Tonal hierarchy in non-diatonic music
There are two motivations for this question: I'm trying to distill the essence of what tonality in common-practice music really means, and I'd like as a compositional resource to port some aspects of common-practice sound to music written in non-diatonic scales.
What would it mean to have "tonality" in any key but a major or minor scale? I understand that tonality involves hierarchy, so whole-tone and other completely symmetric sets aren't great for expressing tonal idioms, but what about the hexatonic, pentatonic, or acoustic scale? Would it simply be a matter of using the usual tonal relationships but adding a few sharps/flats, or could you devise something that is both tonal and substantially different from diatonic tonality? For instance, could one have a system of tonality that revolved around a relationship other than tonic-dominant? Could one devise prolongational structures or music susceptible to Schenkerian-reductions that create completely different assumptions about tonal relationships?
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Keep in mind that simple repetition can often give enough emphasis to imbue a harmony with a sense of tonic. When done correctly, the chords can be anything, and even without a dominant, you still sense where tonic is. When this is the case, the typical tonic--dominant polarity of "home" vs. "tension" simplifies to just "home" vs. "not home." We see this happening at least as early as Liszt, some Brahms, etc.
As for some of your further questions in the comments, there are some articles you may want to check out:
"The Problem of Prolongation in Post-Tonal Music" by Joseph Straus, The Journal of Music Theory 31/1 (1987)
"The Problem of Prolongation in "Tonal" Music: Terminology, Perception, and Expressive Meaning" by Steve Larson, The Journal of Music Theory 41/1/ (1997)
And Straus's "Response to Larson" in the same volume.
And Fred Lerdahl's "Issues in Prolongational Theory: A Response to Larson," also in the same volume.
It's tough to summarize all of these in a single post, but basically Straus doesn't believe it exists. But many claim that his logic is circular, and that his stance fails because of it. Still others (Lerdahl) claim that it all boils down to semantics/terminology, and that Straus things it doesn't exist because he's only considering it from a Schenkerian standpoint.
As for prolongational reductions of non-diatonic music, I really recommend you check out Structural Hearing by Felix Salzer; he graphs hundreds of pieces, and they date from the eras of pre-tonality, tonality, and into Stravinsky and beyond.
And speaking of Stravinsky, you may also enjoy "Harmony and Voice Leading in the Music of Stravinsky " by Joseph Straus in Music Theory Spectrum 36/1 (2014).
There are all kinds of other sources, but this seems like a more than adequate start; from there, you can check out bibliographies to see what to consult next.
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