Orchestration can be a bitch, but I just might have figured out how to be a bigger bitch. Mathematics is a great tool for philosophy and reasoning, and I think something in it might just have this problem licked.
Composing music for multiple instruments seems an impossible N-body problem. Making a melody is simple enough, but making 2 melodies, all of whose notes coincide, let alone an orchestra of 60, becomes impossible.
Here's where I come in. "Hello, impossible! Get a gander at this:" Every moment of time in a song has 1 sound. 60 instruments all playing different things will still make 1 total sound for any given moment in time. Our eardrum can't do two things at once. We have processing in our brains to pick apart the sound into instruments, and cougars running through the bush under the masking sound of the waterfall, but what hits our eardrums is 1 sound. Most of this processing has to do with how the sound has changed over time, if a certain set of frequencies remains constant in amplitude, we hear one type of instrument playing a long note for example.
One would think composing a piece with 4 instruments is an O(n^4) problem, Melody 1 x Melody 2 x Melody 3 x Melody 4, but instead it's O(n^2): Melody x Sound. Sound is the total sum of all instruments at a given time. One might even call it Time x Sound instead of Melody x Sound. No matter how many different melodies are playing, only one sound hits our ears by every moment in time. (Weell... there is space too, since we have two ears, panning and all.) So I guess it could be O(n^3): Time x Space x Sound. And usually space is ignored by the composer, since the Orchestra determines where the different instruments sit. You can bet even Mozart would be at a loss if the orchestra playing his music had flutes next to the cellos and strings behind the timpanis. Space is generally not the composer's responsibility, therefore the problem is once again Time x Sound, or O(n^2).
The amazing thing about the O(n^2) is it is independent of how many instruments are in the composition. 5 instrument quintets have O(n^2) compositional complexity. 60 instrument Mahler symphonies have O(n^2) compositional complexity.
We Westerners often divide Sound into two categories: Harmony and Timbre. (Timbre as I define it is the part of a sound after you take away pitch, the difference between two sounds at the same pitch.) Obviously a composer is limited in what Timbre they can produce: you can't make a clarinet sound like a log hitting water. So normally instead of having full freedom of all sound, they instead have a limited set of timbres that they can combine into different sets of pitches called Harmony. Much in the same way composers have control over time, but not space, composers have more control over harmony than timbre. There is that additional element though, a composer must choose what instruments to write for. Thus the problem finally becomes one of O(n^3): Melody x Harmony x Timbre.
Harmony is the sum of all sounds at any given time, another way of thinking about it is: Harmony is what your eardrum is doing at any given time. Melody is how the sound changes over time. Timbre is what instruments you have to work with, and which instrument plays which part of Harmony.
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