Weird and missing notations in the pdf

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Dave Keenan
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Re: Weird and missing notations in the pdf

Post by Dave Keenan »

I confirm that George's 270edo notation is valid, but I note that it uses the map for the equal-temperament called 270g. Which is to say that it uses the second-best mapping of the prime 17. This is OK because 270edo is not 17-odd-limit consistent, although it is {1 3 5 7 9 11 13 15 19}-consistent.

But given that 270 is not 17-odd-limit consistent, why use any symbol for a comma that includes prime 17? Why not use :)~|: for 3 degrees? It still has perfect flag arithmetic (including double-shaft symbols). It just doesn't have the matching flag sequence for the double-shaft complements. That could be restored by using :/|~: for 9 degrees (still perfect flag arithmetic). But that involves prime 23, for which 270edo isn't consistent either.

So, if you want a matching flag sequence you're going to be using inconsistent commas. Is it better to use an inconsistent 17-limit with the "g" map or an inconsistent 23-limit with the patent map? Hard to say.

But if a user doesn't care about matching flag sequences, because they will be using the evo flavour, then they can have a consistent 13-limit notation with :)~|: = 3 and :(|(: = 9.

I guess matching flag sequences are such a big advantage for revo users, and involve such a small compromise for evo users in this case, that we should go with matching flag sequences here. And George's 270g notation uses more athenian symbols and has a lower prime limit than my alternative. So I'll go with it. Thanks George.
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Re: Weird and missing notations in the pdf

Post by Dave Keenan »

I will attempt to design a (accent-free) 128edo notation that is part of a Magenta apotome-fraction notation that will include multiples of 29edo up to maybe 174edo (18 steps per apotome). download/file.php?id=189
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Re: Weird and missing notations in the pdf

Post by רועיסיני »

Dave Keenan wrote: Wed Jun 07, 2023 6:16 pm And George's 270g notation uses more athenian symbols and has a lower prime limit than my alternative. So I'll go with it. Thanks George.
It doesn't just use more Athenian symbols, it uses all the Athenian symbols (+ three more for 4, 8 and 13), so it's a good enough excuse IMO to use a prime which is not so well approximated, especially when the Athenian symbols fall in places that make it possible for the double shafts to recapitulate a subsequence of the single shafts, as is the case here.

Dave Keenan wrote: Wed Jun 07, 2023 10:44 pm I will attempt to design a (accent-free) 128edo notation that is part of a Magenta apotome-fraction notation that will include multiples of 29edo up to maybe 174edo (18 steps per apotome). download/file.php?id=189
Cool, let me know when it's done.
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Re: Weird and missing notations in the pdf

Post by Dave Keenan »

Because 17C rounds to 4 degrees when tempering 3s only, George's 270edo notation doesn't satisfy the requirements you gave here:
viewtopic.php?p=4738#p4738
when it uses :~|(: for 3 steps, unless it uses a secondary comma such as:
7/13C	[12	-7	0	1	0	-1⟩	non-Olympian
or
245C	[0	-5	1	2⟩			Olympian for :,::~|(:
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Re: Weird and missing notations in the pdf

Post by רועיסיני »

I just noticed this too. When I checked the notation for validity I didn't temper the 3s because I thought the error was negligible, my bad.
In that case I think it's better to put 270 on hold for now, because it's not really needed as a superset for smaller tunings.
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Re: Weird and missing notations in the pdf

Post by Dave Keenan »

Just letting you know I'm still working on the magenta AF notation (to include 128edo). It is more difficult than I thought. I definitely can't get beyond 18 steps per apotome (174edo and 181edo) without accents. But that includes 18 different EDOs whose fifth sizes range from 702.76 (+0.81 ¢ error, 181edo) to 704.00 ¢ (+2.04 ¢ error, 75edo and 150edo).

Several pairs of symbols change the size-order of their primary commas (when 3s are tempered) within that range, notably :/|: and :|\: which are used together in existing notations for the magenta EDOs 29, 58, 70, and (in reverse order) 75. See figure 9, page 17 of https://sagittal.org/sagittal.pdf. The existing notation for 75 is rubbish anyway (using :(|\: for the half-apotome) and will probably need to change. Or the boundaries between the magenta range and the neighbouring blue and grey ranges may need to change.

The pairs that cross over (change their size-order) near the middle of the range are:
:)|: × :|~:
:/|: × :|\:
:)|~: × :)|): [Edit]
(:)|~::)/|:)  × (:~|)::/|~: ) [Edit]
:~|\: × (:(|~::(/|:) [Edit]
://|: × :/|\:

Blatant size-reversals that exist across the entire range are:
:)|(: < :)|:
:)~|: < :~|:
:(|: > :(|(: (almost the entire range) [Edit]
://|: > :)//|: [Edit]

Those on the left have positive apotome-slope while those on the right have negative. The cyan hiliting indicates those with the smallest magnitude of apotome slope in each group, and therefore those most likely to maintain a constant apotome fraction over the range of fifth sizes.
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Re: Weird and missing notations in the pdf

Post by Dave Keenan »

Here's a first pass at a set of Magenta notations. I've shown multiple valid symbols for some degrees. The cyan hiliting shows existing notations. The numbers in parenthesis are the numbers of steps per apotome.

29edo													:/|:
(3)														1
58edo						:/|:							:|\:							:/|\:
(6)							1							2							3
70edo					:/|:						:|\:						:/|\:
(7)						1						2						3
75edo				 :~|: :)~|: :|\:				  :/|:					   :(|(: :/|\:					:)/|\: :(|\:
(8)					 1					  2					   3						4
87edo			   :~|: :)~|: :|~:				 :/|:					:(|(: :/|~:				   :/|\:
(9)				   1					 2					3				   4
99edo			 :)|: :~|: :)~|: :|~:			  :~|(: :~~|: :/|:			   :)|): :~|):			    :~|\: ://|: :/|): :/|\:			:)/|\:
(10)				 1				  2				   3				    4				5
104edo		    :)|: :)~|:			   :~|(: :~~|:			  :/|:				 :(|: :(|(:				:/|): :/|\:
(11)			    1			   2				  3				 4				5
116edo		  :)|:				:~|: :~~|:			  :)|):				:(|(:			  :/|): :/|\:			:)/|\:
(12)			  1				2			  3				4			  5				6
128edo		 :)|:			  :~|: :)~|:			   :/|:			    :)/|: :(|(:			:~|\: :/|): :(|~:		 ://|: :/|\:
(13)			 1			  2			   3			    4			5			 6
133, 140edo	:)|:			:~|: :)~|:			:/|:			:)|): :/|~:			:(|(:			://|: :/|): :/|\:		:)/|\:
(14)			1			2			3			4			5			6			7
145edo	    :)|:		   :)~|:			  :|):			 :)|):			:(|(:		    :~|\:		   :/|\:
(15)		    1		   2			  3			 4			5		    6		   7
150, 157edo  :)|: :|~:		 :)~|:		    :~|(: :~~|:		  :)|): :/|: not		:(|(:		   :(|~:			 :/|\:			:)/|\:
(16)		   1			 2		    3		  4   157		5		   6			 7			8
162, 169edo :)|: :|~:			:)~|:		  :~|(: :~~|:	    :/|:		 :/|~: :~|):		    :(|(:		 :/|):		   :/|\: not valid
(17)		  1			2		  3		    4		 5		    6		 7		   8   for 162
174, 181edo :)|:		   :)~|:			:~~|:		 :/|:		   :~|):			:(|(:		 :~|\:		   :/|\:			:)/|\:
(18)		  1		   2			3		 4		   5			6		 7		   8			9

Regarding the layout of the diagram: I found that 209 characters per apotome gives a local minimum of rounding errors in the symbol positions, while allowing a half-apotome to be displayed in 105 characters, which is just under the maximum number of characters before the forum software enforces horizontal scrolling (109 characters). 209 characters per apotome works well because it is divisible by 11 and 19 while 210 (one more) is divisible by 2, 3, 5, 6, 7, 10, 14 and 15, and 208 (one less) is divisible by 2, 4, 8, 13 and 16, and 207 (two less) is divisible by 9. That covers all the integers from 2 to 19 except for 12, 17 and 18. And I found that setting tabs=5, i.e. [pre=5], was a good compromise between minimising the total number of characters and simplifying the mental arithmetic when figuring out how many tabs and spaces are required to achieve a given position.

The 18-step-per-apotome notation also works for 167edo. The 15-step-per-apotome notation also works for 152edo. The 13-step-per-apotome notation also works for 121edo. The 12-step-per-apotome notation also works for 109edo. None of these are currently magenta notations.
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Re: Weird and missing notations in the pdf

Post by Dave Keenan »

Apotome fractions of unaccented symbols over the magenta range.

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Re: Weird and missing notations in the pdf

Post by Dave Keenan »

@רועיסיני Here's my preferred (evo) notation for 128edo (and 121edo):
:)|:	:~|:	:/|:	:(|(:	:~|\:	:/|\:	:(|):	:#:
1	2	3	4	5	6	7	13

The corresponding revo flavour looks like this:
:)|:	:~|:	:/|:	:(|(:	:~|\:	:/|\:	:(|):	:)~||:	:~||(:	:||\:	:)//||:	:(||~:	:/||\:
1	2	3	4	5	6	7	8	9	10	11	12	13
I think that's a very bad revo notation, since no flag combinations are reused (except :/|\: :/||\:) and the double-shaft flag arithmetic is horribly inconsistent. But any attempt to provide matching flag sequences to improve the revo notation makes the evo notation worse. This is all-too-often the case.

What if we just said that double shaft symbols (and by extension X-shaft symbols) don't have fixed rational definitions, i.e. there are no fixed apotome-complementary pairs of symbols, and you can always just make them recapitulate the single-shaft flag combinations, like this:
:)|:	:~|:	:/|:	:(|(:	:~|\:	:/|\:	:(|):	:)|::|:	:~|::|:	:/||:	:(||(:	:~||\:	:/||\:
1	2	3	4	5	6	7	8	9	10	11	12	13
What problems would that cause?

Well you can immediately see one problem. We do not have any symbols whose ASCII versions would be )|| or ~||. I've had to fake them above. Nor do we have ||(.

But we could add them, and their X-shaft counterparts. There was a discussion of adding them, or at least ||(, for a different reason, here: viewtopic.php?f=5&t=256&hilit=limma+fraction&start=80

Assuming we got them added to SMuFL and the Bravura font, what other consequences might such a decision have?
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Re: Weird and missing notations in the pdf

Post by רועיסיני »

Dave Keenan wrote: Sat Jun 17, 2023 1:54 pm What if we just said that double shaft symbols (and by extension X-shaft symbols) don't have fixed rational definitions, i.e. there are no fixed apotome-complementary pairs of symbols, and you can always just make them recapitulate the single-shaft flag combinations, like this:
:)|:	:~|:	:/|:	:(|(:	:~|\:	:/|\:	:(|):	:)|::|:	:~|::|:	:/||:	:(||(:	:~||\:	:/||\:
1	2	3	4	5	6	7	8	9	10	11	12	13
What problems would that cause?
I think that's really going against the spirit of sagittal and a lot of things established so far. The one hard and fast rule in sagittal is that of apotome complements, and I see no reason to break it here. Even the Trojan double shaft accidentals where :||): is used for 66.7¢ although :/||\: - :/|\: + :|): = 83.3¢ and other similar quirks involving :||~: and :/||~: are advertised on the PDF in all their glory, and even the notation for 72, which is one of the first things a newcomer is supposed to learn in sagittal, has inconsistent double shaft flag arithmetic, and 128 approximates the fifth with a worse relative error than 72. Also, 53's notation uses :/|: ://|: :)||(: :||\: :/||\: where no flag combination is reused and this does not prevent it from being a great notation.
I will note that George suggested using :~|(: for 2\128, which makes at least 9 and 11 degrees use the same combinations as 2 and 4. Why didn't you choose it here?
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