I am usually careful to use "overtones" and "timbre" rather than "harmonic series" to describe the sound of musical instruments. Many of them have non-harmonic components, including brass and most wind instruments. Strings can generate beats among the actual strings on the instrument as they are played, etc. Drums are impulse generators and yet do have fundamental'ish terms depending upon how the head is tuned. Piano is tough for many reasons: the percussives when hammers strike strings, there are 2-3 strings per note and those strings are tuned to sound "richer" so are not necessarily exactly the same note, different strings (different material/construction) have slightly different sounds, the sound comes from strings and sound board, the sound can vary along the string since there are nodes and antinodes, and of course that also affects how nearby strings interact, the sound is modified by interaction of the sound waves from various strings within the piano and sound board depending upon the position of the lid, octaves are generally (intentionally) "stretched" at the top and bottom, it is a very wide frequency range to cover, etc. etc. etc.
I made some files years ago with added distortion for people to try. IMD is much easier to hear than THD, but in music any distortion is harder to hear. The rule of tens is not quite right but decades ago I was curious and did some blind tests with friends and customers of the store and it was not too far off. That is, for a pure (single) tone in the midrange or vocal band folk could detect down to about 0.1% distortion with moderately high volume; with a few multiple tones more like 1% depending on how they were spaced, and for musical selections or many tones more like 5% - 10%. Music includes so many harmonic and non-harmonic frequencies anyway that telling what is distortion can be hard until it gets pretty high. And then add the whole loudness curve bit and any general comments about "x% distortion is audible" pretty much fly out the window.
IMO - Don
Interesting comment, Don. Thanks! My question would be whether this kind of blind testing is comparable to ordinary listening. I mean, it seems fully reasonable to me that 5 to 10 percent distortion (not uncommon in speakers) will be undetectable for a lot of people when hearing certain musical pieces. As you say, how does one know what is distortion and what is not? I would nevertheless assume that over time, with prolonged listening, the non-distorting system will sound better on a larger variety of material - solo voices, choir, solo piano, etc. These small segments where the distortion becomes noticeable might then increase the sense of artificiality of the system as a whole. So I still think best practice in audio engineering is to strive for speakers and components with as low distortion as possible.