Square waves more "dissonant" than sawtooth ones?

[Sendy]

http://www.flickr.com/photos/omegatron/ ... 9941546057

Was just geeking around the interweb when I stumbled across this picture, which claims that for some constructed approximation of "consonance" (I think it's an algorythm), intervals played by two sawtooth waves sound more "musical" than that played by two square waves. In effect they have different "sweet spots" for consonant sounding ratios of freqs. I've flicked through "On The Sensations of Tone" and saw a similar chart in there for intervals created with tuning forks (almost sine-based tones).

I understand that odd harmonics and even harmonics are different, square waves are about the most "unnatural" sound you can get (consisting of nothing but discontinuities and pauses, being the "voice of electronics" etc) but at the end of the day if you set up an interval on a synth with two saw waves and then switch to squares, the interval doesn't really change for me. (or rather, it's degree of consonance doesn't seem to change).

But I haven't delved into psychoacoustics that much (at all, really). To me it's more about aesthetics than anything measurable. I like square waves because they remind me of cheap electronics, and the love humans put into them. But then I also like saw waves for their raspiness.

To me the square wave (or odd harmonic series, if you want to forget about phase) sounds more "voluminous" than a saw. It suggests space resonating. That's why it makes a killer bass. The saw wave suggests something else, it's unidirectional, more focussed. On an oscilloscope, saw waves tend to glide over eachother, whereas squares crash into eachother creating abrupt changes. Perhaps this is why history has granted us the "supersaw" and not the "supersquare".

If a square wave isn't tuned into the same intervallic consonances as a saw wave, is there some different set of intervals or maybe even a tuning system which is more suited to square waves?

Please note, this is just some jumbled thoughts which I thought might make an interesting discussion, not a thesis or anything fancy like that

And obviously, square vs saw is NOT a dichotomy in any meaningful way (other than it's a choice you have to make on analog-style synths) but this thought leads more into how we percieve odd harmonics versus a mixture of odd and even, and how timbre and harmony interact.

[Bronto Scorpio]

Square waves are definitely very interesting for all kinds of dissonant cymbal sounds and stuff like that.

Take an 808 cowbell for example, it's basically just two square waves tuned roughly a fifth apart, it sounds very dissonant nontheless (at least with a very short decay time)!
Make the same sound with two saw waves and you'll suddenly clearly percieve the fifth.

I never really thought about the reasons for this, but it's certainly interesting.

Cheers
Dennis

[CableChannel]

I cannot say I would hear a difference in how "harmonic" say a fifth interval sounds with square waves or saw waves. The difference one hears is the type of the wave themselves, how that can be viewed in terms of the overtonestructures and their playing together in intervals, thats pretty theoretical I guess. But I might be wrong.

[Sendy]

See, to me that 808 cowbell has always been the epitomy of fifthness. But I'm trying it right now (in ACE actually) and the "square fifth" definitely sounds more metallic and clangerous than the "sawtooth fifth" which sounds pitifully weedy as a cowbell patch (i.e. with a short envelope).

Could it be also that playing lots of square-wave-voiced video games growing up has conditioned me to hear and exadurate the consonance of squarewaves playing harmonic ratios?

[CableChannel]

@Sendy:

no actually my first post is bullshit, you definitely got a point there. Comes to mind the example of sampled church bells. Try playing chords with them, it doesn't work, BECAUSE of their overtone structure. So the overtone structure definitely influences how intervals sound. Thinking about it now, it is more than obvious, since all overtones also are in some interval relation, and these can in turn be harmonic or disharmonic.

[Sendy]

Good point 123123, bells have a very inharmonic overtone structure, so it's a good example of that. But the harmonics in a square wave are all taken directly from the harmonic series, it's just that the even ones are missing. Pretty much any periodic waveform will work with chords because periodic waveforms cannot have inharmonic partials, but I guess the lack of many of these harmonics in waves like the square sort of "weaken" the harmonicity of it, in the same way that removing the fifth from a dominant seventh chord can cause it to sound less "stable".

[chokehold]

If you have only one note, one tone, then you have one base frequency and the overtones. The base frequency is the note you hear, and the overtones shape how you perceive it - soft, harsh, dissonant - that's called timbre.

Basically, a Square Wave is just a mass of sine waves.
A mass of sinewaves of which any one sine could be the base frequency and the rest the overtones. In very primitive words, "all the overtones are just as loud as the base frequency (=height) and the overtones are so close together (=width) that the result resemble straight lines.

This here is quite interesting and might help you understand what I mean:


A triangle or sawtooth will probably seem less disharmonic than a square wave, because at least some of the overtones are quieter than the base frequency.

EDIT: adding multiple waves to each other (=chords) will add base frequencies to disharmonic overtones of other waves and vice versa in between the single waves. So the louder the overtones of all single elements in the chord are, the more of a disharmonic mess it will probably sound.

[tanabarbier]

Yes but that's the thing, the overtones are not stronger in a square wave than in a sawtooth. Actually if we ignore the phase we can make a sawtooth out of two squares, an octave apart, and half the volume for the higher one. So could it be because of phase?
I think it is an interesting question.

[Nystul]

Yes but that's the thing, the overtones are not stronger in a square wave than in a sawtooth. Actually if we ignore the phase we can make a sawtooth out of two squares, an octave apart, and half the volume for the higher one. So could it be because of phase?
I think it is an interesting question.

Adding a weaker square wave (or anything else?) an octave above the square wave will give us the sound of the second harmonic which is present in most natural sounds but not the square wave.

[tanabarbier]

I am sorry, I don't get what you are saying, or maybe I explained myself badly. I was saying that two square waves an octave appart, with the second one weaker (half the amplitude) would give the same spectra that the one of a sawtooth.
At least it is my understanding of the spectra of square waves and sawteeth (ah, funny word).


It is this part of Chokehold's post I don't get :

A triangle or sawtooth will probably seem less disharmonic than a square wave, because at least some of the overtones are quieter than the base frequency.

Square waves share the same harmonic proportion that sawtooths, 1/n, just an other distribution, and phases differences, isn't that so?

[Sendy]

I am sorry, I don't get what you are saying, or maybe I explained myself badly. I was saying that two square waves an octave appart, with the second one weaker (half the amplitude) would give the same spectra that the one of a sawtooth.

That's not true (though I totally get why it seems that way), the wave you get by summing ever quieter square waves is a staircase waveform (effectively a sawtooth which has been "sample rate reduced" by a factor which is a haromonic of the wave) which tends towards a pure sawtooth as you add more and more square waves. *

To get a decent-enough-for-human-hearing sawtooth out of summing squares, you need at least 8 or so squares. Still, four squares gives you something that was close enough to serve as the "sawtooth" setting on synthesizers which worked via "additive squarewave synthesis".

* fun little side fact, you can actually make a sine wave from summing a series of square waves. Talk about squaring the circle

[tanabarbier]

Yes, I was trying that in bazille, maybe not the best way to try it because I never got a squre squre wave out of it.
What is weird is that when looking at equations it should work, so why not?

[chokehold]

A sine wave goes up and down smoothly, no overtones.

Anything else causes a jump, either from low to high or high to low or just reverses the phase abruptly.

Which is NOT smooth.
Which means distortion.
And distortion is overtones.

The stronger we perceive something as "distorted", the more is its timbre "saturated" with overtones as against the clean sine wave it could be.

There is no harsher distortion than going from -1 to 1 instantly. (EDIT: or the other way round)
Sawtooth and triangle both change the phase a little more relaxed in at least one flank.

So speaking in primitive words, a sawtooth has less distortion = smoother timbre = less disharmonic overtone content, and a triangle even less.

[Tricky-Loops]

I was saying that two square waves an octave appart, with the second one weaker (half the amplitude) would give the same spectra that the one of a sawtooth.

By combining sine waves in additive synthesis, you can construct a saw wave, but to get a saw wave with 2 square waves seems impossible to me...

[tanabarbier]

Ok, I know that, but the theory says that a sawtooth is:

sum (n going from 1 to infinite) of (1/n)*sin (n*f) ; n being a natural

and square (n going from 0 to infinite) of (1/(2n+1))*sin ((2n+1)*f)

They have the same sinewaves inside, only sawtooth has the even one and the square does not (ignoring phases, once more).

What you pointed out has to do with digital/analog, isn't it? I don't think that the difference is there, maybe but I don't get it.