I'm writing my friend a horn solo (with piano accompaniment) and I asked her what her favorite scale was and she said harmonic minor, "because it sounds Egyptian". If I was trying to write a traditional Egyptian song what time signatures and melodic patterns (interval, rhythmic, etc) should be involved? (I've always wanted to do something in irregular time signatures.) I can rephrase this if it's not clear what I'm asking.
A harmonic minor sounds "Egyptian" or "Arabic" depending on how you use it.
Use of the scale is deeply woven into the European classical music tradition. Occurrences of the scale in, for instance, Baroque music are not strikingly "Arabic" in mood. Here is an exmaple. The "Allemande" from J. S. Bach's E minor suite for guitar (BWV 996) has a quick, descending run of 16th notes: B G F# E D# C B A. The harmonic minor arises out of the need to treat the minor key scales with trappings of the major key: the dominant harmony and its requirement for a leading tone (and the melodic minor scale then "closes the gap" formed by moving the seventh degree).
Your friend might be thinking of music based on the Phrygian mode of the harmonic minor, where the root node is the 5th degree (dominant harmony) of the minor. For instance rooted at E in A harmonic minor. Also, experiment with raising the fourth degree (D -> D#). This scale triggers an "exotic" mood in the ear accustomed to Western music because it contains a tritone as well as a major third, as well as a semitone above the root.
It seems that you are serious about wanting to write actual Egyptian music (which is, without a doubt, a huge category spanning from traditional to contemporary) which requires familiarity, rather than just "faking it" with certain approximate scales. Speaking of just scales alone, there are many in the Arabic world, and they are microtonal; equal temperament approximations make a farce out of them.
If you set a very high standard of authenticity for yourself, you could spend years studying before you produce something that ultimately won't be playable with conventionally built and tuned horns and pianos. I would weakly suggest embracing the watered down stereotype as an artform, and treating it honestly.
This question is a little difficult to address for a couple reasons:
1.) A harmonic minor scale is a Western-European approximation of eastern tonal characteristics - much in the same way a pentatonic scale stereo types Eastern-Asian culture.
2.) "Traditional" Egyptian tonal materials do not fit into Western notation, so if you were honestly going to write a traditional piece, you would need to use their notation.
3.) The same applies to rhythm as well - different divisions and phrase lengths that do not easily conform to European notation.
4.) I would strongly caution against trying to write something "traditional" unless you are very knowledgeable in stylistic and cultural characteristics. Otherwise it will immediately come across as a watered-down stereotype.
Here is what I recommend:
Whatever you end up writing, remember that you're writing for horn and piano, which will never sound traditionally Egyptian.
Best of luck.