Weird and missing notations in the pdf

[Dave Keenan]

What if we just said that double shaft symbols (and by extension X-shaft symbols) don't have fixed rational definitions, i.e. there are no fixed apotome-complementary pairs of symbols, and you can always just make them recapitulate the single-shaft flag combinations, like this:

												
1	2	3	4	5	6	7	8	9	10	11	12	13

What problems would that cause?

I think that's really going against the spirit of sagittal and a lot of things established so far. The one hard and fast rule in sagittal is that of apotome complements, and I see no reason to break it here. Even the Trojan double shaft accidentals where is used for 66.7¢ although - + = 83.3¢ and other similar quirks involving and are advertised on the PDF in all their glory, and even the notation for 72, which is one of the first things a newcomer is supposed to learn in sagittal, has inconsistent double shaft flag arithmetic, and 128 approximates the fifth with a worse relative error than 72. Also, 53's notation uses where no flag combination is reused and this does not prevent it from being a great notation.

Thanks for making me see sense.

I will note that George suggested using for 2\128, which makes at least 9 and 11 degrees use the same combinations as 2 and 4. Why didn't you choose it here?

I deliberately did not look at George's notation for 128edo while designing a 128edo notation as part of an overall magenta apotome-fraction system. Nor did I consider apotome complements. I see now, that versus is the only difference between George's notation and mine.

Here are the reasons I did not choose for 2\128:

1. When tempering 3's only, it does not map to 2/13-apotome across the whole range of magenta fifth sizes (not above 703.9 ¢). In particular, it does not map to 2/13-apotome in the case of 121edo which is just barely outside the magenta range and might usefully be included within it in future. is mapped to 3\121 by both the patent map and 3's-only.

2. In the overall magenta scheme, it seems better to reserve for slightly larger apotome fractions like 3/17, 2/11, 3/16 (0.167, 0.182, 0.188). You can see, in the nested arches of symbols above, that we have a size progression: ( )  . I have put brackets around because they reverse this order across a significant part of the magenta range. has the smallest slope of the two. You can see from the chart above, that the natural boundary between and in the magenta range is around 0.17-apotome. 2/13 is 0.154.

3. While not actually producing inconsistent flag arithmetic, the only way to make it consistent is to assign <> = 0 which implies that <> = = 4, when in fact 7/11C (the primary comma for  ) is 5/13-apotome across most of the magenta range (including 128 and 121edos). We'd need <> = 0 because we have = - ( - ) = 5 - (6 - 3) = 2. This is a fairly minor objection since no one is likely to notice this.

4. With instead of the flag-count is not monotonic with size. Again, a small disadvantage.

None of these are fatal to the use of for 2\128, but the only reason I see to prefer it over for 2\128 is to have one less flag combination in the revo flavour. But some people have no interest in the revo flavour.

Of course some people have no interest in evo. Should we recommend different evo and revo single-shaft sequences for the same tuning? I fear I may have set a bad precedent with 130 and 142edo in the pdf. If we don't do that, how should we decide?

I want to say that revo inherently requires a user to learn more symbols than evo, therefore if they are willing to take that hit, can't they take a little more for the sake of the simplest possible evo notation? Or putting it another way, since evo users likely choose evo for it's smaller learning curve, shouldn't we give them the simplest possible evo notation?

[Dave Keenan]

By the way, what do you think of this "evo" versus "revo" terminology? Is it too hard to remember which is which? Should we go back to calling them "mixed" and "pure"? One reason Douglas and I went to "evo" and "revo" was to avoid the implication that "mixed" Sagittal isn't really Sagittal.

Even though we will be retaining fixed definitions for double and X shaft symbols, can you see potential uses for the following additional double-shaft symbols as alternatives to existing single-shaft apotome complements?

Spartan:
= alternative complements of
Promethean:
= alternative complements of
= alternative complements of

Of course the above are just mock-ups. I would properly design the unified glyphs for these and their inverted and X-shaft counterparts and submit them for inclusion in SMuFL and the Bravura font, using 4 spare code points at the end of the Spartan multishaft section and 8 at the end of the Trojan section (immediately preceding the Promethean single-shaft section). The latter would be its own new section entitled "Promethean Sagittal extension (high precision) multi-shaft accidentals supplement". This is because there is no space at the end of the existing Promethean multi-shaft section.

These would be submitted at the same time as the new Stoic Sagittal symbols which will occupy 18 unused code points at the end of the Magrathean section.

[רועיסיני]

Re I see why you didn't choose it, makes sense. The flag arithmetic claim is a little bit weird to me because having = 4 and = 5 would mean a negative value for <>, which I think is even less desirable. The strongest reason IMO is the second reason you provided, although it still may be possible to use for 2/11, 2/12 and 2/13 of the apotome if it makes valid notations. Anyway I'm also fine with the notation you suggested, even though it has no repeating flag combinations in revo.

Re the names evo and revo, it indeed was confusing at first but now after I got used to it it's fine. I think more easy to remember names may be better, but mixed/pure indeed has unwanted connotations. If you'd ask me to invent names for them now, I think I'll choose multi-shaft for revo and multi-accidental for evo, which I think are self explanatory.

About the supplement multi shaft symbols, the only use I can see for them is for in the limma fraction notation, since that would make the double shafts reuse the same flag combinations as the entire single shafts instead of only a subsequence of them, but I'm not sure even that is a good use.

[רועיסיני]

I recently noticed that the "improvement" I suggested for George's notation of 147edo in this post is terrible and no one should use it: Both the values I suggested for 5 degrees and for 1 degree don't agree with the mapping the other symbols establish (which is also the obvious mapping).
About 5 degrees the only symbols whose primary commas round to 5 steps of 147edo when only prime 3 is tempered are , which as I already said contradicts the previously established mapping, and and , which are both 23-limit. I prefer since if it is chosen the desire to repeat the shape of single shaft symbols in the double shafts aligns with flag arithmetic to suggest the use of for 2 degrees, which is also approximated better in 147edo when tempering only the 3s than . About 1 degree I don't think I can easily improve over George's choice of so I suggest the following notation for 147edo:

													
1	2	3	4	5	6	7	8	9	10	11	12	13	14

About 135edo, 's primary comma indeed doesn't temper to 5 steps in the obvious mapping but the only one which tempers to 5 steps with both the obvious mapping and when only tempering the 3s is that of . However, it is really far off when only 3s are tempered as opposed to which is quite precise and also looks like the complement of , which plays really nice with flag arithmetic. The notation also works in 135f (13 has a relative error of more than 44% in 135) as opposed to the one derived from 270's notation which requires fractional mappings to make sense since the symbol for 5 degrees there is . also has the secondary comma of 37/36 which tempers to 5 degrees both when tempering only the 3s and with the obvious mapping, and therefore I'm happy with the notation proposed there.