# Rules governing the creation of complex waveforms through addition of sine waves

I'm interested in synthesizing complex waveforms using sine waves.

I know that when two sine waves which differ slightly in frequency from one another are summed, amplitude modulation (AKA "beating") with a frequency equal to the difference in frequency between the two sine waves ensues. This relationship is very useful to know for an aspiring sound designer such as myself.

I also know that "stable" waveforms can be synthesized from a root frequency sine wave, with whole number harmonics added to it.

The area where I find myself wishing I knew some useful simplifying relationships is where one adds more than two sine waves of non-integer frequency values. For example, I would have no idea how to predict any characteristics of waveform resulting from the summing of three sine waves, of respective frequencies: 130.813, 131.089 and 131.366 Hz. I could of course simply graph the result and look at it, but I can't do this if I am trying to synthesize a complex waveform which I have imagined, whose component sine wave frequencies and phases I do not know.

I'm wondering, are there any other useful simplifying relationships that anyone can run past me?

One particular issue that is stumping me is as follows: when two sine waves, slightly detuned from one another, are added together, the "beating" behavior and frequency of the resultant waveform is easily predictable, by the simple rule described above. However, when a THIRD sine wave, of a frequency close but not equal to the frequency of the other two sine waves is added to the sum of these other two sine waves, I don't know how to predict the amplitude envelope shape/position of the resultant waveform. I've tried looking at wave1+wave2, and wave1+wave3, and wave2+wave3 in hopes of being able to deduce some sort of simple relationship between their easily predictable amplitude evelopes (i.e. "beat" frequencies) and the amplitude envelope of wave1+wave2+wave3, to no avail. It seems that the amplitude envelope of wave1+wave2+wave3 is determined by some complex interaction of waves 1, 2 and 3 that's result I don't know how to predict.

I could of course simply add wave1+wave2+wave3 and look at the resultant waveform, but I am trying to find a more simple way, if such a way exists, to predict what the amplitude envelope of wave1+wave2+wave3 will be.

If there does not exist any simple rule by which to do this, are there any valid rules/relationships that would at least aid in accomplishing the aforementioned task, perhaps by bounding the range of possible sine waves needed to construct a given waveform, or something along these or other lines?

Another type of behavior that I'm trying to understand has to do with a complex waveform that was synthesized using 16 sine oscillators of varying frequency and phase. The perceived pitch of this waveform actually decreases in one region of the waveform... I'm wondering if there are any simplifying rules/relationships which I can use to predict when a certain combination of sine waves will produce such behavior.

SGT
max_planck735 said:
I'm interested in synthesizing complex waveforms using sine waves.

I know that when two sine waves which differ slightly in frequency from one another are summed, amplitude modulation (AKA "beating") with a frequency equal to the difference in frequency between the two sine waves ensues. This relationship is very useful to know for an aspiring sound designer such as myself.

I also know that "stable" waveforms can be synthesized from a root frequency sine wave, with whole number harmonics added to it.

The area where I find myself wishing I knew some useful simplifying relationships is where one adds more than two sine waves of non-integer frequency values. For example, I would have no idea how to predict any characteristics of waveform resulting from the summing of three sine waves, of respective frequencies: 130.813, 131.089 and 131.366 Hz. I could of course simply graph the result and look at it, but I can't do this if I am trying to synthesize a complex waveform which I have imagined, whose component sine wave frequencies and phases I do not know.

I'm wondering, are there any other useful simplifying relationships that anyone can run past me?

One particular issue that is stumping me is as follows: when two sine waves, slightly detuned from one another, are added together, the "beating" behavior and frequency of the resultant waveform is easily predictable, by the simple rule described above. However, when a THIRD sine wave, of a frequency close but not equal to the frequency of the other two sine waves is added to the sum of these other two sine waves, I don't know how to predict the amplitude envelope shape/position of the resultant waveform. I've tried looking at wave1+wave2, and wave1+wave3, and wave2+wave3 in hopes of being able to deduce some sort of simple relationship between their easily predictable amplitude evelopes (i.e. "beat" frequencies) and the amplitude envelope of wave1+wave2+wave3, to no avail. It seems that the amplitude envelope of wave1+wave2+wave3 is determined by some complex interaction of waves 1, 2 and 3 that's result I don't know how to predict.

I could of course simply add wave1+wave2+wave3 and look at the resultant waveform, but I am trying to find a more simple way, if such a way exists, to predict what the amplitude envelope of wave1+wave2+wave3 will be.

If there does not exist any simple rule by which to do this, are there any valid rules/relationships that would at least aid in accomplishing the aforementioned task, perhaps by bounding the range of possible sine waves needed to construct a given waveform, or something along these or other lines?

Another type of behavior that I'm trying to understand has to do with a complex waveform that was synthesized using 16 sine oscillators of varying frequency and phase. The perceived pitch of this waveform actually decreases in one region of the waveform... I'm wondering if there are any simplifying rules/relationships which I can use to predict when a certain combination of sine waves will produce such behavior.
Amplitude modulation is the result of the multiplication and not of the sum of two sinusoids.

hi max_planck735,

I'm interested in a similar problem. Did you ever find a solution to it? Actually, I was looking for an analytic expression to describe the periodic variations that occur when two frequencies are added together, but where the frequencies are not close and are rational fractions of each other. The standard beat frequency is no longer relevant in this case.

Thanks