Perfect fifth: Difference between revisions

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A '''perfect fifth (P5)''' 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. Depending on the specific tuning, it ranges from 686 to 720 [[cent~~]]s~~ ([[7edo|4\7]] to [[5edo|3\5]]).

A '''perfect fifth (P5)''' 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. 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 4\7 and [[24edo|14\24]] (precisely four steps of the diatonic scale and seven steps of the chromatic scale). 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]].

As a concrete [[interval region]], it is typically near 700 ~~[[cent]]s~~ in size, distinct from semidiminished fifths of rougly 650 ~~cents~~. 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 ~~cents~~, which generates a diatonic scale.

As a concrete [[interval region]], it 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 650 to 750 ~~cents~~, which is very wide, but is done to avoid having to make articles for the tiny interval regions that would otherwise be required to cover the entire span of intervals.

This article covers intervals from 650 to 750{{c}}, which is very wide, but is done to avoid having to make articles for the tiny interval regions that would otherwise be required to cover the entire span of intervals.

== 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" fourth 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:

* 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 [[edos]].

{| class="wikitable"

|+

~~!Edo~~

~~!3/2~~

~~!Other fifths~~

|-

|5

! Edo

|~~720c~~

! 3/2

|

! Other fifths

|-

| 5

| 720{{c}}

|

|-

|7

| 7

|~~686c~~

| 686{{c}}

|

|-

|12

| 12

|~~700c~~

| 700{{c}}

|

|-

|15

| 15

|~~720c~~

| 720{{c}}

|

|-

|16

| 16

|~~675c~~

| 675{{c}}

|~~750c~~ ≈ 20/13

| 750{{c}} ≈ 20/13

|-

|17

| 17

|~~706c~~

| 706{{c}}

|

|-

|19

| 19

|~~694c~~

| 694{{c}}

|

|-

|22

| 22

|~~709c~~

| 709{{c}}

|~~654c~~ ≈ 22/15

| 654{{c}} ≈ 22/15

|-

|24

| 24

|~~700c~~

| 700{{c}}

|~~750c~~ ≈ 20/13, ~~650c~~ ≈ 22/15

| 750{{c}} ≈ 20/13, 650{{c}} ≈ 22/15

|-

|25

| 25

|~~720c~~

| 720{{c}}

|~~672c~~ ≈ 40/27

| 672{{c}} ≈ 40/27

|-

|26

| 26

|~~692c~~

| 692{{c}}

|~~738c~~ ≈ 32/21, 20/13

| 738{{c}} ≈ 32/21, 20/13

|-

|27

| 27

|~~711c~~

| 711{{c}}

|~~666c~~ ≈ 22/15

| 666{{c}} ≈ 22/15

|-

|29

| 29

|~~704c~~

| 704{{c}}

|~~745c~~ ≈ 20/13, ~~663c~~ ≈ 22/15

| 745{{c}} ≈ 20/13, 663{{c}} ≈ 22/15

|-

|31

| 31

|~~697c~~

| 697{{c}}

|~~736c~~ ≈ 32/21, ~~659c~~ ≈ 22/15

| 736{{c}} ≈ 32/21, 659{{c}} ≈ 22/15

|-

|34

| 34

|~~706c~~

| 706{{c}}

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

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

|-

|41

| 41

|~~702c~~

| 702{{c}}

|~~732c~~ ≈ 32/21, ~~674c~~ ≈ 40/27

| 732{{c}} ≈ 32/21, 674{{c}} ≈ 40/27

|-

|53

| 53

|~~702c~~

| 702{{c}}

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

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

|}

@@ Line 1: / Line 1: @@
 {{About|the [[interval region]]|the just perfect fifth|3/2}}
-A '''perfect fifth (P5)''' 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. Depending on the specific tuning, it ranges from 686 to 720 [[cent]]s ([[7edo|4\7]] to [[5edo|3\5]]).
+A '''perfect fifth (P5)''' 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. 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 4\7 and [[24edo|14\24]] (precisely four steps of the diatonic scale and seven steps of the chromatic scale). 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]].
-As a concrete [[interval region]], it is typically near 700 [[cent]]s in size, distinct from semidiminished fifths of rougly 650 cents. 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 cents, which generates a diatonic scale.
+As a concrete [[interval region]], it 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 650 to 750 cents, which is very wide, but is done to avoid having to make articles for the tiny interval regions that would otherwise be required to cover the entire span of intervals.
+This article covers intervals from 650 to 750{{c}}, which is very wide, but is done to avoid having to make articles for the tiny interval regions that would otherwise be required to cover the entire span of intervals.
 == 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" fourth 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:
-* 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 [[edos]].
 {| class="wikitable"
-|+
-!Edo
-!3/2
-!Other fifths
 |-
-|5
+! Edo
-|720c
+! 3/2
-|
+! Other fifths
+|-
+| 5
+| 720{{c}}
+|
 |-
-|7
+| 7
-|686c
+| 686{{c}}
 |
 |-
-|12
+| 12
-|700c
+| 700{{c}}
 |
 |-
-|15
+| 15
-|720c
+| 720{{c}}
 |
 |-
-|16
+| 16
-|675c
+| 675{{c}}
-|750c ≈ 20/13
+| 750{{c}} ≈ 20/13
 |-
-|17
+| 17
-|706c
+| 706{{c}}
 |
 |-
-|19
+| 19
-|694c
+| 694{{c}}
 |
 |-
-|22
+| 22
-|709c
+| 709{{c}}
-|654c ≈ 22/15
+| 654{{c}} ≈ 22/15
 |-
-|24
+| 24
-|700c
+| 700{{c}}
-|750c ≈ 20/13, 650c ≈ 22/15
+| 750{{c}} ≈ 20/13, 650{{c}} ≈ 22/15
 |-
-|25
+| 25
-|720c
+| 720{{c}}
-|672c ≈ 40/27
+| 672{{c}} ≈ 40/27
 |-
-|26
+| 26
-|692c
+| 692{{c}}
-|738c ≈ 32/21, 20/13
+| 738{{c}} ≈ 32/21, 20/13
 |-
-|27
+| 27
-|711c
+| 711{{c}}
-|666c ≈ 22/15
+| 666{{c}} ≈ 22/15
 |-
-|29
+| 29
-|704c
+| 704{{c}}
-|745c ≈ 20/13, 663c ≈ 22/15
+| 745{{c}} ≈ 20/13, 663{{c}} ≈ 22/15
 |-
-|31
+| 31
-|697c
+| 697{{c}}
-|736c ≈ 32/21, 659c ≈ 22/15
+| 736{{c}} ≈ 32/21, 659{{c}} ≈ 22/15
 |-
-|34
+| 34
-|706c
+| 706{{c}}
-|742c ≈ 20/13, 671c ≈ 40/27, 22/15
+| 742{{c}} ≈ 20/13, 671{{c}} ≈ 40/27, 22/15
 |-
-|41
+| 41
-|702c
+| 702{{c}}
-|732c ≈ 32/21, 674c ≈ 40/27
+| 732{{c}} ≈ 32/21, 674{{c}} ≈ 40/27
 |-
-|53
+| 53
-|702c
+| 702{{c}}
-|748c ≈ 20/13, 724c ≈ 32/21, 679c ≈ 40/27, 657c ≈ 22/15
+| 748{{c}} ≈ 20/13, 724{{c}} ≈ 32/21, 679{{c}} ≈ 40/27, 657{{c}} ≈ 22/15
 |}