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Re: Perfect harmony: A mathematical analysis of four historical tunings

I haven't read the paper Frédéric Maintenant refers to in his post below, and I'm not sure that I would be properly qualified to pass judgment on it, in any case. But I'm a bit concerned about the elitist ageism it seems to reflect.

Many scientific minds have produced exquisite and immensely productive thinking at a very early age. Indeed, many minds seem to lose their creative élan after an extended adolescence -- a bit like some recent athletes. And -- in this case, sticking to the French -- writers from Rimbaud to Sagan certainly made their mark before 20.

As we all know, Mozart composed many early pieces, notably, at age 11,the piano concerti K. 37, 39. 40 and 41. In 1775, at age 19, Mozart composed all five of his violin concerti. Anyone interested in following the prodigious progression of the early piano concerti, can do so most enjoyably by listening to the first CDs of the 12-CD Volume 7 of the Philips Complete Mozart Edition (442507-2).

For a less ipseitic example, consider Mendelssohn's 12 String Symphonies, all of which he wrote at 14. I prefer Mendelssohn as an example because he was so sensitive to, and appreciative of, the music of other composers and did much to make it accessible.

Then we have the American composer Donald Keats, currently professor at the University of Denver. Thirty years ago, in the fall of 1964, I made a passing remark to Keats about "student compositions." Keats smiled and pointed out that his two String Quartets were, at the time, the first string quartets to be published by Boosey & Hawkes since Stravinsky. "Needless to say," he added, "I didn't tell the publisher that I wrote them while still a student because Boosey & Hawkes doesn't publish 'student works.'"

So, for whom does "the use of word such as 'harmony' or 'mathematical analysis' [NOT] have the same signification whether you are at high school or in a research team"? Probably not for those who aren't being creatively productive at a relatively early age. When genuinely good thinking -- or, for that matter, good writing or good composing or good performance -- occurs at age 15, is it any different than genuinely good thinking or good writing or good composing or good performance at age 79? Ironically, some people appear no more cognitively capable of producing memorable coherent musical form at 79 than they were at age 15. (Memorability, I suggest, is, or ought to be, a key concept in musical cognition research.)

Just a reaction -- and a thought.

Denis Donovan


Dear friends,

I found a little bit suspect that JASA published a paper written by a
researcher when he was 15. With no disrespect to Michael F. Page, I am not
sure that the use of word such as "harmony" or "mathematical analysis" have
the same signification whether you are at high school or in a research team.
However I must admit that I have only read the abstract so far.

Perfect harmony: A mathematical analysis of four historical tunings
Michael F. Page
The Pingry School, Martinsville Road, Martinsville, New Jersey 08836

(Received 30 September 2003; revised 13 July 2004; accepted 14 July 2004)

In Western music, a musical interval defined by the frequency ratio of two
notes is generally considered consonant when the ratio is composed of small
integers. Perfect harmony or an "ideal just scale," which has no exact
solution, would require the division of an octave into 12 notes, each of
which would be used to create six other consonant intervals. The purpose of
this study is to analyze four well-known historical tunings to evaluate how
well each one approximates perfect harmony. The analysis consists of a
general evaluation in which all consonant intervals are given equal
weighting and a specific evaluation for three preludes from Bach's
"Well-Tempered Clavier," for which intervals are weighted in proportion to
the duration of their occurrence. The four tunings, 5-limit just intonation,
quarter-comma meantone temperament, well temperament (Werckmeister III), and
equal temperament, are evaluated by measures of centrality, dispersion,
distance, and dissonance. When all keys and consonant intervals are equally
weighted, equal temperament demonstrates the strongest performance across a
variety of measures, although it is not always the best tuning. Given C as
the starting note for each tuning, equal temperament and well temperament
perform strongly for the three "Well-Tempered Clavier" preludes examined.
©2004 Acoustical Society of America.

Award of $250 for Excellence in a Written Report [2003 Olympics of Science
Fairs]:
Perfect Musical Harmony: A Mathematical Analysis of Four Historical Tunings
Michael F. Page, age 15, The Pingry School, Martinsville, New Jersey


Best

Frédéric Maintenant