I built a prototype keyboard/sound synthesizer using a chain of 13 astable multivibrator circuits whose outputs are connected to an audio amplifier chip (LM386) and speaker, all powered off a 9V DC battery.
Each individual circuit gets tuned to one of the 13 frequencies in a musical octave (C5, C#, D, etc. up to C6) by varying a fine-tune trimpot that is in series with specific resistor values and which get the oscillation into the ballpark frequency.
The oscillation is the classic BJT astable multivibrator you can see in Figure 1 here and which is explained in this article.
The prototype stays correctly in tune for a short period (up to a day).
You can hear what it sounds like here. (Safe to start at 0:49s — Wadsworth's constant 😉 )
What I can't figure out is why the circuit appears to get spontaneously detuned, i.e. one or more of the individual circuits end up with frequencies that are different from what they were tuned to (checked against an o'scope and a reference piano).
The frequency deviation of the detuning is typically 2-5%, which is audibly noticeable (e.g. C5 at 523Hz might wander to 540Hz or 510Hz). Interestingly, the detuning never occurs while playing. But several hours afterwards, the keys no longer sound the same.
I had originally thought maybe the trimmer pots were mechanically relaxing by themselves. To eliminate this I replaced the trimmer pots to try to "lock in" the specific frequencies based on resistor values alone so that no variability was left in the design.
But the de-tuning problem persists even after replacing the trimpots with fixed resistor values.
Before: 13-key analog synthesizer with fixed resistor values
Resolution: Thanks all for the useful feedback, digital design ideas, and historical context to better understand the challenges of a pure analog design. All the answers were excellent. I've accepted ToddWilcox's answer as I get from it that (a) detuning is an expected part of pure analog designs, (b) the artistry lies in how to establish a slick way of tuning the instrument quickly.
To solve the immediate problem, I've put trimmer pots (1-2K ohms) back in the design to give 2-5% tuneability to each key. It takes a couple of minutes at the start of playing to tune up the 13 oscillators, after which they stay in tune for several hours at a time. See new image below.
Will post the results of the experiments using wall-wart, fresh batteries. The digital designs (using digital divider and/or 555 timer chips) are interesting, and would potentially compress the size significantly. Future updates can be found at the project page here.
After: 13-key analog synthesizer with trimmer pots (1-2k ohm) for tune-ability
Best Answer
Temperature changes, as mentioned in the other answer.
I'm adding an answer here because, as a musician, I prefer the sound of oscillators that are 100% analog over a design based on:
EEs on this Stack might comment endlessly that I scientifically couldn't be able to hear the difference. Believe me when I say my wallet dearly wishes that I couldn't hear the difference, but I can, and it's not subtle.
Anyway, major 100% analog synth manufacturers such as Moog Music and Sequential Circuits (formerly DSI) have solved this problem in different ways over the years. The old-school solution requires user intervention and frequent tuning. The original Moog Minimoog (AKA "Model D" after its most popular variant) had a crystal oscillator circuit built in that was not part of the signal path, but would create a stable 440 Hz tone. You turn on the 440Hz crystal tone, then play an A on the keyboard, and then turn the Master Tuning knob to re-tune the synth by ear. This was practical because the Minimoog was/is (it's been reissued with some technological improvements) a monosynth. Once you've tuned the bank of three oscillators all together, you're done. The Minimoog also has several adjustment trim pots to calibrate it to the various control voltages, which includes the ability to make sure the different oscillators are in tune and track with each other.
The Sequential Circuits Prophet 5 is a different thing. All of the audio generation and signal path are analog and prone to drift, and in a way, a similar process is used as to the Minimoog for tuning, but instead of the user listening to a crystal oscillator tone and manually tuning the analog oscillators, the Prophet 5 featured microprocessor controlled automatic tuning calibration. According to one source, tuning took about 15 seconds after the Tune button was pressed.
One reason why an automatic tuning system was necessary for the Prophet 5 was that instead of being a monophonic 3 oscillator synth, it was polyphonic with 5 voices of 2 oscillators each, for a total of ten oscillators. As drift could happen in the middle of a show, a fairly quick way to re-tune the synth was required to make it useful to musicians.
So, what I'm suggesting is if you are building your own oscillators in order to get that 100% analog tone, you'll want to come up with some tuning mechanism. You also might have to play with oscillator designs to try to make them as thermally stable as possible.
If I were heading down this road, I would start with the Moog method and make sure I know how to design a master tune knob that I can use to quickly re-tune the synth and work to get a design that is stable for at least an hour in a typical home room. Then I might look at "graduating" to tacking on a microprocessor that can electrically compare the oscillators to the reference crystal and automatically adjust the tuning knob.
Today, both Sequential Circuits and Moog Music have real-time microprocessor-controlled tuning adjustment in the Prophet 6 and Model D Reissue products, and Sequential even offers an additional control which lets you control how well the microprocessor maintains the tuning, to get some vintage-style oscillator drift in the sound.
More about the Prophet 5 design
One way the oscillators for the Prophet 5 were made more stable was by using analog integrated circuits that had as much of a complete oscillator as possible on one chip. That meant that all the components on the chip changed temperature together (at least closer together than discrete components).
There was also "on-chip temperature compensation circuitry". I'm not sure exactly what that involves, but my guess is that it's circuit design that uses on-chip components to make actual voltage drifts due to chip temperature "cancel out", as much as possible.
Page 2-19 of the Prophet 5 Service Manual is very interesting on this topic: https://medias.audiofanzine.com/files/sequentialcircuitsprophet-5servicemanual-text-470674.pdf
And I found an interesting paper on analog temperature compensation circuit designs for crystal oscillators: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.11.2410&rep=rep1&type=pdf