Chords of echidna: Difference between revisions

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Below are listed the [[Dyadic_chord|dyadic chords]] of 11-limit [[Diaschismic_family#Echidna|echidna temperament]]. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering by 540/539 are labeled swetismic, by 896/891 pentacircle, and by 176/175 valinorsmic.  
Below are listed the [[dyadic chord]]s of the [[11-limit]] [[echidna]] temperament. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering by [[540/539]] are labeled [[swetismic chords|swetismic]], by [[896/891]] [[pentacircle chords|pentacircle]], and by [[176/175]] [[valinorsmic chords|valinorsmic]].  


The normal mapping for echidna is ech = [<2 1 9 2 12|, <0 3 -6 5 -7|]. From this we may derive a val v = ech[1] + 100 ech[2] = <2 301 -591 502 -688| which we may use to sort and normalize the chords of echidna. Under "Chord" is listed the chord, normalized to start from zero, in the mapping by v, and the next column gives a transversal which tempers to the chord by echidnic tempering. Finally, the Graham complexity is listed in the last column.
The normal mapping for echidna is ech = {{mapping| 2 1 9 2 12 | 0 3 -6 5 -7 }}. From this we may derive a val v = ech[1] + 100 ech[2] = {{val| 2 301 -591 502 -688 }} which we may use to sort and normalize the chords of echidna. Under "Chord" is listed the chord, normalized to start from zero, in the mapping by v, and the next column gives a transversal which tempers to the chord by echidnic tempering. Finally, the Graham complexity is listed in the last column.


Echidna has MOS of size 6, 8, 14, 22, 36, 58 and 80. It may be seen that a modest number of chords are available to 14 notes, but that 22 has many more, and that may be a good choice as a scale with which to explore the harmonies of echidna. If for some reason you needed pentads and hexads, 36 notes would be fine.
Echidna has [[mos]] of size 6, 8, 14, 22, 36, 58 and 80. It may be seen that a modest number of chords are available to 14 notes, but that 22 has many more, and that may be a good choice as a scale with which to explore the harmonies of echidna. If for some reason you needed pentads and hexads, 36 notes would be fine.


=Triads=
== Triads ==


{| class="wikitable"
{| class="wikitable"
|-
|-
| | Number
! Number
| | Chord
! Chord
| | Transversal
! Transversal
| | Type
! Type
| | Complexity
! Complexity
|-
|-
| | 1
| 1
| | 0-100-199
| 0-100-199
| | 1-9/7-7/6
| 1-9/7-7/6
| | swetismic
| swetismic
| | 4
| 4
|-
|-
| | 2
| 2
| | 0-101-199
| 0-101-199
| | 1-20/11-7/6
| 1-20/11-7/6
| | swetismic
| swetismic
| | 4
| 4
|-
|-
| | 3
| 3
| | 0-100-299
| 0-100-299
| | 1-9/7-3/2
| 1-9/7-3/2
| | utonal
| utonal
| | 6
| 6
|-
|-
| | 4
| 4
| | 0-199-299
| 0-199-299
| | 1-7/6-3/2
| 1-7/6-3/2
| | otonal
| otonal
| | 6
| 6
|-
|-
| | 5
| 5
| | 0-199-498
| 0-199-498
| | 1-7/6-7/4
| 1-7/6-7/4
| | utonal
| utonal
| | 10
| 10
|-
|-
| | 6
| 6
| | 0-299-498
| 0-299-498
| | 1-3/2-7/4
| 1-3/2-7/4
| | otonal
| otonal
| | 10
| 10
|-
|-
| | 7
| 7
| | 0-100-596
| 0-100-596
| | 1-9/7-9/8
| 1-9/7-9/8
| | utonal
| utonal
| | 12
| 12
|-
|-
| | 8
| 8
| | 0-299-596
| 0-299-596
| | 1-3/2-9/8
| 1-3/2-9/8
| | ambitonal
| ambitonal
| | 12
| 12
|-
|-
| | 9
| 9
| | 0-498-596
| 0-498-596
| | 1-7/4-9/8
| 1-7/4-9/8
| | otonal
| otonal
| | 12
| 12
|-
|-
| | 10
| 10
| | 0-101-597
| 0-101-597
| | 1-20/11-8/5
| 1-20/11-8/5
| | valinorsmic
| valinorsmic
| | 12
| 12
|-
|-
| | 11
| 11
| | 0-498-597
| 0-498-597
| | 1-7/4-8/5
| 1-7/4-8/5
| | valinorsmic
| valinorsmic
| | 12
| 12
|-
|-
| | 12
| 12
| | 0-100-696
| 0-100-696
| | 1-9/7-16/11
| 1-9/7-16/11
| | pentacircle
| pentacircle
| | 14
| 14
|-
|-
| | 13
| 13
| | 0-101-696
| 0-101-696
| | 1-20/11-16/11
| 1-20/11-16/11
| | otonal
| otonal
| | 14
| 14
|-
|-
| | 14
| 14
| | 0-596-696
| 0-596-696
| | 1-9/8-16/11
| 1-9/8-16/11
| | pentacircle
| pentacircle
| | 14
| 14
|-
|-
| | 15
| 15
| | 0-597-696
| 0-597-696
| | 1-8/5-16/11
| 1-8/5-16/11
| | utonal
| utonal
| | 14
| 14
|-
|-
| | 16
| 16
| | 0-299-894
| 0-299-894
| | 1-3/2-6/5
| 1-3/2-6/5
| | utonal
| utonal
| | 18
| 18
|-
|-
| | 17
| 17
| | 0-597-894
| 0-597-894
| | 1-8/5-6/5
| 1-8/5-6/5
| | otonal
| otonal
| | 18
| 18
|-
|-
| | 18
| 18
| | 0-101-993
| 0-101-993
| | 1-20/11-12/11
| 1-20/11-12/11
| | otonal
| otonal
| | 20
| 20
|-
|-
| | 19
| 19
| | 0-299-993
| 0-299-993
| | 1-3/2-12/11
| 1-3/2-12/11
| | utonal
| utonal
| | 20
| 20
|-
|-
| | 20
| 20
| | 0-696-993
| 0-696-993
| | 1-16/11-12/11
| 1-16/11-12/11
| | otonal
| otonal
| | 20
| 20
|-
|-
| | 21
| 21
| | 0-894-993
| 0-894-993
| | 1-6/5-12/11
| 1-6/5-12/11
| | utonal
| utonal
| | 20
| 20
|-
|-
| | 22
| 22
| | 0-100-1093
| 0-100-1093
| | 1-9/7-7/5
| 1-9/7-7/5
| | swetismic
| swetismic
| | 22
| 22
|-
|-
| | 23
| 23
| | 0-199-1093
| 0-199-1093
| | 1-7/6-7/5
| 1-7/6-7/5
| | utonal
| utonal
| | 22
| 22
|-
|-
| | 24
| 24
| | 0-498-1093
| 0-498-1093
| | 1-7/4-7/5
| 1-7/4-7/5
| | utonal
| utonal
| | 22
| 22
|-
|-
| | 25
| 25
| | 0-597-1093
| 0-597-1093
| | 1-8/5-7/5
| 1-8/5-7/5
| | otonal
| otonal
| | 22
| 22
|-
|-
| | 26
| 26
| | 0-894-1093
| 0-894-1093
| | 1-6/5-7/5
| 1-6/5-7/5
| | otonal
| otonal
| | 22
| 22
|-
|-
| | 27
| 27
| | 0-993-1093
| 0-993-1093
| | 1-12/11-7/5
| 1-12/11-7/5
| | swetismic
| swetismic
| | 22
| 22
|-
|-
| | 28
| 28
| | 0-101-1192
| 0-101-1192
| | 1-20/11-14/11
| 1-20/11-14/11
| | otonal
| otonal
| | 24
| 24
|-
|-
| | 29
| 29
| | 0-199-1192
| 0-199-1192
| | 1-7/6-14/11
| 1-7/6-14/11
| | utonal
| utonal
| | 24
| 24
|-
|-
| | 30
| 30
| | 0-498-1192
| 0-498-1192
| | 1-7/4-14/11
| 1-7/4-14/11
| | utonal
| utonal
| | 24
| 24
|-
|-
| | 31
| 31
| | 0-596-1192
| 0-596-1192
| | 1-9/8-14/11
| 1-9/8-14/11
| | pentacircle
| pentacircle
| | 24
| 24
|-
|-
| | 32
| 32
| | 0-597-1192
| 0-597-1192
| | 1-8/5-14/11
| 1-8/5-14/11
| | valinorsmic
| valinorsmic
| | 24
| 24
|-
|-
| | 33
| 33
| | 0-696-1192
| 0-696-1192
| | 1-16/11-14/11
| 1-16/11-14/11
| | otonal
| otonal
| | 24
| 24
|-
|-
| | 34
| 34
| | 0-993-1192
| 0-993-1192
| | 1-12/11-14/11
| 1-12/11-14/11
| | otonal
| otonal
| | 24
| 24
|-
|-
| | 35
| 35
| | 0-1093-1192
| 0-1093-1192
| | 1-7/5-14/11
| 1-7/5-14/11
| | utonal
| utonal
| | 24
| 24
|-
|-
| | 36
| 36
| | 0-100-1193
| 0-100-1193
| | 1-9/7-9/5
| 1-9/7-9/5
| | utonal
| utonal
| | 24
| 24
|-
|-
| | 37
| 37
| | 0-299-1193
| 0-299-1193
| | 1-3/2-9/5
| 1-3/2-9/5
| | utonal
| utonal
| | 24
| 24
|-
|-
| | 38
| 38
| | 0-596-1193
| 0-596-1193
| | 1-9/8-9/5
| 1-9/8-9/5
| | utonal
| utonal
| | 24
| 24
|-
|-
| | 39
| 39
| | 0-597-1193
| 0-597-1193
| | 1-8/5-9/5
| 1-8/5-9/5
| | otonal
| otonal
| | 24
| 24
|-
|-
| | 40
| 40
| | 0-894-1193
| 0-894-1193
| | 1-6/5-9/5
| 1-6/5-9/5
| | otonal
| otonal
| | 24
| 24
|-
|-
| | 41
| 41
| | 0-1093-1193
| 0-1093-1193
| | 1-7/5-9/5
| 1-7/5-9/5
| | otonal
| otonal
| | 24
| 24
|-
|-
| | 42
| 42
| | 0-100-1292
| 0-100-1292
| | 1-9/7-18/11
| 1-9/7-18/11
| | utonal
| utonal
| | 26
| 26
|-
|-
| | 43
| 43
| | 0-101-1292
| 0-101-1292
| | 1-20/11-18/11
| 1-20/11-18/11
| | otonal
| otonal
| | 26
| 26
|-
|-
| | 44
| 44
| | 0-199-1292
| 0-199-1292
| | 1-7/6-18/11
| 1-7/6-18/11
| | swetismic
| swetismic
| | 26
| 26
|-
|-
| | 45
| 45
| | 0-299-1292
| 0-299-1292
| | 1-3/2-18/11
| 1-3/2-18/11
| | utonal
| utonal
| | 26
| 26
|-
|-
| | 46
| 46
| | 0-596-1292
| 0-596-1292
| | 1-9/8-18/11
| 1-9/8-18/11
| | utonal
| utonal
| | 26
| 26
|-
|-
| | 47
| 47
| | 0-696-1292
| 0-696-1292
| | 1-16/11-18/11
| 1-16/11-18/11
| | otonal
| otonal
| | 26
| 26
|-
|-
| | 48
| 48
| | 0-993-1292
| 0-993-1292
| | 1-12/11-18/11
| 1-12/11-18/11
| | otonal
| otonal
| | 26
| 26
|-
|-
| | 49
| 49
| | 0-1093-1292
| 0-1093-1292
| | 1-7/5-18/11
| 1-7/5-18/11
| | swetismic
| swetismic
| | 26
| 26
|-
|-
| | 50
| 50
| | 0-1192-1292
| 0-1192-1292
| | 1-14/11-18/11
| 1-14/11-18/11
| | otonal
| otonal
| | 26
| 26
|-
|-
| | 51
| 51
| | 0-1193-1292
| 0-1193-1292
| | 1-9/5-18/11
| 1-9/5-18/11
| | utonal
| utonal
| | 26
| 26
|}
|}


=Tetlist=
== Tetrads ==


{| class="wikitable"
{| class="wikitable"
|-
|-
| | Number
! Number
| | Chord
! Chord
| | Transversal
! Transversal
| | Type
! Type
| | Complexity
! Complexity
|-
|-
| | 1
| 1
| | 0-100-199-299
| 0-100-199-299
| | 1-9/7-7/6-3/2
| 1-9/7-7/6-3/2
| | swetismic
| swetismic
| | 6
| 6
|-
|-
| | 2
| 2
| | 0-199-299-498
| 0-199-299-498
| | 1-7/6-3/2-7/4
| 1-7/6-3/2-7/4
| | ambitonal
| ambitonal
| | 10
| 10
|-
|-
| | 3
| 3
| | 0-100-299-596
| 0-100-299-596
| | 1-9/7-3/2-9/8
| 1-9/7-3/2-9/8
| | utonal
| utonal
| | 12
| 12
|-
|-
| | 4
| 4
| | 0-299-498-596
| 0-299-498-596
| | 1-3/2-7/4-9/8
| 1-3/2-7/4-9/8
| | otonal
| otonal
| | 12
| 12
|-
|-
| | 5
| 5
| | 0-100-596-696
| 0-100-596-696
| | 1-9/7-9/8-16/11
| 1-9/7-9/8-16/11
| | pentacircle
| pentacircle
| | 14
| 14
|-
|-
| | 6
| 6
| | 0-101-597-696
| 0-101-597-696
| | 1-20/11-8/5-16/11
| 1-20/11-8/5-16/11
| | valinorsmic
| valinorsmic
| | 14
| 14
|-
|-
| | 7
| 7
| | 0-101-696-993
| 0-101-696-993
| | 1-20/11-16/11-12/11
| 1-20/11-16/11-12/11
| | otonal
| otonal
| | 20
| 20
|-
|-
| | 8
| 8
| | 0-299-894-993
| 0-299-894-993
| | 1-3/2-6/5-12/11
| 1-3/2-6/5-12/11
| | utonal
| utonal
| | 20
| 20
|-
|-
| | 9
| 9
| | 0-100-199-1093
| 0-100-199-1093
| | 1-9/7-7/6-7/5
| 1-9/7-7/6-7/5
| | swetismic
| swetismic
| | 22
| 22
|-
|-
| | 10
| 10
| | 0-199-498-1093
| 0-199-498-1093
| | 1-7/6-7/4-7/5
| 1-7/6-7/4-7/5
| | utonal
| utonal
| | 22
| 22
|-
|-
| | 11
| 11
| | 0-498-597-1093
| 0-498-597-1093
| | 1-7/4-8/5-7/5
| 1-7/4-8/5-7/5
| | valinorsmic
| valinorsmic
| | 22
| 22
|-
|-
| | 12
| 12
| | 0-597-894-1093
| 0-597-894-1093
| | 1-8/5-6/5-7/5
| 1-8/5-6/5-7/5
| | otonal
| otonal
| | 22
| 22
|-
|-
| | 13
| 13
| | 0-894-993-1093
| 0-894-993-1093
| | 1-6/5-12/11-7/5
| 1-6/5-12/11-7/5
| | swetismic
| swetismic
| | 22
| 22
|-
|-
| | 14
| 14
| | 0-101-199-1192
| 0-101-199-1192
| | 1-20/11-7/6-14/11
| 1-20/11-7/6-14/11
| | swetismic
| swetismic
| | 24
| 24
|-
|-
| | 15
| 15
| | 0-199-498-1192
| 0-199-498-1192
| | 1-7/6-7/4-14/11
| 1-7/6-7/4-14/11
| | utonal
| utonal
| | 24
| 24
|-
|-
| | 16
| 16
| | 0-498-596-1192
| 0-498-596-1192
| | 1-7/4-9/8-14/11
| 1-7/4-9/8-14/11
| | pentacircle
| pentacircle
| | 24
| 24
|-
|-
| | 17
| 17
| | 0-101-597-1192
| 0-101-597-1192
| | 1-20/11-8/5-14/11
| 1-20/11-8/5-14/11
| | valinorsmic
| valinorsmic
| | 24
| 24
|-
|-
| | 18
| 18
| | 0-498-597-1192
| 0-498-597-1192
| | 1-7/4-8/5-14/11
| 1-7/4-8/5-14/11
| | valinorsmic
| valinorsmic
| | 24
| 24
|-
|-
| | 19
| 19
| | 0-101-696-1192
| 0-101-696-1192
| | 1-20/11-16/11-14/11
| 1-20/11-16/11-14/11
| | otonal
| otonal
| | 24
| 24
|-
|-
| | 20
| 20
| | 0-596-696-1192
| 0-596-696-1192
| | 1-9/8-16/11-14/11
| 1-9/8-16/11-14/11
| | pentacircle
| pentacircle
| | 24
| 24
|-
|-
| | 21
| 21
| | 0-597-696-1192
| 0-597-696-1192
| | 1-8/5-16/11-14/11
| 1-8/5-16/11-14/11
| | valinorsmic
| valinorsmic
| | 24
| 24
|-
|-
| | 22
| 22
| | 0-101-993-1192
| 0-101-993-1192
| | 1-20/11-12/11-14/11
| 1-20/11-12/11-14/11
| | otonal
| otonal
| | 24
| 24
|-
|-
| | 23
| 23
| | 0-696-993-1192
| 0-696-993-1192
| | 1-16/11-12/11-14/11
| 1-16/11-12/11-14/11
| | otonal
| otonal
| | 24
| 24
|-
|-
| | 24
| 24
| | 0-199-1093-1192
| 0-199-1093-1192
| | 1-7/6-7/5-14/11
| 1-7/6-7/5-14/11
| | utonal
| utonal
| | 24
| 24
|-
|-
| | 25
| 25
| | 0-498-1093-1192
| 0-498-1093-1192
| | 1-7/4-7/5-14/11
| 1-7/4-7/5-14/11
| | utonal
| utonal
| | 24
| 24
|-
|-
| | 26
| 26
| | 0-597-1093-1192
| 0-597-1093-1192
| | 1-8/5-7/5-14/11
| 1-8/5-7/5-14/11
| | valinorsmic
| valinorsmic
| | 24
| 24
|-
|-
| | 27
| 27
| | 0-993-1093-1192
| 0-993-1093-1192
| | 1-12/11-7/5-14/11
| 1-12/11-7/5-14/11
| | swetismic
| swetismic
| | 24
| 24
|-
|-
| | 28
| 28
| | 0-100-299-1193
| 0-100-299-1193
| | 1-9/7-3/2-9/5
| 1-9/7-3/2-9/5
| | utonal
| utonal
| | 24
| 24
|-
|-
| | 29
| 29
| | 0-100-596-1193
| 0-100-596-1193
| | 1-9/7-9/8-9/5
| 1-9/7-9/8-9/5
| | utonal
| utonal
| | 24
| 24
|-
|-
| | 30
| 30
| | 0-299-596-1193
| 0-299-596-1193
| | 1-3/2-9/8-9/5
| 1-3/2-9/8-9/5
| | utonal
| utonal
| | 24
| 24
|-
|-
| | 31
| 31
| | 0-299-894-1193
| 0-299-894-1193
| | 1-3/2-6/5-9/5
| 1-3/2-6/5-9/5
| | ambitonal
| ambitonal
| | 24
| 24
|-
|-
| | 32
| 32
| | 0-597-894-1193
| 0-597-894-1193
| | 1-8/5-6/5-9/5
| 1-8/5-6/5-9/5
| | otonal
| otonal
| | 24
| 24
|-
|-
| | 33
| 33
| | 0-100-1093-1193
| 0-100-1093-1193
| | 1-9/7-7/5-9/5
| 1-9/7-7/5-9/5
| | swetismic
| swetismic
| | 24
| 24
|-
|-
| | 34
| 34
| | 0-597-1093-1193
| 0-597-1093-1193
| | 1-8/5-7/5-9/5
| 1-8/5-7/5-9/5
| | otonal
| otonal
| | 24
| 24
|-
|-
| | 35
| 35
| | 0-894-1093-1193
| 0-894-1093-1193
| | 1-6/5-7/5-9/5
| 1-6/5-7/5-9/5
| | otonal
| otonal
| | 24
| 24
|-
|-
| | 36
| 36
| | 0-100-199-1292
| 0-100-199-1292
| | 1-9/7-7/6-18/11
| 1-9/7-7/6-18/11
| | swetismic
| swetismic
| | 26
| 26
|-
|-
| | 37
| 37
| | 0-101-199-1292
| 0-101-199-1292
| | 1-20/11-7/6-18/11
| 1-20/11-7/6-18/11
| | swetismic
| swetismic
| | 26
| 26
|-
|-
| | 38
| 38
| | 0-100-299-1292
| 0-100-299-1292
| | 1-9/7-3/2-18/11
| 1-9/7-3/2-18/11
| | utonal
| utonal
| | 26
| 26
|-
|-
| | 39
| 39
| | 0-199-299-1292
| 0-199-299-1292
| | 1-7/6-3/2-18/11
| 1-7/6-3/2-18/11
| | swetismic
| swetismic
| | 26
| 26
|-
|-
| | 40
| 40
| | 0-100-596-1292
| 0-100-596-1292
| | 1-9/7-9/8-18/11
| 1-9/7-9/8-18/11
| | utonal
| utonal
| | 26
| 26
|-
|-
| | 41
| 41
| | 0-299-596-1292
| 0-299-596-1292
| | 1-3/2-9/8-18/11
| 1-3/2-9/8-18/11
| | utonal
| utonal
| | 26
| 26
|-
|-
| | 42
| 42
| | 0-100-696-1292
| 0-100-696-1292
| | 1-9/7-16/11-18/11
| 1-9/7-16/11-18/11
| | pentacircle
| pentacircle
| | 26
| 26
|-
|-
| | 43
| 43
| | 0-101-696-1292
| 0-101-696-1292
| | 1-20/11-16/11-18/11
| 1-20/11-16/11-18/11
| | otonal
| otonal
| | 26
| 26
|-
|-
| | 44
| 44
| | 0-596-696-1292
| 0-596-696-1292
| | 1-9/8-16/11-18/11
| 1-9/8-16/11-18/11
| | pentacircle
| pentacircle
| | 26
| 26
|-
|-
| | 45
| 45
| | 0-101-993-1292
| 0-101-993-1292
| | 1-20/11-12/11-18/11
| 1-20/11-12/11-18/11
| | otonal
| otonal
| | 26
| 26
|-
|-
| | 46
| 46
| | 0-299-993-1292
| 0-299-993-1292
| | 1-3/2-12/11-18/11
| 1-3/2-12/11-18/11
| | ambitonal
| ambitonal
| | 26
| 26
|-
|-
| | 47
| 47
| | 0-696-993-1292
| 0-696-993-1292
| | 1-16/11-12/11-18/11
| 1-16/11-12/11-18/11
| | otonal
| otonal
| | 26
| 26
|-
|-
| | 48
| 48
| | 0-100-1093-1292
| 0-100-1093-1292
| | 1-9/7-7/5-18/11
| 1-9/7-7/5-18/11
| | swetismic
| swetismic
| | 26
| 26
|-
|-
| | 49
| 49
| | 0-199-1093-1292
| 0-199-1093-1292
| | 1-7/6-7/5-18/11
| 1-7/6-7/5-18/11
| | swetismic
| swetismic
| | 26
| 26
|-
|-
| | 50
| 50
| | 0-993-1093-1292
| 0-993-1093-1292
| | 1-12/11-7/5-18/11
| 1-12/11-7/5-18/11
| | swetismic
| swetismic
| | 26
| 26
|-
|-
| | 51
| 51
| | 0-101-1192-1292
| 0-101-1192-1292
| | 1-20/11-14/11-18/11
| 1-20/11-14/11-18/11
| | otonal
| otonal
| | 26
| 26
|-
|-
| | 52
| 52
| | 0-199-1192-1292
| 0-199-1192-1292
| | 1-7/6-14/11-18/11
| 1-7/6-14/11-18/11
| | swetismic
| swetismic
| | 26
| 26
|-
|-
| | 53
| 53
| | 0-596-1192-1292
| 0-596-1192-1292
| | 1-9/8-14/11-18/11
| 1-9/8-14/11-18/11
| | pentacircle
| pentacircle
| | 26
| 26
|-
|-
| | 54
| 54
| | 0-696-1192-1292
| 0-696-1192-1292
| | 1-16/11-14/11-18/11
| 1-16/11-14/11-18/11
| | otonal
| otonal
| | 26
| 26
|-
|-
| | 55
| 55
| | 0-993-1192-1292
| 0-993-1192-1292
| | 1-12/11-14/11-18/11
| 1-12/11-14/11-18/11
| | otonal
| otonal
| | 26
| 26
|-
|-
| | 56
| 56
| | 0-1093-1192-1292
| 0-1093-1192-1292
| | 1-7/5-14/11-18/11
| 1-7/5-14/11-18/11
| | swetismic
| swetismic
| | 26
| 26
|-
|-
| | 57
| 57
| | 0-100-1193-1292
| 0-100-1193-1292
| | 1-9/7-9/5-18/11
| 1-9/7-9/5-18/11
| | utonal
| utonal
| | 26
| 26
|-
|-
| | 58
| 58
| | 0-299-1193-1292
| 0-299-1193-1292
| | 1-3/2-9/5-18/11
| 1-3/2-9/5-18/11
| | utonal
| utonal
| | 26
| 26
|-
|-
| | 59
| 59
| | 0-596-1193-1292
| 0-596-1193-1292
| | 1-9/8-9/5-18/11
| 1-9/8-9/5-18/11
| | utonal
| utonal
| | 26
| 26
|-
|-
| | 60
| 60
| | 0-1093-1193-1292
| 0-1093-1193-1292
| | 1-7/5-9/5-18/11
| 1-7/5-9/5-18/11
| | swetismic
| swetismic
| | 26
| 26
|}
|}


=Pentlist=
== Pentads ==


{| class="wikitable"
{| class="wikitable"
|-
|-
| | Number
! Number
| | Chord
! Chord
| | Transversal
! Transversal
| | Type
! Type
| | Complexity
! Complexity
|-
|-
| | 1
| 1
| | 0-101-597-696-1192
| 0-101-597-696-1192
| | 1-20/11-8/5-16/11-14/11
| 1-20/11-8/5-16/11-14/11
| | valinorsmic
| valinorsmic
| | 24
| 24
|-
|-
| | 2
| 2
| | 0-101-696-993-1192
| 0-101-696-993-1192
| | 1-20/11-16/11-12/11-14/11
| 1-20/11-16/11-12/11-14/11
| | otonal
| otonal
| | 24
| 24
|-
|-
| | 3
| 3
| | 0-199-498-1093-1192
| 0-199-498-1093-1192
| | 1-7/6-7/4-7/5-14/11
| 1-7/6-7/4-7/5-14/11
| | utonal
| utonal
| | 24
| 24
|-
|-
| | 4
| 4
| | 0-498-597-1093-1192
| 0-498-597-1093-1192
| | 1-7/4-8/5-7/5-14/11
| 1-7/4-8/5-7/5-14/11
| | valinorsmic
| valinorsmic
| | 24
| 24
|-
|-
| | 5
| 5
| | 0-100-299-596-1193
| 0-100-299-596-1193
| | 1-9/7-3/2-9/8-9/5
| 1-9/7-3/2-9/8-9/5
| | utonal
| utonal
| | 24
| 24
|-
|-
| | 6
| 6
| | 0-597-894-1093-1193
| 0-597-894-1093-1193
| | 1-8/5-6/5-7/5-9/5
| 1-8/5-6/5-7/5-9/5
| | otonal
| otonal
| | 24
| 24
|-
|-
| | 7
| 7
| | 0-100-199-299-1292
| 0-100-199-299-1292
| | 1-9/7-7/6-3/2-18/11
| 1-9/7-7/6-3/2-18/11
| | swetismic
| swetismic
| | 26
| 26
|-
|-
| | 8
| 8
| | 0-100-299-596-1292
| 0-100-299-596-1292
| | 1-9/7-3/2-9/8-18/11
| 1-9/7-3/2-9/8-18/11
| | utonal
| utonal
| | 26
| 26
|-
|-
| | 9
| 9
| | 0-100-596-696-1292
| 0-100-596-696-1292
| | 1-9/7-9/8-16/11-18/11
| 1-9/7-9/8-16/11-18/11
| | pentacircle
| pentacircle
| | 26
| 26
|-
|-
| | 10
| 10
| | 0-101-696-993-1292
| 0-101-696-993-1292
| | 1-20/11-16/11-12/11-18/11
| 1-20/11-16/11-12/11-18/11
| | otonal
| otonal
| | 26
| 26
|-
|-
| | 11
| 11
| | 0-100-199-1093-1292
| 0-100-199-1093-1292
| | 1-9/7-7/6-7/5-18/11
| 1-9/7-7/6-7/5-18/11
| | swetismic
| swetismic
| | 26
| 26
|-
|-
| | 12
| 12
| | 0-101-199-1192-1292
| 0-101-199-1192-1292
| | 1-20/11-7/6-14/11-18/11
| 1-20/11-7/6-14/11-18/11
| | swetismic
| swetismic
| | 26
| 26
|-
|-
| | 13
| 13
| | 0-101-696-1192-1292
| 0-101-696-1192-1292
| | 1-20/11-16/11-14/11-18/11
| 1-20/11-16/11-14/11-18/11
| | otonal
| otonal
| | 26
| 26
|-
|-
| | 14
| 14
| | 0-596-696-1192-1292
| 0-596-696-1192-1292
| | 1-9/8-16/11-14/11-18/11
| 1-9/8-16/11-14/11-18/11
| | pentacircle
| pentacircle
| | 26
| 26
|-
|-
| | 15
| 15
| | 0-101-993-1192-1292
| 0-101-993-1192-1292
| | 1-20/11-12/11-14/11-18/11
| 1-20/11-12/11-14/11-18/11
| | otonal
| otonal
| | 26
| 26
|-
|-
| | 16
| 16
| | 0-696-993-1192-1292
| 0-696-993-1192-1292
| | 1-16/11-12/11-14/11-18/11
| 1-16/11-12/11-14/11-18/11
| | otonal
| otonal
| | 26
| 26
|-
|-
| | 17
| 17
| | 0-199-1093-1192-1292
| 0-199-1093-1192-1292
| | 1-7/6-7/5-14/11-18/11
| 1-7/6-7/5-14/11-18/11
| | swetismic
| swetismic
| | 26
| 26
|-
|-
| | 18
| 18
| | 0-993-1093-1192-1292
| 0-993-1093-1192-1292
| | 1-12/11-7/5-14/11-18/11
| 1-12/11-7/5-14/11-18/11
| | swetismic
| swetismic
| | 26
| 26
|-
|-
| | 19
| 19
| | 0-100-299-1193-1292
| 0-100-299-1193-1292
| | 1-9/7-3/2-9/5-18/11
| 1-9/7-3/2-9/5-18/11
| | utonal
| utonal
| | 26
| 26
|-
|-
| | 20
| 20
| | 0-100-596-1193-1292
| 0-100-596-1193-1292
| | 1-9/7-9/8-9/5-18/11
| 1-9/7-9/8-9/5-18/11
| | utonal
| utonal
| | 26
| 26
|-
|-
| | 21
| 21
| | 0-299-596-1193-1292
| 0-299-596-1193-1292
| | 1-3/2-9/8-9/5-18/11
| 1-3/2-9/8-9/5-18/11
| | utonal
| utonal
| | 26
| 26
|-
|-
| | 22
| 22
| | 0-100-1093-1193-1292
| 0-100-1093-1193-1292
| | 1-9/7-7/5-9/5-18/11
| 1-9/7-7/5-9/5-18/11
| | swetismic
| swetismic
| | 26
| 26
|}
|}


=Hexad=
== Hexads ==


{| class="wikitable"
{| class="wikitable"
|-
|-
| | Number
! Number
| | Chord
! Chord
| | Transversal
! Transversal
| | Type
! Type
| | Complexity
! Complexity
|-
|-
| | 1
| 1
| | 0-101-696-993-1192-1292
| 0-101-696-993-1192-1292
| | 1-20/11-16/11-12/11-14/11-18/11
| 1-20/11-16/11-12/11-14/11-18/11
| | otonal
| otonal
| | 26
| 26
|-
|-
| | 2
| 2
| | 0-100-299-596-1193-1292
| 0-100-299-596-1193-1292
| | 1-9/7-3/2-9/8-9/5-18/11
| 1-9/7-3/2-9/8-9/5-18/11
| | utonal
| utonal
| | 26
| 26
|}
|}
[[Category:Lists of chords]]
[[Category:Lists of chords]]
[[Category:Dyadic chords]]
[[Category:Dyadic chords]]
[[Category:11-limit]]
[[Category:11-limit]]
[[Category:Echidna]]
[[Category:Echidna]]

Latest revision as of 08:42, 15 September 2026

Below are listed the dyadic chords of the 11-limit echidna temperament. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering by 540/539 are labeled swetismic, by 896/891 pentacircle, and by 176/175 valinorsmic.

The normal mapping for echidna is ech = [⟨2 1 9 2 12], ⟨0 3 -6 5 -7]]. From this we may derive a val v = ech[1] + 100 ech[2] = ⟨2 301 -591 502 -688] which we may use to sort and normalize the chords of echidna. Under "Chord" is listed the chord, normalized to start from zero, in the mapping by v, and the next column gives a transversal which tempers to the chord by echidnic tempering. Finally, the Graham complexity is listed in the last column.

Echidna has mos of size 6, 8, 14, 22, 36, 58 and 80. It may be seen that a modest number of chords are available to 14 notes, but that 22 has many more, and that may be a good choice as a scale with which to explore the harmonies of echidna. If for some reason you needed pentads and hexads, 36 notes would be fine.

Triads

Number Chord Transversal Type Complexity
1 0-100-199 1-9/7-7/6 swetismic 4
2 0-101-199 1-20/11-7/6 swetismic 4
3 0-100-299 1-9/7-3/2 utonal 6
4 0-199-299 1-7/6-3/2 otonal 6
5 0-199-498 1-7/6-7/4 utonal 10
6 0-299-498 1-3/2-7/4 otonal 10
7 0-100-596 1-9/7-9/8 utonal 12
8 0-299-596 1-3/2-9/8 ambitonal 12
9 0-498-596 1-7/4-9/8 otonal 12
10 0-101-597 1-20/11-8/5 valinorsmic 12
11 0-498-597 1-7/4-8/5 valinorsmic 12
12 0-100-696 1-9/7-16/11 pentacircle 14
13 0-101-696 1-20/11-16/11 otonal 14
14 0-596-696 1-9/8-16/11 pentacircle 14
15 0-597-696 1-8/5-16/11 utonal 14
16 0-299-894 1-3/2-6/5 utonal 18
17 0-597-894 1-8/5-6/5 otonal 18
18 0-101-993 1-20/11-12/11 otonal 20
19 0-299-993 1-3/2-12/11 utonal 20
20 0-696-993 1-16/11-12/11 otonal 20
21 0-894-993 1-6/5-12/11 utonal 20
22 0-100-1093 1-9/7-7/5 swetismic 22
23 0-199-1093 1-7/6-7/5 utonal 22
24 0-498-1093 1-7/4-7/5 utonal 22
25 0-597-1093 1-8/5-7/5 otonal 22
26 0-894-1093 1-6/5-7/5 otonal 22
27 0-993-1093 1-12/11-7/5 swetismic 22
28 0-101-1192 1-20/11-14/11 otonal 24
29 0-199-1192 1-7/6-14/11 utonal 24
30 0-498-1192 1-7/4-14/11 utonal 24
31 0-596-1192 1-9/8-14/11 pentacircle 24
32 0-597-1192 1-8/5-14/11 valinorsmic 24
33 0-696-1192 1-16/11-14/11 otonal 24
34 0-993-1192 1-12/11-14/11 otonal 24
35 0-1093-1192 1-7/5-14/11 utonal 24
36 0-100-1193 1-9/7-9/5 utonal 24
37 0-299-1193 1-3/2-9/5 utonal 24
38 0-596-1193 1-9/8-9/5 utonal 24
39 0-597-1193 1-8/5-9/5 otonal 24
40 0-894-1193 1-6/5-9/5 otonal 24
41 0-1093-1193 1-7/5-9/5 otonal 24
42 0-100-1292 1-9/7-18/11 utonal 26
43 0-101-1292 1-20/11-18/11 otonal 26
44 0-199-1292 1-7/6-18/11 swetismic 26
45 0-299-1292 1-3/2-18/11 utonal 26
46 0-596-1292 1-9/8-18/11 utonal 26
47 0-696-1292 1-16/11-18/11 otonal 26
48 0-993-1292 1-12/11-18/11 otonal 26
49 0-1093-1292 1-7/5-18/11 swetismic 26
50 0-1192-1292 1-14/11-18/11 otonal 26
51 0-1193-1292 1-9/5-18/11 utonal 26

Tetrads

Number Chord Transversal Type Complexity
1 0-100-199-299 1-9/7-7/6-3/2 swetismic 6
2 0-199-299-498 1-7/6-3/2-7/4 ambitonal 10
3 0-100-299-596 1-9/7-3/2-9/8 utonal 12
4 0-299-498-596 1-3/2-7/4-9/8 otonal 12
5 0-100-596-696 1-9/7-9/8-16/11 pentacircle 14
6 0-101-597-696 1-20/11-8/5-16/11 valinorsmic 14
7 0-101-696-993 1-20/11-16/11-12/11 otonal 20
8 0-299-894-993 1-3/2-6/5-12/11 utonal 20
9 0-100-199-1093 1-9/7-7/6-7/5 swetismic 22
10 0-199-498-1093 1-7/6-7/4-7/5 utonal 22
11 0-498-597-1093 1-7/4-8/5-7/5 valinorsmic 22
12 0-597-894-1093 1-8/5-6/5-7/5 otonal 22
13 0-894-993-1093 1-6/5-12/11-7/5 swetismic 22
14 0-101-199-1192 1-20/11-7/6-14/11 swetismic 24
15 0-199-498-1192 1-7/6-7/4-14/11 utonal 24
16 0-498-596-1192 1-7/4-9/8-14/11 pentacircle 24
17 0-101-597-1192 1-20/11-8/5-14/11 valinorsmic 24
18 0-498-597-1192 1-7/4-8/5-14/11 valinorsmic 24
19 0-101-696-1192 1-20/11-16/11-14/11 otonal 24
20 0-596-696-1192 1-9/8-16/11-14/11 pentacircle 24
21 0-597-696-1192 1-8/5-16/11-14/11 valinorsmic 24
22 0-101-993-1192 1-20/11-12/11-14/11 otonal 24
23 0-696-993-1192 1-16/11-12/11-14/11 otonal 24
24 0-199-1093-1192 1-7/6-7/5-14/11 utonal 24
25 0-498-1093-1192 1-7/4-7/5-14/11 utonal 24
26 0-597-1093-1192 1-8/5-7/5-14/11 valinorsmic 24
27 0-993-1093-1192 1-12/11-7/5-14/11 swetismic 24
28 0-100-299-1193 1-9/7-3/2-9/5 utonal 24
29 0-100-596-1193 1-9/7-9/8-9/5 utonal 24
30 0-299-596-1193 1-3/2-9/8-9/5 utonal 24
31 0-299-894-1193 1-3/2-6/5-9/5 ambitonal 24
32 0-597-894-1193 1-8/5-6/5-9/5 otonal 24
33 0-100-1093-1193 1-9/7-7/5-9/5 swetismic 24
34 0-597-1093-1193 1-8/5-7/5-9/5 otonal 24
35 0-894-1093-1193 1-6/5-7/5-9/5 otonal 24
36 0-100-199-1292 1-9/7-7/6-18/11 swetismic 26
37 0-101-199-1292 1-20/11-7/6-18/11 swetismic 26
38 0-100-299-1292 1-9/7-3/2-18/11 utonal 26
39 0-199-299-1292 1-7/6-3/2-18/11 swetismic 26
40 0-100-596-1292 1-9/7-9/8-18/11 utonal 26
41 0-299-596-1292 1-3/2-9/8-18/11 utonal 26
42 0-100-696-1292 1-9/7-16/11-18/11 pentacircle 26
43 0-101-696-1292 1-20/11-16/11-18/11 otonal 26
44 0-596-696-1292 1-9/8-16/11-18/11 pentacircle 26
45 0-101-993-1292 1-20/11-12/11-18/11 otonal 26
46 0-299-993-1292 1-3/2-12/11-18/11 ambitonal 26
47 0-696-993-1292 1-16/11-12/11-18/11 otonal 26
48 0-100-1093-1292 1-9/7-7/5-18/11 swetismic 26
49 0-199-1093-1292 1-7/6-7/5-18/11 swetismic 26
50 0-993-1093-1292 1-12/11-7/5-18/11 swetismic 26
51 0-101-1192-1292 1-20/11-14/11-18/11 otonal 26
52 0-199-1192-1292 1-7/6-14/11-18/11 swetismic 26
53 0-596-1192-1292 1-9/8-14/11-18/11 pentacircle 26
54 0-696-1192-1292 1-16/11-14/11-18/11 otonal 26
55 0-993-1192-1292 1-12/11-14/11-18/11 otonal 26
56 0-1093-1192-1292 1-7/5-14/11-18/11 swetismic 26
57 0-100-1193-1292 1-9/7-9/5-18/11 utonal 26
58 0-299-1193-1292 1-3/2-9/5-18/11 utonal 26
59 0-596-1193-1292 1-9/8-9/5-18/11 utonal 26
60 0-1093-1193-1292 1-7/5-9/5-18/11 swetismic 26

Pentads

Number Chord Transversal Type Complexity
1 0-101-597-696-1192 1-20/11-8/5-16/11-14/11 valinorsmic 24
2 0-101-696-993-1192 1-20/11-16/11-12/11-14/11 otonal 24
3 0-199-498-1093-1192 1-7/6-7/4-7/5-14/11 utonal 24
4 0-498-597-1093-1192 1-7/4-8/5-7/5-14/11 valinorsmic 24
5 0-100-299-596-1193 1-9/7-3/2-9/8-9/5 utonal 24
6 0-597-894-1093-1193 1-8/5-6/5-7/5-9/5 otonal 24
7 0-100-199-299-1292 1-9/7-7/6-3/2-18/11 swetismic 26
8 0-100-299-596-1292 1-9/7-3/2-9/8-18/11 utonal 26
9 0-100-596-696-1292 1-9/7-9/8-16/11-18/11 pentacircle 26
10 0-101-696-993-1292 1-20/11-16/11-12/11-18/11 otonal 26
11 0-100-199-1093-1292 1-9/7-7/6-7/5-18/11 swetismic 26
12 0-101-199-1192-1292 1-20/11-7/6-14/11-18/11 swetismic 26
13 0-101-696-1192-1292 1-20/11-16/11-14/11-18/11 otonal 26
14 0-596-696-1192-1292 1-9/8-16/11-14/11-18/11 pentacircle 26
15 0-101-993-1192-1292 1-20/11-12/11-14/11-18/11 otonal 26
16 0-696-993-1192-1292 1-16/11-12/11-14/11-18/11 otonal 26
17 0-199-1093-1192-1292 1-7/6-7/5-14/11-18/11 swetismic 26
18 0-993-1093-1192-1292 1-12/11-7/5-14/11-18/11 swetismic 26
19 0-100-299-1193-1292 1-9/7-3/2-9/5-18/11 utonal 26
20 0-100-596-1193-1292 1-9/7-9/8-9/5-18/11 utonal 26
21 0-299-596-1193-1292 1-3/2-9/8-9/5-18/11 utonal 26
22 0-100-1093-1193-1292 1-9/7-7/5-9/5-18/11 swetismic 26

Hexads

Number Chord Transversal Type Complexity
1 0-101-696-993-1192-1292 1-20/11-16/11-12/11-14/11-18/11 otonal 26
2 0-100-299-596-1193-1292 1-9/7-3/2-9/8-9/5-18/11 utonal 26
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