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by RICHARD C. HEYSER There have been many science fiction stories with plots involving worlds of differing dimensionality. My favorite among these is a perennial science-fiction classic called "Flatland, A Romance of Many Dimensions." Written in 1884 by E.A. Abbott, a schoolmaster, it is still available, in its seventh edition in paperback (1). What does that have to do with audio? Well, if you know the story, and accept some of the concepts I am about to present, you might come to agree with me that we now live in an audio flatland. Written more than 20 years before Einstein's first paper on relativity, "Flatland" is pure fantasy and makes no pretense of application to human affairs. Yet in an abstract sense, the subject matter of Abbott's book anticipated some of the conceptual problems which three-dimensional humans might encounter in dealing with the four-dimensional universe of relativity. This has been pointed out by several scientists in popular books discussing the space-time continuum. (2, 3). The central difficulty is in demonstrating to people, whose total conceptual structure is geared to one level of dimensionality, that higher dimensionality exists and is capable of being under stood.
The allegories are numerous, and Abbott's "Flatland" is one of the better known of these. ..I will not spoil the story by telling the plot. It is a delightful little book and, like its subject matter, can be read at several levels of conceptualization. But I will impose on you the following mind-expanding thought. We can quite easily grasp the idea of a two dimensional sub-world that is imbedded in our three-dimensional world. In mathematical language, a subspace is an easy thing to imagine because it is usually generated by denying one or more of a higher dimensional set of at tributes to the lower dimensional space. As an example, this piece of paper on which these words are printed can be thought of as a two-dimensional subspace. Left-right and up-down are the dimensions of the printed page, but in-out is not. The concept of in-out is denied to any being who "lived" in the two-dimensional world of this page. The third dimension, which we call height, is denied this being because he cannot stay completely within the confines of his space and touch all parts of our three-dimensional space, so long as his is a subspace. That is easy to under stand. But suppose his was not a subspace. Suppose his space was our space, but the difference was that he saw it as two-dimensional while we saw it as three-dimensional. In order to visualize, as a three-dimensional person, how this might come about, suppose the sheet of paper was enormously large and we began folding it back and forth to compress the paper to some reasonable volume in our space. The math is sloppy, but it is the thought I want to get across. If all we did was fold and compress the paper, we would never alter the fact that it was a subspace, but we can begin to see that the property we call height would in fact begin appearing as an attribute within the two-dimensional subspace. Depending on the way we folded the paper, things that moved up, as we saw them, might appear to shift in lateral position in our two-dimensional friend's point of view. The being who lived on the paper would now see the third dimension. But he would not recognize it as a dimension unless we told him what it was. He would, instead, observe it as some weird property joining relation ships in his two-dimensional existence. Now, let's do it again. Only this time we "superior" three-dimensional beings see the plight of a one-dimensional being. He lives on a string and does not even know what up-down means. All he knows is what we call left-right. But his string world is so very long that we begin rolling it up like a ball of twine to concentrate it in a small region of our space for observation. Finally, even though this one-dimensional space has no width, when it is so crammed together that no part of our three-dimensional space is farther than some previously agreed upon small distance from some part of the string, we can agree that he "touches" everything we do. He actually sees three dimensions. But he does not ob serve the properties as a thing he might call "dimension." Instead he sees it as certain relationships among his one-dimensional observations. What to us is a vertical line might appear to him as an array of disjointed coordinate locations. Now the mind expanding part ... the part that is new. Two beings can observe and describe everything that happens, but do so from the viewpoints of different dimensionality. I do not mean to imply that there are four-dimensional beings or two-dimensional beings among us. The point I wish to convey is that there is no preferred frame of reference for any observation, either for number of dimensions or units of measurement. There is an additional consideration to this geometric concept of Alternatives, one that has far reaching consequences. Although the folding of the two-dimensional paper and twining of the one-dimensional string were allegorical, they do illustrate that the concepts of continuity and "between ness" do not have to be preserved when we change dimensionality. A trajectory of smoothly continuous values in a higher dimensional alternative, for example, might show up in a lower dimensional alternative as a discontinuous set which may disappear at certain places and reappear elsewhere without being found at intervening locations. A being who is accustomed to sensations perceived in a particular dimensionality and frame of reference might form certain concepts about that situation which "make sense." Any attempt to convey to such a being the possibility of a higher dimensional version of his universe will be a most difficult chore because it cannot make sense if one tries to explain it in terms of the coordinates of that lower dimensional space. This was the conceptual problem in Abbott's "Flatland," except that now we are not talking about a lower dimensional subspace imbedded in a higher dimensional space. We are con fronting the problem of a lower dimensional Alternative to a higher dimensional space. Things can disappear from a subspace and not be found anywhere in it, but that is not the case with Alternatives. Things do not disappear in Alternatives, they show up as other geometric configurations. The point I wish to convey is that if we discover some seemingly bizarre behavior that does not seem to make sense in our otherwise orderly view, we might be able to reconcile such behavior by converting to an alternative system of coordinates--possibly at a different level of dimensionality. These are new ideas and take get ting used to. And yes, by George, they do have application to audio and subjective perception. But allow me to continue with some of the fundamental concepts before swinging into sound.
Suppose we have a good technical description of something. We have a mathematical description nailed down. There are no hidden variables. Our description will involve certain cause and effect relationships among parameters. If we now set up a physical observation in those parameters and if we have not left anything out, then nature will oblige us by operating in consonance with those parameters. This does not mean that nature prefers these parameters. Nature does not give a darn what parameters we choose. All it means is that we were consistent in setting up a model. Suppose that we wish to take another view using other parameters. We wish to see the world through a different window. How many windows are available? As many as we care to find. That is what I call the Principle of Alternatives. We already discussed two of the windows in previous articles. One window is measured in units called time, while the other window is measured in units called frequency. It makes no difference whether we look through the time window or the frequency window, we see all there is to see of the same thing. Only the way it appears is different.
Now let's see what relevance this technofreak talk has in sound reproduction. Did you ever hook an oscilloscope onto the output terminals of a power amplifier and watch the voltage wave form while listening to the sound coming out of the loudspeaker connected to the same terminals? Or hook a 'scope to the output of a microphone while you listened? What you see on the oscilloscope is a representation of a one-dimensional signal. It is volts as a function of time. But when you listened to that signal, what did you perceive in the way of dimensions? More than one, I will wager, if you aligned that perception with prior experience of the way things sound. Sound has a "where." That is, it is located in physical space with respect to us. That is at least three dimensions right there. Sound also has a property which I will simply call "tone." Pitch, timbre, and the things we measure in units of pitch are expressions of tone. "Where" a sound source originates and what "tone" that source has are independent properties. Whether a clarinet is stage left, stage center, or stage right does not dictate what musical notes are going to be playable on that clarinet. So "tone" is somehow in dependent of "where" and can rank as at least an additional dimension. Then there is "when" a sound occurs. Think about it a bit. That is another possible dimension. Then, there is a "how much," or intensity, which is not a property precisely dependent on the other things which I call dimension. All in all, if we add up those proper ties generally agreed to be independent attributes, we find that the least number of dimensions we can get away with in a subjective description of sound is five. And where is all (or most) of that higher dimensional information in that silly one-dimensional waveform we view on an oscilloscope? It is there. But just as the three dimensions of space viewed from a one-dimensional ball of twine, the higher dimensionality of perception is encoded as the relationships existing between properties in the one-dimensional waveform. Yes, yes, I know about two channels for stereo, four for quadraphonic, and all that. But right now we are on the ground floor and trying to tie certain properties of subjective perception with other properties we now measure in objective analysis. I imagine many of us at one time or another have had the experience of finding that due to a technical error we had been listening to a two-channel mono feed when we thought we were hearing stereo. And like the optical illusion which, once recognized, never looks the same again, we find the subjective dimensionality collapsed to an "obvious" mono program when the deception is discovered. But we had been fooled ... that one-dimensional pro gram had supported a whole stereo illusion.
The audio technologist who measures things in the frequency domain resides in a linear world of one dimension. He is a Flatlander. There is nothing wrong with that. So long as the de vice is linear, or essentially linear, the audio Flatlander sees everything there is to see. His window happens to look out onto a one-dimensional universe. The prime audio problem ("how can we measure what we hear?") arises when this Flatlander tries to convey measures of fidelity to the being who sees things through a higher dimensional window. Neither one sees something the other does not. But that which appears essentially perfect to a Flatlander, may or may not be essentially perfect when viewed in a higher dimension. The reason for this, as we have discussed, lies in the fact that if these are genuine Alternatives (different ways of describing the same thing), then the map between them is a geometric transformation. Distributions are mapped into distributions. A simple, elemental place in one Alternative will appear as some geometric distribution (or configuration) in the other Alternative. As an aside, I must point out that the concept of "place" is perhaps better understood in terms of this idea of geometric distribution, or configuration, or figure. The concept of "point," or the concept of "line," or of any special type of "figure" is not fundamental to the establishment of a valid geometry. That is a very difficult thing to recognize, accustomed as we are to the strong arm methods of teaching mathematics to generations who couldn't care less about mathematics. Only recently have we begun to explore distributions as a Theory of Generalized Functions, in order to bring light to a badly illuminated part of our understanding. The Dirac delta is the prime example of a distribution that can begin giving meaning to the "place" where something can be found. Unfortunately, some popular discussions about the so-called delta "function" as applied to audio have fallen into the trap of trying to explain it in heuristic terms, such as ... "existing only at a point but having unit area." Such an explanation is no explanation at all, because generalized functions cannot be assigned values at isolated points. The audio Flatlander, viewing the frequency response of a loudspeaker, amplifier, cartridge, or whatever, can not possibly make "sense" out of the protestations of a higher dimensional being that the Flatland ranking of distortion does not always correlate with that being's ranking of distortion. It does not make sense to the Flatlander as long as he uses his own coordinate system. Unless it is disclosed to him, the Flatlander has no way of knowing that higher dimensional "places" show up as geometric properties in Flatland. The converse is also true; each place in Flatland may appear as some spread of values in the higher dimensionality. The problem becomes enormously compounded when either the Flatlander or the higher dimensional being set up test figures to check for the possibility of geometric warping. The test figure is usually chosen to be that which is easiest to measure within a particular frame of reference. The test figure which a Flatlander might choose does not necessarily correspond to a test figure which might be chosen for any other dimensionality. So our audio Flatlander might set up a test figure which represents a perfect concentration of a unit volume of material at a well-defined "place." We might call it a delta function corresponding to unit energy at one frequency. (Another audio Flatlander, living in the one-dimensional time domain, would perceive that particular test figure as a "wave" extending over the whole of the time domain. He would call it a sine wave.) The frequency domain Flatlander then checks for geometric warping by determining how much material appears at other "places" when the test figure is processed. The Flatlander might set up a standard of warping, such as: If no more than one thousandth of one percent of the volume of the test figure can be found at any place other than the original location, the geometric warping will be essentially nil. It would seem logical to presume that if one found two percent of the volume out of place, the warping would be greater than if one only found two-hundredths percent out of place. So the audio Flatlander can go about checking for geometric warping by placing test figures at various locations in his space. But what might a higher dimension al being perceive? First of all, the Flatlander's test figure might have no particular significance in the view through the higher dimensional window. How might you feel if you looked out of your living room window and saw colored lights flashing across the sky like a giant aurora borealis display? If you asked what was going on, you would get a reply that a one-dimensional being was testing the universe for linearity. "See there," he would say, "that's a perfect signal at coordinate location 47." And you would see a steady green glow with ripples of red slowly moving across the sky. The Flatlander's test figures are things that he can understand. In the Flatlander's world, a test figure corresponding to something of significance to a higher dimensional being might appear hopelessly complicated. If, in looking out of your window, you had said, "Hey, knock off the silly lights; if you want to check for geometric warping, use this meter rod." And you hold the meter rod up. After a brief pause the Flatlander would reply, "You're nuts, all I see is a blurring of edges and a purple glow." Silly example? Perhaps, but the math, or more properly the geometry, is reasonably illustrated by such an example. One-dimension and five-dimension beings cannot agree on the subtler aspects of scene distortion because each sees the view through different windows.
If you remember our discussion of the end product of audio, the query "how do we measure what we hear?" translates to "how do we measure the illusion we perceive?" This brief discussion has been our deepest penetration so far into the abstract. But the problem it addresses is of the utmost practicality. Perhaps we cannot mea sure the illusion of sound, but we might be able one day to grasp some of its structural properties, as a perceived higher dimensional experience. In this case I am attempting to con vey a reason why conventional distortion measurements, such as harmonic and intermodulation, need not necessarily correlate with our subjective impressions of distortion. Geometric warping of a perceived illusion and geometric warping of a lower dimensional test signal are both distortions. But they are distortions expressed in terms of the framework of the reference system against which they are measured. If those frames of reference are not the same, whether they differ in dimensionality or some other way, we cannot automatically rank them as equivalent. It may happen in a gross measurements that three is greater than two, and two is greater than one for certain types of distortion as ex pressed in either reference system. But somewhere along the line, we are going to get into difficulty quantifying subtler forms of distortion with such gross equivalents. But whether we are talking about a distortionless situation or one that is badly distorted, the deeper geometric property I want you to consider is that of the possible dimensionality which is involved in any particular situation. We have only begun to touch on this subject and I hope to expand on it in later discussions, but the next time you hear an argument between a technologist and a golden ear about the audibility of certain types of distortion, think of this: Is the technologist really a Flatlander, and is it possible they do not agree because they each have a view through a different window?
1. E. A. Abbott, "Flatland, A Romance of Many Dimensions," Dover Publications, New York, 1952. 2. A. Eddington, "Space, Time, and Gravitation," Cambridge at the University Press, 1968 (p.57). 3. C. Lanczos, "Space Through the Ages," Academic Press, New York, 1970 (p.88). ( Also see: Speaker Impedance: More Complex than One Number by Richard C. Heyser (June 1984) = = = = |

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