Why call a major second a diminished third?
In the Hungarian minor scale we have the following intervals, expressed here in semitones, between the scale's notes (I to VIII):
I II III IV V VI VII (I) / VIII
2 1 3 1 1 3 1
If we take the C note as the base, we have:
c d d# f# g ab b c
2 1 3 1 1 3 1
If we take the trichord starting at position 4, [F♯, A♭, C], we have a chord consisting of a major second (F♯ to A♭: 2 semitones) and a major third (A♭ to C: 4 semitones).
I've been told that this major second is actually considered to be a diminished third. What is the reason for not calling it a major second? Why not accept this chord as the secundal-tertian hybrid it seems to be?
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In a heptatonic scale, the interval between an arbitrary note and the note that is two steps further in the scale is always a third. The interval between any two adjacent notes is a second, and two seconds always make a third.
You wrote it yourself: it's the interval between F# and Ab. If it were a second, the Ab should be a G#. But the rule is that in a heptatonic scale, you get each note name (and its alteration) only once. That's also why you don't have a D# (as you wrote) but an Eb in the C Hungarian minor scale:
C D Eb F# G Ab B
It's because an interval is a measurement that is dependent on two parameters the letter distance between two notes and the semitones distance between two notes.
Any F to any A must be some kind of 3rd because the letter distance is that of a third. The quality of the 3rd is determined by the distance in semitones For F♯ to A♭ since the distance is 1 lower than a minor 3rd the resulting interval is a diminished 3rd. The naming schema of specific internals and the generalities of are shown in this answer.
Spelling matters greatly in music which is why intervals take letter name into account and this question and answer go more into depth on why it matters.
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