Category Archives: Philosophy

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.

Zen Buddhist Joke

How many Zen Buddhists does it take to screw in a lightbulb?

Four: One to screw in the bulb;

One to not screw in the bulb;

One to both screw in the bulb and not screw in the bulb.

And one to neither screw in the bulb, nor not screw in the bulb.

Is funny because Buddhism aims at enlightenment — something that occurs by the simultaneous destruction and non-destruction of the darkness. Ah you got it: I hear one hand clapping.

Science is the Opposite of Religion

Some years ago, my daughter came back from religions school and asked for a definition of science. I told her that science was the opposite of religion. I didn’t mean to insult religion or science; the big bang for one thing, strongly suggests there is a God -creator, and quantum mechanics suggests (to me) that there is a God -maintainer, but religion deals with other things beyond a belief in God, and I meant to point out that every basic of how science looks at things finds its opposite in religion.

Science is based on reproducibility and lack of meaning: if you do the same experiment over and over, you’ll always get the same result as you did before and the same result as anyone else — when the results are measured to some good, statistical norm. The meaning for the observation? that’s a meaningless question. Religion is based on the centrality of drawing meaning, and the centrality of non-reproducible, one-time events: creation, the exodus from Egypt, the resurrection of Jesus, the birth of Zeus, etc. A religious believer is one who changes his or her life based on the lesson of these; to him, a non-believer is one who draws no meaning, or needs reproducible events.

Science also requires that anyone will get the same result if they do the same process. Thus, chemistry class results don’t depend on the age, sex, or election of the students. Any student who mixes the prescribed two chemicals and heats to a certain temperature is expected to get the same result. The same applies to measures of the size of the universe, or its angular momentum or age. In religion, it is fundamentally important what sex you are, how old you are, who your parents were, or what you are thinking at the time. If the right person says “hoc es corpus” over wine and wafers, they change; if not, they do not. If the right person opens the door to heaven, or closes it, it matters in religion.

A main aspect of all religion is prayer; the idea that what you are thinking or saying changes things on high and here below. In science, we only consider experiments where the words said over the experiment have no effect. Another aspect of religion is tchuvah (regret, repentance); the idea is that thoughts can change the effect of actions, at least retroactively. Science tends to ignore repentance, because they lack the ability to measure things that work backwards in time, and because the scientific instruments we have currently do not take measurements on the soul to see if the repentance had any effect. Basically, the science-universe is only populated with those things which can be measured or reproducibly affected, and that pretty much excludes the soul. That the soul does not exist in the science universe doesn’t mean it doesn’t exist.

Another main aspect of religion is morality: you’re supposed to do the right thing. Morality varies from one religion to another, and you may think the other fellow’s religion has a warped morality, but at least there is one in all religions. In science, for better or worse, there is no apparent morality, either to man or to the universe. Based on science, the universe will end, either by a bang or a whimper, and in that void of end it would seem that killing a mouse is about as important as killing a person. No religion I know of sees the universe ending in either cold or hot death; as a result. Consistent with this, they all see murder is a sin against God. This difference is a big plus for religion, IMHO. That man sees murder as a true evil is either a sign that religion is true, or that it isn’t depending on the value you put on life. Another example of the moral divide: Scientists, especially academics, tend to be elitists. Their morality, such as it is, values great minds and great projects over the humble and stupid. Classical religion sees the opposite; it promoting the elevation of the poor, weak, and humble. There is no fundamental way to tell which one is right, and I tend to think that both are right in their own, mirror-image universes.

It is now worthwhile to consider what each universe sees as wisdom. An Explanation in the universe of science has everything to do with utility and not any internal sense of having understood, as such. I understand something only to the extent I predict that thing or can do something based on the knowledge. in religion, the motivation for all activity is always just understanding — typically of God on the bone-deep level. This difference shows up very clearly in dealing with quantum mechanics. To a scientist, the quantum world is fundamentally a door from religion because it is basically non-understandable but very useful. Religion totally ignores quantum mechanics for the same reason: it’s non-understandable, but very precise and useful. Anything you can’t understand is meaningless to them (literally), and useful is mostly defined in terms of building the particular religion; I think this is a mistake on many levels. I note that looking for disproof is the glory-work of all science development, but the devil’s work of every religion. A religious leader will grab on to statistical findings that suggest that his type of prayer cures people, but will always reject disproof, e.g. evidence that someone else’s prayers works better, or that his prayer does nothing at all. Each religion is thus in a war with the other, each trying to build belief, while not removing it. Science is the opposite. Religion starts with the answer and accepts any support it can; fundamental change is considered a bad thing in religion. The opposite is so with science; disproof is considered “progress,” and change is good.

These are not minor aspects of science and religion, by the way, but these are the fundamental basics of each, as best I can tell. History, politics, and psychology seem to be border-line areas, somewhere between science and religion. The differences do not reflect a lack in these fields, but just a recognition that each works according to its own logic and universe.

My hope in life is to combine science and religion to the extent possible, but find that supporting science in any form presents difficulties when I have to speak to others in the religious community, my daughter’s teachers among them. As an example of the problem that come up, my sense is that the big bang is a fine proof of creation and should be welcomed by all (most) religious people. I think its a sign that there is a creator when science says everything came from nothing, 14,000,000 years ago. Sorry to say, the religious leaders I’ve met reject the big bang, and claim you can’t believe in anything that happened 14,000,000,000 years ago. So long as science shows no evidence of a bearded observer at the center, they are not interested. Scientists, too have trouble with the bang, I find. It’s a one-time event that they can’t quite explain away (Steven Hawking keeps trying). The only sane approach I’ve found is to keep blogging, and otherwise leave each to its area. There seems to be little reason to expect communal agreement.

by Robert E. Buxbaum, Apr. 7, 2013. For some further thoughts, see here.