This short table shows the frequencies of the notes on the scale for the octave starting with middle C:
Equal Hammond Telharmonium Pari-E Electronic C=256 A=435 A=425 Meantone Just C 261.62557 261.54 85 104 261.62 320 80 261.607 261.51 478 256 258.65 252.71 C 254.21 264 256 277.18263 277.07 71 82 277.19 356 84 277.179 277.16 451 271.22 274.03 267.73 C# 265.63 D 293.66477 293.70 67 73 293.66 440 98 293.645 293.43 426 287.35 290.33 283.65 D 284.21 297 288 311.12698 311.11 105 108 311.12 352 74 311.124 310.95 402 304.44 307.59 300.52 Eb 304.11 E 329.62756 329.60 103 100 329.70 494 98 329.606 329.82 379 322.54 325.88 318.39 E 317.76 330 320 F 349.22823 349.09 84 77 349.20 315 59 349.226 349.16 358 341.72 345.26 337.32 F 340 352 341.33 369.99442 370 74 64 369.95 362 64 369.972 369.82 338 362.04 365.79 357.38 F# 355.27 G 391.99544 392 98 80 391.89 725 121 391.994 391.85 319 383.57 387.54 378.63 G 380.13 396 384 415.30470 415.14 96 74 415.32 508 80 415.281 415.28 301 406.37 410.59 401.15 A 440 440 88 64 440 444 66 440 440.14 284 430.54 435 425 A 425 440 426.67 466.16376 466.09 67 46 466.01 456 64 466.139 466.42 268 456.14 460.87 450.27 Bb 454.74 B 493.88330 493.71 108 70 493.85 672 89 493.885 494.07 253 483.26 488.27 477.05 B 475.16 495 480
The first column shows the frequencies of the notes as actually used in the equal-tempered scale at the current concert pitch of A=440 Hz. The second column shows the approximations to those frequencies used in the classic Hammond organ using tonewheels, followed by the number of teeth on the gears that drive the tonewheels, and the third does the same for one of the designs used on the Telharmonium of Thaddeus Cahill. The frequencies used are based on A=440 Hz, to best serve purposes of comparison, but it is possible that the Telharmonium instead used C=256 Hz as its pitch, because the gearing ratio for C is simpler, and because this pitch is cited in the first Telharmonium patent.
The tonewheel principle of the Telharmonium was combined with electronic amplification not once, but twice (at least) by other inventors before Laurens Hammond achieved success! First, in Canada (more specifically, Belleville, Ontario), Morse Robb invented the Wave organ, and then again Richard H. Ranger, of Newark, New Jersey, invented the Rangertone organ.
The Rangertone name was used again, for an early magnetic tape recorder, and there were also the famous NBC chimes; as well, Vassar College added a 32-foot rank to its pipe organ using equipment derived from the Rangertone organ.
The Robb Wave organ used cylinders with teeth shaped after different types of waveform, so it anticipated later electronic organs which used photoelectricity with transparent disks with various wave shapes marked out upon them.
The fourth column shows a hypothetical set of pitches for the Pari-E electronic organ. This uses an interesting principle that simplifies the construction of a tonewheel organ.
The tonewheel assembly of a Pari-E organ includes twelve identical drums which consist of joined tonewheels with 2, 4, 8, 16... 128 and 256 teeth. These are connected, each one to the next, by gears of similar size. From this, I can assume that the rotational speed, from one drum to the next, varies so as to cause it to generate a pitch different by one semitone (instead of, for example, a fifth, which would involve halving the number of teeth in corresponding positions on the drum when an octave boundary is crossed) from that of the preceding one.
The ratio of the twelfth root of two can be very closely approximated by gear ratios of 89:84 and 107:101; these are approximations of about equal closeness, but with errors on opposite sides: they can be derived from continued fractions as shown below:
____
/ 1
12/ 2 = 1.05946309435929526456... = 1 + ---------------
\/ 1
17 - --------
5.469...
and so one can replace 5.469... with either 5 or 6, getting 89:84 in the first case, and 107:101 in the second.
If one uses six pairs of gears with the closer 89:84 ratio along one side of the tone generator, and five pairs with the 107:101 ratio to counterbalance their small error, one can achieve the remarkably accurate approximation to the equal tempered scale shown in the table above.
The fifth column shows the frequencies that are used in many typical electronic organs that use a "top octave generator". The divisors used are shows after the frequencies produced; actually, the divisor for C will be half that shown, or 236, so that the keyboard can include an additional C after the top octave. The input being divided might be 2 MHz; the frequencies shown in the table for the middle octave can be derived by dividing 125 kHz by the divisors shown.
The sixth column shows the lower frequencies for the "scientific" scale of pitches. The seventh column shows the frequencies for a standard of A=435 Hz, as this was fixed at one time as an official standard in France, and is occasionally used at the present time for tuning organs. The eighth column shows the frequencies for a standard of A=425 Hz, as this is close to that in use during much of the classical period. The ninth column shows quarter-comma meantone tuning with A=425 Hz; the tenth column shows the just intonation pitches of Helmholtz which gave rise to the exact numerical value of the current concert pitch, and the eleventh shows just intonation pitches starting from C=256 Hz.