Chords of magic
Below is a complete list of the 11-odd-limit dyadic chords of 11-limit magic 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 major third generators above the root.
Typing the chords requires consideration of the fact that magic conflates 10/9 and 11/10 and so also 9/5 and 20/11. 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 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 10/9 and 9/5.
Chords requiring tempering only by 225/224 are labeled marvel, by 245/243 sensamagic, by 100/99 ptolemismic, by 896/891 pentacircle, by 385/384 keenanismic, and by 540/539 swetismic. Those requiring any two of 100/99, 225/224 or 896/891 are labeled apollo, any two of 100/99, 245/243 or 540/539 octarod, any two of 245/243, 896/891 or 385/384 sensamagic11, any two of 225/224, 385/384, or 540/539 marvel11. Chords requiring both 100/99 and 385/384 are labeled keemic. Finally, anything requiring three independent commas among those discussed above is labeled magic.
Magic has mos scales of 7, 10, 13, 16, 19, and 22 notes. It may be seen that even the 7-note mos is not without a few harmonic resources, and the larger ones do much better.
Kite Giedraitis has named the chords using arrows (ups and downs), as described in Kite's thoughts on pergens. The pergen is (P8, P12/5) fifth-of-a-twelfth, #37 in the list of pergens. One up is 19 generators, octave-reduced. The generator is vM3 = 380 ¢ + c/5, where c is the amount in cents the tempered fifth exceeds 700 ¢. The enharmonic unison is ^5dd2, thus ^5C = Bx. To simplify the chord names, slashes (lifts and drops) are also used. One lift is -22 generators, octave-reduced. Thus /1 = −25G + 3G = m2 + ^^d8 = ^^d2. Thus a lift equals two ups minus a tempered pythagorean comma, so /C = ^^Dbb, \C = vvB#, ^^C = /B#, and vvC = \Dbb. The cents values of sharps, ups and lifts vary greatly, as this table shows. Note that if the fifth is wider than 22edo's fifth, a lift will actually be descending. Furthermore, if the fifth is narrower than 19edo's, an up will be descending.
| Sharp | Up | Lift | How to convert the notation to the edo | |
|---|---|---|---|---|
| 19edo | 1\19 = 61 ¢ | 0\19 = 0 ¢ | 1\19 = 61 ¢ | Ignore the arrows, treat slashes as sharps/flats |
| 22edo | 3\22 = 164 ¢ | 1\22 = 55 ¢ | 0\22 = 0 ¢ | Ignore the slashes |
| 41edo | 4\41 = 117 ¢ | 1\41 = 29 ¢ | 1\41 = 29 ¢ | Treat slashes as arrows |
| 60edo | 5\60 = 100 ¢ | 1\60 = 20 ¢ | 2\60 = 40 ¢ | Treat slashes as double arrows |
| Rank-2 | 100 ¢ + 7c | 20 ¢ + 3.8c | 40 ¢ − 4.4c | N/a |
In magic, 5/4 = vM3, 7/4 = \m7 and 11/8 = vvA4. Thus an up is ~81/80 and a lift is ~64/63. This may not be true for other (P8, P12/5) temperaments. Therefore, the ratios in the following table are specific to magic, but the chord names apply to any (P8, P12/5) temperament.
| Genspan | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | … | 18 | … | 20 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cents (41edo) | 0 | 380 | 761 | 1141 | 322 | 702 | 1083 | 263 | 644 | 1024 | 205 | 585 | 966 | 146 | … | 849 | … | 410 |
| Ratio | 1/1 | 5/4 | 14/9 | 27/14 | 6/5 | 3/2 | 15/8 | 7/6 | 16/11 | 9/5 | 9/8 | 7/5 | 7/4 | 12/11 | … | 18/11 | … | 14/11 |
| Interval | P1 | vM3 | vvA5 \m6 |
^^d8 /M7 |
^m3 | P5 | vM7 | vvA2 \m3 |
^^d5 /A4 |
^m7 | M2 | vA4 ^\d5 |
vvA6 \m7 |
^^m2 /A1 |
… | ^^m6 /A5 |
… | M3 |
| Note (in C) | C | vE | vvG# \Ab |
^^Cb /B |
^Eb | G | vB | vvD# \Eb |
^^Gb /F# |
^Bb | D | vF# ^\Gb |
vvA# \Bb |
^^Db /C# |
… | ^^Ab /G# |
… | E |
|
Todo: complete table |
Triads
| # | Generators | Transversal | Type | Comments | Kite's name |
|---|---|---|---|---|---|
| 1 | 0–1–2 | 1–5/4–14/9 | Marvel | Cv(vv#5) | |
| 2 | 0–2–4 | 1–6/5–14/9 | Sensamagic | C^m(vv#5) | |
| 3 | 0–1–5 | 1–5/4–3/2 | Otonal | 4:5:6 | Cv |
| 4 | 0–4–5 | 1–6/5–3/2 | Utonal | 1/(6:5:4) | C^m |
| 5 | 0–2–7 | 1–7/6–14/9 | Utonal | 1/(9:7:6) | C/ |
| 6 | 0–5–7 | 1–7/6–3/2 | Otonal | 6:7:9 | C\m |
| 7 | 0–1–8 | 1–5/4–16/11 | Keenanismic | Cv(^^b5) | |
| 8 | 0–4–8 | 1–6/5–16/11 | Ptolemismic | C^m(^^b5) | |
| 9 | 0–7–8 | 1–7/6–16/11 | Keenanismic | C\m(^^b5) | |
| 10 | 0–1–9 | 1–5/4–20/11 | Utonal | Cv^7no5 | |
| 11 | 0–2–9 | 1–14/9–9/5 | Sensamagic | C^m7(vv#5)no3 | |
| 12 | 0–4–9 | 1–6/5–9/5 | Otonal | 6:9:10 | C^m7no5 or Cv6no3 |
| 13 | 0–5–9 | 1–3/2–9/5 | Utonal | 1/(9:6:5) | C^m7no3 |
| 14 | 0–7–9 | 1–7/6–9/5 | Sensamagic | C\mv7no5 | |
| 15 | 0–8–9 | 1–16/11–20/11 | Otonal | 1–5/4–11/8 | Cv(\b5) |
| 16 | 0–1–10 | 1–9/8–5/4 | Otonal | Cv,9no5 | |
| 17 | 0–2–10 | 1–9/8–14/9 | Pentacircle | C2(vv#5) | |
| 18 | 0–5–10 | 1–9/8–3/2 | Ambitonal | 6:8:9, 8:9:12 | C2 |
| 19 | 0–8–10 | 1–9/8–16/11 | Pentacircle | C2(^^b5) | |
| 20 | 0–9–10 | 1–9/8–9/5 | Utonal | C^9no35 or C^7sus2no5 | |
| 21 | 0–1–11 | 1–5/4–7/5 | Marvel | Cv(^\b5) | |
| 22 | 0–2–11 | 1–7/5–14/9 | Utonal | 1–9/7–9/5 | C/,^7no5 |
| 23 | 0–4–11 | 1–6/5–7/5 | Otonal | 5:6:7 | C^m(^\b5) |
| 24 | 0–7–11 | 1–7/6–7/5 | Utonal | 1/(7:6:5) | C\m(^\b5) |
| 25 | 0–9–11 | 1–7/5–9/5 | Otonal | 1–9/7–10/7 | C/(^b5) |
| 26 | 0–10–11 | 1–9/8–7/5 | Marvel | 1–5/4–16/9 | Cv,7no5 |
| 27 | 0–1–12 | 1–5/4–7/4 | Otonal | 4:5:7 | Cv,\7no5 |
| 28 | 0–2–12 | 1–14/9–7/4 | Utonal | 1–9/8–9/7 | C/,9no5 |
| 29 | 0–4–12 | 1–6/5–7/4 | Keenanismic | C^m\7 | |
| 30 | 0–5–12 | 1–3/2–7/4 | Otonal | 4:6:7 | C\7no3 |
| 31 | 0–7–12 | 1–7/6–7/4 | Utonal | 1/(12:8:7) | C\m7no5 |
| 32 | 0–8–12 | 1–16/11–7/4 | Keenanismic | 1–6/5–11/8 | C^m(\b5) |
| 33 | 0–10–12 | 1–9/8–7/4 | Otonal | C\7sus2 | |
| 34 | 0–11–12 | 1–7/5–7/4 | Utonal | 1/(10:8:7) | C\7(^\b5)no3 |
| 35 | 0–1–13 | 1–12/11–5/4 | Keenanismic | ||
| 36 | 0–2–13 | 1–12/11–14/9 | Swetismic | 1–9/7–7/5 | C/(^\b5) |
| 37 | 0–4–13 | 1–12/11–6/5 | Utonal | ||
| 38 | 0–5–13 | 1–12/11–3/2 | Utonal | C^^b2 | |
| 39 | 0–8–13 | 1–12/11–16/11 | Otonal | 1–11/8–3/2 | Cvv#4 |
| 40 | 0–9–13 | 1–12/11–20/11 | Otonal | ||
| 41 | 0–11–13 | 1–12/11–7/5 | Swetismic | ||
| 42 | 0–12–13 | 1–12/11–7/4 | Keenanismic | ||
| 43 | 0–5–18 | 1–3/2–18/11 | Utonal | ||
| 44 | 0–7–18 | 1–7/6–18/11 | Swetismic | ||
| 45 | 0–8–18 | 1–16/11–18/11 | Otonal | ||
| 46 | 0–9–18 | 1–18/11–9/5 | Utonal | ||
| 47 | 0–10–18 | 1–9/8–18/11 | Utonal | ||
| 48 | 0–11–18 | 1–7/5–18/11 | Swetismic | ||
| 49 | 0–13–18 | 1–12/11–18/11 | Otonal | ||
| 50 | 0–2–20 | 1–14/11–14/9 | Utonal | ||
| 51 | 0–7–20 | 1–7/6–14/11 | Utonal | ||
| 52 | 0–8–20 | 1–14/11–16/11 | Otonal | ||
| 53 | 0–9–20 | 1–14/11–20/11 | Otonal | ||
| 54 | 0–10–20 | 1–9/8–14/11 | Pentacircle | ||
| 55 | 0–11–20 | 1–14/11–7/5 | Utonal | ||
| 56 | 0–12–20 | 1–14/11–7/4 | Utonal | ||
| 57 | 0–13–20 | 1–12/11–14/11 | Otonal | ||
| 58 | 0–18–20 | 1–14/11–18/11 | Otonal |
Tetrads
| # | Generators | Transversal | Type | Comments | Kite's name |
|---|---|---|---|---|---|
| 1 | 0–1–2–9 | 1–5/4–14/9–9/5 | Magic | Cv^7(vv#5) | |
| 2 | 0–2–4–9 | 1–6/5–14/9–9/5 | Sensamagic | C^m7(vv#5) | |
| 3 | 0–1–5–9 | 1–5/4–3/2–9/5 | Ptolemismic | Cv^7 | |
| 4 | 0–4–5–9 | 1–6/5–3/2–9/5 | Ambitonal | 10:12:15:18, 12:15:18:20 9-odd-limit ASS |
C^m7 or Cv6 |
| 5 | 0–2–7–9 | 1–7/6–14/9–9/5 | Sensamagic | 1–9/7–3/2–7/3 | C/,vv#9 |
| 6 | 0–5–7–9 | 1–7/6–3/2–9/5 | Sensamagic | C\m^7 | |
| 7 | 0–1–8–9 | 1–5/4–16/11–9/5 | Keemic | Cv^7(^^b5) | |
| 8 | 0–4–8–9 | 1–6/5–16/11–9/5 | Ptolemismic | C^m7(^^b5) | |
| 9 | 0–7–8–9 | 1–7/6–16/11–9/5 | Magic | C\m^7(^^b5) | |
| 10 | 0–1–2–10 | 1–9/8–5/4–14/9 | Apollo | Cv,9(vv#5) | |
| 11 | 0–1–5–10 | 1–9/8–5/4–3/2 | Otonal | 4:5:6:9 | Cv,9 |
| 12 | 0–1–8–10 | 1–9/8–5/4–16/11 | Sensamagic11 | Cv,9(^^b5) | |
| 13 | 0–1–9–10 | 1–9/8–5/4–9/5 | Ptolemismic | Cv^7,9no5 or Cv9(^7)no5 | |
| 14 | 0–2–9–10 | 1–9/8–14/9–9/5 | Sensamagic11 | C^9(vv#5)no3 or C^7(vv#5)sus2 | |
| 15 | 0–5–9–10 | 1–9/8–3/2–9/5 | Utonal | 1/(9:6:5:4) | C^9no3 or C^7sus2 or C2,^7 |
| 16 | 0–8–9–10 | 1–9/8–16/11–9/5 | Apollo | ||
| 17 | 0–1–2–11 | 1–5/4–7/5–14/9 | Marvel | ||
| 18 | 0–2–4–11 | 1–6/5–7/5–14/9 | Sensamagic | ||
| 19 | 0–2–7–11 | 1–7/6–7/5–14/9 | Utonal | 1/(9:7:6:5) | |
| 20 | 0–1–9–11 | 1–5/4–7/5–9/5 | Apollo | ||
| 21 | 0–2–9–11 | 1–7/5–14/9–9/5 | Sensamagic | ||
| 22 | 0–4–9–11 | 1–6/5–7/5–9/5 | Otonal | 6:7:9:10 | C^m7(^\b5) or C\mv6 |
| 23 | 0–7–9–11 | 1–7/6–7/5–9/5 | Sensamagic | ||
| 24 | 0–1–10–11 | 1–9/8–5/4–7/5 | Marvel | ||
| 25 | 0–2–10–11 | 1–9/8–7/5–14/9 | Apollo | ||
| 26 | 0–9–10–11 | 1–9/8–7/5–9/5 | Marvel | ||
| 27 | 0–1–2–12 | 1–5/4–14/9–7/4 | Marvel | ||
| 28 | 0–2–4–12 | 1–6/5–14/9–7/4 | Sensamagic11 | ||
| 29 | 0–1–5–12 | 1–5/4–3/2–7/4 | Otonal | 4:5:6:7 | Cv,\7 |
| 30 | 0–4–5–12 | 1–6/5–3/2–7/4 | Keenanismic | C^m\7 | |
| 31 | 0–2–7–12 | 1–7/6–14/9–7/4 | Utonal | C\m7(vv#5) | |
| 32 | 0–5–7–12 | 1–7/6–3/2–7/4 | Ambitonal | 12:14:18:21, 14:18:21:24 9-odd-limit ASS |
C\m7 |
| 33 | 0–1–8–12 | 1–5/4–16/11–7/4 | Keenanismic | ||
| 34 | 0–4–8–12 | 1–6/5–16/11–7/4 | Keemic | ||
| 35 | 0–7–8–12 | 1–7/6–16/11–7/4 | Keenanismic | C\m7(^^b5) | |
| 36 | 0–1–10–12 | 1–9/8–5/4–7/4 | Otonal | 4:5:7:9 | |
| 37 | 0–2–10–12 | 1–9/8–14/9–7/4 | Pentacircle | ||
| 38 | 0–5–10–12 | 1–9/8–3/2–7/4 | Otonal | 4:6:7:9 | C2\7 or C\7sus2 or C\9no3 |
| 39 | 0–8–10–12 | 1–9/8–16/11–7/4 | Sensamagic11 | ||
| 40 | 0–1–11–12 | 1–5/4–7/5–7/4 | Marvel | ||
| 41 | 0–2–11–12 | 1–7/5–14/9–7/4 | Utonal | 1/(9:7:5:4) | |
| 42 | 0–4–11–12 | 1–6/5–7/5–7/4 | Keenanismic | ||
| 43 | 0–7–11–12 | 1–7/6–7/5–7/4 | Utonal | 1/(12:10:8:7) | C\m7(^\b5) or C^m/6 |
| 44 | 0–10–11–12 | 1–9/8–7/5–7/4 | Marvel | ||
| 45 | 0–1–2–13 | 1–12/11–5/4–14/9 | Marvel11 | ||
| 46 | 0–2–4–13 | 1–12/11–6/5–14/9 | Octarod | ||
| 47 | 0–1–5–13 | 1–12/11–5/4–3/2 | Keenanismic | ||
| 48 | 0–4–5–13 | 1–12/11–6/5–3/2 | Utonal | ||
| 49 | 0–1–8–13 | 1–12/11–5/4–16/11 | Keenanismic | ||
| 50 | 0–4–8–13 | 1–12/11–6/5–16/11 | Ptolemismic | ||
| 51 | 0–1–9–13 | 1–12/11–5/4–9/5 | Keemic | ||
| 52 | 0–2–9–13 | 1–12/11–14/9–9/5 | Octarod | ||
| 53 | 0–4–9–13 | 1–12/11–6/5–9/5 | Ptolemismic | ||
| 54 | 0–5–9–13 | 1–12/11–3/2–9/5 | Ptolemismic | ||
| 55 | 0–8–9–13 | 1–12/11–16/11–20/11 | Otonal | ||
| 56 | 0–1–11–13 | 1–12/11–5/4–7/5 | Marvel11 | ||
| 57 | 0–2–11–13 | 1–12/11–7/5–14/9 | Swetismic | ||
| 58 | 0–4–11–13 | 1–12/11–6/5–7/5 | Octarod | ||
| 59 | 0–9–11–13 | 1–12/11–7/5–9/5 | Octarod | ||
| 60 | 0–1–12–13 | 1–12/11–5/4–7/4 | Keenanismic | ||
| 61 | 0–2–12–13 | 1–12/11–14/9–7/4 | Marvel11 | ||
| 62 | 0–4–12–13 | 1–12/11–6/5–7/4 | Keemic | ||
| 63 | 0–5–12–13 | 1–12/11–3/2–7/4 | Keenanismic | ||
| 64 | 0–8–12–13 | 1–12/11–16/11–7/4 | Keenanismic | ||
| 65 | 0–11–12–13 | 1–12/11–7/5–7/4 | Marvel11 | ||
| 66 | 0–5–7–18 | 1–7/6–3/2–18/11 | Swetismic | ||
| 67 | 0–7–8–18 | 1–7/6–16/11–18/11 | Marvel11 | ||
| 68 | 0–5–9–18 | 1–3/2–18/11–9/5 | Utonal | ||
| 69 | 0–7–9–18 | 1–7/6–18/11–9/5 | Octarod | ||
| 70 | 0–8–9–18 | 1–16/11–18/11–20/11 | Otonal | ||
| 71 | 0–5–10–18 | 1–9/8–3/2–18/11 | Utonal | ||
| 72 | 0–8–10–18 | 1–9/8–16/11–18/11 | Pentacircle | ||
| 73 | 0–9–10–18 | 1–9/8–18/11–9/5 | Utonal | ||
| 74 | 0–7–11–18 | 1–7/6–7/5–18/11 | Swetismic | ||
| 75 | 0–9–11–18 | 1–7/5–18/11–9/5 | Octarod | ||
| 76 | 0–10–11–18 | 1–9/8–7/5–18/11 | Marvel11 | ||
| 77 | 0–5–13–18 | 1–12/11–3/2–18/11 | Ambitonal | ||
| 78 | 0–8–13–18 | 1–12/11–16/11–18/11 | Otonal | ||
| 79 | 0–9–13–18 | 1–12/11–18/11–20/11 | Otonal | ||
| 80 | 0–11–13–18 | 1–12/11–7/5–18/11 | Swetismic | ||
| 81 | 0–2–7–20 | 1–7/6–14/11–14/9 | Utonal | ||
| 82 | 0–7–8–20 | 1–7/6–14/11–16/11 | Keenanismic | ||
| 83 | 0–2–9–20 | 1–14/11–14/9–9/5 | Octarod | ||
| 84 | 0–7–9–20 | 1–7/6–14/11–9/5 | Octarod | ||
| 85 | 0–8–9–20 | 1–14/11–16/11–20/11 | Otonal | ||
| 86 | 0–2–10–20 | 1–9/8–14/11–14/9 | Pentacircle | ||
| 87 | 0–8–10–20 | 1–9/8–14/11–16/11 | Pentacircle | ||
| 88 | 0–9–10–20 | 1–9/8–14/11–9/5 | Apollo | ||
| 89 | 0–2–11–20 | 1–7/5–14/11–14/9 | Utonal | ||
| 90 | 0–7–11–20 | 1–7/6–14/11–7/5 | Utonal | ||
| 91 | 0–9–11–20 | 1–7/5–14/11–9/5 | Ptolemismic | ||
| 92 | 0–10–11–20 | 1–9/8–14/11–7/5 | Apollo | ||
| 93 | 0–2–12–20 | 1–14/11–14/9–7/4 | Utonal | ||
| 94 | 0–7–12–20 | 1–7/6–14/11–7/4 | Utonal | ||
| 95 | 0–8–12–20 | 1–14/11–16/11–7/4 | Keenanismic | ||
| 96 | 0–10–12–20 | 1–9/8–14/11–7/4 | Pentacircle | ||
| 97 | 0–11–12–20 | 1–14/11–7/5–7/4 | Utonal | ||
| 98 | 0–2–13–20 | 1–12/11–14/11–14/9 | Swetismic | ||
| 99 | 0–8–13–20 | 1–12/11–14/11–16/11 | Otonal | ||
| 100 | 0–9–13–20 | 1–12/11–14/11–20/11 | Otonal | ||
| 101 | 0–11–13–20 | 1–12/11–14/11–7/5 | Octarod | ||
| 102 | 0–12–13–20 | 1–12/11–14/11–7/4 | Keenanismic | ||
| 103 | 0–7–18–20 | 1–7/6–14/11–18/11 | Swetismic | ||
| 104 | 0–8–18–20 | 1–14/11–16/11–18/11 | Otonal | ||
| 105 | 0–9–18–20 | 1–14/11–18/11–20/11 | Otonal | ||
| 106 | 0–10–18–20 | 1–9/8–14/11–18/11 | Pentacircle | ||
| 107 | 0–11–18–20 | 1–14/11–7/5–18/11 | Octarod | ||
| 108 | 0–13–18–20 | 1–12/11–14/11–18/11 | Otonal |
Pentads
| # | Generators | Transversal | Type | Comments | Kite's name |
|---|---|---|---|---|---|
| 1 | 0–1–2–9–10 | 1–9/8–5/4–14/9–9/5 | Magic | ||
| 2 | 0–1–5–9–10 | 1–9/8–5/4–3/2–9/5 | Ptolemismic | Cv9(^7) | |
| 3 | 0–1–8–9–10 | 1–9/8–5/4–16/11–9/5 | Magic | ||
| 4 | 0–1–2–9–11 | 1–5/4–7/5–14/9–9/5 | Magic | ||
| 5 | 0–2–4–9–11 | 1–6/5–7/5–14/9–9/5 | Sensamagic | ||
| 6 | 0–2–7–9–11 | 1–7/6–7/5–14/9–9/5 | Sensamagic | ||
| 7 | 0–1–2–10–11 | 1–9/8–5/4–7/5–14/9 | Apollo | ||
| 8 | 0–1–9–10–11 | 1–9/8–5/4–7/5–9/5 | Apollo | ||
| 9 | 0–2–9–10–11 | 1–9/8–7/5–14/9–9/5 | Magic | ||
| 10 | 0–1–2–10–12 | 1–9/8–5/4–14/9–7/4 | Apollo | ||
| 11 | 0–1–5–10–12 | 1–9/8–5/4–3/2–7/4 | Otonal | 4:5:6:7:9 | Cv9(\7) |
| 12 | 0–1–8–10–12 | 1–9/8–5/4–16/11–7/4 | Sensamagic11 | ||
| 13 | 0–1–2–11–12 | 1–5/4–7/5–14/9–7/4 | Marvel | ||
| 14 | 0–2–4–11–12 | 1–6/5–7/5–14/9–7/4 | Sensamagic11 | ||
| 15 | 0–2–7–11–12 | 1–7/6–7/5–14/9–7/4 | Utonal | 1/(24:20:16:14:9) | C/9(^7) |
| 16 | 0–1–10–11–12 | 1–9/8–5/4–7/5–7/4 | Marvel | ||
| 17 | 0–2–10–11–12 | 1–9/8–7/5–14/9–7/4 | Apollo | ||
| 18 | 0–1–2–9–13 | 1–12/11–5/4–14/9–9/5 | Magic | ||
| 19 | 0–2–4–9–13 | 1–12/11–6/5–14/9–9/5 | Octarod | ||
| 20 | 0–1–5–9–13 | 1–12/11–5/4–3/2–9/5 | Keemic | ||
| 21 | 0–4–5–9–13 | 1–12/11–6/5–3/2–9/5 | Ptolemismic | ||
| 22 | 0–1–8–9–13 | 1–12/11–5/4–16/11–9/5 | Keemic | ||
| 23 | 0–4–8–9–13 | 1–12/11–6/5–16/11–9/5 | Ptolemismic | ||
| 24 | 0–1–2–11–13 | 1–12/11–5/4–7/5–14/9 | Marvel11 | ||
| 25 | 0–2–4–11–13 | 1–12/11–6/5–7/5–14/9 | Octarod | ||
| 26 | 0–1–9–11–13 | 1–12/11–5/4–7/5–9/5 | Magic | ||
| 27 | 0–2–9–11–13 | 1–12/11–7/5–14/9–9/5 | Octarod | ||
| 28 | 0–4–9–11–13 | 1–12/11–6/5–7/5–9/5 | Octarod | ||
| 29 | 0–1–2–12–13 | 1–12/11–5/4–14/9–7/4 | Marvel11 | ||
| 30 | 0–2–4–12–13 | 1–12/11–6/5–14/9–7/4 | Magic | ||
| 31 | 0–1–5–12–13 | 1–12/11–5/4–3/2–7/4 | Keenanismic | ||
| 32 | 0–4–5–12–13 | 1–12/11–6/5–3/2–7/4 | Keemic | ||
| 33 | 0–1–8–12–13 | 1–12/11–5/4–16/11–7/4 | Keenanismic | ||
| 34 | 0–4–8–12–13 | 1–12/11–6/5–16/11–7/4 | Keemic | ||
| 35 | 0–1–11–12–13 | 1–12/11–5/4–7/5–7/4 | Marvel11 | ||
| 36 | 0–2–11–12–13 | 1–12/11–7/5–14/9–7/4 | Marvel11 | ||
| 37 | 0–4–11–12–13 | 1–12/11–6/5–7/5–7/4 | Magic | ||
| 38 | 0–5–7–9–18 | 1–7/6–3/2–18/11–9/5 | Octarod | ||
| 39 | 0–7–8–9–18 | 1–7/6–16/11–18/11–9/5 | Magic | ||
| 40 | 0–5–9–10–18 | 1–9/8–3/2–18/11–9/5 | Utonal | 1/(24:20:16:11:9) | |
| 41 | 0–8–9–10–18 | 1–9/8–16/11–18/11–9/5 | Apollo | ||
| 42 | 0–7–9–11–18 | 1–7/6–7/5–18/11–9/5 | Octarod | ||
| 43 | 0–9–10–11–18 | 1–9/8–7/5–18/11–9/5 | Magic | ||
| 44 | 0–5–9–13–18 | 1–3/2–12/11–18/11–9/5 | Ptolemismic | ||
| 45 | 0–8–9–13–18 | 1–12/11–16/11–18/11–20/11 | Otonal | 4:5:6:9:11 | |
| 46 | 0–9–11–13–18 | 1–7/5–12/11–18/11–9/5 | Octarod | ||
| 47 | 0–2–7–9–20 | 1–7/6–14/11–14/9–9/5 | Octarod | ||
| 48 | 0–7–8–9–20 | 1–7/6–14/11–16/11–9/5 | Magic | ||
| 49 | 0–2–9–10–20 | 1–9/8–14/11–14/9–9/5 | Magic | ||
| 50 | 0–8–9–10–20 | 1–9/8–14/11–16/11–9/5 | Apollo | ||
| 51 | 0–2–7–11–20 | 1–7/6–7/5–14/11–14/9 | Utonal | 1/(24:20:14:11:9) | |
| 52 | 0–2–9–11–20 | 1–14/11–7/5–14/9–9/5 | Octarod | ||
| 53 | 0–7–9–11–20 | 1–7/6–14/11–7/5–9/5 | Octarod | ||
| 54 | 0–2–10–11–20 | 1–9/8–14/11–7/5–14/9 | Apollo | ||
| 55 | 0–9–10–11–20 | 1–9/8–14/11–7/5–9/5 | Apollo | ||
| 56 | 0–2–7–12–20 | 1–7/6–14/11–14/9–7/4 | Utonal | 1/(24:16:14:11:9) | |
| 57 | 0–7–8–12–20 | 1–7/6–14/11–16/11–7/4 | Keenanismic | ||
| 58 | 0–2–10–12–20 | 1–9/8–14/11–14/9–7/4 | Pentacircle | ||
| 59 | 0–8–10–12–20 | 1–9/8–14/11–16/11–7/4 | Sensamagic11 | ||
| 60 | 0–2–11–12–20 | 1–14/11–7/5–14/9–7/4 | Utonal | 1/(20:16:14:11:9) | |
| 61 | 0–7–11–12–20 | 1–7/6–14/11–7/5–7/4 | Utonal | 1/(24:20:16:14:11) | |
| 62 | 0–10–11–12–20 | 1–9/8–14/11–7/5–7/4 | Apollo | ||
| 63 | 0–2–9–13–20 | 1–12/11–14/11–14/9–9/5 | Octarod | ||
| 64 | 0–8–9–13–20 | 1–12/11–14/11–16/11–20/11 | Otonal | 4:5:6:7:11 | |
| 65 | 0–2–11–13–20 | 1–12/11–14/11–7/5–14/9 | Octarod | ||
| 66 | 0–9–11–13–20 | 1–12/11–14/11–7/5–9/5 | Octarod | ||
| 67 | 0–2–12–13–20 | 1–12/11–14/11–14/9–7/4 | Marvel11 | ||
| 68 | 0–8–12–13–20 | 1–12/11–14/11–16/11–7/4 | Keenanismic | ||
| 69 | 0–11–12–13–20 | 1–12/11–14/11–7/5–7/4 | Magic | ||
| 70 | 0–7–8–18–20 | 1–7/6–14/11–16/11–18/11 | Marvel11 | ||
| 71 | 0–7–9–18–20 | 1–7/6–14/11–18/11–9/5 | Octarod | ||
| 72 | 0–8–9–18–20 | 1–14/11–16/11–18/11–20/11 | Otonal | 4:5:7:9:11 | |
| 73 | 0–8–10–18–20 | 1–9/8–14/11–16/11–18/11 | Pentacircle | ||
| 74 | 0–9–10–18–20 | 1–9/8–14/11–18/11–9/5 | Apollo | ||
| 75 | 0–7–11–18–20 | 1–7/6–14/11–7/5–18/11 | Octarod | ||
| 76 | 0–9–11–18–20 | 1–14/11–7/5–18/11–9/5 | Octarod | ||
| 77 | 0–10–11–18–20 | 1–9/8–14/11–7/5–18/11 | Magic | ||
| 78 | 0–8–13–18–20 | 1–12/11–14/11–16/11–18/11 | Otonal | 4:6:7:9:11 | |
| 79 | 0–9–13–18–20 | 1–12/11–14/11–18/11–20/11 | Otonal | 5:6:7:9:11 | |
| 80 | 0–11–13–18–20 | 1–12/11–14/11–7/5–18/11 | Octarod |
Hexads
| # | Generators | Transversal | Type | Comment |
|---|---|---|---|---|
| 1 | 0–1–2–9–10–11 | 1–9/8–5/4–7/5–14/9–9/5 | Magic | |
| 2 | 0–1–2–10–11–12 | 1–9/8–5/4–7/5–14/9–7/4 | Apollo | |
| 3 | 0–1–2–9–11–13 | 1–12/11–5/4–7/5–14/9–9/5 | Magic | |
| 4 | 0–2–4–9–11–13 | 1–12/11–6/5–7/5–14/9–9/5 | Octarod | |
| 5 | 0–1–2–11–12–13 | 1–12/11–5/4–7/5–14/9–7/4 | Marvel11 | |
| 6 | 0–2–4–11–12–13 | 1–12/11–6/5–7/5–14/9–7/4 | Magic | |
| 7 | 0–2–7–9–11–20 | 1–7/6–14/11–7/5–14/9–9/5 | Octarod | |
| 8 | 0–2–9–10–11–20 | 1–9/8–14/11–7/5–14/9–9/5 | Magic | |
| 9 | 0–2–7–11–12–20 | 1–14/11–7/6–7/5–14/9–7/4 | Utonal | 1/(24:20:16:14:11:9) |
| 10 | 0–2–10–11–12–20 | 1–9/8–14/11–7/5–14/9–7/4 | Apollo | |
| 11 | 0–2–9–11–13–20 | 1–12/11–14/11–7/5–14/9–9/5 | Octarod | |
| 12 | 0–2–11–12–13–20 | 1–12/11–14/11–7/5–14/9–7/4 | Magic | |
| 13 | 0–7–8–9–18–20 | 1–7/6–14/11–16/11–18/11–9/5 | Magic | |
| 14 | 0–8–9–10–18–20 | 1–9/8–14/11–16/11–18/11–9/5 | Apollo | |
| 15 | 0–7–9–11–18–20 | 1–7/6–14/11–7/5–18/11–9/5 | Octarod | |
| 16 | 0–9–10–11–18–20 | 1–9/8–14/11–7/5–18/11–9/5 | Magic | |
| 17 | 0–8–9–13–18–20 | 1–12/11–14/11–16/11–18/11–20/11 | Otonal | 4:5:6:7:9:11 |
| 18 | 0–9–11–13–18–20 | 1–12/11–14/11–7/5–18/11–9/5 | Octarod |