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Once again I've come across some ambiguous concepts,which I'm not sure if I've got their meanings right.

It says that a sine wave has a pale timbre whereas,for example,timbres built from a square wave are penetrating.Is it talking about their penetration depth or there's some other meaning behind this?

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A sine wave (at least an ideal sine wave) is truly only a single frequency, and has no harmonic content beyond the fundamental. This gives it an extremely simple timbre that is indeed rather dull or pale. Square waves still have the fundamental frequency, but they also have many harmonic partials above it—specifically the odd partials, such as an octave and a fifth above, 2 octaves and a major third above, 2 octaves and a very flat minor 7th above, 3 octaves and a major second above, etc. It isn't infinite of course, but it a very rich timbre, especially compared to a sine wave. A completely unfiltered square wave is fairly harsh, though with a hollow component. With lopass filtering you can get a bunch of woody sounds and sounds reminiscent of a clarinet (which is an instrument built around mostly the odd partials).

A sawtooth wave is even harmonically richer than a square wave, and massively richer than a sine wave. It has all partials—odd and even—above the fundamental (octave, octave+P5, 2 octaves, 2 octaves+M3, 2 octaves+P5, 2 octaves+m7, 3 octaves, 3 octaves+M2, etc.). This makes for an especially harsh, penetrating sound when left unfiltered, even more intense than a square wave. But the timbre is fuller and not as hollow or potentially cold as the square wave timbre.

And with that, you have the three basic waves of additive and subtractive synthesis (a triangle wave is just a square wave with quieter upper partials). You can look up the harmonic overtone series on this site if you need more information about what it is and its sonic effects.

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  • "triangle wave is just a square wave with quieter upper partials" - or couldn't it be viewed as a sine wave with a little bit of upper partials added in? To me a triangle sounds closer to sine than square, but every ear is different. – Todd Wilcox Nov 13 '15 at 16:41
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    @ToddWilcox Well, how it sounds may be subjective, but I'm just talking about the actual mathematical construct. In a square wave, the amplitude of the (odd only!) partials is inversely proportional to their frequency—i.e., their amplitude is 1/f. In a triangle wave, the amplitude of the (still odd only) partials drops off much faster—inversely proportional to the square of their frequency, 1/f^2. The upshot is: if you put a lopass filter on a square wave you can create a triangle wave. No amount of filtering can turn a sine wave into a triangle. But agreed, the triangle sound is quite pure. – Pat Muchmore Nov 13 '15 at 16:55
  • @ToddWilcox every sound can be described as a (possibly infinite) sum of sines, and every periodic sound by a discrete sum of sines. Saying that a signal can be described as a sine wave with a little bit of upper partials added in isn’t saying much, at least until you define a little bit precisely. – Édouard Nov 14 '15 at 12:32
  • All I was trying to say was that in some of the synthesizers I have (DSI Tempest being an example), you don't always have a sine wave available, and instead of a sine wave you use a triangle wave, because sound-wise they are almost the same thing. So from a perspective of building sounds on an actual analog subtractive synth, it helps to think of triangles as almost sine waves, as opposed to thinking them as more like square waves. – Todd Wilcox Nov 14 '15 at 15:48
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Don't put too much meaning into the specific words used to describe the sounds, they are, necessarily, ambiguous metaphors. "Talking about music is like dancing about architecture".

In this case, the difference in timbre is due to the much greater harmonic content of the square wave signal. A sine signal has just a single harmonic component. A square wave has multiple overtones at odd integer multiples (3x, 5x, 7x...) of the fundamental, with their amplitude decreasing like 1/f. If you care about the numerical/mathematical description of sounds, the analysis of the the spread, and intensity of the harmonics is important (probably the most important) aspect of the sound that affects timbre.

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Looking at diagrams or reading written descriptions of sounds will be of little help unless you listen to examples of the waveforms with your own ears.

A sine wave is more like the sound of a flute. A square wave is more like the sound of a kazoo. But don't take my word for it. Find examples and listen to them.

A pure sine wave is like the atomic building block of all sounds. Acoustic theory says that all the other (complex) single waveforms actually consist of a combination of a number of pure sine waves (of different frequencies, amplitudes, and phase relationships). Since all acoustic sounds with a periodic waveform can be "deconstructed" into a number of constituent sine waves, this means, according to theory, that the sound of any pitched or tuned acoustic musical instrument can be approximated by combining a number of sine waves together in the proper proportions (which are varied over time). This theory is the basis of a technique used by some electronic music synthesizers called additive synthesis.

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words such as "pale" and "penetrating" as descriptives for waveforms is specious at best. I could put support to words such as "simple/complex," "thin/rich."

Pale and penetrating seem to describe more about the sensibilities of the person describing the sound, than the sound itself.

A loud 4k sine wave can be extremely "penetrating" while a particular square wave might actually be "round and fuzzy." So, while the words do evoke certain timbres, they don't really serve well to objectively describe waveforms.

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A sine wave doesn't have a "timbre", similar to a single point on a paper not having a contour. Sine waves are the "Eigenfunctions" of linear time-invariant systems which are the overwhelming number of components in any sound transmission.

If you put a sine wave into a non-distorting amplifier, you'll get a sine wave out. When you fiddle with the tonal controls, you'll still get a sine wave out. There is no way to distinguish fiddling with the tonal controls with fiddling with the volume control: either way you'll only affect the amplitude of what remains a single sine wave.

If you put a sine wave generator in your throat and form vowels, again there will be nothing except a sine wave of varying volume. There will be no way to actually distinguish the vowels.

Flutes are rather sine-heavy, not having many sound components above the fundamental note being played. The human voice box rather works with a pulse train which has a high content of overtones (for good vocal closure, higher even than a square-shaped wave), so you can use your mouth to form vowels and the shape of the mouth leaves revognizable shapes in the overtone spectrum of the voice rather than just influencing loudness at a single frequency. Reed instruments of various kinds mimic this kind of sound production, making it possible to shape the overtone spectrum into very characteristic timbres by shaping the airways appropriately.

Bowed string instruments start with a saw-tooth kind of tone generator.

With electronic instruments, square-shaped tones can easily be generated (and they are approximated even when starting with clean sounds by overdriving amplifier stages like distortion pedals do) and have even higher overtone content than triangular wave forms but less so than pulse trains (a perfect pulse train has all harmonics with the same strength as the fundamental whereas harmonics of square waveforms taper off inversely with the frequency, and those of triangular waveforms taper off inversely with the square of the frequency).

Square waves (and more so pulse trains) have "too much" of a timbre to be usefully employed, like the color white (which contains actually colors of all the spectrum). You use them more as raw material and shape them with electronic filters (technically related to tone controls on an amplifier or graphical equalizers) into more characteristic timbres.

For example, high-class accordions (which produce their tones by free reeds like a harmonica) may feature a tone chamber for some reed banks, a "cassotto", making for a characteristic mellower sound. It only makes sense with comparatively high-quality reeds since those have the smallest air gaps and consequently the sharpest overtone spectrum. Since those are much more expensive as some additional wood, nowadays you can get cassotto registers even in comparatively cheap instruments where the overall impression is "somewhat more muffled" rather than the sort of vowelizing effect that this was originally built for.

I digress.

At any rate, starting electronically, squares waves are the simplest reasonably rich timbre building blocks (not a useful finished timbre, mind you, but a good building block). You can turn up the overall harmonic content at the cost of sound energy by playing with the duty-cycle and making the "square waves" asymmetrical (note that your tweeters in particular will not share the impression of a decrease in sound energy, and blowing the same energy through them as through a large bass woofer is not likely going to make them happy once you try for some more solid volume).

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