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Because a melody depends on pitch and rhythm. Although there are a limited number of relationships that can occur between notes, rhythm, in theory, could be infinite. You could have a melody pattern that only repeats after several hours....or years for that matter.

there's a finite number of chord sequences, but not of melodies. That's why you can sue someone for stealing your tune, but loads of songs share bits that sound similar.

but the number would be so large its unlikely it could ever be reached. there'd be 12! (1x2x3x..x11x12) ways of arranging the 12 notes in western music, which you'd then have to multiply by a ridiculously huge number to account for all the different ways in which you could hold particular notes and pause between others. even discounting all the "non musical" permutations of the 12 note scale (though schoenberg might disagree) you'd have a ridiculously huge number. and that's only accounting for using each note once before returning to it. there's also microtonal notes and non western tunings to account for. may well be possible to work out a finite number, but it would be so huge you may as well say infinite

definitely not all been used for the second question.

but probably relatively few that sound like a melody, and lots of them would be the same relationships just with a different starting point note so would be the same even though they strictly aren't, so no and yes. But im quite interested in other tuning systems since I saw this schools program on channel 4 about music, the western scale is relatively new and impure, it is just an octave split into 12 notes and means that it you cant have a note and its fifth (the 2nd most harmonious relationsip after an octave of a note) just an approximation of it (but for some reason this means you can shift keys without retuning), there are other musical systems, I think the indian system has 17 notes, arabic 24, so there is plenty to play around with but still the most pleasing relationships are probably similar in each system and well explored by now

Eine Kleine Nachte Musik
Boredom
Zunguzungguguzungguzeng

This thread has helped me get some ideas together to research for my essay :)

because there can't be a truly infinite number of ways to combine notes, pitch, tone, rhythm etc
it might be a very large finite number but mathematically eventually it will reach it's limit

and there's no other melody that seems to work that well but the Eels one

I reckon although technically infinite someone could come up with a ballpark figure if they used a few assumptions based on what most people understand melody as, there is a lot of tiny variation and flourishes but for most it is the main pattern behind that. Although technically it could be infinitely long, really it has to be short enough for short term memory, I reckon anything longer than 4 measures could be chunked as part of a series of melodies rather than a single one, so set that as the max and it will contain shorter ones within it. Then there is the issue of the infinite number of notes that can be played in a measure, I think really anything quicker than 16th notes would no longer be recongnised as melody it would either blend into one sustained note if they are the same note, or be dismissed as tiny variation to the main pattern if different notes, people will just pickout the main notes. So say 16 notes x 4 measures, at 4/4 would be 64 slots in a melody. So if there are 12 notes, plus one extra for silence, that is 13 possible notes for each slot. I have no idea how to work out number of combinations, would it be 13 notes to the power of 64 slots? If it is divide that by 12 to get rid of duplicate melodies in different keys (not sure if that would work?), repeat for a number of different time signatures, I guess there would be 16 if limiting to no notes quicker than 16ths (but not sure) and lots of overlap between them, ones with fractions in I reckon would just be percieved as close enough to one of the other ones. I guess there is still the problem of there really being more than 12 notes, but I assume it is possible to know the resolution of human hearing and know within what range people perceive pitches as different, and run the whole thing again based on that number. But then there is also the issue that the same note in different octaves will feel different, so you would have to multiply the number of notes by the number of octaves people can hear. If someone went through all that, I reckon it would cover the vast majority of possible melodies. I dont really know what I am talking about though, should probably abadon reply