Is there a systematic way to name rhythms and melodies?

You could name a shape by the number of sides it has, and chemical compounds have systematic names. Is there something like that for rhythms and melodies?

What features of a rhythm are the most important to include in its name?

You could name them based on existing nomenclature, like what Native Instruments did for their Guitar Rig presets; e.g. a Hendrix-inspired theme might be called Purple Jimi, or an AC/DC style theme might be called Highway Voltage, and so on.

For tunes without a specific influence, I would go with mood-evoking naming conventions, or ones which indicate when the tune might be played (e.g. "Mysterious Woods", "Battle Fanfare", "Epic Hero Death Scene").

There is nothing systematically established, no. Its pretty much impossible to fully transfer an artistic thought process into a specific classification. You can get fairly close however, particularly when dealing with smaller pieces, or using relative genre-defining terminology. For starters,I would take a look at the work by the music genome project (along with its sister project pandora(.com).

If you want a more "specific" approach, are working with smaller "riffs" rather than entire pieces, and don't mind the occasional overlapping classifications, you could simply resort to more general indicators. For instance, you can determine the BPS (beats per second) of a music, the octave and/or scale. Additionally, You could combine that with a broad spectrum genre-defining term as MGP does, or for riffs just define a pattern such a scale. Doing this could yield a name like "26bpsJazzScaleCmin".

edit: minor language fixes.

[Edited by - PaulCesar on October 8, 2010 11:07:53 PM]

I don't tend to find myself storing or naming individual melodies, but if I did, I'd include the scale, the first note, and the last note - this should be enough information to allow me to guess at which melodies would follow on well from which others. I would probably describe the notes in terms of the scale degree rather than the absolute note because melodies are likely to get transposed anyway. Similarly if it was a chord sequence I might call it "iv sus2 to i" rather than "Csus2 to Fm". (See http://en.wikipedia.org/wiki/Chord_%28music%29#Notation)

As for rhythm, the time signature is essential, the tempo almost as much so. Then I tend to identify the instrumentation used, and finally a subjective indication of 'quirkiness' (which is usually about how much syncopation is in there or how it differs from the norms of the genre). I also distinguish between normal rhythms and fills.

Result:

I started looking at riffs and got to chords. The sort of things I am naming are very simple. For now I am using the scale note numbers to name accompany. So a bass of do so do so mi so do so would get a name Bass_15153515. This also shows that there are 8 beats, so I know it is for 4/4, whereas a Bass_153535 is for 3/4. For some rhythmic bass I name them by S=strong or W=weak, i.e. Bass_ s__ws___, Bass_SwwwSwwS, Bass_wwswSwSw, Bass_swswSwSw. Maybe this shouldn't be coded as a bass pattern but a volume pattern.

For the lead melodies I name them by their meaning. I may not need a systematic way to name lead melodies because I am only assigning a melody with the same meaning to a scene. I might need to systematically name transition, so that the game could decide what to play when transiting from one meaning to another.

Right now there is no transition melody. If melody 1 has chord pattern: I V vi IV V and melody 2 has chord pattern IV V iii vi, the game just plays melody 2 right after melody 1 and it doesn't sound weird.

One property I need to mark in a lead melody is whether it requires its accompany to change on the last bar. So for melody 1 if I assign it to use Bass_15153515 and specify it to change on the last bar, the game would play:

Bass_15153515, Bass_15153515, Bass_15153515, Bass_1513, Bass_1513

This seems like a property of the melody because it should lead up at the end depends on whether the melody is leading up to the next melody as it ends.

Rhythm Changes, Coltrane Changes, Jazz-Blues, Plagal Candence, Secondary Dominant, Appoggiatura, Pedal Point, Counterpoint... these are just some of the already established terms I came up with off the top of my head and that's not even getting into percussive classification. Read up on some music theory, not only is categorizing possible, it's already been done extensively.