> You can reduce this by using another window function - but then, 
> strictly speaking, you're multiplying your signal by a window, which 
> convolves its spectrum with the spectrum of the window. In other words, 
> by windowing the signal you're blurring its spectrum.

Usually a slightly blurred spectrum is the preferred outcome.  

> So far, everything I've ever learned about DSP seems to hinge on one 
> single equation:
> 
>   sin X + sin Y = 2 cos (X-Y)/2 sin (X+Y)/2
>
> This says many things. But most obviously, it says that the sum of any 
> two waves is also the product of two *other* words, and vice versa. So 
> the question is, at what frequency does a wave become "sound" rather 
> than "variation in sound"?

20 Hz?
 
> The *other* question, of course, is "how many frequency bands do I need?"

Experiment!

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> (I have no idea how you measure signal energy...)

You mean how to assign a number to "loudness" given the waveform?  I'd guess 
that taking the RMS over a few cycles followed by some log scaling would be 
a good start, but the ear is complex...

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scott wrote:
>> You can reduce this by using another window function - but then, 
>> strictly speaking, you're multiplying your signal by a window, which 
>> convolves its spectrum with the spectrum of the window. In other 
>> words, by windowing the signal you're blurring its spectrum.
> 
> Usually a slightly blurred spectrum is the preferred outcome.

Preferable to phantom frequencies, sure...

>> This says many things. But most obviously, it says that the sum of any 
>> two waves is also the product of two *other* words, and vice versa. So 
>> the question is, at what frequency does a wave become "sound" rather 
>> than "variation in sound"?
> 
> 20 Hz?

That's approximately the lowest frequency that the human ear can 
mechanically detect. There's no particular reason to expect this to be 
in any way related to the rate of change in frequency spectrum that the 
human mind can percieve. (One is related to the physics of the ear, the 
other is related to the internal operation of the brain...)

>> The *other* question, of course, is "how many frequency bands do I need?"
> 
> Experiment!

Science: It works, bitches! :-D

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>> 20 Hz?
>
> That's approximately the lowest frequency that the human ear can 
> mechanically detect. There's no particular reason to expect this to be in 
> any way related to the rate of change in frequency spectrum that the human 
> mind can percieve.

Modulate a 1 kHz tone with a N Hz square wave.  At what value of N do you 
cease to be able to recognise the 1 kHz tone turning on and off?  I suspect 
it's somewhere around 20 Hz, you could easily figure it out with a sound 
editor program like Audacity.

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scott wrote:

> Modulate a 1 kHz tone with a N Hz square wave.  At what value of N do 
> you cease to be able to recognise the 1 kHz tone turning on and off? 

Yes, this is the experiment to perform.

> I suspect it's somewhere around 20 Hz, you could easily figure it out with 
> a sound editor program like Audacity.

I'd probably use my synthesizer actually, but anyway... ;-)

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Invisible wrote:
> at what frequency does a wave become "sound" rather 
> than "variation in sound"?

Around 10Hz to 12Hz.

-- 
Darren New, San Diego CA, USA (PST)
   Linux: Now bringing the quality and usability of
   open source desktop apps to your personal electronics.

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scott wrote:
> suspect it's somewhere around 20 Hz, 

Closer to 10Hz.  Between 10Hz and 20Hz, you detect sufficiently loud sound 
with different organs, such as your eyeballs vibrating or your sinuses 
resonating. If you play a tone in that range, you get the "spooky" feeling, 
or unaccountably sad, or mystical feelings, or things like that, depending 
on the frequency.

-- 
Darren New, San Diego CA, USA (PST)
   Linux: Now bringing the quality and usability of
   open source desktop apps to your personal electronics.

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On 4/30/2010 10:52 AM, Darren New wrote:

> Closer to 10Hz. Between 10Hz and 20Hz, you detect sufficiently loud
> sound with different organs, such as your eyeballs vibrating or your
> sinuses resonating. If you play a tone in that range, you get the
> "spooky" feeling, or unaccountably sad, or mystical feelings, or things
> like that, depending on the frequency.

I know with my headphones (Hybrid canalphones, actually) I can detect 
down to around 10hz, anything lower is inaudible. I designed my 
headphone amp with a ~15hz corner frequency high pass in it's input 
stage (AC coupling, to get rid of any DC bias before feeding the signal 
to the op-amp)

It sounds beautiful, very rich tones.

There is something about a big pipe organ with extremely low registers. 
Occasionally at church they'll pull what I think is the 16' stop, and it 
sounds incredible, there's something about that deep rumble the big 
pipes put out...

-- 
~Mike

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Mike Raiford wrote:
> There is something about a big pipe organ with extremely low registers. 

That "something about" is exactly what I'm talking about. :-) That's the 
subsonics kicking in.

-- 
Darren New, San Diego CA, USA (PST)
   Linux: Now bringing the quality and usability of
   open source desktop apps to your personal electronics.

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Mike Raiford wrote:

> There is something about a big pipe organ with extremely low registers. 
> Occasionally at church they'll pull what I think is the 16' stop, and it 
> sounds incredible, there's something about that deep rumble the big 
> pipes put out...

On a normal church organ, 16' is the lowest available stop. This is due 
mainly to size considerations. (Not to mention that building large pipes 
requires a lot of metal, and therefore costs money.)

A cathedral organ, however, would usually have at least one 32' stop.

Exactly two pipe organs on the face of the Earth have a 64' stop. But 
then, really, that's not a note, it's a small earthquake! o_O

(I should maybe point out that a 16' stop doesn't necessarily contain 
any pipes that are 16' long. There are ways to make the pipes shorter 
[while muffling the tone they generate]. Or the stop might simply not 
contain all the notes of the scale; only the *lowest* pipe would need to 
be 16' long; the next just one octave up only needs to be _half_ that size!)

I leave it as an exercise for the over-interested reader to figure out 
what the fundamental frequency of a 32' pipe is. (Remember that the 
fundamental wavelength is 2x the pipe length - or 4x if it's a closed pipe!)

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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