You are your own sculpture; Be art.
Here I am wearing a sculpture I made, called Gilroy. The Idea is based on the drawings of Kilroy made during WW2, but to make things spookier the eyes follow you as shown in this video. I suspect that the original drawings were made to discredit the Nazi’s by undermining the sense that they brought order and were the inevitable power in the area.
Feb. 2013 – March, 2015
No one quite knows how nerve cells learn stuff. It is incorrectly thought that you can not get new nerves in the brain, nor that you can get brain cells to grow out further, but people have made new nerve cells, and when I was a professor at Michigan State, a Physiology colleague and I got brain and sensory nerves to grow out axons by pulling on them without the use of drugs.
I had just moved to Michigan State as a fresh PhD (Princeton) as an assistant professor of chemical engineering. Steve Heidemann was a few years ahead of me, a Physiology professor PhD from Princeton. We were both new Yorkers. He had been studying nerve structure, and wondered about how the growth cone makes nerves grow out axons (the axon is the long, stringy part of the nerve). A thought was that nerves were structured as Snelson-Fuller tensegrity structures, but it was not obvious how that would relate to growth or anything else. A Snelson-Fuller structure is shown below the structure stands erect not by compression, as in a pyramid or igloo, but rather because tension in the wires helps lift the metal pipes, and puts them in compression. The nerve cell, shown further below is similar with actin-protein as the outer, tensed skin, and a microtubule-protein core as the compress pipes.
A Snelson-Fuller tensegrity sculpture in the graduate college courtyard at Princeton, an inspiration for our work.
Biothermodynamics was pretty basic 30 years ago (It still is today), and it was incorrectly thought that objects were more stable when put in compression. It didn’t take too much thermodynamics on my part to show otherwise, and so I started a part-time career in cell physiology. Consider first how mechanical force should affect the Gibbs free energy, G, of assembled microtubules. For any process at constant temperature and pressure, ∆G = work. If force is applied we expect some elastic work will be put into the assembled Mts in an amount ∫f dz, where f is the force at every compression, and ∫dz is the integral of the distance traveled. Assuming a small force, or a constant spring, f = kz with k as the spring constant. Integrating the above, ∆G = ∫kz dz = kz2; ∆G is always positive whether z is positive or negative, that is the microtubule is most stable with no force, and is made less stable by any force, tension or compression.
A cell showing what appears to be tensegrity. The microtubules (green) surrounded by actin (red). In nerves Heidemann and I showed actin is in tension the microtubules in compression.
Assuming that microtubules in the nerve- axon are generally in compression as in the Snelson-Fuller structure, then pulling on the axon could potentially reduce the compression. Normally, this is done by a growth cone, we posited, but we could also do it by pulling. In either case, a decrease in the compression of the assembled microtubules should favor microtubule assembly.
To calculate the rates, I used absolute rate theory, something I’d learned from Dr. Mortimer Kostin, a most-excellent thermodynamics professor. I assumed that the free energy of the monomer was unaffected by force, and that the microtubules were in pseudo- equilibrium with the monomer. Growth rates were predicted to be proportional to the decrease in G, and the prediction matched experimental data.
Our few efforts to cure nerve disease by pulling did not produce immediate results; it turns out to by hard to pull on nerves in the body. Still, we gained some publicity, and a variety of people seem to have found scientific and/or philosophical inspiration in this sort of tensegrity model for nerve growth. I particularly like this review article by Don Ingber in Scientific American. A little more out there is this view of consciousness life and the fate of the universe (where I got the cell picture). In general, tensegrity structures are more tough and flexible than normal construction. A tensegrity structure will bend easily, but rarely break. It seems likely that your body is held together this way, and because of this you can carry heavy things, and still move with flexibility. It also seems likely that bones are structured this way; as with nerves; they are reasonably flexible, and can be made to grow by pulling.
Now that I think about it, we should have done more theoretical or experimental work in this direction. I imagine that pulling on the nerve also affects the stability of the actin network by affecting the chain configuration entropy. This might slow actin assembly, or perhaps not. It might have been worthwhile to look at new ways to pull, or at bone growth. In our in-vivo work we used an external magnetic field to pull. We might have looked at NASA funding too, since it’s been observed that astronauts grow in outer space by a solid inch or two, and their bodies deteriorate. Presumably, the lack of gravity causes the calcite in the bones to grow, making a person less of a tensegrity structure. The muscle must grow too, just to keep up, but I don’t have a theory for muscle.
Robert Buxbaum, February 2, 2014. Vaguely related to this, I’ve written about architecture, art, and mechanical design.
Three score days ago, The Harrisburg Patriot & Union retracted its unflattering 1863 review of Lincoln’s Gettysburg address. But this retraction deserves more attention, I think, than that the editors reconsidered. The Patriot & Union was a Republican journal; it carried an accurate account of the speech, and so it’s worthwhile to ask why its editors labeled this great speech, “silly remarks”, deserving “a veil of oblivion”; “without sense.” Clearly the editors saw a serious lack that we do not see today. It’s worth asking then, what made them think it was silly and lacking in sense?
Lincoln in 1863; The Union & Patriot has retracted their review of this Gettysburg speech — in the fullness of time, they’ve come to reconsider their original review.
Lincoln spoke a few words in honor of the dead, but Edward Everett spoke on this topic for two hours before Lincoln rose. This lack does not appear to be what bothered the editors: “To say of Mr. Everett’s oration that it rose to the height which the occasion demanded, or to say of the President’s remarks that they fell below our expectations, would be alike false. Neither the orator nor the jester surprised or deceived us. Whatever may be Mr. Everett’s failings he does not lack sense – whatever may be the President’s virtues, he does not possess sense. Mr. Everett failed as an orator, because the occasion was a mockery, and he knew it, and the President succeeded, because he acted naturally, without sense and without constraint, in a panorama which was gotten up more for his benefit and the benefit of his party than for the glory of the nation and the honor of the dead.” The editors came to Gettysburg (I think) to hear Lincoln to hear things that only LIncoln could provide — his real thoughts on slavery and an update on his efforts at peace. As best I can tell, it was in these areas that they saw “a veil of oblivion.” Even so, for them to call this address, “silly remarks” there must be more going on. Here are my thoughts.
Lincoln had freed southern slaves a few months earlier by the emancipation proclamation, but no one knew their status; there had been a riot over this a few days previous. Did Lincoln claim equality for these ex-slaves, and if not, what were his thoughts on the extent of their in-equality. They were confiscated as war booty; would Lincoln return them to their owners after the war was over? If so, they were not free at all. Along with this, what was Lincoln doing to end the war? It was far from clear that the North could win in 1863. Lee had many victories, and now England had entered in support of the Confederacy. In my opinion, it was Ericsson’s Monitors that allowed the North to stop the British and win, but it appears that, in 1863, only the British navy realized that their power had been neutralized, and the south was lost.
The North’s Monitor, right, fights the Confederate Merrimac, left, to a draw over control of Norfolk harbor. Ericsson turned out two Monitor ships per month. In my opinion is was these ships that stopped the British and won the war.
Lincoln was cryptically brief when it came to slavery or peace: 271 words. About half the speech is devoted to the brave men who struggled here; the other half speaks of “the Nation,” or the “government.” Not the United States, the Union, the North, the South, but an undefined entity that Lincoln claims came into existence 70 years earlier, in 1776. Most educated people would have said that 1776 created no nation or government, only a confederation of independent states as described by the articles of confederation. Under these articles, these 13 states could only act by consensus and had the right to leave at will. To the extent that anyone held the South was bound now, it was because of the Constitution, signed ten years later, but Lincoln does not mention the Constitution at all– perhaps because most Democrats, understood the Constitution to allow departure. Also, to the extent the Constitution mentions slavery, it’s not to promote equality, but to give each slave 3/5 the vote-power of a free man. If “created equal” is to come from anywhere, it’s the Declaration, but most people understood the intent of the Declaration differently from the vision Lincoln now presented.
As far as most people understood it, The Declaration claimed the God-given right to separate from England and gain us a measure of self-rule — something that the South now claimed for itself, but Lincoln opposed. Further, we claimed in The Declaration, that British mis-management made the separation necessary, and listed the abhorrent offenses including suspension of habeas corpus, and the confiscation of property without process of law — things Lincoln was doing even now. Even the introductory phrase, created equal, was not understood to imply that everyone was equal. Rather, as Stephen Douglass pointed out in their 1858 Chicago debate, we’d created a nation “by the white man, for the benefit of the white man, to be administered by white men, in such a manner as they should determine.”
Ulysses Grant had a slave he freed in 1859; his wife held slaves till 1865. Lee freed his in 1862.
Where was Lincoln coming from? What was he saying that November day? It’s been speculated that Lincoln was proposing a secular religion of administered freedom. There appears to be some legitimacy here, but more I suspect Lincoln was referring to the UNANIMITY requirement behind the Declaration — by agreement all the states had to agree to independence, or we would all stay bound to Britain. If we had to unanimously bind ourselves, we must have unanimously bound ourselves to some shared vision of the union or democracy, -presumably that all were created equal. Five years earlier, William Herndon, Lincoln’s law partner, had given Lincoln a book of sermons by Theodore Parker, a Boston Unitarian. That volume includes the following section marked by Lincoln in reference to what the unanimous binding entailed: “Democracy is direct self-government, over all the people, for all the people, by all the people.” Whether Lincoln was now speaking in direct reference to this line, or more-likely, as I suspect, to a more general refutation of the claims of Southern separation and of Douglas’s 1858 white man claim, Lincoln’s understanding of the import of the Declaration was one that few understood or agreed with. The North still had slaves — Grant’s wife for example, and there was no obvious desire for a new birth of freedom, just an end to the war. Lincoln’s words thus must have sounded like gobbledygook to the majority of learned ears.
Based on the events and issues of the time, and the un-obvious point of the speech, I’d say the editors were justified in their ill review. Further, the issues that bothered them then, abuse of power, citizen and states’ rights, remain as relevant today as ever. Do the current editors see any import of the 9th and 10th amendment limiting the power of federal government? If so, what. Thus, I’m a bit disappointed that the Union & Patriot retracted its review of Lincoln’s short speech with nothing more than claiming to see things differently today. We stand on LIncoln’s shoulders now, and though we see the nation, and the Declaration, through his eyes, their issues remain, and the original review gives perspective on the nation as it looked at a very different time. Thus, while I understand the editors desire to look correct in retrospect, I’d prefer if the current editors would have left the review, or at least addressed the points that bothered their earlier colleagues. It’s a needed discussion. When every person thinks alike, nobody thinks very much.
January 6, 2014 by Robert E. Buxbaum, a doctor of Philosophy (in Chemical Engineering). Here is a translation of the Address into Jive. And into yeshivish. I’ve also written an essay on a previous retraction (regarding GM food). If Lincoln had a such a long address, how did he ever get mail?
Back when I taught materials science to chemical engineers, I used the following poem to teach my aesthetic for the strength target for product design:
The secret to design, as the parson explained, is that the weakest part must withstand the strain. And if that part is to withstand the test, then it must be made as strong as all the rest. (by R.E. Buxbaum, based on “The Wonderful, One-hoss Shay, by Oliver Wendell Holmes, 1858).
My thought was, if my students had no idea what good mechanical design looked like, they’d never be able to it well. I wanted them to realize that there is always a weakest part of any device or process for every type of failure. Good design accepts this and designs everything else around it. You make sure that the device will fail at a part of your choosing, when it fails, preferably one that you can repair easily and cheaply (a fuse, or a door hinge), and which doesn’t cause too much mayhem when it fails. Once this failure part is chosen and in place, I taught that the rest should be stronger, but there is no point in making any other part of that failure chain significantly stronger than the weakest link. Thus for example, once you’ve decided to use a fuse of a certain amperage, there is no point in making the rest of the wiring take more than 2-3 times the amperage of the fuse.
This is an aesthetic argument, of course, but it’s important for a person to know what good work looks like (to me, and perhaps to the student) — beyond just by compliments from the boss or grades from me. Some day, I’ll be gone, and the boss won’t be looking. There are other design issues too: If you don’t know what the failure point is, make a prototype and test it to failure, and if you don’t like what you see, remodel accordingly. If you like the point of failure but decide you really want to make the device stronger or more robust, be aware that this may involve strengthening that part only, or strengthening the entire chain of parts so they are as failure resistant as this part (the former is cheaper).
I also wanted to teach that there are many failure chains to look out for: many ways that things can wrong beyond breaking. Check for failure by fire, melting, explosion, smell, shock, rust, and even color change. Color change should not be ignored, BTW; there are many products that people won’t use as soon as they look bad (cars, for example). Make sure that each failure chain has it’s own known, chosen weak link. In a car, the paint on a car should fade, chip, or peel some (small) time before the metal underneath starts rusting or sagging (at least that’s my aesthetic). And in the DuPont gun-powder mill below, one wall should be weaker so that the walls should blow outward the right way (away from traffic).Be aware that human error is the most common failure mode: design to make things acceptably idiot-proof.
Dupont powder mills had a thinner wall and a stronger wall so that, if there were an explosion, it would blow out ‘safely.’ This mill has a second wall to protect workers. The thinner wall must be strong enough to stand up to wind and rain; the stronger walls should stand up to all likely explosions.
Related to my aesthetic of mechanical strength, I tried to teach an aesthetic of cost, weight, appearance, and green: Choose materials that are cheaper, rather than more expensive; use less weight rather than more if both ways worked equally well. Use materials that look better if you’ve got the choice, and use recyclable materials. These all derive from the well-known axiom, omit needless stuff. Or, as William of Occam put it, “Entia non sunt multiplicanda sine necessitate.” As an aside, I’ve found that, when engineers use Latin, we look smart: “lingua bona lingua motua est.” (a good language is a dead language) — it’s the same with quoting 19th century poets, BTW: dead 19th century poets are far better than undead ones, but I digress.
Use of recyclable materials gets you out of lots of problems relative to materials that must be disposed of. E.g. if you use aluminum insulation (recyclable) instead of ceramic fiber, you will have an easier time getting rid of the scrap. As a result, you are not as likely to expose your workers (or you) to mesothelioma, or similar disease. You should not have to pay someone to haul away excess or damaged product; a scraper will oblige, and he may even pay you for it if you have enough. Recycling helps cash flow with decommissioning too, when money is tight. It’s better to find your $1 worth of scrap is now worth $2 instead of discovering that your $1 worth of garbage now costs $2 to haul away. By the way, most heat loss is from black body radiation, so aluminum foil may actually work better than ceramics of the same thermal conductivity.
Buildings can be recycled too. Buy them and sell them as needed. Shipping containers make for great lab buildings because they are cheap, strong, and movable. You can sell them off-site when you’re done. We have a shipping container lab building, and a shipping container storage building — both worth more now than when I bought them. They are also rather attractive with our advertising on them — attractive according to my design aesthetic. Here’s an insight into why chemical engineers earn more than chemists; and insight into the difference between mechanical engineering and civil engineering. Here’s an architecture aesthetic. Here’s one about the scientific method.
Robert E. Buxbaum, October 31, 2013
How many surrealists does it take to screw in a lightbulb.
The fish.
Surrealism aims to show the reality that exceeds realism; the dream-like absurd that is beyond the rational, common-sensical and practical. Beyond control engineering.
And you know “How many engineers would it take to screw in a lightbulb?” —- “Minimally two, and it would have to be a very large lightbulb.”
Even if the insights of surrealism are common-place, for example, that the eye is a false mirror of the world, I like is that they become real (if the surrealist is talented.)
False Mirror by Magritte; the idea, I suppose is that we see what’s already within us.
“The greatest obstacle to discovering the shape of the earth, the continents, and the oceans was not ignorance but the illusion of knowledge.” What I particularly like is the falseness of the mirror is shown as both false and true. The world is rarely this or that. Another insight / joke.
We all have masks, especially with those we love.
I imagine most I could make second-rate surrealistic works. The way to know your work is second rate it’s beautiful and insightful, but not funny.
Creation of Man-the-militant. Kuksi. It’s well done, and interesting (a retake on Michelangelo), but it’s not funny. See my cartoon in mechanical v civil engineers joke.
And then there is bad modern art. You could argue that this isn’t surreal, but some sort of other modern art, or post modern art. But that’s all false: it’s just bad art.
Bad modern art: little skill, little meaning, no humor. If you have to ask: “is it art?” It usually isn’t.
If you buy something like this, and put it in your corporate headquarters lobby, the joke’s on you, and the artist is laughing his or her way to the bank. Here is a link to why surrealism should be funny, And why architecture should not be (someone’s got to live in that joke).
R. E. Buxbaum, August 5, 2013
Robert Leighton, from the New Yorker,
Is funny because …. there’s an Escher-like impossible structure and a dirty word (ass, tee hee). Besides that, this joke highlights a fundamental conflict between the architect and the client (customer): what is good architecture?
Typically the customer whats a home or office that “looks nice”, “doesn’t cost too much”, and “works,” perhaps as an advertisement for the company. Often the architect wants to make a statement for him/herself, or wants to produce a work of art. Left to their own, architects can produce expensive monuments that no one can live in.
A wonderful (horrible) case concerns The Cooper Union, my alma mater, and more-or-less the only free college in America. The Cooper Union was founded by an inventive mechanic, Peter Cooper, see my biography, who invented jello, and rolled steel, laid the transatlantic cable, founded AT&T, and managed to give free education to a century and a half of students. The trustees of the school tore down the old, serviceable building, sold the land, and built a $270,000,000 dollar monstrosity. Hailed by the New York Times as great architecture, it bankrupted the school, and is unusable for the sort of hands-on education that Peter Cooper devised.
In hopes of attracting a rich donor, Cooper Union sold its engineering building and borrowed $175 million to erect this replacement. No donor materialized, and, with it, a 155-year-old policy of free tuition.
Here’s a surrealist joke, an engineer joke, and a thought on control engineering. Here too is a sculpture I put on top of my building; the eyes follow you.
R.E. Buxbaum, July 8, 2013; I do consulting on hydrogen, and my company makes hydrogen products.
We recently put a sculpture on our roof: Gilroy, or “Mr Hydrogen.” It’s a larger version of a creepy face sculpture I’d made some moths ago. Like it, and my saber-toothed tiger, the eyes follow you. A worry about this version: is there enough keeping it from blowing down on the cars? Anyone who puts up a large structure must address this worry, but I’m a professional engineer with a PhD from Princeton, so my answer is a bit different from most.
Gilroy (Mr Hydrogen) sculpture on roof of REB Research & Consulting. The eyes follow you. Aim is that it should withstand 50 mph winds.
The main force on most any structure is the wind (the pyramids are classic exceptions). Wind force is generally proportional to the exposed area and to the wind-speed squared: something called form-drag or quadratic drag. Since force is related to wind-speed, I start with some good statistics for wind speed, shown in the figure below for Detroit where we are.
The highest Detroit wind speeds are typically only 16 mph, but every few years the winds are seen to reach 23 mph. These are low relative to many locations: Detroit has does not get hurricanes and rarely gets tornadoes. Despite this, I’ve decided to brace the sculpture to withstand winds of 50 mph, or 22.3 m/s. On the unlikely chance there is a tornado, I figure there would be so much other flotsam that I would not have to answer about losing my head. (For why Detroit does not get hurricanes or tornadoes, see here. If you want to know why tornadoes lift things, see here).
The maximum area Gilroy presents is 1.5 m2. The wind force is calculated by multiplying this area by the kinetic energy loss per second 1/2ρv2, times a form factor. F= (Area)*ƒ* 1/2ρv2, where ρ is the density of air, 1.29Kg/m3, and v is velocity, 22.3 m/s. The form factor, ƒ, is about 1.25 for this shape: ƒ is found to be 1.15 for a flat plane, and 1.1 to 1.3 a rough sphere or ski-jumper. F = 1.5*1.25* (1/2 *1.29*22.32) = 603 Nt = 134 lb.; pressure is this divided by area. Since the weight is only about 40 lbs, I find I have to tie down the sculpture. I’ve done that with a 150 lb rope, tying it to a steel vent pipe.
Wind speed for Detroit month by month. Used to calculate the force. From http://weatherspark.com/averages/30042/Detroit-Michigan-United-States
It is possible that there’s a viscous lift force too, but it is likely to be small given the blunt shape and the flow Reynolds number: 3190. There is also the worry that Gilroy might fall apart from vibration. Gilroy is made of 3/4″ plywood, treated for outdoor use and then painted, but the plywood is held together with 25 steel screws 4″ long x 1/4″ OD. Screws like this will easily hold 134 lbs of steady wind force, but a vibrating wind will cause fatigue in the metal (bend a wire often enough and it falls apart). I figure I can leave Gilroy up for a year or so without worry, but will then go up to replace the screws and check if I have to bring him/ it down.
In the meantime, I’ll want to add a sign under the sculpture: “REB Research, home of Mr Hydrogen” I want to keep things surreal, but want to be safe and make sales.
by Robert E. Buxbaum, June 21, 2013
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, 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.
So how many surrealists does it take to screw in a lightbulb? The fish.
by R. E. Buxbaum, June 14, 2013
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.
R.E. Buxbaum, May 20 2013 (edited Sept 23, 2013) — I’m not very musical, but my children are.
Here’s the latest REB Research sculpture: a saber-toothed tiger:
Saber-toothed Tiger sculpture at REB Research; the face follows you. A bit of our 3 foot geodesic dome is shown in the foreground.
It’s face follows you (somewhat); It was inspired by my recent visit to Princeton Univ — they had lots of tiger statues, but none that looked eerie enough as you walked by. Click here for: YouTube movie.
Normally, by the way, REB Research makes hydrogen generators and other hydrogen stuff. May 1, 2013
