Chords of huygens: Difference between revisions

Latest revision as of 23:07, 4 September 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 huygens also conflates 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.

(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, by any two of 99/98, 176/175, or 225/224 minerva, and by any two of 126/125, 176/175, or 441/440 thrush. A chord requiring any three independent commas from those discussed above is labeled huygens.

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 huygens[7], six times in huygens[12], and 13 times in huygens[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.

Huygens 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 huygens itself is hardly explored. 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.

11-limit huygens's genchain
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 – 10/9 – 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 – 10/9 – 5/3 – 7/5	starling	1/1 – 10/9 – 7/5 – 5/3	1/1 – 5/4 – 3/2 – 9/5	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	marvel	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 – 16/9	C9no5
7	0 – 3 – 4 – 6	1 – 5/3 – 5/4 – 7/5	starling	1/1 – 5/4 – 7/5 – 5/3	1/1 – 10/9 – 4/3 – 8/5	1/1 – 6/5 – 10/7 – 9/5	1/1 – 6/5 – 3/2 – 5/3	Cm7(b5) or Cm6
8	0 – 2 – 4 – 8	1 – 10/9 – 5/4 – 14/9	marvel	1/1 – 10/9 – 5/4 – 14/9	1/1 – 9/8 – 7/5 – 9/5	1/1 – 5/4 – 8/5 – 16/9	1/1 – 9/7 – 10/7 – 8/5	Caug,9
9	0 – 2 – 6 – 8	1 – 10/9 – 7/5 – 14/9	starling	1/1 – 10/9 – 7/5 – 14/9	1/1 – 5/4 – 7/5 – 9/5	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	marvel	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	Cvm,6 or 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	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	Caug,v7
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 – 8/7 – 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 – 11/7 – 11/10	valinorsmic	1/1 – 11/10 – 5/4 – 11/7	1/1 – 8/7 – 10/7 – 20/11	1/1 – 5/4 – 8/5 – 7/4	1/1 – 14/11 – 7/5 – 8/5
27	0 – 6 – 8 – 14	1 – 7/5 – 14/9 – 11/10	mothwellsmic	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 – 8/7 – 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	valinorsmic	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 – 10/7 – 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 – 10/9 – 7/5 – 11/9	thrush	1/1 – 10/9 – 11/9 – 7/5	1/1 – 11/10 – 5/4 – 9/5	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	werckismic	1/1 – 11/9 – 7/5 – 14/9	1/1 – 8/7 – 14/11 – 18/11	1/1 – 10/9 – 10/7 – 7/4	1/1 – 9/7 – 11/7 – 9/5
34	0 – 2 – 10 – 16	1 – 10/9 – 7/4 – 11/9	werckismic	1/1 – 10/9 – 11/9 – 7/4	1/1 – 11/10 – 11/7 – 9/5	1/1 – 10/7 – 18/11 – 20/11	1/1 – 8/7 – 14/11 – 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 – 11/7 – 7/4 – 11/9	werckismic	1/1 – 11/9 – 11/7 – 7/4	1/1 – 9/7 – 10/7 – 18/11	1/1 – 10/9 – 14/11 – 14/9	1/1 – 8/7 – 7/5 – 9/5
37	0 – 6 – 14 – 16	1 – 7/5 – 11/10 – 11/9	werckismic	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	thrush	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 – 11/7 – 11/8	werckismic	1/1 – 9/8 – 11/8 – 11/7	1/1 – 11/9 – 7/5 – 16/9	1/1 – 8/7 – 16/11 – 18/11	1/1 – 14/11 – 10/7 – 7/4
51	0 – 4 – 8 – 18	1 – 5/4 – 11/7 – 11/8	valinorsmic	1/1 – 5/4 – 11/8 – 11/7	1/1 – 11/10 – 5/4 – 8/5	1/1 – 8/7 – 16/11 – 20/11	1/1 – 14/11 – 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	valinorsmic	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	werckismic	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 – 10/9 – 5/3 – 5/4 – 7/5	starling	1/1 – 10/9 – 5/4 – 7/5 – 5/3	1/1 – 9/8 – 5/4 – 3/2 – 9/5	1/1 – 10/9 – 4/3 – 8/5 – 16/9	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 – 9/8 – 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	marvel	1/1 – 9/8 – 5/4 – 7/5 – 7/4	1/1 – 10/9 – 5/4 – 14/9 – 16/9	1/1 – 9/8 – 7/5 – 8/5 – 9/5	1/1 – 5/4 – 10/7 – 8/5 – 16/9	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	marvel	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 – 16/9	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	minerva	1/1 – 11/10 – 5/4 – 7/5 – 14/9	1/1 – 8/7 – 9/7 – 10/7 – 20/11	1/1 – 9/8 – 5/4 – 8/5 – 7/4	1/1 – 10/9 – 10/7 – 14/9 – 16/9	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 – 8/7 – 14/11 – 8/5 – 20/11	1/1 – 9/8 – 7/5 – 8/5 – 7/4	1/1 – 5/4 – 10/7 – 11/7 – 16/9	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 – 8/7 – 10/7 – 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	minerva	1/1 – 11/10 – 7/5 – 14/9 – 7/4	1/1 – 14/11 – 10/7 – 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 – 10/9 – 7/5 – 14/9 – 11/9	thrush	1/1 – 10/9 – 11/9 – 7/5 – 14/9	1/1 – 11/10 – 5/4 – 7/5 – 9/5	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 – 10/9 – 7/5 – 7/4 – 11/9	thrush	1/1 – 10/9 – 11/9 – 7/5 – 7/4	1/1 – 11/10 – 5/4 – 11/7 – 9/5	1/1 – 8/7 – 10/7 – 18/11 – 20/11	1/1 – 5/4 – 10/7 – 8/5 – 7/4	1/1 – 8/7 – 14/11 – 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	thrush	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 – 11/7 – 7/4 – 11/10 – 11/9	thrush	1/1 – 11/10 – 11/9 – 11/7 – 7/4	1/1 – 10/9 – 7/5 – 8/5 – 20/11	1/1 – 9/7 – 10/7 – 18/11 – 9/5	1/1 – 10/9 – 14/11 – 7/5 – 14/9	1/1 – 8/7 – 5/4 – 7/5 – 9/5
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 – 11/7 – 11/8	thrush	1/1 – 9/8 – 5/4 – 11/8 – 11/7	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 – 14/11 – 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 – 11/7 – 11/10 – 11/8	valinorsmic	1/1 – 11/10 – 5/4 – 11/8 – 11/7	1/1 – 8/7 – 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 – 14/11 – 7/5 – 8/5 – 7/4
0 – 4 – 10 – 14 – 18	1 – 5/4 – 7/4 – 11/10 – 11/8	valinorsmic	1/1 – 11/10 – 5/4 – 11/8 – 7/4	1/1 – 8/7 – 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 – 10/7 – 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 – 10/9 – 14/9 – 11/9 – 11/8	euterpe	1/1 – 10/9 – 11/9 – 11/8 – 14/9	1/1 – 11/10 – 11/9 – 7/5 – 9/5	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	thrush	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	minerva
3	0 – 2 – 6 – 8 – 10 – 16	1 – 9/8 – 7/5 – 14/9 – 7/4 – 11/9	huygens
4	0 – 6 – 8 – 10 – 14 – 16	1 – 7/5 – 14/9 – 7/4 – 11/10 – 11/9	huygens
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	huygens
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-	huygens
10	0 – 8 – 14 – 16 – 17 – 18	1 – 11/7 – 11/10 – 11/9 – 11/6 – 11/8	utonal

Chords of huygens: Difference between revisions

Latest revision as of 23:07, 4 September 2025

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@@ Line 1: / Line 1: @@
-Below are listed the [[Dyadic_chord|dyadic chords]] of 11-limit [[Meantone_family#Septimal meantone-Unidecimal meantone aka Huygens|meantone temperament]]. By "meantone" is meant 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.
+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 huygens also conflates 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.
 (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.
+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, by any two of 99/98, 176/175, or 225/224 minerva, and by any two of 126/125, 176/175, or 441/440 thrush. A chord requiring any three independent commas from those discussed above is labeled huygens.
-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 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 huygens[7], six times in huygens[12], and 13 times in huygens[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 ({{nowrap|{{dash|1/1, 6/5, 3/2}}}}, or 10:12:15) is the second inversion of the generated {{nowrap|{{dash|0, 3, 4}}}} chord.
@@ Line 11: / Line 11: @@
 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.
+Huygens 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 huygens itself is hardly explored. 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 {{nowrap|^1 {{=}} d2}} and {{nowrap|^C {{=}} Dbb}}. One up represents ~64/63, ~45/44, and ~40/39. If ''c'' is the difference between 700{{cent}} and the size of the generator in cents, then one up is equal to 12''c''.
 {| class="wikitable"
-|+ style="font-size: 105%;" | 11-limit meantone's genchain
+|+ style="font-size: 105%;" | 11-limit huygens's genchain
 |-
 ! Genspan
@@ Line 591: / Line 591: @@
 | {{dash|'''1/1, 5/4, 3/2, 5/3'''}}
 | {{dash|1/1, 6/5, 4/3, 8/5}}
-| {{dash|1/1, 9/8, 4/3, 5/3}}
+| {{dash|1/1, 10/9, 4/3, 5/3}}
 | {{dash|'''1/1, 6/5, 3/2, 9/5'''}}
 | C6 <u>or</u> Cm7
@@ Line 607: / Line 607: @@
 | 5
 | {{dash|0, 2, 3, 6}}
-| {{dash|1, 9/8, 5/3, 7/5}}
+| {{dash|1, 10/9, 5/3, 7/5}}
-| erato
+| starling
-| {{dash|1/1, 9/8, 7/5, 5/3}}
+| {{dash|1/1, 10/9, 7/5, 5/3}}
-| {{dash|'''1/1, 5/4, 3/2, 16/9'''}}
+| {{dash|'''1/1, 5/4, 3/2, 9/5'''}}
 | {{dash|1/1, 6/5, 10/7, 8/5}}
 | {{dash|1/1, 6/5, 4/3, 5/3}}
@@ Line 618: / Line 618: @@
 | {{dash|0, 2, 4, 6}}
 | {{dash|1, 9/8, 5/4, 7/5}}
-| erato
+| marvel
 | {{dash|1/1, 9/8, 5/4, 7/5}}
 | {{dash|'''1/1, 10/9, 5/4, 16/9'''}}
 | {{dash|1/1, 9/8, 8/5, 9/5}}
-| {{dash|1/1, 10/7, 8/5, 9/5}}
+| {{dash|1/1, 10/7, 8/5, 16/9}}
 | C9no5
 |-
@@ Line 628: / Line 628: @@
 | {{dash|0, 3, 4, 6}}
 | {{dash|1, 5/3, 5/4, 7/5}}
-| erato
+| starling
 | {{dash|1/1, 5/4, 7/5, 5/3}}
-| {{dash|1/1, 9/8, 4/3, 8/5}}
+| {{dash|1/1, 10/9, 4/3, 8/5}}
-| {{dash|'''1/1, 6/5, 10/7, 16/9'''}}
+| {{dash|'''1/1, 6/5, 10/7, 9/5'''}}
 | {{dash|'''1/1, 6/5, 3/2, 5/3'''}}
 | Cm7(b5) <u>or</u> Cm6
@@ Line 637: / Line 637: @@
 | 8
 | {{dash|0, 2, 4, 8}}
-| {{dash|1, 9/8, 5/4, 14/9}}
+| {{dash|1, 10/9, 5/4, 14/9}}
-| erato
+| marvel
-| {{dash|'''1/1, 9/8, 5/4, 14/9'''}}
+| {{dash|'''1/1, 10/9, 5/4, 14/9'''}}
-| {{dash|1/1, 10/9, 7/5, 16/9}}
+| {{dash|1/1, 9/8, 7/5, 9/5}}
-| {{dash|1/1, 5/4, 8/5, 9/5}}
+| {{dash|1/1, 5/4, 8/5, 16/9}}
 | {{dash|1/1, 9/7, 10/7, 8/5}}
 | Caug,9
@@ Line 647: / Line 647: @@
 | 9
 | {{dash|0, 2, 6, 8}}
-| {{dash|1, 9/8, 7/5, 14/9}}
+| {{dash|1, 10/9, 7/5, 14/9}}
-| erato
+| starling
-| {{dash|1/1, 9/8, 7/5, 14/9}}
+| {{dash|1/1, 10/9, 7/5, 14/9}}
-| {{dash|'''1/1, 5/4, 7/5, 16/9'''}}
+| {{dash|'''1/1, 5/4, 7/5, 9/5'''}}
 | {{dash|1/1, 10/9, 10/7, 8/5}}
 | {{dash|1/1, 9/7, 10/7, 9/5}}
@@ Line 658: / Line 658: @@
 | {{dash|0, 4, 6, 8}}
 | {{dash|1, 5/4, 7/5, 14/9}}
-| erato
+| marvel
 | {{dash|1/1, 5/4, 7/5, 14/9}}
 | {{dash|1/1, 9/8, 5/4, 8/5}}
@@ Line 669: / Line 669: @@
 | {{dash|1, 3/2, 5/3, 7/6}}
 | otonal
-| {{dash|1/1, 7/6, 3/2, 5/3}}
+| '''{{dash|1/1, 7/6, 3/2, 5/3}}'''
 | {{dash|1/1, 9/7, 10/7, 12/7}}
 | {{dash|1/1, 10/9, 4/3, 14/9}}
 | {{dash|'''1/1, 6/5, 7/5, 9/5'''}}
-| Cm7(vb5)
+| Cvm,6 <u>or</u> Cm7(vb5)
 |-
 | 12
@@ Line 733: / Line 733: @@
 | {{dash|1/1, 5/4, 10/7, 8/5}}
 | {{dash|1/1, 8/7, 9/7, 8/5}}
-| C7(#5) <u>or</u> Caug7
+| Caug7
 |-
 | 18
@@ Line 759: / Line 759: @@
 | {{dash|1, 5/4, 14/9, 7/4}}
 | marvel
-| {{dash|1/1, 5/4, 14/9, 7/4}}
+| '''{{dash|1/1, 5/4, 14/9, 7/4}}'''
 | {{dash|1/1, 5/4, 7/5, 8/5}}
 | {{dash|1/1, 9/8, 9/7, 8/5}}
 | {{dash|1/1, 8/7, 10/7, 16/9}}
-|
+| Caug,v7
 |-
 | 21
@@ Line 810: / Line 810: @@
 | minerva
 | {{dash|1/1, 11/10, 5/4, 7/5}}
-| {{dash|1/1, 9/8, 14/11, 20/11}}
+| {{dash|1/1, 8/7, 14/11, 20/11}}
 | {{dash|1/1, 9/8, 8/5, 7/4}}
 | {{dash|1/1, 10/7, 11/7, 16/9}}
@@ Line 817: / Line 817: @@
 | 26
 | {{dash|0, 4, 8, 14}}
-| {{dash|1, 5/4, 14/9, 11/10}}
+| {{dash|1, 5/4, 11/7, 11/10}}
-| minerva
+| valinorsmic
-| {{dash|1/1, 11/10, 5/4, 14/9}}
+| {{dash|1/1, 11/10, 5/4, 11/7}}
-| {{dash|1/1, 9/8, 10/7, 20/11}}
+| {{dash|1/1, 8/7, 10/7, 20/11}}
 | {{dash|1/1, 5/4, 8/5, 7/4}}
-| {{dash|1/1, 9/7, 7/5, 8/5}}
+| {{dash|1/1, 14/11, 7/5, 8/5}}
 |
 |-
@@ Line 828: / Line 828: @@
 | {{dash|0, 6, 8, 14}}
 | {{dash|1, 7/5, 14/9, 11/10}}
-| euterpe
+| mothwellsmic
 | {{dash|1/1, 11/10, 7/5, 14/9}}
 | {{dash|1/1, 14/11, 10/7, 20/11}}
@@ Line 840: / Line 840: @@
 | valinorsmic
 | {{dash|1/1, 11/10, 5/4, 7/4}}
-| {{dash|1/1, 25/22, 8/5, 20/11}}
+| {{dash|1/1, 8/7, 8/5, 20/11}}
 | {{dash|1/1, 7/5, 8/5, 7/4}}
 | {{dash|'''1/1, 8/7, 5/4, 10/7'''}}
@@ Line 848: / Line 848: @@
 | {{dash|0, 6, 10, 14}}
 | {{dash|1, 7/5, 7/4, 11/10}}
-| minerva
+| valinorsmic
 | {{dash|1/1, 11/10, 7/5, 7/4}}
 | {{dash|1/1, 14/11, 8/5, 20/11}}
@@ Line 860: / Line 860: @@
 | minerva
 | {{dash|1/1, 11/10, 14/9, 7/4}}
-| {{dash|1/1, 7/5, 8/5, 20/11}}
+| {{dash|1/1, 10/7, 8/5, 20/11}}
 | {{dash|1/1, 9/8, 9/7, 7/5}}
 | {{dash|1/1, 8/7, 5/4, 16/9}}
@@ Line 867: / Line 867: @@
 | 31
 | {{dash|0, 2, 6, 16}}
-| {{dash|1, 9/8, 7/5, 11/9}}
+| {{dash|1, 10/9, 7/5, 11/9}}
-| meantone
+| thrush
-| {{dash|1/1, 9/8, 11/9, 7/5}}
+| {{dash|1/1, 10/9, 11/9, 7/5}}
-| {{dash|1/1, 11/10, 5/4, 16/9}}
+| {{dash|1/1, 11/10, 5/4, 9/5}}
 | {{dash|1/1, 8/7, 18/11, 20/11}}
 | {{dash|1/1, 10/7, 8/5, 7/4}}
@@ Line 888: / Line 888: @@
 | {{dash|0, 6, 8, 16}}
 | {{dash|1, 7/5, 14/9, 11/9}}
-| euterpe
+| werckismic
 | {{dash|1/1, 11/9, 7/5, 14/9}}
 | {{dash|1/1, 8/7, 14/11, 18/11}}
-| {{dash|1/1, 9/8, 10/7, 7/4}}
+| {{dash|1/1, 10/9, 10/7, 7/4}}
 | {{dash|1/1, 9/7, 11/7, 9/5}}
 |
@@ Line 897: / Line 897: @@
 | 34
 | {{dash|0, 2, 10, 16}}
-| {{dash|1, 9/8, 7/4, 11/9}}
+| {{dash|1, 10/9, 7/4, 11/9}}
-| euterpe
+| werckismic
-| {{dash|1/1, 9/8, 11/9, 7/4}}
+| {{dash|1/1, 10/9, 11/9, 7/4}}
-| {{dash|1/1, 11/10, 14/9, 16/9}}
+| {{dash|1/1, 11/10, 11/7, 9/5}}
 | {{dash|1/1, 10/7, 18/11, 20/11}}
-| {{dash|1/1, 8/7, 9/7, 7/5}}
+| {{dash|1/1, 8/7, 14/11, 7/5}}
 |
 |-
@@ Line 917: / Line 917: @@
 | 36
 | {{dash|0, 8, 10, 16}}
-| {{dash|1, 14/9, 7/4, 11/9}}
+| {{dash|1, 11/7, 7/4, 11/9}}
-| euterpe
+| werckismic
-| {{dash|1/1, 11/9, 14/9, 7/4}}
+| {{dash|1/1, 11/9, 11/7, 7/4}}
-| {{dash|1/1, 14/11, 10/7, 18/11}}
+| {{dash|1/1, 9/7, 10/7, 18/11}}
-| {{dash|1/1, 9/8, 9/7, 11/7}}
+| {{dash|1/1, 10/9, 14/11, 14/9}}
-| {{dash|1/1, 8/7, 7/5, 16/9}}
+| {{dash|1/1, 8/7, 7/5, 9/5}}
 |
 |-
@@ Line 928: / Line 928: @@
 | {{dash|0, 6, 14, 16}}
 | {{dash|1, 7/5, 11/10, 11/9}}
-| euterpe
+| werckismic
 | {{dash|1/1, 11/10, 11/9, 7/5}}
 | {{dash|1/1, 10/9, 14/11, 20/11}}
@@ Line 948: / Line 948: @@
 | {{dash|0, 10, 14, 16}}
 | {{dash|1, 7/4, 11/10, 11/9}}
-| meantone
+| thrush
 | {{dash|1/1, 11/10, 11/9, 7/4}}
 | {{dash|1/1, 10/9, 8/5, 20/11}}
@@ Line 1,057: / Line 1,057: @@
 | 50
 | {{dash|0, 2, 8, 18}}
-| {{dash|1, 9/8, 14/9, 11/8}}
+| {{dash|1, 9/8, 11/7, 11/8}}
-| euterpe
+| werckismic
-| {{dash|1/1, 9/8, 11/8, 14/9}}
+| {{dash|1/1, 9/8, 11/8, 11/7}}
 | {{dash|1/1, 11/9, 7/5, 16/9}}
 | {{dash|1/1, 8/7, 16/11, 18/11}}
-| {{dash|1/1, 9/7, 10/7, 7/4}}
+| {{dash|1/1, 14/11, 10/7, 7/4}}
 |
 |-
 | 51
 | {{dash|0, 4, 8, 18}}
-| {{dash|1, 5/4, 14/9, 11/8}}
+| {{dash|1, 5/4, 11/7, 11/8}}
-| minerva
+| valinorsmic
-| {{dash|1/1, 5/4, 11/8, 14/9}}
+| {{dash|1/1, 5/4, 11/8, 11/7}}
 | {{dash|1/1, 11/10, 5/4, 8/5}}
 | {{dash|1/1, 8/7, 16/11, 20/11}}
-| {{dash|1/1, 9/7, 8/5, 7/4}}
+| {{dash|1/1, 14/11, 8/5, 7/4}}
 |
 |-
@@ Line 1,168: / Line 1,168: @@
 | {{dash|0, 10, 14, 18}}
 | {{dash|1, 7/4, 11/10, 11/8}}
-| minerva
+| valinorsmic
 | {{dash|1/1, 11/10, 11/8, 7/4}}
 | {{dash|1/1, 5/4, 8/5, 20/11}}
@@ Line 1,198: / Line 1,198: @@
 | {{dash|0, 10, 16, 18}}
 | {{dash|1, 7/4, 11/9, 11/8}}
-| euterpe
+| werckismic
 | {{dash|1/1, 11/9, 11/8, 7/4}}
 | {{dash|1/1, 9/8, 10/7, 18/11}}
@@ Line 1,288: / Line 1,288: @@
 |-
 | {{dash|0, 2, 3, 4, 6}}
-| {{dash|1, 9/8, 5/3, 5/4, 7/5}}
+| {{dash|1, 10/9, 5/3, 5/4, 7/5}}
-| erato
+| starling
-| {{dash|1/1, 9/8, 5/4, 7/5, 5/3  }}
+| {{dash|1/1, 10/9, 5/4, 7/5, 5/3  }}
-| {{dash|1/1, 10/9, 5/4, 3/2, 16/9  }}
+| {{dash|1/1, 9/8, 5/4, 3/2, 9/5}}
-| {{dash|1/1, 10/9, 4/3, 8/5, 9/5  }}
+| {{dash|1/1, 10/9, 4/3, 8/5, 16/9}}
 | {{dash|1/1, 6/5, 10/7, 8/5, 9/5  }}
 | {{dash|1/1, 6/5, 4/3, 3/2, 5/3  }}
@@ Line 1,301: / Line 1,301: @@
 | {{dash|1/1, 9/8, 5/4, 7/5, 14/9  }}
 | {{dash|1/1, 10/9, 5/4, 7/5, 16/9  }}
-| {{dash|1/1, 10/9, 5/4, 8/5, 9/5  }}
+| {{dash|1/1, 9/8, 5/4, 8/5, 9/5  }}
 | {{dash|1/1, 10/9, 10/7, 8/5, 9/5  }}
 | {{dash|1/1, 9/7, 10/7, 8/5, 9/5  }}
@@ Line 1,316: / Line 1,316: @@
 | {{dash|0, 2, 4, 6, 10}}
 | {{dash|1, 9/8, 5/4, 7/5, 7/4}}
-| erato
+| marvel
 | {{dash|1/1, 9/8, 5/4, 7/5, 7/4  }}
 | {{dash|1/1, 10/9, 5/4, 14/9, 16/9  }}
-| {{dash|1/1, 10/9, 7/5, 8/5, 9/5  }}
+| {{dash|1/1, 9/8, 7/5, 8/5, 9/5  }}
-| {{dash|1/1, 5/4, 10/7, 8/5, 9/5  }}
+| {{dash|1/1, 5/4, 10/7, 8/5, 16/9}}
 | {{dash|1/1, 8/7, 9/7, 10/7, 8/5  }}
 |-
@@ Line 1,343: / Line 1,343: @@
 | {{dash|0, 4, 6, 8, 10}}
 | {{dash|1, 5/4, 7/5, 14/9, 7/4}}
-| erato
+| marvel
 | {{dash|1/1, 5/4, 7/5, 14/9, 7/4  }}
 | {{dash|1/1, 9/8, 5/4, 7/5, 8/5  }}
-| {{dash|1/1, 10/9, 5/4, 10/7, 9/5  }}
+| {{dash|1/1, 10/9, 5/4, 10/7, 16/9}}
 | {{dash|1/1, 9/8, 9/7, 8/5, 9/5  }}
 | {{dash|1/1, 8/7, 10/7, 8/5, 16/9  }}
@@ Line 1,361: / Line 1,361: @@
 | {{dash|0, 4, 6, 8, 14}}
 | {{dash|1, 5/4, 7/5, 14/9, 11/10}}
-| meantone
+| minerva
 | {{dash|1/1, 11/10, 5/4, 7/5, 14/9  }}
-| {{dash|1/1, 9/8, 14/11, 10/7, 20/11 }}
+| {{dash|1/1, 8/7, 9/7, 10/7, 20/11 }}
-| {{dash|1/1, 10/9, 5/4, 8/5, 7/4  }}
+| {{dash|1/1, 9/8, 5/4, 8/5, 7/4  }}
-| {{dash|1/1, 10/9, 10/7, 11/7, 9/5  }}
+| {{dash|1/1, 10/9, 10/7, 14/9, 16/9}}
 | {{dash|1/1, 9/7, 7/5, 8/5, 9/5  }}
 |-
@@ Line 1,372: / Line 1,372: @@
 | minerva
 | {{dash|1/1, 11/10, 5/4, 7/5, 7/4  }}
-| {{dash|1/1, 9/8, 14/11, 8/5, 20/11 }}
+| {{dash|1/1, 8/7, 14/11, 8/5, 20/11 }}
-| {{dash|1/1, 10/9, 7/5, 8/5, 7/4  }}
+| {{dash|1/1, 9/8, 7/5, 8/5, 7/4  }}
-| {{dash|1/1, 5/4, 10/7, 11/7, 9/5  }}
+| {{dash|1/1, 5/4, 10/7, 11/7, 16/9}}
 | {{dash|1/1, 8/7, 5/4, 10/7, 8/5  }}
 |-
@@ Line 1,381: / Line 1,381: @@
 | minerva
 | {{dash|1/1, 11/10, 5/4, 14/9, 7/4  }}
-| {{dash|1/1, 9/8, 7/5, 8/5, 20/11 }}
+| {{dash|1/1, 8/7, 10/7, 8/5, 20/11 }}
 | {{dash|1/1, 5/4, 7/5, 8/5, 7/4  }}
 | {{dash|1/1, 9/8, 9/7, 7/5, 8/5  }}
@@ Line 1,388: / Line 1,388: @@
 | {{dash|0, 6, 8, 10, 14}}
 | {{dash|1, 7/5, 14/9, 7/4, 11/10}}
-| meantone
+| minerva
 | {{dash|1/1, 11/10, 7/5, 14/9, 7/4  }}
-| {{dash|1/1, 14/11, 7/5, 8/5, 20/11 }}
+| {{dash|1/1, 14/11, 10/7, 8/5, 20/11 }}
 | {{dash|1/1, 10/9, 5/4, 10/7, 11/7  }}
 | {{dash|1/1, 9/8, 9/7, 7/5, 9/5  }}
@@ Line 1,396: / Line 1,396: @@
 |-
 | {{dash|0, 2, 6, 8, 16}}
-| {{dash|1, 9/8, 7/5, 14/9, 11/9}}
+| {{dash|1, 10/9, 7/5, 14/9, 11/9}}
-| meantone
+| thrush
-| {{dash|1/1, 9/8, 11/9, 7/5, 14/9  }}
+| {{dash|1/1, 10/9, 11/9, 7/5, 14/9  }}
-| {{dash|1/1, 11/10, 5/4, 7/5, 16/9  }}
+| {{dash|1/1, 11/10, 5/4, 7/5, 9/5}}
 | {{dash|1/1, 8/7, 14/11, 18/11, 20/11 }}
 | {{dash|1/1, 10/9, 10/7, 8/5, 7/4  }}
@@ Line 1,405: / Line 1,405: @@
 |-
 | {{dash|0, 2, 6, 10, 16}}
-| {{dash|1, 9/8, 7/5, 7/4, 11/9}}
+| {{dash|1, 10/9, 7/5, 7/4, 11/9}}
-| meantone
+| thrush
-| {{dash|1/1, 9/8, 11/9, 7/5, 7/4  }}
+| {{dash|1/1, 10/9, 11/9, 7/5, 7/4  }}
-| {{dash|1/1, 11/10, 5/4, 14/9, 16/9  }}
+| {{dash|1/1, 11/10, 5/4, 11/7, 9/5}}
 | {{dash|1/1, 8/7, 10/7, 18/11, 20/11 }}
 | {{dash|1/1, 5/4, 10/7, 8/5, 7/4  }}
-| {{dash|1/1, 8/7, 9/7, 7/5, 8/5  }}
+| {{dash|1/1, 8/7, 14/11, 7/5, 8/5  }}
 |-
 | {{dash|0, 2, 8, 10, 16}}
@@ Line 1,442: / Line 1,442: @@
 | {{dash|0, 6, 10, 14, 16}}
 | {{dash|1, 7/5, 7/4, 11/10, 11/9}}
-| meantone
+| thrush
 | {{dash|1/1, 11/10, 11/9, 7/5, 7/4  }}
 | {{dash|1/1, 10/9, 14/11, 8/5, 20/11 }}
@@ Line 1,450: / Line 1,450: @@
 |-
 | {{dash|0, 8, 10, 14, 16}}
-| {{dash|1, 14/9, 7/4, 11/10, 11/9}}
+| {{dash|1, 11/7, 7/4, 11/10, 11/9}}
-| meantone
+| thrush
-| {{dash|1/1, 11/10, 11/9, 14/9, 7/4  }}
+| {{dash|1/1, 11/10, 11/9, 11/7, 7/4  }}
 | {{dash|1/1, 10/9, 7/5, 8/5, 20/11 }}
-| {{dash|1/1, 14/11, 10/7, 18/11, 9/5  }}
+| {{dash|1/1, 9/7, 10/7, 18/11, 9/5  }}
-| {{dash|1/1, 9/8, 9/7, 7/5, 11/7  }}
+| {{dash|1/1, 10/9, 14/11, 7/5, 14/9}}
-| {{dash|1/1, 8/7, 5/4, 7/5, 16/9  }}
+| {{dash|1/1, 8/7, 5/4, 7/5, 9/5}}
 |-
 | {{dash|0, 1, 3, 9, 17}}
@@ Line 1,486: / Line 1,486: @@
 |-
 | {{dash|0, 2, 4, 8, 18}}
-| {{dash|1, 9/8, 5/4, 14/9, 11/8}}
+| {{dash|1, 9/8, 5/4, 11/7, 11/8}}
-| meantone
+| thrush
-| {{dash|1/1, 9/8, 5/4, 11/8, 14/9  }}
+| {{dash|1/1, 9/8, 5/4, 11/8, 11/7}}
 | {{dash|1/1, 10/9, 11/9, 7/5, 16/9  }}
 | {{dash|1/1, 11/10, 5/4, 8/5, 9/5  }}
 | {{dash|1/1, 8/7, 16/11, 18/11, 20/11 }}
-| {{dash|1/1, 9/7, 10/7, 8/5, 7/4  }}
+| {{dash|1/1, 14/11, 10/7, 8/5, 7/4  }}
 |-
 | {{dash|0, 1, 2, 10, 18}}
@@ Line 1,558: / Line 1,558: @@
 |-
 | {{dash|0, 4, 8, 14, 18}}
-| {{dash|1, 5/4, 14/9, 11/10, 11/8}}
+| {{dash|1, 5/4, 11/7, 11/10, 11/8}}
-| minerva
+| valinorsmic
-| {{dash|1/1, 11/10, 5/4, 11/8, 14/9  }}
+| {{dash|1/1, 11/10, 5/4, 11/8, 11/7}}
-| {{dash|1/1, 9/8, 5/4, 10/7, 20/11 }}
+| {{dash|1/1, 8/7, 5/4, 10/7, 20/11 }}
 | {{dash|1/1, 11/10, 5/4, 8/5, 7/4  }}
 | {{dash|1/1, 8/7, 16/11, 8/5, 20/11 }}
-| {{dash|1/1, 9/7, 7/5, 8/5, 7/4  }}
+| {{dash|1/1, 14/11, 7/5, 8/5, 7/4  }}
 |-
 | {{dash|0, 4, 10, 14, 18}}
 | {{dash|1, 5/4, 7/4, 11/10, 11/8}}
-| minerva
+| valinorsmic
 | {{dash|1/1, 11/10, 5/4, 11/8, 7/4  }}
-| {{dash|1/1, 9/8, 5/4, 8/5, 20/11 }}
+| {{dash|1/1, 8/7, 5/4, 8/5, 20/11 }}
 | {{dash|1/1, 11/10, 7/5, 8/5, 7/4  }}
 | {{dash|1/1, 14/11, 16/11, 8/5, 20/11 }}
@@ Line 1,579: / Line 1,579: @@
 | minerva
 | {{dash|1/1, 11/10, 11/8, 14/9, 7/4  }}
-| {{dash|1/1, 5/4, 7/5, 8/5, 20/11 }}
+| {{dash|1/1, 5/4, 10/7, 8/5, 20/11 }}
 | {{dash|1/1, 8/7, 14/11, 16/11, 8/5  }}
 | {{dash|1/1, 9/8, 9/7, 7/5, 7/4  }}
@@ Line 1,585: / Line 1,585: @@
 |-
 | {{dash|0, 2, 8, 16, 18}}
-| {{dash|1, 9/8, 14/9, 11/9, 11/8}}
+| {{dash|1, 10/9, 14/9, 11/9, 11/8}}
 | euterpe
-| {{dash|1/1, 9/8, 11/9, 11/8, 14/9  }}
+| {{dash|1/1, 10/9, 11/9, 11/8, 14/9  }}
-| {{dash|1/1, 11/10, 11/9, 7/5, 16/9  }}
+| {{dash|1/1, 11/10, 11/9, 7/5, 9/5}}
 | {{dash|1/1, 9/8, 14/11, 18/11, 20/11 }}
 | {{dash|1/1, 8/7, 16/11, 18/11, 16/9  }}
@@ Line 1,622: / Line 1,622: @@
 | {{dash|0, 10, 14, 16, 18}}
 | {{dash|1, 7/4, 11/10, 11/9, 11/8}}
-| meantone
+| thrush
 | {{dash|1/1, 11/10, 11/9, 11/8, 7/4  }}
 | {{dash|1/1, 10/9, 5/4, 8/5, 20/11 }}
@@ Line 1,691: / Line 1,691: @@
 | {{dash|0, 4, 6, 8, 10, 14}}
 | {{dash|1, 5/4, 7/5, 14/9, 7/4, 11/10}}
-| meantone
+| minerva
 |-
 | 3
 | {{dash|0, 2, 6, 8, 10, 16}}
 | {{dash|1, 9/8, 7/5, 14/9, 7/4, 11/9}}
-| meantone
+| huygens
 |-
 | 4
 | {{dash|0, 6, 8, 10, 14, 16}}
 | {{dash|1, 7/5, 14/9, 7/4, 11/10, 11/9}}
-| meantone
+| huygens
 |-
 | 5
@@ Line 1,711: / Line 1,711: @@
 | {{dash|0, 2, 4, 8, 10, 18}}
 | {{dash|1, 9/8, 5/4, 14/9, 7/4, 11/8}}
-| meantone
+| huygens
 |-
 | 7
@@ Line 1,726: / Line 1,726: @@
 | {{dash|0, 8, 10, 14, 16, 18}}
 | {{dash|1, 14/9, 7/4, 11/10, 11/9, 11/8-}}
-| meantone
+| huygens
 |-
 | 10

Chords of huygens: Difference between revisions

Latest revision as of 23:07, 4 September 2025

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