Subject: Re: [HM] Mathematics and Music
From: Clark Kimberling (ck6@cedar.evansville.edu)
Date: Thu Jul 27 2000 - 09:59:40 EDT
Thanks, several readers, for locating Hardy's remark, "Chess problems are
the hymn-tunes of mathematics" in A Mathematician's Apology. Hardy goes
on to say that "A chess problem is genuine mathematics, but it is in some
way 'trivial' mathematics..." Thus, we surmise, a hymn-tune is genuine
music, but 'trivial'...
Hardy is focusing on the relative significance of chess-problems within
mathematics and hymn-tunes within music. I'll return to this focus.
But first, let it be noted that Hardy takes for granted the main thing
that chess-problems and hymn-tunes share: combinatorialness. For both,
there are sets of rules. To compose a hymn-tune, you obey certain
combinatorial rules about melody, harmony, and rhythm. (Writing without
rules, it is said, is like playing tennis without a net.) Hardy's
reference to hymn-tunes would mean more if there were evidence that he was
familiar with the "rules" of their construction.
Now let's turn to the relative significance of hymn-tunes -
Two years ago, a wonderful work was published. It is Nicholas Temperley's
THE HYMN TUNE INDEX (Clarendon Press Oxford). In four volumes, this index
gives the publication history from 1535 to 1820 of more than 17,000 tunes.
Unlike chess problems, each hymn tune has a name. For example, the great
tune HELMSLEY, found in many modern hymnals, was published in 165
collections indexed in HTI (and many more after 1820). A tune like this
(especially after Vaughan Williams arranged it in 1906) is, among other
things, a combinatorial gem, and, I assert, it occupies a larger place in
music than any chess problem does in mathematics.
HTI is compactly organized, and the tunes are identified not only by name,
but by mathematically formulated incipits. When I first saw HTI, I
thought: this must have been put together by a mathematician - indeed, a
measure-theorist. It turns out that Temperley, a professor of music at
the University of Illinois, has, in fact, published some mathematics (in
Hardy's corner).
Getting back to Hardy's reference to hymn-tunes, my main point is that
hymn-tunes are not a good analogy for chess-problems. To push a bit
further:
1. Hymn-tunes are recognized within music literature (e.g., in Grove's
Dictionary; in lists of composer's works); are chess-problems
comparably recognized within mathematics literature?
2. Great composers used hymn-tunes within symphonies - what great
mathematician developed a chess-problem into a major theorem, in which
the chess-problem remains a main theme?
3. Dozens of themes by major composers became hymn-tunes (HTI, vol. I,
38-42); what major theorems have produced chess-problems?
Clark Kimberling
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