I've written most of the content of my microtonal music theory textbook, and now I have to go back edit things until they're clear, informative, and enjoyable to read. I thought that my chapter on Intervals And Tuning Systems was the next one to fix, so I started rewriting it in my head. Unfortunately, the chapter actually starts out pretty strong. So I'm not sure what to do with this text. I might find some other place to work it in. Right now, I just don't want to lose it.
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What are intervals? This shouldn't a big mystery; they're standard objects of western music theory, but lots of people think about them incorrectly, especially in the microtonal music community.
I'm going to give you a complicated but fairly complete definition of intervals and then we'll spend the rest of the chapter understanding it in detail.
Intervals are vectors abstracted from the prime factorization of frequency ratios. They are the natural domain in which we perform most operations of musical composition. They exist in a one-to-one correspondence with pitches of sheet music like "C#3", and they naturally represent the signed distance between pitches. They have musically significant names like "perfect fifth" and "minor seventh" which can be calculated from their vector coordinates, and these names tell us things about their consonance, how they participate in scales and chords, and how they transform into one another through standard operations like inversion and augmentation. Intervals are also the objects which tuning systems act on, i.e. tune, to produce frequencies and frequency ratios. Finally, intervals are a psycho-acoustically convenient representation of sound combination which allow independent aspects of musical objects to be manipulated independently, and allow dependent aspects of musical objects to be manipulated consistently.
Intervals, if you haven't figured out from the preceding paragraph, are not the same as frequency ratios, and they're not the same as "numbers of steps" in 12 tone equal temperament or other equal temperaments. The microtonal theorists who don't understand this are constantly saying nonsensical things, like that they have changed a frequency ratio into another one, or made two different ratios equal - as though rational numbers like 7/4 and 8/5 were clay in their hands to be molded and mixed. If this book achieves just one thing, I hope that it corrects the language of microtonal music theory to be free of arithmetical errors, simply by teaching the standard western music theory of intervals - and a small extension of this theory for representing music which is non-standard in the West.
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