I believe Isaac Newton did some work with this as well; unfortunately I don't have a source at hand. Leonard Williams -----Original Message----- From: Rainer <[email protected]> To: Lute net <[email protected]> Sent: Sun, Jul 28, 2019 10:16 am Subject: [LUTE] "Equal" temoerament By the way, a few minutes ago I calculated the first terms of the continued fraction expansion of the twelfth root of 2 (which is infinite and non-periodic-the continued fraction and the decimal fraction :)). This gives in a precisely defined meaning the "best" approximations of 2^(1/12) in rational numbers. The first approximations are: 1 not very useful for tuning :) 17/16 quite good 18/17 Much better - Galilei was lucky here 89/84 Too complicated already A stack of 12 semitones at 18/17 gives an octave of 1.985559952 Best wishes, Rainer PS As far as I know the first who clearly states that the twelfth root of two should be used was Hendrik Stevin in his "Van de Spiegheling der Singconst" written before 1608 but not published until 1894. To get on or off this list see list information at [1]http://www.cs.dartmouth.edu/~wbc/lute-admin/index.html
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