Square waves more "dissonant" than sawtooth ones?

[stringtapper]

Leaving aside the cultural aspects of defining consonance and dissonance and looking at it purely in terms acoustic/psychoacoustics one reason that fifths with saw waves might sound more consonant to someone than fifths with square waves has to do with the coincidence of the overtones of each of the two fundamentals forming the interval of a fifth.

If you have a sawtooth at 100Hz then it will have overtones in increments of 100: 200Hz, 300Hz, 400Hz, 500Hz, 600Hz, etc.

Add a fifth above (in just tuning to keep numbers manageable) at 150Hz and you will get multiples of 150: 300Hz, 450Hz, 600Hz, 750Hz, 900Hz, etc.

Notice that every other overtone of the sawtooth wave at 150Hz will coincide with an overtone in the wave at 100Hz (e.g. the second harmonic of the 100Hz wave coincides with the first harmonic of the 150Hz wave at 300Hz). These coincidences might strengthen the perceived cohesion of the two sounds heard together.

Take a square wave at 100Hz and you get the odd harmonics: 300Hz, 500Hz, 700Hz, 900Hz, 1100Hz, etc.

Add the fifth above at 150Hz and the harmonics are 450Hz, 750Hz, 1050Hz, 1350Hz, 1650Hz, etc.

With Just tuned fifths each partial lies exactly half way between two adjacent overtones from the other wave. No coincidence between overtones.

[stringtapper]

Here's an old article on this subject of consonance and how it is affected by "critical bandwidth."

http://pubman.mpdl.mpg.de/pubman/item/e ... l_1965.pdf

Critical bandwidth is the point at which the beating between two detuned pitches become harder for the human ear to discern and start to sound like one "thing". This area also causes a sense of what they call "roughness", i.e. contributing to the sense of "dissonance". The critical bandwidth as Helmholtz held was about 30-40Hz, roughly(!) the interval of a minor third.

So the two square waves in fifths sounding more "dissonant" is mostly likely a result of their upper partials exceeding the critical bandwidth by being offset by 50Hz (in the example I gave in my last post).

[xx JPRacer xx]

I'm no expert but I think the reason a square wave is more dissonant is because of the lack of even harmonics. Even harmonics are octaves apart from the fundamental, they're in perfect tune, they solidify the harmony.

Odd harmonics form ratios that are more complex and detuned from the fundamental in equal tempered tuning.

Saw waves have odd harmonics too but the added even (octave) harmonics help solidifying the consonance.

Add two square waves together and you end up with a lot of frequency that are just too detune (from equal tempered tuning) from the fundamental and with no even harmonics to compensate/solidify consonance.

[JumpingJackFlash]

I'm no expert but I think the reason a square wave is more dissonant is because of the lack of even harmonics. Even harmonics are octaves apart from the fundamental, they're in perfect tune, they solidify the harmony.

The word "dissonant" in this context is slightly misleading.

In the real world for example, the even numbered harmonics are suppressed on the clarinet and you hear mostly the odd numbered ones. Does that make a clarinet more dissonant? - No, it is still capable of playing music that is completely consonant (that is, within key).

What is at issue here is timbre.

You hear lots of harmonics when a trumpet plays, but hardly any when a flute plays. They have a different timbre. But both can play the same music (suitably transposed)- you can have a lovely sounding melody played on a trumpet and a jarring melody played on a flute or vice versa, but of course they sound very different.

[jof]

is there some different set of intervals or maybe even a tuning system which is more suited to square waves?

The Bohlen-Pierce scale is based on odd integer ratios.

[Sendy]

Thanks for all the replies, some really interesting discussion here.

[xoxos]

change "more dissonant" to "less harmonic" in a sense, by being less.

as harmonics are more isolated and more clearly perceptible, they're more likely to stand out if they do something naughty.

[xx JPRacer xx]

change "more dissonant" to "less harmonic" in a sense, by being less.

as harmonics are more isolated and more clearly perceptible, they're more likely to stand out if they do something naughty.

Yes, that's kind of what I've said. The even, perfectly tuned octaves harmonics of a saw wave reinforce the overall harmony and kind of mask the not so perfect/detune odd harmonics.

[Tricky-Loops]

A square wave compared to a saw wave HARMONICALLY is like a chili soup where you've poured some water in to make it less hot (and less delicious)...

[Fox Crazy]

I wish I didn't blow at math because all of this is incredibly fascinating.

I have a bit of a question, but I'm not sure how answerable it is without getting insanely technical. Mostly, I was just wondering, with all the talk of a staircase pattern being built out of square waves to simulate a saw wave, how all of this relates to bit reduction or bit crushing? Because I've noticed some methods of this seem to create staircases out of the waveform, and in doing so the actual sound of it gains some square like properties.

[Sendy]

I wish I didn't blow at math because all of this is incredibly fascinating.

I have a bit of a question, but I'm not sure how answerable it is without getting insanely technical. Mostly, I was just wondering, with all the talk of a staircase pattern being built out of square waves to simulate a saw wave, how all of this relates to bit reduction or bit crushing? Because I've noticed some methods of this seem to create staircases out of the waveform, and in doing so the actual sound of it gains some square like properties.

Yeah, exactly, it's pretty mathy to give any concrete answer. It's almost asif samplerate reduction breaks things down into their respective "squarewave harmonics". Each squarewave that's removed takes steps out of the waveform and removes harmonics from the saw spectrum, until all that's left are the odd ones.

I recommend getting a good 4-osc or more synth like Zebra and mess around with stacking squarewaves, watching the spectral and waveform changes it makes to the sound. It's fun stuff Just remember the waves have to be phase locked, at octaves and drop off in volume the higher they go, for it to work.

[Sendy]

A square wave compared to a saw wave HARMONICALLY is like a chili soup where you've poured some water in to make it less hot (and less delicious)...

Bah, a squarewave is worth TWO of those pesky sawtooths!

[Fox Crazy]

Yeah, exactly, it's pretty mathy to give any concrete answer. It's almost asif samplerate reduction breaks things down into their respective "squarewave harmonics". Each squarewave that's removed takes steps out of the waveform and removes harmonics from the saw spectrum, until all that's left are the odd ones.

I recommend getting a good 4-osc or more synth like Zebra and mess around with stacking squarewaves, watching the spectral and waveform changes it makes to the sound. It's fun stuff Just remember the waves have to be phase locked, at octaves and drop off in volume the higher they go, for it to work.

This is pretty much what I noticed. I was actually playing around with FM8, and looking at its "spect" panel. It has a useful digital feature, which is more or less a form of bit reduction from what I understand. What's interesting is that it has a more profound effect on the mid range than the low range, and oddly enough it actually creates a few high harmonics that aren't there in a normal saw wave.

You can create a rather interesting sound by cranking the digital up all the way and setting it to 64 voice unison and medium detune. You get this wide, really robotic vocal sort of sound, especially if you start playing around with modulation.

I also tried the stacking square waves a little while ago and got something pretty close to a saw wave. You start to get the saw sound at about 3 waves, and it really starts to sound right at about 5. At around 4 waves you get something that's similar to the digital feature, but with more high harmonics. So more or less, it's exactly what was said would be the case.

Fun stuff.

Bah, a squarewave is worth TWO of those pesky sawtooths!

I think we can all agree that synth is awesome, be it subtractive analog squares and saws, weird FM stuff or a damn Hammond Organ, AKA the first additive synthesizer.

With that said, I do think the good old square wave doesn't get enough credit. Maybe that's just my love for the ye olde PSG sound found in 8 and 16 bit game consoles, though.

[Shabdahbriah]

Interesting thread, which could only benefit (IMHO) by viewing:

"The Super Position Test"