Given that an Octatonic Scale has 8 notes, what are the names of the scale intervals e.g. for a diminished scale?

Question

Assuming the C° (half/whole) scale (C D♭ E♭ E G♭ G A B♭ C) comprises the intervals P1, m2, m3, d4, d5, d6, and m7, what is the correct name/interval descriptions for B♭ & C (the 8th & 9th tones)?

This question is about identifying intervals. It is not unrelated to the question on notation. However, it does not ask about notation.

I am grateful in any event for the answer provided.

@Richard Got it, I think! However my original listing (P1, m2, m3, M3, d5, P5, M6, m7, P8) omits a 4th.

As per your suggestion: C° (half/whole): C-D♭-E♭-E-F#-G-A-B♭-C (P1-m2-m3-M3-A4-P5-M6-m7-P8), so d5 may be become an A4.

Many thanks!

Answer 1

I think you might be conflating intervals with scale degrees. Count intervals as notes above a pitch, not as members within a scale.

Your interval listing should be P1, m2, m3, M3, d5, P5, M6, m7, P8.¹ Let's look at our first disagreement: for C to E, you have d4 and I have M3. C to E must be some type of third, because counting up from C we get (C–D–E = 1–2...) three. It doesn't matter what scale we're in; even though this E might be the fourth scale degree, it's still a third above C, and thus the interval will be understood as a (major) third.

As such, the B♭ and C up top will be a standard m7 and P8, even though they are scale-degrees 8 and 9 (or 1), respectively.

¹And technically, your scale is better written as C–D♭–E♭–E–F♯–G–A–B♭–C. This is so that we have at least one of every note name, instead of having two types of E, two types of G, and no F (as you had). See also Notating the diminished Scale