Tetrachordal otonal scale: Difference between revisions

Latest revision as of 21:51, 26 October 2026

Tetrachordal otonal scales (TO(S)) are a type of octave-periodic scale built by taking concatenating two harmonic series segments that overlap at some points. They are a unique mix of tetrachords and overtone scales that exhibit omnitetrachordality to some extent, and a type of NEJI (Near-equivalent Just Intonation).

The concept is an extensio of what had already been theorized by Erv Wilson, which he named diaphonic cycle.^[1]

Definitions

To build a TO scale, do the following:

Take two harmonic series segments (rows): 2k::4k and 3k::6k. The 3k::6k is called the bottom row.
The rows will overlap exactly at the points of the harmonic series segment k::2k. The set of all TO scales for each k is called category k. Category 0 and 1 do not have any meaningful TO scales.
Any point where the two rows overlap is called a transition point. A transition point at 4/3 or 3/2 is called perfect, between 4/3 and 3/2 imperfect. Outside 4/3 ≤ c ≤ 3/2 the scale the transition is non-tetrachordal, and between that range, the TO scale can be built and thus they are optionally called tetrachordal transition points. In this article, all transition points are tetrachordal.
To build a TO scale, build a scale by starting on the bottom row and transitioning to the top row at a transition point between 4/3 and 3/2 inclusive.

Properties

Category x's amount of transition points can be derived with the following formula:[math]\displaystyle{ J\left(x\right)=1+\operatorname{floor}\left(\frac{x}{6}\right)-I\left(\frac{\operatorname{mod}\left(x,6\right)-1}{6}\right) \text{for} \space I\left(x\right)=1-\operatorname{ceil}\left(x\right)+\operatorname{floor}\left(x\right) }[/math] Which for each category x = 2,3,4… the list of tetrachordial transition points starts with 1,1,1,1,2,1,2,2,2…
The amount of perfect transition points of a scale can be derived with [math]\displaystyle{ K\left(x\right)=I\left(\frac{x}{2}\right)+I\left(\frac{x}{3}\right) }[/math]. Thus the amount of imperfect transition points is [math]\displaystyle{ J\left(x\right)-K\left(x\right) }[/math].
Clean scales are scales with that transition only once, from the bottom to the top row, and their amount is equal to the number of transition points [math]\displaystyle{ J(x) }[/math].
The total number of scales per category is [math]\displaystyle{ 2^{J\left(x\right)-1} }[/math], and the amount of non-clean scales (otherwise called convoluted) is thus [math]\displaystyle{ 2^{J\left(x\right)-1}-J\left(x\right) }[/math].
Category 0 and 1 are excluded, as they get nonsensical results like category 1 having 0.5 scales.
The amount of notes a scale of a given category x has [math]\displaystyle{ S(x) }[/math] falls between the ranges of these functions:[math]\displaystyle{ 2x+1+\operatorname{floor}\left(\frac{x-1}{3}\right) ≤ S(x) ≤ 2x+1+\operatorname{floor}\left(\frac{x-2}{2}\right) }[/math] And the amount of notes are closely related to the scale tree of diatonic scales.
The prime limits of the TOs of a category k sit between the biggest primes smaller than the 3 times the smallest and biggest transition point ratios.
All TOs are chiral, for which the right hand version is TO, the left hand is TU.

The data can be summarized in this table:

[math]\displaystyle{ x }[/math]	Transition points Number of clean scales	Perfect TP	Imperfect TP	Total scales	Convoluted scales	Note ranges		p-limits
2	1	1	0	1	0	5		7
3	1	1	0	1	0	7		11
4	1	1	0	1	0	10		17
5	1	0	1	1	0	12		19
6	2	2	0	2	0	14	15	23
7	1	0	1	1	0	17		29
8	2	1	1	2	0	19	20	31
9	2	1	1	2	0	21	22	31	37
10	2	1	1	2	0	24	25	37	43
11	2	0	2	2	0	26	27	43	47
12	3	2	1	4	1	28	30	47	53
13	2	0	2	2	0	31	32	47	53
14	3	1	2	4	1	33	35	53	61
15	3	1	2	4	1	35	37	59	61
16	3	1	2	4	1	38	40	61	71

Simple TOs can be visualized as edo or MOS detempers, which makes them particularly useful either for NEJI targets or essential tempering.

List of categories

For every provided TO, the EFR and used segment are written out explicitly. Names in italic are names never seen before on the wiki, declared here for specificity. Only TOs are listed: to get the respective TUs, retrovert the scale.

For categories with more than one scale, append letters from a up to whatever is necessary, which works for categories from 2 to 31 except 30 (using letters a-p). After that, append numbers.

Category 2 - Septimal minor pentatonic

12:14:16:18:21:24: TO 2.

First usable TO, the first prime TO, and the only one that is Strict variety 3 and a generator sequence.

It is naturally a 5edo detemper in the 2.3.7 subgroup, and either a semaphore (1L 4s 2|2) or archy detemper (2L 3s 3|1).

7/6	4/3	3/2	7/4
8/7	9/7	3/2	12/7
9/8	21/16	3/2	7/4
7/6	4/3	14/9	16/9
8/7	4/3	32/21	12/7

Category 3

18:20:22:24:27:30:33:36: TO 3.

Second prime TO. It is naturally a 7edo detemper in the 2.3.5.11 subgroup, and either a porcupine (1L 6s 3|3) or dicot/mothra detemper (3L 4s 5|1 #5).

10/9	11/9	4/3	3/2	5/3	11/6
11/10	6/5	27/20	3/2	33/20	9/5
12/11	27/22	15/11	3/2	18/11	20/11
9/8	5/4	11/8	3/2	5/3	11/6
10/9	11/9	4/3	40/27	44/27	16/9
11/10	6/5	4/3	22/15	8/5	9/5
12/11	40/33	4/3	16/11	18/11	20/11

Category 4

24:26:28:30:32:34:36:39:42:45:48: TO 4.

First composite TO, containing TO 2 as a subset. It is naturally a 10edo detemper in the 2.3.5.7.13 subgroup, a negridetemper (1L 8s 4|4) if you exclude 17/12, and a pajarous detemper, 2L 8s 6|2 #7 scale.

13/12	7/6	5/4	4/3	17/12	3/2	13/8	7/4	15/8
14/13	15/13	16/13	17/13	18/13	3/2	21/13	45/26	24/13
15/14	8/7	17/14	9/7	39/28	3/2	45/28	12/7	13/7
16/15	17/15	6/5	13/10	7/5	3/2	8/5	26/15	28/15
17/16	9/8	39/32	21/16	45/32	3/2	13/8	7/4	15/8
18/17	39/34	21/17	45/34	24/17	26/17	28/17	30/17	32/17
13/12	7/6	5/4	4/3	13/9	14/9	5/3	16/9	17/9
14/13	15/13	16/13	4/3	56/39	20/13	64/39	68/39	24/13
15/14	8/7	26/21	4/3	10/7	32/21	34/21	12/7	13/7
16/15	52/45	56/45	4/3	64/45	68/45	8/5	26/15	28/15

Category 5

30:32:34:36:38:40:42:45:48:51:54:57:60: TO 5.

15::20. Third prime TO. Its extension is naturally a very close 12edo detemper in the 2.3.5.17.19 subgroup, and a diaschismic detemper (10L 2s 8|2 b4) scale.

16/15	17/15	6/5	19/15	4/3	7/5	3/2	8/5	17/10	9/5	19/10
17/16	9/8	19/16	5/4	21/16	45/32	3/2	51/32	27/16	57/32	15/8
18/17	19/17	20/17	21/17	45/34	24/17	3/2	27/17	57/34	30/17	32/17
19/18	10/9	7/6	5/4	4/3	17/12	3/2	19/12	5/3	16/9	17/9
20/19	21/19	45/38	24/19	51/38	27/19	3/2	30/19	32/19	34/19	36/19
21/20	9/8	6/5	51/40	27/20	57/40	3/2	8/5	17/10	9/5	19/10
15/14	8/7	17/14	9/7	19/14	10/7	32/21	34/21	12/7	38/21	40/21
16/15	17/15	6/5	19/15	4/3	64/45	68/45	8/5	76/45	16/9	28/15
17/16	9/8	19/16	5/4	4/3	17/12	3/2	19/12	5/3	7/4	15/8
18/17	19/17	20/17	64/51	4/3	24/17	76/51	80/51	28/17	30/17	32/17
19/18	10/9	32/27	34/27	4/3	38/27	40/27	14/9	5/3	16/9	17/9
20/19	64/57	68/57	24/19	4/3	80/57	28/19	30/19	32/19	34/19	36/19

Category 6

It is the last category to feature no imperfect transition points and the first category to feature two transition points, and thus two scales. Last category to feature TO constant structures. It is a 14c-edo and 15edo detemper in the 2.3.5.7.11.19.23 subgroup (add-17 or add-13), and nautilus detemper (14L 1s 6|8).

TO 6a

36:38:40:42:44:46:48:51:54:57:60:63:66:69:72.

19/18	10/9	7/6	11/9	23/18	4/3	17/12	3/2	19/12	5/3	7/4	11/6	23/12
20/19	21/19	22/19	23/19	24/19	51/38	27/19	3/2	30/19	63/38	33/19	69/38	36/19
21/20	11/10	23/20	6/5	51/40	27/20	57/40	3/2	63/40	33/20	69/40	9/5	19/10
22/21	23/21	8/7	17/14	9/7	19/14	10/7	3/2	11/7	23/14	12/7	38/21	40/21
23/22	12/11	51/44	27/22	57/44	15/11	63/44	3/2	69/44	18/11	19/11	20/11	21/11
24/23	51/46	27/23	57/46	30/23	63/46	33/23	3/2	36/23	38/23	40/23	42/23	44/23
17/16	9/8	19/16	5/4	21/16	11/8	23/16	3/2	19/12	5/3	7/4	11/6	23/12
18/17	19/17	20/17	21/17	22/17	23/17	24/17	76/51	80/51	28/17	88/51	92/51	32/17
19/18	10/9	7/6	11/9	23/18	4/3	38/27	40/27	14/9	44/27	46/27	16/9	17/9
20/19	21/19	22/19	23/19	24/19	4/3	80/57	28/19	88/57	92/57	32/19	34/19	36/19
21/20	11/10	23/20	6/5	19/15	4/3	7/5	22/15	23/15	8/5	17/10	9/5	19/10
22/21	23/21	8/7	76/63	80/63	4/3	88/63	92/63	32/21	34/21	12/7	38/21	40/21
23/22	12/11	38/33	40/33	14/11	4/3	46/33	16/11	17/11	18/11	19/11	20/11	21/11
24/23	76/69	80/69	28/23	88/69	4/3	32/23	34/23	36/23	38/23	40/23	42/23	44/23

TO 6b

36:38:40:42:44:46:48:50:52:54:57:60:63:66:69:72.

19/18	10/9	7/6	11/9	23/18	4/3	25/18	13/9	3/2	19/12	5/3	7/4	11/6	23/12
20/19	21/19	22/19	23/19	24/19	25/19	26/19	27/19	3/2	30/19	63/38	33/19	69/38	36/19
21/20	11/10	23/20	6/5	5/4	13/10	27/20	57/40	3/2	63/40	33/20	69/40	9/5	19/10
22/21	23/21	8/7	25/21	26/21	9/7	19/14	10/7	3/2	11/7	23/14	12/7	38/21	40/21
23/22	12/11	25/22	13/11	27/22	57/44	15/11	63/44	3/2	69/44	18/11	19/11	20/11	21/11
24/23	25/23	26/23	27/23	57/46	30/23	63/46	33/23	3/2	36/23	38/23	40/23	42/23	44/23
25/24	13/12	9/8	19/16	5/4	21/16	11/8	23/16	3/2	19/12	5/3	7/4	11/6	23/12
26/25	27/25	57/50	6/5	63/50	33/25	69/50	36/25	38/25	8/5	42/25	44/25	46/25	48/25
27/26	57/52	15/13	63/52	33/26	69/52	18/13	19/13	20/13	21/13	22/13	23/13	24/13	25/13
19/18	10/9	7/6	11/9	23/18	4/3	38/27	40/27	14/9	44/27	46/27	16/9	50/27	52/27
20/19	21/19	22/19	23/19	24/19	4/3	80/57	28/19	88/57	92/57	32/19	100/57	104/57	36/19
21/20	11/10	23/20	6/5	19/15	4/3	7/5	22/15	23/15	8/5	5/3	26/15	9/5	19/10
22/21	23/21	8/7	76/63	80/63	4/3	88/63	92/63	32/21	100/63	104/63	12/7	38/21	40/21
23/22	12/11	38/33	40/33	14/11	4/3	46/33	16/11	50/33	52/33	18/11	19/11	20/11	21/11
24/23	76/69	80/69	28/23	88/69	4/3	32/23	100/69	104/69	36/23	38/23	40/23	42/23	44/23

Category 7

42:44:46:48:50:52:54:56:58:60:63:66:69:72:75:78:81:84: TO 7. The last category to feature only one scale.

Fourth prime TO. TU 7 is surprisingly a subharmonic interpolation of TO 3. It is naturally a 17edo detemper in the no-5,17,19,add-25-29-limit subgroup, surfacing as a strange liese extension detemper (2L 15s 15|1 10b).

It is also the biggest category to feature only one scale.

22/21	23/21	8/7	25/21	26/21	9/7	4/3	29/21	10/7	3/2	11/7	23/14	12/7	25/14	13/7	27/14
23/22	12/11	25/22	13/11	27/22	14/11	29/22	15/11	63/44	3/2	69/44	18/11	75/44	39/22	81/44	21/11
24/23	25/23	26/23	27/23	28/23	29/23	30/23	63/46	33/23	3/2	36/23	75/46	39/23	81/46	42/23	44/23
25/24	13/12	9/8	7/6	29/24	5/4	21/16	11/8	23/16	3/2	25/16	13/8	27/16	7/4	11/6	23/12
26/25	27/25	28/25	29/25	6/5	63/50	33/25	69/50	36/25	3/2	39/25	81/50	42/25	44/25	46/25	48/25
27/26	14/13	29/26	15/13	63/52	33/26	69/52	18/13	75/52	3/2	81/52	21/13	22/13	23/13	24/13	25/13
28/27	29/27	10/9	7/6	11/9	23/18	4/3	25/18	13/9	3/2	14/9	44/27	46/27	16/9	50/27	52/27
29/28	15/14	9/8	33/28	69/56	9/7	75/56	39/28	81/56	3/2	11/7	23/14	12/7	25/14	13/7	27/14
30/29	63/58	33/29	69/58	36/29	75/58	39/29	81/58	42/29	44/29	46/29	48/29	50/29	52/29	54/29	56/29
21/20	11/10	23/20	6/5	5/4	13/10	27/20	7/5	22/15	23/15	8/5	5/3	26/15	9/5	28/15	29/15
22/21	23/21	8/7	25/21	26/21	9/7	4/3	88/63	92/63	32/21	100/63	104/63	12/7	16/9	116/63	40/21
23/22	12/11	25/22	13/11	27/22	14/11	4/3	46/33	16/11	50/33	52/33	18/11	56/33	58/33	20/11	21/11
24/23	25/23	26/23	27/23	28/23	88/69	4/3	32/23	100/69	104/69	36/23	112/69	116/69	40/23	42/23	44/23
25/24	13/12	9/8	7/6	11/9	23/18	4/3	25/18	13/9	3/2	14/9	29/18	5/3	7/4	11/6	23/12
26/25	27/25	28/25	88/75	92/75	32/25	4/3	104/75	36/25	112/75	116/75	8/5	42/25	44/25	46/25	48/25
27/26	14/13	44/39	46/39	16/13	50/39	4/3	18/13	56/39	58/39	20/13	21/13	22/13	23/13	24/13	25/13
28/27	88/81	92/81	32/27	100/81	104/81	4/3	112/81	116/81	40/27	14/9	44/27	46/27	16/9	50/27	52/27

Category 8

It contains TO 2 and TO 4. With supersets of TO 4, it adds many new interesting intervals with odds 9, 11, 15, 21, 25, 29, 31, 35 along its modes. It is also the last category to fit completely within the 31-limit. TO 8a adds 23, TO 8b adds 17.

The scales can be seen wilsec detemper (negri weak extension) 19L 1s 10|9, but it is a very innacurate realization that only 19edo through patent val can faithfully represent (well enough for TO 8a albeit badly for TO 8b as it tempers together 17/12 and 35/24). Thus, it's better seen as just a linear interpolation of TO 4 in the 2.3.5.7.11.13.17.29.31 subgroup.

TO 8a

48:50:52:54:56:58:60:62:64:66:69:72:75:78:81:84:87:90:93:96. Last TO to be a constant structure

25/24	13/12	9/8	7/6	29/24	5/4	31/24	4/3	11/8	23/16	3/2	25/16	13/8	27/16	7/4	29/16	15/8	31/16
26/25	27/25	28/25	29/25	6/5	31/25	32/25	33/25	69/50	36/25	3/2	39/25	81/50	42/25	87/50	9/5	93/50	48/25
27/26	14/13	29/26	15/13	31/26	16/13	33/26	69/52	18/13	75/52	3/2	81/52	21/13	87/52	45/26	93/52	24/13	25/13
28/27	29/27	10/9	31/27	32/27	11/9	23/18	4/3	25/18	13/9	3/2	14/9	29/18	5/3	31/18	16/9	50/27	52/27
29/28	15/14	31/28	8/7	33/28	69/56	9/7	75/56	39/28	81/56	3/2	87/56	45/28	93/56	12/7	25/14	13/7	27/14
30/29	31/29	32/29	33/29	69/58	36/29	75/58	39/29	81/58	42/29	3/2	45/29	93/58	48/29	50/29	52/29	54/29	56/29
31/30	16/15	11/10	23/20	6/5	5/4	13/10	27/20	7/5	29/20	3/2	31/20	8/5	5/3	26/15	9/5	28/15	29/15
32/31	33/31	69/62	36/31	75/62	39/31	81/62	42/31	87/62	45/31	3/2	48/31	50/31	52/31	54/31	56/31	58/31	60/31
33/32	69/64	9/8	75/64	39/32	81/64	21/16	87/64	45/32	93/64	3/2	25/16	13/8	27/16	7/4	29/16	15/8	31/16
23/22	12/11	25/22	13/11	27/22	14/11	29/22	15/11	31/22	16/11	50/33	52/33	18/11	56/33	58/33	20/11	62/33	64/33
24/23	25/23	26/23	27/23	28/23	29/23	30/23	31/23	32/23	100/69	104/69	36/23	112/69	116/69	40/23	124/69	128/69	44/23
25/24	13/12	9/8	7/6	29/24	5/4	31/24	4/3	25/18	13/9	3/2	14/9	29/18	5/3	31/18	16/9	11/6	23/12
26/25	27/25	28/25	29/25	6/5	31/25	32/25	4/3	104/75	36/25	112/75	116/75	8/5	124/75	128/75	44/25	46/25	48/25
27/26	14/13	29/26	15/13	31/26	16/13	50/39	4/3	18/13	56/39	58/39	20/13	62/39	64/39	22/13	23/13	24/13	25/13
28/27	29/27	10/9	31/27	32/27	100/81	104/81	4/3	112/81	116/81	40/27	124/81	128/81	44/27	46/27	16/9	50/27	52/27
29/28	15/14	31/28	8/7	25/21	26/21	9/7	4/3	29/21	10/7	31/21	32/21	11/7	23/14	12/7	25/14	13/7	27/14
30/29	31/29	32/29	100/87	104/87	36/29	112/87	4/3	40/29	124/87	128/87	44/29	46/29	48/29	50/29	52/29	54/29	56/29
31/30	16/15	10/9	52/45	6/5	56/45	58/45	4/3	62/45	64/45	22/15	23/15	8/5	5/3	26/15	9/5	28/15	29/15
32/31	100/93	104/93	36/31	112/93	116/93	40/31	4/3	128/93	44/31	46/31	48/31	50/31	52/31	54/31	56/31	58/31	60/31

TO 8b

48:50:52:54:56:58:60:62:64:66:68:70:72:75:78:81:84:87:90:93:96.

25/24	13/12	9/8	7/6	29/24	5/4	31/24	4/3	11/8	17/12	35/24	3/2	25/16	13/8	27/16	7/4	29/16	15/8	31/16
26/25	27/25	28/25	29/25	6/5	31/25	32/25	33/25	34/25	7/5	36/25	3/2	39/25	81/50	42/25	87/50	9/5	93/50	48/25
27/26	14/13	29/26	15/13	31/26	16/13	33/26	17/13	35/26	18/13	75/52	3/2	81/52	21/13	87/52	45/26	93/52	24/13	25/13
28/27	29/27	10/9	31/27	32/27	11/9	34/27	35/27	4/3	25/18	13/9	3/2	14/9	29/18	5/3	31/18	16/9	50/27	52/27
29/28	15/14	31/28	8/7	33/28	17/14	5/4	9/7	75/56	39/28	81/56	3/2	87/56	45/28	93/56	12/7	25/14	13/7	27/14
30/29	31/29	32/29	33/29	34/29	35/29	36/29	75/58	39/29	81/58	42/29	3/2	45/29	93/58	48/29	50/29	52/29	54/29	56/29
31/30	16/15	11/10	17/15	7/6	6/5	5/4	13/10	27/20	7/5	29/20	3/2	31/20	8/5	5/3	26/15	9/5	28/15	29/15
32/31	33/31	34/31	35/31	36/31	75/62	39/31	81/62	42/31	87/62	45/31	3/2	48/31	50/31	52/31	54/31	56/31	58/31	60/31
33/32	17/16	35/32	9/8	75/64	39/32	81/64	21/16	87/64	45/32	93/64	3/2	25/16	13/8	27/16	7/4	29/16	15/8	31/16
34/33	35/33	12/11	25/22	13/11	27/22	14/11	29/22	15/11	31/22	16/11	50/33	52/33	18/11	56/33	58/33	20/11	62/33	64/33
35/34	18/17	75/68	39/34	81/68	21/17	87/68	45/34	93/68	24/17	25/17	26/17	27/17	28/17	29/17	30/17	31/17	32/17	33/17
36/35	15/14	39/35	81/70	6/5	87/70	9/7	93/70	48/35	10/7	52/35	54/35	8/5	58/35	12/7	62/35	64/35	66/35	68/35
25/24	13/12	9/8	7/6	29/24	5/4	31/24	4/3	25/18	13/9	3/2	14/9	29/18	5/3	31/18	16/9	11/6	17/9	35/18
26/25	27/25	28/25	29/25	6/5	31/25	32/25	4/3	104/75	36/25	112/75	116/75	8/5	124/75	128/75	44/25	136/75	28/15	48/25
27/26	14/13	29/26	15/13	31/26	16/13	50/39	4/3	18/13	56/39	58/39	20/13	62/39	64/39	22/13	68/39	70/39	24/13	25/13
28/27	29/27	10/9	31/27	32/27	100/81	104/81	4/3	112/81	116/81	40/27	124/81	128/81	44/27	136/81	140/81	16/9	50/27	52/27
29/28	15/14	31/28	8/7	25/21	26/21	9/7	4/3	29/21	10/7	31/21	32/21	11/7	34/21	5/3	12/7	25/14	13/7	27/14
30/29	31/29	32/29	100/87	104/87	36/29	112/87	4/3	40/29	124/87	128/87	44/29	136/87	140/87	48/29	50/29	52/29	54/29	56/29
31/30	16/15	10/9	52/45	6/5	56/45	58/45	4/3	62/45	64/45	22/15	68/45	14/9	8/5	5/3	26/15	9/5	28/15	29/15
32/31	100/93	104/93	36/31	112/93	116/93	40/31	4/3	128/93	44/31	136/93	140/93	48/31	50/31	52/31	54/31	56/31	58/31	60/31

Beyond Category 8

There are infinite categories with infinitely many TOs, but TOs grow increasingly more complex in prime palette and in interval count. As such, we present here only the TOs whose palettes fit within the 31-limit.

TO 9a

54:56:58:60:62:64:66:68:70:72:75:78:81:84:87:90:93:96:99:102:105:108. The biggest TO to feature a palette below the 31-limit, with harmonics 9-18 as a subset, as well as TO 3.

No patent val supports an equalized tuning; 19edo, the closest match, tempers together 34/27 and 35/27, as well as 17/9 and 35/128. The closest match is 22edo, which either of 22 or 22f vals provide the basic tuning for 1L 19s.

28/27	29/27	10/9	31/27	32/27	11/9	34/27	35/27	4/3	25/18	13/9	3/2	14/9	29/18	5/3	31/18	16/9	11/6	17/9	35/18
29/28	15/14	31/28	8/7	33/28	17/14	5/4	9/7	75/56	39/28	81/56	3/2	87/56	45/28	93/56	12/7	99/56	51/28	15/8	27/14
30/29	31/29	32/29	33/29	34/29	35/29	36/29	75/58	39/29	81/58	42/29	3/2	45/29	93/58	48/29	99/58	51/29	105/58	54/29	56/29
31/30	16/15	11/10	17/15	7/6	6/5	5/4	13/10	27/20	7/5	29/20	3/2	31/20	8/5	33/20	17/10	7/4	9/5	28/15	29/15
32/31	33/31	34/31	35/31	36/31	75/62	39/31	81/62	42/31	87/62	45/31	3/2	48/31	99/62	51/31	105/62	54/31	56/31	58/31	60/31
33/32	17/16	35/32	9/8	75/64	39/32	81/64	21/16	87/64	45/32	93/64	3/2	99/64	51/32	105/64	27/16	7/4	29/16	15/8	31/16
34/33	35/33	12/11	25/22	13/11	27/22	14/11	29/22	15/11	31/22	16/11	3/2	17/11	35/22	18/11	56/33	58/33	20/11	62/33	64/33
35/34	18/17	75/68	39/34	81/68	21/17	87/68	45/34	93/68	24/17	99/68	3/2	105/68	27/17	28/17	29/17	30/17	31/17	32/17	33/17
36/35	15/14	39/35	81/70	6/5	87/70	9/7	93/70	48/35	99/70	51/35	3/2	54/35	8/5	58/35	12/7	62/35	64/35	66/35	68/35
25/24	13/12	9/8	7/6	29/24	5/4	31/24	4/3	11/8	17/12	35/24	3/2	14/9	29/18	5/3	31/18	16/9	11/6	17/9	35/18
26/25	27/25	28/25	29/25	6/5	31/25	32/25	33/25	34/25	7/5	36/25	112/75	116/75	8/5	124/75	128/75	44/25	136/75	28/15	48/25
27/26	14/13	29/26	15/13	31/26	16/13	33/26	17/13	35/26	18/13	56/39	58/39	20/13	62/39	64/39	22/13	68/39	70/39	24/13	25/13
28/27	29/27	10/9	31/27	32/27	11/9	34/27	35/27	4/3	112/81	116/81	40/27	124/81	128/81	44/27	136/81	140/81	16/9	50/27	52/27
29/28	15/14	31/28	8/7	33/28	17/14	5/4	9/7	4/3	29/21	10/7	31/21	32/21	11/7	34/21	5/3	12/7	25/14	13/7	27/14
30/29	31/29	32/29	33/29	34/29	35/29	36/29	112/87	4/3	40/29	124/87	128/87	44/29	136/87	140/87	48/29	50/29	52/29	54/29	56/29
31/30	16/15	11/10	17/15	7/6	6/5	56/45	58/45	4/3	62/45	64/45	22/15	68/45	14/9	8/5	5/3	26/15	9/5	28/15	29/15
32/31	33/31	34/31	35/31	36/31	112/93	116/93	40/31	4/3	128/93	44/31	136/93	140/93	48/31	50/31	52/31	54/31	56/31	58/31	60/31
33/32	17/16	35/32	9/8	7/6	29/24	5/4	31/24	4/3	11/8	17/12	35/24	3/2	25/16	13/8	27/16	7/4	29/16	15/8	31/16
34/33	35/33	12/11	112/99	116/99	40/33	124/99	128/99	4/3	136/99	140/99	16/11	50/33	52/33	18/11	56/33	58/33	20/11	62/33	64/33
35/34	18/17	56/51	58/51	20/17	62/51	64/51	22/17	4/3	70/51	24/17	25/17	26/17	27/17	28/17	29/17	30/17	31/17	32/17	33/17
36/35	16/15	116/105	8/7	124/105	128/105	44/35	136/105	4/3	48/35	10/7	52/35	54/35	8/5	58/35	12/7	62/35	64/35	66/35	68/35

References

↑ 1. WILSON, Erv. Anaphoria [online]. [Accessed 26 October 2026 19:00 UTC+2]. Available from: https://www.anaphoria.com/diaphonicset.pdf

[1] 1. WILSON, Erv. Anaphoria [online]. [Accessed 26 October 2026 19:00 UTC+2]. Available from: https://www.anaphoria.com/diaphonicset.pdf

[1]

Tetrachordal otonal scale: Difference between revisions

Eufalesio (talk | contribs)

Latest revision as of 21:51, 26 October 2026

Contents

Definitions

Properties

List of categories

Category 2 - Septimal minor pentatonic

Category 3

Category 4

Category 5

Category 6

TO 6a

TO 6b

Category 7

Category 8

TO 8a

TO 8b

Beyond Category 8

TO 9a

References

Navigation menu

@@ Line 1: / Line 1: @@
-Harmonic segments along fifths (HSAF) are a type of octave-[[periodic scale]] built by taking a specific [[harmonic series segment]] that ends in 4/3 and repeating times 3/2. They are a unique mix of [[tetrachord]]<nowiki/>s and [[overtone scales]] that exhibit [[omnitetrachordality]] to some extent, and a type of [[NEJI]] (Near-equivalent Just Intonation).
+Tetrachordal otonal scales (TO(S)) are a type of octave-[[periodic scale]] built by taking concatenating two harmonic series segments that overlap at some points. They are a unique mix of [[tetrachord]]<nowiki/>s and [[overtone scales]] that exhibit [[omnitetrachordality]] to some extent, and a type of [[NEJI]] (Near-equivalent Just Intonation).
+The concept is an extensio of what had already been theorized by [[Erv Wilson]], which he named ''diaphonic cycle''.<ref>1. WILSON, Erv. ''Anaphoria'' [online]. [Accessed 26 October 2026 19:00 UTC+2]. Available from: https://www.anaphoria.com/diaphonicset.pdf</ref>
 == Definitions ==
-For a scale to be a HSAF, it must satisfy the following:
+To build a TO scale, do the following:
-* Its [[period]] is the octave.
-* Its prime mode always contains 4/3 and 3/2.
-* Its prime mode is built like a harmonic series segment up to 4/3, and that same harmonic segment is repeated transposed by 3/2 until it reaches the octave.
-* HSAF 1 is 6:8:9:12 with [[delta signature]] +2,+1,+2. Except for HSAF 1, the delta signature of any HSAF k is +2,+2,+2,+2,[…],+3,+3,+3,[…]. If the HSAF is even-indexed, the +3's start right at 3/2. If the HSAF is odd-indexed, at the step right before 3/2.
-HSAFs are a countable set of scales, and thus can be indexed.
+* Take two harmonic series segments (rows): 2k::4k and 3k::6k. The 3k::6k is called the '''bottom row.'''
+* The rows will overlap exactly at the points of the harmonic series segment k::2k. The set of all TO scales for each k is called '''category k.''' Category 0 and 1 do not have any meaningful TO scales.
+* Any point where the two rows overlap is called a '''transition point'''. A transition point at 4/3 or 3/2 is called '''perfect''', between 4/3 and 3/2 '''imperfect'''. Outside 4/3 ≤ c ≤ 3/2 the scale the transition is '''non-tetrachordal,''' and between that range, the TO scale can be built and thus they are optionally called '''tetrachordal''' transition points. In this article, all transition points are tetrachordal.
+* To build a TO scale, build a scale by starting on the bottom row and transitioning to the top row at a transition point between 4/3 and 3/2 inclusive.
 == Properties ==
-* Harmonic segments conforming HSAFs up to the 13th* are [[strictly proper]] [[constant structure]]<nowiki/>s.
+* Category x's amount of transition points can be derived with the following formula:<math>J\left(x\right)=1+\operatorname{floor}\left(\frac{x}{6}\right)-I\left(\frac{\operatorname{mod}\left(x,6\right)-1}{6}\right) \text{for} \space I\left(x\right)=1-\operatorname{ceil}\left(x\right)+\operatorname{floor}\left(x\right)</math> Which for each category x = 2,3,4… the list of tetrachordial transition points starts with 1,1,1,1,2,1,2,2,2…
-* HSAF k is a subset of the 6k::12k harmonic segment.
+* The amount of perfect transition points of a scale can be derived with <math>K\left(x\right)=I\left(\frac{x}{2}\right)+I\left(\frac{x}{3}\right)</math>. Thus the amount of imperfect transition points is <math>J\left(x\right)-K\left(x\right)</math>.
-* HSAF k contains harmonics k–2k as a subset.
+* '''Clean scales''' are scales with that transition only once, from the bottom to the top row, and their amount is equal to the number of transition points <math>J(x)</math>.
-* Because they are formed from harmonic series segments, they are chiral scales*, whose left-handed counterparts are built from [[subharmonic series]] segments. The left-handed versions are '''not''' HSAFs per se. They are only the chiral version of the HSAF, which you might call a SSAF, for Subharmonic Segment [...].
+* The total number of scales per category is <math>2^{J\left(x\right)-1}</math>, and the amount of non-clean scales (otherwise called ''convoluted'') is thus <math>2^{J\left(x\right)-1}-J\left(x\right)</math>.
+* Category 0 and 1 are excluded, as they get nonsensical results like category 1 having 0.5 scales.
+* The amount of notes a scale of a given category x has <math>S(x)</math> falls between the ranges of these functions:<math>2x+1+\operatorname{floor}\left(\frac{x-1}{3}\right) ≤ S(x) ≤ 2x+1+\operatorname{floor}\left(\frac{x-2}{2}\right) </math> And the amount of notes are closely related to the scale tree of diatonic scales.
+* The prime limits of the TOs of a category k sit between the biggest primes smaller than the 3 times the smallest and biggest transition point ratios.
+* All TOs are chiral, for which the right hand version is TO, the left hand is TU.
+The data can be summarized in this table:
+{| class="wikitable"
+|-
+! <math>x</math> !! Transition points
+Number of clean scales
+! Perfect TP !! Imperfect TP !! Total scales !! Convoluted scales !! colspan="2" | Note ranges
+! colspan="2" |p-limits
+|-
+| 2 || 1 || 1 || 0 || 1 || 0 || colspan="2" | 5
+| colspan="2" |7
+|-
+| 3 || 1 || 1 || 0 || 1 || 0 || colspan="2" | 7
+| colspan="2" |11
+|-
+| 4 || 1 || 1 || 0 || 1 || 0 || colspan="2" | 10
+| colspan="2" |17
+|-
+| 5 || 1 || 0 || 1 || 1 || 0 || colspan="2" | 12
+| colspan="2" |19
+|-
+| 6 || 2 || 2 || 0 || 2 || 0 || 14 || 15
+| colspan="2" |23
+|-
+| 7 || 1 || 0 || 1 || 1 || 0 || colspan="2" | 17
+| colspan="2" |29
+|-
+| 8 || 2 || 1 || 1 || 2 || 0 || 19 || 20
+| colspan="2" |31
+|-
+| 9 || 2 || 1 || 1 || 2 || 0 || 21 || 22
+|31
+|37
+|-
+| 10 || 2 || 1 || 1 || 2 || 0 || 24 || 25
+|37
+|43
+|-
+| 11 || 2 || 0 || 2 || 2 || 0 || 26 || 27
+|43
+|47
+|-
+| 12 || 3 || 2 || 1 || 4 || 1 || 28 || 30
+|47
+|53
+|-
+| 13 || 2 || 0 || 2 || 2 || 0 || 31 || 32
+|47
+|53
+|-
+| 14 || 3 || 1 || 2 || 4 || 1 || 33 || 35
+|53
+|61
+|-
+| 15 || 3 || 1 || 2 || 4 || 1 || 35 || 37
+|59
+|61
+|-
+| 16 || 3 || 1 || 2 || 4 || 1 || 38 || 40
+|61
+|71
+|}
+Simple TOs can be visualized as edo or MOS detempers, which makes them particularly useful either for NEJI targets or essential tempering.
-<nowiki>*</nowiki>except the first
+== List of categories ==
+For every provided TO, the [[EFR]] and used segment are written out explicitly. Names in italic are names never seen before on the wiki, declared here for specificity. Only TOs are listed: to get the respective TUs, retrovert the scale.
-Simple HSAFs can be visualized as edo or MOS detempers for simple HSAFs, which makes them particularly useful either for NEJI targets or essential tempering.
+For categories with more than one scale, append letters from a up to whatever is necessary, which works for categories from 2 to 31 except 30 (using letters a-p). After that, append numbers.
-== List of HSAFs 1-8 ==
+=== Category 2 - Septimal minor pentatonic ===
-For every provided HSAF, the [[EFR]] and used segment are written out explicitly. Names in italic are names never seen before on the wiki, declared here for specificity.
+<code>12:14:16:18:21:24</code>: TO 2.
-=== HSAF/SSAF 1 - Pythagorean Trial ===
+First usable TO, the first prime TO, and the only one that is [[Strict variety]] 3 and a [[generator sequence]].
-<code>6:8:9:12</code>. Harmonic segment: '''3:4'''.
-First meaningful HSAF albeit a trivial case, and the only HSAF that is also a [[MOS scale]] ([[2L 1s]]), and thus [[achiral]].
+It is naturally a [[5edo]] detemper in the [[2.3.7 subgroup]], and either a semaphore ([[1L 4s]] 2|2) or [[archy]] detemper ([[2L 3s]] 3|1).
 {| class="wikitable" data-darkreader-inline-color=""
+!7/6
 !4/3
 !3/2
-! rowspan="3" |2
+!7/4
+! rowspan="5" |2
 |-
-|<small>9/8</small>
+|<small>8/7</small>
+|<small>9/7</small>
 |<small>3/2</small>
+|<small>12/7</small>
 |-
-|<small>4/3</small>
+|<small>9/8</small>
-|<small>16/9</small>
-|}
-=== HSAF 2 - Septimal minor pentatonic ===
-<code>12:14:16:18:21:24</code>. Harmonic segment: '''6::8'''.
-First usable HSAF, the first prime HSAF, and the only one that is [[Strict variety]] 3 and a [[generator sequence]].
-It is naturally a [[5edo]] detemper in the [[2.3.7 subgroup]], and either a semaphore ([[1L 4s]] 2|2) or [[archy]] detemper ([[2L 3s]] 3|1).
-{| class="wikitable" data-darkreader-inline-color=""
-!7/6
-!4/3
-!3/2
-!7/4
-! rowspan="5" |2
-|-
-|<small>8/7</small>
-|<small>9/7</small>
-|<small>3/2</small>
-|<small>12/7</small>
-|-
-|<small>9/8</small>
 |<small>21/16</small>
 |<small>3/2</small>
@@ Line 75: / Line 123: @@
 |}
-=== HSAF 3 ===
+=== Category 3 ===
-<code>18:20:22:24:27:30:33:36</code>. Harmonic segment: '''9::12'''.
+<code>18:20:22:24:27:30:33:36</code>: TO 3.
-Second prime HSAF. It is naturally a [[7edo]] detemper in the [[2.3.5.11 subgroup]], and either a [[porcupine]] ([[1L 6s]] 3|3) or [[dicot]]/[[mothra]] detemper ([[3L 4s]] 5|1 #5).
+Second prime TO. It is naturally a [[7edo]] detemper in the [[2.3.5.11 subgroup]], and either a [[porcupine]] ([[1L 6s]] 3|3) or [[dicot]]/[[mothra]] detemper ([[3L 4s]] 5|1 #5).
 {| class="wikitable" data-darkreader-inline-color=""
 !'''10/9'''
@@ Line 131: / Line 179: @@
 |}
-=== HSAF 4 ===
+=== Category 4 ===
-<code>24:26:28:30:32:34:36:39:42:45:48</code>. Harmonic segment: '''12::16'''.
+<code>24:26:28:30:32:34:36:39:42:45:48</code>: TO 4.
-First composite HSAF, containing HSAF 2 as a subset. It is naturally a [[10edo]] detemper in the [[2.3.5.7.13 subgroup]], a [[negri]]<nowiki/>detemper ([[1L 8s]] 4|4) if you exclude 17/12, and a [[pajarous]] detemper, 2L 8s 6|2 #7 scale.
+First composite TO, containing TO 2 as a subset. It is naturally a [[10edo]] detemper in the [[2.3.5.7.13 subgroup]], a [[negri]]<nowiki/>detemper ([[1L 8s]] 4|4) if you exclude 17/12, and a [[pajarous]] detemper, 2L 8s 6|2 #7 scale.
 {| class="wikitable" data-darkreader-inline-color=""
 !'''13/12'''
@@ Line 238: / Line 286: @@
 |}
-=== HSAF 5 ===
+=== Category 5 ===
-<code>30:32:34:36:38:40:42:45:48:51:54:57:60</code>. Harmonic segment: '''12::16'''.
+<code>30:32:34:36:38:40:42:45:48:51:54:57:60</code>: TO 5.
-'''15::20'''. It is the third prime HSAF. Its extension is naturally a very close [[12edo]] detemper in the [[2.3.5.17.19 subgroup]], and a [[diaschismic]] detemper ([[10L 2s]] 8|2 b4) scale.
+'''15::20'''. Third prime TO. Its extension is naturally a very close [[12edo]] detemper in the [[2.3.5.17.19 subgroup]], and a [[diaschismic]] detemper ([[10L 2s]] 8|2 b4) scale.
 {| class="wikitable" data-darkreader-inline-color=""
 !16/15
@@ Line 389: / Line 437: @@
 |}
-=== HSAF 6 ===
+=== Category 6 ===
-<code>36:38:40:42:44:46:48:50:52:54:57:60:63:66:69:72</code>. Harmonic segment: '''18::24'''.
+It is the last category to feature no imperfect transition points and the first category to feature two transition points, and thus two scales. Last category to feature TO [[constant structure]]<nowiki/>s. It is a [[14edo|14c-edo]] and [[15edo]] detemper in the 2.3.5.7.11.19.23 subgroup (add-17 or add-13), and [[nautilus]] detemper ([[14L 1s]] 6|8).
-Last HSAF to be a constant structure, and first HSAF to contain two different (non-1) HSAFs as subsets. It is a [[15edo]]<nowiki/>detemper in the 2.3.5.7.11.19.23 subgroup, and [[nautilus]] detemper ([[1L 12s]] 6|6 and [[14L 1s]] 6|8).
+==== TO 6a ====
+<code>36:38:40:42:44:46:48:51:54:57:60:63:66:69:72</code>.
 {| class="wikitable" data-darkreader-inline-color=""
+|-
 !19/18
 !10/9
@@ Line 400: / Line 450: @@
 !23/18
 !4/3
-!25/18
+!17/12
-!13/9
 !3/2
 !19/12
@@ Line 408: / Line 457: @@
 !11/6
 !23/12
-! rowspan="15" |2
+! rowspan="14" |2
 |-
 |<small>20/19</small>
@@ Line 415: / Line 464: @@
 |<small>23/19</small>
 |<small>24/19</small>
-|<small>25/19</small>
+|<small>51/38</small>
-|<small>26/19</small>
 |<small>27/19</small>
 |<small>3/2</small>
@@ Line 429: / Line 477: @@
 |<small>23/20</small>
 |<small>6/5</small>
-|<small>5/4</small>
+|<small>51/40</small>
-|<small>13/10</small>
 |<small>27/20</small>
 |<small>57/40</small>
@@ Line 443: / Line 490: @@
 |<small>23/21</small>
 |<small>8/7</small>
-|<small>25/21</small>
+|<small>17/14</small>
-|<small>26/21</small>
 |<small>9/7</small>
 |<small>19/14</small>
@@ Line 457: / Line 503: @@
 |<small>23/22</small>
 |<small>12/11</small>
-|<small>25/22</small>
+|<small>51/44</small>
-|<small>13/11</small>
 |<small>27/22</small>
 |<small>57/44</small>
@@ Line 471: / Line 516: @@
 |-
 |<small>24/23</small>
-|<small>25/23</small>
+|<small>51/46</small>
-|<small>26/23</small>
 |<small>27/23</small>
 |<small>57/46</small>
@@ Line 485: / Line 529: @@
 |<small>44/23</small>
 |-
-|<small>25/24</small>
+|<small>17/16</small>
-|<small>13/12</small>
 |<small>9/8</small>
 |<small>19/16</small>
@@ Line 500: / Line 543: @@
 |<small>23/12</small>
 |-
-|<small>26/25</small>
+|<small>18/17</small>
-|<small>27/25</small>
+|<small>19/17</small>
-|<small>57/50</small>
+|<small>20/17</small>
-|<small>6/5</small>
+|<small>21/17</small>
-|<small>63/50</small>
+|<small>22/17</small>
-|<small>33/25</small>
+|<small>23/17</small>
-|<small>69/50</small>
+|<small>24/17</small>
-|<small>36/25</small>
+|<small>76/51</small>
-|<small>38/25</small>
+|<small>80/51</small>
-|<small>8/5</small>
+|<small>28/17</small>
-|<small>42/25</small>
+|<small>88/51</small>
-|<small>44/25</small>
+|<small>92/51</small>
-|<small>46/25</small>
+|<small>32/17</small>
-|<small>48/25</small>
-|-
-|<small>27/26</small>
-|<small>57/52</small>
-|<small>15/13</small>
-|<small>63/52</small>
-|<small>33/26</small>
-|<small>69/52</small>
-|<small>18/13</small>
-|<small>19/13</small>
-|<small>20/13</small>
-|<small>21/13</small>
-|<small>22/13</small>
-|<small>23/13</small>
-|<small>24/13</small>
-|<small>25/13</small>
 |-
 |<small>19/18</small>
@@ Line 542: / Line 569: @@
 |<small>46/27</small>
 |<small>16/9</small>
-|<small>50/27</small>
+|<small>17/9</small>
-|<small>52/27</small>
 |-
 |<small>20/19</small>
@@ Line 556: / Line 582: @@
 |<small>92/57</small>
 |<small>32/19</small>
-|<small>100/57</small>
+|<small>34/19</small>
-|<small>104/57</small>
 |<small>36/19</small>
 |-
@@ Line 570: / Line 595: @@
 |<small>23/15</small>
 |<small>8/5</small>
-|<small>5/3</small>
+|<small>17/10</small>
-|<small>26/15</small>
 |<small>9/5</small>
 |<small>19/10</small>
@@ Line 584: / Line 608: @@
 |<small>92/63</small>
 |<small>32/21</small>
-|<small>100/63</small>
+|<small>34/21</small>
-|<small>104/63</small>
 |<small>12/7</small>
 |<small>38/21</small>
@@ Line 598: / Line 621: @@
 |<small>46/33</small>
 |<small>16/11</small>
-|<small>50/33</small>
+|<small>17/11</small>
-|<small>52/33</small>
 |<small>18/11</small>
 |<small>19/11</small>
@@ Line 612: / Line 634: @@
 |<small>4/3</small>
 |<small>32/23</small>
-|<small>100/69</small>
+|<small>34/23</small>
-|<small>104/69</small>
 |<small>36/23</small>
 |<small>38/23</small>
@@ Line 621: / Line 642: @@
 |}
-=== HSAF 7 ===
+==== TO 6b ====
-<code>42:44:46:48:50:52:54:56:58:60:63:66:69:72:75:78:81:84</code>'''.''' Harmonic segment: '''21::28'''.
+<code>36:38:40:42:44:46:48:50:52:54:57:60:63:66:69:72</code>.
-The fourth prime HSAF. SSAF 7 is surprisingly a subharmonic interpolation of HSAF 3. It is naturally a [[17edo]] detemper in the no-5,17,19,add-25-29-limit subgroup, surfacing as a strange liese extension detemper ([[2L 15s]] 15|1 10b).
 {| class="wikitable" data-darkreader-inline-color=""
-!22/21
+!19/18
-!23/21
+!10/9
-!8/7
+!7/6
-!25/21
+!11/9
-!26/21
+!23/18
-!9/7
 !4/3
-!29/21
+!25/18
-!10/7
+!13/9
 !3/2
-!11/7
+!19/12
-!23/14
+!5/3
-!12/7
+!7/4
-!25/14
+!11/6
-!13/7
+!23/12
-!27/14
+! rowspan="15" |2
-! rowspan="17" |2
 |-
-|<small>23/22</small>
+|<small>20/19</small>
-|<small>12/11</small>
+|<small>21/19</small>
-|<small>25/22</small>
+|<small>22/19</small>
-|<small>13/11</small>
+|<small>23/19</small>
-|<small>27/22</small>
+|<small>24/19</small>
-|<small>14/11</small>
+|<small>25/19</small>
-|<small>29/22</small>
+|<small>26/19</small>
-|<small>15/11</small>
+|<small>27/19</small>
-|<small>63/44</small>
 |<small>3/2</small>
-|<small>69/44</small>
+|<small>30/19</small>
-|<small>18/11</small>
+|<small>63/38</small>
-|<small>75/44</small>
+|<small>33/19</small>
-|<small>39/22</small>
+|<small>69/38</small>
-|<small>81/44</small>
+|<small>36/19</small>
-|<small>21/11</small>
 |-
-|<small>24/23</small>
+|<small>21/20</small>
-|<small>25/23</small>
+|<small>11/10</small>
-|<small>26/23</small>
+|<small>23/20</small>
-|<small>27/23</small>
+|<small>6/5</small>
-|<small>28/23</small>
+|<small>5/4</small>
-|<small>29/23</small>
+|<small>13/10</small>
-|<small>30/23</small>
+|<small>27/20</small>
-|<small>63/46</small>
+|<small>57/40</small>
-|<small>33/23</small>
 |<small>3/2</small>
-|<small>36/23</small>
+|<small>63/40</small>
-|<small>75/46</small>
+|<small>33/20</small>
-|<small>39/23</small>
+|<small>69/40</small>
-|<small>81/46</small>
+|<small>9/5</small>
-|<small>42/23</small>
+|<small>19/10</small>
-|<small>44/23</small>
 |-
-|<small>25/24</small>
+|<small>22/21</small>
-|<small>13/12</small>
+|<small>23/21</small>
-|<small>9/8</small>
+|<small>8/7</small>
-|<small>7/6</small>
+|<small>25/21</small>
-|<small>29/24</small>
+|<small>26/21</small>
-|<small>5/4</small>
+|<small>9/7</small>
-|<small>21/16</small>
+|<small>19/14</small>
-|<small>11/8</small>
+|<small>10/7</small>
-|<small>23/16</small>
 |<small>3/2</small>
-|<small>25/16</small>
+|<small>11/7</small>
-|<small>13/8</small>
+|<small>23/14</small>
-|<small>27/16</small>
+|<small>12/7</small>
-|<small>7/4</small>
+|<small>38/21</small>
-|<small>11/6</small>
+|<small>40/21</small>
-|<small>23/12</small>
 |-
-|<small>26/25</small>
+|<small>23/22</small>
-|<small>27/25</small>
+|<small>12/11</small>
-|<small>28/25</small>
+|<small>25/22</small>
-|<small>29/25</small>
+|<small>13/11</small>
-|<small>6/5</small>
+|<small>27/22</small>
-|<small>63/50</small>
+|<small>57/44</small>
-|<small>33/25</small>
+|<small>15/11</small>
-|<small>69/50</small>
+|<small>63/44</small>
-|<small>36/25</small>
 |<small>3/2</small>
-|<small>39/25</small>
+|<small>69/44</small>
-|<small>81/50</small>
+|<small>18/11</small>
-|<small>42/25</small>
+|<small>19/11</small>
-|<small>44/25</small>
+|<small>20/11</small>
-|<small>46/25</small>
+|<small>21/11</small>
-|<small>48/25</small>
 |-
-|<small>27/26</small>
+|<small>24/23</small>
-|<small>14/13</small>
+|<small>25/23</small>
-|<small>29/26</small>
+|<small>26/23</small>
-|<small>15/13</small>
+|<small>27/23</small>
-|<small>63/52</small>
+|<small>57/46</small>
-|<small>33/26</small>
+|<small>30/23</small>
-|<small>69/52</small>
+|<small>63/46</small>
-|<small>18/13</small>
+|<small>33/23</small>
-|<small>75/52</small>
 |<small>3/2</small>
-|<small>81/52</small>
+|<small>36/23</small>
-|<small>21/13</small>
+|<small>38/23</small>
-|<small>22/13</small>
+|<small>40/23</small>
-|<small>23/13</small>
+|<small>42/23</small>
-|<small>24/13</small>
+|<small>44/23</small>
-|<small>25/13</small>
 |-
-|<small>28/27</small>
+|<small>25/24</small>
-|<small>29/27</small>
+|<small>13/12</small>
-|<small>10/9</small>
+|<small>9/8</small>
-|<small>7/6</small>
+|<small>19/16</small>
-|<small>11/9</small>
+|<small>5/4</small>
-|<small>23/18</small>
+|<small>21/16</small>
-|<small>4/3</small>
+|<small>11/8</small>
-|<small>25/18</small>
+|<small>23/16</small>
-|<small>13/9</small>
 |<small>3/2</small>
-|<small>14/9</small>
+|<small>19/12</small>
-|<small>44/27</small>
+|<small>5/3</small>
-|<small>46/27</small>
+|<small>7/4</small>
-|<small>16/9</small>
+|<small>11/6</small>
-|<small>50/27</small>
+|<small>23/12</small>
-|<small>52/27</small>
 |-
-|<small>29/28</small>
+|<small>26/25</small>
-|<small>15/14</small>
+|<small>27/25</small>
-|<small>9/8</small>
+|<small>57/50</small>
-|<small>33/28</small>
+|<small>6/5</small>
-|<small>69/56</small>
+|<small>63/50</small>
-|<small>9/7</small>
+|<small>33/25</small>
-|<small>75/56</small>
+|<small>69/50</small>
-|<small>39/28</small>
+|<small>36/25</small>
-|<small>81/56</small>
+|<small>38/25</small>
-|<small>3/2</small>
+|<small>8/5</small>
-|<small>11/7</small>
+|<small>42/25</small>
-|<small>23/14</small>
+|<small>44/25</small>
-|<small>12/7</small>
+|<small>46/25</small>
-|<small>25/14</small>
+|<small>48/25</small>
-|<small>13/7</small>
-|<small>27/14</small>
 |-
-|<small>30/29</small>
+|<small>27/26</small>
-|<small>63/58</small>
+|<small>57/52</small>
-|<small>33/29</small>
+|<small>15/13</small>
-|<small>69/58</small>
+|<small>63/52</small>
-|<small>36/29</small>
+|<small>33/26</small>
-|<small>75/58</small>
+|<small>69/52</small>
-|<small>39/29</small>
+|<small>18/13</small>
-|<small>81/58</small>
+|<small>19/13</small>
-|<small>42/29</small>
+|<small>20/13</small>
-|<small>44/29</small>
+|<small>21/13</small>
-|<small>46/29</small>
+|<small>22/13</small>
-|<small>48/29</small>
+|<small>23/13</small>
-|<small>50/29</small>
+|<small>24/13</small>
-|<small>52/29</small>
+|<small>25/13</small>
-|<small>54/29</small>
-|<small>56/29</small>
 |-
-|<small>21/20</small>
+|<small>19/18</small>
-|<small>11/10</small>
+|<small>10/9</small>
-|<small>23/20</small>
+|<small>7/6</small>
-|<small>6/5</small>
+|<small>11/9</small>
-|<small>5/4</small>
+|<small>23/18</small>
-|<small>13/10</small>
+|<small>4/3</small>
-|<small>27/20</small>
+|<small>38/27</small>
-|<small>7/5</small>
+|<small>40/27</small>
-|<small>22/15</small>
+|<small>14/9</small>
-|<small>23/15</small>
+|<small>44/27</small>
-|<small>8/5</small>
+|<small>46/27</small>
-|<small>5/3</small>
+|<small>16/9</small>
-|<small>26/15</small>
+|<small>50/27</small>
-|<small>9/5</small>
+|<small>52/27</small>
-|<small>28/15</small>
-|<small>29/15</small>
 |-
-|<small>22/21</small>
+|<small>20/19</small>
-|<small>23/21</small>
+|<small>21/19</small>
-|<small>8/7</small>
+|<small>22/19</small>
-|<small>25/21</small>
+|<small>23/19</small>
-|<small>26/21</small>
+|<small>24/19</small>
-|<small>9/7</small>
 |<small>4/3</small>
-|<small>88/63</small>
+|<small>80/57</small>
-|<small>92/63</small>
+|<small>28/19</small>
-|<small>32/21</small>
+|<small>88/57</small>
-|<small>100/63</small>
+|<small>92/57</small>
-|<small>104/63</small>
+|<small>32/19</small>
-|<small>12/7</small>
+|<small>100/57</small>
-|<small>16/9</small>
+|<small>104/57</small>
-|<small>116/63</small>
+|<small>36/19</small>
-|<small>40/21</small>
 |-
-|<small>23/22</small>
+|<small>21/20</small>
-|<small>12/11</small>
+|<small>11/10</small>
-|<small>25/22</small>
+|<small>23/20</small>
-|<small>13/11</small>
+|<small>6/5</small>
-|<small>27/22</small>
+|<small>19/15</small>
-|<small>14/11</small>
 |<small>4/3</small>
-|<small>46/33</small>
+|<small>7/5</small>
-|<small>16/11</small>
+|<small>22/15</small>
-|<small>50/33</small>
+|<small>23/15</small>
-|<small>52/33</small>
+|<small>8/5</small>
-|<small>18/11</small>
+|<small>5/3</small>
-|<small>56/33</small>
+|<small>26/15</small>
-|<small>58/33</small>
+|<small>9/5</small>
-|<small>20/11</small>
+|<small>19/10</small>
-|<small>21/11</small>
 |-
-|<small>24/23</small>
+|<small>22/21</small>
-|<small>25/23</small>
+|<small>23/21</small>
-|<small>26/23</small>
+|<small>8/7</small>
-|<small>27/23</small>
+|<small>76/63</small>
-|<small>28/23</small>
+|<small>80/63</small>
-|<small>88/69</small>
 |<small>4/3</small>
-|<small>32/23</small>
+|<small>88/63</small>
-|<small>100/69</small>
+|<small>92/63</small>
-|<small>104/69</small>
+|<small>32/21</small>
-|<small>36/23</small>
+|<small>100/63</small>
-|<small>112/69</small>
+|<small>104/63</small>
-|<small>116/69</small>
+|<small>12/7</small>
-|<small>40/23</small>
+|<small>38/21</small>
-|<small>42/23</small>
+|<small>40/21</small>
-|<small>44/23</small>
 |-
-|<small>25/24</small>
+|<small>23/22</small>
-|<small>13/12</small>
+|<small>12/11</small>
-|<small>9/8</small>
+|<small>38/33</small>
-|<small>7/6</small>
+|<small>40/33</small>
-|<small>11/9</small>
+|<small>14/11</small>
-|<small>23/18</small>
 |<small>4/3</small>
-|<small>25/18</small>
+|<small>46/33</small>
-|<small>13/9</small>
+|<small>16/11</small>
-|<small>3/2</small>
+|<small>50/33</small>
-|<small>14/9</small>
+|<small>52/33</small>
-|<small>29/18</small>
+|<small>18/11</small>
-|<small>5/3</small>
+|<small>19/11</small>
-|<small>7/4</small>
+|<small>20/11</small>
-|<small>11/6</small>
+|<small>21/11</small>
-|<small>23/12</small>
 |-
-|<small>26/25</small>
+|<small>24/23</small>
-|<small>27/25</small>
+|<small>76/69</small>
-|<small>28/25</small>
+|<small>80/69</small>
-|<small>88/75</small>
+|<small>28/23</small>
-|<small>92/75</small>
+|<small>88/69</small>
-|<small>32/25</small>
 |<small>4/3</small>
-|<small>104/75</small>
+|<small>32/23</small>
-|<small>36/25</small>
+|<small>100/69</small>
-|<small>112/75</small>
+|<small>104/69</small>
-|<small>116/75</small>
+|<small>36/23</small>
-|<small>8/5</small>
+|<small>38/23</small>
-|<small>42/25</small>
+|<small>40/23</small>
-|<small>44/25</small>
+|<small>42/23</small>
-|<small>46/25</small>
+|<small>44/23</small>
-|<small>48/25</small>
+|}
-|-
-|<small>27/26</small>
+=== Category 7 ===
-|<small>14/13</small>
+<code>42:44:46:48:50:52:54:56:58:60:63:66:69:72:75:78:81:84</code>''':''' TO 7. The last category to feature only one scale.
-|<small>44/39</small>
-|<small>46/39</small>
+Fourth prime TO. TU 7 is surprisingly a subharmonic interpolation of TO 3. It is naturally a [[17edo]] detemper in the no-5,17,19,add-25-29-limit subgroup, surfacing as a strange liese extension detemper ([[2L 15s]] 15|1 10b).
-|<small>16/13</small>
-|<small>50/39</small>
+It is also the biggest category to feature only one scale.
-|<small>4/3</small>
+{| class="wikitable" data-darkreader-inline-color=""
-|<small>18/13</small>
+!22/21
-|<small>56/39</small>
+!23/21
-|<small>58/39</small>
+!8/7
-|<small>20/13</small>
+!25/21
-|<small>21/13</small>
+!26/21
-|<small>22/13</small>
+!9/7
-|<small>23/13</small>
+!4/3
-|<small>24/13</small>
+!29/21
-|<small>25/13</small>
+!10/7
-|-
+!3/2
-|<small>28/27</small>
+!11/7
-|<small>88/81</small>
+!23/14
-|<small>92/81</small>
+!12/7
-|<small>32/27</small>
+!25/14
-|<small>100/81</small>
+!13/7
-|<small>104/81</small>
+!27/14
-|<small>4/3</small>
+! rowspan="17" |2
-|<small>112/81</small>
+|-
-|<small>116/81</small>
+|<small>23/22</small>
-|<small>40/27</small>
+|<small>12/11</small>
-|<small>14/9</small>
+|<small>25/22</small>
-|<small>44/27</small>
+|<small>13/11</small>
-|<small>46/27</small>
+|<small>27/22</small>
-|<small>16/9</small>
+|<small>14/11</small>
-|<small>50/27</small>
+|<small>29/22</small>
-|<small>52/27</small>
+|<small>15/11</small>
-|}
+|<small>63/44</small>
+|<small>3/2</small>
-=== HSAF 8 ===
+|<small>69/44</small>
-<code>48:50:52:54:56:58:60:62:64:66:68:70:72:75:78:81:84:87:90:93:96</code>. Harmonic segment: '''24::32.'''
+|<small>18/11</small>
+|<small>75/44</small>
-It contains HSAF 2 and HSAF 4. As a superset of HSAF 4, it adds many new interesting intervals with odds 9, 11, 15, 21, 25, 29, 31, 35 along its modes. It is also the last HSAF to fit within the [[31-limit]].
+|<small>39/22</small>
+|<small>81/44</small>
-The raw scale or its extension could be seen as a [[wilsec]] detemper (negri weak extension) [[19L 1s]] 10|9, but it is a very innacurate realization that only [[19edo]] through patent val can faithfully represent (albeit badly as it tempers together 17/12 and 35/24). Thus, it's better seen as just a linear interpolation of HSAF 4 in the 2.3.5.7.11.13.17.29.31 subgroup.
+|<small>21/11</small>
-{| class="wikitable" data-darkreader-inline-color=""
+|-
-!25/24
+|<small>24/23</small>
-!13/12
+|<small>25/23</small>
-!9/8
+|<small>26/23</small>
-!7/6
+|<small>27/23</small>
-!29/24
+|<small>28/23</small>
-!5/4
+|<small>29/23</small>
-!31/24
+|<small>30/23</small>
-!4/3
+|<small>63/46</small>
-!11/8
+|<small>33/23</small>
-!17/12
+|<small>3/2</small>
-!35/24
+|<small>36/23</small>
-!3/2
+|<small>75/46</small>
-!25/16
+|<small>39/23</small>
-!13/8
+|<small>81/46</small>
-!27/16
+|<small>42/23</small>
-!7/4
+|<small>44/23</small>
-!29/16
-!15/8
-!31/16
-! rowspan="20" |2
 |-
-|<small>26/25</small>
+|<small>25/24</small>
-|<small>27/25</small>
+|<small>13/12</small>
-|<small>28/25</small>
+|<small>9/8</small>
-|<small>29/25</small>
+|<small>7/6</small>
-|<small>6/5</small>
+|<small>29/24</small>
-|<small>31/25</small>
+|<small>5/4</small>
-|<small>32/25</small>
+|<small>21/16</small>
+|<small>11/8</small>
+|<small>23/16</small>
+|<small>3/2</small>
+|<small>25/16</small>
+|<small>13/8</small>
+|<small>27/16</small>
+|<small>7/4</small>
+|<small>11/6</small>
+|<small>23/12</small>
+|-
+|<small>26/25</small>
+|<small>27/25</small>
+|<small>28/25</small>
+|<small>29/25</small>
+|<small>6/5</small>
+|<small>63/50</small>
 |<small>33/25</small>
-|<small>34/25</small>
+|<small>69/50</small>
-|<small>7/5</small>
 |<small>36/25</small>
 |<small>3/2</small>
@@ Line 960: / Line 961: @@
 |<small>81/50</small>
 |<small>42/25</small>
-|<small>87/50</small>
+|<small>44/25</small>
-|<small>9/5</small>
+|<small>46/25</small>
-|<small>93/50</small>
 |<small>48/25</small>
 |-
@@ Line 969: / Line 969: @@
 |<small>29/26</small>
 |<small>15/13</small>
-|<small>31/26</small>
+|<small>63/52</small>
-|<small>16/13</small>
 |<small>33/26</small>
-|<small>17/13</small>
+|<small>69/52</small>
-|<small>35/26</small>
 |<small>18/13</small>
 |<small>75/52</small>
@@ Line 979: / Line 977: @@
 |<small>81/52</small>
 |<small>21/13</small>
-|<small>87/52</small>
+|<small>22/13</small>
-|<small>45/26</small>
+|<small>23/13</small>
-|<small>93/52</small>
 |<small>24/13</small>
 |<small>25/13</small>
@@ Line 988: / Line 985: @@
 |<small>29/27</small>
 |<small>10/9</small>
-|<small>31/27</small>
+|<small>7/6</small>
-|<small>32/27</small>
 |<small>11/9</small>
-|<small>34/27</small>
+|<small>23/18</small>
-|<small>35/27</small>
 |<small>4/3</small>
 |<small>25/18</small>
@@ Line 998: / Line 993: @@
 |<small>3/2</small>
 |<small>14/9</small>
-|<small>29/18</small>
+|<small>44/27</small>
-|<small>5/3</small>
+|<small>46/27</small>
-|<small>31/18</small>
 |<small>16/9</small>
 |<small>50/27</small>
@@ Line 1,007: / Line 1,001: @@
 |<small>29/28</small>
 |<small>15/14</small>
-|<small>31/28</small>
+|<small>9/8</small>
-|<small>8/7</small>
 |<small>33/28</small>
-|<small>17/14</small>
+|<small>69/56</small>
-|<small>5/4</small>
 |<small>9/7</small>
 |<small>75/56</small>
@@ Line 1,017: / Line 1,009: @@
 |<small>81/56</small>
 |<small>3/2</small>
-|<small>87/56</small>
+|<small>11/7</small>
-|<small>45/28</small>
+|<small>23/14</small>
-|<small>93/56</small>
 |<small>12/7</small>
 |<small>25/14</small>
@@ Line 1,026: / Line 1,017: @@
 |-
 |<small>30/29</small>
-|<small>31/29</small>
+|<small>63/58</small>
-|<small>32/29</small>
 |<small>33/29</small>
-|<small>34/29</small>
+|<small>69/58</small>
-|<small>35/29</small>
 |<small>36/29</small>
 |<small>75/58</small>
@@ Line 1,036: / Line 1,025: @@
 |<small>81/58</small>
 |<small>42/29</small>
-|<small>3/2</small>
+|<small>44/29</small>
-|<small>45/29</small>
+|<small>46/29</small>
-|<small>93/58</small>
 |<small>48/29</small>
 |<small>50/29</small>
@@ Line 1,045: / Line 1,033: @@
 |<small>56/29</small>
 |-
-|<small>31/30</small>
+|<small>21/20</small>
-|<small>16/15</small>
 |<small>11/10</small>
-|<small>17/15</small>
+|<small>23/20</small>
-|<small>7/6</small>
 |<small>6/5</small>
 |<small>5/4</small>
@@ Line 1,055: / Line 1,041: @@
 |<small>27/20</small>
 |<small>7/5</small>
-|<small>29/20</small>
+|<small>22/15</small>
-|<small>3/2</small>
+|<small>23/15</small>
-|<small>31/20</small>
 |<small>8/5</small>
 |<small>5/3</small>
@@ Line 1,065: / Line 1,050: @@
 |<small>29/15</small>
 |-
-|<small>32/31</small>
+|<small>22/21</small>
-|<small>33/31</small>
+|<small>23/21</small>
-|<small>34/31</small>
+|<small>8/7</small>
-|<small>35/31</small>
+|<small>25/21</small>
-|<small>36/31</small>
+|<small>26/21</small>
-|<small>75/62</small>
+|<small>9/7</small>
-|<small>39/31</small>
+|<small>4/3</small>
-|<small>81/62</small>
+|<small>88/63</small>
-|<small>42/31</small>
+|<small>92/63</small>
-|<small>87/62</small>
+|<small>32/21</small>
-|<small>45/31</small>
+|<small>100/63</small>
-|<small>3/2</small>
+|<small>104/63</small>
-|<small>48/31</small>
+|<small>12/7</small>
-|<small>50/31</small>
+|<small>16/9</small>
-|<small>52/31</small>
+|<small>116/63</small>
-|<small>54/31</small>
+|<small>40/21</small>
-|<small>56/31</small>
-|<small>58/31</small>
-|<small>60/31</small>
 |-
-|<small>33/32</small>
+|<small>23/22</small>
-|<small>17/16</small>
+|<small>12/11</small>
-|<small>35/32</small>
+|<small>25/22</small>
-|<small>9/8</small>
+|<small>13/11</small>
-|<small>75/64</small>
+|<small>27/22</small>
-|<small>39/32</small>
+|<small>14/11</small>
-|<small>81/64</small>
+|<small>4/3</small>
-|<small>21/16</small>
+|<small>46/33</small>
-|<small>87/64</small>
+|<small>16/11</small>
-|<small>45/32</small>
+|<small>50/33</small>
-|<small>93/64</small>
-|<small>3/2</small>
-|<small>25/16</small>
-|<small>13/8</small>
-|<small>27/16</small>
-|<small>7/4</small>
-|<small>29/16</small>
-|<small>15/8</small>
-|<small>31/16</small>
-|-
-|<small>34/33</small>
-|<small>35/33</small>
-|<small>12/11</small>
-|<small>25/22</small>
-|<small>13/11</small>
-|<small>27/22</small>
-|<small>14/11</small>
-|<small>29/22</small>
-|<small>15/11</small>
-|<small>31/22</small>
-|<small>16/11</small>
-|<small>50/33</small>
 |<small>52/33</small>
 |<small>18/11</small>
@@ Line 1,122: / Line 1,082: @@
 |<small>58/33</small>
 |<small>20/11</small>
-|<small>62/33</small>
+|<small>21/11</small>
-|<small>64/33</small>
 |-
-|<small>35/34</small>
+|<small>24/23</small>
-|<small>18/17</small>
+|<small>25/23</small>
-|<small>75/68</small>
+|<small>26/23</small>
-|<small>39/34</small>
+|<small>27/23</small>
-|<small>81/68</small>
+|<small>28/23</small>
-|<small>21/17</small>
+|<small>88/69</small>
-|<small>87/68</small>
+|<small>4/3</small>
-|<small>45/34</small>
+|<small>32/23</small>
-|<small>93/68</small>
+|<small>100/69</small>
-|<small>24/17</small>
+|<small>104/69</small>
-|<small>25/17</small>
+|<small>36/23</small>
-|<small>26/17</small>
+|<small>112/69</small>
-|<small>27/17</small>
+|<small>116/69</small>
-|<small>28/17</small>
+|<small>40/23</small>
-|<small>29/17</small>
+|<small>42/23</small>
-|<small>30/17</small>
+|<small>44/23</small>
-|<small>31/17</small>
-|<small>32/17</small>
-|<small>33/17</small>
-|-
-|<small>36/35</small>
-|<small>15/14</small>
-|<small>39/35</small>
-|<small>81/70</small>
-|<small>6/5</small>
-|<small>87/70</small>
-|<small>9/7</small>
-|<small>93/70</small>
-|<small>48/35</small>
-|<small>10/7</small>
-|<small>52/35</small>
-|<small>54/35</small>
-|<small>8/5</small>
-|<small>58/35</small>
-|<small>12/7</small>
-|<small>62/35</small>
-|<small>64/35</small>
-|<small>66/35</small>
-|<small>68/35</small>
 |-
 |<small>25/24</small>
@@ Line 1,169: / Line 1,105: @@
 |<small>9/8</small>
 |<small>7/6</small>
-|<small>29/24</small>
+|<small>11/9</small>
-|<small>5/4</small>
+|<small>23/18</small>
-|<small>31/24</small>
 |<small>4/3</small>
 |<small>25/18</small>
@@ Line 1,179: / Line 1,114: @@
 |<small>29/18</small>
 |<small>5/3</small>
-|<small>31/18</small>
+|<small>7/4</small>
-|<small>16/9</small>
 |<small>11/6</small>
-|<small>17/9</small>
+|<small>23/12</small>
-|<small>35/18</small>
 |-
 |<small>26/25</small>
 |<small>27/25</small>
 |<small>28/25</small>
-|<small>29/25</small>
+|<small>88/75</small>
-|<small>6/5</small>
+|<small>92/75</small>
-|<small>31/25</small>
 |<small>32/25</small>
 |<small>4/3</small>
@@ Line 1,198: / Line 1,130: @@
 |<small>116/75</small>
 |<small>8/5</small>
-|<small>124/75</small>
+|<small>42/25</small>
-|<small>128/75</small>
 |<small>44/25</small>
-|<small>136/75</small>
+|<small>46/25</small>
-|<small>28/15</small>
 |<small>48/25</small>
 |-
 |<small>27/26</small>
 |<small>14/13</small>
-|<small>29/26</small>
+|<small>44/39</small>
-|<small>15/13</small>
+|<small>46/39</small>
-|<small>31/26</small>
 |<small>16/13</small>
 |<small>50/39</small>
@@ Line 1,217: / Line 1,146: @@
 |<small>58/39</small>
 |<small>20/13</small>
-|<small>62/39</small>
+|<small>21/13</small>
-|<small>64/39</small>
 |<small>22/13</small>
-|<small>68/39</small>
+|<small>23/13</small>
-|<small>70/39</small>
 |<small>24/13</small>
 |<small>25/13</small>
 |-
 |<small>28/27</small>
-|<small>29/27</small>
+|<small>88/81</small>
-|<small>10/9</small>
+|<small>92/81</small>
-|<small>31/27</small>
 |<small>32/27</small>
 |<small>100/81</small>
@@ Line 1,236: / Line 1,162: @@
 |<small>116/81</small>
 |<small>40/27</small>
-|<small>124/81</small>
+|<small>14/9</small>
-|<small>128/81</small>
 |<small>44/27</small>
-|<small>136/81</small>
+|<small>46/27</small>
-|<small>140/81</small>
 |<small>16/9</small>
 |<small>50/27</small>
 |<small>52/27</small>
+|}
+=== Category 8 ===
+It contains TO 2 and TO 4. With supersets of TO 4, it adds many new interesting intervals with odds 9, 11, 15, 21, 25, 29, 31, 35 along its modes. It is also the last category to fit completely within the [[31-limit]]. TO 8a adds 23, TO 8b adds 17.
+The scales can be seen [[wilsec]] detemper (negri weak extension) [[19L 1s]] 10|9, but it is a very innacurate realization that only [[19edo]] through patent val can faithfully represent (well enough for TO 8a albeit badly for TO 8b as it tempers together 17/12 and 35/24). Thus, it's better seen as just a linear interpolation of TO 4 in the 2.3.5.7.11.13.17.29.31 subgroup.
+==== TO 8a ====
+<code>48:50:52:54:56:58:60:62:64:66:69:72:75:78:81:84:87:90:93:96</code>. Last TO to be a constant structure
+{| class="wikitable"
 |-
-|<small>29/28</small>
+!25/24
-|<small>15/14</small>
+!13/12
-|<small>31/28</small>
+!9/8
-|<small>8/7</small>
+!7/6
-|<small>25/21</small>
+!29/24
-|<small>26/21</small>
+!5/4
-|<small>9/7</small>
+!31/24
-|<small>4/3</small>
+!4/3
-|<small>29/21</small>
+!11/8
-|<small>10/7</small>
+!23/16
-|<small>31/21</small>
+!3/2
-|<small>32/21</small>
+!25/16
-|<small>11/7</small>
+!13/8
-|<small>34/21</small>
+!27/16
-|<small>5/3</small>
+!7/4
-|<small>12/7</small>
+!29/16
-|<small>25/14</small>
+!15/8
-|<small>13/7</small>
+!31/16
-|<small>27/14</small>
+! rowspan="19" |2
 |-
-|<small>30/29</small>
+|<small>26/25</small>
-|<small>31/29</small>
+|<small>27/25</small>
-|<small>32/29</small>
+|<small>28/25</small>
-|<small>100/87</small>
+|<small>29/25</small>
-|<small>104/87</small>
-|<small>36/29</small>
-|<small>112/87</small>
-|<small>4/3</small>
-|<small>40/29</small>
-|<small>124/87</small>
-|<small>128/87</small>
-|<small>44/29</small>
-|<small>136/87</small>
-|<small>140/87</small>
-|<small>48/29</small>
-|<small>50/29</small>
-|<small>52/29</small>
-|<small>54/29</small>
-|<small>56/29</small>
-|-
-|<small>31/30</small>
-|<small>16/15</small>
-|<small>10/9</small>
-|<small>52/45</small>
 |<small>6/5</small>
-|<small>56/45</small>
+|<small>31/25</small>
-|<small>58/45</small>
+|<small>32/25</small>
-|<small>4/3</small>
+|<small>33/25</small>
-|<small>62/45</small>
+|<small>69/50</small>
-|<small>64/45</small>
+|<small>36/25</small>
-|<small>22/15</small>
+|<small>3/2</small>
-|<small>68/45</small>
+|<small>39/25</small>
-|<small>14/9</small>
+|<small>81/50</small>
-|<small>8/5</small>
+|<small>42/25</small>
-|<small>5/3</small>
+|<small>87/50</small>
-|<small>26/15</small>
 |<small>9/5</small>
-|<small>28/15</small>
+|<small>93/50</small>
-|<small>29/15</small>
+|<small>48/25</small>
 |-
-|<small>32/31</small>
+|<small>27/26</small>
-|<small>100/93</small>
+|<small>14/13</small>
-|<small>104/93</small>
+|<small>29/26</small>
-|<small>36/31</small>
+|<small>15/13</small>
-|<small>112/93</small>
+|<small>31/26</small>
-|<small>116/93</small>
+|<small>16/13</small>
-|<small>40/31</small>
+|<small>33/26</small>
+|<small>69/52</small>
+|<small>18/13</small>
+|<small>75/52</small>
+|<small>3/2</small>
+|<small>81/52</small>
+|<small>21/13</small>
+|<small>87/52</small>
+|<small>45/26</small>
+|<small>93/52</small>
+|<small>24/13</small>
+|<small>25/13</small>
+|-
+|<small>28/27</small>
+|<small>29/27</small>
+|<small>10/9</small>
+|<small>31/27</small>
+|<small>32/27</small>
+|<small>11/9</small>
+|<small>23/18</small>
 |<small>4/3</small>
-|<small>128/93</small>
+|<small>25/18</small>
-|<small>44/31</small>
+|<small>13/9</small>
-|<small>136/93</small>
+|<small>3/2</small>
-|<small>140/93</small>
+|<small>14/9</small>
-|<small>48/31</small>
+|<small>29/18</small>
-|<small>50/31</small>
+|<small>5/3</small>
-|<small>52/31</small>
+|<small>31/18</small>
-|<small>54/31</small>
+|<small>16/9</small>
-|<small>56/31</small>
+|<small>50/27</small>
+|<small>52/27</small>
+|-
+|<small>29/28</small>
+|<small>15/14</small>
+|<small>31/28</small>
+|<small>8/7</small>
+|<small>33/28</small>
+|<small>69/56</small>
+|<small>9/7</small>
+|<small>75/56</small>
+|<small>39/28</small>
+|<small>81/56</small>
+|<small>3/2</small>
+|<small>87/56</small>
+|<small>45/28</small>
+|<small>93/56</small>
+|<small>12/7</small>
+|<small>25/14</small>
+|<small>13/7</small>
+|<small>27/14</small>
+|-
+|<small>30/29</small>
+|<small>31/29</small>
+|<small>32/29</small>
+|<small>33/29</small>
+|<small>69/58</small>
+|<small>36/29</small>
+|<small>75/58</small>
+|<small>39/29</small>
+|<small>81/58</small>
+|<small>42/29</small>
+|<small>3/2</small>
+|<small>45/29</small>
+|<small>93/58</small>
+|<small>48/29</small>
+|<small>50/29</small>
+|<small>52/29</small>
+|<small>54/29</small>
+|<small>56/29</small>
+|-
+|<small>31/30</small>
+|<small>16/15</small>
+|<small>11/10</small>
+|<small>23/20</small>
+|<small>6/5</small>
+|<small>5/4</small>
+|<small>13/10</small>
+|<small>27/20</small>
+|<small>7/5</small>
+|<small>29/20</small>
+|<small>3/2</small>
+|<small>31/20</small>
+|<small>8/5</small>
+|<small>5/3</small>
+|<small>26/15</small>
+|<small>9/5</small>
+|<small>28/15</small>
+|<small>29/15</small>
+|-
+|<small>32/31</small>
+|<small>33/31</small>
+|<small>69/62</small>
+|<small>36/31</small>
+|<small>75/62</small>
+|<small>39/31</small>
+|<small>81/62</small>
+|<small>42/31</small>
+|<small>87/62</small>
+|<small>45/31</small>
+|<small>3/2</small>
+|<small>48/31</small>
+|<small>50/31</small>
+|<small>52/31</small>
+|<small>54/31</small>
+|<small>56/31</small>
 |<small>58/31</small>
 |<small>60/31</small>
+|-
+|<small>33/32</small>
+|<small>69/64</small>
+|<small>9/8</small>
+|<small>75/64</small>
+|<small>39/32</small>
+|<small>81/64</small>
+|<small>21/16</small>
+|<small>87/64</small>
+|<small>45/32</small>
+|<small>93/64</small>
+|<small>3/2</small>
+|<small>25/16</small>
+|<small>13/8</small>
+|<small>27/16</small>
+|<small>7/4</small>
+|<small>29/16</small>
+|<small>15/8</small>
+|<small>31/16</small>
+|-
+|<small>23/22</small>
+|<small>12/11</small>
+|<small>25/22</small>
+|<small>13/11</small>
+|<small>27/22</small>
+|<small>14/11</small>
+|<small>29/22</small>
+|<small>15/11</small>
+|<small>31/22</small>
+|<small>16/11</small>
+|<small>50/33</small>
+|<small>52/33</small>
+|<small>18/11</small>
+|<small>56/33</small>
+|<small>58/33</small>
+|<small>20/11</small>
+|<small>62/33</small>
+|<small>64/33</small>
+|-
+|<small>24/23</small>
+|<small>25/23</small>
+|<small>26/23</small>
+|<small>27/23</small>
+|<small>28/23</small>
+|<small>29/23</small>
+|<small>30/23</small>
+|<small>31/23</small>
+|<small>32/23</small>
+|<small>100/69</small>
+|<small>104/69</small>
+|<small>36/23</small>
+|<small>112/69</small>
+|<small>116/69</small>
+|<small>40/23</small>
+|<small>124/69</small>
+|<small>128/69</small>
+|<small>44/23</small>
+|-
+|<small>25/24</small>
+|<small>13/12</small>
+|<small>9/8</small>
+|<small>7/6</small>
+|<small>29/24</small>
+|<small>5/4</small>
+|<small>31/24</small>
+|<small>4/3</small>
+|<small>25/18</small>
+|<small>13/9</small>
+|<small>3/2</small>
+|<small>14/9</small>
+|<small>29/18</small>
+|<small>5/3</small>
+|<small>31/18</small>
+|<small>16/9</small>
+|<small>11/6</small>
+|<small>23/12</small>
+|-
+|<small>26/25</small>
+|<small>27/25</small>
+|<small>28/25</small>
+|<small>29/25</small>
+|<small>6/5</small>
+|<small>31/25</small>
+|<small>32/25</small>
+|<small>4/3</small>
+|<small>104/75</small>
+|<small>36/25</small>
+|<small>112/75</small>
+|<small>116/75</small>
+|<small>8/5</small>
+|<small>124/75</small>
+|<small>128/75</small>
+|<small>44/25</small>
+|<small>46/25</small>
+|<small>48/25</small>
+|-
+|<small>27/26</small>
+|<small>14/13</small>
+|<small>29/26</small>
+|<small>15/13</small>
+|<small>31/26</small>
+|<small>16/13</small>
+|<small>50/39</small>
+|<small>4/3</small>
+|<small>18/13</small>
+|<small>56/39</small>
+|<small>58/39</small>
+|<small>20/13</small>
+|<small>62/39</small>
+|<small>64/39</small>
+|<small>22/13</small>
+|<small>23/13</small>
+|<small>24/13</small>
+|<small>25/13</small>
+|-
+|<small>28/27</small>
+|<small>29/27</small>
+|<small>10/9</small>
+|<small>31/27</small>
+|<small>32/27</small>
+|<small>100/81</small>
+|<small>104/81</small>
+|<small>4/3</small>
+|<small>112/81</small>
+|<small>116/81</small>
+|<small>40/27</small>
+|<small>124/81</small>
+|<small>128/81</small>
+|<small>44/27</small>
+|<small>46/27</small>
+|<small>16/9</small>
+|<small>50/27</small>
+|<small>52/27</small>
+|-
+|<small>29/28</small>
+|<small>15/14</small>
+|<small>31/28</small>
+|<small>8/7</small>
+|<small>25/21</small>
+|<small>26/21</small>
+|<small>9/7</small>
+|<small>4/3</small>
+|<small>29/21</small>
+|<small>10/7</small>
+|<small>31/21</small>
+|<small>32/21</small>
+|<small>11/7</small>
+|<small>23/14</small>
+|<small>12/7</small>
+|<small>25/14</small>
+|<small>13/7</small>
+|<small>27/14</small>
+|-
+|<small>30/29</small>
+|<small>31/29</small>
+|<small>32/29</small>
+|<small>100/87</small>
+|<small>104/87</small>
+|<small>36/29</small>
+|<small>112/87</small>
+|<small>4/3</small>
+|<small>40/29</small>
+|<small>124/87</small>
+|<small>128/87</small>
+|<small>44/29</small>
+|<small>46/29</small>
+|<small>48/29</small>
+|<small>50/29</small>
+|<small>52/29</small>
+|<small>54/29</small>
+|<small>56/29</small>
+|-
+|<small>31/30</small>
+|<small>16/15</small>
+|<small>10/9</small>
+|<small>52/45</small>
+|<small>6/5</small>
+|<small>56/45</small>
+|<small>58/45</small>
+|<small>4/3</small>
+|<small>62/45</small>
+|<small>64/45</small>
+|<small>22/15</small>
+|<small>23/15</small>
+|<small>8/5</small>
+|<small>5/3</small>
+|<small>26/15</small>
+|<small>9/5</small>
+|<small>28/15</small>
+|<small>29/15</small>
+|-
+|<small>32/31</small>
+|<small>100/93</small>
+|<small>104/93</small>
+|<small>36/31</small>
+|<small>112/93</small>
+|<small>116/93</small>
+|<small>40/31</small>
+|<small>4/3</small>
+|<small>128/93</small>
+|<small>44/31</small>
+|<small>46/31</small>
+|<small>48/31</small>
+|<small>50/31</small>
+|<small>52/31</small>
+|<small>54/31</small>
+|<small>56/31</small>
+|<small>58/31</small>
+|<small>60/31</small>
+|}
+==== TO 8b ====
+<code>48:50:52:54:56:58:60:62:64:66:68:70:72:75:78:81:84:87:90:93:96</code>.
+{| class="wikitable" data-darkreader-inline-color=""
+!25/24
+!13/12
+!9/8
+!7/6
+!29/24
+!5/4
+!31/24
+!4/3
+!11/8
+!17/12
+!35/24
+!3/2
+!25/16
+!13/8
+!27/16
+!7/4
+!29/16
+!15/8
+!31/16
+! rowspan="20" |2
+|-
+|<small>26/25</small>
+|<small>27/25</small>
+|<small>28/25</small>
+|<small>29/25</small>
+|<small>6/5</small>
+|<small>31/25</small>
+|<small>32/25</small>
+|<small>33/25</small>
+|<small>34/25</small>
+|<small>7/5</small>
+|<small>36/25</small>
+|<small>3/2</small>
+|<small>39/25</small>
+|<small>81/50</small>
+|<small>42/25</small>
+|<small>87/50</small>
+|<small>9/5</small>
+|<small>93/50</small>
+|<small>48/25</small>
+|-
+|<small>27/26</small>
+|<small>14/13</small>
+|<small>29/26</small>
+|<small>15/13</small>
+|<small>31/26</small>
+|<small>16/13</small>
+|<small>33/26</small>
+|<small>17/13</small>
+|<small>35/26</small>
+|<small>18/13</small>
+|<small>75/52</small>
+|<small>3/2</small>
+|<small>81/52</small>
+|<small>21/13</small>
+|<small>87/52</small>
+|<small>45/26</small>
+|<small>93/52</small>
+|<small>24/13</small>
+|<small>25/13</small>
+|-
+|<small>28/27</small>
+|<small>29/27</small>
+|<small>10/9</small>
+|<small>31/27</small>
+|<small>32/27</small>
+|<small>11/9</small>
+|<small>34/27</small>
+|<small>35/27</small>
+|<small>4/3</small>
+|<small>25/18</small>
+|<small>13/9</small>
+|<small>3/2</small>
+|<small>14/9</small>
+|<small>29/18</small>
+|<small>5/3</small>
+|<small>31/18</small>
+|<small>16/9</small>
+|<small>50/27</small>
+|<small>52/27</small>
+|-
+|<small>29/28</small>
+|<small>15/14</small>
+|<small>31/28</small>
+|<small>8/7</small>
+|<small>33/28</small>
+|<small>17/14</small>
+|<small>5/4</small>
+|<small>9/7</small>
+|<small>75/56</small>
+|<small>39/28</small>
+|<small>81/56</small>
+|<small>3/2</small>
+|<small>87/56</small>
+|<small>45/28</small>
+|<small>93/56</small>
+|<small>12/7</small>
+|<small>25/14</small>
+|<small>13/7</small>
+|<small>27/14</small>
+|-
+|<small>30/29</small>
+|<small>31/29</small>
+|<small>32/29</small>
+|<small>33/29</small>
+|<small>34/29</small>
+|<small>35/29</small>
+|<small>36/29</small>
+|<small>75/58</small>
+|<small>39/29</small>
+|<small>81/58</small>
+|<small>42/29</small>
+|<small>3/2</small>
+|<small>45/29</small>
+|<small>93/58</small>
+|<small>48/29</small>
+|<small>50/29</small>
+|<small>52/29</small>
+|<small>54/29</small>
+|<small>56/29</small>
+|-
+|<small>31/30</small>
+|<small>16/15</small>
+|<small>11/10</small>
+|<small>17/15</small>
+|<small>7/6</small>
+|<small>6/5</small>
+|<small>5/4</small>
+|<small>13/10</small>
+|<small>27/20</small>
+|<small>7/5</small>
+|<small>29/20</small>
+|<small>3/2</small>
+|<small>31/20</small>
+|<small>8/5</small>
+|<small>5/3</small>
+|<small>26/15</small>
+|<small>9/5</small>
+|<small>28/15</small>
+|<small>29/15</small>
+|-
+|<small>32/31</small>
+|<small>33/31</small>
+|<small>34/31</small>
+|<small>35/31</small>
+|<small>36/31</small>
+|<small>75/62</small>
+|<small>39/31</small>
+|<small>81/62</small>
+|<small>42/31</small>
+|<small>87/62</small>
+|<small>45/31</small>
+|<small>3/2</small>
+|<small>48/31</small>
+|<small>50/31</small>
+|<small>52/31</small>
+|<small>54/31</small>
+|<small>56/31</small>
+|<small>58/31</small>
+|<small>60/31</small>
+|-
+|<small>33/32</small>
+|<small>17/16</small>
+|<small>35/32</small>
+|<small>9/8</small>
+|<small>75/64</small>
+|<small>39/32</small>
+|<small>81/64</small>
+|<small>21/16</small>
+|<small>87/64</small>
+|<small>45/32</small>
+|<small>93/64</small>
+|<small>3/2</small>
+|<small>25/16</small>
+|<small>13/8</small>
+|<small>27/16</small>
+|<small>7/4</small>
+|<small>29/16</small>
+|<small>15/8</small>
+|<small>31/16</small>
+|-
+|<small>34/33</small>
+|<small>35/33</small>
+|<small>12/11</small>
+|<small>25/22</small>
+|<small>13/11</small>
+|<small>27/22</small>
+|<small>14/11</small>
+|<small>29/22</small>
+|<small>15/11</small>
+|<small>31/22</small>
+|<small>16/11</small>
+|<small>50/33</small>
+|<small>52/33</small>
+|<small>18/11</small>
+|<small>56/33</small>
+|<small>58/33</small>
+|<small>20/11</small>
+|<small>62/33</small>
+|<small>64/33</small>
+|-
+|<small>35/34</small>
+|<small>18/17</small>
+|<small>75/68</small>
+|<small>39/34</small>
+|<small>81/68</small>
+|<small>21/17</small>
+|<small>87/68</small>
+|<small>45/34</small>
+|<small>93/68</small>
+|<small>24/17</small>
+|<small>25/17</small>
+|<small>26/17</small>
+|<small>27/17</small>
+|<small>28/17</small>
+|<small>29/17</small>
+|<small>30/17</small>
+|<small>31/17</small>
+|<small>32/17</small>
+|<small>33/17</small>
+|-
+|<small>36/35</small>
+|<small>15/14</small>
+|<small>39/35</small>
+|<small>81/70</small>
+|<small>6/5</small>
+|<small>87/70</small>
+|<small>9/7</small>
+|<small>93/70</small>
+|<small>48/35</small>
+|<small>10/7</small>
+|<small>52/35</small>
+|<small>54/35</small>
+|<small>8/5</small>
+|<small>58/35</small>
+|<small>12/7</small>
+|<small>62/35</small>
+|<small>64/35</small>
+|<small>66/35</small>
+|<small>68/35</small>
+|-
+|<small>25/24</small>
+|<small>13/12</small>
+|<small>9/8</small>
+|<small>7/6</small>
+|<small>29/24</small>
+|<small>5/4</small>
+|<small>31/24</small>
+|<small>4/3</small>
+|<small>25/18</small>
+|<small>13/9</small>
+|<small>3/2</small>
+|<small>14/9</small>
+|<small>29/18</small>
+|<small>5/3</small>
+|<small>31/18</small>
+|<small>16/9</small>
+|<small>11/6</small>
+|<small>17/9</small>
+|<small>35/18</small>
+|-
+|<small>26/25</small>
+|<small>27/25</small>
+|<small>28/25</small>
+|<small>29/25</small>
+|<small>6/5</small>
+|<small>31/25</small>
+|<small>32/25</small>
+|<small>4/3</small>
+|<small>104/75</small>
+|<small>36/25</small>
+|<small>112/75</small>
+|<small>116/75</small>
+|<small>8/5</small>
+|<small>124/75</small>
+|<small>128/75</small>
+|<small>44/25</small>
+|<small>136/75</small>
+|<small>28/15</small>
+|<small>48/25</small>
+|-
+|<small>27/26</small>
+|<small>14/13</small>
+|<small>29/26</small>
+|<small>15/13</small>
+|<small>31/26</small>
+|<small>16/13</small>
+|<small>50/39</small>
+|<small>4/3</small>
+|<small>18/13</small>
+|<small>56/39</small>
+|<small>58/39</small>
+|<small>20/13</small>
+|<small>62/39</small>
+|<small>64/39</small>
+|<small>22/13</small>
+|<small>68/39</small>
+|<small>70/39</small>
+|<small>24/13</small>
+|<small>25/13</small>
+|-
+|<small>28/27</small>
+|<small>29/27</small>
+|<small>10/9</small>
+|<small>31/27</small>
+|<small>32/27</small>
+|<small>100/81</small>
+|<small>104/81</small>
+|<small>4/3</small>
+|<small>112/81</small>
+|<small>116/81</small>
+|<small>40/27</small>
+|<small>124/81</small>
+|<small>128/81</small>
+|<small>44/27</small>
+|<small>136/81</small>
+|<small>140/81</small>
+|<small>16/9</small>
+|<small>50/27</small>
+|<small>52/27</small>
+|-
+|<small>29/28</small>
+|<small>15/14</small>
+|<small>31/28</small>
+|<small>8/7</small>
+|<small>25/21</small>
+|<small>26/21</small>
+|<small>9/7</small>
+|<small>4/3</small>
+|<small>29/21</small>
+|<small>10/7</small>
+|<small>31/21</small>
+|<small>32/21</small>
+|<small>11/7</small>
+|<small>34/21</small>
+|<small>5/3</small>
+|<small>12/7</small>
+|<small>25/14</small>
+|<small>13/7</small>
+|<small>27/14</small>
+|-
+|<small>30/29</small>
+|<small>31/29</small>
+|<small>32/29</small>
+|<small>100/87</small>
+|<small>104/87</small>
+|<small>36/29</small>
+|<small>112/87</small>
+|<small>4/3</small>
+|<small>40/29</small>
+|<small>124/87</small>
+|<small>128/87</small>
+|<small>44/29</small>
+|<small>136/87</small>
+|<small>140/87</small>
+|<small>48/29</small>
+|<small>50/29</small>
+|<small>52/29</small>
+|<small>54/29</small>
+|<small>56/29</small>
+|-
+|<small>31/30</small>
+|<small>16/15</small>
+|<small>10/9</small>
+|<small>52/45</small>
+|<small>6/5</small>
+|<small>56/45</small>
+|<small>58/45</small>
+|<small>4/3</small>
+|<small>62/45</small>
+|<small>64/45</small>
+|<small>22/15</small>
+|<small>68/45</small>
+|<small>14/9</small>
+|<small>8/5</small>
+|<small>5/3</small>
+|<small>26/15</small>
+|<small>9/5</small>
+|<small>28/15</small>
+|<small>29/15</small>
+|-
+|<small>32/31</small>
+|<small>100/93</small>
+|<small>104/93</small>
+|<small>36/31</small>
+|<small>112/93</small>
+|<small>116/93</small>
+|<small>40/31</small>
+|<small>4/3</small>
+|<small>128/93</small>
+|<small>44/31</small>
+|<small>136/93</small>
+|<small>140/93</small>
+|<small>48/31</small>
+|<small>50/31</small>
+|<small>52/31</small>
+|<small>54/31</small>
+|<small>56/31</small>
+|<small>58/31</small>
+|<small>60/31</small>
+|}
+== Beyond Category 8 ==
+There are infinite categories with infinitely many TOs, but TOs grow increasingly more complex in prime palette and in interval count. As such, we present here only the TOs whose palettes fit within the 31-limit.
+==== TO 9a ====
+<code>54:56:58:60:62:64:66:68:70:72:75:78:81:84:87:90:93:96:99:102:105:108</code>. The biggest TO to feature a palette below the 31-limit, with [[harmonics 9-18]] as a subset, as well as TO 3.
+No patent val supports an equalized tuning; 19edo, the closest match, tempers together 34/27 and 35/27, as well as 17/9 and 35/128. The closest match is 22edo, which either of 22 or 22f vals provide the basic tuning for [[1L 19s]].
+{| class="wikitable"
+|-
+!28/27
+!29/27
+!10/9
+!31/27
+!32/27
+!11/9
+!34/27
+!35/27
+!4/3
+!25/18
+!13/9
+!3/2
+!14/9
+!29/18
+!5/3
+!31/18
+!16/9
+!11/6
+!17/9
+!35/18
+! rowspan="21" |2
+|-
+|<small>29/28</small>
+|<small>15/14</small>
+|<small>31/28</small>
+|<small>8/7</small>
+|<small>33/28</small>
+|<small>17/14</small>
+|<small>5/4</small>
+|<small>9/7</small>
+|<small>75/56</small>
+|<small>39/28</small>
+|<small>81/56</small>
+|<small>3/2</small>
+|<small>87/56</small>
+|<small>45/28</small>
+|<small>93/56</small>
+|<small>12/7</small>
+|<small>99/56</small>
+|<small>51/28</small>
+|<small>15/8</small>
+|<small>27/14</small>
+|-
+|<small>30/29</small>
+|<small>31/29</small>
+|<small>32/29</small>
+|<small>33/29</small>
+|<small>34/29</small>
+|<small>35/29</small>
+|<small>36/29</small>
+|<small>75/58</small>
+|<small>39/29</small>
+|<small>81/58</small>
+|<small>42/29</small>
+|<small>3/2</small>
+|<small>45/29</small>
+|<small>93/58</small>
+|<small>48/29</small>
+|<small>99/58</small>
+|<small>51/29</small>
+|<small>105/58</small>
+|<small>54/29</small>
+|<small>56/29</small>
+|-
+|<small>31/30</small>
+|<small>16/15</small>
+|<small>11/10</small>
+|<small>17/15</small>
+|<small>7/6</small>
+|<small>6/5</small>
+|<small>5/4</small>
+|<small>13/10</small>
+|<small>27/20</small>
+|<small>7/5</small>
+|<small>29/20</small>
+|<small>3/2</small>
+|<small>31/20</small>
+|<small>8/5</small>
+|<small>33/20</small>
+|<small>17/10</small>
+|<small>7/4</small>
+|<small>9/5</small>
+|<small>28/15</small>
+|<small>29/15</small>
+|-
+|<small>32/31</small>
+|<small>33/31</small>
+|<small>34/31</small>
+|<small>35/31</small>
+|<small>36/31</small>
+|<small>75/62</small>
+|<small>39/31</small>
+|<small>81/62</small>
+|<small>42/31</small>
+|<small>87/62</small>
+|<small>45/31</small>
+|<small>3/2</small>
+|<small>48/31</small>
+|<small>99/62</small>
+|<small>51/31</small>
+|<small>105/62</small>
+|<small>54/31</small>
+|<small>56/31</small>
+|<small>58/31</small>
+|<small>60/31</small>
+|-
+|<small>33/32</small>
+|<small>17/16</small>
+|<small>35/32</small>
+|<small>9/8</small>
+|<small>75/64</small>
+|<small>39/32</small>
+|<small>81/64</small>
+|<small>21/16</small>
+|<small>87/64</small>
+|<small>45/32</small>
+|<small>93/64</small>
+|<small>3/2</small>
+|<small>99/64</small>
+|<small>51/32</small>
+|<small>105/64</small>
+|<small>27/16</small>
+|<small>7/4</small>
+|<small>29/16</small>
+|<small>15/8</small>
+|<small>31/16</small>
+|-
+|<small>34/33</small>
+|<small>35/33</small>
+|<small>12/11</small>
+|<small>25/22</small>
+|<small>13/11</small>
+|<small>27/22</small>
+|<small>14/11</small>
+|<small>29/22</small>
+|<small>15/11</small>
+|<small>31/22</small>
+|<small>16/11</small>
+|<small>3/2</small>
+|<small>17/11</small>
+|<small>35/22</small>
+|<small>18/11</small>
+|<small>56/33</small>
+|<small>58/33</small>
+|<small>20/11</small>
+|<small>62/33</small>
+|<small>64/33</small>
+|-
+|<small>35/34</small>
+|<small>18/17</small>
+|<small>75/68</small>
+|<small>39/34</small>
+|<small>81/68</small>
+|<small>21/17</small>
+|<small>87/68</small>
+|<small>45/34</small>
+|<small>93/68</small>
+|<small>24/17</small>
+|<small>99/68</small>
+|<small>3/2</small>
+|<small>105/68</small>
+|<small>27/17</small>
+|<small>28/17</small>
+|<small>29/17</small>
+|<small>30/17</small>
+|<small>31/17</small>
+|<small>32/17</small>
+|<small>33/17</small>
+|-
+|<small>36/35</small>
+|<small>15/14</small>
+|<small>39/35</small>
+|<small>81/70</small>
+|<small>6/5</small>
+|<small>87/70</small>
+|<small>9/7</small>
+|<small>93/70</small>
+|<small>48/35</small>
+|<small>99/70</small>
+|<small>51/35</small>
+|<small>3/2</small>
+|<small>54/35</small>
+|<small>8/5</small>
+|<small>58/35</small>
+|<small>12/7</small>
+|<small>62/35</small>
+|<small>64/35</small>
+|<small>66/35</small>
+|<small>68/35</small>
+|-
+|<small>25/24</small>
+|<small>13/12</small>
+|<small>9/8</small>
+|<small>7/6</small>
+|<small>29/24</small>
+|<small>5/4</small>
+|<small>31/24</small>
+|<small>4/3</small>
+|<small>11/8</small>
+|<small>17/12</small>
+|<small>35/24</small>
+|<small>3/2</small>
+|<small>14/9</small>
+|<small>29/18</small>
+|<small>5/3</small>
+|<small>31/18</small>
+|<small>16/9</small>
+|<small>11/6</small>
+|<small>17/9</small>
+|<small>35/18</small>
+|-
+|<small>26/25</small>
+|<small>27/25</small>
+|<small>28/25</small>
+|<small>29/25</small>
+|<small>6/5</small>
+|<small>31/25</small>
+|<small>32/25</small>
+|<small>33/25</small>
+|<small>34/25</small>
+|<small>7/5</small>
+|<small>36/25</small>
+|<small>112/75</small>
+|<small>116/75</small>
+|<small>8/5</small>
+|<small>124/75</small>
+|<small>128/75</small>
+|<small>44/25</small>
+|<small>136/75</small>
+|<small>28/15</small>
+|<small>48/25</small>
+|-
+|<small>27/26</small>
+|<small>14/13</small>
+|<small>29/26</small>
+|<small>15/13</small>
+|<small>31/26</small>
+|<small>16/13</small>
+|<small>33/26</small>
+|<small>17/13</small>
+|<small>35/26</small>
+|<small>18/13</small>
+|<small>56/39</small>
+|<small>58/39</small>
+|<small>20/13</small>
+|<small>62/39</small>
+|<small>64/39</small>
+|<small>22/13</small>
+|<small>68/39</small>
+|<small>70/39</small>
+|<small>24/13</small>
+|<small>25/13</small>
+|-
+|<small>28/27</small>
+|<small>29/27</small>
+|<small>10/9</small>
+|<small>31/27</small>
+|<small>32/27</small>
+|<small>11/9</small>
+|<small>34/27</small>
+|<small>35/27</small>
+|<small>4/3</small>
+|<small>112/81</small>
+|<small>116/81</small>
+|<small>40/27</small>
+|<small>124/81</small>
+|<small>128/81</small>
+|<small>44/27</small>
+|<small>136/81</small>
+|<small>140/81</small>
+|<small>16/9</small>
+|<small>50/27</small>
+|<small>52/27</small>
+|-
+|<small>29/28</small>
+|<small>15/14</small>
+|<small>31/28</small>
+|<small>8/7</small>
+|<small>33/28</small>
+|<small>17/14</small>
+|<small>5/4</small>
+|<small>9/7</small>
+|<small>4/3</small>
+|<small>29/21</small>
+|<small>10/7</small>
+|<small>31/21</small>
+|<small>32/21</small>
+|<small>11/7</small>
+|<small>34/21</small>
+|<small>5/3</small>
+|<small>12/7</small>
+|<small>25/14</small>
+|<small>13/7</small>
+|<small>27/14</small>
+|-
+|<small>30/29</small>
+|<small>31/29</small>
+|<small>32/29</small>
+|<small>33/29</small>
+|<small>34/29</small>
+|<small>35/29</small>
+|<small>36/29</small>
+|<small>112/87</small>
+|<small>4/3</small>
+|<small>40/29</small>
+|<small>124/87</small>
+|<small>128/87</small>
+|<small>44/29</small>
+|<small>136/87</small>
+|<small>140/87</small>
+|<small>48/29</small>
+|<small>50/29</small>
+|<small>52/29</small>
+|<small>54/29</small>
+|<small>56/29</small>
+|-
+|<small>31/30</small>
+|<small>16/15</small>
+|<small>11/10</small>
+|<small>17/15</small>
+|<small>7/6</small>
+|<small>6/5</small>
+|<small>56/45</small>
+|<small>58/45</small>
+|<small>4/3</small>
+|<small>62/45</small>
+|<small>64/45</small>
+|<small>22/15</small>
+|<small>68/45</small>
+|<small>14/9</small>
+|<small>8/5</small>
+|<small>5/3</small>
+|<small>26/15</small>
+|<small>9/5</small>
+|<small>28/15</small>
+|<small>29/15</small>
+|-
+|<small>32/31</small>
+|<small>33/31</small>
+|<small>34/31</small>
+|<small>35/31</small>
+|<small>36/31</small>
+|<small>112/93</small>
+|<small>116/93</small>
+|<small>40/31</small>
+|<small>4/3</small>
+|<small>128/93</small>
+|<small>44/31</small>
+|<small>136/93</small>
+|<small>140/93</small>
+|<small>48/31</small>
+|<small>50/31</small>
+|<small>52/31</small>
+|<small>54/31</small>
+|<small>56/31</small>
+|<small>58/31</small>
+|<small>60/31</small>
+|-
+|<small>33/32</small>
+|<small>17/16</small>
+|<small>35/32</small>
+|<small>9/8</small>
+|<small>7/6</small>
+|<small>29/24</small>
+|<small>5/4</small>
+|<small>31/24</small>
+|<small>4/3</small>
+|<small>11/8</small>
+|<small>17/12</small>
+|<small>35/24</small>
+|<small>3/2</small>
+|<small>25/16</small>
+|<small>13/8</small>
+|<small>27/16</small>
+|<small>7/4</small>
+|<small>29/16</small>
+|<small>15/8</small>
+|<small>31/16</small>
+|-
+|<small>34/33</small>
+|<small>35/33</small>
+|<small>12/11</small>
+|<small>112/99</small>
+|<small>116/99</small>
+|<small>40/33</small>
+|<small>124/99</small>
+|<small>128/99</small>
+|<small>4/3</small>
+|<small>136/99</small>
+|<small>140/99</small>
+|<small>16/11</small>
+|<small>50/33</small>
+|<small>52/33</small>
+|<small>18/11</small>
+|<small>56/33</small>
+|<small>58/33</small>
+|<small>20/11</small>
+|<small>62/33</small>
+|<small>64/33</small>
+|-
+|<small>35/34</small>
+|<small>18/17</small>
+|<small>56/51</small>
+|<small>58/51</small>
+|<small>20/17</small>
+|<small>62/51</small>
+|<small>64/51</small>
+|<small>22/17</small>
+|<small>4/3</small>
+|<small>70/51</small>
+|<small>24/17</small>
+|<small>25/17</small>
+|<small>26/17</small>
+|<small>27/17</small>
+|<small>28/17</small>
+|<small>29/17</small>
+|<small>30/17</small>
+|<small>31/17</small>
+|<small>32/17</small>
+|<small>33/17</small>
+|-
+|<small>36/35</small>
+|<small>16/15</small>
+|<small>116/105</small>
+|<small>8/7</small>
+|<small>124/105</small>
+|<small>128/105</small>
+|<small>44/35</small>
+|<small>136/105</small>
+|<small>4/3</small>
+|<small>48/35</small>
+|<small>10/7</small>
+|<small>52/35</small>
+|<small>54/35</small>
+|<small>8/5</small>
+|<small>58/35</small>
+|<small>12/7</small>
+|<small>62/35</small>
+|<small>64/35</small>
+|<small>66/35</small>
+|<small>68/35</small>
 |}
-== Beyond HSAF 8 ==
+== References ==
-There are infinite HSAFs as stated before, but HSAFs grow increasingly more complex in prime palette and in interval count the more times the fourth is split. As such, we present here only the first 8 HSAFs as their palettes fit within the 31-limit.
+<references />

[math]\displaystyle{ x }[/math]	Transition points Number of clean scales	Perfect TP	Imperfect TP	Total scales	Convoluted scales	Note ranges		p-limits
2	1	1	0	1	0	5		7
3	1	1	0	1	0	7		11
4	1	1	0	1	0	10		17
5	1	0	1	1	0	12		19
6	2	2	0	2	0	14	15	23
7	1	0	1	1	0	17		29
8	2	1	1	2	0	19	20	31
9	2	1	1	2	0	21	22	31	37
10	2	1	1	2	0	24	25	37	43
11	2	0	2	2	0	26	27	43	47
12	3	2	1	4	1	28	30	47	53
13	2	0	2	2	0	31	32	47	53
14	3	1	2	4	1	33	35	53	61
15	3	1	2	4	1	35	37	59	61
16	3	1	2	4	1	38	40	61	71

[math]\displaystyle{ x }[/math]	Transition points Number of clean scales	Perfect TP	Imperfect TP	Total scales	Convoluted scales	Note ranges		p-limits
2	1	1	0	1	0	5		7
3	1	1	0	1	0	7		11
4	1	1	0	1	0	10		17
5	1	0	1	1	0	12		19
6	2	2	0	2	0	14	15	23
7	1	0	1	1	0	17		29
8	2	1	1	2	0	19	20	31
9	2	1	1	2	0	21	22	31	37
10	2	1	1	2	0	24	25	37	43
11	2	0	2	2	0	26	27	43	47
12	3	2	1	4	1	28	30	47	53
13	2	0	2	2	0	31	32	47	53
14	3	1	2	4	1	33	35	53	61
15	3	1	2	4	1	35	37	59	61
16	3	1	2	4	1	38	40	61	71

Tetrachordal otonal scale: Difference between revisions

Eufalesio (talk | contribs)

Latest revision as of 21:51, 26 October 2026

Definitions

Properties

List of categories

Category 2 - Septimal minor pentatonic

Category 3

Category 4

Category 5

Category 6

TO 6a

TO 6b

Category 7

Category 8

TO 8a

TO 8b

Beyond Category 8

TO 9a

References

Navigation menu

Search

[math]\displaystyle{ x }[/math]	Transition points Number of clean scales	Perfect TP	Imperfect TP	Total scales	Convoluted scales	Note ranges		p-limits
2	1	1	0	1	0	5		7
3	1	1	0	1	0	7		11
4	1	1	0	1	0	10		17
5	1	0	1	1	0	12		19
6	2	2	0	2	0	14	15	23
7	1	0	1	1	0	17		29
8	2	1	1	2	0	19	20	31
9	2	1	1	2	0	21	22	31	37
10	2	1	1	2	0	24	25	37	43
11	2	0	2	2	0	26	27	43	47
12	3	2	1	4	1	28	30	47	53
13	2	0	2	2	0	31	32	47	53
14	3	1	2	4	1	33	35	53	61
15	3	1	2	4	1	35	37	59	61
16	3	1	2	4	1	38	40	61	71