Sine wave
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If we break down the components into smaller pieces for better understanding when working with music and sound design, the first and most basic topic to consider is sine wave. We humans perceive the invisible phenomenon of air density changes in nature as sound through our ears. The smallest unit of sound is a sine wave with one period. We may say that composition is the macroscopic way to compose sine wave, and sound design is the microscopic way to compose sine wave.

A sine wave is the only wave among the various waveforms that has a single frequency component, that is, it is a pure sound that contains no overtones or incidental layers of sound, but in a strict sense, pure sine waves do not exist in nature.
In other words, sound composed of sine waves has the potential to develop a sound world unique to electronic music that cannot be realized in the acoustic world.

An especially interesting part of the generation of accurate sine waves with stable pitch and volume, made possible by digital technology, is the phenomenon of interference. Do you remember studying in high school physics class that when sine waves of adjacent frequencies are superimposed on each other, the difference in frequency becomes a beat, and the rhythm of the sound is created? When stable sine waves are used, this rhythm is mechanically accurate, so it is possible to play inhuman rhythms like techno with beats generated spontaneously by the difference in frequencies. Ryoji Ikeda is a representative example of the application of this phenomenon in his works.

The sound of various musical instruments, such as the sound of a piano, a guitar, or a violin, contains overtones. An overtone is also a state in which multiple sine waves are superimposed. Various unique tones are created when sounds with integer multiples of two or three times the frequency of the fundamental sound are superimposed, and they are all sine waves of different frequencies.

The method of synthesis that creates sound by layering overtones with sine waves is called additive synthesis. Even square waves and sawtooth waves, familiar to those who have used synthesizers, can be reproduced by stacking sine waves. However, a huge number of sine waves are required to reproduce all kinds of tones. Because of the increased computational complexity, electronic instruments and computers until the 20th century rarely implemented and commercialized sine waves, despite the simplicity of the principle.

Download Max Patch basicwavefoorm.maxpat

Now that we have published a program that allows you to listen to sounds using Cycling '74 Max, let's take a look at the relationship between the waveform and the overtone structure most frequently employed in the oscillator portion of synthesizers while actually running the program. If you do not yet have Max, please download it from the download link at the bottom of the page. You cannot edit the program without using the paid version, but you can run the distributed program and make sounds while changing the parameters.

Sine wave does not contain any overtones, as can be seen from the spectrogram, which displays the volume of each frequency The overtones of sawtooth waves include both even and odd harmonics, and the volume of the nth harmonic (a sine wave with a frequency N times that of the lowest reference note) has the characteristic of becoming 1/N, for example, the volume of the 8th harmonic is 1/8. As a tone, it is used to create brass or string-like sounds.

Triangule wave is a waveform that contains only odd harmonics, with the volume of the Nth-order harmonic being 1/N squared. Since the higher the order of overtones, the lower the volume, it is often used to create bass tones in electronic music or soft flute-like tones.

Like triangle wave, square wave contains only odd harmonics, but since the volume of the Nth harmonic is 1/N, it may be thought of as a triangle wave with a louder version of the harmonics. It is used to create sounds like woodwinds, but it can also be filtered to create a dubstep-like wobble bass, detuned to create a neo-grime lead-like tone, or used as a standard chiptune tone since many old video game sounds were made with square waves, It may be the waveform with the greatest range of applications.

Let's take a look at a prior example of a work that was conceptually created using sine waves.

The SINE WAVE ORCHESTRA (SWO) is a project that works exclusively with sine waves that was launched in 2002 by Ken Furudate, Kazuhiro Jo, Daisuke Ishida and Mizuki Noguchi.

The work "stay" is an installation work in which a device that emits sine waves, the frequency of which can be freely set by the viewer, is placed in any position on wires installed at the venue, creating a more complex space as the exhibition progresses.

Ryoji Ikeda creates site-specific works around the world using digital technology to the utmost limit, making full use of the physical characteristics of sound, its relationship to human perception, time, and space. His works are characterized by his composition of data by reducing light and sound to the smallest unit of pixels, white noise, and sine waves, and structuring them based on a variety of concepts.

The idea of additive synthesis can be traced back to medieval organs, long before the invention of synthesizers and electronic technology.
The mechanism of creating various tones by changing the air flowing through the physical pipes with numerous organ stops and their combination type may be said to be a synthesizer before the advent of electronic technology.

The interesting thing is that the organ, which can be called the oldest synthesizer in existence since the Middle Ages, is still in use around the world in the 21st century, and works for the organ are still being written and attracting people today. It is simply amazing to see the longevity of the instrument before the advent of electronic technology.

There are not many synthesizers that use additive synthesis of sine waves to create tones, but several exist, both in software and hardware.
Digital additive synthesis is now capable of producing not only integer overtones with integer frequency ratios, but also metallic tones with a large number of overtones in complex ratios, as represented by the Razor and Odessa tones, a characteristic of digital additive synthesis now that machine power has become more ample.

Native Instruments Razor

Razor Official Page

Demo Movie

These modern digital additive synthesizers allow more than 1000 overtones to be set, making it possible to create tones with rich overtones, especially in the low frequency range. The 100th overtone of 40 Hz is 4000 Hz, a frequency that can be perceived by the human ear. You can imagine the importance of the number of overtones. On the other hand, analog additive synthesis, such as Verbos' Harmonic Oscillator, which can only be set to 8 overtones, also has a different appeal from digital synthesis.

Verbos Electronics Harmonic Oscillator

Harmonic Oscillator Official Page

Demo Movie

Also, a method called resynth, which analyzes the momentary overtone structure of the input sound and reconstructs the sound with a free pitch using a set of sine waves, such as Make Noise Spectraphon, may be considered a kind of additive synthesis.

Spectraphon Official Page

Demo Movie

Download Max Patch sinewave.maxpat

Download Ableton M4L device sinewave.amxd

If we review that each and every partial in the overtone structure is a sine wave, it also opens up the possibility of transforming it into various elements of music by freely designing sine wave setting without being bound by the concept of overtones. This kind of thinking is also implemented in the 4ms modular synthesizer Ensemble Oscillator, a 4ms modular synthesizer. It could be polyphonic music with sine waves, it could be chords, it could be overtones, or it could be a detuned tone with many slightly pitch-shifted sine waves placed around a reference frequency.

Here we have a maxpat with six oscillators, left and right panning, and volume control. The structure is simple, but you can experiment with the variety of sounds you can create with this alone. The simplicity of this product will reflect your tastes and preferences. Also, note the similarity between the process of timbre formation through the independent and time-varying overtones of real-world live instruments without stagnation, and the polyphonic chants of the 16th century, which consisted of multiple voices, each of which was composed independently of the other and varied over time. You can imagine the possibility that composition and sound design may actually follow basically the same path, just from a macro or micro perspective of sound phenomena.