# What is the size of a doubly diminished second

What is the size of a doubly diminished second? Let's take E# and Fb as an example.

One argument, courtesy @Divizna, is as follows. A doubly diminished n-th should be smaller by two semitones than the minor (or perfect) n-th, so the size of a doubly diminished second is -1 semitones because a minor 2nd is 1 semitone.

But how could the size of an interval be negative? What does that even mean? The above argument undoubtedly works for thirds, fourths, and larger intervals, but seconds?

Another argument is as follows. Per the above, a diminished second is of size 0, but then further diminution of the interval would have swapped the higher and lower tones, so you should increase the size, so the size is 1 semitone. For instance, E to Fb is a diminished second, but E# is higher than Fb (at least in 12TET, in other tuning system where the two tones do exist one might be able to compare their respective pitches to get either the same or a different result) while E is lower than F.

But the statement "E# is higher than Fb" is in reference to their respective pitches, or rather, frequencies, and if we instead consider the placement of these notes on the staff, the opposite would be true: E# is lower (in staff position) than Fb. Then the swap shouldn't have taken place, and we should just keep subtracting 1 semitone from the size of the interval, hence a doubly diminished second has the size of -1 semitone.

Yet another argument, assuming 12TET: E# and F have the same pitch (frequency), while Fb and E have the same pitch (frequency). So the interval between E# and Fb should have the same size as that between E and F, which is 1 semitone.

But enharmonic does not mean identical, and indeed there are case where it makes a difference. For exmaple, a C# major triad has an E#, not an F; E# to F is a diminished second, not a perfect prime. So it is not clear how this problem wouldn't arise in the above argument.

So, what is the size (how many semitones) of a doubly diminished second? 1? -1? Or are there any other possibility I've missed?

• Or, does a doubly diminished interval actually exist? We need an example, maybe in real music, or at least in note names, or is this absolute pure theory?
– Tim
Commented Feb 11 at 15:37
• I'll just add that this arose from the comments under this question, in case anyone wants to see the discussion. As I said there, a negative size means that the upper tone (Fb) ends up below the base tone (E#). Commented Feb 11 at 15:44
• @Tim I believe It is purely theoretical. I'm not really aware of any tonal context where E# and Fb appear together or in quick succession. Commented Feb 12 at 2:18