SouthSIDE Glen
independentrecording.net
Hey mattr, yeah, I know you were just referencing someone else. I didn't mean to make it sound like I was correcting *you*. My apologies if it came out that wayThe analogies are always poor, I know, I was just quoting from the other thread for the possible benefit or musings of others
I think where most people stumble up is when they can't understand how everything below the nyquist frequency can be faithfully reproduced (assuming everything above the nyquist frequency was filtered so no aliasing has occurred), and have a notion that having more samples (which they visualize as 'data points' on the waveforms) will mean a more accurate reproduction - the word 'smoother' also pops up from time to time. I think part of this problem and the way people visualise digital sampling as being 'jagged' is because you only really get to see the blocky dot-to-dot "waveforms" of most editors and not the smooth interpolations and reconstructions that actually occur during digital-analogue conversion. Intersample peaks are usually a good starting point for explaining this. I recall seeing some good images accompanied by some basic but insightful explanations of how the signal is reconstructed on a website somewhere and if I can find it then I'll post a link.
.I think a big part of the misconception is the "connect-the-dots" or "stairstep" illustrations that most lay books use to illustrate the concept of sample rate. With such pictures it only makes sense that the more "dots" or "stairs" you have, the "smoother' the reproduction will be.
The problem is that at real sample rates, there's nowhere near as many "dots" as they imply in those diagrams. For example, a 20kHz sine wave samples at 44.1k will only have slightly more than two samples or dots per wave. there's no way to connect those dots based upon those dots alone to come even close to guarantying accurately redrawing that sine (the same would be true for a triangle, sawtooth or square wave, BTW). Even at 1/4th the signal frequency, at 5k - which I don't think anyone seriously doubts that 44.1k can handle just fine without breaking a sweat - we're only talking 8 or 9 samples per wave. A far cry what what they usually show in those illustrations, and still nowhere near enough to draw a very smooth or accurate wave by connecting the dots or building a staircase. That''s because that's not how digital reconstruction via Nyquist-Shannon actually works.
----
Also, just a bit more about Harvey's square wave example - which was a good point to bring up. The thing to remember is that a square wave (also true to a degree of sawtooth and triangle waves) is so loaded with harmonic components in the form of higher-frequency sine waves that one cannot consider their fundamental frequency alone when looking at the issue of bandwidth limiting as required by Nyquist-Shannon. When you get a *true* square wave of high enough frequency, like Harvey illustrated, it contains components that are going to exist beyond the Nyquist frequency. For this reason, Nyquist-Shannon cannot apply to such waves with 100% accuracy *unless* the sampling frequency is raised to infinity, which is impractical.
Harvey did bring up a borderline example popular amongst many proponents of higher sampling rates of a square of sufficiently high frequency where one could and should see a significant difference between the square sampled at 44.1 and one sampled at, say 88.2 or 96. This is true, because of the complex and extended harmonic structure of the square wave takes it well beyond the fundamental frequency. This can on first blush appear to be a hole in Nyquist-Shannon and a real reason for extended sample rates.
It's not a hole in N-S theory, because of the theory's stipulation for bandwidth limiting. No rules are broken, no holes punched. Nyquist does not claim to be able to reproduce a square wave with a 7.5k fundamental exactly, because the fundamental is not a super quorum portion of the actual square wave's frequency content.
And as far a it being a "real reason" to justify higher sample rates, I reiterate that the likelihood of finding anything close to true square wave (or any other such significantly harmonically-complex waveform) at such high frequencies of any significant amplitude - even if running a synth direct* - is not only astronomically small, but does not address the OP question here, nor does it come close to explaining the much larger audible differences that proponents can hear between sampling rates.
*I grew up on analog synths; mostly Arp, but also Moog, PAIA, Oberheim Sequential Circuits and others, and regularly ran their outputs through an oscilloscope. I never did find one that could generate a square wave that any reasonable carpenter or EE could consider anything close to square. And as far as the VSTis, I've only looked at a couple, but they as often as not get what the original synth actually did wrong (the Arturia sawtooth for the ARP 2600 is a joke, for example), and what they do get right is usually no more 'pure" in shape than the original.
Now there many be some scientific tone generators and test gear that can do a better job, but I have yet to see one of those used as a high-frequency musical instrument.
G.
, the limited response characteristics of the recoding chain, the natural geometric decrease in energy as one climbs the frequency ladder, the natural lack of high-frequency complex waves like squares, sawtooths and such, and the other distortions natural to the music production process, sample rate seems so far down the list of factors as to be something I personally don't lose any sleep over. 
