Tuesday, January 18, 2011

Music in the "Mathematical View of the Universe"

I was attracted to Stephen Shankland’s Deep Tech column for CNET News this morning by its title, “Wolfram education apps raise teaching dilemma;”  but I quickly discovered that my personal interest departed significantly from the primary substance of his article.  Here is the core of that primary substance:

Wolfram Research got its start with the hard-core Mathematica software, itself an offshoot of Stephen Wolfram's attempt to explain his mathematical view of the universe embodied in his book, A New Kind of Science. It was therefore fitting that the company's "knowledge engine," Wolfram Alpha, took a rigorous approach to facts and data.

So perhaps it shouldn't have been a surprise that the company's first mobile application to use Alpha was similarly tailored for a refined audience and came with a correspondingly expensive price tag of $50. No doubt displeased with the response, Wolfram shortly after decided to "focus on ubiquity" and cut the price to $2.

Now Wolfram is showing signs that indicate a deeper understanding of consumer sensibilities, announcing new iOS applications called Wolfram Course Assistants to help students with algebra, calculus, and music theory. They tap into Alpha's Mathematica abilities behind the scenes, but they're focused, packaged, and reasonably priced at $2 for algebra and music theory and $3 for calculus.

Like just about everyone of my generation with an interest in computers, I have known about Mathematica for some time.  Indeed, I was an undergraduate at the Massachusetts Institute of Technology when its pioneering predecessor, MATHLAB, first surfaced at The MITRE Corporation.  There is no questioning that Mathematica was a great leap forward from MATHLAB;  but, for the sort of research I was doing, I never needed to take that leap.  Most of my work involved symbol manipulation but had little to do with the symbolic forms of higher mathematics.  LISP was sufficient to serve my needs, and I am happy to say that I have never lost my capacity for expressing my computational needs in this programming language.

On the other hand those same research activities have set me apart from Wolfram’s belief in that “mathematical view of the universe” ever since I first encountered it.  The more I learned about the impact of technologies, particularly information technology, on human behavior in the “real world,” the more skeptical I became about that view and the more inclined I was to dismiss it as outmoded deleterious Enlightenment thinking.  The piece on A New Kind of Science in The New York Review reinforced my skepticism;  and my basic reaction to the launch of Wolfram Alpha was your classic “Whiskey Tango Foxtrot moment.”

Nevertheless, I was no more surprised than Shankland to see Wolfram venture into the wonderful world of apps;  nor was I surprised that this would launch a debate similar to the one that arose when sophisticated pocket calculators first started appearing in classrooms.  In this debate I continue to side with Seymour Papert, who, in his pioneering work on LOGO software as an educational tool, wrote a perceptive paper arguing that it was more important to teach children to “be mathematicians” than to teach them mathematics.  I continue to believe that a good teacher can, indeed, teach students of any age to be mathematicians;  and I accept the premise that software like Mathematica can contribute to that teaching process.  Does it matter that bad teachers may misuse Mathematica?  Of course it does, just as it matters that they may misuse pocket calculators, slide rules, and, for that matter, pencil and paper.

Where I really get uneasy, however, is the way in which Wolfram and his colleagues have now used apps to fit music theory into that “mathematical view of the universe.”  This led me to examine in more detail than Shankland provided just how that “fit” was established by the Music Theory Course Assistant app.  On the surface it seems to provide a useful path of entry to what one might call the “lexical basics” of music theory:

  • Hear and view accidentals and octaves anywhere on the staff
  • Choose from both common scales and hundreds of more advanced scales
  • Explore triads and major, minor, and seventh chords
  • Input up to four chords and hear their progression
  • Learn how to identify music intervals by their name and what they sound like
  • Find interval inversions for every interval type
  • Reference musical terms like "melody" and "fortissimo" in the abbreviated music dictionary

Nevertheless, there are certain fundamental matters that really need to be covered from the very beginning.  The most important of these pertain to listening, and I have to wonder whether or not they are being properly honored.

The most challenging of these is the matter of enharmonic intervals.  These are the intervals that sound the same when they are played on the keyboard of an instrument using equal-tempered tuning;  and this tends to cultivate the misconception that enharmonics are nothing more than intervals spelled in different ways.  However, those who practice music know that this is about as specious as the proposition that homonyms are words that mean the same thing spelled in different ways.  It is thus important that even the earliest stages of ear training recognize that these differences involve more than spelling and that students begin to learn the logic behind what the proper spelling is.  The mere fact that the interval training portion of the Course Assistant includes a keyboard display leaves me worrying that any sound synthesis provided by the software never gets beyond equal-tempered tuning and may ultimately mislead the beginning student.
This then leads to the related question of how this software handles those “more advanced scales.”  The sample display on the Web site indicates that one of these categories is “Grecian Scales.”  Our knowledge of the music of Ancient Greece is very limited, particularly when it comes to how (if at all) it was notated and how it actually sounded.  (The Wikipedia entry does not do a bad job at summarizing the current state of knowledge in this regard.)  We know from sources like Aristeides and Ptolemy that the “music theory” of the Greeks was based on the interval of the perfect fourth, the interval between the third and fourth overtones in the natural harmonic series.  This interval was divided into 60 equal parts.  On the piano keyboard the perfect fourth consists of five semitones, which means that an equal-tempered semitone would be divided into 12 equal parts.  Therefore, the entire octave would be divided into 144 equal parts, making it twice as refined as the EUTERPE system I developed for my own experiments in microtonality.  Now it turns out that the term “enharmonic” can be traced back to the Greeks;  but, given their refinement of the perfect fourth, it goes without saying that it refers to something more sophisticated than different ways of spelling the same interval!  If the Wolfram app recognizes this sophistication, I shall be surprised;  and I shall be even more surprised if the audio it provides is good enough to help the student recognize it too!

All this may be a classic example of using a sledgehammer to crack a walnut, as one of my teachers used to say.  However, the real purpose of the sledgehammer is not to pick nits over microtonal intervals but to generalize Papert’s precept.  Teaching music is all about teach the students to be musicians.  One must start with approaches to making music;  and, through those approaches, one may introduce music theory for its capacity to facilitate how we can talk about making music.  Such conversations can probably benefit from an easily consulted dictionary;  and it would certainly be nice to have such a dictionary on a portable digital device (as it would be for a dictionary of the English language).  It does not appear that the Wolfram app was intended to be such a dictionary;  and, if it can serve as one, consulting it may not be as easy as we would expect from an English dictionary.

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