Rules for Adding Specific Intervals
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
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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.
...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.
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