I have recently encountered something on the web that piqued my curiosity.

For a short time, some Hammond tonewheel organs utilized a second tone generator unit which produced slightly different frequencies than the main one, as a way of producing a chorus effect.

The service manual for Hammond organs does not give the details of the gear ratios used in this unit, but it does note the following:

Some notes are sharp, and others flat, by about the same amount;

For frequencies 56 to 67, the difference in frequency is 0.8%;

For frequencies 68 to 91, the difference in frequency is 0.4%.

Middle C is frequency 37, and the frequency numbers increase by one for each semitone higher in pitch a tone is.

The following diagram illustrates the frequency numbers, from 1 to 91, used in the service guides for the Hammond organ:

Also note that in this diagram, white keys change to yellow, black keys change to blue, and brown keys change to green, to indicate frequencies 10, 20, 30... up to 90.

Note that the additional drawbars of the later models of the Hammond tonewheel organ are also illustrated. As well, while the keyboards illustrating the connections of the frequencies for the original drawbars have the keys corresponding to frequencies above 91 shown in a different color, to indicate that foldback is taking place, since the later models with those two additional drawbars had tone generators which went up to frequency 96, foldback is indicated for those drawbars above that frequency instead.

Some of the higher frequencies were not generated by the tonewheels; just using frequencies that the tonewheels did generate an octave or two away, instead, is called "foldback". Some models of the Hammond organ also omitted frequencies 1 through 12, or just frequencies 1 through 9, from the tone generator, and thus there was foldback at the bottom of the range as well as the top. (This is oversimplified; in some cases, frequencies 1 through 12 were not omitted, but they were produced by "complex tonewheels", which produced a sound with harmonics already present, for use with the pedals only.)

The Hammond organ had stops from 16' to 1', and it had the normal 61-key organ keyboard; this, nominally, would require 61 frequencies, one for each note, plus an additional 48 frequencies: 12 frequencies to extend the range downwards by one octave, as the 16' stop would require, being one octave below the nominal pitches of the keys, considered to be those associated with the 8' stop, and 36 frequencies to extend the range upwards by three octaves, as the 1' stop requires.

Given what we know of the frequencies used for the regular frequency generator in a Hammond organ:

Equal Hammond C 261.63 261.54 85 104 277.18 277.07 71 82 D 293.66 293.70 67 73 311.13 311.11 105 108 E 329.63 329.60 103 100 F 349.23 349.09 84 77 369.99 370 74 64 G 392.00 392 98 80 415.30 415.14 96 74 A 440 440 88 64 466.16 466.09 67 46 B 493.88 493.71 108 70

is it possible to hazard a guess (of course, there are extant Model BC Hammond organs out there which could be examined, even if I hadn't initially been able to find the information on the web) as to how producing the chorus tones, 0.4% or 0.8% higher and lower, was done at a reasonable cost?

Only the highest seven frequencies produced by the tone generator of the Hammond organ, frequencies 85 to 91, were produced using 192-tooth tonewheels, so this doesn't account for the change from 0.8% to 0.4%, eliminating one possible way to achieve a difference in frequency offsets.

It is noted, as well, that while the gears moved at twelve different speeds in the main tone generator, they were on twenty-four driveshafts, not twelve, with pairs of driveshafts moving at the same speed. In the chorus generator, it is specifically noted that the driveshafts move at twenty-four different speeds. Each driveshaft has two gears on it, thus producing two frequencies.

Since twelve frequencies (from 56 to 67) are produced with the 0.8% offset, and twenty-four frequencies (from 68 to 91) are produced with the 0.4% offset, one possibility that might suggest itself is the change in offsets is produced by the driveshaft speeds. However, the twenty-four frequencies with 0.4% offsets were all produced by dual tonewheels, and each driveshaft had one dual tonewheel and one regular tonewheel for the other twelve frequencies on it.

A web site in tribute to the organist Ken Griffin shows a photograph of some of the gears in an actual chorus tone generator. In the ones pictured, the dual tonewheels seem to have a one-tooth difference between their two parts, and the other tonewheel seems to have either about one-half or about one-quarter the number of teeth.

A dual tonewheel with approximately 125 teeth would produce an 0.8% difference between the two component frequencies it produces, suitable for producing +0.4% and -0.4%. But how would one then produce either +0.8% or -0.8% from the other tonewheel?

For example, if the dual tonewheel were composed of two parts with 124 and 125 teeth, and the plain tonewheel had 63 teeth (half of 126), then, given the gear ratio driving those tonewheels being such as to produce -0.4% from 124 teeth, and +0.4% from 125 teeth, then, for the lower octave, 63 teeth would produce not +0.8%, but +1.2%. That is because 63 teeth are twice as far away from 62 teeth as 125 teeth are from 124 teeth. What is desired, instead, is something that is one-and-a-half times as far away. Which certainly is attainable; two-thirds of 126 teeth is 84 teeth, and so 85 teeth would produce +0.8% for a note a perfect fifth below the one involving the dual tonewheels.

But a manual for several models of the Hammond organ, including those with a chorus tone generator, gives positions for the magnets which do not allow for such a trick to be used.

According to the book "Beauty in the B", the reason for using 192-tooth tonewheels, moved over to the driveshafts associated with notes a fifth higher, instead of 256-tooth tonewheels, was not, as I thought, because the magnetic pickup mechanism would not get a signal from a 256-tooth tonewheel. Instead, it was because 256-tooth tonewheels could not be machined at the time the Model A organ was made. In that case, if improving technology allowed 256-tooth tonewheels to be made when the chorus tone generator was designed, then one could have a dual tonewheel with 249 and 251 teeth, giving tones with -0.4% and +0.4% compared to 250 teeth... and a single tonewheel with either 124 teeth, for -0.8%, or 126 teeth, for +0.8%, and there would be no problem. It is only because I thought that tonewheels with around 256 teeth, instead of around 128 teeth or 192 teeth, were not possible that I came to the conclusion that the chorus tone generator could not actually be made as specified.

However, photographs of an actual chorus tone generator to make it appear that the dual tonewheel does have in the range of 128 teeth, which leads to the problem I am considering here being a real problem.

Possible gear ratios that could be used as the basis for a chorus tone generator operating in that fashion could be:

Equal Hammond Hammond + 1.2% -0.4% +0.4% Hammond - 1.2% -0.4% +0.4% 32 126 127 63 127 128 (Teeth on tonewheel) DG DN DG DN DG DN G 392.00 392 98 80 396.52 57 92 390.33 393.42 387.36 91 74 390.44 393.52 415.30 415.14 96 74 420.2 65 99 413.64 416.92 410.33 99 76 413.59 416.84 A 440 440 88 64 445.22 48 69 438.26 441.74 434.48 80 58 437.93 441.38 466.16 466.09 67 46 471.58 70 95 464.21 467.89 460.83 79 54 464.49 468.15 B 493.88 493.71 108 70 499.73 57 73 491.92 495.82 488.03 110 71 491.90 495.77 C 523.25 523.08 85 104 529.38 67 81 521.11 525.25 517.16 110 67 521.27 525.37 554.37 554.15 71 82 560.9 78 89 552.13 556.52 547.83 120 69 552.17 556.52 D 587.33 587.4 67 73 594.29 65 70 585 589.64 580.26 105 57 584.87 589.47 622.25 622.22 105 108 629.51 60 61 619.67 624.59 614.76 121 62 619.64 624.52 E 659.26 659.2 103 100 667.04 74 71 656.62 661.83 651.72 120 58 656.9 662.07 F 698.46 698.18 84 77 706.49 85 77 695.45 700.97 690.32 103 47 695.8 701.28 739.99 740 74 64 748.68 62 53 736.98 742.83 731.25 130 56 737.05 742.86 56-67 80-91 56-67 68-79 (Frequencies covered) (/2) (/8) (/2) (/4)

The chorus tone generator appears to be divided into two halves, and so one driveshaft could be spinning at half the rate of the other. But pairs of tonewheels where the plain tonewheel has about 1/4 the teeth of the dual one, and about 1/2 the teeth of the dual one, appear to alternate on the same driveshaft.

The manual for a number of Hammond organ models, including the model BC and others with a chorus tone generator, shows the locations of the magnetic pickups for the tonewheels, and thus an attempt to reconstruct how it is designed can be made:

Tone Generator Hypothetical Chorus Tone Generator --------------------------------------------- ------------------------------ | ______|______ | | ______|______ | | | | ______|______ | 61--|--->(______64_____) | | (_______2_____)<---|-- 1 | | | (___126/127___)<---|-- 68 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [___104___][____85___][___104___] | | [____57___][____92___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 13--|--->(_______4_____) | | (______32_____)<---|-- 49 | | | (______63_____)<---|-- 56 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | | (_____________) | | (_______8_____)<---|-- 25 | | | (______32_____)<---|-- 56 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [___104___][____85___][___104___] | | [____91___][____74___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 37--|--->(______16_____) | | (_____128_____)<---|-- 73 | | | (___127/128___)<---|-- 80 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | 68--|--->(______64_____) | | (_______2_____)<---|-- 8 | | | (___126/127___)<---|-- 75 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [____80___][____98___][____80___] | | [____65___][____70___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 20--|--->(_______4_____) | | (______32_____)<---|-- 56 | | | (______63_____)<---|-- 63 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | 87--|--->(_____192_____) | | (_______8_____)<---|-- 32 | | | (______32_____)<---|-- 63 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [____80___][____98___][____80___] | | [___105___][____57___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 44--|--->(______16_____) | | (_____128_____)<---|-- 80 | | | (___127/128___)<---|-- 87 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | 63--|--->(______64_____) | | (_______2_____)<---|-- 3 | | | (___126/127___)<---|-- 70 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [____73___][____67___][____73___] | | [____48___][____69___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 15--|--->(_______4_____) | | (______32_____)<---|-- 51 | | | (______63_____)<---|-- 58 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | | (_____________) | | (_______8_____)<---|-- 27 | | | (______32_____)<---|-- 58 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [____73___][____67___][____73___] | | [____80___][____58___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 39--|--->(______16_____) | | (_____128_____)<---|-- 75 | | | (___127/128___)<---|-- 82 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | 70--|--->(______64_____) | | (_______2_____)<---|-- 10 | | | (___126/127___)<---|-- 77 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [____64___][____88___][____64___] | | [____74___][____71___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 22--|--->(_______4_____) | | (______32_____)<---|-- 58 | | | (______63_____)<---|-- 65 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | 89--|--->(_____192_____) | | (_______8_____)<---|-- 34 | | | (______32_____)<---|-- 65 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [____64___][____88___][____64___] | | [___120___][____58___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 46--|--->(______16_____) | | (_____128_____)<---|-- 82 | | | (___127/128___)<---|-- 89 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | 65--|--->(______64_____) | | (_______2_____)<---|-- 5 | | | (___126/127___)<---|-- 72 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [___100___][___103___][___100___] | | [____57___][____73___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 17--|--->(_______4_____) | | (______32_____)<---|-- 53 | | | (______63_____)<---|-- 60 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | | (_____________) | | (_______8_____)<---|-- 29 | | | (______32_____)<---|-- 60 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [___100___][___103___][___100___] | | [___110___][____71___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 41--|--->(______16_____) | | (_____128_____)<---|-- 77 | | | (___127/128___)<---|-- 84 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | 72--|--->(______64_____) | | (_______2_____)<---|-- 12 | | | (___126/127___)<---|-- 79 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [____70___][___108___][____70___] | | [____62___][____53___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 24--|--->(_______4_____) | | (______32_____)<---|-- 60 | | | (______63_____)<---|-- 67 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | 91--|--->(_____192_____) | | (_______8_____)<---|-- 36 | | | (______32_____)<---|-- 67 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [____70___][___108___][____70___] | | [___130___][____56___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 48--|--->(______16_____) | | (_____128_____)<---|-- 84 | | | (___127/128___)<---|-- 91 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | 67--|--->(______64_____) | | (_______2_____)<---|-- 7 | | | (___126/127___)<---|-- 74 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [____64___][____74___][____64___] | | [____78___][____89___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 19--|--->(_______4_____) | | (______32_____)<---|-- 55 | | | (______63_____)<---|-- 62 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | 86--|--->(_____192_____) | | (_______8_____)<---|-- 31 | | | (______32_____)<---|-- 62 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [____64___][____74___][____64___] | | [___120___][____69___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 43--|--->(______16_____) | | (_____128_____)<---|-- 79 | | | (___127/128___)<---|-- 86 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | 62--|--->(______64_____) | | (_______2_____)<---|-- 2 | | | (___126/127___)<---|-- 69 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [____82___][____71___][____82___] | | [____65___][____99___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 14--|--->(_______4_____) | | (______32_____)<---|-- 50 | | | (______63_____)<---|-- 57 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | | (_____________) | | (_______8_____)<---|-- 26 | | | (______32_____)<---|-- 57 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [____82___][____71___][____82___] | | [____99___][____76___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 38--|--->(______16_____) | | (_____128_____)<---|-- 74 | | | (___127/128___)<---|-- 81 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | 69--|--->(______64_____) | | (_______2_____)<---|-- 9 | | | (___126/127___)<---|-- 76 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [____74___][____96___][____74___] | | [____60___][____61___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 21--|--->(_______4_____) | | (______32_____)<---|-- 57 | | | (______63_____)<---|-- 64 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | 88--|--->(_____192_____) | | (_______8_____)<---|-- 33 | | | (______32_____)<---|-- 64 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [____74___][____96___][____74___] | | [___121___][____62___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 45--|--->(______16_____) | | (_____128_____)<---|-- 81 | | | (___127/128___)<---|-- 88 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | 64--|--->(______64_____) | | (_______2_____)<---|-- 4 | | | (___126/127___)<---|-- 71 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [___108___][___105___][___108___] | | [____70___][____95___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 16--|--->(_______4_____) | | (______32_____)<---|-- 52 | | | (______63_____)<---|-- 59 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | | (_____________) | | (_______8_____)<---|-- 28 | | | (______32_____)<---|-- 59 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [___108___][___105___][___108___] | | [____79___][____54___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 40--|--->(______16_____) | | (_____128_____)<---|-- 76 | | | (___127/128___)<---|-- 83 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | 71--|--->(______64_____) | | (_______2_____)<---|-- 11 | | | (___126/127___)<---|-- 78 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [____46___][____67___][____46___] | | [____85___][____77___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 23--|--->(_______4_____) | | (______32_____)<---|-- 59 | | | (______63_____)<---|-- 66 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | 90--|--->(_____192_____) | | (_______8_____)<---|-- 35 | | | (______32_____)<---|-- 66 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [____46___][____67___][____46___] | | [___103___][____47___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 47--|--->(______16_____) | | (_____128_____)<---|-- 83 | | | (___127/128___)<---|-- 90 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | 66--|--->(______64_____) | | (_______2_____)<---|-- 6 | | | (___126/127___)<---|-- 73 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [____77___][____84___][____77___] | | [____67___][____81___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 18--|--->(_______4_____) | | (______32_____)<---|-- 54 | | | (______63_____)<---|-- 61 | | | | | | | | | | | |---------------------------------------------| |------------------------------| | ______|______ | | ______|______ | | | | ______|______ | 85--|--->(_____192_____) | | (_______8_____)<---|-- 30 | | | (______32_____)<---|-- 61 | ___|_|___ ___|_|___ ___|_|___ | | ___|_|___ ___|_|___ | | [____77___][____84___][____77___] | | [___110___][____67___] | | _____|_|_____ | | _____|_|_____ | | | | _____|_|_____ | 42--|--->(______16_____) | | (_____128_____)<---|-- 78 | | | (___127/128___)<---|-- 85 | | | | | | | | | | | --------------------------------------------- ------------------------------

Of course, if one used dual tonewheels with about 192 teeth instead of 128 teeth, then one could approximate 0.4% and 0.8% by 0.3% and 0.9%, so the fact that the big change has to be three times the small change could still be reconciled with keeping the frequency deviations in the ballpark of the original goal.

So perhaps the design was really based on gear ratios like these:

Equal Hammond Hammond - 0.9% -0.3% +0.3% Hammond + 0.9% -0.3% +0.3% 44 177 178 90 178 179 (Teeth on tonewheel) DG DN DG DN DG DN G 392.00 392 98 80 388.67 53 60 390.88 393.08 395.33 47 107 390.93 393.13 415.30 415.14 96 74 411.79 73 78 414.13 416.47 418.81 47 101 414.16 416.48 A 440 440 88 64 436.27 117 118 438.75 441.23 443.66 35 71 438.73 441.2 466.16 466.09 67 46 462.22 104 99 464.85 467.47 470.15 35 67 464.93 467.54 B 493.88 493.71 108 70 489.68 69 62 492.46 495.24 497.87 26 47 492.34 495.11 C 523.25 523.08 85 104 518.81 79 67 521.75 524.70 527.59 17 29 521.72 524.66 554.37 554.15 71 82 550 110 88 553.13 556.25 560.2 61 98 553.98 557.09 D 587.33 587.4 67 73 582.35 90 68 585.66 588.97 593.62 62 94 587.02 590.32 622.25 622.22 105 108 616 119 85 619.5 623 628.92 58 83 621.93 625.42 E 659.26 659.2 103 100 653.71 104 70 657.43 661.14 666.23 57 77 658.83 662.53 F 698.46 698.18 84 77 692.59 85 54 696.53 700.46 705.68 69 88 697.84 701.76 739.99 740 74 64 733.33 100 60 737.5 741.67 747.69 54 65 739.38 743.54 56-67 80-91 56-67 68-79 (Frequencies covered) (/2) (/8) (/2) (/4)

with the frequency difference specification just nudged a bit in order to confuse those who would attempt to imitate them?

Still, there may be possibilities I'm overlooking.

One possibility is that the simple tonewheels are chosen so that the offsets would be +0.4% and -0.4% instead of +1.2% and -1.2%, but the fact that the simple tonewheels are driven by two separate gears is used. So the frequencies are chosen so that one compound tonewheel provides +0.1% and -0.7%, and the other compound tonewheel provides +0.7% and -0.1%; choose the two simple tonewheels to get +0.7% and -0.7%, and you have the right effect.

And if *asymmetrical* offsets are allowed, then a light dawns!

Keep all the compound tonewheels close to symmetric offsets of +0.4% and -0.4%.

But choose the simple tonewheels so that, for example, *one* is at +1.2%, but the
*other* is at -0.4%. Then, the two chorus tones are separated by 1.6%, just as if they were
+0.8% and -0.8%, and only the position of the regular tone between them is changed.

And finally, would there be a way to add a feature similar to the chorus tone generator to a Pari-E organ without having to substantially increase its weight?

The Pari-E organ has tonewheels for eight octaves on drums, each drum having tonewheels with 2, 4, 8, 16, 32, 64, 128, and 256 teeth. The drums are driven at speeds reflecting the equal-tempered scale. I had noted that a very close approximation to that scale could be achieved by alternating between the gear ratios 89:84 and 107:101 to represent a semitone.

On the equal-tempered scale, seven semitones differ from a perfect fifth as a 3:2 ratio by only 0.11%. Four semitones, however, differ from a major third as a 5:4 ratio by 0.79%, which is close to one of the ratios that we're looking for.

So, if the chorus tones can be offset by plus or minus 0.1%, +0.4% and -0.4% tones could be produced by a second unit with only five drums.

Five successive notes in the octave could be produced by tonewheels with 45 teeth on each drum.

Then tonewheels with 60 teeth on each drum would produce the next five notes, and finally tonewheels with 80 teeth on the first two drums would produce the last two notes of the octave.

Four semitones are 0.79% greater than a 5:4 ratio.

Put a tonewheel with 36 teeth on the fifth drum, and that takes care of the higher frequency for the first note in the octave.

Tonewheels with 48 teeth on each drum would handle the next five notes, and then tonewheels with 64 teeth on each drum would handle the next five notes. One note is left over to handle in another way, since one can't divide 64 by three: but there is an obvious way to do that, simply put a tonewheel with 72 teeth on the fourth drum.

Doubling the number of teeth on all the tonewheels allows another octave to be handled, providing two octaves with -0.4% and +0.4% like the Hammond chorus tone generator. The second unit would be driven about 0.4% slower so that normal frequencies and frequencies that are 0.8% higher due to the major third discrepancy would serve that purpose.

Frequencies 0.8% higher due to the major third discrepancy could be produced on the main drums using tonewheels with 10, 20, 40, 80, or 160 teeth, depending on the octave for which they are desired. The only thing that remains is frequencies that are 0.8% lower.

Six semitones correspond to the square root of two. 42/32 is 0.78% less than the square root of two, so tonewheels with 21 or 42 teeth, offset by six positions, could handle that task.