## Limma fraction and apotome fraction accidentals

Dave Keenan
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### Re: Limma fraction and apotome fraction accidentals

[This was originally posted in the Microtonal Music and Theory facebook group in response to Cryptic Ruse's post (I can't type or pronounce the name he goes by now). It is preserved here at the request of Joe Monzo. Thanks Joe.]

Congratulations on finishing the draft. It may well lead to improvements in Sagittal ET notations. I'd be happy to wait for a more polished version, but hopefully these comments based on your original post will help.

The Spartan subset of Sagittal, that consists of 8 pairs of simple symbols, presently notates 47 of the most commonly-used ETs from 5 to 72. Those that require additional symbols are 14, 20, 30, 32, 37, 44, 47, 48, 49, 52, 54, 55, 58, 59, 60, 61, 66, 67, 68, 70, 71. With some notable exceptions, most of these ETs have rarely been used by anyone, except possibly in exercises of the kind: "I'm going to prove I can write something even in this awful ET".

I assume you mean your notation is "ASCII-friendly" in the way that Sagittal is ASCII friendly, and not in the way that HEWM is ASCII-friendly. i.e. I hope you just mean it can be unambiguously represented in ASCII, and you're not proposing to use ASCII symbols on the staff. HEWM's use of + and - is particularly problematic, unless they are heavily modified so that the horizontal strokes become Bosanquet's comma slashes, as in Erv Wilson's version.

I understand your point about those ETs that are inconsistent regarding certain combinations of primes, and so may in some cases have multiple usable mappings.

When standardising Sagittal notations for ETs we deliberately tried to avoid using symbols for sets of primes that were not mutually consistent. So Sagittal already attempts to be agnostic about the mappings of primes which could map in multiple ways. We may well have failed to do so in some cases. We'd be grateful if you could point these out, preferably on the Sagittal forum where they will still be findable in weeks (and years) to come, unlike these ephemeral facebook discussions.

I'm surprised that you include 72-ET in this class of ETs. I would have thought there was no argument about its 11-limit mapping, combined with the fact that 11-limit symbols are quite sufficient to notate it.

It seems to me that when an ET _does_ have a definite best mapping for some small set of low primes, then it is not an advantage, but a disadvantage to fail to indicate it in the notation. Being agnostic in these cases seems to me to just be ignoring the facts. However, even if there is a good reason to ignore such facts, you are free to do so no matter what the notation is. You are always free to ignore the fact that a certain symbol has a certain meaning when notating JI, and simply treat it as the symbol for so many degrees of that ET.

You claim to be "drastically reduc[ing] the number of accidentals that must be learned". Firstly I would say that since only 8 pairs of Sagittals are required to notate 52 ETs, including the most popular by far, reducing from 8 to 6 is hardly "drastically reducing". But it's worse than that.

Since you're not notating JI (or large ETs) you're not actually reducing the number of symbols to be learned at all. You're increasing it. Unless of course you're repurposing Sagittal symbols, which would be fine if it's done in a logically consistent manner, or unless the user of your notation can guarantee they will never in their life need to learn a JI (or large ET) notation.

There is no "ambiguity or uncertainty in using Sagittal to notate ETs ... that support multiple useful mappings within a variety of prime limits". In Sagittal, the ET composer is not required to "determine which mapping to use or what limit the temperament is conceived as tempering". She can simply look up the standard notation for the ET.

Of course Sagittal also ensures that all pitches can be notated in alphabetical order (with enharmonic spellings), and that no more than one accidental symbol (two if you want to keep existing sharps and flats) will ever be necessary to notate a given pitch (however you're free to use more, e.g. in one-symbol-per-prime JI notation).

And of course existing Sagittal ET notations also respect subsets such as 17 and 34, since accidentals are defined based on harmonic relationships rather than ET steps. The ultimate example is the mutual consistency of all the 12n-ET notations. But if you can find subsets we've missed, please let us know.

Dave Keenan
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### Re: Limma fraction and apotome fraction accidentals

Here's a thought, in the spirit of brainstorming. In a fractional-apotome-fractional-limma type notation, for badly-behaved ETs, we might repurpose the otherwise-very-rarely-used 19-schisma flag to indicate that we are using such a notation.
I earlier suggested the following symbols.
apotome          limma
full     sharp           paosharp
half     pakai or vai    phai or fai
quarter   tai             pai
But it was rightly noted that their meaning as ratios of 5, 7 and 11 are rather too well known and would be misleading.
So one possibility is
apotome            limma
full     sharp            no accidental, change of nominal (as pointed out by Cryptic Ruse)
half     prakai or vrai    phrai
quarter  ratai              prai

Dave Keenan
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### Re: Limma fraction and apotome fraction accidentals

Of course you can use the standard sharp and flat for the apotome too.

ASCII for those could just ignore the left scroll and make it clear in the context, that you're using fractional 3-comma notation, and use the following:
.
apotome    limma
full    # b        no accidental, change of nominal (as pointed out by Cryptic Ruse)
half    ^ v        = _
quarter f t        / \

cryptic.ruse
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Joined: Tue Sep 22, 2015 4:38 am

### Re: Limma fraction and apotome fraction accidentals

I can dig it. The ASCII symbols I was using differ only on the 1/4-apotome [I was using / for raising, \ for lowering] and the limma accidentals [I was using > for raising by 1/2 limma, < for lowering, and ) for raising by 1/2 limma, ( for lowering]. Of course, I'm not married to these symbols, and the parentheses in particular can be problematic for plain-English writing. I recall George suggesting some limma accidentals in a facebook comment that used parentheses and vertical lines of some sort, and I might prefer those over the = _ pair, if only because those two symbols might have a slightly higher chance of confusion.

I'll dig into the Sagittal symbol set and see if I can't find some alternatives worth considering, but otherwise these should work nicely. I appreciate the suggestions! I think the 19-schisma flag to denote the difference is not a bad idea, though in practice I would probably also issue a written warning just to be safe.
Last edited by Dave Keenan on Thu Oct 06, 2016 8:17 pm, edited 1 time in total.
Reason: Accidentally hit the edit icon instead of the quote icon. Sorry. The pitfalls of being admin. I put it back how it was.

Dave Keenan
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Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
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### Re: Limma fraction and apotome fraction accidentals

I'm glad you like the symbols, Igliashon. The more I look at them, the more they makes sense to me too. In one sense they are the most common symbols used in Sagittal, the same Spartan symbols used for 11-limit JI-based notations for EDOs, and so they have the same size order, and the same approximate size when the fifths are near just. And yet, by having that extra little left scroll they cannot be confused with them. The existing higher-prime-comma meanings of these left-scroll symbols are very rarely used because they notate complex ratios. and are way up in the Promethean (high precision JI) extension. is in the Trojan (12n-edo) extension, but it is only used to notate 108-edo and 120-edo.

And the more I investigate your idea of apotome and limma fraction notations for small EDOs (particularly those with bad fifths), the more I love it. I think it was brilliant of you to think that it might be possible to do it with only four symbol pairs in addition to conventional sharps and flats -- half and quarter apotome symbols and half and quarter limma symbols.

I would not have thought it possible. But since you told us that it is, I'm guessing this is how you do it (although you would be using ASCII placeholders rather than Sagittals). I've shown the compound accidentals in the left-right order they would appear on the staff:

1/5, 1/6, 1/7 apotome	1/4, 1/5 limma
1/4, 1/3, 2/7 apotome	1/3, 2/5 limma
2/5, 3/7 apotome	1/2 limma
1/2 apotome		2/3, 3/5 limma
3/5, 4/7 apotome
3/4, 2/3, 5/7 apotome
4/5, 5/6, 6/7 apotome
1 apotome
As it happens, there are two other Promethean-extension symbols with left scrolls, and , that would allow us to extend the system as follows:

1/8, 1/9, 1/10 apotome,		1/7 limma
1/5, 1/6, 1/7, 2/9 apotome	1/4, 1/5, 1/6, 2/7 limma
1/4, 1/3, 2/7, 3/10 apotome	1/3, 2/5, 3/7 limma
2/5, 3/7, 3/8, 4/9 apotome	1/2 limma
1/2 apotome			2/3, 3/5, 4/7 limma
3/5, 4/7, 5/8, 5/9 apotome	3/4, 4/5, 5/6, 5/7 limma
3/4, 2/3, 5/7, 7/10 apotome
4/5, 5/6, 6/7, 7/9 apotome
7/8, 8/9, 9/10 apotome
1 apotome
The 1/8-apotome symbol allows us to notate EDOs with up to 10 steps per apotome, the smallest being 71-edo, where the previous set only allows up to 7 steps per apotome, the smallest being 42-edo. But I freely admit that whose long ASCII is )~| , is very hard to distinguish from whose long ASCII is )/| .

The 3/4-limma symbol allows for some enharmonic spelling in the limma-notated EDOs 23, 28 and 35, and it replaces in apotome-notated EDOs and thereby eliminates one case of accidentals altering in opposite directions.

Dave Keenan