Linear temperament notation diagrams

I've been looking at diagrams of linear temperaments with the X axis representing pitch and the Y axis the generator count. These are similar to some of Erv Wilson's generalized keyboard diagrams. It seems reasonable to use these diagrams as a blueprint for a notation system. The pitches form a regular grid that can be divided into a series of slanted columns. In each column, the pitch closest to the vertical center of the diagram can be given a name, and the other pitches in the column can be named by using accidentals. For example, here's a diagram based on the "lemba" temperament [<2, 2, 5, 6|, <0, 3, -1, -1|]:

http://www.io.com/~hmiller/png/pan-lemba.png

From the diagram you can see that there are five slanting columns in each period (half octave), implying a notation with 10 nominals per octave. You could also use the slanting "rows", which would be 6 per octave, or the diagonal between them (16 per octave). The spacing of the circles can be adjusted, producing grids with different numbers of columns, but there are certain spacing values that work best for each temperament (without leaving large empty gaps between rows or columns).

Since this is a temperament, you could notate it by using the just intervals: the 10-note MOS might be notated 1/1 16/15 8/7 5/4 21/16 7/5 25/16 8/5 7/4 15/8 2/1. But the problem with this is that a basic note like 5/4 would be notated using a pitch not in the basic scale, or even the larger 26-note MOS (32/27, which is -9 generators in lemba) plus an accidental. So a temperament like this could use its own notation. Lemba works well with the 26-note pan-alphabetic notation, since it's close to 26-ET. The 10-note MOS can be notated with 26-PAN as D L T U N O P I J R D. I'm using # and b as generic accidentals for convenience, but in most cases it would probably be better to pick an appropriate Sagittal accidental for each temperament.

This sounds like a nice plan in general, but there are problems, as illustrated with the "helmholtz" temperament [<1, 2, -1|, <0, -1, 8|] (i.e., 5-limit schismic).

http://www.io.com/~hmiller/png/pan-helmholtz.png

The notes of helmholtz[29] are labeled, along with some of the simpler JI equivalents. The obvious way to notate this is with the 12 columns. You could use the upward-slanting rows to label 5 notes per octave (D E G A C), but you'd need triple accidentals in some cases. And that's just for the 29-note MOS; the 53-note MOS would be worse. You could also use the downward-slanting rows, labeling the 7 notes D E F G A B C. This runs into the problem that D# is higher than Eb, Dx is higher than E, and you still need double sharps and flats in some cases, but that's probably a good enough compromise for this temperament if you don't need more than 29 notes. But look what happens if you want to label the 12 columns. The problem is that the middle column has two pitches at the same distance from the center line. So you either have to pick one of these arbitrarily and label it, or label both of them. Neither one of those seems like a particularly good option, so you're stuck with the 7-note option or the larger 17-note one.

But in addition to linear temperaments, this system can just as easily be used for tunings that don't have a specifically defined mapping of primes to generators, like Wilson's golden horagram scales. Some of those have two pitches that are pretty close to the same distance from a just interval, one slightly higher and the other one slightly lower; so they're like inconsistent ET's in that respect.