Is there a set of rules determining the quality after adding specific intervals?

For example, m3 + M3 = P5. It seems obvious that the number of the interval is addition but minus 1 because the bottom note is "counted twice", i.e. 3 + 3 - 1 = 5. But I'm curious if there is a general way to describe how the quality of the interval changes, (and why those rules would work). So I made a chart trying to find any patterns (I only did half of it since it should be symmetrical since order of addition shouldn't matter, and I left out the number because that is much more trivial to calculate).

   m2 M2|m3 M3|P4|P5|m6 M6|m7 M7
m2|d  m |d  P |d |m |d  m |d  P
M2|   M |P  A |P |M |m  M |P  A
m3|     |d  P |m |m |d  P |m  M
M3|     |   A |M |M |P  A |M  A
P4|     |     |m |P |m  M |m  M
P5|     |     |  |M |m  M |P  A
m6|     |     |  |  |d  P |d  P
M6|     |     |  |  |   A |P  A
m7|     |     |  |  |     |d  M
M7|     |     |  |  |     |   A
  • It's hard to know what you are trying to do. Why in the first place are you "adding" intervals? Does it have a musical motivation? Like building chords by stacking 3rds? If you can articulate the cause for this action it may be easier to describe a rule for it. – ggcg Apr 10 at 15:45

Looking at the chart it seems like M + m = P if the resulting interval is a 4, 5, or 8; = M if the result is greater than an octave and not a perfect interval; and = m if the result is less than an octave and not a perfect interval. M + M = A unless the result is 2 or 6, otherwise it is major. m + m = d unless the result = 2 or 6, otherwise it is minor. Lastly, any interval + P = quality of the first interval.

I might have made a mistake somewhere, and there might be a more concise way to say this as well.

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  • 2
    A second plus a fourth equals a fifth...which unfortunately can only be perfect, augmented, or diminished. This is despite a second being major or minor and a fourth being perfect. Ergo, your "any interval + P = quality of the first interval" rule is wrong in at least one place. – Dekkadeci Dec 22 '19 at 11:30
  • 1
    Also, where you say "unless the result is 2 or 6" what you actually mean is something more like "unless you add a second to a second or a sixth" as the resulting intervals are not seconds or sixths (but instead thirds and sevenths). In addition, the "any interval+P=quality of the first" fails for combinations of perfect intervals. P5+P5=M9, while P4+P4=m7. If you allow intervals greater than an octave to be added, things may get even more complicated. I don't know that there's a set of concise rules here that can cover all cases. – Athanasius Dec 26 '19 at 18:35
  • Yeah I think I've pretty much given up on finding a rule. It's fairly easy for me to imagine the C major scale and add them like that, in addition to the rule that the numbers are added. – awe lotta Dec 26 '19 at 23:29

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