It sounds to me like that "hum" is really the frequency of the 100 point tone generator playing non stop?

If you wiggle the ribbon cable between the MPU board and sound board at each end, does the tone go away?

Do you have a logic probe?

Ok, can you hook up the logic probe and tell us what it indicates at these chip output pins on the SB-100 sound board:

U3 pin 9 (play 10 point tone) =
U3 pin 10 (play 100 point tone) =
U3 pin 15 (play 1000 point tone) =
U3 pin 16 (play 10000 point tone) =

U7 pin 3 (10 point tone out) =
U7 pin 6 (100 point tone out) =
U7 pin 8 (1000 point tone out) =
U7 pin 11 (10000 point tone out) =

Cheers, I should have asked you to probe the input pins of U3 too.
Under normal conditions those four output pins on U3 you probed should be low and only go high to play the respective tone. You can see that pin 10 of U3 is high which is enabling the 100 point tone to play constantly.

The pin 10 output of U3 is controlled by the pin 6 input of U3.
Probe pin 6 of U3, it should be pulsing. If it is then U3 is faulty. If it isn't pulsing you have a bad connection coming from pin 7 of the ribbon cable to pin 6 of U3.

Let us know what you find.