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Tutorial about synthesis of sound: the properties of sound and types of waveforms.

Understanding sound

Sound can be defined as the displacement of successive pressure waves across the air - or another medium - which can be perceived by the hearing organs, and it is made of three different components that define how it is perceived. Thereby, we will see with detail these components to later learn how to "sculpt" sound in an electronic synthesizer.

Frequency is the number of waves or cycles that a certain sound produces every second, and is measured in cycles per second or Hertz (Hz). Therefore, a 50 Hz sound emits 50 cycles in a single second. As a matter of convenience, when frequency reaches 1000 or more cycles per second its magnitude is usually indicated in kiloHertz (kHz), namely, thousands of Hertz. The human ear can perceive a frequency range from 20 Hz to 20kHz, approximately. Musical notes are based on the frequency of the sound that musical instruments produce. In the world of music, sound can be classified into three categories attending to its frequency:

Bass: from 10 Hz to 200 Hz
Mid: from 200 Hz to 3 kHz
Treble: from 3 kHz to 20 kHz

Frequency correlates with the pitch that a hearer perceives. The following are usual frequency ranges for some common musical instruments:

Kick drum: from 20 to 150 Hz
Bass: from 20 to 250 Hz
Piano: from 80 to 4500 Hz
Snare drum: from 100 to 200 Hz
Cymbal: from 300 to 600 Hz

The following picture shows all the notes (or pitches) that can be found in the keyboard of a piano, organ, synthesizer or sampler. Note that the notes of an octave have twice the frequency of the equivalent notes of the previous octave; for example, the frequency of C2 is twice the frequency of C1 and half the frequency of C3.

Piano keyboard pitch table

Amplitude is the loudness level of a sound and is measured in deciBels (dB), unit which represents the tenth part of a Bel (B), root unit of logarithmic nature used to express the relationship between two magnitudes: a magnitude that is studied and a magnitude that serves as reference. Each successive Bel in the scale multiplies by ten times the power over the reference magnitude, which has a value of 0 Bel. Since the Bel is too large to be used with ease the deciBel is used instead. An amplitude with a value of 0 dB represents a non perceivable sound, whereas an amplitude with a value of 10 dB represents a sound that is ten times louder; the loudness of a sound is multiplied by ten everytime that amplitude increases its value in 10 dB. We have to be careful for an excessive amplitude in the sound can damage our hearing organs.

The following picture shows how frequency and amplitude relate to a waveform; frequency is conceptualized as an horizontal dimension and amplitude as a vertical dimension. The two waveforms have the same amplitude, so a listener would perceive a similar loudness in both cases; however, frequency varies, so a listener would perceive a lower pitch in the upper waveform and a higher pitch in the lower one. The distance between each wave is called cycle. The topmost area of a wave is called peak and the lowermost area is called valley.

Waveform amplitude and frequency

Timbre is the characteristic of a particular sound which could be defined as its personality, or its color, if we wanted to make a comparison with visual arts. Timbre is defined by the source of a sound and the way this one was generated; it is not directly related with frequency, as two sounds having the same frequency can have very different timbres. A musical instrument is designed for having its own distinctive timbre and therefore its own personality. Each note in a musical instrument has its own frequency but they all have the same timbre, whereas different instruments can play the same notes but with different timbres.

Timbres are composed of multiple waveforms which combined together form a more complex waveform. Musical tones comprise numerous sine waves having different frequencies and amplitudes, and their pitch is defined by their fundamental frequency, which is the lowest frequency present in any tone. The higher frequencies that accompany the fundamental frequency and form the timbre of the sound are called overtones or, less often, upperpartials, and those overtones which are multiples of the fundamental frequency are called harmonics; taking as example a tone which has a fundamental frequency of 1 kHz, its second harmonic would be 2 kHz, its third harmonic would be 3 kHz, and so forth...

Harmonics change the timbre of a sound without affecting its pitch. The number of harmonics in a tone is variable; tones which have a higher number of harmonics sound brighter or more defined than tones which have a lesser number of harmonics. Harmonics are an essential concept in the field of sound generation and manipulation, and hence they will be recalled further in this tutorial. For now, we will dive deeper into the concept of timbre and for such purpose we have to learn about the different types of waveforms.

Types of waveforms

In a technical sense we refer to sound as waveform. Every sound is either a single waveform or multiple different waveforms which combined together form a certain sonic timbre. Each of the basic types of waveforms has its own sonic quality, and a basic understanding of how each sounds like is essential for the purpose of creating the desired timbre through synthesis. The following picture shows how the basic types of waveforms look like in a graphical representation. As in the previous example, the horizontal dimension represents time and the vertical dimension represents amplitude. If these waveforms were generated by an electronic oscillator the amplitude would be controlled by voltage, and the shape of the diverse waveforms would be determined by how the oscillator varies voltage on each cycle.

Waveform types

Sine waves are useful for creating deep warm basses or smooth lead lines. They can be used to create whistles, layered with kick drums to give a deep subby effect. The sine wave is a pure waveform and its harmonic content is fundamental; this means that this is the most basic type of waveform and that it has no harmonics, but only the tone of the fundamental frequency (called pitch). Almost every other waveform consists of a certain amount of combined sine waves, all having different frequencies and amplitudes. A waveform which does not change its timbre over time is made up of sine waves which are multiples of the fundamental frequency (the natural harmonic series).

Audio example of a sine wave

Square waves are great for brass and deeper wind instruments, and are usually used in combination with other types of waveform because they are quite strong and hard on their own. Square waves can be generated by adding a certain amount of sine waves with decreasing volume (each harmonic being quieter than its previous). However, the square wave contains only the odd numbered harmonics.

Audio example of a square wave

Triangle waves are great for bell or wind instruments. Like square waves, they contain only the odd harmonics of the fundamental frequency, but differ in that the volume of each added harmonic drops faster.

Audio example of a triangle wave

Sawtooth waves have a buzzy, bright and edgy sonic quality, and are suitable for creating strings, brass instruments, trance pads and leads or electro basses, among others. A sawtooth wave can be made by adding a series of sine waves at different frequencies and amplitudes, and the frequency of the first and loudest sine wave is what we perceive as the fundamental frequency of the resulting wave. Each of the other, progressively quieter, sine waves have frequencies which are integer multiples of the fundamental frequency.

Audio example of a sawtooth wave

Noise waves are randomly changing, chaotic signals which contain an endless number of sine waves of any possible frequency and different amplitudes. However, randomness will always have specific statistical properties, which will give the noise its specific character or timbre. If the amplitude of sine waves is uniform the noise sounds very bright and in this case it is called white noise. If the amplitude of sine waves decreases as their frequency rise, the noise sounds much warmer; if it decreases following a curve of about -6 dB per octave, it is called pink noise, and if it decreases following a curve of about -12 dB per octave it is called brown noise.

Due to its bright quality, white noise is used in the synthesis of elements such as hi-hats, crashes or cymbals. Pink noise is great for synthesizing ocean waves and warm ethereal pads, whereas brown noise is suitable for synthesizing thunder sounds and deep, bursting claps. In overall, all the types can be used in diverse ways for attaining different sonic textures and effects, being specially useful when used along with the aforementioned regular-shaped waveforms. The possibilities of noise waves are endless and therefore the largest part of modern synthesizers have included them as a prime sound generation source.

The following pictures show the result of adding two sine waves together; in the first example waves of different frequencies form a new wave which has different shape and timbral properties, whereas in the second example waves of equal frequency but opposite amplitude nullify each other, giving silence as result.

Addition of waveforms

To program a new sound in a synthesizer, the first step would be to select the right combination of waveforms in the oscillators (OSC), which would give a starting point approximate the timbre of the desired sound. Oscillators are the components which generate the simple waveforms which should be combined and further processed to create complex waveforms. The following picture shows the controls of the oscillator bank of the FreeMoog VST analogue synthesizer. The controls to the right allow to choose six different types of waveforms for each of the three oscillators. As it is easy to understand, each oscillator present in the bank can exponentially increase the amount of possible combinations.

FreeMoog VST oscillator bank

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