Naming Rank-2 Scales
Mike Battaglia Method
See Modal UDP notation.
Kite Giedraitis method
Mode numbers provide a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like Modal UDP notation, it starts with the convention of using some-temperament-name[some-number] to create a generator-chain, and adds a way to number each mode uniquely.
Jake Freivald method
My goals for numbering the modes are to make it as simple as possible for people to identify and use the modes they're talking about. As such, desired characteristics include
(1) as little knowledge needed as possible, to help the less-sophisticated user,
(2) reasonably intuitive if possible,
(3) easy to remember and check your own work, and therefore
(3a) biased toward major being the "right" answer for meantone, and
(4) extensibility of the method beyond MOS.
I've created a method that uses step sizes, generally identified as s for small, M for medium, and L for large. (That would need to be extended for scales with more than three sizes of steps, of course, but the principle remains the same.)
Once a mode 1 has been identified, each mode is counted up by steps from the root of mode 1.
For example, using my method starting on C for meantone:
mode 1 is C major (LLsLLLs)
mode 2 is D dorian (LsLLLsL)
mode 3 is E phrygian (sLLLsLL)
mode 4 is F lydian (LLLsLLs)
mode 5 is G mixolydian (LLsLLsL)
mode 6 is A minor (LsLLsLL)
mode 7 is B locrian (sLLsLLL)
We'll start with MOS scales. For MOS scales, we'll only have s and L steps sizes.
Inversion seems important for intuition. If 5L+2s is LLsLLLs, 5s+2L should be ssLsssL.
What if the algorithm were something like this:
Start with whatever you have more of, L or s. Put the smallest cluster of those you can at the beginning. Then alternate, using the smallest clusters of steps you can until it's done.
For 5L+2s, which is some rotation of LLsLLsL, I know I have to start with L. The smallest cluster I have is LL, so I get LLsLLLs.
For 1L+ys where y>1, I know I start with s, which means all of them go at the front and L goes at the end: s..sL. xL+1s is L..Ls.
For 4L+5s, which is sLsLsLsLs, it seems a little more complicated -- but not much. 5>4, so I have to start with s. I see one place where there are two s's together, so that has to go before the last L: sLsLsLssL
For 9L+4s, which is sLLsLLsLLsLLL, we start with L. The longest string of L's I have is three, so mode 1 is LLsLLsLLsLLLs.
For 7L+8s, which is sLsLsLsLsLsLsLs, we start with s and put the string ss just before the last L: mode 1 for this structure (e.g., for porcupine) is sLsLsLsLsLsLssL.
For an MOS like 3L+3s, make it as much "like meantone major" as you can: L to start, and a small leading tone: LsLsLs.
Note the things I *don't* need to know to do this: I don't have to know what a generator is, what mappings are, what utonality or otonality is, or a host of other things. I won't get confused by seeing intervals that don't map to JI well (e.g., phi). This is, in fact, just basic string manipulation.
I also have built-in checks: I know that if I start and end with the same step size that I'm doing something wrong, and using the technique for meantone gives me the diatonic major scale LLsLLLs, or CDEFGABC.
Extending to non-MOS
My suggestion is that (a) you still start with the step size that occurs the largest number of times, and (b) you still push the largest cluster of that as far out as possible
If the scale is made of xL+ys+zM, and any two of x, y, or z are the same, and they're larger than the third number, then the scale starts with the larger of the steps -- just like meantone major starts with larger steps. In other words, if it's 3L+3s+2M, we start with L. If it's 2L+3s+3M, we start with M.
(Note that the word "scale" is ambiguous in some of what follows. I don't think we'll get away from that in real life, so I'm just pushing on, even where it sounds weird.)
I'm going to start with some of the scales Kite has already used on the wiki page he created.
The harmonic minor scale is already structured this way: A B C D E F G# A is MsMMsLs. There are the same number of M and s steps, so M goes first. The one-step cluster of M goes first, and the two-M cluster goes second. The harmonic minor scale is mode 1 of this scale. The phyrigian dominant scale, which features in a lot of world music (I think of it as the "Hava Nagila scale"), is mode 5 of this scale.
Melodic minor ascending is also already structured that way: A B C D E F# G# A is LsLLLLs.
Double harmonic minor (never heard of this -- I learn something new every day on this list) is A B C D# E F G# A, or MsLssLs. Mode 1 will start with a small step and have the largest cluster of s's last in the scale: sMsLssL. Double harmonic minor is thus mode 2 of this scale.
Double harmonic major (never heard of this either) is A Bb C# D E F G# A, or sLsMssLs. Start with the smallest cluster of small steps, and this has to be sMssLssL. Double harmonic major is mode 7 of this scale.
Hungarian gypsy minor is A B C D# E F G A, or MsLssMM. We have the same number of s's and M's, so we start the mode with M. After that decision, there are no more to be made: It's MMMsLss. Hungarian gypsy minor is mode 3 of this scale.
What Kite calls "a pentatonic scale" on the wiki page he just made is C D E G A#, or LLssL. That would be LLLss, and Kite's pentatonic would be mode 2 of that scale.
None of these scales have had a problem that I'm about to address and resolve. To wit:
Let's pick a rank-3 scale: 1/1 - 9/8 - 5/4 - 4/3 - 3/2 - 5/3 - 15/8 - 2/1. (Note that this is not a temperament at all.) That's LMsLMLs. With three L's and only two M's and s's, L has to go first. But there are no larger or smaller clusters of L's! They only come one at a time. So let's pick the mode that has the longest string of non-L's to go first. (That pushes the last L out as far as it will go.) This gives us LMsLMLs, which is our original scale, which is also a convenient touchpoint, in my opinion.
This also works for Kite's question about meantone, which is LMsMLLML. There are more L's than any other step size, so the scale has to start with L. The L's are in equal clusters of two, so there's no obvious way to pick which one goes first: mode 1 must start with LL. So let's again pick the mode that has the longest string of non-L's to go after that first LL: Mode 1 of meantone is LLMsMLLM. (That's C - D - E - F - F# - G - A - B - C, or something like it.)
Let's try something harder: the rank-3 scale minerva, which I found through Graham's temperament finder. Since there are four step sizes, I'm going to label them LMms, where the capital M is larger than the small m. With steps of 113, 113, 87, 113, 87, 99, 113, 87, 113, 87, 113, and 73, that's LLmLmMLmLmLs, 6L+1M+4m+1s. L has to go first. There's a cluster of two L's in the string, and I push that as close to the end as possible: LmMLmLmLsLLm. Graham is showing mode 10 of this scale in his temperament finder.
NOTE: NO collapsing genchains. NO generator knowledge needed. No mapping knowledge (or indeed mapping at all) required. Extensible to higher ranks without problems. It doesn't matter whether the scale is a temperament at all.