I'm interested in making a bass kazoo (for obvious reasons). This is about the most scholarly article I've found on the internet about how these instruments work.
Would a longer or wider resonating chamber have a lower pitch?
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A kazoo is a membranophone, meaning that its sound is produced by a vibrating membrane. In this sense, it's an evolutionary cousin to something like the timpani (another membranophone).
This means that the general characteristics you get with different timpani sizes apply to the kazoo: the larger the vibrating membrane, the deeper the tone (generally speaking).
There's one catch, of course, with comparing it to the timpani: a timpani player can control the tension on the drum head, thus altering the pitches. Obviously a kazoo player doesn't have this ability.
Otherwise, the kazoo player sings/hums into the kazoo, so I don't see how construction of the kazoo would really alter that aspect too much. (Unless it was big enough for the player to start buzzing into it like a brass player, but that's a different story!) The important aspect is the size of the vibrating membrane.
I've played kazoo before, and I don't think the dimensions of the kazoo will actually change the pitch of the emitted sound. Here's why:
If you have a kazoo in hand, try singing a note, then playing that same note on the kazoo. Then try saying "doo" and then putting your mouth up to the kazoo. The note pitch shouldn't change.
Why? I think it has to do with the fact that the kazoo produces sound via air vibrating the membrane, so the only thing that can make the pitch change is the frequency of the input sound. However, you could change the timbre of the kazoo by changing the dimensions, I bet.
The best way to make a bass kazoo is to get a kazoo, then find a bass voice to play it.