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I'm interested in learning about how human beings perceive qualities of sound.

We know that the human ear hears sound when air vibrates against the eardrums at different frequencies and amplitudes. Frequency and amplitude are perhaps the two most basic properties of the waveform of a sound.

I also understand that sound has a quality called timbre, which describes the difference in quality of sounds of the same frequency and amplitude. I seem to have misplaced it, but I had a reference that explains that timbre is the result of "overtones", which are softer, higher frequency sounds that accompany the frequency that the human ear perceives, which is called the "fundamental frequency". Wikipedia explains that a mathematical technique called the "Fourier Transform" can be used to compute the frequencies of the various overtones of a sound. I plan to study this in greater detail in the days to come.

However, there is something that I do not understand regarding timbre.

It seems to me that the definition of timbre that uses "overtones" to describe qualitative differences between sounds of the same frequency and amplitude does not take into account the shape of the waveform.

Here's a link to a YouTube video that demonstrates the difference in the sound produced by a sine wave, a square wave, and a saw-tooth wave.

Perceptually, these sounds have the same frequency and the same amplitude, but they sound completely different. I don't see how there are any overtones of any frequency other than the fundamental frequency involved in the demonstration, so it appears to me that in addition to the overtones, the shape of the waveform itself must also influence the perceived quality of the sound.

If not timbre, then what word describes the quality of a sound based on the shape of its waveform?

  • Square waves and triangles by definition contain all possible overtones, at very high levels, and this is what determines the shape of the wave. And it isn't just a matter of overtones. There are also the initial transients to be considered; for example, the scratching sound from a violin bow, or the click of a guitar plectrum. – user207421 May 6 at 2:21
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    @user207421 Sawtooth waves contain all overtones at levels 1/n (where n is the number of the overtone, i.e. 1, 1/2, 1/3, 1/4...). Square waves contain only odd-numbered overtones, at levels 1/n. Triangle waves contain only odd-numbered overtones at levels 1/n^2 (i.e. 1, 1/9, 1/25, 1/49...). – Your Uncle Bob May 6 at 5:22
  • "overtones, which are softer, higher frequency sounds" - not necessarily softer. The bottom notes of a piano have little or no fundamental frequency. We perceive the low pitch due to the combination of the louder harmonics. – jwvh May 6 at 9:44
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It seems to me that the definition of timbre that uses "overtones" to describe qualitative differences between sounds of the same frequency and amplitude does not take into account the shape of the waveform.

The shape of the waveform is different because of the differences in the levels of the overtones.

Or, to put it another way - the shape of the waveform is a representation of the sound observed in the time domain. The frequency content (in terms of overtones) is a representation of the sound in the frequency domain.

Have a play with https://meettechniek.info/additional/additive-synthesis.html. You will see (and hear) that as you change the levels of the overtones, the shape of the wave and the audible timbre of the sound all change too. The three things (timbre, waveshape, and overtones) are linked.

I don't see how there are any overtones of any frequency other than the fundamental frequency involved in the demonstration

The square wave and the saw-tooth wave do have overtones. Have a search for e.g. 'overtones of square wave' (or 'harmonics of square wave'). It's the presence of those overtones that make them sound different.

Again, just to be clear - the square wave doesn't have overtones 'as well as' being square; the overtones are the reason why it's square.

If not timbre, then what word describes the quality of a sound based on the shape of its waveform?

'Timbre' isn't a bad word, but I don't think it's specifically based on the shape of the waveform - in fact, the human ear divides sound up by frequency, so it hears the overtones more than it hears the shape. It might just be better to say that the following things are linked:

  • Timbre - the subjective quality of a sound as perceived by the listener
  • Waveshape - the time-domain behaviour of the sound
  • Harmonic Content/Overtones/Spectral content - the frequency-domain 'signature' of the sound
  • Formants are perhaps also a useful property when describing timbre. Perceiving peaks and troughs in certain frequency ranges is perhaps a more realistic way of describing how we distinguish between different voices and instruments, than perceiving waveform shapes or individual harmonics. – Your Uncle Bob May 5 at 21:16
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    Yep. OP may be confused because waveform is a direct consequence of overtones and harmonic content. Well written. Take my +1! – user45266 May 6 at 2:09
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    I just fixed the missing YouTube link. – Daniel Allen Langdon May 6 at 3:23
  • Here's what I don't understand. If I look at an A4 440 hz sawtooth wave, I see that there are 440 crests hitting my hear every second. If I understand correctly, to say that there are "overtones" is to say that there is a wave of some higher frequency in the waveform, but I don't see if it is in anyway possible for more than 440 crests to hit my ear every second. Can you explain that to me? – Daniel Allen Langdon May 6 at 4:02
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    @DanielAllenLangdon - Basically, only the sine wave has no overtones. All other sound waves are made by adding together some number (possibly infinite) of sine waves with different frequencies. The period of the combined waves determines the frequency of the resulting note. You can easily make a 440 Hz note by adding together a 440-Hz wave and an 880-Hz wave - the resulting waveform is still 440 Hz, despite looking wavier. The 880-Hz component wave is called an overtone. – Dekkadeci May 6 at 5:03
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  • The reason that you only see one frequency in these synthesized waveforms is that the overtones are all integer multiples of the fundamental, so every frame of one period of the fundamental frequency looks just like any other. The type of overtones (odd or even), and their power relative to the fundamental are what determines the shape. If you view a waveform where the overtones are not integer multiples of the fundamental, or the amplitudes of the overtones decay at different rates (such as with a stringed instrument), the waveform will appear to evolve over time, and no single frame can characterize the waveform.
  • The human ear directly senses the amplitude of each overtone, but (unlike Fourier analysis) cannot detect the relative phases of the overtones, so there are some waveforms that sound the same but look quite different on an oscilloscope.
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It may be helpful to realize that square waves (and triangle waves) are idealized phenomena and cannot actually occur. They would require summing an infinite number of sine waves to make perfect.

As such, it may be useful to look at approximations. There you can more easily see all the frequencies. Here is a decent set from Mathworld:

enter image description here

As they add more overtones, each approximation gets closer and closer to the square wave. And, with more overtones, you see more bumps.

  • This answer is missing the word "sum". – Tobia Tesan May 6 at 17:10
  • @TobiaTesan Your comment is a bit unclear. Is this what you mean? (see edit) – trlkly May 6 at 17:19
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I believe you are looking for the terms attack and decay transients. Decay transients in particular make up the sound signatures we recognize as different musical instruments.

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    Transients do form a big part of the perceived timbre, but they aren't necessarily a separate thing to overtones - you can consider the transients in the frequency domain too. – topo Reinstate Monica May 6 at 8:55

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