source file: m1453.txt Date: Sat, 20 Jun 1998 12:10:22 -0500 Subject: Ratios, Cents, LucyTuning, Pi and E From: "Benjamin Tubb" The following info is for anyone interested. Comments are welcome. :Lucy Tuning based on A=440 Hz :Chromatic/Enharmonic Scale 10^-4 10^-5 :Note Offset Cents Ratio 0 A 0.000 0.0000 1/1 1/1 1 A# -31.550 68.4500 129/124 = 1 Bb 22.535 122.5350 161/150 600/559 2 B -9.014 190.9860 67/60 584/523 2 Cb 45.070 245.0700 250/217 = 3 B# -40.564 259.4360 115/99 194/167 3 C 13.521 313.5210 163/136 = 4 C# -18.028 381.9720 101/81 298/239 4 Db 36.057 436.0570 256/199 = 5 D 4.507 504.5070 91/68 716/535 6 D# -27.042 572.9580 181/130 795/571 6 Eb 27.042 627.0420 260/181 701/488 7 E -4.507 695.4930 133/89 535/358 7 Fb 49.579 749.5790 387/251 = 8 E# -36.057 763.9430 199/128 = 8 F 18.028 818.0280 239/149 = 9 F# -13.521 886.4790 272/163 = 9 Gb 40.564 940.5640 167/97 1200/697 10 G 9.014 1009.0140 120/67 593/292 11 G# -22.535 1077.4650 300/161 559/300 11 Ab 31.550 1131.5500 248/129 967/503 *Offsets are cents from 12tet "equivalents" scale steps Based on two intervals related to Pi: Large 2^(1/(2*Pi)) = 1.1166329 or 190.9859 cents Small (2/(2^(1/(2*Pi)))^5)^(1/2) = 1.073344 or 122.5354 cents : Error Tolerance Levels for Ratios :Given Value 10^-4 10^-7 10^-10 :Pi 3.141593 355/113 104348/33215 312689/99532 :E 2.718282 193/71 25946/9545 566827/208524 :Pi^E 22.459158 2201/98 44267/1971 1474556/65655 :E^Pi 23.140693 1481/64 10691/462 6324467/273305 >From those values, to then reduce those ratios (to their lowest "octave" using powers of 2) to produce a "ratio" value between 1 and 2 for use in computing a cents conversion for 12tet, I get the following results: : Error Tolerance Levels for Cents :Given Value 10^-4 10^-7 10^-10 :Pi/2 1.05707963 781.7955024 781.7953555 781.7953553 :E/2 1.3591409 531.2519013 531.2339789 531.2340491 :(Pi^E)/16 1.4036974 587.0442215 587.078274 587.0783022 :(E^Pi)/16 1.4462933 638.8174495 638.8321708 638.8321702 ------------- Benjamin Tubb brtubb@cybertron.com http://home.cybertron.com/~brtubb/theory.html