This month, Martin Russ looks at Ratios, and how to recreate those over-used chime/bell timbres, so popular in the recent past.

So far in this series we've looked at how to get the best out of FM synthesizers, by examining how the elements in this method of synthesis work together to produce different sounds. We are now turning the corner into the final straight, and so we need to spend some time on a subject hitherto virtually ignored - Ratios. Whereas most FM tutorial books place a great deal of emphasis on the correct choice of Ratios between Operator frequencies, I believe that the most important aspect to get right first is the Envelope - which is why, so far, we have concentrated our programming efforts on the Envelopes. After the Envelopes, Output Levels and Ratios are the most important factors affecting our sound, so let's investigate Ratios:

THE RATIOS

FM theory says that the Ratio is the major parameter determining the basic tonal colour of an FM sound. This is analogous to the 'XR' effect in cars - you take a basic building block, and depending upon what you add to it, it ends up somewhere between a standard 'L' model or a posey 'XR3' model. You could say that the letter is the major parameter affecting the perceived value of the car.

Escorting us (this month's entry in the awful pun competition!) quickly away from cars and back to FM, the Algorithm is the starting point from which we can develop a timbre. Unlike most other synthesis methods where the basic waveforms are fixed, in FM you can build up your own sound by combining Operators in Algorithms, and the choice of Ratios between Operator frequencies is the key to controlling the resulting waveshapes. With only a single Operator, you get a rather boring sine wave - the 'L' model - whilst a stack of four Operators with many Ratios hidden inside can produce 'XR' type sophisticated timbres. The permutations are almost limitless...

So where to begin? All the Ratios values in 'Vanilla' are set to 1.00, and this is a good starting point for further investigations. (The parameters for our 'Vanilla' DX starter voice are displayed in any of the past episodes of this series.) Select 'Vanilla' and enter Edit mode. Turn off Operators 3 to 6 and listen to Ops 1 and 2 (a simple FM pair). Now change the Frequency Ratio of Operator 2 (the Modulator), listening for the 'islands of harmony' where the tone sounds right (my music teacher has a lot to answer for!). Don't be afraid to try both the Coarse and Fine controls!

(Short pause for bursts of strident cacophony, muttered curses, knocks on the wall from the neighbours, giggles from the rest of the band, and all those other embarrassing things a keyboard player has to put up with.) Like most people before you, you will now have discovered that there are far more bad settings than good ones for Frequency Ratios! I hope you actually found a few settings where things sounded OK - so where were they? Here is a list of Ratios that I reckon are perhaps worth a further look:

Of these, 3.50 has a certain chime-like ring to it, although it also reminds me of big, heavy bells rather more than light, delicate chimes. Restore Operator 2 to a Ratio of 1.00, and now try varying the Coarse Frequency of Operator 1 - the Carrier - from 0.50 to 8.00. You will find that as you increase the Ratio, the sound gets thinner and more twangy in character.

Store all these timbres away in your mind for future use! Finally, try changing both the Ratios; again, altering just the Coarse control should give you enough possibilities. I chose the following pairs of Frequencies as my selections for chimes:

Please note that there is nothing mysterious or unusual in these values - they just have the correct sound to my ears. Some Ratios, like 0.50 and 8.00, sound too 'glassy', whilst others, like 4.00 and 1.00, are totally out of context. The most important element in this investigation is that you have done it, and you should now have some sort of feel for the basic range of raw timbres.

Having found the Ratios by experiment, the rest should be easy. There are no prizes for guessing that we need a percussive Envelope rather than the sustaining Envelope supplied by 'Vanilla'. The percussive Envelopes used in Parts 2 and 3 of this series are the kind of thing we need, so here to remind you is a similar Envelope, but with a faster decay and some sustain added:

Apart from the faster attack rate, this Envelope bears some similarity to last month's Brass Envelope as well - notice the sustain level. Copying this to the two Operators produces a good, sharp start and a long release; although the latter could also be achieved by reducing Rate 4 and compensating with some reverb, if you prefer. Whenever I programme a long release on a sound, I always consider what happens to that sound as it dies away - if the timbre stays the same then I make the Modulator's Rate 4 longer than that of the Carriers; if the sound mellows as it decays, I do the reverse.

Having decided that chimes probably do not lose their edge as they fade, I reduced Rate 4 of Operator 2 to a value of 34, ie. just on the limits of audibility - adjust it to your own taste if you wish.

Rather than waste all this programming effort on a pair of Operators, I decided to capitalise on it by choosing Algorithm 6 as the basis of the chime sound. This features three Operator pairs with a big Feedback loop. (Actually, as I will not be using any Feedback, Algorithm 5 would have been just as good). Once you have copied the Envelopes from the first pair of Operators to the other four, you can assign the Ratios using the set of values above. I used the first three sets, although again there is much scope for the adventurous explorer. Since most percussive instruments decay faster at the top of their range, I imposed a Rate Scaling of 2 on everything.

Listening to the sound at this point, I felt that it decayed rather too quickly, as well as sounding a little too 'big' for comfort. To take care of the first problem, I increased the decay time of Op 1 to 27, and that of Op 2 to 20. To thin out the sound I added some Velocity Sensitivity on the second and third pairs, setting a value of 1 for Ops 3 and 4, and 2 for Ops 5 and 6. This achieved the necessary trimming. Finally, I adjusted the Detune parameters, and brought the Levels of the Carriers up to 99 to enhance the ringing tone.

The final sound parameters are shown on the accompanying screen dump. As usual, I searched high and low for a suitable title for this sound - 'SOSchimes' seemed to be obscure enough! For the 4-Operator enthusiasts out there, I have reduced the number of Operator pairs to two, but the Ratios and everything else apply in exactly the same way. The accompanying printout of the 4-Operator sound should make everything clear.

By now you should be well acquainted with the techniques for altering voices, so you might like to try editing the Velocity Sensitivity to suit your own playing style — I tend to prefer a very light touch with lots of sensitivity. As always, remember to isolate the Operators you want to concentrate on by turning the unwanted Operators off, and carry out 'before' and 'after' comparisons with the DX's Compare button until you are happy with the changes. You could also try using your own favourite Ratios to tailor the sound to personal taste. Have fun!

THE THEORY

How many of you observed that, in keeping with my goal of changing only what needs to be changed, I have not altered the Output Levels at all, nor have I delved into Level Scaling? In fact, the 'SOSchimes' voice, like all the others we've created in 'Practically FM', is just a careful combination of the right Envelope with the right timbre.

The timbre is a result of two variables - the Ratio sets the basic starting point, but the Output Level provides the control and the changes. By using the Envelope associated with the Output, you can control the way the sound changes through time, from bright to mellow as you wish. Earlier parts of this series looked at some of the effects of changing the Output Level; this part shows some of the effects of changing the Ratios instead. Notice that the Ratios cannot be changed in real time, except by using the data entry slider on your synth, whereas the Output Levels can be varied by using the Envelopes. The Frequency Ratios fix the point about which you can vary the timbre with the Output Level.

With the Envelopes we are using for 'SOSchimes', there is not much variation in timbre as the sound decays - to investigate more synthetic effects try reducing Level 3 in the Modulators (Operators 2, 4 and 6) to 0, and set Rates 2 and 3 to 20. The quick way to achieve this uses the EG Copy function - can you figure it out? (You set Level 3 of EG2 to 0, and then copy it to EG4 and EG6.) It sounds better with the Modulator Output Levels set to 95. A held chord now decays from a sharp, harpsichord-like start to a pipe-organish sustain.

Don't forget that in all the experimentation so far, I have tended to leave the Output Level fixed and varied the Ratios instead. Our discussion of the way that the Envelope affects the timbre also points the way for the Output Level. As you increase the Output Level of a Modulator, the resulting sound will grow wider and wider in bandwidth - like opening up a filter. Low Modulator Output Levels will produce sounds with simple pure tones, whilst high values will produce bright sounds bursting with harmonics.

At this juncture we require some words of explanation about Ratios. Technically, the Ratio normally referred to in FM terminology (as used in Radio) means the Ratio between two frequencies. In the DX7, you can see the Operator frequencies defined as Ratios, rather than by their actual frequency measured in Hertz. The Ratio display behaves very similarly to the drawbar or footage markings on an organ (remember what else Yamaha make!), and so it makes sense to refer to the frequencies as relative Ratios - the DX7 always refers them to an internal reference with a frequency of 1.00, as displayed in Ratio mode. So, for the first pair in our 'SOSchimes' example voice, the Ratio between the two Operator frequencies is 2.00 to 4.00, or 1 to 2. In other words, the Modulator is twice the frequency of the Carrier. This saves a great deal of calculation on our part, which might be necessary if the DX7's LCD display read something like '440' and '1760', as it would if it dealt in Hertz.

For a Carrier frequency of 1.00, increasing the Modulator frequency results in a series of tones, each thinner and more metallic than those before. Above a Ratio of 16.00, the sound begins to appear bottle-like or 'glassy', and some notes begin to sound different from their neighbours as the effects of aliasing begin to appear (in much the same way as in a digital sampler). For a fixed Modulator frequency of 1.00, increasing the Carrier frequency makes the sound thinner and twangy, with the fundamental frequency disappearing in an effect much like that of a high-pass filter.

But don't take my word for all this, there is no substitute for finding out yourself. Explore the available timbres using simple 2-Operator pairs at first, then try Algorithms with two Modulators or two Carriers. You will find that for most purposes, altering the Coarse Frequency will give a good 'palate' of Frequency Ratios, and therefore a wide range of timbral variation. You will almost never need to use the Fine Frequency control, except perhaps for 0.99 and 0.50 type values. Incidentally, one of the 'tricks of the trade' on analogue synthesizers was to tune the oscillators to octaves or fifths, and you can do this with pairs of Operators by choosing Ratios of 1.00 and 1.00 for one pair, and 2.00 and 2.00, or 0.50 and 0.50 for the other, respectively. With the DX7 being a 6-Operator synth, you have more choices for this sort of thing - your territory is large and will need a lot of exploring!

As a sort of guide to the range of available sounds, Yamaha kindly provided the ROM preset voices which come with all FM synthesizers. Careful examination of the Ratios used in any of these voices should give you some ideas for the sort of values to try. So let's investigate some of these patches and see what we can unearth...

You will find that certain programming approaches repeat themselves - for instance, setting all Ratios to 1.00, as in the string and brass sounds, is very popular. Variations on the theme of a Carrier with a higher frequency than the Modulator provide most of the guitar/twangy sounds, whilst the piano sound is created by the reverse arrangement, with Carriers at 1.00 and Modulators at odd Ratios above, such as 3.00, 5.00, 7.00 etc. Electric pianos tend to use even higher Ratios - both odd and even (11.00, 13.00, 14.00, 16.00 etc). Drums and percussion tend to exploit the Fine Frequencies, with dissonant Ratios like 0.87 and 0.78 being used, as well as Fixed Frequencies (a whole series in itself, I'm afraid!). Flutes and other 'breathy' instruments use Feedback combined with Ratios like 7.04 or 1.53 lurking high in a stack of three or four Operators. And to come full circle, chimes and bells use paralleled medium sized Ratios, as in this month's example.

If I was counting hints and tips, then the above paragraph must be one of the most prolific and myth-destroying so far. [You will have to widen your door-frames at this rate, Martin! - Ed.] I'll leave decisions like which Envelopes to use to your own judgement, since I am sure that by now you have a good idea of what to choose. Can FM really be as simple as this? Why didn't Yamaha tell us all this when the DX7 first came out?

Next month I will conclude this series with a review of the programming techniques described so far, and a look at a slightly special DX voice!

• The 6-Operator voice dump and envelope display is from Lawrence Wilkes' DXPERTprogram for the Atari ST. A shareware disk version of this featuring just the Librarian functions is available for £7.

Contact SOS Shareware, (Contact Details).

• The 4-Operator dump and parameter table is from a new product covering all Yamaha 4-Operator FM synths except the FB01 (and including the DX11) called the Soundbits 4X4.

Contact Soundbits Software UK, (Contact Details).