Technically, this is less about form and more about orchestration .
People can use instruments to denote a shift or division or a new section in the piece, though they don't have to.
Orchestration is the art of timbre.
When instruments enter one at a time, their entrances are described as "staggered".
The fact that the instruments enter halfway through the piece is an orchestrational one.
Canons and Rounds do not apply in this context if the staggered voices do not also enter through imitation as well.
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Development , or buildup?
Crescendo is when the music gets louder.
Development is nearer to what you ask.
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Though this technique does occur in "composed" music - Ravel's "Bolero" and (in reverse)
Haydn's "Farewell" symphony
- it's more typical of music "constructed" in a sequencer.
Because a sequencer makes it easy to work in this way
- set up a drum loop, add a bass, add a guitar
- you're right
, we hear this done far too often!
I don't think there's an accepted term for it.
"Layering" perhaps.
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The closest term I found for this was a canon .
Not exactly the same, but pretty close.
A canon pretty much when one voice has the melody and some duration later another voice comes in playing the same melody and then the pattern continues for more voices.
A round is also a type of canon that may be slightly closer to what you are thinking of.
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I think you can call this a " Mannheim Crescendo ".
I also have a teacher who use a french expression to name it, a "crescendo d'orchestre", but I don't find sources for this denomination.
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I don't think you'll find an exact term for this, but what you are referring to is "texture."
In the type of piece you describe, the texture starts "thin" and then layers or builds to a "dense" texture.
For the reverse(at the end of the piece)
you could describe the texture as "thinning out" or "layering out.
" If there is one instrument, the texture is monophonic, two is biphonic, and 3 or more polyphonic.
More info: texture on Wikipedia.
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Many songs will start with only a few number of instruments, and after a specified amount of time has passed a new instrument is added, and this is repeated until all instruments for the song are playing its rhythms. Sometimes this is also done when finishing the song, but in reverse order.  Does this style of opening a piece has a name?
What is it called when a song starts with few instruments, and gradually adds one at a time?
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When instruments enter one at a time, their entrances are described as "staggered".  Other suggestions for the same event are development, canon and Mannheim Crescendo. Many believe there is no exact term. It is possible that there is no accepted term for this, but some suggestions include "development", "canon" or "Mannheim Crescendo".
There are a number of different terms for this, such as "development", canon" or "Mannheim Crescendo". It is unclear that there is an accepted term for this.