How to avoid using relays/analog switches in a musical sequencer?
As I see it, you want to select one of (how many? 8?) adjustable resistors into a timing circuit to control pitch. They are shown as 100R in one schematic and 100K in the other; IMO 100R is far too low (given that the other timing resistor is 10K.)

Now diode switching is possible but it's tricky to get it working, especially with large amplitude LF signals. Essentially you need enough excess voltage to keep all the "off" channels firmly off - their diodes reverse biassed, - and the "on" channel should approximate a constant current through the diode whatever the signal voltage. 

That's because you don't want the switching current to affect the voltage you are transmitting. So the switching current must be (a) greater than the maximum possible negative signal current (or the diode turns off) and (b) be independent of the signal voltage (or you add some distortion to the signal).

Which either means additional transistors acting as constant current circuits, or relatively high switching voltages and high value series resistors. It's going to be difficult to achieve for this application.

An alternative is an 8:1 analog switch - like the CD4051 or faster lower voltage 74HCT4051 connecting 8 inputs to one output. It's controlled by a 3 bit binary counter (counting from 0 to 7). Internally the switches are like the 4066 analog switches, but in a more convenient package. If you need more than 8 channels, an "Enable" input turns the whole chip off or on so you can connect several to your input (the 555 "Discharge" pin)