Summation of Similar Terms

As a digital music instrument design project that endeavours to coalesce audience's devices' sound synthesis and loudspeaker capabilities, it garnered the name DXΣ -- an homage to the Yamaha DX7, but inflected to reference the mathematical process of summation, in which the addition of a series of similar terms is denoted by "Σ" - the Greek capital letter sigma. For example:

$$\sum_{n=1}^{6}2n=(2\times1)+(2\times2)+(2\times3)+...+(2\times6)$$

By specifying an index of summation (in this case, $n$), a lower bound ( $1$ ), an upper bound ( $6$ ), and a function to iterate ( $2n$ ), capital-sigma notation describes an algorithm by which a series of similar terms can be generated and summed together.

Distributed Synthesis

This mathematical process is formally analagous to the control flow of distributed synthesis, a novel form of sound synthesis wherein a series of signals, as specified by a central function, are synthesised by a number of peripheral audience devices, which emit the signals via their loudspeakers into the air of a shared physical space where they are summed together.

Periodic Functions

If we can define a periodic function as being any function $f(x)$ for which the following holds true:

$$f(x + T) = f(x)$$

... we might also understand the period of that function to be $T$.

Harmonics

If we consider:

$$y = A sin(ωx + φ)$$

... where $A$, $ω$, and $φ$ are the amplitude, frequency, and phase of a sinusoidal harmonic, we might understand the following:

$$\sum_{ω=1}^{∞}\frac{1}{ω}sin(ωx+φ)$$

... to describe a sawtooth wave function, and:

$$\sum_{ω=1}^{6}\frac{1}{ω}sin(ωx+φ)$$

... to describe the first six harmonics of just such a sawtooth.

Audio Worklet

An implementation of such a wave function in AudioWorkletProcessor, might look like this:

process (_inputs, outputs, parameters) {

    const out = outputs[0][0]

    for (let frame = 0; frame < out.length; frame++) {

        let sig = 0

        const freq   = deparameterise (parameters.freq,   frame)
        const amp    = deparameterise (parameters.amp,    frame)
        const bright = deparameterise (parameters.bright, frame)

        let bright_dec = (bright * 5) + 1

        for (let i = 1; i <= 6; i++) {
            const b_amp = Math.min (bright_dec, 1)

            sig += Math.sin (this.phase * Math.PI * 2 * i) * (amp / i) * b_amp

            bright_dec -= 1
            bright_dec = Math.max (bright_dec, 0)
        }

        this.phase += this.inc * freq
        this.phase %= 1
        out[frame] = sig
    }

    return this.alive
}

Find an implementation of this code → here

Learn more about how AudioWorklet works → here