%% scale_length.txt
%% Created by Laurence D. Finston (LDF) Fri 08 Jul 2022 04:52:28 AM CEST
%% * (1) Copyright and License.
%%%% Copyright (C) 2022 Laurence Finston
%%%% You can redistribute this file and/or modify
%%%% it under the terms of the GNU General Public License as published by
%%%% the Free Software Foundation; either version 3 of the License, or
%%%% (at your option) any later version.
%%%% This file is distributed in the hope that it will be useful,
%%%% but WITHOUT ANY WARRANTY; without even the implied warranty of
%%%% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
%%%% GNU General Public License for more details.
%%%% You should have received a copy of the GNU General Public License
%%%% along with this file; if not, write to the Free Software
%%%% Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
%%%% The author can be contacted at:
%%%% Laurence Finston
%%%% Laurence.Finston@gmx.de
\input eplain
\input epsf
\input colordvi
%% * (1)
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\begingroup
\advance\hsize by -12cm
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\parskip=.5\baselineskip
\centerline{{\largebx Scale Length Example}}
\vskip\baselineskip
\small
Copyright (C) 2022 Laurence Finston
\setbox0=\hbox{Last updated:\quad}
\leavevmode\hbox to \wd0{Created:\hfil}July 8, 2022
\leavevmode\box0 July 10, 2022 %% Updated
You can redistribute this file and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 3 of the License, or
(at your option) any later version.
This file is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with This file; if not, write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
See the GNU Free Documentation License for the copying conditions
that apply to this document.
You should have received a copy of the GNU Free Documentation License
along with this file; if not, write to the Free Software
Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
The author can be contacted at:
Laurence Finston\hfil\break
{\smalltt Laurence.Finston@gmx.de}
\endgroup
\vskip2\baselineskip
{\bf See following page.}
\vfil\eject
\centerline{{\bf Scale Length Example}}
\vskip2\baselineskip
\leftline{\epsffile{scale_length001.eps}}
\vskip\baselineskip
\leftline{\epsffile{scale_length002.eps}}
\vskip\baselineskip
\leftline{\epsffile{scale_length003.eps}}
\bye
%% using namespace std;
%% int
%% main(int argc, char *arv[])
%% {
%% printf("Entering `main'.\n");
%% float f = powf(2.0F, 1.0F/12.0F);
%% double d = pow(2.0D, 1.0D/12.0D);
%% long double ld = powl(2.0L, 1.0L/12.0L);
%% cout << setprecision(128)
%% << "f == " << f << endl
%% << "d == " << d << endl
%% << "ld == " << ld << endl;
%% // f == 1.05946314334869384765625
%% // d == 1.0594630943592953098431053149397484958171844482421875
%% // (length "1.0594630943592953098431053149397484958171844482421875") == 54
%% // ld == 1.05946309435929526452345450504566315430565737187862396240234375
%% // (length "1.05946309435929526452345450504566315430565737187862396240234375") == 64
%% // Emacs-Lisp 1.0594630943592953
%% return 0;
%% }
%% (length "0594630943592953")16
%% %% This is a bit more technical, in case anyone is interested in the
%% details (and doesn't know this already). %% The reason for using the
%% 12th root of 2 is in order to lay out a scale using equal temperament.
%% The reason for using the proportion 18/17 was because of the
%% impracticality of doing it using direct calculation or a better
%% approximation using the ratio of larger whole numbers before there
%% were pocket calculators and computers. Nowadays, of course, we can
%% let the computer do the calculations for us.
%% So, why did I use
%% 1.0595? Well, just in order to be able to use the values in the book
%% by Benade. It would have been just as easy to let the computer use a
%% better approximation. %% So, what would these approximations be? If
%% I evaluate the Emacs-Lisp expression (expt 2 (/ 1 12.0)) in an Emacs
%% buffer, I get the value 1.0594630943592953 with 16 places after the
%% decimal point. It might be possible to get more decimal places, but I
%% haven't bothered to look them up because Emacs-Lisp isn't really
%% letting me get "close to the machine". One way to do this is to use
%% the programming language C. %% In C, corresponding to the
%% possibilites the hardware has of performing arithmetic, there are
%% three kinds of "floating point" numbers for representing numbers with
%% places to the right of the decimal point. They are called "float",
%% "double" and "long double" and they have increasing degrees of
%% precision.
%% float f = powf(2.0F, 1.0F/12.0F);
%% double d = pow(2.0D, 1.0D/12.0D);
%% long double ld = powl(2.0L, 1.0L/12.0L);
%% cout << setprecision(128)
%% << "f == " << f << endl
%% << "d == " << d << endl
%% << "ld == " << ld << endl;
%% // f == 1.05946314334869384765625
%% // d == 1.0594630943592953098431053149397484958171844482421875
%% // (length "1.0594630943592953098431053149397484958171844482421875") == 54
%% // ld == 1.05946309435929526452345450504566315430565737187862396240234375
%% // (length "1.05946309435929526452345450504566315430565737187862396240234375") == 64
%% // Emacs-Lisp 1.0594630943592953
\bye
%% ** (2)
%% * (1) Local variables for Emacs.
%% Local Variables:
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