**Principles
of Audio-Rate Frequency Modulation
**

**Terms:**
audio rate, modulation, carrier, modulator, sidebands, reflected
sidebands, Bessel function, carrier:modulator ratio, linear
(Chowning) or exponential FM.

This article explains the phenomenon of audio-rate frequency modulation of sound, which was explored and used compositionally by John Chowning of Stanford University around 1970. His discoveries eventually lead to the design and release of the Yamaha DX-7 family of instruments, one of the most successful synthesizers of all time.

**sub-audio-rate
frequency modulation**

If the output of
an oscillator is applied to control the frequency of another
oscillator, frequency modulation (FM) will result. The oscillator
providing the control source is referred to as the modulator, the
oscillator providing the signal is referred to as the carrier. If
the modulating oscillator is tuned below audio-rate (or
approximately 20 Hz), sub-audio frequency modulation or **vibrato**
will result. Very simply put, as the modulating waveform rises
(increases in amplitude), so too will the frequency of the
carrier; as it falls, so too will the frequency of the carrier.
The rate of the vibrato is determined by the modulator's
frequency, the depth of the vibrato (or how far above and below
its center frequency the carrier will be pushed) is determined by
the modulator's amplitude and the shape of the vibrato is
determined by the modulator's waveform. It is important at the
outset of the discussion to realize that the modulator is not
part of the signal path--it is never heard directly, only its
effect on the carrier frequency.

**audio-rate
frequency modulation**

When the rate of
the modulating oscillator is tuned above 20 Hz, or at an *audio
rate*, very interesting things happen to the sound.
Additional frequencies called **sidebands **appear
symmetrically around the carrier frequency. Those above the
carrier frequency are called *upper sidebands* and below, *lower
sidebands*. In essence, as will be seen below, some of the
energy of the carrier frequency is being stolen to create these
additional frequencies.

**'Chowning'
FM**

Both the exact
frequencies and the relative strength of the sidebands are
predictable using digital technology where all parameters can be
precisely controlled. The bulk of our discussion will deal with
classic or Chowning FM, named after its greatest proponent.
Simple Chowning FM uses the most basic sine wave, which produces
no other frequencies apart from its fundamental, as both the
carrier and modulating waveform. Indeed, one of the beauties of
Chowning FM is its ability to do so much from two very simple
waves. The other qualification of Chowning FM is that the
modulation be *linear*, whereby the carrier is pushed an
equal number of cycles per second above and below its center
frequency. *Exponential* FM, where the carrier is pushed
up and down an equal musical interval (therefore more Hz up than
down) drifts upwards in its pitch axis as the modulation depth is
increased . Linear FM allows the strength of modulation to be
increased without the perceived center frequency rising.

**calculating
sideband frequencies**

Sideband frequencies can be calculated with the following formula where Cƒ=carrier frequency, Mƒ=modulator frequency, n=all positive integers including 0:

**C**_{ƒ} ± n**M**_{ƒ} **[n=0,1,2,3...]**

OR (for those
with math anxiety)

**the carrier
frequency (Cƒ) plus and minus all the integer multiples of
the modulating frequency**** (Mƒ)**

OR (for those with really serious math anxiety)

Cƒ, Cƒ + Mƒ, Cƒ - Mƒ, Cƒ +
2Mƒ, Cƒ - 2Mƒ, Cƒ + 3Mƒ, Cƒ -
3Mƒ, etc. to infinity and beyond

For example, a carrier frequency of 400 and a modulating
frequency of 50 will produce a spectrum for

n=0 of 400 Hz (400 + (0 * 50))

n=1 of 450 Hz (400 + (1 * 50)) and 350 Hz (400 - (1 * 50))

n=2 of 500 Hz (400 + (2 * 50)) and 300 Hz (400 - (2 * 50))

etc.

A graph of this example's sideband pairs, related by color, appears below.:

Another way of calculating sidebands, useful when the carrier and modulator are maintaining a constant frequency relationship is through a ratio of carrier to modulating frequency (C:M). For example, a Cƒ of 100 and an Mƒ of 200 would produce a C:M ratio of 1:2 (click here to see how to reduce C:M ratios to their "normal form"). We'll see below that integer ratios that have a carrier value of 1 have certain properties. We could calculate the upper sidebands for this ratio in relation to the carrier frequency as C+M, C+2M, C+3M, C+4M, etc. For our 1:2 example, the first upper sideband would be 1+2=3, the second would be 1 + (2 * 2)=5. If you worked this out in Hz, you would quickly come to the conclusion that these are the odd numbered partials of the carrier frequency. We calculate the lower sidebands similarly as C-M, C-2M, C-3M, C-4M or in our 1:2 example, -1, -3, -5, etc.

**reflected
sidebands**

What to do with these
negative values? Using our two methods of calculating sidebands,
say we had a carrier ƒ of 200 Hz and a modulator ƒ of
400 Hz -- that would give us our 1:2 C:M ratio. If we calculate
the n=1 pair, we get an upper sideband of 600, but a lower
sideband of -200 using our first method or a relative frequency
of -1 to the carrier. These sidebands in the negative domain are
called **reflected sidebands **because they bounce
back from zero at their absolute value 180 degrees out of phase
with their sideband partners. So for both methods of calculating
frequency, we simply remove the minus sign, expressed
mathematically as absolute or /-200/. If these frequencies do not
bounce back on top of other frequencies, then the phase reversal
is inaudible. However, as is particularly true in harmonic
spectra, when they do bounce back on top of other partials, phase
cancellation or summation has a great impact on the timbre. For
example, if you had a positive sideband at 400 Hz and a negative
sideband at 400 Hz but half the strength of the positive one,
only half the amplitude of the positive one will survive. If both
were at equal strength, neither would be heard since they would
completely cancel each other out. If they were both positive or
both negative, they would be summed. In our example above of a
Cƒ=200 Hz and Mƒ=400 Hz, the lower sideband of the n=1
pair would reflect back on the carrier frequency (n=0), or
/C-M/=C (/-1/=1). Who will survive will be a mystery to be solved
below when we can calculate the relative strength of each.

**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).

**computing
the sideband strengths • the modulation index (I)**

As with ordinary complex
waveforms, the timbre perceived by the listener is determined not
only by the frequencies present, but also their relative
strengths. The upper and lower sideband of each sideband pair has
the same strength. In order to calculate the strength of
each sideband pair relative to the others, we must first look at the prime
factor which determines it. As we have seen above, when the
carrier is modulated, its frequency rises and falls with the
amplitude of the modulating wave. The greater the amplitudes of
the modulating wave are at its peaks, the greater the maximum
distance the carrier is pushed off its center frequency. At
sub-audio rate, we would perceive this as the depth of a vibrato.
When using a pure sine wave and linear modulation, these peaks
will be an equal number of Hz above and below the carrier's
center frequency. The number of cycles above or below the center
frequency is called the **peak deviation **(or p.d.,
or delta () ƒ). As the *amplitude* of the
modulating wave is increased or decreased by some means, perhaps
using an envelope generator, so too does the peak deviation
change. It is this parameter, the changing strength of the
modulating wave, that allows us to create dynamic, time-varying
spectra of a sort very different from subtractive filtering and
one that can, under certain circumstances, mimic the complexity of
real-world sound characteristics using only two oscillators.

To compute how the
strength of the sideband pairs change over time as the strength
of the modulating wave is varied, we divide the peak deviation by
the modulating frequency to produce a value called the **modulation
index **or simply **I****. **

If none of the modulating
wave is permitted to reach the carrier, the peak deviation and
the value of **I****
**will be zero, since no
modulation will be taking place. As the amplitude of the
modulating wave increases, the carrier is pushed farther and
farther off its center frequency and the value of **I**** **also increases. The effect of an
increasing **I****
**is different for
each sideband pair. In our first formula for predicting sideband
frequencies, we used all the integer values of n (Cƒ ±
nMƒ). A sideband pair calculated with a particular value of
n can be call a pair of the n^{th }order.

As **I**** **increases, each sideband pair follows
its own path of increasing and decreasing strength called a **Bessel
function. **The Bessel function curves followed are
different for each of the n-order sidebands--one of the things
that makes frequency modulation so interesting. (To trig students,
these are called *Bessel functions of the first kind of order
n*; to non-trig students, it's more like Close Encounters of
the Third Kind.) Below is a graph of the first seven orders
(beginning with 0) of sideband pairs, showing their relative
strength on the vertical axis as **I****
**increases on the
horizontal axis.

Note that at * I *= 0 (i.e. no modulation), the carrier (red, n=0)
is at full strength. As

Here are two
examples of the spectra produced for fixed values of * I*, computed by simply looking at the vertical
example lines above. The first value of

The second
example shows a higher value of * I*, which also includes some negative
strengths.

In general, as * I* increases, we can infer that more and more
frequencies become audible. This can be a real problem for
digital synthesis, where the upper sidebands may reach the
Nyquist frequency (see digital audio) and alias. FM is

What happens to those mysterious lower sidebands that reflect at their absolute value 180 degrees out of phase? Well, take their order's Bessel function above and invert it. If its strength would normally be in the positive domain, it will be of equal value in the negative domain. This adds greatly to the interest of a dynamically changing harmonic spectrum where sidebands are likely to foldback on top of other sidebands because of the increased complexity of phase cancellation and summation.

Below is the same mod index as example 2, but with a carrier frequency plotted low enough so that the lower sidebands, starting with the n=4 pair reflect back on top of existing sidebands. In this case, the lower n=2 (green) and n=4 (purple) will sum, n=1(blue) and n=5 (orange) will just about cancel each other out and n=0 (red) and n=6 (dark green) will fight it out, but the stronger n=0 will be heard, but reduced by the value of n=6. The lower n=3 (yellow) plots out at 0 Hz and so is not heard at all.

**two FM
audio examples**

Here are two audio examples. The first has a C:M ratio of 1:2 creating a harmonic spectrum, the second a C:M ratio of 1:1.31 creating an inharmonic spectrum. In both examples, the modulation index is increased slowly over 10 seconds from 0 (no modulation) to 15. Play these several times and focus on different frequencies as they move through their Bessel functions. Start with the carrier frequency (the first frequency you will hear). Listen as it immediately begins to lose strength, completely disappears and then reappears. Listen to the effects of phase cancellation in the first harmonic example, which will have far fewer discreet frequencies than the inharmonic one.

**Click here to play harmonic
example (C:M = 1:2, carrier frequency = 200 Hz)**

**Click here to play the
inharmonic example (C:M = 1:1.31, carrier frequency=150 Hz**

**some
variations on FM**

Many things can
be done to create more complex spectra with FM. The DX-7 was
built around the idea of both double-carrier FM, in which a
single modulator controls two carriers, tuned differently. This
allows the creation of formant areas not possible with single FM.
Also, stacks of modulators, where a modulator was itself
modulated, could either produce wildly complex spectra if tuned
inharmonically or produce weighted spectra, which could create a
more realist bass. This helped with one of FM's greatest
drawbacks--the strength of the upper and lower sidebands are
equal, but our human hearing requirings more energy in the lower
frequencies to be considered as equally loud as the higher
frequencies. Therefore, single FM always seemed weighted to the
treble, particularly at higher values of * I*. Another interesting idea is to
modulate the modulation index itself, providing a rapid timbral
shift. or to low-frequency modulate the modulator or carrier,
changing the C:M ratio and therefore the frequencies of the
sidebands for some very nice effects.

**suggested
listening examples**

To hear
audio-rate FM used with a high level of artistry, there can be no
better source than the works of John Chowning himself. Highly
recommended are *Stria (1976), Turenas (1972) and PhonČ
(1981). *Barry Truax was another pioneering FM composer with *Arras,
Androgyny, Wave Edge, Solar Ellipse, *and *Sonic Landscape
No. 3*.

**suggested
reading**

J. Chowning, "The
Synthesis of Complex Audio Spectra by Means of Frequency
Modulation," Journal of the Audio Engineering Society 21(7),
1973; reprinted in Computer Music

Journal 1(2), 1977; reprinted in Foundations of Computer Music,
C. Roads and J. Strawn (eds.). MIT Press, 1985.

B. Truax, "Organizational Techniques for C:M Ratios in
Frequency Modulation", Computer Music Journal, 1(4), 1978,
pp. 39-45; reprinted in Foundations of Computer Music, C. Roads
and J. Strawn (eds.). MIT Press, 1985.

C. Dodge and T. Jerse, Computer Music, 2nd Ed., Schirmer Books,
1997.

F. Richard Moore, Elements of Computer Music, Prentice Hall,
1990.

**finding the C:M ratio ****normal
form**

The concept of
the *normal form* for a C:M ratio has been used for a long
time. It is useful for predicting which C:M ratios will produce
the same sidebands, but it is **not** useful for
predicting their relative strengths or phases. If the value of M
in a ratio is less than twice the value of C, it is not in normal
form, but can be reduced to normal form by applying the
operation: C = /C - M/. What this means is that you subtract M
from C (ignoring any minus sign) and treat the result as the new
C value. You keep doing this (often several times) until the
ratio satisfies the normal form criterion.

For example, take the C:M ratio of 3:2. Take 3 - 2 and get 1.
That is the new value of C (keep the old value of M), so the new
ratio will be 1:2. How is this possible--how can 3:2 produce the
same sidebands as 1:2? Let's try it out with 300:200 Hz as our
3:2 ratio and 100:200 Hz as our 1:2 ratio.

3:2 sidebands | 1:2 sidebands | |

n=0 | 300 | 100 |

n=1 | 100, 500 | /-100/, 300 |

n=2 | /-100/, 700 | /-300/, 500 |

n=3 | /-300/, 900 | /-500/, 700 |

So you can see
they produce the same frequencies, but with sidebands of
different orders and different reflections. Therefore, the way
these frequencies react to changing values of * I* will be completely different. But some
interesting things can be deduced using normal form. A C:M ratio
is in normal form when the carrier is the fundamental in the
spectrum it produces, as in our 1:2 example above -- 100 Hz

* Return to CECM Home Page.*

This document is prepared and maintained by the Indiana
University School of Music

Center for Electronic and Computer Music

Prof. Jeffrey Hass

Last updated: 10 November 2001

URL: http://www.indiana.edu/~emusic/fm/fm.htm

Comments: cecm@indiana.edu

†Copyright 1995-2001, Jeffrey Hass and The Trustees of
Indiana University