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| en =  
| en =  
| es =  
| es =  
| ja =  
| ja =3/2
| ko =  
| ko =  
| ro = 3/2 (ro)
| ro = 3/2 (ro)
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{{Wikipedia|Perfect fifth}}
{{Wikipedia|Perfect fifth}}


'''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. Only the [[2/1|octave]] and the [[3/1|tritave]] have smaller numbers.
'''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 [[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.


== Properties ==
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.  
For harmonic [[timbre|timbres]], 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's easy to hear if it's mistuned.  


== Usage ==
== 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.
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.


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 [[Wikipedia:Diatonic scale #Iteration of the fifth|Pythagorean diatonic scale]]. This scale is also the 7 natural notes of all "pyth-spine" notations, in which all uninflected notes are pythogorean, such as [[HEJI]], [[Sagittal notation|Sagittal]], [[Ups and downs notation|ups and downs]], [[FJS]] and [[color notation]].
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]].


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]].
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 ===
=== In regular temperament theory ===
Because 3/2 has very low [[harmonic entropy]], 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:
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 such that the major third generated by stacking four fifths is closer to (or even identical to) 5/4. The minor 3rd generated by stacking three fourths is closer to 6/5.
[[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 7th 16/9 approximates 7/4 instead of 9/5.
[[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.  


[[Schismatic]] temperament flattens the fifth very slightly such that the ''diminished'' fourth generated by stacking eight fourths approximates 5/4. 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}}}}).
[[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 ==
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.
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.
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"
Line 44: Line 45:
! [[Edo]]
! [[Edo]]
! class="unsortable" | Deg\edo
! class="unsortable" | Deg\edo
! Absolute <br>error ([[Cent|¢]])
! Absolute <br>error ([[cent|¢]])
! Relative <br>error (%)
! Relative <br>error (%)
! &#x2195;
! &#x2195;
! class="unsortable" | Equally accurate <br>multiples
! class="unsortable" | Equally accurate <br>multiples
|-
|-
|  [[12edo|12]]  ||  7\12  || 1.955 || 1.955 || &darr; || [[24edo|14\24]], [[36edo|21\36]]
|  [[12edo|12]]  ||  7\12  || 1.955 || 1.955 || ↓ || [[24edo|14\24]], [[36edo|21\36]]
|-
|-
|  [[17edo|17]]  ||  10\17  || 3.927 || 5.564 || &uarr; ||  
|  [[17edo|17]]  ||  10\17  || 3.927 || 5.564 || ↑ ||  
|-
|-
|  [[29edo|29]]  ||  17\29  || 1.493 || 3.609 || &uarr; ||  
|  [[29edo|29]]  ||  17\29  || 1.493 || 3.609 || ↑ ||  
|-
|-
|  [[41edo|41]]  ||  24\41  || 0.484 || 1.654 || &uarr; || [[82edo|48\82]], [[123edo|72\123]], [[164edo|96\164]]
|  [[41edo|41]]  ||  24\41  || 0.484 || 1.654 || ↑ || [[82edo|48\82]], [[123edo|72\123]], [[164edo|96\164]]
|-
|-
|  [[53edo|53]]  ||  31\53  || 0.068 || 0.301 || &darr; || [[106edo|62\106]], [[159edo|93\159]]
|  [[53edo|53]]  ||  31\53  || 0.068 || 0.301 || ↓ || [[106edo|62\106]], [[159edo|93\159]]
|-
|-
|  [[65edo|65]]  ||  38\65  || 0.416 || 2.256 || &darr; || [[130edo|76\130]], [[195edo|114\195]]
|  [[65edo|65]]  ||  38\65  || 0.416 || 2.256 || ↓ || [[130edo|76\130]], [[195edo|114\195]]
|-
|-
|  [[70edo|70]]  ||  41\70  || 0.902 || 5.262 || &uarr; ||  
|  [[70edo|70]]  ||  41\70  || 0.902 || 5.262 || ↑ ||  
|-
|-
|  [[77edo|77]]  ||  45\77  || 0.656 || 4.211 || &darr; ||  
|  [[77edo|77]]  ||  45\77  || 0.656 || 4.211 || ↓ ||  
|-
|-
|  [[89edo|89]]  ||  52\89  || 0.831 || 6.166 || &darr; ||  
|  [[89edo|89]]  ||  52\89  || 0.831 || 6.166 || ↓ ||  
|-
|-
|  [[94edo|94]]  ||  55\94  || 0.173 || 1.352 || &uarr; || [[188edo|110\188]]
|  [[94edo|94]]  ||  55\94  || 0.173 || 1.352 || ↑ || [[188edo|110\188]]
|-
|-
| [[111edo|111]] ||  65\111 || 0.748 || 6.916 || &uarr; ||  
| [[111edo|111]] ||  65\111 || 0.748 || 6.916 || ↑ ||  
|-
|-
| [[118edo|118]] ||  69\118 || 0.260 || 2.557 || &darr; ||  
| [[118edo|118]] ||  69\118 || 0.260 || 2.557 || ↓ ||  
|-
|-
| [[135edo|135]] ||  79\135 || 0.267 || 3.006 ||&uarr; ||  
| [[135edo|135]] ||  79\135 || 0.267 || 3.006 || ↑ ||  
|-
|-
| [[142edo|142]] ||  83\142 || 0.547 || 6.467 || &darr; ||  
| [[142edo|142]] ||  83\142 || 0.547 || 6.467 || ↓ ||  
|-
|-
| [[147edo|147]] ||  86\147 || 0.086 || 1.051 || &uarr; ||  
| [[147edo|147]] ||  86\147 || 0.086 || 1.051 || ↑ ||  
|-
|-
| [[171edo|171]] || 100\171 || 0.200 || 2.859 || &darr; ||  
| [[171edo|171]] || 100\171 || 0.200 || 2.859 || ↓ ||  
|-
|-
| [[176edo|176]] || 103\176 || 0.318 || 4.660 || &uarr; ||  
| [[176edo|176]] || 103\176 || 0.318 || 4.660 || ↑ ||  
|-
|-
| [[183edo|183]] || 107\183 || 0.316 || 4.814 || &darr; ||  
| [[183edo|183]] || 107\183 || 0.316 || 4.814 || ↓ ||  
|-
|-
| [[200edo|200]] || 117\200 || 0.045 || 0.750 || &uarr; ||  
| [[200edo|200]] || 117\200 || 0.045 || 0.750 || ↑ ||  
|}
|}


Edos can be classified by their approximation of 3/2 as:
Edos can be classified by their approximation of 3/2 as:
* '''Superflat''' edos have fifths narrower than {{nowrap|4\7 {{=}} ~686{{c}}}}
* '''Superflat''' edos have fifths narrower than {{nowrap| 4\7 {{=}} ~686{{c}} }}
* '''Perfect''' edos have fifths of exactly 4\7
* '''Perfect''' edos have fifths of exactly 4\7
* '''Diatonic''' edos have fifths between 4\7 and {{nowrap|3\5 {{=}} 720{{c}}}}
* '''Diatonic''' edos have fifths between 4\7 and {{nowrap| 3\5 {{=}} 720{{c}} }}
* '''Pentatonic''' have fifths of exactly 3\5
* '''Pentatonic''' have fifths of exactly 3\5
* '''Supersharp''' edos have fifths wider than 3\5
* '''Supersharp''' edos have fifths wider than 3\5
Line 101: Line 102:
! Degree
! Degree
! Cents
! Cents
! Edo Category
! Edo category
! Error (¢)
! Error (¢)
|-
|-
Line 113: Line 114:
| 4\7
| 4\7
| 685.714
| 685.714
| perfect edo
| Perfect edo
| −16.241
| −16.241
|-
|-
Line 119: Line 120:
| 5\8
| 5\8
| 750.000
| 750.000
| supersharp edo
| Supersharp edo
|  +48.045
|  +48.045
|-
|-
Line 125: Line 126:
| 5\9
| 5\9
| 666.667
| 666.667
| superflat edo
| Superflat edo
| −35.288
| −35.288
|-
|-
Line 131: Line 132:
| 6\10
| 6\10
| 720.000
| 720.000
| pentatonic edo
| Pentatonic edo
|  +18.045
|  +18.045
|-
|-
Line 137: Line 138:
| 6\11
| 6\11
| 654.545
| 654.545
| superflat edo
| Superflat edo
| −47.41
| −47.41
|-
|-
Line 143: Line 144:
| 7\12
| 7\12
| 700.000
| 700.000
| diatonic edo
| Diatonic edo
| −1.955
| −1.955
|-
|-
Line 149: Line 150:
| 8\13
| 8\13
| 738.462
| 738.462
| supersharp edo
| Supersharp edo
|  +36.507
|  +36.507
|-
|-
Line 155: Line 156:
| 8\14
| 8\14
| 685.714
| 685.714
| perfect edo
| Perfect edo
| −16.241
| −16.241
|-
|-
Line 161: Line 162:
| 9\15
| 9\15
| 720.000
| 720.000
| pentatonic edo
| Pentatonic edo
|  +18.045
|  +18.045
|-
|-
Line 167: Line 168:
| 9\16
| 9\16
| 675.000
| 675.000
| superflat edo
| Superflat edo
| −26.955
| −26.955
|-
|-
Line 173: Line 174:
| 10\17
| 10\17
| 705.882
| 705.882
| diatonic edo
| Diatonic edo
|  +3.927
|  +3.927
|-
|-
Line 179: Line 180:
| 11\18
| 11\18
| 733.333
| 733.333
| supersharp edo
| Supersharp edo
|  +31.378
|  +31.378
|-
|-
Line 185: Line 186:
| 11\19
| 11\19
| 694.737
| 694.737
| diatonic edo
| Diatonic edo
| −7.218
| −7.218
|-
|-
Line 191: Line 192:
| 12\20
| 12\20
| 720.000
| 720.000
| pentatonic edo
| Pentatonic edo
|  +18.045
|  +18.045
|-
|-
Line 197: Line 198:
| 12\21
| 12\21
| 685.714
| 685.714
| perfect edo
| Perfect edo
| −16.241
| −16.241
|-
|-
Line 203: Line 204:
| 13\22
| 13\22
| 709.091
| 709.091
| diatonic edo
| Diatonic edo
|  +7.136
|  +7.136
|-
|-
Line 209: Line 210:
| 13\23
| 13\23
| 678.261
| 678.261
| superflat edo
| Superflat edo
| −23.694
| −23.694
|-
|-
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| 14\24
| 14\24
| 700.000
| 700.000
| diatonic edo
| Diatonic edo
| −1.955
| −1.955
|-
|-
Line 221: Line 222:
| 15\25
| 15\25
| 720.000
| 720.000
| pentatonic edo
| Pentatonic edo
|  +18.045
|  +18.045
|-
|-
Line 227: Line 228:
| 15\26
| 15\26
| 692.308
| 692.308
| diatonic edo
| Diatonic edo
| −9.647
| −9.647
|-
|-
Line 233: Line 234:
| 16\27
| 16\27
| 711.111
| 711.111
| diatonic edo
| Diatonic edo
|  +9.156
|  +9.156
|-
|-
Line 239: Line 240:
| 16\28
| 16\28
| 685.714
| 685.714
| perfect edo
| Perfect edo
| −16.241
| −16.241
|-
|-
Line 245: Line 246:
| 17\29
| 17\29
| 703.448
| 703.448
| diatonic edo
| Diatonic edo
|  +1.493
|  +1.493
|-
|-
Line 251: Line 252:
| 18\30
| 18\30
| 720.000
| 720.000
| pentatonic edo
| Pentatonic edo
|  +18.045
|  +18.045
|-
|-
Line 257: Line 258:
| 18\31
| 18\31
| 696.774
| 696.774
| diatonic edo
| Diatonic edo
| −5.181
| −5.181
|}
|}
== As a dyad ==
{{Infobox Chord
| 2:3
| ColorName=5
| debug=1
}}
'''2:3''' is a 3-limit [[dyad]], known as the ''five chord'' (as in C5 not V) or ''power chord''. It is used to create an open, stable, and powerful sound in contemporary western music. It is also indispensable in certain musical genres such as [[African music #Equiheptatonic tunings|mbira music]] and late medieval music. In the latter, when voiced as '''1:2:3''', it is known as the ''trine'', a very common closing chord.
=== Notable voicings ===
{| 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 ==
Line 266: Line 311:
* [[Edf]] – tunings which equally divide 3/2
* [[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]]
[[Category:Taxicab-2 intervals]]
Retrieved from "https://en.xen.wiki/w/3/2"