Frequency Ratio Mysteries

Many have wondered what the heck is up with the seemingly random assortment of frequency ratios, especially if you're writing an editor program for the thing and you want a simple, closed formula that you can just plug the coarse and fine settings into and calculate the resulting frequency ratio:

I tried to figure this out when I was originally writing my own editor, the TX81Z Programmer, but I was unsuccessful and turned to using a lookup table for the frequency ratios.

I was recently emailing with Sean Luke about this and it revitalized my interest in this problem. With Sean's help, I feel like we got pretty far towards figuring out a set of closed formulas. As you'll see, it's pretty unlikely that there is any kind of simple solution to this problem, but maybe these explorations will shed some light on the structure of these frequency ratios.

The Frequency Ratio Lookup Table

Since I couldn't figure out a formula when writing my editor, I was left no alternative but to make a table of all the coarse and fine value combinations by hand. Here it is in (coarse, fine) order with the addition of a delta column, which is the difference in ratio from the previous row, when there is one.

(The first four coarse groups only have 8 valid values; the TX81Z ignores fine values higher than 7 for coarse values less than 4.)

When I looked over this table, I noticed 4 different groups of coarse values that have very similar deltas (colored):

Coarse	Fine	Ratio	Delta	Coarse	Fine	Ratio	Delta	Coarse	Fine	Ratio	Delta	Coarse	Fine	Ratio	Delta
0	0	0.50		16	0	5.00		32	0	10.38		48	0	15.70
0	1	0.56	0.06	16	1	5.06	0.06	32	1	10.48	0.10	48	1	15.79	0.09
0	2	0.62	0.06	16	2	5.12	0.06	32	2	10.59	0.11	48	2	15.89	0.10
0	3	0.68	0.06	16	3	5.18	0.06	32	3	10.70	0.11	48	3	15.99	0.10
0	4	0.75	0.07	16	4	5.25	0.07	32	4	10.81	0.11	48	4	16.09	0.10
0	5	0.81	0.06	16	5	5.31	0.06	32	5	10.92	0.11	48	5	16.19	0.10
0	6	0.87	0.06	16	6	5.37	0.06	32	6	11.02	0.10	48	6	16.28	0.09
0	7	0.93	0.06	16	7	5.43	0.06	32	7	11.13	0.11	48	7	16.38	0.10
0	8			16	8	5.50	0.07	32	8	11.24	0.11	48	8	16.48	0.10
0	9			16	9	5.56	0.06	32	9	11.35	0.11	48	9	16.58	0.10
0	10			16	10	5.62	0.06	32	10	11.46	0.11	48	10	16.68	0.10
0	11			16	11	5.68	0.06	32	11	11.56	0.10	48	11	16.77	0.09
0	12			16	12	5.75	0.07	32	12	11.67	0.11	48	12	16.87	0.10
0	13			16	13	5.81	0.06	32	13	11.78	0.11	48	13	16.97	0.10
0	14			16	14	5.87	0.06	32	14	11.89	0.11	48	14	17.07	0.10
0	15			16	15	5.93	0.06	32	15	12.00	0.11	48	15	17.17	0.10
1	0	0.71		17	0	5.19		33	0	10.99		49	0	16.96
1	1	0.79	0.08	17	1	5.29	0.10	33	1	11.08	0.09	49	1	17.00	0.04
1	2	0.88	0.09	17	2	5.40	0.11	33	2	11.18	0.10	49	2	17.09	0.09
1	3	0.96	0.08	17	3	5.51	0.11	33	3	11.28	0.10	49	3	17.18	0.09
1	4	1.05	0.09	17	4	5.62	0.11	33	4	11.38	0.10	49	4	17.27	0.09
1	5	1.14	0.09	17	5	5.73	0.11	33	5	11.48	0.10	49	5	17.36	0.09
1	6	1.23	0.09	17	6	5.83	0.10	33	6	11.57	0.09	49	6	17.44	0.08
1	7	1.32	0.09	17	7	5.94	0.11	33	7	11.67	0.10	49	7	17.53	0.09
1	8			17	8	6.05	0.11	33	8	11.77	0.10	49	8	17.62	0.09
1	9			17	9	6.16	0.11	33	9	11.87	0.10	49	9	17.71	0.09
1	10			17	10	6.27	0.11	33	10	11.97	0.10	49	10	17.80	0.09
1	11			17	11	6.37	0.10	33	11	12.06	0.09	49	11	17.88	0.08
1	12			17	12	6.48	0.11	33	12	12.16	0.10	49	12	17.97	0.09
1	13			17	13	6.59	0.11	33	13	12.26	0.10	49	13	18.06	0.09
1	14			17	14	6.70	0.11	33	14	12.36	0.10	49	14	18.15	0.09
1	15			17	15	6.81	0.11	33	15	12.46	0.10	49	15	18.24	0.09
2	0	0.78		18	0	5.65		34	0	11.00		50	0	17.27
2	1	0.88	0.10	18	1	5.72	0.07	34	1	11.06	0.06	50	1	17.36	0.09
2	2	0.98	0.10	18	2	5.81	0.09	34	2	11.12	0.06	50	2	17.46	0.10
2	3	1.07	0.09	18	3	5.90	0.09	34	3	11.18	0.06	50	3	17.56	0.10
2	4	1.17	0.10	18	4	5.99	0.09	34	4	11.25	0.07	50	4	17.66	0.10
2	5	1.27	0.10	18	5	6.08	0.09	34	5	11.31	0.06	50	5	17.76	0.10
2	6	1.37	0.10	18	6	6.16	0.08	34	6	11.37	0.06	50	6	17.85	0.09
2	7	1.47	0.10	18	7	6.25	0.09	34	7	11.43	0.06	50	7	17.95	0.10
2	8			18	8	6.34	0.09	34	8	11.50	0.07	50	8	18.05	0.10
2	9			18	9	6.43	0.09	34	9	11.56	0.06	50	9	18.15	0.10
2	10			18	10	6.52	0.09	34	10	11.62	0.06	50	10	18.25	0.10
2	11			18	11	6.60	0.08	34	11	11.68	0.06	50	11	18.34	0.09
2	12			18	12	6.69	0.09	34	12	11.75	0.07	50	12	18.44	0.10
2	13			18	13	6.78	0.09	34	13	11.81	0.06	50	13	18.54	0.10
2	14			18	14	6.87	0.09	34	14	11.87	0.06	50	14	18.64	0.10
2	15			18	15	6.96	0.09	34	15	11.93	0.06	50	15	18.74	0.10
3	0	0.87		19	0	6.00		35	0	11.30		51	0	17.30
3	1	0.97	0.10	19	1	6.06	0.06	35	1	11.36	0.06	51	1	17.40	0.10
3	2	1.08	0.11	19	2	6.12	0.06	35	2	11.45	0.09	51	2	17.51	0.11
3	3	1.18	0.10	19	3	6.18	0.06	35	3	11.54	0.09	51	3	17.62	0.11
3	4	1.29	0.11	19	4	6.25	0.07	35	4	11.63	0.09	51	4	17.73	0.11
3	5	1.40	0.11	19	5	6.31	0.06	35	5	11.72	0.09	51	5	17.84	0.11
3	6	1.51	0.11	19	6	6.37	0.06	35	6	11.80	0.08	51	6	17.94	0.10
3	7	1.62	0.11	19	7	6.43	0.06	35	7	11.89	0.09	51	7	18.05	0.11
3	8			19	8	6.50	0.07	35	8	11.98	0.09	51	8	18.16	0.11
3	9			19	9	6.56	0.06	35	9	12.07	0.09	51	9	18.27	0.11
3	10			19	10	6.62	0.06	35	10	12.16	0.09	51	10	18.38	0.11
3	11			19	11	6.68	0.06	35	11	12.24	0.08	51	11	18.48	0.10
3	12			19	12	6.75	0.07	35	12	12.33	0.09	51	12	18.59	0.11
3	13			19	13	6.81	0.06	35	13	12.42	0.09	51	13	18.70	0.11
3	14			19	14	6.87	0.06	35	14	12.51	0.09	51	14	18.81	0.11
3	15			19	15	6.93	0.06	35	15	12.60	0.09	51	15	18.92	0.11
4	0	1.00		20	0	6.28		36	0	12.00		52	0	18.37
4	1	1.06	0.06	20	1	6.37	0.09	36	1	12.06	0.06	52	1	18.41	0.04
4	2	1.12	0.06	20	2	6.47	0.10	36	2	12.12	0.06	52	2	18.50	0.09
4	3	1.18	0.06	20	3	6.57	0.10	36	3	12.18	0.06	52	3	18.59	0.09
4	4	1.25	0.07	20	4	6.67	0.10	36	4	12.25	0.07	52	4	18.68	0.09
4	5	1.31	0.06	20	5	6.77	0.10	36	5	12.31	0.06	52	5	18.77	0.09
4	6	1.37	0.06	20	6	6.86	0.09	36	6	12.37	0.06	52	6	18.85	0.08
4	7	1.43	0.06	20	7	6.96	0.10	36	7	12.43	0.06	52	7	18.94	0.09
4	8	1.50	0.07	20	8	7.06	0.10	36	8	12.50	0.07	52	8	19.03	0.09
4	9	1.56	0.06	20	9	7.16	0.10	36	9	12.56	0.06	52	9	19.12	0.09
4	10	1.62	0.06	20	10	7.26	0.10	36	10	12.62	0.06	52	10	19.21	0.09
4	11	1.68	0.06	20	11	7.35	0.09	36	11	12.68	0.06	52	11	19.29	0.08
4	12	1.75	0.07	20	12	7.45	0.10	36	12	12.75	0.07	52	12	19.38	0.09
4	13	1.81	0.06	20	13	7.55	0.10	36	13	12.81	0.06	52	13	19.47	0.09
4	14	1.87	0.06	20	14	7.65	0.10	36	14	12.87	0.06	52	14	19.56	0.09
4	15	1.93	0.06	20	15	7.75	0.10	36	15	12.93	0.06	52	15	19.65	0.09
5	0	1.41		21	0	6.92		37	0	12.11		53	0	18.84
5	1	1.49	0.08	21	1	7.02	0.10	37	1	12.21	0.10	53	1	18.93	0.09
5	2	1.58	0.09	21	2	7.13	0.11	37	2	12.32	0.11	53	2	19.03	0.10
5	3	1.67	0.09	21	3	7.24	0.11	37	3	12.43	0.11	53	3	19.13	0.10
5	4	1.76	0.09	21	4	7.35	0.11	37	4	12.54	0.11	53	4	19.23	0.10
5	5	1.85	0.09	21	5	7.46	0.11	37	5	12.65	0.11	53	5	19.33	0.10
5	6	1.93	0.08	21	6	7.56	0.10	37	6	12.75	0.10	53	6	19.42	0.09
5	7	2.02	0.09	21	7	7.67	0.11	37	7	12.86	0.11	53	7	19.52	0.10
5	8	2.11	0.09	21	8	7.78	0.11	37	8	12.97	0.11	53	8	19.62	0.10
5	9	2.20	0.09	21	9	7.89	0.11	37	9	13.08	0.11	53	9	19.72	0.10
5	10	2.29	0.09	21	10	8.00	0.11	37	10	13.19	0.11	53	10	19.82	0.10
5	11	2.37	0.08	21	11	8.10	0.10	37	11	13.29	0.10	53	11	19.91	0.09
5	12	2.46	0.09	21	12	8.21	0.11	37	12	13.40	0.11	53	12	20.01	0.10
5	13	2.55	0.09	21	13	8.32	0.11	37	13	13.51	0.11	53	13	20.11	0.10
5	14	2.64	0.09	21	14	8.43	0.11	37	14	13.62	0.11	53	14	20.21	0.10
5	15	2.73	0.09	21	15	8.54	0.11	37	15	13.73	0.11	53	15	20.31	0.10
6	0	1.57		22	0	7.00		38	0	12.56		54	0	19.03
6	1	1.66	0.09	22	1	7.06	0.06	38	1	12.65	0.09	54	1	19.13	0.10
6	2	1.76	0.10	22	2	7.12	0.06	38	2	12.75	0.10	54	2	19.24	0.11
6	3	1.86	0.10	22	3	7.18	0.06	38	3	12.85	0.10	54	3	19.35	0.11
6	4	1.96	0.10	22	4	7.25	0.07	38	4	12.95	0.10	54	4	19.46	0.11
6	5	2.06	0.10	22	5	7.31	0.06	38	5	13.05	0.10	54	5	19.57	0.11
6	6	2.15	0.09	22	6	7.37	0.06	38	6	13.14	0.09	54	6	19.67	0.10
6	7	2.25	0.10	22	7	7.43	0.06	38	7	13.24	0.10	54	7	19.78	0.11
6	8	2.35	0.10	22	8	7.50	0.07	38	8	13.34	0.10	54	8	19.89	0.11
6	9	2.45	0.10	22	9	7.56	0.06	38	9	13.44	0.10	54	9	20.00	0.11
6	10	2.55	0.10	22	10	7.62	0.06	38	10	13.54	0.10	54	10	20.11	0.11
6	11	2.64	0.09	22	11	7.68	0.06	38	11	13.63	0.09	54	11	20.21	0.10
6	12	2.74	0.10	22	12	7.75	0.07	38	12	13.73	0.10	54	12	20.32	0.11
6	13	2.84	0.10	22	13	7.81	0.06	38	13	13.83	0.10	54	13	20.43	0.11
6	14	2.94	0.10	22	14	7.87	0.06	38	14	13.93	0.10	54	14	20.54	0.11
6	15	3.04	0.10	22	15	7.93	0.06	38	15	14.03	0.10	54	15	20.65	0.11
7	0	1.73		23	0	7.07		39	0	12.72		55	0	19.78
7	1	1.83	0.10	23	1	7.13	0.06	39	1	12.77	0.05	55	1	19.82	0.04
7	2	1.94	0.11	23	2	7.22	0.09	39	2	12.86	0.09	55	2	19.91	0.09
7	3	2.05	0.11	23	3	7.31	0.09	39	3	12.95	0.09	55	3	20.00	0.09
7	4	2.16	0.11	23	4	7.40	0.09	39	4	13.04	0.09	55	4	20.09	0.09
7	5	2.27	0.11	23	5	7.49	0.09	39	5	13.13	0.09	55	5	20.18	0.09
7	6	2.37	0.10	23	6	7.57	0.08	39	6	13.21	0.08	55	6	20.26	0.08
7	7	2.48	0.11	23	7	7.66	0.09	39	7	13.30	0.09	55	7	20.35	0.09
7	8	2.59	0.11	23	8	7.75	0.09	39	8	13.39	0.09	55	8	20.44	0.09
7	9	2.70	0.11	23	9	7.84	0.09	39	9	13.48	0.09	55	9	20.53	0.09
7	10	2.81	0.11	23	10	7.93	0.09	39	10	13.57	0.09	55	10	20.62	0.09
7	11	2.91	0.10	23	11	8.01	0.08	39	11	13.65	0.08	55	11	20.70	0.08
7	12	3.02	0.11	23	12	8.10	0.09	39	12	13.74	0.09	55	12	20.79	0.09
7	13	3.13	0.11	23	13	8.19	0.09	39	13	13.83	0.09	55	13	20.88	0.09
7	14	3.24	0.11	23	14	8.28	0.09	39	14	13.92	0.09	55	14	20.97	0.09
7	15	3.35	0.11	23	15	8.37	0.09	39	15	14.01	0.09	55	15	21.06	0.09
8	0	2.00		24	0	7.85		40	0	13.00		56	0	20.41
8	1	2.06	0.06	24	1	7.94	0.09	40	1	13.06	0.06	56	1	20.50	0.09
8	2	2.12	0.06	24	2	8.04	0.10	40	2	13.12	0.06	56	2	20.60	0.10
8	3	2.18	0.06	24	3	8.14	0.10	40	3	13.18	0.06	56	3	20.70	0.10
8	4	2.25	0.07	24	4	8.24	0.10	40	4	13.25	0.07	56	4	20.80	0.10
8	5	2.31	0.06	24	5	8.34	0.10	40	5	13.31	0.06	56	5	20.90	0.10
8	6	2.37	0.06	24	6	8.43	0.09	40	6	13.37	0.06	56	6	20.99	0.09
8	7	2.43	0.06	24	7	8.53	0.10	40	7	13.43	0.06	56	7	21.09	0.10
8	8	2.50	0.07	24	8	8.63	0.10	40	8	13.50	0.07	56	8	21.19	0.10
8	9	2.56	0.06	24	9	8.73	0.10	40	9	13.56	0.06	56	9	21.29	0.10
8	10	2.62	0.06	24	10	8.83	0.10	40	10	13.62	0.06	56	10	21.39	0.10
8	11	2.68	0.06	24	11	8.92	0.09	40	11	13.68	0.06	56	11	21.48	0.09
8	12	2.75	0.07	24	12	9.02	0.10	40	12	13.75	0.07	56	12	21.58	0.10
8	13	2.81	0.06	24	13	9.12	0.10	40	13	13.81	0.06	56	13	21.68	0.10
8	14	2.87	0.06	24	14	9.22	0.10	40	14	13.87	0.06	56	14	21.78	0.10
8	15	2.93	0.06	24	15	9.32	0.10	40	15	13.93	0.06	56	15	21.88	0.10
9	0	2.82		25	0	8.00		41	0	13.84		57	0	20.76
9	1	2.90	0.08	25	1	8.06	0.06	41	1	13.94	0.10	57	1	20.86	0.10
9	2	2.99	0.09	25	2	8.12	0.06	41	2	14.05	0.11	57	2	20.97	0.11
9	3	3.08	0.09	25	3	8.18	0.06	41	3	14.16	0.11	57	3	21.08	0.11
9	4	3.17	0.09	25	4	8.25	0.07	41	4	14.27	0.11	57	4	21.19	0.11
9	5	3.26	0.09	25	5	8.31	0.06	41	5	14.38	0.11	57	5	21.30	0.11
9	6	3.34	0.08	25	6	8.37	0.06	41	6	14.48	0.10	57	6	21.40	0.10
9	7	3.43	0.09	25	7	8.43	0.06	41	7	14.59	0.11	57	7	21.51	0.11
9	8	3.52	0.09	25	8	8.50	0.07	41	8	14.70	0.11	57	8	21.62	0.11
9	9	3.61	0.09	25	9	8.56	0.06	41	9	14.81	0.11	57	9	21.73	0.11
9	10	3.70	0.09	25	10	8.62	0.06	41	10	14.92	0.11	57	10	21.84	0.11
9	11	3.78	0.08	25	11	8.68	0.06	41	11	15.02	0.10	57	11	21.94	0.10
9	12	3.87	0.09	25	12	8.75	0.07	41	12	15.13	0.11	57	12	22.05	0.11
9	13	3.96	0.09	25	13	8.81	0.06	41	13	15.24	0.11	57	13	22.16	0.11
9	14	4.05	0.09	25	14	8.87	0.06	41	14	15.35	0.11	57	14	22.27	0.11
9	15	4.14	0.09	25	15	8.93	0.06	41	15	15.46	0.11	57	15	22.38	0.11
10	0	3.00		26	0	8.48		42	0	14.00		58	0	21.20
10	1	3.06	0.06	26	1	8.54	0.06	42	1	14.06	0.06	58	1	21.23	0.03
10	2	3.12	0.06	26	2	8.63	0.09	42	2	14.12	0.06	58	2	21.32	0.09
10	3	3.18	0.06	26	3	8.72	0.09	42	3	14.18	0.06	58	3	21.41	0.09
10	4	3.25	0.07	26	4	8.81	0.09	42	4	14.25	0.07	58	4	21.50	0.09
10	5	3.31	0.06	26	5	8.90	0.09	42	5	14.31	0.06	58	5	21.59	0.09
10	6	3.37	0.06	26	6	8.98	0.08	42	6	14.37	0.06	58	6	21.67	0.08
10	7	3.43	0.06	26	7	9.07	0.09	42	7	14.43	0.06	58	7	21.76	0.09
10	8	3.50	0.07	26	8	9.16	0.09	42	8	14.50	0.07	58	8	21.85	0.09
10	9	3.56	0.06	26	9	9.25	0.09	42	9	14.56	0.06	58	9	21.94	0.09
10	10	3.62	0.06	26	10	9.34	0.09	42	10	14.62	0.06	58	10	22.03	0.09
10	11	3.68	0.06	26	11	9.42	0.08	42	11	14.68	0.06	58	11	22.11	0.08
10	12	3.75	0.07	26	12	9.51	0.09	42	12	14.75	0.07	58	12	22.20	0.09
10	13	3.81	0.06	26	13	9.60	0.09	42	13	14.81	0.06	58	13	22.29	0.09
10	14	3.87	0.06	26	14	9.69	0.09	42	14	14.87	0.06	58	14	22.38	0.09
10	15	3.93	0.06	26	15	9.78	0.09	42	15	14.93	0.06	58	15	22.47	0.09
11	0	3.14		27	0	8.65		43	0	14.10		59	0	21.98
11	1	3.23	0.09	27	1	8.75	0.10	43	1	14.18	0.08	59	1	22.07	0.09
11	2	3.33	0.10	27	2	8.86	0.11	43	2	14.27	0.09	59	2	22.17	0.10
11	3	3.43	0.10	27	3	8.97	0.11	43	3	14.36	0.09	59	3	22.27	0.10
11	4	3.53	0.10	27	4	9.08	0.11	43	4	14.45	0.09	59	4	22.37	0.10
11	5	3.63	0.10	27	5	9.19	0.11	43	5	14.54	0.09	59	5	22.47	0.10
11	6	3.72	0.09	27	6	9.29	0.10	43	6	14.62	0.08	59	6	22.56	0.09
11	7	3.82	0.10	27	7	9.40	0.11	43	7	14.71	0.09	59	7	22.66	0.10
11	8	3.92	0.10	27	8	9.51	0.11	43	8	14.80	0.09	59	8	22.76	0.10
11	9	4.02	0.10	27	9	9.62	0.11	43	9	14.89	0.09	59	9	22.86	0.10
11	10	4.12	0.10	27	10	9.73	0.11	43	10	14.98	0.09	59	10	22.96	0.10
11	11	4.21	0.09	27	11	9.83	0.10	43	11	15.06	0.08	59	11	23.05	0.09
11	12	4.31	0.10	27	12	9.94	0.11	43	12	15.15	0.09	59	12	23.15	0.10
11	13	4.41	0.10	27	13	10.05	0.11	43	13	15.24	0.09	59	13	23.25	0.10
11	14	4.51	0.10	27	14	10.16	0.11	43	14	15.33	0.09	59	14	23.35	0.10
11	15	4.61	0.10	27	15	10.27	0.11	43	15	15.42	0.09	59	15	23.45	0.10
12	0	3.46		28	0	9.00		44	0	14.13		60	0	22.49
12	1	3.56	0.10	28	1	9.06	0.06	44	1	14.22	0.09	60	1	22.59	0.10
12	2	3.67	0.11	28	2	9.12	0.06	44	2	14.32	0.10	60	2	22.70	0.11
12	3	3.78	0.11	28	3	9.18	0.06	44	3	14.42	0.10	60	3	22.81	0.11
12	4	3.89	0.11	28	4	9.25	0.07	44	4	14.52	0.10	60	4	22.92	0.11
12	5	4.00	0.11	28	5	9.31	0.06	44	5	14.62	0.10	60	5	23.03	0.11
12	6	4.10	0.10	28	6	9.37	0.06	44	6	14.71	0.09	60	6	23.13	0.10
12	7	4.21	0.11	28	7	9.43	0.06	44	7	14.81	0.10	60	7	23.24	0.11
12	8	4.32	0.11	28	8	9.50	0.07	44	8	14.91	0.10	60	8	23.35	0.11
12	9	4.43	0.11	28	9	9.56	0.06	44	9	15.01	0.10	60	9	23.46	0.11
12	10	4.54	0.11	28	10	9.62	0.06	44	10	15.11	0.10	60	10	23.57	0.11
12	11	4.64	0.10	28	11	9.68	0.06	44	11	15.20	0.09	60	11	23.67	0.10
12	12	4.75	0.11	28	12	9.75	0.07	44	12	15.30	0.10	60	12	23.78	0.11
12	13	4.86	0.11	28	13	9.81	0.06	44	13	15.40	0.10	60	13	23.89	0.11
12	14	4.97	0.11	28	14	9.87	0.06	44	14	15.50	0.10	60	14	24.00	0.11
12	15	5.08	0.11	28	15	9.93	0.06	44	15	15.60	0.10	60	15	24.11	0.11
13	0	4.00		29	0	9.42		45	0	15.00		61	0	23.55
13	1	4.06	0.06	29	1	9.51	0.09	45	1	15.06	0.06	61	1	23.64	0.09
13	2	4.12	0.06	29	2	9.61	0.10	45	2	15.12	0.06	61	2	23.74	0.10
13	3	4.18	0.06	29	3	9.71	0.10	45	3	15.18	0.06	61	3	23.84	0.10
13	4	4.25	0.07	29	4	9.81	0.10	45	4	15.25	0.07	61	4	23.94	0.10
13	5	4.31	0.06	29	5	9.91	0.10	45	5	15.31	0.06	61	5	24.04	0.10
13	6	4.37	0.06	29	6	10.00	0.09	45	6	15.37	0.06	61	6	24.13	0.09
13	7	4.43	0.06	29	7	10.10	0.10	45	7	15.43	0.06	61	7	24.23	0.10
13	8	4.50	0.07	29	8	10.20	0.10	45	8	15.50	0.07	61	8	24.33	0.10
13	9	4.56	0.06	29	9	10.30	0.10	45	9	15.56	0.06	61	9	24.43	0.10
13	10	4.62	0.06	29	10	10.40	0.10	45	10	15.62	0.06	61	10	24.53	0.10
13	11	4.68	0.06	29	11	10.49	0.09	45	11	15.68	0.06	61	11	24.62	0.09
13	12	4.75	0.07	29	12	10.59	0.10	45	12	15.75	0.07	61	12	24.72	0.10
13	13	4.81	0.06	29	13	10.69	0.10	45	13	15.81	0.06	61	13	24.82	0.10
13	14	4.87	0.06	29	14	10.79	0.10	45	14	15.87	0.06	61	14	24.92	0.10
13	15	4.93	0.06	29	15	10.89	0.10	45	15	15.93	0.06	61	15	25.02	0.10
14	0	4.24		30	0	9.89		46	0	15.55		62	0	24.22
14	1	4.31	0.07	30	1	9.95	0.06	46	1	15.59	0.04	62	1	24.32	0.10
14	2	4.40	0.09	30	2	10.04	0.09	46	2	15.68	0.09	62	2	24.43	0.11
14	3	4.49	0.09	30	3	10.13	0.09	46	3	15.77	0.09	62	3	24.54	0.11
14	4	4.58	0.09	30	4	10.22	0.09	46	4	15.86	0.09	62	4	24.65	0.11
14	5	4.67	0.09	30	5	10.31	0.09	46	5	15.95	0.09	62	5	24.76	0.11
14	6	4.75	0.08	30	6	10.39	0.08	46	6	16.03	0.08	62	6	24.86	0.10
14	7	4.84	0.09	30	7	10.48	0.09	46	7	16.12	0.09	62	7	24.97	0.11
14	8	4.93	0.09	30	8	10.57	0.09	46	8	16.21	0.09	62	8	25.08	0.11
14	9	5.02	0.09	30	9	10.66	0.09	46	9	16.30	0.09	62	9	25.19	0.11
14	10	5.11	0.09	30	10	10.75	0.09	46	10	16.39	0.09	62	10	25.30	0.11
14	11	5.19	0.08	30	11	10.83	0.08	46	11	16.47	0.08	62	11	25.40	0.10
14	12	5.28	0.09	30	12	10.92	0.09	46	12	16.56	0.09	62	12	25.51	0.11
14	13	5.37	0.09	30	13	11.01	0.09	46	13	16.65	0.09	62	13	25.62	0.11
14	14	5.46	0.09	30	14	11.10	0.09	46	14	16.74	0.09	62	14	25.73	0.11
14	15	5.55	0.09	30	15	11.19	0.09	46	15	16.83	0.09	62	15	25.84	0.11
15	0	4.71		31	0	10.00		47	0	15.57		63	0	25.95
15	1	4.80	0.09	31	1	10.06	0.06	47	1	15.67	0.10	63	1	26.05	0.10
15	2	4.90	0.10	31	2	10.12	0.06	47	2	15.78	0.11	63	2	26.16	0.11
15	3	5.00	0.10	31	3	10.18	0.06	47	3	15.89	0.11	63	3	26.27	0.11
15	4	5.10	0.10	31	4	10.25	0.07	47	4	16.00	0.11	63	4	26.38	0.11
15	5	5.20	0.10	31	5	10.31	0.06	47	5	16.11	0.11	63	5	26.49	0.11
15	6	5.29	0.09	31	6	10.37	0.06	47	6	16.21	0.10	63	6	26.59	0.10
15	7	5.39	0.10	31	7	10.43	0.06	47	7	16.32	0.11	63	7	26.70	0.11
15	8	5.49	0.10	31	8	10.50	0.07	47	8	16.43	0.11	63	8	26.81	0.11
15	9	5.59	0.10	31	9	10.56	0.06	47	9	16.54	0.11	63	9	26.92	0.11
15	10	5.69	0.10	31	10	10.62	0.06	47	10	16.65	0.11	63	10	27.03	0.11
15	11	5.78	0.09	31	11	10.68	0.06	47	11	16.75	0.10	63	11	27.13	0.10
15	12	5.88	0.10	31	12	10.75	0.07	47	12	16.86	0.11	63	12	27.24	0.11
15	13	5.98	0.10	31	13	10.81	0.06	47	13	16.97	0.11	63	13	27.35	0.11
15	14	6.08	0.10	31	14	10.87	0.06	47	14	17.08	0.11	63	14	27.46	0.11
15	15	6.18	0.10	31	15	10.93	0.06	47	15	17.19	0.11	63	15	27.57	0.11

Putting Them Together

The colors highlight groups of coarse values with similar delta values. If these groups are collected and arranged in strictly ascending order, we have:

Order	Coarse	Fine	Ratio	Delta	Coarse	Fine	Ratio	Delta	Coarse	Fine	Ratio	Delta	Coarse	Fine	Ratio	Delta
0	0	0	0.50		1	0	0.71		2	0	0.78		3	0	0.87
1	0	1	0.56	0.06	1	1	0.79	0.08	2	1	0.88	0.10	3	1	0.97	0.10
2	0	2	0.62	0.06	1	2	0.88	0.09	2	2	0.98	0.10	3	2	1.08	0.11
3	0	3	0.68	0.06	1	3	0.96	0.08	2	3	1.07	0.09	3	3	1.18	0.10
4	0	4	0.75	0.07	1	4	1.05	0.09	2	4	1.17	0.10	3	4	1.29	0.11
5	0	5	0.81	0.06	1	5	1.14	0.09	2	5	1.27	0.10	3	5	1.40	0.11
6	0	6	0.87	0.06	1	6	1.23	0.09	2	6	1.37	0.10	3	6	1.51	0.11
7	0	7	0.93	0.06	1	7	1.32	0.09	2	7	1.47	0.10	3	7	1.62	0.11
8	4	0	1.00	0.07	5	0	1.41	0.09	6	0	1.57	0.10	7	0	1.73	0.11
9	4	1	1.06	0.06	5	1	1.49	0.08	6	1	1.66	0.09	7	1	1.83	0.10
10	4	2	1.12	0.06	5	2	1.58	0.09	6	2	1.76	0.10	7	2	1.94	0.11
11	4	3	1.18	0.06	5	3	1.67	0.09	6	3	1.86	0.10	7	3	2.05	0.11
12	4	4	1.25	0.07	5	4	1.76	0.09	6	4	1.96	0.10	7	4	2.16	0.11
13	4	5	1.31	0.06	5	5	1.85	0.09	6	5	2.06	0.10	7	5	2.27	0.11
14	4	6	1.37	0.06	5	6	1.93	0.08	6	6	2.15	0.09	7	6	2.37	0.10
15	4	7	1.43	0.06	5	7	2.02	0.09	6	7	2.25	0.10	7	7	2.48	0.11
16	4	8	1.50	0.07	5	8	2.11	0.09	6	8	2.35	0.10	7	8	2.59	0.11
17	4	9	1.56	0.06	5	9	2.20	0.09	6	9	2.45	0.10	7	9	2.70	0.11
18	4	10	1.62	0.06	5	10	2.29	0.09	6	10	2.55	0.10	7	10	2.81	0.11
19	4	11	1.68	0.06	5	11	2.37	0.08	6	11	2.64	0.09	7	11	2.91	0.10
20	4	12	1.75	0.07	5	12	2.46	0.09	6	12	2.74	0.10	7	12	3.02	0.11
21	4	13	1.81	0.06	5	13	2.55	0.09	6	13	2.84	0.10	7	13	3.13	0.11
22	4	14	1.87	0.06	5	14	2.64	0.09	6	14	2.94	0.10	7	14	3.24	0.11
23	4	15	1.93	0.06	5	15	2.73	0.09	6	15	3.04	0.10	7	15	3.35	0.11
24	8	0	2.00	0.07	9	0	2.82	0.09	11	0	3.14	0.10	12	0	3.46	0.11
25	8	1	2.06	0.06	9	1	2.90	0.08	11	1	3.23	0.09	12	1	3.56	0.10
26	8	2	2.12	0.06	9	2	2.99	0.09	11	2	3.33	0.10	12	2	3.67	0.11
27	8	3	2.18	0.06	9	3	3.08	0.09	11	3	3.43	0.10	12	3	3.78	0.11
28	8	4	2.25	0.07	9	4	3.17	0.09	11	4	3.53	0.10	12	4	3.89	0.11
29	8	5	2.31	0.06	9	5	3.26	0.09	11	5	3.63	0.10	12	5	4.00	0.11
30	8	6	2.37	0.06	9	6	3.34	0.08	11	6	3.72	0.09	12	6	4.10	0.10
31	8	7	2.43	0.06	9	7	3.43	0.09	11	7	3.82	0.10	12	7	4.21	0.11
32	8	8	2.50	0.07	9	8	3.52	0.09	11	8	3.92	0.10	12	8	4.32	0.11
33	8	9	2.56	0.06	9	9	3.61	0.09	11	9	4.02	0.10	12	9	4.43	0.11
34	8	10	2.62	0.06	9	10	3.70	0.09	11	10	4.12	0.10	12	10	4.54	0.11
35	8	11	2.68	0.06	9	11	3.78	0.08	11	11	4.21	0.09	12	11	4.64	0.10
36	8	12	2.75	0.07	9	12	3.87	0.09	11	12	4.31	0.10	12	12	4.75	0.11
37	8	13	2.81	0.06	9	13	3.96	0.09	11	13	4.41	0.10	12	13	4.86	0.11
38	8	14	2.87	0.06	9	14	4.05	0.09	11	14	4.51	0.10	12	14	4.97	0.11
39	8	15	2.93	0.06	9	15	4.14	0.09	11	15	4.61	0.10	12	15	5.08	0.11
40	10	0	3.00	0.07	14	0	4.24	0.10	15	0	4.71	0.10	17	0	5.19	0.11
41	10	1	3.06	0.06	14	1	4.31	0.07	15	1	4.80	0.09	17	1	5.29	0.10
42	10	2	3.12	0.06	14	2	4.40	0.09	15	2	4.90	0.10	17	2	5.40	0.11
43	10	3	3.18	0.06	14	3	4.49	0.09	15	3	5.00	0.10	17	3	5.51	0.11
44	10	4	3.25	0.07	14	4	4.58	0.09	15	4	5.10	0.10	17	4	5.62	0.11
45	10	5	3.31	0.06	14	5	4.67	0.09	15	5	5.20	0.10	17	5	5.73	0.11
46	10	6	3.37	0.06	14	6	4.75	0.08	15	6	5.29	0.09	17	6	5.83	0.10
47	10	7	3.43	0.06	14	7	4.84	0.09	15	7	5.39	0.10	17	7	5.94	0.11
48	10	8	3.50	0.07	14	8	4.93	0.09	15	8	5.49	0.10	17	8	6.05	0.11
49	10	9	3.56	0.06	14	9	5.02	0.09	15	9	5.59	0.10	17	9	6.16	0.11
50	10	10	3.62	0.06	14	10	5.11	0.09	15	10	5.69	0.10	17	10	6.27	0.11
51	10	11	3.68	0.06	14	11	5.19	0.08	15	11	5.78	0.09	17	11	6.37	0.10
52	10	12	3.75	0.07	14	12	5.28	0.09	15	12	5.88	0.10	17	12	6.48	0.11
53	10	13	3.81	0.06	14	13	5.37	0.09	15	13	5.98	0.10	17	13	6.59	0.11
54	10	14	3.87	0.06	14	14	5.46	0.09	15	14	6.08	0.10	17	14	6.70	0.11
55	10	15	3.93	0.06	14	15	5.55	0.09	15	15	6.18	0.10	17	15	6.81	0.11
56	13	0	4.00	0.07	18	0	5.65	0.10	20	0	6.28	0.10	21	0	6.92	0.11
57	13	1	4.06	0.06	18	1	5.72	0.07	20	1	6.37	0.09	21	1	7.02	0.10
58	13	2	4.12	0.06	18	2	5.81	0.09	20	2	6.47	0.10	21	2	7.13	0.11
59	13	3	4.18	0.06	18	3	5.90	0.09	20	3	6.57	0.10	21	3	7.24	0.11
60	13	4	4.25	0.07	18	4	5.99	0.09	20	4	6.67	0.10	21	4	7.35	0.11
61	13	5	4.31	0.06	18	5	6.08	0.09	20	5	6.77	0.10	21	5	7.46	0.11
62	13	6	4.37	0.06	18	6	6.16	0.08	20	6	6.86	0.09	21	6	7.56	0.10
63	13	7	4.43	0.06	18	7	6.25	0.09	20	7	6.96	0.10	21	7	7.67	0.11
64	13	8	4.50	0.07	18	8	6.34	0.09	20	8	7.06	0.10	21	8	7.78	0.11
65	13	9	4.56	0.06	18	9	6.43	0.09	20	9	7.16	0.10	21	9	7.89	0.11
66	13	10	4.62	0.06	18	10	6.52	0.09	20	10	7.26	0.10	21	10	8.00	0.11
67	13	11	4.68	0.06	18	11	6.60	0.08	20	11	7.35	0.09	21	11	8.10	0.10
68	13	12	4.75	0.07	18	12	6.69	0.09	20	12	7.45	0.10	21	12	8.21	0.11
69	13	13	4.81	0.06	18	13	6.78	0.09	20	13	7.55	0.10	21	13	8.32	0.11
70	13	14	4.87	0.06	18	14	6.87	0.09	20	14	7.65	0.10	21	14	8.43	0.11
71	13	15	4.93	0.06	18	15	6.96	0.09	20	15	7.75	0.10	21	15	8.54	0.11
72	16	0	5.00	0.07	23	0	7.07	0.11	24	0	7.85	0.10	27	0	8.65	0.11
73	16	1	5.06	0.06	23	1	7.13	0.06	24	1	7.94	0.09	27	1	8.75	0.10
74	16	2	5.12	0.06	23	2	7.22	0.09	24	2	8.04	0.10	27	2	8.86	0.11
75	16	3	5.18	0.06	23	3	7.31	0.09	24	3	8.14	0.10	27	3	8.97	0.11
76	16	4	5.25	0.07	23	4	7.40	0.09	24	4	8.24	0.10	27	4	9.08	0.11
77	16	5	5.31	0.06	23	5	7.49	0.09	24	5	8.34	0.10	27	5	9.19	0.11
78	16	6	5.37	0.06	23	6	7.57	0.08	24	6	8.43	0.09	27	6	9.29	0.10
79	16	7	5.43	0.06	23	7	7.66	0.09	24	7	8.53	0.10	27	7	9.40	0.11
80	16	8	5.50	0.07	23	8	7.75	0.09	24	8	8.63	0.10	27	8	9.51	0.11
81	16	9	5.56	0.06	23	9	7.84	0.09	24	9	8.73	0.10	27	9	9.62	0.11
82	16	10	5.62	0.06	23	10	7.93	0.09	24	10	8.83	0.10	27	10	9.73	0.11
83	16	11	5.68	0.06	23	11	8.01	0.08	24	11	8.92	0.09	27	11	9.83	0.10
84	16	12	5.75	0.07	23	12	8.10	0.09	24	12	9.02	0.10	27	12	9.94	0.11
85	16	13	5.81	0.06	23	13	8.19	0.09	24	13	9.12	0.10	27	13	10.05	0.11
86	16	14	5.87	0.06	23	14	8.28	0.09	24	14	9.22	0.10	27	14	10.16	0.11
87	16	15	5.93	0.06	23	15	8.37	0.09	24	15	9.32	0.10	27	15	10.27	0.11
88	19	0	6.00	0.07	26	0	8.48	0.11	29	0	9.42	0.10	32	0	10.38	0.11
89	19	1	6.06	0.06	26	1	8.54	0.06	29	1	9.51	0.09	32	1	10.48	0.10
90	19	2	6.12	0.06	26	2	8.63	0.09	29	2	9.61	0.10	32	2	10.59	0.11
91	19	3	6.18	0.06	26	3	8.72	0.09	29	3	9.71	0.10	32	3	10.70	0.11
92	19	4	6.25	0.07	26	4	8.81	0.09	29	4	9.81	0.10	32	4	10.81	0.11
93	19	5	6.31	0.06	26	5	8.90	0.09	29	5	9.91	0.10	32	5	10.92	0.11
94	19	6	6.37	0.06	26	6	8.98	0.08	29	6	10.00	0.09	32	6	11.02	0.10
95	19	7	6.43	0.06	26	7	9.07	0.09	29	7	10.10	0.10	32	7	11.13	0.11
96	19	8	6.50	0.07	26	8	9.16	0.09	29	8	10.20	0.10	32	8	11.24	0.11
97	19	9	6.56	0.06	26	9	9.25	0.09	29	9	10.30	0.10	32	9	11.35	0.11
98	19	10	6.62	0.06	26	10	9.34	0.09	29	10	10.40	0.10	32	10	11.46	0.11
99	19	11	6.68	0.06	26	11	9.42	0.08	29	11	10.49	0.09	32	11	11.56	0.10
100	19	12	6.75	0.07	26	12	9.51	0.09	29	12	10.59	0.10	32	12	11.67	0.11
101	19	13	6.81	0.06	26	13	9.60	0.09	29	13	10.69	0.10	32	13	11.78	0.11
102	19	14	6.87	0.06	26	14	9.69	0.09	29	14	10.79	0.10	32	14	11.89	0.11
103	19	15	6.93	0.06	26	15	9.78	0.09	29	15	10.89	0.10	32	15	12.00	0.11
104	22	0	7.00	0.07	30	0	9.89	0.11	33	0	10.99	0.10	37	0	12.11	0.11
105	22	1	7.06	0.06	30	1	9.95	0.06	33	1	11.08	0.09	37	1	12.21	0.10
106	22	2	7.12	0.06	30	2	10.04	0.09	33	2	11.18	0.10	37	2	12.32	0.11
107	22	3	7.18	0.06	30	3	10.13	0.09	33	3	11.28	0.10	37	3	12.43	0.11
108	22	4	7.25	0.07	30	4	10.22	0.09	33	4	11.38	0.10	37	4	12.54	0.11
109	22	5	7.31	0.06	30	5	10.31	0.09	33	5	11.48	0.10	37	5	12.65	0.11
110	22	6	7.37	0.06	30	6	10.39	0.08	33	6	11.57	0.09	37	6	12.75	0.10
111	22	7	7.43	0.06	30	7	10.48	0.09	33	7	11.67	0.10	37	7	12.86	0.11
112	22	8	7.50	0.07	30	8	10.57	0.09	33	8	11.77	0.10	37	8	12.97	0.11
113	22	9	7.56	0.06	30	9	10.66	0.09	33	9	11.87	0.10	37	9	13.08	0.11
114	22	10	7.62	0.06	30	10	10.75	0.09	33	10	11.97	0.10	37	10	13.19	0.11
115	22	11	7.68	0.06	30	11	10.83	0.08	33	11	12.06	0.09	37	11	13.29	0.10
116	22	12	7.75	0.07	30	12	10.92	0.09	33	12	12.16	0.10	37	12	13.40	0.11
117	22	13	7.81	0.06	30	13	11.01	0.09	33	13	12.26	0.10	37	13	13.51	0.11
118	22	14	7.87	0.06	30	14	11.10	0.09	33	14	12.36	0.10	37	14	13.62	0.11
119	22	15	7.93	0.06	30	15	11.19	0.09	33	15	12.46	0.10	37	15	13.73	0.11
120	25	0	8.00	0.07	35	0	11.30	0.11	38	0	12.56	0.10	41	0	13.84	0.11
121	25	1	8.06	0.06	35	1	11.36	0.06	38	1	12.65	0.09	41	1	13.94	0.10
122	25	2	8.12	0.06	35	2	11.45	0.09	38	2	12.75	0.10	41	2	14.05	0.11
123	25	3	8.18	0.06	35	3	11.54	0.09	38	3	12.85	0.10	41	3	14.16	0.11
124	25	4	8.25	0.07	35	4	11.63	0.09	38	4	12.95	0.10	41	4	14.27	0.11
125	25	5	8.31	0.06	35	5	11.72	0.09	38	5	13.05	0.10	41	5	14.38	0.11
126	25	6	8.37	0.06	35	6	11.80	0.08	38	6	13.14	0.09	41	6	14.48	0.10
127	25	7	8.43	0.06	35	7	11.89	0.09	38	7	13.24	0.10	41	7	14.59	0.11
128	25	8	8.50	0.07	35	8	11.98	0.09	38	8	13.34	0.10	41	8	14.70	0.11
129	25	9	8.56	0.06	35	9	12.07	0.09	38	9	13.44	0.10	41	9	14.81	0.11
130	25	10	8.62	0.06	35	10	12.16	0.09	38	10	13.54	0.10	41	10	14.92	0.11
131	25	11	8.68	0.06	35	11	12.24	0.08	38	11	13.63	0.09	41	11	15.02	0.10
132	25	12	8.75	0.07	35	12	12.33	0.09	38	12	13.73	0.10	41	12	15.13	0.11
133	25	13	8.81	0.06	35	13	12.42	0.09	38	13	13.83	0.10	41	13	15.24	0.11
134	25	14	8.87	0.06	35	14	12.51	0.09	38	14	13.93	0.10	41	14	15.35	0.11
135	25	15	8.93	0.06	35	15	12.60	0.09	38	15	14.03	0.10	41	15	15.46	0.11
136	28	0	9.00	0.07	39	0	12.72	0.12	44	0	14.13	0.10	47	0	15.57	0.11
137	28	1	9.06	0.06	39	1	12.77	0.05	44	1	14.22	0.09	47	1	15.67	0.10
138	28	2	9.12	0.06	39	2	12.86	0.09	44	2	14.32	0.10	47	2	15.78	0.11
139	28	3	9.18	0.06	39	3	12.95	0.09	44	3	14.42	0.10	47	3	15.89	0.11
140	28	4	9.25	0.07	39	4	13.04	0.09	44	4	14.52	0.10	47	4	16.00	0.11
141	28	5	9.31	0.06	39	5	13.13	0.09	44	5	14.62	0.10	47	5	16.11	0.11
142	28	6	9.37	0.06	39	6	13.21	0.08	44	6	14.71	0.09	47	6	16.21	0.10
143	28	7	9.43	0.06	39	7	13.30	0.09	44	7	14.81	0.10	47	7	16.32	0.11
144	28	8	9.50	0.07	39	8	13.39	0.09	44	8	14.91	0.10	47	8	16.43	0.11
145	28	9	9.56	0.06	39	9	13.48	0.09	44	9	15.01	0.10	47	9	16.54	0.11
146	28	10	9.62	0.06	39	10	13.57	0.09	44	10	15.11	0.10	47	10	16.65	0.11
147	28	11	9.68	0.06	39	11	13.65	0.08	44	11	15.20	0.09	47	11	16.75	0.10
148	28	12	9.75	0.07	39	12	13.74	0.09	44	12	15.30	0.10	47	12	16.86	0.11
149	28	13	9.81	0.06	39	13	13.83	0.09	44	13	15.40	0.10	47	13	16.97	0.11
150	28	14	9.87	0.06	39	14	13.92	0.09	44	14	15.50	0.10	47	14	17.08	0.11
151	28	15	9.93	0.06	39	15	14.01	0.09	44	15	15.60	0.10	47	15	17.19	0.11
152	31	0	10.00	0.07	43	0	14.10	0.09	48	0	15.70	0.10	51	0	17.30	0.11
153	31	1	10.06	0.06	43	1	14.18	0.08	48	1	15.79	0.09	51	1	17.40	0.10
154	31	2	10.12	0.06	43	2	14.27	0.09	48	2	15.89	0.10	51	2	17.51	0.11
155	31	3	10.18	0.06	43	3	14.36	0.09	48	3	15.99	0.10	51	3	17.62	0.11
156	31	4	10.25	0.07	43	4	14.45	0.09	48	4	16.09	0.10	51	4	17.73	0.11
157	31	5	10.31	0.06	43	5	14.54	0.09	48	5	16.19	0.10	51	5	17.84	0.11
158	31	6	10.37	0.06	43	6	14.62	0.08	48	6	16.28	0.09	51	6	17.94	0.10
159	31	7	10.43	0.06	43	7	14.71	0.09	48	7	16.38	0.10	51	7	18.05	0.11
160	31	8	10.50	0.07	43	8	14.80	0.09	48	8	16.48	0.10	51	8	18.16	0.11
161	31	9	10.56	0.06	43	9	14.89	0.09	48	9	16.58	0.10	51	9	18.27	0.11
162	31	10	10.62	0.06	43	10	14.98	0.09	48	10	16.68	0.10	51	10	18.38	0.11
163	31	11	10.68	0.06	43	11	15.06	0.08	48	11	16.77	0.09	51	11	18.48	0.10
164	31	12	10.75	0.07	43	12	15.15	0.09	48	12	16.87	0.10	51	12	18.59	0.11
165	31	13	10.81	0.06	43	13	15.24	0.09	48	13	16.97	0.10	51	13	18.70	0.11
166	31	14	10.87	0.06	43	14	15.33	0.09	48	14	17.07	0.10	51	14	18.81	0.11
167	31	15	10.93	0.06	43	15	15.42	0.09	48	15	17.17	0.10	51	15	18.92	0.11
168	34	0	11.00	0.07	46	0	15.55	0.13	50	0	17.27	0.10	54	0	19.03	0.11
169	34	1	11.06	0.06	46	1	15.59	0.04	50	1	17.36	0.09	54	1	19.13	0.10
170	34	2	11.12	0.06	46	2	15.68	0.09	50	2	17.46	0.10	54	2	19.24	0.11
171	34	3	11.18	0.06	46	3	15.77	0.09	50	3	17.56	0.10	54	3	19.35	0.11
172	34	4	11.25	0.07	46	4	15.86	0.09	50	4	17.66	0.10	54	4	19.46	0.11
173	34	5	11.31	0.06	46	5	15.95	0.09	50	5	17.76	0.10	54	5	19.57	0.11
174	34	6	11.37	0.06	46	6	16.03	0.08	50	6	17.85	0.09	54	6	19.67	0.10
175	34	7	11.43	0.06	46	7	16.12	0.09	50	7	17.95	0.10	54	7	19.78	0.11
176	34	8	11.50	0.07	46	8	16.21	0.09	50	8	18.05	0.10	54	8	19.89	0.11
177	34	9	11.56	0.06	46	9	16.30	0.09	50	9	18.15	0.10	54	9	20.00	0.11
178	34	10	11.62	0.06	46	10	16.39	0.09	50	10	18.25	0.10	54	10	20.11	0.11
179	34	11	11.68	0.06	46	11	16.47	0.08	50	11	18.34	0.09	54	11	20.21	0.10
180	34	12	11.75	0.07	46	12	16.56	0.09	50	12	18.44	0.10	54	12	20.32	0.11
181	34	13	11.81	0.06	46	13	16.65	0.09	50	13	18.54	0.10	54	13	20.43	0.11
182	34	14	11.87	0.06	46	14	16.74	0.09	50	14	18.64	0.10	54	14	20.54	0.11
183	34	15	11.93	0.06	46	15	16.83	0.09	50	15	18.74	0.10	54	15	20.65	0.11
184	36	0	12.00	0.07	49	0	16.96	0.13	53	0	18.84	0.10	57	0	20.76	0.11
185	36	1	12.06	0.06	49	1	17.00	0.04	53	1	18.93	0.09	57	1	20.86	0.10
186	36	2	12.12	0.06	49	2	17.09	0.09	53	2	19.03	0.10	57	2	20.97	0.11
187	36	3	12.18	0.06	49	3	17.18	0.09	53	3	19.13	0.10	57	3	21.08	0.11
188	36	4	12.25	0.07	49	4	17.27	0.09	53	4	19.23	0.10	57	4	21.19	0.11
189	36	5	12.31	0.06	49	5	17.36	0.09	53	5	19.33	0.10	57	5	21.30	0.11
190	36	6	12.37	0.06	49	6	17.44	0.08	53	6	19.42	0.09	57	6	21.40	0.10
191	36	7	12.43	0.06	49	7	17.53	0.09	53	7	19.52	0.10	57	7	21.51	0.11
192	36	8	12.50	0.07	49	8	17.62	0.09	53	8	19.62	0.10	57	8	21.62	0.11
193	36	9	12.56	0.06	49	9	17.71	0.09	53	9	19.72	0.10	57	9	21.73	0.11
194	36	10	12.62	0.06	49	10	17.80	0.09	53	10	19.82	0.10	57	10	21.84	0.11
195	36	11	12.68	0.06	49	11	17.88	0.08	53	11	19.91	0.09	57	11	21.94	0.10
196	36	12	12.75	0.07	49	12	17.97	0.09	53	12	20.01	0.10	57	12	22.05	0.11
197	36	13	12.81	0.06	49	13	18.06	0.09	53	13	20.11	0.10	57	13	22.16	0.11
198	36	14	12.87	0.06	49	14	18.15	0.09	53	14	20.21	0.10	57	14	22.27	0.11
199	36	15	12.93	0.06	49	15	18.24	0.09	53	15	20.31	0.10	57	15	22.38	0.11
200	40	0	13.00	0.07	52	0	18.37	0.13	56	0	20.41	0.10	60	0	22.49	0.11
201	40	1	13.06	0.06	52	1	18.41	0.04	56	1	20.50	0.09	60	1	22.59	0.10
202	40	2	13.12	0.06	52	2	18.50	0.09	56	2	20.60	0.10	60	2	22.70	0.11
203	40	3	13.18	0.06	52	3	18.59	0.09	56	3	20.70	0.10	60	3	22.81	0.11
204	40	4	13.25	0.07	52	4	18.68	0.09	56	4	20.80	0.10	60	4	22.92	0.11
205	40	5	13.31	0.06	52	5	18.77	0.09	56	5	20.90	0.10	60	5	23.03	0.11
206	40	6	13.37	0.06	52	6	18.85	0.08	56	6	20.99	0.09	60	6	23.13	0.10
207	40	7	13.43	0.06	52	7	18.94	0.09	56	7	21.09	0.10	60	7	23.24	0.11
208	40	8	13.50	0.07	52	8	19.03	0.09	56	8	21.19	0.10	60	8	23.35	0.11
209	40	9	13.56	0.06	52	9	19.12	0.09	56	9	21.29	0.10	60	9	23.46	0.11
210	40	10	13.62	0.06	52	10	19.21	0.09	56	10	21.39	0.10	60	10	23.57	0.11
211	40	11	13.68	0.06	52	11	19.29	0.08	56	11	21.48	0.09	60	11	23.67	0.10
212	40	12	13.75	0.07	52	12	19.38	0.09	56	12	21.58	0.10	60	12	23.78	0.11
213	40	13	13.81	0.06	52	13	19.47	0.09	56	13	21.68	0.10	60	13	23.89	0.11
214	40	14	13.87	0.06	52	14	19.56	0.09	56	14	21.78	0.10	60	14	24.00	0.11
215	40	15	13.93	0.06	52	15	19.65	0.09	56	15	21.88	0.10	60	15	24.11	0.11
216	42	0	14.00	0.07	55	0	19.78	0.13	59	0	21.98	0.10	62	0	24.22	0.11
217	42	1	14.06	0.06	55	1	19.82	0.04	59	1	22.07	0.09	62	1	24.32	0.10
218	42	2	14.12	0.06	55	2	19.91	0.09	59	2	22.17	0.10	62	2	24.43	0.11
219	42	3	14.18	0.06	55	3	20.00	0.09	59	3	22.27	0.10	62	3	24.54	0.11
220	42	4	14.25	0.07	55	4	20.09	0.09	59	4	22.37	0.10	62	4	24.65	0.11
221	42	5	14.31	0.06	55	5	20.18	0.09	59	5	22.47	0.10	62	5	24.76	0.11
222	42	6	14.37	0.06	55	6	20.26	0.08	59	6	22.56	0.09	62	6	24.86	0.10
223	42	7	14.43	0.06	55	7	20.35	0.09	59	7	22.66	0.10	62	7	24.97	0.11
224	42	8	14.50	0.07	55	8	20.44	0.09	59	8	22.76	0.10	62	8	25.08	0.11
225	42	9	14.56	0.06	55	9	20.53	0.09	59	9	22.86	0.10	62	9	25.19	0.11
226	42	10	14.62	0.06	55	10	20.62	0.09	59	10	22.96	0.10	62	10	25.30	0.11
227	42	11	14.68	0.06	55	11	20.70	0.08	59	11	23.05	0.09	62	11	25.40	0.10
228	42	12	14.75	0.07	55	12	20.79	0.09	59	12	23.15	0.10	62	12	25.51	0.11
229	42	13	14.81	0.06	55	13	20.88	0.09	59	13	23.25	0.10	62	13	25.62	0.11
230	42	14	14.87	0.06	55	14	20.97	0.09	59	14	23.35	0.10	62	14	25.73	0.11
231	42	15	14.93	0.06	55	15	21.06	0.09	59	15	23.45	0.10	62	15	25.84	0.11
232	45	0	15.00	0.07	58	0	21.20	0.14	61	0	23.55	0.10	63	0	25.95	0.11
233	45	1	15.06	0.06	58	1	21.23	0.03	61	1	23.64	0.09	63	1	26.05	0.10
234	45	2	15.12	0.06	58	2	21.32	0.09	61	2	23.74	0.10	63	2	26.16	0.11
235	45	3	15.18	0.06	58	3	21.41	0.09	61	3	23.84	0.10	63	3	26.27	0.11
236	45	4	15.25	0.07	58	4	21.50	0.09	61	4	23.94	0.10	63	4	26.38	0.11
237	45	5	15.31	0.06	58	5	21.59	0.09	61	5	24.04	0.10	63	5	26.49	0.11
238	45	6	15.37	0.06	58	6	21.67	0.08	61	6	24.13	0.09	63	6	26.59	0.10
239	45	7	15.43	0.06	58	7	21.76	0.09	61	7	24.23	0.10	63	7	26.70	0.11
240	45	8	15.50	0.07	58	8	21.85	0.09	61	8	24.33	0.10	63	8	26.81	0.11
241	45	9	15.56	0.06	58	9	21.94	0.09	61	9	24.43	0.10	63	9	26.92	0.11
242	45	10	15.62	0.06	58	10	22.03	0.09	61	10	24.53	0.10	63	10	27.03	0.11
243	45	11	15.68	0.06	58	11	22.11	0.08	61	11	24.62	0.09	63	11	27.13	0.10
244	45	12	15.75	0.07	58	12	22.20	0.09	61	12	24.72	0.10	63	12	27.24	0.11
245	45	13	15.81	0.06	58	13	22.29	0.09	61	13	24.82	0.10	63	13	27.35	0.11
246	45	14	15.87	0.06	58	14	22.38	0.09	61	14	24.92	0.10	63	14	27.46	0.11
247	45	15	15.93	0.06	58	15	22.47	0.09	61	15	25.02	0.10	63	15	27.57	0.11

As you can see if you look carefully, they stitch together pretty seamlessly. The delta between the last value of a coarse group and the first value of the next group is more or less the same as the deltas within the groups.

Graphing the Groups

Linear Functions?

So they seem to form 4 linear functions, with some minor anomalies. The worst ones are in group 1 where it seems that the set of values where fine = 0 increase at a faster rate than the others, and end up differing by .6 at the bottom of the table. For example, at order=231 it leaps 0.14 from 21.06 to 21.20 and then creeps up 0.03 to 21.23. If 21.20 would have been 21.14 instead, it would have had deltas of 0.08 and 0.09, which fits in with the rest of the pattern.

One can only speculate why these aren't completely linear. If these nonlinearities didn't exist, then we could model the ratios with these 4 simple functions:

Rounding the calculated ratios down to 2 decimal places and comparing them side-by-side with the actual display values gives (error values of 0.0 have been erased to make the errors more visible):

Order	Coarse	Fine	Display Ratio	Calc'ed Ratio	Error	Coarse	Fine	Display Ratio	Calc'ed Ratio	Error	Coarse	Fine	Display Ratio	Calc'ed Ratio	Error	Coarse	Fine	Display Ratio	Calc'ed Ratio	Error
0	0	0	0.50	0.50		1	0	0.71	0.71		2	0	0.78	0.78		3	0	0.87	0.87
1	0	1	0.56	0.56		1	1	0.79	0.79		2	1	0.88	0.87	-0.01	3	1	0.97	0.97
2	0	2	0.62	0.62		1	2	0.88	0.88		2	2	0.98	0.97	-0.01	3	2	1.08	1.08
3	0	3	0.68	0.68		1	3	0.96	0.97	0.01	2	3	1.07	1.07		3	3	1.18	1.19	0.01
4	0	4	0.75	0.75		1	4	1.05	1.06	0.01	2	4	1.17	1.17		3	4	1.29	1.30	0.01
5	0	5	0.81	0.81		1	5	1.14	1.15	0.01	2	5	1.27	1.27		3	5	1.40	1.41	0.01
6	0	6	0.87	0.87		1	6	1.23	1.23		2	6	1.37	1.36	-0.01	3	6	1.51	1.51
7	0	7	0.93	0.93		1	7	1.32	1.32		2	7	1.47	1.46	-0.01	3	7	1.62	1.62
8	4	0	1.00	1.00		5	0	1.41	1.41		6	0	1.57	1.56	-0.01	7	0	1.73	1.73
9	4	1	1.06	1.06		5	1	1.49	1.50	0.01	6	1	1.66	1.66		7	1	1.83	1.84	0.01
10	4	2	1.12	1.12		5	2	1.58	1.59	0.01	6	2	1.76	1.76		7	2	1.94	1.95	0.01
11	4	3	1.18	1.18		5	3	1.67	1.67		6	3	1.86	1.85	-0.01	7	3	2.05	2.05
12	4	4	1.25	1.25		5	4	1.76	1.76		6	4	1.96	1.95	-0.01	7	4	2.16	2.16
13	4	5	1.31	1.31		5	5	1.85	1.85		6	5	2.06	2.05	-0.01	7	5	2.27	2.27
14	4	6	1.37	1.37		5	6	1.93	1.94	0.01	6	6	2.15	2.15		7	6	2.37	2.38	0.01
15	4	7	1.43	1.43		5	7	2.02	2.03	0.01	6	7	2.25	2.25		7	7	2.48	2.49	0.01
16	4	8	1.50	1.50		5	8	2.11	2.11		6	8	2.35	2.35		7	8	2.59	2.59
17	4	9	1.56	1.56		5	9	2.20	2.20		6	9	2.45	2.44	-0.01	7	9	2.70	2.70
18	4	10	1.62	1.62		5	10	2.29	2.29		6	10	2.55	2.54	-0.01	7	10	2.81	2.81
19	4	11	1.68	1.68		5	11	2.37	2.38	0.01	6	11	2.64	2.64		7	11	2.91	2.92	0.01
20	4	12	1.75	1.75		5	12	2.46	2.47	0.01	6	12	2.74	2.74		7	12	3.02	3.03	0.01
21	4	13	1.81	1.81		5	13	2.55	2.56	0.01	6	13	2.84	2.84		7	13	3.13	3.14	0.01
22	4	14	1.87	1.87		5	14	2.64	2.64		6	14	2.94	2.93	-0.01	7	14	3.24	3.24
23	4	15	1.93	1.93		5	15	2.73	2.73		6	15	3.04	3.03	-0.01	7	15	3.35	3.35
24	8	0	2.00	2.00		9	0	2.82	2.82		11	0	3.14	3.13	-0.01	12	0	3.46	3.46
25	8	1	2.06	2.06		9	1	2.90	2.91	0.01	11	1	3.23	3.23		12	1	3.56	3.57	0.01
26	8	2	2.12	2.12		9	2	2.99	3.00	0.01	11	2	3.33	3.33		12	2	3.67	3.68	0.01
27	8	3	2.18	2.18		9	3	3.08	3.08		11	3	3.43	3.42	-0.01	12	3	3.78	3.78
28	8	4	2.25	2.25		9	4	3.17	3.17		11	4	3.53	3.52	-0.01	12	4	3.89	3.89
29	8	5	2.31	2.31		9	5	3.26	3.26		11	5	3.63	3.62	-0.01	12	5	4.00	4.00
30	8	6	2.37	2.37		9	6	3.34	3.35	0.01	11	6	3.72	3.72		12	6	4.10	4.11	0.01
31	8	7	2.43	2.43		9	7	3.43	3.44	0.01	11	7	3.82	3.82		12	7	4.21	4.22	0.01
32	8	8	2.50	2.50		9	8	3.52	3.52		11	8	3.92	3.92		12	8	4.32	4.32
33	8	9	2.56	2.56		9	9	3.61	3.61		11	9	4.02	4.01	-0.01	12	9	4.43	4.43
34	8	10	2.62	2.62		9	10	3.70	3.70		11	10	4.12	4.11	-0.01	12	10	4.54	4.54
35	8	11	2.68	2.68		9	11	3.78	3.79	0.01	11	11	4.21	4.21		12	11	4.64	4.65	0.01
36	8	12	2.75	2.75		9	12	3.87	3.88	0.01	11	12	4.31	4.31		12	12	4.75	4.76	0.01
37	8	13	2.81	2.81		9	13	3.96	3.96		11	13	4.41	4.41		12	13	4.86	4.86
38	8	14	2.87	2.87		9	14	4.05	4.05		11	14	4.51	4.50	-0.01	12	14	4.97	4.97
39	8	15	2.93	2.93		9	15	4.14	4.14		11	15	4.61	4.60	-0.01	12	15	5.08	5.08
40	10	0	3.00	3.00		14	0	4.24	4.23	-0.01	15	0	4.71	4.70	-0.01	17	0	5.19	5.19
41	10	1	3.06	3.06		14	1	4.31	4.32	0.01	15	1	4.80	4.80		17	1	5.29	5.30	0.01
42	10	2	3.12	3.12		14	2	4.40	4.41	0.01	15	2	4.90	4.90		17	2	5.40	5.41	0.01
43	10	3	3.18	3.18		14	3	4.49	4.49		15	3	5.00	5.00		17	3	5.51	5.51
44	10	4	3.25	3.25		14	4	4.58	4.58		15	4	5.10	5.09	-0.01	17	4	5.62	5.62
45	10	5	3.31	3.31		14	5	4.67	4.67		15	5	5.20	5.19	-0.01	17	5	5.73	5.73
46	10	6	3.37	3.37		14	6	4.75	4.76	0.01	15	6	5.29	5.29		17	6	5.83	5.84	0.01
47	10	7	3.43	3.43		14	7	4.84	4.85	0.01	15	7	5.39	5.39		17	7	5.94	5.95	0.01
48	10	8	3.50	3.50		14	8	4.93	4.93		15	8	5.49	5.49		17	8	6.05	6.05
49	10	9	3.56	3.56		14	9	5.02	5.02		15	9	5.59	5.58	-0.01	17	9	6.16	6.16
50	10	10	3.62	3.62		14	10	5.11	5.11		15	10	5.69	5.68	-0.01	17	10	6.27	6.27
51	10	11	3.68	3.68		14	11	5.19	5.20	0.01	15	11	5.78	5.78		17	11	6.37	6.38	0.01
52	10	12	3.75	3.75		14	12	5.28	5.29	0.01	15	12	5.88	5.88		17	12	6.48	6.49	0.01
53	10	13	3.81	3.81		14	13	5.37	5.37		15	13	5.98	5.98		17	13	6.59	6.59
54	10	14	3.87	3.87		14	14	5.46	5.46		15	14	6.08	6.07	-0.01	17	14	6.70	6.70
55	10	15	3.93	3.93		14	15	5.55	5.55		15	15	6.18	6.17	-0.01	17	15	6.81	6.81
56	13	0	4.00	4.00		18	0	5.65	5.64	-0.01	20	0	6.28	6.27	-0.01	21	0	6.92	6.92
57	13	1	4.06	4.06		18	1	5.72	5.73	0.01	20	1	6.37	6.37		21	1	7.02	7.03	0.01
58	13	2	4.12	4.12		18	2	5.81	5.82	0.01	20	2	6.47	6.47		21	2	7.13	7.14	0.01
59	13	3	4.18	4.18		18	3	5.90	5.90		20	3	6.57	6.57		21	3	7.24	7.24
60	13	4	4.25	4.25		18	4	5.99	5.99		20	4	6.67	6.66	-0.01	21	4	7.35	7.35
61	13	5	4.31	4.31		18	5	6.08	6.08		20	5	6.77	6.76	-0.01	21	5	7.46	7.46
62	13	6	4.37	4.37		18	6	6.16	6.17	0.01	20	6	6.86	6.86		21	6	7.56	7.57	0.01
63	13	7	4.43	4.43		18	7	6.25	6.26	0.01	20	7	6.96	6.96		21	7	7.67	7.68	0.01
64	13	8	4.50	4.50		18	8	6.34	6.34		20	8	7.06	7.06		21	8	7.78	7.78
65	13	9	4.56	4.56		18	9	6.43	6.43		20	9	7.16	7.15	-0.01	21	9	7.89	7.89
66	13	10	4.62	4.62		18	10	6.52	6.52		20	10	7.26	7.25	-0.01	21	10	8.00	8.00
67	13	11	4.68	4.68		18	11	6.60	6.61	0.01	20	11	7.35	7.35		21	11	8.10	8.11	0.01
68	13	12	4.75	4.75		18	12	6.69	6.70	0.01	20	12	7.45	7.45		21	12	8.21	8.22	0.01
69	13	13	4.81	4.81		18	13	6.78	6.78		20	13	7.55	7.55		21	13	8.32	8.32
70	13	14	4.87	4.87		18	14	6.87	6.87		20	14	7.65	7.65		21	14	8.43	8.43
71	13	15	4.93	4.93		18	15	6.96	6.96		20	15	7.75	7.74	-0.01	21	15	8.54	8.54
72	16	0	5.00	5.00		23	0	7.07	7.05	-0.02	24	0	7.85	7.84	-0.01	27	0	8.65	8.65
73	16	1	5.06	5.06		23	1	7.13	7.14	0.01	24	1	7.94	7.94		27	1	8.75	8.76	0.01
74	16	2	5.12	5.12		23	2	7.22	7.22		24	2	8.04	8.04		27	2	8.86	8.86
75	16	3	5.18	5.18		23	3	7.31	7.31		24	3	8.14	8.14		27	3	8.97	8.97
76	16	4	5.25	5.25		23	4	7.40	7.40		24	4	8.24	8.23	-0.01	27	4	9.08	9.08
77	16	5	5.31	5.31		23	5	7.49	7.49		24	5	8.34	8.33	-0.01	27	5	9.19	9.19
78	16	6	5.37	5.37		23	6	7.57	7.58	0.01	24	6	8.43	8.43		27	6	9.29	9.30	0.01
79	16	7	5.43	5.43		23	7	7.66	7.67	0.01	24	7	8.53	8.53		27	7	9.40	9.41	0.01
80	16	8	5.50	5.50		23	8	7.75	7.75		24	8	8.63	8.63		27	8	9.51	9.51
81	16	9	5.56	5.56		23	9	7.84	7.84		24	9	8.73	8.72	-0.01	27	9	9.62	9.62
82	16	10	5.62	5.62		23	10	7.93	7.93		24	10	8.83	8.82	-0.01	27	10	9.73	9.73
83	16	11	5.68	5.68		23	11	8.01	8.02	0.01	24	11	8.92	8.92		27	11	9.83	9.84	0.01
84	16	12	5.75	5.75		23	12	8.10	8.11	0.01	24	12	9.02	9.02		27	12	9.94	9.95	0.01
85	16	13	5.81	5.81		23	13	8.19	8.19		24	13	9.12	9.12		27	13	10.05	10.05
86	16	14	5.87	5.87		23	14	8.28	8.28		24	14	9.22	9.22		27	14	10.16	10.16
87	16	15	5.93	5.93		23	15	8.37	8.37		24	15	9.32	9.31	-0.01	27	15	10.27	10.27
88	19	0	6.00	6.00		26	0	8.48	8.46	-0.02	29	0	9.42	9.41	-0.01	32	0	10.38	10.38
89	19	1	6.06	6.06		26	1	8.54	8.55	0.01	29	1	9.51	9.51		32	1	10.48	10.49	0.01
90	19	2	6.12	6.12		26	2	8.63	8.63		29	2	9.61	9.61		32	2	10.59	10.59
91	19	3	6.18	6.18		26	3	8.72	8.72		29	3	9.71	9.71		32	3	10.70	10.70
92	19	4	6.25	6.25		26	4	8.81	8.81		29	4	9.81	9.80	-0.01	32	4	10.81	10.81
93	19	5	6.31	6.31		26	5	8.90	8.90		29	5	9.91	9.90	-0.01	32	5	10.92	10.92
94	19	6	6.37	6.37		26	6	8.98	8.99	0.01	29	6	10.00	10.00		32	6	11.02	11.03	0.01
95	19	7	6.43	6.43		26	7	9.07	9.07		29	7	10.10	10.10		32	7	11.13	11.13
96	19	8	6.50	6.50		26	8	9.16	9.16		29	8	10.20	10.20		32	8	11.24	11.24
97	19	9	6.56	6.56		26	9	9.25	9.25		29	9	10.30	10.30		32	9	11.35	11.35
98	19	10	6.62	6.62		26	10	9.34	9.34		29	10	10.40	10.39	-0.01	32	10	11.46	11.46
99	19	11	6.68	6.68		26	11	9.42	9.43	0.01	29	11	10.49	10.49		32	11	11.56	11.57	0.01
100	19	12	6.75	6.75		26	12	9.51	9.52	0.01	29	12	10.59	10.59		32	12	11.67	11.68	0.01
101	19	13	6.81	6.81		26	13	9.60	9.60		29	13	10.69	10.69		32	13	11.78	11.78
102	19	14	6.87	6.87		26	14	9.69	9.69		29	14	10.79	10.79		32	14	11.89	11.89
103	19	15	6.93	6.93		26	15	9.78	9.78		29	15	10.89	10.88	-0.01	32	15	12.00	12.00
104	22	0	7.00	7.00		30	0	9.89	9.87	-0.02	33	0	10.99	10.98	-0.01	37	0	12.11	12.11
105	22	1	7.06	7.06		30	1	9.95	9.96	0.01	33	1	11.08	11.08		37	1	12.21	12.22	0.01
106	22	2	7.12	7.12		30	2	10.04	10.04		33	2	11.18	11.18		37	2	12.32	12.32
107	22	3	7.18	7.18		30	3	10.13	10.13		33	3	11.28	11.28		37	3	12.43	12.43
108	22	4	7.25	7.25		30	4	10.22	10.22		33	4	11.38	11.37	-0.01	37	4	12.54	12.54
109	22	5	7.31	7.31		30	5	10.31	10.31		33	5	11.48	11.47	-0.01	37	5	12.65	12.65
110	22	6	7.37	7.37		30	6	10.39	10.40	0.01	33	6	11.57	11.57		37	6	12.75	12.76	0.01
111	22	7	7.43	7.43		30	7	10.48	10.48		33	7	11.67	11.67		37	7	12.86	12.86
112	22	8	7.50	7.50		30	8	10.57	10.57		33	8	11.77	11.77		37	8	12.97	12.97
113	22	9	7.56	7.56		30	9	10.66	10.66		33	9	11.87	11.87		37	9	13.08	13.08
114	22	10	7.62	7.62		30	10	10.75	10.75		33	10	11.97	11.96	-0.01	37	10	13.19	13.19
115	22	11	7.68	7.68		30	11	10.83	10.84	0.01	33	11	12.06	12.06		37	11	13.29	13.30	0.01
116	22	12	7.75	7.75		30	12	10.92	10.93	0.01	33	12	12.16	12.16		37	12	13.40	13.41	0.01
117	22	13	7.81	7.81		30	13	11.01	11.01		33	13	12.26	12.26		37	13	13.51	13.51
118	22	14	7.87	7.87		30	14	11.10	11.10		33	14	12.36	12.36		37	14	13.62	13.62
119	22	15	7.93	7.93		30	15	11.19	11.19		33	15	12.46	12.45	-0.01	37	15	13.73	13.73
120	25	0	8.00	8.00		35	0	11.30	11.28	-0.02	38	0	12.56	12.55	-0.01	41	0	13.84	13.84
121	25	1	8.06	8.06		35	1	11.36	11.37	0.01	38	1	12.65	12.65		41	1	13.94	13.95	0.01
122	25	2	8.12	8.12		35	2	11.45	11.45		38	2	12.75	12.75		41	2	14.05	14.05
123	25	3	8.18	8.18		35	3	11.54	11.54		38	3	12.85	12.85		41	3	14.16	14.16
124	25	4	8.25	8.25		35	4	11.63	11.63		38	4	12.95	12.94	-0.01	41	4	14.27	14.27
125	25	5	8.31	8.31		35	5	11.72	11.72		38	5	13.05	13.04	-0.01	41	5	14.38	14.38
126	25	6	8.37	8.37		35	6	11.80	11.81	0.01	38	6	13.14	13.14		41	6	14.48	14.49	0.01
127	25	7	8.43	8.43		35	7	11.89	11.89		38	7	13.24	13.24		41	7	14.59	14.59
128	25	8	8.50	8.50		35	8	11.98	11.98		38	8	13.34	13.34		41	8	14.70	14.70
129	25	9	8.56	8.56		35	9	12.07	12.07		38	9	13.44	13.44		41	9	14.81	14.81
130	25	10	8.62	8.62		35	10	12.16	12.16		38	10	13.54	13.53	-0.01	41	10	14.92	14.92
131	25	11	8.68	8.68		35	11	12.24	12.25	0.01	38	11	13.63	13.63		41	11	15.02	15.03	0.01
132	25	12	8.75	8.75		35	12	12.33	12.33		38	12	13.73	13.73		41	12	15.13	15.13
133	25	13	8.81	8.81		35	13	12.42	12.42		38	13	13.83	13.83		41	13	15.24	15.24
134	25	14	8.87	8.87		35	14	12.51	12.51		38	14	13.93	13.93		41	14	15.35	15.35
135	25	15	8.93	8.93		35	15	12.60	12.60		38	15	14.03	14.02	-0.01	41	15	15.46	15.46
136	28	0	9.00	9.00		39	0	12.72	12.69	-0.03	44	0	14.13	14.12	-0.01	47	0	15.57	15.57
137	28	1	9.06	9.06		39	1	12.77	12.78	0.01	44	1	14.22	14.22		47	1	15.67	15.68	0.01
138	28	2	9.12	9.12		39	2	12.86	12.86		44	2	14.32	14.32		47	2	15.78	15.78
139	28	3	9.18	9.18		39	3	12.95	12.95		44	3	14.42	14.42		47	3	15.89	15.89
140	28	4	9.25	9.25		39	4	13.04	13.04		44	4	14.52	14.52		47	4	16.00	16.00
141	28	5	9.31	9.31		39	5	13.13	13.13		44	5	14.62	14.61	-0.01	47	5	16.11	16.11
142	28	6	9.37	9.37		39	6	13.21	13.22	0.01	44	6	14.71	14.71		47	6	16.21	16.22	0.01
143	28	7	9.43	9.43		39	7	13.30	13.30		44	7	14.81	14.81		47	7	16.32	16.32
144	28	8	9.50	9.50		39	8	13.39	13.39		44	8	14.91	14.91		47	8	16.43	16.43
145	28	9	9.56	9.56		39	9	13.48	13.48		44	9	15.01	15.01		47	9	16.54	16.54
146	28	10	9.62	9.62		39	10	13.57	13.57		44	10	15.11	15.10	-0.01	47	10	16.65	16.65
147	28	11	9.68	9.68		39	11	13.65	13.66	0.01	44	11	15.20	15.20		47	11	16.75	16.76	0.01
148	28	12	9.75	9.75		39	12	13.74	13.74		44	12	15.30	15.30		47	12	16.86	16.86
149	28	13	9.81	9.81		39	13	13.83	13.83		44	13	15.40	15.40		47	13	16.97	16.97
150	28	14	9.87	9.87		39	14	13.92	13.92		44	14	15.50	15.50		47	14	17.08	17.08
151	28	15	9.93	9.93		39	15	14.01	14.01		44	15	15.60	15.59	-0.01	47	15	17.19	17.19
152	31	0	10.00	10.00		43	0	14.10	14.10		48	0	15.70	15.69	-0.01	51	0	17.30	17.30
153	31	1	10.06	10.06		43	1	14.18	14.19	0.01	48	1	15.79	15.79		51	1	17.40	17.41	0.01
154	31	2	10.12	10.12		43	2	14.27	14.27		48	2	15.89	15.89		51	2	17.51	17.51
155	31	3	10.18	10.18		43	3	14.36	14.36		48	3	15.99	15.99		51	3	17.62	17.62
156	31	4	10.25	10.25		43	4	14.45	14.45		48	4	16.09	16.09		51	4	17.73	17.73
157	31	5	10.31	10.31		43	5	14.54	14.54		48	5	16.19	16.18	-0.01	51	5	17.84	17.84
158	31	6	10.37	10.37		43	6	14.62	14.63	0.01	48	6	16.28	16.28		51	6	17.94	17.95	0.01
159	31	7	10.43	10.43		43	7	14.71	14.71		48	7	16.38	16.38		51	7	18.05	18.05
160	31	8	10.50	10.50		43	8	14.80	14.80		48	8	16.48	16.48		51	8	18.16	18.16
161	31	9	10.56	10.56		43	9	14.89	14.89		48	9	16.58	16.58		51	9	18.27	18.27
162	31	10	10.62	10.62		43	10	14.98	14.98		48	10	16.68	16.67	-0.01	51	10	18.38	18.38
163	31	11	10.68	10.68		43	11	15.06	15.07	0.01	48	11	16.77	16.77		51	11	18.48	18.49	0.01
164	31	12	10.75	10.75		43	12	15.15	15.15		48	12	16.87	16.87		51	12	18.59	18.59
165	31	13	10.81	10.81		43	13	15.24	15.24		48	13	16.97	16.97		51	13	18.70	18.70
166	31	14	10.87	10.87		43	14	15.33	15.33		48	14	17.07	17.07		51	14	18.81	18.81
167	31	15	10.93	10.93		43	15	15.42	15.42		48	15	17.17	17.17		51	15	18.92	18.92
168	34	0	11.00	11.00		46	0	15.55	15.51	-0.04	50	0	17.27	17.26	-0.01	54	0	19.03	19.03
169	34	1	11.06	11.06		46	1	15.59	15.59		50	1	17.36	17.36		54	1	19.13	19.13
170	34	2	11.12	11.12		46	2	15.68	15.68		50	2	17.46	17.46		54	2	19.24	19.24
171	34	3	11.18	11.18		46	3	15.77	15.77		50	3	17.56	17.56		54	3	19.35	19.35
172	34	4	11.25	11.25		46	4	15.86	15.86		50	4	17.66	17.66		54	4	19.46	19.46
173	34	5	11.31	11.31		46	5	15.95	15.95		50	5	17.76	17.75	-0.01	54	5	19.57	19.57
174	34	6	11.37	11.37		46	6	16.03	16.04	0.01	50	6	17.85	17.85		54	6	19.67	19.68	0.01
175	34	7	11.43	11.43		46	7	16.12	16.12		50	7	17.95	17.95		54	7	19.78	19.78
176	34	8	11.50	11.50		46	8	16.21	16.21		50	8	18.05	18.05		54	8	19.89	19.89
177	34	9	11.56	11.56		46	9	16.30	16.30		50	9	18.15	18.15		54	9	20.00	20.00
178	34	10	11.62	11.62		46	10	16.39	16.39		50	10	18.25	18.24	-0.01	54	10	20.11	20.11
179	34	11	11.68	11.68		46	11	16.47	16.48	0.01	50	11	18.34	18.34		54	11	20.21	20.22	0.01
180	34	12	11.75	11.75		46	12	16.56	16.56		50	12	18.44	18.44		54	12	20.32	20.32
181	34	13	11.81	11.81		46	13	16.65	16.65		50	13	18.54	18.54		54	13	20.43	20.43
182	34	14	11.87	11.87		46	14	16.74	16.74		50	14	18.64	18.64		54	14	20.54	20.54
183	34	15	11.93	11.93		46	15	16.83	16.83		50	15	18.74	18.74		54	15	20.65	20.65
184	36	0	12.00	12.00		49	0	16.96	16.92	-0.04	53	0	18.84	18.83	-0.01	57	0	20.76	20.76
185	36	1	12.06	12.06		49	1	17.00	17.00		53	1	18.93	18.93		57	1	20.86	20.86
186	36	2	12.12	12.12		49	2	17.09	17.09		53	2	19.03	19.03		57	2	20.97	20.97
187	36	3	12.18	12.18		49	3	17.18	17.18		53	3	19.13	19.13		57	3	21.08	21.08
188	36	4	12.25	12.25		49	4	17.27	17.27		53	4	19.23	19.23		57	4	21.19	21.19
189	36	5	12.31	12.31		49	5	17.36	17.36		53	5	19.33	19.32	-0.01	57	5	21.30	21.30
190	36	6	12.37	12.37		49	6	17.44	17.44		53	6	19.42	19.42		57	6	21.40	21.40
191	36	7	12.43	12.43		49	7	17.53	17.53		53	7	19.52	19.52		57	7	21.51	21.51
192	36	8	12.50	12.50		49	8	17.62	17.62		53	8	19.62	19.62		57	8	21.62	21.62
193	36	9	12.56	12.56		49	9	17.71	17.71		53	9	19.72	19.72		57	9	21.73	21.73
194	36	10	12.62	12.62		49	10	17.80	17.80		53	10	19.82	19.82		57	10	21.84	21.84
195	36	11	12.68	12.68		49	11	17.88	17.89	0.01	53	11	19.91	19.91		57	11	21.94	21.95	0.01
196	36	12	12.75	12.75		49	12	17.97	17.97		53	12	20.01	20.01		57	12	22.05	22.05
197	36	13	12.81	12.81		49	13	18.06	18.06		53	13	20.11	20.11		57	13	22.16	22.16
198	36	14	12.87	12.87		49	14	18.15	18.15		53	14	20.21	20.21		57	14	22.27	22.27
199	36	15	12.93	12.93		49	15	18.24	18.24		53	15	20.31	20.31		57	15	22.38	22.38
200	40	0	13.00	13.00		52	0	18.37	18.33	-0.04	56	0	20.41	20.40	-0.01	60	0	22.49	22.49
201	40	1	13.06	13.06		52	1	18.41	18.41		56	1	20.50	20.50		60	1	22.59	22.59
202	40	2	13.12	13.12		52	2	18.50	18.50		56	2	20.60	20.60		60	2	22.70	22.70
203	40	3	13.18	13.18		52	3	18.59	18.59		56	3	20.70	20.70		60	3	22.81	22.81
204	40	4	13.25	13.25		52	4	18.68	18.68		56	4	20.80	20.80		60	4	22.92	22.92
205	40	5	13.31	13.31		52	5	18.77	18.77		56	5	20.90	20.89	-0.01	60	5	23.03	23.03
206	40	6	13.37	13.37		52	6	18.85	18.85		56	6	20.99	20.99		60	6	23.13	23.13
207	40	7	13.43	13.43		52	7	18.94	18.94		56	7	21.09	21.09		60	7	23.24	23.24
208	40	8	13.50	13.50		52	8	19.03	19.03		56	8	21.19	21.19		60	8	23.35	23.35
209	40	9	13.56	13.56		52	9	19.12	19.12		56	9	21.29	21.29		60	9	23.46	23.46
210	40	10	13.62	13.62		52	10	19.21	19.21		56	10	21.39	21.39		60	10	23.57	23.57
211	40	11	13.68	13.68		52	11	19.29	19.30	0.01	56	11	21.48	21.48		60	11	23.67	23.68	0.01
212	40	12	13.75	13.75		52	12	19.38	19.38		56	12	21.58	21.58		60	12	23.78	23.78
213	40	13	13.81	13.81		52	13	19.47	19.47		56	13	21.68	21.68		60	13	23.89	23.89
214	40	14	13.87	13.87		52	14	19.56	19.56		56	14	21.78	21.78		60	14	24.00	24.00
215	40	15	13.93	13.93		52	15	19.65	19.65		56	15	21.88	21.88		60	15	24.11	24.11
216	42	0	14.00	14.00		55	0	19.78	19.74	-0.04	59	0	21.98	21.97	-0.01	62	0	24.22	24.22
217	42	1	14.06	14.06		55	1	19.82	19.82		59	1	22.07	22.07		62	1	24.32	24.32
218	42	2	14.12	14.12		55	2	19.91	19.91		59	2	22.17	22.17		62	2	24.43	24.43
219	42	3	14.18	14.18		55	3	20.00	20.00		59	3	22.27	22.27		62	3	24.54	24.54
220	42	4	14.25	14.25		55	4	20.09	20.09		59	4	22.37	22.37		62	4	24.65	24.65
221	42	5	14.31	14.31		55	5	20.18	20.18		59	5	22.47	22.47		62	5	24.76	24.76
222	42	6	14.37	14.37		55	6	20.26	20.26		59	6	22.56	22.56		62	6	24.86	24.86
223	42	7	14.43	14.43		55	7	20.35	20.35		59	7	22.66	22.66		62	7	24.97	24.97
224	42	8	14.50	14.50		55	8	20.44	20.44		59	8	22.76	22.76		62	8	25.08	25.08
225	42	9	14.56	14.56		55	9	20.53	20.53		59	9	22.86	22.86		62	9	25.19	25.19
226	42	10	14.62	14.62		55	10	20.62	20.62		59	10	22.96	22.96		62	10	25.30	25.30
227	42	11	14.68	14.68		55	11	20.70	20.70		59	11	23.05	23.05		62	11	25.40	25.40
228	42	12	14.75	14.75		55	12	20.79	20.79		59	12	23.15	23.15		62	12	25.51	25.51
229	42	13	14.81	14.81		55	13	20.88	20.88		59	13	23.25	23.25		62	13	25.62	25.62
230	42	14	14.87	14.87		55	14	20.97	20.97		59	14	23.35	23.35		62	14	25.73	25.73
231	42	15	14.93	14.93		55	15	21.06	21.06		59	15	23.45	23.45		62	15	25.84	25.84
232	45	0	15.00	15.00		58	0	21.20	21.15	-0.05	61	0	23.55	23.54	-0.01	63	0	25.95	25.95
233	45	1	15.06	15.06		58	1	21.23	21.23		61	1	23.64	23.64		63	1	26.05	26.05
234	45	2	15.12	15.12		58	2	21.32	21.32		61	2	23.74	23.74		63	2	26.16	26.16
235	45	3	15.18	15.18		58	3	21.41	21.41		61	3	23.84	23.84		63	3	26.27	26.27
236	45	4	15.25	15.25		58	4	21.50	21.50		61	4	23.94	23.94		63	4	26.38	26.38
237	45	5	15.31	15.31		58	5	21.59	21.59		61	5	24.04	24.04		63	5	26.49	26.49
238	45	6	15.37	15.37		58	6	21.67	21.67		61	6	24.13	24.13		63	6	26.59	26.59
239	45	7	15.43	15.43		58	7	21.76	21.76		61	7	24.23	24.23		63	7	26.70	26.70
240	45	8	15.50	15.50		58	8	21.85	21.85		61	8	24.33	24.33		63	8	26.81	26.81
241	45	9	15.56	15.56		58	9	21.94	21.94		61	9	24.43	24.43		63	9	26.92	26.92
242	45	10	15.62	15.62		58	10	22.03	22.03		61	10	24.53	24.53		63	10	27.03	27.03
243	45	11	15.68	15.68		58	11	22.11	22.11		61	11	24.62	24.62		63	11	27.13	27.13
244	45	12	15.75	15.75		58	12	22.20	22.20		61	12	24.72	24.72		63	12	27.24	27.24
245	45	13	15.81	15.81		58	13	22.29	22.29		61	13	24.82	24.82		63	13	27.35	27.35
246	45	14	15.87	15.87		58	14	22.38	22.38		61	14	24.92	24.92		63	14	27.46	27.46
247	45	15	15.93	15.93		58	15	22.47	22.47		61	15	25.02	25.02		63	15	27.57	27.57
	Cumulative Error:				0.00	Cumulative Error:				0.88	Cumulative Error:				0.70	Cumulative Error:				0.54

The Inexplicable Errors

So that is a lot of errors, and there's no tweaking of these linear formulas that can be done to fix them (the bounds are correct, and only one line exists between two points).

My first thought is that it could be due to floating point limitations that made it not able to represent the number exactly. Some of the errors seem really large for a representation error, but it could easily be the case that representation errors happened early in some long calculation process and then got magnified by multiplication or other transformation.

But if you look at the error column of the table, you can see there's behavior that can't really be explained by floating point error. There's periodicity in the errors and there's also a gradual diminishing in groups 1, 2 and 3, with the strange exception of the fine=0 values in group 1, whose errors are actually increasing, and these exceptions themselves have a bizarre exception at order=152, where it suddenly becomes 100% accurate, and then resumes it downward course where it left off. It's also interesting that groups 1 and 3 would be exactly the same if it weren't for the strange fine=0 behavior of group 1.

There are solutions to a certain differential equation that have been shown to predict the amplitudes of sidebands in FM synthesis called Bessel functions. I'm not an expert in FM synthesis, but I've read John Chowning's book, FM Theory and Applications, and he doesn't mention them in the context of measuring frequencies, but it's not a completely rigorous text on FM theory, so maybe they do have something to do with it. After all, the periodicity and the diminishing with increasing (coarse, fine) values are kind of suggestive of the shape of Bessel function graphs.

More Lookup Tables

Even supposing that we've corrected the formulas and eliminated the need for the lookup table of frequency ratios, we've created the need for two other lookup tables:

We really only need to look up the orders for the first value in each coarse group since we can just add the fine value to it to get the exact order. So that reduces the table to 1/16 of the size:

Coarse	Group	Order (Fine=0)
0	0	0
1	1	0
2	2	0
3	3	0
4	0	8
5	1	8
6	2	8
7	3	8
8	0	24
9	1	24
10	0	40
11	2	24
12	3	24
13	0	56
14	1	40
15	2	40
16	0	72
17	3	40
18	1	56
19	0	88
20	2	56
21	3	56
22	0	104
23	1	72
24	2	72
25	0	120
26	1	88
27	3	72
28	0	136
29	2	88
30	1	104
31	0	152
32	3	88
33	2	104
34	0	168
35	1	120
36	0	184
37	3	104
38	2	120
39	1	136
40	0	200
41	3	120
42	0	216
43	1	152
44	2	136
45	0	232
46	1	168
47	3	136
48	2	152
49	1	184
50	2	168
51	3	152
52	1	200
53	2	184
54	3	168
55	1	216
56	2	200
57	3	184
58	1	232
59	2	216
60	3	200
61	2	232
62	3	216
63	3	232

It might be more insightful seeing it with the coarse_to_group table inverted and is actually more compact because we can now derive both the group and the order from the position of the coarse value in the list.

We could attempt to go further and find formulas for the 2 outstanding lookup tables:

So, even if the 4 linear formulas are repaired (e.g. with a polynomial of a sufficient number, potentially 248, of terms), there's still the problem of figuring out which formula is applicable for any given coarse value so the proper 0-intercept and increment can be applied, and, since the coarse values along one line don't follow a linear pattern (as can be seen in the group_to_coarse table above), the problem of figuring out the index of the point on the line where the (coarse, fine) pair falls so that it can used as a multipier for the increment.

If anyone wants to pursue this any further, they are welcome to inform me of their results.

Coarse	Group	Order (Fine=0)
0	0	0
1	1	0
2	2	0
3	3	0
4	0	8
5	1	8
6	2	8
7	3	8
8	0	24
9	1	24
10	0	40
11	2	24
12	3	24
13	0	56
14	1	40
15	2	40
16	0	72
17	3	40
18	1	56
19	0	88
20	2	56
21	3	56
22	0	104
23	1	72
24	2	72
25	0	120
26	1	88
27	3	72
28	0	136
29	2	88
30	1	104
31	0	152
32	3	88
33	2	104
34	0	168
35	1	120
36	0	184
37	3	104
38	2	120
39	1	136
40	0	200
41	3	120
42	0	216
43	1	152
44	2	136
45	0	232
46	1	168
47	3	136
48	2	152
49	1	184
50	2	168
51	3	152
52	1	200
53	2	184
54	3	168
55	1	216
56	2	200
57	3	184
58	1	232
59	2	216
60	3	200
61	2	232
62	3	216
63	3	232

Coarse	Group	Order (Fine=0)
0	0	0
1	1	0
2	2	0
3	3	0
4	0	8
5	1	8
6	2	8
7	3	8
8	0	24
9	1	24
10	0	40
11	2	24
12	3	24
13	0	56
14	1	40
15	2	40
16	0	72
17	3	40
18	1	56
19	0	88
20	2	56
21	3	56
22	0	104
23	1	72
24	2	72
25	0	120
26	1	88
27	3	72
28	0	136
29	2	88
30	1	104
31	0	152
32	3	88
33	2	104
34	0	168
35	1	120
36	0	184
37	3	104
38	2	120
39	1	136
40	0	200
41	3	120
42	0	216
43	1	152
44	2	136
45	0	232
46	1	168
47	3	136
48	2	152
49	1	184
50	2	168
51	3	152
52	1	200
53	2	184
54	3	168
55	1	216
56	2	200
57	3	184
58	1	232
59	2	216
60	3	200
61	2	232
62	3	216
63	3	232