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Re: Help me understand why sound is one dimensional

Reply #25
Also, the Fourier transform can be used to convert audio data to a frequency domain representation, which is a 2D array of frequency bins that vary in intensity over time, so in that sense audio could be considered 2D. :D
The FFT of a signal returns a function which itself returns a two-dimensinal value for each input component. It is therefore one-dimensional as well.

Re: Help me understand why sound is one dimensional

Reply #26
could it just be that a lot is lost between what he's trying to say and what you think he said?
because I have a hard time imagining a math professor saying something along the lines of:  sound has only one dimension.
even considering a punctual source or punctual recording position, we're still very much getting amplitude over time.
just here to contradict myself like a boss. amplitude over time is actually a one dimensional signal. Saratoga is right and reading his post got the me from more than 20 years ago instantly punching today's me in the face. I first went for a 2 dimensions space for the graph showing a signal, but time is the only independent variable in the function.

sorry math guy, sorry younger me. I'm a fool who forgot all his math.

Re: Help me understand why sound is one dimensional

Reply #27
I looked-up "dimension" and I found this:
Quote
Dimension

The word dimension is related to the word measure. It is used in two ways in geometry.
•It is used to specify a measurement:
"What are the dimensions of this rectangle?" or "Build a rectangular prism that has these dimensions."

•It is also used to count the (mutually perpendicular) directions that an object can be measured.
"A rectangle has two dimensions" or "This is a three-dimensional figure" or "How many dimensions does a point have?"

"Length is a one-dimensional measure, but area is a two-dimensional measure."
Counting mutually perpendicular directions an object can be measured
•A point has zero dimensions: there is nothing to measure; a point just specifies a location, but has no size.
•A line] segment has one dimension: we can measure its length, but it has no width or thickness or any other measurable feature.
•A rectangle has two dimensions: we can measure its length and, perpendicular to that, its width. The interior of a triangle or oval is also two-dimensional. Though we don't think of these as having "length" or "height," they cover a region that has extent in not just one direction but two.

Nothing surprising there...  It's all "common sense" ... So I'm back to 2-dimensions....    Sound can't be represented as a straight-line.

We can (and do) represent audio (or approximate audio) as a "1-dimensional array".  But, that's an array full of many one-dimensional values.   It can't be represented as one-single line or graphed/mapped in one-dimension.      A continuous wave is an infinite number of one-dimensional values...     So, I'm still at 2-dimensions.  

Re: Help me understand why sound is one dimensional

Reply #28
So a waveform is an area?  No, it is not.

Perhaps you should be researching degrees of freedom, as likening a sound wave to a geometric shape is a mistake.

Who says an array cannot have infinite values separated in time by an infinitesimal amount?

So, I'm still at 2-dimensions.
Hurrah!
Is 24-bit/192kHz good enough for your lo-fi vinyl, or do you need 32/384?

Re: Help me understand why sound is one dimensional

Reply #29
PS:  Let's have a look at the generic equation for a line...

y=A*x + B

You have a y and an x where A and B are constants.  It must be two dimensions, right?
Is 24-bit/192kHz good enough for your lo-fi vinyl, or do you need 32/384?

Re: Help me understand why sound is one dimensional

Reply #30
PPS:  Why not look at how a wave drives a speaker...

It moves in and out following the path of a what?  A straight line; very good!

Must be two dimensions, right?
Is 24-bit/192kHz good enough for your lo-fi vinyl, or do you need 32/384?

Re: Help me understand why sound is one dimensional

Reply #31
I looked-up "dimension" and I found this:
Quote
Dimension

The word dimension is related to the word measure. It is used in two ways in geometry.
•It is used to specify a measurement:
"What are the dimensions of this rectangle?" or "Build a rectangular prism that has these dimensions."

•It is also used to count the (mutually perpendicular) directions that an object can be measured.
"A rectangle has two dimensions" or "This is a three-dimensional figure" or "How many dimensions does a point have?"

"Length is a one-dimensional measure, but area is a two-dimensional measure."
Counting mutually perpendicular directions an object can be measured
•A point has zero dimensions: there is nothing to measure; a point just specifies a location, but has no size.
•A line] segment has one dimension: we can measure its length, but it has no width or thickness or any other measurable feature.
•A rectangle has two dimensions: we can measure its length and, perpendicular to that, its width. The interior of a triangle or oval is also two-dimensional. Though we don't think of these as having "length" or "height," they cover a region that has extent in not just one direction but two.

Nothing surprising there...  It's all "common sense" ... So I'm back to 2-dimensions....

The second and third definitions are the appropriate ones, and would give you an answer of 1 dimension.

 
 
We can (and do) represent audio (or approximate audio) as a "1-dimensional array".  But, that's an array full of many one-dimensional values.

The values of a 1 dimensional array are zero dimensional points.  If the values were 1 dimensional, you would have a 2D array, which is more formally called a matrix.

   It can't be represented as one-single line or graphed/mapped in one-dimension.

The output of a line out jack is literally a 1D linear array of speaker displacements.  If you could not represent sound as a linear sequence of driver positions, speakers would not work.

     A continuous wave is an infinite number of one-dimensional values...     So, I'm still at 2-dimensions.  

Off by  one error.  YOu have zero dimensional values so you'are at 1 dimension when you accumulate them into a series or vector.

Re: Help me understand why sound is one dimensional

Reply #32
The output of a line out jack is literally a 1D linear array of speaker displacements.  If you could not represent sound as a linear sequence of driver positions, speakers would not work.
Conversely, what is the motion of a diaphragm in a microphone that generates a waveform?
Is 24-bit/192kHz good enough for your lo-fi vinyl, or do you need 32/384?

Re: Help me understand why sound is one dimensional

Reply #33
I still think the confusion is down to the rather inappropriate usage of the term "sound". It doesn't determine what realm we're discussing in here right now.
Things like waveforms and time domain signals are one dimensional. Signal transformation through things like FFT is also one dimensional but returns a two-dimensional value.

Sound propagation through air is a multivariate problem, where sound is propagated in three dimensions, changes through time, and returns a sound pressure for each point at any given point, that's a four-dimensional function.

Similar things happen when looking at RF signal propagation through space.

So, perhaps it's a good idea to first define the kinda frame we're discussing here?

Re: Help me understand why sound is one dimensional

Reply #34
I still think the confusion is down to the rather inappropriate usage of the term "sound". It doesn't determine what realm we're discussing in here right now.

From the replies to this thread, I think the real problem is people not understanding what dimensions are.

Things like waveforms and time domain signals are one dimensional. Signal transformation through things like FFT is also one dimensional but returns a two-dimensional value.

I'm surprised how many people don't realize this, but the FFT returns the same number of dimensions as you put in, so a 1D function has a 1D FFT. It's a linear transform, so you get the same number of points and dimensions between domains. I suspect that the misconception you have hear (that frequency is somehow of higher dimensionality than time in spite of them having inverse units) is related to the general confusion most people have on this topic.

Sound propagation through air is a multivariate problem, where sound is propagated in three dimensions, changes through time, and returns a sound pressure for each point at any given point, that's a four-dimensional function.

People have such a limited ability to grasp what a 3D sound field is that most do not realize they even exist. The perception of sound is basically 1D with a bit of stereo, and this is the thing people are talking about. And they do mean to say 1D, they're just not sure what the words mean and are expressing themselves incorrectly.

Re: Help me understand why sound is one dimensional

Reply #35
Sound implies the ability to hear, so we're back to where we started.
Is 24-bit/192kHz good enough for your lo-fi vinyl, or do you need 32/384?

Re: Help me understand why sound is one dimensional

Reply #36
Things like waveforms and time domain signals are one dimensional. Signal transformation through things like FFT is also one dimensional but returns a two-dimensional value.

I'm surprised how many people don't realize this, but the FFT returns the same number of dimensions as you put in, so a 1D function has a 1D FFT. It's a linear transform, so you get the same number of points and dimensions between domains. I suspect that the misconception you have hear (that frequency is somehow of higher dimensionality than time in spite of them having inverse units) is related to the general confusion most people have on this topic.
Well, a Fourier Transform (discrete or continuous) returns a complex-valued function, which real and imaginary components can be mapped onto a 2D plane. Complex numbers by definition extend the one-dimensional number line in \mathbb{R}, to the two-dimensional comlpex-plane \mathbb{C}. That's what I was referring to. Then again, judging by how things were discussed here, I doubt most participants have grasped the concept of complex numbers...

People have such a limited ability to grasp what a 3D sound field is that most do not realize they even exist. The perception of sound is basically 1D with a bit of stereo, and this is the thing people are talking about. And they do mean to say 1D, they're just not sure what the words mean and are expressing themselves incorrectly.
Well, I'm not sure we're talking about perception of sound here. When Op started the thread, I thought we talk about sound as in the way signals are expressed. Hence I tried to clarify in that direction. Perhaps I should've given this a pass.

Re: Help me understand why sound is one dimensional

Reply #37
Things like waveforms and time domain signals are one dimensional. Signal transformation through things like FFT is also one dimensional but returns a two-dimensional value.

I'm surprised how many people don't realize this, but the FFT returns the same number of dimensions as you put in, so a 1D function has a 1D FFT. It's a linear transform, so you get the same number of points and dimensions between domains. I suspect that the misconception you have hear (that frequency is somehow of higher dimensionality than time in spite of them having inverse units) is related to the general confusion most people have on this topic.
Well, a Fourier Transform (discrete or continuous) returns a complex-valued function, which real and imaginary components can be mapped onto a 2D plane. Complex numbers by definition extend the one-dimensional number line in \mathbb{R}, to the two-dimensional comlpex-plane \mathbb{C}. That's what I was referring to.

To try and make my point above clear, I'll reiterate that an FFT maps a (possibly complex valued) input to another (possibly complex) output of equal dimensionality. It is not correct that a (possibly complex) input of dimensionality 1 can map to an output of dimension anything but 1.

More importantly, being complex valued in this case is distinct from defining a plane as you are assuming. Recall that the FFT of a real valued function (such as a series of sampled points of a transducer in time) must be symmetric about zero. This means N unique values would seem to map to N/2 values - which would make the transform non invertible. Instead it actually maps to N/2 complex value pairs, or more simply just N values. Thus it's actually a mapping from N to N and therefore invertible.

Finally getting back to my original point, all of this math could be avoided by just defining dimensionality correctly.


People have such a limited ability to grasp what a 3D sound field is that most do not realize they even exist. The perception of sound is basically 1D with a bit of stereo, and this is the thing people are talking about. And they do mean to say 1D, they're just not sure what the words mean and are expressing themselves incorrectly.
Well, I'm not sure we're talking about perception of sound here. When Op started the thread, I thought we talk about sound as in the way signals are expressed.

Seems like a distinction without a difference given that they're expressed similarly 99.99999% of the time.

Re: Help me understand why sound is one dimensional

Reply #38
This is very similar to a previous posting, but with a simplified view of things:

At any given point, the sound pressure is a 1 dimensional signal vs time.
Given two ears, there are two sound pressure values vs. time measured -- each is still 1 dimension.

But, with human perception, the two sound pressure values can be processed into a spatial (partially accurate two, perhaps three) form.  The various cues to the spatial location are complex -- timing, spectral/frequency response hints, and probably other things.  So, those two single dimensional values from two ears can sometimes provide more information than two simple scalar values vs. time.

 

Re: Help me understand why sound is one dimensional

Reply #39
Interesting thread! I'm not a mathematician, just an IT guy with barely enough math knowledge to be dangerous, but hear me out.

It always seemed to me that a waveform is something that can only be approximated. Unless it's generated by a function, that is (like a sine wave.) In music, the waveforms can only be approximated since the underlying sounds are very complex and chaotic. The actual complexity of a waveform makes it a very fractal-like structure. As such, it might make more sense to assign a fractal dimensionality to waveforms, which probably lies somewhere between 1 and 2?

Re: Help me understand why sound is one dimensional

Reply #40
More importantly, being complex valued in this case is distinct from defining a plane as you are assuming. Recall that the FFT of a real valued function (such as a series of sampled points of a transducer in time) must be symmetric about zero. This means N unique values would seem to map to N/2 values - which would make the transform non invertible. Instead it actually maps to N/2 complex value pairs, or more simply just N values. Thus it's actually a mapping from N to N and therefore invertible.
Ah right, this is actually a good point. For some reason I ended up going more into complex numbers as composite values.

A two-dimensional DFT also results in a two-dimensional frequency domain function which is also complex-valued, as it is with image processing. I was probably focusing too much on explaining that a complex number has two components (which is more of a consideration in programming).

Btw. FYE: MRI machines return a (2D) frequency domain, the picture has to be IFFT'd to reconstruct the image.

The two pictures I've attached is Mr. Joseph Fourier, and Mr. Fourier, fourier-transformed. Should be noted, that's the combined image with both magnitude and phase merged, and centered around 50% gray for 0.

Re: Help me understand why sound is one dimensional

Reply #41
Interesting thread! I'm not a mathematician, just an IT guy with barely enough math knowledge to be dangerous, but hear me out.

It always seemed to me that a waveform is something that can only be approximated. Unless it's generated by a function, that is (like a sine wave.) In music, the waveforms can only be approximated since the underlying sounds are very complex and chaotic. The actual complexity of a waveform makes it a very fractal-like structure. As such, it might make more sense to assign a fractal dimensionality to waveforms, which probably lies somewhere between 1 and 2?
No.

As you said, a simple sine can be accurately expressed through it's coefficients, like frequency, amplitude, phase-shift and bias.
The same is true for any higher order signals, you just add more and more of the sinusoidal components (all signals can be expressed as a sum of sinusoids). For some signals, that number of components is infinite, those can only be approximated at a certain cut-off.

However, this is not a problem in practical applications, since prior to sampling, you filter the signal below the max frequency, according to the Shannon-Nyquist sampling theorem, to avoid artifacts through under-sampling.

Assuming your sampling frequency is 48kHz (as most sound cards do), the maximum frequency is 24kHz, according to the Shannon-Nyquist theorem. All signals sampled by that device are accurately reproduce-able, up to the frequency of 24kHz. In practice, the sound card must therefore low-pass the analog sound at 24kHz. This low-passed signal can now be exactly reproduced, from the sampled data. Keep in mind, that we're discussing sound in the human aural spectrum here. Anything above 20kHz is inaudible, anyhow. The higher frequency cut-off is mainly to allow the filter some headroom, as no filter is abrupt.
Also note, this is us discussing the ADCs ability to sample sounds. As an ADC has a technical limit at which it can reproduce a signal, it will simply refuse to output signals of higher sampling rates.

Also, keep in mind that this is also exactly true in the RF spectrum. you can easily build a device to sample in the 10GHz range and, reproduce whatever signal at ½f of that (<5GHz, in this example).

The idea of dimensionality as you tried to infer, is a non-sequitur, and is kinda in-line with what Saratoga mentions, how dimensionality is misunderstood in this context. I must admit, me explaining complex planes as being two dimensional was kinda misleading, too. I should've kept this out of the discussion.

It is also exactly not a fractal, or fractal-like! Fractals are also by definition functions, which are not differentiable at any point however signals are very much differentiable at allmost all points.
Fractals are highly structured, the exact opposite of noise! However in analog sampled signals (sound waves in particular) we're dealing specifically with noise, which is by definition, un-structured. Fractals are the very essence of patters, while noise is the absence of patterns. Especially in music the wave forms are extremely well-formed and highly harmonic, compared to fractals.
Fractals are essentially a form of symmetry.

However, there is fractal-like music or perhaps less so music, but fractal-like tones: The Shepard Tone is an Example that comes to mind:
https://www.youtube.com/watch?v=BzNzgsAE4F0

One attribute of fractals is self-similarity, the Shepard Tone essentially implements that, but obviously the non-differentiability isn't there.

For some reason I just forgot the rest of my ideas I should go to bed.

Re: Help me understand why sound is one dimensional

Reply #42
When people talk about the mysterious additional phase on an sound pressure wave, it can only be relative to another signal.  Pressure is effectively measured at a single point in space.

The idea of phase is a relative time relationship at a given frequency (when talking about sine type waves.)  So -- phase is kind of meaningless for a sine wave when talking about hearing -- each ear simply hears a single complicated sound wave. There CAN be a matter of phase relationships, but not absolute phase.  When one speaks of something like A*sin and B*cos constituents of a signal, it is mostly just a way to shift through a full 360degrees given the same frequency or mixing multiple such consituents of multiple frequencies.  But, given a single signal in a single location, there is only that signal with whatever internal phase relationships that it might appear to have.  But, the signal itself is still a scalar pressure level.   Please explain 'phase' when it comes to pressure? Are we starting to talk about dimensions beyond 3+time?  If so, that is beyond me, because pressure is measured at a single point unless you are doing some kind of array scheme and somehow map that to space.  A good example of adding perceived phase is with human perception and more than one sensor (e.g. 2 ears.)  The hearing system uses more information than just the phase shift between the two ears to create the spatial image.

John


Re: Help me understand why sound is one dimensional

Reply #44
This wasn't correct about fractals either ... but I think this is getting increasingly irrelevant to the OP's issue. (That need not be a bad thing, morphing threads isn't uncommon ...) If we get down to the molecular level, and define the "sound" in a gas medium to be the position (x,y,z) of each particle at each time - i.e. two variables (t,n) as input, and three variables (x,y,z) as output - then one could certainly discuss whether the "dimension" is 2 or 5 of a number in between, but neither answer will help explaining an analogy to (motion?) pictures. At least not to me.

I kinda suspect that the OP's friend sees only one particular model of sound, and which isn't sufficiently consistent with any model of picture to get any idea on the key differences.  (Are they in the physical phenomenon, or in what could possibly reach the brain through our sensory organs? I.e., is inaudible sound still sound and different shades of infrared still a picture?)
Memento: this is Hydrogenaudio. Do not assume good faith.

Re: Help me understand why sound is one dimensional

Reply #45
Interesting thread! I'm not a mathematician, just an IT guy with barely enough math knowledge to be dangerous, but hear me out.

It always seemed to me that a waveform is something that can only be approximated. Unless it's generated by a function, that is (like a sine wave.) In music, the waveforms can only be approximated since the underlying sounds are very complex and chaotic.
Have you considered how waveforms not generated by a function are created?  It isn't like they are created by some algorithm used to approximate sound.

The actual complexity of a waveform makes it a very fractal-like structure. As such, it might make more sense to assign a fractal dimensionality to waveforms, which probably lies somewhere between 1 and 2?
Fractal-like?  That's quite a leap.
Is 24-bit/192kHz good enough for your lo-fi vinyl, or do you need 32/384?

Re: Help me understand why sound is one dimensional

Reply #46
All this problem lies in the definition of dimension, as it varies with the context from the classical 3 spatial dimensions + time up to the multiple dimension in quantum physics. Sometimes you can argue the existence of hidden or omitted dimensions.

In the case you can argue that audio is 2 dimensional because it needs 2 elements to exist, magnitude (pressure) and time, after all audio is the information conveyed by the difference in pressure over time, even if its commonly represented as 1d vector because of its simplicity in this representation it exist a hidden dimension that is time because each position in the vector represents concrete time. It's possible to represent each point of audio as list of [pressure, time], this representation can led to variable sample rate audio, something exotic but plausible.

Also I can tell that normally a 1D vector is actually a 2D object because we normally omit the magnitude (weight, dimension) of the element in the vector to define its dimension. Basically we have a 1D vector of 1D element making it a 2D object. Or we can represent a vector by a list of [magnitude, position] demonstrating that a vector is a 1D representation of a list of 2D objects.

The devil of this discussion is hidden in the language and concepts.







Re: Help me understand why sound is one dimensional

Reply #47
Thanks for not making me feel quite so silly. My friend is a maths professor, and I believe his assertion. I just don't understand how you can have that type of information in a single dimension (time).
Perhaps you can get him to clarify.
Is 24-bit/192kHz good enough for your lo-fi vinyl, or do you need 32/384?

Re: Help me understand why sound is one dimensional

Reply #48
Thanks for not making me feel quite so silly. My friend is a maths professor, and I believe his assertion. I just don't understand how you can have that type of information in a single dimension (time).
Perhaps you can get him to clarify.
Or at least try  :o
Memento: this is Hydrogenaudio. Do not assume good faith.

Re: Help me understand why sound is one dimensional

Reply #49
space is not a dimension ?


 
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