Re: [LAD] twice as loud

On Tue, 2010-07-27 at 15:44 -0500, Charles Henry wrote:
> On Tue, Jul 27, 2010 at 12:38 PM, Ralf Mardorf
> <ralf.mardorf@email-addr-hidden-dsl.net> wrote:
>
> > It's not impossible. I guess nobody is able to note, let's say, 10 000
> > pictures a second as single steps for a movie, of course you and I
> > aren't able to note it for just 30 pictures a second. But I don't
> > believe in digital audio math, on the niveau we reached until today.
> > Btw. I don't have knowledge of this math, I'm just listening and have
> > long time experience with doing analog recordings.
> Ah! That's just a bandwidth limitation, but it's a rather good
> example for the mathematically inclined. Let's say we're just talking
> about the set of sounds that are 1 second long or less.
>
> I'd like to show that the human auditory system performs a significant
> reduction in the dimensionality of sounds. Start with sets of signals
> on [0,1] that have finite energy and power: s(t) on [0,1] is finite,
> and the integral of s(t)^2*dt on [0,1] is also finite.
>
> Q: So, how many dimensions do we start out with?
> A: infinite--this is one example of a Hilbert space. The
> dimensionality is clear by application of Fourier series. We can
> represent functions in this space with a series of orthogonal
> functions (sines and cosines), but to represent *all* functions in
> this space, the series has to be infinitely long.
>
> Q: Now suppose we limit the bandwidth to 200 kHz. How many
> dimensions do we need?
> A: 400,000. By Nyquist's sampling theorem, we need 400,000 samples
> to represent continuous signals up to 200 kHz. Either by
> sampling/reconstruction or Fourier Series, we can show that our space
> is homeomorphic to R^400,000.
>
> So, your own example shows that if we increase our bandwidth
> arbitrarily high, we can't tell the difference anymore. The auditory
> system is bandwidth limited in this way--typical rule of thumb is
> about 20kHz of bandwidth. We represent these continuous sounds with
> samples at a rate more than twice the bandwidth. So typically, we
> sample at 40kHz and above. Real acoustic sounds can have a lot of
> extra frequencies above 20kHz, so sample at higher rates to reduce
> aliasing of those frequencies onto the auditory band. No further
> increases in quality can be obtained by sampling at faster rates.
>
> Regardless, it's a gigantic number of dimensions. The essence of
> psychology is the study of mental representations. How can each of
> those things be represented in the mind? The problem becomes, what is
> the smallest integer-dimensional space into which we can embed the
> space of all sounds? This is not a problem that has been solved, nor
> do I prescribe how to take such a measurement.
>
> But finding such a result is the *exact* problem to solve in
> psychoacoustic coding. It's reducing a problem from a set which takes
> a large number of points to represent all possibilities to a set which
> takes the fewest number of them.

Full ACK again. My example was for video and film, of cause we only need
30 pics/second, but we might be completely save with 10000 pics/second.

For digital audio there still is an issue regarding to the
transformation D --> a, A --> D.

For my ears I've got some perfect masterings at 32 KHz DAT long play and
I'm fine with every 48 KHz standard DAT recording I ever made.
I'm not fine with any 96 KHz recording done with my sound cards.

So, the reasons obviously depend to hardware, regarding to 'my DAT
recorders are better, than my sound card is', but for the same DAT
recorder doing the mastering sometimes was fine when using just 32 KHz
long play and sometimes I need to use 48 KHz, even CD quality wasn't
good enough. NOT BECAUSE 'twice as loud', but because of the complete
quality, e.g. loudness ratio for high and low frequencies.

I never heard or read about such issues, but I had such effects when
doing a mastering at my home studio.

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Received on Wed Jul 28 00:15:06 2010