4:5:6: Difference between revisions

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{{Infobox Chord|ColorName=yo or y}}
{{Infobox Chord|ColorName=yo or y}}
'''4:5:6''' is an otonal [[major triad]], known as the '''just major triad''', '''classical major triad''', or '''Ptolemaic major triad'''. It is among the most consonant triads, and it is among the most common triads in music.  
'''4:5:6''' is an otonal [[major triad]], known as the '''just major triad''', '''classical major triad''', or '''Ptolemaic major triad'''. It is the most consonant triad, and it is the most common triad in music. In close voicing root position, it is an [[Delta-rational chord #Isoharmonic chord|isoharmonic chord]]. It occurs as a major triad in the [[meantone]] diatonic scale, as the major third formed by stacking four [[3/2|perfect fifths]] octave reduced, [[81/64]], is equated with [[5/4]]. It serves as the fundamental major root chord in classical music, and approximating is key in [[5-limit]] temperaments. This chord is approximated with all intervals having less than 20{{c}} of error in [[edo]]s {{EDOs|12, 15, 19, 22, 24, 26, 27}}, etc.


Its second rotation, 3:4:5, has a lower [[harmonic entropy]] and is sometimes considered to be more consonant (for example, by [[Lériendil]]). However, 3:4:5 is not typically considered to be consonant in counterpoint and some derived theories (but not necessarily modern pop or xenharmonic theory), as it is not only [[rooted interval|nonrooted]] but the fourth above the root contrasts and therefore wants to move to the missing major third. It is also not considered to be stable in [[Shasavistic theory]] unlike 4:5:6.  
This chord is formed by stacking [[5/4]] and [[6/5]] in that order. Swapping the order of these intervals results in [[10:12:15]], the classical minor triad. The minor triad can also be obtained from 4:5:6 by flattening the third by [[25/24]], the diptolemaic chromatic semitone.


== Rotations around the octave ==
== Audio of close voicings ==
[[File:SculpEufaDem3-4-5-onD.mp3|none|thumb|3:4:5, "2nd inversion"]]
[[File:SculpEufaDem4-5-6-onD.mp3|none|thumb|4:5:6, Root position]]
[[File:SculpEufaDem4-5-6-onD.mp3|none|thumb|4:5:6, "Root position"]]
[[File:SculpEufaDem5-6-8-onD.mp3|none|thumb|5:6:8, 1st inversion]][[File:SculpEufaDem3-4-5-onD.mp3|none|thumb|3:4:5, 2nd inversion]]
[[File:SculpEufaDem5-6-8-onD.mp3|none|thumb|5:6:8, "1st inversion"]]


== Voicings and rotations around two octaves ==
== Approximation by edos ==
{| class="wikitable sortable" style="text-align:center;"
[[7edo]] contains a rough approximation to 4:5:6, which equates the 5/4 and 6/5 steps. The smallest edo to approximate it with acceptable accuracy is [[12edo]], with [[19edo]] being the next edo to improve on it.
|+
{{chord edo approximation}}
! style="width:90px;" {{diagonal split header|Rotation|Voicing}}
 
! style="width:90px;" | Root
== Notable voicings ==
! style="width:90px;" | '3
Voicings are arranged from simple to complex using [[Wilson norm]]. AOV and CAOV stand for [[Odd limit #Proposed extensions|all-odd voicing]] and ''condensed'' AOV respectively. Numbers in '''bold''' denote doubled pitches. This list is only a brief overview, see [[Voicings of 4:5:6]] for a more comprehensive list and audio examples.
! style="width:90px;" | '''<nowiki/>'5'''
 
! style="width:90px;" | '3'5
{| class="wikitable"
! Voices
! [[EFR]]
! [[Kite's thoughts on hi-lo notation|Hi-lo name]]
! Special properties
|-
| rowspan="4" |3 voices
| 1:3:5
| hi3loR
| AOV, [[Delta-rational chord #Isoharmonic chord|isoharmonic]]
|-
| 2:3:5
| hi3
| CAOV
|-
| 3:4:5
| lo5
| 1st inversion, isoharmonic
|-
| 4:5:6
| basic
| isoharmonic
|-
|-
! style="width:90px;" | On 1
| rowspan="4" |4 voices
| style="width:90px;" data-sort-value="004-005-006" | ''4:5:6''
| 2:3:'''4''':5
| style="width:90px;" data-sort-value="004-005-012" | 4:5:12
| hi3add8
| style="width:90px;" data-sort-value="002-003-005" | 2:3:5
| isoharmonic
| style="width:90px;" data-sort-value="002-005-006" | 2:5:6
|-
|-
! style="width:90px;" | On 3
| 2:'''4''':5:6
| style="width:90px;" data-sort-value="003-004-005" | 3:8:10
| addloR
| style="width:90px;" data-sort-value="003-004-005" | ''3:4:5''
|  
| style="width:90px;" data-sort-value="003-005-008" | 3:5:8
| style="width:90px;" data-sort-value="003-004-010" | 3:4:10
|-
|-
! style="width:90px;" | On 5
| 3:4:5:'''6'''
| style="width:90px;" data-sort-value="005-006-016" | 5:6:16
| addlo5
| style="width:90px;" data-sort-value="005-012-016" | 5:12:16
| isoharmonic
| style="width:90px;" data-sort-value="005-008-012" | 5:8:12
|-
| style="width:90px;" data-sort-value="005-006-008" | ''5:6:8''
| 4:5:6:'''8'''
| add8
|
|}
|}
== Related chords ==
Melodic inversion: 1/(6:5:4) = [[10:12:15]].


Plausible [[chord homonym|homonyms]]: None.


== Related chords ==
Lower limit soundalikes: [[64:81:96]] (3-limit)
Chords related to this triad (5-limit except where noted):
 
* [[2:3:5]] - expanded voicing
Notable extensions (5-limit except where noted):
* [[10:12:15]] - chord inversion
* [[12:15:18:20]] – adds 5/3
* [[12:15:18:20]] - adds 5/3
* [[8:10:12:15]] – adds 15/8
* [[8:10:12:15]] - adds 15/8
* [[36:45:54:64]] – adds 16/9
* [[36:45:54:64]] - adds 16/9
* [[20:25:30:36]] – adds 9/5
* [[20:25:30:36]] - adds 9/5
* [[4:5:6:7]] – adds 7/4 (7-limit)
* [[4:5:6:7]] - adds 7/4 (7-limit)


[[Category:Major triads|#]] <!-- 1-digit first number -->
[[Category:Major triads|#]] <!-- 1-digit first number -->
[[Category:Major triads|#]] <!-- 1-digit first number -->

Latest revision as of 23:53, 31 May 2026

Chord information
Harmonics 4:5:6
Subharmonics 1/(15:12:10)
Intervals from root 1/1–5/4–3/2
Cents from root 0¢–386¢–702¢
Step intervals 5/4, 6/5
Step cents 386¢, 316¢
Color name yo or y
Prime limit 5
Genus 3⋅5 (15)
Intervallic odd limit 5
Otonal odd limit 5
Utonal odd limit 15
Consistent edos (d ≥ 2) 3edo*, 12edo*, 15edo*, 19edo**, …

4:5:6 is an otonal major triad, known as the just major triad, classical major triad, or Ptolemaic major triad. It is the most consonant triad, and it is the most common triad in music. In close voicing root position, it is an isoharmonic chord. It occurs as a major triad in the meantone diatonic scale, as the major third formed by stacking four perfect fifths octave reduced, 81/64, is equated with 5/4. It serves as the fundamental major root chord in classical music, and approximating is key in 5-limit temperaments. This chord is approximated with all intervals having less than 20 ¢ of error in edos 12, 15, 19, 22, 24, 26, 27, etc.

This chord is formed by stacking 5/4 and 6/5 in that order. Swapping the order of these intervals results in 10:12:15, the classical minor triad. The minor triad can also be obtained from 4:5:6 by flattening the third by 25/24, the diptolemaic chromatic semitone.

Audio of close voicings

4:5:6, Root position
5:6:8, 1st inversion
3:4:5, 2nd inversion

Approximation by edos

7edo contains a rough approximation to 4:5:6, which equates the 5/4 and 6/5 steps. The smallest edo to approximate it with acceptable accuracy is 12edo, with 19edo being the next edo to improve on it.

Edo approximations for 4:5:6 ▶
intervals: 5/4, 3/2 · ≤ 60edo, RMS rel. error ≤ 15%
  Edo Steps Cents (¢) Absolute errors (¢) RMS (¢) RMS (%)
▶ 7 0  2  4 0.00 342.86 685.71 0.00 -43.46 -16.24 17.93 10.46
▶ 12 0  4  7 0.00 400.00 700.00 0.00 +13.69  -1.96 6.96 6.96
▶ 15 0  5  9 0.00 400.00 720.00 0.00 +13.69 +18.04 7.69 9.61
▶ 16 0  5  9 0.00 375.00 675.00 0.00 -11.31 -26.96 11.05 14.74
▶ 19 0  6 11 0.00 378.95 694.74 0.00  -7.37  -7.22 3.44 5.44
▶ 22 0  7 13 0.00 381.82 709.09 0.00  -4.50  +7.14 4.79 8.78
▶ 24 0  8 14 0.00 400.00 700.00 0.00 +13.69  -1.96 6.96 13.92
▶ 27 0  9 16 0.00 400.00 711.11 0.00 +13.69  +9.16 5.69 12.81
▶ 31 0 10 18 0.00 387.10 696.77 0.00  +0.78  -5.18 2.65 6.84
▶ 34 0 11 20 0.00 388.24 705.88 0.00  +1.92  +3.93 1.60 4.54
▶ 38 0 12 22 0.00 378.95 694.74 0.00  -7.37  -7.22 3.44 10.89
▶ 41 0 13 24 0.00 380.49 702.44 0.00  -5.83  +0.48 2.87 9.80
▶ 43 0 14 25 0.00 390.70 697.67 0.00  +4.38  -4.28 3.54 12.68
▶ 46 0 15 27 0.00 391.30 704.35 0.00  +4.99  +2.39 2.04 7.81
▶ 49 0 16 29 0.00 391.84 710.20 0.00  +5.52  +8.25 3.43 14.01
▶ 50 0 16 29 0.00 384.00 696.00 0.00  -2.31  -5.96 2.45 10.21
▶ 53 0 17 31 0.00 384.91 701.89 0.00  -1.41  -0.07 0.65 2.86
▶ 56 0 18 33 0.00 385.71 707.14 0.00  -0.60  +5.19 2.60 12.13
▶ 58 0 19 34 0.00 393.10 703.45 0.00  +6.79  +1.49 2.91 14.08
▶ 60 0 19 35 0.00 380.00 700.00 0.00  -6.31  -1.96 2.64 13.20

Notable voicings

Voicings are arranged from simple to complex using Wilson norm. AOV and CAOV stand for all-odd voicing and condensed AOV respectively. Numbers in bold denote doubled pitches. This list is only a brief overview, see Voicings of 4:5:6 for a more comprehensive list and audio examples.

Voices EFR Hi-lo name Special properties
3 voices 1:3:5 hi3loR AOV, isoharmonic
2:3:5 hi3 CAOV
3:4:5 lo5 1st inversion, isoharmonic
4:5:6 basic isoharmonic
4 voices 2:3:4:5 hi3add8 isoharmonic
2:4:5:6 addloR
3:4:5:6 addlo5 isoharmonic
4:5:6:8 add8

Related chords

Melodic inversion: 1/(6:5:4) = 10:12:15.

Plausible homonyms: None.

Lower limit soundalikes: 64:81:96 (3-limit)

Notable extensions (5-limit except where noted):

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