Why does music synthesizer built from chain of astable multivibrator circuits get "out-of-tune" after a few hours?
You have clearly built a circuit that is fully analog in nature and produces a frequency in each oscillator that is dependent upon various factors such as:


Changes to the voltage level of the power supply to oscillator.
Changes of the Vbe level of the transistors with temperature.
Changes of the values of resistors over time and temperature.
Changes of the values of the capacitors over time and temperature.
Changes in capacitor dielectric characteristics in astable oscillator configuration.
Stray circuit behavior changes due to things nearby the prototype.
Position of the moon relative to the sun as viewed from earth.


There are ways to build circuits that do not have as much drift in operational frequency. They are designed to eliminate or cancel out the various effects enumerated above. One conventional way is to design a circuit that uses as single higher frequency oscillator based upon a close tolerance crystal. Then the use of digital counters are used to divide this frequency down to the desired frequency for each note in the scale. 

To show the value of a digital circuit approach I created a small spreadsheet that shows the octave of musical notes from C5 through C6. (The nominal frequencies are values taken from a chart found on Google and not computed in the spreadsheet with scale formulas from the A[440] reference). 

Using a crystal frequency of 22.1184 MHz (which is a common MCU frequency used in the 8-bit embedded business) you can see that with an integer digital divide factor for each note that the generated frequency is very close to the desired nominal.