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I know that starting from a perfect interval and shortening the interval by two semitones gives a doubly-diminished interval. But what about starting with a major interval?

For instance, starting with a major 6th, I have 9 semitones. If I shorten it to 8 semitones, I have a minor 6th, and 7 semitones is a diminished 6th. Would 6 semitones (3 less than were we started) be a double diminished 6th, or a triply diminished sixth?

In other words, is the "numeric name" of a diminished interval based on how many semitones it is below the "reference" interval (major or perfect), or how many semitones it is starting from singly diminished?

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    Why would that need to happen? Technically I don't think it would be necessary to move the same name note down that far.
    – Tim
    Jul 11, 2014 at 17:10
  • I have no idea why, but it is possible and I've seen references to it. For instance A# to Eb is a doubly-diminished 5th. I guess I'm just assuming that the same applies to Major intervals, and even if it's not commonly used, it is at least sensical on a technical level.
    – brianmearns
    Jul 11, 2014 at 17:14
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    It happens, though admittedly rarely, in order to indicate rarer intervallic functions, usually in a complex contrapuntal framework. For example, Ligeti has need occasionally in Musica Ricercata to indicate a resolution on Bb and E, so he precedes it with Cb and D#, which is a doubly-augmented second. People argue about whether it's appropriate, but some composers and editors think it is.
    – Pat Muchmore
    Jul 11, 2014 at 17:45

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Yep, correct. I think it's easiest to picture the perfect quality of 4ths, 5ths, 8ves, etc. as being taken over by two possibilities (M and m) in the other intervals. In other words, once you've compressed or expanded beyond the central quality(ies) ({P} or {m,M}) then the diminished and augmented stuff functions in the same way.

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