file-name: t2dx-fm.htm • site-name: Synthesizers, Music & Television • by T. Yahaya Abdullah • site: yala.freeservers.com/ • www.geocities.com/SunsetStrip/Underground/2288 • yala0.i8.com/ • www.angelfire.com/in2/yala/ • yala.freevar.com/ •
General • Converting between FM synths • Enharmonic Detuned Frequencies • about DX Operators • DX Spectrum Amplitudes • Modulation Index Calculation • DX Out - Mod.Index Table • Deviation • Bessel Functions • Spectrum Amplitudes • Two or more Modulators • FM Bessel Tables • Miscellaneous • Recommended Reading • A Personal History of FM •
Conventions Used
* = multiplication [ 2 * 3 = 6] ^ = to-the-power-of [ 2 ^ {3} = 8, curly brackets {} show the power] PI used is 3.14159265358979 [Circumference of Circle = 2*PI*radius]
Converting between FM synths
Direct conversion between the different FM synths can be a bit tricky. Below are some conversion tables for the various FM synths. Please note that there are limitations to conversion (eg Conversion from 6-op to 4-op ~or~ from a complex Rate/Level envelope to ADSDR may not be ideal).
MODULATION OUTPUT CONVERSION Orig : 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 X : 3 6 11 15 19 23 28 33 38 43 48 53 58 63 68 CX : 21 32 42 49 54 59 64 69 74 79 84 89 94 99 104 continued... Orig : 80 82 84 86 88 90 91 92 93 94 95 96 97 98 99 X : 73 75 77 79 81 83 84 85 86 87 88 89 90 91 92 CX : 109 111 113 115 117 119 120 121 122 123 124 125 126 127 127 ENVELOPE PARAMETERS Attack (A) ~ Rate (R) Conversion DX-7 R: 15 21 27 34 40 47 54 60 67 74 80 85 89 93 96 99 DX-21 A: 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 Decay (D) ~ Rate (R) Conversion DX-7 R: 10 16 21 27 33 39 45 51 57 63 69 75 81 87 93 99 DX-21 D: 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 Sustain (S) ~ Level (L) Conversion DX-7 L: 35 39 44 48 53 57 62 66 71 75 80 84 89 93 99 DX-21 S: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Release (R bottom) ~ Rate (R top) Conversion DX-7 R: 21 27 32 38 43 49 54 60 65 71 76 82 87 94 99 DX-21 R: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Certain parameters like Feedback and Frequency are the same (although Frequency may take some fiddling to convert). However, most other parameters are not the same.
Enharmonic Detuned Frequencies
The frequency of an operator is dependent on 3 parameters (1) the Coarse Frequency, (2) the Fine Frequency, and (3) Detune. So far, we have dealt mainly with the Coarse Frequency which is the main integer ratio of the base frequency (eg M:C = 2:1). We have also looked at Detune and its effects where (a) detuning the Carrier shifts the entire spectrum, and (b) detuning the Modulator changes the separation between the sidebands.
FM synths will have some form of Fine Frequency. The Fine Frequency allows the selection of non-integer multiples of the base frequency. Normally, this would yield something clangorous or enharmonic. This brings another dimension into FM sounds because a new set of overtones are introduced. Typically, this would be used for bell-type or percussion sounds. You can also obtain very unique and strange timbres too (a great source of experimentation).
Specifically for the original DX-7 (and DX-9) range, the Fine Frequency can also be used to obtain "extra" detuning for the operator. Basically the Fine Frequency had a range of 0~99 where each increment increased the Frequency by 1 percent. The table below shows the frequencies which are within 0.05 of a Coarse frequency; achieved by a combination of the CF [Coarse Frequency selected] and FF [Fine Frequency increments].
Freq CF FF Freq CF FF Freq CF FF Freq CF FF Freq CF FF 0.505 0.5 1 4.95 3 65 11.97 7 71 18.96 12 58 25.95 15 73 0.510 0.5 2 4.96 4 24 11.97 9 33 18.98 13 46 25.96 22 18 0.515 0.5 3 4.98 3 66 11.99 11 9 19.03 11 73 25.99 23 13 0.520 0.5 4 5.01 3 67 12.04 7 72 19.04 14 36 26.01 17 53 0.525 0.5 5 5.04 4 26 ---- -- -- 19.04 16 19 26.03 19 37 0.530 0.5 6 5.04 3 68 12.95 7 85 19.04 17 12 26.04 14 86 0.535 0.5 7 5.05 5 1 12.96 9 44 19.05 15 27 26.04 21 24 0.540 0.5 8 ---- -- -- 12.96 8 62 ---- -- -- ---- -- -- 0.545 0.5 9 5.95 5 19 12.96 12 8 19.95 19 5 26.98 19 42 0.550 0.5 10 5.96 4 49 12.98 11 18 19.95 15 33 27.02 14 93 ---- --- -- 5.97 3 99 13.02 7 86 19.98 18 11 27.03 17 59 0.950 0.5 90 6.04 4 51 13.04 8 63 20.02 11 82 27.04 16 69 0.955 0.5 91 6.05 5 21 13.05 9 45 20.02 13 54 27.04 26 4 0.960 0.5 92 ---- -- -- ---- -- -- 20.02 14 43 ---- -- -- 0.965 0.5 93 6.95 5 39 13.95 9 55 20.04 12 67 28.05 17 65 0.970 0.5 94 6.96 6 16 13.97 11 27 ---- -- -- 28.05 15 87 0.975 0.5 95 6.96 4 74 14.04 12 17 20.96 16 31 ---- -- -- 0.980 0.5 96 7.02 6 17 14.04 9 56 21.01 11 91 28.95 15 93 0.985 0.5 97 7.04 4 76 14.04 13 8 ---- -- -- 28.96 16 81 0.990 0.5 98 7.05 5 41 ---- -- -- 21.96 12 83 28.98 18 61 0.995 0.5 99 ---- -- -- 14.95 13 15 21.96 18 22 28.98 21 38 1.01 1 1 7.95 5 59 14.96 11 36 21.97 13 69 28.98 23 26 1.02 1 2 7.96 4 99 14.96 8 87 21.98 14 57 29.04 24 21 1.03 1 3 7.98 6 33 14.98 14 7 22.04 19 16 29.04 22 32 1.04 1 4 7.98 7 14 15.03 9 67 22.05 15 47 ---- -- -- 1.05 1 5 8.04 6 34 15.04 8 88 22.05 21 5 29.96 28 7 ---- --- -- 8.05 5 61 ---- -- -- ---- -- -- 29.97 27 11 1.95 1 95 8.05 7 15 15.95 11 45 22.95 15 53 30.02 19 58 1.96 1 96 ---- -- -- 15.96 12 33 22.95 17 35 30.03 21 43 1.97 1 97 8.95 5 79 15.96 14 14 22.96 14 64 ---- -- -- 1.98 1 98 8.96 7 28 15.99 13 23 22.99 19 21 30.96 18 72 1.99 1 99 8.96 8 12 16.02 9 78 23.01 13 77 30.96 24 29 2.02 2 1 9.03 7 29 16.05 15 7 23.04 12 92 30.97 19 63 2.04 2 2 9.04 8 13 ---- -- -- 23.04 16 44 31.02 22 41 ---- --- -- 9.05 5 81 16.95 15 13 23.04 18 28 31.03 29 7 2.96 2 48 ---- -- -- 16.96 16 6 ---- -- -- 31.04 16 94 2.98 2 49 9.95 5 99 17.01 9 89 23.97 17 41 31.05 27 15 3.02 2 51 9.96 6 66 17.03 13 31 23.98 22 9 31.05 23 35 3.03 3 1 9.99 9 11 17.04 12 42 24.05 13 85 3.04 2 52 10.01 7 43 17.05 11 55 ---- -- -- ---- --- -- 10.02 6 67 ---- -- -- 24.96 13 92 3.96 2 98 ---- -- -- 18.02 17 6 24.96 16 56 3.96 3 32 10.96 8 37 18.04 11 64 24.96 24 4 3.98 2 99 10.98 6 83 24.99 17 47 3.99 3 33 10.98 9 22 24.99 21 19 4.02 3 34 10.99 7 57 25.02 18 39 4.04 4 1 11.04 6 84 25.05 15 67 4.05 3 35 11.04 8 38
These "near integer" or "extra detuned" frequencies are not available for the other DX-21 variants nor the CX variants.
about DX Operators
A DX-Synth actually calculates the entire algorithm and operator arrangement and outputs the final waveform calculation into a D/A converter for conversion into voltage. The entire process is handled digitally (ie it's one massive calculation).
Each operator has 2 inputs : (1) Pitch Data input, and (2) Modulation Data input. Both information is handled by an input buffer which supplies this information to an Oscillator (Waveform calculator). This Waveform calculation is further processed by an Amplifier (Magnitude calculator) which is controlled from an Envelope Generator. The destination of the final out put depends on the position of the operator in an algorithm. If it is a modulator, then the information is passed down the chain. If it is a carrier, then the information is ready for D/A conversion (actually it's not quite ready as there may be more carriers in the algorithm which then need to be summed together).
Modulation Index Calculation
We already know about how sidebands are generated from a M:C combination. The amplitudes of each order of Sideband is determined by the Modulator's Output level. Before we can work out the amplitudes, we need to convert the Modulator's Output into a reference calculation known as Modulation Index.
The first step is to convert Output level into a TL number (doesn't apply to CX-5 and FB-01... see below). The reason for this is that the Output is non-linear and actually goes through a look-up table. So to get a proper linear output, we have to look-up the correct TL number [TL numbers have a negative relationship to the output levels].
For all Output values beyond 19, use the formula:- [TL] = 99 - [Out] For Output values from 0 to 19, use this table:-
DX.Out 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 TL.val 127 122 118 114 110 107 104 102 100 98 96 94 92 90 88 86 85 84 82 81
Now we can convert it into Modulation Index. The formula for conversion differs for each ganeration of FM-synth and are as follows:-
For the classic/ original range of DX-synths ~ DX-1/DX-5/DX-7/DX-9/TX-7.
MODULATION INDEX I = (PI) * 2 to-the-power-of ( (33/16) - (TL/8) )
For the next generation DX-synths ~ DX-21/DX-27/DX-100.
MODULATION INDEX I = (8 x PI) * 2 to-the-power-of ( -TL / 8 )
For the computer range CX-5/FB-01 (No conversion into TL numbers).
MODULATION INDEX I = (8 x PI) * 2 to-the-power-of ( (OUT-135) / 8)
NOTE - In case you want to convert from "I" back into TL values (or OUT values), the trick is to take LOGS to-the-base 2 on each side of the equation.
NOTE - Don't forget that the output is controlled from an Envelope Generator. This means that changes in Level settings over time (ie an envelope shape) results in a change in "I" (Mod Index). The original DX-synths have envelope Levels from 0~99.
DX Output ~to~ Modulation Index Table
Below is the table of Modulation Indices for each of the FM synth types:-
------Modulation Indices------- ------Modulation Indices------- -Mod Indx- OUT TL --[DX7]-- --[DX21]- --[CX5]-- OUT TL --[DX7]-- --[DX21]- --[CX5]-- OUT --[CX5]-- 0 127 0.000218 0.000418 0.000209 50 49 0.188025 0.360107 0.015915 100 1.211250 1 122 0.000337 0.000645 0.000228 51 48 0.205043 0.392699 0.017355 101 1.320877 2 118 0.000476 0.000912 0.000249 52 47 0.223601 0.428241 0.018926 102 1.440427 3 114 0.000674 0.001290 0.000271 53 46 0.243838 0.467001 0.020639 103 1.570796 4 110 0.000952 0.001824 0.000296 54 45 0.265907 0.509268 0.022507 104 1.712966 5 107 0.001235 0.002366 0.000322 55 44 0.289974 0.555360 0.024544 105 1.868002 6 104 0.001602 0.003068 0.000352 56 43 0.316219 0.605625 0.026765 106 2.037071 7 102 0.001905 0.003648 0.000383 57 42 0.344839 0.660439 0.029188 107 2.221441 8 100 0.002265 0.004339 0.000418 58 41 0.376050 0.720213 0.031829 108 2.422499 9 98 0.002694 0.005160 0.000456 59 40 0.410085 0.785398 0.034710 109 2.641754 10 96 0.003204 0.006136 0.000497 60 39 0.447201 0.856483 0.037852 110 2.880853 11 94 0.003810 0.007297 0.000542 61 38 0.487676 0.934001 0.041277 111 3.141593 12 92 0.004531 0.008678 0.000591 62 37 0.531815 1.018535 0.045013 112 3.425931 13 90 0.005388 0.010319 0.000645 63 36 0.579948 1.110721 0.049087 113 3.736004 14 88 0.006408 0.012272 0.000703 64 35 0.632438 1.211250 0.053530 114 4.074142 15 86 0.007620 0.014594 0.000767 65 34 0.689679 1.320877 0.058375 115 4.442883 16 85 0.008310 0.015915 0.000836 66 33 0.752100 1.440427 0.063658 116 4.844998 17 84 0.009062 0.017355 0.000912 67 32 0.820171 1.570796 0.069420 117 5.283508 18 82 0.010776 0.020639 0.000995 68 31 0.894403 1.712966 0.075703 118 5.761706 19 81 0.011752 0.022507 0.001085 69 30 0.975353 1.868002 0.082555 119 6.283185 20 79 0.013975 0.026765 0.001183 70 29 1.063630 2.037071 0.090027 120 6.851862 21 78 0.015240 0.029188 0.001290 71 28 1.159897 2.221441 0.098175 121 7.472009 22 77 0.016619 0.031829 0.001407 72 27 1.264876 2.422499 0.107060 122 8.148283 23 76 0.018123 0.034710 0.001534 73 26 1.379357 2.641754 0.116750 123 8.885766 24 75 0.019764 0.037852 0.001673 74 25 1.504200 2.880853 0.127317 124 9.689996 25 74 0.021552 0.041277 0.001824 75 24 1.640341 3.141593 0.138840 125 10.567016 26 73 0.023503 0.045013 0.001989 76 23 1.788805 3.425931 0.151406 126 11.523413 27 72 0.025630 0.049087 0.002169 77 22 1.950706 3.736004 0.165110 127 12.566371 28 71 0.027950 0.053530 0.002366 78 21 2.127260 4.074142 0.180053 29 70 0.030480 0.058375 0.002580 79 20 2.319793 4.442883 0.196350 30 69 0.033238 0.063658 0.002813 80 19 2.529752 4.844998 0.214121 31 68 0.036247 0.069420 0.003068 81 18 2.758714 5.283508 0.233500 32 67 0.039527 0.075703 0.003346 82 17 3.008399 5.761706 0.254634 33 66 0.043105 0.082555 0.003648 83 16 3.280683 6.283185 0.277680 34 65 0.047006 0.090027 0.003979 84 15 3.577610 6.851862 0.302812 35 64 0.051261 0.098175 0.004339 85 14 3.901411 7.472009 0.330219 36 63 0.055900 0.107060 0.004731 86 13 4.254519 8.148283 0.360107 37 62 0.060960 0.116750 0.005160 87 12 4.639586 8.885766 0.392699 38 61 0.066477 0.127317 0.005627 88 11 5.059505 9.689996 0.428241 39 60 0.072494 0.138840 0.006136 89 10 5.517429 10.567016 0.467001 40 59 0.079055 0.151406 0.006691 90 9 6.016799 11.523413 0.509268 41 58 0.086210 0.165110 0.007297 91 8 6.561366 12.566371 0.555360 42 57 0.094012 0.180053 0.007957 92 7 7.155220 13.703724 0.605625 43 56 0.102521 0.196350 0.008678 93 6 7.802823 14.944017 0.660439 44 55 0.111800 0.214121 0.009463 94 5 8.509039 16.296566 0.720213 45 54 0.121919 0.233500 0.010319 95 4 9.279172 17.771532 0.785398 46 53 0.132954 0.254634 0.011253 96 3 10.119009 19.379993 0.856483 47 52 0.144987 0.277680 0.012272 97 2 11.034858 21.134032 0.934001 48 51 0.158110 0.302812 0.013383 98 1 12.033598 23.046825 1.018535 49 50 0.172420 0.330219 0.014594 99 0 13.122731 25.132741 1.110721
Note - When Output=0, it should really be Index=Zero. But these are the equations given and so we'll use these figures anyway.
Deviation
Before we calculate the amplitudes of the sidebands, you might want to know what the Modulation Index actually is.
FM is a Modulator modulating the Pitch of a Carrier. Think of the Carrier being a "centre frequency" and think of the Modulation as "shifting the carrier to up and down in terms of pitch" (frequency). This occurs over time at the rate "M" (the Modulator frequency). This shifting of the carrier up and down is sinusoidal over time (since the Modulator is a Sine wave).
The Deviation is the difference between Carrier and the the lowest or highest instantaneous frequency. Let's say the Carrier is at 100Hz. The shifting could cause the Carrier to oscillate between say 90Hz (lowest freq) and 110Hz (highest freq). The width of oscillation from the Carrier to either extreme is the Deviation, in this case, 10Hz.
The relationship to Modulation Index is as follows:-
"d" = Deviation d "I" = Modulation Index I = --- ~or~ d = I * M "M" = Modulator Frequency M
So we can think of the Modulation Index as the "change in pitch" relative to the Modulator pitch (frequency).
Bessel Functions
"Order" refers to the distance of any Sideband from the Carrier expressed in multiples of "M". So C+M and C-M are the first order sidebands, C+2M and C-2M are the second order sidebands... etc.
Given you know "I" (Mod.Index), you can calculate the amplitudes of each Sideband generated by "C" and "M" using a Bessel Function. We use the letter "J" to denote a Bessel function, and they are ordered as J0, J1, J2, J3, J4 etc
Bessel function J0(I) = Amplitude for the Carrier, "C".
Bessel function Jn(I) = Amplitude for the Sidebands at "C + nM" and "C - nM" (where n = 1, 2, 3, 4... etc ).
Jn(I) (the nth order of "J") is a Bessel function of "I" (index). The Bessel function is expressed as:- inf (-1)^{k} * (index/2)^{n+2k} Jn(index) = SUM --------------------------- k=0 k! * (n + k)! Also: Jn+1(index) = (2n/index) * Jn(index) - Jn-1(index)
I shall not even pretend to understand the logic for Bessel functions. However, note that the Bessel function is in 2 parts: The sum (sigma) portion and the algebra portion. This means that you start with k=0 in the algebra portion, then do it again for k=1... then k=2, up to k=infinity. Jn(index) is the sum of all these numbers. Basically, each increase in "k" takes the sum one step closer to the final answer. Thankfully, there comes a point where further increase in "k" becomes fairly insignificant. Let's look at an example:-
Calculating the zero order (n=0), where Modulation Index = 1
k=0, (-1)^{0} * (1/2)^{0+2*0} 1 ------------------------ = ------ = 1 0! * (0+0)! 1 k=1, (-1)^{1} * (1/2)^{0+2*1} -0.25 ------------------------ = ------ = -0.25 1! * (0+1)! 1 k=2, (-1)^{2} * (1/2)^{0+2*2} 0.0625 ------------------------ = ------ = 0.015625 2! * (0+2)! 4 Sum for k=0 to 2, we obtain 1 - 0.25 + 0.015625 = 0.765625 If we were carry on with more "k", we should get 0.765197684 eventually.
Fortunately, some spreadsheets include this feature [in Excel, the formula is "=BesselJ(index,order)" - you will need to activate the Analysis Toolpack under Tools/AddIns]. The Bessel function tables can be found near the end of this article.
Spectrum Amplitudes
Having calculated the Bessel functions, you now have to assign the Jn(index) to the relevant harmonics as follows:-
Freq : C C+M C+2M C+3M C+4M C+5M C+6M etc Amplitude: J0(I) J1(I) J2(I) J3(I) J4(I) J5(I) J6(I) etc Freq : C-M C-2M C-3M C-4M C-5M C-6M etc Amplitude: J1(I) J2(I) J3(I) J4(I) J5(I) J6(I) etc
For a graphical representation of Bessel functions, see FM Spectrum Graphs (contains animated GIFs). It displays the Amplitudes of each Order as the Modulation Index is increased.
IMPORTANT - Where the Sidebands are reflected (ie "C - nM" becomes negative), the phase is inverted. Instead of treating it as a negative frequency with amplitude Jn(index), you should treat it as a positive frequency with a negative amplitude Jn(index). In short, reflected Sidebands = inverted phase = negative amplitude.
This is especially important if you are dealing with conicidental refected Sidebands. If you refer to the previous article "FM Synthesis", the M:C Series of 1:1 and 2:1 (and their permutations) have reflected Sidebands which are coincidental with the non-reflected Sidebands. Where the Sidebands are coincidental, just remember that the reflected Sidebands are phase inverted and hence will have negative values.
Here's an example of M:C = 2:3 on a DX-7 with Out=75:- Freq : 3 5 7 9 11 13 etc Amplitude: 43% 57% 27% 8% 2% 0% ignore below 0.5% Freq : 1 1 3 5 7 etc Amplitude: 57% -27% -8% -2% -0% ignore below 0.5% The resultant spectrum would be:- Freq : 1 2 3 4 5 6 7 8 9 10 11 etc Amplitude: 30% - 35% - 55% - 27% - 8% - 2% etc
Note - Bessel functions can in themselves yield negative values (phase inverted). For a DX-7, when Out > 79, some orders are negative. In short, you need to be a bit careful in assigning amplitude values.
Note - It is important to establish for yourself the number of decimal places to use. This is to define what value of amplitude of Sidebands are significant. I would recommend using either 2 or 3 decimal places (beyond which the Sidebands can be disregarded). The following table gives values below which there are no significant Sidebands (ie they result in Carrier only) :-
No significant Sidebands index DX7 Out DX21 Out CX5 Out Using 2 decimal places 0.009 17 12 44 Using 3 decimal places 0.0009 4 2 17
Most of the examples in this article use 2 decimal places as significant (ie anything below 0.005 or 0.5% is ignored).
For a graphical representation of FM Spectrums, see FM Spectrum Graphs (contains animated GIFs). It displays the amplitudes for various M:C combinations as the Modulation Index is increased. It also shows the effects of reflected Sidebands for coincident and non-coincident M:C combinations.
Two or more Modulators
So far, we have only dealt with M:C ; single sine-modulator to single sine-carrier. When there are two Modulators, they can either be "Two-Into-One" (M1 + M2 : C) or "In-Series" (M2 : M1: C).
M1 + M2 : C (ie two separate Modulators)
M1--->-+->-C M2--->-+
Basically, you will end up with "M1:C" and "M2:C" added together.
Let's try a DX-7 example with M1+M2:C with 3 + 1 : 1 and let's use M1 Out=70 and M2 Out=80
For M1:C = 3 : 1 with M1 Out=70, we get:- Freq : 1 4 7 10 13 16 etc Amplitude: 74% 46% 13% 2% 0% 0% less than 0.5% Freq : 2 5 8 11 14 etc Amplitude: -46% -13% -2% -0% -0% less than 0.5% For M2:C = 1 : 1 with M2 Out=80, we get:- Freq : 1 2 3 4 5 6 7 etc Amplitude: -6% 49% 45% 22% 7% 2% 0% ignore Freq : 0 1 2 3 4 5 etc Amplitude: 49% -45% -22% -7% -2% -0% ignore The resultant spectrum would be adding the results:- Freq : 1 2 3 4 5 6 7 8 9 10 11 12 13 etc Amp. M1:C : 74% -46% - 46% -13% - 13% -2% - 2% -0% - 0% etc Amp. M2:C : -51% 27% 38% 20% 7% 2% 0% - - - - - - etc Amplitude : 23% -19% 38% 66% -6% 2% 13% -2% - 2% 0% - 0% etc
In reality, the overall amplitudes would be factored-down as part of the algorithm calculation so as not to overload the Carrier's input.
Note that it is "convenient" to add up the amplitudes for high amounts of modulation (which results in significant Sidebands) but this is not always correct for the Carrier amplitude. The correct way to approach this is really to think of how much energy is taken from the Carrier and transferred to the Sidebands. For example in M2+M1:C, using M2 Out=17 would result in a Carrier amplitude of 100% and 1st-order Sidebands with amplitudes of less than 1%. In this case, adding up "M2:C" (Carrier amplitude only) with "M1:C" (Carrier and Sidebands) is not representative of what is happening. Don't forget that the Carrier exists unchanged (ie 100%) when there is no modulator.
M2 : M1 : C (ie two Modulators in series)
M2--->---M1--->---C
This one is a lot more complicated. Basically, "M2:M1" will produce one complex waveform and each sine-frequency (in the harmonic spectrum) will act as a sine-modulator into "C". Let's try a simplified DX-7 example with M2:M2:C with 3 : 2 : 3 and let's use M2 Out=75 and M1 Out=90.
For M2:M1 = 3 : 2 with M2 Out=75, we get J0=43%, J1=57%, J2=27%, J3=8%, and J4=2%. We can predict the outcome as being:- Freq: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 etc Ampl: -57% 43% - -27% 57% - -8% 27% - -2% 8% - -0% 2% etc
So each of these sine-frequencies acts a modulators into "C".
WARNING! The next few calculations make certain assumptions about the inner-workings of the DX-chipsets. Quite frankly, I am not exactly sure of this part.
Let's normalise these sine-frequencies as a factor of M1's output (M1 Out=90).
Freq : F1 F2 F4 F5 F7 F8 F10 F11 F13 F14 Ampl : -57% 43% -27% 57% -8% 27% -2% 8% -0% 2% Ampl*90: -51.3 38.7 -24.3 51.3 -7.2 24.3 -1.8 7.2 - 1.8
At this point I would disregard any "Ampl*M1" below 35 as I have precalculated that they would be too small to produce any significant Sidebands (ie Carrier only) for 2 decimal places.
Next, we have to convert them into TL numbers linearly. Plus, I'm going to round them to the nearest integer.
TL(F1) would be 127 - (-51.3*127/99) = -61
TL(F2) would be 127 - (38.7*127/99) = 77
TL(F5) would be 127 - (51.3*127/99) = 61
If you compare these TL numbers with the Mod.Index table, we can look up the equivalent Output value (ie Out equivalents are -38, 22 and 38 respectively).
For F1:C = 1:3 with TL(F1)= -61 (similar to DX-Out=-38) Freq : 1 2 3 4 5 6 etc Amplitude: -0% -3% -100% -3% -0% -0% ignore For F2:C = 1:3 with TL(F2)= 77 (similar to DX-Out=22) Freq : 1 2 3 4 5 6 etc Amplitude: 1% - 100% - 1% - ignore For F5:C = 5:1 with TL(F5)= 61 (similar to DX-Out=38):- Freq : 1 2 3 4 5 6 7 8 etc Amplitude: - -1% 100% - - - -0% 1% ignore This results in Freq : 1 2 3 4 5 6 7 8 etc Amplitude: 1% -4% 100% -3% 1% -0% -0% 1% ignore
In this case, there really that much to add up especially since the Carriers are hardly affected (apart from one inverted Carrier in F1:C). As previously mentioned, we have to look at the process as the modulation taking energy away from the Carrier to produce Sidebands. Adding up convenient for Sidebands but not always applicable for the Carrier amplitude.
Using a different M2:M1 would yield a totally result. Also using different modulation outputs would change the Sideband amplitudes greatly. I purposely chose a small output for M2:M1 otherwise the number of significant Sinewave components would be very great indeed. As you can appreciate, the "In-Series" modulators calculation can become very complicated and perhaps confusing too. Furthermore, the results may not quite be what you imagined.
More Modulators and FM Algorithms
The algorithms available on FM synths can be fairly elaborate but, in general, you can analyse them into combinations of "two-into-one" or "in-series". This makes the calculation of frequencies and their amplitudes far too difficult. Perhaps this is why this kind of information is usually not presented at all. You need to ask yourself if this is really worth all the effort.
However, I would recommend scanning through the tables below to get a feel for the numbers of orders generated by the different outputs.
Actual DX algorithms can be found in article Synthesizer Layouts.
FM Bessel Tables
Below are Bessel tables for the DX-7 type, the DX-21 type and the CX-5 type synths. The DX-7 table is calculated to 6 decimal places to illustrate how the amplitudes quickly diminish into insignificance as we progress up the orders. As such, for the DX-21 type and CX-5 type synths, the tables are calculated to 4 decimal places.
Every table contains some small rounding errors. However, the obvious errors will be near Out=Zero where J0=1 and all other orders should be Zero. This arises from the Mod.Index imprecisions.
DX-7 Output level ~to~ Bessel function [Jn] Table - up to 6 decimal places.
OUT --J0-- --J1-- OUT --J0-- --J1-- --J2-- OUT --J0-- --J1-- --J2-- 0 1.000000 0.000109 8 0.999999 0.001133 0.000001 18 0.999971 0.005388 0.000015 1 1.000000 0.000168 9 0.999998 0.001347 0.000001 19 0.999965 0.005876 0.000017 2 1.000000 0.000238 10 0.999997 0.001602 0.000001 20 0.999951 0.006987 0.000024 3 1.000000 0.000337 11 0.999996 0.001905 0.000002 21 0.999942 0.007620 0.000029 4 1.000000 0.000476 12 0.999995 0.002265 0.000003 22 0.999931 0.008309 0.000035 5 1.000000 0.000618 13 0.999993 0.002694 0.000004 23 0.999918 0.009061 0.000041 6 0.999999 0.000801 14 0.999990 0.003204 0.000005 24 0.999902 0.009881 0.000049 7 0.999999 0.000952 15 0.999985 0.003810 0.000007 25 0.999884 0.010776 0.000058 16 0.999983 0.004155 0.000009 26 0.999862 0.011751 0.000069 17 0.999979 0.004531 0.000010 27 0.999836 0.012814 0.000082 28 0.999805 0.013974 0.000098 OUT --JO-- --JI-- --J2-- --J3-- OUT --J0-- --J1-- --J2-- --J3-- --J4-- 29 0.999768 0.015238 0.000116 0.000001 42 0.997792 0.046954 0.001104 0.000017 30 0.999724 0.016617 0.000138 0.000001 43 0.997374 0.051193 0.001313 0.000022 31 0.999672 0.018120 0.000164 0.000001 44 0.996878 0.055813 0.001561 0.000029 --J4-- 32 0.999609 0.019760 0.000195 0.000001 45 0.996287 0.060846 0.001856 0.000038 0.000001 33 0.999536 0.021547 0.000232 0.000002 46 0.995586 0.066330 0.002206 0.000049 0.000001 34 0.999448 0.023497 0.000276 0.000002 47 0.994752 0.072303 0.002623 0.000063 0.000001 35 0.999343 0.025622 0.000328 0.000003 48 0.993760 0.078808 0.003118 0.000082 0.000002 36 0.999219 0.027939 0.000391 0.000004 49 0.992582 0.085890 0.003707 0.000107 0.000002 37 0.999071 0.030466 0.000464 0.000005 50 0.991181 0.093598 0.004406 0.000138 0.000003 38 0.998896 0.033220 0.000552 0.000006 51 0.989517 0.101984 0.005237 0.000179 0.000005 39 0.998687 0.036223 0.000657 0.000008 52 0.987540 0.111103 0.006224 0.000232 0.000006 40 0.998438 0.039497 0.000781 0.000010 53 0.985191 0.121015 0.007395 0.000301 0.000009 41 0.998143 0.043065 0.000928 0.000013 54 0.982401 0.131782 0.008786 0.000390 0.000013 OUT --JO-- --JI-- --J2-- --J3-- --J4-- --J5-- --J6-- --J7-- --J8-- 55 0.979089 0.143468 0.010437 0.000505 0.000018 0.000001 56 0.975157 0.156141 0.012395 0.000655 0.000026 0.000001 57 0.970492 0.169869 0.014718 0.000848 0.000037 0.000001 58 0.964958 0.184721 0.017469 0.001098 0.000052 0.000002 59 0.958397 0.200763 0.020728 0.001422 0.000073 0.000003 60 0.950624 0.218057 0.024585 0.001840 0.000103 0.000005 61 0.941421 0.236661 0.029144 0.002381 0.000146 0.000007 62 0.930533 0.256617 0.034527 0.003079 0.000205 0.000011 --J6-- 63 0.917666 0.277953 0.040876 0.003979 0.000290 0.000017 0.000001 64 0.902478 0.300670 0.048351 0.005140 0.000408 0.000026 0.000001 65 0.884575 0.324739 0.057135 0.006634 0.000575 0.000040 0.000002 66 0.863508 0.350080 0.067432 0.008554 0.000810 0.000061 0.000004 67 0.838770 0.376556 0.079469 0.011019 0.001139 0.000094 0.000006 --J7-- 68 0.809791 0.403950 0.093493 0.014175 0.001601 0.000144 0.000011 0.000001 69 0.775944 0.431938 0.109763 0.018208 0.002247 0.000221 0.000018 0.000001 70 0.736553 0.460072 0.128544 0.023345 0.003149 0.000338 0.000030 0.000002 71 0.690906 0.487735 0.150091 0.029867 0.004405 0.000517 0.000050 0.000004 --J8-- 72 0.638284 0.514114 0.174623 0.038109 0.006150 0.000789 0.000084 0.000008 0.000001 73 0.578004 0.538153 0.202292 0.048475 0.008565 0.001201 0.000140 0.000014 0.000001 74 0.509483 0.558520 0.233131 0.061428 0.011893 0.001824 0.000232 0.000025 0.000002 75 0.432336 0.573565 0.266988 0.077491 0.016455 0.002762 0.000384 0.000046 0.000005 76 0.346495 0.581302 0.303438 0.097224 0.022672 0.004169 0.000634 0.000082 0.000009 77 0.252390 0.579415 0.341666 0.121186 0.031077 0.006266 0.001043 0.000148 0.000018 78 0.151156 0.565312 0.380338 0.149858 0.042340 0.009370 0.001708 0.000265 0.000036 79 0.044890 0.536256 0.417441 0.183534 0.057259 0.013929 0.002784 0.000473 0.000070 80 -0.063063 0.489599 0.450135 0.222147 0.076747 0.020556 0.004511 0.000840 0.000136 81 -0.167841 0.423182 0.474638 0.265020 0.101760 0.030074 0.007254 0.001481 0.000262 82 -0.262887 0.335919 0.486208 0.310548 0.133154 0.043537 0.011565 0.002593 0.000504 83 -0.339957 0.228591 0.479313 0.355815 0.171433 0.062226 0.018242 0.004500 0.000960 84 -0.389527 0.104837 0.448134 0.396206 0.216341 0.087561 0.028407 0.007722 0.001810 85 -0.401787 -0.027801 0.387535 0.425130 0.266273 0.120875 0.043550 0.013077 0.003375 86 -0.368493 -0.157054 0.294664 0.434090 0.317518 0.162956 0.065501 0.021791 0.006206 87 -0.285797 -0.265834 0.171203 0.413436 0.363460 0.213275 0.096226 0.035608 0.011220 88 -0.157918 -0.333651 0.026027 0.354228 0.394047 0.268832 0.137294 0.056799 0.019872 89 -0.000902 -0.340427 -0.122499 0.251618 0.396125 0.322743 0.188827 0.087942 0.034318 90 0.155265 -0.273345 -0.246126 0.109719 0.355539 0.363009 0.247787 0.131181 0.057448 91 0.268997 -0.136238 -0.310524 -0.053066 0.261998 0.372510 0.305734 0.186643 0.092507 92 0.297214 0.041363 -0.285652 -0.201051 0.117061 0.331933 0.346843 0.249756 0.141833 93 0.214839 0.201891 -0.163090 -0.285497 -0.056443 0.227627 0.348168 0.307821 0.204132 94 0.039470 0.273200 0.024744 -0.261568 -0.209184 0.064898 0.285453 0.337667 0.270113 95 -0.153442 0.204107 0.197435 -0.118998 -0.274380 -0.117558 0.147691 0.308554 0.317842 96 -0.249334 0.013629 0.252028 0.085997 -0.201036 -0.244935 -0.041017 0.196292 0.312595 97 -0.164935 -0.182076 0.131935 0.229901 -0.006930 -0.234925 -0.205964 0.010948 0.219853 98 0.055158 -0.221098 -0.091905 0.190548 0.186913 -0.066288 -0.241998 -0.175035 0.038361 99 0.213943 -0.043921 -0.220636 -0.023333 0.209968 0.151335 -0.094645 -0.237883 -0.159141 continuing across... OUT --J9-- --J10- --J11- --J12- --J13- --J14- --J15- --J16- --J17- 76 0.000001 77 0.000002 78 0.000004 --J10- 79 0.000009 0.000001 80 0.000019 0.000002 --J11- 81 0.000041 0.000006 0.000001 82 0.000086 0.000013 0.000002 --J12- 83 0.000180 0.000030 0.000005 0.000001 84 0.000373 0.000069 0.000011 0.000002 --J13- 85 0.000765 0.000155 0.000028 0.000005 0.000001 86 0.001548 0.000344 0.000069 0.000013 0.000002 --J14- 87 0.003087 0.000755 0.000166 0.000033 0.000006 0.000001 --J15- 88 0.006043 0.001629 0.000395 0.000087 0.000018 0.000003 0.000001 89 0.011576 0.003447 0.000921 0.000223 0.000049 0.000010 0.000002 --J16- 90 0.021585 0.007126 0.002102 0.000561 0.000137 0.000031 0.000006 0.000001 --J17- 91 0.038937 0.014309 0.004679 0.001381 0.000372 0.000092 0.000021 0.000005 0.000001 92 0.067400 0.027723 0.010090 0.003301 0.000982 0.000268 0.000068 0.000016 0.000004 93 0.110760 0.051376 0.020924 0.007621 0.002515 0.000760 0.000212 0.000055 0.000013 94 0.170241 0.090014 0.041333 0.016850 0.006194 0.002077 0.000641 0.000184 0.000049 95 0.239498 0.146743 0.076787 0.035311 0.014543 0.005437 0.001865 0.000591 0.000175 96 0.297977 0.217456 0.131820 0.069136 0.032157 0.013487 0.005164 0.001821 0.000596 97 0.307828 0.282275 0.203778 0.123993 0.065898 0.031273 0.013455 0.005308 0.001936 98 0.226040 0.299753 0.272153 0.197801 0.122344 0.066539 0.032479 0.014432 0.005900 99 0.043850 0.219288 0.290361 0.267496 0.198860 0.126504 0.071062 0.035952 0.016607 continuing across even further... OUT --J18- --J19- --J20- --J21- --J22- --J23- --J24- --J25- --J26- 92 0.000001 --J19- 93 0.000003 0.000001 --J20- 94 0.000012 0.000003 0.000001 --J21- 95 0.000048 0.000013 0.000003 0.000001 --J22- 96 0.000182 0.000052 0.000014 0.000004 0.000001 --J23- 97 0.000658 0.000209 0.000063 0.000018 0.000005 0.000001 --J24- 98 0.002237 0.000792 0.000263 0.000082 0.000024 0.000007 0.000002 --J25- --J26- 99 0.007075 0.002801 0.001038 0.000361 0.000119 0.000037 0.000011 0.000003 0.000001
Note - Actually at Out=0, J0=1 and J1=Zero=J2=J3 etc but the imprecision of Mod.Index previously plus some calculation roundings will cause some small errors. The numbers near Out=Zero tend to be a bit out.
I recommend using only up to 3 decimal places, otherwise you'll go mad. Even at Out=95, you'll be calculating to 15 orders of Sidebands.
DX-21 Output level ~to~ Bessel function [Jn] Table - up to 4 decimal places.
OUT -JO-- --J1-- --J2-- --J3-- OUT --J0-- --J1-- --J2-- --J3-- --J4-- --J5-- --J6-- 0 1.0000 0.0002 0.0000 32 0.9986 0.0378 0.0007 0.0000 1 1.0000 0.0003 0.0000 33 0.9983 0.0412 0.0009 0.0000 2 1.0000 0.0005 0.0000 34 0.9980 0.0450 0.0010 0.0000 3 1.0000 0.0006 0.0000 35 0.9976 0.0490 0.0012 0.0000 4 1.0000 0.0009 0.0000 36 0.9971 0.0535 0.0014 0.0000 5 1.0000 0.0012 0.0000 37 0.9966 0.0583 0.0017 0.0000 6 1.0000 0.0015 0.0000 38 0.9960 0.0635 0.0020 0.0000 --J4-- 7 1.0000 0.0018 0.0000 39 0.9952 0.0693 0.0024 0.0001 0.0000 8 1.0000 0.0022 0.0000 40 0.9943 0.0755 0.0029 0.0001 0.0000 9 1.0000 0.0026 0.0000 41 0.9932 0.0823 0.0034 0.0001 0.0000 10 1.0000 0.0031 0.0000 42 0.9919 0.0897 0.0040 0.0001 0.0000 11 1.0000 0.0036 0.0000 43 0.9904 0.0977 0.0048 0.0002 0.0000 12 1.0000 0.0043 0.0000 44 0.9886 0.1064 0.0057 0.0002 0.0000 13 1.0000 0.0052 0.0000 45 0.9864 0.1160 0.0068 0.0003 0.0000 14 1.0000 0.0061 0.0000 46 0.9839 0.1263 0.0081 0.0003 0.0000 15 0.9999 0.0073 0.0000 47 0.9808 0.1375 0.0096 0.0004 0.0000 16 0.9999 0.0080 0.0000 48 0.9772 0.1497 0.0114 0.0006 0.0000 17 0.9999 0.0087 0.0000 --J3-- 49 0.9729 0.1629 0.0135 0.0007 0.0000 18 0.9999 0.0103 0.0001 0.0000 50 0.9678 0.1772 0.0160 0.0010 0.0000 --J5-- 19 0.9999 0.0113 0.0001 0.0000 51 0.9618 0.1926 0.0190 0.0012 0.0001 0.0000 20 0.9998 0.0134 0.0001 0.0000 52 0.9547 0.2092 0.0226 0.0016 0.0001 0.0000 21 0.9998 0.0146 0.0001 0.0000 53 0.9462 0.2272 0.0268 0.0021 0.0001 0.0000 22 0.9997 0.0159 0.0001 0.0000 54 0.9362 0.2465 0.0317 0.0027 0.0002 0.0000 23 0.9997 0.0174 0.0002 0.0000 55 0.9244 0.2671 0.0376 0.0035 0.0002 0.0000 24 0.9996 0.0189 0.0002 0.0000 56 0.9104 0.2891 0.0445 0.0045 0.0003 0.0000 25 0.9996 0.0206 0.0002 0.0000 57 0.8939 0.3125 0.0526 0.0058 0.0005 0.0000 26 0.9995 0.0225 0.0003 0.0000 58 0.8745 0.3373 0.0621 0.0075 0.0007 0.0000 --J6-- 27 0.9994 0.0245 0.0003 0.0000 59 0.8516 0.3632 0.0732 0.0097 0.0010 0.0001 0.0000 28 0.9993 0.0268 0.0004 0.0000 60 0.8248 0.3902 0.0862 0.0125 0.0014 0.0001 0.0000 29 0.9991 0.0292 0.0004 0.0000 61 0.7935 0.4179 0.1013 0.0161 0.0019 0.0002 0.0000 30 0.9990 0.0318 0.0005 0.0000 62 0.7570 0.4460 0.1188 0.0206 0.0027 0.0003 0.0000 31 0.9988 0.0347 0.0006 0.0000 63 0.7146 0.4740 0.1390 0.0264 0.0037 0.0004 0.0000 OUT --J0-- --J1-- --J2-- --J3-- --J4-- --J5-- --J6-- --J7-- --J8-- --J9-- --J10- 64 0.6655 0.5011 0.1620 0.0337 0.0052 0.0006 0.0001 0.0000 65 0.6091 0.5265 0.1881 0.0430 0.0073 0.0010 0.0001 0.0000 66 0.5448 0.5489 0.2173 0.0546 0.0101 0.0015 0.0002 0.0000 67 0.4720 0.5668 0.2497 0.0690 0.0140 0.0022 0.0003 0.0000 --J8-- 68 0.3905 0.5785 0.2849 0.0869 0.0193 0.0034 0.0005 0.0001 0.0000 69 0.3004 0.5817 0.3224 0.1086 0.0266 0.0051 0.0008 0.0001 0.0000 70 0.2026 0.5741 0.3611 0.1349 0.0363 0.0077 0.0013 0.0002 0.0000 --J9-- 71 0.0985 0.5528 0.3992 0.1661 0.0493 0.0114 0.0022 0.0004 0.0001 0.0000 72 -0.0091 0.5153 0.4346 0.2022 0.0664 0.0169 0.0035 0.0006 0.0001 0.0000 73 -0.1162 0.4590 0.4637 0.2431 0.0885 0.0249 0.0057 0.0011 0.0002 0.0000 --J10- 74 -0.2171 0.3822 0.4824 0.2876 0.1166 0.0362 0.0092 0.0020 0.0004 0.0001 0.0000 75 -0.3042 0.2846 0.4854 0.3335 0.1514 0.0521 0.0145 0.0034 0.0007 0.0001 0.0000 76 -0.3688 0.1684 0.4671 0.3770 0.1931 0.0740 0.0228 0.0059 0.0013 0.0003 0.0000 77 -0.4009 0.0390 0.4218 0.4126 0.2409 0.1032 0.0352 0.0101 0.0025 0.0005 0.0001 78 -0.3912 -0.0938 0.3452 0.4327 0.2921 0.1408 0.0536 0.0169 0.0046 0.0011 0.0002 79 -0.3333 -0.2152 0.2364 0.4281 0.3417 0.1872 0.0797 0.0279 0.0084 0.0022 0.0005 80 -0.2268 -0.3062 0.1004 0.3891 0.3814 0.2407 0.1154 0.0451 0.0150 0.0043 0.0011 81 -0.0815 -0.3457 -0.0494 0.3084 0.3996 0.2966 0.1618 0.0710 0.0262 0.0084 0.0024 82 0.0797 -0.3164 -0.1895 0.1848 0.3820 0.3455 0.2178 0.1080 0.0446 0.0159 0.0050 83 0.2203 -0.2124 -0.2879 0.0291 0.3157 0.3728 0.2777 0.1575 0.0733 0.0291 0.0101 84 0.2961 -0.0495 -0.3105 -0.1318 0.1951 0.3596 0.3297 0.2178 0.1153 0.0515 0.0200 85 0.2700 0.1282 -0.2357 -0.2544 0.0314 0.2880 0.3541 0.2806 0.1717 0.0870 0.0380 86 0.1354 0.2530 -0.0733 -0.2890 -0.1394 0.1521 0.3261 0.3281 0.2377 0.1386 0.0685 87 -0.0616 0.2572 0.1195 -0.2034 -0.2569 -0.0278 0.2255 0.3324 0.2982 0.2045 0.1161 88 -0.2206 0.1190 0.2452 -0.0178 -0.2562 -0.1937 0.0562 0.2634 0.3243 0.2721 0.1811 89 -0.2309 -0.0940 0.2131 0.1746 -0.1139 -0.2609 -0.1330 0.1099 0.2785 0.3119 0.2527 90 -0.0623 -0.2294 0.0225 0.2372 0.1010 -0.1671 -0.2460 -0.0891 0.1377 0.2804 0.3002 91 0.1575 -0.1545 -0.1821 0.0966 0.2282 0.0487 -0.1894 -0.2296 -0.0664 0.1451 0.2742 92 0.2029 0.0799 -0.1913 -0.1357 0.1318 0.2127 0.0233 -0.1922 -0.2197 -0.0643 0.1352 93 -0.0027 0.2063 0.0303 -0.1982 -0.1099 0.1394 0.2032 0.0238 -0.1809 -0.2175 -0.0810 94 -0.1935 0.0342 0.1977 0.0143 -0.1924 -0.1088 0.1257 0.2013 0.0473 -0.1549 -0.2184 95 -0.0559 -0.1824 0.0353 0.1904 0.0290 -0.1774 -0.1288 0.0904 0.2000 0.0896 -0.1092 96 0.1751 -0.0423 -0.1794 0.0053 0.1811 0.0694 -0.1452 -0.1594 0.0301 0.1842 0.1410 97 0.0135 0.1734 0.0030 -0.1728 -0.0520 0.1531 0.1245 -0.0825 -0.1791 -0.0531 0.1338 98 -0.1604 -0.0470 0.1563 0.0741 -0.1370 -0.1217 0.0842 0.1655 0.0163 -0.1542 -0.1368 99 0.1120 -0.1109 -0.1208 0.0917 0.1427 -0.0462 -0.1611 -0.0307 0.1440 0.1223 -0.0564 continuing across... OUT --J11- --J12- --J13- --J14- --J15- --J16- --J17- --J18- --J19- --J20- --J21- 78 0.0000 79 0.0001 0.0000 --J13- 80 0.0003 0.0001 0.0000 81 0.0006 0.0001 0.0000 --J14- 82 0.0014 0.0004 0.0001 0.0000 --J15- 83 0.0031 0.0009 0.0002 0.0001 0.0000 84 0.0069 0.0021 0.0006 0.0002 0.0000 --J16- 85 0.0146 0.0050 0.0016 0.0005 0.0001 0.0000 --J17- 86 0.0296 0.0114 0.0040 0.0013 0.0004 0.0001 0.0000 --J18- 87 0.0568 0.0246 0.0096 0.0034 0.0011 0.0003 0.0001 0.0000 --J19- 88 0.1017 0.0499 0.0218 0.0086 0.0031 0.0010 0.0003 0.0001 0.0000 --J20- 89 0.1664 0.0938 0.0465 0.0207 0.0084 0.0031 0.0011 0.0003 0.0001 0.0000 --J21- 90 0.2407 0.1593 0.0911 0.0462 0.0211 0.0088 0.0034 0.0012 0.0004 0.0001 0.0000 91 0.2913 0.2358 0.1590 0.0932 0.0487 0.0231 0.0100 0.0040 0.0015 0.0005 0.0002 92 0.2617 0.2849 0.2373 0.1652 0.1004 0.0545 0.0269 0.0122 0.0051 0.0020 0.0007 93 0.1090 0.2415 0.2789 0.2437 0.1777 0.1130 0.0643 0.0332 0.0158 0.0070 0.0029 94 -0.1131 0.0657 0.2098 0.2691 0.2525 0.1957 0.1318 0.0793 0.0434 0.0218 0.0102 95 -0.2125 -0.1539 0.0047 0.1608 0.2486 0.2589 0.2176 0.1573 0.1012 0.0590 0.0316 96 -0.0387 -0.1849 -0.1903 -0.0704 0.0886 0.2076 0.2541 0.2383 0.1885 0.1313 0.0825 97 0.1798 0.0533 -0.1192 -0.2000 -0.1458 -0.0069 0.1353 0.2246 0.2473 0.2200 0.1691 98 0.0355 0.1707 0.1422 -0.0102 -0.1546 -0.1911 -0.1106 0.0278 0.1541 0.2263 0.2386 99 -0.1672 -0.0900 0.0813 0.1741 0.1126 -0.0396 -0.1631 -0.1810 -0.0962 0.0356 0.1528 continuing even further across... OUT --J22- --J23- --J24- --J25- --J26- --J27- --J28- --J29- --J30- --J31- --J32- 91 0.0001 0.0000 92 0.0003 0.0001 0.0000 --J25- 93 0.0011 0.0004 0.0001 0.0000 --J26- --J27- 94 0.0045 0.0018 0.0007 0.0003 0.0001 0.0000 --J28- --J29- 95 0.0157 0.0073 0.0032 0.0013 0.0005 0.0002 0.0001 0.0000 --J30- --J31- 96 0.0475 0.0253 0.0126 0.0059 0.0026 0.0011 0.0004 0.0002 0.0001 0.0000 --J32- 97 0.1161 0.0725 0.0418 0.0225 0.0113 0.0054 0.0024 0.0010 0.0004 0.0002 0.0001 98 0.2086 0.1596 0.1099 0.0694 0.0406 0.0222 0.0114 0.0056 0.0026 0.0011 0.0005 99 0.2198 0.2319 0.2048 0.1591 0.1118 0.0722 0.0434 0.0244 0.0130 0.0065 0.0031 continuing yet even further across... OUT --J33- --J34- --J35- --J36- 97 0.0000 98 0.0002 0.0001 0.0000 99 0.0014 0.0006 0.0003 0.0001
Note - Actually at Out=0, J0=1 and J1=Zero=J2=J3 etc but the imprecision of Mod.Index previously plus some calculation roundings will cause some small errors. The numbers near Out=Zero tend to be a bit out.
Because the DX-21 type synths can attain Mod.Index=25, the number of Sidebands can go up to very high orders.
CX-5 Output level ~to~ Bessel function [Jn] Table - up to 4 decimal places.
OUT --J0-- --J1-- --J2-- OUT --J0-- --J2-- --J3-- --J4-- --J5-- --J6-- 0 1.0000 0.0001 0.0000 47 1.0000 0.0061 0.0000 1 1.0000 0.0001 0.0000 48 1.0000 0.0067 0.0000 2 1.0000 0.0001 0.0000 49 0.9999 0.0073 0.0000 3 1.0000 0.0001 0.0000 50 0.9999 0.0080 0.0000 4 1.0000 0.0001 0.0000 51 0.9999 0.0087 0.0000 5 1.0000 0.0002 0.0000 52 0.9999 0.0095 0.0000 --J4-- 6 1.0000 0.0002 0.0000 53 0.9999 0.0103 0.0001 0.0000 7 1.0000 0.0002 0.0000 54 0.9999 0.0113 0.0001 0.0000 8 1.0000 0.0002 0.0000 55 0.9998 0.0123 0.0001 0.0000 9 1.0000 0.0002 0.0000 56 0.9998 0.0134 0.0001 0.0000 10 1.0000 0.0002 0.0000 57 0.9998 0.0146 0.0001 0.0000 11 1.0000 0.0003 0.0000 58 0.9997 0.0159 0.0001 0.0000 12 1.0000 0.0003 0.0000 59 0.9997 0.0174 0.0002 0.0000 13 1.0000 0.0003 0.0000 60 0.9996 0.0189 0.0002 0.0000 14 1.0000 0.0004 0.0000 61 0.9996 0.0206 0.0002 0.0000 15 1.0000 0.0004 0.0000 62 0.9995 0.0225 0.0003 0.0000 16 1.0000 0.0004 0.0000 63 0.9994 0.0245 0.0003 0.0000 17 1.0000 0.0005 0.0000 64 0.9993 0.0268 0.0004 0.0000 18 1.0000 0.0005 0.0000 65 0.9991 0.0292 0.0004 0.0000 19 1.0000 0.0005 0.0000 66 0.9990 0.0318 0.0005 0.0000 20 1.0000 0.0006 0.0000 67 0.9988 0.0347 0.0006 0.0000 21 1.0000 0.0006 0.0000 68 0.9986 0.0378 0.0007 0.0000 22 1.0000 0.0007 0.0000 69 0.9983 0.0412 0.0009 0.0000 23 1.0000 0.0008 0.0000 70 0.9980 0.0450 0.0010 0.0000 24 1.0000 0.0008 0.0000 71 0.9976 0.0490 0.0012 0.0000 25 1.0000 0.0009 0.0000 72 0.9971 0.0535 0.0014 0.0000 26 1.0000 0.0010 0.0000 73 0.9966 0.0583 0.0017 0.0000 27 1.0000 0.0011 0.0000 74 0.9960 0.0635 0.0020 0.0000 --J5-- 28 1.0000 0.0012 0.0000 75 0.9952 0.0693 0.0024 0.0001 0.0000 29 1.0000 0.0013 0.0000 76 0.9943 0.0755 0.0029 0.0001 0.0000 30 1.0000 0.0014 0.0000 77 0.9932 0.0823 0.0034 0.0001 0.0000 31 1.0000 0.0015 0.0000 78 0.9919 0.0897 0.0040 0.0001 0.0000 32 1.0000 0.0017 0.0000 79 0.9904 0.0977 0.0048 0.0002 0.0000 33 1.0000 0.0018 0.0000 80 0.9886 0.1064 0.0057 0.0002 0.0000 34 1.0000 0.0020 0.0000 81 0.9864 0.1160 0.0068 0.0003 0.0000 35 1.0000 0.0022 0.0000 82 0.9839 0.1263 0.0081 0.0003 0.0000 36 1.0000 0.0024 0.0000 83 0.9808 0.1375 0.0096 0.0004 0.0000 37 1.0000 0.0026 0.0000 84 0.9772 0.1497 0.0114 0.0006 0.0000 38 1.0000 0.0028 0.0000 85 0.9729 0.1629 0.0135 0.0007 0.0000 39 1.0000 0.0031 0.0000 86 0.9678 0.1772 0.0160 0.0010 0.0000 --J6-- 40 1.0000 0.0033 0.0000 87 0.9618 0.1926 0.0190 0.0012 0.0001 0.0000 41 1.0000 0.0036 0.0000 88 0.9547 0.2092 0.0226 0.0016 0.0001 0.0000 42 1.0000 0.0040 0.0000 89 0.9462 0.2272 0.0268 0.0021 0.0001 0.0000 43 1.0000 0.0043 0.0000 90 0.9362 0.2465 0.0317 0.0027 0.0002 0.0000 44 1.0000 0.0047 0.0000 91 0.9244 0.2671 0.0376 0.0035 0.0002 0.0000 45 1.0000 0.0052 0.0000 92 0.9104 0.2891 0.0445 0.0045 0.0003 0.0000 46 1.0000 0.0056 0.0000 93 0.8939 0.3125 0.0526 0.0058 0.0005 0.0000 94 0.8745 0.3373 0.0621 0.0075 0.0007 0.0000 OUT --J0-- --J1-- --J2-- --J3-- --J4-- --J5-- --J6-- --J7-- --J8-- --J9-- --J10- 95 0.8516 0.3632 0.0732 0.0097 0.0010 0.0001 0.0000 96 0.8248 0.3902 0.0862 0.0125 0.0014 0.0001 0.0000 97 0.7935 0.4179 0.1013 0.0161 0.0019 0.0002 0.0000 98 0.7570 0.4460 0.1188 0.0206 0.0027 0.0003 0.0000 99 0.7146 0.4740 0.1390 0.0264 0.0037 0.0004 0.0000 --J7-- 100 0.6655 0.5011 0.1620 0.0337 0.0052 0.0006 0.0001 0.0000 101 0.6091 0.5265 0.1881 0.0430 0.0073 0.0010 0.0001 0.0000 102 0.5448 0.5489 0.2173 0.0546 0.0101 0.0015 0.0002 0.0000 103 0.4720 0.5668 0.2497 0.0690 0.0140 0.0022 0.0003 0.0000 --J8-- 104 0.3905 0.5785 0.2849 0.0869 0.0193 0.0034 0.0005 0.0001 0.0000 105 0.3004 0.5817 0.3224 0.1086 0.0266 0.0051 0.0008 0.0001 0.0000 106 0.2026 0.5741 0.3611 0.1349 0.0363 0.0077 0.0013 0.0002 0.0000 --J9-- 107 0.0985 0.5528 0.3992 0.1661 0.0493 0.0114 0.0022 0.0004 0.0001 0.0000 108 -0.0091 0.5153 0.4346 0.2022 0.0664 0.0169 0.0035 0.0006 0.0001 0.0000 109 -0.1162 0.4590 0.4637 0.2431 0.0885 0.0249 0.0057 0.0011 0.0002 0.0000 --J10- 110 -0.2171 0.3822 0.4824 0.2876 0.1166 0.0362 0.0092 0.0020 0.0004 0.0001 0.0000 111 -0.3042 0.2846 0.4854 0.3335 0.1514 0.0521 0.0145 0.0034 0.0007 0.0001 0.0000 112 -0.3688 0.1684 0.4671 0.3770 0.1931 0.0740 0.0228 0.0059 0.0013 0.0003 0.0000 113 -0.4009 0.0390 0.4218 0.4126 0.2409 0.1032 0.0352 0.0101 0.0025 0.0005 0.0001 114 -0.3912 -0.0938 0.3452 0.4327 0.2921 0.1408 0.0536 0.0169 0.0046 0.0011 0.0002 115 -0.3333 -0.2152 0.2364 0.4281 0.3417 0.1872 0.0797 0.0279 0.0084 0.0022 0.0005 116 -0.2268 -0.3062 0.1004 0.3891 0.3814 0.2407 0.1154 0.0451 0.0150 0.0043 0.0011 117 -0.0815 -0.3457 -0.0494 0.3084 0.3996 0.2966 0.1618 0.0710 0.0262 0.0084 0.0024 118 0.0797 -0.3164 -0.1895 0.1848 0.3820 0.3455 0.2178 0.1080 0.0446 0.0159 0.0050 119 0.2203 -0.2124 -0.2879 0.0291 0.3157 0.3728 0.2777 0.1575 0.0733 0.0291 0.0101 120 0.2961 -0.0495 -0.3105 -0.1318 0.1951 0.3596 0.3297 0.2178 0.1153 0.0515 0.0200 121 0.2700 0.1282 -0.2357 -0.2544 0.0314 0.2880 0.3541 0.2806 0.1717 0.0870 0.0380 122 0.1354 0.2530 -0.0733 -0.2890 -0.1394 0.1521 0.3261 0.3281 0.2377 0.1386 0.0685 123 -0.0616 0.2572 0.1195 -0.2034 -0.2569 -0.0278 0.2255 0.3324 0.2982 0.2045 0.1161 124 -0.2206 0.1190 0.2452 -0.0178 -0.2562 -0.1937 0.0562 0.2634 0.3243 0.2721 0.1811 125 -0.2309 -0.0940 0.2131 0.1746 -0.1139 -0.2609 -0.1330 0.1099 0.2785 0.3119 0.2527 126 -0.0623 -0.2294 0.0225 0.2372 0.1010 -0.1671 -0.2460 -0.0891 0.1377 0.2804 0.3002 127 0.1575 -0.1545 -0.1821 0.0966 0.2282 0.0487 -0.1894 -0.2296 -0.0664 0.1451 0.2742 continuing across... OUT --J11- --J12- --J13- --J14- --J15- --J16- --J17- --J18- --J19- --J20- --J21- --J22- 114 0.0000 115 0.0001 0.0000 --J13- 116 0.0003 0.0001 0.0000 117 0.0006 0.0001 0.0000 --J14- 118 0.0014 0.0004 0.0001 0.0000 --J15- 119 0.0031 0.0009 0.0002 0.0001 0.0000 120 0.0069 0.0021 0.0006 0.0002 0.0000 --J16- 121 0.0146 0.0050 0.0016 0.0005 0.0001 0.0000 --J17- 122 0.0296 0.0114 0.0040 0.0013 0.0004 0.0001 0.0000 --J18- 123 0.0568 0.0246 0.0096 0.0034 0.0011 0.0003 0.0001 0.0000 --J19- 124 0.1017 0.0499 0.0218 0.0086 0.0031 0.0010 0.0003 0.0001 0.0000 --J20- 125 0.1664 0.0938 0.0465 0.0207 0.0084 0.0031 0.0011 0.0003 0.0001 0.0000 --J21- 126 0.2407 0.1593 0.0911 0.0462 0.0211 0.0088 0.0034 0.0012 0.0004 0.0001 0.0000 --J22- 127 0.2913 0.2358 0.1590 0.0932 0.0487 0.0231 0.0100 0.0040 0.0015 0.0005 0.0002 0.0001
Note - Actually at Out=0, J0=1 and J1=Zero=J2=J3 etc but the imprecision of Mod.Index previously plus some calculation roundings will cause some small errors. The numbers near Out=Zero tend to be a bit out.
The range of Bessel values for the CX-5 type synths are similar to the DX-7 type but the Mod.Index curve is slightly different.
Recommended Reading
Here are some links and material I've found for FM Synthesis and related information.
http://www.sfu.ca/sca/Manuals/fm/FM_Tutorial.html
http://www.sfu.ca/~truax/fmtut.html
http://www.esm.rochester.edu/www/onlinedocs/allan.cs/chapter3.html
http://cis.poly.edu/cs240/notes6.htm
http://www.telecommunication.msu.edu/classes/tc201/slides/fm/index.htm
http://www.neuroinformatik.ruhr-uni-bochum.de/ini/PEOPLE/heja/sy-prog/node55.html
http://ccrma-www.stanford.edu/CCRMA/Software/clm/clm-manual/fm.html
I haven't personally read this stuff, but they come highly recommended.
The Synthesis of Complex Audio Spectra by Means of Frequency Modulation [John M Chowning & Max Mathews] - Audio Engineering Society Journal - Vol.21/ No.7 (1973): pg 526-534.
FM Theory and Applications by Musicians for Musicians [John M Chowning & Dave Bristow] - Yamaha Music Foundation, Tokyo (1986) ISBN 4-636-17482-8
The Simulation of Natural Instrument Tones using Frequency Modulation with a Complex Modulating Wave [Bill Schottstaedt] - Computer Music Journal - Vol.1, No.4 (1977) : pg 46-50.
A Derivation of the Spectrum of FM with a Complex Modulating Wave [Marc LeBrun] - Computer Music Journal - Vol.1/ No.4 (1977): pg 51-52.
The F.M. legend - a personal history
For me, it all started one day in mid'1984 when I walked into my friendly neighbourhood music-inst shop (actually, it was Soho Soundhouse, London) and there was a buzz in the air. The sales-guy says "You're here to try out the DX-7, right?". I, of course, didn't know what he was talking about and enquired about the price. It was out of my reach (obviously). Not to be deterred, the sales-guy instead plonks me in front of a Yamaha DX-9 and hands me some headphones. I started playing and... aaaahh, heaven!
It's hard to describe what I heard (bearing in mind it's my first time hearing F.M. synthesis). You have to understand that, up to this point, synths were all about strings and brass. Occasionally, you'd have a few plinky plonky xylo-sounds (heck, the MKS-10 was considered realistic) but percussives, vibes, pianos etc were elusive (ie non-existent). But right there in front of me, in the form of a DX-9, was the holy grail. And, to make matters worse, there weren't any knobs or sliders or anything in fact which gave any inkling as to how this synth worked.
I was hooked! I took the plunge and bought a DX-7 in Nov'84 and later a CX-5 in Jan'85. Unfortunately, programming these FM synths was an absolute nightmare. Nothing was fast and nothing was easy. It really wasn't intuitive at all and the manual wasn't exactly that helpful. But I was determined to master this beast. Learning to program the DX-7 was a slow and tedious process.
But one day in late 1984, humanity was saved by a fellow synth-enthusiast called Tony Wride. Fed-up and tired with struggling alone with his DX-7, he mooted the idea of a "DX-club" in a letter to the magazine "Electronics And Music Maker". This caused a huge stirring of support from the public (DX synth owners), the media (music mags) and Yamaha too. Thus was born the DX-Owners Club.
It was the DX-Owners Club which took FM programming to new heights. Via its newsletter, we began sharing patches/ sounds (one patch called "Wurlitzer" was really popular) and programming techniques (excellent articles by Ken Campbell). Part of Tony Wride's vision was also to have a network of co-ordinators who anyone could telephone for help and advice (you'll find listed under Area Co-ordinator for London W2 is "Yahaya 01-221-5314" which is me).
Beyond the popular newsletter (these typed-up/ hand-drawn photocopy newsletters were inspirational), the club also organised get-together seminars bringing programmers together to meet experts like Dave Bristow to discuss FM in depth (I remember that fixed-frequency operators was a big topic). FM ruled and the DX-7 was king... Life was good!
But as life goes on, reality takes its hold... Tony's job in the RAF gave him less time to run the club. Eventually, the club was handed over to (surprise surprise) Yamaha who appointed Martin Tennant (not to be confused with Martin Russ) to run the re-named X-Series Owners Club. With Yamaha's backing (ie money), the newsletter became a regular magazine (with pictures and all) and everything was taken to a more polished level.
The X-Series Owners Club was good... but with the change of management of the club, came a change in objectivity. You see, us members may all be dx-synth enthusiasts but we didn't work for Yamaha (the old newsletter would include info on non-Yamaha products as well). I remember an interesting session where Yamaha was launching their DX-5 while Tony was happily proposing to just add a TX-7 to a DX-7 (ie half the cost). Ah, well! Nevermind.
As far as I know, the magazine continued until around mid'1987.
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