The simplest oscillator with sync. Power it with a 9 to 12v dc wall wart. You can plug it right into a cheap set of powered speakers to test it, but I recommend adding the volume control with AC coupler. The sync function isn't necessary, but it adds a lot to the sounds that are possible to make. Vids are below with build guides.
Annotated schematic below so you know what's what
Add this to control the volume and make the output oscillate around zero rather than 0 to 12v.
Below is the full schematic for an oscillator with sync and sync rate LED indicator. It contains the volume knob and all that jazz. The 15uf cap on the far left stabilizes the power supply for a much cleaner sound.
This video has an overview of functions and the schematic. There is a demo of the synth being played in the last half.
This one's more in depth if you have never tried to make one before. Includes breadboard basics, a simple practice circuit, and step by step instructions for the build.
A passive low pass filter adds a nice touch of tone adjustment and is very easy to incorporate.
Create a vactrol based LFO to modulate the passive low pass filter. It's like having a third hand!
Dual Drone Light Theremin
The schematic below has two oscillators, a low pass filter, and a summing amp to control the mix of the two oscillators. A light dependent resistor act as a light theremin to control the pitch with any light source. Try using unconventional sources such as electric candles and toys to get some crazy sounds.
Expandable Module- all the projects in one
This module contains sync, low pass filter with lfo, a clock out, and an ohm in.
The ohm in jack allows you to use any source of resistance you can imagine. A banana worked surprisingly well. I made a stylophone to use as a keyboard that can be tuned to eight different notes. A VHS tape makes a rudimentary pitch ribbon strip.