I have an instrument called a Huaca that is a three chambered clay vessel flute. So as a wind instrument it plays three note chords. I have always thought of it as a fixed pitch instrument as it is not designed to play in multiple keys. It is mostly great in one key and can adapt to another by focusing the playing of certain notes but basically one key. Whearas the piano can play in all the keys so I have not thought of it as a fixed pitch instrument (even though the notes are fixed). I now see that it is considered a fixed pitch instrument. What would I call my huaca if not a fixed pitch instrument? What complicates it is that I can bend tones by shading the finger over the hole. Lastly I have tuned them to just intonation or what I call proportional harmony. Essentially how do you all differentiate between a fixed pitch instrument and a non fixed pitch (fretless strings?) and help me understand the nuances. Thanks.
A fixed pitch instrument is one that cannot be tuned. It has nothing to do with whether you can play in more or fewer keys. For instance, I play the pennywhistle. My whistle is a single piece of aluminum tubing, and can not be tuned. It is a fixed pitch instrument. But I could spend more money and buy a pennywhistle with a tuning slide. With either instrument, it only playes comfortably in a couple of keys, but with a tuning slide, I could tune to A = 440 hz, or A = 430 hz.
A piano can be tuned, but it needs an expert, specific tools, and a fair amount of time. So while you could argue that it's less fixed pitch than your clay flute, for practical purposes it's fixed because if you sit down at a piano, you can not tune the instrument so that the keys are at a different pitch.
A guitar, violin, or concert flute (for example) can be tuned by the musician in under a minute, and will be tuned before each performance. These are not fixed pitch instruments. If they play with a fixed pitch instrument, everyone will tune to that instrument.