# How to synthesize non-pitched sounds? How pitched is a sound in general?

Non pitched sounds are sounds which are not only composed of a fundamental frequency and a distribution of its harmonics. For instance, a sound composed of a sum of a sin wave at 440Hz and a sin at 500Hz would be considered as non-pitched if I understand well.

As a consequence, these sounds are non periodics. Let's forget about envelopes and LFOs, as they make your sound non periodic, but on much longer timescales.

How is it possible to synthesize these kind of sounds? Obviously, additive synthesis (and unison which I think of a subset of additive synthesis) would work, but not subtractive synthesis. I guess FM would work also if modulator and carrier frequencies do not share a common divider.

Are there other methods to synthesize such kind of sounds?

Also (and my main interest in there), is there a criteria for how "pitched" such a sound would sound? Unison is done by adding waves which very close in pitches, thus sounding as the average pitch, but for more extreme cases (for instance `A*sin @ 440 + B*sin @ 500`) would sound as a A if `B` is small compared to `A`. How long would that stands when increasing `B`?

Edit: let's change 500 Hz to sqrt(2)*440, then these two frequencies do not have a common divider and the sum of the two is non-periodical.

• Many non-pitched sounds are combinations of all frequencies,, eg white noise. Commented Jun 24, 2020 at 12:13
• You are wrong: a sum of 440 and 500 Hz produces 440, 500, 60, and 940 Hz components. They are periodic. I don't see any strict definition of "non-pitched", tho' often things like the white-ish noise blast from a cymbal qualifies. As to how to synthesize -- a noisy analog tank circuit is a good source, or an untuned-FM receiver. Commented Jun 24, 2020 at 12:56
• @Tom_C That pattern most certainly does repeat. I'm not sure where you're getting your information from. We are talking about Fourier Transform, not a Fourier Series expansion. Commented Jun 24, 2020 at 13:33
• @CarlWitthoft For even a clearer case, take `f0` and `sqrt(2)*f0`. There is no way a sum of two sin waves at these frequencies can repeat.
– Tom
Commented Jun 24, 2020 at 13:39
• Note that in the well tempered system, all intervals other than octaves are irrational and hence non-periodic. In particular, multiplying by sqrt(2) will raise the note by an augmented fourth / diminished fifth. So, play A4 and D#4 and you will get 440Hz and 440*sqrt(2) Hz. This is a fairly dissonant interval but it is not unpitched. Unpitched instruments, such as many percussion ones, are not just non-periodic but a broad spread of frequencies. Commented Jun 25, 2020 at 12:43

Non pitched sounds are sounds which are not only composed of a fundamental frequency and a distribution of its harmonics.

That's quite a strict definition of 'non-pitched'. Most sounds that are essentially perceived as pitched by humans nevertheless contain energy at frequencies that are not exactly at the multiples of a fundamental - possibly because the harmonics are stretched (often the case with strings), or because there's some aspect of the driving mechanism that causes turbulent or chaotic behavior (e.g. flutes).

For instance, a sound composed of a sum of a sin wave at 440Hz and a sin at 500Hz would be considered as non-pitched if I understand well. As a consequence, these sounds are non periodics.

Well, a sound consisting of a sin wave at 440Hz and a sin at 500Hz is definitely periodic - the highest common factor of those sine waves is 20Hz, so that's the frequency of the repeating waveform. Would your example would actually be heard as a pitched sound at 20Hz with only the 22nd and 25th harmonics? that frequency difference is on the edge of being within the critical band - if I generate those two tones, I can hear it as two tones (if I try), or indeed as a kind of rough sound of indeterminate pitch.

But imagine that you had sines of 40, 60, 80.... and so on. That would essentially be a similar example, but I'd imagine that would be heard as a sound at 20Hz.

How is it possible to synthesize these kind of sounds? Obviously, additive synthesis (and unison which I think of a subset of additive synthesis) would work, but not subtractive synthesis.

Subtractive synthesis doesn't help you if you are limited to one periodic oscillator, but if you have multiple detunable oscillators, it seems that you can move towards your definition of 'non-pitched' - especially if you have ring mod, FM, oscillator sync etc. available, which will generate more 'inharmonic' partials.

Also (and my main interest in there), is there a criteria for how "pitched" such a sound would sound? Unison is done by adding waves which very close in pitches, thus sounding as the average pitch, but for more extreme cases (for instance Asin @ 440 + Bsin @ 500) would sound as a A if B is small compared to A. How long would that stands when increasing B?

I think the simplest answer is that 'pitchedness' isn't a very well-defined concept, so there isn't really a strict criteria. It's not a single-dimensional problem either, because you are dealing not only with how 'pitched' something sounds, but also how distinguishable two different sounds are - again, see https://en.wikipedia.org/wiki/Critical_band/.

• Comments are not for extended discussion; this conversation has been moved to chat. Commented Jun 25, 2020 at 10:53

I think you are confusing multiple concepts. As piiperi Reinstate Monica stated, the example you give would be 2 pitches rather than a non-pitched sound.

The term non-pitched usually refers to percussion instruments, percussive sounds, scratching, and other "sounds" for which it is difficult to define a fundamental. I think you got that part right. But the example of two sine waves would be perceived as an interval not a non pitched sound. This is because our brain is capable of doing FFT, or some equivalent process, and discovering the "tones" present in a complex signal. Actually the ear may have something to do with the process too, as the nerves within it are probably "tuned" to pick up specific frequencies.

To address your question about how two tones would be perceived when they have different amplitudes I'd ask you how you perceive the instruments in a band or orchestra when one is playing quiet while the other is loud? You would hear two things, two tones, at two volumes. This is a common experience. Using MATLAB, SCILAB, or any other off the shelf high level programming language you can make a wave file from sample of

A1 * sin(2pi f1 t) + A2 * sin(2pi f2 t)

sampling t as time. You can adjust the A1, and A2 as well as f1 and f2 as answer your own question. Humans have limits to their detection abilities. When one amplitude is too small perhaps that tone will not be picked up by the ear. When the two tones are close in frequency you will likely not be able to distinguish them at all, you may hear beats from the amplitude modulation. You cannot not stop the envelop from forming with sig proc tricks. Your synthesis will generate the envelope. If f2 is a multiple of f1 you may not hear an interval but your brain might start to form the perception that this is a single tone with different timbre. this is a tricky phenomenon to evaluate. Many musicians in their training are exposed to the art of listening for harmonics (I have been indoctrinated to that). With practice it is alleged that the trained ear can separate the harmonics in a complex tone. I am NOT talking about distinguishing 440Hz from 500Hz which is probably easy if it is bigger than a 1/2 step, but hearing the octave, octave and a 5th, etc, in a single plucked guitar note, or hammered piano note. I'd be curious to hear from others here if they are successful in doing this. In the case when two notes are closer than the pitch discrimination ability of the listener we typically hear the average tone, sum tone. And this is NOT a non-pitched sound, it is very well pitched.

Moving on to timbre and tone. What makes a guitar sound like a guitar and a violin sound like a violin (and not each other) is the harmonic content of the wave produced. In both cases that is "likely" dominated by the physics string. The body construction can dampen or amplify various harmonics by the starting point is the string. The key feature for any instrument is the method of attack. This is what produces the initial set of harmonics in the wave form. Bowed, hammered, plucked, at various locations with various angles and pressures all produce different tones. Without the attack everything would be a dull sine wave buzz. Even turning on a sine wave generator producing an infinite series of harmonics due to the hard edge rise of the wave at t = 0. Once the tone is produced by the instrument damping will cause the harmonics to decay over time, usually higher harmonics decay faster because the damping force is proportional to velocity, which is proportional to frequency in the frequency domain. Thus, over time any instrument excited in any manner will eventually ring at the fundamental, or lowest note in the excitation spectrum (may not be the true fundamental). The rate of decay depends on several factors but lone story short each instrument will have a characteristic decay profile. Thus, any "pitched" sound from an instrument will have an attack, sustain, and decay portion to the envelope.

If you wanted to synthesize two guitars playing a two part harmony, an interval, then you would need to generate the full spectrum for each guitar given its fundamental, then add the two results.

Anyone can do a fairly accurate job these days synthesizing musical instruments using Fourier analysis and some type of computer s/w. You need to know enough physics to model it correctly without data, but you can get real data and use that as a characteristic sample to measure key parameters. A very poor person's version of a synthesized guitar string would just be the FFT of a plucked string over the entire duration of the sound. This would sound like crap! Because the synthesis mixes all the physics of attack, sustain, and decay into one wave that is constant over time. To do the job correctly one needs to chop the time domain window up in to small slices and analyze the content of each window, building up a time dependent set of parameters. If you have a really high fidelity model of the instrument you can predict this behavior in time. I have done this in my career, synthesizing sound from sources based on physics based simulations.

So, why describe all this? Because I think that the parameters of your question make it unanswerable. You want us to ignore the envelope for now and just focus on the tone within it. One could argue that the initial "Attack" of any instrument (the first few milliseconds or hundredths of a second) are always non-pitched. By the same token everything has natural resonance frequencies. If you whack a bar the initial "POP" sound of that whack is "non-pitched" but in time you may hear the faint ringing of a resonance frequency left over, which is in fact pitched. Non-pitched sounds are, by their very nature, broad band in the frequency domain. So you CANNOT ignore the envelop. If your desire is to synthesize percussive instruments, or scratching of a bow on a cello, or the sound of an over driven woodwind instrument then I'd say you can absolutely do that with FFT and iFFT techniques, but it would require detailed analysis of the time evolution of the envelop. The initial rise is very important for this.

• What makes a guitar sound like a guitar and a violin sound like a violin (and not each other) is the harmonic content of the wave produced. This is not true, as you say yourself a couple of paragraphs later. If this were true, then a cheap keyboard from 1980 would have a violin timbre that actually sounded like a violin.
– user9480
Commented Jun 25, 2020 at 0:04
• Harmonic content is a function of time.
– user50691
Commented Jun 25, 2020 at 0:10

Simultaneous sine waves at 440 Hz and 500 Hz are some kind of a two-note chord, or an interval at least, something like a major second. It's definitely pitched, because there are two distinct constant pitches.

If you cannot identify a pitch, then the sound is unpitched. I don't know if silence counts as a sound, even though there's a song about it, but at least the following are unpitched:

• very short transient snaps
• wide-range noises where the spectrum is filled with frequencies
• fast wide sweeps up or down (but not repeating up-and-down)

Periodic repeating sweeping frequencies such as a vibrato are pitched, and the pitch of such a sound is identified somewhere in the middle of the sweeping range.

For short transient snaps and sweeps, use fast envelopes. If your synth has a noise genarator, use that to get noise.

You can synthesize a wide-band noise in an FM synth with operator feedback modulation, if the feedback level can be set high enough. Even a simple sine wave becomes a chaotic noise if you crank the feedback, like in some Yamaha OPL chips you can have 200% or 400% feedback (if I understood the meaning of the settings correctly).

• How would you classify Gage's 4:33?
– user50691
Commented Jun 24, 2020 at 13:49
• @ggcg - it's certainly not music to my ears!
– Tim
Commented Jun 24, 2020 at 14:59
• @Tim, right on. But from what I've read the performances get very loud!
– user50691
Commented Jun 24, 2020 at 15:01
• I'd forgotten about the dynamics markings. Sorry.
– Tim
Commented Jun 24, 2020 at 15:10
• I think the dynamics were improvised.
– user50691
Commented Jun 24, 2020 at 16:21