4:5:6

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*, …
	* 2 ≤ d < 4; 4 ≤ d < 8; * 8 ≤ d < 16; …

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):

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)