Monday, January 07, 2019

Letter from a reader: “What’s so bad about randomness?”

[The best part of publishing a book has been getting feedback from readers who report their own experience as it relates to what I wrote about. With permission, I want to share this letter which I received the other day from Dave Hurwitz, a classical music critic.]

Dear Dr. Hossenfelder,

I hope that I am not bothering you, but I just wanted to write to tell you how much I am enjoying your book “Lost in Math.” I haven’t quite finished it yet, but I was so taken with it that I thought I might write to let you know anyway. I am about as far away from theoretical physics as it’s possible to be: I am a classical music critic and independent musical scholar, and I support myself working in real estate; but I am a very serious follower of the popular scientific literature, and I was so impressed by your directness, literacy, and ability to make complex topics digestible and entertaining for the general reader.

I am also very much in sympathy with your point of view. Even though I don’t understand the math, it often seems to me that so much of what theoretical physicists are doing amounts to little more than a sort of high-end gematria – numerology with a kind of mystical value assigned to mathematical coincidence or consistency, or, as you (they) call it, “beauty.” I cringe whenever I hear these purely aesthetic judgments applied to theoretical speculation about the nature of reality, based primarily on the logic of the underlying math. And don’t get me wrong: I like math. Personally, I have no problem with the idea that the laws governing the universe may not be elegant and tidy, and I see no reason why they should be. They are what they are, and that’s all. What’s so bad about randomness? It’s tough enough trying to figure out what they are without assigning to them purely subjective moral or aesthetic values (or giving these undue weight in guiding the search).

It may interest you to know that something similar seems to be infecting current musicology, and I am sure many other academic fields. Discussion of specific musical works often hinges on standardized and highly technical versions of harmonic analysis, mostly because the language and methodology have been systematized and everyone agrees on how to do it – but what it actually means, how it creates meaning or expressiveness, is anyone’s guess. It is assumed to be important, but there is no demonstrable causal connection between the correctness of the analysis and the qualitative values assigned to it. It all comes down to a kind of circular reasoning: the subjective perception of “beauty” drives the search for a coherent musical substructure which, not surprisingly, once described is alleged to justify the original assumption of “beauty.” If you don’t “get” physicists today, then I don’t “get” musicologists.

Anyway, I’m sorry to take up so much of your time, but I just wanted to note that what you see – the kind of reasoning that bothers you so much – has its analogues way beyond the field of theoretical physics. I take your point that scientists, perhaps, should know better, but the older I get the more I realize two things: first, human nature is the same everywhere, and second, as a consequence, it’s precisely the people who ought to know better that, for the most part, seldom do. I thank you once again for making your case so lucidly and incisively.

Best regards,

Dave Hurwitz
ClassicsToday.com

40 comments:

  1. Dave Hurwitz:

    I don't really understand music that well, but I have been wondering for some time whether Schoenberg's music is aesthetically successful because of his theories of atonality and tone rows, or despite them. If the second possibility is the case, it gave an enormous handicap to all the composers who followed Schoenberg and tried to develop his techniques.

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  2. Dave wrote: If you don’t “get” physicists today, then I don’t “get” musicologists . . . the subjective perception of “beauty” drives the search for a coherent musical substructure which, not surprisingly, once described is alleged to justify the original assumption of “beauty.”

    I'm a fan of musicologist Robert Greenberg's lectures. He can get technical, but he doesn't get caught up in that "circular reasoning" you refer to, and he's critical of musicologists who do. For example, I recall in one lecture he said, "Music is not math and math is not music. Math is math and music is music." (Greenberg's way of saying "don't get lost in the math.")

    Here's a Greenberg quote:

    “We are hardwired to hear and make music. Yes, we will sigh with pleasure when we hear a favorite theme played by an orchestra, and who hasn’t felt a stab of nostalgia, or even brushed away a tear, when hearing a song reminiscent of youth or a lost love? However, such exquisite moments notwithstanding, the musical experience represents something far deeper. Broadly defined, music is sound in time. Sound is nothing less than our perception of the vibrations, the movement, of the universe around us. Music is an intensification, a crystallization, a celebration, a glorification, of that movement and those vibrations. Pretty heady stuff. Far beyond spoken language—which, with its sounds in time, might rightly be considered a low-end sort of music—music is a universal language; one need not speak Ashanti in order to groove to West African drumming; or German in order to be emotionally flayed by Beethoven; or English to totally freak when listening to Bruce Springsteen. Say it with flowers? Nah. If you really want to get your expressive point across, say it with music.”

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  3. Peter wrote: I have been wondering for some time whether Schoenberg's music is aesthetically successful because of his theories of atonality and tone rows, or despite them.

    Schoenberg's Piano Concerto is aesthetically appealing, don't you think?

    Musicologist Robert Greenberg discusses Schoenberg and other modern composers in his lecture course "Great Music of the Twentieth Century."

    Here's a blurb on Schoenberg:

    "There are few names and/or phrases more likely to terrify your average concertgoer than “Arnold Schoenberg” and the “12-Tone Method”.

    Well, it’s time for this madness to stop. Because there has never been – in the whole, magnificent history of Western music – a more talented, more important, more misunderstood and ultimately more underappreciated great composer than Arnold Schoenberg.

    So why all the fear and loathing? Because he is blamed – unfairly and inappropriately – for being an unrepentant modernist who single-handedly turned the art of music composition into a dry, meaningless, intellectualized exercise in sonic acrostics."

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  4. "The people who ought to know better ... seldom do." What a beautiful turn of phrase.

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  5. https://www.youtube.com/watch?v=XVSRm80WzZk
    … "The Twilight Zone" theme was loudly condemned as music, yet it remains evocative. The best John Cage did was "4'33," yet he was acclaimed.

    Science had empirical standards. Science corrupted into business models wherein “research” garners headlines (False News!), futures of penury and salt. A Tiger Team should smash skulls between two gold bricks. Shout “diversity” in counterpoint.

    https://www.youtube.com/watch?v=A-fyWc6Mpd8
    … It's not something you walk out humming. “Ode to interfacial impedance mismatch.”

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  6. Dave Hurwitz: “... human nature is the same everywhere, and second, as a consequence, it’s precisely the people who ought to know better that, for the most part, seldom do.

    Well said.

    sean s.

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  7. The analogy with music has some interest for me because I also write music, for myself. (I took a number of courses as a graduate student, including an honours course in composition.) What has struck me about music is that composition is, with some string exceptions, fashionable. People write music in the style of someone else around the same time period. Obviously Arnold Schoenberg was an exception, although how much you like him depends on your own musical feelings. But the question then is, how much of scientific theory is based on fashion, on "I must try to be in accord with the rest"? Obviously physicists cannot just go anywhere - as an example, relativity has to be respected because of its agreement with observation, but there are a number of other things in science where the agreement with observation is obtained with some assignable constants and computational gymnastics, the details of which we are not given. I often wonder whether some of the theory could not have taken a different form had it been developed a different way. Quantum mechanics comes to mind here. But are we ever interested in revisiting previous theory? Or do we overly respect fashion?

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  8. I wouldn't use Schönbergs approach - trying to reach new borders by neglecting inherited harmonical/melodical possibility frameworks - as an analogy to the physics beauty discussion.
    Rather step back to Pythagoras, who claimed beauty for natural number based interval relations - failing, and understand how >2000 years later finding out that the 12th square root of 2 did good to music - even to Schönberg, who in fact worked then with the non pythagorean frequencies based on "ugly" relations. It all depends, how "beauty" is defined. There may always be regularity from a "higher" point of view.

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  9. Shoenberg sounds ugly, no matter how you try to rationalize it. Academic musicologists like it the same way academic physicists like being lost in math. It’s the thing to do. If beauty is in the ear of the beholder then take a poll and see who thinks 12 tone isn’t ugly. That’s why people don’t listen to it.

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  10. In my book, I use Schoenberg as an illustration for how expertise can shift perceptions of beauty.

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  11. I read your book and now I think you should read mine.
    I have built a model that is beautiful, symmetrical, and physical.
    No magic. A fair amount of math, but I don't think you'll get lost!
    You will freak out. Guaranteed!

    Let's do physics!

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  12. "classical music critic"

    I'm pretty sure that he is a classical-music critic. Maybe he is also a classical critic, I don't know.

    I recently read something about a book directed at small telescope owners. It was corrected to small-telescope owners.

    There are elementary-particles physicists, and there are elementary particle physicists.

    What have high energy physicists been smoking?

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  13. "Music is not math and math is not music. Math is math and music is music."

    “Information is not knowledge.
    Knowledge is not wisdom.
    Wisdom is not truth.
    Truth is not beauty.
    Beauty is not love.
    Love is not music.
    Music is THE BEST.”

    ---Frank Zappa

    For the record (geddit?), I'm not a fan of Zappa's music, but have respect for him as a musician (and as a person).

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  14. "Arnold Schoenberg was an exception, although how much you like him depends on your own musical feelings."

    Reminds me of Mark Twain's quip: Wagner's music is better than it sounds.

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  15. "Well, it’s time for this madness to stop. Because there has never been – in the whole, magnificent history of Western music – a more talented, more important, more misunderstood and ultimately more underappreciated great composer than Arnold Schoenberg."

    "Shoenberg sounds ugly, no matter how you try to rationalize it. Academic musicologists like it the same way academic physicists like being lost in math. It’s the thing to do. If beauty is in the ear of the beholder then take a poll and see who thinks 12 tone isn’t ugly. That’s why people don’t listen to it."

    I'm decidedly in the second camp. Put Schönberg up there with Jackson Pollack and other post-modernists. The only art involved is the art of convincing people that their products are actually worth (in any sense) anything.

    Look up the biography of Nat Tate to get an idea of the self-delusion in this community.

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  16. So now I'm curious about Arnold Schönberg's music. Listening to it on Spotify now ...

    sean s.

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  17. To Mr. Shor:

    First, many thanks to Sabine for posting my email, and to everyone who has responded to it. Regarding Schoenberg and the questions posed: I would hesitate to call atonal or twelve-tone (serial) composition a "theory." As it applies to individual works, it is a technique, a tool, and I believe I can say safely that no piece of music (or work of art in general) succeeds aesthetically despite the technique used in its construction.

    As to Schoenberg more generally, he is a very difficult composer to speak of in generalities because he never ceased to evolve--to the point where almost no two works share the same sound. It really is remarkable; but to speak of him sensibly requires a knowledge of his entire output (or most of it), and that takes an amount of time that most people are unwilling to spend, especially if they find much of the music disagreeable. Schoenberg's music is challenging and difficult, all of it, angular and hard-edged in a way similar to the manner in which Cubist painting is challenging to those who prefer illustrative realism. I find his tonal music as uncompromising as his atonal music, sometimes more so because of the expectations it creates in the listener. But as Sabine points out in her book, what to some may look (or sound) ugly initially may in time become a new standard of beauty, especially if it turns out to be true. Schoenberg's music has that potential once you get to know it. Love it or hate it, it is relentlessly, intensely honest and sincere. It is "in your face" music that demands to be taken on its own terms, but I have found that it rewards you when you do.

    The discussion of Schoenberg almost exclusively in harmonic terms--tonal v. atonal or serial--highlights the degree to which most of us think of music as harmony, in the form of tonal melody, to the exclusion of practically all other musical constituents, such as timbre, texture, rhythm, dynamics, and so forth. Works such as the Five Pieces for Orchestra challenge us to consider music in this wider sense. I would not say that all of Schoenberg's music is equally successful, which is only to be expected in a composer of such a questing spirit; and for the same reason I would never consider his approach to composition to be "theoretical." Schoenberg was, in many ways, the most extreme of all the late Romantics, and we need to distinguish his musical output from his (often superbly argued) pedagogical efforts to systematize and explain what he was trying to do.

    Dave Hurwitz

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  18. After reading this letter I was looking on the Arxiv for a paper I read some years ago on the relationship between sequences of dyads and triads with orbifolds. I read this in Science around 10 years ago. I tried looking it up to find the author named something like Tiniencho, at MIT as I recall. Anyway there are a number of papers I ran across on this type of subject.

    Music by Schoenberg is not often what I listen to. The same holds I guess for his understudy Berg and related composers such as Pierre Boulez. In some way I see classical music as having "over intellectualized" composition forms. This occurred at the time jazz and then rock took off. Generally people play and listen to music not to try to figure out its intricacies, but more to be carried away by it.

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  19. sshawnuff wrote: Rather step back to Pythagoras

    Good point and great example of how people got lost in math trying to create beautiful models for music.

    The math for music is beautiful in some parts and messy in other parts. When I first learned this I remember feeling disappointed that music wasn't perfect, that it had some kind of defect. Maybe the entire universe has some kind of defect. :-)

    Regarding Schoenberg (and other "weird" music), music has always been an important part of my life and my musical tastes have developed over a lifetime. I worked my way up to Schoenberg. My grandparents never learned to like rock-n-roll. In the 1970's, their idea of modern popular music was Barbra Streisand.

    I'll never forget the first time I watched the film 2001: A Space Odyssey in 1968. When I heard the music for the monolith, I was emotionally and acoustically terrified. If it was Kubrick's intention to scare the bejesus out of me, he succeeded. Ten years later I could enjoy and appreciate Gyorgy Ligeti's Requiem.

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  20. Ira wrote: Shoenberg sounds ugly, no matter how you try to rationalize it. Academic musicologists like it the same way academic physicists like being lost in math.

    In music, I don't tell anyone that what they find beautiful is ugly, or what they find ugly is beautiful, as if there is some kind of objective standard. And it's downright wrong to suggest that Schoenberg can be enjoyed only by pretentious musicologists.

    I know lots of Asian people who think cheese of any kind is disgusting. It's a cultural thing, of course. If Schoenberg was cheese, he'd be Limburger. I happen to like Schoenberg and Limburger. :-)

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  21. "Those who know don't have the words to tell
    And the ones with the words don't know too well."
    ~Bruce Cockburn,
    ~"Burden Of The Angel/Beast",
    ~Dart to the Heart (1994)

    As for knowledge increasing beauty, certainly it increases appreciation.

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  22. Phillip wrote: For the record (geddit?), I'm not a fan of Zappa's music, but have respect for him as a musician (and as a person).

    For the CD, I'm not a fan of Zappa's music either.

    I don't know enough about Zappa to say I respect him. I read that he was a very difficult man to get along with. I also read that he didn't care much for The Beatles and even said he preferred The Monkees to The Beatles. That's almost enough to make me *not* respect him. :-)

    That being said, in the Sgt. Pepper movie, Zappa would have been a much better Mean Mr. Mustard than Frankie Howerd or a much better Maxwell than Steve Martin.

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  23. Phillip wrote: the only art involved is the art of convincing people that their products are actually worth (in any sense) anything.

    Listen to Half-Wolf Dances Mad in Moonlight by modernist and minimalist composer Terry Riley and tell me it doesn't move you and blow your mind. There are other ways to get high besides drugs. :-)

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  24. David wrote: Schoenberg's music has that potential once you get to know it.

    I hope that the skeptics here will accept your point.

    It's funny: In another discussion someone said that scientists were being corrupted by the Templeton Foundation. In this discussion, there seems to be a suggestion that musicologists are being corrupted by Schoenberg and are misleading the public about the value of such music. Not only is the emperor not wearing any clothes, he's also playing horrible music. :-)

    Do you have a favorite Schoenberg piece?

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  25. The premise of this thread is that out-of-the-box thinking is highly desirable but is antithetical to both current scientific and systematic aesthetic judgments. Be careful what you wish for; you just might get it. At this current juncture, where is out-of-the-box thinking primarily happening? As witnessed by countless YouTube presentations, there are thousands upon thousands of vagabond inventors and experimenters at work in their home labs exploring strange and puzzling observations that they are struggling to come to grips with. Most will fail to get anywhere. But like the proverbial infinite monkey theorem which states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, such as the complete works of William Shakespeare, a huge crowd of scattershot experimenters secretly working tirelessly for decades blindly using trial and error methodology one of their number will stumble upon a set of breakthrough cosmological principles that will advance science by a century or two.

    It has happened. At the end of this month, a product will be introduced to the marketplace that all of science will considered scientifically impossible. One of these experimenters will emerge and distinguish himself from the horde. He will have bypassed science altogether and directly try his invention in the marketplace.

    I stand now before you and prognosticate. In the imminent future, decision makers both great and humble will beat a path to the sacred halls of science and ask how this invention works. It is inevitable when forced to confront the mindless grandeur of the trial and error method that has over eons served mankind so well. I am interested in how science now shackled by its lust for beauty and addicted with the logic and discipline of mathematics will resolve and accept this paradigm shift of centuries when irrevocably forced to confront it.



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  26. Dave Hurwitz' is letter beautifully composed. What a marvelous bit of writing.

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  27. What’s so bad about randomness?

    I'm sorely tempted to invoke Easley Blackwood in some fashion.

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  28. To Mr. Mason: I have many favorite pieces by Schoenberg, among them:

    Pierrot Lunaire
    Erwartung
    Five Pieces for Orchestra
    Variations for Orchestra
    Piano Concerto
    Concerto for String Quartet
    Cello Concerto
    A Survivor from Warsaw
    Moses und Aron
    Serenade
    Piano Pieces Opp 19 & 23
    His wonderfully demented arrangement of the Brahms Piano Quartet in G minor

    I confess that I do not listen to Schoenberg very often; I have to be "in the mood," but it seems that when I do listen I find the experience all the more refreshing as a result.

    It may interest you to know that music even has its controversial equivalent to the "fine tuning" in physics that Sabine describes so well in her book: this is "vibrato," the slight variance from absolute pitch used to color a note and make it more expressive and pleasing to the ear. I've written about this pretty extensively as it concerns the theory and practice of "historically informed performance" (HIP). Many proponents of HIP are like those theoretical physicists who insist on notions of "naturalness" and maintain a certain puritanical standard of mathematical consistency. One of the most intelligent comments on vibrato was made by none other than Schoenberg. He wrote, with respect to string playing:

    "Whereas the ‘open’ string shuts off its far end with a hard, ‘stiff’ piece of wood, giving a ‘sharp demarcation’, in the case of the ‘stopped’ note this marking-off is done by the ‘soft’, ‘movable’ finger, giving less sharp demarcation. So absence of vibrato will not mean a pure tone, because of this indefinite demarcation. The note need not actually be out of tune, but its intonation is unconvincing. There will, in addition, be a vague tremor on the part of the finger. So to touch up the impurity of this lifeless tone one uses vibrato…. This makes the tone ‘living’, ‘interesting’, ‘lively’, ‘warm’, and all the rest of it."

    So let's hear it for "fine tuning," and the realization that reality need not conform to our standards of purity or subjective dislike of messiness!

    Dave Hurwitz

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  29. David wrote: I've written about [vibrato] pretty extensively

    How interesting. I have a tendency to overthink things, but I haven't given much thought to vibrato because it seems so obvious. When I hear a particularly effective use of vibrato, I marvel at how evocative it is. Why do slight variations in pitch seem to affect our mesolimbic dopamine system? I just accept that it does. :-)

    So Schoenberg said that "absence of vibrato will not mean a pure tone." I agree. And it occurs to me that there is still vibrato even when there isn't an intentional vibrato. Whether it's a string instrument, a trumpet, or a voice, there is vibrato because humans aren't perfect frequency generators. It also occurs to me that the equal temperament tuning of a piano, which creates beats, might be considered a kind of vibrato. There is honky-tonk piano tuning, where strings are deliberately tuned a couple of beats off, which creates a vibrato feel.

    A couple of months ago I watched the Freddie Mercury movie, which prompted me to read some background on him. There was a lot of discussion about his masterful use of vibrato. There were groups of *scientists* who studied his vibrato! One thing that surprised me was the conclusion that he was a baritone, notwithstanding the fact that he usually sang in the tenor range. I'm inclined to say he was a baritone *and* a tenor. Why can't he be both?

    Sometimes I don't like the vibrato of certain performers. One example is Stevie Nicks. In her earlier years with Fleetwood Mac her vibrato was distinctive and evocative. For some reason, it seems she decided to double down on it, and now it's excessive. Too much of a good thing, I suppose.

    By the way, I play trumpet, piano and guitar, just for fun. I was a total band and orchestra nerd in school.

    Thanks for your Schoenberg favorites. I haven't heard many of them.

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  30. Axil;

    At the end of this month, a product will be introduced to the marketplace that all of science will considered scientifically impossible.

    And if this product does not live up to the hype, I’ll be unsurprised. Standing by for the Big Announcement.

    sean s.

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  31. I’ve been listening off-and-on to Schoenberg yesterday and today (on Spotify). His works are of many styles and I didn’t find any particularly distasteful. My tastes are eclectic, so I’m already habituated to considering wildly different music.

    Still struggle with country-western tho’ ...

    ; )

    sean s.

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  32. To Mr. Mason:

    Yes, perception of vibrato is a very subjective thing. There is in fact a type of vibrato produced by acoustic beats--mainly on the organ by the Vox Celeste stops. These consist of two ranks of pipes tuned slightly sharp against each other.

    What interests me here, though, as it applies to Sabine's book, is the fact that in vibrato we have a case where the lack of tonal purity (strictly speaking) actually assists with good intonation and in sounding more pure to most human ears than an attempt to play without vibrato, for reasons that Schoenberg has partly identified.

    In other words, it is the impurity, or messiness, that sounds more "natural" and "beautiful," always with the proviso that the impurity must be controlled in various ways to create the proper aesthetically pleasing effect (unlike the current state of Stevie Nicks' voice, for example). Psychological studies have shown that the most pleasing vibrato is in fact limited in range and quite inaudible to most people, and is simply perceived as warm, accurate timbre. You don't hear it, but it's there all the same.

    Might it not be the same in physics--that nature admits "messiness" or lack of precision as part of the natural order of things, and certain excessively orderly mathematical theories are unable to capture this because of the purely subjective premise of "beauty" out of which they are constructed? Isn't the probabilistic nature of quantum mechanics an example of much the same thing? I personally love the idea that at a certain fundamental level, all matter is merely the crystallization of a range of possibilities (or aesthetically speaking, impurities), and that beauty and naturalness arise out of and embody inherent uncertainties and imperfections. I suppose it sounds philosophical but that, for me, is freedom.

    Anyway, these are some of the questions that struck me while reading Sabine's splendidly provocative book.

    Dave Hurwitz

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  33. David wrote: There is in fact a type of vibrato produced by acoustic beats--mainly on the organ by the Vox Celeste stops.

    I'll have to look for that on the internet. In jazz, sometimes instruments are tuned as much as a quarter-tone apart, and I generally like the effect. Of course that same tuning can have disastrous effects as anyone who's heard a school band can attest.

    Impurity is essential in music, which might explain why so many composers and performers are impure (ha ha). As you know, pure sine waves sound pretty sterile and unnatural.

    I'll have to give some thought to messy music and messy physics. Using your analogy, Sabine is describing scientists who think pure sine waves sound beautiful and natural tones sound dissonant. A model of the universe could be composed of pure sine waves, but it wouldn't sound the same as the one we've got.

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  34. David wrote: these are some of the questions that struck me while reading Sabine's splendidly provocative book.

    Okay, I'll ask: Was music playing in the background as you read Sabine's book? You're a music critic so it's a reasonable question. :-)

    I *can't* listen to music when I'm reading or working. My attention will always want to go toward the music. I can easily ignore a TV as long as there's no music.

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  35. Whereas the ‘open’ string shuts off its far end with a hard, ‘stiff’ piece of wood, giving a ‘sharp demarcation’, in the case of the ‘stopped’ note this marking-off is done by the ‘soft’, ‘movable’ finger, giving less sharp demarcation.

    Charles Mingus played the fretless bass for a reason, I think. I highly recommend Black Saint and the Sinner Lady for a start, which comes with the bonus of the liner notes' having been written by his psychiatrist at Bellevue. I've heard that he never made much money despite the genius.

    https://www.wikipedia.org/wiki/Charles_Mingus

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  36. As you know, pure sine waves sound pretty sterile and unnatural.

    O RLY? Maybe it's my experience in Howard Sanndroff's electronic music lab, or having been a shortwave enthusiast, or listening on VHF to lake buoys, or an ancient undergraduate degree in physics, but they sound awfully natural to me. Then again, I suppose there's really no accounting for taste.

    https://www.youtube.com/watch?v=uIuJTWS2uvY

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  37. Did you hear about the musicologist whose PhD thesis title was "The Role of Rock in Rock and Roll" ?

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  38. Otto wrote: [pure sine waves] sound awfully natural to me . . . I suppose there's really no accounting for taste

    As far as I know there's no natural source of pure sine waves, but I could be wrong. Maybe there is, say, some insect or bird that naturally generates sine waves. So I'll concede that I don't actually know if pure sine waves are unnatural.

    In any case, maybe your real point is that sine waves don't sound "sterile" to your ears; they sound aesthetically and musically pleasing.

    In the YouTube clip you provided, I was expecting some nice piece of electronic music that demonstrates the natural, pleasing musical potential of sine waves. But all you gave me was square waves and sine waves. I'm an engineer so I'm all too familiar with oscilloscopes and waveforms.

    If the sound of sine waves is an acquired taste, I'm sorry to say I've never acquired it. It's my negative gain and your negative loss, I'm sure.

    In the 1970's some of my friends said that the theme from Close Encounters of the Third Kind is a splendid example of sinusoidal music. It's splendid all right, but it doesn't contain even a single sinusoidal note. They're tone deaf, I reckon.

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  39. As far as I know there's no natural source of pure sine waves, but I could be wrong.

    I file wind as natural. One of my apartments over the years opened onto a back alley that was an Aeolian harp.

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  40. To Mr. Mason:

    Actually, music was not playing as I read Sabine's book. I mostly read it on the train traveling between New York and Connecticut. It made the miles fly by and what seemed like relativistic speeds.

    Dave

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