Is a ♭ 3 a minor 3rd, a ♭♭3 a diminished 3rd, and ♭5 a diminished 5th?

Question

I just need some clarification. Am I getting chords notated this way - 1-5-♭7-9-11, 1-3-5-♭7-♯11, 1-3-5-♭7-9-13, 1-4-5-♭7-9-13 - right?

    3   == M3
    5   == P5
    ♭7  == m7
    ♭♭7 == d7 (?)
    ♭5  == d5 (?) 
    ♯5  == A5
    ♯9  == A9

Answer 1

Every one of your intervals looks right, but I'll explain a little more. When people use numbers to represent notes like above, they always do it in reference to the major scale. So the interval they correlate to is:

1 - P1
2 - M2
3 - M3
4 - P4
5 - P5
6 - M6
7 - M7
8 - P8
9 - M9
10 - M10
11 - P11
12 - P12
13 - M13
14 - M14
15 - P15
...

The sharps and flats obviously alter the intervals in there respected ways with sharps raising the interval and flats lowering them.

For perfect intervals, the intervals go as followed when raised or lowered:

♭♭ - ♭ - natural -  ♯  - ♯♯
dd - d -    P    -  A  - AA

For major intervals, the intervals go as followed when raised or lowered:

♭♭♭ - ♭♭ - ♭ - natural -  ♯  - ♯♯
dd - d  - m -   M    -   A  - AA