Can all the promethean symbols be realized consistently in the same edo?

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רועיסיני
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Re: Can all the promethean symbols be realized consistently in the same edo?

Post by רועיסיני »

Dave Keenan wrote: Thu Jun 08, 2023 10:16 pm
If I temper only the 3s, the way I thought EDO notations should be checked for validity (which here is very similar to not temper anything) I get that the first comma is indeed > 13.5\400 (only by a small amount) but the other comma rounds to 13\400 and both round to 32\971.
That's a very good point. I'm now confused about which I should be doing. Ideally they would give the same result, but when the error in some prime is large enough, of course they do not. But then a non-patent map may give the same result as tempering only prime 3. But maybe not the same map for all the EDO's symbol's commas. I feel there should be one map for all symbol commas for an EDO.
I think that commas that justify an EDO notation should satisfy both of those requirements:
  1. validity – when 3 is tempered, they map to the correct EDO degrees. That is a "local" requirement for each comma.
  2. consistency/coherence – there is a map that gives for each comma the number of steps the corresponding accidental is notating. That is a "global" requirement that makes the entire notation "stick together".
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Re: Can all the promethean symbols be realized consistently in the same edo?

Post by Dave Keenan »

רועיסיני wrote: Thu Jun 08, 2023 10:37 pm I think that commas that justify an EDO notation should satisfy both of those requirements:
  1. validity – when 3 is tempered, they map to the correct EDO degrees. That is a "local" requirement for each comma.
  2. consistency/coherence – there is a map that gives for each comma the number of steps the corresponding accidental is notating. That is a "global" requirement that makes the entire notation "stick together".
That makes perfect sense. I'm embarrassed that George and I never formulated the requirement in that way. Although I believe almost all of our EDO notations would meet that requirement (except maybe some desperate cases where we cheated a little to get any notation at all), we may have missed out on some opportunities to use a non-patent map. What we usually did, when some comma was inconsistent with the patent map, was to simply not use that comma.
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Re: Can all the promethean symbols be realized consistently in the same edo?

Post by רועיסיני »

Dave Keenan wrote: Fri Jun 09, 2023 9:51 am What we usually did, when some comma was inconsistent with the patent map, was to simply not use that comma.
You can also strengthen the second requirement and say that the map has to be the patent map, but that may exclude some notations for inconsistent edos which use commas for products of primes that are close to a half integer step from the same side or for quotients of primes that are close to a half integer step from different sides. The best example I can give right now is George's proposed notation for 135 which had to give 5 a fractional value, but there are probably some agreed notations out there that are consistent with an integer non-patent map.
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Re: Can all the promethean symbols be realized consistently in the same edo?

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רועיסיני wrote: Wed Jun 07, 2023 3:08 am Unrelated to the notations, I noticed that my finding that the symbols cannot be realised consistently in the same EDO means that the schisminas that vanish when you add flags can be added to each other to make an exact octave. I don't have an example, but I think it has to include 400(2:|\: - :|\\:) + 971(:~|: + :|\: - :~|\:).
Indeed, one such example, that comes from the minimal set I gave in the original post, is
836(:)|(:-:)|:-:|(:) - 265(:)|~:-:)|:-:|~:) + 1012(:~~|:-2:~|:) + 1053(:~|:+:|):-:~|):) + 971(:~|:+:|\:-:~|\:) - 706(:(|~:-:(|:-:|~:) + 935(:|\):-:|\:-:|):) - 400(:|\\:-2:|\:)
which evaluates to
(31363135)836 / (531441531392)265 * (2604236826040609)1012 * (58325831)1053 * (35203519)971 / (247808247779)706 * (1476502530314763950080)935 / (272629233272588800)400 = 2
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Re: Can all the promethean symbols be realized consistently in the same edo?

Post by רועיסיני »

In fact, there are 14 (single shaft) Prometheans whose primary commas don't equal the sum of their flags' - :)|(:, :)~|:, :~|(:, :~~|:, :)|~:, :~|):, :~|\:, :)//|:, :(|~:, :(/|:, :)/|\:, :|\):, :|\\: and :)|\\:.
These are 14 sagittals but there are only 9 primes up to 23, so there have to be at least 5 (and in fact there are exactly 5) dependencies which are linearly independent from each other. Removing either :~|\: or :|\\: makes the set compatible with some edo (971 and 400i respectively), so their schisminas cannot be linear combinations of other sagittals'. :(/|: and :|\): vanish the same schismina (in opposite directions) so that's one dependency, and it's possible to complete it to a full basis of the dependencies using these four:
(:~|(:-:~|:-:|(:) = (:~~|:-2:~|:) + (:(|:+:/|:-:(/|:)
(:)|(:-:)|:-:|(:) = (:)//|:-:)|:-2:/|:) + (:)|:+:/|:+:|\:-:)/|\:) + (:)|\\:-:)|:-2:|\:)
(:)|:+:~|:-:)~|:) + (:)|\\:-:)|:-2:|\:) = (:~|:+:|):-:~|):) + (:(|:+:/|:-:(/|:)
(:)|~:-:)|:-:|~:) + (:)|:+:/|:+:|\:-:)/|\:) = (:(|~:-:(|:-:|~:) + (:(|:+:/|:-:(/|:)

That means that from the original expression of an octave as a sum of schisminas we can get many more, for example:
836(:)|(:-:)|:-:|(:) - 265(:)|~:-:)|:-:|~:) + 1012(:~~|:-2:~|:) + 1053(:~|:+:|):-:~|):) + 971(:~|:+:|\:-:~|\:) - 706(:(|~:-:(|:-:|~:) + 935(:(|:+:/|:-:(/|:) - 400(:|\\:-2:|\:)
836(:)|(:-:)|:-:|(:) + 935(:~|(:-:~|:-:|(:) - 265(:)|~:-:)|:-:|~:) + 77(:~~|:-2:~|:) + 1053(:~|:+:|):-:~|):) + 971(:~|:+:|\:-:~|\:) - 706(:(|~:-:(|:-:|~:) - 400(:|\\:-2:|\:)
836(:)|(:-:)|:-:|(:) + 1012(:~~|:-2:~|:) + 1053(:~|:+:|):-:~|):) + 971(:~|:+:|\:-:~|\:) - 971(:(|~:-:(|:-:|~:) + 670(:(|:+:/|:-:(/|:) + 265(:)|:+:/|:+:|\:-:)/|\:) - 400(:|\\:-2:|\:)

רועיסיני wrote: Fri Jun 09, 2023 8:03 pm ... there are probably some agreed notations out there that are consistent with an integer non-patent map.
Indeed there are - the notations for 60edo and 108edo use the symbol for :(|: for a value that is not the same as the value its comma is tempered to in their patent vals - 60edo uses it for 1 degree but tempers the comma out while 108 uses it for 2 but tempers it to 1. These edos' notations are consistent, however, with 60d, 60e and 108e.
You may claim that it's just trojan being trojan, but:
  1. This comma's common trojan value is defined to be 1\60edo.
  2. The first trojan edo where the commas cannot agree with any map is 216edo, where there are a lot of other problems, like an invalid use of 3 commas including the 5-comma, a blatant size reversal (:)/|: < :/|:) and misleading flag arithmetic (:/|: + :|)::/|):).
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Re: Can all the promethean symbols be realized consistently in the same edo?

Post by Dave Keenan »

I don't understand the significance of schisminas summing to the octave, unless it is of purely mathematical interest to you. I suppose a basis for the Promethean schisminas might have been relevant to the decisions about what ratios to assign to the Magrathean (tina) accents. You may find that we have assigned accents to the schisminas of your basis.
See viewtopic.php?f=10&t=430&p=2714#tina

Well found re 60edo and 108edo. I think 1\60 got :(|: because 7/11C was valid for, and considered the best choice for, 2\120.

Yes, the 216edo notation is poorly justified, but we'd never heard of anyone using any 12n notation past 192edo, so we weren't too worried. It took a hit for the team. Its flag arithmetic failure comes because we wanted to preserve 1\72 = :(|: [Edit: :/|:] and have the same symbol for the identical pitches 3\216 and 1\72.

Having fixed capture zones is a very strict requirement to place on an apotome-fraction notation for any fifth size other than pure Pythagorean. And of course there are many competing criteria for a good notation system.

Any apotome-fraction notations for colours other than grey and orange on the periodic table will not need to obey such strict criteria as SymbolFor(2\58) = SymbolFor(1\29). They will only need to have a total order of symbols (as close to the Pythagorean order as possible) such that it is not violated by any individual EDO in that colour.
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Re: Can all the promethean symbols be realized consistently in the same edo?

Post by רועיסיני »

Dave Keenan wrote: Sun Jun 11, 2023 1:36 pm I don't understand the significance of schisminas summing to the octave, unless it is of purely mathematical interest to you.
It's just a different equivalent way of looking at things, that may be more intuitive to some people. It means that things you neglect may aggregate into something large and bite you if you're not careful.

Dave Keenan wrote: Sun Jun 11, 2023 1:36 pm I suppose a basis for the Promethean schisminas might have been relevant to the decisions about what ratios to assign to the Magrathean (tina) accents. You may find that we have assigned accents to the schisminas of your basis.
See viewtopic.php?f=10&t=430&p=2714#tina
It does not seem like so, at least not for the eight I wrote explicitly. Of course, they all are some number of tinas (the number of degrees they are mapped to in 8539edo) and any default tina number value can be expressed as a sum or a difference of the Promethean schisminas because every 23-limit interval can, but the intervals themselves don't seem to be the same.

Dave Keenan wrote: Sun Jun 11, 2023 1:36 pm Well found re 60edo and 108edo
Thanks!

Dave Keenan wrote: Sun Jun 11, 2023 1:36 pm Yes, the 216edo notation is poorly justified, but we'd never heard of anyone using any 12n notation past 192edo, so we weren't too worried. It took a hit for the team. Its flag arithmetic failure comes because we wanted to preserve 1\72 = :(|: and have the same symbol for the identical pitches 3\216 and 1\72.
I suppose you meant :/|: and not :(|: there...
Anyway, I gave 216 here only as a possible counter argument that someone could make, like "Trojan is not really focused on JI because it's just assigning symbols to capture zones to make the flags look nice, there are even Trojan notations that cannot be justified by any val so the fact that 60's is not justified by the patient val is not that important". Of course I don't blame you or anyone for any seemingly bad choice that was made in the design of the Trojan system, especially for large EDOs. I understand the need of a consistent 12R notation and of course when you also try to fit a JI-based system where symbols are made from a small number of components to this task some compromises have to be made. I agree that Trojan is a very good JI based system for 12n ≤ 144 edos, and to a lesser extent also up to 192 and even 204edo.
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Re: Can all the promethean symbols be realized consistently in the same edo?

Post by Dave Keenan »

רועיסיני wrote: Mon Jun 12, 2023 1:53 am It's just a different equivalent way of looking at things, that may be more intuitive to some people. It means that things you neglect may aggregate into something large and bite you if you're not careful.
Ah. OK.
It does not seem like so, at least not for the eight I wrote explicitly. Of course, they all are some number of tinas (the number of degrees they are mapped to in 8539edo) and any default tina number value can be expressed as a sum or a difference of the Promethean schisminas because every 23-limit interval can, but the intervals themselves don't seem to be the same.
OK. Thanks. I see that the primary-commas for 1 to 9 tinas, with the 5-schisma, cannot form a basis for all of the Promethean schisminas because prime 23 doesn't occur among the accents' commas and :(|~: uses a 23-limit flag <:|~:> while its primary comma is only 19-limit.
I suppose you meant :/|: and not :(|: there...
Yes, I did. Sorry. Edit note added above. As you would know, :/|: is only valid as 2 degrees of 216edo (which would give correct flag arithmetic), not 3 degrees as we have it.
Anyway, I gave 216 here only as a possible counter argument that someone could make, like "Trojan is not really focused on JI because it's just assigning symbols to capture zones to make the flags look nice, there are even Trojan notations that cannot be justified by any val so the fact that 60's is not justified by the patient val is not that important".
There is a good deal of truth in that. Anyone who uses the Trojan notation for anything other than 12n-edos ≤ 144edo probably doesn't care very much about JI. Anyone is welcome to use a more JI-based (or accented Trojan) notation for the larger 12n-edos.

We only aim to give (at least) one standard notation for each EDO that we think anyone might want to use, so that people who don't want to get into the math can obtain something immediately "off the shelf".
Of course I don't blame you or anyone for any seemingly bad choice that was made in the design of the Trojan system, especially for large EDOs. I understand the need of a consistent 12R notation and of course when you also try to fit a JI-based system where symbols are made from a small number of components to this task some compromises have to be made. I agree that Trojan is a very good JI based system for 12n ≤ 144 edos, and to a lesser extent also up to 192 and even 204edo.
Thanks.
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