Inharmonic Series

In Harmonic Series, we build a sound consisting only of specific harmonics, i.e., a sound wave consistent of many frequencies which are whole-number multiples of the fundamental. This gave us an harmonic tone.

We added a slight frequency modulation to simulate a vibrato effect to make the sound more natural. Furthermore, we increased the steepness of the envelope for increasing frequency to emulate the natural phenomenon of faster disappearing high frequencies. In the context of additive synthesis, inharmonics refer to frequencies that are not whole-number multiples of the fundamental frequency.

What sounds contain a lot of inharmonic content? Many percussive instruments, like bells, gongs, or xylophones, produce inharmonic overtones. We could use additive synthesis with inharmonics to emulate these types of sounds. Moreover, by carefully tuning the inharmonics, we could create complex, evolving pad sounds. These could have a more dissonant or out of tune character compared to pads created with harmonics. Often metallic sounds, such as those produced by a struck piece of metal, have a high degree of inharmonicity. Compare, for instance, the sound of a Glockenspiel:

If you’re creating experimental or atonal music, inharmonics could be used to create novel, unusual, or dissonant sounds. Finally if we are creating experimental or atonal music, inharmonics could be used to create novel, unusual, or dissonant sounds.

Let us now synthesize a sound consistent of many inharmonics.

(
SynthDef(\inharm, {
  var freqs, sig, env;
  freqs = \freq.kr(100) * [0.7, 0.8, 1.2, 1.8, 2.23];

  sig = SinOsc.ar(freqs *.x [1.0, 1.2]);
  env = EnvGen.ar(
    Env([0, 1.0, 0], [0.1, 1]), 
    doneAction: Done.freeSelf);
  sig = sig * env * freqs.size.reciprocal;
	sig = Splay.ar(sig);
  Out.ar(0, sig);
}).add;
)

Synth(\inharm, [freq: 500]);

Remember, the key to additive synthesis is balancing and shaping the volume and envelopes of the individual sine waves. You’ll often need to adjust the volumes and envelopes of the inharmonics over time, creating a shifting, evolving sound. This can be quite a complex process, but it allows for a high degree of sonic control and creativity. So let us develop the above example further and introduce individual envelopes:

(
SynthDef(\inharm, {
  var freqs, sig, env, amps, envs;
  freqs = [0.88, 0.92, 1.2, 1.8, 2.23] *.x [1.0, 1.2, 0.91];
  freqs = freqs.sort * \freq.kr(100);
  amps = {rand(0.3, 1.0)}!freqs.size;
  amps = amps.normalizeSum;
  amps = amps.sort;

  sig = SinOsc.ar(freqs);

  envs = freqs.collect {
    arg element, index;
    var env = EnvGen.ar(Env(
      [0, 1.0, 0.3, 0],
      [ExpRand(0.01, 0.03), Rand(0.3, 1.15), ExpRand(0.2, 0.34)],
      [Rand(3, 0.4), Rand(-1, 1), Rand(-3, -0.4)]
    ));
    env**(index*\envp.kr(1.3)+1);
  };

  sig = sig * amps * envs;
  sig = Splay.ar(sig);
  DetectSilence.ar(sig, doneAction: Done.freeSelf);
  Out.ar(0, sig);
}).add;
)

Please note that in the second line of the SynthDef, we utilize an adverb to multiply each element in the left list with every element in the right one. Furthermore, we organize frequencies and amplitudes in a way that gives more gain to higher frequencies. However, we also ensure these high frequencies decay faster using the formula env**(index*\envp.kr(1.3)+1).

Let us use a pattern to play an inharmonic “melody”.

(
Pbindef(\melody,
  \instrument, \inharm,
  \dur, Pseq(0.25!6 ++ [Rest(2.0)], inf),
  \scale, Scale.minor,
  \root, 2,
  \octave, 5,
  \degree, Pseq([0, 3, 5, 3, 4, 0, Rest()], inf)
).play;
)

The result does not sound amazing but we can hear a metallic tendency.