What are those small notes below other ones?
When I search for &quot;tiny notes&quot; in the Internet, it says to me &quot;appoggiature&quot;.  But they are placed just before the main note, not immediately below, like in my picture above.
So...
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Music: Practice & Theory Stack Exchange
Small notes can have a lot of different meanings. In this case this would most likely be so called cue notes. These are small notes included in usually parts and vocal scores, not to be played by the musician, but to give some context to what the other voices are playing. Usually this is done to facilitate finding the cue, although in this case I suppose they are there to give the group an idea of what the other voices are playing at the same time. That being said, this is the first time I’m encountering this use. Maybe you could share with us what composer, piece and edition this is?

Small notes can also designate grace notes, alternative notes, on polyphonic instruments optional notes, less important notes (e.g. to emphasize the melody) ...

One other possibility that @Lazy didn't mention is the notes to be played by other Trumpet players if there are more than one trumpet available, or if there are no players for other instruments in a similar range (such as Cornets, Flugelhorns, or French horns).

Again, as @Lazy is saying, it would have been helpful to provide the composer, the title, and a link to the full score of what could have been an arrangement for brass band / marching band, so we can know from the score what those small notes mean.

The cue here is that this is notated for C trumpet, a comparatively rare instrument. If it were for violin, chances would be that the section was expected (if large enough) to split into multiple voices. However, the last chord contains 4 notes, the preceding ones 3 notes. If it was a multi-instrument arrangement, it would not change numbers of notes as lackadaisically.

So hunting for other hints, this is the top staff of a system and in treble clef. That gives us a strong clue about the unusual choice of "C trumpet" (well, apart from the piece presumptively being in C Major though the passage's chords are kind of unusual): that is the only trumpet notated at concert pitch, meaning that its staff can double as the treble staff of a piano.

And that is what we are likely seeing here: a top staff played either by C trumpet (in which case you leave off the small notes) or by piano solo (in which case you play all notes) or by both (in which case the piano may play the small notes while probably leaving out the big notes to give the trumpet more room for its interpretation).

That would mean that the next lower staff would likely be in bass clef and written in a manner that can be played reasonably on a piano. If it can also be played on a bass instrument (and indicated as such), this staff is likely to contain only single notes, or at least a well-distinguishable (possibly by other notes being in small print) single-note voice.

This kind of piano-or-else notation is pretty common for musical tunes where the soloist will typically practice with an accompanist (who plays piano) while the full performance would want more separate instruments.

No, not appoggiaturas in this case, or any other type of grace note.

This appears to be the top line of a musical score, probably for some sort of band. It would have been very useful to show us more of it! But I think we can be sure that the small notes indicate what other instruments are playing, maybe the other trumpet parts. Maybe the first trumpet player directs the band. In old-style brass bands the conductor, even if not playing themselves, often worked from a fully-cues Solo Cornet part (though I don't think this is quite what is happening here). Maybe the intention is 'if only one player, play the top notes, if more, fill in the harmony with the others'.

If it was the upper stave of a piece for solo trumpet and piano, I'd expect to see a brace in front of the staves linking it to the stave below.

We're fishing here, with too little information, though! Perhaps @user103031 could replace the picture with one showing the whole page?