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:

ratio = coarse_fine_to_ratio(coarse, fine)   # where (0 <= coarse <= 63) and (0 <= fine <= 15)

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):

CoarseFineRatioDelta CoarseFineRatioDelta CoarseFineRatioDelta CoarseFineRatioDelta
0 0 0.50 16 0 5.00 32 010.38 48 015.70
0 1 0.560.06 16 1 5.060.06 32 110.480.10 48 115.790.09
0 2 0.620.06 16 2 5.120.06 32 210.590.11 48 215.890.10
0 3 0.680.06 16 3 5.180.06 32 310.700.11 48 315.990.10
0 4 0.750.07 16 4 5.250.07 32 410.810.11 48 416.090.10
0 5 0.810.06 16 5 5.310.06 32 510.920.11 48 516.190.10
0 6 0.870.06 16 6 5.370.06 32 611.020.10 48 616.280.09
0 7 0.930.06 16 7 5.430.06 32 711.130.11 48 716.380.10
0 8 16 8 5.500.07 32 811.240.11 48 816.480.10
0 9 16 9 5.560.06 32 911.350.11 48 916.580.10
010 1610 5.620.06 321011.460.11 481016.680.10
011 1611 5.680.06 321111.560.10 481116.770.09
012 1612 5.750.07 321211.670.11 481216.870.10
013 1613 5.810.06 321311.780.11 481316.970.10
014 1614 5.870.06 321411.890.11 481417.070.10
015 1615 5.930.06 321512.000.11 481517.170.10
1 0 0.71 17 0 5.19 33 010.99 49 016.96
1 1 0.790.08 17 1 5.290.10 33 111.080.09 49 117.000.04
1 2 0.880.09 17 2 5.400.11 33 211.180.10 49 217.090.09
1 3 0.960.08 17 3 5.510.11 33 311.280.10 49 317.180.09
1 4 1.050.09 17 4 5.620.11 33 411.380.10 49 417.270.09
1 5 1.140.09 17 5 5.730.11 33 511.480.10 49 517.360.09
1 6 1.230.09 17 6 5.830.10 33 611.570.09 49 617.440.08
1 7 1.320.09 17 7 5.940.11 33 711.670.10 49 717.530.09
1 8 17 8 6.050.11 33 811.770.10 49 817.620.09
1 9 17 9 6.160.11 33 911.870.10 49 917.710.09
110 1710 6.270.11 331011.970.10 491017.800.09
111 1711 6.370.10 331112.060.09 491117.880.08
112 1712 6.480.11 331212.160.10 491217.970.09
113 1713 6.590.11 331312.260.10 491318.060.09
114 1714 6.700.11 331412.360.10 491418.150.09
115 1715 6.810.11 331512.460.10 491518.240.09
2 0 0.78 18 0 5.65 34 011.00 50 017.27
2 1 0.880.10 18 1 5.720.07 34 111.060.06 50 117.360.09
2 2 0.980.10 18 2 5.810.09 34 211.120.06 50 217.460.10
2 3 1.070.09 18 3 5.900.09 34 311.180.06 50 317.560.10
2 4 1.170.10 18 4 5.990.09 34 411.250.07 50 417.660.10
2 5 1.270.10 18 5 6.080.09 34 511.310.06 50 517.760.10
2 6 1.370.10 18 6 6.160.08 34 611.370.06 50 617.850.09
2 7 1.470.10 18 7 6.250.09 34 711.430.06 50 717.950.10
2 8 18 8 6.340.09 34 811.500.07 50 818.050.10
2 9 18 9 6.430.09 34 911.560.06 50 918.150.10
210 1810 6.520.09 341011.620.06 501018.250.10
211 1811 6.600.08 341111.680.06 501118.340.09
212 1812 6.690.09 341211.750.07 501218.440.10
213 1813 6.780.09 341311.810.06 501318.540.10
214 1814 6.870.09 341411.870.06 501418.640.10
215 1815 6.960.09 341511.930.06 501518.740.10
3 0 0.87 19 0 6.00 35 011.30 51 017.30
3 1 0.970.10 19 1 6.060.06 35 111.360.06 51 117.400.10
3 2 1.080.11 19 2 6.120.06 35 211.450.09 51 217.510.11
3 3 1.180.10 19 3 6.180.06 35 311.540.09 51 317.620.11
3 4 1.290.11 19 4 6.250.07 35 411.630.09 51 417.730.11
3 5 1.400.11 19 5 6.310.06 35 511.720.09 51 517.840.11
3 6 1.510.11 19 6 6.370.06 35 611.800.08 51 617.940.10
3 7 1.620.11 19 7 6.430.06 35 711.890.09 51 718.050.11
3 8 19 8 6.500.07 35 811.980.09 51 818.160.11
3 9 19 9 6.560.06 35 912.070.09 51 918.270.11
310 1910 6.620.06 351012.160.09 511018.380.11
311 1911 6.680.06 351112.240.08 511118.480.10
312 1912 6.750.07 351212.330.09 511218.590.11
313 1913 6.810.06 351312.420.09 511318.700.11
314 1914 6.870.06 351412.510.09 511418.810.11
315 1915 6.930.06 351512.600.09 511518.920.11
4 0 1.00 20 0 6.28 36 012.00 52 018.37
4 1 1.060.06 20 1 6.370.09 36 112.060.06 52 118.410.04
4 2 1.120.06 20 2 6.470.10 36 212.120.06 52 218.500.09
4 3 1.180.06 20 3 6.570.10 36 312.180.06 52 318.590.09
4 4 1.250.07 20 4 6.670.10 36 412.250.07 52 418.680.09
4 5 1.310.06 20 5 6.770.10 36 512.310.06 52 518.770.09
4 6 1.370.06 20 6 6.860.09 36 612.370.06 52 618.850.08
4 7 1.430.06 20 7 6.960.10 36 712.430.06 52 718.940.09
4 8 1.500.07 20 8 7.060.10 36 812.500.07 52 819.030.09
4 9 1.560.06 20 9 7.160.10 36 912.560.06 52 919.120.09
410 1.620.06 2010 7.260.10 361012.620.06 521019.210.09
411 1.680.06 2011 7.350.09 361112.680.06 521119.290.08
412 1.750.07 2012 7.450.10 361212.750.07 521219.380.09
413 1.810.06 2013 7.550.10 361312.810.06 521319.470.09
414 1.870.06 2014 7.650.10 361412.870.06 521419.560.09
415 1.930.06 2015 7.750.10 361512.930.06 521519.650.09
5 0 1.41 21 0 6.92 37 012.11 53 018.84
5 1 1.490.08 21 1 7.020.10 37 112.210.10 53 118.930.09
5 2 1.580.09 21 2 7.130.11 37 212.320.11 53 219.030.10
5 3 1.670.09 21 3 7.240.11 37 312.430.11 53 319.130.10
5 4 1.760.09 21 4 7.350.11 37 412.540.11 53 419.230.10
5 5 1.850.09 21 5 7.460.11 37 512.650.11 53 519.330.10
5 6 1.930.08 21 6 7.560.10 37 612.750.10 53 619.420.09
5 7 2.020.09 21 7 7.670.11 37 712.860.11 53 719.520.10
5 8 2.110.09 21 8 7.780.11 37 812.970.11 53 819.620.10
5 9 2.200.09 21 9 7.890.11 37 913.080.11 53 919.720.10
510 2.290.09 2110 8.000.11 371013.190.11 531019.820.10
511 2.370.08 2111 8.100.10 371113.290.10 531119.910.09
512 2.460.09 2112 8.210.11 371213.400.11 531220.010.10
513 2.550.09 2113 8.320.11 371313.510.11 531320.110.10
514 2.640.09 2114 8.430.11 371413.620.11 531420.210.10
515 2.730.09 2115 8.540.11 371513.730.11 531520.310.10
6 0 1.57 22 0 7.00 38 012.56 54 019.03
6 1 1.660.09 22 1 7.060.06 38 112.650.09 54 119.130.10
6 2 1.760.10 22 2 7.120.06 38 212.750.10 54 219.240.11
6 3 1.860.10 22 3 7.180.06 38 312.850.10 54 319.350.11
6 4 1.960.10 22 4 7.250.07 38 412.950.10 54 419.460.11
6 5 2.060.10 22 5 7.310.06 38 513.050.10 54 519.570.11
6 6 2.150.09 22 6 7.370.06 38 613.140.09 54 619.670.10
6 7 2.250.10 22 7 7.430.06 38 713.240.10 54 719.780.11
6 8 2.350.10 22 8 7.500.07 38 813.340.10 54 819.890.11
6 9 2.450.10 22 9 7.560.06 38 913.440.10 54 920.000.11
610 2.550.10 2210 7.620.06 381013.540.10 541020.110.11
611 2.640.09 2211 7.680.06 381113.630.09 541120.210.10
612 2.740.10 2212 7.750.07 381213.730.10 541220.320.11
613 2.840.10 2213 7.810.06 381313.830.10 541320.430.11
614 2.940.10 2214 7.870.06 381413.930.10 541420.540.11
615 3.040.10 2215 7.930.06 381514.030.10 541520.650.11
7 0 1.73 23 0 7.07 39 012.72 55 019.78
7 1 1.830.10 23 1 7.130.06 39 112.770.05 55 119.820.04
7 2 1.940.11 23 2 7.220.09 39 212.860.09 55 219.910.09
7 3 2.050.11 23 3 7.310.09 39 312.950.09 55 320.000.09
7 4 2.160.11 23 4 7.400.09 39 413.040.09 55 420.090.09
7 5 2.270.11 23 5 7.490.09 39 513.130.09 55 520.180.09
7 6 2.370.10 23 6 7.570.08 39 613.210.08 55 620.260.08
7 7 2.480.11 23 7 7.660.09 39 713.300.09 55 720.350.09
7 8 2.590.11 23 8 7.750.09 39 813.390.09 55 820.440.09
7 9 2.700.11 23 9 7.840.09 39 913.480.09 55 920.530.09
710 2.810.11 2310 7.930.09 391013.570.09 551020.620.09
711 2.910.10 2311 8.010.08 391113.650.08 551120.700.08
712 3.020.11 2312 8.100.09 391213.740.09 551220.790.09
713 3.130.11 2313 8.190.09 391313.830.09 551320.880.09
714 3.240.11 2314 8.280.09 391413.920.09 551420.970.09
715 3.350.11 2315 8.370.09 391514.010.09 551521.060.09
8 0 2.00 24 0 7.85 40 013.00 56 020.41
8 1 2.060.06 24 1 7.940.09 40 113.060.06 56 120.500.09
8 2 2.120.06 24 2 8.040.10 40 213.120.06 56 220.600.10
8 3 2.180.06 24 3 8.140.10 40 313.180.06 56 320.700.10
8 4 2.250.07 24 4 8.240.10 40 413.250.07 56 420.800.10
8 5 2.310.06 24 5 8.340.10 40 513.310.06 56 520.900.10
8 6 2.370.06 24 6 8.430.09 40 613.370.06 56 620.990.09
8 7 2.430.06 24 7 8.530.10 40 713.430.06 56 721.090.10
8 8 2.500.07 24 8 8.630.10 40 813.500.07 56 821.190.10
8 9 2.560.06 24 9 8.730.10 40 913.560.06 56 921.290.10
810 2.620.06 2410 8.830.10 401013.620.06 561021.390.10
811 2.680.06 2411 8.920.09 401113.680.06 561121.480.09
812 2.750.07 2412 9.020.10 401213.750.07 561221.580.10
813 2.810.06 2413 9.120.10 401313.810.06 561321.680.10
814 2.870.06 2414 9.220.10 401413.870.06 561421.780.10
815 2.930.06 2415 9.320.10 401513.930.06 561521.880.10
9 0 2.82 25 0 8.00 41 013.84 57 020.76
9 1 2.900.08 25 1 8.060.06 41 113.940.10 57 120.860.10
9 2 2.990.09 25 2 8.120.06 41 214.050.11 57 220.970.11
9 3 3.080.09 25 3 8.180.06 41 314.160.11 57 321.080.11
9 4 3.170.09 25 4 8.250.07 41 414.270.11 57 421.190.11
9 5 3.260.09 25 5 8.310.06 41 514.380.11 57 521.300.11
9 6 3.340.08 25 6 8.370.06 41 614.480.10 57 621.400.10
9 7 3.430.09 25 7 8.430.06 41 714.590.11 57 721.510.11
9 8 3.520.09 25 8 8.500.07 41 814.700.11 57 821.620.11
9 9 3.610.09 25 9 8.560.06 41 914.810.11 57 921.730.11
910 3.700.09 2510 8.620.06 411014.920.11 571021.840.11
911 3.780.08 2511 8.680.06 411115.020.10 571121.940.10
912 3.870.09 2512 8.750.07 411215.130.11 571222.050.11
913 3.960.09 2513 8.810.06 411315.240.11 571322.160.11
914 4.050.09 2514 8.870.06 411415.350.11 571422.270.11
915 4.140.09 2515 8.930.06 411515.460.11 571522.380.11
10 0 3.00 26 0 8.48 42 014.00 58 021.20
10 1 3.060.06 26 1 8.540.06 42 114.060.06 58 121.230.03
10 2 3.120.06 26 2 8.630.09 42 214.120.06 58 221.320.09
10 3 3.180.06 26 3 8.720.09 42 314.180.06 58 321.410.09
10 4 3.250.07 26 4 8.810.09 42 414.250.07 58 421.500.09
10 5 3.310.06 26 5 8.900.09 42 514.310.06 58 521.590.09
10 6 3.370.06 26 6 8.980.08 42 614.370.06 58 621.670.08
10 7 3.430.06 26 7 9.070.09 42 714.430.06 58 721.760.09
10 8 3.500.07 26 8 9.160.09 42 814.500.07 58 821.850.09
10 9 3.560.06 26 9 9.250.09 42 914.560.06 58 921.940.09
1010 3.620.06 2610 9.340.09 421014.620.06 581022.030.09
1011 3.680.06 2611 9.420.08 421114.680.06 581122.110.08
1012 3.750.07 2612 9.510.09 421214.750.07 581222.200.09
1013 3.810.06 2613 9.600.09 421314.810.06 581322.290.09
1014 3.870.06 2614 9.690.09 421414.870.06 581422.380.09
1015 3.930.06 2615 9.780.09 421514.930.06 581522.470.09
11 0 3.14 27 0 8.65 43 014.10 59 021.98
11 1 3.230.09 27 1 8.750.10 43 114.180.08 59 122.070.09
11 2 3.330.10 27 2 8.860.11 43 214.270.09 59 222.170.10
11 3 3.430.10 27 3 8.970.11 43 314.360.09 59 322.270.10
11 4 3.530.10 27 4 9.080.11 43 414.450.09 59 422.370.10
11 5 3.630.10 27 5 9.190.11 43 514.540.09 59 522.470.10
11 6 3.720.09 27 6 9.290.10 43 614.620.08 59 622.560.09
11 7 3.820.10 27 7 9.400.11 43 714.710.09 59 722.660.10
11 8 3.920.10 27 8 9.510.11 43 814.800.09 59 822.760.10
11 9 4.020.10 27 9 9.620.11 43 914.890.09 59 922.860.10
1110 4.120.10 2710 9.730.11 431014.980.09 591022.960.10
1111 4.210.09 2711 9.830.10 431115.060.08 591123.050.09
1112 4.310.10 2712 9.940.11 431215.150.09 591223.150.10
1113 4.410.10 271310.050.11 431315.240.09 591323.250.10
1114 4.510.10 271410.160.11 431415.330.09 591423.350.10
1115 4.610.10 271510.270.11 431515.420.09 591523.450.10
12 0 3.46 28 0 9.00 44 014.13 60 022.49
12 1 3.560.10 28 1 9.060.06 44 114.220.09 60 122.590.10
12 2 3.670.11 28 2 9.120.06 44 214.320.10 60 222.700.11
12 3 3.780.11 28 3 9.180.06 44 314.420.10 60 322.810.11
12 4 3.890.11 28 4 9.250.07 44 414.520.10 60 422.920.11
12 5 4.000.11 28 5 9.310.06 44 514.620.10 60 523.030.11
12 6 4.100.10 28 6 9.370.06 44 614.710.09 60 623.130.10
12 7 4.210.11 28 7 9.430.06 44 714.810.10 60 723.240.11
12 8 4.320.11 28 8 9.500.07 44 814.910.10 60 823.350.11
12 9 4.430.11 28 9 9.560.06 44 915.010.10 60 923.460.11
1210 4.540.11 2810 9.620.06 441015.110.10 601023.570.11
1211 4.640.10 2811 9.680.06 441115.200.09 601123.670.10
1212 4.750.11 2812 9.750.07 441215.300.10 601223.780.11
1213 4.860.11 2813 9.810.06 441315.400.10 601323.890.11
1214 4.970.11 2814 9.870.06 441415.500.10 601424.000.11
1215 5.080.11 2815 9.930.06 441515.600.10 601524.110.11
13 0 4.00 29 0 9.42 45 015.00 61 023.55
13 1 4.060.06 29 1 9.510.09 45 115.060.06 61 123.640.09
13 2 4.120.06 29 2 9.610.10 45 215.120.06 61 223.740.10
13 3 4.180.06 29 3 9.710.10 45 315.180.06 61 323.840.10
13 4 4.250.07 29 4 9.810.10 45 415.250.07 61 423.940.10
13 5 4.310.06 29 5 9.910.10 45 515.310.06 61 524.040.10
13 6 4.370.06 29 610.000.09 45 615.370.06 61 624.130.09
13 7 4.430.06 29 710.100.10 45 715.430.06 61 724.230.10
13 8 4.500.07 29 810.200.10 45 815.500.07 61 824.330.10
13 9 4.560.06 29 910.300.10 45 915.560.06 61 924.430.10
1310 4.620.06 291010.400.10 451015.620.06 611024.530.10
1311 4.680.06 291110.490.09 451115.680.06 611124.620.09
1312 4.750.07 291210.590.10 451215.750.07 611224.720.10
1313 4.810.06 291310.690.10 451315.810.06 611324.820.10
1314 4.870.06 291410.790.10 451415.870.06 611424.920.10
1315 4.930.06 291510.890.10 451515.930.06 611525.020.10
14 0 4.24 30 0 9.89 46 015.55 62 024.22
14 1 4.310.07 30 1 9.950.06 46 115.590.04 62 124.320.10
14 2 4.400.09 30 210.040.09 46 215.680.09 62 224.430.11
14 3 4.490.09 30 310.130.09 46 315.770.09 62 324.540.11
14 4 4.580.09 30 410.220.09 46 415.860.09 62 424.650.11
14 5 4.670.09 30 510.310.09 46 515.950.09 62 524.760.11
14 6 4.750.08 30 610.390.08 46 616.030.08 62 624.860.10
14 7 4.840.09 30 710.480.09 46 716.120.09 62 724.970.11
14 8 4.930.09 30 810.570.09 46 816.210.09 62 825.080.11
14 9 5.020.09 30 910.660.09 46 916.300.09 62 925.190.11
1410 5.110.09 301010.750.09 461016.390.09 621025.300.11
1411 5.190.08 301110.830.08 461116.470.08 621125.400.10
1412 5.280.09 301210.920.09 461216.560.09 621225.510.11
1413 5.370.09 301311.010.09 461316.650.09 621325.620.11
1414 5.460.09 301411.100.09 461416.740.09 621425.730.11
1415 5.550.09 301511.190.09 461516.830.09 621525.840.11
15 0 4.71 31 010.00 47 015.57 63 025.95
15 1 4.800.09 31 110.060.06 47 115.670.10 63 126.050.10
15 2 4.900.10 31 210.120.06 47 215.780.11 63 226.160.11
15 3 5.000.10 31 310.180.06 47 315.890.11 63 326.270.11
15 4 5.100.10 31 410.250.07 47 416.000.11 63 426.380.11
15 5 5.200.10 31 510.310.06 47 516.110.11 63 526.490.11
15 6 5.290.09 31 610.370.06 47 616.210.10 63 626.590.10
15 7 5.390.10 31 710.430.06 47 716.320.11 63 726.700.11
15 8 5.490.10 31 810.500.07 47 816.430.11 63 826.810.11
15 9 5.590.10 31 910.560.06 47 916.540.11 63 926.920.11
1510 5.690.10 311010.620.06 471016.650.11 631027.030.11
1511 5.780.09 311110.680.06 471116.750.10 631127.130.10
1512 5.880.10 311210.750.07 471216.860.11 631227.240.11
1513 5.980.10 311310.810.06 471316.970.11 631327.350.11
1514 6.080.10 311410.870.06 471417.080.11 631427.460.11
1515 6.180.10 311510.930.06 471517.190.11 631527.570.11

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 CoarseFineRatioDelta CoarseFineRatioDelta CoarseFineRatioDelta CoarseFineRatioDelta
0 0 0 0.50 1 0 0.71 2 0 0.78 3 0 0.87
1 0 1 0.560.06 1 1 0.790.08 2 1 0.880.10 3 1 0.970.10
2 0 2 0.620.06 1 2 0.880.09 2 2 0.980.10 3 2 1.080.11
3 0 3 0.680.06 1 3 0.960.08 2 3 1.070.09 3 3 1.180.10
4 0 4 0.750.07 1 4 1.050.09 2 4 1.170.10 3 4 1.290.11
5 0 5 0.810.06 1 5 1.140.09 2 5 1.270.10 3 5 1.400.11
6 0 6 0.870.06 1 6 1.230.09 2 6 1.370.10 3 6 1.510.11
7 0 7 0.930.06 1 7 1.320.09 2 7 1.470.10 3 7 1.620.11
8 4 0 1.000.07 5 0 1.410.09 6 0 1.570.10 7 0 1.730.11
9 4 1 1.060.06 5 1 1.490.08 6 1 1.660.09 7 1 1.830.10
10 4 2 1.120.06 5 2 1.580.09 6 2 1.760.10 7 2 1.940.11
11 4 3 1.180.06 5 3 1.670.09 6 3 1.860.10 7 3 2.050.11
12 4 4 1.250.07 5 4 1.760.09 6 4 1.960.10 7 4 2.160.11
13 4 5 1.310.06 5 5 1.850.09 6 5 2.060.10 7 5 2.270.11
14 4 6 1.370.06 5 6 1.930.08 6 6 2.150.09 7 6 2.370.10
15 4 7 1.430.06 5 7 2.020.09 6 7 2.250.10 7 7 2.480.11
16 4 8 1.500.07 5 8 2.110.09 6 8 2.350.10 7 8 2.590.11
17 4 9 1.560.06 5 9 2.200.09 6 9 2.450.10 7 9 2.700.11
18 410 1.620.06 510 2.290.09 610 2.550.10 710 2.810.11
19 411 1.680.06 511 2.370.08 611 2.640.09 711 2.910.10
20 412 1.750.07 512 2.460.09 612 2.740.10 712 3.020.11
21 413 1.810.06 513 2.550.09 613 2.840.10 713 3.130.11
22 414 1.870.06 514 2.640.09 614 2.940.10 714 3.240.11
23 415 1.930.06 515 2.730.09 615 3.040.10 715 3.350.11
24 8 0 2.000.07 9 0 2.820.09 11 0 3.140.10 12 0 3.460.11
25 8 1 2.060.06 9 1 2.900.08 11 1 3.230.09 12 1 3.560.10
26 8 2 2.120.06 9 2 2.990.09 11 2 3.330.10 12 2 3.670.11
27 8 3 2.180.06 9 3 3.080.09 11 3 3.430.10 12 3 3.780.11
28 8 4 2.250.07 9 4 3.170.09 11 4 3.530.10 12 4 3.890.11
29 8 5 2.310.06 9 5 3.260.09 11 5 3.630.10 12 5 4.000.11
30 8 6 2.370.06 9 6 3.340.08 11 6 3.720.09 12 6 4.100.10
31 8 7 2.430.06 9 7 3.430.09 11 7 3.820.10 12 7 4.210.11
32 8 8 2.500.07 9 8 3.520.09 11 8 3.920.10 12 8 4.320.11
33 8 9 2.560.06 9 9 3.610.09 11 9 4.020.10 12 9 4.430.11
34 810 2.620.06 910 3.700.09 1110 4.120.10 1210 4.540.11
35 811 2.680.06 911 3.780.08 1111 4.210.09 1211 4.640.10
36 812 2.750.07 912 3.870.09 1112 4.310.10 1212 4.750.11
37 813 2.810.06 913 3.960.09 1113 4.410.10 1213 4.860.11
38 814 2.870.06 914 4.050.09 1114 4.510.10 1214 4.970.11
39 815 2.930.06 915 4.140.09 1115 4.610.10 1215 5.080.11
40 10 0 3.000.07 14 0 4.240.10 15 0 4.710.10 17 0 5.190.11
41 10 1 3.060.06 14 1 4.310.07 15 1 4.800.09 17 1 5.290.10
42 10 2 3.120.06 14 2 4.400.09 15 2 4.900.10 17 2 5.400.11
43 10 3 3.180.06 14 3 4.490.09 15 3 5.000.10 17 3 5.510.11
44 10 4 3.250.07 14 4 4.580.09 15 4 5.100.10 17 4 5.620.11
45 10 5 3.310.06 14 5 4.670.09 15 5 5.200.10 17 5 5.730.11
46 10 6 3.370.06 14 6 4.750.08 15 6 5.290.09 17 6 5.830.10
47 10 7 3.430.06 14 7 4.840.09 15 7 5.390.10 17 7 5.940.11
48 10 8 3.500.07 14 8 4.930.09 15 8 5.490.10 17 8 6.050.11
49 10 9 3.560.06 14 9 5.020.09 15 9 5.590.10 17 9 6.160.11
50 1010 3.620.06 1410 5.110.09 1510 5.690.10 1710 6.270.11
51 1011 3.680.06 1411 5.190.08 1511 5.780.09 1711 6.370.10
52 1012 3.750.07 1412 5.280.09 1512 5.880.10 1712 6.480.11
53 1013 3.810.06 1413 5.370.09 1513 5.980.10 1713 6.590.11
54 1014 3.870.06 1414 5.460.09 1514 6.080.10 1714 6.700.11
55 1015 3.930.06 1415 5.550.09 1515 6.180.10 1715 6.810.11
56 13 0 4.000.07 18 0 5.650.10 20 0 6.280.10 21 0 6.920.11
57 13 1 4.060.06 18 1 5.720.07 20 1 6.370.09 21 1 7.020.10
58 13 2 4.120.06 18 2 5.810.09 20 2 6.470.10 21 2 7.130.11
59 13 3 4.180.06 18 3 5.900.09 20 3 6.570.10 21 3 7.240.11
60 13 4 4.250.07 18 4 5.990.09 20 4 6.670.10 21 4 7.350.11
61 13 5 4.310.06 18 5 6.080.09 20 5 6.770.10 21 5 7.460.11
62 13 6 4.370.06 18 6 6.160.08 20 6 6.860.09 21 6 7.560.10
63 13 7 4.430.06 18 7 6.250.09 20 7 6.960.10 21 7 7.670.11
64 13 8 4.500.07 18 8 6.340.09 20 8 7.060.10 21 8 7.780.11
65 13 9 4.560.06 18 9 6.430.09 20 9 7.160.10 21 9 7.890.11
66 1310 4.620.06 1810 6.520.09 2010 7.260.10 2110 8.000.11
67 1311 4.680.06 1811 6.600.08 2011 7.350.09 2111 8.100.10
68 1312 4.750.07 1812 6.690.09 2012 7.450.10 2112 8.210.11
69 1313 4.810.06 1813 6.780.09 2013 7.550.10 2113 8.320.11
70 1314 4.870.06 1814 6.870.09 2014 7.650.10 2114 8.430.11
71 1315 4.930.06 1815 6.960.09 2015 7.750.10 2115 8.540.11
72 16 0 5.000.07 23 0 7.070.11 24 0 7.850.10 27 0 8.650.11
73 16 1 5.060.06 23 1 7.130.06 24 1 7.940.09 27 1 8.750.10
74 16 2 5.120.06 23 2 7.220.09 24 2 8.040.10 27 2 8.860.11
75 16 3 5.180.06 23 3 7.310.09 24 3 8.140.10 27 3 8.970.11
76 16 4 5.250.07 23 4 7.400.09 24 4 8.240.10 27 4 9.080.11
77 16 5 5.310.06 23 5 7.490.09 24 5 8.340.10 27 5 9.190.11
78 16 6 5.370.06 23 6 7.570.08 24 6 8.430.09 27 6 9.290.10
79 16 7 5.430.06 23 7 7.660.09 24 7 8.530.10 27 7 9.400.11
80 16 8 5.500.07 23 8 7.750.09 24 8 8.630.10 27 8 9.510.11
81 16 9 5.560.06 23 9 7.840.09 24 9 8.730.10 27 9 9.620.11
82 1610 5.620.06 2310 7.930.09 2410 8.830.10 2710 9.730.11
83 1611 5.680.06 2311 8.010.08 2411 8.920.09 2711 9.830.10
84 1612 5.750.07 2312 8.100.09 2412 9.020.10 2712 9.940.11
85 1613 5.810.06 2313 8.190.09 2413 9.120.10 271310.050.11
86 1614 5.870.06 2314 8.280.09 2414 9.220.10 271410.160.11
87 1615 5.930.06 2315 8.370.09 2415 9.320.10 271510.270.11
88 19 0 6.000.07 26 0 8.480.11 29 0 9.420.10 32 010.380.11
89 19 1 6.060.06 26 1 8.540.06 29 1 9.510.09 32 110.480.10
90 19 2 6.120.06 26 2 8.630.09 29 2 9.610.10 32 210.590.11
91 19 3 6.180.06 26 3 8.720.09 29 3 9.710.10 32 310.700.11
92 19 4 6.250.07 26 4 8.810.09 29 4 9.810.10 32 410.810.11
93 19 5 6.310.06 26 5 8.900.09 29 5 9.910.10 32 510.920.11
94 19 6 6.370.06 26 6 8.980.08 29 610.000.09 32 611.020.10
95 19 7 6.430.06 26 7 9.070.09 29 710.100.10 32 711.130.11
96 19 8 6.500.07 26 8 9.160.09 29 810.200.10 32 811.240.11
97 19 9 6.560.06 26 9 9.250.09 29 910.300.10 32 911.350.11
98 1910 6.620.06 2610 9.340.09 291010.400.10 321011.460.11
99 1911 6.680.06 2611 9.420.08 291110.490.09 321111.560.10
100 1912 6.750.07 2612 9.510.09 291210.590.10 321211.670.11
101 1913 6.810.06 2613 9.600.09 291310.690.10 321311.780.11
102 1914 6.870.06 2614 9.690.09 291410.790.10 321411.890.11
103 1915 6.930.06 2615 9.780.09 291510.890.10 321512.000.11
104 22 0 7.000.07 30 0 9.890.11 33 010.990.10 37 012.110.11
105 22 1 7.060.06 30 1 9.950.06 33 111.080.09 37 112.210.10
106 22 2 7.120.06 30 210.040.09 33 211.180.10 37 212.320.11
107 22 3 7.180.06 30 310.130.09 33 311.280.10 37 312.430.11
108 22 4 7.250.07 30 410.220.09 33 411.380.10 37 412.540.11
109 22 5 7.310.06 30 510.310.09 33 511.480.10 37 512.650.11
110 22 6 7.370.06 30 610.390.08 33 611.570.09 37 612.750.10
111 22 7 7.430.06 30 710.480.09 33 711.670.10 37 712.860.11
112 22 8 7.500.07 30 810.570.09 33 811.770.10 37 812.970.11
113 22 9 7.560.06 30 910.660.09 33 911.870.10 37 913.080.11
114 2210 7.620.06 301010.750.09 331011.970.10 371013.190.11
115 2211 7.680.06 301110.830.08 331112.060.09 371113.290.10
116 2212 7.750.07 301210.920.09 331212.160.10 371213.400.11
117 2213 7.810.06 301311.010.09 331312.260.10 371313.510.11
118 2214 7.870.06 301411.100.09 331412.360.10 371413.620.11
119 2215 7.930.06 301511.190.09 331512.460.10 371513.730.11
120 25 0 8.000.07 35 011.300.11 38 012.560.10 41 013.840.11
121 25 1 8.060.06 35 111.360.06 38 112.650.09 41 113.940.10
122 25 2 8.120.06 35 211.450.09 38 212.750.10 41 214.050.11
123 25 3 8.180.06 35 311.540.09 38 312.850.10 41 314.160.11
124 25 4 8.250.07 35 411.630.09 38 412.950.10 41 414.270.11
125 25 5 8.310.06 35 511.720.09 38 513.050.10 41 514.380.11
126 25 6 8.370.06 35 611.800.08 38 613.140.09 41 614.480.10
127 25 7 8.430.06 35 711.890.09 38 713.240.10 41 714.590.11
128 25 8 8.500.07 35 811.980.09 38 813.340.10 41 814.700.11
129 25 9 8.560.06 35 912.070.09 38 913.440.10 41 914.810.11
130 2510 8.620.06 351012.160.09 381013.540.10 411014.920.11
131 2511 8.680.06 351112.240.08 381113.630.09 411115.020.10
132 2512 8.750.07 351212.330.09 381213.730.10 411215.130.11
133 2513 8.810.06 351312.420.09 381313.830.10 411315.240.11
134 2514 8.870.06 351412.510.09 381413.930.10 411415.350.11
135 2515 8.930.06 351512.600.09 381514.030.10 411515.460.11
136 28 0 9.000.07 39 012.720.12 44 014.130.10 47 015.570.11
137 28 1 9.060.06 39 112.770.05 44 114.220.09 47 115.670.10
138 28 2 9.120.06 39 212.860.09 44 214.320.10 47 215.780.11
139 28 3 9.180.06 39 312.950.09 44 314.420.10 47 315.890.11
140 28 4 9.250.07 39 413.040.09 44 414.520.10 47 416.000.11
141 28 5 9.310.06 39 513.130.09 44 514.620.10 47 516.110.11
142 28 6 9.370.06 39 613.210.08 44 614.710.09 47 616.210.10
143 28 7 9.430.06 39 713.300.09 44 714.810.10 47 716.320.11
144 28 8 9.500.07 39 813.390.09 44 814.910.10 47 816.430.11
145 28 9 9.560.06 39 913.480.09 44 915.010.10 47 916.540.11
146 2810 9.620.06 391013.570.09 441015.110.10 471016.650.11
147 2811 9.680.06 391113.650.08 441115.200.09 471116.750.10
148 2812 9.750.07 391213.740.09 441215.300.10 471216.860.11
149 2813 9.810.06 391313.830.09 441315.400.10 471316.970.11
150 2814 9.870.06 391413.920.09 441415.500.10 471417.080.11
151 2815 9.930.06 391514.010.09 441515.600.10 471517.190.11
152 31 010.000.07 43 014.100.09 48 015.700.10 51 017.300.11
153 31 110.060.06 43 114.180.08 48 115.790.09 51 117.400.10
154 31 210.120.06 43 214.270.09 48 215.890.10 51 217.510.11
155 31 310.180.06 43 314.360.09 48 315.990.10 51 317.620.11
156 31 410.250.07 43 414.450.09 48 416.090.10 51 417.730.11
157 31 510.310.06 43 514.540.09 48 516.190.10 51 517.840.11
158 31 610.370.06 43 614.620.08 48 616.280.09 51 617.940.10
159 31 710.430.06 43 714.710.09 48 716.380.10 51 718.050.11
160 31 810.500.07 43 814.800.09 48 816.480.10 51 818.160.11
161 31 910.560.06 43 914.890.09 48 916.580.10 51 918.270.11
162 311010.620.06 431014.980.09 481016.680.10 511018.380.11
163 311110.680.06 431115.060.08 481116.770.09 511118.480.10
164 311210.750.07 431215.150.09 481216.870.10 511218.590.11
165 311310.810.06 431315.240.09 481316.970.10 511318.700.11
166 311410.870.06 431415.330.09 481417.070.10 511418.810.11
167 311510.930.06 431515.420.09 481517.170.10 511518.920.11
168 34 011.000.07 46 015.550.13 50 017.270.10 54 019.030.11
169 34 111.060.06 46 115.590.04 50 117.360.09 54 119.130.10
170 34 211.120.06 46 215.680.09 50 217.460.10 54 219.240.11
171 34 311.180.06 46 315.770.09 50 317.560.10 54 319.350.11
172 34 411.250.07 46 415.860.09 50 417.660.10 54 419.460.11
173 34 511.310.06 46 515.950.09 50 517.760.10 54 519.570.11
174 34 611.370.06 46 616.030.08 50 617.850.09 54 619.670.10
175 34 711.430.06 46 716.120.09 50 717.950.10 54 719.780.11
176 34 811.500.07 46 816.210.09 50 818.050.10 54 819.890.11
177 34 911.560.06 46 916.300.09 50 918.150.10 54 920.000.11
178 341011.620.06 461016.390.09 501018.250.10 541020.110.11
179 341111.680.06 461116.470.08 501118.340.09 541120.210.10
180 341211.750.07 461216.560.09 501218.440.10 541220.320.11
181 341311.810.06 461316.650.09 501318.540.10 541320.430.11
182 341411.870.06 461416.740.09 501418.640.10 541420.540.11
183 341511.930.06 461516.830.09 501518.740.10 541520.650.11
184 36 012.000.07 49 016.960.13 53 018.840.10 57 020.760.11
185 36 112.060.06 49 117.000.04 53 118.930.09 57 120.860.10
186 36 212.120.06 49 217.090.09 53 219.030.10 57 220.970.11
187 36 312.180.06 49 317.180.09 53 319.130.10 57 321.080.11
188 36 412.250.07 49 417.270.09 53 419.230.10 57 421.190.11
189 36 512.310.06 49 517.360.09 53 519.330.10 57 521.300.11
190 36 612.370.06 49 617.440.08 53 619.420.09 57 621.400.10
191 36 712.430.06 49 717.530.09 53 719.520.10 57 721.510.11
192 36 812.500.07 49 817.620.09 53 819.620.10 57 821.620.11
193 36 912.560.06 49 917.710.09 53 919.720.10 57 921.730.11
194 361012.620.06 491017.800.09 531019.820.10 571021.840.11
195 361112.680.06 491117.880.08 531119.910.09 571121.940.10
196 361212.750.07 491217.970.09 531220.010.10 571222.050.11
197 361312.810.06 491318.060.09 531320.110.10 571322.160.11
198 361412.870.06 491418.150.09 531420.210.10 571422.270.11
199 361512.930.06 491518.240.09 531520.310.10 571522.380.11
200 40 013.000.07 52 018.370.13 56 020.410.10 60 022.490.11
201 40 113.060.06 52 118.410.04 56 120.500.09 60 122.590.10
202 40 213.120.06 52 218.500.09 56 220.600.10 60 222.700.11
203 40 313.180.06 52 318.590.09 56 320.700.10 60 322.810.11
204 40 413.250.07 52 418.680.09 56 420.800.10 60 422.920.11
205 40 513.310.06 52 518.770.09 56 520.900.10 60 523.030.11
206 40 613.370.06 52 618.850.08 56 620.990.09 60 623.130.10
207 40 713.430.06 52 718.940.09 56 721.090.10 60 723.240.11
208 40 813.500.07 52 819.030.09 56 821.190.10 60 823.350.11
209 40 913.560.06 52 919.120.09 56 921.290.10 60 923.460.11
210 401013.620.06 521019.210.09 561021.390.10 601023.570.11
211 401113.680.06 521119.290.08 561121.480.09 601123.670.10
212 401213.750.07 521219.380.09 561221.580.10 601223.780.11
213 401313.810.06 521319.470.09 561321.680.10 601323.890.11
214 401413.870.06 521419.560.09 561421.780.10 601424.000.11
215 401513.930.06 521519.650.09 561521.880.10 601524.110.11
216 42 014.000.07 55 019.780.13 59 021.980.10 62 024.220.11
217 42 114.060.06 55 119.820.04 59 122.070.09 62 124.320.10
218 42 214.120.06 55 219.910.09 59 222.170.10 62 224.430.11
219 42 314.180.06 55 320.000.09 59 322.270.10 62 324.540.11
220 42 414.250.07 55 420.090.09 59 422.370.10 62 424.650.11
221 42 514.310.06 55 520.180.09 59 522.470.10 62 524.760.11
222 42 614.370.06 55 620.260.08 59 622.560.09 62 624.860.10
223 42 714.430.06 55 720.350.09 59 722.660.10 62 724.970.11
224 42 814.500.07 55 820.440.09 59 822.760.10 62 825.080.11
225 42 914.560.06 55 920.530.09 59 922.860.10 62 925.190.11
226 421014.620.06 551020.620.09 591022.960.10 621025.300.11
227 421114.680.06 551120.700.08 591123.050.09 621125.400.10
228 421214.750.07 551220.790.09 591223.150.10 621225.510.11
229 421314.810.06 551320.880.09 591323.250.10 621325.620.11
230 421414.870.06 551420.970.09 591423.350.10 621425.730.11
231 421514.930.06 551521.060.09 591523.450.10 621525.840.11
232 45 015.000.07 58 021.200.14 61 023.550.10 63 025.950.11
233 45 115.060.06 58 121.230.03 61 123.640.09 63 126.050.10
234 45 215.120.06 58 221.320.09 61 223.740.10 63 226.160.11
235 45 315.180.06 58 321.410.09 61 323.840.10 63 326.270.11
236 45 415.250.07 58 421.500.09 61 423.940.10 63 426.380.11
237 45 515.310.06 58 521.590.09 61 524.040.10 63 526.490.11
238 45 615.370.06 58 621.670.08 61 624.130.09 63 626.590.10
239 45 715.430.06 58 721.760.09 61 724.230.10 63 726.700.11
240 45 815.500.07 58 821.850.09 61 824.330.10 63 826.810.11
241 45 915.560.06 58 921.940.09 61 924.430.10 63 926.920.11
242 451015.620.06 581022.030.09 611024.530.10 631027.030.11
243 451115.680.06 581122.110.08 611124.620.09 631127.130.10
244 451215.750.07 581222.200.09 611224.720.10 631227.240.11
245 451315.810.06 581322.290.09 611324.820.10 631327.350.11
246 451415.870.06 581422.380.09 611424.920.10 631427.460.11
247 451515.930.06 581522.470.09 611525.020.10 631527.570.11

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

A graph will make this more clear:

Zoom in by dragging. Zoom out by double-clicking. Pan by shift-dragging.

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:

group0ratio = 0.0625 * order(coarse, fine) + 0.50
group1ratio = 0.088105 * order(coarse, fine) + 0.71
group2ratio = 0.098145 * order(coarse, fine) + 0.78
group3ratio = 0.108105 * order(coarse, fine) + 0.87

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 CoarseFineDisplay
Ratio
Calc'ed
Ratio
Error CoarseFineDisplay
Ratio
Calc'ed
Ratio
Error CoarseFineDisplay
Ratio
Calc'ed
Ratio
Error CoarseFineDisplay
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 0.970.01 2 3 1.07 1.07 3 3 1.18 1.190.01
4 0 4 0.75 0.75 1 4 1.05 1.060.01 2 4 1.17 1.17 3 4 1.29 1.300.01
5 0 5 0.81 0.81 1 5 1.14 1.150.01 2 5 1.27 1.27 3 5 1.40 1.410.01
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 1.500.01 6 1 1.66 1.66 7 1 1.83 1.840.01
10 4 2 1.12 1.12 5 2 1.58 1.590.01 6 2 1.76 1.76 7 2 1.94 1.950.01
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 1.940.01 6 6 2.15 2.15 7 6 2.37 2.380.01
15 4 7 1.43 1.43 5 7 2.02 2.030.01 6 7 2.25 2.25 7 7 2.48 2.490.01
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
18 410 1.62 1.62 510 2.29 2.29 610 2.55 2.54-0.01 710 2.81 2.81
19 411 1.68 1.68 511 2.37 2.380.01 611 2.64 2.64 711 2.91 2.920.01
20 412 1.75 1.75 512 2.46 2.470.01 612 2.74 2.74 712 3.02 3.030.01
21 413 1.81 1.81 513 2.55 2.560.01 613 2.84 2.84 713 3.13 3.140.01
22 414 1.87 1.87 514 2.64 2.64 614 2.94 2.93-0.01 714 3.24 3.24
23 415 1.93 1.93 515 2.73 2.73 615 3.04 3.03-0.01 715 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 2.910.01 11 1 3.23 3.23 12 1 3.56 3.570.01
26 8 2 2.12 2.12 9 2 2.99 3.000.01 11 2 3.33 3.33 12 2 3.67 3.680.01
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 3.350.01 11 6 3.72 3.72 12 6 4.10 4.110.01
31 8 7 2.43 2.43 9 7 3.43 3.440.01 11 7 3.82 3.82 12 7 4.21 4.220.01
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
34 810 2.62 2.62 910 3.70 3.70 1110 4.12 4.11-0.01 1210 4.54 4.54
35 811 2.68 2.68 911 3.78 3.790.01 1111 4.21 4.21 1211 4.64 4.650.01
36 812 2.75 2.75 912 3.87 3.880.01 1112 4.31 4.31 1212 4.75 4.760.01
37 813 2.81 2.81 913 3.96 3.96 1113 4.41 4.41 1213 4.86 4.86
38 814 2.87 2.87 914 4.05 4.05 1114 4.51 4.50-0.01 1214 4.97 4.97
39 815 2.93 2.93 915 4.14 4.14 1115 4.61 4.60-0.01 1215 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 4.320.01 15 1 4.80 4.80 17 1 5.29 5.300.01
42 10 2 3.12 3.12 14 2 4.40 4.410.01 15 2 4.90 4.90 17 2 5.40 5.410.01
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 4.760.01 15 6 5.29 5.29 17 6 5.83 5.840.01
47 10 7 3.43 3.43 14 7 4.84 4.850.01 15 7 5.39 5.39 17 7 5.94 5.950.01
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 1010 3.62 3.62 1410 5.11 5.11 1510 5.69 5.68-0.01 1710 6.27 6.27
51 1011 3.68 3.68 1411 5.19 5.200.01 1511 5.78 5.78 1711 6.37 6.380.01
52 1012 3.75 3.75 1412 5.28 5.290.01 1512 5.88 5.88 1712 6.48 6.490.01
53 1013 3.81 3.81 1413 5.37 5.37 1513 5.98 5.98 1713 6.59 6.59
54 1014 3.87 3.87 1414 5.46 5.46 1514 6.08 6.07-0.01 1714 6.70 6.70
55 1015 3.93 3.93 1415 5.55 5.55 1515 6.18 6.17-0.01 1715 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 5.730.01 20 1 6.37 6.37 21 1 7.02 7.030.01
58 13 2 4.12 4.12 18 2 5.81 5.820.01 20 2 6.47 6.47 21 2 7.13 7.140.01
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 6.170.01 20 6 6.86 6.86 21 6 7.56 7.570.01
63 13 7 4.43 4.43 18 7 6.25 6.260.01 20 7 6.96 6.96 21 7 7.67 7.680.01
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 1310 4.62 4.62 1810 6.52 6.52 2010 7.26 7.25-0.01 2110 8.00 8.00
67 1311 4.68 4.68 1811 6.60 6.610.01 2011 7.35 7.35 2111 8.10 8.110.01
68 1312 4.75 4.75 1812 6.69 6.700.01 2012 7.45 7.45 2112 8.21 8.220.01
69 1313 4.81 4.81 1813 6.78 6.78 2013 7.55 7.55 2113 8.32 8.32
70 1314 4.87 4.87 1814 6.87 6.87 2014 7.65 7.65 2114 8.43 8.43
71 1315 4.93 4.93 1815 6.96 6.96 2015 7.75 7.74-0.01 2115 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 7.140.01 24 1 7.94 7.94 27 1 8.75 8.760.01
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 7.580.01 24 6 8.43 8.43 27 6 9.29 9.300.01
79 16 7 5.43 5.43 23 7 7.66 7.670.01 24 7 8.53 8.53 27 7 9.40 9.410.01
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 1610 5.62 5.62 2310 7.93 7.93 2410 8.83 8.82-0.01 2710 9.73 9.73
83 1611 5.68 5.68 2311 8.01 8.020.01 2411 8.92 8.92 2711 9.83 9.840.01
84 1612 5.75 5.75 2312 8.10 8.110.01 2412 9.02 9.02 2712 9.94 9.950.01
85 1613 5.81 5.81 2313 8.19 8.19 2413 9.12 9.12 271310.0510.05
86 1614 5.87 5.87 2314 8.28 8.28 2414 9.22 9.22 271410.1610.16
87 1615 5.93 5.93 2315 8.37 8.37 2415 9.32 9.31-0.01 271510.2710.27
88 19 0 6.00 6.00 26 0 8.48 8.46-0.02 29 0 9.42 9.41-0.01 32 010.3810.38
89 19 1 6.06 6.06 26 1 8.54 8.550.01 29 1 9.51 9.51 32 110.4810.490.01
90 19 2 6.12 6.12 26 2 8.63 8.63 29 2 9.61 9.61 32 210.5910.59
91 19 3 6.18 6.18 26 3 8.72 8.72 29 3 9.71 9.71 32 310.7010.70
92 19 4 6.25 6.25 26 4 8.81 8.81 29 4 9.81 9.80-0.01 32 410.8110.81
93 19 5 6.31 6.31 26 5 8.90 8.90 29 5 9.91 9.90-0.01 32 510.9210.92
94 19 6 6.37 6.37 26 6 8.98 8.990.01 29 610.0010.00 32 611.0211.030.01
95 19 7 6.43 6.43 26 7 9.07 9.07 29 710.1010.10 32 711.1311.13
96 19 8 6.50 6.50 26 8 9.16 9.16 29 810.2010.20 32 811.2411.24
97 19 9 6.56 6.56 26 9 9.25 9.25 29 910.3010.30 32 911.3511.35
98 1910 6.62 6.62 2610 9.34 9.34 291010.4010.39-0.01 321011.4611.46
99 1911 6.68 6.68 2611 9.42 9.430.01 291110.4910.49 321111.5611.570.01
100 1912 6.75 6.75 2612 9.51 9.520.01 291210.5910.59 321211.6711.680.01
101 1913 6.81 6.81 2613 9.60 9.60 291310.6910.69 321311.7811.78
102 1914 6.87 6.87 2614 9.69 9.69 291410.7910.79 321411.8911.89
103 1915 6.93 6.93 2615 9.78 9.78 291510.8910.88-0.01 321512.0012.00
104 22 0 7.00 7.00 30 0 9.89 9.87-0.02 33 010.9910.98-0.01 37 012.1112.11
105 22 1 7.06 7.06 30 1 9.95 9.960.01 33 111.0811.08 37 112.2112.220.01
106 22 2 7.12 7.12 30 210.0410.04 33 211.1811.18 37 212.3212.32
107 22 3 7.18 7.18 30 310.1310.13 33 311.2811.28 37 312.4312.43
108 22 4 7.25 7.25 30 410.2210.22 33 411.3811.37-0.01 37 412.5412.54
109 22 5 7.31 7.31 30 510.3110.31 33 511.4811.47-0.01 37 512.6512.65
110 22 6 7.37 7.37 30 610.3910.400.01 33 611.5711.57 37 612.7512.760.01
111 22 7 7.43 7.43 30 710.4810.48 33 711.6711.67 37 712.8612.86
112 22 8 7.50 7.50 30 810.5710.57 33 811.7711.77 37 812.9712.97
113 22 9 7.56 7.56 30 910.6610.66 33 911.8711.87 37 913.0813.08
114 2210 7.62 7.62 301010.7510.75 331011.9711.96-0.01 371013.1913.19
115 2211 7.68 7.68 301110.8310.840.01 331112.0612.06 371113.2913.300.01
116 2212 7.75 7.75 301210.9210.930.01 331212.1612.16 371213.4013.410.01
117 2213 7.81 7.81 301311.0111.01 331312.2612.26 371313.5113.51
118 2214 7.87 7.87 301411.1011.10 331412.3612.36 371413.6213.62
119 2215 7.93 7.93 301511.1911.19 331512.4612.45-0.01 371513.7313.73
120 25 0 8.00 8.00 35 011.3011.28-0.02 38 012.5612.55-0.01 41 013.8413.84
121 25 1 8.06 8.06 35 111.3611.370.01 38 112.6512.65 41 113.9413.950.01
122 25 2 8.12 8.12 35 211.4511.45 38 212.7512.75 41 214.0514.05
123 25 3 8.18 8.18 35 311.5411.54 38 312.8512.85 41 314.1614.16
124 25 4 8.25 8.25 35 411.6311.63 38 412.9512.94-0.01 41 414.2714.27
125 25 5 8.31 8.31 35 511.7211.72 38 513.0513.04-0.01 41 514.3814.38
126 25 6 8.37 8.37 35 611.8011.810.01 38 613.1413.14 41 614.4814.490.01
127 25 7 8.43 8.43 35 711.8911.89 38 713.2413.24 41 714.5914.59
128 25 8 8.50 8.50 35 811.9811.98 38 813.3413.34 41 814.7014.70
129 25 9 8.56 8.56 35 912.0712.07 38 913.4413.44 41 914.8114.81
130 2510 8.62 8.62 351012.1612.16 381013.5413.53-0.01 411014.9214.92
131 2511 8.68 8.68 351112.2412.250.01 381113.6313.63 411115.0215.030.01
132 2512 8.75 8.75 351212.3312.33 381213.7313.73 411215.1315.13
133 2513 8.81 8.81 351312.4212.42 381313.8313.83 411315.2415.24
134 2514 8.87 8.87 351412.5112.51 381413.9313.93 411415.3515.35
135 2515 8.93 8.93 351512.6012.60 381514.0314.02-0.01 411515.4615.46
136 28 0 9.00 9.00 39 012.7212.69-0.03 44 014.1314.12-0.01 47 015.5715.57
137 28 1 9.06 9.06 39 112.7712.780.01 44 114.2214.22 47 115.6715.680.01
138 28 2 9.12 9.12 39 212.8612.86 44 214.3214.32 47 215.7815.78
139 28 3 9.18 9.18 39 312.9512.95 44 314.4214.42 47 315.8915.89
140 28 4 9.25 9.25 39 413.0413.04 44 414.5214.52 47 416.0016.00
141 28 5 9.31 9.31 39 513.1313.13 44 514.6214.61-0.01 47 516.1116.11
142 28 6 9.37 9.37 39 613.2113.220.01 44 614.7114.71 47 616.2116.220.01
143 28 7 9.43 9.43 39 713.3013.30 44 714.8114.81 47 716.3216.32
144 28 8 9.50 9.50 39 813.3913.39 44 814.9114.91 47 816.4316.43
145 28 9 9.56 9.56 39 913.4813.48 44 915.0115.01 47 916.5416.54
146 2810 9.62 9.62 391013.5713.57 441015.1115.10-0.01 471016.6516.65
147 2811 9.68 9.68 391113.6513.660.01 441115.2015.20 471116.7516.760.01
148 2812 9.75 9.75 391213.7413.74 441215.3015.30 471216.8616.86
149 2813 9.81 9.81 391313.8313.83 441315.4015.40 471316.9716.97
150 2814 9.87 9.87 391413.9213.92 441415.5015.50 471417.0817.08
151 2815 9.93 9.93 391514.0114.01 441515.6015.59-0.01 471517.1917.19
152 31 010.0010.00 43 014.1014.10 48 015.7015.69-0.01 51 017.3017.30
153 31 110.0610.06 43 114.1814.190.01 48 115.7915.79 51 117.4017.410.01
154 31 210.1210.12 43 214.2714.27 48 215.8915.89 51 217.5117.51
155 31 310.1810.18 43 314.3614.36 48 315.9915.99 51 317.6217.62
156 31 410.2510.25 43 414.4514.45 48 416.0916.09 51 417.7317.73
157 31 510.3110.31 43 514.5414.54 48 516.1916.18-0.01 51 517.8417.84
158 31 610.3710.37 43 614.6214.630.01 48 616.2816.28 51 617.9417.950.01
159 31 710.4310.43 43 714.7114.71 48 716.3816.38 51 718.0518.05
160 31 810.5010.50 43 814.8014.80 48 816.4816.48 51 818.1618.16
161 31 910.5610.56 43 914.8914.89 48 916.5816.58 51 918.2718.27
162 311010.6210.62 431014.9814.98 481016.6816.67-0.01 511018.3818.38
163 311110.6810.68 431115.0615.070.01 481116.7716.77 511118.4818.490.01
164 311210.7510.75 431215.1515.15 481216.8716.87 511218.5918.59
165 311310.8110.81 431315.2415.24 481316.9716.97 511318.7018.70
166 311410.8710.87 431415.3315.33 481417.0717.07 511418.8118.81
167 311510.9310.93 431515.4215.42 481517.1717.17 511518.9218.92
168 34 011.0011.00 46 015.5515.51-0.04 50 017.2717.26-0.01 54 019.0319.03
169 34 111.0611.06 46 115.5915.59 50 117.3617.36 54 119.1319.13
170 34 211.1211.12 46 215.6815.68 50 217.4617.46 54 219.2419.24
171 34 311.1811.18 46 315.7715.77 50 317.5617.56 54 319.3519.35
172 34 411.2511.25 46 415.8615.86 50 417.6617.66 54 419.4619.46
173 34 511.3111.31 46 515.9515.95 50 517.7617.75-0.01 54 519.5719.57
174 34 611.3711.37 46 616.0316.040.01 50 617.8517.85 54 619.6719.680.01
175 34 711.4311.43 46 716.1216.12 50 717.9517.95 54 719.7819.78
176 34 811.5011.50 46 816.2116.21 50 818.0518.05 54 819.8919.89
177 34 911.5611.56 46 916.3016.30 50 918.1518.15 54 920.0020.00
178 341011.6211.62 461016.3916.39 501018.2518.24-0.01 541020.1120.11
179 341111.6811.68 461116.4716.480.01 501118.3418.34 541120.2120.220.01
180 341211.7511.75 461216.5616.56 501218.4418.44 541220.3220.32
181 341311.8111.81 461316.6516.65 501318.5418.54 541320.4320.43
182 341411.8711.87 461416.7416.74 501418.6418.64 541420.5420.54
183 341511.9311.93 461516.8316.83 501518.7418.74 541520.6520.65
184 36 012.0012.00 49 016.9616.92-0.04 53 018.8418.83-0.01 57 020.7620.76
185 36 112.0612.06 49 117.0017.00 53 118.9318.93 57 120.8620.86
186 36 212.1212.12 49 217.0917.09 53 219.0319.03 57 220.9720.97
187 36 312.1812.18 49 317.1817.18 53 319.1319.13 57 321.0821.08
188 36 412.2512.25 49 417.2717.27 53 419.2319.23 57 421.1921.19
189 36 512.3112.31 49 517.3617.36 53 519.3319.32-0.01 57 521.3021.30
190 36 612.3712.37 49 617.4417.44 53 619.4219.42 57 621.4021.40
191 36 712.4312.43 49 717.5317.53 53 719.5219.52 57 721.5121.51
192 36 812.5012.50 49 817.6217.62 53 819.6219.62 57 821.6221.62
193 36 912.5612.56 49 917.7117.71 53 919.7219.72 57 921.7321.73
194 361012.6212.62 491017.8017.80 531019.8219.82 571021.8421.84
195 361112.6812.68 491117.8817.890.01 531119.9119.91 571121.9421.950.01
196 361212.7512.75 491217.9717.97 531220.0120.01 571222.0522.05
197 361312.8112.81 491318.0618.06 531320.1120.11 571322.1622.16
198 361412.8712.87 491418.1518.15 531420.2120.21 571422.2722.27
199 361512.9312.93 491518.2418.24 531520.3120.31 571522.3822.38
200 40 013.0013.00 52 018.3718.33-0.04 56 020.4120.40-0.01 60 022.4922.49
201 40 113.0613.06 52 118.4118.41 56 120.5020.50 60 122.5922.59
202 40 213.1213.12 52 218.5018.50 56 220.6020.60 60 222.7022.70
203 40 313.1813.18 52 318.5918.59 56 320.7020.70 60 322.8122.81
204 40 413.2513.25 52 418.6818.68 56 420.8020.80 60 422.9222.92
205 40 513.3113.31 52 518.7718.77 56 520.9020.89-0.01 60 523.0323.03
206 40 613.3713.37 52 618.8518.85 56 620.9920.99 60 623.1323.13
207 40 713.4313.43 52 718.9418.94 56 721.0921.09 60 723.2423.24
208 40 813.5013.50 52 819.0319.03 56 821.1921.19 60 823.3523.35
209 40 913.5613.56 52 919.1219.12 56 921.2921.29 60 923.4623.46
210 401013.6213.62 521019.2119.21 561021.3921.39 601023.5723.57
211 401113.6813.68 521119.2919.300.01 561121.4821.48 601123.6723.680.01
212 401213.7513.75 521219.3819.38 561221.5821.58 601223.7823.78
213 401313.8113.81 521319.4719.47 561321.6821.68 601323.8923.89
214 401413.8713.87 521419.5619.56 561421.7821.78 601424.0024.00
215 401513.9313.93 521519.6519.65 561521.8821.88 601524.1124.11
216 42 014.0014.00 55 019.7819.74-0.04 59 021.9821.97-0.01 62 024.2224.22
217 42 114.0614.06 55 119.8219.82 59 122.0722.07 62 124.3224.32
218 42 214.1214.12 55 219.9119.91 59 222.1722.17 62 224.4324.43
219 42 314.1814.18 55 320.0020.00 59 322.2722.27 62 324.5424.54
220 42 414.2514.25 55 420.0920.09 59 422.3722.37 62 424.6524.65
221 42 514.3114.31 55 520.1820.18 59 522.4722.47 62 524.7624.76
222 42 614.3714.37 55 620.2620.26 59 622.5622.56 62 624.8624.86
223 42 714.4314.43 55 720.3520.35 59 722.6622.66 62 724.9724.97
224 42 814.5014.50 55 820.4420.44 59 822.7622.76 62 825.0825.08
225 42 914.5614.56 55 920.5320.53 59 922.8622.86 62 925.1925.19
226 421014.6214.62 551020.6220.62 591022.9622.96 621025.3025.30
227 421114.6814.68 551120.7020.70 591123.0523.05 621125.4025.40
228 421214.7514.75 551220.7920.79 591223.1523.15 621225.5125.51
229 421314.8114.81 551320.8820.88 591323.2523.25 621325.6225.62
230 421414.8714.87 551420.9720.97 591423.3523.35 621425.7325.73
231 421514.9314.93 551521.0621.06 591523.4523.45 621525.8425.84
232 45 015.0015.00 58 021.2021.15-0.05 61 023.5523.54-0.01 63 025.9525.95
233 45 115.0615.06 58 121.2321.23 61 123.6423.64 63 126.0526.05
234 45 215.1215.12 58 221.3221.32 61 223.7423.74 63 226.1626.16
235 45 315.1815.18 58 321.4121.41 61 323.8423.84 63 326.2726.27
236 45 415.2515.25 58 421.5021.50 61 423.9423.94 63 426.3826.38
237 45 515.3115.31 58 521.5921.59 61 524.0424.04 63 526.4926.49
238 45 615.3715.37 58 621.6721.67 61 624.1324.13 63 626.5926.59
239 45 715.4315.43 58 721.7621.76 61 724.2324.23 63 726.7026.70
240 45 815.5015.50 58 821.8521.85 61 824.3324.33 63 826.8126.81
241 45 915.5615.56 58 921.9421.94 61 924.4324.43 63 926.9226.92
242 451015.6215.62 581022.0322.03 611024.5324.53 631027.0327.03
243 451115.6815.68 581122.1122.11 611124.6224.62 631127.1327.13
244 451215.7515.75 581222.2022.20 611224.7224.72 631227.2427.24
245 451315.8115.81 581322.2922.29 611324.8224.82 631327.3527.35
246 451415.8715.87 581422.3822.38 611424.9224.92 631427.4627.46
247 451515.9315.93 581522.4722.47 611525.0225.02 631527.5727.57
Cumulative Error:0.00Cumulative Error:0.88Cumulative Error:0.70Cumulative 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.

The chances of this being resolved conclusively are pretty slim, as it's extremely likely that everyone at Yamaha has moved on:

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:

CoarseGroupOrder
(Fine=0)
000
110
220
330
408
518
628
738
8024
9124
10040
11224
12324
13056
14140
15240
16072
17340
18156
19088
20256
21356
220104
23172
24272
250120
26188
27372
280136
29288
301104
310152
32388
332104
340168
351120
360184
373104
382120
391136
400200
413120
420216
431152
442136
450232
461168
473136
482152
491184
502168
513152
521200
532184
543168
551216
562200
573184
581232
592216
603200
612232
623216
633232

Tables into Code

So, the table at The Frequency Ratio Table has been compressed into the following code (Python):

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

coeffs = [
        [ 0.50,  0.0625   ],                                         # group 0
        [ 0.71,  0.088105 ],   # 0.50 + 0.21,  0.0625 + 0.025605     # group 1
        [ 0.78,  0.098145 ],   # 0.71 + 0.07,  0.088105 + 0.01004    # group 2
        [ 0.87,  0.108105 ],   # 0.78 + 0.09,  0.098145 + 0.00996    # group 3
    ]

def calcedRatio(coarse, fine):
    if coarse < 4 and fine >= 8:
        return None
    group = coarse_to_group[coarse][0]
    order = coarse_to_group[coarse][1] + fine
    ratio = coeffs[group][0] + coeffs[group][1] * order
    return int(ratio * 100) / 100

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.

group_to_coarse = [
        0,  4,  8, 10, 13, 16, 19, 22, 25, 28, 31, 34, 36, 40, 42, 45,  # group 0
        1,  5,  9, 14, 18, 23, 26, 30, 35, 39, 43, 46, 49, 52, 55, 58,  # group 1
        2,  6, 11, 15, 20, 24, 29, 33, 38, 44, 48, 50, 53, 56, 59, 61,  # group 2
        3,  7, 12, 17, 21, 27, 32, 37, 41, 47, 51, 54, 57, 60, 62, 63,  # group 3
    ]

coeffs = [
        [ 0.50,  0.0625   ],                                         # group 0
        [ 0.71,  0.088105 ],   # 0.50 + 0.21,  0.0625 + 0.025605     # group 1
        [ 0.78,  0.098145 ],   # 0.71 + 0.07,  0.088105 + 0.01004    # group 2
        [ 0.87,  0.108105 ],   # 0.78 + 0.09,  0.098145 + 0.00996    # group 3
    ]

def calcedRatio(coarse, fine):
    if coarse < 4 and fine >= 8:
        return None
    skip = 8 if coarse >= 4 else 0
    coarse_index = group_to_coarse.index(coarse)
    group = int(coarse_index / 16)
    order = (coarse_index - group * 16) * 16 - skip + fine
    ratio = coeffs[group][0] + coeffs[group][1] * order
    return int(ratio * 100) / 100

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.


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mgregory22@gmail.com
Last modified Sept 17, 2017