Tag Archives: physics

Physics of no fear, no fall ladders

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I recently achieved a somewhat mastery over my fear of heights while working on the flat roof of our lab building / factory. I decided to fix the flat roof of our hydrogen engineering company, REB Research (with help from employees), and that required me to climb some 20 feet to the roof to do some work myself and inspect the work of others. I was pretty sure we could tar the roof cheaper and better than the companies we’d used in the past, and decided that the roof  should be painted white over the tar or that silvered tar should be used — see why. So far the roof is holding up pretty well (looks good, no leaks) and my summer air-conditioning bills were lowered as well.

Perhaps the main part of overcoming my fear of heights was practice, but another part was understanding the physics of what it takes to climb a tall ladder safely. Once I was sure I knew what to do, I was far less afraid. As Emil Faber famously said, “Knowledge is good.”

me on tall ladder

Me on tall ladder and forces. It helps to use the step above the roof, and to have a ladder that extends 3-4′ feet past roof level

One big thing I learned (and this isn’t physics), was to not look down, especially when you are going down the ladder. It’s best to look at the ladder and make sure your hands and feet are going where they should. The next trick I learned was to use a tall ladder — one that I could angle at 20° and extends 4 feet above the roof, see figure. Those 4 feet gave me something to hold on to, and something to look at while going on and off the ladder. I found I preferred to go to or from the roof from a rung that was either at the level of the roof, or a half-step above (see figure). By contrast, I found it quite scary to step on a ladder rung that was significantly below roof level even when I had an extended ladder. I bought my ladder from Acme Ladder of Capital St. in Oak Park; a fiberglass ladder, light weight and rot-proof.

I preferred to set the ladder level (with the help of a shim if needed) at an angle about 20° to the wall, see figure. At this angle, I felt certain the ladder would not tip over from the wind or my motion, and that it would not slip at the bottom, see calculations below.

if the force of the wall acts at right angles to the ladder (mostly horizontally), the wall force will depend only on the lever angle and the center of mass for me and the ladder. It will be somewhat less than the total weight of me and the ladder times sin 20°. Since sin 20° is 0.342, I’ll say the wall force will be less than 30% of the total weight, about 65lb. The wall force provides some lift to the ladder, 34.2% of the wall force, about 22 lb, or 10% of the total weight. Mostly, the wall provides horizontal force, 65 lb x cos 20°, or about 60 lbs. This is what keeps the ladder from tipping backward if I make a sudden motion, and this is the force that must be restrained by friction from the ladder feet. At a steeper angle the anti-tip force would be less, but the slip tendency would be less too.

The rest of the total weight of me and the ladder, the 90% of the weight that is not supported by the roof, rests on the ground. This is called the “normal force,” the force in the vertical direction from the ground. The friction force, what keeps the ladder from slipping out while I’m on it, is this “normal force” times the ‘friction factor’ of the ground. The bottom of my ladder has rubber pads, suggesting a likely friction factor of .8, and perhaps more. As the normal force will be about 90% of the total weight, the slip-restraining force is calculated to be at least 72% of this weight, more than double the 28% of weight that the wall pushes with. The difference, some 44% of the weight (100 lbs or so) is what keeps the ladder from slipping, even when I get on and off the ladder. I find that I don’t need a person on the ground for physics reasons, but sometimes found it helped to steady my nerves, especially in a strong wind.

Things are not so rosy if you use a near vertical ladder, with <10° to the wall, or a widely inclined one, >40°. The vertical ladder can tip over, and the widely inclined ladder can slip at the bottom, especially if you climb past the top of the roof or if your ladder is on a slippery surface without rubber feet.

Robert E. Buxbaum Nov 20, 2013. For a visit to our lab, see here. For some thoughts on wind force, and comments on Engineering aesthetics. I owe to Th. Roosevelt the manly idea that overcoming fear is a worthy achievement. Here he is riding a moose. Here are some advantages of our hydrogen generators for gas chromatography.

Calculus is taught wrong, and is often wrong

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The high point of most people’s college math is The Calculus. Typically this is a weeder course that separates the science-minded students from the rest. It determines which students are admitted to medical and engineering courses, and which will be directed to english or communications — majors from which they can hope to become lawyers, bankers, politicians, and spokespeople (the generally distrusted). While calculus is very useful to know, my sense is that it is taught poorly: it is built up on a year of unnecessary pre-calculus and several shady assumptions that were not necessary for the development, and that are not generally true in the physical world. The material is presented in a way that confuses and turns off many of the top students — often the ones most attached to the reality of life.

The most untenable assumption in calculus teaching, in my opinion, are that the world involves continuous functions. That is, for example, that at every instant in time an object has one position only, and that its motion from point to point is continuous, defining a slow-changing quantity called velocity. That is, every x value defines one and only one y value, and there is never more than a small change in y at the limit of a small change in X. Does the world work this way? Some parts do, others do not. Commodity prices are not really defined except at the moment of sale, and can jump significantly between two sales a micro-second apart. Objects do not really have one position, the quantum sense, at any time, but spread out, sometimes occupying several positions, and sometimes jumping between positions without ever occupying the space in-between.

These are annoying facts, but calculus works just fine in a discontinuous world — and I believe that a discontinuous calculus is easier to teach and understand too. Consider the fundamental law of calculus. This states that, for a continuous function, the integral of the derivative of changes equals the function itself (nearly incomprehensible, no?) Now consider the same law taught for a discontinuous group of changes: the sum of the changes that take place over a period equals the total change. This statement is more general, since it applies to discrete and continuous functions, and it’s easier to teach. Any idiot can see that this is true. By contrast, it takes weeks of hard thinking to see that the integral of all the derivatives equals the function — and then it takes more years to be exposed to delta functions and realize that the statement is still true for discrete change. Why don’t we teach so that people will understand? Teach discrete first and then smooth as a special case where the discrete changes happen at a slow rate. Is calculus taught this way to make us look smart, or because we want this to be a weeder course?

Because most students are not introduced to discrete change, they are in a very poor position  to understand, or model, activities that are discreet, like climate change or heart rate. Climate only makes sense year to year, as day-to-day behavior is mostly affected by seasons, weather, and day vs night. We really want to model the big picture and leave out the noise by considering each day or year as a whole, keeping track of the average temperature for noon on September 21, for example. Similarly with heart rate, the rate has no meaning if measured every microsecond; it’s only meaning is as a measure of the time between beats. If we taught calculus in terms of discrete functions, our students would be in a better place to deal with these things, and in a better place to deal with total discontinuous behaviors, like chaos and fractals, an important phenomena when dealing with economics, for example.

A fundamental truth of quantum mechanics is that there is no defined speed and position of an object at any given time. Students accept this, but (because they are used to continuous change) they come to wonder how it is that over time energy is conserved. It’s simple, quantum motion involves a gross discrete changes in position that leaves energy conserved by the end, but where an item goes from here to there without ever having to be in the middle. This helps explain the old joke about Heisenberg and his car.

Calculus-based physics is taught in terms of limits and the mean value theorem: that if x is the position of a thing at any time, t then the derivative of these positions, the velocity, will approach ∆x/∆t more and more as ∆x and ∆t become more tightly defined. When this is found to be untrue in a quantum sense, the remnant of the belief in it hinders them when they try to solve real world problems. Normal physics is the limit of quantum physics because velocity is really a macroscopic ratio of difference in position divided by macroscopic difference in time. Because of this, it is obvious that the sum of these differences is the total distance traveled even when summed over many simultaneous paths. A feature of electromagnetism, Green’s theorem becomes similarly obvious: the sum effect of a field of changes is the total change. It’s only confusing if you try to take the limits to find the exact values of these change rates at some infinitesimal space.

This idea is also helpful in finance, likely a chaotic and fractal system. Finance is not continuous: just because a stock price moved from $1 to $2 per share in one day does not mean that the price was ever $1.50 per share. While there is probably no small change in sales rate caused by a 1¢ change in sales price at any given time, this does not mean you won’t find it useful to consider the relation between the sales of a product. Though the details may be untrue, the price demand curve is still very useful (but unjustified) abstraction.

This is not to say that there are not some real-world things that are functions and continuous, but believing that they are, just because the calculus is useful in describing them can blind you to some important insights, e.g. of phenomena where the butterfly effect predominates. That is where an insignificant change in one place (a butterfly wing in China) seems to result in a major change elsewhere (e.g. a hurricane in New York). Recognizing that some conclusions follow from non-continuous math may help students recognize places where some parts of basic calculus allies, while others do not.

Dr. Robert Buxbaum (my thanks to Dr. John Klein for showing me discrete calculus).

How to make a simple time machine

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I’d been in science fairs from the time I was in elementary school until 9th grade, and  usually did quite well. One trick: I always like to do cool, unexpected things. I didn’t have money, but tried for the gee-whiz factor. Sorry to say, the winning ideas of my youth are probably old hat, but here’s a project that I never got to do, but is simple and cheap and good enough to win today. It’s a basic time machine, or rather a quantum eraser — it lets you go back in time and erase something.

The first thing you should know is that the whole aspect of time rests on rather shaky footing in modern science. It is possible therefore that antimatter, positrons say, are just regular matter moving backwards in time.

The trick behind this machine is the creation of entangled states, an idea that Einstein and Rosen proposed in the 1930s (they thought it could not work and thus disproved quantum mechanics, turned out the trick works). The original version of the trick was this: start with a particle that splits in half at a given, known energy. If you measure the energy of either of the halves of the particle they are always the same, assuming the source particle starts at rest. The thing is, if you start with the original particle at absolute zero and were to measure the position of one half, and the velocity of the other, you’d certainly know the position and velocity of the original particle. Actually, you should not need to measure the velocity, since that’s fixed by they energy of the split, but we’re doing it just to be sure. Thing is quantum mechanics is based on the idea that you can not know both the velocity and position, even just before the split. What happens? If you measure the position of one half the velocity of the other changes, but if you measure the velocity of both halves it is the same, and this even works backward in time. QM seems to know if you intend to measure the position, and you measure an odd velocity even before you do so. Weird. There is another trick to making time machines, one found in Einstein’s own relativity by Gödel. It involves black holes, and we’re not sure if it works since we’ve never had a black hole to work with. With the QM time machine you’re never able to go back in time before the creation of the time machine.

To make the mini-version of this time machine, we’re going to split a few photons and play with the halves. This is not as cool as splitting an elephant, or even a proton, but money don’t grow on trees, and costs go up fast as the mass of the thing being split increases. You’re not going back in time more than 10 attoseconds (that’s a hundredth of a femtosecond), but that’s good enough for the science fair judges (you’re a kid, and that’s your lunch money at work). You’ll need a piece of thick aluminum foil, a sharp knife or a pin, a bright lamp, superglue (or, in a pinch, Elmer’s), a polarizing sunglass lens, some colored Saran wrap or colored glass, a shoe-box worth of cardboard, and wood + nails  to build some sort of wooden frame to hold everything together. Make your fixture steady and hard to break; judges are clumsy. Use decent wood (judges don’t like splinters). Keep spares for the moving parts in case someone breaks them (not uncommon). Ideally you’ll want to attach some focussing lenses a few inches from the lamp (a small magnifier or reading glass lens will do). You’ll want to lay the colored plastic smoothly over this lens, away from the lamp heat.

First make a point light source: take the 4″ square of shoe-box cardboard and put a quarter-inch hole in it near the center. Attach it in front of your strong electric light at 6″ if there is no lens, or at the focus if there is a lens. If you have no lens, you’ll want to put the Saran over this cardboard.

Take two strips of aluminum foil about 6″ square and in the center of each, cut two slits perhaps 4 mm long by .1 mm wide, 1 mm apart from each other near the middle of both strips. Back both strips with some cardboard with a 1″ hole in the middle (use glue to hold it there). Now take the sunglass lens; cut two strips 2 mm x 10 mm on opposite 45° diagonals to the vertical of the lens. Confirm that this is a polarized lens by rotating one against the other; at some rotation the pieces of sunglass, the pair should be opaque, at 90° it should be fairly clear. If this is not so, get a different sunglass.

Paste these two strips over the two slits on one of the aluminum foil sheets with a drop of super-glue. The polarization of the sunglasses is normally up and down, so when these strips are glued next to one another, the polarization of the strips will be opposing 45° angles. Look at the point light source through both of your aluminum foils (the one with the polarized filter and the one without); they should look different. One should look like two pin-points (or strips) of light. The other should look like a fog of dots or lines.

The reason for the difference is that, generally speaking a photon passes through two nearby slits as two entangled halves, or its quantum equivalent. When you use the foil without the polarizers, the halves recombine to give an interference pattern. The result with the polarization is different though since polarization means you can (in theory at least) tell the photons apart. The photons know this and thus behave like they were not two entangled halves, but rather like they passed either through one slit or the other. Your device will go back in time after the light has gone through the holes and will erase this knowledge.

Now cut another 3″ x 3″ cardboard square and cut a 1/4″ hole in the center. Cut a bit of sunglass lens, 1/2″ square and attach it over the hole of this 3×3″ cardboard square. If you view the aluminum square through this cardboard, you should be able to make one hole or the other go black by rotating this polarized piece appropriately. If it does not, there is a problem.

Set up the lamp (with the lens) on one side so that a bright light shines on the slits. Look at the light from the other side of the aluminum foil. You will notice that the light that comes through the foil with the polarized film looks like two dots, while the one that comes through the other one shows a complex interference pattern; putting the other polarizing lens in front of the foil or behind it does not change the behavior of the foil without the polarizing filters, but if done right it will change things if put behind the other foil, the one with the filters.

Robert Buxbaum, of the future.

yet another quantum joke

Why do you get more energy from a steak than from the same amount of hamburger?

 

Hamburger is steak in the ground state.

 

Is funny because….. it’s a pun on the word ground. Hamburger is ground-up meat, of course, but the reference to a ground state also relates to a basic discovery of quantum mechanics (QM): that all things exist in quantized energy states. The lowest of these is called the ground state, and you get less energy out of a process if you start with things at this ground state. Lasers, as an example, get their energy by electrons being made to drop to their ground state at the same time; you can’t get any energy from a laser if the electrons start in the ground state.

The total energy of a thing can be thought of as having a kinetic and a potential energy part. The potential energy is usually higher the more an item moves from its ideal (lowest potential point). The kinetic energies of though tends to get lower when more space is available because, from Heisenberg uncertainty, ∆l•∆v=h. That is, the more space there is, the less uncertainty of speed, and thus the less kinetic energy other things being equal. The ground energy state is the lowest sum of potential and kinetic energy, and thus all things occupy a cloud of some size, even at the ground state. Without this size, the world would cease to exist. Atoms would radiate energy, and shrink until they vanished.

In grad school, I got into understanding thermodynamics, transport phenomena, and quantum mechanics, particularly involving hydrogen. This lead to my hydrogen production and purification inventions, what my company sells.

Click here for a quantum cartoon on waves and particles, an old Heisenberg joke, or a joke about how many quantum mechanicians it takes to change a lightbulb.

R. E. Buxbaum, July 16, 2013. I once claimed that the unseen process that maintains existence could be called God; this did not go well with the religious.

 

Another Quantum Joke, and Schrödinger’s waves derived

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Quantum mechanics joke. from xkcd.

Quantum mechanics joke. from xkcd.

Is funny because … it’s is a double entente on the words grain (as in grainy) and waves, as in Schrödinger waves or “amber waves of grain” in the song America (Oh Beautiful). In Schrödinger’s view of the quantum world everything seems to exist or move as a wave until you observe it, and then it always becomes a particle. The math to solve for the energy of things is simple, and thus the equation is useful, but it’s hard to understand why,  e.g. when you solve for the behavior of a particle (atom) in a double slit experiment you have to imagine that the particle behaves as an insubstantial wave traveling though both slits until it’s observed. And only then behaves as a completely solid particle.

Math equations can always be rewritten, though, and science works in the language of math. The different forms appear to have different meaning but they don’t since they have the same practical predictions. Because of this freedom of meaning (and some other things) science is the opposite of religion. Other mathematical formalisms for quantum mechanics may be more comforting, or less, but most avoid the wave-particle duality.

The first formalism was Heisenberg’s uncertainty. At the end of this post, I show that it is identical mathematically to Schrödinger’s wave view. Heisenberg’s version showed up in two quantum jokes that I explained (beat into the ground), one about a lightbulb  and one about Heisenberg in a car (also explains why water is wet or why hydrogen diffuses through metals so quickly).

Yet another quantum formalism involves Feynman’s little diagrams. One assumes that matter follows every possible path (the multiple universe view) and that time should go backwards. As a result, we expect that antimatter apples should fall up. Experiments are underway at CERN to test if they do fall up, and by next year we should finally know if they do. Even if anti-apples don’t fall up, that won’t mean this formalism is wrong, BTW: all identical math forms are identical, and we don’t understand gravity well in any of them.

Yet another identical formalism (my favorite) involves imagining that matter has a real and an imaginary part. In this formalism, the components move independently by diffusion, and as a result look like waves: exp (-it) = cost t + i sin t. You can’t observe the two parts independently though, only the following product of the real and imaginary part: (the real + imaginary part) x (the real – imaginary part). Slightly different math, same results, different ways of thinking of things.

Because of quantum mechanics, hydrogen diffuses very quickly in metals: in some metals quicker than most anything in water. This is the basis of REB Research metal membrane hydrogen purifiers and also causes hydrogen embrittlement (explained, perhaps in some later post). All other elements go through metals much slower than hydrogen allowing us to make hydrogen purifiers that are effectively 100% selective. Our membranes also separate different hydrogen isotopes from each other by quantum effects (big things tunnel slower). Among the uses for our hydrogen filters is for gas chromatography, dynamo cooling, and to reduce the likelihood of nuclear accidents.

Dr. Robert E. Buxbaum, June 18, 2013.

To see Schrödinger’s wave equation derived from Heisenberg for non-changing (time independent) items, go here and note that, for a standing wave there is a vibration in time, though no net change. Start with a version of Heisenberg uncertainty: h =  λp where the uncertainty in length = wavelength = λ and the uncertainty in momentum = momentum = p. The kinetic energy, KE = 1/2 p2/m, and KE+U(x) =E where E is the total energy of the particle or atom, and U(x) is the potential energy, some function of position only. Thus, p = √2m(E-PE). Assume that the particle can be described by a standing wave with a physical description, ψ, and an imaginary vibration you can’t ever see, exp(-iωt). And assume this time and space are completely separable — an OK assumption if you ignore gravity and if your potential fields move slowly relative to the speed of light. Now read the section, follow the derivation, and go through the worked problems. Most useful applications of QM can be derived using this time-independent version of Schrödinger’s wave equation.

Do antimatter apples fall up?

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by Dr. Robert E. Buxbaum,

The normal view of antimatter is that it’s just regular matter moving backwards in time. This view helps explain why antimatter has the same mass as regular matter, but has the opposite charge, spin, etc. An antiproton has the same mass as a proton because it is a proton. In our (forward) time-frame the anti-proton appears to be attracted by a positive plate and repelled by a negative one because, when you are going backward in time, attraction looks like repulsion.

In this view, the reason that antimatter particles annihilate when they come into contact with matter –sometimes– is that the annihilation is nothing more than the matter particle (or antimatter) switching direction in time. In our (forward) time-frame it looks like two particles met and both disappeared leaving nothing but photons (light). But in the time reversal view, shown in the figure below, there is only one normal matter particle. In the figure, this particle (solid line) comes from the left, and meets a photon, a wiggly line who’s arrow shows it traveling backwards in time. The normal proton then reverses in time, giving off a photon, another wiggly line. I’d alluded to this in my recent joke about an antimatter person at a bar, but there is also a famous poem.

proton-antiproton

This time reverse approach is best tested using entropy, the classical “arrow of time.” The best way to tell you can tell you are going forward in time is to drop an ice-cube into a hot cup of coffee and produce a warm cup of diluted coffee. This really only shows that you and the cup are moving in the same direction — both forward or both backward, something we’ll call forward. If you were moving in the opposite direction in time, e.g. you had a cup of anti-coffee that was moving backward in time relative to you, you could pull an anti -ice cube out of it, and produce a steaming cup of stronger anti-coffee.

We can not do the entropy test of time direction yet because it requires too much anti matter, but we can use another approach to test the time-reverse idea: gravity. You can make a very small drop of antimatter using only a few hundred atoms. If the antimatter drop is really going backwards in time, it should not fall on the floor and splatter, but should fly upward off the floor and coalesce. The Laboratory at CERN has just recently started producing enough atoms of anti-hydrogen to allow this test. So far the atoms are too hot but sometime in 2014 they expect to cool the atoms, some 300 atoms of anti hydrogen, into a drop or two. They will then see if the drop falls down or up in gravity. The temperature necessary for this study is about 1/100,000 of a degree K.

The anti-time view of antimatter is still somewhat controversial. For it to work, light must reside outside of time, or must move forward and backward in time with some ease. This makes some sense since light travels “at the speed of light,” and is thus outside of time. In the figure, the backwards moving photon would look like a forward on moving in the other direction (left). In a future post I hope to give instructions for building a simple, quantum time machine that uses the fact that light can move backwards in time to produce an event eraser — a device that erases light events in the present. It’s a somewhat useful device, if only for a science fair demonstration. Making one to work on matter would be much harder, and may be impossible if the CERN experiments don’t work out.

It becomes a little confusing how to deal with entropy in a completely anti-time world, and it’s somewhat hard to see why, in this view of time, there should be so little antimatter in the universe and so much matter: you’d expect equal amounts of both. As I have strong feelings for entropy, I’d posted a thought explanation for this some months ago imagining anti matter as normal forward-time matter, and posits the existence of an undiscovered particle that interacts with its magnetism to make matter more stable than antimatter. To see how it works, recall the brainteaser about a tribe that always speaks lies and another that always speaks truth. (I’m not the first to think of this explanation).

If the anti hydrogen drop at CERN is seen to fall upwards, but entropy still works in the positive direction as in my post (i.e. drops still splatter, and anti coffee cools like normal coffee), it will support a simple explanation for dark energy — the force that prevents the universe from collapsing. Dark energy could be seen to result from the antigravity of antimatter. There would have to be large collections of antimatter somewhere, perhaps anti-galaxies isolated from normal galaxies, that would push away the positive matter galaxies while moving forward in time and entropy. If the antigalaxies were close to normal galaxies they would annihilate at the edges, and we’d see lots of photons, like in the poem. Whatever they find at CERN, the future will be interesting. And if time travel turns out to be the norm, the past will be more interesting than it was.

Musical Color and the Well Tempered Scale

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by R. E. Buxbaum, (the author of all these posts)

I first heard J. S. Bach’s Well Tempered Clavier some 35 years ago and was struck by the different colors of the different scales. Some were dark and scary, others were light and enjoyable. All of them worked, but each was distinct, though I could not figure out why. That Bach was able to write in all the keys without retuning was a key innovation of his. In his day, people tuned in fifths, a process that created gaps (called wolf) that prevented useful composition in affected keys.

We don’t know exactly how Bach tuned his instruments as he had no scientific way to describe it; we can guess that it was more uniform than the temper produced by tuning in fifths, but it probably was not quite equally spaced. Nowadays electronic keyboards are tuned to 12 equally spaced frequencies per octave through the use of frequency counters.  Starting with the A below “middle C”, A4, tuned at 440 cycles/second (the note symphonies tune to), each note is programmed to vibrate at a wavelength that is lower or higher than one next to it by a factor of the twelfth root of two, 12√2= 1.05946. After 12 multiples of this size, the wavelength has doubled or halved and there is an octave. This is called equal tempering.

Currently, many non-electric instruments are also tuned this way.  Equally tempering avoids all wolf, but makes each note equally ill-tempered. Any key can be transposed to another, but there are no pure harmonies because 12√2 is an irrational number (see joke). There is also no color or feel to any given key except that which has carried over historically in the listeners’ memory. It’s sad.

I’m going to speculate that J.S. Bach found/ favored a way to tune instruments where all of the keys were usable, and OK sounding, but where some harmonies are more perfect than others. Necessarily this means that some harmonies will be less-perfect. There should be no wolf gaps that would sound so bad that Bach could not compose and transpose in every key, but since there is a difference, each key will retain a distinct color that JS Bach explored in his work — or so I’ll assume.

Pythagoras found that notes sound best together when the vibrating lengths are kept in a ratio of small numbers. Consider the tuning note, A4, the A below middle C; this note vibrates a column of air .784 meters long, about 2.5 feet or half the length of an oboe. The octave notes for A4 are called A3 and A5. They vibrate columns of air 2x as long and 1/2 as long as the original. They’re called octaves because they’re eight white keys away from A4. Keyboards add 4 black notes per octave so octaves are always 12 notes away. Keyboards are generally tuned so octaves are always 12 keys away. Based on Pythagoras, a reasonable presumption is that J.S Bach tuned every non-octave note so that it vibrates an air column similar to the equal tuning ratio, 12√2 = 1.05946, but whose wavelength was adjusted, in some cases to make ratios of small, whole numbers with the wavelength for A4.

Aside from octaves, the most pleasant harmonies are with notes whose wavelength is 3/2 as long as the original, or 2/3 as long. The best harmonies with A4 (0.784 m) will be with notes with wavelengths (3/2)*0.784 m long, or (2/3)*0.784m long. The first of these is called D3 and the other is E4. A4 combines with D3 to make a chord called D-major, the so-called “the key of glory.” The Hallelujah chorus, Beethoven’s 9th (Ode to Joy), and Mahler’s Titan are in this key. Scriabin believed that D-major had a unique color, gold, suggesting that the pure ratios were retained.

A combines with E (plus a black note C#) to make a chord called A major. Songs in this key sound (to my ear) robust, cheerful and somewhat pompous; Here, in A-major is: Dancing Queen by ABBA, Lady Madonna by the Beatles; Prelude and Fugue in A major by JS Bach. Scriabin believed that A-major was green.

A4 also combines with E and a new white note, C3, to make a chord called A minor. Since E4 and E3 vibrate at 2/3 and 4/3 the wavelength of A4 respectively, I’ll speculate that Bach tuned C3 to 5/3 the length of A4; 5/3*.0784m =1.307m long. Tuned his way, the ratio of wavelengths in the A minor chord are 3:4:5. Songs in A minor tend to be edgy and sort-of sad: Stairway to heaven, Für Elise, “Songs in A Minor sung by Alicia Keys, and PDQ Bach’s Fugue in A minor. I’m going to speculate the Bach tuned this to 1.312 m (or thereabouts), roughly half-way between the wavelength for a pure ratio and that of equal temper.

The notes D3 and E3 will not sound particularly good together. In both pure ratios and equal tempers their wavelengths are in a ratio of 3/2 to 4/3, that is a ratio of 9 to 8. This can be a tensional transition, but it does not provide a satisfying resolution to my, western ears.

Now for the other white notes. The next white key over from A4 is G3, two half-tones longer that for A4. For equal tuning, we’d expect this note to vibrate a column of air 1.059462 = 1.1225 times longer than A4. The most similar ratio of small whole numbers is 9/8 = 1.1250, and we’d already generated one before between D and E. As a result, we may expect that Bach tuned G3 to a wavelength 9/8*0.784m = .88 meters.

For equal tuning, the next white note, F3, will vibrate an air column 1.059464 = 1.259 times as long as the A4 column. Tuned this way, the wavelength for F3 is 1.259*.784 = .988m. Alternately, since 1.259 is similar to 5/4 = 1.25, it is reasonable to tune F3 as (5/4)*.784 = .980m. I’ll speculate that he split the difference: .984m. F, A, and C combine to make a good harmony called the F major chord. The most popular pieces in F major sound woozy and not-quite settled in my opinion, perhaps because of the oddness of the F tuning. See, e.g. the Jeopardy theme song, “My Sweet Lord,” Come together (Beetles), Beethoven’s Pastoral symphony (Movement 1, “Awakening of cheerful feelings upon arrival in the country”). Scriabin saw F-major as bright blue.

We’ve only one more white note to go in this octave: B4, the other tension note to A4. Since the wavelengths for G3 was 9/8 as long as for A4, we can expect the wavelength for B4 will be 8/9 as long. This will be dissonant to A4, but it will go well with E3 and E4 as these were 2/3 and 4/3 of A4 respectively. Tuned this way, B4 vibrates a column 1.40 m. When B, in any octave, is combined with E it’s called an E chord (E major or E minor); it’s typically combined with a black key, G-sharp (G#). The notes B, E vibrate at a ratio of 4 to 3. J.S. Bach called the G#, “H” allowing him to spell out his name in his music. When he played the sequence BACH, he found B to A created tension; moving to C created harmony with A, but not B, while the final note, G# (H) provided harmony for C and the original B. Here’s how it works on cello; it’s not bad, but there is no grand resolution. The Promenade from “Pictures at an Exhibition” is in E.

The black notes go somewhere between the larger gaps of the white notes, and there is a traditional confusion in how to tune them. One can tune the black notes by equal temper  (multiples of 21/12), or set them exactly in the spaces between the white notes, or tune them to any alternate set of ratios. A popular set of ratios is found in “Just temper.” The black note 6 from A4 (D#) will have wavelength of 0.784*26/12= √2 *0.784 m =1.109m. Since √2 =1.414, and that this is about 1.4= 7/5, the “Just temper” method is to tune D# to 1.4*.784m =1.098m. If one takes this route, other black notes (F#3 and C#3) will be tuned to ratios of 6/5, and 8/5 times 0.784m respectively. It’s possible that J.S. Bach tuned his notes by Just temper, but I suspect not. I suspect that Bach tuned these notes to fall in-between Just Temper and Equal temper, as I’ve shown below. I suspect that his D#3 might vibrated at about 1.104 m, half way between Just and Equal temper. I would not be surprised if Jazz musicians tuned their black notes more closely to the fifths of Just temper: 5/5 6/5, 7/5, 8/5 (and 9/5?) because jazz uses the black notes more, and you generally want your main chords to sound in tune. Then again, maybe not. Jimmy Hendrix picked the harmony D#3 with A (“Diabolus”, the devil harmony) for his Purple Haze; it’s also used for European police sirens.

To my ear, the modified equal temper is more beautiful and interesting than the equal temperament of todays electronic keyboards. In either temper music plays in all keys, but with an un-equal temper each key is distinct and beautiful in its own way. Tuning is engineering, I think, rather than math or art. In math things have to be perfect; in art they have to be interesting, and in engineering they have to work. Engineering tends to be beautiful its way. Generally, though, engineering is not perfect.

Summary of air column wave-lengths, measured in meters, and as a ratio to that for A4. Just Tempering, Equal Tempering, and my best guess of J.S. Bach's Well Tempered scale.

Summary of air column wave-lengths, measured in meters, and as a ratio to that for A4. Just Tempering, Equal Tempering, and my best guess of J.S. Bach’s Well Tempered scale.

R.E. Buxbaum, May 20 2013 (edited Sept 23, 2013) — I’m not very musical, but my children are.

The Gift of Chaos

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Many, if not most important engineering systems are chaotic to some extent, but as most college programs don’t deal with this behavior, or with this type of math, I thought I might write something on it. It was a big deal among my PhD colleagues some 30 years back as it revolutionized the way we looked at classic problems; it’s fundamental, but it’s now hardly mentioned.

Two of the first freshman engineering homework problems I had turn out to have been chaotic, though I didn’t know it at the time. One of these concerned the cooling of a cup of coffee. As presented, the coffee was in a cup at a uniform temperature of 70°C; the room was at 20°C, and some fanciful data was presented to suggest that the coffee cooled at a rate that was proportional the difference between the (changing) coffee temperature and the fixed room temperature. Based on these assumptions, we predicted exponential cooling with time, something that was (more or less) observed, but not quite in real life. The chaotic part in a real cup of coffee, is that the cup develops currents that move faster and slower. These currents accelerate heat loss, but since they are driven by the temperature differences within the cup they tend to speed up and slow down erratically. They accelerate when the cup is not well stirred, causing new stir, and slow down when it is stirred, and the temperature at any point is seen to rise and fall in an almost rhythmic fashion; that is, chaotically.

While it is impossible to predict what will happen over a short time scale, there are some general patterns. Perhaps the most remarkable of these is self-similarity: if observed over a short time scale (10 seconds or less), the behavior over 10 seconds will look like the behavior over 1 second, and this will look like the behavior over 0.1 second. The only difference being that, the smaller the time-scale, the smaller the up-down variation. You can see the same thing with stock movements, wind speed, cell-phone noise, etc. and the same self-similarity can occur in space so that the shape of clouds tends to be similar at all reasonably small length scales. The maximum average deviation is smaller over smaller time scales, of course, and larger over large time-scales, but not in any obvious way. There is no simple proportionality, but rather a fractional power dependence that results in these chaotic phenomena having fractal dependence on measure scale. Some of this is seen in the global temperature graph below.

Global temperatures measured from the antarctic ice showing stable, cyclic chaos and self-similarity.

Global temperatures measured from the antarctic ice showing stable, cyclic chaos and self-similarity.

Chaos can be stable or unstable, by the way; the cooling of a cup of coffee was stable because the temperature could not exceed 70°C or go below 20°C. Stable chaotic phenomena tend to have fixed period cycles in space or time. The world temperature seems to follow this pattern though there is no obvious reason it should. That is, there is no obvious maximum and minimum temperature for the earth, nor any obvious reason there should be cycles or that they should be 120,000 years long. I’ll probably write more about chaos in later posts, but I should mention that unstable chaos can be quite destructive, and quite hard to prevent. Some form of chaotic local heating seems to have caused battery fires aboard the Dreamliner; similarly, most riots, famines, and financial panics seem to be chaotic. Generally speaking, tight control does not prevent this sort of chaos, by the way; it just changes the period and makes the eruptions that much more violent. As two examples, consider what would happen if we tried to cap a volcano, or provided  clamp-downs on riots in Syria, Egypt or Ancient Rome.

From math, we know some alternate ways to prevent unstable chaos from getting out of hand; one is to lay off, another is to control chaotically (hard to believe, but true).

 

Joke about antimatter and time travel

I’m sorry we don’t serve antimatter men here.

Antimatter man walks into a bar.

Is funny because … in quantum-physics there is no directionality in time. Thus an electron can change directions in time and then appears to the observer as a positron, an anti electron that has the same mass as a normal electron but the opposite charge and an opposite spin, etc. In physics, the reason electrons and positrons appear to annihilate is that it’s there was only one electron to begin with. That electron started going backwards in time so it disappeared in our forward-in-time time-frame.

The thing is, time is quite apparent on a macroscopic scales. It’s one of the most apparent aspects of macroscopic existence. Perhaps the clearest proof that time is flowing in one direction only is entropy. In normal life, you can drop a glass and watch it break whenever you like, but you can not drop shards and expect to get a complete glass. Similarly, you know you are moving forward in time if you can drop an ice cube into a hot cup of coffee and make it luke-warm. If you can reach into a cup of luke-warm coffee and extract an ice cube to make it hot, you’re moving backwards in time.

It’s also possible that gravity proves that time is moving forward. If an anti apple is just a normal apple that is moving backwards in time, then I should expect that, when I drop an anti-apple, I will find it floats upward. On the other hand, if mass is inherently a warpage of space-time, it should fall down. Perhaps when we understand gravity we will also understand how quantum physics meets the real world of entropy.

Joke re: SI pressure

Einstein, Newton, and the two Pascal brothers are playing hide and seek. Einstein has his eyes covered and is counting. The two Pascal bothers run and hide but Isaac Newton does not. He draws a square around him in the dust and stands waiting. When Einstein finishes counting he says, “I see you Sir Isaac standing there.” “No you don’t.” says Newton. “You see two Pascals: there’s one Newton in half a square meter area.