Chords of porcupine
Below are listed the 15-odd-limit dyadic chords of 11-limit porcupine temperament that do not have generator steps 7 or 13 as dyads. Typing the chords requires consideration of the fact that porcupine conflates 10/9, 11/10 and 12/11 and also 16/9 and 7/4. 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 a chord is essentially tempered, the chord is analyzed in terms of the transversals 11/10, 6/5, 16/11 and 7/4. Chords that require only 64/63 tempering are marked archytas, by 100/99 ptolemismic, by 121/120 biyatismic, by 176/175 valinorsmic, and by 385/384 keenanismic. Chords that require 64/63 and 176/175 tempering are marked ares, 100/99 and 385/384 tempered chords are supermagic, and 176/175 and 385/384 tempered chords are marked zeus. Chords that receive tempering by three independent commas above are labeled porcupine.
The transversal is in generator order. This is useful because it tells how common the chords are: For instance, a chord that appears on the sixth generation will appear exactly once in Porcupine[7], twice in porcupine[8], and nine times in Porcupine[15].
The "As generated" column takes the intervals that were generated and places them in size order. The 1st and 2nd inversion (and so on) columns show the inversions of those generated tones. Note that this gives different results than you might be used to: the major chord (1/1–5/4–3/2, or 4:5:6) is the second inversion of the generated 0–2–5 chord.
Though we are used to thinking of 4:5:6 as the definitive "major chord", with all inversions coming from that, there is nothing definitive about calling these lists below "chord" or "inversion". That is just the way the generators came out.
The bolded inversions are named using ups and downs as described on the Pergen page. The pergen is (P8, P4/3) third-of-a-4th, #7 in the notation guide for rank-2 pergens. One up is -7 generators, octave-reduced, which is a third-sharp. Thus ^3C = C# and the enharmonic interval is v3A1. The generator is vM2 = 167¢ - c/3, where c is the amount in cents the tempered fifth exceeds 700¢. ^1 = 33¢ + 2.33c. In 22edo, ^1 = 1\22 = 54.5¢.
In porcupine, 5/4 = vM3, 7/4 = m7 and 11/8 = ^4. Thus ^1 equals ~81/80 and ~33/32. This may not be true for other (P8, P4/3) temperaments. So the ratios in the table below are specific to Porcupine, but the chord names apply to any (P8, P4/3) temperament.
| Genspan | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cents (22edo) | 0 | 164 | 327 | 491 | 655 | 818 | 982 | 1145 | 109 | 273 | 436 | 600 | 764 | 927 | 1091 |
| Ratio | 1/1 | 10/9 11/10 |
6/5 11/9 |
4/3 | 16/11 | 8/5 | 16/9 7/4 |
48/25 160/81 |
16/15 21/20 |
7/6 | 14/11 | 7/5 | 14/9 | 28/15 | |
| Interval | P1 | vM2 | ^m3 | P4 | v5 | ^m6 | m7 | v8 | ^m2 | m3 | v4 | ^b5 | m6 | vm7 | ^d8 |
| Note (in C) | C | vD | ^Eb | F | vG | ^Ab | Bb | vC | ^Db | Eb | vF | ^Gb | Ab | vBb | ^Cb |
 |
Todo: complete table, research Both tetrads and pentads are incomplete. Add the missing chords. |
Triads
| Chord | Transversal | Type | As generated | 1st inversion | 2nd inversion | Name |
|---|---|---|---|---|---|---|
| 0-1-2 | 1-11/10-6/5 | otonal | 1/1-11/10-6/5 | 1/1-12/11-20/11 | 1/1-5/3-11/6 | C^mv9no5 |
| 0-1-3 | 1-10/9-4/3 | otonal | 1/1-10/9-4/3 | 1/1-6/5-9/5 | 1/1-3/2-5/3 | C^m7no5 |
| 0-2-3 | 1-11/9-4/3 | otonal | 1/1-11/9-4/3 | 1/1-12/11-18/11 | 1/1-3/2-11/6 | C^m7no3 |
| 0-1-4 | 1-12/11-16/11 | otonal | 1/1-12/11-16/11 | 1/1-4/3-11/6 | 1/1-11/8-3/2 | C^4 |
| 0-2-4 | 1-6/5-22/15 | otonal | 1/1-6/5-22/15 | 1/1-11/9-5/3 | 1/1-15/11-18/11 | C^m(v5) |
| 0-3-4 | 1-4/3-22/15 | otonal | 1/1-4/3-22/15 | 1/1-11/10-3/2 | 1/1-15/11-20/11 | Cv2 |
| 0-1-5 | 1-11/10-8/5 | otonal | 1/1-11/10-8/5 | 1/1-16/11-9/5 | 1/1-5/4-11/8 | C^7(v5)no3 |
| 0-2-5 | 1-6/5-8/5 | otonal | 1/1-6/5-8/5 | 1/1-4/3-5/3 | 1/1-5/4-3/2 | Cv |
| 0-3-5 | 1-4/3-8/5 | utonal | 1/1-4/3-8/5 | 1/1-6/5-3/2 | 1/1-5/4-5/3 | C^m |
| 0-4-5 | 1-22/15-8/5 | otonal | 1/1-22/15-8/5 | 1/1-12/11-15/11 | 1/1-5/4-11/6 | Cv^7no5 |
| 0-1-6 | 1-10/9-16/9 | otonal | 1/1-10/9-16/9 | 1/1-8/5-9/5 | 1/1-9/8-5/4 | Cv,9no5 |
| 0-2-6 | 1-11/9-16/9 | otonal | 1/1-11/9-16/9 | 1/1-16/11-5/3 | 1/1-9/8-11/8 | Csus2(v5) |
| 0-3-6 | 1-4/3-16/9 | ambitonal | 1/1-4/3-16/9 | 1/1-4/3-3/2 | 1/1-9/8-3/2 | C4 or C2 |
| 0-4-6 | 1-16/11-16/9 | utonal | 1/1-16/11-16/9 | 1/1-11/9-11/8 | 1/1-9/8-5/3 | C^m^4no5 |
| 0-5-6 | 1-8/5-16/9 | utonal | 1/1-8/5-16/9 | 1/1-10/9-5/4 | 1/1-9/8-9/5 | C^m9no35 |
| 0-2-8 | 1-6/5-16/15 | otonal | 1/1-16/15-6/5 | 1/1-9/8-15/8 | 1/1-5/3-16/9 | CvM9no35 |
| 0-3-8 | 1-4/3-16/15 | ambitonal | 1/1-16/15-4/3 | 1/1-5/4-15/8 | 1/1-3/2-8/5 | CvM7no5 |
| 0-4-8 | 1-22/15-16/15 | otonal | 1/1-16/15-22/15 | 1/1-11/8-15/8 | 1/1-15/11-16/11 | C^4(v5) |
| 0-5-8 | 1-8/5-16/15 | ambitonal | 1/1-16/15-8/5 | 1/1-3/2-15/8 | 1/1-5/4-4/3 | CvM7no3 |
| 0-6-8 | 1-16/9-16/15 | utonal | 1/1-16/15-16/9 | 1/1-5/3-15/8 | 1/1-9/8-6/5 | C^m,9no5 |
| 0-1-9 | 1-11/10-7/6 | valinorsmic | 1/1-11/10-7/6 | 1/1-16/15-20/11 | 1/1-12/7-15/8 | Cmv9no5 |
| 0-3-9 | 1-4/3-7/6 | otonal | 1/1-7/6-4/3 | 1/1-8/7-12/7 | 1/1-3/2-7/4 | C7no3 |
| 0-4-9 | 1-16/11-7/6 | keenanismic | 1/1-7/6-16/11 | 1/1-5/4-12/7 | 1/1-11/8-8/5 | Cv,6no5 |
| 0-5-9 | 1-8/5-7/6 | keenanismic | 1/1-7/6-8/5 | 1/1-11/8-12/7 | 1/1-5/4-16/11 | Cv(v5) |
| 0-6-9 | 1-7/4-7/6 | utonal | 1/1-7/6-7/4 | 1/1-3/2-12/7 | 1/1-8/7-4/3 | Cm7no5 |
| 0-8-9 | 1-16/15-7/6 | valinorsmic | 1/1-16/15-7/6 | 1/1-11/10-15/8 | 1/1-12/7-20/11 | Cm^b9no5 |
| 0-1-10 | 1-12/11-14/11 | otonal | 1/1-12/11-14/11 | 1/1-7/6-11/6 | 1/1-11/7-12/7 | Cm^7no5 |
| 0-2-10 | 1-6/5-14/11 | valinorsmic | 1/1-6/5-14/11 | 1/1-16/15-5/3 | 1/1-11/7-15/8 | CvM7(^5)no3 |
| 0-4-10 | 1-16/11-14/11 | otonal | 1/1-14/11-16/11 | 1/1-8/7-11/7 | 1/1-11/8-7/4 | C7(^4)no5 |
| 0-5-10 | 1-8/5-14/11 | valinorsmic | 1/1-14/11-8/5 | 1/1-5/4-11/7 | 1/1-5/4-8/5 | Cv^b6 |
| 0-6-10 | 1-7/4-14/11 | utonal | 1/1-14/11-7/4 | 1/1-11/8-11/7 | 1/1-8/7-16/11 | C7(v4)no5 |
| 0-8-10 | 1-16/15-14/11 | valinorsmic | 1/1-16/15-14/11 | 1/1-6/5-15/8 | 1/1-11/7-5/3 | C^mvM7 |
| 0-9-10 | 1-7/6-14/11 | utonal | 1/1-7/6-14/11 | 1/1-12/11-12/7 | 1/1-11/7-11/6 | Cm,v11no5 |
| 0-1-11 | 1-11/10-7/5 | otonal | 1/1-11/10-7/5 | 1/1-14/11-20/11 | 1/1-10/7-11/7 | C^7(v4)no5 |
| 0-2-11 | 1-6/5-7/5 | otonal | 1/1-6/5-7/5 | 1/1-7/6-5/3 | 1/1-10/7-12/7 | C^m(vv5) |
| 0-3-11 | 1-4/3-7/5 | archytas | 1/1-4/3-7/5 | 1/1-16/15-3/2 | 1/1-10/7-15/8 | C^b2 |
| 0-5-11 | 1-8/5-7/5 | otonal | 1/1-7/5-8/5 | 1/1-8/7-10/7 | 1/1-5/4-7/4 | Cv,7no5 |
| 0-6-11 | 1-7/4-7/5 | utonal | 1/1-7/5-7/4 | 1/1-5/4-10/7 | 1/1-8/7-8/5 | C7(vv5)no3 |
| 0-8-11 | 1-16/15-7/5 | archytas | 1/1-16/15-7/5 | 1/1-4/3-15/8 | 1/1-10/7-3/2 | CvM7(4) |
| 0-9-11 | 1-7/6-7/5 | utonal | 1/1-7/6-7/5 | 1/1-6/5-12/7 | 1/1-10/7-5/3 | Cm(vv5) |
| 0-10-11 | 1-14/11-7/5 | utonal | 1/1-14/11-7/5 | 1/1-11/10-11/7 | 1/1-10/7-20/11 | Cv4(vv5) |
| 0-1-12 | 1-10/9-14/9 | otonal | 1/1-10/9-14/9 | 1/1-7/5-9/5 | 1/1-9/7-10/7 | C,^^11no5 |
| 0-2-12 | 1-11/9-14/9 | otonal | 1/1-11/9-14/9 | 1/1-14/11-18/11 | 1/1-9/7-11/7 | C(^5) |
| 0-3-12 | 1-4/3-14/9 | otonal | 1/1-4/3-14/9 | 1/1-7/6-3/2 | 1/1-9/7-12/7 | Cm |
| 0-4-12 | 1-16/11-14/9 | keenanismic | 1/1-16/11-14/9 | 1/1-16/15-11/8 | 1/1-9/7-15/8 | C,vM7no5 |
| 0-6-12 | 1-16/9-14/9 | otonal | 1/1-14/9-16/9 | 1/1-8/7-9/7 | 1/1-9/8-7/4 | C9no35 |
| 0-8-12 | 1-16/15-14/9 | keenanismic | 1/1-16/15-14/9 | 1/1-16/11-15/8 | 1/1-9/7-11/8 | C,^11no5 |
| 0-9-12 | 1-7/6-14/9 | utonal | 1/1-7/6-14/9 | 1/1-4/3-12/7 | 1/1-9/7-3/2 | C |
| 0-10-12 | 1-14/11-14/9 | utonal | 1/1-14/11-14/9 | 1/1-11/9-11/7 | 1/1-9/7-18/11 | C,v6no5 |
| 0-11-12 | 1-7/5-14/9 | utonal | 1/1-7/5-14/9 | 1/1-10/9-10/7 | 1/1-9/7-9/5 | C,^7no5 |
| 0-2-14 | 1-6/5-28/15 | otonal | 1/1-6/5-28/15 | 1/1-14/9-5/3 | 1/1-15/14-9/7 | |
| 0-3-14 | 1-4/3-28/15 | otonal | 1/1-4/3-28/15 | 1/1-7/5-3/2 | 1/1-15/14-10/7 | |
| 0-4-14 | 1-22/15-28/15 | otonal | 1/1-22/15-28/15 | 1/1-14/11-15/11 | 1/1-15/14-11/7 | |
| 0-5-14 | 1-8/5-28/15 | otonal | 1/1-8/5-28/15 | 1/1-7/6-5/4 | 1/1-15/14-12/7 | |
| 0-6-14 | 1-7/4-28/15 | utonal | 1/1-7/4-28/15 | 1/1-16/15-8/7 | 1/1-15/14-15/8 | |
| 0-8-14 | 1-16/15-28/15 | otonal | 1/1-16/15-28/15 | 1/1-7/4-15/8 | 1/1-15/14-8/7 | |
| 0-9-14 | 1-7/6-28/15 | utonal | 1/1-7/6-28/15 | 1/1-8/5-12/7 | 1/1-15/14-5/4 | |
| 0-10-14 | 1-14/11-28/15 | utonal | 1/1-14/11-28/15 | 1/1-22/15-11/7 | 1/1-15/14-15/11 | |
| 0-11-14 | 1-7/5-28/15 | utonal | 1/1-7/5-28/15 | 1/1-4/3-10/7 | 1/1-15/14-3/2 | |
| 0-12-14 | 1-14/9-28/15 | utonal | 1/1-14/9-28/15 | 1/1-6/5-9/7 | 1/1-15/14-5/3 |
Tetrads
| Chord | Transversal | Type | As generated | First inversion | Second inversion | Third inversion | Name |
|---|---|---|---|---|---|---|---|
| 0-1-2-3 | 1-10/9-11/9-4/3 | otonal | 1/1-10/9-11/9-4/3 | 1/1-11/10-6/5-9/5 | 1/1-12/11-18/11-20/11 | 1/1-3/2-5/3-11/6 | Cv6^7no3 |
| 0-1-2-4 | 1-11/10-11/9-22/15 | utonal | 1/1-11/10-11/9-22/15 | 1/1-10/9-4/3-20/11 | 1/1-6/5-18/11-9/5 | 1/1-15/11-3/2-5/3 | Cv6(^4) |
| 0-1-3-4 | 1-10/9-4/3-22/15 | ptolemismic | 1/1-10/9-4/3-22/15 | 1/1-6/5-4/3-9/5 | 1/1-11/10-3/2-5/3 | 1/1-15/11-3/2-20/11 | Cv2v6 |
| 0-1-2-5 | 1-11/10-6/5-8/5 | otonal | 1/1-11/10-6/5-8/5 | 1/1-12/11-16/11-20/11 | 1/1-4/3-5/3-11/6 | 1/1-5/4-11/8-3/2 | Cv^4 |
| 0-1-3-5 | 1-11/10-4/3-8/5 | ptolemismic | 1/1-11/10-4/3-8/5 | 1/1-6/5-16/11-20/11 | 1/1-6/5-3/2-5/3 | 1/1-5/4-11/8-5/3 | C^mv6 |
| 0-1-4-5 | 1-11/10-16/11-8/5 | biyatismic | 1/1-11/10-16/11-8/5 | 1/1-4/3-16/11-20/11 | 1/1-11/10-11/8-3/2 | 1/1-5/4-11/8-20/11 | C^4v9 |
| 0-2-3-5 | 1-6/5-4/3-8/5 | ambitonal | 1/1-6/5-4/3-8/5 | 1/1-10/9-4/3-5/3 | 1/1-6/5-3/2-9/5 | 1/1-5/4-3/2-5/3 | Cv6 or C^m7 |
| 0-2-4-5 | 1-6/5-16/11-8/5 | ptolemismic | 1/1-6/5-16/11-8/5 | 1/1-6/5-4/3-5/3 | 1/1-11/10-11/8-5/3 | 1/1-5/4-3/2-9/5 | Cv^7 |
| 0-2-4-6 | 1-6/5-16/11-7/4 | supermagic | 1/1-6/5-16/11-7/4 | 1/1-6/5-16/11-5/3 | 1/1-6/5-11/8-5/3 | 1/1-8/7-11/8-5/3 | C^m,7(v5) or C^mv6^11no5 |
| 0-3-6-9 | 1-4/3-7/4-7/6 | archytas | 1/1/1-7/6-4/3-7/4 | 1/1-8/7-3/2-12/7 | 1/1-4/3-3/2-7/4 | 1/1-8/7-4/3-3/2 | C7sus4 |
| 0-3-9-12 | 1-4/3-7/6-14/9 | archytas | 1/1-7/6-4/3-14/9 | 1/1-7/6-3/2-7/4 | 1/1-9/8-4/3-12/7 | 1/1-9/7-3/2-12/7 | Cm7 or C6 |
| 0-4-8-12 | 1-16/11-16/15-14/9 | zeus | 1/1-16/15-16/11-14/9 | 1/1-15/11-16/11-15/8 | 1/1-16/15-11/8-22/15 | 1/1-9/7-11/8-15/8 | C,vM7^11no5 |
Pentads
| Chord | Transversal | Type | Name |
|---|---|---|---|
| 0-1-2-3-6 | 1-10/9-11/9-4/3-16/9 | otonal | Cv,9^11 |
| 0-2-3-4-6 | 1-6/5-4/3-16/11-16/9 | supermagic | C^m,7,11(v5) or C4^7v9 or C^4v6,9 |
| 0-3-4-5-6 | 1-4/3-16/11-8/5-16/9 | utonal | C^mv9,11 |
| 0-2-4-6-8 | 1-6/5-16/11-7/4-16/15 | porcupine | C^m,7(v5) or C^mv6^11no5 |
| 0-3-6-9-12 | 1-4/3-7/4-7/6-14/9 | archytas | C9(4) or C6,9 or Cm7,11 |