You don't understand.
He was lucky that the continued fraction expansion of 2^(1/12) started with such
"simple" fractions.
Rainer
On 28.07.2019 16:53, Ron Andrico wrote:
<Galilei was lucky here>
Galilei arrived at the best approximation with the information and
tools available to him at the time. No other system could be more
appropriate to the (evolving) music of his time. And he had a grasp of
the physical realities of the lute, as well as taste.
RA
__________________________________________________________________
From: [email protected]
<[email protected]> on behalf of Rainer
<[email protected]>
Sent: Sunday, July 28, 2019 2:15 PM
To: Lute net <[email protected]>
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.
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