Article Discussion: Re: "Can One Hear the Sound of a Theorem"

Can One Hear the Sound of a Theorem?  Rob Schneiderman
Notices of the American Mathematical Society August 2011
http://www.ams.org/notices/201107/rtx110700929p.pdf

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In this blog, I believe I find myself generally having a positive tone. Normally I am writing about musicians and concerts, and have been having a great time, and, as I understand it, the general practice of critiquing music, perhaps more so in North America than in Europe, but nonetheless, is to acknowledge what has been given and achieved, and only then perhaps state which elements one might take exception to.

However, from my philosophical studies, I've learned also that the world of Acadamia is happy to enter battles of opinions rather more overtly, without mincing words.

Rob Schneiderman's article is of this nature, and my response to it is equally so.

I originally sent my response to the editor of the AMS, who wrote back to say I would have been invited to submit it for publication, but that I'd written too late. I blame summer/fall musician travels - the mind is elsewhere.

In any case, here goes. I will dig in. Oh, and yes, I include my credentials at the end.

Dear AMS Editor,

I am mystified as to what the point of Schneidermann's rather contrarian article "Can One Hear the Sounds of a Theorem" in the August issue might be, other than that it is very difficult to prove a rigorous correlation between mathematics and music. Certainly, his own examples don't hold up to his kind of scrutiny, any more than the examples of others he takes such pains to shoot down. The article reads more like a rant than a discussion or constructive criticism, and lacks a basic knowledge in the philosophy of aesthetic theory.
Take for example his attempt to distinguish between music/mathematics from other arts and sciences, via assigning "intrinsic meaning". Schneiderman realizes that this is a problematic concept (without a meaning-giver, ie. a person, is there actually meaning?) - and tries to get out of it by whittling down "intrinsic meaning" to "degrees of intrinsic meaning" ... but if a thing doesn't exist, how can you have degrees of it? As anyone who has studied the problems surrounding both formalism and hermeneutics knows, the concept of meaning is very difficult if not impossible to extract from human qualities and experiences. Equally problematic is his assertion regarding reference, that "both mathematics and music frequently do refer to the natural world". Reference, as philosophy of language tells us, is as troublesome a concept as meaning, and in taking the viewpoint that music and mathematics can refer, Schneiderman suddenly adopts a hermeneutic viewpoint that is completely unsupported by anything else in his article, and actually contradicted by his formalist stance elsewhere. Just as unsupported is his opinion that music is somehow more abstract than other art forms, which he defends solely on the basis of his own intuition, "I stand by the claim," and an attempt to describe his definition as "what is special to mathematics and music is that their content is capable of being expressed entirely in terms of their own raw material, namely, logical thought and audible sound". However, this latter sentence is merely a definition of abstraction, which exists in all the other art forms - painting as expressed in terms of interplays of light, dance as expressed in terms of motion, poetry as expressed in uniqueness of form, which may contain jibberish - the Jabberwocky poem of that great mathematician/writer Lewis Carroll comes to mind. Schneidermann tries to get out his dilemma by stating that abstracted poetry is music - but this is just an easy way out: "abstracted anything is mathematics or music", and takes us to a circular definition.
In the end, it appears that he is making the claim that "patterns that do not rely on sensory perception" are the core of his special categorization of mathematics and music. But again, this definition contradicts what he wrote earlier, having included "audible sound" as a "raw material" of music. Setting aside that discrepancy (and the floodgate of the artforms that the "raw materials" admission allows, as discussed above), it is unclear how he is establishing that there is anything that is especially non-sensory about music, other than his claim that it is somehow mathematical.

Another distinction he is missing is that between analysis and creativity, two opposites in terms of attitude. One similiarity between creating/discovering a musical piece and developing a theory along a branch of mathematics is that one sees how far an idea can go. The difference is that mathematical rules are less subjective, most will agree with what is proper or improper procedure, whereas in music one can decide at any moment to do anything - to create - and the result will be either more or less pleasing to various audiences, or up for argument. He dimisses all music that is inspired by mathematics - but where is the mention of some of the greatest "mathematical" composers, for example, Milton Babbitt, Iannis Xenakis, or even Pierre Boulez? There is a distinction between being inspired by and using mathematical constructs in musical composition, and creating new forms utilizing them (as so many 20th-and 21st-century composers do and with great success), and adhering to a rigorous one-to-one mapping between a mathematical concept and some (arbitrarily) chosen correspondence in the sound world, as appears to be the thing Schneiderman is railing against. The beauty and elegance of a composer's meaningful and convincing message using complicated mathematical constructs, is precisely that, on the basis of what appears to touch people, it need not be via one-to-one correspondence, and that we can feel a strong message even if the mathematical idea utilized is hidden. His example of Bach's crab canon is apt here - where the correspondence is too strong, the message we feel is that it is a little too perfect: it works, it tells us something, but there is something not entirely natural-feeling about it. And that is a difference between mathematics and music - we are supposed to be able to work out and understand mathematics to the last degree, and perfection of symmetry can be an end, but music is meant to leave something inexplicable, and also tell us something about ourselves - why do we find ourselves attracted to one form or another?

Perhaps his thesis is actually, that correlation between mathematics and music is merely metaphorical - for precisely the reason that mathematics does not take "raw material" to fill its place-holders, and music does. But if he wishes to claim that music is as pattern-oriented as mathematics, in order to support his claim that music is of an abstract nature, then he has again taken the leap of faith he criticizes in the other authors. Yet perhaps it is precisely this leap of faith that we must take in order to get the "what, if anything, do-music-and-math-have-in-common" project off the ground at all. That would be an interesting idea to develop further - and amazingly, would lead us not to need to reform the school curriculum anywhere as violently as he suggests, for his currently narrow view of what he considers "correct" in this regard would, I posit for the reasons I've mentioned above, open up to many more possibilities.

Sincerely,

Claudia Schaer

Faculty, Bloomingdale School of Music, New York City

D.M.A. Stony Brook University, Violin Performance
Graduate Certificate Equivalent - Philosophy and Music

Aesthetics and Logic studies at Columbia University
MM, BM, The Juilliard School, Violin Performance

Concert Review - Bramwell Tovey conducts the Calgary Philharmonic Orchestra, with Andrew McCandless, trumpet. Sat. Apr.14, 2012

There's both something special, and something very simple, about going to hear your hometown symphony. You remember your thrill as a child, perhaps studying with members of the orchestra, of looking up to the professionals who seem to magically perform impossible feats every moment. You also remember when you started learning about the world outside of your immediate periphery, and became curious to hear live what others transmit as legend, or what you've gleaned from a library's recording. And then, you grow up, and realize how vital a role the orchestra plays in your community, how fortunate your city is to have one, and come to understand the commitment, care, and yes, sacrifice, that every member of the orchestra makes, day in and day out, to keep its existence continuing.

So when my good friend said she had two tickets courtesy of a corporate sponsor, and did I want to go, we had a look at the programme, and decided, "Sure, looks interesting, why not." Very nice that we had among the best seats in the house - prime parquet.

The programme was titled "From the New World" and consisted of Copland's Quiet City, a new trumpet concerto (2009) by also-conductor Bramwell Tovey, and, of course, Dvorak's New World Symphony.

As is the new way in many orchestral concerts, the evening began with an acknowledgement of the main sponsors. At worst, this can frame the concert as background music for an advertisement for these companies. At best, as was tonight's delivery, it serves as an equally grateful and proud introduction to the enjoyment that is to follow, a recognition of the work and support that, as the cliché so aptly states, makes the evening possible.

A succinct and well-crafted intro by the CPO's Principle Trumpet "Emeritus" Howard Engstrom took us from Calgary into the world of Aaron Copland - early Americana portrayed in youthful innocence. The solos were lovely, gently played by Adam Zinatelli, CPO's Principle Trumpet , and David Sussman, CPO's English Horn, and the piece kept its spell from beginning to end.

Bramwell Tovey's trumpet concerto Songs from the Paradise Saloon, required a veritable family of trumpets, and soloist Andrew McCandless, Toronto Symphony's Principle Trumpet, took advantage of this very visual conversation-piece in his introduction. Appearing thoroughly at home with addressing a large crowd, his speech started down a road that well could have ended in a ditch of not-quite-successful stand-up-comedy - a few remarks in, he noted having no idea whether the concert would go well tonight, and then noted his use of self-deprecation  ... and then, with the audience already warmed up to him, he steered it straight into very-successful and witty stand-up comedy zone, accompanied zestfully by Tovey, with great rapport. When Tovey pointed out McCandless's second use of self-deprecation, the latter responded in mock exasperation, "Just let me handle this, ok??", and sighed to the audience, "Conductors!" The duo's hilarious improv culminated in airing dirty laundry regarding the correct pronunciation of "cornet", which McCandless informed us that Tovey, a Brit, had corrected him of "in front of two thousand people" during his introduction at yesterday's concert. Tovey meanwhile glanced pointedly at his wristwatch, and then in his part of the introduction wished us a "Happy Titanic Anniversary", explaining that all that could be said about the concerto had been said.
As it turned out, that wasn't quite the case: the concerto is extracted from Tovey's opera The Inventor, about notorious 19th-century Canadian criminal "Sandy" Keith, and Tovey's descriptions of the scenes of debauchery were a colourful and helpful lead-in to the music. My friend (not a musician) and I (a musician) agreed that this way of learning about the music was thoroughly entertaining, and made for much better evening than either not knowing what the music was intended to be about (potentially shocking anyone expecting more demure or Mozartean classical music), or having to read programme notes to find out the plot (I personally prefer to read the notes on the way home, after the concert). This might be the place for an enthusiastic "Two thumbs up!!"

The music, and the performance, were no less praiseworthy. I was deeply impressed by the concerto, which displayed a mastery of style and all the old-time enterprises of harmony, counterpoint, rhythm, and orchestration, coming together in songs with fresh sounds, blends, and expression to suit the scene and characters of the saloon. There is a great deal of focus in modern classical composition to create anew, but to do so with full knowledge, understanding, and ability of the old, shows a rare and complete musician. I felt myself becoming a Bramwell Tovey fan.
Andrew McCandless played flawlessly and with verve, fighting the orchestra's thick sonority bravely, weaving spikily in and out of the texture. He is in some ways at the mercy of the orchestra, and finding flexibility is not easy, but then perhaps that is not the point - he never gets to play the theme but dances around it in variations. He is, as it were, re-inventing the reality around him. 

In terms of trumpet show-casing, the evening offered an interesting juxtaposition: Zinatelli had the more exposed role, but played in more subdued hues, while McCandless had the brilliant and more technical role, as well as the longer piece, but was in a sense granted less spotlight in relation to the orchestra. And so the question of "which city, Toronto or Calgary, has the better Principal Trumpet", could be neatly avoided, not always a given in such guest situations.

The best-known work on the programme was of course Dvorák's New World Symphony. It needed no introduction. But it needed a little reminder of traditions: the Calgary audience, under music director Roberto Minczuk's now 6-year tenure, has been encouraged to clap between movements, and did so enthusiastically after the first movement. Maestro Tovey handled it just as I would hope - he gave a nod of appreciation for the sentiment, but made it clear with another gesture that he wished to continue, for the movements to be allowed to relate to one another uninterruptedly. (Presumably those few who clapped after the 2nd movement appreciated his reaction so much they wished to elicit it again - but this time I daresay he frowned.) He was careful to take only a short enough break between the third and fourth, to avoid any applause there. But the immediate and heartfelt standing ovation at the end was fully taken in and absolutely deserved - it was a wonderful performance. The orchestra sounded among the best I've heard it - beautifully blended, with a clear path of interpretation throughout, warm and meaningful. The very special second movement in particular shone in splendid beauty. It was a very satisfying performance indeed.
It also reminded me of just how much power is vested in the conductor. In contrast to tonight, I had been rather disheartened by a previous CPO performance of Brahms' 4th symphony, with a different conductor, which to my ears had been overly driven and harsh, (reminding me for some reason of undercooked risotto, or overbrushed- and too-sparkling-whitened- teeth). It does not take musician-ears to feel something amiss in the emotion: the audience remained seated and the clapping was distinctly polite, ending after just one curtain call. As Brahms and Dvorák were contemporaries with similarities in their writing, and a Brahms symphony certainly no less of a masterwork than Dvorák's, the cause was clearly the performance, and I had had some hesitation at the prospect of subjecting myself to an interpretation so counter my taste again.
So it was especially delightful to find just the opposite tonight - to find myself thrilled, and the whole house too. The conductor is, after all, the guiding light - leading the rehearsals, and channelling his interpretation's electricity into gestures to unite the musicians, taking into account all that we have to offer as well. 

It was thoroughly satisfying to see my home-city's now-colleagues playing their best, to experience this excellent performance. Sipping tea with my friend afterwards, I found myself telling the waiter - "We were down the street at the symphony tonight - you've never been? It was wonderful, you should go!" 

And so, regarding the much-talked-about future of classical music and of reaching new audiences, I can't help feeling that it takes BOTH of two key ingredients: new innovative presentation, AND the old traditional highest standards of quality.

We all feel it. Or as the CPO slogan goes, we "Feel It Live".

Creating the Future, Present, and Past

This year, I have had the privilege of being a guest participant in Ray Anderson's improvisation class at Stony Brook University. As a “good student”, the thing for me to say, which is also really true: it's been a wonderful experience, and I've learned a lot. As a free spirit, I also have a lot more to say.
The most intriguing observation for me is how the class navigates the realm of emotion, and of pure next-ness, which perhaps might be termed subconscious emotion. When improvising in a group, we may have a general emotion in mind, and start from there, but in the moment itself, when it's time to do something, somehow the next sound appears. It is this connection between the two, the letting what one feels in the moment tie to and directly influence what happens next, that was central to the magic that happened in the class.
Apart from Ray, a great jazz player, we were a group of classical musicians, and as is often the case in our reverent tradition, we take pieces as already completed and learn how to execute them; it is still rare, though this is evolving as musical institutions look back to how the music we play was actually created, for our education to feature freestyle classical jam sessions, and a strong focus on both improvisation and composition for instrumentalists. So we heard a great deal from Ray about “turning off the inner critic”, to free up from comparing to the enormous shoulders of the greats of the past and present. It's about getting up the courage to walk a new course without recourse to a map. He actually had a session entirely on ugly sounds, encouraging them, to get out the fear of making them. In this class, better to speak and have the result be un-aesthetically-pleasing, than not speak at all – and as it turns out, in art there are no wrong answers, everything can go somewhere, or nowhere if we prefer.
Since we were exploring entirely free-style, we removed all constraints of style. It is true that constraints sometimes give more options than pure freedom, but our aim here was not to create a perfect polka, or a judicious smooth jazz jam, or a sonorous sonata. The aim was to link the visceral emotion of creation, response, and sound, with something coming out of our instruments. As a result, we found ourselves talking much more about situations, than about music. We'd have a follower and a responder, a soloist and an accompanist, a joiner and an abstainer. What those would sound like was left up to us. And somehow, we found strings of sounds, however imprecise, that would convey what we felt in the moment, much as a baby might make sounds, without knowing a language, that somehow seem appropriate to her in her situation. After that, it was inventing situations, either explicitly, or as we went along, that would create the framework we played within and without. In one very interesting experiment, each one of us “conducted” the ensemble, with gestures to distinguish desired articulations, dynamics, solos, rhythm, and anything else we could come up with. It left room for the players to improvise around the indications – a solo could be anything – and, in being on the spot, it let the “conductor” try out whatever came to mind, which wound up being an interesting reflection of the very personal unique types of patterns each of us gravitate towards. Indeed the structure of each improvisation in the end reflected and perhaps exaggerated each of our very distinct personalities, and in this realization we could take what we learned about our tendencies, and apply our increased self-awareness towards a greater freedom in improvisational decision-making, and refresh potential habits with views of new directions.
In composition, there is much focus on “creating a language”; in these improvisation sessions, I had the opportunity to get my feet wet in the pre-language stage. My conclusion: it is great fun to play with the ethers, from which life congeals and springs up!