Catastrophe Theory and its Effect on Audio (Part II) by Richard C. Heyser (Apr. 1979)

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“Away, you scullion! you rampallion! you fustilarian! I'll tickle your catastrophe.”

--King Henry IV, Part II, II, i.

In the previous article [coming soon. –Ed.], we discussed the elementary basis of Catastrophe Theory and suggested that it may be applicable to problems in the perception of sound. In this discussion, I would like to present a simple example to show how this can be done.

Catastrophe theory, if you remember, is a mathematical basis for modeling certain simple patterns of response that can be expected under the influence of conflicting drives. This is a general nonlinear theory which can be applied to the analysis of equipment and to the study of human behavior.

The theory gets its name from the fact that sudden and dramatic alterations in behavior, response catastrophes, can be predicted within its framework. What makes this attractive, from the standpoint of our perception of sound quality, is the structured analysis which it brings to bear on problems involving the emotional reaction of the listener.

Suppose we now consider a very simple and straightforward problem in audio: How might one's opinion of the quality of sound reproduction from an existing audio system be modified under the influence of two factors, the amount of live music one hears and the amount of reproduced music one listens to from this system? First, let us postulate a scenario that draws only on our observations of human nature. Presumably, if the owner of an audio system likes music, he will continue to indulge himself by acquiring new records and listening to reproduced sound. If there are no interfering factors which can reveal imperfections in quality, there is no drive to modify the opinion of the present audio system. If the listener never goes to a live concert, he is probably satisfied with the music heard at home.

The listener probably never thinks about the audio system and would be perfectly satisfied with music heard from a table-model radio.

But suppose this person goes to a live concert. The clarity of live sound, its dynamic range, and its full use of frequency will enhance the pleasure of his musical experience. And, indeed, that IS the music. If, very shortly after leaving the live concert, our friend plays a record of the same program on his audio system, he will probably note imperfections intruding on the music. Maybe the record noise did not bother him before; now it intrudes. The first level of dissatisfaction sets in.

If our friend never goes to another live concert, the memory will fade and eventually the old "hi fi" will no longer bother enjoyment of the music. He will remember that the reproduction is not perfect, but he is listening through the imperfections to the music and they will not bother him.

If, on the other hand, there is a larger percentage of time spent on listening to live music, there is a good chance that one night, when he comes home and puts a record on the turntable, it will suddenly dawn on him how lousy the sound really is. He no longer likes the sound of music played on his system. From that point on, the degree of discontent will grow in proportion to the ratio of live vs. reproduced sound that is heard. If he mostly attends live concerts and only rarely plays records at home, his knowledge of what live music sounds like will increase the discontent he has with the quality of his reproduced sound.

If he drops off in the amount of live concert attendance, but maintains a small, steady diet of listening to reproduced music, his discontent will, at first, slowly diminish. But without his awareness, there will suddenly come a time when he is so caught up in the music that he never once thinks about the record scratch that bothered him so much just a few evenings before.

His opinion switched from moderate dislike to moderate like. If the ratio of live to reproduced sound continues to diminish, he will again resort to a condition of satisfaction with reproduced music.

Fig. 1 -- If there are two control factors, 1) the amount of experience with live music and 2) the amount of experience with listening to one's home audio equipment, then the degree to which we approve of the quality of audio reproduction from that equipment is a response that must lie on the behavior Manifold M.

Cusp Catastrophe

Admittedly this scenario is quite simplistic. But the behavior is not out of line with human reaction. Let us once again set up the same problem, but this time use catastrophe theory to anticipate behavior.

There are two control factors: Amount of experience with live music and amount of experience with reproduced sound. There is only one response we wish to consider, the degree to which the listener approves of the quality of audio reproduction.

Two factors and one response define a three-dimensional behavior space. This three-dimensional behavior space is sketched in Fig. 1. From our previous discussion we know that the behavior manifold, the location of all stable responses under unchanging factors, will be a subspace with the same number of dimensions as there are control factors. The manifold M, is a two-dimensional surface.

This surface, as we discussed last time, forms a folded shape of the type shown here. The horizontal plane, C, represents the given coordinates of live vs. reproduced listening. We use the letter C because this represents the Control subspace within the higher-dimensional behavior space. It is also referred to as the Parameter space in some mathematical literature. In this figure, I have dropped the position of plane C down below the behavior manifold for illustrative clarity. It makes no difference where the plane C is located relative to M because our interest lies in the projected "shadow" of M onto the control space. By separating M from C, we can readily observe what goes on.

The orientation of coordinate axes.

on the plane C depends upon the nature of the factors which they represent. The derivation which we presented in our prior discussion developed the concept from a behavior potential which would give coordinates u and v, shown dotted in Fig. 1. In contemporary literature, the axes u and v are referred to as the splitting factor and the normal factor, respectively. These coordinates would be used for situations in which the response under consideration is normally influenced by a single factor in a smooth, continuous manner, while, above a certain threshold, the action of the second factor is to set up a trigger condition where slight changes in the normal factor precipitates larger than normal changes in response. The start-up conditions in a free-sunning multivibrator are examples of this; a perfectly balanced circuit could not oscillate when voltage is applied, but offset symmetry

the splitting factor can allow circuit noise above a certain level to start oscillations that build up to full-limit cycles.

Fig. 2--A flatlander, living on the control space C, will encounter mysterious, unmarked boundaries, b1 and b2 which will cause him to experience abrupt changes in behavior when passed in one direction, but not when passed in the opposite direction.

When there are conflicting factors which pretty much compete in their contributions to response, then there is a little bit of control and a little bit of splitting in each of them. These axes are then rotated relative to u and v, as shown in Fig. 1 by the solid lines marked by the capital letters U and V. In the case of audio listening, I will assume that the conflicting factors of live music experience and reproduced music experience are of this latter type. This does not mean that they are rotated 45 degrees with respect to nor mal and splitting, but that they have some amount of rotation.

Flatland The projection of the behavior fold in M onto the surface C is called the Bifurcation Set, and this curve is symbolized here by the letter b. It is called bifurcation, or two-pronged fork, because two different kinds of behavior occur when we move our location away from this line. This is the boundary of precipitous behavior in terms of the control factors. This curve has a sharp point which forms a cusp, and that is why the particular type of behavior pattern associated with this type of precipitous response is called a Cusp Catastrophe.

In order to understand how our listening emotions enter the picture, refer to Fig. 2. Imagine that we are flatlanders living on the surface C. We are moved about our flatworld under the influence of two factors, and our position within flatland is marked by the coordinates U and V. Our emotional feelings alter with our position in flatland. As we move along the trajectory marked (a), our feelings smoothly and continuously change with our coordinate location. When, in our wanderings, we cross back to the coordinate location shown here as (1), we duplicate the emotions we previously experienced when passing this same place.

But when our trajectory crosses the magic boundary b, we suffer a dramatic and sudden change in emotion. Just before we got to this boundary we were content and liked our state. At the moment we touched this boundary, our state flipped to that of a strong dislike. Our emotions suffered a catastrophic change.

Seeking to restore our status we loop back to position (2) which we had just before encountering this magic boundary and which we knew was a position of content. But our emotions do not come back to what they were.

Now, at position (2) we still have a feeling of strong discontent.

Baffled, we retrace our path until suddenly, at another magic boundary, b2, we catastrophically jump back in emotion state to our previous condition. We had previously passed this second magic boundary going in another direction and nothing had happened; now, coming back across it, our emotions are dramatically altered.

Living life as a flatlander, unable to comprehend forces outside our world, we would probably attribute this mischief to divine influence and might develop some interesting theories to explain why these things should happen to us.

We might even develop a technocult of flatland surveyors who, through ever finer instruments and more glorious linear mathematics, seek to quantify the measure of the geometry of flatland. Of course, these technocultists might be so burdened down with the weight of their precision apparatus that they cannot stray far from low curvature regions where no catastrophic changes occur. Rumors of catastrophes might reach their ears, but no technocultist would ever accept the existence of such magic nonsense, since it was not only inconsistent with their linear mathematics, but could not be discerned by their survey instruments. In order to ease the fears of the perceptofreaks, who believe in such magic nonsense, the survey instruments are constantly being improved to measure imperfections to an ever finer resolution.

But a flatlander falling off a cliff takes little comfort from knowing that the science minds of flatland, who do not believe in the existence of cliffs, had developed a new flat measuring rod capable of resolving a nanometer.

It is not that the science minds are wrong; they just do not happen to be where the action is. They are under the wrong lamppost. If this situation sounds a little bit like our own problems in audio flatland, the resemblance is not coincidental.

Fig. 3--As our exposure to live music and reproduced music changes, we trace out a path on the behavior Manifold M. Whenever our experience takes us past folds in this surface, our opinion of the sound quality which our audio system provides will take a catastrophic jump.

What no one in flatland can realize is that his fate depends on higher dimensional influences. Let us go back to Fig. 1. The dramatic change called a catastrophe is a symbolic falling off a cliff in a higher dimensional space.

The catastrophe map, shown here by the capital X, is the process of projecting the shadow of the actual position of response on the behavior surface M onto an apparent position in terms of the control space C.

Journey on a Manifold

Let us take a journey on the manifold M. This journey is shown by the dashed line in Fig. 3 and starts out at the place marked "satisfied." Our altitude marks our attitude. Our height above the plane C (flatland) is a measure of response to the control parameters. The higher we are on M, the more we dislike the sound of the audio system. We are driven up and down this surface by the control parameters. We start this journey at "satisfied," the position of which is determined by control coordinates U. and V. At "satisfied" we have Vo units of listening to our audio system and U. units of listening to live music. Out of enjoyment of music, we begin to listen to more reproduced sound and begin our journey on the manifold M. The more we listen to reproduced sound, the more that sound becomes our standard of performance. This drives our location on manifold M to a lower height, which means we become more satisfied with the sound of our audio system ... or, looked at another way, the less we think about the quality of reproduced sound.

Then, around coordinates U, and V, we being to go to more like concerts.

Our trajectory now takes a sharp change of direction back up the manifold. With increasing live music experience, our opinion of the old "hi fi" begins to drop, until somewhere around coordinates U2 and V2 we cross the magic boundary b,. All of a sudden we experience a disillusionment catastrophe ... our opinion changes from "like" to "dislike." The reason for this is that in order to remain on the surface of stable response, M, we had to jump from the lower sheet to the upper sheet where our trajectory took us past the fold. Under small changes in factors, we must take a big jump in response in order to stay on the manifold of stable response. When we approach a fold under smooth progressive drives, there is no way we can find ourselves on the inner sheet of M. If we slack off on the ratio of live to reproduced listening, our opinion of reproduced sound quality will not snap back until we cross the boundary b2 at position U3 and V3. Then, as we cross this boundary, our opinion will fall off the cliff, and we will suffer what I have referred to here as an acceptance catastrophe. We are back on the original trajectory and must accumulate a bit more live-listening experience before again experiencing a disillusionment catastrophe.

If, on the other hand, we simply slack off in both live and reproduced listening experience, we are passing along the path called here as "familiarity." The old habit patterns slowly take hold and we again will find ourselves at a "satisfied" status, back where we started.

Buy By

It is rather startling to contemplate the richness of emotional reaction which is revealed by even this naive catastrophe model. Each of us, I am sure, would like to believe that he is master of his own behavior under conflicting factors. But the trend in behavior pattern which is disclosed by Thom's theory reveals the existence of an inexorable machine which we ought to be aware of. Knowledge of this machine introduces a new factor into the game and raises the dimensionality to a higher level. It is a case of forewarned being forearmed; once we know that participation in a situation with two factors and one response yields a cusp catastrophe, we can introduce that knowledge as a new control factor and avoid the cusp. But our cleverness could also be our undoing since we may have changed the situation to one of higher dimensionality.

One of the situations in which knowledge of this elementary catastrophe can be of value is in the purchase of audio equipment. If instead of a live listening versus reproduced listening, we were to label the control factors:

Listening to Brand

A versus listening to Brand B, we can sense how a clever salesperson could walk an unsuspecting customer up the manifold to sufficient strength of opinion to trigger purchase of a component.

Suppose you had decided Brand B sounded pretty good and was an excellent match to your bank account.

About the time you show signs of being ready to purchase Brand B, the clever salesperson lets you hear just a brief bit of sound from a more expensive Brand A. By this time you had disclosed which kind of music you like and had expressed satisfaction with the way Brand B reproduced that sound. So, quite by "accident," Brand A is punched up on that music.

You are at point U1 and V1 and suddenly the introduction of a better sound stops your downward plunge on the opinion manifold and pulls you in a new direction of upward motion.

You like the music and your curiosity makes you want to hear a bit more.

The smallest dissatisfaction with Brand B starts to set in, and a clever salesperson knows that if you can be persuaded to listen long enough you will get "hooked" on the better sound of Brand A. A good salesperson will not force you to listen to Brand A; you said what your purchase limit was and Brand B was at that limit. So the trap is sprung to let you sell yourself.

A simple A-B comparison switch is all it takes, with you the unsuspecting driver of the machine when you are allowed to switch the music back and forth between the two competing systems. Any increase in relative exposure to the sound of A versus B will inexorably drive you upward on the manifold. If you trigger a disillusionment catastrophe, the deed is all but done.

Once you are on the uppermost sheet of the manifold, it is likely that you will subconsciously place the switch in the A position for an increasingly longer time than in the B position. You are driving yourself higher on the manifold. By that time the salesperson is mentally computing his commission on the sale of Brand A. Maybe you do not dig math. Maybe the idea of topological manifolds in a behavior space does nothing to you.

But just knowing of the existence of such things can save your wallet from a needless onslaught the next time you go shopping for audio equipment.

At the very least, you can be aware of emotional forces which can be set into motion to present you with tempting bait. Once you grab such bait, the hook is sunk, and it is your own struggle which sets the barb in deeper and pulls you into the purchase of a component you did not previously want to buy.

All Is Not Gold That Listens

Quite obviously, the more interesting situations arise when there are a multiplicity of factors, some conflicting and others of a splitting nature.

We all recognize that emotional bias definitely plays a role in the reaction we have to conflicting circumstances.

Our individual perception of quality involves a delicate balance of conflicting factors, including our own involvement with one or more of those factors. The designer of a particular audio product may be a poor judge, from the standpoint of detached objectivity, of the relative merits of that product.

And it must be admitted that the ratio of lead to gold in the ear of the listener is somehow related to the personal involvement which that listener has with the product being heard. This apparent rupture of objectivity, as perceived in the frame of reference of others, may occur without conscious awareness of the occurrence.

It is also possible that even in the presence of an emotional bias something can happen which will "change our mind" and alter the response we have to a given situation. Beauty, it is said, is in the eye of the beholder. But we all know that events can occur which catastrophically alter our perceptions even in the presence of prior strength of opinion.

In our next discussion we will consider another common audio situation which involves a higher dimensional catastrophe.

[...] the foil which will re-radiate the interference field."

Foil shielding material is optionally available with two-way, pressure-sensitive adhesive backing. This offers a simple way of attaching the material to the device to be shielded.

Decisions as to choice of sheet or foil, its thickness, and the number of layers depend in large part on the degree of attenuation sought. (Sheet material may also be chosen because of its rigidity, so as to preserve the shape of the shield, enable it to be secured in place, etc.) While formulas are available to assist in selection of the proper shielding material, gauge, and/or number of layers, these formulas yield only approximate results. A trial and error procedure is often necessary.

Here an a.c. magnetic field evaluator probe, such as the one made by Perfection Mica, can be helpful to the engineer, technician, or hobbyist. In conjunction with an a.c. voltmeter or oscilloscope, the probe can measure the intensity of a magnetic field before and after shielding, thus indicating the degree of attenuation actually achieved. "With this necessary information," states Perfection Mica, "magnetic shielding material type and gauge can be intelligently selected and shield design optimized."

How Shielding Material Operates

A magnetic shield effective in the audio range performs its functions in two ways, depending on frequency of the interference field: (1) For a stationary field (zero frequency) produced by a permanent magnet, d.c. current, or the earth, the shield offers a path with low reluctance that is, with low magnetic resistance. Thus the shield acts as a shunt, diverting the field from the sensitive device. At low audio frequencies, the shield behaves in much the same manner. At higher frequencies, however, the interference field produces eddy currents in the shield material. These currents in turn produce an electromagnetic field that opposes the interference field.

The specific shielding materials that we shall next discuss, namely those made by Perfection Mica, consist of an iron-nickel alloy. As previously stated, they are effective from d.c. to about 50,000 Hz; for higher frequencies, other shielding materials, such as copper, are required.

Perfection Mica's two basic types of shielding materials are:

1. Co-Netic. This has very high relative permeability, which is a measure of material's ability to provide path for magnetic lines of force. The higher the permeability, the greater is the shielding effect. Relative permeability is the permeability of a material relative to that of a vacuum. Air has a relative permeability of about 1. Co-Netic has a maximum relative permeability of about 500,000. For interference fields of weak to moderate strength, Co-Netic can prove an effective barrier when properly used, with a single layer typically providing about 40 dB attenuation of the field. Unfortunately, Co-Netic has a relatively low saturation level. That is, it cannot effectively serve as a barrier to strong magnetic fields. Beyond a point, it can no longer provide a path for magnetic lines of force.

2. Netic. This has much lower relative permeability than Co-Netic, but a much higher saturation level. While its permeability is only about 1/100th that of Co-Netic, it can accept more than 100 times as great a magnetizing force before reaching saturation. Hence, when dealing with strong magnetic fields, use of Netic is indicated; however, a single layer will provide only about 1/10th as much attenuation as does Co-Netic.

To achieve a desired degree of attenuation, several layers of shielding material may be needed. For an intense field, the most effective shielding tends to result from a combination of Netic and Co-Netic. The rule to be followed is to place the Netic material closest to the generating device, and the Co-Netic closest to the sensitive device. For example, if combination shielding were to be placed around a transformer, the Netic layer would be on the inside and the Co-Netic layer on the outside; if a combination shield were to be placed around a tape head, the Co-Netic would be on the inside and the Netic on the outside.

Both are available in sheet and foil form, each in a variety of gauges, widths, and lengths. For ease of handling (cutting, shaping, etc.), foil would probably be more advantageous to the audio hobbyist, although sheet has the advantages of rigidity and greater attenuation owing to its greater thickness.

Foil has another important advantage to the hobbyist: It eliminates the possible need for annealing in order to preserve the magnetic properties of the shielding material after it is worked into the desired shape. Annealing consists of heating the material to an appropriate temperature in an appropriate environment and cooling it at an appropriate rate requiring facilities seldom available to the amateur.

Also see: Phase, Time, Ears & Tape (Apr. 1979)

(Source: Audio magazine, April 1979; Richard C. Heyser)

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