Chords of superpyth: Difference between revisions

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{{Editable user page| Please complete the description and tables. This page is planned to be moved to main space once completed.}}
{{Breadcrumb|Superpyth}}
Below are listed the [[11-odd-limit]] [[dyadic chord]]s of [[11-limit]] [[superpyth|superpyth temperament]]. Typing the chords requires consideration of the fact that superpyth conflates [[9/8]] with [[8/7]], and [[11/10]] with [[10/9]]. If a transversal can be found which shows the chord to be essentially just, that transversal is listed along with a typing as otonal, utonal, or ambitonal. If the chord is essentially tempered, it is analyzed in terms of the transversal that employs 8/7 and 10/9.
Below is a complete list of all [[11-odd-limit]] [[dyadic chord]]s in [[11-limit]] [[superpyth|superpyth temperament]]. Note that there are many common chords, for example [[8:10:12:15]], which are not listed; in this case due to [[15/8]] not being in the 11-odd-limit. Every chord listed has multiple [[chord #Inversion|inversions]]; only one is listed, that being the inversion where all notes are a nonnegative number of perfect fifth [[generator]]s above the root.


Chords essentially tempered by [[64/63]] are labeled [[archytas chords|archytas]], by [[100/99]] [[ptolemismic chords|ptolemismic]], by [[176/175]] [[valinorsmic chords|valinorsmic]], by [[245/243]] [[sensamagic chords|sensamagic]], by [[540/539]] [[swetismic chords|swetismic]]. Chords that require any two of 64/63, 100/99 and 176/175 tempering are marked [[ares chords|ares]]. Chords that require any two of 100/99, 245/243 and 540/539 tempering are marked [[octarod chords|octarod]]. Chords that require 176/175 and 540/539 tempering are marked [[guanyin chords|guanyin]].  
If a chord is [[dyadic chord #Essentially tempered dyadic chord|essentially just]], it is classified as [[otonal]] if it is best analyzed in terms of the [[harmonic series]], [[utonal]] if best analyzed in terms of the [[subharmonic series]], and [[ambitonal]] if equally well analyzed by either. If a chord is [[dyadic chord #Essentially tempered dyadic chord|essentially tempered]], it is classified based on which [[comma]]s are needed to define the chord. Chords essentially tempered by [[64/63]] are labeled [[archytas chords|archytas]], by [[100/99]] [[ptolemismic chords|ptolemismic]], by [[176/175]] [[valinorsmic chords|valinorsmic]], by [[245/243]] [[sensamagic chords|sensamagic]], and by [[540/539]] [[swetismic chords|swetismic]]. Chords that require any two of 64/63, 100/99 and 176/175 to vanish are labeled [[ares chords|ares]]. Finally, chords that require any two of 100/99, 245/243 and 540/539 to vanish are labeled [[octarod chords|octarod]].  
 
Typing the chords requires consideration of the fact that superpyth conflates [[9/8]] with [[8/7]], and [[11/10]] with [[10/9]]. If a transversal can be found which shows the chord to be essentially just, that transversal is listed along with a typing as otonal, utonal, or ambitonal. However, sometimes multiple such transversals exist, in which case the chord is a [[plurichord]], and the type is given for all possible interpretations. If the chord is essentially tempered, it is analyzed in terms of the transversal that requires the minimum amount of commas to be tempered out; if there is a tie between multiple transversals, it is analyzed in terms of the transversal which employs 8/7 or 10/9 above the root.
 
Superpyth is generated by a sharp [[~]][[3/2]] between [[22edo|13\22]] (709.{{Overline|09}}[[{{c}}]]) and [[27edo|16\27]] (711.{{Overline|11}}{{c}}), and generates [[mos]] scales of the patterns [[2L 3s]] (pentic), [[5L 2s]] (diatonic), [[5L 7s]] (p-chromatic), [[5L 12s]], [[5L 17s]], and [[22L 5s]]. The pentic and diatonic scales contain some chords in the [[2.3.7 subgroup|2.3.7]] [[subgroup]], though the 12-note chromatic scale is needed to properly utilize intervals of [[5/1|5]], and intervals of [[11/1|11]] don't become common until the 17- and 22-note scales. Superpyth has hardly been explored in the 11-limit, and full 7-limit superpyth has not been explored much either, so these mos scales would be a great place to start such explorations.


== Triads ==
== Triads ==
{| class="wikitable"
{| class="wikitable sortable center-1"
|-
|-
! #
! #
! Generators
! class="unsortable" | Generators
! Transversal
! class="unsortable" | Transversal
! Type
! Type
! class="unsortable" | Comments
|-
|-
| 1
| 1
Line 16: Line 21:
| 1–9/8–3/2
| 1–9/8–3/2
| Ambitonal
| Ambitonal
| [[6:8:9]], [[8:9:12]]
|-
|-
| 2
| 2
Line 21: Line 27:
| 1–3/2–12/7
| 1–3/2–12/7
| Utonal
| Utonal
| [[14:21:24|1/(12:8:7)]]
|-
|-
| 3
| 3
Line 26: Line 33:
| 1–8/7–12/7
| 1–8/7–12/7
| Otonal
| Otonal
| [[4:6:7]]
|-
|-
| 4
| 4
Line 31: Line 39:
| 1–9/7–3/2
| 1–9/7–3/2
| Utonal
| Utonal
| [[14:18:21|1/(9:7:6)]]
|-
|-
| 5
| 5
Line 36: Line 45:
| 1–8/7–9/7
| 1–8/7–9/7
| Otonal/utonal
| Otonal/utonal
| 7:8:9~1/(9:8:7)
|-
|-
| 6
| 6
Line 41: Line 51:
| 1–9/7–12/7
| 1–9/7–12/7
| Otonal
| Otonal
| [[6:7:9]]
|-
|-
| 7
| 7
Line 46: Line 57:
| 1–10/9–12/7
| 1–10/9–12/7
| Sensamagic
| Sensamagic
|
|-
|-
| 8
| 8
Line 51: Line 63:
| 1–10/9–9/7
| 1–10/9–9/7
| Sensamagic
| Sensamagic
|
|-
|-
| 9
| 9
Line 56: Line 69:
| 1–3/2–5/3
| 1–3/2–5/3
| Otonal
| Otonal
| [[6:9:10]]
|-
|-
| 10
| 10
Line 61: Line 75:
| 1–9/7–5/3
| 1–9/7–5/3
| Sensamagic
| Sensamagic
|
|-
|-
| 11
| 11
Line 66: Line 81:
| 1–10/9–5/3
| 1–10/9–5/3
| Utonal
| Utonal
| [[10:15:18|1/(9:6:5)]]
|-
|-
| 12
| 12
Line 71: Line 87:
| 1–5/4–3/2
| 1–5/4–3/2
| Otonal
| Otonal
| [[4:5:6]]
|-
|-
| 13
| 13
Line 76: Line 93:
| 1–9/8–5/4
| 1–9/8–5/4
| Otonal
| Otonal
|
|-
|-
| 14
| 14
Line 81: Line 99:
| 1–10/9–5/4
| 1–10/9–5/4
| Utonal
| Utonal
|
|-
|-
| 15
| 15
Line 86: Line 105:
| 1–5/4–5/3
| 1–5/4–5/3
| Utonal
| Utonal
| [[10:12:15|1/(6:5:4)]]
|-
|-
| 16
| 16
Line 91: Line 111:
| 1–8/7–10/7
| 1–8/7–10/7
| Otonal
| Otonal
| [[4:5:7]]
|-
|-
| 17
| 17
Line 96: Line 117:
| 1–10/7–12/7
| 1–10/7–12/7
| Otonal
| Otonal
| [[5:6:7]]
|-
|-
| 18
| 18
Line 101: Line 123:
| 1–9/7–10/7
| 1–9/7–10/7
| Otonal
| Otonal
|
|-
|-
| 19
| 19
Line 106: Line 129:
| 1–10/9–10/7
| 1–10/9–10/7
| Utonal
| Utonal
|
|-
|-
| 20
| 20
Line 111: Line 135:
| 1–10/7–5/3
| 1–10/7–5/3
| Utonal
| Utonal
| [[30:35:42|1/(7:6:5)]]
|-
|-
| 21
| 21
Line 116: Line 141:
| 1–5/4–10/7
| 1–5/4–10/7
| Utonal
| Utonal
| [[28:35:40|1/(10:8:7)]]
|-
|-
| 22
| 22
Line 121: Line 147:
| 1–11/9–12/7
| 1–11/9–12/7
| Swetismic
| Swetismic
|
|-
|-
| 23
| 23
Line 126: Line 153:
| 1–10/9–11/9
| 1–10/9–11/9
| Otonal/utonal
| Otonal/utonal
| 9:10:11~1/(11:10:9)
|-
|-
| 24
| 24
Line 131: Line 159:
| 1–11/9–10/7
| 1–11/9–10/7
| Swetismic
| Swetismic
|
|-
|-
| 25
| 25
Line 136: Line 165:
| 1–3/2–11/6
| 1–3/2–11/6
| Otonal
| Otonal
|
|-
|-
| 26
| 26
Line 141: Line 171:
| 1–9/7–11/6
| 1–9/7–11/6
| Swetismic
| Swetismic
|
|-
|-
| 27
| 27
Line 146: Line 177:
| 1–11/10–11/6
| 1–11/10–11/6
| Utonal
| Utonal
|
|-
|-
| 28
| 28
Line 151: Line 183:
| 1–5/3–11/6
| 1–5/3–11/6
| Otonal
| Otonal
|
|-
|-
| 29
| 29
Line 156: Line 189:
| 1–10/7–11/6
| 1–10/7–11/6
| Swetismic
| Swetismic
|
|-
|-
| 30
| 30
Line 161: Line 195:
| 1–11/9–11/6
| 1–11/9–11/6
| Utonal
| Utonal
|
|-
|-
| 31
| 31
Line 166: Line 201:
| 1–11/8–3/2
| 1–11/8–3/2
| Otonal
| Otonal
|
|-
|-
| 32
| 32
Line 171: Line 207:
| 1–9/8–11/8
| 1–9/8–11/8
| Otonal
| Otonal
|
|-
|-
| 33
| 33
Line 176: Line 213:
| 1–11/10–11/8
| 1–11/10–11/8
| Utonal
| Utonal
|
|-
|-
| 34
| 34
Line 181: Line 219:
| 1–11/8–5/3
| 1–11/8–5/3
| Ptolemismic
| Ptolemismic
|
|-
|-
| 35
| 35
Line 186: Line 225:
| 1–5/4–11/8
| 1–5/4–11/8
| Otonal
| Otonal
|
|-
|-
| 36
| 36
Line 191: Line 231:
| 1–11/9–11/8
| 1–11/9–11/8
| Utonal
| Utonal
|
|-
|-
| 37
| 37
Line 196: Line 237:
| 1–11/8–11/6
| 1–11/8–11/6
| Utonal
| Utonal
|
|-
|-
| 38
| 38
Line 201: Line 243:
| 1–8/7–11/7
| 1–8/7–11/7
| Otonal
| Otonal
|
|-
|-
| 39
| 39
Line 206: Line 249:
| 1–11/7–12/7
| 1–11/7–12/7
| Otonal
| Otonal
|
|-
|-
| 40
| 40
Line 211: Line 255:
| 1–9/7–11/7
| 1–9/7–11/7
| Otonal
| Otonal
|
|-
|-
| 41
| 41
Line 216: Line 261:
| 1–11/10–11/7
| 1–11/10–11/7
| Utonal
| Utonal
|
|-
|-
| 42
| 42
Line 221: Line 267:
| 1–5/4–11/7
| 1–5/4–11/7
| Valinorsmic
| Valinorsmic
|
|-
|-
| 43
| 43
Line 226: Line 273:
| 1–10/7–11/7
| 1–10/7–11/7
| Otonal
| Otonal
|
|-
|-
| 44
| 44
Line 231: Line 279:
| 1–11/9–11/7
| 1–11/9–11/7
| Utonal
| Utonal
|
|-
|-
| 45
| 45
Line 236: Line 285:
| 1–11/7–11/6
| 1–11/7–11/6
| Utonal
| Utonal
|
|-
|-
| 46
| 46
Line 241: Line 291:
| 1–11/8–11/7
| 1–11/8–11/7
| Utonal
| Utonal
|
|}
|}


== Tetrads ==
== Tetrads ==
{| class="wikitable"
{| class="wikitable sortable center-1"
|-
|-
! #
! #
! Generators
! class="unsortable" | Generators
! Transversal
! class="unsortable" | Transversal
! Type
! Type
! class="unsortable" | Comments
|-
|-
| 1
| 1
|  
| 0–1–2–3
| 1–3/2–8/7–12/7
| 1–8/7–3/2–12/7
| Archytas
|  
|  
|-
|-
| 2
| 2
|  
| 0–1–2–4
| 1–3/2–8/7–9/7
| 1–9/8–9/7–3/2
|  
| Utonal
| [[28:36:42:63|1/(9:7:6:4)]]
|-
|-
| 3
| 3
|  
| 0–1–3–4
| 1–3/2–12/7–9/7
| 1–9/7–3/2–12/7
|  
| Ambitonal
| [[12:14:18:21]], [[14:18:21:24]]<br>[[9-odd-limit]] [[ASS]]
|-
|-
| 4
| 4
|  
| 0–2–3–4
| 1–8/7–12/7–9/7
| 1–8/7–9/7–12/7
|  
| Otonal
| [[4:6:7:9]]
|-
|-
| 5
| 5
|  
| 0–3–4–7
| 1–12/7–9/7–10/9
| 1–10/9–9/7–12/7
| Sensamagic
|  
|  
|-
|-
| 6
| 6
|  
| 0–1–4–8
| 1–3/2–9/7–5/3
| 1–9/7–3/2–5/3
| Sensamagic
|  
|  
|-
|-
| 7
| 7
|  
| 0–4–7–8
| 1–9/7–10/9–5/3
| 1–9/7–10/9–5/3
| Sensamagic
|  
|  
|-
|-
| 8
| 8
|  
| 0–1–2–9
| 1–3/2–8/7–5/4
| 1–9/8–5/4–3/2
|  
| Otonal
| [[4:5:6:9]]
|-
|-
| 9
| 9
|  
| 0–1–8–9
| 1–3/2–5/3–5/4
| 1–5/4–3/2–5/3
|  
| Ambitonal
| [[10:12:15:18]], [[12:15:18:20]]<br>9-odd-limit ASS
|-
|-
| 10
| 10
|  
| 0–7–8–9
| 1–10/9–5/3–5/4
| 1–10/9–5/4–5/3
|  
| Utonal
| [[20:30:36:45|1/(9:6:5:4)]]
|-
|-
| 11
| 11
|  
| 0–2–3–11
| 1–8/7–12/7–10/7
| 1–8/7–10/7–12/7
|  
| Otonal
| [[4:5:6:7]]
|-
|-
| 12
| 12
|  
| 0–2–4–11
| 1–8/7–9/7–10/7
| 1–8/7–9/7–10/7
|  
| Otonal
| [[4:5:7:9]]
|-
|-
| 13
| 13
|  
| 0–3–4–11
| 1–12/7–9/7–10/7
| 1–9/7–10/7–12/7
|  
| Otonal
| [[6:7:9:10]]
|-
|-
| 14
| 14
|  
| 0–3–7–11
| 1–12/7–10/9–10/7
| 1–10/9–10/7–12/7
| Sensamagic
|  
|  
|-
|-
| 15
| 15
|  
| 0–4–7–11
| 1–9/7–10/9–10/7
| 1–10/9–9/7–10/7
| Sensamagic
|  
|  
|-
|-
| 16
| 16
|  
| 0–4–8–11
| 1–9/7–5/3–10/7
| 1–9/7–10/7–5/3
| Sensamagic
|  
|  
|-
|-
| 17
| 17
|  
| 0–7–8–11
| 1–10/9–5/3–10/7
| 1–10/9–10/7–5/3
|  
| Utonal
| [[70:90:105:126|1/(9:7:6:5)]]
|-
|-
| 18
| 18
|  
| 0–2–9–11
| 1–8/7–5/4–10/7
| 1–8/7–5/4–10/7
| Archytas/valinorsmic
|  
|  
|-
|-
| 19
| 19
|  
| 0–7–9–11
| 1–10/9–5/4–10/7
| 1–10/9–5/4–10/7
|  
| Utonal
| [[140:180:252:315|1/(9:7:5:4)]]
|-
|-
| 20
| 20
|  
| 0–8–9–11
| 1–5/3–5/4–10/7
| 1–5/4–10/7–5/3
|  
| Utonal
| [[70:84:105:120|1/(12:10:8:7)]]
|-
|-
| 21
| 21
|  
| 0–3–7–14
| 1–12/7–10/9–11/9
| 1–11/10–11/9–12/7
| Swetismic
|  
|  
|-
|-
| 22
| 22
|  
| 0–3–11–14
| 1–12/7–10/7–11/9
| 1–11/9–10/7–12/7
| Swetismic
|  
|  
|-
|-
| 23
| 23
|  
| 0–7–11–14
| 1–10/9–10/7–11/9
| 1–11/10–11/9–10/7
| Swetismic
|  
|  
|-
|-
| 24
| 24
|  
| 0–1–4–15
| 1–3/2–9/7–11/6
| 1–9/7–3/2–11/6
| Swetismic
|  
|  
|-
|-
| 25
| 25
|  
| 0–4–7–15
| 1–9/7–10/9–11/6
| 1–10/9–9/7–11/6
| Octarod
|  
|  
|-
|-
| 26
| 26
|  
| 0–1–8–15
| 1–3/2–5/3–11/6
| 1–3/2–5/3–11/6
| Otonal
|  
|  
|-
|-
| 27
| 27
|  
| 0–4–8–15
| 1–9/7–5/3–11/6
| 1–9/7–5/3–11/6
| Octarod
|  
|  
|-
|-
| 28
| 28
|  
| 0–7–8–15
| 1–10/9–5/3–11/6
| 1–10/9–5/3–11/6
| Ptolemismic
|  
|  
|-
|-
| 29
| 29
|  
| 0–4–11–15
| 1–9/7–10/7–11/6
| 1–9/7–10/7–11/6
| Swetismic
|  
|  
|-
|-
| 30
| 30
|  
| 0–7–11–15
| 1–10/9–10/7–11/6
| 1–10/9–10/7–11/6
| Octarod
|  
|  
|-
|-
| 31
| 31
|  
| 0–8–11–15
| 1–5/3–10/7–11/6
| 1–10/7–5/3–11/6
| Octarod
|  
|  
|-
|-
| 32
| 32
|  
| 0–7–14–15
| 1–10/9–11/9–11/6
| 1–11/10–11/9–11/6
| Utonal
|  
|  
|-
|-
| 33
| 33
|  
| 0–11–14–15
| 1–10/7–11/9–11/6
| 1–11/9–10/7–11/6
| Swetismic
|  
|  
|-
|-
| 34
| 34
|  
| 0–1–2–16
| 1–3/2–8/7–11/8
| 1–9/8–11/8–3/2
| Otonal
|  
|  
|-
|-
| 35
| 35
|  
| 0–1–8–16
| 1–3/2–5/3–11/8
| 1–11/8–3/2–5/3
| Ptolemismic
|  
|  
|-
|-
| 36
| 36
|  
| 0–7–8–16
| 1–10/9–5/3–11/8
| 1–10/9–5/3–11/8
| Ptolemismic
|  
|  
|-
|-
| 37
| 37
|  
| 0–1–9–16
| 1–3/2–5/4–11/8
| 1–5/4–11/8–3/2
| Otonal
|  
|  
|-
|-
| 38
| 38
|  
| 0–2–9–16
| 1–8/7–5/4–11/8
| 1–9/8–5/4–11/8
| Otonal
|  
|  
|-
|-
| 39
| 39
|  
| 0–7–9–16
| 1–10/9–5/4–11/8
| 1–10/9–5/4–11/8
| Ptolemismic/valinorsmic
|  
|  
|-
|-
| 40
| 40
|  
| 0–8–9–16
| 1–5/3–5/4–11/8
| 1–5/4–11/8–5/3
| Ptolemismic
|  
|  
|-
|-
| 41
| 41
|  
| 0–9–14–16
| 1–10/9–11/9–11/8
| 1–11/10–11/9–11/8
| Utonal
|  
|  
|-
|-
| 42
| 42
|  
| 0–1–15–16
| 1–3/2–11/6–11/8
| 1–11/8–3/2–11/6
|  
| Ambitonal
| 11-odd-limit ASS
|-
|-
| 43
| 43
|  
| 0–7–15–16
| 1–10/9–11/6–11/8
| 1–11/10–11/8–11/6
| Utonal
|  
|  
|-
|-
| 44
| 44
|  
| 0–8–15–16
| 1–5/3–11/6–11/8
| 1–11/8–5/3–11/6
| Ptolemismic
|  
|  
|-
|-
| 45
| 45
|  
| 0–14–15–16
| 1–11/9–11/6–11/8
| 1–11/9–11/8–11/6
| Utonal
|  
|  
|-
|-
| 46
| 46
|  
| 0–2–3–18
| 1–8/7–12/7–11/7
| 1–8/7–11/7–12/7
| Otonal
|  
|  
|-
|-
| 47
| 47
|  
| 0–2–4–18
| 1–8/7–9/7–11/7
| 1–8/7–9/7–11/7
| Otonal
|  
|  
|-
|-
| 48
| 48
|  
| 0–3–4–18
| 1–12/7–9/7–11/7
| 1–9/7–11/7–12/7
| Otonal
|  
|  
|-
|-
| 49
| 49
|  
| 0–3–7–18
| 1–12/7–10/9–11/7
| 1–10/9–11/7–12/7
| Octarod
|  
|  
|-
|-
| 50
| 50
|  
| 0–4–7–18
| 1–9/7–10/9–11/7
| 1–11/10–9/7–11/7
| Swetismic
|  
|  
|-
|-
| 51
| 51
|  
| 0–2–9–18
| 1–8/7–5/4–11/7
| 1–8/7–5/4–11/7
| Valinorsmic
|  
|  
|-
|-
| 52
| 52
|  
| 0–7–9–18
| 1–10/9–5/4–11/7
| 1–11/10–5/4–11/7
| Valinorsmic
|  
|  
|-
|-
| 53
| 53
|  
| 0–2–11–18
| 1–8/7–10/7–11/7
| 1–8/7–10/7–11/7
| Otonal
|  
|  
|-
|-
| 54
| 54
|  
| 0–3–11–18
| 1–12/7–10/7–11/7
| 1–10/7–11/7–12/7
| Otonal
|  
|  
|-
|-
| 55
| 55
|  
| 0–4–11–18
| 1–9/7–10/7–11/7
| 1–9/7–10/7–11/7
| Otonal
|  
|  
|-
|-
| 56
| 56
|  
| 0–7–11–18
| 1–10/9–10/7–11/7
| 1–10/9–10/7–11/7
| Ptolemismic
|  
|  
|-
|-
| 57
| 57
|  
| 0–9–11–18
| 1–5/4–10/7–11/7
| 1–5/4–10/7–11/7
| Valinorsmic
|  
|  
|-
|-
| 58
| 58
|  
| 0–3–14–18
| 1–12/7–11/9–11/7
| 1–11/9–11/7–12/7
| Swetismic
|  
|  
|-
|-
| 59
| 59
|  
| 0–7–14–18
| 1–10/9–11/9–11/7
| 1–11/10–11/9–11/7
| Utonal
|  
|  
|-
|-
| 60
| 60
|  
| 0–11–14–18
| 1–10/7–11/9–11/7
| 1–11/9–10/7–11/7
| Swetismic
|  
|  
|-
|-
| 61
| 61
|  
| 0–4–15–18
| 1–9/7–11/6–11/7
| 1–9/7–11/7–11/6
| Swetismic
|  
|  
|-
|-
| 62
| 62
|  
| 0–7–15–18
| 1–10/9–11/6–11/7
| 1–11/10–11/7–11/6
| Utonal
|  
|  
|-
|-
| 63
| 63
|  
| 0–11–15–18
| 1–10/7–11/6–11/7
| 1–10/7–11/7–11/6
| Swetismic
|  
|  
|-
|-
| 64
| 64
|  
| 0–14–15–18
| 1–11/9–11/6–11/7
| 1–11/9–11/7–11/6
| Utonal
|  
|  
|-
|-
| 65
| 65
|  
| 0–2–16–18
| 1–8/7–11/8–11/7
| 1–8/7–11/8–11/7
| Archytas
|  
|  
|-
|-
| 66
| 66
|  
| 0–7–16–18
| 1–10/9–11/8–11/7
| 1–11/10–11/8–11/7
| Utonal
|  
|  
|-
|-
| 67
| 67
|  
| 0–9–16–18
| 1–5/4–11/8–11/7
| 1–5/4–11/8–11/7
| Valinorsmic
|  
|  
|-
|-
| 68
| 68
|  
| 0–14–16–18
| 1–11/9–11/8–11/7
| 1–11/9–11/8–11/7
| Utonal
|  
|  
|-
|-
| 69
| 69
|  
| 0–15–16–18
| 1–11/6–11/8–11/7
| 1–11/8–11/7–11/6
| Utonal
|  
|  
|}
|}


== Pentads ==
== Pentads ==
{| class="wikitable"
{| class="wikitable sortable center-1"
|-
|-
! #
! #
! Generators
! class="unsortable" | Generators
! Transversal
! class="unsortable" | Transversal
! Type
! Type
! class="unsortable" | Comments
|-
|-
| 1
| 1
|  
| 0–1–2–3–4
| 1–3/2–8/7–12/7–9/7
| 1–8/7–9/7–3/2–12/7
| Archytas
|  
|  
|-
|-
| 2
| 2
|  
| 0–2–3–4–11
| 1–8/7–12/7–9/7–10/7
| 1–8/7–9/7–10/7–12/7
|  
| Otonal
| [[4:5:6:7:9]]
|-
|-
| 3
| 3
|  
| 0–3–4–7–11
| 1–12/7–9/7–10/9–10/7
| 1–10/9–9/7–10/7–12/7
| Sensamagic
|  
|  
|-
|-
| 4
| 4
|  
| 0–4–7–8–11
| 1–9/7–10/9–5/3–10/7
| 1–10/9–9/7–10/7–5/3
| Sensamagic
|  
|  
|-
|-
| 5
| 5
|  
| 0–7–8–9–11
| 1–10/9–5/3–5/4–10/7
| 1–10/9–5/4–10/7–5/3
|  
| Utonal
| [[210:252:315:360:560|1/(24:20:16:14:9)]]
|-
|-
| 6
| 6
|  
| 0–3–7–11–14
| 1–12/7–10/9–10/7–11/9
| 1–10/9–11/9–10/7–12/7
| Octarod
|  
|  
|-
|-
| 7
| 7
|  
| 0–1–4–8–15
| 1–3/2–9/7–5/3–11/6
| 1–9/7–3/2–5/3–11/6
| Octarod
|  
|  
|-
|-
| 8
| 8
|  
| 0–4–7–8–15
| 1–9/7–10/9–5/3–11/6
| 1–10/9–9/7–5/3–11/6
| Octarod
|  
|  
|-
|-
| 9
| 9
|  
| 0–4–7–11–15
| 1–9/7–10/9–10/7–11/6
| 1–9/7–10/9–10/7–11/6
| Octarod
|  
|  
|-
|-
| 10
| 10
|  
| 0–4–8–11–15
| 1–9/7–5/3–10/7–11/6
| 1–9/7–5/3–10/7–11/6
| Octarod
|  
|  
|-
|-
| 11
| 11
|  
| 0–7–8–11–15
| 1–10/9–5/3–10/7–11/6
| 1–10/9–5/3–10/7–11/6
| Octarod
|  
|  
|-
|-
| 12
| 12
|  
| 0–7–11–14–15
| 1–10/9–10/7–11/9–11/6
| 1–10/9–11/9–10/7–11/6
| Octarod
|  
|  
|-
|-
| 13
| 13
|  
| 0–1–2–9–16
| 1–3/2–8/7–5/4–11/8
| 1–9/8–5/4–11/8–3/2
|  
| Otonal
| [[4:5:6:9:11]]
|-
|-
| 14
| 14
|  
| 0–1–8–9–16
| 1–3/2–5/3–5/4–11/8
| 1–5/4–11/8–3/2–5/3
| Ptolemismic
|  
|  
|-
|-
| 15
| 15
|  
| 0–7–8–9–16
| 1–10/9–5/3–5/4–11/8
| 1–10/9–5/3–5/4–11/8
| Ptolemismic
|  
|  
|-
|-
| 16
| 16
|  
| 0–1–8–15–16
| 1–3/2–5/3–11/6–11/8
| 1–11/8–3/2–5/3–11/6
| Ptolemismic
|  
|  
|-
|-
| 17
| 17
|  
| 0–7–8–15–16
| 1–10/9–5/3–11/6–11/8
| 1–10/9–11/8–5/3–11/6
| Ptolemismic
|  
|  
|-
|-
| 18
| 18
|  
| 0–7–14–15–16
| 1–10/9–11/9–11/6–11/8
| 1–11/10–11/9–11/8–11/6
|  
| Utonal
| [[330:396:495:720:880|1/(24:20:16:11:9)]]
|-
|-
| 19
| 19
|  
| 0–2–3–4–18
| 1–8/7–12/7–9/7–11/7
| 1–8/7–9/7–11/7–12/7
|  
| Otonal
| [[4:6:7:9:11]]
|-
|-
| 20
| 20
|  
| 0–3–4–7–18
| 1–12/7–9/7–10/9–11/7
| 1–10/9–9/7–11/7–12/7
| Octarod
|  
|  
|-
|-
| 21
| 21
|  
| 0–2–3–11–18
| 1–8/7–12/7–10/7–11/7
| 1–8/7–10/7–11/7–12/7
|  
| Otonal
| [[4:5:6:7:11]]
|-
|-
| 22
| 22
|  
| 0–2–4–11–18
| 1–8/7–9/7–10/7–11/7
| 1–8/7–9/7–10/7–11/7
|  
| Otonal
| [[4:5:7:9:11]]
|-
|-
| 23
| 23
|  
| 0–3–4–11–18
| 1–12/7–9/7–10/7–11/7
| 1–9/7–10/7–11/7–12/7
|  
| Otonal
| [[5:6:7:9:11]]
|-
|-
| 24
| 24
|  
| 0–3–7–11–18
| 1–12/7–10/9–10/7–11/7
| 1–12/7–10/9–10/7–11/7
| Octarod
|  
|  
|-
|-
| 25
| 25
|  
| 0–4–7–11–18
| 1–9/7–10/9–10/7–11/7
| 1–10/9–9/7–10/7–11/7
| Octarod
|  
|  
|-
|-
| 26
| 26
|  
| 0–2–9–11–18
| 1–8/7–5/4–10/7–11/7
| 1–8/7–5/4–10/7–11/7
| Valinorsmic
|  
|  
|-
|-
| 27
| 27
|  
| 0–7–9–11–18
| 1–10/9–5/4–10/7–11/7
| 1–10/9–5/4–10/7–11/7
| Ares
|  
|  
|-
|-
| 28
| 28
|  
| 0–3–7–14–18
| 1–12/7–10/9–11/9–11/7
| 1–10/9–11/9–11/7–12/7
| Octarod
|  
|  
|-
|-
| 29
| 29
|  
| 0–3–11–14–18
| 1–12/7–10/7–11/9–11/7
| 1–11/9–10/7–11/7–12/7
| Swetismic
|  
|  
|-
|-
| 30
| 30
|  
| 0–7–11–14–18
| 1–10/9–10/7–11/9–11/7
| 1–10/9–11/9–10/7–11/7
| Octarod
|  
|  
|-
|-
| 31
| 31
|  
| 0–4–7–15–18
| 1–9/7–10/9–11/6–11/7
| 1–10/9–9/7–11/7–11/6
| Octarod
|  
|  
|-
|-
| 32
| 32
|  
| 0–4–11–15–18
| 1–9/7–10/7–11/6–11/7
| 1–9/7–10/7–11/7–11/6
| Octarod
|  
|  
|-
|-
| 33
| 33
|  
| 0–7–11–15–18
| 1–10/9–10/7–11/6–11/7
| 1–10/9–10/7–11/7–11/6
| Octarod
|  
|  
|-
|-
| 34
| 34
|  
| 0–7–14–15–18
| 1–10/9–11/9–11/6–11/7
| 1–11/10–11/9–11/7–11/6
|  
| Utonal
| [[1155:1386:1980:2520:3080|1/(24:20:14:11:9)]]
|-
|-
| 35
| 35
|  
| 0–11–14–15–18
| 1–10/7–11/9–11/6–11/7
| 1–11/9–10/7–11/7–11/6
| Octarod
|  
|  
|-
|-
| 36
| 36
|  
| 0–2–9–16–18
| 1–8/7–5/4–11/8–11/7
| 1–8/7–5/4–11/8–11/7
| Ares
|  
|  
|-
|-
| 37
| 37
|  
| 0–7–9–16–18
| 1–10/9–5/4–11/8–11/7
| 1–11/10–5/4–11/8–11/7
| Valinorsmic
|  
|  
|-
|-
| 38
| 38
|  
| 0–7–14–16–18
| 1–10/9–11/9–11/8–11/7
| 1–11/10–11/9–11/8–11/7
|  
| Utonal
| [[924:1155:1320:2016:2464|1/(20:16:14:11:9)]]
|-
|-
| 39
| 39
|  
| 0–7–15–16–18
| 1–10/9–11/6–11/8–11/7
| 1–11/10–11/8–11/7–11/6
|  
| Utonal
| [[770:924:1155:1320:1680|1/(24:20:16:14:11)]]
|-
|-
| 40
| 40
|  
| 0–14–15–16–18
| 1–11/9–11/6–11/8–11/7
| 1–11/9–11/8–11/7–11/6
|  
| Utonal
| [[462:693:792:1008:1232|1/(24:16:14:11:9)]]
|}
|}


== Hexads ==
== Hexads ==
{| class="wikitable"
{| class="wikitable sortable center-1"
|-
|-
! #
! #
! Generators
! class="unsortable" | Generators
! Transversal
! class="unsortable" | Transversal
! Type
! Type
! class="unsortable" | Comments
|-
|-
| 1
| 1
| 0–4–7–8–11–15
| 0–4–7–8–11–15
| 1–9/7–10/9–5/3–10/7–11/6
| 1–10/9–9/7–10/7–5/3–11/6
| Octarod
|  
|  
|-
|-
Line 823: Line 987:
| 1–8/7–9/7–10/7–11/7–12/7
| 1–8/7–9/7–10/7–11/7–12/7
| Otonal
| Otonal
| [[4:5:6:7:9:11]]
|-
|-
| 3
| 3
| 0–3–4–7–11–18
| 0–3–4–7–11–18
| 1–12/7–9/7–10/9–10/7–11/7
| 1–10/9–9/7–10/7–11/7–12/7
| Octarod
|  
|  
|-
|-
| 4
| 4
| 0–3–7–11–14–18
| 0–3–7–11–14–18
| 1–12/7–10/9–10/7–11/9–11/7
| 1–10/9–11/9–10/7–11/7–12/7
| Octarod
|  
|  
|-
|-
| 5
| 5
| 0–4–7–11–15–18
| 0–4–7–11–15–18
| 1–9/7–10/9–10/7–11/6–11/7
| 1–10/9–9/7–10/7–11/7–11/6
| Octarod
|  
|  
|-
|-
Line 842: Line 1,010:
| 0–7–11–14–15–18
| 0–7–11–14–15–18
| 1–10/9–10/7–11/9–11/6–11/7
| 1–10/9–10/7–11/9–11/6–11/7
| Octarod
|  
|  
|-
|-
Line 848: Line 1,017:
| 1–11/10–11/9–11/8–11/7–11/6
| 1–11/10–11/9–11/8–11/7–11/6
| Utonal
| Utonal
| [[2310:2772:3465:3960:5040:6160|1/(24:20:16:14:11:9)]]
|}
|}


[[Category:Todo:expand]]
[[Category:Superpyth]]
[[Category:Lists of chords]]
[[Category:Dyadic chords]]
[[Category:Triads]]
[[Category:Tetrads]]
[[Category:Pentads]]
[[Category:Hexads]]

Latest revision as of 01:22, 1 February 2026

< Superpyth

Below is a complete list of all 11-odd-limit dyadic chords in 11-limit superpyth temperament. Note that there are many common chords, for example 8:10:12:15, which are not listed; in this case due to 15/8 not being in the 11-odd-limit. Every chord listed has multiple inversions; only one is listed, that being the inversion where all notes are a nonnegative number of perfect fifth generators above the root.

If a chord is essentially just, it is classified as otonal if it is best analyzed in terms of the harmonic series, utonal if best analyzed in terms of the subharmonic series, and ambitonal if equally well analyzed by either. If a chord is essentially tempered, it is classified based on which commas are needed to define the chord. Chords essentially tempered by 64/63 are labeled archytas, by 100/99 ptolemismic, by 176/175 valinorsmic, by 245/243 sensamagic, and by 540/539 swetismic. Chords that require any two of 64/63, 100/99 and 176/175 to vanish are labeled ares. Finally, chords that require any two of 100/99, 245/243 and 540/539 to vanish are labeled octarod.

Typing the chords requires consideration of the fact that superpyth conflates 9/8 with 8/7, and 11/10 with 10/9. If a transversal can be found which shows the chord to be essentially just, that transversal is listed along with a typing as otonal, utonal, or ambitonal. However, sometimes multiple such transversals exist, in which case the chord is a plurichord, and the type is given for all possible interpretations. If the chord is essentially tempered, it is analyzed in terms of the transversal that requires the minimum amount of commas to be tempered out; if there is a tie between multiple transversals, it is analyzed in terms of the transversal which employs 8/7 or 10/9 above the root.

Superpyth is generated by a sharp ~3/2 between 13\22 (709.09 ¢) and 16\27 (711.11 ¢), and generates mos scales of the patterns 2L 3s (pentic), 5L 2s (diatonic), 5L 7s (p-chromatic), 5L 12s, 5L 17s, and 22L 5s. The pentic and diatonic scales contain some chords in the 2.3.7 subgroup, though the 12-note chromatic scale is needed to properly utilize intervals of 5, and intervals of 11 don't become common until the 17- and 22-note scales. Superpyth has hardly been explored in the 11-limit, and full 7-limit superpyth has not been explored much either, so these mos scales would be a great place to start such explorations.

Triads

# Generators Transversal Type Comments
1 0–1–2 1–9/8–3/2 Ambitonal 6:8:9, 8:9:12
2 0–1–3 1–3/2–12/7 Utonal 1/(12:8:7)
3 0–2–3 1–8/7–12/7 Otonal 4:6:7
4 0–1–4 1–9/7–3/2 Utonal 1/(9:7:6)
5 0–2–4 1–8/7–9/7 Otonal/utonal 7:8:9~1/(9:8:7)
6 0–3–4 1–9/7–12/7 Otonal 6:7:9
7 0–3–7 1–10/9–12/7 Sensamagic
8 0–4–7 1–10/9–9/7 Sensamagic
9 0–1–8 1–3/2–5/3 Otonal 6:9:10
10 0–4–8 1–9/7–5/3 Sensamagic
11 0–7–8 1–10/9–5/3 Utonal 1/(9:6:5)
12 0–1–9 1–5/4–3/2 Otonal 4:5:6
13 0–2–9 1–9/8–5/4 Otonal
14 0–7–9 1–10/9–5/4 Utonal
15 0–8–9 1–5/4–5/3 Utonal 1/(6:5:4)
16 0–2–11 1–8/7–10/7 Otonal 4:5:7
17 0–3–11 1–10/7–12/7 Otonal 5:6:7
18 0–4–11 1–9/7–10/7 Otonal
19 0–7–11 1–10/9–10/7 Utonal
20 0–8–11 1–10/7–5/3 Utonal 1/(7:6:5)
21 0–9–11 1–5/4–10/7 Utonal 1/(10:8:7)
22 0–3–14 1–11/9–12/7 Swetismic
23 0–7–14 1–10/9–11/9 Otonal/utonal 9:10:11~1/(11:10:9)
24 0–11–14 1–11/9–10/7 Swetismic
25 0–1–15 1–3/2–11/6 Otonal
26 0–4–15 1–9/7–11/6 Swetismic
27 0–7–15 1–11/10–11/6 Utonal
28 0–8–15 1–5/3–11/6 Otonal
29 0–11–15 1–10/7–11/6 Swetismic
30 0–14–15 1–11/9–11/6 Utonal
31 0–1–16 1–11/8–3/2 Otonal
32 0–2–16 1–9/8–11/8 Otonal
33 0–7–16 1–11/10–11/8 Utonal
34 0–8–16 1–11/8–5/3 Ptolemismic
35 0–9–16 1–5/4–11/8 Otonal
36 0–14–16 1–11/9–11/8 Utonal
37 0–15–16 1–11/8–11/6 Utonal
38 0–2–18 1–8/7–11/7 Otonal
39 0–3–18 1–11/7–12/7 Otonal
40 0–4–18 1–9/7–11/7 Otonal
41 0–7–18 1–11/10–11/7 Utonal
42 0–9–18 1–5/4–11/7 Valinorsmic
43 0–11–18 1–10/7–11/7 Otonal
44 0–14–18 1–11/9–11/7 Utonal
45 0–15–18 1–11/7–11/6 Utonal
46 0–16–18 1–11/8–11/7 Utonal

Tetrads

# Generators Transversal Type Comments
1 0–1–2–3 1–8/7–3/2–12/7 Archytas
2 0–1–2–4 1–9/8–9/7–3/2 Utonal 1/(9:7:6:4)
3 0–1–3–4 1–9/7–3/2–12/7 Ambitonal 12:14:18:21, 14:18:21:24
9-odd-limit ASS
4 0–2–3–4 1–8/7–9/7–12/7 Otonal 4:6:7:9
5 0–3–4–7 1–10/9–9/7–12/7 Sensamagic
6 0–1–4–8 1–9/7–3/2–5/3 Sensamagic
7 0–4–7–8 1–9/7–10/9–5/3 Sensamagic
8 0–1–2–9 1–9/8–5/4–3/2 Otonal 4:5:6:9
9 0–1–8–9 1–5/4–3/2–5/3 Ambitonal 10:12:15:18, 12:15:18:20
9-odd-limit ASS
10 0–7–8–9 1–10/9–5/4–5/3 Utonal 1/(9:6:5:4)
11 0–2–3–11 1–8/7–10/7–12/7 Otonal 4:5:6:7
12 0–2–4–11 1–8/7–9/7–10/7 Otonal 4:5:7:9
13 0–3–4–11 1–9/7–10/7–12/7 Otonal 6:7:9:10
14 0–3–7–11 1–10/9–10/7–12/7 Sensamagic
15 0–4–7–11 1–10/9–9/7–10/7 Sensamagic
16 0–4–8–11 1–9/7–10/7–5/3 Sensamagic
17 0–7–8–11 1–10/9–10/7–5/3 Utonal 1/(9:7:6:5)
18 0–2–9–11 1–8/7–5/4–10/7 Archytas/valinorsmic
19 0–7–9–11 1–10/9–5/4–10/7 Utonal 1/(9:7:5:4)
20 0–8–9–11 1–5/4–10/7–5/3 Utonal 1/(12:10:8:7)
21 0–3–7–14 1–11/10–11/9–12/7 Swetismic
22 0–3–11–14 1–11/9–10/7–12/7 Swetismic
23 0–7–11–14 1–11/10–11/9–10/7 Swetismic
24 0–1–4–15 1–9/7–3/2–11/6 Swetismic
25 0–4–7–15 1–10/9–9/7–11/6 Octarod
26 0–1–8–15 1–3/2–5/3–11/6 Otonal
27 0–4–8–15 1–9/7–5/3–11/6 Octarod
28 0–7–8–15 1–10/9–5/3–11/6 Ptolemismic
29 0–4–11–15 1–9/7–10/7–11/6 Swetismic
30 0–7–11–15 1–10/9–10/7–11/6 Octarod
31 0–8–11–15 1–10/7–5/3–11/6 Octarod
32 0–7–14–15 1–11/10–11/9–11/6 Utonal
33 0–11–14–15 1–11/9–10/7–11/6 Swetismic
34 0–1–2–16 1–9/8–11/8–3/2 Otonal
35 0–1–8–16 1–11/8–3/2–5/3 Ptolemismic
36 0–7–8–16 1–10/9–5/3–11/8 Ptolemismic
37 0–1–9–16 1–5/4–11/8–3/2 Otonal
38 0–2–9–16 1–9/8–5/4–11/8 Otonal
39 0–7–9–16 1–10/9–5/4–11/8 Ptolemismic/valinorsmic
40 0–8–9–16 1–5/4–11/8–5/3 Ptolemismic
41 0–9–14–16 1–11/10–11/9–11/8 Utonal
42 0–1–15–16 1–11/8–3/2–11/6 Ambitonal 11-odd-limit ASS
43 0–7–15–16 1–11/10–11/8–11/6 Utonal
44 0–8–15–16 1–11/8–5/3–11/6 Ptolemismic
45 0–14–15–16 1–11/9–11/8–11/6 Utonal
46 0–2–3–18 1–8/7–11/7–12/7 Otonal
47 0–2–4–18 1–8/7–9/7–11/7 Otonal
48 0–3–4–18 1–9/7–11/7–12/7 Otonal
49 0–3–7–18 1–10/9–11/7–12/7 Octarod
50 0–4–7–18 1–11/10–9/7–11/7 Swetismic
51 0–2–9–18 1–8/7–5/4–11/7 Valinorsmic
52 0–7–9–18 1–11/10–5/4–11/7 Valinorsmic
53 0–2–11–18 1–8/7–10/7–11/7 Otonal
54 0–3–11–18 1–10/7–11/7–12/7 Otonal
55 0–4–11–18 1–9/7–10/7–11/7 Otonal
56 0–7–11–18 1–10/9–10/7–11/7 Ptolemismic
57 0–9–11–18 1–5/4–10/7–11/7 Valinorsmic
58 0–3–14–18 1–11/9–11/7–12/7 Swetismic
59 0–7–14–18 1–11/10–11/9–11/7 Utonal
60 0–11–14–18 1–11/9–10/7–11/7 Swetismic
61 0–4–15–18 1–9/7–11/7–11/6 Swetismic
62 0–7–15–18 1–11/10–11/7–11/6 Utonal
63 0–11–15–18 1–10/7–11/7–11/6 Swetismic
64 0–14–15–18 1–11/9–11/7–11/6 Utonal
65 0–2–16–18 1–8/7–11/8–11/7 Archytas
66 0–7–16–18 1–11/10–11/8–11/7 Utonal
67 0–9–16–18 1–5/4–11/8–11/7 Valinorsmic
68 0–14–16–18 1–11/9–11/8–11/7 Utonal
69 0–15–16–18 1–11/8–11/7–11/6 Utonal

Pentads

# Generators Transversal Type Comments
1 0–1–2–3–4 1–8/7–9/7–3/2–12/7 Archytas
2 0–2–3–4–11 1–8/7–9/7–10/7–12/7 Otonal 4:5:6:7:9
3 0–3–4–7–11 1–10/9–9/7–10/7–12/7 Sensamagic
4 0–4–7–8–11 1–10/9–9/7–10/7–5/3 Sensamagic
5 0–7–8–9–11 1–10/9–5/4–10/7–5/3 Utonal 1/(24:20:16:14:9)
6 0–3–7–11–14 1–10/9–11/9–10/7–12/7 Octarod
7 0–1–4–8–15 1–9/7–3/2–5/3–11/6 Octarod
8 0–4–7–8–15 1–10/9–9/7–5/3–11/6 Octarod
9 0–4–7–11–15 1–9/7–10/9–10/7–11/6 Octarod
10 0–4–8–11–15 1–9/7–5/3–10/7–11/6 Octarod
11 0–7–8–11–15 1–10/9–5/3–10/7–11/6 Octarod
12 0–7–11–14–15 1–10/9–11/9–10/7–11/6 Octarod
13 0–1–2–9–16 1–9/8–5/4–11/8–3/2 Otonal 4:5:6:9:11
14 0–1–8–9–16 1–5/4–11/8–3/2–5/3 Ptolemismic
15 0–7–8–9–16 1–10/9–5/3–5/4–11/8 Ptolemismic
16 0–1–8–15–16 1–11/8–3/2–5/3–11/6 Ptolemismic
17 0–7–8–15–16 1–10/9–11/8–5/3–11/6 Ptolemismic
18 0–7–14–15–16 1–11/10–11/9–11/8–11/6 Utonal 1/(24:20:16:11:9)
19 0–2–3–4–18 1–8/7–9/7–11/7–12/7 Otonal 4:6:7:9:11
20 0–3–4–7–18 1–10/9–9/7–11/7–12/7 Octarod
21 0–2–3–11–18 1–8/7–10/7–11/7–12/7 Otonal 4:5:6:7:11
22 0–2–4–11–18 1–8/7–9/7–10/7–11/7 Otonal 4:5:7:9:11
23 0–3–4–11–18 1–9/7–10/7–11/7–12/7 Otonal 5:6:7:9:11
24 0–3–7–11–18 1–12/7–10/9–10/7–11/7 Octarod
25 0–4–7–11–18 1–10/9–9/7–10/7–11/7 Octarod
26 0–2–9–11–18 1–8/7–5/4–10/7–11/7 Valinorsmic
27 0–7–9–11–18 1–10/9–5/4–10/7–11/7 Ares
28 0–3–7–14–18 1–10/9–11/9–11/7–12/7 Octarod
29 0–3–11–14–18 1–11/9–10/7–11/7–12/7 Swetismic
30 0–7–11–14–18 1–10/9–11/9–10/7–11/7 Octarod
31 0–4–7–15–18 1–10/9–9/7–11/7–11/6 Octarod
32 0–4–11–15–18 1–9/7–10/7–11/7–11/6 Octarod
33 0–7–11–15–18 1–10/9–10/7–11/7–11/6 Octarod
34 0–7–14–15–18 1–11/10–11/9–11/7–11/6 Utonal 1/(24:20:14:11:9)
35 0–11–14–15–18 1–11/9–10/7–11/7–11/6 Octarod
36 0–2–9–16–18 1–8/7–5/4–11/8–11/7 Ares
37 0–7–9–16–18 1–11/10–5/4–11/8–11/7 Valinorsmic
38 0–7–14–16–18 1–11/10–11/9–11/8–11/7 Utonal 1/(20:16:14:11:9)
39 0–7–15–16–18 1–11/10–11/8–11/7–11/6 Utonal 1/(24:20:16:14:11)
40 0–14–15–16–18 1–11/9–11/8–11/7–11/6 Utonal 1/(24:16:14:11:9)

Hexads

# Generators Transversal Type Comments
1 0–4–7–8–11–15 1–10/9–9/7–10/7–5/3–11/6 Octarod
2 0–2–3–4–11–18 1–8/7–9/7–10/7–11/7–12/7 Otonal 4:5:6:7:9:11
3 0–3–4–7–11–18 1–10/9–9/7–10/7–11/7–12/7 Octarod
4 0–3–7–11–14–18 1–10/9–11/9–10/7–11/7–12/7 Octarod
5 0–4–7–11–15–18 1–10/9–9/7–10/7–11/7–11/6 Octarod
6 0–7–11–14–15–18 1–10/9–10/7–11/9–11/6–11/7 Octarod
7 0–7–14–15–16–18 1–11/10–11/9–11/8–11/7–11/6 Utonal 1/(24:20:16:14:11:9)
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