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Every interval can also have an augmented (and diminished) version.

Is that true about augmented (and diminished?) unisons? As in C>C♯, and C>C♭? I ask, as C>D♭ is m2, as is C>B, but that's only the same in sound and one semitone distance between. Intervals have two criteria - note names and distance apart.

Edit - as requested. My question asked both about aug. and dim. unison. And - it's elicited some interesting thoughts - hasn't it?

Tim
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    Yes, C -> C♯ is an augmented unison, and C -> C♭ is a diminished unison. What is the question here? Melodic intervals also have another criteria: direction. – Elements In Space Feb 08 '24 at 12:18
  • @ElementsInSpace - question - do they exist? Is that what they're called given the examples in the question? Can't find them in my theory books. Don't comment - answer! – Tim Feb 08 '24 at 12:31
  • To some extent, it really is as simple as “augmented“ meaning “made bigger” and “diminished“ meaning “made smaller.“ So you can have “doubly augmented“ like C -> C##. There’s no reasonable use for “triple sharps,“ but if there were, I suppose there could be such a thing as “triply augmented.“ – Andy Bonner Feb 08 '24 at 14:07
  • @Andy Bonner I think Bbb - Cx would be a triply augmented 2nd? No need for a triple sharp! – Laurence Feb 08 '24 at 14:18
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    @ElementsInSpace The interval C - C♭ is an augmented unison. You can augment an interval up or down. In both cases an increase in distance has happened. Examples of augmented thirds: C - E♯ and C♭ - E. You can also dimish intervals up and down. Examples of diminished thirds: B - D♭ and B♯ - D. There is one interval which can not be diminished and that is a unison because you can not decrease the distance. – Lars Peter Schultz Feb 08 '24 at 15:35
  • One important point here is the why. We choose enharmonic spellings for a reason, often having to do with where the melody is going or what happened to a chord. I can imagine a reason to spell an augmented unison rather than a minor 2nd (e.g. one voice holds a long C; another starts on C and then moves to D by way of C#). But as many are pointing out by now, I can't conceive of a "diminished unison," since you can't "make it smaller." – Andy Bonner Feb 08 '24 at 15:38
  • @LarsPeterSchultz What name would you give to the interval between E# and Fb? And what size would you say the interval is? I dare you to find a different answer than "that's a doubly diminished second with a size of -1 semitone" that wouldn't lead to inconsistencies. (And no, don't tell me that doesn't exist. E# and Fb are both perfectly good notes, surely there is an interval between the two.) And if we can diminish a second to a negative size, why not a prime? The thing with diminished unison isn't that it's impossible, but that it isn't distinct from augmented unison. – Divizna Feb 08 '24 at 15:43
  • @Divizna OK I see your point. At least C to C♭ is clearly augmented. The size of E♯ - Fb? Good question. In this case I can see the point of talking about negative intervals. How would I play it on a violin? If the sequence was like this F♯ - E♯ - Fb - Eb I would probably play the E♯ rather sharp close to F♯ and the Fb rather flat close to Eb. Thus the distance between E♯ - Fb would be bigger than on a piano while the semitones F♯ - E♯ and Fb - Eb would be very narrow. But performance depends on the circumstances. I would probably choose to not bother about what to call the interval E♯ - Fb. – Lars Peter Schultz Feb 08 '24 at 16:19
  • Does this answer your question? Does a diminished first exist? – Aaron Feb 08 '24 at 16:50
  • @Aaron - I couldn't find reference to aug or dim so thought I'd combine them in the question. So, half, yes, half no. – Tim Feb 08 '24 at 17:15
  • @Tim Sorry to disappoint, I don't think it's elicited any interesting new thoughts for me. But when I noticed it was you asking (which was only after I posted my answer, oops), I was really wondering why, because I'm quite sure you didn't need to ask about this. – Divizna Feb 08 '24 at 17:28
  • @Divizna I would argue the size of an interval is not well-defined except by the method of its name, like augmented third, perfect fourth, doubly diminished second, etc. In all other ways, any possible inconsistency is due to some preconceived notion that the size you defined has to have some certain relationship with the standard names of the intervals (examples as above). – Divide1918 Feb 08 '24 at 17:30
  • @Divide1918 You mean it's only a preconceived notion that a doubly diminished n-th should be smaller by two semitones than the minor (or perfect) n-th? I'd rather think it follows straight from the definition of "diminished". – Divizna Feb 08 '24 at 17:38
  • @Tim Augmented unison is address in what's the purpose behind these alternate namings for intervals?. – Aaron Feb 08 '24 at 19:51
  • @Divizna By inconsistencies you actually meant that the size of the interval and the name of the interval has to correspond in a certain way, following certain rules. The problem is, you decided what those rules are, so that you can say that "doubly diminished second with a size of -1 semitone" totally makes sense, and that any other possibility is inconsistent. But for instance why is "-1 semitone" considered consistent then? This looks rather arbitrary to me, as if you picked the set of rules that would grant your version as the only "correct" version. – Divide1918 Feb 09 '24 at 12:56
  • @Divizna "A doubly diminished n-th should be smaller by two semitones than the minor (or perfect) n-th" does follow from the definition of diminished intervals for thirds and larger intervals, but not unisons or seconds. Once you get to size 0, further diminution of the interval would have swapped the higher and lower tones, so you should increase the size, unless you want to include the direction of the intervals, in a way that a perfect fourth might have a size of either 5 or -5 semitones. – Divide1918 Feb 09 '24 at 13:11
  • @Divide1918 "By how much is the upper of the pair higher than the lower?" Okay, first of all, which of the two is which? Pretty straightforward when it comes to the minor second between E and F. E is lower, obviously. But whoa, when we're dealing with E# and Fb, which of the two is lower now? Conceptually, E# is lower - that's why you find it on the first line and Fb in the space above it. But pitch-wise, it's higher than Fb. Assigning a negative size to the interval allows to neatly describe this bizarre trait. E# is lower, Fb is upper and it's higher by -1 semitone. – Divizna Feb 09 '24 at 14:00
  • @Divizna "Conceptually, E# is lower" How? Their placement in staff notation also does not qualify as any justification to me. E is lower than E#, but they are on the same place in staff notation. Referring to pitch seems like a more reasonable conceptual way to compare E# and Fb to me: in 12TET, E#=F, Fb=E, and F is higher than E, so E# is higher than Fb. Nevertheless I would love to hear more details on your conceptual way of comparing E# and Fb, that is, what does "conceptual" mean exactly, and how does that apply to our present case? – Divide1918 Feb 11 '24 at 11:16
  • @Divide1918 For the same reason why you won't see Db in the A chord, A# in the scale of D minor, why we have distinct terms for major third and diminished fourth, and the E# note even exists. Because enharmonic is not considered identical, and this is really not something that I'm making up. I suggest you post your puzzlement as a question; there are people around who are much better able to give you an in-depth explanation (or point you to some good resource) than me, and I really don't like being singled out as the one person who has to personally provide it. – Divizna Feb 11 '24 at 12:15
  • @Divizna I agree that enharmonic != identical, so in that light I would have to clarify my use of "=", which only means that the two tones have the same frequency in 12TET with reference to the same reference pitch (e.g. A4=440Hz). The spelling of the note may be different in different context, but under the above tuning assumption, they are the same frequency. And the terminology of M3 vs d4 comes from a diatonic tonal context; as in one of the answers to this question, in a fully chromatic context the two intervals have no difference. – Divide1918 Feb 11 '24 at 12:37
  • @Divizna That being said, thank you for your suggestions and input thus far. – Divide1918 Feb 11 '24 at 12:46

3 Answers3

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Augmented unison is definitely a thing. C and C#. Nothing weird about it.

Diminished unison, well... would be kind of the same as augmented unison, only you'd think of the two tones in reverse - instead of "from C, a semitone up to C#", or even "from C#, a semitone down to C" (which is still the same augmented unison), you'd now have "from C#, a negative semitone up to C".

Divizna
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Sure, the system extends to unisons. An augmented unison is straightforward, you just decide to spell a semitone as C to C♯ rather than C to D♭. The diminished unison may be more of a theoretical entity, it's difficult to see why anyone would want to see C - C♭ as a diminished unison UP - but the system allows it.

(Possibly - just possibly - this sort of analysis might open the door to new and interesting music. Sort of like the square root of -1 in mathematics and physics. Unreal, but useful. I'm listening...)

Other opinions are available.

"Some theorists do not allow the diminished unison because the C flat lies below the C natural and this breaks the[ir] rule that all dyadic intervals are named from the lower note."

https://simple.wikipedia.org/wiki/User:Sonia/Diminished_unison

Laurence
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    The interval C - C♭ is an augmented unison which should be rather clear I would say. Augment means "make greater" or "increase" and the interval C - C♭ is in fact bigger than C - C. I mean it doesn't matter whether you augment upwards or you augment downwards. – Lars Peter Schultz Feb 08 '24 at 16:27
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    @LarsPeterSchultz I HOPE you're just fooling with us! But as your comment got a +1, guess I'd better put you right. In music interval-speak, 'augment' means 'bigger upward'. – Laurence Feb 08 '24 at 20:31
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    No fooling here. I have never seen a music theory book that would restrict augmented intervals to only mean bigger upwards. That would make no sense. An augmented interval is defined as an interval that is one half step larger than a perfect or a major interval. The enlargement can be up or down. An example of an augmented third could be A♭ - C♯. On this particular augmented third you don't even know whether it is enlarged upwards or downwards. Like is it A♭ - C that is changed upwards to A♭ - C♯ or is it A - C♯ that is changed downwards to A♭ - C♯? It is an augmented third either way. – Lars Peter Schultz Feb 08 '24 at 21:30
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    @LarsPeterSchultz There is a problem of translation and language-dependent terminology, actually: In Czech for instance, the only equivalent of "augmented" is really "zvětšený" (literally "enlarged") and it always is meant upwards (unless you specify the direction in which you mean the interval). And we quite carefully distinguish "zvětšený" and "zmenšený" (literally "diminished"). I would never call C - Cb augmented, to me that's just simply diminished. – yo' Feb 09 '24 at 11:58
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    @Laurence Dimished first is not a theory, it appear in music already in baroque, and in a way even sooner, and it's the key ingredient to a descending passus durisucullus. – yo' Feb 09 '24 at 11:58
  • @yo' Ale nekecej. – Divizna Feb 09 '24 at 14:35
  • For the record, "zvětšený" is most often used for objects that have more than one dimension (for example, a photograph, or a snowball, or a lymph node) and the word generally implies that they're enlarged in all dimensions and all directions. What's more, the overviews I've seen about it always (AFAIR) explicitly stated that an interval can be augmented both up or down - and yes, they were written in Czech. – Divizna Feb 11 '24 at 12:44
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There are three things to state when describing an interval. Firstly, major/minor/perfect/diminished/augmented. Secondly a number. Thirdly a direction (up/down). So we can talk about a major third upward such as C upward to E, or a major third downward such as C downward to A♭. This allows us to think mathematically and describe a "diminished unison upward" such as C to C♭, but the interval C to C♭ would be better described as an augmented unison downward.

This is one of many contexts where we do not use negative expressions even though they might make mathematical sense.

Peter
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  • Generally speaking, intervals are worked out, thus labelled, from lower to higher notes. – Tim Feb 09 '24 at 10:31
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    @Tim, not if they are played consecutively. Then the labelling goes from the first note. Agreed if the notes are simultaneous though, in which case there is no need to specify direction. – Peter Feb 09 '24 at 10:34
  • @Peter Yes, but I would be inclined to not say "augmented first down" and prefer "dimished first" or simply "flatten" when talking about the situation :) – yo' Feb 09 '24 at 13:47
  • @yo', I would totally avoid using "diminished first", as it gives a negative outcome - the presumed upward interval is actually downward. If C-natural is followed by C-flat it is an augmented first downward. If you play C-natural and C-flat simultaneously it is an augmented first with C-flat as the base note. However I would prefer to call the interval a chromatic semitone. – Peter Feb 11 '24 at 09:09