%% scale_length_4.ldf %% Created by Laurence D. Finston (LDF) Sat 16 Jul 2022 04:51:35 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 %% * (1) Top verbatim_metapost "prologues := 3;\noutputtemplate := \"%j_%3c.eps\";"; input "plainldf.lmc"; pen medium_pen; medium_pen := pencircle scaled (.5mm, .5mm, .5mm); pen small_pen; small_pen := pencircle scaled (.333mm, .333mm, .333mm); pen dot_pen; dot_pen := pencircle scaled (2.5mm, 2.5mm, 2.5mm); pen tiny_pen; tiny_pen := pencircle scaled (.175mm, .175mm, .175mm); pickup medium_pen; numeric wd; numeric n[]; picture v[]; numeric alpha; alpha := 2 root 12; numeric ref_fret; ref_fret := 24; %% With float, there are only 23 places to the right %% of the decimal point. %% LDF 2022.07.10. %% % message "alpha:"; % show alpha fixed 23; %% * (1) macro werckmeister_three; point_vector pv; def werckmeister_three {numeric sscale_length} = begingroup; numeric uunit_size; uunit_size := sscale_length / (2 * 8.4430); boolean do_labels; do_labels := false; % true point p[]; path q[]; message "Entering `werkmeister_three'."; message "sscale_length = " & decimal sscale_length; message "uunit_size = " & decimal uunit_size; numeric a[]; a0 := 1; a1 := 256/243; a2 := (64/81); a2 *= (sqrt 2); a3 := 32/27; a4 := 256/243; a4 *= 2 root 4; a5 := 4/3; a6 := 1024/729; a7 := 8 root 4; %% 8 == 2 pow 3; a7 *= 8/9; a8 := 128/81; a9 := 1024/729; a9 *= 2 root 4; a10 := 16/9; a11 := 2 root 4; a11 *= 128/81; a12 := 2; p0 := origin; if do_labels: dotlabel.lft("$p_0$", p0); fi label.top("Nut", p0) shifted (0, .25cm); draw (p0 shifted (-.25cm, 0)) -- (p0 shifted (.25cm, 0)); label.lft("\setbox0=\hbox{\quad 0}Unison\hbox to \wd0{\hss}", p0) shifted (-.25cm, 0); q0 += --; q0 += p0; a_inverse0 := 1; numeric shift_val; shift_val := .25cm; for i = 1 upto 12: message "a" & decimal i & " = " & decimal a[i]; a_inverse[i] := 1/a[i]; message "a_inverse" & decimal i & " = " & decimal a_inverse[i]; p[i] := p[i-1] shifted (0, -uunit_size * a_inverse[i]); q0 += p[i]; if do_labels: dotlabel.lft("$p_{" & decimal i & "}$", p[i]); fi endfor; p13 := p12 shifted (0, -2cm); label.bot("Saddle, " & decimal sscale_length & "\Thinspace cm", p13) shifted (0, -.5cm); label.lft("Minor $2^{\rm{nd}}$\quad 1", p1 shifted (-shift_val, 0)); label.lft("Major $2^{\rm{nd}}$\quad 2", p2 shifted (-shift_val, 0)); label.lft("Minor $3^{\rm{rd}}$\quad 3", p3 shifted (-shift_val, 0)); label.lft("Major $3^{\rm{rd}}$\quad 4", p4 shifted (-shift_val, 0)); label.lft("$4^{\rm{th}}$\quad 5", p5 shifted (-shift_val, 0)); label.lft("Diminished $5^{\rm{th}}$\quad 6", p6 shifted (-shift_val, 0)); label.lft("$5^{\rm{th}}$\quad 7", p7 shifted (-shift_val, 0)); label.lft("Minor $6^{\rm{th}}$\quad 8", p8 shifted (-shift_val, 0)); label.lft("Major $6^{\rm{th}}$\quad 9", p9 shifted (-shift_val, 0)); label.lft("Minor $7^{\rm{th}}$\quad 10", p10 shifted (-shift_val, 0)); label.lft("Major $7^{\rm{th}}$\quad 11", p11 shifted (-shift_val, 0)); label.lft("Octave\quad 12", p12 shifted (-shift_val, 0)); label.rt("$1/1 = " & decimal a0 & " = " & decimal magnitude(ypart p0) & "\Thinspace\rm{cm}$", p0) shifted (shift_val, 0); label.rt("$256/243 = " & decimal a1 & " = " & decimal magnitude(ypart p1) & "\Thinspace\rm{cm}$", p1) shifted (shift_val, 0); label.rt("$64/81\sqrt 2 = " & decimal a2 & " = " & decimal magnitude(ypart p2) & "\Thinspace\rm{cm}$", p2) shifted (shift_val, 0); label.rt("$32/27 = " & decimal a3 & " = " & decimal magnitude(ypart p3) & "\Thinspace\rm{cm}$", p3) shifted (shift_val, 0); label.rt("$256/243\root 4 \of 2 = " & decimal a4 & " = " & decimal magnitude(ypart p4) & "\Thinspace\rm{cm}$", p4) shifted (shift_val, 0); label.rt("$4/3 = " & decimal a5 & " = " & decimal magnitude(ypart p5) & "\Thinspace\rm{cm}$", p5) shifted (shift_val, 0); label.rt("$1024/729 = " & decimal a6 & " = " & decimal magnitude(ypart p6) & "\Thinspace\rm{cm}$", p6) shifted (shift_val, 0); label.rt("$8/9\root 4 \of {2^3} = " & decimal a7 & " = " & decimal magnitude(ypart p7) & "\Thinspace\rm{cm}$", p7) shifted (shift_val, 0); label.rt("$128/81 = " & decimal a8 & " = " & decimal magnitude(ypart p8) & "\Thinspace\rm{cm}$", p8) shifted (shift_val, 0); label.rt("$1024/729\root 4 \of 2 = " & decimal a9 & " = " & decimal magnitude(ypart p9) & "\Thinspace\rm{cm}$", p9) shifted (shift_val, 0); label.rt("$16/9 = " & decimal a10 & " = " & decimal magnitude(ypart p10) & "\Thinspace\rm{cm}$", p10) shifted (shift_val, 0); label.rt("$128/81\root 4 \of 2 = " & decimal a11 & " = " & decimal magnitude(ypart p11) & "\Thinspace\rm{cm}$", p11) shifted (shift_val, 0); label.rt("$2/1 = " & decimal a12 & " = " & decimal magnitude(ypart p12) & "\Thinspace\rm{cm}$", p12) shifted (shift_val, 0); %% ** (2) Scale (Ruler) p14 := p0 shifted (9cm, 0); p15 := p14 shifted (0, -ceil(scale_length / 2)); draw p14 -- p15 with_pen small_pen; for i = 0 upto ceil(sscale_length / 2): draw (p14 shifted (-1cm, 0) -- p14) shifted (0, -i) with_pen small_pen; label.rt(decimal i & "cm", p14 shifted (0, -i)) shifted (.5shift_val, 0); endfor; for i = -.5 step -.5 until floor((-sscale_length / 2)): draw (p14 shifted (-.5cm, 0) -- p14) shifted (0, i) with_pen small_pen; endfor; for i = -.1 step -.1 until floor((-sscale_length / 2)): draw (p14 shifted (-.25cm, 0) -- p14) shifted (0, i) with_pen small_pen; endfor; %% ** (2) Pure intervals (harmonics) p16 := p0 shifted (0, -sscale_length); p212 := mediate(p0, p16, 1/2); %% Octave p207 := mediate(p0, p16, 1/3); %% Perfect 5th p204 := mediate(p0, p16, 1/5); %% Maj. 3rd p205 := mediate(p0, p16, 1/4); %% 4th p202 := mediate(p0, p16, 1/9); %% 2nd p209 := mediate(p0, p16, 2/5); %% 6th p211 := mediate(p0, p16, 7/15); %% Maj. 7th numeric u; %% 5th u := ((magnitude(magnitude(ypart p207) - magnitude(ypart p7))) * 100) / (magnitude(ypart p207)); drawdot p207 with_color green with_pen dot_pen; label.lrt("$ 5^{\rm{th}}$\hskip-.25ex, $3/2$, $" & decimal -ypart p207 & "\Thinspace\rm{cm}$, $" & "\Delta " & decimal magnitude(ypart p207 - ypart p7) & "\Thinspace\rm{cm}" & "=" & "\Delta" & decimal u & "\%" & "$", p207) shifted (.25cm, -6pt) with_color dark_green; draw p207 shifted (-1.25mm, 0) -- p207 shifted (1.25mm, 0) with_pen tiny_pen with_color magenta; %% Maj. 3rd u := ((magnitude(magnitude(ypart p204) - magnitude(ypart p4))) * 100) / (magnitude(ypart p204)); drawdot p204 with_color green with_pen dot_pen; label.urt("$ 3^{\rm{rd}}$\hskip-.25ex, $5/4$, $" & decimal -ypart p204 & "\Thinspace\rm{cm}$, " & "$\Delta " & decimal magnitude(ypart p204 - ypart p4) & "\Thinspace\rm{cm} =\Thinspace \Delta" & decimal u & "\%$" , p204) shifted (.25cm, 0pt) with_color dark_green; draw p204 shifted (-1.25mm, 0) -- p204 shifted (1.25mm, 0) with_pen tiny_pen with_color magenta; %% 4th u := ((magnitude(magnitude(ypart p205) - magnitude(ypart p5))) * 100) / (magnitude(ypart p205)); drawdot p205 with_color green with_pen dot_pen; label.lrt("$ 4^{\rm{th}}$\hskip-.25ex, $3/2$, $" & decimal -ypart p205 & "\Thinspace\rm{cm}$, $" & "\Delta " & decimal magnitude(ypart p205 - ypart p5) & "\Thinspace\rm{cm}" & "=" & "\Delta" & decimal u & "\%" & "$", p205) shifted (.25cm, -6pt) with_color dark_green; draw p205 shifted (-1.25mm, 0) -- p205 shifted (1.25mm, 0) with_pen tiny_pen with_color magenta; %% 2nd u := ((magnitude(magnitude(ypart p202) - magnitude(ypart p2))) * 100) / (magnitude(ypart p202)); drawdot p202 with_color green with_pen dot_pen; label.lrt("$ 2^{\rm{nd}}$\hskip-.25ex, $9/8$, $" & decimal -ypart p202 & "\Thinspace\rm{cm}$, $" & "\Delta " & decimal magnitude(ypart p202 - ypart p2) & "\Thinspace\rm{cm}" & "=" & "\Delta" & decimal u & "\%" & "$", p202) shifted (.25cm, -6pt) with_color dark_green; draw p202 shifted (-1.25mm, 0) -- p202 shifted (1.25mm, 0) with_pen tiny_pen with_color magenta; %% 6th u := ((magnitude(magnitude(ypart p209) - magnitude(ypart p9))) * 100) / (magnitude(ypart p209)); drawdot p209 with_color green with_pen dot_pen; label.urt("$6^{\rm{th}}$\hskip-.25ex, $5/3$, $" & decimal -ypart p209 & "\Thinspace\rm{cm}$, $" & "\Delta " & decimal magnitude(ypart p209 - ypart p9) & "\Thinspace\rm{cm}" & "=" & "\Delta" & decimal u & "\%" & "$", p209) shifted (.25cm, 0pt) with_color dark_green; draw p209 shifted (-1.25mm, 0) -- p209 shifted (1.25mm, 0) with_pen tiny_pen with_color magenta; %% Maj. 7th u := ((magnitude(magnitude(ypart p211) - magnitude(ypart p11))) * 100) / (magnitude(ypart p211)); drawdot p211 with_color green with_pen dot_pen; label.urt("Maj.~$7^{\rm{th}}$\hskip-.25ex, $5/3$, $" & decimal -ypart p211 & "\Thinspace\rm{cm}$, $" & "\Delta " & decimal magnitude(ypart p211 - ypart p11) & "\Thinspace\rm{cm}" & "=" & "\Delta" & decimal u & "\%" & "$", p211) shifted (.25cm, 1pt) with_color dark_green; draw p211 shifted (-1.25mm, 0) -- p211 shifted (1.25mm, 0) with_pen tiny_pen with_color magenta; %% Octave drawdot p212 with_color green with_pen dot_pen; label.lrt("Octave", p212) shifted (.25cm, -6pt) with_color dark_green; %% ** (2) draw q0; drawarrow p12 -- p13; for i = 1 upto 12: if not do_labels: draw (p[i] shifted (-.25cm, 0)) -- (p[i] shifted (.25cm, 0)) with_pen small_pen; fi endfor; %% ** (2) message "Exiting `werkmeister_three'."; endgroup; enddef; %% * (1) Fig. 0. Dummy figure. beginfig(0); drawdot origin; endfig with_projection parallel_x_y; %% * (1) point p[]; beginfig(1); numeric scale_length; scale_length := 45cm; werckmeister_three {scale_length}; endfig with_projection parallel_x_y; end_mp; end; %% ** (2) End of 3DLDF code. %% * (1) Emacs-Lisp code for use in indirect buffers when using the %% GNU Emacs editor. The local variable list is not evaluated when an %% indirect buffer is visited, %% %% so it's necessary to evaluate the %% following s-expression in order to use the facilities normally %% accessed via the local variables list. %% \initials{LDF 2004.02.12}. %% (progn (metapost-mode) (outline-minor-mode t) (setq fill-column 80)) %% * (1) Local variables for Emacs. %% Local Variables: %% mode:MetaPost %% eval:(outline-minor-mode t) %% eval:(read-abbrev-file abbrev-file-name) %% outline-regexp:"%% [*\f]+" %% End: