Category Archives: Philosophy

The Scientific Method isn’t the method of scientists

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A linchpin of middle school and high-school education is teaching ‘the scientific method.’ This is the method, students are led to believe, that scientists use to determine Truths, facts, and laws of nature. Scientists, students are told, start with a hypothesis of how things work or should work, they then devise a set of predictions based on deductive reasoning from these hypotheses, and perform some critical experiments to test the hypothesis and determine if it is true (experimentum crucis in Latin). Sorry to say, this is a path to error, and not the method that scientists use. The real method involves a few more steps, and follows a different order and path. It instead follows the path that Sherlock Holmes uses to crack a case.

The actual method of Holmes, and of science, is to avoid beginning with a hypothesis. Isaac Newton claimed: “I never make hypotheses” Instead as best we can tell, Newton, like most scientists, first gathered as much experimental evidence on a subject as possible before trying to concoct any explanation. As Holmes says (Study in Scarlet): “It is a capital mistake to theorize before you have all the evidence. It biases the judgment.”

It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts (Holmes, Scandal in Bohemia).

Holmes barely tolerates those who hypothesize before they have all the data: “It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts.” (Scandal in Bohemia).

Then there is the goal of science. It is not the goal of science to confirm some theory, model, or hypothesis; every theory probably has some limited area where it’s true. The goal for any real-life scientific investigation is the desire to explain something specific and out of the ordinary, or do something cool. Similarly, with Sherlock Holmes, the start of the investigation is the arrival of a client with a specific, unusual need – one that seems a bit outside of the normal routine. Similarly, the scientist wants to do something: build a bigger bridge, understand global warming, or how DNA directs genetics; make better gunpowder, cure a disease, or Rule the World (mad scientists favor this). Once there is a fixed goal, it is the goal that should direct the next steps: it directs the collection of data, and focuses the mind on the wide variety of types of solution. As Holmes says: , “it’s wise to make one’s self aware of the potential existence of multiple hypotheses, so that one eventually may choose one that fits most or all of the facts as they become known.” It’s only when there is no goal, that any path will do. 

In gathering experimental data (evidence), most scientists spend months in the less-fashionable sections of the library, looking at the experimental methods and observations of others, generally from many countries, collecting any scrap that seems reasonably related to the goal at hand. I used 3 x5″ cards to catalog this data and the references. From many books and articles, one extracts enough diversity of data to be able to look for patterns and to begin to apply inductive logic. “The little things are infinitely the most important” (Case of Identity). You have to look for patterns in the data you collect. Holmes does not explain how he looks for patterns, but this skill is innate in most people to a greater or lesser extent. A nice set approach to inductive logic is called the Baconian Method, it would be nice to see schools teach it. If the author is still alive, a scientist will try to contact him or her to clarify things. In every SH mystery, Holmes does the same and is always rewarded. There is always some key fact or observation that this turns up: key information unknown to the original client.

Based on the facts collected one begins to create the framework for a variety of mathematical models: mathematics is always involved, but these models should be pretty flexible. Often the result is a tree of related, mathematical models, each highlighting some different issue, process, or problem. One then may begin to prune the tree, trying to fit the known data (facts and numbers collected), into a mathematical picture of relevant parts of this tree. There usually won’t be quite enough for a full picture, but a fair amount of progress can usually be had with the application of statistics, calculus, physics, and chemistry. These are the key skills one learns in college, but usually the high-schooler and middle schooler has not learned them very well at all. If they’ve learned math and physics, they’ve not learned it in a way to apply it to something new, quite yet (it helps to read the accounts of real scientists here — e.g. The Double Helix by J. Watson).

Usually one tries to do some experiments at this stage. Homes might visit a ship or test a poison, and a scientist might go off to his, equally-smelly laboratory. The experiments done there are rarely experimenti crucae where one can say they’ve determined the truth of a single hypothesis. Rather one wants to eliminated some hypotheses and collect data to be used to evaluate others. An answer generally requires that you have both a numerical expectation and that you’ve eliminated all reasonable explanations but one. As Holmes says often, e.g. Sign of the four, “when you have excluded the impossible, whatever remains, however improbable, must be the truth”. The middle part of a scientific investigation generally involves these practical experiments to prune the tree of possibilities and determine the coefficients of relevant terms in the mathematical model: the weight or capacity of a bridge of a certain design, the likely effect of CO2 on global temperature, the dose response of a drug, or the temperature and burn rate of different gunpowder mixes. Though not mentioned by Holmes, it is critically important in science to aim for observations that have numbers attached.

The destruction of false aspects and models is a very important part of any study. Francis Bacon calls this act destruction of idols of the mind, and it includes many parts: destroying commonly held presuppositions, avoiding personal preferences, avoiding the tendency to see a closer relationship than can be justified, etc.

In science, one eliminates the impossible through the use of numbers and math, generally based on your laboratory observations. When you attempt to the numbers associated with our observations to the various possible models some will take the data well, some poorly; and some twill not fit the data at all. Apply the deductive reasoning that is taught in schools: logical, Boolean, step by step; if some aspect of a model does not fit, it is likely the model is wrong. If we have shown that all men are mortal, and we are comfortable that Socrates is a man, then it is far better to conclude that Socrates is mortal than to conclude that all men but Socrates is mortal (Occam’s razor). This is the sort of reasoning that computers are really good at (better than humans, actually). It all rests on the inductive pattern searches similarities and differences — that we started with, and very often we find we are missing a piece, e.g. we still need to determine that all men are indeed mortal, or that Socrates is a man. It’s back to the lab; this is why PhDs often take 5-6 years, and not the 3-4 that one hopes for at the start.

More often than not we find we have a theory or two (or three), but not quite all the pieces in place to get to our goal (whatever that was), but at least there’s a clearer path, and often more than one. Since science is goal oriented, we’re likely to find a more efficient than we fist thought. E.g. instead of proving that all men are mortal, show it to be true of Greek men, that is for all two-legged, fairly hairless beings who speak Greek. All we must show is that few Greeks live beyond 130 years, and that Socrates is one of them.

Putting numerical values on the mathematical relationship is a critical step in all science, as is the use of models — mathematical and otherwise. The path to measure the life expectancy of Greeks will generally involve looking at a sample population. A scientist calls this a model. He will analyze this model using statistical model of average and standard deviation and will derive his or her conclusions from there. It is only now that you have a hypothesis, but it’s still based on a model. In health experiments the model is typically a sample of animals (experiments on people are often illegal and take too long). For bridge experiments one uses small wood or metal models; and for chemical experiments, one uses small samples. Numbers and ratios are the key to making these models relevant in the real world. A hypothesis of this sort, backed by numbers is publishable, and is as far as you can go when dealing with the past (e.g. why Germany lost WW2, or why the dinosaurs died off) but the gold-standard of science is predictability.  Thus, while we a confident that Socrates is definitely mortal, we’re not 100% certain that global warming is real — in fact, it seems to have stopped though CO2 levels are rising. To be 100% sure you’re right about global warming we have to make predictions, e.g. that the temperature will have risen 7 degrees in the last 14 years (it has not), or Al Gore’s prediction that the sea will rise 8 meters by 2106 (this seems unlikely at the current time). This is not to blame the scientists whose predictions don’t pan out, “We balance probabilities and choose the most likely. It is the scientific use of the imagination” (Hound of the Baskervilles). The hope is that everything matches; but sometimes we must look for an alternative; that’s happened rarely in my research, but it’s happened.

You are now at the conclusion of the scientific process. In fiction, this is where the criminal is led away in chains (or not, as with “The Woman,” “The Adventure of the Yellow Face,” or of “The Blue Carbuncle” where Holmes lets the criminal free — “It’s Christmas”). For most research the conclusion includes writing a good research paper “Nothing clears up a case so much as stating it to another person”(Memoirs). For a PhD, this is followed by the search for a good job. For a commercial researcher, it’s a new product or product improvement. For the mad scientist, that conclusion is the goal: taking over the world and enslaving the population (or not; typically the scientist is thwarted by some detail!). But for the professor or professional research scientist, the goal is never quite reached; it’s a stepping stone to a grant application to do further work, and from there to tenure. In the case of the Socrates mortality work, the scientist might ask for money to go from country to country, measuring life-spans to demonstrate that all philosophers are mortal. This isn’t as pointless and self-serving as it seems, Follow-up work is easier than the first work since you’ve already got half of it done, and you sometimes find something interesting, e.g. about diet and life-span, or diseases, etc. I did some 70 papers when I was a professor, some on diet and lifespan.

One should avoid making some horrible bad logical conclusion at the end, by the way. It always seems to happen that the mad scientist is thwarted at the end; the greatest criminal masterminds are tripped by some last-minute flaw. Similarly the scientist must not make that last-mistep. “One should always look for a possible alternative, and provide against it” (Adventure of Black Peter). Just because you’ve demonstrated that  iodine kills germs, and you know that germs cause disease, please don’t conclude that drinking iodine will cure your disease. That’s the sort of science mistakes that were common in the middle ages, and show up far too often today. In the last steps, as in the first, follow the inductive and quantitative methods of Paracelsus to the end: look for numbers, (not a Holmes quote) check how quantity and location affects things. In the case of antiseptics, Paracelsus noticed that only external cleaning helped and that the help was dose sensitive.

As an example in the 20th century, don’t just conclude that, because bullets kill, removing the bullets is a good idea. It is likely that the trauma and infection of removing the bullet is what killed Lincoln, Garfield, and McKinley. Theodore Roosevelt was shot too, but decided to leave his bullet where it was, noticing that many shot animals and soldiers lived for years with bullets in them; and Roosevelt lived for 8 more years. Don’t make these last-minute missteps: though it’s logical to think that removing guns will reduce crime, the evidence does not support that. Don’t let a leap of bad deduction at the end ruin a line of good science. “A few flies make the ointment rancid,” said Solomon. Here’s how to do statistics on data that’s taken randomly.

Dr. Robert E. Buxbaum, scientist and Holmes fan wrote this, Sept 2, 2013. My thanks to Lou Manzione, a friend from college and grad school, who suggested I reread all of Holmes early in my PhD work, and to Wikiquote, a wonderful site where I found the Holmes quotes; the Solomon quote I knew, and the others I made up.

Mechanical Engineer v Civil Engineer Joke

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What’s the difference between a mechanical engineer and a civil engineer?

 

 

Mechanical engineers make weapons; civil engineers make targets.

 

 

Is funny because ….. it’s sort of true. Much of engineering is war-related, and always was. In earlier times, an engineer was someone who made engines of war: catapults, battering rams, and the like. Nowadays, mechanical engineers are the main designers for tanks, cannons, and ships. A civil engineer is one whose projects have civilian applications. But as these projects have military uses (roads, ports, offices, and bridges, for example), civilian projects are major targets for an opposing army.

An observation about war and peace: if you are really at making peacetime products, you’re a hero in your country and outside; if you design weapons, you are vilified by the enemy and likely to become a prisoner in your own land. Consider the designers of the atom bomb in the US, Russia, Israel, India, or Iran. They can’t go abroad, and are likely suspect at home. The leaders have to worry that these scientists will give the same weapons to their enemies (it’s happened) or that they will not be dedicated enough to make the next iteration of the weapon (ditto).

My advice: specialize items for peacetime or civilian use if you can. Those who make better cars, music, art or architecture are welcome everywhere; advances in death usually rebound on the inventor. Here’s a joke comparing chemists and chemical engineers, a piece on a favorite car engine advance, on perfect tuning of musical instruments, an architecture joke, and a control engineer joke. People like civil engineers.

What sort of guy does a king keep locked in the castle dungeon — not the common thief.  #wordstothewise.

R. E. Buxbaum, August 1, 2013. I’m a chemical engineer, who makes hydrogen stuff and consults, mostly for peace-time use.

Crime: US vs UK and Canada

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The US has a lot of guns and a lot of murders compared to England, Canada, and most of Europe. This is something Piers Morgan likes to point out to Americans who then struggle to defend the wisdom of gun ownership and the 2nd Amendment: “How do you justify 4.8 murders/year per 100,000 population when there are only 1.6/year per 100,000 in Canada, 1.2/year per 100,000 in the UK, and 1.0/year per 100,000 in Australia — countries with few murders and tough anti-gun laws?,” he asks. What Piers doesn’t mention, is that these anti-gun countries have far higher contact crime (assault) rates than the US, see below.

Contact Crime Per Country

Contact crime rates for 17 industrialized countries. From the Dutch Ministry of Justice. Click here for details about the survey and a breakdown of crimes.

The differences narrow somewhat when considering most violent crimes, but we still have far fewer than Canada and the UK. Canada has 963/year per 100,000 “most violent crimes,” while the US has 420/year per 100,000. “Most violent crimes” here are counted as: “murder and non-negligent manslaughter,” “forcible rape,” “robbery,” and “aggravated assault” (FBI values). England and Wales classify crimes somewhat differently, but have about two times the US rate, 775/year per 100,000, if “most violent crimes” are defined as: “violence against the person, with injury,” “most serious sexual crime,” and “robbery.”

It is possible that the presence of guns protects Americans from general crime while making murder more common, but it’s also possible that gun ownership is a murder deterrent too. Our murder rate is 1/5 that of Mexico, 1/4 that of Brazil, and 1/3 that of Russia; all countries with strong anti-gun laws but a violent populous. Perhaps the US (Texan) penchant for guns is what keeps Mexican gangs on their, gun-control side of the border. Then again, it’s possible that guns neither increase nor decrease murder rates, so that changing our laws would not have any major effect. Switzerland (a country with famously high gun ownership) has far fewer murders than the US and about 1/2 the rate of the UK: 0.7 murders/ year per 100,000. Japan, a country with low gun ownership has hardly any crime of any sort — not even littering. As in the zen buddhist joke, change comes from within.

Homicide rate per country

Homicide rate per country

One major theory for US violence was that drugs and poverty were the causes. Remove these by stricter anti-drug laws and government welfare, and the violent crime would go away. Sorry to say, it has not happened; worse yet, murder rates are highest in cities like Detroit where welfare is a way of life, and where a fairly high fraction of the population is in prison for drugs.

I suspect that our welfare payments have hurt Detroit as much as they’ve helped, and that Detroit’s higher living wage, has made it hard for people to find honest work. Stiff drug penalties have not helped Detroit either, and may contribute to making crimes more violent. As Thomas More pointed out in the 1500s, if you are going to prison for many years for a small crime, you’re more likely to use force to avoid risk capture. Perhaps penalties would work better if they were smaller.

Charity can help a city, i think, and so can good architecture. I’m on the board of two charities that try to do positive things, and I plant trees in Detroit (sometimes).

R. E. Buxbaum, July 10, 2013. To make money, I sell hydrogen generators: stuff I invented, mostly.

Control engineer joke

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What made the control engineer go crazy?

 

He got positive feedback.

Is funny because …… it’s a double entente, where both meanings are true: (1) control engineers very rarely get compliments (positive feedback); the aim of control is perfection, something that’s unachievable for a dynamic system (and generally similar to near perfection: the slope at a maximum is zero). Also (2) systems go unstable if the control feedback is positive. This can happen if the controller was set backwards, but more usually happens when the response is too fast or too extreme. Positive feedback pushes a system further to error and the process either blows up, or (more commonly) goes wildly chaotic, oscillating between two or more “strange attractor” states.

It seems to me that hypnosis, control-freak love, and cult behaviors are the result of intentionally produced positive feedback. Palsies, economic cycles, and global warming are more likely the result of unintentional positive feedback. In each case, the behavior is oscillatory chaotic.

The normal state of Engineering is lack of feedback. Perhaps this is good because messed up feedback leads to worse results. From xykd.

Our brains give little reliable feedback on how well they work, but that may be better than strong, immediate feedback, as that could lead to bipolar instability. From xkcd. For more on this idea, see Science and Sanity, by Alfred Korzbski (mini youtube)

Control engineers tend to be male (85%), married (80%), happy people (at least they claim to be happy). Perhaps they know that near-perfection is close enough for a complex system in a dynamic world, or that one is about as happy as believes ones-self to be. It also helps that control engineer salaries are about $95,000/ year with excellent benefits and low employment turnover.

Here’s a chemical engineer joke I made up, and an older engineering joke. If you like, I’ll be happy to consult with you on the behavior of your processes.

By Dr. Robert E. Buxbaum, July 4, 2013

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.

Surrealism Jokes

What is it that is red and white, polka-dotted, filled with moisture, and hangs from trees in the winter?

 

Unity

 

Is funny because …… it’s more true than truth. Whatever claims to be unity must include the red and white, polka-dotted, moist items that hang from trees. Otherwise it wouldn’t be unity. Surrealism jokes should not be confused with Zen Jokes. Eg this. and that.  As a practical matter, you can tell surrealists from Buddhists because surrealists are drunks and have hair. And you know why Dali wore a mustache?

 

To pass unobserved

Dali's mustache without dali; notice how the mustache obscures the man.

Dali’s mustache without Dali, from Dali’s Mustache, the only book (to my knowledge) about a part of an artist. There are many books about Picasso, for example, but none about his left foot.

See how it’s true. The mustache takes the place of the man, standing in for him, or here the lack of him. Surrealism sees the absurd dream realism that is beyond the surd. “If you act the genius you will be one.” See? It even speaks for him, when needed.

Dali and his mustache agree, they love art for art's sake.

Dali and his mustache agree, they love art for art’s sake.

So how many surrealists does it take to screw in a lightbulb?  The fish.

by R. E. Buxbaum, June 14, 2013

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.

Religion vs Philosophy joke

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“A philosopher is a blind man in a dark room looking for a black cat that isn’t there. A theologian is the man who finds it.” ~ H. L. Mencken

The distinction joke here is more sad than funny, I would say. It speaks to the inability of people to grapple with the big questions of their life in any really rational way. We’d like to be able to communicate directly with God, and have him speak back, but we can’t quite, and at some level we’d be too small for the interaction. We’d like to be able to stop evil with our religion, by holding up a cross, say, or by squirting holy water, but we can’t. I suspect it’s better that way, but sad. We’d like to know how and why the universe came to be, and what happens after death, but our best rational efforts are helpless. All of this is as they should be, says the philosopher, and he’s right, but it’s sad that it is and that he is. And then the theologian (rabbi, priest, imam) says he’s got all the answers and all the powers too. It’s too sad for words.

The philosopher in this joke is (I imagine) a PhD scientist, like me. While rational thought is great, and a PhD scientist can actually predict quite a lot that will happen in some cases, we have no real clue as to why things happen — except in terms of other things that we can’t explain: forces, gravity, electrons. It seems clear that the answer to the big-issue questions can not be found in science or rational philosophy. Nor can science deal well with one-time events like the creation of the universe, or unmeasurable items like where the apparent zero-point energy of quantum mechanics comes from. Untestable, one time events are the basis of religion and not science: science is the opposite of religion.

We thus turn to the theologian. In a sense, he has the answer: it’s God, Jesus, Jihad, prayer… Perhaps these words mean the same thing, or perhaps something different. A theologian can talk about this for hours. He has all the answers, but when he’s done, he’s left them as incomprehensible as before. Likely he is as confused as we are, but he doesn’t know it, or show it. While something like God does seem to underly the concept of time, or creation (the big bang), a one-word answer, like “God” isn’t really an answer. Even though there appears to be a God, God doesn’t seem contained within the word — he’s not there. And calling “God” doesn’t give us the power we’d want: it does not save the drowning, or cure disease.

Though the theologian will likely tell you miracle stories, and show you a pretty picture: long-haired Jesus, seated Zeus, or a dancing woman with the head of an elephant, that’s God and it isn’t. The reason people believe the theologian, is optimism: we hope he knows, though we know he doesn’t. Besides, the theologian has a costume and an audience, and that helps. He keeps on talking till he wears the audience down. Eventually we believe he sees the black cat in the dark room called God. Eventually we don’t care that he can’t do anything on the physical plane. Theologians work in pairs to increase their believability: one tells you the other is much smarter and holier than you; the other one tells you the same about the first. Eventually, you believe them both — or at least you believe you are stupider and eviler than they are.

A wise and good philosopher or theologian is very hard to find. He doesn’t talk too much, and instead lets his fine example do the teaching. He does charity and justice (Gen. 18:18) and makes good lemonade from the lemons life gives him. He will admit that he doesn’t really know which set of words and bows actually open up God’s warehouse (or if any are particularly effective) “God speaks within a cloud” (Ex. 40:34, etc.); “[His] thoughts are not our thoughts,” (Is. 55:8, etc.). “No man can see my face and live” (Ex. 33:20).

What percentage of leaders are like this? “In a thousand, I have found one leader of men”, says Solomon (Eccles 7:28). “The other 999 follow after the women” (Groucho Marx).

My hope with this blog post is not to diminish the good of rabbis, priests, or other theologians, but rather that you will not finish reading the post thinking you are stupid or evil for not understanding your theologian’s many words. Also, I can hope that you will seek justice, help the downtrodden, and make yourself into something of value. Then again, you might be tempted to run off to a bad theologian — to someone who will encourage you to pray long and hard, and who will get you to pay him for a picture of God that only he can provide — that is, for his special picture of the black cat, in the dark room, that can never be photographed.

Robert E. Buxbaum; Amateur philosopher, and maker of a good glass of lemonade.

Zen and the hotdog vendor (a joke)

What did the Zen master ask from the hot dog vendor?

“Can you make me one — with everything?”

The vendor (so the story goes) replied “That will be $1.50.” The Master handed him $10 and the vendor handed him a hot-dog and said, “change comes from within.” (thought you’d like to know).

If you think this is funny, you may also like my previous Zen joke or (for all I know), my recent personal relationship cartoon.