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.
• 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.

The data can be summarized in this table:

[math]\displaystyle{ x }[/math]	Tetrachordal transition points Number of clean scales	Perfect TP	Imperfect TP	Total scales	Convoluted scales	Note ranegs		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.

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	2
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:

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	2
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.

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	2
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.

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	2
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. 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	2
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	2
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.

The 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	2
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 8-0 adds 23, TO 8-1 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 8-0 albeit badly for TO 8-1 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	2
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	2
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 scale to feature a palette below the 31-limit. 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	2
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