source file: mills2.txt Subject: Consistency limits From: COUL@ezh.nl (Manuel Op de Coul) I have created a list of equal temperaments up to about 650 tones per octave with their consistency limits as was explained by Paul Erlich's post of 13 June 1996. However it's not restricted to integer divisions, all scale step sizes that give a different consistency limit were calculated. Initially I thought this could make a nice graph, step size horizontally, limit vertically, but the function behaves very wildly and there are very many points. Not included in the file are points where both left and right side limits are smaller than 5. This is the filename: ftp://ella.mills.edu/ccm/tuning/papers/consist_limits.txt Another larger file ftp://ella.mills.edu/ccm/tuning/papers/cons_limit_bounds.txt contains the step size bounds for each consistency limit of 3 to 18, also up to 650-tET. The first list looks like this: Limit at right side (larger step) Step size in cents Number of tones per octave Limit at left side (smaller step) Uniqueness limits for this temperament 4 589.572 2.03537 5 3 2 5 543.415 2.20825 2 3 2 2 422.656 2.83918 5 4 4 5 413.594 2.90139 6 4 4 6 387.530 3.09653 7 4 4 7 371.508 3.23007 4 4 4 3 320.000 3.75000 7 3 2 7 308.735 3.88682 8 3 2 8 295.424 4.06195 5 3 2 5 293.296 4.09142 4 3 2 4 242.288 4.95278 9 4 4 9 241.594 4.96699 10 4 4 10 233.004 5.15010 6 4 4 6 229.774 5.22251 5 4 4 5 228.571 5.25000 3 4 4 2 200.205 5.99383 5 4 4 5 200.126 5.99622 8 4 4 8 199.770 6.00691 7 4 4 7 199.218 6.02355 3 4 4 3 177.777 6.75000 5 5 4 5 177.254 6.76992 6 5 4 6 168.867 7.10616 4 5 4 4 150.611 7.96752 6 4 4 6 149.050 8.05098 7 4 4 7 145.454 8.25000 3 4 4 4 135.917 8.82887 7 6 6 7 135.843 8.83369 8 6 6 8 132.110 9.08328 6 6 6 6 131.998 9.09104 5 6 6 5 131.169 9.14848 2 6 6 2 122.706 9.77941 5 5 4 5 121.645 9.86475 8 5 4 8 119.717 10.0236 9 5 4 9 118.566 10.1209 4 5 4 3 110.676 10.8423 6 5 4 6 110.375 10.8719 4 5 4 4 101.320 11.8436 6 6 6 6 100.562 11.9329 10 6 6 10 99.2594 12.0895 11 6 6 The second list runs like this: 3 2400.00 0.50000 - 1403.91 0.85475 3 1267.97 0.94639 - 800.000 1.50000 3 760.782 1.57732 - 543.415 2.20825 3 480.000 2.50000 - 467.970 2.56426 3 422.656 2.83918 - 345.810 3.47011 3 342.857 3.50000 - 292.608 4.10104 etc. Manuel Op de Coul coul@ezh.nl ------------------------------ Topic No. 7 Date: Thu, 11 Jul 96 09:58 EST From: PAULE To: tuning Subject: RE: Another post from McLaren Message-ID: <73960711145837/0005695065PK2EM@MCIMAIL.COM> >As I've pointed out in prior posts, the strict >extreme hard-line Pythagorean viewpoint >fails when it encounters reality, since >the 3:5:7 triad demonstrably sounds less >consonant than the 4:5:6 triad--yet the >integers of the 3:5:7 triad are obviously >smaller. Why are so many things obvious to Brian that are not obvious to the rest of us?