Gear Ratio

The difference in the number of teeth between the Driving Gears and the Driven Gears is the gear ratio. With the driving gears rotating at a fixed speed of 20 revolutions per second, the difference in the number of teeth between the two gears means that the driven gears, and the tonewheels connected to them, will spin at at a speed set by the gear ratio. Since the relationship between semitones is the twelfth root of two and gears can only have an integral number of teeth, Hammond selected gear ratios that were as close as possible to the correct values. With the number of "teeth" on the Tonewheels doubling every octave, the minimum number of different gear ratios needed is twelve:

Note Driving Driven    Ratio
C       85    104   0.817307692
C#      71     82   0.865853659
D       67     73   0.917808219
D#      70     72   0.972222222
E       69     67   1.029850746
F       54     99   0.545454545
F#      37     64   0.578125000
G       49     80   0.612500000
G#      48     74   0.648648649
A       66     96   0.687500000
A#      67     92   0.728260870
B       54     70   0.771428571

In the patent organ, Hammond had shown two octaves of 16 toothed tonewheels, with different gear ratios for the second set of tonewheels.

Note that these ratios produce a very close *approximation* of the equal tempered scale. They are *NOT* exact. These ratios are used in each of the first seven octaves. The octave produced is dependent on the number of teeth on the ToneWheels. In 60 Hz organs, the motor turns at 1200 RPM / 60Hz or 20 revolutions per second. The lowest octave has tonewheels with 2 teeth. The first 7 tones in the highest octave (C through F#) are produced by tonewheels with 192 teeth. For these 7 tones to be correct with only 192 teeth, they use the gear ratios for notes F through B.

The formula to calculate the exact frequency produced is:

(Motor RPM/60)*Teeth in Toneweel*(Teeth in Driving Gear/Teeth in Driven Gear)

So for the A above middle C on a 60 Hz organ:  (1200/60) * 16 * (88/64) = 440 Hz

This A tonewheel frequency actually influenced the establishment of the commonly accepted concert pitch standard as 440 Hz. Historically, concert A has wandered between 430-450 Hz. The famous Stroboconn tuner was introduced in 1936 and copied Hammond's use of an AC mains driven synchronous motor spinning an optical disk that also set the A above middle C at 440 Hz. Within a couple of years, Stroboconn tuners were being used globally to help tune concert orchestras. By 1939 enough orchestras had set their concert pitch to the A of the Stroboconn (and the Hammond) that the international concert pitch standard of A = 440 Hz was established.

None of the twelve semitones on the Hammond are exactly equal tempered in their relationship to each other. The notes which are the farthest off pitch in the first seven octaves are the G#. They are .69 cents flat from the correct pitch. The top half octave is farther off pitch due to the number of teeth on the tonewheel not equaling 256. Using 192 teeth, generating the correct pitch from C7 to F#7 requires using the gear ratios for F through B. In the top half octave, the C# is the farthest off pitch, about 1.93 cents sharp:

For C#-4434:                   (1200/60) * 192 * (74/64) = 4440 Hz