Ratios, Cents, LucyTuning, Pi and E

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

Aline Surman wrote:
> As it is now, the
> actual notes are theoretical, since no one (no human, anyway) can hear
> them in reality...Hstick

I take it then that you are pondering cases where the pitches would
go ultrasonic if not brought down into the orginal octave? If so, then
it's an impossible question to answer, since there's no basis in
everyday experience within which to answer the question.

But it's fun to imagine "what if" sorts of answers!