Interesting instruments
ANS synthesizer
https://warmplace.ru/soft/ans/
Theremin
A theremin senses a nearby hand and uses the distance to it for pitch.
And, often, a second antenna and hand for volume.
The antenna are often physically relatively distant that movement near one is not relevant to the other.
The shapes of the antenna are somewhat secondary, but the loop actually acts more like a plate, which gives slightly easier control (when close).
The fact that antenna are mounted perpendicular seems less for field reasons and more for so that movement meant for one antenna is easier to separate mentally for a human player (verify).
In the original design, the pitch itself comes from heterodyning oscillators.
This isn't the only way to do it, but at the time an efficient and reasonably stable one(verify), more stable than the cheap imitations decades later.
But it was also that it predated transistors, and radio tech at the time could not really generate stable low-frequency baseband frequencies, without actually having it come from beating of two higher-frequency ones (...which creates the sum and difference. Usually a lowpass is used to attenuate the sum).
You could call it a beat frequency oscillator - and is indeed related to BFOs in radio use.
In these solid-state days you can do the generation from a simpler oscillator.
The oscillators that you heterodyne are on the order of 100 to 1000 kHz.
Basically because if you heterodyne, there are a few reasons that combine to make that the sensible range.
One, the difference from the antenna is a handful of picofarads, and you might want a few octaves of audible pitch change. To do that at baseband you need an impractically huge inductor, and more reasonable component choice for the desired few khz of difference place the actual frequency at a few hundred kHz.
In other words, heterodyning turns out to be one practical choice for sensing, even when you do have modern components for the sound generation part.
If using that directly in an (LC) oscillator, this is the range you get decent control in,
and which combined with a similar-pitched fixed oscillator gives decent response in the audible range. (also determines its sound, since it's modulation of two sine waves)
Part of the trick of a decently playable theremin is getting good linearity - that is, a decent linear response in pitch to what you consider constant motion - and not to capacitance, because there is (a lot) more change in capacitance near the antenna than further away.
If used fairly directly, high pitches are so close to the antenna that vibrato is hard, and low pitches are large gestures.
There's some confusion over whether there's RF going on.
Heterodyning comes from radio tech, the antenna being a radio-like antenna suggests RF, the frequencies used are in the lower AM radio range, and in the classical design, the capacitance involves active LC oscillators.
All of which suggests RF, but in the end, the sensing is mostly just capacitive sensing.
See also:
Simpler theremins
You can sense distance with capacitive sensing, you can make beeps in response to whatever value that gets you.
For example, Doepfer A178 [1] is basically just a capacitive sensor,
but more specifically, an analog one that turns differences in resonance into a CV signal.
It too has the "more sensitive nearby" quality of the simpler design.
Also it's sensitive to the rest of your rack, people have even put it in a separate enclosure.
Other theremins
An optical theremin is a very simple design, usually a photoresistor (a.k.a. Light Dependent Resistor) to control the speed at which a 555 oscillates.
It reacts to the amount of light, not distance as in a real theremin.
It's even harder to control in a musical way, but it's fun for a few bucks.
https://www.popsci.com/diy/article/2008-04/build-pocket-theremin-cheap
In the realm of cheap, an ultrasonic distance sensor is arguably a more workable alternative, because it's not too hard to get linearity from it, and you probably involve some more components (e.g. a microcontroller) that gives you a little more control over the range of tones it makes.
However, since the resolution is on the order of 2mm, and the practical range is ~50cm, which are on a useful order but not very precise and thereby not very stable.
CV theremin
An antenna input that needs to only output CV doesn't need all that oscillating and heterodyning.
It can be a basic LC circuit.
Ondes-martenot
https://nl.wikipedia.org/wiki/Ondes-Martenot
Trautonium
Celeste
A Celeste looks like a piano, but has chime bars instead, and in a resonating enclosure that makes it softer (particularly the lower range).
It sounds like a cross between a piano and a xylophone or maybe glockenspiel.
The best introductory example, that many know already, may be the melody from Sugar Plum Faery.
The lower range, due to its softer nature, can be used to add to the bass, percussion, and others.
Hammond
The sound of a Hammond is characteristic, and comes from a few specific bits of engineering.
One is the tonewheels, which are metal gears being sensed magnetically.
Emulating the tonewheels is possible, but more work than you'ld think.
The tonewheels themselves will not produce perfect sine waves. Mostly, yes, but there are harmonics.
The synchronization between played notes is relevant, because it is mechanical and thereby entirely fixed:
There is a mechanical drive shaft that drives one gear per key,
which drives a tonewheel, which is picked up via magnetic induction.
This means that while different keys go at different speeds, and there seems to be intentional detuning to avoid overly strong beating(verify), the phase differences are constant.
That means that e.g. sampling individual keys and preproducing them in response to MIDI producing per-key samples will have (due to note timing) different phase relations every time you play it, unless you specifically model it as addition from the same thing.
You can mostly hide this by adding vibrato (because most will imply constantly altering phase anyway), which is close enough for various uses,
Note also that the leslie speaker does both vibrato (FM) and tremolo (AM), and at the same speed (the rotation speed) but with their peak noticeability out of phase.
Since most are tuned to equal temperament, they will still beat noticeably.
Note also that it's only nearly TET, because it has to be the nearest gear ratio
(the equal-temperament step is an irrational number).
Waterphone
Waterphones are a metal resonant things (ideophones), with thin bars that can be played with a violin-style bow.
They are mostly known for eerie sounds in movies, what with the metallic sound, them not being tuned, and the water in the bowl amounting changing the resonance to effectively detune them a little on the fly.
Hurdy gurdy
A cranked string instrument, with a fairly specific high-pitched droney sound.
Depending on model, you can do chords and such.