## 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

Thanks for that. You'll find boundaries between the colours defined here:
viewtopic.php?f=5&t=256&p=843#table

They are estimated as fifth-errors in cents, which could be converted to group numbers. They can only be made more precise by attempting to devise an apotome-fraction notation for each colour (except Gold and Orange, for which this has already been done, and Rose which must remain a limma-fraction notation, and Black and White which must remain subset notations).

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

For @cmloegcmluin.

Only the single-shaft sagittals, as needed for the Evo (mixed) notation, are shown on the Periodic table. The multi-shaft apotome-complements, for the Revo (pure) notation, must be looked up elsewhere, such as in figure 13 on page 24 of http://sagittal.org/sagittal.pdf. But the multi-shaft symbols for the Revo limma-fraction notation cannot be looked up there.

Here are the full Revo limma-fraction notations. We repurpose as the limma symbol. So F can always be spelled  E in these notations. And we use as the limma-complement of  . is the limma-complement of  , and is the half limma.

Final limma-fraction notations:
Steps	0	1	2	3	4	5	6
7-edo
9,14-edo
16,21-edo
23,28,33-edo
35,40-edo
47-edo
Steps	0	1	2	3	4	5	6

[Edit: Deleted 11 and 26.]

Note that, unlike an Evo apotome-fraction notation, an Evo limma-fraction notation does not use conventional sharps or flats. It uses no sharps or flats, and uses only the single-shaft symbols above. So it has fewer alternative spellings than the Revo variant.

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

Thank you very much Dave. This is completely clear to me now. It's not much different from the earlier work of yours on this thread to which you had linked me recently, along with some caveats, but it's always nice to have a version without caveats!

It's interesting how the Revo notation is limited (limmated?!) here where Evo has the advantage. Although I suppose we can't blame it on any shortcomings of Sagittal to make things fair between Evo and Revo; if anything, it's the lack of a historical/conventional symbol for a limma! I suppose if anyone were to know something about that, it'd be you.

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

Didn't you, like Joe Monzo on facebook recently, just (accidentally) make a case against the use of the terms Evo and Revo, by getting them swapped? Didn't you mean to say that Revo has the advantage in the case of limma-fraction notation, because Evo has a more "limmated" range?

No, I've never heard of a limma symbol outside of Sagittal. Not much call for it as, like an 8:9 major whole-tone, it can be notated by a change of nominal (possibly with sharps or flats).

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

Dave Keenan wrote:
Wed Jun 17, 2020 12:06 pm
Didn't you, like Joe Monzo on facebook recently, just (accidentally) make a case against the use of the terms Evo and Revo, by getting them swapped? Didn't you mean to say that Revo has the advantage in the case of limma-fraction notation, because Evo has a more "limmated" range.
Omigosh! I sure did. Particularly frustrating, since I was just catching up on that very thread today and drafting an email to you referencing it...

I still believe in Evo and Revo.
No, I've never heard of a limma symbol outside of Sagittal. Not much call for it as, like an 8:9 major whole-tone, it can be notated by a change of nominal (possibly with sharps or flats).
That’s what I figured.

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

11,16,21,26-edo
I thought 11-EDO was only given a subset notation (of 22). I'm looking at the Periodic Table of EDOs.

And 26-EDO is not a limma-fraction notation. It can get away with sharps/flats and double-sharps/double-flats alone.

Have either of these changed?

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

My bad. Periodic table is correct. I just failed to edit off 11 and 26 from the earlier table I copied above. Fixed now.

volleo6144
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### Re: A proposal to simplify the notation of EDOs with bad fifths

Dave Keenan wrote:
Wed Jun 17, 2020 1:04 am
They are estimated as fifth-errors in cents, which could be converted to group numbers. They can only be made more precise by attempting to devise an apotome-fraction notation for each colour (except Gold and Orange, for which this has already been done, and Rose which must remain a limma-fraction notation, and Black and White which must remain subset notations).
I ... forgot to ask: What exactly makes an apotome-fraction or limma-fraction notation not work for specific colors? (I assume this is what you mean by "attempting to devise [a] notation for each color"?)

What's so special about a 692.2¢ fifth that makes it the boundary between purple and rose?
Is the boundary between black and gold just 800.0001¢ (4\6 is gold, but anything more is black)?
Is the boundary between rose and white just 654.5454¢ (6\11 is rose, but anything less is white)?

Also I just ... noticed: because of the linearity of the ♯=X lines on the 7xx¢ side and the EF=Y lines on the 6xx¢ side of the periodic table, the smaller EDOs for the integral group numbers are really just a distraction from the underlying pattern:
---
group -2/7 is division by 0
-¼ 01\0¼ = 04\01 (4 octaves)
-⅛ 1½\1⅛ = 04\03 (12edo M10)

00 02\02 = 01\01 (02b = 01)
01 06\09 = 04\06 (09b = 06)
02 10\16 = 05\08 (16b = 08)
03 14\23         (23b)
04 18\30 = 03\05 (30)
05 22\37
06 26\44 = 13\22 (44)
07 30\51 = 10\17 (51)
08 34\58 = 17\29 (58)
09 38\65
9¼ F6\Q7
9½ 80\D7
9¾ G4\S1

10 42\72 = 07\12 (72)

A¼ G5\S3
A½ 81\D9
A¾ F9\R3
11 39\67
12 36\62 = 18\31 (62)
13 33\57 = 11\19 (57)
14 30\52 = 15\26 (52)
15 27\47
16 24\42 = 04\07 (42b = 07)
17 21\37         (37b)
18 18\32 = 09\16 (32b = 16)
19 15\27 = 05\09 (27b = 09)
20 12\22 = 06\11 (22b = 11)
21 09\17         (17b)
22 06\12 = 01\02 (12bb = 02)
23 03\07         (07bb)
24 00\02         (02bb = 01b)

O⅛ -⅜\1⅜ = -3\11
O¼ -¾\0¾ = -1\01 (-1 octave)
group 24+2/5 is division by 0
Group 10-X is (42-4X) steps of (72-7X)
Group 10+X is (42-3X) steps of (72-5X)

Group number for a fifth above 700¢ is the number of (tempered) apotomes in a (tempered) minor seventh
Group number for a fifth below 700¢ as a similar fraction has a denominator of a (tempered) fourfold-augmented second [-47 30>
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Dave Keenan
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### Re: A proposal to simplify the notation of EDOs with bad fifths

volleo6144 wrote:
Sun Jun 28, 2020 5:08 am
I ... forgot to ask: What exactly makes an apotome-fraction or limma-fraction notation not work for specific colors? (I assume this is what you mean by "attempting to devise [a] notation for each color"?)
Sorry for the delay in responding. I'll talk in terms of apotome-fraction notations, but the same goes if you substitute "limma" for "apotome". In any case, we already have the only limma-fraction notation we need, so any future notation of this type will be apotome-fraction.

By the way, thanks for correcting my pronunciation of "a-POT-o-me". I had been pronouncing it "AP-o-toam" like "microtome" all these years!

An apotome-fraction notation that "works" over a given range of fifth-sizes and up to some maximum number of steps per apotome, is a sequence of symbols without accents, beginning with the natural and continuing with distinct upward single-shaft symbols, with boundaries between them defined as some (possibly irrational) positive fraction of the tempered apotome, covering up to at least the half-apotome, such that every EDO in that range has a distinct symbol for each number of steps up to the half-apotome, and every symbol represents a comma whose tempered size rounds to the number of steps it represents in every such EDO, where that comma is preferably the primary comma for that symbol, but may be a secondary comma which is the primary comma for an accented version of that symbol.

As I think @cmloegcmluin once said, but I failed to appreciate at the time: Trojan (orange) is the epitome of an apotome-fraction notation. Gilbert and Sullivan? "He is the very model of a modern major general".

Finding that one cannot get such a notation to work, may just mean that we need to change the range of fifth sizes for a colour, from where my brief preliminary investigations suggested they should be. It may also mean that we're trying too hard in terms of steps per apotome and may just need to abandon some of the higher-numbered EDOs in the range (when we're pushing beyond 72-edo).

I note that an ideal situation would be to have notations overlap slightly, so that for EDOs near the boundary between colours, it wouldn't matter which of the two apotome-fraction notations you applied to it, they would both give the same notation for the EDO. The notation for 59edo is an example of this. Notice that although 59edo uses the gold notation, it has exactly the same notation as 66-edo, which is green and has the same number of steps to the apotome "#=9".
What's so special about a 692.2¢ fifth that makes it the boundary between purple and rose?
Nothing. Any value between 15\26 ≈ 690.91 ¢ and 19\33 ≈ 692.3 ¢ would do just as well. But if you ask, "Would 59b (34\59) be purple or rose?" then I'd say: Try both notations and see which ones work (if any). When you look at the limma-fraction (rose) notation, as set out a few posts back, you'll see it doesn't have enough symbols to do 7 steps to the limma. But I haven't checked whether /|) as 35M (purple) is valid as (i.e. has tempered value that rounds to) 1 step of 59b.
Is the boundary between black and gold just 800.0001¢ (4\6 is gold, but anything more is black)?
Is the boundary between rose and white just 654.5454¢ (6\11 is rose, but anything less is white)?
Probably. But again, the question is: Can you make the gold notation or rose notation "work" for something outside those bounds.
Also I just ... noticed: because of the linearity of the ♯=X lines on the 7xx¢ side and the EF=Y lines on the 6xx¢ side of the periodic table, the smaller EDOs for the integral group numbers are really just a distraction from the underlying pattern:
...
Cool.

volleo6144
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### Re: A proposal to simplify the notation of EDOs with bad fifths

Dave Keenan wrote:
Wed Jul 01, 2020 11:04 am
By the way, thanks for correcting my pronunciation of "a-POT-o-me". I had been pronouncing it "AP-o-toam" like "microtome" all these years!
Yeah, me too.
What's so special about a 692.2¢ fifth that makes it the boundary between purple and rose?
Nothing. Any value between 15\26 ≈ 690.91 ¢ and 19\33 ≈ 692.3 ¢ would do just as well. But if you ask, "Would 59b (34\59) be purple or rose?" then I'd say: Try both notations and see which ones work (if any). When you look at the limma-fraction (rose) notation, as set out a few posts back, you'll see it doesn't have enough symbols to do 7 steps to the limma. But I haven't checked whether as 35M (purple) is valid as (i.e. has tempered value that rounds to) 1 step of 59b.
= 35:36 has the tempered value of a tempered whole-tone minus 32:35, which, in 59b, is 9\59 minus 32:35 (7.73\59), which is 1.37\59, so yeah, it is valid as 1\59b.

It turns out that, if ||( was an actual symbol in Revo, it would complete the matching shaft sequences for the Rose notation: , with being ||(.
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