# frequency modulation synthesis

# links
* [an introduction to fm](https://ccrma.stanford.edu/software/snd/snd/fm.html)
* [frequency modulation synthesis](http://qcpages.qc.cuny.edu/~howe/music726.1/FMSynthesis.html)
* [wikipedia: frequency modulation synthesis](https://en.wikipedia.org/wiki/Frequency_modulation_synthesis)
* [wikipedia: frequency modulation](https://en.wikipedia.org/wiki/Frequency_modulation)
* [formulas for frequency modulation](http://alumni.media.mit.edu/~gan/Gan/Education/NUS/Physics/MScThesis/Chapter2.html)

# notes
* modulating the frequency of one signal by another
* vibrato is slow frequency modulation below 20 hz
* relatively rich sounds through the ease of creating time-varying spectra
* the modulator frequency defines the location of spectral components
* the modulator amplitude defines the amplitudes, number of spectral components, and bandwidth
* because only the frequency changes, the waveform looks level and not spikey. it looks distinctly different from a waveform created by summing the corresponding partials
* the overall amplitude stays practically the same regardless of bandwidth
* partials with negative frequencies are possible, which in this context means they will have an inverted phase. these partials may interfere with the positive frequency partials
* there can be multiple carriers and modulators, and the general name for each is operator
* modulation depth: frequency deviation from the carrier frequency
* modulation index or index of modulation
  * carrier-deviation-frq / mod-frq
  * example: carrier 200, modulator 100, modulator peak amplitude 300 (-300 to 300), modulation index 6 (600 / 100)
* if the modulation were not be around the center then the center frequency would vary with the modulation index

# operator configurations
* feedback: has a more even partial number and amplitude as index changes
* multiple carrier
  * superimposes the spectra of each carrier
  * can create formant regions, additionally with differing decay times per formant region
* multiple modulator
  * parallel
    * sidebands: c +- (i * m1) +- (k * m2)
    * like each produced sideband is modulated as a carrier by the other modulator
    * wave equation: a * sin(c(t) + i1 sin(m1 t) + i2 * sin(m2 t))
  series
    In series MM FM the modulating sine wave M1 is itself modulated by M2
    amplitude SMMFMt = A × sin {Ct + [I1 × sin(M1t + [I2 × sin(M2t)])]}.

# partial frequencies
~~~
carrier-frequency - partial-index * modulator-frequency
carrier-frequency + partial-index * modulator-frequency
~~~

# partial amplitudes
bessel(partial-index, modulator-amplitude / modulator-frequency)

the first parameter is the order of the [bessel function](/computer/guides/bessel.html) and the second argument is the modulation index

[online calculator](/dynamic/fm-partials)

# time domain waveform
fm(t) = carrier-amplitude * sin(t * carrier-frequency + modulator-amplitude * sin(t * modulator-frequency))

alternative:

fm(t) = sum(n, -infinity, infinity, bessel(n, modulator-amplitude / modulator-frequency) * sin(t * carrier-frequency + n * t * modulator-frequency))