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With a couple of friends I am designing a synthesizer on an altera FPGA (cyclone II, the one on the DE2 board). We almost wrapped it up nicely but there's a problem that bugs us.
The synth can generate four kinds of sound waves: sinusoidal, square, triangular and sawtooth. The user (the player) can change the wave while he's using the synth of course. In our first solution the perceived volume changed greatly between the various waves, square being the loudest and sin being the lowest.
We calculated the normalised power of each kind of wave and added a corrective factor for the amplitude: $$\sqrt{\frac{2}{3}}\rightarrow \text{sine wave}\\ \sqrt{\frac{1}{3}}\rightarrow \text{square wave}\\ 1\rightarrow \text{triangular wave}\\ 1\rightarrow \text{sawtooth wave}$$

The problem is that the volumes still don't seem right, the sine wave seems still way lower than the others. We were a bit unsure about the power of the triangular wave, but our calculations should be correct. We then thought that our first assumption was wrong.

Question: is it correct to assume that the perceived volume of a sound wave is proportional to its power intended as the power defined in signal theory books, i.e.:

$$P_x=\frac{1}{T}\int_0^T|x(\alpha)|^2d\alpha$$ of course \$x(t)\$ must be periodic of period T.
If that is true: did we miscalculate something? We are listening to the produced sounds through a guitar amplifier so the band probably is not that wide, but that would lead to a lower square wave and a louder sinusoid if total power is the same.

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2 Answers 2

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You need to account for the ear by using the Fletcher Munsen curve (spelling may not be perfect but google will show the graph).

Basically all sound levels for different frequencies have been empirically captured onto the graph.

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Andy's answer is quite right, I just want to add that a guitar amplifier's speaker is generally designed to emphasize the mid- to high-frequencies, and most guitar speakers have a very pronounced resonant peak in the 1-2kHz range.

See frequency response graph at the bottom of this page for a typical example.

You could do a sine sweep to hear this for yourself; a constant-amplitude sine wave applied to your guitar speaker will appear to increase in volume in the 1-10kHz range and decrease as you move away from that range.

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