742c~~ ≈ 20/~~13, 671c~~ ≈ ~~40/27,~~ 22/15

| {{nowrap|736{{c}} ≈ 32/21|659{{c}} ≈ 22/15}}

|-

|41

| 34

|~~702c~~

| 706{{c}}

|~~732c~~ ≈ 32/~~21, 674c~~ ≈ 40/27

| {{nowrap|742{{c}} ≈ 20/13|671{{c}} ≈ 40/27, 22/15}}

|-

|53

| 41

|~~702c~~

| 702{{c}}

|~~748c~~ ≈ 20/13~~, 724c~~ ≈ 32/21~~, 679c~~ ≈ 40/27~~, 657c~~ ≈ 22/15

| {{nowrap|732{{c}} ≈ 32/21|674{{c}} ≈ 40/27}}

|-

| 53

| 702{{c}}

| {{nowrap|748{{c}} ≈ 20/13|724{{c}} ≈ 32/21|679{{c}} ≈ 40/27|657{{c}} ≈ 22/15}}

|}

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The simplest perfect 5th ratio is 3/2. The following notable temperaments are generated by it:

===Temperaments that use 3/2 as a generator===

=== Temperaments that use 3/2 as a generator ===

* [[Meantone]], the temperament flattening 3/2 such that four 3/2s stack to [[5/4]]

*[[Meantone]], the temperament flattening 3/2 such that four 3/2s stack to [[5/4]]

* [[Schismatic]], the temperament slightly sharpening 3/2 such that nine 3/2s stack to [[6/5]]

*[[Schismatic]], the temperament slightly sharpening 3/2 such that nine 3/2s stack to [[6/5]]

* [[Superpyth]], the temperament sharpening 3/2 such that four 3/2s stack to [[9/7]]

*[[Superpyth]], the temperament sharpening 3/2 such that four 3/2s stack to [[9/7]]

* [[Compton]], the temperament of the Pythagorean comma, equivalent to 12edo

*[[Compton]], the temperament of the Pythagorean comma, equivalent to 12edo

* [[Mavila]], the temperament flattening 3/2 such that four 3/2s stack to [[6/5]]

*[[Mavila]], the temperament flattening 3/2 such that four 3/2s stack to [[6/5]]

* Various historical [[well temperament]]s generated by tempered 4/3s or 3/2s, equivalent to 12edo as compton and meantone

*Various historical [[~~Well~~ temperament~~|well temperaments~~]] generated by tempered 4/3s or 3/2s, equivalent to 12edo as compton and meantone

@@ Line 1: / Line 1: @@
 {{About|the [[interval region]]|the just perfect fifth|3/2}}
-A '''perfect fifth (P5)''' is an interval that is near 700 [[cent]]s in size, distinct from diminished fifths (a type of [[tritone]], about 600 cents). A rough tuning range for the perfect fifth is about 650 to 750 [[cents]], though this is extremely wide; some might prefer to restrict it to around 670-730 cents. Another common range is the even stricter [[diatonic range]], from ~686 to 720 cents, which corresponds to [[diatonic perfect fifth]]s that can be used to generate a [[5L 2s|diatonic scale]].
+{{Wikipedia}}
+A '''perfect fifth''' ('''P5'''), is the large and most common of the two fifths – intervals spanning 5 degrees or 4 scale steps in the diatonic scale. It is found between the 1st and 5th degrees of all diatonic modes except Locrian. Another diatonic interval around the same size is the '''diminished sixth''' ('''d6'''). More generally, an interval close to 700 cents can be called a perfect fifth.
+== As an interval region ==
+{{Infobox interval region
+| Name = Perfect fifth
+| Cents lower = 686
+| Cents lower wide = 650
+| Cents upper = 720
+| Cents upper wide = 750
+| JI intervals = 3/2
+| MOSes = [[5L 2s]], [[2L 5s]], [[5L 3s]], [[7L 2s]], [[2L 7s]]
+| Complement = [[Perfect fourth]]
+| Lower region = [[Semidiminished&nbsp;fifth]] <br> [[Tritone]]
+| Higher region = [[Minor&nbsp;sixth]]
+}}
+As a concrete [[interval region]], a perfect fifth is typically near 700{{c}} in size, distinct from semidiminished fifths of rougly 650{{c}}. A rough tuning range for the perfect fifth is about 670 to 730 [[cents]] according to [[Margo Schulter]]'s theory of interval regions. Another common range is the stricter range from 686 to 720{{c}}, which generates a diatonic scale.
+This article covers intervals from 660 to 750{{c}}, but intervals between 650 and 660 cents have been "grandfathered in" due to the fact that superfourths and subfifths were not originally given their own articles.
+=== In mos scales ===
+Intervals between 654 and 750{{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="6" | Mos
+|-
+| 720–750{{c}}
+| rowspan="5" | [[1L&nbsp;1s]]
+| rowspan="5" | [[2L&nbsp;1s]]
+| [[3L&nbsp;2s]]
+| colspan="3" | [[5L&nbsp;3s]]
+|-
+| 700–720{{c}}
+| rowspan="4" | [[2L&nbsp;3s]]
+| rowspan="2" | [[5L&nbsp;2s]]
+| colspan="2" | [[5L&nbsp;7s]]
+|-
+| 686–700{{c}}
+| colspan="2" | [[7L&nbsp;5s]]
+|-
+| 667–686{{c}}
+| rowspan="2" | [[2L&nbsp;5s]]
+| colspan="2" | [[7L&nbsp;2s]]
+|-
+| 654–667{{c}}
+| [[2L&nbsp;7s]]
+| [[9L&nbsp;2s]]
+|}
+== As a diatonic interval category ==
+{{Infobox
+| Title = Diatonic perfect fifth
+| Header 1 = MOS | Data 1 = [[5L&nbsp;2s]]
+| Header 2 = Other names | Data 2 = Perfect 4-diastep
+| Header 3 = Generator span | Data 3 = +1 generator
+| Header 4 = Tuning range | Data 4 = 686–720{{c}}
+| Header 5 = Basic tuning | Data 5 = 700{{c}}
+| Header 6 = Function on root | Data 6 = Dominant
+| Header 7 = Interval regions | Data 7 = Perfect fifth
+| Header 8 = Associated just intervals | Data 8 = [[3/2]]
+| Header 9 = Octave complement | Data 9 = [[Perfect fourth]]
+}}
+A perfect fourth is an interval that spans four steps of the [[5L 2s|diatonic]] scale with a perfect quality, i.e. the quality that exists in all but one modes. It is a [[generator]] of the diatonic scale. Depending on the specific tuning, it ranges from 686 to 720{{cent}} ([[7edo|4\7]] to [[5edo|3\5]]).
+In [[just intonation]], the just perfect fifth is [[3/2]]. Other intervals are also classified as perfect fifths, sometimes called ''wolf fifths'' or ''imperfect fifths'', if they are reasonably mapped to four steps of the diatonic scale and seven steps of the chromatic scale, or formally 4\7 and [[24edo|14\24]]. The use of 24edo's 14\24 as the mapping criteria here rather than [[12edo]]'s 7\12 better captures the characteristics of many intervals in the [[11-limit|11-]] and [[13-limit]].
+In [[TAMNAMS]], this interval is called the '''perfect 4-diastep'''.
 == In just intonation ==
-The only "perfect" fourth in JI is the '''Pythagorean perfect fifth''' of [[3/2]], about 702 cents in size, which corresponds to the MOS-based interval category of the diatonic perfect fifth, and is the generator for Pythagorean tuning and the diatonic scale. However, various "out of tune" fifths exist, such as the '''Pythagorean wolf fifth [[262144/177147]]''', which is flat of 3/2 by one [[Pythagorean comma]], and is about 678 cents in size.
+The only "perfect" fifth in JI is the Pythagorean perfect fifth of [[3/2]], about 702{{c}} in size, which corresponds to the mos-based interval category of the diatonic perfect fifth, and is the generator for Pythagorean tuning and the diatonic scale. However, various "out of tune" fifths exist, such as the Pythagorean wolf fifth [[262144/177147]], which is flat of 3/2 by one [[Pythagorean comma]], and is about 678{{c}} in size.
-Other "out of tune" fifths in higher [[Prime limit|limits]] include:
+Other "out of tune" fifths in higher [[prime limit|limits]] include:
-*The 5-limit '''grave fifth''' is a ratio of 40/27, and is about 680 cents
+* The 5-limit '''grave fifth''' is a ratio of 40/27, and is about 680{{c}}
-*The 7-limit '''superfifth''' is a ratio of 32/21, and is about 729 cents.
+* The 7-limit '''superfifth''' is a ratio of 32/21, and is about 729{{c}}.
-*The 11-limit '''diminished fifth''' is a ratio of 22/15, and is about 663 cents.
+* The 11-limit '''diminished fifth''' is a ratio of 22/15, and is about 663{{c}}.
-**There is also an 11-limit '''acute fifth,''' which is a ratio of 50/33, and is about 720 cents.
+** There is also an 11-limit '''acute fifth,''' which is a ratio of 50/33, and is about 720{{c}}.
-*The 13-limit '''ultrafifth''' is a ratio of 20/13, and is about 746 cents, but it might be better analyzed as an [[Minor sixth|inframinor sixth]]. Despite that, it is also here for completeness.
+* The 13-limit '''ultrafifth''' is a ratio of 20/13, and is about 746{{c}}, but it might be better analyzed as an [[minor sixth|inframinor sixth]]. Despite that, it is also here for completeness.
+== In edos ==
+The following table lists the best tuning of 3/2, as well as other fifths if present, in various significant [[edo]]s.
-== In EDOs ==
-The following table lists the best tuning of 3/2, as well as other fifths if present, in various significant [[EDOs]].
 {| class="wikitable"
-|+
-!EDO
-!3/2
-!Other fifths
 |-
-|5
+! Edo
-|720c
+! 3/2
-|
+! Other fifths
 |-
-|7
+| 5
-|686c
+| 720{{c}}
 |
 |-
-|12
+| 7
-|700c
+| 686{{c}}
 |
 |-
-|15
+| 12
-|720c
+| 700{{c}}
 |
 |-
-|16
+| 15
-|675c
+| 720{{c}}
-|750c ≈ 20/13
+|
 |-
-|17
+| 16
-|706c
+| 675{{c}}
-|
+| {{nowrap|750{{c}} ≈ 20/13}}
 |-
-|19
+| 17
-|694c
+| 706{{c}}
 |
 |-
-|22
+| 19
-|709c
+| 694{{c}}
-|654c ≈ 22/15
+|
 |-
-|24
+| 22
-|700c
+| 709{{c}}
-|750c ≈ 20/13, 650c ≈ 22/15
+| {{nowrap|654{{c}} ≈ 22/15}}
 |-
-|25
+| 24
-|720c
+| 700{{c}}
-|672c ≈ 40/27
+| {{nowrap|750{{c}} ≈ 20/13|650{{c}} ≈ 22/15}}
 |-
-|26
+| 25
-|692c
+| 720{{c}}
-|738c ≈ 32/21, 20/13
+| {{nowrap|672{{c}} ≈ 40/27}}
 |-
-|27
+| 26
-|711c
+| 692{{c}}
-|666c ≈ 22/15
+| {{nowrap|738{{c}} ≈ 32/21, 20/13}}
 |-
-|29
+| 27
-|704c
+| 711{{c}}
-|745c ≈ 20/13, 663c ≈ 22/15
+| {{nowrap|666{{c}} ≈ 22/15}}
 |-
-|31
+| 29
-|697c
+| 704{{c}}
-|736c ≈ 32/21, 659c ≈ 22/15
+| {{nowrap|745{{c}} ≈ 20/13|663{{c}} ≈ 22/15}}
 |-
-|34
+| 31
-|706c
+| 697{{c}}
-|742c ≈ 20/13, 671c ≈ 40/27, 22/15
+| {{nowrap|736{{c}} ≈ 32/21|659{{c}} ≈ 22/15}}
 |-
-|41
+| 34
-|702c
+| 706{{c}}
-|732c ≈ 32/21, 674c ≈ 40/27
+| {{nowrap|742{{c}} ≈ 20/13|671{{c}} ≈ 40/27, 22/15}}
 |-
-|53
+| 41
-|702c
+| 702{{c}}
-|748c ≈ 20/13, 724c ≈ 32/21, 679c ≈ 40/27, 657c ≈ 22/15
+| {{nowrap|732{{c}} ≈ 32/21|674{{c}} ≈ 40/27}}
+|-
+| 53
+| 702{{c}}
+| {{nowrap|748{{c}} ≈ 20/13|724{{c}} ≈ 32/21|679{{c}} ≈ 40/27|657{{c}} ≈ 22/15}}
 |}
@@ Line 92: / Line 165: @@
 The simplest perfect 5th ratio is 3/2. The following notable temperaments are generated by it:
-===Temperaments that use 3/2 as a generator===
+=== Temperaments that use 3/2 as a generator ===
+* [[Meantone]], the temperament flattening 3/2 such that four 3/2s stack to [[5/4]]
-*[[Meantone]], the temperament flattening 3/2 such that four 3/2s stack to [[5/4]]
+* [[Schismatic]], the temperament slightly sharpening 3/2 such that nine 3/2s stack to [[6/5]]
-*[[Schismatic]], the temperament slightly sharpening 3/2 such that nine 3/2s stack to [[6/5]]
+* [[Superpyth]], the temperament sharpening 3/2 such that four 3/2s stack to [[9/7]]
-*[[Superpyth]], the temperament sharpening 3/2 such that four 3/2s stack to [[9/7]]
+* [[Compton]], the temperament of the Pythagorean comma, equivalent to 12edo
-*[[Compton]], the temperament of the Pythagorean comma, equivalent to 12edo
+* [[Mavila]], the temperament flattening 3/2 such that four 3/2s stack to [[6/5]]
-*[[Mavila]], the temperament flattening 3/2 such that four 3/2s stack to [[6/5]]
+* Various historical [[well temperament]]s generated by tempered 4/3s or 3/2s, equivalent to 12edo as compton and meantone
-*Various historical [[Well temperament|well temperaments]] generated by tempered 4/3s or 3/2s, equivalent to 12edo as compton and meantone
 {{Navbox intervals}}