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{{Infobox Interval
{{interwiki
| de =
| en =
| es =
| ja =3/2
| ko =
| ro = 3/2 (ro)
}}
{{Infobox interval
| Name = just perfect fifth
| Name = just perfect fifth
| Color name = w5, wa 5th
| Color name = w5, wa 5th
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{{Wikipedia|Perfect fifth}}
{{Wikipedia|Perfect fifth}}


'''3/2''', the '''just perfect fifth''', is the second largest [[superparticular]] [[interval]], spanning the distance between the 2nd and 3rd harmonics. It is an interval with low [[harmonic entropy]], and therefore high consonance. In composition, the presence of perfect fifths can provide a "ground" upon which unusual intervals may be placed while still sounding structurally coherent. Not only that, but there are many other uses of 3/2, and thus, systems excluding perfect fifths can thus sound more "xenharmonic". On a harmonic instrument, the third harmonic is usually the loudest one that is not an octave double of the fundamental, with 3/2 itself being the [[octave reduced]] form of this interval. Variations of the perfect fifth (whether just or not) appear in most music of the world. Treatment of the perfect fifth as consonant historically precedes treatment of the major third – specifically [[5/4]] – as consonant. 3/2 is the simplest [[just intonation]] interval to be very well approximated by [[12edo]], after the [[octave]].
'''3/2''', the '''just perfect fifth''', is a very [[consonance|consonant]] interval, due to the numerator and denominator of its ratio being very small numbers, with only the [[2/1|octave]] and the [[3/1|tritave]] having smaller numbers. As such, it is very important in western music and many musical traditions, and approximating it is key in systems like [[12edo]] and other [[edo]]s.


Producing a chain of just perfect fifths yields [[Pythagorean tuning]]. Such a chain does not close at a circle, but continues infinitely. Nevertheless, even in Xenharmonic circles, the common label "perfect" for this interval retains value in at least some of the [[moment of symmetry]] scales created by this tuning – specifically in the [[TAMNAMS]] system – due to it being an interval that can be thought of as a multiple of the period plus or minus 0 or 1 generators.  An example of such a scale is the familiar [[Wikipedia: Diatonic scale #Iteration of the fifth|Pythagorean diatonic scale]].
For harmonic [[timbre]]s, the loudest harmonics are usually the second and third ones (2/1 and 3/1). 3/2 is the interval between these two harmonics (which incidentally makes 3/2 [[superparticular]]). Thus 3/2 is easy to tune by ear, and it is easy to hear if it is mistuned.  


Meanwhile, [[meantone]] temperaments are systems which flatten the perfect fifth such that the major third generated by four fifths upward be closer to 5/4 – or, in the case of [[quarter-comma meantone]] (see also [[31edo]]), identical. In such systems, and in common practice theory, the perfect fifth consists of four diatonic semitones and three chromatic semitones. In [[12edo]], and hence in most discussions these days, this is simplified to seven semitones, which is fitting seeing as 12edo is a system which flattens the perfect fifth by about 2 cents so that the circle close at 12 tones. On the other hand, in just intonation, the perfect fifth consists of four just diatonic semitones of [[16/15]], three just chromatic semitones of [[25/24]], and two syntonic commas of [[81/80]], and is the just perfect fifth of 3/2.
== Usage ==
Variations of the perfect fifth (whether [[just]] or tempered) appear in most [[approaches to musical tuning|music of the world]]. [[Historical temperaments|Historically]], European music treated the perfect fifth as consonant long before it treated the major third—specifically [[5/4]]—as consonant. In the present day, the dominant tuning [[12edo]] approximates 3/2 very accurately.


Then there is the possibility of [[schismatic]] temperaments, which flatten the perfect fifth such that an approximated 5/4 is generated by stacking eight fifths downwards; however, without a notation system that properly accounts for the syntonic comma (such as [[Syntonic-Rastmic Subchroma notation]]), the 5/4 will be invariably classified as a diminished fourth due to being enharmonic with [[8192/6561]], and this in turn results in common chords such as conventional [[Wikipedia: Major chord|Major]] and [[Wikipedia: Minor chord|Minor]] triads being awkward to notate.
A [[chain of fifths|chain of just perfect fifths]] generates [[Pythagorean tuning]]. The chain continues indefinitely and theoretically never returns to the starting note. A chain that ends at seven notes generates the historically important Pythagorean [[5L 2s|diatonic]] scale. This scale is also the 7 natural notes of all "pyth-spine" notations, in which all uninflected notes are Pythagorean, such as [[HEJI]], [[Sagittal notation|Sagittal]], [[ups and downs notation|ups and downs]], [[FJS]] and [[color notation]].


Some better (compared to 12edo) approximations of the perfect fifth are [[29edo]], [[41edo]], and [[53edo]].  Of the aforementioned systems, the latter is particularly noteworthy in regards to [[telicity]] as while 12edo is a 2-strong 3-2 telic system, 53edo is a 3-strong 3-2 telic system.
Music using unusual intervals can be very disorienting. The presence of perfect fifths can provide a "ground" that make it less so. Some composers deliberately use tunings that lack fifths, to make their music sound more [[xenharmonic]].
 
=== In regular temperament theory ===
Because 3/2 is a very simple and concordant interval, it is still recognizable even when heavily tempered. Often it is tempered so that an octave-reduced stack of fourths or fifths approximates some other interval. Some examples:
 
[[Meantone]] temperament flattens the fifth from just (to around 695–700 cents) such that the major third generated by stacking four fifths is closer to (or even identical to) 5/4. The minor third generated by stacking three fourths is closer to 6/5.
 
[[Superpyth]] temperaments ''sharpen'' the fifth from just so that the major third is closer to 9/7 and the minor third is closer to 7/6. Thus the minor seventh 16/9 approximates 7/4 instead of 9/5.
 
[[Schismic]] temperament adjusts the fifth such that the ''diminished fourth'' generated by stacking eight fourths approximates 5/4. As this is already a close approximation, the tuning of the fifth can be varied around its just tuning, but is most accurately flattened by a tiny amount. Thus a triad with 5/4 is written as {{nowrap|{{dash|C, F♭, G}}}} (unless the notation has accidentals for [[81/80]], e.g. {{nowrap|{{dash|C, vE, G}}}}).
 
* Garibaldi temperament is an extension of schismic that sharpens the fifth so that the small interval between the major third and diminished fourth can also be used to create simple 7-limit intervals.


== Approximations by edos ==
== Approximations by edos ==
The following [[edo]]s (up to 200) contain good approximations<ref>error magnitude below 7, both, absolute (in ¢) and relative (in r¢)</ref> of the interval 3/2. Errors are given by magnitude, the arrows in the table show if the edo representation is sharp (&uarr;) or flat (&darr;).
12edo approximates 3/2 to within only 2{{c}}. [[29edo]], [[41edo]], and [[53edo]] are even more accurate. In regards to [[telicity]], while 12edo is a 2-strong 3-2 telic system, 53edo is notably a 3-strong 3-2 telic system.
 
The following edos (up to 200) approximate 3/2 to within both 7{{c}} and 7%. Errors are unsigned so that the table can be sorted by them. The arrow column indicates a sharp (↑) or flat (↓) fifth.


{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
|-
! [[Edo]]
! [[Edo]]
! class="unsortable" | deg\edo
! class="unsortable" | Deg\edo
! Absolute<br>Error ([[Cent|¢]])
! Absolute <br>error ([[cent|¢]])
! Relative<br>Error ([[Relative cent|r¢]])
! Relative <br>error (%)
! &#8597;
! &#x2195;
! class="unsortable" | Equally acceptable multiples <ref>Super-edos up to 200 within the same error tolerance</ref>
! class="unsortable" | Equally accurate <br>multiples
|-
|  [[12edo|12]]  ||  7\12  || 1.9550 || 1.9550 || &darr; || [[24edo|14\24]], [[36edo|21\36]]
|-
|-
|  [[17edo|17]]  || 10\17 || 3.9274 || 5.5637 || &uarr; ||  
|  [[12edo|12]]  ||   7\12 || 1.955 || 1.955 || ↓ || [[24edo|14\24]], [[36edo|21\36]]
|-
|-
|  [[29edo|29]]  ||  17\29 || 1.4933 || 3.6087 || &uarr; ||  
|  [[17edo|17]]  ||  10\17  || 3.927 || 5.564 || ↑ ||  
|-
|-
|  [[41edo|41]]  ||  24\41 || 0.4840 || 1.6537 || &uarr; || [[82edo|48\82]], [[123edo|72\123]], [[164edo|96\164]]
|  [[29edo|29]]  ||  17\29 || 1.493 || 3.609 || ↑ ||  
|-
|-
|  [[53edo|53]]  ||  31\53 || 0.0682 || 0.3013 || &darr; || [[106edo|62\106]], [[159edo|93\159]]
|  [[41edo|41]]  ||  24\41 || 0.484 || 1.654 || ↑ || [[82edo|48\82]], [[123edo|72\123]], [[164edo|96\164]]
|-
|-
|  [[65edo|65]]  ||  38\65 || 0.4165 || 2.2563 || &darr; || [[130edo|76\130]], [[195edo|114\195]]
|  [[53edo|53]]  ||  31\53 || 0.068 || 0.301 || ↓ || [[106edo|62\106]], [[159edo|93\159]]
|-
|-
|  [[70edo|70]]  ||  41\70 || 0.9021 || 5.2625 || &uarr; ||  
|  [[65edo|65]]  ||  38\65 || 0.416 || 2.256 || ↓ || [[130edo|76\130]], [[195edo|114\195]]
|-
|-
|  [[77edo|77]]  ||  45\77 || 0.6563 || 4.2113 || &darr; ||  
|  [[70edo|70]]  ||  41\70 || 0.902 || 5.262 || ↑ ||  
|-
|-
|  [[89edo|89]]  ||  52\89 || 0.8314 || 6.1663 || &darr; ||  
|  [[77edo|77]]  ||  45\77 || 0.656 || 4.211 || ↓ ||  
|-
|-
|  [[94edo|94]]  ||  55\94 || 0.1727 || 1.3525 || &uarr; || [[188edo|110\188]]
|  [[89edo|89]]  ||  52\89 || 0.831 || 6.166 || ↓ ||  
|-
|-
| [[111edo|111]] ||  65\111 || 0.7477 || 6.9162 || &uarr; ||  
| [[94edo|94]] ||  55\94  || 0.173 || 1.352 || ↑ || [[188edo|110\188]]
|-
|-
| [[118edo|118]] ||  69\118 || 0.2601 || 2.5575 || &darr; ||  
| [[111edo|111]] ||  65\111 || 0.748 || 6.916 || ↑ ||  
|-
|-
| [[135edo|135]] ||  79\135 || 0.2672 || 3.0062 || &uarr; ||  
| [[118edo|118]] ||  69\118 || 0.260 || 2.557 || ↓ ||  
|-
|-
| [[142edo|142]] ||  83\142 || 0.5466 || 6.4675 || &darr; ||  
| [[135edo|135]] ||  79\135 || 0.267 || 3.006 || ↑ ||  
|-
|-
| [[147edo|147]] ||  86\147 || 0.0858 || 1.0512 || &uarr; ||  
| [[142edo|142]] ||  83\142 || 0.547 || 6.467 || ↓ ||  
|-
|-
| [[171edo|171]] || 100\171 || 0.2006 || 2.8588 || &darr; ||  
| [[147edo|147]] || 86\147 || 0.086 || 1.051 || ↑ ||  
|-
|-
| [[176edo|176]] || 103\176 || 0.3177 || 4.6600 || &uarr; ||  
| [[171edo|171]] || 100\171 || 0.200 || 2.859 || ↓ ||  
|-
|-
| [[183edo|183]] || 107\183 || 0.3157 || 4.8138 || &darr; ||  
| [[176edo|176]] || 103\176 || 0.318 || 4.660 || ↑ ||  
|-
|-
| [[200edo|200]] || 117\200 || 0.0450 || 0.7500 || &uarr; ||  
| [[183edo|183]] || 107\183 || 0.316 || 4.814 || ↓ ||  
|-
|-
| [[200edo|200]] || 117\200 || 0.045 || 0.750 || ↑ ||
|}
|}


<references/>
Edos can be classified by their approximation of 3/2 as:
* '''Superflat''' edos have fifths narrower than {{nowrap| 4\7 {{=}} ~686{{c}} }}
* '''Perfect''' edos have fifths of exactly 4\7
* '''Diatonic''' edos have fifths between 4\7 and {{nowrap| 3\5 {{=}} 720{{c}} }}
* '''Pentatonic''' have fifths of exactly 3\5
* '''Supersharp''' edos have fifths wider than 3\5
 
{| class="wikitable sortable"
{| class="wikitable sortable"
|+Comparison of edo approximations of 3/2 and "fifth classes" (from 5edo to 31edo)
|+ style="font-size: 105%;" | Comparison of the fifths of edos 5 to 31
|-
! Edo
! Edo
! Degree
! Degree
! Cents
! Cents
! Fifth Category
! Edo category
! Error (¢)
! Error (¢)
|-
|-
Line 80: Line 108:
| 3\5
| 3\5
| 720.000
| 720.000
| pentatonic edo
| Pentatonic edo
| +18.045
| +18.045
|-
|-
| [[7edo]]
| [[7edo]]
| 4\7
| 4\7
| 685.714
| 685.714
| perfect edo
| Perfect edo
| -16.241
| −16.241
|-
|-
| [[8edo]]
| [[8edo]]
| 5\8
| 5\8
| 750.000
| 750.000
| supersharp edo
| Supersharp edo
| +48.045
| +48.045
|-
|-
| [[9edo]]
| [[9edo]]
| 5\9
| 5\9
| 666.667
| 666.667
| superflat edo
| Superflat edo
| -35.288
| −35.288
|-
|-
| [[10edo]]
| [[10edo]]
| 6\10
| 6\10
| 720.000
| 720.000
| pentatonic edo
| Pentatonic edo
| +18.045
| +18.045
|-
|-
| [[11edo]]
| [[11edo]]
| 6\11
| 6\11
| 654.545
| 654.545
| superflat edo
| Superflat edo
| -47.41
| −47.41
|-
|-
| [[12edo]]
| [[12edo]]
| 7\12
| 7\12
| 700.000
| 700.000
| diatonic edo
| Diatonic edo
| -1.955
| −1.955
|-
|-
| [[13edo]]
| [[13edo]]
| 8\13
| 8\13
| 738.462
| 738.462
| supersharp edo
| Supersharp edo
| +36.507
| +36.507
|-
|-
| [[14edo]]
| [[14edo]]
| 8\14
| 8\14
| 685.714
| 685.714
| perfect edo
| Perfect edo
| -16.241
| −16.241
|-
|-
| [[15edo]]
| [[15edo]]
| 9\15
| 9\15
| 720.000
| 720.000
| pentatonic edo
| Pentatonic edo
| +18.045
| +18.045
|-
|-
| [[16edo]]
| [[16edo]]
| 9\16
| 9\16
| 675.000
| 675.000
| superflat edo
| Superflat edo
| -26.955
| −26.955
|-
|-
| [[17edo]]
| [[17edo]]
| 10\17
| 10\17
| 705.882
| 705.882
| diatonic edo
| Diatonic edo
| +3.927
| +3.927
|-
|-
| [[18edo]]
| [[18edo]]
| 11\18
| 11\18
| 733.333
| 733.333
| supersharp edo
| Supersharp edo
| +31.378
| +31.378
|-
|-
| [[19edo]]
| [[19edo]]
| 11\19
| 11\19
| 694.737
| 694.737
| diatonic edo
| Diatonic edo
| -7.218
| −7.218
|-
|-
| [[20edo]]
| [[20edo]]
| 12\20
| 12\20
| 720.000
| 720.000
| pentatonic edo
| Pentatonic edo
| +18.045
| +18.045
|-
|-
| [[21edo]]
| [[21edo]]
| 12\21
| 12\21
| 685.714
| 685.714
| perfect edo
| Perfect edo
| -16.241
| −16.241
|-
|-
| [[22edo]]
| [[22edo]]
| 13\22
| 13\22
| 709.091
| 709.091
| diatonic edo
| Diatonic edo
| +7.136
| +7.136
|-
|-
| [[23edo]]
| [[23edo]]
| 13\23
| 13\23
| 678.261
| 678.261
| superflat edo
| Superflat edo
| -23.694
| −23.694
|-
|-
| [[24edo]]
| [[24edo]]
| 14\24
| 14\24
| 700.000
| 700.000
| diatonic edo
| Diatonic edo
| -1.955
| −1.955
|-
|-
| [[25edo]]
| [[25edo]]
| 15\25
| 15\25
| 720.000
| 720.000
| pentatonic edo
| Pentatonic edo
| +18.045
| +18.045
|-
|-
| [[26edo]]
| [[26edo]]
| 15\26
| 15\26
| 692.308
| 692.308
| diatonic edo
| Diatonic edo
| -9.647
| −9.647
|-
|-
| [[27edo]]
| [[27edo]]
| 16\27
| 16\27
| 711.111
| 711.111
| diatonic edo
| Diatonic edo
| +9.156
| +9.156
|-
|-
| [[28edo]]
| [[28edo]]
| 16\28
| 16\28
| 685.714
| 685.714
| perfect edo
| Perfect edo
| -16.241
| −16.241
|-
|-
| [[29edo]]
| [[29edo]]
| 17\29
| 17\29
| 703.448
| 703.448
| diatonic edo
| Diatonic edo
| +1.493
| +1.493
|-
|-
| [[30edo]]
| [[30edo]]
| 17\30
| 18\30
| 720.000
| 720.000
| pentatonic edo
| Pentatonic edo
| +18.045
| +18.045
|-
|-
| [[31edo]]
| [[31edo]]
| 18\31
| 18\31
| 696.774
| 696.774
| diatonic edo
| Diatonic edo
| -5.181
| −5.181
|}
|}


* The many and various 3/2 approximations in different edos can be classified as (after [[Kite Giedraitis]]):
== As a dyad ==
** '''superflat''' edo – fifth is narrower than 686 cents.
{{Infobox Chord|2:3|ColorName=5|debug=1}}
** '''perfect''' edo – fifth is 686 cents wide (and 4/7 steps).
'''2:3''' is a 3-limit [[dyad]], known as the '''five chord''' (as in C5 not V), or as the '''power chord'''. This dyad is indispensable in certain musical genres such as [[African music #Equiheptatonic tunings|mbira music]] and late medieval music. In the latter, when voiced as hi5add8, it's known as the '''trine''', a very common closing chord.
** '''diatonic''' edo – fifth is between 686.1 - 719.9 cents wide.
 
** '''pentatonic''' edo – fifth is exactly 720 cents wide.
=== Notable voicings ===
** '''supersharp''' edo – fifth is wider than 720 cents.
{| class="wikitable"
|+
! Voices
! [[EFR]]
! [[Kite's thoughts on hi-lo notation|Hi-lo name]]
! Special properties
|-
| rowspan="3" | 2 voices
| 1:3
| hi5
| AOV ([[Odd limit #Proposed extensions|all-odd voicing]])
|-
| 2:3
| basic
| CAOV (condensed AOV)
|-
| 3:4
| lo5
| 1st inversion
|-
| rowspan="3" |3 voices
| 1:2:3
| hi5add8
| The trine
|-
| 2:3:4
| add8
|
|-
| 3:4:6
| addlo5
| 2:3:4 melodically inverted
|}
{{Clear}}


== See also ==
== See also ==
* [[4/3]] – its [[octave complement]]
* [[4/3]] – its [[octave complement]]
* [[Fifth complement]]
* [[Fifth complement]]
* [[Edf]] – tunings which equally divide 3/2
* [[Gallery of just intervals]]
* [[Gallery of just intervals]]
* {{OEIS| A060528 }} – sequence of edos with increasingly better approximations of 3/2 (and by extension 4/3)
* {{OEIS|A060528}} – sequence of edos with increasingly better approximations of 3/2 (and by extension 4/3)
* {{OEIS| A005664 }} – denominators of the convergents to log<sub>2</sub>(3)
* {{OEIS|A005664}} – denominators of the convergents to log<sub>2</sub>(3)
* {{OEIS| A206788 }} – denominators of the semiconvergents to log<sub>2</sub>(3)
* {{OEIS|A206788}} – denominators of the semiconvergents to log<sub>2</sub>(3)


[[Category:Fifth]]
[[Category:Fifth]]
[[Category:Taxicab-2 intervals]]
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