D&D 5E - A question about the Entertainer Background and proficiencies

Secondly, no, I'd disagree with you, I'm afraid.

The rules say "Each type of musical instrument requires a separate proficiency".

Your Performance proficiency covers "music, dance, acting, storytelling or some other form of entertainment". The "natural language" wording indicates that you, as a player, make a choice as to where your talent lies, IMO.

As instrument proficiencies are a separate thing, I view that "Music" in this (second quote) context, if the character has not got proficiency in a musical instrument, would mean song, or perhaps virtuoso whistling, or percussion (beat out a crazy syncopated rhythm on that bar counter, daddy-o), or playing the spoons (perhaps the latter would qualify as "other form of entertainment", to be fair, along with puppetry, shadow puppetry, mime, stand up comedy, etc).

Example 1: I can sing (Performance proficiency), but cannot play an instrument. If I sang a song (Performance check), I'd get my proficiency bonus. If, however, I attempted to accompany myself on guitar, my lack of skill would counteract my proficiency bonus (in reality, I'd make such a jangling sonic nightmare that no matter how in tune my voice was, I'd still be booed off the stage and quite possibly shivved as I left, but in-game I'd rule that you lose the proficiency bonus).

Example 2: I can both play the guitar (Tool Proficiency) and sing (Performance). If I strummed my guitar in my performance, either as an accompaniment to my song, or on its own as an instrumental, I'd get my proficiency bonus. But put a violin in my hands and I'd make a sound like a dozen cats being castrated. Again with the booing and the shiv. I'd need Tool Proficiency in both instruments.

Example 3: I can play guitar (Tool Proficiency) but cannot sing (no Performance proficiency). The audience are spellbound by my intricate fretwork if I play an instrumental. But if I sing along my voice may come out flat and out of tune, I may honk and squeak like a goose humping a mouse: either way it would counteract my beautiful guitar-playing bonus.

The Entertainer background is not the only way a character can acquire the Tool Proficiency with a given instrument. Particularly where a race or class provide a Tool Proficiency that the chosen Background, whatever it is, already provides (or just by player choice for colour), a PC may choose to have proficiency in an instrument without proficiency in Performance - for example, the gruff drill sergeant who his troops would never imagine goes off on his own to the hills above camp and in seclusion plays the most heartbreakingly beautiful flute in memory of his lost love, but would never ever be seen dead giving a public performance. Or, perhaps, he is a Warlock who communes with his Patron by calling it up using the flute.

Mechanically, a player may hand-wave things, saying "I do some music", but the entertainer background is one of the more colourful backgrounds and deserves a bit more RP attention...eg, "I sing a lament to the lost Elven village of Shothimintheda'ark" or "I'll get up and play a lively reel like The Innkeeper's Daughter, that should get the crowd going".

Ymmv.

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