# Introduction to Computer Music: Volume One

### 6. Principles of Audio-rate Frequency Modulation | page 5

Harmonic vs. inharmonic spectra and finding the fundamental frequency

If C and M are both integers (N), a ratio of 1:N will be harmonic but missing the partial numbers which are multiples of N, as in our 1:2 example above, which was missing all the even-numbered partials. Theoretically, any C-to-M ratio that is reducible to integers will produce sidebands that can be seen as harmonically related. If either the carrier or modulator frequency is an irrational number, then the spectrum will be inharmonic. Some integer ratios are very close to irrational, such as Chowning's favorite 1:1.31 or 100:131 as integers. The result for the listener, who will not be able to fuse the sound into a harmonic one, will for practical purposes be inharmonic. The nature of these inharmonic spectra, which have at least twice the frequency components of the harmonic spectra with no phase cancellation, give FM synthesis a wide palette of bright, vibrant timbres, including many bell-like possibilities. Many of these inharmonic spectra can have sidebands that reflect close to, but not on top of existing sidebands, providing the opportunity for shimmering, chorusing-type effects with certain ratios. Below you can see that the reflected sidebands do not reflect to positions midway between the non-reflected sidebands, thereby creating an inharmonic spectrum. A little further tweaking of the C:M ratio below could put these reflected sidebands closer to, but not directly on top of the non-reflected ones, creating a chorusing effect.

For harmonic spectra, there will usually be an implied fundamental frequency, though as we will see below, it may not always be audible. The carrier frequency is not necessarily the fundamental frequency. For the carrier to be the fundamental, M must be greater than or equal to 2 * C, or else be a 1:1 ratio. If, using the ratio method, C and M are integers that have no common factors (i.e. they have been reduced to their lowest form, 2:4 ->1:2), then the fundamental frequency will be the carrier frequency(in Hz)/C, which should also equal the modulating frequency(in Hz)/M. For example, 100 Hz/1 or 200 Hz/2 will both give the fundamental of a 100 Hz:200 Hz or 1:2 C:M ratio.

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