When I play a note on my synthesizer or my guitar, it is exactly one frequency plus the overtones and subfrequencies which are doubles and halves of the tone.
But when I listen to a vocalist, it sounds like the vocals are not just "one frequency" but that the voice has many different frequencies at the same time.
Is this something that can be confirmed or am I just hearing it wrong?
Every instrument has its unique voice - except, perhaps, synthesiser, which has the propensity to produce many different voices.
The main difference between each is the number of overtones or harmonics produced by that instrument, along with the fundamental ADSR envelope, which is often variable. Listen to a trumpet and a violin play the same pitch, and the difference is recognisable immediately.
Vocals have a unique set of overtones, but also the ability to make different vowel, and to a degree, consonant sounds. Also, quite capable of reproducing some instrumental sounds too. So, where does that leave us?
Answer to header question, many, variable.
You're working from a false premise. There's inharmonicity in every real instrument's sound. Partly in the source, partly in the playback environment. Let me know when you design a system that will really deliver a mixture of harmonic pure sine waves to the ear!
https://en.wikipedia.org/wiki/Inharmonicity
But maybe less in an instrument like electric guitar than in a human voice. Which is why guitarists typically add effects and choose deliberately distorting amps and speakers, where a vocalist often prefers 'clean' amplification.
When I play a note on my synthesizer or my guitar, it is exactly one frequency plus the overtones and subfrequencies which are doubles and halves of the tone.
As per Tetsujin's comment, that's not always the case; there will often be inharmonic partials, variations in the frequency of individual partials, and regions of energy that perhaps don't have a distinct frequency. But you're right it is also often possible to create sounds that stay close to perfectly periodic (and therefore have only harmonic overtones) at least for a short time.
when I listen to a vocalist, it sounds like the vocals are not just "one frequency" but that the voice has many different frequencies at the same time.
The human voice also tends to be an unstable sound - the singer may be deliberately varying the air pressure, pitch, resonances of the vocal tract and mouth (which will emphasize different frequencies within the note) - or these may be varying involuntarily. As well as the basic frequency of the sung note, there may be a periodic vibrato taking place at a different (much lower, non-'audio') frequency, and if the vocalist is making a growling sound, you may hear the aryepiglottic folds vibrating at yet another frequency. Additionally, there is a lot of turbulence and 'noise' in a human vocal sound.
However, though the human voice is complex, there isn't any essential characteristic of it that can't be found in other instruments or sounds; I don't think you're hearing the human voice wrong, but perhaps you haven't noticed the subtle complexities in other sounds yet.
