# 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
``````
• 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

...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.

• 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
• 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