The basics of synthesis
The basics of synthesis
1) The harmonic series
Pitched musical instruments such as stringed, wind and keyboard instruments, tuned to a particular tuning system, e.g. concert pitch create sound waves that resonate and decay in a mathematical relationship to the note first played. It is the interaction with the surrounding air that creates audible sound waves that travel away from the instrument. The typical spacing of these resonances, results in frequencies that are usually integer multiples, or harmonics, of the lowest frequency. These multiples form the harmonic series.
But, enough of that boring stuff! Put simply, a note played on say, a guitar will vibrate in such a way that it creates audible sound waves at a higher frequency until they fade to silence. Still confused? Hit the middle C on a piano and as well as hearing the note you played - the C, you will also hear a note half a pitch above it and a quarter above it and a sixteenth above it and so on. The note you play is called the fundamental as shown on the picture below, that illustrates all the other harmonics above that note.
A feature of this harmonic series is that each note above the fundamental will sound quieter until there is just silence and therefore the fundamental will appear to be the loudest note. Also, we can perceive the pitch of the fundmental even if it is missing, but beware, the brain can also be fooled into perceiving the wrong fundamental pitch, but let's not go there yet.
Each of the harmonics are directly related to each other and sound harmonious, but depending on the structure of the instrument, particular harmonics may be emphasised, adding musical colour. For example, pucked instruments will decay faster than bowed instruments where the bow sustains the vibration.
Partials are really thought of as additional harmonics that occur between principle harmonics and the term Inharmonicity is the extent to which a partial deviates from the closest ideal harmonic, these deviations are measured in cents or one-hundredths for each partial. Wow, that's confusing isn't it, well lets just think of these partials as additional musical colour that distinguishes one musical instrument from another, and why people pay big money for a Martin acoustic guitar and don't just content themselves with a cheap chinese import.
Some pitched instruments, particularly percussion instruments like a vibraphone, have non-harmonic partials, that make a rich sound to the ear.
Finally, overtones refer to any partial except the lowest and does not mean that the instrument is either harmonic nor inharmonicity in sound, it simply is a term that excludes the fundamental. Therefore the first overtone is the second partial.
If that is confusing, look at the following chart that shows the fundamental and harmonics of various instruments including the human voice. This illustrates the huge amount of additional musical information the brain hears or body feels when a particular instrument is played. Note that the male voice has a potentially huge range and con be extended further by basso profundo and whistling, that can be found on YouTube, if you're curious.
Music Ranges https://www.independentrecording.net/irn/resources/freqchart/main_display.htm
Electronic instruments, such as synthesizers, can play a pure frequency with no overtones. At the same time, they can be used to create frequencies and sound characteristics to simulate other instruments, or combine various instrumental textures and random noises together to create a unique sound.
2) The harmonic content of a sine, triangle, square and sawtooth waveform
The graph below, illustrates the four periodic sound waves mentioned above. They are called periodic because they repeat the same waveform shape again and again. These are the main wave forms used in synthesis.
A waveform consists of many different sine waves, and the slowest is the frequency of the note, with the slowest sine wave being the fundamental harmonic. Put another way, a basic sine wave has not harmonics and consists only of the fundamental.
A sine wave
All sounds are made up of sine waves and the harmonic content of a waveform, is a feature of frequencies of all the sine waves that are present within it and how loud they are.
A triangle wave
A triangle wave, as the name suggests looks triangular and the wave falls slowly then rises slowly. Odd harmonics only, are present in the triangle wave and the higher harmonics are of low amplitude.
A square wave
The square wave only has odd harmonics with the first sine wave, the third, the fifth and so on. A square wave is similar to a ramp wave, (see below) but with every even numbered harmonic missing. This gives it sound hollow by comparison with the saw tooth or ramp wave shown below.
A sawtooth wave
So named because it looks like the teeth of a saw. There are actually two sorts of sawtooth wave that can be created: a ramp wave or rising sawtooth, that rises slowly then falls quickly, and this is the most common of the two, or a sawtooth wave that falls slowly. The rising sawtooth wave includes all harmonics, including the fundamental harmonic, and then a second sine wave, half as loud but twice as fast. The third one is a third as loud, but three times the speed. The ramp waveform has a rich timbre with plenty of harmonics.
The waveforms above, are the periodic waveforms most often produced by the oscillators of synthesisers, however it is possible to synthesize virtually any shape of waveform when experimenting with synthesizers. It is claimed that the human ear gets bored with harmonic as they become too predictable. Using lots of with variations such as quasi-periodic waveforms, can overcome this boredom, however the instability of analogue circuitry is making a comeback. long live analogue I say, but on the other hand, digital synthesizers are so user friendly, it is hard to resist them.
More info can be found on the link below, which also gives you the chance to hear the sounds these wave forms produce.
https://www.thedawstudio.com/Tips/Soundwaves.html