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Warning: this is purely a theoretical question, most likely won't have any practical uses. The reason I'm asking is because I'm trying to express musical intervals programatically. Expressing them as semitones is not an option, as I want my interval representation to carry semantic information about what the intention with the interval was.

If my understanding is correct of intervals they can be either:

  • Perfect (unison, 4th, 5th, octave) - these can be diminished/augmented to express an interval one semitone up or down, or doubly diminished/augmented to express an interval 2 semitones up or down.
  • Imperfect (2nd, 3rd, 6th, 7th) - these can be either minor/major which have one semitone difference, or diminished/augmented which further move the interval up or down.

So far the most extreme interval alteration is doubly diminished/augmented.

Can we have triple, quadruple or even more diminished/augmented intervals? Does it make any musical sense to do this? Would any musician recognize such an interval (be that classical or other)? Are there practical examples?

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    Makes sense musically, but lacks a common symbol set. In microtonal theory triple and higher-order accidentals are used sometimes, see for example here systems beyond the reach of the 35 notes that can be achieved by combining 7 base notes with 5 accidentals (neutral=none inclusively): en.xen.wiki/w/Golden_meantone
    – Wolf
    Dec 9 '20 at 10:25
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    @Wolf thanks, gonna check this out, I haven't considered microtonal music, but let's see how it translates! Dec 9 '20 at 10:34
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    Nit#2: If your getting into terms like double augmented, why not use "imperfect" instead of "other intervals?" That would be the music theory term. Dec 9 '20 at 21:48
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    @MichaelCurtis haha I didn't know that's a thing, but now I do :) Dec 10 '20 at 21:56
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In theory, if you accept no limit to the multiplicity of chromatic alteration, there is no limit to the multiplicity of augmentation or diminution. So, for example, C-octuple-flat to E-septuple-sharp is a quindecuply augmented third.

In practice, you're never going to see that sort of thing except if someone is trying to make a point. Doubly augmented intervals are already fairly rare. Does it make musical sense to exceed that? In general, it does not, which is why you generally don't see such intervals.

As a programmer, and assuming that you're not targeting highly experimental composition, I would start with 35 possibilities as suggested by Wolf (A through G and double flat through double sharp). If you need to support larger chromatic alteration, you can add it later.

Do note that if you're planning to map the pitches to frequency ratios, and you're not using a twelve-tone temperament, identifying the interval quality isn't sufficient to identify the frequency ratios. For example, in 5-limit just intonation, there are two different major seconds, which implies at least three minor seconds.

So, while you may have 35 letter-plus-accidental notes, you'll have more pitches. Even limiting yourself to C major and only the seven letters with no accidentals, you need to have two different versions of A and possibly two different versions of D.

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they exist. For example the interval f-double-flat up to b-double-sharp is a 5x augmented fourth. You can’t tell by listening of course, and if you had to read it it would stop you dead in your tracks. As an aside, the composer Morton Feldman used multi augmented/diminished intervals because of the awkward feel it gave to the performance (e.g. successive melody notes b-sharp c-flat a-double-flat g-sharp)

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From a programming perspective, if you allow pieces to be transposed, I would allow for the possibility of creating "nonsensical" notes and intervals like triple sharps, even if one would normally want to rework them to more reasonable enharmonic equivalents before trying to perform them. If one has a piece of music which includes a sequence of chromatic key changes from A major to Bb, Cb, C, Db, and D major, and wishes to transpose it so that it starts in Ab major, performing such a transposition while keeping the same relative keys would yield the key sequence Ab, Bbb, Cbb, Cb, Dbb, and Db. Obviously one wouldn't want to try to perform a piece written in such keys; one would instead replace the keys with the enharmonic equivalents Ab, A, Bb, Cb, C, and Db. If, however, one were to subsequently transpose the piece to start on E major, the enharmonically-transposed form would become E, E#, F#, G, G#, A while the transposing either the original or the "nonsensical" form would have yielded the sequence E, F, Gb, G, Ab, A.

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  • Hey, thanks for the answer, I modeled my notes exactly like this, they can be sharpened/flattened to taste :) But we are talking about 2 different things, I'm ok with notes, intervals are the question. If we model it with geometry, I think of notes as points and intervals as vectors to translate them with on an 1d line (not sure if I make sense). Dec 10 '20 at 22:00
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    @BalázsÉdes: If the notes Abbb and B### exist, then the interval between them will be a sextuply-augmented second, rather than a minor sixth, since the interval "number" is determined solely by the letter part of the note name. Since A and B are consecutive, any form of B will be some kind of second away from any form of A.
    – supercat
    Dec 10 '20 at 22:15
  • @BalázsÉdes what precisely do you mean by "interval"? There are three distinct ways to to express intervals, two of which supercat has mentioned (half-step count or quality-plus-degree, as in "major second"). The third is as a ratio of frequencies. This third method is necessary because in some tuning systems, or really every tuning system other than equal temperament, any given quality-plus-degree interval can have more than one size.
    – phoog
    Dec 10 '20 at 23:06
  • @BalázsÉdes: If one regards notes as being a (letter name; half-step number) pair, then intervals can be treated a (difference in letter name; difference in half steps) pair. For example, if one selects middle C as being letter 0, half-step 0, then the F# above middle C would be letter +3, half-step +6, and the Gb above middle C would be letter +4, half-step +6, and the C## above those would be letter +7 half-step +14. The interval from the F# to the C## would be letter +4 half-step +8, i.e. an augmented fifth. From the Gb to the C## would be letter +3, half-step +8, i.e. a triply-aug. 4th.
    – supercat
    Dec 10 '20 at 23:16
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    @BalázsÉdes: Exactly, except that I wouldn't describe the difference as being "on the C major scale", but rather "[non-German] letter names, ignoring accidentals" (from what I understand, German terminology uses the name B to refer to what is elsewhere called Bb, and H to refer to what is elsewhere called B natural).
    – supercat
    Dec 11 '20 at 15:43
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...Expressing them as semitones is not an option, as I want my interval representation to carry semantic information about what the intention with the interval was.

I'm not entire sure what you mean. But in music the difference between mere semitone count and the "semantic" meaning of an interval, the musical meaning of an interval, comes from how the interval is spelled and placed within a tonality. This is not a theoretical nit pick, it has practical application and gets to the very heart of understand an interval in a tonal context, the "semantic" meaning.

But first the answer to your question: yes, you can have double diminished and augmented intervals. There really is much to it other than you can make such intervals and they will be impractical in most cases, because they could be rewritten enharmonically as something much easier to understand. You could have a double diminished fifth C G𝄫 but it is probably easier in most cases to call it a perfect fourth C F.

You can continue the concept with triple, quadruple, etc. But it will be the same situation as the doubled cases. It's not very practical.

How you account for these intervals depends what you are doing with your program. On the one hand this is all a numbers game of counting semitones and then mapping to the gamut of letters ABCDEFG using accidentals. It just a mix of counting bases, with a funny symbol set of ABCDEFG♮♯♭𝄪𝄫, otherwise it's just number and theoretically infinite.

About the "semantic" aspect and semitone count. You could have a programming object with properties or methods for expressing them, like interval.type(), interval.quality(), interval.spelling(), interval.semitones(), etc.

Possibly you could even have a function to get a kind of musical function if the interval was placed into a key. Give a key of C and an interval of A♭4 F♯5, invert it to a tertian stack of F♯ A♭ the root is an altered F, the "function" is an altered subdominant ♯iv. Such a function is tricky, for a number of reason, but it could be an interesting experiment. The important thing is something like this would be necessary to truly get the "semantic" meaning of an interval. Intervals don't have such meaning, at least in tonal music, until they are placed within a key.

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