A proposal to simplify the notation of EDOs with bad fifths

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

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.

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

Dave Keenan wrote:Well George, we've slept on it for 9 months now, and there has recently been a request on one of the facebook groups, for someone to add the corresponding Sagittal notation(s) to every EDO entry in the Xenharmonic Wiki. So we really ought to decide whether these will become the new standard notations for these poor-fifth EDOs, and update figures 8 and 9 on pages 16 and 17 of Sagittal.pdf (the updated Xenharmonikon journal article) accordingly.

Since no one else is arguing, I suggest that we both attempt to come up with reasons why the existing notations for these EDOs should not be changed, or should be changed in ways different from this proposal. i.e. play devil's advocate. For this purpose, it is useful to repost this diagram.

There are eleven existing native-fifth notations that would change under this proposal. These can be grouped as follows. You should locate each group on the above diagram.
Near-superpythagorean (amber): 27, 49 (also includes 54 (2x27) and 71, which don't presently have native fifth notations)
Near-meantone: (red) 26, 45, 64 (also includes 52 (2x26), which doesn't presently have a native fifth notation)
Narrow fifths with one step per apotome (red): 33, 40, 47
Mavila, -1 step per apotome (red): 9, 16, 23

I note that 27 is not simplified by this proposal, since 1\27 changes from the spartan to the non-spartan . Nor is 26 simplified, as it goes from being notated only with sharps and flats (apotomes), to requiring spartan symbols (for limma fractions).

One could argue that the blue area on the diagram (JI-based notations) should be expanded to include the first two categories above. This would change the boundaries, in fifth sizes, from +-7.5 c of just, to +-10 c of just. We might continue to show how apotome and limma fraction notations can be defined for those with fifth errors between 7.5 c and 10 c, but we need not list them as the standard native-fifth notations for those EDOs (the first two categories above).
This is the first installment of my response, which will cover all of the near-superpythagorean divisions (27, 49, 54, and 71) and some additional adjoining divisions (32, 37, 42, and 59). In evaluating these (one by one), I have examined the fractional apotome bad-5ths notation for each one to determine whether it could be replaced by a simpler JI-based notation. For reference, I have also tabulated the prime 3 error both in absolute (cents) and relative (% of a degree) terms, as well as the prime-limit consistency. In additional, I have listened to some of these in Scala to judge whether a JI-based simplification is worthwhile.

Here goes:

27 is 9-limit consistent (prime 3 has 9.156 cents or 20.60% error) and has a valid 5-comma (and 11M-diesis) of 1 degree and valid 13L-diesis (and 13M-diesis) of 2 degrees (7C vanishes). I recommend a simplification of the bad-5ths notation:
replacing it with the following JI notation:
27: (JI notation)
This is not the JI notation that we formerly agreed on, since I have replaced 13M with 13L , which directly notates 13/8.

49 is 7-limit consistent (prime 3 has 8.249 cents or 33.68% error) and has a valid 5-comma of 2 degrees and valid 11M-diesis of 3 degrees (7C vanishes). I recommend keeping the original 11-limit notation,
49: (JI notation)
which is simpler than the bad-5ths notation:

54 is 5-limit inconsistent (prime 3 has 9.156 cents or 41.20% error); the 7-comma vanishes, and the 5-comma and 11M-diesis are both 3 degrees. It does not appear that the bad-5ths proposal can be simplified.
or as subset of 108

71 is 5-limit consistent (prime 3 has 7.904 cents or 46.77% error) and has a valid 5-comma of 3 degrees, 11M-diesis of 4 degrees, and 13M-diesis of 5 degrees. However, although it has a valid 7-comma of 1 degree, it is severely 7-limit inconsistent, making it impractical to use the 7-comma symbol in the notation. Therefore, I recommend that the bad-5ths proposal be used.
or as subset of 142

32 is 5-limit inconsistent (prime 3 has 10.545 cents or 28.12% error); the 7-comma vanishes, and the 5-comma and 11M-diesis are both 2 degrees. It does not appear that the bad-5ths proposal can be simplified.
or as subset of 96

37 is 7-limit consistent (prime 3 has 11.559 cents or 35.64% error); the 7-comma vanishes, and the 5-comma and 11M-diesis are both 2 degrees. It does not appear that the bad-5ths proposal can be simplified.

42 is 7-limit consistent (prime 3 has 12.331 cents or 43,16% error); the 7-comma vanishes, and the 5-comma and 11M-diesis are both 2 degrees. It does not appear that the bad-5ths proposal can be simplified.
or as subset of 84

59 is 7-limit consistent (prime 3 has 9.909 cents or 48.72% error); the 7-comma vanishes, and the 5-comma and 11M-diesis are both 3 degrees. It does not appear that the bad-5ths proposal can be simplified.
or as subset of 118

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

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.

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

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.

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

Dave Keenan wrote: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".
Before I look into examining the merits of JI-based notations for 54 and 71, I observe that these two could be excluded if the criterion for prime-3 error were a combination of absolute (<10 cents) and relative values (<35% or <40% of a degree). I also noted that 27 and 49 are both 7-limit consistent; although that's another thing that sets these apart from most of the other bad-5th divisions, apparently it isn't necessary to make that an additional condition:

27: 9.156 cents, 20.60%, 9-limit consistent
49: 8.249 cents, 33.68%, 7-limit consistent
54: 9.156 cents, 41.20%
71: 7.904 cents, 46.77%

Others that would be excluded:
32: 10.545 cents, 28.12%
37: 11.559 cents, 35.64%
42: 12.331 cents, 43.16%
52: 9.647 cents, 41.81%
59: 9.909 cents, 48.72%
64: 8.205 cents, 43.76%

Two others would then be candidates for JI notation (also 7-limit consistent):
26: 9.647 cents, 20.90%, 13-limit consistent
45: 8.622 cents, 32.33%, 7-limit consistent

So here's my writeup on those last two:

26 is 13-limit consistent (prime 3 has -9.647 cents or -20.90% error) and has a valid apotome of 1 degree. I recommend the simple notation:
26:

45 is 7-limit consistent (prime 3 has -8.622 cents or -32.33% error) and has a valid apotome of 2 degrees. I recommend that although (as 1deg45) does not give the best 11/8 (due to 11-limit inconsistency), it actually gives the best 11/6, 11/7, and 11/9 and results in good 11-limit (and 13-limit) chords (notably 6:7:9:11:13):
45:

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

Thanks George.

I note for other readers, that although we are summarising these notations using multi-shaft sagittals, each such symbol can be replaced by an equivalent combination of conventional sharps and flats with a single-shaft sagittal.
George Secor wrote: Before I look into examining the merits of JI-based notations for 54 and 71, I observe that these two could be excluded if the criterion for prime-3 error were a combination of absolute (<10 cents) and relative values (<35% or <40% of a degree).
Nice try. But you'd also exclude (from JI-based notations) an infinite number of larger EDOs—about 30% of all EDOs—beginning with 66 78 83 90 95 on the wide side and 69 76 81 88 93 100 on the narrow side. We could not make apotome-fraction or limma-fraction notations for EDOs with so many steps to the apotome or limma.

You should get the idea if you look at the green and magenta regions in the diagram in this post
viewtopic.php?p=457#p457
that shows EDOs having relative errors greater than 25%, as this is equivalent to being 1:3:9 inconsistent.
Others that would be excluded:
32: 10.545 cents, 28.12%
37: 11.559 cents, 35.64%
42: 12.331 cents, 43.16%
52: 9.647 cents, 41.81%
59: 9.909 cents, 48.72%
64: 8.205 cents, 43.76%
Right. So we're looking at narrow fifths now.
Two others would then be candidates for JI notation (also 7-limit consistent):
26: 9.647 cents, 20.90%, 13-limit consistent
45: 8.622 cents, 32.33%, 7-limit consistent

So here's my writeup on those last two:

26 is 13-limit consistent (prime 3 has -9.647 cents or -20.90% error) and has a valid apotome of 1 degree. I recommend the simple notation:
26:
I agree. This is of course the existing standard notation for 26-edo.
45 is 7-limit consistent (prime 3 has -8.622 cents or -32.33% error) and has a valid apotome of 2 degrees. I recommend that although (as 1deg45) does not give the best 11/8 (due to 11-limit inconsistency), it actually gives the best 11/6, 11/7, and 11/9 and results in good 11-limit (and 13-limit) chords (notably 6:7:9:11:13):
45:
I note that the existing standard notation for 45 is:
45:
And I note that in its primary role as the 35-diesis (36/35) actually does map to 1°45 whereas as the 11-diesis (33/32) maps to 2°45.

And if we want the simple criterion of prime_3_error<10c to be preserved, in this (narrow-fifths) case we need to also find simple JI-based notations for 52 and 64.

We already have a simple JI-based notation for 64-edo. It's the same as the one for 50 and 57.
64:

And the existing standard notation for 45 turns out also to be valid for 52:
52:

The above set of JI-based notations for 26 45 52 and 64 also constitutes a consistent apotome-fraction notation for all EDOs with fifths between 7.5c and 10c narrow.
1/3 and 1/2 apotome
2/3 apotome
1 apotome
4/3 and 3/2 apotome
5/3 apotome
2 apotomes

I note that implementing JI-based notations for everything with less than a 10c prime-3 error, simplifies the apotome-fraction and limma-fraction notation for those EDOs with worse fifths, because now we only need to cater for a maximum of 9 steps to the apotome instead of 10, and 6 steps to the limma instead of 7.

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

Dave Keenan wrote:Thanks George.

I note for other readers, that although we are summarising these notations using multi-shaft sagittals, each such symbol can be replaced by an equivalent combination of conventional sharps and flats with a single-shaft sagittal.
George Secor wrote: Before I look into examining the merits of JI-based notations for 54 and 71, I observe that these two could be excluded if the criterion for prime-3 error were a combination of absolute (<10 cents) and relative values (<35% or <40% of a degree).
Nice try. But you'd also exclude (from JI-based notations) an infinite number of larger EDOs—about 30% of all EDOs—beginning with 66 78 83 90 95 on the wide side and 69 76 81 88 93 100 on the narrow side. We could not make apotome-fraction or limma-fraction notations for EDOs with so many steps to the apotome or limma.

You should get the idea if you look at the green and magenta regions in the diagram in this post
viewtopic.php?p=457#p457
that shows EDOs having relative errors greater than 25%, as this is equivalent to being 1:3:9 inconsistent.
<Groan> Please excuse me if I let off some steam at this point, because I had the impression that we already had native-5th notations for all (or nearly all) of the divisions not identified as "bad-5th". Instead, I see that we have very few (if any) native-5th notations for the (more than 20) 1:3:9-inconsistent divisions in the green and magenta regions in the referenced diagram. It really seems like a waste of time for us to devise native-5th notations for those divisions, most of which nobody will ever use. After all, why would anyone spend their time scavenging through the microtonal garbage heap when there are so many other good divisions to explore? Certainly nobody is going to refret a guitar to one of these.

So when are we going to tackle the garbage heap?

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

George Secor wrote: <Groan> Please excuse me if I let off some steam at this point, because I had the impression that we already had native-5th notations for all (or nearly all) of the divisions not identified as "bad-5th". Instead, I see that we have very few (if any) native-5th notations for the (more than 20) 1:3:9-inconsistent divisions in the green and magenta regions in the referenced diagram. It really seems like a waste of time for us to devise native-5th notations for those divisions, most of which nobody will ever use. After all, why would anyone spend their time scavenging through the microtonal garbage heap when there are so many other good divisions to explore? Certainly nobody is going to refret a guitar to one of these.

So when are we going to tackle the garbage heap?
Relax. I don't think we need to come up with native-fifth notations (whether JI-based, apotome-fraction or limma-fraction) for any 1:3:9-inconsistent division bigger than some number. What should that number be?

44-edo is presently the smallest division we have not given any kind of native-fifth notation for. By the way, in sagittal.pdf we say that 44-edo should be notated as a subset of 176-edo. I don't understand why we don't say "as a subset of 132-edo" instead. I think we should correct that.

I'll repeat the diagram we're referencing:

Of the 21 green and magentas, we have native-fifth notations (given in sagittal.pdf) for only 3 of them, namely magentas 57, 62 and 69. If we added native-fifth notations (JI-based) for only 3 more, namely greens 44, 61 and 66, we'd have them for all EDOs up to 72, which is something we have been asked for in the past.

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

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

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

Dave Keenan wrote:
George Secor wrote: <Groan> Please excuse me if I let off some steam at this point, because I had the impression that we already had native-5th notations for all (or nearly all) of the divisions not identified as "bad-5th". Instead, I see that we have very few (if any) native-5th notations for the (more than 20) 1:3:9-inconsistent divisions in the green and magenta regions in the referenced diagram. It really seems like a waste of time for us to devise native-5th notations for those divisions, most of which nobody will ever use. After all, why would anyone spend their time scavenging through the microtonal garbage heap when there are so many other good divisions to explore? Certainly nobody is going to refret a guitar to one of these.

So when are we going to tackle the garbage heap?
Relax. I don't think we need to come up with native-fifth notations (whether JI-based, apotome-fraction or limma-fraction) for any 1:3:9-inconsistent division bigger than some number. What should that number be?
72 is as good a cutoff as any. As I said before, I don't think anyone is going to refret a guitar to any division this complex.
Dave Keenan wrote:44-edo is presently the smallest division we have not given any kind of native-fifth notation for. By the way, in sagittal.pdf we say that 44-edo should be notated as a subset of 176-edo. I don't understand why we don't say "as a subset of 132-edo" instead. I think we should correct that.
There are two reasons for making it a subset of 176-EDO:
1) 176 is a much better division than 132; and
2) The notation for 88-EDO is as a subset of 176, so the tones common to 44 and 88 would be notated alike.

I saw your other message, and I'll be responding to it soon.