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A proposal to simplify the notation of EDOs with bad fifths - Page 3 - The Sagittal forum

A proposal to simplify the notation of EDOs with bad fifths

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Dave Keenan
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Re: A proposal to simplify the notation of EDOs with bad fifths

Postby Dave Keenan » Mon Jul 16, 2018 3:52 pm

Here's another way of looking at the small-EDO notation problem. I've added lines of constant fifth-size. A JI-based notation can also be an apotome-fraction notation that's designed for fifths of a specific size (or a small range of sizes), so there's no real distinction between apotome-fraction and JI-based notations. Only the limma-fraction notations (the red region) are fundamentally different.

Small Edos 2.png

While apotome-fraction notations are possible for 6 8 13 18, and a limma-fraction notation is possible for 11, I think the default notations for those should be subset notations (of 12, 24, 26, 36 and 22 respectively).

For 9 16 23, I think sagittal.pdf should give (a) limma-fraction notations, (b) subset notations and (c) mavila-(linear temperament)-based notations, however I think the default notation should be limma-fraction, as it should be for 7 14 21 28 35.

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Re: A proposal to simplify the notation of EDOs with bad fifths

Postby George Secor » Thu Jul 19, 2018 3:02 am


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Re: A proposal to simplify the notation of EDOs with bad fifths

Postby Dave Keenan » Thu Jul 19, 2018 8:17 pm

So, to summarise: Of all those EDOs having fifths more than 7.5c wide, and therefore able to be notated using the above apotome-fraction notation, you find that 27 and 49 are not simplified thereby, and so should retain, as their preferred notation, their existing* JI-based notations.

I have checked your data and considered possible alternatives, and so far I agree with your choices. But it's a shame that, with these choices, there is no longer a simple description for which EDOs have preferred apotome-fraction notations. If we could find simple enough JI-based notations for 54 and 71, then the description would be simply "fifths more than 10c wide".

71-edo is 1:3:5:11:13:19 consistent. It has essentially the same fifth-size as 49-edo, so the size reversal of :|\: and :/|: is just as appropriate (or inappropriate). So what's wrong with this notation?
71: :|\: :)|: :/|: :/|\: :(|\: :(|): :||\: :(||~: :/||: :/||\: (JI-based), compared with
71: :)|: :)~|: :/|: :/|\: :(|\: :(|): :||\: :~||\: :(||~: :/||\: (apotome-fraction (with 13L instead of 13M))

*I agree with your suggestion that when a 13 diesis symbol is used for the half-apotome, it can be the larger one, symbolised by :(|\:, not :/|):. This is independent of the choice between apotome-fraction and JI-based notations. It affects not only 27-edo, but also the the JI-based notations for 51 68 75 (but not 45) and the apotome-fraction notations for 6, 13, 10, 20, 30, 37, 54, 71. My reason for preferring :(|\: now, is that since we defined it as the symbol for 13 in the one-symbol-per-prime notation, it more strongly suggests 13 than :/|): which suggests 35. But flag arithmetic should also be considered.

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Re: A proposal to simplify the notation of EDOs with bad fifths

Postby Dave Keenan » Fri Jul 20, 2018 12:05 am

54-edo is far more difficult to find a JI-based notation for. It is 1:3:7:11:13:17 consistent, but, as you say, the 7-comma is zero degrees and so is notationally-useless. So are the 17-comma and 17-kleisma, as they are the same size as the 11-diesis (3 degrees). At least the 13L-diesis is 4 degrees (half the apotome). And so the difference between primes 11 and 13 gives us one degree as 143C. But we have nothing consistent for 2 degrees.

The list of pathetic candidates for 2°54 are:
(accents would be dropped)
:)|):  7:19
:/ /|::'::': 5:13
:~~|:  11:49
:'::)|(: 11:35k
:/ /|::': 5*5*7
:.::|):  77

The least worst choice is probably :/ /|: as it is a Spartan symbol and is valid as two of its secondary commas, although not its primary.

54: :)~|: :/ /|: :/|\: :(|\: :(|): :)||(: :~||\: :/||\: (JI-based)
54: :)|::)~|::/|::(|\::||\: :~||\: :(||~: :/||\: (apotome-fraction (with 13M replaced by 13L))

One would have to define one's criteria for "simplicity" in some detail to argue whether either of the above is simpler than the other. But the JI-based notation does at least have fewer non-spartan symbols.

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Re: A proposal to simplify the notation of EDOs with bad fifths

Postby George Secor » Fri Jul 20, 2018 9:02 am


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Re: A proposal to simplify the notation of EDOs with bad fifths

Postby Dave Keenan » Sat Jul 21, 2018 12:07 am


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Re: A proposal to simplify the notation of EDOs with bad fifths

Postby George Secor » Sat Jul 21, 2018 6:15 am


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Re: A proposal to simplify the notation of EDOs with bad fifths

Postby Dave Keenan » Sat Jul 21, 2018 9:13 am


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Re: A proposal to simplify the notation of EDOs with bad fifths

Postby Dave Keenan » Sat Jul 21, 2018 11:31 am

I agree that 1:3:9 inconsistency (or equivalently: more than 25% relative error in the fifth) is another kind of "bad fifth", and such divisions do not require a native fifth notation. They can make do with a subset notation. But we would like to give, at least the low numbered ones, a native fifth notation of some kind (whether JI-based, ap-frac or lim-frac), when this can be made simple enough. Where "simple enough" means: using only Spartan symbols, plus (in order of decreasing simplicity/increasing prime-limit) kai :|\:, slai :)~|: and rai :)|:.

I suggest we tackle this in two stages. The first stage is nearly complete and consists of agreeing, for each division from 5 to 72, what is its best-if-any JI-based notation, its best-if-any apotome-fraction notation, its best-if-any limma-fraction notation and its best-if-any subset notation. I feel we should include all such notations in sagittal.pdf. These four notation types could be abbreviated JB AF LF and SS. And I note that in many cases JB = AF.

The second stage would be to decide which single notation to call the "preferred" or "default" notation for each division. This is the one that we would want to be selected in Scala when the user types SET NOTATION SA<n> where <n> is the number of the division.

I think it is very desirable that the boundaries between the regions where the four different notation types are preferred, should consist of straight lines on the above diagram, and with a good deal of reflective symmetry about the line of pythagoreans (just fifths). The divisions for which JB = AF make this easier since the boundary between these types then becomes somewhat arbitrary. Examples of straight lines are: steps per apotome, steps per limma, steps per octave, absolute fifth-error, relative fifth error.

I have responded to your recent proposals for new JI-based notations for some divisions—in most cases accepting them. I'd appreciate if you would address those I have not (yet) accepted, and address my recent JI notation proposals, including changing :/|): to :(|\: everywhere that it is used as a half-apotome with a 13-limit meaning, including in the AF notation for wide fifths.

I note that 44-edo is 1:3:5:11:13-consistent, and the following 1:3:5:11:13 JI-based notation is valid:
44: :)~|: :/|: :(|\: :||\: :~||\: :/||\:
It also happens to be the same as the apotome-fraction notation used for other divisions having 6 steps to the apotome, 30 and 37. Its other neighbouring 6-step-to-the-apotome division is 51-edo. 51's standard notation differs in using the 7-comma symbol :|): for 1 step. This would not be valid in 44-edo as the 7-comma vanishes there.

I will return to 61 and 66 later, but for now I want to flag two possible native-fifth notations for 61 that need to be investigated for validity.
61: :|\: :/|: :/|\: :(|\: :(|): :||\::/||: :/||\: possible JB same as 68-edo
61: :)|::)~|: :/|:  :(|\: :||\: :~||\: :(||~: :/||\: possible AF or JB

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Re: A proposal to simplify the notation of EDOs with bad fifths

Postby George Secor » Sat Jul 21, 2018 12:33 pm



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