Digital Music Couldn't Exist Without the Fourier Transform
gizmodo.com
gizmodo.com
JPEG uses DCT in particular because it has the nice property that the "top left" corner of the block will contain the DC offset (since cosine of 0 is 1) and the coefficients near the top corner correspond to half-wave and full cycles which gets you most of the way to simple gradients of color across the block with the right coefficients. So for most areas of an image only the top left coefficients will be significant. By using a zig-zag pattern for each block we are grouping the largest values to the front and zeroes to the back, which when coupled with RLE makes the rows of zero in each block a very compact, further-compressable representation.
Meanwhile, a fourier transform gives you imaginary magnitudes for frequencies which corresponds to the phase shift that is most appropriate for that frequency to match most strongly (as opposed to be aligned at the corner/beginning of the integral window). Not useful in an image format where you won't get the transformed magnitudes all nice and grouped for you. This is useful in audio compression where we care to find the location of transients that correspond to note attacks, percussion strikes, etc. Note that even in MP3 this is only used to drive the psychoacoustical model that decides the frame type and where to allocate the bits; the audio data itself is processed out of the time domain by overlapped DCT just like Ogg Vorbis.
The author should familiarize himself with the history of digital audio, and its milestones such as the development of the compact disc format: long before MP3, Ogg and the popularization of the Internet, and its use for media streaming.
At its bare bones, digital audio requires time domain sampling and reconstruction, sandwiched between some filters that can be analog. It requires understanding of the Nyquist limit, which can be purely in time domain terms (sampling frequently enough to avoid an temporal aliasing ambiguity in the reconstruction).
Digital synthesis of music can also be as simple as playing recorded samples in loops, and scaling them in the time domain for various pitches (or changing the sample rate, or both), which doesn't require Fourier.
Those analog anti-aliasing and anti-imaging filters are designed using LTI systems theory, that fundamentally rely on the Fourier tranform to reason about their transfer functions. The Nyquist-Shannon sampling theorem was proven using the fourier transform. Without the fourier transform, you need to rely entirely upon time domain representations of signals, and perform your analysis using tedious convolutions. You can't use a spectrum analyzer to examine the signal to noise ratio of your CD player. While it's true that digital music could technically exist without the fourier, there is no way in hell it would be as pervasive as it is today.
Microsoft's ADPCM was always 4 bits encode of the 16-bit sample. The distorted sound in some files was due to reducing the sampling rate.
https://support.microsoft.com/en-us/kb/89879
"ADPCM stores the value differences between two adjacent PCM samples and makes some assumptions that allow data reduction. Because of these assumptions, low frequencies are properly reproduced, but any high frequencies tend to get distorted. The distortion is easily audible in 11 kHz ADPCM files, but becomes more difficult to discern with higher sampling rates, and is virtually impossible to recognize with 44 kHz ADPCM files."
I've already linked this article and it has even more details, highly recommended.
The Microsoft article gets a few things wrong. The distorted sound is not due to reducing the sample rate. The distorted sound comes from taking a perfectly-good 11k file and then ADPCM compressing it. This is obviously due to throwing away information on each sample as part of the encoding process, not anything due to sample rate. (Of course it sounds better at higher sample rates. More data, more better.)
ADPCM for telephony seldom even hit 11k rates. 6000 and 8000Hz ADPCM files are common. (And nope, not 16 bit either.)
I'd be of course happy to hear something more about the work you did.
I think the article says, though, that fourier analysis on the music is necessary for digital audio.
Yeah, we could have done digital audio without FFTs. Woulda been infeasible though given the circumstances.
ADPCM. No FFT. Worked on my 1991 PC:
https://support.microsoft.com/en-us/kb/89879
"22 kHz mono ADPCM can be decompressed real-time (that is, while playing) on a 386SX/16 megahertz CPU"
Note: 16 MHz 386SX was effectively slower than the 286 of the same speed.
I had all the Beatles albums on one CD, all encoded with ADPCM, and the sound was not bad.
If I remember correctly, the guys at Bell Labs had also some server filled with the digital music. But I can't locate the story (maybe somebody remembers, was it Ritchie or Thompson?) and don't remember when actually.
That being said, it's a poor linkbait title for a superficial article that's missing nuance, rigor, and causality. Anyone who takes interest in this subject should read Wikipedia or an actual textbook, lest they feed their Dunning-Kruger.
Optics too. People understood geometrically how a lens focuses reflected light to produce an image before the Fourier transform view on it was revealed.
(http://en.wikipedia.org/wiki/Fourier_optics#Fourier_transfor...)
Ugh, I see a trend starting here:
"This is the first in a new experimental series called Favored Equation. Each month, we’ll dive into a piece of math which makes your life easier in some way without you even realizing."
This is a spin on http://objectsobjectsobjects.com/ by The Atlantic:
"Object lessons: An ongoing series about the hidden lives of ordinary things."
I'm a fan of this type of writing. But when Sagan and Feynman did it -- hell when pornographers did it with OMNI Magazine -- it wasn't quite so rough around the edges.
I'm now a month into arguing with some ex-Gawker hack at The Atlantic over quotes like "New effects can change a guitarist’s playing ability completely" and a declaration that the transistor was invented in the 1960s. No corrections or retractions imminent.
No interest in battling the newer, younger, even-less-experienced Gawker editor too.
Effects can and do change the guitarist's playing ability. Distortion or compression, for example, generally lowers the threshold for making a note sound clearly (or at least clear enough). The result is a smoother sounding performance which can give the player more confidence when then actually results in a better performance. It hardly matters how you get to the end waveforms if they are indeed sounding good.
But more than interpreting "ability" to mean a degree of technical skill, certain effects can completely change the way a good guitar player approaches the instrument creatively, particularly if they're listening closely and reacting (not playing from muscle memory).
"Put a little reverb on it" is a good way to comfort a singer or a musician and perhaps coax a better performance out of them. But the effect itself does not change the skill level of the performer.
If the article were discussing production techniques I'd agree with you. But it says things like "Guitar effects have modified their users" and gives a comical explanation of how a rotating speaker works so I think the author is just nuts.
Guitar distortion reduces the need to mute adjacent strings, a very difficult thing to master, because only the strongest tone comes through.
Or, given equal skill, distortion meant you could jump around, play writhing on your back with your tongue, play with your guitar on fire, which you absolutely cannot playing without distortion.
A previous poster said that compression helps. Well, no. Compression reduces dynamic range. If there's a soft squeak between loud notes, a compressor makes the soft squeak louder and the loud note quieter. That's the definition of a compressor.
And none of this changes the fundamental skill of the performer, in the same way that an Instagram filter does not change the fundamental beauty of the object. Sure, presentation is important and changing your resume font might even get you a better coding job. But changing the font doesn't make you a better coder.
The only way you could quantify the type of skill you seem to be referring to is to entirely remove any degree of expression or improvisation (or effects) and boil it down to the raw performance data. You may then succeed in determining who is objectively a "better" musician but you've lost all the aspects that make a Beatles song based on a I IV V chord progression sound different than one by the Velvet Underground, or anyone else.
EDIT: also, yes to some degree that is how harmonic distortion works, though the particular harmonics and amplitudes of those harmonics vary widely and there is often some filtering added to reign in those harmonics in a particular way. Distortion also effectively compresses the signal. Sometimes it starts oscillating and generating notes that aren't even being played. The point is that if you're playing with the distortion (not just laying it on top) then it is changing how you play.
Instagram filters don't make people better photographers; AutoTune doesn't make people better singers. Hell, the Abbey Road mic collection didn't make John and Paul better singers.
By your logic you can flip anything around and make the author's argument: Shitty British industrial towns modified their users. Beer and the tiny stage at CBGB modified their users. Cheap Seattle heroin and terrible weather modified their users.
Is any artistic pursuit highly dependent on the mood of the performer and her implicit capacity to make you feel? Of course.
But the author did not say "playing a guitar through a stomp box changes the way you play it" or "the Edge's signature delay effect created a new sound that nobody had heard -- or felt -- before." He said "guitar effects have modified their users" and "new effects can change a guitarist's playing ability completely."
If you read the entire article you'll see the author suffers from an bad case of "word salad" and these are not meant as debatable nuances. He's suffering from dysgraphia, ignorance of the subject matter as a whole, and a really bad editor.
Which is why he defines "clipping" with a phrase lifted from the Wikipedia article for digital clipping, applies it to slicing a speaker with a razor blade, then claims rotating motors were picked up by speakers in a Leslie after being picked up by "coiled magnets" in a guitar pickup.
"""But add them together, and that pleasant sounding chord actually looks altogether more messy, like this:"""
No, it doesn't look like this, at all.
FFT is closer to what's going on with Cochlear Implants. Which both suck for some of the same reasons.
I think it's just a total different approach to compression that doesn't use Fourier.
The theory around any family of functions useful for compression/feature detection (like wavelets) is going to have the property that they define a Hilbert basis. And the idea of how to conceive of such families of functions and their potential required them being generalized from the specific cases of the Fourier and Laplacian transformations. Moreover wavelets have properties/tradeoffs defined in terms of time and frequency which are couched in terms and based on theories that are derived from this early complex analysis.