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EDOs with multiple prime mappings - Page 3 - The Sagittal forum

EDOs with multiple prime mappings

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Re: EDOs with multiple prime mappings

Postby Dave Keenan » Wed Oct 05, 2016 7:11 pm

Last edited by Dave Keenan on Sun Oct 09, 2016 9:11 pm, edited 1 time in total.
Reason: Added a correction where I mistakenly equated the 5-comma and the half-apotome in 22-edo. It's really 1/3-apotome.

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Re: EDOs with multiple prime mappings

Postby Dave Keenan » Thu Oct 06, 2016 1:14 am

One of the things I'm struggling with here, in regard to revising some of the standard notations to use 3-limit fractional comma notations, is how and where to make the transition from the JI-based notations (primes greater than 3 allowed, fractional commas not allowed) to the apotome-fraction notations.

The further transition from apotome-fractions to limma-fractions seems obvious enough: Only use limma fraction symbols for EDOs where the apotome is zero or negative (6, 11, 9, 16, 23, 7, 14, 21, 28, 35).

Ideally such a transition (JI to fractional-3) would somehow be gradual and seamless. It's difficult to imagine how that could happen, but it could be that the maximum prime allowed could be stepped down gradually. In any case it's worth keeping the idea of a smooth transition in the back of our minds in case something occurs to us.

But whether the change is gradual or abrupt, it's not totally clear to me what property of the EDO should determine the JI-ness or otherwise of its notation. Our first pass at this was that it would be determined by the size of the EDO -- large EDOs have JI Notations, small EDOs have fractional-3-limit notations. But that doesn't work because I think 22-edo and 29-edo should definitely keep their existing 5-limit notation, while 42-edo and 47-edo don't deserve anything above the 3-limit.

People may of course choose to use a fractional 3-limit notation for 22 and 29 -- they both divide the apotome into 3 parts -- but this would not be their standard notation.

So my current suggestion is to base it on the error in the fifth. Error greater than 10 cents gets a fractional-3-limit notation. The 10 cent cutoff seems somewhat arbitrary. Reducing it to 9 cents would add 26 and 52 on the narrow side and 27, 54 and 59 on the wide side, as EDOs requiring fractional-3-limit notations. With a 10 cent cutoff, 47-edo has the greatest number of steps to the apotome (7 steps). With a 9 cent cutoff, 59-edo has 9 steps to the apotome, requiring more fractional-apotome symbols. One might argue that 26-edo is a meantone and so deserves a JI-based notation, however, like 12 and 19, it only has 1 step to the apotome, so its standard notation is the same whether JI-based or fractional-apotome.

I toyed briefly with the idea of making the notation fractional-3-limit whenever the EDO was not 1,3,9-consistent, i.e. whenever 2 fifths are not the best approximation to 4:9. But that occurs whenever the error in the fifth is more than 1/4 of a step, and so it does not have any upper limit in ET numbers, and therefore no upper limit on the number of steps per apotome requiring symbols. So I currently prefer a fifth-error cutoff, and 10 cents seems right to me.

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Re: EDOs with multiple prime mappings

Postby Dave Keenan » Thu Oct 06, 2016 11:38 pm

I'm now favouring a 7.5 cent error in the fifth as the transition from JI-based to fractional-apotome for the standard EDO notations. That means fractional-apotome notations for these EDOs:
Narrow fifths: 47 40 33, 26, 52, 45, 64
Wide fifths: 71, 49, 27, 54, 59, 32 37 42 5 10 15 20 25 30 18 13 8 6

And fractional limma notations for these:
Very narrow fifths: 11 9 16 23 7 14 21 28 35
(In previous posts I mistakenly included 6-edo among the fractional-limma notations.)

The reason for the change is that I learned that we have suitable Sagittal symbols to notate up to 10 steps in the apotome (as required by 71-edo). See

Even if we had the symbols to do so, I wouldn't want to extend the apotome-fraction notation to EDOs with fifth-error any less than 7.5 c, because the next to be included would be 19-edo and its multiples (on the narrow fifths side) and and 22-edo and its multiples (on the wide fifths side). I feel strongly that they should keep their JI-based notations as their standard notation.

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Re: EDOs with multiple prime mappings

Postby cryptic.ruse » Sun Oct 09, 2016 1:16 pm

I have yet to catch up on your latest replies in this thread, as I've been traveling for business this weekend and have elected to focus on actually finishing the list of enharmonic equivalents in my proposal. I have finally completed it, although I confess much of it was typed out on airplanes, in the backseats of cars, and after long days. As such, errors are likely, as are minor inconsistencies in formatting. Hopefully they are few enough that they do not completely wreck the entire edifice. It was quite a slog!

I'll try it posted here enclosed in [ pre ] tags; if that does not display correctly, I'll render a .pdf and upload it.

Note that I include a symbol glossary for each ET, listing every symbol and the number of steps it maps to. For a select few ETs, I included an "extra" mapping that violates the formula, but which may be useful under certain circumstances nevertheless (usually for backwards-compatibility with smaller ETs). Nevertheless, here is every ET from 5 through 72, notated with five symbol pairs in total (counting the Apotome):
Apotome fractions:
Whole: # b
1/2: ^ v
1/4: / \
Limma fractions:
1/2: < >
1/4: ( )

I should note that this is not an exhaustive list of all enharmonic equivalents; whenever there are multiple accidentals, there are multiple ways to specify a single pitch with a single nominal. This makes it difficult to count all the combined accidentals that may crop up, since I have not exhaustively searched for circumstances when one spelling may be more "correct" than another (for instance, when one might wish to use F^^^ in place of F# in 22edo). I am not sure if you've spent much time exhausting the different ways Standard Sagittal accidentals may or may not combine in each ET.

Anyway, here it is!


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Re: EDOs with multiple prime mappings

Postby Dave Keenan » Sun Oct 09, 2016 3:36 pm


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Re: EDOs with multiple prime mappings

Postby cryptic.ruse » Sun Oct 09, 2016 3:59 pm


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Re: EDOs with multiple prime mappings

Postby Dave Keenan » Wed Oct 12, 2016 4:17 pm


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Re: EDOs with multiple prime mappings

Postby Dave Keenan » Wed Oct 12, 2016 7:06 pm

Some questions after having perused your notations:

1. (Mentioned above) When the limma is zero or negative, why not notate using apotome fractions only, instead of resorting to second-best fifths or subset notations?

2. When the apotome is zero or negative, why not notate using limma fractions only, instead of allowing :#: to lower pitch and :b: to raise it? That way madness lies.

3. Why use multiple symbols when a single symbol will do?

I expect your answer to all of these is that there aren't enough symbols, and you don't want to add any more, because the whole point was to minimise the symbol count.

But all three can be achieved without adding any more symbols, except in the case of six largish EDOs: 54, 59, 61, 66, 68, 71 (those with more than 7 steps to the apotome). We'll ignore those for the moment.

The algorithm works by first dividing EDOs into two classes -- those that have a positive apotome and those that do not. The non-positive-apotome class consists of the nine EDOs: 7, 9, 11, 14, 16, 21, 23, 28, 35. We notate positive-apotome EDOs using only apotome fractions and we notate non-positive-apotome EDOs using only limma fractions.

I expect you would have noticed that in Pythagorean tuning (3-limit JI), a limma is very close to 4/5ths of an apotome, which means a 1/2-limma is 2/5-apotome and 1/4-limma is 1/5-apotome. So when we're notating positive-apotome EDOs we repurpose the limma-fraction symbols as apotome-fraction symbols (using your ASCII placeholders):

e.g.
[Use the horizontal scroll bar at the bottom of the post to see the rest of 70-edo -->]

And when we're notating non-positive-apotome EDOs we repurpose the apotome-fraction symbols as limma-fraction symbols:

e.g.

Those with more than 7 steps to the apotome require one more symbol-pair -- call it the 1/8-apotome pair. I'll use curly braces as placeholders for them, } up, { down.

e.g.
[Use the horizontal scroll bar below to see the rest of 71-edo -->]

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Re: EDOs with multiple prime mappings

Postby Dave Keenan » Wed Oct 12, 2016 10:56 pm

Hi Cryptic. I expect you have a table like this that you have referred to while constructing these notations. This one shows EDOs from 5 to 100 arranged according to their steps per apotome (# or b) and steps per limma (B:C or E:F). Steps per whole tone (C:D, D:E, F:G, G:A, A:B) can be obtained by adding the preceding two figures.

I've coloured it to shown my proposed classes of EDO notation.

EdoNotationClasses.gif
[Edit: 66-edo should be blue on the diagram]
Last edited by Dave Keenan on Thu Oct 13, 2016 2:34 pm, edited 1 time in total.
Reason: Added a note that 66-edo should be blue on the diagram.

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Re: EDOs with multiple prime mappings

Postby Dave Keenan » Thu Oct 13, 2016 12:20 am

I see I'm in trouble, with purely apotome-fraction notations for 40 and 47. Not really a problem for mixed Sagittal, since I can use triple sharps and flats, but there are no equivalent symbols in pure-Sagittal. So a limma-fraction notation might be preferable.

Apotome fraction notations


Limma fraction notations

For 47-edo, I need to add 1/6-limma as one of the allowed meanings for the 1/4-limma symbol.


Perhaps all the EDOs with fifths narrower than meantone should have limma-fraction notations. So all the orange EDOs on the upper-right of my diagram would be coloured red, like those above them. Let's see if we can do 64-edo, which has 7 steps to the limma and 10 steps to the whole-tone. To do that, I need to repurpose the curly-brace 1/8-apotome symbols } { for 1/7-limma.

[Use the horizontal scroll bar at the bottom of the post to see the rest of 64-edo -->]

That doesn't quite get us there. We can do it if we add a 3/4-limma symbol. I'll use square brackets, ] up, [ down.

[Use the horizontal scroll bar at the bottom of the post to see the rest of 64-edo -->]

The 3/4-limma symbols ] and [ can also replace the combinations #< and b> as 3/5-apotome symbols.


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