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
  • 2
    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.
    – user50691
    Apr 10, 2020 at 15:45

2 Answers 2


...I'm curious if there is a general way to describe how the quality of the interval changes

I don't think there is a "trick" to just take qualities and "add" them to get a quality for the sum. m + m = d works for m3 + m3 = d5 and a bunch of other combinations, but not for m3 + m7 = m9 just as your chart demonstrates.

The musical way to do it is two calculations, the interval type, then the half step count...

m3 + m3 = ?

Interval type: 3 + 3 (-1) = 5

Half steps: 3 + 3 = 6

A5 = 8
P5 = 7
d5 = 6

m3 + m3 = d5

Getting the qualified interval is tricky, because theoretically you can augment/diminish intervals as many times as you want. (A double diminished unison isn't practical, but theoretically it perfectly fine to notate.)

You could have a half step table of practical intervals like...

A5 = 8
P5 = 7
d5 =6

...or you could make a kind of function for perfect and major intervals and then calculate minor, diminished, and augmented as a difference. A computer function might be something like...

P5 = 7
M3 = 4

...then m3 + m3 is handled as: 3+3-1=5 interval type 5, half step size 3+3 is 6, P5 size is 7, 7-6=1 for one "diminished", it's a d, a d5. m3 + m3 = C, Eb, Gb = d5.

m2 + m2: 2+2-1=3 interval type 3, half step size 1+1 is 2, M3 size is 4, 4-2=2 or just for major/minor 4-2(-1)=1, zero is m, one or greater is the number of diminished, one diminish in this case d, it's a d3. m2 + m2 = C, Db, Ebb = d3.

You would do a similar calculation for augmented but you need to juggle positive/negative, take absolute values, etc.

Double diminished, triple diminished, double augmented, etc. would be labeled like dd, ddd, AA, etc.

It's a pain in the neck to write out the exact steps because in essence the musical system combines the interval types from the diatonic scale of base 7 with half step counting which is base 12. Things are just not straight forward when calculating different bases. It's a bit like asking what is 1:00 + 2:30? 150 minutes... and maybe 3:30 on the clock face.


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.

  • 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, 2019 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, 2019 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, 2019 at 23:29

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