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Approximating the Equal-Tempered Scale

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 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.


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