Topic: Help me understand why sound is one dimensional (Read 5054 times)
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## Help me understand why sound is one dimensional

##### 2018-07-15 14:22:21
I try to understand my friends argument that sound, and thus music, is one dimensional. An image consists of a matrix of numbers placed on two axis. In my limited understanding, I would assume sound also has at least two axis, one representing time. My friend makes the claim that there only is one. I tried googling this to no avail. Anyone up for the task?

## Re: Help me understand why sound is one dimensional

##### Reply #1 – 2018-07-15 14:34:41
Yeah that's a weird assertion, since the most common (scientific) representation of music/sound is the waveform, which as you mention, contains time.

Of course the actual question is whether that is the minimum amount of dimensions - this is depending on type of representation. Though it seems intuitively true that an arbitrary piece of sound/music can't be represented by a single value.

## Re: Help me understand why sound is one dimensional

##### Reply #2 – 2018-07-15 14:45:50
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).

## Re: Help me understand why sound is one dimensional

##### Reply #3 – 2018-07-15 18:07:24
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.

## Re: Help me understand why sound is one dimensional

##### Reply #4 – 2018-07-15 18:07:58
If the propagation medium can't be considered a dimension then you can imagine it as a line that changes length over time, now is that 1 + 1 or 2d ... or
PANIC: CPU 1: Cache Error (unrecoverable - dcache data) Eframe = 0x90000000208cf3b8
NOTICE - cpu 0 didn't dump TLB, may be hung

## Re: Help me understand why sound is one dimensional

##### Reply #5 – 2018-07-15 18:30:32
IMHO it is 2 dimensional.
X=time Y=amplitude

## Re: Help me understand why sound is one dimensional

##### Reply #6 – 2018-07-15 18:47:21
IMHO it is 2 dimensional.
X=time Y=amplitude

The function y=f(x) is one dimensional. A two dimensional function would have the form y=f(x1,x2).

## Re: Help me understand why sound is one dimensional

##### Reply #7 – 2018-07-15 19:14:44
A waveform is one dimensional.  Sound in a non-anechoic environment is a different story entirely.  Then there is our perception of sound...
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 #8 – 2018-07-15 19:19:33
I think it's confusing because you look at the graph of the function, that is a two dimensional image.

Perhaps it helps to look at it these ways:
1. What you measure is the amplitude. To help you see the change over time you draw the graph. But if you move all your measurements over each other (compress the time axis to a point) you'll see that you only get the amplitude in one dimension.

2. If you want to convert analog sound to digital you only need a 1d DAC because that one dimension is enough to capture the sound.

3. With the "sound is two dimensional" argumentation you'll have that you watch 3d TV as you have a two dimensional picture that changes over time. You could even argue that the picture on the wall is three dimensional.

The first time I stumbled over this was a one dimensional potential well in physics in school. It took quite some time to wrap my head around this. Why it's only one dimensional although it looks so two dimensional

## Re: Help me understand why sound is one dimensional

##### Reply #9 – 2018-07-15 21:05:39
Think of it this way: which way do your eardrums move?  Is it back and forth, or do they also move side to side?
Ed Seedhouse
VA7SDH

## Re: Help me understand why sound is one dimensional

##### Reply #10 – 2018-07-15 21:17:08
Quote
I would assume sound also has at least two axis, one representing time. My friend makes the claim that there only is one.
Did you ask your friend about time?    Sound is an air-pressure wave* and you can't have a wave without time.

* A philosopher would say sound is the perception of an air-pressure wave.    You know... that old question, "If a tree fall in the forest and no one is there to hear it, does it make a sound?"     But, I'm more practical...   I say heck yes!    I put a tape recorder in the forest and I recorded the sound!

## Re: Help me understand why sound is one dimensional

##### Reply #11 – 2018-07-15 22:48:32
An image consists of a matrix of numbers placed on two axis. In my limited understanding, I would assume sound also has at least two axis, one representing time.
If that is your idea of an image, then I would say that a point sound source - or a "point" as a model for an eardrum - would be 0D rather tham 1D ...
But rather than claiming "0D", I would say that your model of an "image" might be wrong or at least not in line with your model of soumd.
Each point in the image carries a compound of (time-) frequencies. So: if you insist on "time" in a sound waveform, why don't you insist on time in the light waveform?

There are at least two answers to that latter question. 1: In how the human eye projects colour down to a triplet. But that is how humans work, not what is emitted. 2: In that you think that sound changes over time; "music", not just "chord". But then the analogy should be motion picture rather than image.
Memento: this is Hydrogenaudio. Do not assume good faith.

## Re: Help me understand why sound is one dimensional

##### Reply #12 – 2018-07-15 23:54:55
Any representation of data which consists of 2 coordinates for each data point is 2-dimensional. In order to uniquely specify a point, you must give 2 coordinates or there would be confusion.  A signal waveform must have 2 coordinates, or else the numerous times a signal has the same amplitude would be indistinguishable from one another.

A 1-dimensional data representation would have only the single number to unambiguously specify each point. One example of this would be the x- or y-coordinate axis by itself.  A 1-dimensional representation of data is almost useless by itself; it only becomes useful when used as a reference for another data set (value on a number line).  Once it is referenced to 2 other data sets (such as the x and y coordinates of a signal existing in time) it is then a 2-dimensional data set.

Your math professor friend is either confused, or is confusing you with his explanation.

## Re: Help me understand why sound is one dimensional

##### Reply #13 – 2018-07-16 03:08:22
Yes, in terms of waveforms signals in the time domain are one dimensional.

If you think of a discrete-time signal - which I assume you're more familiar with - for each sample in a stream of samples, each one has a value. So, your projection function is s(x) -> y, where s(x) is your sample list, x the sample, and y the sample value at point x.

A two-dimensional plot of a function like this. i.e. mapping y for every x, is a mapping. Mappings may or may not produce additional "dimensions" for perception's sake.

If you think of an image, you're thinking of a matrix of values, each corresponding to a pixel (discussing this in terms of bitmaps is a bit easier, here). So, each column (we'll call that x) and each row (we'll call that y), maps to a color (let's call that z).

Assuming we have a grayscale picture, you can map each (x,y) point in that matrix to its pixel value as brightness, giving you that image. However you might think of it as a contour plot, mapping each value of each (x,y) point on axis z, that would result in a 3-dimensional plot. Both are valid, however depending on how we understand the mapping, he might get an image in one instance, or a 3-dimensional plot in another.

Now, let's take a step back from discrete-time signals. I'm assuming you're familiar with the sinusoidal function sin(x).
now, let's define a function, which is quite simply: f(x) = sin(x). Now, as we can see from the function definition and the function signature, that it takes only one variable, in this case x. The return value of f(x), we might call y, giving us: y = sin(x). We can now plot this function on a 2D plane, but notice what we're doing here: we're plotting the input and output values of a one-dimensional function onto a 2d plane!

A two-dimensional function might look like this: f(x,y) = sin(x) * y. If we want we can assign the return value of f(x,y) to z, we can think of x, y, and z as coordinates in a 3D plot, however, that depends on how we want to map that function. It is important to note, that both our first and this second example, the function map to a scalar value, however this is not always necessarily a requirement.

Things like FFT, returns a two-dimensional number for each set of input numbers.

Harking back to the one-dimensional-ness of a discrete-time signal, like PCM, we can think of the entire audio file or whatever, as a string of values, like a list. Each item in that list can be addressed by only one coordinate: it's time index. Say our audio chunk is one hundred samples long, and lets call that list 's'. Now, let's use some list/array notation here, so with something like s{x} I can get the value of s at position x. To plot each value, we might say: y = s{x}, so each point x in s is returned and plotted on the y axis. Note, that we're not addressing that point with x and y, y is merely the result or value at position x! Instead we address the value of y ONLY with x! Furthermore (the f(x) = sin(x) being a good example here), note that for each input of x, we get a defined and single result. However, the result would return an array of values if we'd try to map in reverse!

If you have further questions, feel free to ask further questions.

Any representation of data which consists of 2 coordinates for each data point is 2-dimensional. In order to uniquely specify a point, you must give 2 coordinates or there would be confusion.
Incorrect.
Case in point: f(x) = sin(x)
You're confusing coordinate and return value. The return value is defined by its coordinate, which in case of a simple sinusoid, is only one-dimensional. You can map input coordinates and its return values onto a 2D plane, sure, but this doesn't make the function two-dimensional.

A signal waveform must have 2 coordinates, or else the numerous times a signal has the same amplitude would be indistinguishable from one another.
Incorrect.
Case in point: f(x) = sin(x)
Periodic functions like sinusoids, are perfectly fine having the same return value every 2π. While the function has periodically the same return value, doesn't mean the signal or whatever other function is invalid. In case of continuous-time domain signals, this is pretty much the only way this actually works. However even lower order functions express this behaviour: f(x) = x², returns 2 for both x = 2 and x = -2. Yet, we can trivially plot these functions on a 2D plane.

A 1-dimensional data representation would have only the single number to unambiguously specify each point. One example of this would be the x- or y-coordinate axis by itself.  A 1-dimensional representation of data is almost useless by itself; it only becomes useful when used as a reference for another data set (value on a number line).  Once it is referenced to 2 other data sets (such as the x and y coordinates of a signal existing in time) it is then a 2-dimensional data set.
Incorrect.
Another simple way of thinking of 1-dimensional structures is strings:
Let's define the string S = "Hello, World". Let's further assume, that the first value starts with 0.
In that case I can access each letter by only one dimension: S{0} = 'H', S{1} = 'e', S{2} = 'l', S{3} = 'l', and so forth.
Note that even though S{2} and S{3} have the same return value (both times 'l') this doesn't invalidate the data structure being one-dimensional.

Your math professor friend is either confused, or is confusing you with his explanation.
No, it makes perfect sense, and is pretty much defined in every single school-type math book, like that. I believe the confusion is down to understand mappings and representations, and actual dimensions of a function.

You can also plot a 3D image on a 2D plane (such as your monitor rendering a 3D object in a game), it does work, because we can either use projection (in that case a rendering pipeline), or some other means (like mapping f(x) = x² onto a piece of paper). Drawing a 3D cube on a piece of paper, doesn't make the cube 2D, neither does drawing the function f(x) = x² as y = x² onto a piece of paper two-dimensional.

## Re: Help me understand why sound is one dimensional

##### Reply #14 – 2018-07-16 04:22:08
If that is your idea of an image, then I would say that a point sound source - or a "point" as a model for an eardrum - would be 0D rather tham 1D ...
But rather than claiming "0D", I would say that your model of an "image" might be wrong or at least not in line with your model of soumd.
Each point in the image carries a compound of (time-) frequencies. So: if you insist on "time" in a sound waveform, why don't you insist on time in the light waveform?

There are at least two answers to that latter question. 1: In how the human eye projects colour down to a triplet. But that is how humans work, not what is emitted. 2: In that you think that sound changes over time; "music", not just "chord". But then the analogy should be motion picture rather than image.
I'm not sure this is helping, but that aside, the "0D" is a bit conflated in mathematics, rather if you think of something having no dimension, they're simply non-dimensional, or: scalar. A point has no dimensional attributes. A point might be addressed by coordinates and return a scalar, or higher-dimension value.

For instance you can map one 2D space into another, a common example is the conversion of polar coordinates into Cartesian coordinates, and back. Cartesian coordinates are points defined by x and y, while polar coordinates are defined by r and θ (where θ is an angle).

To convert from polar to Cartesian, you'd do: f(r, θ) = {r * cos(θ), y * sin(θ)} → {x, y}
To convert from Cartesian to polar, you'd do: g(x, y) = {sqrt(x² + y²), atan(y/x)} → {r, θ}
I.e. both functions take two values, and return two values, one 2D point, returns a 2D number.

In terms of an RGB color bitmap image, you could say that each x and y coordinate for each pixel, returns three values: r, g, b. Of course we can map this number onto a linear scale (since most are limited for all color spaces), but theoretically the color plane is infinite, and cannot be linearized like we do in a fixed color gamut, like 24-bit color, etc. So in these terms, the pixel coordinates in a color picture, return a three-dimensional number value. In case we have a grayscale image, where each pixel is just one number, each pixel coordinates return a scalar value.

Each higher-order value, can be composed of an arbitrary number of dimensions, including scalar. In cases of an RGB color image, each two dimensional pixel coordinate, of which each component is scalar, maps to a three-dimensional value, where each component of that value is a scalar as well.

A point has no length or area or volume. A single point only defines itself. A line is defined by at least two points in n-dimensional space, it may have a length, but no area or volume. A plane is defined by at least three points, which are not on the same gradient as the other two. Planes may have areas, but no volume. And finally a space needs at least four points, etc.
Higher order objects also exist, things like hypercubes, in 4D space, etc. Anything of a higher order than a point, is a set of points.

I believe this is kinda where the confusion of Op comes from. Plotting a waveform is essentially a function that maps all values of a 1-dimensional discrete function into a 2-dimensional discrete plane, where each valid point in the mapped function is assigned one color, and each invalid point no color (background).

Having said that, the statement "Sound is one-dimensional" is incredibly ambiguous. In terms of signal definitions it is, but in terms of propagation in space it isn't. So, yeah...

## Re: Help me understand why sound is one dimensional

##### Reply #15 – 2018-07-16 04:24:46
f(x) = x², returns 2 for both x = 2 and x = -2
It returns 4 damnit! It's getting late...

## Re: Help me understand why sound is one dimensional

##### Reply #16 – 2018-07-16 05:07:02
Or put very simply:

A linear array of data has one dimension as it requires only a single number to specify which piece of data is being considered. The data contained within the array is irrelevant to the dimensionality of the array itself: It could be a sequence of air pressure measurements (aka sound,) 3D models for a game, video files, forum posts, or a mix of any and all of that. All that matters is that each piece of data is referenced by a single number.

## Re: Help me understand why sound is one dimensional

##### Reply #17 – 2018-07-16 06:46:55
Thanks for all the feedback. I really appreciate people taking the time to try to make sense of this. Unfortunately, I dont have the necessary mathematical background to understand a lot of the math equations, even though I have tried my best in this case. Also, good to see that its not just me who finds this confusing. Thanks for the lengthy explanation, polemon. As I clearly am not very math-inclined, I think both your explanation in plain English as well as Rotareneg (and Rumbah) makes sense to me here,

Or put very simply:

A linear array of data has one dimension as it requires only a single number to specify which piece of data is being considered. The data contained within the array is irrelevant to the dimensionality of the array itself:
.

The initial conversation started by my friend pointing out that "its interesting how we are so much better at frequency separation using our ears then through vision. Even though sound is one dimensional, we are still better at separating two different sounds (like two different notes or instruments) then we are at two different frequencies (or wavelengths (I dont know if they are comparable and can be used interchanged)) through vision, as they merge to form a separate color." My position was that we indeed are good at separating wavelengths into individual colors, but he disagreed. But thats for a different topic, I suppose. Also, this was after a few beers so I might be quoting it incorrectly. Sounds like a fun conversation to have at a music festival, right?

So to sum up how I understand it... Sound is considered one dimensional since it at any given point in time only can have one value.  Sorry if this became an ELI5-type of situation, but I am glad to see that this has started a discussion among others.

## Re: Help me understand why sound is one dimensional

##### Reply #18 – 2018-07-16 07:51:28
The initial conversation started by my friend pointing out that "its interesting how we are so much better at frequency separation using our ears then through vision. Even though sound is one dimensional, we are still better at separating two different sounds (like two different notes or instruments) then we are at two different frequencies (or wavelengths (I dont know if they are comparable and can be used interchanged)) through vision, as they merge to form a separate color." My position was that we indeed are good at separating wavelengths into individual colors, but he disagreed. But thats for a different topic, I suppose. Also, this was after a few beers so I might be quoting it incorrectly. Sounds like a fun conversation to have at a music festival, right?

So to sum up how I understand it... Sound is considered one dimensional since it at any given point in time only can have one value.  Sorry if this became an ELI5-type of situation, but I am glad to see that this has started a discussion among others.
It is considered one-dimensional because it is a one-dimensional function. I.e. the variable determining the value (amplitude or level) is referenced by only one, scalar value, in our case time. And as we all know, time is one-dimensional, in our perception of the world, anyhow.

What your friend says however is true when it comes to frequency separation, between vision and hearing. However, we have to consider the spectrum. We have to consider these as a fraction of our aural or visual frame. Also, human hearing and vision isn't linear. A change in tone between 50Hz and 60Hz is quite noticeable, while a change between 3000Hz and 3010Hz isn't.

It's also important to note, that it's quite difficult to discern mixed signals in aurally. Add 1kHz, 1.1kHz, and 1.2kHz, and people will have a hard time, separating these three frequencies from a single sound chunk. However in vision, our brain has only three different frequencies to work with, and mixes them rather nicely to create a color gamut. So much so, that it's easier for us to describe a color by it's hue, brightness, and saturation, rather than their red, green, and blue color values. Also, our vision isn't linear either. Our perception of blue is much weaker than green and red. Also, resolution also doesn't line up nicely either. Most of our highest sharpness is in the green. Also, the the spectral width as well as the distance between the frequency responses isn't linear or equal either. The frequency responses of the blue and green cone cells are much closer to each other, than the green and red cone cells. And to make matters worse, our perception in low-light condition changes yet again, this is because rod-cells tend to respond to blue-ish light more than further down the spectrum. In low light conditions, we see "better" with green-blue-ish light, while in bright light conditions, we see better in green/reddish light.

## Re: Help me understand why sound is one dimensional

##### Reply #19 – 2018-07-16 11:40:03
Let me throw in another viewing angle just to confuse things a bit further.

Sound is a pressure wave in space. That's its physical definition. Since space is three-dimensional, and wave propagation introduces time, which is one more dimension, we could say that sound is four-dimensional. In mathematical terms, it is a function of four variables, whose value is the pressure at an arbitrary point in this four-dimensional space. The variables are x, y and z for the space coordinates of the point, and t for the time.

This doesn't yet take perception into account. Human sound perception works by having two pressure sensors (the ear drums) at some place within the space where the sound happens. For each ear, this means that x, y and z are fixed, and t remains the only free variable. Effectively, for one ear, sound becomes a one-dimensional function, whose only dimension is time.

Having two ears, then, means that perceived sound is two one-dimensional functions. Or, since the variable t is the same for both, one could say that perceived sound is a one-dimensional function with two results.

It is actually a bit more complicated than that, since part of human perception relies on the movement of the head while listening. One can get some subconscious clues by observing the changes in perception when the head position changes. This means that x, y and z aren't really fixed, but change with time. But we can ignore this most of the time.

## Re: Help me understand why sound is one dimensional

##### Reply #20 – 2018-07-16 14:12:54
Though it seems intuitively true that an arbitrary piece of sound/music can't be represented by a single value.
If you only look for finite precision, then anything at all can be represented by a single value, even a video.
Any file (its contents, to be precise) on a filesystem is just a single number with a lot of digits.
Keep calm and opusenc --bitrate 128

## Re: Help me understand why sound is one dimensional

##### Reply #21 – 2018-07-16 20:06:33
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.

## Re: Help me understand why sound is one dimensional

##### Reply #22 – 2018-07-16 20:31:25
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,

The discrete Fourier transform of a 1D function is also one dimensional. This includes the Fourier transform of audio.

## Re: Help me understand why sound is one dimensional

##### Reply #23 – 2018-07-16 22:03:46
Ok, to be more specific: a short-time Fourier transform, as used to generate a spectrogram, is 2D.

## Re: Help me understand why sound is one dimensional

##### Reply #24 – 2018-07-16 22:12:49
Ok, to be more specific: a short-time Fourier transform, as used to generate a spectrogram, is 2D.

The short time Fourier transform of audio is a way to make a 2D image by tiling the 1D frequency domain data into a picture. Creating a 2, 3, 4... N-D picture of the 1D frequency domain vector doesn't make it 2D or anything else. It just means you made a picture.