Perfect harmony: A mathematical analysis of four historical tunings

["f.maintenant" <f.maintenant@xxxxxxxxxxxx>]

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