FM synthesis: recipes from two sine waves
Two sine waves and two numbers. Ratio decides which harmonics can exist; modulation index decides how many of them carry energy.
Every lesson so far started with a rich waveform and took something away. FM starts with two sine waves, the emptiest sound there is, and puts harmonics in. One sine is the carrier, and it is the note you hear. The other is the modulator, and its output never reaches the speaker. It is wired into the carrier's frequency instead, pushing that frequency up and down hundreds of times a second. Lesson 6 already drew the line this crosses. An oscillator aimed at pitch below about 20 Hz is vibrato. The same wiring at 440 Hz is far too fast to hear as movement, and what arrives instead is a set of new frequencies standing around the carrier. They are sidebands, and they are the harmonics of the new sound.
The whole patch the toy below builds. The dashed line is modulation: the modulator's output goes into the carrier's frequency input, marked f, never into the audio path. The carrier is fixed at 220 Hz; RATIO sets the modulator's frequency as a multiple of it, and INDEX scales how far the carrier is pushed.
Where the sidebands land
Sidebands arrive in pairs, one above the carrier and one below, spaced at whole multiples of the modulator's frequency. Written out, they sit at c ± k·m for k = 1, 2, 3 and upward. The toy fixes the carrier at 220 Hz and draws the first eight pairs as amber ticks, so the prediction sits beside the green bars it is predicting. Set the ratio to 2.00 and the modulator runs at 440 Hz. The first pair is 660 Hz and −220 Hz; the second, 1100 Hz and −660 Hz.
Those negative numbers are real, and they are the part most explanations skip. A sideband below the carrier keeps subtracting past zero, and −220 Hz is heard as 220 Hz: it folds onto the positive axis and lands on top of whatever is already there. The toy folds them when it draws its ticks, because the audio folds them too. Fold the ratio-2.00 list and the spectrum collapses onto 220, 660, 1100, 1540, 1980, 2420, 2860 and 3300 Hz — the odd multiples of 220 Hz, which lesson 2 assigned to the square wave and the clarinet, and why the toy's second preset carries that name.
Ratio picks the recipe
The ratio between the two frequencies decides which multiples survive the fold, so it decides the family the sound belongs to. At 1.00 the modulator matches the carrier at 220 Hz and the sidebands land on every multiple of it: 220, 440, 660, 880 and on up, plus a term at 0 Hz that you cannot hear. That is the sawtooth family and the toy's first preset. At 2.00 only the odd multiples survive. At 3.00 you get every multiple except the 3rd, 6th, 9th and 12th, which leaves a thin reedy tone close to a narrow pulse. Higher whole numbers thin the set further.
Predicted sidebands for the toy's 220 Hz carrier at two of its ratio settings, over a faint grid of the 220 Hz harmonic series. At 2.00 every mark sits on a grid line and the comb is evenly spaced. At 3.50 the spacing alternates and four of the nine marks (550, 990, 2090 and 2530 Hz) land halfway between two harmonics. Positions only: bar heights carry no information.
A ratio that is not a whole number pushes marks off the harmonic grid, and the ear reads that as struck metal rather than as a note. Ratio 3.50 is the clearest case the slider can reach. Its marks land at 1, 2.5, 4.5, 6, 8, 9.5, 11.5, 13 and 15 times the carrier, so four of the nine sit between harmonics. Bells and gongs have partials arranged much like this, which is how FM took over that corner of sound design in the 1980s. The slider steps in units of 0.25, so every ratio it reaches still lands the spectrum on some regular grid; at 3.50 that grid is 110 Hz, half the carrier. Leaving every grid takes the FINE control below.
Modulation amount is brightness
Ratio decides which frequencies can appear. The modulation index decides how many of them carry real energy. It scales how far the modulator pushes the carrier: the toy sets that peak swing to the index times the modulator's frequency. On the clarinet preset, ratio 2.00 and index 2, the modulator runs at 440 Hz and the carrier's 220 Hz is being shoved 880 Hz either side. At index 0 there is no swing and the carrier plays as a plain sine. Raising it takes energy out of the carrier and spreads it outward, and a workable rule of thumb is that about index + 1 sideband pairs come out loud enough to matter: index 1 gives two, index 4 gives five, index 8 fills all eight the toy predicts. This is FM's version of opening a filter, and it is why a dull FM patch almost always needs the modulator's level raised first.
Two sine waves and nothing else. The amber ticks are the predicted sideband positions for the current settings; the green bars are the analyser reading the real output. Start on the 2:1 clarinet preset and count bars against ticks. Then set index to 0 and raise it slowly, watching the centre bar shrink as the outer ones grow. Then walk ratio from 1.00 to 8.00 in its 0.25 steps and sort the settings by ear into ones that sound like a note and ones that sound like a bell. Finish on FINE.
FINE: the last step to a bell
FINE adds up to 20 Hz to the modulator's frequency in 0.5 Hz steps, on top of whatever the ratio set. It is a small number next to 440 Hz and it does something out of proportion to its size. At ratio 2.00 the upper sideband of one pair and the lower sideband of the next land on the same frequency, which is what produced that clean odd-harmonic stack. Add 7 Hz of FINE and each coincidence separates by 7 Hz. Two tones 7 Hz apart beat seven times a second, so the spectrum begins to shimmer. Take FINE to its 20 Hz maximum and the beating is too fast to count and reads as roughness.
The offset compounds upward: the kth sideband is shifted by k times the FINE amount, so the top of the spectrum drifts further off the grid than the bottom. That is worth separating from what a tuning knob does. Moving the carrier slides the whole spectrum sideways and keeps its spacing; moving the modulator stretches the spacing itself. The toy only offers the second, since its carrier is nailed to 220 Hz. Its bell preset is ratio 3.50 with FINE at +7.0, putting the modulator at 777 Hz and the partials at 220, 557, 997, 1334, 1774 Hz and upward. None is a whole-number multiple of the 220 Hz note.
The Digitone II's FM TONE machine is this lesson with four operators instead of two, grouped C, A and B (B1 and B2), with ALGO picking one of eight fixed routings between them. C always works as a carrier and its ratio is held mostly to integers, because it carries the note; A gets a wider ratio list on purpose, so it can reach the inharmonic relationships this lesson used for bells. Each modulator's LEV is its index, and it runs through an attack/decay/end envelope, so brightness moves across the note the way a struck piano string's does. DTUN offsets operators A and B2 and is where the FINE experiment lives: subtle up to about 64, heavy above it. One notation warning — the manual prints ratios as carrier:modulator, so this lesson's ratio of 2 appears there as 1:2. FM DRUM is the same engine with three operators and a pitch sweep.