Yamaha DX11 Digital Programmable Algorithm Synthesizer

OSC Frequency Ratios

You may have wondered how the strange-looking oscillator frequency ratios (which can be chosen using the “frequency coarse” and “frequency fine” settings) are defined. It’s actually all explicable with a (more or less) simple algorithm – plus some inexplicable exceptions.

The basic idea is: use integer multiples of the four base ratios (rb) 1.00, 1.41 ≈ √( 2 ), 1.57 ≈ π/2, and 1.73 ≈ √( 3 ) as coarse starting points (selected by the “frequency coarse” setting, c = 0…63, as explained below), and add 15 intermediate fine steps between these multiples (selected by the “frequency fine” setting, f = 0…15). So, in summary, we have something like

where the (coarse) multiplier, m, and the base ratio, rb, are derived (see below) from the “frequency coarse” parameter, c.

The 64 “coarse” starting points can be obtained if you multiply each of the 4 base ratios rb = 1.00, 1.41, 1.57, and 1.73 by 16 factors or multipliers m (0…15); 4 × 16 = 64. But since a multiplier of m = 0 wouldn’t make sense (no such thing as a frequency 0 here), the 0 is replaced by ½, so the 16 multipliers are in fact m = ½, 1, 2, 3, …, 15.

These 64 coarse frequency ratios r = m × rb are then sorted by magnitude, resulting in a list L of 64 ratios from 0.50 to 25.95, which can be selected by the “frequency coarse” setting c = 0…63 – simply use c as index to the list: r = L[c].

To illustrate this point, we start with 4 separate lists of length 16 for each base factor:

and sort all these numbers into our long list L:

which gives us the frequency ratios without fine tuning r = L[c].

To these, we add the “fine” intermediate frequencies f × rb / 16, i. e. r = L[c] + f × rb(c) / 16, f = 0…15, where it is important to choose the correct base ratio rb depending on c, which I have therefore denoted by rb(c). So, there is some additional book-keeping necessary (e. g., an additional list Rb of length 64) that provides these base ratios as rb(c) = Rb[c]:

(Each entry in this list corresponds to the entry in the list L with the same index. Note that the ordering here becomes pretty mixed up starting at the bold entry.)

This is it – more or less. Now there are some simple and some inexplicable exceptions:

  1. For the 4 lowest frequency ratios (based on multiplier m = ½), only fine-tuning by f = 0…7 is used (because then, the next multiple (m = 1) is already reached for each base ratio – makes sense!).
  2. Rounding is done by simply cutting after 2 decimals (appears acceptable).
  3. The ratios based on rb = 1.41 without(!) additional fine-tuning, i. e. for f = 0, are modified by some inexplicable logic (WTF!?):
    0.705→0.710, 4.23→4.24, 5.64→5.65, 7.05→7.07, 8.46→8.48, 9.87→9.89, 11.28→11.30, 12.69→12.72, 15.51→15.55, 16.92→16.96, 18.33→18.37, 19.74→19.78, 21.15→21.20.
    And it’s only these numbers for f = 0, all others for f = 1…15 are based on the correctly calculated ratios. (If you should have any explanation for this, please send me an e-mail message to <olaf@dtrx.de>.)
  4. The ratio for m = ½ and rb = 1.73 is differently rounded: 0.86→0.87.

Based on these rules, a table of all frequency ratios can be calculated using the short python script shown below.

Table of Frequency Ratios

The following 16×64 table contains all 992 available frequency ratios with the 64 coarse settings, c, from top to bottom and the 16 fine settings, f, from left to right. (Table can be scrolled, if too wide.)

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

Reverse Lookup Table of Frequency Ratios

The following table provides the “frequency coarse” (c = 0…63) and “frequency fine” (f = 0…15) parameters for all available frequency ratios r; formatting of each entry is: “r : (cf)”.

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

The Python Script

The following short python script was used to calculate and print the two tables above.

def osc_freq_ratio(c, f, warn=True):
    """Calculate Yamaha DX11 OSC frequency ratios from coarse c, fine f."""

    # "base" ratios r_b: (about) 1, sqrt(2), pi/2, sqrt(3)
    R_b = (1.00, 1.41, 1.57, 1.73)
    # for "WTF!?" correction (only) for r_b = 1.41 at (fine) f = 0
    C_wtf = (0.5, 0, 0, 1, 1, 2, 2, 2, 2, 3, 0, 4, 4, 4, 4, 5)
    # generate list of coarse frequency ratios and sort it
    L = list()
    for n in range(16):
        m = (n if n>0 else 1/2) # multiplier
        for r_b in R_b:
            L.append((m*r_b, r_b, C_wtf[n]/100))
    L = sorted(L)
    # calculate frequency ratios r = m * r_b + f * r_b/16
    r = L[c][0] + f * L[c][1] / 16
    # apply "special" corrections for some f = 0 values
    if f == 0:
        if L[c][1] == 1.41: 
            r += L[c][2] # apply "WTF!?" correction
        if c == 3:
            r = 0.87 # whyever
    # "round" by cutting after 2 decimals
    # add 0.0001 to compensate for floating-point number inaccuracies
    return_string = f"{int(r * 100 + 0.0001) / 100:5.2f}"
    # check input parameters
    if c < 4 and f > 7:
        if warn: print("\nWARNING: invalid combination: c < 4, f > 7.")
        return_string = '   - '
    return return_string

# print table of all frequency ratios
freq_list = list()
for c in range(64):
    for f in range(16):
        freq = osc_freq_ratio(c,f, warn=False)
        print(freq, end=' ')
        if not '-' in freq:
            freq_list.append((freq, c, f))
    print('')
        
# print reverse lookup table of all frequency ratios
freq_list_sorted = sorted(freq_list)
n_freq = len(freq_list_sorted)
cols = 4
for n in range(n_freq // cols):
    for m in range(cols):
        f = freq_list_sorted[n +  m * n_freq // cols]
        print(f"{f[0]}: ({f[1]:2d},{f[2]:2d})", end='     ')
    print('')