4:5:6:7: Difference between revisions

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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 ==
== Audio of close voicings ==
[[File:SculpEufaDem4-5-6-7-onD.mp3|none|thumb|4:5:6:7 "Root position"]]
[[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:SculpEufaDem5-6-7-8-onD.mp3|none|thumb|5:6:7:8, 1st inversion]]
[[Category:Dominant seventh chords|#]]
[[Category:Dominant seventh chords|#]]
[[File:SculpEufaDem6-7-8-10-onD.mp3|none|thumb|6:7:8:10 "2nd 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"]]
[[File:SculpEufaDem7-8-10-12-onD.mp3|none|thumb|7:8:10:12, 3rd inversion]]


== Voicings and rotations around two octaves ==
== Notable voicings ==
{| class="wikitable sortable" style="text-align:center;"
AOV and CAOV stand for [[Odd limit#Proposed extensions|all-odd voicing]] and ''condensed'' AOV respectively.
{| class="wikitable"
|+
|+
! style="width:120px;" {{diagonal split header|Rotation|Voicing}}
!Voices
! style="width:120px;" | Root
!Voicing
! 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
! style="width:120px;" | '3'7
! style="width:120px;" | '''<nowiki/>'5'7'''
! style="width:120px;" | '3'5'7
|-
|-
! style="width:120px;" | On 1
| rowspan="5" |4 voices
| data-sort-value="004-005-006-007" | ''4:5:6:7''
|1:3:5:7
| data-sort-value="004-005-007-012" | 4:5:7:12
|hi37loR
| data-sort-value="004-006-007-010" | 4:6:7:10
|AOV, [[Isodifferential chord|isodifferential]]
| 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
|2:3:5:7
| data-sort-value="006-007-016-020" | 6:7:16:20
|hi37
| data-sort-value="003-004-005-007" | 3:4:5:7
|CAOV
| 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
|3:4:5:7
| data-sort-value="005-006-007-016" | 5:6:7:16
|lo5
| data-sort-value="005-007-012-016" | 5:7:12:16
|
| 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
|4:5:6:7
| data-sort-value="007-016-020-024" | 7:16:20:24
|basic
| data-sort-value="007-012-016-020" | 7:12:16:20
|isodifferential
| data-sort-value="007-010-016-024" | 7:10:16:24
|-
| data-sort-value="007-008-010-012" | ''7:8:10:12''
|4:6:7:10
| data-sort-value="007-010-012-016" | 7:10:12:16
|hi3
| 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: 7:6:5:4 = [[60:70:84:105]] = 1/1 - 7/6 - 7/5 - 7/4, and its homonym 12:10:8:7 = [[70:84:105:120]] = 1/1 - 6/5 - 3/2 - 12/7.


== Related chords ==
Plausible [[Chord homonym|homonyns]]: none.
Chords related to this triad (7-limit except where noted):
 
* [[60:70:84:105]] – chord inversion
Notable extensions (7-limit):
* [[4:5:6:7:9]] – adds 9/4 (two octave spanning pentad)
* [[4:5:6:7:9]] – adds 9/4
* [[12:14:18:21]] – substitutes 5/4 with 7/6
* [[12:15:18:21:28]] – adds 7/3 to make the 7-limit Hendrix chord
Notable restrictions:
 
* [[4:5:6]]
* [[4:5:7]]
* [[4:6:7]]
* [[5:6:7]]


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

Revision as of 16:17, 20 November 2025

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.

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

AOV and CAOV stand for all-odd voicing and condensed AOV respectively.

Voices Voicing 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: 7:6:5:4 = 60:70:84:105 = 1/1 - 7/6 - 7/5 - 7/4, and its homonym 12:10:8:7 = 70:84:105:120 = 1/1 - 6/5 - 3/2 - 12/7.

Plausible homonyns: none.

Notable extensions (7-limit):

Notable restrictions:

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