## Limma fraction and apotome fraction accidentals

Dave Keenan
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### Limma fraction and apotome fraction accidentals

On the facebook "22-Tone workgroup", Cryptic Ruse mentioned that he was working on a non-JI-based general notation system for EDOs 72 and below, based on accidentals for fractions of the Pythagorean apotome (7 fifths, octave-reduced) and Pythagorean limma (5 fourths, octave-reduced), and asked what Sagittal accidentals would be most suitable for 1/4-apotome, 1/2-limma and 1/4-limma. The Sagittal apotome symbol is of course (or ) and the medium-precision 1/2-apotome is . My choice for a 1/4-apotome symbol is the 7-comma symbol because the exact relationship + = is highly suggestive of + = although this is only approximate in standard Sagittal.

With pure fifths, the difference between the apotome (116.7 c) and the limma (90.2 c) is the Pythagorean comma or 3-comma (23.5 c). This is symbolised exactly as in Sagittal, so the limma is exactly . But we wouldn't use diacritics for EDOs of this size. We'd just use for the limma. The 1/4-limma (22.6 c) happens to be between the 5-comma and the 3-comma in size, so we'd use the 5-comma symbol for the 1/4-limma, and the double-5-comma symbol for the 1/2-limma.
Summary
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apotome          limma
full     sharp           paosharp
half     pakai or vai    phai or fai
quarter   tai             pai

cam.taylor
Posts: 51
Joined: Thu Sep 03, 2015 11:55 am

### Re: Limma fraction and apotome fraction accidentals

Excellent. These were my picks for the limma-based and apotome-based accidentals too, but I wasn't 100% confident so I suggested Cryptic ask you or George. Good to see I was on the right track.

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

### Re: Limma fraction and apotome fraction accidentals

Very good. These do seem to make sense. Can they be written in single-character ascii? Relatedly, is there a resource that's basically just a glossary of all the Sagittal accidentals in full, mixed, multi-character ascii, and simplified ascii, as well as the ratios they represent? I haven't seen a comprehensive glossary like this yet.

I am a little bit concerned about using the 7-comma (64/63) for the 1/4-apotome. There are a few superpyth temperaments that need the 1/4 apotome accidental to be fully notated, and it may confuse some folks to use a symbol for a comma that vanishes. Specifically, 27, 32, 34, 37, 42, 44, 49, and 54 all have 64/63 vanishing on their best 7-limit mappings, and all would require a 1/4-apotome accidental. I think 20edo and 25edo also will have this problem. Being that fractional-apotome accidentals are primarily of use when fifths get sharper than 700¢, we are most likely to find them in ETs pushing into superpyth territory where 64/63 will vanish. Sometimes (as in 34) we can get around this by using the half-limma, but often we can't. So I would think a 5- or 11-limit comma would be a better choice, perhaps?

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

### Re: Limma fraction and apotome fraction accidentals

Hmm...looks like my comment didn't post. Apologies if it comes through and this post is redundant!

64/63 for the 1/4-apotome is problematic because the 1/4-apotome is very frequently useful for ETs in the superpyth family, i.e. where 64/63 vanishes under the best 7-limit mapping. This includes the following ETs: 20, 25, 27, 32, 34, 37, 42, 44, 49, and 54, as far as I can tell. I think a 5-limit or 11-limit comma might be a better choice if an appropriate one can be found.

William Lynch
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Joined: Mon Sep 21, 2015 9:27 pm

### Re: Limma fraction and apotome fraction accidentals

Note yet cryptic, trying to get there, once we get to a good point we can create an entire glossary on the xenwiki to notate every single tuning that we use in sagittal, how it all works, and how to type it in ascii. This is my goal here, I want to create such a guide once I gather enough information.

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

Sorry Cryptic. Every new member's first post is moderated, to stop spammers. And with me being in the eastern Australia time zone there can be long delays for western hemisphere first-time posters. I have just now emailed a potential western hemisphere moderator.

There is a glossary such as you describe, although it does not include symbols with diacritics. Its character code columns are out of date since SMuFL and the Bravura font. And some entries in its short-ASCII column are now out of date since George's recent Sagispeak post where we changed them to agree with the sagispeak pronunciation. It gives only the primary comma ratio and its prime-exponent-vector. Secondary commas for a symbol (such as the 13-M-diesis meaning of ) are really [Edit: not "really" but "often"] just primary comma meanings of accented versions of the same symbol.

It is called the "Sagittal-2 Character Map" and you will find it on the Sagittal website in 3 different formats.

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

### Re: Limma fraction and apotome fraction accidentals

Thanks Dave. I found the character map, but unfortunately you have all the ratios represented by accidentals given in name form, not ratio form, and I'm at a loss to translate their meanings without yet another glossary, as I am not familiar with all these names.

Are there accidentals for any of the following ratios? 36/35, 56/55, 54/55, or 78/77?

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

The ratios (numerator and denominator) and prime-exponent vectors are all there as well. I suspect you didn't scroll far enough.

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

Hi Cryptic,

Ratios for a few of the accented symbols can be found here:
http://sagittal.org/SagittalJI.gif
which you've probably seen.

But you have proven yourself worthy of access to this secret Olympian-level file, which gives every ratio that has an exact single-shaft representation in Sagittal. As with the other sources, some of the short-ASCII is now out-of-date.
http://sagittal.org/sag_ji4.par

Just joking about it being secret (but not about you being worthy). It's actually the file that Scala uses when you SET NOTATION SAJI4, and so it is included when you download and install Scala. It's not really intended for human consumption, but it is tab-delimited text.

cam.taylor
Posts: 51
Joined: Thu Sep 03, 2015 11:55 am

### Re: Limma fraction and apotome fraction accidentals

This is amazing. And should be wonderfully helpful.