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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|compound fifth]] 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 [[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.


== 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¢. [[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¢ 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"
|-
|-
! [[Edo]]
! [[Edo]]
! class="unsortable" | deg\edo
! class="unsortable" | Deg\edo
! Absolute<br>Error ([[Cent|¢]])
! Absolute <br>error ([[cent|¢]])
! Relative
! Relative <br>error (%)
Error (%)
! &#x2195;
! &#x2195;
! class="unsortable" | Equally accurate
! class="unsortable" | Equally accurate <br>multiples
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 103: Line 102:
! Degree
! Degree
! Cents
! Cents
! Edo Category
! Edo category
! Error (¢)
! Error (¢)
|-
|-
Line 115: Line 114:
| 4\7
| 4\7
| 685.714
| 685.714
| perfect edo
| Perfect edo
| −16.241
| −16.241
|-
|-
Line 121: Line 120:
| 5\8
| 5\8
| 750.000
| 750.000
| supersharp edo
| Supersharp edo
|  +48.045
|  +48.045
|-
|-
Line 127: Line 126:
| 5\9
| 5\9
| 666.667
| 666.667
| superflat edo
| Superflat edo
| −35.288
| −35.288
|-
|-
Line 133: Line 132:
| 6\10
| 6\10
| 720.000
| 720.000
| pentatonic edo
| Pentatonic edo
|  +18.045
|  +18.045
|-
|-
Line 139: Line 138:
| 6\11
| 6\11
| 654.545
| 654.545
| superflat edo
| Superflat edo
| −47.41
| −47.41
|-
|-
Line 145: Line 144:
| 7\12
| 7\12
| 700.000
| 700.000
| diatonic edo
| Diatonic edo
| −1.955
| −1.955
|-
|-
Line 151: Line 150:
| 8\13
| 8\13
| 738.462
| 738.462
| supersharp edo
| Supersharp edo
|  +36.507
|  +36.507
|-
|-
Line 157: Line 156:
| 8\14
| 8\14
| 685.714
| 685.714
| perfect edo
| Perfect edo
| −16.241
| −16.241
|-
|-
Line 163: Line 162:
| 9\15
| 9\15
| 720.000
| 720.000
| pentatonic edo
| Pentatonic edo
|  +18.045
|  +18.045
|-
|-
Line 169: Line 168:
| 9\16
| 9\16
| 675.000
| 675.000
| superflat edo
| Superflat edo
| −26.955
| −26.955
|-
|-
Line 175: Line 174:
| 10\17
| 10\17
| 705.882
| 705.882
| diatonic edo
| Diatonic edo
|  +3.927
|  +3.927
|-
|-
Line 181: Line 180:
| 11\18
| 11\18
| 733.333
| 733.333
| supersharp edo
| Supersharp edo
|  +31.378
|  +31.378
|-
|-
Line 187: Line 186:
| 11\19
| 11\19
| 694.737
| 694.737
| diatonic edo
| Diatonic edo
| −7.218
| −7.218
|-
|-
Line 193: Line 192:
| 12\20
| 12\20
| 720.000
| 720.000
| pentatonic edo
| Pentatonic edo
|  +18.045
|  +18.045
|-
|-
Line 199: Line 198:
| 12\21
| 12\21
| 685.714
| 685.714
| perfect edo
| Perfect edo
| −16.241
| −16.241
|-
|-
Line 205: Line 204:
| 13\22
| 13\22
| 709.091
| 709.091
| diatonic edo
| Diatonic edo
|  +7.136
|  +7.136
|-
|-
Line 211: Line 210:
| 13\23
| 13\23
| 678.261
| 678.261
| superflat edo
| Superflat edo
| −23.694
| −23.694
|-
|-
Line 217: Line 216:
| 14\24
| 14\24
| 700.000
| 700.000
| diatonic edo
| Diatonic edo
| −1.955
| −1.955
|-
|-
Line 223: Line 222:
| 15\25
| 15\25
| 720.000
| 720.000
| pentatonic edo
| Pentatonic edo
|  +18.045
|  +18.045
|-
|-
Line 229: Line 228:
| 15\26
| 15\26
| 692.308
| 692.308
| diatonic edo
| Diatonic edo
| −9.647
| −9.647
|-
|-
Line 235: Line 234:
| 16\27
| 16\27
| 711.111
| 711.111
| diatonic edo
| Diatonic edo
|  +9.156
|  +9.156
|-
|-
Line 241: Line 240:
| 16\28
| 16\28
| 685.714
| 685.714
| perfect edo
| Perfect edo
| −16.241
| −16.241
|-
|-
Line 247: Line 246:
| 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
|-
|-
Line 259: 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 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.
=== 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 268: Line 307:
* [[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]]

Latest revision as of 00:34, 2 April 2026

Interval information
Ratio 3/2
Factorization 2-1 × 3
Monzo [-1 1⟩
Size in cents 701.955¢
Name just perfect fifth
Color name w5, wa 5th
FJS name [math]\displaystyle{ \text{P5} }[/math]
Special properties superparticular,
reduced,
reduced harmonic
Tenney norm (log2 nd) 2.58496
Weil norm (log2 max(n, d)) 3.16993
Wilson norm (sopfr(nd)) 5

[sound info]
Open this interval in xen-calc
English Wikipedia has an article on:

3/2, the just perfect fifth, is a very consonant interval, due to the numerator and denominator of its ratio being very small numbers, with only the octave and the 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 edos.

For harmonic 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 is easy to hear if it is mistuned.

Usage

Variations of the perfect fifth (whether just or tempered) appear in most music of the world. 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 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 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, 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.

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 C – F♭ – G (unless the notation has accidentals for 81/80, e.g. 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

12edo approximates 3/2 to within only 2 ¢. 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 ¢ and 7%. Errors are unsigned so that the table can be sorted by them. The arrow column indicates a sharp (↑) or flat (↓) fifth.

Edo Deg\edo Absolute
error (¢) 		Relative
error (%) 		↕ Equally accurate
multiples
12 7\12 1.955 1.955 ↓ 14\24, 21\36
17 10\17 3.927 5.564 ↑
29 17\29 1.493 3.609 ↑
41 24\41 0.484 1.654 ↑ 48\82, 72\123, 96\164
53 31\53 0.068 0.301 ↓ 62\106, 93\159
65 38\65 0.416 2.256 ↓ 76\130, 114\195
70 41\70 0.902 5.262 ↑
77 45\77 0.656 4.211 ↓
89 52\89 0.831 6.166 ↓
94 55\94 0.173 1.352 ↑ 110\188
111 65\111 0.748 6.916 ↑
118 69\118 0.260 2.557 ↓
135 79\135 0.267 3.006 ↑
142 83\142 0.547 6.467 ↓
147 86\147 0.086 1.051 ↑
171 100\171 0.200 2.859 ↓
176 103\176 0.318 4.660 ↑
183 107\183 0.316 4.814 ↓
200 117\200 0.045 0.750 ↑

Edos can be classified by their approximation of 3/2 as:

  • Superflat edos have fifths narrower than 4\7 = ~686 ¢
  • Perfect edos have fifths of exactly 4\7
  • Diatonic edos have fifths between 4\7 and 3\5 = 720 ¢
  • Pentatonic have fifths of exactly 3\5
  • Supersharp edos have fifths wider than 3\5
Comparison of the fifths of edos 5 to 31
Edo Degree Cents Edo category Error (¢)
5edo 3\5 720.000 Pentatonic edo +18.045
7edo 4\7 685.714 Perfect edo −16.241
8edo 5\8 750.000 Supersharp edo +48.045
9edo 5\9 666.667 Superflat edo −35.288
10edo 6\10 720.000 Pentatonic edo +18.045
11edo 6\11 654.545 Superflat edo −47.41
12edo 7\12 700.000 Diatonic edo −1.955
13edo 8\13 738.462 Supersharp edo +36.507
14edo 8\14 685.714 Perfect edo −16.241
15edo 9\15 720.000 Pentatonic edo +18.045
16edo 9\16 675.000 Superflat edo −26.955
17edo 10\17 705.882 Diatonic edo +3.927
18edo 11\18 733.333 Supersharp edo +31.378
19edo 11\19 694.737 Diatonic edo −7.218
20edo 12\20 720.000 Pentatonic edo +18.045
21edo 12\21 685.714 Perfect edo −16.241
22edo 13\22 709.091 Diatonic edo +7.136
23edo 13\23 678.261 Superflat edo −23.694
24edo 14\24 700.000 Diatonic edo −1.955
25edo 15\25 720.000 Pentatonic edo +18.045
26edo 15\26 692.308 Diatonic edo −9.647
27edo 16\27 711.111 Diatonic edo +9.156
28edo 16\28 685.714 Perfect edo −16.241
29edo 17\29 703.448 Diatonic edo +1.493
30edo 18\30 720.000 Pentatonic edo +18.045
31edo 18\31 696.774 Diatonic edo −5.181

As a dyad

Chord information
Harmonics 2:3
Subharmonics 1/(3:2)
Intervals from root 1/1–3/2
Cents from root 0¢–702¢
Step intervals 3/2
Step cents 702¢
Color name 5
Prime limit 3
Genus 3 (3)
Intervallic odd limit 3
Otonal odd limit 3
Utonal odd limit 3
Consistent edos (d ≥ 3) 5edo**, 7edo**, 10edo*, 12edo****, …

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 mbira music and late medieval music. In the latter, when voiced as hi5add8, it's known as the trine, a very common closing chord.

Notable voicings

Voices EFR Hi-lo name Special properties
2 voices 1:3 hi5 AOV (all-odd voicing)
2:3 basic CAOV (condensed AOV)
3:4 lo5 1st inversion
3 voices 1:2:3 hi5add8 The trine
2:3:4 add8
3:4:6 addlo5 2:3:4 melodically inverted

See also

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