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{{Infobox Chord|4:5:6:7|ColorName=yo zo-7 or y,z7, har-7 or h7}}
{{Infobox Chord|4:5:6:7|ColorName=yo zo-7 or y,z7, har-7 or h7}}
{{Wikipedia|Harmonic seventh chord}}
{{Wikipedia|Harmonic seventh chord}}
'''4:5:6:7''', the ''harmonic seventh chord'', is the simplest [[tetrad]] in [[7-limit]] harmony. It is often used as a tuning target for the [[dominant seventh chord]] in barbershop music (→ [[Wikipedia: Harmonic seventh chord #Barbershop seventh ]]), and also for the German augmented sixth chord in [[septimal meantone]].  
'''4:5:6:7''', the ''harmonic seventh chord'', is the simplest [[tetrad]] in [[7-limit]] harmony. It is often used as a tuning target for the [[dominant seventh chord]] in barbershop music (→ [[Wikipedia: Harmonic seventh chord #Barbershop seventh]]), and also for the German augmented sixth chord in [[septimal meantone]].  


It is a [[dyadic chord]] in the [[7-odd-limit]], with its most complex interval a [[7/5]] tritone. It is the [[octave reduction|octave-reduced]] version of the first four odd harmonics, 1:3:5:7, or the first eight harmonics, 1::8. It is the fundamental otonal consonance of the 7-odd-limit. The utonal minor version of this chord is [[70:84:105:120|1–6/5–3/2–12/7]], sometimes called the ''subharmonic sixth chord''. The harmonic seventh chord can be modified by inflecting the [[5/4]] down by [[25/24]] and [[7/4]] down by [[49/48]] to get the subharmonic sixth chord. The intervals [[25/24]] and [[49/48]] thus serve as chromas, and they are equated when [[50/49]] is tempered out, such as in [[pajara]].
It is a [[dyadic chord]] in the [[7-odd-limit]], with its most complex interval a [[7/5]] tritone. It is the [[octave reduction|octave-reduced]] version of the first four odd harmonics, 1:3:5:7, or the first eight harmonics, 1::8. It is the fundamental otonal consonance of the 7-odd-limit. The utonal minor version of this chord is [[70:84:105:120|1–6/5–3/2–12/7]], sometimes called the ''subharmonic sixth chord''. The harmonic seventh chord can be modified by inflecting the [[5/4]] down by [[25/24]] and [[7/4]] down by [[49/48]] to get this chord, or by inflecting both up by [[36/35]] to get the ''subharmonic seventh chord'' [[70:90:105:126|1–9/7–3/2–9/5]].


== Rotations around the octave ==
Despite being harmonically simple, this chord may sound unresolved because it is so similar to the dominant seventh chord. However, this means it can be used as one, with a 4:5:6:7 chord on the dominant being [[3/2]]–[[15/8]]–[[9/4]]–[[21/8]] above the tonic, which is octave-equivalent to [[16/15|15/16]]–[[9/8]]–[[21/16]]–[[3/2]]. The 15/16 can step up by 16/15 to reach [[1/1]], and the 21/16 down by [[21/20]] to reach 5/4, resolving to the [[4:5:6]] chord on the tonic.
[[File:SculpEufaDem4-5-6-7-onD.mp3|none|thumb|4:5:6:7 "Root position"]]
 
[[File:SculpEufaDem5-6-7-8-onD.mp3|none|thumb|5:6:7:8 "1st inversion"]]
{{chord edo approximation}}
[[Category:Dominant seventh chords|#]]
 
[[File:SculpEufaDem6-7-8-10-onD.mp3|none|thumb|6:7:8:10 "2nd inversion"]]
== Audio of close voicings ==
[[File:SculpEufaDem7-8-10-12-onD.mp3|none|thumb|7:8:10:12 "3rd inversion"]]
[[File:SculpEufaDem4-5-6-7-onD.mp3|none|thumb|4:5:6:7, Root position]]
[[File:SculpEufaDem5-6-7-8-onD.mp3|none|thumb|5:6:7:8, 1st inversion]]
[[File:SculpEufaDem6-7-8-10-onD.mp3|none|thumb|6:7:8:10, 2nd inversion]]
[[File:SculpEufaDem7-8-10-12-onD.mp3|none|thumb|7:8:10:12, 3rd inversion]]
 
== Notable voicings ==
Sorted by [[Wilson norm]]. AOV and CAOV stand for [[Odd limit #Proposed extensions|all-odd voicing]] and ''condensed'' AOV respectively. This list is only a brief overview, see [[Voicings of 4:5:6:7]] for a more comprehensive list.


== Voicings and rotations around two octaves ==
{| class="wikitable"
{| class="wikitable sortable" style="text-align:center;"
|+
|+
! style="width:120px;" {{diagonal split header|Rotation|Voicing}}
! Voices
! style="width:120px;" | Root
! [[EFR]]
! style="width:120px;" | '3
! [[Kite's thoughts on hi-lo notation|Hi-lo name]]
! style="width:120px;" | '5
! Special properties
! style="width:120px;" | '7
|-
! style="width:120px;" | '3'5
| rowspan="5" |4 voices
! style="width:120px;" | '3'7
| 1:3:5:7
! style="width:120px;" | '''<nowiki/>'5'7'''
| hi37loR
! style="width:120px;" | '3'5'7
| AOV, [[Isodifferential chord|isodifferential]]
|-
|-
! style="width:120px;" | On 1
| 2:3:5:7
| data-sort-value="004-005-006-007" | ''4:5:6:7''
| hi37
| data-sort-value="004-005-007-012" | 4:5:7:12
| CAOV
| data-sort-value="004-006-007-010" | 4:6:7:10
| data-sort-value="004-005-006-014" | 4:5:6:14
| data-sort-value="004-007-010-012" | 4:7:10:12
| data-sort-value="004-005-012-014" | 4:5:12:14
| data-sort-value="002-003-005-007" | 2:3:5:7
| data-sort-value="002-005-006-007" | 2:5:6:7
|-
|-
! style="width:120px;" | On 3
| 3:4:5:7
| data-sort-value="006-007-016-020" | 6:7:16:20
| lo5
| data-sort-value="003-004-005-007" | 3:4:5:7
|  
| data-sort-value="006-007-010-016" | 6:7:10:16
| data-sort-value="003-007-008-010" | 3:7:8:10
| data-sort-value="003-004-007-010" | 3:4:7:10
| data-sort-value="006-007-008-010" | ''6:7:8:10''
| data-sort-value="003-005-007-008" | 3:5:7:8
| data-sort-value="006-007-008-020" | 6:7:8:20
|-
|-
! style="width:120px;" | On 5
| 4:5:6:7
| data-sort-value="005-006-007-016" | 5:6:7:16
| basic
| data-sort-value="005-007-012-016" | 5:7:12:16
| Isodifferential
| data-sort-value="005-008-012-014" | 5:8:12:14
| data-sort-value="005-006-014-016" | 5:6:14:16
| data-sort-value="005-006-008-014" | 5:6:8:14
| data-sort-value="005-012-014-016" | 5:12:14:16
| data-sort-value="005-007-008-012" | 5:7:8:12
| data-sort-value="005-006-007-008" | ''5:6:7:8''
|-
|-
! style="width:120px;" | On 7
| 4:6:7:10
| data-sort-value="007-016-020-024" | 7:16:20:24
| hi3
| data-sort-value="007-012-016-020" | 7:12:16:20
|  
| data-sort-value="007-010-016-024" | 7:10:16:24
| data-sort-value="007-008-010-012" | ''7:8:10:12''
| data-sort-value="007-010-012-016" | 7:10:12:16
| data-sort-value="007-008-010-024" | 7:8:10:24
| data-sort-value="007-008-012-020" | 7:8:12:20
| data-sort-value="007-008-020-024" | 7:8:20:24
|}
|}


Enumerations in italics map to an existing octave-reduced rotation.
== Related chords ==
Melodic inversion: 1/(7:6:5:4) = [[60:70:84:105]] = 1–7/6–7/5–7/4, and its homonym 1/(12:10:8:7) = [[70:84:105:120]] = 1–6/5–3/2–12/7.
 
Plausible [[chord homonym|homonyms]]: none.


== Related chords ==
Notable extensions (7-limit):
Chords related to this triad (7-limit except where noted):
* [[4:5:6:7:9]] – adds 9/4
* [[60:70:84:105]] – chord inversion
* [[12:15:18:21:28]] – adds 7/3 to make the 7-limit Hendrix chord
* [[4:5:6:7:9]] – adds 9/4 (two octave spanning pentad)
 
* [[12:14:18:21]] – substitutes 5/4 with 7/6
Notable restrictions:
* [[4:5:6]]
* [[4:5:7]]
* [[4:6:7]]
* [[5:6:7]]


[[Category:Dominant seventh chords|#]]
[[Category:German sixth chords|#]] <!-- 1-digit first number -->
[[Category:German sixth chords|#]] <!-- 1-digit first number -->

Latest revision as of 04:29, 26 May 2026

Chord information
Harmonics 4:5:6:7
Subharmonics 1/(105:84:70:60)
Intervals from root 1/1–5/4–3/2–7/4
Cents from root 0¢–386¢–702¢–969¢
Step intervals 5/4, 6/5, 7/6
Step cents 386¢, 316¢, 267¢
Color names yo zo-7 or y,z7
har-7 or h7
Prime limit 7
Genus 3⋅5⋅7 (105)
Intervallic odd limit 7
Otonal odd limit 7
Utonal odd limit 105
Consistent edos (d ≥ 2) 31edo*, 41edo*, 68edo*, 72edo*
English Wikipedia has an article on:

4:5:6:7, the harmonic seventh chord, is the simplest tetrad in 7-limit harmony. It is often used as a tuning target for the dominant seventh chord in barbershop music (→ Wikipedia: Harmonic seventh chord #Barbershop seventh), and also for the German augmented sixth chord in septimal meantone.

It is a dyadic chord in the 7-odd-limit, with its most complex interval a 7/5 tritone. It is the octave-reduced version of the first four odd harmonics, 1:3:5:7, or the first eight harmonics, 1::8. It is the fundamental otonal consonance of the 7-odd-limit. The utonal minor version of this chord is 1–6/5–3/2–12/7, sometimes called the subharmonic sixth chord. The harmonic seventh chord can be modified by inflecting the 5/4 down by 25/24 and 7/4 down by 49/48 to get this chord, or by inflecting both up by 36/35 to get the subharmonic seventh chord 1–9/7–3/2–9/5.

Despite being harmonically simple, this chord may sound unresolved because it is so similar to the dominant seventh chord. However, this means it can be used as one, with a 4:5:6:7 chord on the dominant being 3/2–15/8–9/4–21/8 above the tonic, which is octave-equivalent to 15/16–9/8–21/16–3/2. The 15/16 can step up by 16/15 to reach 1/1, and the 21/16 down by 21/20 to reach 5/4, resolving to the 4:5:6 chord on the tonic.


Edo approximations for 4:5:6:7 ▶
intervals: 5/4, 3/2, 7/4 · ≤ 60edo, RMS rel. error ≤ 15%
  Edo Steps Cents (¢) Absolute errors (¢) RMS (¢) RMS (%)
▶ 10 0  3  6  8 0.00 360.00 720.00  960.00 0.00 -26.31 +18.04  -8.83 15.99 13.33
▶ 12 0  4  7 10 0.00 400.00 700.00 1000.00 0.00 +13.69  -1.96 +31.17 13.25 13.25
▶ 15 0  5  9 12 0.00 400.00 720.00  960.00 0.00 +13.69 +18.04  -8.83 10.72 13.40
▶ 19 0  6 11 15 0.00 378.95 694.74  947.37 0.00  -7.37  -7.22 -21.46 7.78 12.32
▶ 22 0  7 13 18 0.00 381.82 709.09  981.82 0.00  -4.50  +7.14 +12.99 6.69 12.26
▶ 27 0  9 16 22 0.00 400.00 711.11  977.78 0.00 +13.69  +9.16  +8.95 4.96 11.17
▶ 31 0 10 18 25 0.00 387.10 696.77  967.74 0.00  +0.78  -5.18  -1.08 2.30 5.94
▶ 37 0 12 22 30 0.00 389.19 713.51  972.97 0.00  +2.88 +11.56  +4.15 4.26 13.15
▶ 41 0 13 24 33 0.00 380.49 702.44  965.85 0.00  -5.83  +0.48  -2.97 2.54 8.67
▶ 46 0 15 27 37 0.00 391.30 704.35  965.22 0.00  +4.99  +2.39  -3.61 3.17 12.14
▶ 50 0 16 29 40 0.00 384.00 696.00  960.00 0.00  -2.31  -5.96  -8.83 3.38 14.08
▶ 53 0 17 31 43 0.00 384.91 701.89  973.58 0.00  -1.41  -0.07  +4.76 2.34 10.34
▶ 58 0 19 34 47 0.00 393.10 703.45  972.41 0.00  +6.79  +1.49  +3.59 2.55 12.32

Audio of close voicings

4:5:6:7, Root position
5:6:7:8, 1st inversion
6:7:8:10, 2nd inversion
7:8:10:12, 3rd inversion

Notable voicings

Sorted by Wilson norm. AOV and CAOV stand for all-odd voicing and condensed AOV respectively. This list is only a brief overview, see Voicings of 4:5:6:7 for a more comprehensive list.

Voices EFR Hi-lo name Special properties
4 voices 1:3:5:7 hi37loR AOV, isodifferential
2:3:5:7 hi37 CAOV
3:4:5:7 lo5
4:5:6:7 basic Isodifferential
4:6:7:10 hi3

Related chords

Melodic inversion: 1/(7:6:5:4) = 60:70:84:105 = 1–7/6–7/5–7/4, and its homonym 1/(12:10:8:7) = 70:84:105:120 = 1–6/5–3/2–12/7.

Plausible homonyms: none.

Notable extensions (7-limit):

Notable restrictions:

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