Tetrachordal otonal scale

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).

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.
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.

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

List of TOs 2-8

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.

TO 2 - Septimal minor pentatonic

12:14:16:18:21:24. Harmonic segment: 6::8.

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

TO 3

18:20:22:24:27:30:33:36. Harmonic segment: 9::12.

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

TO 4

24:26:28:30:32:34:36:39:42:45:48. Harmonic segment: 12::16.

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

TO 5

30:32:34:36:38:40:42:45:48:51:54:57:60. Harmonic segment: 15::20.

15::20. It is the 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

TO 6

36:38:40:42:44:46:48:50:52:54:57:60:63:66:69:72. Harmonic segment: 18::24.

Last TO to be a constant structure, and first TO to contain two different (non-1) TOs as subsets. It is a 15edodetemper in the 2.3.5.7.11.19.23 subgroup, and nautilus detemper (1L 12s 6|6 and 14L 1s 6|8).

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

TO 7

42:44:46:48:50:52:54:56:58:60:63:66:69:72:75:78:81:84. Harmonic segment: 21::28.

The fourth prime TO. SSAF 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).

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

TO 8

48:50:52:54:56:58:60:62:64:66:68:70:72:75:78:81:84:87:90:93:96. Harmonic segment: 24::32.

It contains TO 2 and TO 4. As a superset 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 TO to fit within the 31-limit.

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 TO 4 in the 2.3.5.7.11.13.17.29.31 subgroup.

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 TO 8

There are infinite TOs as stated before, but TOs 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 TOs as their palettes fit within the 31-limit.