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In music theory when naming intervals, a lowered diminished interval is a doubly diminished interval (dd) and a raised augmented interval is a doubly augmented interval (AA). Ex a C# to a Gb would be a doubly diminished 5th (dd5) and a Cb to a F# would be a doubly augmented 4th (AA4). However, these intervals can be converted to their enharmonic equivalent so the make more sense so instead of having a C# to a Gb you could write it as C# to F# and have it be a P4 and instead of having a Cb to F# you could write a B to F# and have it be a P5.

I have never actually seen a naturally occurring doubly diminished or doubly augmented interval I have only isolated examples in theory books. Is there ever any scenario where any doubly diminished or doubly augmented intervals are used in melodies or chords?

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5 Answers 5

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The last piece in Ligeti's 'Musica Ricercata' is a good example of double diminished everything. But these intervals in the piece occur only because of poyphony. I'm quite sure that if you dig up some early or very late Shostakovich you'll be able to find examples of more harmonic and melodic (rather than polyphonic) use of double diminished intervals. Shostakovich is a very frequent user of diminished and double diminished intervals.

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Excellent, there they are! Including the inversion of the exact same interval I hypothesized about in my answer, Gb–C# doubly-augmented fourth implying resolution to F–D. Thanks so much for providing an example, I'm going to show it to my students next week! –  Pat Muchmore Feb 23 at 14:20

Doubly-diminished intervals will seldom occur in a piece whose tonal center remains in one key. Weird things can happen when pieces modulate, however, especially if the most "natural" way of writing the modulation would result in an excessive numbers of sharps or flats. Consider, for example, a piece of music that starts in E major and modulates upwards twice times by a whole step; each modulation is preceded by a V7 of the original key. If the second key is written as Gb major, then the old-V7 to new-I transition will be

    B  D#  F#  A
    Gb Db  Gb  Bb

The interval from D# to Db will be a doubly-augmented prime. If there were only one modulations, one could avoid the doubly-augmented prime by modulating into F# rather than Gb, but that require that one either modulate from F# to Ab (posing the same problem as modulating from E to Gb) or else modulate to G#--a key with eight sharps.

In general, harmonies are worked into key signatures in such a way as to avoid doubly-augmented or doubly-diminished intervals, but every once in awhile, key signatures need to "go around the bend" from lots of sharps to lots of flats or vice versa, and such transitions can make it necessary to use some wonky intervals. Augmented and diminished seconds, thirds, sixths, and sevenths are very common in such cases; even those cases don't often cause doubly-augmented or diminished intervals, however. Certain required modulations, however, can compel their usage.

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Db to D# is a doubly-augmented prime, not diminished. In fact, diminished primes are impossible, because moving either note is always expanding the interval, never contracting. –  Pat Muchmore Aug 2 at 21:42
    
@PatMuchmore: You're quite right of course (corrected I think). A transition from D# to Db could be a doubly-diminished or doubly-augmented octave, depending upon direction. I know I've heard music which modulated with an old-key V7 going to the tonic of the key a major second higher, so although the choice of starting key was contrived to cause "trouble", the chord sequence was "real". I found it surprisingly difficult to create any sort of doubly-augmented or diminished interval, since just about all the intervals which could get "double-whammied" were either major intervals which got... –  supercat Aug 3 at 15:36
    
...reduced twice (and thus became only singly-diminished) or minor intervals that got expanded twice (and thus only singly-augmented). I have actually encountered a doubly-diminished second (pitch moving the opposite direction of the staff note) in print (choral sheet music); there was a section of about eight bars that was in Cb, but two non-consecutive bars within that section notated pitches enharmonically as though it was in a "sharps" key. I'm sure it was "accidental", but it was definitely weird having the staff notes go one way and the pitches go the other. –  supercat Aug 3 at 15:42

Besides for user181381's excellent example, I was looking around last week and found that altered chords may have a doubly augmented or doubly diminished interval in them.

Let's look at a C7b5#9 and a C7#5b9

C7b5#9 - C  E  Gb  Bb  D#

C7#5b9 - C  E  G#  Bb  Db 

In the case of a C7b5#9, the interval between Gb and D# is a doubly augmented 5th and the Gb wants to go to an F(tonic) and the D# wants to go to an E(tonic's 7th aka leading tone). In the case of a C7#5b9, the interval between G# and Db is a doubly diminished 5th and the G# wants to go to an A(the 3rd of tonic) and the Db wants to go to a C(the 5th of tonic).

So these intervals just help lead back to tonic and function as an extension of a dominant chord.

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In standard functional harmony, diminished intervals only naturally occur between the 4th and 7th scale degrees, which is typically found in a vii° or V7 harmony. Double-diminished or double-augmented intervals don't occur anywhere in this system in its most basic version. Much of our Western music is based on this harmonic system, so intervals of this type are naturally much less common.

But the more inventive and expansive harmony becomes, and the more chromatic a piece of music is, the more likely you are to find intervals of this sort. Usually they arise out of linear motion than a vertical, simultaneous chord with a functional harmonic structure, though.

I recall seeing lots of chromaticism of this type (leading to the intervals you seek) when I was analyzing the music of Scriabin and Berg in school. That's a good place to start.

Unfortunately I can't think of any functional harmonic chord structure or progression that contains a doubly-diminished or doubly-augmented interval.

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I haven't seen them in an actual piece, but I'm hoping somebody else on here has, because I'd be very interested. However, I can think of a functional reason to use your example, C#-Gb; it would make sense as a chromatic dissonance that resolves to a D-F minor third. Doubly diminished sevenths such as C#-Bbb would imply "resolution" to a o5, in this case D-Ab. They should absolutely never be used unless there's a specific implied resolution like this, and there's a pretty strong case to be made that they shouldn't be used even then because they're just too unfamiliar to a performer. Personally, I think there's something to be said for truly showing the actual function despite odd intervals (for example, I often use augmented 3rds in my own music, implying an outward resolution to a P5), but I just don't know if I could bring myself to throw a doubly diminished or augmented interval at somebody.

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I've seen one somewhere, but can't remember where. Probably in some "innovative" 20th century music somewhere. –  BobRodes Feb 21 at 15:12

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