and sines of madness.

In the last two projects sound was “sculpted” with subtractive synthesis —using some basic patterns and tools for filtering frequencies and modulating amplitudes. It was more like working with clay. Molding and pinching. This next approach constructs sound from individual sine waves.

Sine waves are the “atoms” of sound. They are pure frequencies, simple vibrations.

If you want to find the secrets of the universe, think in terms of energy, frequency and vibration. -Nikola Tesla

Sines hold an intimate relationship with right triangles and circles. This makes some intuitive sense as circles are the quintessential symbol of periodic time. The seasons, a watch hand, the cycles of life the rotation of the planets around the sun. A sine itself is the relation between two sides of a right triangle (the hypotenuse and opposite angles), and when this relationship is plotted around the circumference of a circle it produces a wave, beautifully illustrated here.

The cosine in the illustration is the relation between another two sides of the triangle: the hypotenuse and adjacent angles. The final formula completing the foundations of trignometry is the tangent, or relation between the last unpaired sides of the triangle, the adjacent and the opposite.

Sine and Cosine can be used to plot a rotation around a circle (Sine for the y value and Cosine for the x) and they also underlie important building blocks of nature, like the spiral, which can be created by plotting these same points around an expanding or contracting radius. (So, what’s a tangent good for? With a clinometer app or apparatus, it can determine the height of something that you can see the top of, given the angle to the top and the distance from the base of the thing.)

In the 19th century, Joseph Fourier demonstrated that sines and cosines formed building blocks of all periodic phenomenon. A map for everything that happens in waves or vibrations can be expressed as a composition of these simple relationships. Even gnarly waves. His theorem was crucial to the development of a number of sciences: optics, signal processing, information theory, quantum mechanics and more. If all matter can be expressed as particles and/or waves, as is currently believed, it makes sense such a theorem would have broad application. ^{1}

To get a better feel for how simple sines and cosines can under-wire more complex phenomenon, this project uses a handful of pure sine waves to compose the sound of a bell.

The first pass at this was taking an existing bell sound and looking at its spectrogram, which uses Fourier analysis to break down the frequencies of the sound:

A spectrogram reveals frequencies that are more active over time. It’s indispensable for understanding sound and can be found as a tool in the open-source Audacity program, Adobe’s Audition or the standalone and multiplatform Sonic Visualizer.

From first glance at view above, the bell sound appears to be broken down into a handful of dominant frequencies. In reconstructing sound, the lowest frequency on the bottom of the graph is called the fundamental frequency and the frequencies above it are called overtones and relate to the fundamental frequency in ratios, either harmonically (multiples of the fundamental) or in more complex and fractional ways.

The fundamental frequency of the bell appears to be in the range of 500-700hz, and the overtones at the 1400 and 2100 and 2600 spectra. The 1400 overtone is about the same duration as the fundamental but with slightly less amplitude and the next two, 2100 and 2600 are a bit shorter in duration with quite a bit less amplitude.

The first, simple tool constructed to experiment with additive synthesis generates 4 simultaneous sine waves of selectable frequencies and each has envelope controls for duration and amplitude (as well as attack and decay for a little more control over the shape of the sound volume.) Playing with values within the fundamental and overtone spectra, and emulating their amplitudes with envelopes, can simulate a variety of sounds. Playing with similar ratios to the sample can even create a “family” of bell sounds.

The bell tool can be found here. A short video on how to use it and some results is here. The tool was built with HTML5, Javascript and Web Audio using pixi.js to write the graphics and tone.js to put together the audio graphs.

“Profound study of nature is the most fertile source of mathematical discoveries” -Joseph Fourier

References

Trigonometric Delights by Eli Maor

Notes:

- We don’t even have time to get into another amazing application of sines and cosines called Euler’s formula, which physicist Richard Feynman called the “jewel” and “the most remarkable formula in mathematics.” ↩