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 the otonal [[major triad]], also known as the '''classical''' '''major triad''', '''Ptolemaic major triad''', or more generally - albeit less precisely - as the '''just 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 most consonant rotation, 3:4:5, is not typically considered to be consonant, at least in more classical tonal music, where the fourth above the root is considered a dissonance. Acoustically speaking however, 3:4:5 has the least [[harmonic entropy]].  
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"]]


== 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.
{{chord edo approximation}}
== Notable voicings ==
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.
{| 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
|-
| rowspan="4" |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 ==
== Related chords ==
Chords related to this triad (5-limit except where noted):
Melodic inversion: 1/(6:5:4) = [[10:12:15]].
* 2:3:5 - open voicing
 
* [[10:12:15]] - chord inversion
Plausible [[chord homonym|homonyms]]: None.
* [[12:15:18:20]] - adds 5/3
 
* [[8:10:12:15]] - adds 15/8
Lower limit soundalikes: [[64:81:96]] (3-limit)
* [[36:45:54:64]] - adds 16/9
 
* [[20:25:30:36]] - adds 9/5
Notable extensions (5-limit except where noted):
* [[4:5:6:7]] - adds 7/4 (7-limit)
* [[12:15:18:20]] – adds 5/3
* [[8:10:12:15]] – adds 15/8
* [[36:45:54:64]] – adds 16/9
* [[20:25:30:36]] – adds 9/5
* [[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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