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{{About|the [[interval region]]|the just perfect fourth|4/3}}
{{About|the [[interval region]]|the just perfect fourth|4/3}}
A '''perfect fourth (P4)''' is an interval that spans three 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 480 to 514 [[cent]]s ([[5edo|2\5]] to [[7edo|3\7]]).   
A '''perfect fourth (P4)''' is an interval that spans three 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 480 to 514{{cent}} ([[5edo|2\5]] to [[7edo|3\7]]).   


In [[just intonation]], the just perfect fourth is [[4/3]]. Other intervals are also classified as perfect fourths, sometimes called '''wolf fourths''' or '''imperfect fourths''', if they are reasonably mapped to 3\7 and [[24edo|10\24]] (precisely three steps of the diatonic scale and five steps of the chromatic scale). The use of 24edo's 10\24 as the mapping criteria here rather than [[12edo]]'s 5\12 better captures the characteristics of many intervals in the [[11-limit|11-]] and [[13-limit]].  
In [[just intonation]], the just perfect fourth is [[4/3]]. Other intervals are also classified as perfect fourths, sometimes called '''wolf fourths''' or '''imperfect fourths''', if they are reasonably mapped to 3\7 and [[24edo|10\24]] (precisely three steps of the diatonic scale and five steps of the chromatic scale). The use of 24edo's 10\24 as the mapping criteria here rather than [[12edo]]'s 5\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 500 cents in size, distinct from the [[semiaugmented fourth]] of roughly 550 cents. A rough tuning range for the perfect fourth is about 470 to 530 cents according to [[Margo Schulter]]'s theory of interval regions. Another common range is the stricter range from 480 to 514 cents, which generates a diatonic scale.
As a concrete [[interval region]], it is typically near 500{{c}} in size, distinct from the [[semiaugmented fourth]] of roughly 550{{c}}. A rough tuning range for the perfect fourth is about 470 to 530{{c}} according to [[Margo Schulter]]'s theory of interval regions. Another common range is the stricter range from 480 to 514{{c}}, which generates a diatonic scale.


This article covers intervals from 450 to 550 cents, in order to cover the range of intervals without needing extra articles for subfourths or superfourths.  
This article covers intervals from 450 to 550{{c}}, in order to cover the range of intervals without needing extra articles for subfourths or superfourths.  


== In just intonation ==
== In just intonation ==
=== By prime limit ===
=== By prime limit ===
The only "perfect" fourth in JI is the '''Pythagorean perfect fourth''' of [[4/3]], about 498 cents in size, which corresponds to the MOS-based interval category of the diatonic perfect fourth and is the octave complement of the perfect fifth of [[3/2]]. However, various "out of tune" fourths exist, such as the '''Pythagorean wolf fourth''' [[177147/131072]], which is sharp of 4/3 by one [[Pythagorean comma]], and is about 522 cents in size.
The only "perfect" fourth in JI is the '''Pythagorean perfect fourth''' of [[4/3]], about 498{{c}} in size, which corresponds to the MOS-based interval category of the diatonic perfect fourth and is the octave complement of the perfect fifth of [[3/2]]. However, various "out of tune" fourths exist, such as the '''Pythagorean wolf fourth''' [[177147/131072]], which is sharp of 4/3 by one [[Pythagorean comma]], and is about 522{{c}} in size.


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


* The 5-limit '''acute fourth''' is a ratio of 27/20, and is about 520 cents
* The 5-limit '''acute fourth''' is a ratio of 27/20, and is about 520{{c}}
* The 7-limit '''subfourth''' is a ratio of 21/16, and is about 471 cents.
* The 7-limit '''subfourth''' is a ratio of 21/16, and is about 471{{c}}.
* The 11-limit '''augmented fourth''' is a ratio of 15/11, and is about 537 cents.
* The 11-limit '''augmented fourth''' is a ratio of 15/11, and is about 537{{c}}.
** There is also an 11-limit '''grave fourth,''' which is a ratio of 33/25, and is about 480 cents.
** There is also an 11-limit '''grave fourth,''' which is a ratio of 33/25, and is about 480{{c}}.
* The 13-limit '''infrafourth''' is a ratio of 13/10, and is about 454 cents, but it might be better analyzed as an [[Major third|ultramajor third]]. Despite that, it is also here for completeness.
* The 13-limit '''infrafourth''' is a ratio of 13/10, and is about 454{{c}}, but it might be better analyzed as an [[Major third|ultramajor third]]. Despite that, it is also here for completeness.


=== By delta ===
=== By delta ===
{| class="wikitable"
{| class="wikitable"
|+
!Delta 1
!Cents
!Delta 3
!Cents
!Delta 4
!Cents
!Delta 5
!Cents
!Delta 6
!Cents
|-
|-
|4/3
! Delta 1
|498c
! Cents
|13/10
! Delta 3
|454c
! Cents
|15/11
! Delta 4
|537c
! Cents
|19/14
! Delta 5
|529c
! Cents
|23/17
! Delta 6
|523c
! Cents
|-
|-
|
| 4/3
|
| 498{{c}}
|
| 13/10
|
| 454{{c}}
|17/13
| 15/11
|464c
| 537{{c}}
|21/16
| 19/14
|471c
| 529{{c}}
|25/19
| 23/17
|475c
| 523{{c}}
|-
|-
|
|  
|
|  
|
|  
|
|  
|
| 17/13
|
| 464{{c}}
|
| 21/16
|
| 471{{c}}
|
| 25/19
|
| 475{{c}}
|}
|}


Line 71: Line 60:
The following table lists the best tuning of 4/3, as well as other fourths if present, in various significant [[edos]].
The following table lists the best tuning of 4/3, as well as other fourths if present, in various significant [[edos]].
{| class="wikitable"
{| class="wikitable"
|+
!Edo
!4/3
!Other fourths
|-
|-
|5
! Edo
|480c
! 4/3
|
! Other fourths
|-
|-
|7
| 5
|514c
| 480{{c}}
|
|  
|-
|-
|12
| 7
|500c
| 514{{c}}
|
|  
|-
|-
|15
| 12
|480c
| 500{{c}}
|
|  
|-
|-
|16
| 15
|525c
| 480{{c}}
|450c ≈ 13/10
|  
|-
|-
|17
| 16
|494c
| 525{{c}}
|
| 450{{c}} ≈ 13/10
|-
|-
|19
| 17
|506c
| 494{{c}}
|
|  
|-
|-
|22
| 19
|491c
| 506{{c}}
|545c ≈ 15/11
|  
|-
|-
|24
| 22
|500c
| 491{{c}}
|450c ≈ 13/10, 550c ≈ 15/11
| 545{{c}} ≈ 15/11
|-
|-
|25
| 24
|480c
| 500{{c}}
|528c ≈ 27/20
| 450{{c}} ≈ 13/10, 550{{c}} ≈ 15/11
|-
|-
|26
| 25
|508c
| 480{{c}}
|462c ≈ 21/16, 13/10
| 528{{c}} ≈ 27/20
|-
|-
|27
| 26
|489c
| 508{{c}}
|533c ≈ 15/11
| 462{{c}} ≈ 21/16, 13/10
|-
|-
|29
| 27
|496c
| 489{{c}}
|455c ≈ 13/10, 537c ≈ 15/11
| 533{{c}} ≈ 15/11
|-
|-
|31
| 29
|503c
| 496{{c}}
|464c ≈ 21/16, 541c ≈ 15/11
| 455{{c}} ≈ 13/10, 537{{c}} ≈ 15/11
|-
|-
|34
| 31
|494c
| 503{{c}}
|458c ≈ 13/10, 529c ≈ 27/20, 15/11
| 464{{c}} ≈ 21/16, 541{{c}} ≈ 15/11
|-
|-
|41
| 34
|498c
| 494{{c}}
|468c ≈ 21/16, 526c ≈ 27/20
| 458{{c}} ≈ 13/10, 529{{c}} ≈ 27/20, 15/11
|-
|-
|53
| 41
|498c
| 498{{c}}
|452c ≈ 13/10, 476c ≈ 21/16, 521c ≈ 27/20, 543c ≈ 15/11
| 468{{c}} ≈ 21/16, 526{{c}} ≈ 27/20
|-
| 53
| 498{{c}}
| 452{{c}} ≈ 13/10, 476{{c}} ≈ 21/16, 521{{c}} ≈ 27/20, 543{{c}} ≈ 15/11
|}
|}


Line 154: Line 143:
* [[Meantone]], the temperament sharpening 4/3 such that three 4/3s stack to [[6/5]]
* [[Meantone]], the temperament sharpening 4/3 such that three 4/3s stack to [[6/5]]
* [[Mavila]], the temperament sharpening 4/3 such that three 4/3s stack to [[6/5|5/4]]
* [[Mavila]], the temperament sharpening 4/3 such that three 4/3s stack to [[6/5|5/4]]
* Various historical [[Well temperament|well temperaments]] generated by tempered 4/3s or 3/2s, equivalent to 12edo as compton and meantone
* Various historical [[Well temperament]]s generated by tempered 4/3s or 3/2s, equivalent to 12edo as compton and meantone


{{Navbox intervals}}
{{Navbox intervals}}

Revision as of 19:29, 26 February 2025

This page is about the interval region. For the just perfect fourth, see 4/3.

A perfect fourth (P4) is an interval that spans three steps of the 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 480 to 514 ¢ (2\5 to 3\7).

In just intonation, the just perfect fourth is 4/3. Other intervals are also classified as perfect fourths, sometimes called wolf fourths or imperfect fourths, if they are reasonably mapped to 3\7 and 10\24 (precisely three steps of the diatonic scale and five steps of the chromatic scale). The use of 24edo's 10\24 as the mapping criteria here rather than 12edo's 5\12 better captures the characteristics of many intervals in the 11- and 13-limit.

As a concrete interval region, it is typically near 500 ¢ in size, distinct from the semiaugmented fourth of roughly 550 ¢. A rough tuning range for the perfect fourth is about 470 to 530 ¢ according to Margo Schulter's theory of interval regions. Another common range is the stricter range from 480 to 514 ¢, which generates a diatonic scale.

This article covers intervals from 450 to 550 ¢, in order to cover the range of intervals without needing extra articles for subfourths or superfourths.

In just intonation

By prime limit

The only "perfect" fourth in JI is the Pythagorean perfect fourth of 4/3, about 498 ¢ in size, which corresponds to the MOS-based interval category of the diatonic perfect fourth and is the octave complement of the perfect fifth of 3/2. However, various "out of tune" fourths exist, such as the Pythagorean wolf fourth 177147/131072, which is sharp of 4/3 by one Pythagorean comma, and is about 522 ¢ in size.

Other "out of tune" fourths in higher limits include:

  • The 5-limit acute fourth is a ratio of 27/20, and is about 520 ¢
  • The 7-limit subfourth is a ratio of 21/16, and is about 471 ¢.
  • The 11-limit augmented fourth is a ratio of 15/11, and is about 537 ¢.
    • There is also an 11-limit grave fourth, which is a ratio of 33/25, and is about 480 ¢.
  • The 13-limit infrafourth is a ratio of 13/10, and is about 454 ¢, but it might be better analyzed as an ultramajor third. Despite that, it is also here for completeness.

By delta

Delta 1 Cents Delta 3 Cents Delta 4 Cents Delta 5 Cents Delta 6 Cents
4/3 498 ¢ 13/10 454 ¢ 15/11 537 ¢ 19/14 529 ¢ 23/17 523 ¢
17/13 464 ¢ 21/16 471 ¢ 25/19 475 ¢

In edos

The following table lists the best tuning of 4/3, as well as other fourths if present, in various significant edos.

Edo 4/3 Other fourths
5 480 ¢
7 514 ¢
12 500 ¢
15 480 ¢
16 525 ¢ 450 ¢ ≈ 13/10
17 494 ¢
19 506 ¢
22 491 ¢ 545 ¢ ≈ 15/11
24 500 ¢ 450 ¢ ≈ 13/10, 550 ¢ ≈ 15/11
25 480 ¢ 528 ¢ ≈ 27/20
26 508 ¢ 462 ¢ ≈ 21/16, 13/10
27 489 ¢ 533 ¢ ≈ 15/11
29 496 ¢ 455 ¢ ≈ 13/10, 537 ¢ ≈ 15/11
31 503 ¢ 464 ¢ ≈ 21/16, 541 ¢ ≈ 15/11
34 494 ¢ 458 ¢ ≈ 13/10, 529 ¢ ≈ 27/20, 15/11
41 498 ¢ 468 ¢ ≈ 21/16, 526 ¢ ≈ 27/20
53 498 ¢ 452 ¢ ≈ 13/10, 476 ¢ ≈ 21/16, 521 ¢ ≈ 27/20, 543 ¢ ≈ 15/11

In temperaments

The simplest perfect 4th ratio is 4/3. The following notable temperaments are generated by it:

Temperaments that use 4/3 as a generator

  • Compton, the temperament of the Pythagorean comma, equivalent to 12edo
  • Archy, the temperament flattening 4/3 such that three 4/3s stack to 7/6
  • Meantone, the temperament sharpening 4/3 such that three 4/3s stack to 6/5
  • Mavila, the temperament sharpening 4/3 such that three 4/3s stack to 5/4
  • Various historical Well temperaments generated by tempered 4/3s or 3/2s, equivalent to 12edo as compton and meantone


View • Talk • EditInterval classification
Seconds and thirds Unison • Comma and diesis • Semitone • Neutral second • Major second • (Interseptimal second-third) • Minor third • Neutral third • Major third
Fourths and fifths (Interseptimal third-fourth) • Perfect fourth • Superfourth • Tritone • Subfifth • Perfect fifth • (Interseptimal fifth-sixth)
Sixths and sevenths Minor sixth • Neutral sixth • Major sixth • (Interseptimal sixth-seventh) • Minor seventh • Neutral seventh • Major seventh • Octave
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
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