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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
Line 6: Line 14:
{{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. 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.
 
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.  
 
== 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.
 
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]].


Producing a chain of just perfect fifths yields [[Pythagorean tuning]]. Since log<sub>2</sub>(3) is an irrational number, a chain of just fifths continues indefinitely and will never returns to the starting note in either direction. 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&mdash;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]].
=== 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:


Meanwhile, [[meantone]] temperaments flatten the perfect fifth such that the major third generated by four fifths upward be closer to 5/4&mdash;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 5-limit just intonation, the just 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]].
[[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.


There are also [[superpyth]] (or "superpythagorean") temperaments, which ''sharpen'' the fifth from just so that the interval generated by four fifths upwards is closer to 9/7 and the interval generated by three fifths downnward is closer to 7/6. This also means that intervals such as A&ndash;G or C&ndash;B&#x266D; approximate 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.  


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 [[ups and downs notation]] or [[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.
[[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}}}}).


Some tunings which have better (in terms of closeness to just intonation) approximations of the perfect fifth than in 12edo are [[29edo]], [[41edo]], and [[53edo]]. Of the aforementioned systems, 53edo 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.
* 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]]
|  [[12edo|12]]  ||  7\12  || 1.955 || 1.955 || ↓ || [[24edo|14\24]], [[36edo|21\36]]
|-
|-
|  [[17edo|17]]  ||  10\17  || 3.9274 || 5.5637 || &uarr; ||  
|  [[17edo|17]]  ||  10\17  || 3.927 || 5.564 || ↑ ||  
|-
|-
|  [[29edo|29]]  ||  17\29  || 1.4933 || 3.6087 || &uarr; ||  
|  [[29edo|29]]  ||  17\29  || 1.493 || 3.609 || ↑ ||  
|-
|-
|  [[41edo|41]]  ||  24\41  || 0.4840 || 1.6537 || &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  || 0.0682 || 0.3013 || &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  || 0.4165 || 2.2563 || &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  || 0.9021 || 5.2625 || &uarr; ||  
|  [[70edo|70]]  ||  41\70  || 0.902 || 5.262 || ↑ ||  
|-
|-
|  [[77edo|77]]  ||  45\77  || 0.6563 || 4.2113 || &darr; ||  
|  [[77edo|77]]  ||  45\77  || 0.656 || 4.211 || ↓ ||  
|-
|-
|  [[89edo|89]]  ||  52\89  || 0.8314 || 6.1663 || &darr; ||  
|  [[89edo|89]]  ||  52\89  || 0.831 || 6.166 || ↓ ||  
|-
|-
|  [[94edo|94]]  ||  55\94  || 0.1727 || 1.3525 || &uarr; || [[188edo|110\188]]
|  [[94edo|94]]  ||  55\94  || 0.173 || 1.352 || ↑ || [[188edo|110\188]]
|-
|-
| [[111edo|111]] ||  65\111 || 0.7477 || 6.9162 || &uarr; ||  
| [[111edo|111]] ||  65\111 || 0.748 || 6.916 || ↑ ||  
|-
|-
| [[118edo|118]] ||  69\118 || 0.2601 || 2.5575 || &darr; ||  
| [[118edo|118]] ||  69\118 || 0.260 || 2.557 || ↓ ||  
|-
|-
| [[135edo|135]] ||  79\135 || 0.2672 || 3.0062 || &uarr; ||  
| [[135edo|135]] ||  79\135 || 0.267 || 3.006 || ↑ ||  
|-
|-
| [[142edo|142]] ||  83\142 || 0.5466 || 6.4675 || &darr; ||  
| [[142edo|142]] ||  83\142 || 0.547 || 6.467 || ↓ ||  
|-
|-
| [[147edo|147]] ||  86\147 || 0.0858 || 1.0512 || &uarr; ||  
| [[147edo|147]] ||  86\147 || 0.086 || 1.051 || ↑ ||  
|-
|-
| [[171edo|171]] || 100\171 || 0.2006 || 2.8588 || &darr; ||  
| [[171edo|171]] || 100\171 || 0.200 || 2.859 || ↓ ||  
|-
|-
| [[176edo|176]] || 103\176 || 0.3177 || 4.6600 || &uarr; ||  
| [[176edo|176]] || 103\176 || 0.318 || 4.660 || ↑ ||  
|-
|-
| [[183edo|183]] || 107\183 || 0.3157 || 4.8138 || &darr; ||  
| [[183edo|183]] || 107\183 || 0.316 || 4.814 || ↓ ||  
|-
|-
| [[200edo|200]] || 117\200 || 0.0450 || 0.7500 || &uarr; ||  
| [[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 81: 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''' edos have fifths narrower than 686 cents.
{{Infobox Chord|2:3|ColorName=5|debug=1}}
** '''Perfect''' or '''heptatonic''' edos have fifths 685{{frac|4|7}} 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''' edos have fifths between 685{{frac|4|7}} and 720 cents wide.
 
** '''Pentatonic''' have fifths exactly 720 cents wide.
=== Notable voicings ===
** '''Supersharp''' edos have fifths 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]]
* [[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]]

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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