Generic Additive Synthesis

Julian Rohrhuber and Juan Sebastián Lach Lau.

A generalisation of additive synthesis, as described in our article Generic Additive Synthesis. Hints from the Early Foundational Crisis in Mathematics for Experiments in Sound Ontology

Here we provide some examples from the article and will add more in the future.

A few simple examples, using SuperCollider:

// simple case of concatenative frequency (phase) modulation (p. 273)
(
Ndef(\g, {
	var combinator = { |a, b| a <> b }; // operator: function composition
	var c = { |i| LFTri.ar(1 / i, 1 / i).range(-1, 1).max(0) }; // spectrum: time varying
	var g = { |x, i|
		SinOsc.ar(i * 40, x * 3) // basis: sine function that takes a modulated phase argument
	};
	var n = (1..12); // number of operands
	n.inject(0, { |x, i| // inject is also known as left fold. Base case is 0 here
		g.(x, i) * c.(i) + (x * (1 - c.(i))) // this is so we get 0 as the neutral element
	}) * 0.1
}).play;
)

// generic additive synthesis with a sine basis and addition (p. 271)
(
Ndef(\g, {

	var combinator = { |a, b| a + b }; // operator: just binary sum here
	var c = { |i| 1 / ((i % 7) + (i % 8)  + (i % 11) + 1) }; // spectrum: jagged static shape
	var g = { |i|
		SinOsc.ar(110 * i) * c.(i); // basis: sine function
	};
	var z = (1..30); // number of operands
	var set = z.collect { |i| g.(i) }; // sequence
	 // combine and scale output:
	set.reduce(combinator) ! 2 * (1 / z.size)

}).play;
)


// generic additive synthesis with a more complicated basis and a product combinator  (p. 271)
(
Ndef(\g, {
	var combinator = { |a, b| a * b }; // operator: product function
	var c = { |i| 8 / i }; // spectrum: linear
	var g = { |i| // basis: phase modulated pulses
		var cn = c.(i);
		var y1 = SinOsc.ar(120 * i, SinOsc.ar(cn * 10 * i) * (1/i));
		var y2 = LFPulse.kr(cn, 0, SinOsc.ar(cn * i, i, 0.2, 0.3));
		y1 * y2 * cn + 1
	};
	var n = (1..12); // number of operands
	var set = n.collect { |i| g.(i) }; // sequence
	LeakDC.ar(set.reduce(combinator) * (0.01 / n.size)).tanh * 0.1 ! 2 // tanh projects the final output into range
}).play;
)

Modifications of the examples from the article:

// generic additive synthesis with a sine basis and addition, interactively control the spectral shape
(
Ndef(\g, {
	var x = MouseX.kr(0, 50);
	var y = MouseY.kr(0, 10);
	var combinator = { |a, b| a + b }; // operator: just binary sum here
	var c = { |i|  // spectrum: jagged dynamic shape
		i = i * y + x; 
		1 / (((i % 7) + (i % 8)  + (i % 11)).max(0) + 1) 
	}; 
	var g = { |i|
		SinOsc.ar(50 * i) * (100 / i) * c.(i); // basis: sine function
	};
	var z = (1..100); // number of operands
	var set = z.collect { |i| g.(i) }; // sequence
	 // combine and scale output:
	set.reduce(combinator) ! 2 * (1 / z.size)

}).play;
)

Another example, using the Steno embedded language:

t = Steno.push(8); // a small 8 channel spectrum

// definition of a spectrum composition operator G
(
t.filter(\G, { |in, envir|
	var r = \rotate.kr(0);
	in = in.collect { |x| PanAz.ar(in.size, x, 2*r/in.size) }.sum;
	SinOsc.ar((100 / (\index.kr + 1)), in * 2)
});

// the last node mixes down to stereo
t.filter('.', { |in|
	Splay.ar(in) 
})
);


--GGGGG.

t.setGlobal(\index, { |i| i + 1 });