Geodesic
The Twelve Magnificents
Zbornik radova, Tome 21 (2024) no. 29, p. 23

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DOI

We present the history of the theory and practice of musical scales, which culminated in today's universal 12-tone scale. Then we prove that this 12-part division of the octave is optimal. The proof is an application of the theory of continued fractions. Drobisch in \emph{Über musikalische Tonbestimmung und Temperatur} (1852) used continued fractions for subdividing the octave into intervals. It is not clear whether he was the first to use this approach; e.g. Euler was dealing with mathematics of temperament and also wrote a tract on continued fractions, \emph{De Fractionibus Continuis Dissertatio} (1737).sikic.pdf
DOI : 10.18485/mi_sanu_zr.2024.29.21.ch2
Classification : 00A65
Keywords: interval, scale, tuning, temperament, continued fraction 
Zvonimir Šikić. The Twelve Magnificents. Zbornik radova, Tome 21 (2024) no. 29, p. 23 . doi: 10.18485/mi_sanu_zr.2024.29.21.ch2
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     title = {The {Twelve} {Magnificents}},
     journal = {Zbornik radova},
     pages = {23 },
     year = {2024},
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     doi = {10.18485/mi_sanu_zr.2024.29.21.ch2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.18485/mi_sanu_zr.2024.29.21.ch2/}
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