With that large a range of candidate selections, the simple drop-down you show would, I think, be "clunky" even on a desktop.
In this case, especially on a mobile, I think you need to somewhat separate the functions of "showing the currently selected items" from the "selection picker" to add new items. Probably a good starting point is how the StackExchange sites handles picking of tags when asking questions.
On a desktop, there is less separation: the tags are accumulated in the same box where you can begin typing the name of a new tag. At that point, a panel appears with partial matches which you can select from (or type more to refine the search). On a mobile, the input box where type tag's name is separated from the display of current tags.
As an alternative (or in addition) to typing, you could have a "New Item" button/icon that brings up a categorised list (e.g. "Woodwind", "Brass", etc.) that the user can use to "drill-down" to the instrument of choice.
As implemented by SO, users can't "browse" tags (although they can start typing and see the "top six" of a narrowing selection of tags that contain that text). However, it should be possible to allow a more comprehensive browse/search facility in the "tag window" should you desire, e.g. show more/all matches and allow the user to scroll through them.
I think my key point is that trying to provide "browsing through a list" in a "traditional" drop-down is doomed to failure/clunkiness... like SO, the tag picker/browser/search box should be a separate entity of some kind:
On a desktop, like like SO, it could be a "pop-up" window. Given the nature of pop-ups (particularly the ease of them "popping-off") you probably wouldn't want to use this option if you added much more complexity.
If you wanted more complex functions on a desktop you could make the "tag selector" part of the main page (that either is always there, or pops into existence on demand). You could then, for instance, add scroll-bars to allow browsing of all (matching) tags.
On a mobile, if you want more than the current SO implementation offers, you probably want to switch to a "full screen" selector that at least acts as a standalone page (potentially allowing scrolling through matching tags etc.). Whether this is implemented as a separate page, or a full-screen pop-up is more of a technicality.
So: what SO does is an example of how to provide better way of selecting from hundreds or more options. However, you can extend that idea and make a more complex selector if you wish.