Chords of huygens: Difference between revisions
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Below are listed the [[Dyadic_chord|dyadic chords]] of 11-limit [[ | Below are listed the [[Dyadic_chord|dyadic chords]] of 11-limit [[Meantone family#Undecimal meantone (huygens)|huygens temperament]]. Huygens is one of the two extensions of septimal meantone, which itself is the main extension of 5-limit meantone; this is the temperament tempering out 81/80, 99/98, and 126/125. Typing the chords requires consideration of the fact that meantone conflates 9/8 and 10/9 and also 9/7 and 14/11. 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 which employs 9/8 and 16/9. | ||
(For the inversions, there has been no attempt to note any equivalencies. Keep in mind that 10/9 can be thought of as 9/8, 16/9 as 9/5, 9/7 as 14/11, and vice versa.) | (For the inversions, there has been no attempt to note any equivalencies. Keep in mind that 10/9 can be thought of as 9/8, 16/9 as 9/5, 9/7 as 14/11, and vice versa.) | ||
Latest revision as of 21:58, 22 June 2025
Below are listed the dyadic chords of 11-limit huygens temperament. Huygens is one of the two extensions of septimal meantone, which itself is the main extension of 5-limit meantone; this is the temperament tempering out 81/80, 99/98, and 126/125. Typing the chords requires consideration of the fact that meantone conflates 9/8 and 10/9 and also 9/7 and 14/11. 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 which employs 9/8 and 16/9.
(For the inversions, there has been no attempt to note any equivalencies. Keep in mind that 10/9 can be thought of as 9/8, 16/9 as 9/5, 9/7 as 14/11, and vice versa.)
Chords requiring tempering only by 81/80 are labeled didymic, by 99/98 mothwellsmic, by 126/125 starling, by 176/175 valinorsmic, by 225/224 marvel, and by 441/440 werckismic. Chords which require any two of 81/80, 99/98 or 441/440 are labeled euterpe, by and two of 81/80, 126/125, or 225/224 erato, and by any two of 99/98, 176/175, or 225/224 minerva. A chord requiring any three independent commas from those discussed above is labeled meantone.
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 meantone[7], six times in meantone[12], and 13 times in meantone[19].
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: For instance, the minor chord (1/1 – 6/5 – 3/2, or 10:12:15) is the second inversion of the generated 0 – 3 – 4 chord.
Though we're used to thinking of 10:12:15 as the definitive "minor chord", with all inversions coming from that, there is nothing definitive about calling these lists below "chord" or "inversion". That's just the way the generators came out.
Meantone has MOS of size 5, 7, 12, 19, 31, 43 and 74. While one might suppose meantone has been thoroughly explored, this really isn't true in the 7-limit, and it can hardly be said to have been explored at all in the 11-limit. The 19 note MOS would seem to be a good place to start such explorations.
The bolded inversions are named using ups and downs, to avoid aug, dim, double-aug and double-dim intervals. One up is −12 fifths, a descending Pythagorean comma. Thus ^1 = d2 and ^C = Dbb. One up represents ~64/63, ~45/44, and ~40/39. If c is the difference between 700 ¢ and the size of the generator in cents, then one up is equal to 12c.
| Genspan | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cents (31edo) | 0 | 697 | 194 | 890 | 387 | 1084 | 581 | 77 | 774 | 271 | 968 | 465 | 1161 | 658 | 155 | 852 | 348 | 1045 | 542 |
| Ratio | 1/1 | 3/2 | 9/8 10/9 |
5/3 | 5/4 | 15/8 | 45/32 7/5 |
21/20 | 63/40 14/9 |
7/6 | 7/4 | 21/16 13/10 |
35/18 | 13/9 | 11/10 13/12 |
13/8 | 11/9 | 11/6 | 11/8 |
| Interval | P1 | P5 | M2 | M6 | M3 | M7 | A4 vd5 |
A1 vm2 |
A5 vm6 |
A2 vm3 |
A6 vm7 |
A3 v4 |
A7 v8 |
AA4 v5 |
AA1 vM2 |
AA5 vM6 |
AA2 vM3 |
AA6 vM7 |
AA3 vA4 |
| Note (in C) |
C | G | D | A | E | B | F# | C# | G# vAb |
D# vEb |
A# vBb |
E# vF |
B# vC |
Fx vG |
Cx vD |
Gx vA |
Dx vE |
Ax vB |
Ex vF# |
TODO: complete the tables
Triads
| # | Chord | Transversal | Type | As generated | 1st inversion | 2nd inversion | Name |
|---|---|---|---|---|---|---|---|
| 1 | 0 – 1 – 2 | 1 – 3/2 – 9/8 | ambitonal | 1/1 – 9/8 – 3/2 | 1/1 – 4/3 – 16/9 | 1/1 – 4/3 – 3/2 | C2 or C4 |
| 2 | 0 – 1 – 3 | 1 – 3/2 – 5/3 | otonal | 1/1 – 3/2 – 5/3 | 1/1 – 10/9 – 4/3 | 1/1 – 6/5 – 9/5 | Cm7no5 |
| 3 | 0 – 2 – 3 | 1 – 10/9 – 5/3 | utonal | 1/1 – 10/9 – 5/3 | 1/1 – 3/2 – 9/5 | 1/1 – 6/5 – 4/3 | C7no3 |
| 4 | 0 – 1 – 4 | 1 – 3/2 – 5/4 | otonal | 1/1 – 5/4 – 3/2 | 1/1 – 6/5 – 8/5 | 1/1 – 4/3 – 5/3 | C |
| 5 | 0 – 2 – 4 | 1 – 9/8 – 5/4 | otonal | 1/1 – 9/8 – 5/4 | 1/1 – 10/9 – 16/9 | 1/1 – 8/5 – 9/5 | Cadd9no5 |
| 6 | 0 – 3 – 4 | 1 – 5/3 – 5/4 | utonal | 1/1 – 5/4 – 5/3 | 1/1 – 4/3 – 8/5 | 1/1 – 6/5 – 3/2 | Cm |
| 7 | 0 – 2 – 6 | 1 – 9/8 – 7/5 | marvel | 1/1 – 9/8 – 7/5 | 1/1 – 5/4 – 16/9 | 1/1 – 10/7 – 8/5 | C7no5 |
| 8 | 0 – 3 – 6 | 1 – 5/3 – 7/5 | starling | 1/1 – 7/5 – 5/3 | 1/1 – 6/5 – 10/7 | 1/1 – 6/5 – 5/3 | Cdim |
| 9 | 0 – 4 – 6 | 1 – 5/4 – 7/5 | marvel | 1/1 – 5/4 – 7/5 | 1/1 – 9/8 – 8/5 | 1/1 – 10/7 – 16/9 | C(vb5) |
| 10 | 0 – 2 – 8 | 1 – 10/9 – 14/9 | otonal | 1/1 – 10/9 – 14/9 | 1/1 – 7/5 – 9/5 | 1/1 – 9/7 – 10/7 | C2(#5) or C^(b5) |
| 11 | 0 – 4 – 8 | 1 – 5/4 – 14/9 | marvel | 1/1 – 5/4 – 14/9 | 1/1 – 5/4 – 8/5 | 1/1 – 9/7 – 8/5 | Caug |
| 12 | 0 – 6 – 8 | 1 – 7/5 – 14/9 | utonal | 1/1 – 7/5 – 14/9 | 1/1 – 10/9 – 10/7 | 1/1 – 9/7 – 9/5 | C2(b5) or C^,7no5 |
| 13 | 0 – 1 – 9 | 1 – 3/2 – 7/6 | otonal | 1/1 – 7/6 – 3/2 | 1/1 – 9/7 – 12/7 | 1/1 – 4/3 – 14/9 | Cvm |
| 14 | 0 – 3 – 9 | 1 – 5/3 – 7/6 | otonal | 1/1 – 7/6 – 5/3 | 1/1 – 10/7 – 12/7 | 1/1 – 6/5 – 7/5 | Cdim7no3 or Cdim(v5) |
| 15 | 0 – 6 – 9 | 1 – 7/5 – 7/6 | utonal | 1/1 – 7/6 – 7/5 | 1/1 – 6/5 – 12/7 | 1/1 – 10/7 – 5/3 | Cvdim(v5) or Cdim7no5 |
| 16 | 0 – 8 – 9 | 1 – 14/9 – 7/6 | utonal | 1/1 – 7/6 – 14/9 | 1/1 – 4/3 – 12/7 | 1/1 – 9/7 – 3/2 | C^ |
| 17 | 0 – 1 – 10 | 1 – 3/2 – 7/4 | otonal | 1/1 – 3/2 – 7/4 | 1/1 – 7/6 – 4/3 | 1/1 – 8/7 – 12/7 | Cv7no3 |
| 18 | 0 – 2 – 10 | 1 – 9/8 – 7/4 | otonal | 1/1 – 9/8 – 7/4 | 1/1 – 14/9 – 16/9 | 1/1 – 8/7 – 9/7 | Cv9no35 |
| 19 | 0 – 4 – 10 | 1 – 5/4 – 7/4 | otonal | 1/1 – 5/4 – 7/4 | 1/1 – 7/5 – 8/5 | 1/1 – 8/7 – 10/7 | C,v7no5 |
| 20 | 0 – 6 – 10 | 1 – 7/5 – 7/4 | utonal | 1/1 – 7/5 – 7/4 | 1/1 – 5/4 – 10/7 | 1/1 – 8/7 – 8/5 | C(b5) |
| 21 | 0 – 8 – 10 | 1 – 14/9 – 7/4 | utonal | 1/1 – 14/9 – 7/4 | 1/1 – 9/8 – 9/7 | 1/1 – 8/7 – 16/9 | C^,9no5 |
| 22 | 0 – 9 – 10 | 1 – 7/6 – 7/4 | utonal | 1/1 – 7/6 – 7/4 | 1/1 – 3/2 – 12/7 | 1/1 – 8/7 – 4/3 | Cvm7no5 |
| 23 | 0 – 4 – 14 | 1 – 5/4 – 11/10 | valinorsmic | 1/1 – 11/10 – 5/4 | 1/1 – 8/7 – 20/11 | 1/1 – 8/5 – 7/4 | |
| 24 | 0 – 6 – 14 | 1 – 7/5 – 11/10 | otonal | 1/1 – 11/10 – 7/5 | 1/1 – 14/11 – 20/11 | 1/1 – 10/7 – 11/7 | |
| 25 | 0 – 8 – 14 | 1 – 11/7 – 11/10 | utonal | 1/1 – 11/10 – 11/7 | 1/1 – 10/7 – 20/11 | 1/1 – 14/11 – 7/5 | |
| 26 | 0 – 10 – 14 | 1 – 7/4 – 11/10 | valinorsmic | 1/1 – 11/10 – 7/4 | 1/1 – 8/5 – 20/11 | 1/1 – 8/7 – 5/4 | |
| 27 | 0 – 2 – 16 | 1 – 10/9 – 11/9 | otonal | 1/1 – 10/9 – 11/9 | 1/1 – 11/10 – 9/5 | 1/1 – 18/11 – 20/11 | |
| 28 | 0 – 6 – 16 | 1 – 7/5 – 11/9 | werckismic | 1/1 – 11/9 – 7/5 | 1/1 – 8/7 – 18/11 | 1/1 – 10/7 – 7/4 | |
| 29 | 0 – 8 – 16 | 1 – 14/9 – 11/9 | otonal | 1/1 – 11/9 – 14/9 | 1/1 – 14/11 – 18/11 | 1/1 – 9/7 – 11/7 | |
| 30 | 0 – 10 – 16 | 1 – 7/4 – 11/9 | werckismic | 1/1 – 11/9 – 7/4 | 1/1 – 10/7 – 18/11 | 1/1 – 8/7 – 7/5 | |
| 31 | 0 – 14 – 16 | 1 – 11/10 – 11/9 | utonal | 1/1 – 11/10 – 11/9 | 1/1 – 10/9 – 20/11 | 1/1 – 18/11 – 9/5 | |
| 32 | 0 – 1 – 17 | 1 – 3/2 – 11/6 | otonal | 1/1 – 3/2 – 11/6 | 1/1 – 11/9 – 4/3 | 1/1 – 12/11 – 18/11 | |
| 33 | 0 – 3 – 17 | 1 – 5/3 – 11/6 | otonal | 1/1 – 5/3 – 11/6 | 1/1 – 11/10 – 6/5 | 1/1 – 12/11 – 20/11 | |
| 34 | 0 – 8 – 17 | 1 – 11/7 – 11/6 | utonal | 1/1 – 11/7 – 11/6 | 1/1 – 7/6 – 14/11 | 1/1 – 12/11 – 12/7 | |
| 35 | 0 – 9 – 17 | 1 – 7/6 – 11/6 | otonal | 1/1 – 7/6 – 11/6 | 1/1 – 11/7 – 12/7 | 1/1 – 12/11 – 14/11 | |
| 36 | 0 – 14 – 17 | 1 – 11/10 – 11/6 | utonal | 1/1 – 11/10 – 11/6 | 1/1 – 5/3 – 20/11 | 1/1 – 12/11 – 6/5 | |
| 37 | 0 – 16 – 17 | 1 – 11/9 – 11/6 | utonal | 1/1 – 11/9 – 11/6 | 1/1 – 3/2 – 18/11 | 1/1 – 12/11 – 4/3 | |
| 38 | 0 – 1 – 18 | 1 – 3/2 – 11/8 | otonal | 1/1 – 11/8 – 3/2 | 1/1 – 12/11 – 16/11 | 1/1 – 4/3 – 11/6 | |
| 39 | 0 – 2 – 18 | 1 – 9/8 – 11/8 | otonal | 1/1 – 9/8 – 11/8 | 1/1 – 11/9 – 16/9 | 1/1 – 16/11 – 18/11 | |
| 40 | 0 – 4 – 18 | 1 – 5/4 – 11/8 | otonal | 1/1 – 5/4 – 11/8 | 1/1 – 11/10 – 8/5 | 1/1 – 16/11 – 20/11 | |
| 41 | 0 – 8 – 18 | 1 – 11/7 – 11/8 | utonal | 1/1 – 11/8 – 11/7 | 1/1 – 8/7 – 16/11 | 1/1 – 14/11 – 7/4 | |
| 42 | 0 – 9 – 18 | 1 – 7/6 – 11/8 | mothwellsmic | 1/1 – 7/6 – 11/8 | 1/1 – 7/6 – 12/7 | 1/1 – 16/11 – 12/7 | |
| 43 | 0 – 10 – 18 | 1 – 7/4 – 11/8 | otonal | 1/1 – 11/8 – 7/4 | 1/1 – 14/11 – 16/11 | 1/1 – 8/7 – 11/7 | |
| 44 | 0 – 14 – 18 | 1 – 11/10 – 11/8 | utonal | 1/1 – 11/10 – 11/8 | 1/1 – 5/4 – 20/11 | 1/1 – 16/11 – 8/5 | |
| 45 | 0 – 16 – 18 | 1 – 11/9 – 11/8 | utonal | 1/1 – 11/9 – 11/8 | 1/1 – 9/8 – 18/11 | 1/1 – 16/11 – 16/9 | |
| 46 | 0 – 17 – 18 | 1 – 11/6 – 11/8 | utonal | 1/1 – 11/8 – 11/6 | 1/1 – 4/3 – 16/11 | 1/1 – 12/11 – 3/2 |
Tetrads
| # | Chord | Transversal | Type | As generated | 1st inversion | 2nd inversion | 3rd inversion | Name |
|---|---|---|---|---|---|---|---|---|
| 1 | 0 – 1 – 2 – 3 | 1 – 3/2 – 9/8 – 5/3 | didymic | 1/1 – 9/8 – 3/2 – 5/3 | 1/1 – 4/3 – 3/2 – 16/9 | 1/1 – 9/8 – 4/3 – 3/2 | 1/1 – 6/5 – 4/3 – 9/5 | C7sus4 or C4,9 |
| 2 | 0 – 1 – 2 – 4 | 1 – 3/2 – 9/8 – 5/4 | otonal | 1/1 – 9/8 – 5/4 – 3/2 | 1/1 – 10/9 – 4/3 – 16/9 | 1/1 – 6/5 – 8/5 – 9/5 | 1/1 – 4/3 – 3/2 – 5/3 | Cadd9 |
| 3 | 0 – 1 – 3 – 4 | 1 – 3/2 – 5/3 – 5/4 | ambitonal | 1/1 – 5/4 – 3/2 – 5/3 | 1/1 – 6/5 – 4/3 – 8/5 | 1/1 – 9/8 – 4/3 – 5/3 | 1/1 – 6/5 – 3/2 – 9/5 | C6 or Cm7 |
| 4 | 0 – 2 – 3 – 4 | 1 – 10/9 – 5/3 – 5/4 | utonal | 1/1 – 10/9 – 5/4 – 5/3 | 1/1 – 9/8 – 3/2 – 9/5 | 1/1 – 4/3 – 8/5 – 16/9 | 1/1 – 6/5 – 4/3 – 3/2 | C9no3 |
| 5 | 0 – 2 – 3 – 6 | 1 – 9/8 – 5/3 – 7/5 | erato | 1/1 – 9/8 – 7/5 – 5/3 | 1/1 – 5/4 – 3/2 – 16/9 | 1/1 – 6/5 – 10/7 – 8/5 | 1/1 – 6/5 – 4/3 – 5/3 | C7 |
| 6 | 0 – 2 – 4 – 6 | 1 – 9/8 – 5/4 – 7/5 | erato | 1/1 – 9/8 – 5/4 – 7/5 | 1/1 – 10/9 – 5/4 – 16/9 | 1/1 – 9/8 – 8/5 – 9/5 | 1/1 – 10/7 – 8/5 – 9/5 | C9no5 |
| 7 | 0 – 3 – 4 – 6 | 1 – 5/3 – 5/4 – 7/5 | erato | 1/1 – 5/4 – 7/5 – 5/3 | 1/1 – 9/8 – 4/3 – 8/5 | 1/1 – 6/5 – 10/7 – 16/9 | 1/1 – 6/5 – 3/2 – 5/3 | Cm7(b5) or Cm6 |
| 8 | 0 – 2 – 4 – 8 | 1 – 9/8 – 5/4 – 14/9 | erato | 1/1 – 9/8 – 5/4 – 14/9 | 1/1 – 10/9 – 7/5 – 16/9 | 1/1 – 5/4 – 8/5 – 9/5 | 1/1 – 9/7 – 10/7 – 8/5 | Caug,9 |
| 9 | 0 – 2 – 6 – 8 | 1 – 9/8 – 7/5 – 14/9 | erato | 1/1 – 9/8 – 7/5 – 14/9 | 1/1 – 5/4 – 7/5 – 16/9 | 1/1 – 10/9 – 10/7 – 8/5 | 1/1 – 9/7 – 10/7 – 9/5 | C7(vb5) |
| 10 | 0 – 4 – 6 – 8 | 1 – 5/4 – 7/5 – 14/9 | erato | 1/1 – 5/4 – 7/5 – 14/9 | 1/1 – 9/8 – 5/4 – 8/5 | 1/1 – 10/9 – 10/7 – 16/9 | 1/1 – 9/7 – 8/5 – 9/5 | C9(b5)no3 |
| 11 | 0 – 1 – 3 – 9 | 1 – 3/2 – 5/3 – 7/6 | otonal | 1/1 – 7/6 – 3/2 – 5/3 | 1/1 – 9/7 – 10/7 – 12/7 | 1/1 – 10/9 – 4/3 – 14/9 | 1/1 – 6/5 – 7/5 – 9/5 | Cm7(vb5) |
| 12 | 0 – 3 – 6 – 9 | 1 – 5/3 – 7/5 – 7/6 | starling | 1/1 – 7/6 – 7/5 – 5/3 | 1/1 – 6/5 – 10/7 – 12/7 | 1/1 – 6/5 – 10/7 – 5/3 | 1/1 – 6/5 – 7/5 – 5/3 | Cdim7 |
| 13 | 0 – 6 – 8 – 9 | 1 – 7/5 – 14/9 – 7/6 | utonal | 1/1 – 7/6 – 7/5 – 14/9 | 1/1 – 6/5 – 4/3 – 12/7 | 1/1 – 10/9 – 10/7 – 5/3 | 1/1 – 9/7 – 3/2 – 9/5 | C^,7 |
| 14 | 0 – 1 – 2 – 10 | 1 – 3/2 – 9/8 – 7/4 | otonal | 1/1 – 9/8 – 3/2 – 7/4 | 1/1 – 4/3 – 14/9 – 16/9 | 1/1 – 7/6 – 4/3 – 3/2 | 1/1 – 8/7 – 9/7 – 12/7 | Cv7no3 or Cvm,11 |
| 15 | 0 – 1 – 4 – 10 | 1 – 3/2 – 5/4 – 7/4 | otonal | 1/1 – 5/4 – 3/2 – 7/4 | 1/1 – 6/5 – 7/5 – 8/5 | 1/1 – 7/6 – 4/3 – 5/3 | 1/1 – 8/7 – 10/7 – 12/7 | C,v7 |
| 16 | 0 – 2 – 4 – 10 | 1 – 9/8 – 5/4 – 7/4 | otonal | 1/1 – 9/8 – 5/4 – 7/4 | 1/1 – 10/9 – 14/9 – 16/9 | 1/1 – 7/5 – 8/5 – 9/5 | 1/1 – 8/7 – 9/7 – 10/7 | C9(v7)no5 |
| 17 | 0 – 2 – 6 – 10 | 1 – 9/8 – 7/5 – 7/4 | marvel | 1/1 – 9/8 – 7/5 – 7/4 | 1/1 – 5/4 – 14/9 – 16/9 | 1/1 – 5/4 – 10/7 – 8/5 | 1/1 – 8/7 – 9/7 – 8/5 | C7(#5) or Caug7 |
| 18 | 0 – 4 – 6 – 10 | 1 – 5/4 – 7/5 – 7/4 | marvel | 1/1 – 5/4 – 7/5 – 7/4 | 1/1 – 9/8 – 7/5 – 8/5 | 1/1 – 5/4 – 10/7 – 16/9 | 1/1 – 8/7 – 10/7 – 8/5 | C7(b5) |
| 19 | 0 – 2 – 8 – 10 | 1 – 9/8 – 14/9 – 7/4 | didymic | 1/1 – 9/8 – 14/9 – 7/4 | 1/1 – 7/5 – 14/9 – 16/9 | 1/1 – 9/8 – 9/7 – 10/7 | 1/1 – 8/7 – 9/7 – 16/9 | |
| 20 | 0 – 4 – 8 – 10 | 1 – 5/4 – 14/9 – 7/4 | marvel | 1/1 – 5/4 – 14/9 – 7/4 | 1/1 – 5/4 – 7/5 – 8/5 | 1/1 – 9/8 – 9/7 – 8/5 | 1/1 – 8/7 – 10/7 – 16/9 | |
| 21 | 0 – 6 – 8 – 10 | 1 – 7/5 – 14/9 – 7/4 | utonal | 1/1 – 7/5 – 14/9 – 7/4 | 1/1 – 10/9 – 5/4 – 10/7 | 1/1 – 9/8 – 9/7 – 9/5 | 1/1 – 8/7 – 8/5 – 16/9 | C,9(b5) |
| 22 | 0 – 1 – 9 – 10 | 1 – 3/2 – 7/6 – 7/4 | ambitonal | 1/1 – 7/6 – 3/2 – 7/4 | 1/1 – 9/7 – 3/2 – 12/7 | 1/1 – 7/6 – 4/3 – 14/9 | 1/1 – 8/7 – 4/3 – 12/7 | Cvm7 or C^6 |
| 23 | 0 – 6 – 9 – 10 | 1 – 7/5 – 7/6 – 7/4 | utonal | 1/1 – 7/6 – 7/5 – 7/4 | 1/1 – 6/5 – 3/2 – 12/7 | 1/1 – 5/4 – 10/7 – 5/3 | 1/1 – 8/7 – 4/3 – 8/5 | Cvm7(vb5) or Cm^6 or C6(b5) |
| 24 | 0 – 8 – 9 – 10 | 1 – 14/9 – 7/6 – 7/4 | utonal | 1/1 – 7/6 – 14/9 – 7/4 | 1/1 – 4/3 – 3/2 – 12/7 | 1/1 – 9/8 – 9/7 – 3/2 | 1/1 – 8/7 – 4/3 – 16/9 | Cvm7(#5) or C4^6 or C^,9 |
| 25 | 0 – 4 – 6 – 14 | 1 – 5/4 – 7/5 – 11/10 | minerva | 1/1 – 11/10 – 5/4 – 7/5 | 1/1 – 9/8 – 14/11 – 20/11 | 1/1 – 9/8 – 8/5 – 7/4 | 1/1 – 10/7 – 11/7 – 16/9 | |
| 26 | 0 – 4 – 8 – 14 | 1 – 5/4 – 14/9 – 11/10 | minerva | 1/1 – 11/10 – 5/4 – 14/9 | 1/1 – 9/8 – 10/7 – 20/11 | 1/1 – 5/4 – 8/5 – 7/4 | 1/1 – 9/7 – 7/5 – 8/5 | |
| 27 | 0 – 6 – 8 – 14 | 1 – 7/5 – 14/9 – 11/10 | euterpe | 1/1 – 11/10 – 7/5 – 14/9 | 1/1 – 14/11 – 10/7 – 20/11 | 1/1 – 10/9 – 10/7 – 11/7 | 1/1 – 9/7 – 7/5 – 9/5 | C^,7(vb5) |
| 28 | 0 – 4 – 10 – 14 | 1 – 5/4 – 7/4 – 11/10 | valinorsmic | 1/1 – 11/10 – 5/4 – 7/4 | 1/1 – 25/22 – 8/5 – 20/11 | 1/1 – 7/5 – 8/5 – 7/4 | 1/1 – 8/7 – 5/4 – 10/7 | C,^9(b5) |
| 29 | 0 – 6 – 10 – 14 | 1 – 7/5 – 7/4 – 11/10 | minerva | 1/1 – 11/10 – 7/5 – 7/4 | 1/1 – 14/11 – 8/5 – 20/11 | 1/1 – 5/4 – 10/7 – 11/7 | 1/1 – 8/7 – 5/4 – 8/5 | |
| 30 | 0 – 8 – 10 – 14 | 1 – 14/9 – 7/4 – 11/10 | minerva | 1/1 – 11/10 – 14/9 – 7/4 | 1/1 – 7/5 – 8/5 – 20/11 | 1/1 – 9/8 – 9/7 – 7/5 | 1/1 – 8/7 – 5/4 – 16/9 | |
| 31 | 0 – 2 – 6 – 16 | 1 – 9/8 – 7/5 – 11/9 | meantone | 1/1 – 9/8 – 11/9 – 7/5 | 1/1 – 11/10 – 5/4 – 16/9 | 1/1 – 8/7 – 18/11 – 20/11 | 1/1 – 10/7 – 8/5 – 7/4 | |
| 32 | 0 – 2 – 8 – 16 | 1 – 10/9 – 14/9 – 11/9 | otonal | 1/1 – 10/9 – 11/9 – 14/9 | 1/1 – 11/10 – 7/5 – 9/5 | 1/1 – 14/11 – 18/11 – 20/11 | 1/1 – 9/7 – 10/7 – 11/7 | |
| 33 | 0 – 6 – 8 – 16 | 1 – 7/5 – 14/9 – 11/9 | euterpe | 1/1 – 11/9 – 7/5 – 14/9 | 1/1 – 8/7 – 14/11 – 18/11 | 1/1 – 9/8 – 10/7 – 7/4 | 1/1 – 9/7 – 11/7 – 9/5 | |
| 34 | 0 – 2 – 10 – 16 | 1 – 9/8 – 7/4 – 11/9 | euterpe | 1/1 – 9/8 – 11/9 – 7/4 | 1/1 – 11/10 – 14/9 – 16/9 | 1/1 – 10/7 – 18/11 – 20/11 | 1/1 – 8/7 – 9/7 – 7/5 | |
| 35 | 0 – 6 – 10 – 16 | 1 – 7/5 – 7/4 – 11/9 | werckismic | 1/1 – 11/9 – 7/5 – 7/4 | 1/1 – 8/7 – 10/7 – 18/11 | 1/1 – 5/4 – 10/7 – 7/4 | 1/1 – 8/7 – 7/5 – 8/5 | |
| 36 | 0 – 8 – 10 – 16 | 1 – 14/9 – 7/4 – 11/9 | euterpe | 1/1 – 11/9 – 14/9 – 7/4 | 1/1 – 14/11 – 10/7 – 18/11 | 1/1 – 9/8 – 9/7 – 11/7 | 1/1 – 8/7 – 7/5 – 16/9 | |
| 37 | 0 – 6 – 14 – 16 | 1 – 7/5 – 11/10 – 11/9 | euterpe | 1/1 – 11/10 – 11/9 – 7/5 | 1/1 – 10/9 – 14/11 – 20/11 | 1/1 – 8/7 – 18/11 – 9/5 | 1/1 – 10/7 – 11/7 – 7/4 | |
| 38 | 0 – 8 – 14 – 16 | 1 – 11/7 – 11/10 – 11/9 | utonal | 1/1 – 11/10 – 11/9 – 11/7 | 1/1 – 10/9 – 10/7 – 20/11 | 1/1 – 9/7 – 18/11 – 9/5 | 1/1 – 14/11 – 7/5 – 14/9 | |
| 39 | 0 – 10 – 14 – 16 | 1 – 7/4 – 11/10 – 11/9 | meantone | 1/1 – 11/10 – 11/9 – 7/4 | 1/1 – 10/9 – 8/5 – 20/11 | 1/1 – 10/7 – 18/11 – 9/5 | 1/1 – 8/7 – 5/4 – 7/5 | |
| 40 | 0 – 1 – 3 – 17 | 1 – 3/2 – 5/3 – 11/6 | otonal | 1/1 – 3/2 – 5/3 – 11/6 | 1/1 – 10/9 – 11/9 – 4/3 | 1/1 – 11/10 – 6/5 – 9/5 | 1/1 – 12/11 – 18/11 – 20/11 | |
| 41 | 0 – 1 – 9 – 17 | 1 – 3/2 – 7/6 – 11/6 | otonal | 1/1 – 7/6 – 3/2 – 11/6 | 1/1 – 9/7 – 11/7 – 12/7 | 1/1 – 11/9 – 4/3 – 14/9 | 1/1 – 12/11 – 14/11 – 18/11 | |
| 42 | 0 – 3 – 9 – 17 | 1 – 5/3 – 7/6 – 11/6 | otonal | 1/1 – 7/6 – 5/3 – 11/6 | 1/1 – 10/7 – 11/7 – 12/7 | 1/1 – 11/10 – 6/5 – 7/5 | 1/1 – 12/11 – 14/11 – 20/11 | |
| 43 | 0 – 8 – 9 – 17 | 1 – 14/9 – 7/6 – 11/6 | mothwellsmic | 1/1 – 7/6 – 14/9 – 11/6 | 1/1 – 4/3 – 11/7 – 12/7 | 1/1 – 7/6 – 9/7 – 3/2 | 1/1 – 12/11 – 14/11 – 12/7 | |
| 44 | 0 – 8 – 14 – 17 | 1 – 11/7 – 11/10 – 11/6 | utonal | 1/1 – 11/10 – 11/7 – 11/6 | 1/1 – 10/7 – 5/3 – 20/11 | 1/1 – 7/6 – 14/11 – 7/5 | 1/1 – 12/11 – 6/5 – 12/7 | |
| 45 | 0 – 8 – 16 – 17 | 1 – 11/7 – 11/9 – 11/6 | utonal | 1/1 – 11/9 – 11/7 – 11/6 | 1/1 – 9/7 – 3/2 – 18/11 | 1/1 – 7/6 – 14/11 – 14/9 | 1/1 – 12/11 – 4/3 – 12/7 | |
| 46 | 0 – 14 – 16 – 17 | 1 – 11/10 – 11/9 – 11/6 | utonal | 1/1 – 11/10 – 11/9 – 11/6 | 1/1 – 10/9 – 5/3 – 20/11 | 1/1 – 3/2 – 18/11 – 9/5 | 1/1 – 12/11 – 6/5 – 4/3 | |
| 47 | 0 – 1 – 2 – 18 | 1 – 3/2 – 9/8 – 11/8 | otonal | 1/1 – 9/8 – 11/8 – 3/2 | 1/1 – 11/9 – 4/3 – 16/9 | 1/1 – 12/11 – 16/11 – 18/11 | 1/1 – 4/3 – 3/2 – 11/6 | |
| 48 | 0 – 1 – 4 – 18 | 1 – 3/2 – 5/4 – 11/8 | otonal | 1/1 – 5/4 – 11/8 – 3/2 | 1/1 – 11/10 – 6/5 – 8/5 | 1/1 – 12/11 – 16/11 – 20/11 | 1/1 – 4/3 – 5/3 – 11/6 | |
| 49 | 0 – 2 – 4 – 18 | 1 – 9/8 – 5/4 – 11/8 | otonal | 1/1 – 9/8 – 5/4 – 11/8 | 1/1 – 10/9 – 11/9 – 16/9 | 1/1 – 11/10 – 8/5 – 9/5 | 1/1 – 16/11 – 18/11 – 20/11 | |
| 50 | 0 – 2 – 8 – 18 | 1 – 9/8 – 14/9 – 11/8 | euterpe | 1/1 – 9/8 – 11/8 – 14/9 | 1/1 – 11/9 – 7/5 – 16/9 | 1/1 – 8/7 – 16/11 – 18/11 | 1/1 – 9/7 – 10/7 – 7/4 | |
| 51 | 0 – 4 – 8 – 18 | 1 – 5/4 – 14/9 – 11/8 | minerva | 1/1 – 5/4 – 11/8 – 14/9 | 1/1 – 11/10 – 5/4 – 8/5 | 1/1 – 8/7 – 16/11 – 20/11 | 1/1 – 9/7 – 8/5 – 7/4 | |
| 52 | 0 – 1 – 9 – 18 | 1 – 3/2 – 7/6 – 11/8 | mothwellsmic | 1/1 – 7/6 – 11/8 – 3/2 | 1/1 – 7/6 – 9/7 – 12/7 | 1/1 – 12/11 – 16/11 – 12/7 | 1/1 – 4/3 – 14/9 – 11/6 | |
| 53 | 0 – 8 – 9 – 18 | 1 – 14/9 – 7/6 – 11/8 | mothwellsmic | 1/1 – 7/6 – 11/8 – 14/9 | 1/1 – 7/6 – 4/3 – 12/7 | 1/1 – 8/7 – 16/11 – 12/7 | 1/1 – 9/7 – 3/2 – 7/4 | |
| 54 | 0 – 1 – 10 – 18 | 1 – 3/2 – 7/4 – 11/8 | otonal | 1/1 – 11/8 – 3/2 – 7/4 | 1/1 – 12/11 – 14/11 – 16/11 | 1/1 – 7/6 – 4/3 – 11/6 | 1/1 – 8/7 – 11/7 – 12/7 | |
| 55 | 0 – 2 – 10 – 18 | 1 – 9/8 – 7/4 – 11/8 | otonal | 1/1 – 9/8 – 11/8 – 7/4 | 1/1 – 11/9 – 14/9 – 16/9 | 1/1 – 14/11 – 16/11 – 18/11 | 1/1 – 8/7 – 9/7 – 11/7 | |
| 56 | 0 – 4 – 10 – 18 | 1 – 5/4 – 7/4 – 11/8 | otonal | 1/1 – 5/4 – 11/8 – 7/4 | 1/1 – 11/10 – 7/5 – 8/5 | 1/1 – 14/11 – 16/11 – 20/11 | 1/1 – 8/7 – 10/7 – 11/7 | |
| 57 | 0 – 8 – 10 – 18 | 1 – 14/9 – 7/4 – 11/8 | mothwellsmic | 1/1 – 11/8 – 14/9 – 7/4 | 1/1 – 8/7 – 14/11 – 16/11 | 1/1 – 9/8 – 9/7 – 7/4 | 1/1 – 8/7 – 11/7 – 16/9 | |
| 58 | 0 – 9 – 10 – 18 | 1 – 7/6 – 7/4 – 11/8 | mothwellsmic | 1/1 – 7/6 – 11/8 – 7/4 | 1/1 – 7/6 – 3/2 – 12/7 | 1/1 – 14/11 – 16/11 – 12/7 | 1/1 – 8/7 – 4/3 – 11/7 | |
| 59 | 0 – 4 – 14 – 18 | 1 – 5/4 – 11/10 – 11/8 | valinorsmic | 1/1 – 11/10 – 5/4 – 11/8 | 1/1 – 8/7 – 5/4 – 20/11 | 1/1 – 11/10 – 8/5 – 7/4 | 1/1 – 16/11 – 8/5 – 20/11 | |
| 60 | 0 – 8 – 14 – 18 | 1 – 11/7 – 11/10 – 11/8 | utonal | 1/1 – 11/10 – 11/8 – 11/7 | 1/1 – 5/4 – 10/7 – 20/11 | 1/1 – 8/7 – 16/11 – 8/5 | 1/1 – 14/11 – 7/5 – 7/4 | |
| 61 | 0 – 10 – 14 – 18 | 1 – 7/4 – 11/10 – 11/8 | minerva | 1/1 – 11/10 – 11/8 – 7/4 | 1/1 – 5/4 – 8/5 – 20/11 | 1/1 – 14/11 – 16/11 – 8/5 | 1/1 – 8/7 – 5/4 – 11/7 | |
| 62 | 0 – 2 – 16 – 18 | 1 – 9/8 – 11/9 – 11/8 | didymic | 1/1 – 9/8 – 11/9 – 11/8 | 1/1 – 11/10 – 11/9 – 16/9 | 1/1 – 9/8 – 18/11 – 20/11 | 1/1 – 16/11 – 18/11 – 16/9 | |
| 63 | 0 – 8 – 16 – 18 | 1 – 11/7 – 11/9 – 11/8 | utonal | 1/1 – 11/9 – 11/8 – 11/7 | 1/1 – 9/8 – 9/7 – 18/11 | 1/1 – 8/7 – 16/11 – 16/9 | 1/1 – 14/11 – 14/9 – 7/4 | |
| 64 | 0 – 10 – 16 – 18 | 1 – 7/4 – 11/9 – 11/8 | euterpe | 1/1 – 11/9 – 11/8 – 7/4 | 1/1 – 9/8 – 10/7 – 18/11 | 1/1 – 14/11 – 16/11 – 16/9 | 1/1 – 8/7 – 7/5 – 11/7 | |
| 65 | 0 – 14 – 16 – 18 | 1 – 11/10 – 11/9 – 11/8 | utonal | 1/1 – 11/10 – 11/9 – 11/8 | 1/1 – 10/9 – 5/4 – 20/11 | 1/1 – 9/8 – 18/11 – 9/5 | 1/1 – 16/11 – 8/5 – 16/9 | |
| 66 | 0 – 1 – 17 – 18 | 1 – 3/2 – 11/6 – 11/8 | ambitonal | 1/1 – 11/8 – 3/2 – 11/6 | 1/1 – 12/11 – 4/3 – 16/11 | 1/1 – 11/9 – 4/3 – 11/6 | 1/1 – 12/11 – 3/2 – 18/11 | |
| 67 | 0 – 8 – 17 – 18 | 1 – 11/7 – 11/6 – 11/8 | utonal | 1/1 – 11/8 – 11/7 – 11/6 | 1/1 – 8/7 – 4/3 – 16/11 | 1/1 – 7/6 – 14/11 – 7/4 | 1/1 – 12/11 – 3/2 – 12/7 | |
| 68 | 0 – 9 – 17 – 18 | 1 – 7/6 – 11/6 – 11/8 | mothwellsmic | 1/1 – 7/6 – 11/8 – 11/6 | 1/1 – 7/6 – 11/7 – 12/7 | 1/1 – 4/3 – 16/11 – 12/7 | 1/1 – 12/11 – 14/11 – 3/2 | |
| 69 | 0 – 14 – 17 – 18 | 1 – 11/10 – 11/6 – 11/8 | utonal | 1/1 – 11/10 – 11/8 – 11/6 | 1/1 – 5/4 – 5/3 – 20/11 | 1/1 – 4/3 – 16/11 – 8/5 | 1/1 – 12/11 – 6/5 – 3/2 | |
| 70 | 0 – 16 – 17 – 18 | 1 – 11/9 – 11/6 – 11/8 | utonal | 1/1 – 11/9 – 11/8 – 11/6 | 1/1 – 9/8 – 3/2 – 18/11 | 1/1 – 4/3 – 16/11 – 16/9 | 1/1 – 12/11 – 4/3 – 3/2 |
Pentads
| Chord | Transversal | Type | As generated | First inversion | Second inversion | Third inversion | Fourth inversion |
|---|---|---|---|---|---|---|---|
| 0 – 1 – 2 – 3 – 4 | 1 – 3/2 – 9/8 – 5/3 – 5/4 | didymic | 1/1 – 9/8 – 5/4 – 3/2 – 5/3 | 1/1 – 10/9 – 4/3 – 3/2 – 16/9 | 1/1 – 6/5 – 4/3 – 8/5 – 9/5 | 1/1 – 10/9 – 4/3 – 3/2 – 5/3 | 1/1 – 6/5 – 4/3 – 3/2 – 9/5 |
| 0 – 2 – 3 – 4 – 6 | 1 – 9/8 – 5/3 – 5/4 – 7/5 | erato | 1/1 – 9/8 – 5/4 – 7/5 – 5/3 | 1/1 – 10/9 – 5/4 – 3/2 – 16/9 | 1/1 – 10/9 – 4/3 – 8/5 – 9/5 | 1/1 – 6/5 – 10/7 – 8/5 – 9/5 | 1/1 – 6/5 – 4/3 – 3/2 – 5/3 |
| 0 – 2 – 4 – 6 – 8 | 1 – 9/8 – 5/4 – 7/5 – 14/9 | erato | 1/1 – 9/8 – 5/4 – 7/5 – 14/9 | 1/1 – 10/9 – 5/4 – 7/5 – 16/9 | 1/1 – 10/9 – 5/4 – 8/5 – 9/5 | 1/1 – 10/9 – 10/7 – 8/5 – 9/5 | 1/1 – 9/7 – 10/7 – 8/5 – 9/5 |
| 0 – 1 – 2 – 4 – 10 | 1 – 3/2 – 9/8 – 5/4 – 7/4 | otonal | 1/1 – 9/8 – 5/4 – 3/2 – 7/4 | 1/1 – 10/9 – 4/3 – 14/9 – 16/9 | 1/1 – 6/5 – 7/5 – 8/5 – 9/5 | 1/1 – 7/6 – 4/3 – 3/2 – 5/3 | 1/1 – 8/7 – 9/7 – 10/7 – 12/7 |
| 0 – 2 – 4 – 6 – 10 | 1 – 9/8 – 5/4 – 7/5 – 7/4 | erato | 1/1 – 9/8 – 5/4 – 7/5 – 7/4 | 1/1 – 10/9 – 5/4 – 14/9 – 16/9 | 1/1 – 10/9 – 7/5 – 8/5 – 9/5 | 1/1 – 5/4 – 10/7 – 8/5 – 9/5 | 1/1 – 8/7 – 9/7 – 10/7 – 8/5 |
| 0 – 2 – 4 – 8 – 10 | 1 – 9/8 – 5/4 – 14/9 – 7/4 | erato | 1/1 – 9/8 – 5/4 – 14/9 – 7/4 | 1/1 – 10/9 – 7/5 – 14/9 – 16/9 | 1/1 – 5/4 – 7/5 – 8/5 – 9/5 | 1/1 – 9/8 – 9/7 – 10/7 – 8/5 | 1/1 – 8/7 – 9/7 – 10/7 – 16/9 |
| 0 – 2 – 6 – 8 – 10 | 1 – 9/8 – 7/5 – 14/9 – 7/4 | erato | 1/1 – 9/8 – 7/5 – 14/9 – 7/4 | 1/1 – 5/4 – 7/5 – 14/9 – 16/9 | 1/1 – 10/9 – 5/4 – 10/7 – 8/5 | 1/1 – 9/8 – 9/7 – 10/7 – 9/5 | 1/1 – 8/7 – 9/7 – 8/5 – 16/9 |
| 0 – 4 – 6 – 8 – 10 | 1 – 5/4 – 7/5 – 14/9 – 7/4 | erato | 1/1 – 5/4 – 7/5 – 14/9 – 7/4 | 1/1 – 9/8 – 5/4 – 7/5 – 8/5 | 1/1 – 10/9 – 5/4 – 10/7 – 9/5 | 1/1 – 9/8 – 9/7 – 8/5 – 9/5 | 1/1 – 8/7 – 10/7 – 8/5 – 16/9 |
| 0 – 6 – 8 – 9 – 10 | 1 – 7/5 – 14/9 – 7/6 – 7/4 | utonal | 1/1 – 7/6 – 7/5 – 14/9 – 7/4 | 1/1 – 6/5 – 4/3 – 3/2 – 12/7 | 1/1 – 10/9 – 5/4 – 10/7 – 5/3 | 1/1 – 9/8 – 9/7 – 3/2 – 9/5 | 1/1 – 8/7 – 4/3 – 8/5 – 16/9 |
| 0 – 4 – 6 – 8 – 14 | 1 – 5/4 – 7/5 – 14/9 – 11/10 | meantone | 1/1 – 11/10 – 5/4 – 7/5 – 14/9 | 1/1 – 9/8 – 14/11 – 10/7 – 20/11 | 1/1 – 10/9 – 5/4 – 8/5 – 7/4 | 1/1 – 10/9 – 10/7 – 11/7 – 9/5 | 1/1 – 9/7 – 7/5 – 8/5 – 9/5 |
| 0 – 4 – 6 – 10 – 14 | 1 – 5/4 – 7/5 – 7/4 – 11/10 | minerva | 1/1 – 11/10 – 5/4 – 7/5 – 7/4 | 1/1 – 9/8 – 14/11 – 8/5 – 20/11 | 1/1 – 10/9 – 7/5 – 8/5 – 7/4 | 1/1 – 5/4 – 10/7 – 11/7 – 9/5 | 1/1 – 8/7 – 5/4 – 10/7 – 8/5 |
| 0 – 4 – 8 – 10 – 14 | 1 – 5/4 – 14/9 – 7/4 – 11/10 | minerva | 1/1 – 11/10 – 5/4 – 14/9 – 7/4 | 1/1 – 9/8 – 7/5 – 8/5 – 20/11 | 1/1 – 5/4 – 7/5 – 8/5 – 7/4 | 1/1 – 9/8 – 9/7 – 7/5 – 8/5 | 1/1 – 8/7 – 5/4 – 10/7 – 16/9 |
| 0 – 6 – 8 – 10 – 14 | 1 – 7/5 – 14/9 – 7/4 – 11/10 | meantone | 1/1 – 11/10 – 7/5 – 14/9 – 7/4 | 1/1 – 14/11 – 7/5 – 8/5 – 20/11 | 1/1 – 10/9 – 5/4 – 10/7 – 11/7 | 1/1 – 9/8 – 9/7 – 7/5 – 9/5 | 1/1 – 8/7 – 5/4 – 8/5 – 16/9 |
| 0 – 2 – 6 – 8 – 16 | 1 – 9/8 – 7/5 – 14/9 – 11/9 | meantone | 1/1 – 9/8 – 11/9 – 7/5 – 14/9 | 1/1 – 11/10 – 5/4 – 7/5 – 16/9 | 1/1 – 8/7 – 14/11 – 18/11 – 20/11 | 1/1 – 10/9 – 10/7 – 8/5 – 7/4 | 1/1 – 9/7 – 10/7 – 11/7 – 9/5 |
| 0 – 2 – 6 – 10 – 16 | 1 – 9/8 – 7/5 – 7/4 – 11/9 | meantone | 1/1 – 9/8 – 11/9 – 7/5 – 7/4 | 1/1 – 11/10 – 5/4 – 14/9 – 16/9 | 1/1 – 8/7 – 10/7 – 18/11 – 20/11 | 1/1 – 5/4 – 10/7 – 8/5 – 7/4 | 1/1 – 8/7 – 9/7 – 7/5 – 8/5 |
| 0 – 2 – 8 – 10 – 16 | 1 – 9/8 – 14/9 – 7/4 – 11/9 | euterpe | 1/1 – 9/8 – 11/9 – 14/9 – 7/4 | 1/1 – 11/10 – 7/5 – 14/9 – 16/9 | 1/1 – 14/11 – 10/7 – 18/11 – 20/11 | 1/1 – 9/8 – 9/7 – 10/7 – 11/7 | 1/1 – 8/7 – 9/7 – 7/5 – 16/9 |
| 0 – 6 – 8 – 10 – 16 | 1 – 7/5 – 14/9 – 7/4 – 11/9 | euterpe | 1/1 – 11/9 – 7/5 – 14/9 – 7/4 | 1/1 – 8/7 – 14/11 – 10/7 – 18/11 | 1/1 – 10/9 – 5/4 – 10/7 – 7/4 | 1/1 – 9/8 – 9/7 – 11/7 – 9/5 | 1/1 – 8/7 – 7/5 – 8/5 – 16/9 |
| 0 – 6 – 8 – 14 – 16 | 1 – 7/5 – 14/9 – 11/10 – 11/9 | euterpe | 1/1 – 11/10 – 11/9 – 7/5 – 14/9 | 1/1 – 10/9 – 14/11 – 10/7 – 20/11 | 1/1 – 8/7 – 14/11 – 18/11 – 9/5 | 1/1 – 10/9 – 10/7 – 11/7 – 7/4 | 1/1 – 9/7 – 7/5 – 11/7 – 9/5 |
| 0 – 6 – 10 – 14 – 16 | 1 – 7/5 – 7/4 – 11/10 – 11/9 | meantone | 1/1 – 11/10 – 11/9 – 7/5 – 7/4 | 1/1 – 10/9 – 14/11 – 8/5 – 20/11 | 1/1 – 8/7 – 10/7 – 18/11 – 9/5 | 1/1 – 5/4 – 10/7 – 11/7 – 7/4 | 1/1 – 8/7 – 5/4 – 7/5 – 8/5 |
| 0 – 8 – 10 – 14 – 16 | 1 – 14/9 – 7/4 – 11/10 – 11/9 | meantone | 1/1 – 11/10 – 11/9 – 14/9 – 7/4 | 1/1 – 10/9 – 7/5 – 8/5 – 20/11 | 1/1 – 14/11 – 10/7 – 18/11 – 9/5 | 1/1 – 9/8 – 9/7 – 7/5 – 11/7 | 1/1 – 8/7 – 5/4 – 7/5 – 16/9 |
| 0 – 1 – 3 – 9 – 17 | 1 – 3/2 – 5/3 – 7/6 – 11/6 | otonal | 1/1 – 7/6 – 3/2 – 5/3 – 11/6 | 1/1 – 9/7 – 10/7 – 11/7 – 12/7 | 1/1 – 10/9 – 11/9 – 4/3 – 14/9 | 1/1 – 11/10 – 6/5 – 7/5 – 9/5 | 1/1 – 12/11 – 14/11 – 18/11 – 20/11 |
| 0 – 8 – 14 – 16 – 17 | 1 – 11/7 – 11/10 – 11/9 – 11/6 | utonal | 1/1 – 11/10 – 11/9 – 11/7 – 11/6 | 1/1 – 10/9 – 10/7 – 5/3 – 20/11 | 1/1 – 9/7 – 3/2 – 18/11 – 9/5 | 1/1 – 7/6 – 14/11 – 7/5 – 14/9 | 1/1 – 12/11 – 6/5 – 4/3 – 12/7 |
| 0 – 1 – 2 – 4 – 18 | 1 – 3/2 – 9/8 – 5/4 – 11/8 | otonal | 1/1 – 9/8 – 5/4 – 11/8 – 3/2 | 1/1 – 10/9 – 11/9 – 4/3 – 16/9 | 1/1 – 11/10 – 6/5 – 8/5 – 9/5 | 1/1 – 12/11 – 16/11 – 18/11 – 20/11 | 1/1 – 4/3 – 3/2 – 5/3 – 11/6 |
| 0 – 2 – 4 – 8 – 18 | 1 – 9/8 – 5/4 – 14/9 – 11/8 | meantone | 1/1 – 9/8 – 5/4 – 11/8 – 14/9 | 1/1 – 10/9 – 11/9 – 7/5 – 16/9 | 1/1 – 11/10 – 5/4 – 8/5 – 9/5 | 1/1 – 8/7 – 16/11 – 18/11 – 20/11 | 1/1 – 9/7 – 10/7 – 8/5 – 7/4 |
| 0 – 1 – 2 – 10 – 18 | 1 – 3/2 – 9/8 – 7/4 – 11/8 | otonal | 1/1 – 9/8 – 11/8 – 3/2 – 7/4 | 1/1 – 11/9 – 4/3 – 14/9 – 16/9 | 1/1 – 12/11 – 14/11 – 16/11 – 18/11 | 1/1 – 7/6 – 4/3 – 3/2 – 11/6 | 1/1 – 8/7 – 9/7 – 11/7 – 12/7 |
| 0 – 1 – 4 – 10 – 18 | 1 – 3/2 – 5/4 – 7/4 – 11/8 | otonal | 1/1 – 5/4 – 11/8 – 3/2 – 7/4 | 1/1 – 11/10 – 6/5 – 7/5 – 8/5 | 1/1 – 12/11 – 14/11 – 16/11 – 20/11 | 1/1 – 7/6 – 4/3 – 5/3 – 11/6 | 1/1 – 8/7 – 10/7 – 11/7 – 12/7 |
| 0 – 2 – 4 – 10 – 18 | 1 – 9/8 – 5/4 – 7/4 – 11/8 | otonal | 1/1 – 9/8 – 5/4 – 11/8 – 7/4 | 1/1 – 10/9 – 11/9 – 14/9 – 16/9 | 1/1 – 11/10 – 7/5 – 8/5 – 9/5 | 1/1 – 14/11 – 16/11 – 18/11 – 20/11 | 1/1 – 8/7 – 9/7 – 10/7 – 11/7 |
| 0 – 2 – 8 – 10 – 18 | 1 – 9/8 – 14/9 – 7/4 – 11/8 | euterpe | 1/1 – 9/8 – 11/8 – 14/9 – 7/4 | 1/1 – 11/9 – 7/5 – 14/9 – 16/9 | 1/1 – 8/7 – 14/11 – 16/11 – 18/11 | 1/1 – 9/8 – 9/7 – 10/7 – 7/4 | 1/1 – 8/7 – 9/7 – 11/7 – 16/9 |
| 0 – 4 – 8 – 10 – 18 | 1 – 5/4 – 14/9 – 7/4 – 11/8 | minerva | 1/1 – 5/4 – 11/8 – 14/9 – 7/4 | 1/1 – 11/10 – 5/4 – 7/5 – 8/5 | 1/1 – 8/7 – 14/11 – 16/11 – 20/11 | 1/1 – 9/8 – 9/7 – 8/5 – 7/4 | 1/1 – 8/7 – 10/7 – 11/7 – 16/9 |
| 0 – 1 – 9 – 10 – 18 | 1 – 3/2 – 7/6 – 7/4 – 11/8 | mothwellsmic | 1/1 – 7/6 – 11/8 – 3/2 – 7/4 | 1/1 – 7/6 – 9/7 – 3/2 – 12/7 | 1/1 – 12/11 – 14/11 – 16/11 – 12/7 | 1/1 – 7/6 – 4/3 – 14/9 – 11/6 | 1/1 – 8/7 – 4/3 – 11/7 – 12/7 |
| 0 – 8 – 9 – 10 – 18 | 1 – 14/9 – 7/6 – 7/4 – 11/8 | mothwellsmic | 1/1 – 7/6 – 11/8 – 14/9 – 7/4 | 1/1 – 7/6 – 4/3 – 3/2 – 12/7 | 1/1 – 8/7 – 14/11 – 16/11 – 12/7 | 1/1 – 9/8 – 9/7 – 3/2 – 7/4 | 1/1 – 8/7 – 4/3 – 11/7 – 16/9 |
| 0 – 4 – 8 – 14 – 18 | 1 – 5/4 – 14/9 – 11/10 – 11/8 | minerva | 1/1 – 11/10 – 5/4 – 11/8 – 14/9 | 1/1 – 9/8 – 5/4 – 10/7 – 20/11 | 1/1 – 11/10 – 5/4 – 8/5 – 7/4 | 1/1 – 8/7 – 16/11 – 8/5 – 20/11 | 1/1 – 9/7 – 7/5 – 8/5 – 7/4 |
| 0 – 4 – 10 – 14 – 18 | 1 – 5/4 – 7/4 – 11/10 – 11/8 | minerva | 1/1 – 11/10 – 5/4 – 11/8 – 7/4 | 1/1 – 9/8 – 5/4 – 8/5 – 20/11 | 1/1 – 11/10 – 7/5 – 8/5 – 7/4 | 1/1 – 14/11 – 16/11 – 8/5 – 20/11 | 1/1 – 8/7 – 5/4 – 10/7 – 11/7 |
| 0 – 8 – 10 – 14 – 18 | 1 – 14/9 – 7/4 – 11/10 – 11/8 | minerva | 1/1 – 11/10 – 11/8 – 14/9 – 7/4 | 1/1 – 5/4 – 7/5 – 8/5 – 20/11 | 1/1 – 8/7 – 14/11 – 16/11 – 8/5 | 1/1 – 9/8 – 9/7 – 7/5 – 7/4 | 1/1 – 8/7 – 5/4 – 11/7 – 16/9 |
| 0 – 2 – 8 – 16 – 18 | 1 – 9/8 – 14/9 – 11/9 – 11/8 | euterpe | 1/1 – 9/8 – 11/9 – 11/8 – 14/9 | 1/1 – 11/10 – 11/9 – 7/5 – 16/9 | 1/1 – 9/8 – 14/11 – 18/11 – 20/11 | 1/1 – 8/7 – 16/11 – 18/11 – 16/9 | 1/1 – 9/7 – 10/7 – 11/7 – 7/4 |
| 0 – 2 – 10 – 16 – 18 | 1 – 9/8 – 7/4 – 11/9 – 11/8 | euterpe | 1/1 – 9/8 – 11/9 – 11/8 – 7/4 | 1/1 – 11/10 – 11/9 – 14/9 – 16/9 | 1/1 – 9/8 – 10/7 – 18/11 – 20/11 | 1/1 – 14/11 – 16/11 – 18/11 – 16/9 | 1/1 – 8/7 – 9/7 – 7/5 – 11/7 |
| 0 – 8 – 10 – 16 – 18 | 1 – 14/9 – 7/4 – 11/9 – 11/8 | euterpe | 1/1 – 11/9 – 11/8 – 14/9 – 7/4 | 1/1 – 9/8 – 14/11 – 10/7 – 18/11 | 1/1 – 8/7 – 14/11 – 16/11 – 16/9 | 1/1 – 9/8 – 9/7 – 11/7 – 7/4 | 1/1 – 8/7 – 7/5 – 11/7 – 16/9 |
| 0 – 8 – 14 – 16 – 18 | 1 – 11/7 – 11/10 – 11/9 – 11/8 | utonal | 1/1 – 11/10 – 11/9 – 11/8 – 11/7 | 1/1 – 10/9 – 5/4 – 10/7 – 20/11 | 1/1 – 9/8 – 9/7 – 18/11 – 9/5 | 1/1 – 8/7 – 16/11 – 8/5 – 16/9 | 1/1 – 14/11 – 7/5 – 14/9 – 7/4 |
| 0 – 10 – 14 – 16 – 18 | 1 – 7/4 – 11/10 – 11/9 – 11/8 | meantone | 1/1 – 11/10 – 11/9 – 11/8 – 7/4 | 1/1 – 10/9 – 5/4 – 8/5 – 20/11 | 1/1 – 9/8 – 10/7 – 18/11 – 9/5 | 1/1 – 14/11 – 16/11 – 8/5 – 16/9 | 1/1 – 8/7 – 5/4 – 7/5 – 11/7 |
| 0 – 1 – 9 – 17 – 18 | 1 – 3/2 – 7/6 – 11/6 – 11/8 | mothwellsmic | 1/1 – 7/6 – 11/8 – 3/2 – 11/6 | 1/1 – 7/6 – 9/7 – 11/7 – 12/7 | 1/1 – 12/11 – 4/3 – 16/11 – 12/7 | 1/1 – 11/9 – 4/3 – 14/9 – 11/6 | 1/1 – 12/11 – 14/11 – 3/2 – 18/11 |
| 0 – 8 – 9 – 17 – 18 | 1 – 14/9 – 7/6 – 11/6 – 11/8 | mothwellsmic | 1/1 – 7/6 – 11/8 – 14/9 – 11/6 | 1/1 – 7/6 – 4/3 – 11/7 – 12/7 | 1/1 – 8/7 – 4/3 – 16/11 – 12/7 | 1/1 – 7/6 – 9/7 – 3/2 – 7/4 | 1/1 – 12/11 – 14/11 – 3/2 – 12/7 |
| 0 – 8 – 14 – 17 – 18 | 1 – 11/7 – 11/10 – 11/6 – 11/8 | utonal | 1/1 – 11/10 – 11/8 – 11/7 – 11/6 | 1/1 – 5/4 – 10/7 – 5/3 – 20/11 | 1/1 – 8/7 – 4/3 – 16/11 – 8/5 | 1/1 – 7/6 – 14/11 – 7/5 – 7/4 | 1/1 – 12/11 – 6/5 – 3/2 – 12/7 |
| 0 – 8 – 16 – 17 – 18 | 1 – 11/7 – 11/9 – 11/6 – 11/8 | utonal | 1/1 – 11/9 – 11/8 – 11/7 – 11/6 | 1/1 – 9/8 – 9/7 – 3/2 – 18/11 | 1/1 – 8/7 – 4/3 – 16/11 – 16/9 | 1/1 – 7/6 – 14/11 – 14/9 – 7/4 | 1/1 – 12/11 – 4/3 – 3/2 – 12/7 |
| 0 – 14 – 16 – 17 – 18 | 1 – 11/10 – 11/9 – 11/6 – 11/8 | utonal | 1/1 – 11/10 – 11/9 – 11/8 – 11/6 | 1/1 – 10/9 – 5/4 – 5/3 – 20/11 | 1/1 – 9/8 – 3/2 – 18/11 – 9/5 | 1/1 – 4/3 – 16/11 – 8/5 – 16/9 | 1/1 – 12/11 – 6/5 – 4/3 – 3/2 |
Hexads
| Number | Chord | Transversal | Type |
|---|---|---|---|
| 1 | 0 – 2 – 4 – 6 – 8 – 10 | 1 – 9/8 – 5/4 – 7/5 – 14/9 – 7/4 | erato |
| 2 | 0 – 4 – 6 – 8 – 10 – 14 | 1 – 5/4 – 7/5 – 14/9 – 7/4 – 11/10 | meantone |
| 3 | 0 – 2 – 6 – 8 – 10 – 16 | 1 – 9/8 – 7/5 – 14/9 – 7/4 – 11/9 | meantone |
| 4 | 0 – 6 – 8 – 10 – 14 – 16 | 1 – 7/5 – 14/9 – 7/4 – 11/10 – 11/9 | meantone |
| 5 | 0 – 1 – 2 – 4 – 10 – 18 | 1 – 3/2 – 9/8 – 5/4 – 7/4 – 11/8 | otonal |
| 6 | 0 – 2 – 4 – 8 – 10 – 18 | 1 – 9/8 – 5/4 – 14/9 – 7/4 – 11/8 | meantone |
| 7 | 0 – 4 – 8 – 10 – 14 – 18 | 1 – 5/4 – 14/9 – 7/4 – 11/10 – 11/8 | minerva |
| 8 | 0 – 2 – 8 – 10 – 16 – 18 | 1 – 9/8 – 14/9 – 7/4 – 11/9 – 11/8 | euterpe |
| 9 | 0 – 8 – 10 – 14 – 16 – 18 | 1 – 14/9 – 7/4 – 11/10 – 11/9 – 11/8- | meantone |
| 10 | 0 – 8 – 14 – 16 – 17 – 18 | 1 – 11/7 – 11/10 – 11/9 – 11/6 – 11/8 | utonal |