4:5:6:7: Difference between revisions

'''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]].

== Rotations around the octave ==

[[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"]]

== Voicings and rotations around two octaves ==

{| class="wikitable sortable" style="text-align:center;"

! style="width:120px;" {{diagonal split header|Rotation|Voicing}}

! style="width:120px;" | Root

! style="width:120px;" | '3'7

! style="width:120px;" | '''<nowiki/>'5'7'''

! style="width:120px;" | '3'5'7

| data-sort-value="004-005-006-007" | ''4:5:6:7''

| data-sort-value="004-005-007-012" | 4:5:7:12

| 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="002-003-005-007" | 2:3:5:7

| data-sort-value="002-005-006-007" | 2:5:6:7

| data-sort-value="006-007-016-020" | 6:7:16:20

| 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="003-005-007-008" | 3:5:7:8

| data-sort-value="005-006-007-016" | 5:6:7:16

| 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="007-016-020-024" | 7:16:20:24

| 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.

Chords related to this triad (7-limit except where noted):

* [[4:5:6:7:9]] – adds 9/4 (two octave spanning pentad)

* [[12:14:18:21]] – substitutes 5/4 with 7/6