Phase and Harmonics
This month I am going to concentrate on a few crucial areas of synthesis, with a general overview of what has been covered. However, to fully understand what I am talking about, you need to understand, yep, the original concept of sound, but in this instance, with emphasis on harmonics.
First, let me glide over the concept of phase. Whereas I promised we would cover this topic in more detail in reference to filters, I am now forced to bring it up in reference to the overall concept of wave shaping.
Let us take our simplest example, two sine waves, and combine them together. As you might have guessed from Fig1 (top), adding together two identical waves produces the same sound, but louder. But what happens if you start the lower wave halfway through the cycle of the upper one, as in Fig1 (bottom)? They cancel each other out and you end up with silence. This may seem hard to bend your head around, but it makes perfect sense.
As we saw in earlier parts of this tutorial, a single sine wave can be measured by specifying just its frequency and amplitude. But by combining two or more waves you must consider their relative offset. This offset is usually called the 'phase' of one wave with respect to the other. We can then measure these in degrees or time. Don’t worry about this for now. What you do need to be clear about is that the simple offset of one wave against another has very serious implications in sound wave theory.
Combining ‘complex out of phase' signals does not necessarily lead to complete cancellation. Let us take an example that deftly brings us to the topic of harmonics and fundamentals. You should know, from earlier parts of this tutorial, what fundamentals and harmonics are. So if you do not know or are confused at this juncture, then go back and reread the relevant part. Let us take a saw wave as our example. The saw wave has every harmonic present. If the first harmonic (fundamental) lies at 100Hz, the second harmonic will be at 200Hz etc. Adding two of these saw waves, with the fundamentals offset half a cycle means that the fundamentals are cancelled out. But the second harmonics, lying at 200Hz, will be added. The third harmonic will be cancelled out, the fourth harmonic will be reinforced, the fifth harmonic will be cancelled and the sixth reinforced and so on. The result is a waveform with harmonics at 200Hz, 400Hz etc. What we are left with is a saw wave with the amplitude of the original but twice the frequency.
Fourier analysis states that any two complex signals can be described as an infinite number of sine waves that represent all the frequencies present in the signal. So, it follows that any given offset between two identical signals, each frequency will be phase-shifted by a different amount.
You are probably wondering what all this means and what it has to do with synthesis. Well, it is essential that you understand the concept of phase, and what I have avoided is to go into deep and emotional graphs and explanations. The subtle move into the topic of phase is an ongoing diatribe that follows on from the subject of filters in the last part of this tutorial. The final conclusion is to state that filtering leads to changes in phase. The very fact that we are effecting frequencies means that, using the Fourier dude’s thinking, phase is not only created as a by product of audio manipulation, as outlined above, but filtering can also lead to phase changes.
Ok, we have touched on phase and, as stated earlier, I want to now enter the world of harmonics.
I cannot begin to stress how important it is for you to try to bend your head around this subject. You have often heard the word harmonics used, and in almost all cases, it is referred to in just about every area of sound synthesis and design. It is also used in production, mastering, playing/performing, recording etc.
Since we have already touched on this topic, not only in earlier parts of this tutorial, but also in this month’s instalment, you should have a relatively good idea as to what harmonics are. We have also covered the fundamental, so that does not need explaining here.
Harmonic Series - Also know as Overtones
To explain what the harmonics series is, it is best to take the sine wave as our example. The reason for this is quite simple. A Sine wave is the most basic waveform there is. It is the foundation for the harmonic series. We already know that a sine wave has no harmonics. It only has the tone of the fundamental frequency. The fundamental frequency, denoted with the symbol F, is the base or root frequency which we identify as pitch. If you refer back to the frequency chart in the first part of this tutorial, you will find that all notes have a frequency. As an example, A4 equals 440 Hz. If we were to take that as the fundamental, then the 1 st harmonic would be 880 Hz, 2 nd harmonic would be 1320 Hz etc.
Since a sine wave is a pure waveform, i.e. it does not contain any harmonics, we can create other waveforms simply by adding together any number of sine waves, all at different frequencies and amplitudes. Any sound can be created using sine waves at different frequencies and amplitudes. The reverse is also true. Any sound can be broken down into discrete and distinct sine waves at different frequencies and amplitudes. A waveform, that does not change it’s timbre over time, is made up of sine waves which are multiples of the fundamental frequency. This is called the Natural Harmonic Series, self explanatory really. We can now define the harmonics as F, F2, F3, etc. In this case, we have the fundamental (F), the second harmonic (F2) and so on.
As we have already covered the topic of phase and what it does in terms of harmonics, it is then easy to define odd and even harmonics. With the example earlier in this part, we saw that by offsetting by half a cycle, we were left with the 2nd harmonic, 4 th harmonic, etc. This is called even harmonics. It then makes sense to assume that we also have odd harmonics. Both odd and even exist for different types of waveforms. For complex waveforms, we need to break down the waveforms and ascertain their harmonics. This is not crucial for you to understand at this juncture. What is crucial is that you understand the concept of odd and even harmonics.
Saw and Pulse, generally have all the harmonics. Square and Triangle, generally have odd numbered harmonics. However, if you narrow the pulsewidth of the square waveform, even harmonics will appear.
Other areas to note are the amplitudes
As with most analogue synthesizers, two or more oscillators can be mixed together to give a resultant waveform. The resultant waveform is simply the sum of the waveforms (the oscillators). The resultant harmonic series would also be the sum of the harmonics of the oscillators.
What I have not mentioned, so far, is noise.
In essence, noise is a randomly changing, chaotic signal, containing an endless number of sine waves of all possible frequencies with different amplitudes. However randomness will always have specific statistical properties. These will give the noise its specific character or timbre. If the sine waves’ amplitude is uniform, which means every frequency has the same volume, the noise sounds very bright. This type of noise is called white noise. If the amplitude of the sine waves decreases with a curve of about -6 dB per octave when their frequencies rise, the noise sounds much warmer. This is called pink noise. If it decreases with a curve of about -12 dB per octave we call it brown noise. Bet you didn’t know that one, huh, brown noise?
So we have all these funky names for noise, even though you need to understand their characteristics, but what are they used for? White noise is used in the synthesizing of hi-hats, crashes, cymbals etc, and is even used to test certain generators. Pink noise is great for synthesizing ocean waves and the warmer type of ethereal pads. Brown noise is cool for synthesizing thunderous sounds and deep and bursting claps. Of course, they can all be used in varying ways for attaining different textures and results, but the idea is simply for you to get an idea of what they ‘sound’ like.
At the end of the day, it all boils down to maths and physics.
Next time: some more processes and terminology.