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?
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.
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).
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.
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
IMHO it is 2 dimensional.
X=time Y=amplitude
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).
A waveform is one dimensional. Sound in a non-anechoic environment is a different story entirely. Then there is our perception of sound...
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 :)
Think of it this way: which way do your eardrums move? Is it back and forth, or do they also move side to side?
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!
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.
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.
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.
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...
f(x) = x², returns 2 for both x = 2 and x = -2
It returns
4 damnit! It's getting late...
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.
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.
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.
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.
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.
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
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.
Ok, to be more specific: a short-time Fourier transform, as used to generate a spectrogram, is 2D.
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.
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.
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.
I looked-up "dimension" and I found
this (http://thinkmath.edc.org/resources/glossary/dimension):
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.
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!
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?
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?
I looked-up "dimension" and I found this (http://thinkmath.edc.org/resources/glossary/dimension):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.
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?
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?
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.
Sound implies the ability to hear, so we're back to where we started.
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.
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.
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.
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?
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.
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.
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
https://www.youtube.com/watch?v=spUNpyF58BY <-- it even uses sound as examples.
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?)
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.
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.
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.
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
space is not a dimension ?
space is not a dimension ?
A location in space can be described with three dimensions, and space/time (in certain circumstances) 4 dimensions. However, the number of dimensions for a sound pressure measurement is scalar pressure + time. One can mix multiple pressure measurements (along with other intelligence in the data -- incl spatial location of the measurements), then allow the sound to help describe the environment (including the source locations.) This description of the environment is what sonar does (and how the brain calculates the spatial relationships.)
John
space is not a dimension ?
A location in space can be described with three dimensions, and space/time (in certain circumstances) 4 dimensions. However, the number of dimensions for a sound pressure measurement is scalar pressure + time. One can mix multiple pressure measurements (along with other intelligence in the data -- incl spatial location of the measurements), then allow the sound to help describe the environment (including the source locations.) This description of the environment is what sonar does (and how the brain calculates the spatial relationships.)
John
Are you a vacuum cleaner designer ?
(The smallish high RPM motors embedded in the dyson appliances are horribly noisy and is a real pain to endure)
A mathematician should think about the Inverse-square law, no ?
space is not a dimension ?
A location in space can be described with three dimensions, and space/time (in certain circumstances) 4 dimensions. However, the number of dimensions for a sound pressure measurement is scalar pressure + time. One can mix multiple pressure measurements (along with other intelligence in the data -- incl spatial location of the measurements), then allow the sound to help describe the environment (including the source locations.) This description of the environment is what sonar does (and how the brain calculates the spatial relationships.)
John
Are you a vacuum cleaner designer ?
(The smallish high RPM motors embedded in the dyson appliances are horribly noisy and is a real pain to endure)
A mathematician should think about the Inverse-square law, no ?
Not directly related to Sir James Dyson, however I did recently mention his gathering of the dyson.com domain early in the Internet, therby frustrating my own ability to get it back then :-). I am of much less noteriety -- only writing a part of the early FreeBSD kernel code and doing various interesting things at Bell Labs many years ago. Most recently, working on audio processors including a fully functional DolbyA decoder -- (some interesting things now happening in that realm.) Basically, just a geek.
Regarding the inverse-square law -- that can certainly be part of the calculation used to help determine information about the environment given multiple scalar sound measurements vs. time.
John
environment given multiple scalar sound measurements vs. time.
John
I don't understand "scalar" part of the sentence, you mean that we can vectorize the sound propagation with a microphone array... how ?
It can only be performed in the far field IMO (if it is possible)
environment given multiple scalar sound measurements vs. time.
John
I don't understand "scalar" part of the sentence, you mean that we can vectorize the sound propagation with a microphone array... how ?
It can only be performed in the far field IMO (if it is possible)
I am not saying how (by what method), except that information about the environment can be inferred by using multiple sensors. A single sensor with no other information would only provide 1 dimension of data. Perhaps the distance between two impulses can be measured -- but what is the reference point of the two? SO, one sensor basically gives you the ability to detect sound at that point only (no directional information.) With multiiple sensors, things like reflections and signal intensity can start to be used to guess better at directions. I am not claiming exactly what kind of data about the environment that can be figured out. For example, a lot of information can be figured out about earthquakes by simply listening -- not even pinging of any kind. By using more sensors -- more kinds of information (dimensions) can possibly be derived. By explicitly adding a controlled source of some kind, even more information might be able to be obtained. (or, maybe with a controlled source of some kind, the precision might be greater.)
IMHO the 1D point of view don't conceptualize the reality because all audio measurements are performed by people who have the sound field in mind.
IMHO the 1D point of view don't conceptualize the reality because all audio measurements are performed by people who have the sound field in mind.
If a person has a picture of what they expect the sound field to be -- and then uses a singular real data point, aren't they really sustituting 'fake' (not necessarily invalid) effective data points for real ones? No matter what you are imagining, there has to be some kind of framework -- and in space, there are normally 3 dimensions. One can abstractly create more dimensions for some kind of analysis purpose, but in reality there are three dimensions in space. Adding more dimensions isn't just describing space, but rather some kind of hyperspace (no scifi here.)
John
If a person has a picture of what they expect the sound field to be -- and then uses a singular real data point, aren't they really sustituting 'fake' (not necessarily invalid) effective data points for real ones?
The more data points the better which, as I recall, is kind of why we moved from mono to stereo back in the 1950's.
But please note that your ears have only two data points, yet you are able to hear a three dimensional sound field. If two ears can do it, why can't two microphones?
If a person has a picture of what they expect the sound field to be -- and then uses a singular real data point, aren't they really sustituting 'fake' (not necessarily invalid) effective data points for real ones?
The more data points the better which, as I recall, is kind of why we moved from mono to stereo back in the 1950's.
But please note that your ears have only two data points, yet you are able to hear a three dimensional sound field. If two ears can do it, why can't two microphones?
Before reading -- note that there is a difference between 'HEARING' (includes the mental, head and ear characteristics) and 'SOUND' which is the pressure measurement thing.
Two microphones can do it just like two ears -- there is more information than just the simple math/physics of the sound field itself. The brain can process directional hints like frequency response associated with outside ear/head shape, and also use mental imaging.
So, if you are talking about a computer without the brain power of the human/animal brain, then we are stuck with the the simple physics and a priori 'guesses' (based upon expected environment.)
With all of the actual physics (without any mental imaging), plus mental imaging available ot humans, there can be sythesized information that goes beyond the simple physics. However, to be intellectually honest -- that information that the human brain can synthesize, the ability to use the shape/structure of the ears' environment and the hard-core physics of the environment -- that is GREATER than just the two sensors. That is why more information can be gleaned.
So, one has to ask, are they talking about 'SOUND', or are they talking about 'HEARING'. Hearing can gather more information (whether physically real or not), than just SOUND.
Those talking about 'imaging' beyond what can be gathered by math calculations based just on the two point sensors are really talking about HEARING -- not just SOUND. This is probably where the confusion happens.
John
However, to be intellectually honest -- that information that the human brain can synthesize, the ability to use the shape/structure of the ears' environment and the hard-core physics of the environment -- that is GREATER than just the two sensors. That is why more information can be gleaned.
Right. You could put it this way: There are a lot of assumptions built into human hearing. Some of them are learned, others are "wired in". This is a reason why hearing can be fooled and occasionally produces wrong impressions.
Stereo only works because of this. A phantom source isn't anything physical. There is no phantom source in the sound field. It is the post-processing of the signal from our two sensors (ears), which creates (the impression of) the phantom source.
You could say that with stereo, human hearing gets it consistently wrong. Stereo is a way of using a specific deficiency of human hearing to fool it into hearing a 3D sound field that isn't there. We all know that this illusion breaks down easily (sweet spot etc.).
However, to be intellectually honest -- that information that the human brain can synthesize, the ability to use the shape/structure of the ears' environment and the hard-core physics of the environment -- that is GREATER than just the two sensors. That is why more information can be gleaned.
Right. You could put it this way: There are a lot of assumptions built into human hearing. Some of them are learned, others are "wired in". This is a reason why hearing can be fooled and occasionally produces wrong impressions.
Stereo only works because of this. A phantom source isn't anything physical. There is no phantom source in the sound field. It is the post-processing of the signal from our two sensors (ears), which creates (the impression of) the phantom source.
You could say that with stereo, human hearing gets it consistently wrong. Stereo is a way of using a specific deficiency of human hearing to fool it into hearing a 3D sound field that isn't there. We all know that this illusion breaks down easily (sweet spot etc.).
Good -- that is the kind of point that I was trying to make. We all (including me) have this trouble of seperating real (simple) physics from what our minds can do. Normally, to me, the real world is how I perceive it -- but that is actually not true. The same kind of thing applies to audio perception.
All of this said - even though we have clarified the difference between HEARING and SOUND, for the purposes of audio ENJOYMENT, hearing is more of the goal than sound. Sound (I mean, the physics and technology to reproduce) is important -- making a lot of hearing enjoyment possible, so we cannot discredit either.
Where we get in trouble is doing the reverse of 'Sound directly helping to make hearing work' -- which is valid.
Trying to describe the real world (reality) by depending solely on what one simply hears can cause confusion -- as you implied above, perceptual errors.
It is so very nice that English (and probably a lot of other languages) allow us to distinguish between SOUND (the physics) and HEARING (what our brains perceive.) If we don't distinguish between the two, audiophile-land can be very confusing, and is probably a major reason for all of the 'audio-religion', and wine-tasting-esque descriptions of audio out there. There are real technical descriptions available, but until the idea of SOUND and HEARING are seperated, then a lot of people are going to stay confused, but not know it.
John
If we don't distinguish between the two, audiophile-land can be very confusing, and is probably a major reason for all of the 'audio-religion', and wine-tasting-esque descriptions of audio out there. There are real technical descriptions available, but until the idea of SOUND and HEARING are seperated, then a lot of people are going to stay confused, but not know it.
Quite. In my view, the inability (and unwillingness) to acknowledge the role of one's own brain in what they perceive, is among the most prominent reasons for their cluelessnes. It is getting fuelled by the pervasive style of gear reviews, where the item that is reviewed is being put into the active position. Anything that people allegedly hear is said to be the "work" of the item tested. In this manner, even benign pieces of kit like connectors have a "sonic signature", affect spatial perception in mysterious ways (i.e. put instruments in certain places), and such hogwash. It is basically a projection. Something that is a function of one's brain is projected onto a piece of kit.
In theory, if people were responsible and rational, they would judge their gear according to what it does to the signal, not what it does to the hearing. But it would require a level of technical sophistication in the audience that is just unrealistic.
IMHO the 1D point of view don't conceptualize the reality because all audio measurements are performed by people who have the sound field in mind.
If a person has a picture of what they expect the sound field to be -- and then uses a singular real data point, aren't they really sustituting 'fake' (not necessarily invalid) effective data points for real ones? No matter what you are imagining, there has to be some kind of framework -- and in space, there are normally 3 dimensions. One can abstractly create more dimensions for some kind of analysis purpose, but in reality there are three dimensions in space. Adding more dimensions isn't just describing space, but rather some kind of hyperspace (no scifi here.)
John
We have a tension, an intensity and time at the input of the loudspeaker coil, we can also try to link the voltage and the intensity through an imaginary thing called phase... also linked with the coil motion... and the air motion... abstraction is infinitely complex.
I only matter about what i'm imagining when i perform a measurement, the loudspeaker surface is radiating within two modes, a pistonic and radiative motion that generate a complex soundfield composed of a cloud of air of physical elementary particles moving relatively coherentely with the emissive surfaces when you are close to them and doing unpredictible things when you get farther (everyting is also highly impacted by the signals)
You should also add all the reflexions and the air volume when you are in a room, as you are probably awared of expertise is everthing and a clever measurement can validate a very complex theory.
If we don't matter about how many dimensions to use, could you say if there is a way to beat a human brain in this field of competence ?
... i'm not as intelligent as Garry Kasparov, but deeper blue is still biting the dust.
IMHO the 1D point of view don't conceptualize the reality because all audio measurements are performed by people who have the sound field in mind.
If a person has a picture of what they expect the sound field to be -- and then uses a singular real data point, aren't they really sustituting 'fake' (not necessarily invalid) effective data points for real ones? No matter what you are imagining, there has to be some kind of framework -- and in space, there are normally 3 dimensions. One can abstractly create more dimensions for some kind of analysis purpose, but in reality there are three dimensions in space. Adding more dimensions isn't just describing space, but rather some kind of hyperspace (no scifi here.)
John
We have a tension, an intensity and time at the input of the loudspeaker coil, we can also try to link the voltage and the intensity through an imaginary thing called phase... also linked with the coil motion... and the air motion... abstraction is infinitely complex.
I only matter about what i'm imagining when i perform a measurement, the loudspeaker surface is radiating within two modes, a pistonic and radiative motion that generate a complex soundfield composed of a cloud of air of physical elementary particles moving relatively coherentely with the emissive surfaces when you are close to them and doing unpredictible things when you get farther (everyting is also highly impacted by the signals)
You should also add all the reflexions and the air volume when you are in a room, as you are probably awared of expertise is everthing and a clever measurement can validate a very complex theory.
If we don't matter about how many dimensions to use, could you say if there is a way to beat a human brain in this field of competence ?
... i'm not as intelligent as Garry Kasparov, but deeper blue is still biting the dust.
All of the physical stuff being described can be modeled by a competent mechanical/electrical engineering team. Modeled so accurately that a 'virtual' copy of the speaker can likely predict the behavior incredibly accurately (including the enviornment.) All of the things like 'standing waves' on a speaker diaphragm are 'old hat', and can be predicted fairly well.
Where we do have troubles is modeling exactly what the human hearing system (incl internal/external ears, head, brain, etc) perceives. We can predict the frequency resp/phase relasionships of what enters the ear, and even model some of the ear, but that is as far as it goes. Beyond this level is where the 'audiophile' mode becomes much more valid.
For example, I cannot predict what you will hear, and how you will react to a given stimulus. Will it seem 'pretty' to you or seem 'ugly'? I don't know, and won't even attempt to scientificially predict. (I can certainly guess, however :-)).
As an engineer, all I can claim is to be able to parametrically describe or create a design (either of my own creation or someone else.) When in a field where I am competent (and it is NOT speakers), I can create a design that is almost as perfect as the situation (costs, parts availability, software tools, CPU capability) allows. (I am both a full EE, DSP person and operating systems software designer -- all with at least 3 decades experience and successful.) BUT, I'd suck at speaker or microphone design -- I know the general physics, but not the techniques, technology or have any experience.
So real engineers can do a lot, but cannot predict what people can really perceive. I know my limitations, and happily admit them -- otherwise, cannot be credible. I am an engineer, not a able to do a mind meld to really understand what people feel/think.
John
All of the physical stuff being described can be modeled by a competent mechanical/electrical engineering team. Modeled so accurately that a 'virtual' copy of the speaker can likely predict the behavior incredibly accurately (including the enviornment.) All of the things like 'standing waves' on a speaker diaphragm are 'old hat', and can be predicted fairly well.
It is perhaps time to add a new knowledge to your personal culture... it hadn't been done yet, i would be very pleased to see one of that kind of study before my death.
In an effort to stay in touch with the topic, i can say that the sound propagation in our environement (air, room and objects) is infinitely complex and unpredictible by nature, and the 1D point of view is a good explanation for the children but not for an adult.
All of the physical stuff being described can be modeled by a competent mechanical/electrical engineering team. Modeled so accurately that a 'virtual' copy of the speaker can likely predict the behavior incredibly accurately (including the enviornment.) All of the things like 'standing waves' on a speaker diaphragm are 'old hat', and can be predicted fairly well.
It is perhaps time to add a new knowledge to your personal culture... it hadn't been done yet, i would be very pleased to see one of that kind of study before my death.
In an effort to stay in touch with the topic, i can say that the sound propagation in our environement (air, room and objects) is infinitely complex and unpredictible by nature, and the 1D point of view is a good explanation for the children but not for an adult.
The environment can also be included in models. But, at a certain point, it is silly to go into more detail. There are so many kinds of environments, that a generally applicable design will work well.
This is all about the nonsensical 'audiophile' reasoning (and pseudo-wine tasting language) vs. the real state of engineering and research. There are still alot of snake oil salesemen selling inferior product with 'personality' vs. very accurate equipment.
I vote on the side of real, accurate engineering vs. the overly expensive snake oil. Too many people without real engineering backgrounds get taken in by snake oil marketing.
In some cases, some of the snake oil marketing IS backed by sound engineering, but that is really kind of sad. People are being dys-educated into being idiots in the guise of pseudo-tech.
Real world engineering thinking and work is where the real advances are made (or, in the case of a lot of legacy technologies -- e.g. most audio stuff), already have been made. Please refer to the rather curious love of vacuum tubes (definitely legacy, but cute technology.) It is damned hard to get really good distortion and bandwidth on tube gear, but they try (barely succeeding) -- and some make money at it, esp with the snake oil behind it. (I started designing with 'tubes' years ago -- I know what I am talking about.)
My guess is that because audio (in almost all ways) has advanced almost as far as it can -- some of the marketeers who take advantage of the non-engineering-knowing people and instead look backwards. Sure -- 'adequate' quality can be had with 'tubes', but why do it? Marketeering (coined word -- marketing based upon feeling instead of fact, with a bit of hogwash added in), knowing that selling goods based upon technical advantage is now nearly specious (given competent, up-to-date design.)
John
The environment can also be included in models. But, at a certain point, it is silly to go into more detail. There are so many kinds of environments, that a generally applicable design will work well.
All good measuring softwares that i've tested are unable to corellate the measurements and the theory, the calculated room modes are always wrong.
The devil is in the details, and the lack of these details is killing the math models accuracy.
This is all about the nonsensical 'audiophile' reasoning (and pseudo-wine tasting language) vs. the real state of engineering and research. There are still alot of snake oil salesemen selling inferior product with 'personality' vs. very accurate equipment.
I vote on the side of real, accurate engineering vs. the overly expensive snake oil. Too many people without real engineering backgrounds get taken in by snake oil marketing.
In some cases, some of the snake oil marketing IS backed by sound engineering, but that is really kind of sad. People are being dys-educated into being idiots in the guise of pseudo-tech.
Real world engineering thinking and work is where the real advances are made (or, in the case of a lot of legacy technologies -- e.g. most audio stuff), already have been made. Please refer to the rather curious love of vacuum tubes (definitely legacy, but cute technology.) It is damned hard to get really good distortion and bandwidth on tube gear, but they try (barely succeeding) -- and some make money at it, esp with the snake oil behind it. (I started designing with 'tubes' years ago -- I know what I am talking about.)
My guess is that because audio (in almost all ways) has advanced almost as far as it can -- some of the marketeers who take advantage of the non-engineering-knowing people and instead look backwards. Sure -- 'adequate' quality can be had with 'tubes', but why do it? Markettering (coined word -- marketing based upon feeling instead of fact), knowing that selling goods based upon technical advantage is now nearly specious (given competent, up-to-date design.)
John
The only way to go is to calculate and measure by yourself IMHO, the market is saturated of junk product and the good ones are also junk when used inappropriately.
The environment can also be included in models. But, at a certain point, it is silly to go into more detail. There are so many kinds of environments, that a generally applicable design will work well.
All good measuring softwares that i've tested are unable to corellate the measurements and the theory, the calculated room modes are always wrong.
The devil is in the details, and the lack of these details is killing the math models accuracy.
This is all about the nonsensical 'audiophile' reasoning (and pseudo-wine tasting language) vs. the real state of engineering and research. There are still alot of snake oil salesemen selling inferior product with 'personality' vs. very accurate equipment.
I vote on the side of real, accurate engineering vs. the overly expensive snake oil. Too many people without real engineering backgrounds get taken in by snake oil marketing.
In some cases, some of the snake oil marketing IS backed by sound engineering, but that is really kind of sad. People are being dys-educated into being idiots in the guise of pseudo-tech.
Real world engineering thinking and work is where the real advances are made (or, in the case of a lot of legacy technologies -- e.g. most audio stuff), already have been made. Please refer to the rather curious love of vacuum tubes (definitely legacy, but cute technology.) It is damned hard to get really good distortion and bandwidth on tube gear, but they try (barely succeeding) -- and some make money at it, esp with the snake oil behind it. (I started designing with 'tubes' years ago -- I know what I am talking about.)
My guess is that because audio (in almost all ways) has advanced almost as far as it can -- some of the marketeers who take advantage of the non-engineering-knowing people and instead look backwards. Sure -- 'adequate' quality can be had with 'tubes', but why do it? Markettering (coined word -- marketing based upon feeling instead of fact), knowing that selling goods based upon technical advantage is now nearly specious (given competent, up-to-date design.)
John
The only way to go is to calculate and measure by yourself IMHO, the market is saturated of junk product and the good ones are also junk when used inappropriately.
When it comes down to it, I agree with you. It is probably impossible for even an experienced engineer to be able to 100% a-priori detect real vs. exaggerated. (I won't claim that all of the excessive claims are made about bad equipment -- just exaggerated. Not all wonderful claims are exaggerations either.)
So, IMO - you are 100% correct -- paraphrasing your main point: listen for oneself. Decide for oneself.
John
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.
Well. I tried. But as I initially started out with, its quite complicated. But again, happy to see that that there are other people here who are confused. And also, other people who seem to know what they are talking about, and explains it in a way that makes sense. I am in the strange position to where I am certain he is correct, without initially knowing why. I guess his knowledge in the field of mathematics (and my lack of understanding) creates a situation to where it was hard convey and understand the explanation. Everyone involved here has certainly contributed to both my understanding as well as my initial confusion. :)
You could very well ask him for something to be copied/pasted here. It would surprise me if there is not enough of the (relevant) math skills here to catch and explain (or posstibly rectify) the main point.
https://www.youtube.com/watch?v=KJnAy6hzetw&list=PL41692B571DD0AF9B
clever mustache man mentions the one dimension denomination around 2mn of the video for those who "ain't got no time for that". maybe that helps a little? if only thanks to an appeal to MIT's authority^_^.
The function y=f(x) is one dimensional. A two dimensional function would have the form y=f(x1,x2).
Why not y=f(x,t) ?
Related http://www.feynmanlectures.caltech.edu/I_47.html
edit: I guess I'am struggling between basic physics prototype and math representation of it. And somewhere along the path the word 'sound' seems to loose any meaning.
The function y=f(x) is one dimensional. A two dimensional function would have the form y=f(x1,x2).
Why not y=f(x,t) ?
Related http://www.feynmanlectures.caltech.edu/I_47.html
Strictly, the f(x,t) is indeed two dimensional -- 'f' is a function of two variables, where time is one of them. In this case, there is nothing magical about time. Just as (for example), the sin function in sin(a*t) is also a function whose value is a function of time.
John
I'd like to point out that I think much of the confusion here is regarding the usage of the word "dimension". The graph of a one-variable function y = f(x) is indeed two-dimensional (that is, you need two dimensions to draw the function) even if you only have one independent variable x.
To the OP; you say in your first message, "I would assume sound also has at least two axis, one representing time" -- that is absolutely correct, sound is a wave that propagates through space and time, so you need at least one spatial independent variable and the time independent variable; a third variable also exists here, the pressure, which is the dependent variable, which, when the wave equation is solved, gives the pressure as a function of time and space.
Then you say, "My friend makes the claim that there only is one". Could it be possible your friend is thinking of the typical expansion-compression cycle, like that of a Slinky, which goes back and forth in one spatial dimension but otherwise also advances in time?
As others here have pointed out, it'd be useful to see, verbatim, what your friend says, from which we can attempt to deduce what was meant, and if needed, correct what they said.
I'd like to point out that I think much of the confusion here is regarding the usage of the word "dimension". The graph of a one-variable function y = f(x) is indeed two-dimensional (that is, you need two dimensions to draw the function) even if you only have one independent variable x.
To the OP; you say in your first message, "I would assume sound also has at least two axis, one representing time" -- that is absolutely correct, sound is a wave that propagates through space and time, so you need at least one spatial independent variable and the time independent variable; a third variable also exists here, the pressure, which is the dependent variable, which, when the wave equation is solved, gives the pressure as a function of time and space.
Then you say, "My friend makes the claim that there only is one". Could it be possible your friend is thinking of the typical expansion-compression cycle, like that of a Slinky, which goes back and forth in one spatial dimension but otherwise also advances in time?
As others here have pointed out, it'd be useful to see, verbatim, what your friend says, from which we can attempt to deduce what was meant, and if needed, correct what they said.
This is starting to be a 'how many angels on the head of a pin' or a terminology discussion. One can look at these things from too many directions (or dimensions. :-)). Describing a function in the terms of dimensions is confusing because (as previous poster said), it all depends on the context -- so I wouldn't normally even use the term dimension in this context. In a way, describing a function as having dimensions doens't really make much sense because it is a fuzzy meaning (dimensions regarding what?)
Some inputs to the function might not even be continuous - so how is that described to be a dimension in a common sense way?One can say that space has so many dimensions (plus time), but a function having dimensions depends on what is being talked about -- almost ending up being a meaningless term without LOTS of qualification.
There is probably a pure math definition, and that is probably what should be used -- but it is still confusing (and I use LOTS of complicated functions all of the time!!!)
John
I'd like to point out that I think much of the confusion here is regarding the usage of the word "dimension". The graph of a one-variable function y = f(x) is indeed two-dimensional (that is, you need two dimensions to draw the function) even if you only have one independent variable x.
To the OP; you say in your first message, "I would assume sound also has at least two axis, one representing time" -- that is absolutely correct, sound is a wave that propagates through space and time, so you need at least one spatial independent variable and the time independent variable; a third variable also exists here, the pressure, which is the dependent variable, which, when the wave equation is solved, gives the pressure as a function of time and space.
Then you say, "My friend makes the claim that there only is one". Could it be possible your friend is thinking of the typical expansion-compression cycle, like that of a Slinky, which goes back and forth in one spatial dimension but otherwise also advances in time?
As others here have pointed out, it'd be useful to see, verbatim, what your friend says, from which we can attempt to deduce what was meant, and if needed, correct what they said.
This is starting to be a 'how many angels on the head of a pin' or a terminology discussion. One can look at these things from too many directions (or dimensions. :-)). Describing a function in the terms of dimensions is confusing because (as previous poster said), it all depends on the context -- so I wouldn't normally even use the term dimension in this context. In a way, describing a function as having dimensions doens't really make much sense because it is a fuzzy meaning (dimensions regarding what?) Some inputs to the function might not even be continuous - so how is that described to be a dimension in a common sense way?
One can say that space has so many dimensions (plus time), but a function having dimensions depends on what is being talked about -- almost ending up being a meaningless term without LOTS of qualification.
There is probably a pure math definition, and that is probably what should be used -- but it is still confusing (and I use LOTS of complicated functions all of the time!!!)
John
I agree, using "dimension" for a function itself is confusing, which is why I clarified that usually one speaks of geometrical dimensions, in which the dimensions of the graph of a function does make sense.
But yes, until a more verbatim statement from the OP's friend comes, it's probably best to withhold judgement.
I agree, using "dimension" for a function itself is confusing, which is why I clarified that usually one speaks of geometrical dimensions, in which the dimensions of the graph of a function does make sense.
But yes, until a more verbatim statement from the OP's friend comes, it's probably best to withhold judgement.
The way I understand it, the OP's friend must have referred to time as the one dimension. I can't believe that he wanted to restrict sound to one dimension in space - that would have been sound in a tube, for example.
Regarding sound as one-dimensional makes sense if you look at the sound in one particular location in space, for example the position of a microphone, or an ear. Seen this way, sound is a pressure that varies with time, i.e. a function of time that yields pressure. A one- dimensional function in the mathematical sense. After the microphone, the signal is electrical, but still it is a function of time, i.e. a one-dimensional function of time that yields voltage. In general, once you are in the electronics domain, sound is one-dimensional, since it is an electrical signal that varies with time. (More precisely, the electrical signal is only a representation of sound, not sound itself, but that's splitting hairs).
Note that we can't speak of a wave in this context. A wave is something that propagates through space (in one, two or 3 dimensions). Something that depends only on time doesn't propagate, hence can't be a wave. It can be a sample of a wave at some point in space.
The OP's friend probably wanted to highlight this fundamental difference: There is a soundfield in space, a 3D pressure wave, which is actually a four-dimensional phenomenon because it also varies with time. And there is an electronic representation of sound, which is what all our gear (including our ear) deals with, which is merely one or several electrical signals that vary with time.
This goes to show that recording sound is always a very "lossy" activity. You can't capture a soundfield as a whole, you can only capture it at certain locations where you can place a microphone. If you wanted to capture the entire soundfield, you could place microphones in a 3D grid throughout the space the soundfield occupies. But in order to cover the frequencies humans can perceive, the spacing of the microphones in the grid would have to be less than an inch, which makes the endeavor extremely impractical.
@pelmazo: All of that is quite possible, and I agree that once in the electronics domain, the "one-dimensional" nomenclature makes more sense. For me, without any context, the word "sound" implies the physical phenomenon of, as you very neatly elucidated, a pressure wave propagating through space and time, which
can't be, thus, one-dimensional. This means that the OP's friend is
probably talking about something else, quite possibly the signal that is captured by microphones or other devices, as you stated.
Alas, without further clarification from the OP and their friend, I'm not sure we'll get anywhere.
Hold on now.
This is an oversimplification, but lets assume the waveform, which is one-dimensional, is sent through a static signal chain, eventually moving a set of voicecoils in a static environment (in other words, nothing else but the waveform is time-dependent) then the pressure of the medium at any given point in three dimensional space is dependent only on the waveform and is thus also only one-dimensional.
This is an oversimplification, but lets assume the waveform, which is one-dimensional, is sent through a static signal chain, eventually moving a set of voicecoils in a static environment (in other words, nothing else but the waveform is time-dependent) then the pressure of the medium at any given point in three dimensional space is dependent only on the waveform and is thus also only one-dimensional.
You can make every 4-dimensional function one-dimensional by fixing 3 of the free variables. If you fix all spatial variables, leaving only the time free, then you have a one-dimensional function that describes the pressure over time at this fixed point in space. Of course, you can do this for any given point, as you say.
Just for the hell of it, you could instead fix two spatial dimensions and time, and keep the remaining spatial dimension free. You would get, for example, a function that describes pressure along a line parallel to the X axis, for a certain point in time, where the position of the line, i.e. its Y and Z position, is fixed. Picture a stroboscope flash illuminating a vibrating string.
This is a theoretical view, for sure, but I hope the point is clear: A 4-dimensional function can be converted to an "array" of lesser-dimensional functions, one for each combination of the variables that have been eliminated. This doesn't change the fundamental situation.
@pelmazo: All of that is quite possible, and I agree that once in the electronics domain, the "one-dimensional" nomenclature makes more sense. For me, without any context, the word "sound" implies the physical phenomenon of, as you very neatly elucidated, a pressure wave propagating through space and time, which can't be, thus, one-dimensional.
The dimensionality of a sound wave depends on the boundary conditions. 1, 2 or 3D are all possible. Stringed instruments are an obvious example of 1 D propagation. No way to have more than 1D when your waveform is confined to a (subwavelength diameter) line.
...and as I said earlier, and although it doesn't matter, for those who think this is about geometry with time being a fourth dimension, the magneto force pushes the voice coil of a driver in a straight line.
You can make every 4-dimensional function one-dimensional by fixing 3 of the free variables. If you fix all spatial variables, leaving only the time free, then you have a one-dimensional function that describes the pressure over time at this fixed point in space. Of course, you can do this for any given point, as you say.
[...]
This is a theoretical view, for sure, but I hope the point is clear: A 4-dimensional function can be converted to an "array" of lesser-dimensional functions, one for each combination of the variables that have been eliminated. This doesn't change the fundamental situation.
Hopefully the point is clear since it seemed to me that the straw vote was that sound was multidimensional. Heck, I also made the comment in error that an anechoic environment increases the dimension count.
The next question is the dimensionality of how hard-headed the participants are in this discussion.
@pelmazo: All of that is quite possible, and I agree that once in the electronics domain, the "one-dimensional" nomenclature makes more sense. For me, without any context, the word "sound" implies the physical phenomenon of, as you very neatly elucidated, a pressure wave propagating through space and time, which can't be, thus, one-dimensional.
The dimensionality of a sound wave depends on the boundary conditions. 1, 2 or 3D are all possible. Stringed instruments are an obvious example of 1 D propagation. No way to have more than 1D when your waveform is confined to a (subwavelength diameter) line.
This depends highly on what is meant with "dimension"; if it's the count of independent variables, since you always have at least one spatial dimension, and the time dimension, the minimum would be two-dimensional. Now, if you mean how the wave is "confined", that's different altogether.
It is a ginormous understatement that my linear algebra is rusty, but have a lood at the original post:
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?
PS: Ignore the post before this edit at 25 after the hour.
Yeah, I'm really hoping for sizetwo to show up and tell us if they've spoken to their mathematician friend; I really would like to know what it's all about.
@pelmazo: All of that is quite possible, and I agree that once in the electronics domain, the "one-dimensional" nomenclature makes more sense. For me, without any context, the word "sound" implies the physical phenomenon of, as you very neatly elucidated, a pressure wave propagating through space and time, which can't be, thus, one-dimensional.
The dimensionality of a sound wave depends on the boundary conditions. 1, 2 or 3D are all possible. Stringed instruments are an obvious example of 1 D propagation. No way to have more than 1D when your waveform is confined to a (subwavelength diameter) line.
This depends highly on what is meant with "dimension"; if it's the count of independent variables, since you always have at least one spatial dimension, and the time dimension,
You can sound without a time dimension. Standing waves for instance. All comes down to your boundary conditions.
@pelmazo: All of that is quite possible, and I agree that once in the electronics domain, the "one-dimensional" nomenclature makes more sense. For me, without any context, the word "sound" implies the physical phenomenon of, as you very neatly elucidated, a pressure wave propagating through space and time, which can't be, thus, one-dimensional.
The dimensionality of a sound wave depends on the boundary conditions. 1, 2 or 3D are all possible. Stringed instruments are an obvious example of 1 D propagation. No way to have more than 1D when your waveform is confined to a (subwavelength diameter) line.
This depends highly on what is meant with "dimension"; if it's the count of independent variables, since you always have at least one spatial dimension, and the time dimension,
You can sound without a time dimension. Standing waves for instance. All comes down to your boundary conditions.
That's not true. As its name indicates, a standing wave oscillates in time but has a stationary, or standing, spatial dependence; at any chosen point along the wave, the amplitude is constant and depends solely on the location of the chosen point, such that the equation of the resultant wave looks like this (https://wikimedia.org/api/rest_v1/media/math/render/svg/f923d59022b190fc4b2f53d607dc662b7a67de44).
All of this is a consequence of the wave equation (of which the acoustic wave equation, the equation that describes the physics of sound, is a specific case) being a partial differential equation in time and at least one spatial variable, so it's only natural for time to show up in the solutions as an explicit independent variable.
But I feel we digress; until the OP clarifies the situation, I feel we'll only go around in circles, quibbling over minutiae and semantics.
Not an area of particular expertise for me, but doesn't a sound wave propagate and aren't all of spatial dimensional properties then determined by the interaction with the propagation media and not the wave itself?
Thinking further - with an image there can be multiple points for y for a given value of x, so you need to express it as a matrix (a 2 dimensional array) of values for every (x,y) point.
With sound there is only a single point relative to a given axis and thus you only need a one dimensional array.
@pelmazo: All of that is quite possible, and I agree that once in the electronics domain, the "one-dimensional" nomenclature makes more sense. For me, without any context, the word "sound" implies the physical phenomenon of, as you very neatly elucidated, a pressure wave propagating through space and time, which can't be, thus, one-dimensional.
The dimensionality of a sound wave depends on the boundary conditions. 1, 2 or 3D are all possible. Stringed instruments are an obvious example of 1 D propagation. No way to have more than 1D when your waveform is confined to a (subwavelength diameter) line.
This depends highly on what is meant with "dimension"; if it's the count of independent variables, since you always have at least one spatial dimension, and the time dimension,
You can sound without a time dimension. Standing waves for instance. All comes down to your boundary conditions.
That's not true. As its name indicates, a standing wave oscillates in time but has a stationary, or standing, spatial dependence; at any chosen point along the wave, the amplitude is constant and depends solely on the location of the chosen point, such that the equation of the resultant wave looks
If amplitude is solely a function of position than it is not a function of time. A standing wave with 1 spatial dimension is therefore a 1D waveform.
That's not true. As its name indicates, a standing wave oscillates in time but has a stationary, or standing, spatial dependence; at any chosen point along the wave, the amplitude is constant and depends solely on the location of the chosen point, such that the equation of the resultant wave looks like this (https://wikimedia.org/api/rest_v1/media/math/render/svg/f923d59022b190fc4b2f53d607dc662b7a67de44).
Can i answer a question ?
A real standing wave is permanently consuming its own energy in the air, so its amplitude should always be different no ?
@pelmazo: All of that is quite possible, and I agree that once in the electronics domain, the "one-dimensional" nomenclature makes more sense. For me, without any context, the word "sound" implies the physical phenomenon of, as you very neatly elucidated, a pressure wave propagating through space and time, which can't be, thus, one-dimensional.
The dimensionality of a sound wave depends on the boundary conditions. 1, 2 or 3D are all possible. Stringed instruments are an obvious example of 1 D propagation. No way to have more than 1D when your waveform is confined to a (subwavelength diameter) line.
This depends highly on what is meant with "dimension"; if it's the count of independent variables, since you always have at least one spatial dimension, and the time dimension,
You can sound without a time dimension. Standing waves for instance. All comes down to your boundary conditions.
That's not true. As its name indicates, a standing wave oscillates in time but has a stationary, or standing, spatial dependence; at any chosen point along the wave, the amplitude is constant and depends solely on the location of the chosen point, such that the equation of the resultant wave looks
If amplitude is solely a function of position than it is not a function of time. A standing wave with 1 spatial dimension is therefore a 1D waveform.
I suppose that's fair enough.
That's not true. As its name indicates, a standing wave oscillates in time but has a stationary, or standing, spatial dependence; at any chosen point along the wave, the amplitude is constant and depends solely on the location of the chosen point, such that the equation of the resultant wave looks like this (https://wikimedia.org/api/rest_v1/media/math/render/svg/f923d59022b190fc4b2f53d607dc662b7a67de44).
Can i answer a question ?
A real standing wave is permanently consuming its own energy in the air, so its amplitude should always be different no ?
If you mean that friction/drag due to the air will make the wave eventually die out, that's true, but you'll see that textbook derivations of these equations usually neglect these effects of attenuation.
If you mean that friction/drag due to the air will make the wave eventually die out, that's true, but you'll see that textbook derivations of these equations usually neglect these effects of attenuation.
A pseudo periodic function will give the real life taste to the equation.
If amplitude is solely a function of position than it is not a function of time. A standing wave with 1 spatial dimension is therefore a 1D waveform.
It's still two dimensions: position and amplitude (and amplitude requires a second axis or dimension to distinguish, so there's your 2D). Otherwise your wave could have amplitude 0, 1, or whatever and still be considered the "same" wave - and I think your ears would clearly tell you they're not.
You're mixing up all kinds of things: spatial dimensions, spatial and temporal dimensions, dimensions of waveform visualizations, "dimensions of functions" (which doesn't exist) ...
Sound, defined as a mechanical wave, propagates through 3D space over time.
You can cut down spatial dimensions until you have a fixed position where you just measure pressure (an amplitude) over time. This is also what you get when you quantize it. (Even if the time axis is implicit, it's still there.)
That's it.
If amplitude is solely a function of position than it is not a function of time. A standing wave with 1 spatial dimension is therefore a 1D waveform.
It's still two dimensions: position and amplitude (and amplitude requires a second axis or dimension to distinguish, so there's your 2D). Otherwise your wave could have amplitude 0, 1, or whatever and still be considered the "same" wave - and I think your ears would clearly tell you they're not.
Incorrect.
You seem to be unable to understand the difference between projection of a function, and mapping of values. We've tried to explain this to exhaustion, but it seems we're running against a brick wall.
It could be consided one dimensional given that our eardrums have a single axis.