Hendrix chord: Difference between revisions

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== JI interpretations ==
== JI interpretations ==
Interpretations as a [[just intonation]] chord:
Interpretations as a [[just intonation]] chord.
 
<!-- ''The only interpretations allowed in this section are ones where all 'intervals from root'
''have a denominator of 8 or less (to prevent the section becoming too long).'' -->


=== 5-limit ===
=== 5-limit ===
{{Infobox Chord|10:20:25:36:40:48|ColorName=yo gu-7 guqu-9 no-5 or y,g7gq9no5|Root=20}}
{{Infobox Chord|10:20:25:36:40:48|ColorName=yo gu-7 guqu-9 no-5 or y,g7gq9no5|Root=20}}
In the [[5-limit]] it may be tuned as 10:20:25:36:40:48, an extended [[20:25:30:36|major-minor seventh chord]].
In the [[5-limit]] it may be tuned as '''10:20:25:36:40:48''', an extended [[20:25:30:36|major-minor seventh chord]].
 
This chord appears in [[20afdo]] and all higher [[AFDO]]s.
 
According to this interpretation, the 12edo hendrix chord has 30.2 [[cents]] of total error and no more than 17.6 cents error on any single interval.
 
[[EDO]]s with less than 20 cents error on any one interval include: {{EDOs|12, 15, 18, 19, 22, 24, 26, 27, 28...}} and everything bigger than 28.
 
[[EDO]]s up to 100 with successively lower error include: {{EDOs|4, 5, 6, 7, 12, 19, 34, 46, 53}}.
{{clear}}
{{clear}}


=== 7-limit ===
=== 7-limit ===
{{Infobox Chord|6:12:15:21:24:28|ColorName=har-7 zoqu-9 no-5 or h7zq9no5|Root=12}}
{{Infobox Chord|6:12:15:21:24:28|ColorName=har-7 zoqu-9 no-5 or h7zq9no5|Root=12}}
In the [[7-limit]] it may be tuned as 6:12:15:21:24:28, an extended [[4:5:6:7|harmonic seventh chord]].
In the [[7-limit]] it may be tuned as '''6:12:15:21:24:28''', an extended [[4:5:6:7|harmonic seventh chord]].
 
This chord appears in [[12afdo]] and all higher [[AFDO]]s.
 
According to this interpretation, the 12edo hendrix chord has 78.0 [[cents]] of total error and no more than 33.1 cents error on any single interval.
 
[[EDO]]s with less than 35 cents error on any one interval include: {{EDOs|10, 12, 15, 16, 17...}} and everything bigger than 17.
 
[[EDO]]s up to 100 with successively lower error include: {{EDOs|4, 6, 9, 10, 19, 21, 22, 31, 68, 99}}.
{{clear}}
{{clear}}


=== 19-limit ===
=== 19-limit ===
{{Infobox Chord|4:8:10:14:16:19|ColorName=har-7 noqu-9 no-5 or h7,19oq9no5|Root=8}}
{{Infobox Chord|4:8:10:14:16:19|ColorName=har-7 noqu-9 no-5 or h7,19oq9no5|Root=8}}
In the [[19-limit]] it may be tuned as 4:8:10:14:16:19, an extended [[4:5:6:7|harmonic seventh chord]].
In the [[19-limit]] it may be tuned as '''4:8:10:14:16:19''', an extended [[4:5:6:7|harmonic seventh chord]].
 
This chord appears in [[8afdo]] and all higher [[AFDO]]s.
 
According to this interpretation, the 12edo hendrix chord has 47.4 [[cents]] of total error and no more than 33.1 cents error on any single interval.
 
[[EDO]]s with less than 35 cents error on any one interval include: {{EDOs|12, 15, 16, 17...}} and everything bigger than 17.
 
[[EDO]]s up to 100 with successively lower error include: {{EDOs|4, 7, 9, 12, 15, 16, 25, 37, 53, 56, 68, 93}}.
{{clear}}
{{clear}}


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=== Essentially tempered ===
=== Essentially tempered ===
It can also be tuned as an [[essentially tempered chord]] that splits the difference between the [[19/16|19/8]] 10th and the [[7/3]] 10th. This chord tempers out the ''hendrix comma'' of [[57/56]].  
It can also be tuned as an [[essentially tempered chord]] that splits the difference between the [[19/16|19/8]] 10th and the [[7/3]] 10th.  
{{Todo|review|inline=1|text=The paragraph about tempering 57/56 needs more explanation, and may be incorrect.}}
 
This chord tempers out the ''hendrix comma'' of [[57/56]].  
 
[[EDO]]s including this chord are {{EDOs|9, 10, 12, 14, 16, 17, 21, 22, 26, and 31}}.


=== In EDOs ===
{{Todo|review|inline=1|text=The section about tempering 57/56 needs more explanation, and may be incorrect.}}
[[EDO]]s with hendrix chords include the {{EDOs|9, 10, 14, 16, 17, 21, 22, 26, and 31}}.
{{Todo|complete section|inline=1|text=Explain the reasoning for exactly when an EDO is considered to "have" a Hendrix chord and when not.}}


[[Category:Hendrix| ]] <!-- main article -->
[[Category:Hendrix| ]] <!-- main article -->
[[Category:Essentially tempered chords]]
[[Category:Essentially tempered chords]]
[[Category:Just intonation chords]]
[[Category:Just intonation chords]]

Latest revision as of 13:18, 29 May 2025

English Wikipedia has an article on:

The hendrix chord, a 7#9no5 chord originating from 12edo, has several possible interpretations.

JI interpretations

Interpretations as a just intonation chord.


5-limit

Chord information
Harmonics 10:20:25:36:40:48
Intervals from root 1/2 – 1/1 – 5/4 – 9/5 – 2/1 – 12/5
Cents from root -1200¢ 0¢ 386¢ 1018¢ 1200¢ 1516¢
Step intervals 2/1, 5/4, 36/25, 10/9, 6/5
Step cents 1200¢, 386¢, 631¢, 182¢, 316¢
Color name yo gu-7 guqu-9 no-5 or y,g7gq9no5
Prime limit 5
Genus 32 ⋅ 52 (225)
Intervallic odd limit 25
Otonal odd limit 25
Utonal odd limit 75
Consistent edos (d ≥ 1.5) 12edo, 15edo, 19edo**, 31edo, …

In the 5-limit it may be tuned as 10:20:25:36:40:48, an extended major-minor seventh chord.

This chord appears in 20afdo and all higher AFDOs.

According to this interpretation, the 12edo hendrix chord has 30.2 cents of total error and no more than 17.6 cents error on any single interval.

EDOs with less than 20 cents error on any one interval include: 12, 15, 18, 19, 22, 24, 26, 27, 28... and everything bigger than 28.

EDOs up to 100 with successively lower error include: 4, 5, 6, 7, 12, 19, 34, 46, 53.

7-limit

Chord information
Harmonics 6:12:15:21:24:28
Intervals from root 1/2 – 1/1 – 5/4 – 7/4 – 2/1 – 7/3
Cents from root -1200¢ 0¢ 386¢ 969¢ 1200¢ 1467¢
Step intervals 2/1, 5/4, 7/5, 8/7, 7/6
Step cents 1200¢, 386¢, 583¢, 231¢, 267¢
Color name har-7 zoqu-9 no-5 or h7zq9no5
Prime limit 7
Genus 3 ⋅ 5 ⋅ 7 (105)
Intervallic odd limit 15
Otonal odd limit 21
Utonal odd limit 35
Consistent edos (d ≥ 1.5) 10edo*, 12edo, 21edo, 22edo, …

In the 7-limit it may be tuned as 6:12:15:21:24:28, an extended harmonic seventh chord.

This chord appears in 12afdo and all higher AFDOs.

According to this interpretation, the 12edo hendrix chord has 78.0 cents of total error and no more than 33.1 cents error on any single interval.

EDOs with less than 35 cents error on any one interval include: 10, 12, 15, 16, 17... and everything bigger than 17.

EDOs up to 100 with successively lower error include: 4, 6, 9, 10, 19, 21, 22, 31, 68, 99.

19-limit

Chord information
Harmonics 4:8:10:14:16:19
Intervals from root 1/2 – 1/1 – 5/4 – 7/4 – 2/1 – 19/8
Cents from root -1200¢ 0¢ 386¢ 969¢ 1200¢ 1498¢
Step intervals 2/1, 5/4, 7/5, 8/7, 19/16
Step cents 1200¢, 386¢, 583¢, 231¢, 298¢
Color name har-7 noqu-9 no-5 or h7,19oq9no5
Prime limit 19
Genus 5 ⋅ 7 ⋅ 19 (665)
Intervallic odd limit 19
Otonal odd limit 19
Utonal odd limit 665
Consistent edos (d ≥ 1.5) 4edo, 12edo, 16edo*, 25edo*, …

In the 19-limit it may be tuned as 4:8:10:14:16:19, an extended harmonic seventh chord.

This chord appears in 8afdo and all higher AFDOs.

According to this interpretation, the 12edo hendrix chord has 47.4 cents of total error and no more than 33.1 cents error on any single interval.

EDOs with less than 35 cents error on any one interval include: 12, 15, 16, 17... and everything bigger than 17.

EDOs up to 100 with successively lower error include: 4, 7, 9, 12, 15, 16, 25, 37, 53, 56, 68, 93.

Tempered interpretations

Interpretations as a tempered chord.

Essentially tempered

It can also be tuned as an essentially tempered chord that splits the difference between the 19/8 10th and the 7/3 10th.

This chord tempers out the hendrix comma of 57/56.

EDOs including this chord are 9, 10, 12, 14, 16, 17, 21, 22, 26, and 31.

Todo: review

The section about tempering 57/56 needs more explanation, and may be incorrect.

Retrieved from "https://en.xen.wiki/index.php?title=Hendrix_chord&oldid=198063"