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by weinzierl
3606 days ago
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> It’s worth noting that instruments were being tuned to this scale well before the invention of logarithms. I assume it was done by ear or perhaps by geometry, not by algebra. Well temperament wasn't known before 1681, there were predecessors that came close but none used the the twelfth root of two.
Logarithms were introduced by John Napier in 1614 [1], the twelfth root of two
was first calculated by Marin Mersenne in 1636 [2], well temperament was introduced by Werckmeister 1681 [3]. > Around 1600 Simon Stevin did attempt to calculate numerical values for the pitch intervals by decomposing 12th roots into combinations of square and cube roots; his results were not flawless. He had the right idea but not the right math. [1] https://en.wikipedia.org/wiki/John_Napier [2] https://en.wikipedia.org/wiki/Twelfth_root_of_two [3] https://en.wikipedia.org/wiki/Well_temperament |
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My source for the statement that "instruments were being tuned to this scale well before the invention of logarithms" is an article by Edward Dunne and Mark McConnell, "Pianos and Continued Fractions," Mathematics Magazine, Vol. 72, No. 2 (Apr., 1999), pp. 104-115. They write: "Guitars in Spain were evenly tempered at least as early as the fiftheenth century, two hundred years before Bach. And Hermanus Contractus, born 18 July 1013, invented a system of intervalic notation that anticipated equal temperatment."
My source on Simon Stevin's work is a Rudolf Rasch, "Tuning and Temperament," published as Chapter 7 in The Cambridge History of Western Music Theory." Rasch gives an interesting account of the state of root extraction before the advent of logarithms. He attributes the errors to "Stevin’s sometimes rather reckless rounding of digits after the decimal period, which are often truncated rather than rounded."