One rule is that if you have an odd scale degree in a name, like in ".m13", and it is natural, rather than sharpened or flattened, that implies that every other odd scale degree below it is also present (as a natural degree, or a sharpened degree, or a flattened degree, but it has to be there in some form). The chord .m13 happens to have these scale degrees: (1 b3 5 b7 9 11 3). If you add the 13th degree to a .m7 chord, that's not a .m13 chord, it's a .m7add13.
Actually, the rule is a little more complicated than "write the highest odd natural scale degree for which every lower odd scale degree also appears". One quibble is that you would still write a 7 at the end if there were a suspended 2nd or 4th instead of a 3rd. For example, the chord .maj7sus4 doesn't have a 3rd, but we still write a 7 as if the (1 3 5) scale degrees were all there. This isn't as simple algorithmically as writing .sus4add5b7, but it is a common way of writing chords for musicians, and I want my system to look normal to musicians, and not just systematic. Another quibble is that the chord .7 in standard practice has a b7 scale degree, and chords with natural 7th scale degrees have names like maj7 (for a natural 3rd scale degree) and m-maj7 (for a b3). I didn't make this up. Don't blame me.
Another rule is to list altered scale degrees other than (3 5 7) only after the highest natural odd scale degree which you can write. So for example, the chord with scale degrees (1 4 5 b7 b9 13) would be called .7sus4b9add13. Let's walk through it piece by piece: "7" is the chord name for the flat 7th scale degree "b7". It's one of those complicating concessions I'm making to common practice. It would be easier to call it a b7. There are no natural scale degrees above 7 for which every lower scale degree is also present, so this is some kind of 7 chord, and not a 9 chord or an 11 chord or a 13. After that, we just walk up the chord degrees and we describe anything that hasn't yet been specified in the name: the weird 4th instead of a 3rd, the flatted 9th, and finally the added 13.
It would also be tempting to instead call (1 4 5 b7 b9 13) a .13sus4b9no11, but I've decided against using "no X" and "omit X" in my chord names because that would introduce an unavoidable ambiguity in what name some chords should have.
Here is a list of chord patterns in terms of their scale degrees: https://pastebin.com/H1pjhfr9. There are 135 of them! Not bad. It wasn't made by a program and certainly has a few errors, but it's pretty good. I'll verify all of the entries someday.
What can you do with it? Why you can voice chords! The simplest way is to pick a root pitch, and for every scale degree in the pattern, add a pitch to your chord voicing which is a corresponding number of semitones up from that root. What is the corresponding number of semitones for a scale degree? I'll show you! Here's a Python dictionary:
{"1": 0, "b2": 1, "2": 2, "b3": 3, "3": 4, "4": 5, "b5": 6, "5": 7, "b6": 8, "6": 9, "bb7": 9, "b7": 10, "7": 11, "b9": 13, "9": 14, "#9": 15, "11": 17, "#11": 18, "b13": 20, "13": 21}Let's do one together. For the chord quality .6add9, the pattern of scale degrees is (1 3 5 6 9). Translating this to a pattern of semitone intervals using the dictionary, we get [0, 4, 7, 9, 14]. Do you see? Like we know that a 9th scale degree is 14 semitones up from the root, because "9" maps ":" to 14 in the dictionary. If our root pitch is Bb2, then the pitches which are 0 semitones up and 4 semitones up and 7 semitones up ... are {Bb2, D3, F3, G3, C4}, which can be verified by sitting at a piano and counting keys or visualizing a piano in your head or doing some MIDI-like modular arithmetic. And now you know the pitches of Bb.6add9 voiced up from a root pitch of Bb2. You're welcome!
Really I just wanted to share my chord patterns. I'm proud of them and their naming scheme. I hope you find as much use for them as I have.
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