One thing I've noticed in this forum and others like it is that the music people identify as Klingon is not only very Terran, but specifically from a western musical tradition. One of the drawbacks of this is that this music usually uses the 12 tone equal tempered octave scale, which is a modern, western invention, very specific to our culture and species. The fact is that even on Earth, this is but one of many different musical scales available; for example traditional Arabic music uses a scale that is closer to 24 tones. There are some modern musicians who try and break out of the constraints of the 12 tone scale by using so-called "microtones", that is, tones that occur in between the notes of conventional music. The result is music that to modern ears can sometimes sound disturbing, sometimes amusing, but always surprising.
This is where Klingon comes in. if we look at the Klingon language, it was created by Marc Okrand with the specific goal of being not only very different from English and Western languages, but also all human languages. Why shouldn't the same logic apply to Klingon music? Why not go out of our way to create a musical system, from the bottom up, that is uniquely Klingon and un-human in character, using the techniques of microtonal music?
Here's a variety of different examples of microtonal music to give you an idea of the kind of strange sounds that can be achieved using this method:
http://www.youtube.com/watch?v=2PlRSG7Rmmg Syzygys are an avant-garde Japanese group that use a variation of a 43 tone scale devised by 20th century American composer Harry Partch.
http://www.youtube.com/watch?v=EU85bUyDPWs piece by composer Charles Ives (there are three parts, but this is my favourite). Two normal pianos, except one of them is tuned a quarter of a note sharp.
http://www.youtube.com/watch?v=Jq8RIE3iDD4 Dr. Richard Boulanger is a researcher who has worked a lot with the Bohlen-Pierce scale, a scale that has 13 tones to a perfect twelfth. This song in particular seems like it belongs in a Klingon Opera.
http://www.youtube.com/watch?v=0pwO-iqe-l4 This guy retuned his piano to a 19 tone scale. Very cool sounding, in my opinion.
http://www.youtube.com/watch?v=fdzBfTF5os4 24 tone guitar jamming.
http://www.youtube.com/watch?v=IEpqFCRuCL4 an example of "traditional" microtonal music. Gamelan is an ancient Indonesian tradition that uses a combination of 5 tone and 7 tone scales. Probably a bit too peaceful for Klingons.
http://www.youtube.com/watch?v=aJG55p2STCI interesting composition in a 16 tone scale.
http://www.youtube.com/watch?v=UWYggKqsDqw Pachabel's Canon "retuned" to the above mentioned Bohlen-Pierce scale.
http://www.youtube.com/watch?v=S0EqDsumU_Y a 36 tone song. This one never fails to creep me out. I'm not sure even Klingons would be able to listen to this.
William Sethares, a researcher in microtonal music, has even created
this example of an actual Klingon song using microtones, with lyrics in tlhIngan Hol. I'm not sure if his specific scale of 60 notes to an octave is the right fit for Klingons, but it's certainly a start.
Microtonal music can be a little bit difficult to listen to. After all, the whole point is that the music is delibrately out of tune. However I think it is something that should definitely be considered if we ever want to create music that is befitting of the label of "alien". Otherwise all we are doing is recycling Terran ideas of what music should be.