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

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

Post by רועיסיני »

No. If they are assumed to have primary comma values, there is no val (except for the trivial val) that gives all the single-shaft promethean symbols valid flag arithmetic. The reason is quite complicated, and I do not understand this better than "the computer says the only solution for the equations is the trivial solution", but if you want to check by yourself, it seems that :)|:, :|(:, :|~:, :|):, :(|:, :|\:, :)|(:, :)|~:, :~~|:, :~|):, :~|\:, :(|~:, :|\): and :|\\: are one minimal unreconcilable set.
However, it turns out that there are two accidentals that when one of them is removed the promethean set does become consistent. One of them is :|\\:, which if removed leaves a set that forces 400edo with a sharp 23rd harmonic instead of a flat one, and the other is :~|\:, that if removed forces 971edo. That of course prompted me to try and make notations for these edos (which will probably be of interest to someone, since they both have excellent 5ths and 400edo is compatible with cents).

First, for 400edo I had a bit of a challenge. The beginning wasn't hard, but I'm not sure how to represent 14 and 17 degrees. That's the notation I have so far:
:)|:	:|(:	:~|:	:)~|:	:~|(:	:~~|:	:/|:	:)/|:	:|):	:)|):	:|\:	:~|):	:(|(:	?	:)//|:	:/|):	??	:/|\:	:)/|\:	:(|):	??	:(|\:	:)|\\:
1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23
For 14 degrees I have two competing commas:
  1. :~|\:, because it is similar to the apotome complement of :)~|:. The problem with it is that its primary value is 23-limit and isn't a valid comma here. This problem can be solved with a suitable secondary value, which may be 23-limit or not, but I have no idea how to find one.
  2. ://|:, because it's spartan and its primary comma fits the value perfectly. The problem is that it will lead to 24 degrees being notated as :)||(: which is not consistent with flag arithmetic.
But the real problem is for 17 (and therefore also 21) degrees. The obvious accidentals for these degrees are :(|~: and :|\\:, but :|\\: is the exact comma that was removed to make the set compatible with a 400edo val in the first place (and indeed 11*2 ≠ 21).

Second, for 971edo I had kind of an opposite problem, where all the accidentals are mapped to distinct degrees but there are not enough of them, so we have to use accent marks. The schisma maps to 2\971 and the mina to 0, but the JI value for the double mina maps to 1\971, so I devised this Olympian notation:
:``::|:	:,,::)|:	 :)|:	:``::)|:	 :|(:	:``::|(:	 :~|:	 :)|(:	:``::)|(:	 :)~|:	:``::)~|:	:,,::|~:	 :|~:	 :~~|:	:``::~~|:	 :)|~:	 :/|:	:``::/|:	:,,::)/|:	 :)/|:	:``::)/|:	 :|):	:``::|):	:,,::)|):	 :)|):	 :|\:	 :(|:	:``::(|:	 :~|):	 :/|~:	:``::/|~:	 :(|(:	:``::(|(:	 ://|:	:``:://|:	:,,::)//|:	 :)//|:	:``::)//|:	 :/|):	 :(|~:	:``::(|~:	:,,::/|\:	 :/|\:	 :(/|:	:``::(/|:	 :)/|\:	:``::/|\:	 :|\):	 :(|):	:``::(|):	:,,::|\\:	 :|\\:	 :(|\:	:``::(|\:	 :)|\\:	:``::)|\\:
1	 2	 3	 4	 5	 6	 7	 8	 9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	 29	 30	 31	 32	 33	 34	 35	 36	 37	 38	 39	 40	 41	 42	 43	 44	 45	 46	 47	 48	 49	 50	 51	 52	 53	 54	 55	 56
And I guess the places where there is only a need for one accented accidental would have :``: on the lower one in revo as well (which is not the apotome complement of the base accidental of the apotome complement interval). Also, 33 steps could be notated with :~|\: if we can find a valid secondary comma for it.

Any thoughts?
Last edited by רועיסיני on Fri Jun 09, 2023 8:51 pm, edited 1 time in total.
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Re: Can all the promethean symbols be realized consistently in the same edo?

Post by Dave Keenan »

Those are interesting results. I certainly wouldn't have expected any EDO to use all the Promethean symbols. They are too unevenly spaced. They were the result of seeing how far we could push a system with no accents. We looked at every geometrically-viable combination of the 8 flags at-most 3 at a time, and rejected those that were too close in size to others, unless they helped to fill in some gaps in a system consisting of Athenians plus accents.

In case you haven't seen them, these are related:
viewtopic.php?p=4408#p4408
viewtopic.php?f=5&t=413&p=881#p881
viewtopic.php?p=257#p257
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Re: Can all the promethean symbols be realized consistently in the same edo?

Post by Dave Keenan »

רועיסיני wrote: Sun May 28, 2023 6:40 am First, for 400edo I had a bit of a challenge. The beginning wasn't hard, but I'm not sure how to represent 14 and 17 degrees. That's the notation I have so far:
:)|:	:|(:	:~|:	:)~|:	:~|(:	:~~|:	:/|:	:)/|:	:|):	:)|):	:|\:	:~|):	:(|(:	?	:)//|:	:/|):	??	:/|\:	:)/|\:	:(|):	??	:(|\:	:)|\\:
1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23
For 14 degrees I have two competing commas:
  1. :~|\:, because it is similar to the apotome complement of :)~|:. The problem with it is that its primary value is 23-limit and isn't a valid comma here. This problem can be solved with a suitable secondary value, which may be 23-limit or not, but I have no idea how to find one.
  2. ://|:, because it's spartan and its primary comma fits the value perfectly. The problem is that it will lead to 24 degrees being notated as :)||(: which is not consistent with flag arithmetic.
But the real problem is for 17 (and therefore also 21) degrees. The obvious accidentals for these degrees are :(|~: and :|\\:, but :|\\: is the exact comma that was removed to make the set compatible with a 400edo val in the first place (and indeed 11*2 ≠ 21).
"A secondary comma for unaccented symbol X" has come to mean simply "a comma for symbol X-plus-accents, in the Extreme precision (Olympian) JI notation". Also, "a secondary comma for symbol X having only schisma accents" means "a comma for symbol X-plus-mina-accents, in the Extreme precision notation".

There are no accented versions of :~|\: in Olympian, so it has no secondary commas.

Because 400edo is consistent in the 19-odd-limit but not the 23-odd-limit we should not use any commas with 23 in them if it can be avoided. That eliminates the symbols :|~: :~|\: :/|~: .

The symbol ://|: is the only unaccented symbol whose comma is valid as 14 degrees, and it is a common symbol, being Spartan. The fact that its apotome complement doesn't have valid flag arithmetic is far from being a sufficient reason to reject it.

And if you want an unaccented notation for 400edo, there is no reason not to use the only symbols whose commas are valid as 17 and 21 degrees, namely :(|~: and :|\\: even though they fail flag arithmetic. The only alternatives are :'::/|): and :.::(|\: or :.::/|\: and :'::(|): .

I prefer :)|(: to :~|: for 3 degrees because it comes from a less advanced symbol set and notates a lower prime-limit comma, despite the fact that its apotome complement ://||: (35 degrees) has bad flag arithmetic. But then I prefer the evolutionary notation over the revolutionary and so prefer not to use multi-shaft symbols. It would be perfectly valid to use :~|: and its complement :)//||: if you prefer.

So my preferred accent-free notation for 400edo is:
:)|:	:|(:	:)|(:	:)~|:	:~|(:	:~~|:	:/|:	:)/|:	:|):	:)|):	:|\:	:~|):	:(|(:	://|:	:)//|:	:/|):	:(|~:	:/|\:	:)/|\:	:(|):	:|\\:	:(|\:	:)|\\:
1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23

But maybe we should prefer an accented-athenian notation for it, such as:
:'::|:	:|(:	:)|(:	:'::)|(:	:~|(:	:'::~|(:	:/|:	:'::/|:	:|):	:'::|):	:|\:	:'::|\:	:(|(:	://|:	:':://|:	:/|):	:'::/|):	:/|\:	:'::/|\:	:(|):	:.::(|\:	:(|\:
1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22

In either of the above notations, :(|: could have been substituted for :|\: except that its complement :)||~: is much harder to remember (than :/||: ) and :)||~: has bad flag arithmetic.
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Re: Can all the promethean symbols be realized consistently in the same edo?

Post by Dave Keenan »

רועיסיני wrote: Sun May 28, 2023 6:40 am Second, for 971edo I had kind of an opposite problem, where all the accidentals are mapped to distinct degrees but there are not enough of them, so we have to use accent marks. The schisma maps to 2\971 and the mina to 0, but the JI value for the double mina maps to 1\971, so I devised this Olympian notation:
:``::|:	:,,::)|:	 :)|:	:``::)|:	 :|(:	:``::|(:	 :~|:	 :)|(:	:``::)|(:	 :)~|:	:``::)~|:	:,,::|~:	 :|~:	 :~~|:	:``::~~|:	 :)|~:	 :/|:	:``::/|:	:,,::)/|:	 :)/|:	:``::)/|:	 :|):	:``::|):	:,,::)|):	 :)|):	 :|\:	 :(|:	:``::(|:	 :~|):	 :/|~:	:``::/|~:	 :(|(:	:``::(|(:	 ://|:	:``:://|:	:,,::)//|:	 :)//|:	:``::)//|:	 :/|):	 :(|~:	:``::(|~:	:,,::/|\:	 :/|\:	 :(/|:	:``::(/|:	 :)/|\:	:``::/|\:	 :|\):	 :(|):	:``::(|):	:,,::|\\:	 :|\\:	 :(|\:	:``::(|\:	 :)|\\:	:``::)|\\:
1	 2	 3	 4	 5	 6	 7	 8	 9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	 29	 30	 31	 32	 33	 34	 35	 36	 37	 38	 39	 40	 41	 42	 43	 44	 45	 46	 47	 48	 49	 50	 51	 52	 53	 54	 55	 56
And I guess the places where there is only a need for one accented accidental would have :``: on the lower one in revo as well (which is not the apotome complement of the base accidental of the apotome complement interval). Also, 33 steps could be notated with :~|\: if we can find a valid secondary comma for it.
As above, there are no secondary commas for :~|\: .

That's a good notation for 971edo except for the issue you raise about accents and complements. The apotome-complement of an accented symbol always consists of the complement of the base (or core) symbol, with the accent pointing in the opposite direction. e.g. the complement of :``::(/|: must be :,,::|\): not :``::)/|\: (which you mistyped as :``::/|\: ).

It is not really correct to call it an Olympian notation, because many of those accented symbols do not occur in Olympian, and some of those that do, have commas that map to a different number of degrees of 971:
:``::/|: :``::(|: :``:://|: :,,::/|\: :``::)/|\: and their complements.
It would be more correct to call it an "accented Promethean notation".

I note that the unaccented :~|(: corresponds to 12 degrees. I assume you avoided it because its double-shaft complement does not have valid flag arithmetic.
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Re: Can all the promethean symbols be realized consistently in the same edo?

Post by רועיסיני »

Dave Keenan wrote: Tue Jun 06, 2023 2:48 pm Those are interesting results. I certainly wouldn't have expected any EDO to use all the Promethean symbols. They are too unevenly spaced. They were the result of seeing how far we could push a system with no accents. We looked at every geometrically-viable combination of the 8 flags at-most 3 at a time, and rejected those that were too close in size to others, unless they helped to fill in some gaps in a system consisting of Athenians plus accents.

In case you haven't seen them, these are related:
viewtopic.php?p=4408#p4408
viewtopic.php?f=5&t=413&p=881#p881
viewtopic.php?p=257#p257
I have seen the first two but not the last one. I'll check it when I have time.

Dave Keenan wrote: Tue Jun 06, 2023 6:22 pm So my preferred accent-free notation for 400edo is:
:)|:	:|(:	:)|(:	:)~|:	:~|(:	:~~|:	:/|:	:)/|:	:|):	:)|):	:|\:	:~|):	:(|(:	://|:	:)//|:	:/|):	:(|~:	:/|\:	:)/|\:	:(|):	:|\\:	:(|\:	:)|\\:
1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23
It all started by finding large sets of symbols that can be set to degrees of an EDO in a way that agrees with flag arithmetic, so it hurts to see it broken in the end with :|\: and :|\\: (and to a lesser extent with :)||(: or ://||:), but I agree that these came from stronger considerations

Dave Keenan wrote: Tue Jun 06, 2023 7:37 pm That's a good notation for 971edo except for the issue you raise about accents and complements. The apotome-complement of an accented symbol always consists of the complement of the base (or core) symbol, with the accent pointing in the opposite direction.
Cool, I didn't know that extension to the apotome complement rule. It's a bit sad that we get more accents that point in different directions to the accidentals than necessary, but if that's the rule I accept it. Considering that, it may be better to fill 2 degree gaps with lower accents also before :)/|\:, to make the unaccented double-shafts recapitulate a subsequence of the unaccented single shafts. I'll see if it works fully some time in the future.

Dave Keenan wrote: Tue Jun 06, 2023 7:37 pm It is not really correct to call it an Olympian notation, because many of those accented symbols do not occur in Olympian, and some of those that do, have commas that map to a different number of degrees of 971... It would be more correct to call it an "accented Promethean notation".
Okay, so the secondary commas and the terms "Herculran" and "Olympian" are only specific closed sets of symbols, which are listed in the spreadsheet. Good to know! What about things like 119:120 being notated as :~|(: in table 1 of the pdf? How is this phenomenon called? Does it have any validity in EDO and more general regular temperament notations?

Dave Keenan wrote: Tue Jun 06, 2023 7:37 pm I note that the unaccented :~|(: corresponds to 12 degrees. I assume you avoided it because its double-shaft complement does not have valid flag arithmetic.
I think that was completely by accident, since this notation doesn't have double shaft flag arithmetic anyway (for example :|(: is 5 degrees so :/||): is 87 degrees but :(|): + :/|): = 39 + 49 = 88 degrees), so there's no reason not to use :~|(: for 12 degrees.

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(:~|: + :|\: - :~|\:).
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Re: Can all the promethean symbols be realized consistently in the same edo?

Post by Dave Keenan »

רועיסיני wrote: Wed Jun 07, 2023 3:08 am Okay, so the secondary commas and the terms "Herculean" and "Olympian" are only specific closed sets of symbols, which are listed in the spreadsheet. Good to know! What about things like 119:120 being notated as :~|(: in table 1 of the pdf? How is this phenomenon called? Does it have any validity in EDO and more general regular temperament notations?
That is a good question. You have made me realise that, while it is true that "Herculean" and "Olympian" are specific closed sets of symbols, the set of secondary commas for unaccented symbols cannot be limited to the set of Olympian primary commas.

When I wrote that secondary comma "has come to mean" comma for accented version in Olympian, I was acknowledging that it has not always meant that. But you have made me realise that it still cannot mean that, and so there is still some chance of justifying the use of :~|\: for 14°400edo or 33°971edo (14\400 or 33\971).

I can weasel out of the specific example you give, where 119:120 is in table 1 of the pdf but not in Olympian, by claiming that it is not intended to be notated by the given symbol. The evidence for this is that it is called "sum of flags"; it is not given a comma name; and it is set in smaller type than the notated commas. I can argue that it was only included to show that (a) the primary comma is not the same as the comma obtained by combining the commas of its flags, and (b) it is however very close to its sum of flags.

I can further argue that its sum-of-flags comma was not considered to be worth using for notation, because it notated such a complex and therefore unlikely-to-ever-be-used ratio, namely 7×17/5.

I can make the same argument for the other "sum of flags" commas in table 1.

But I cannot make that argument for the case of 29S as a secondary comma for :(|: even though it does not appear in Olympian. We definitely still want to consider 29S as a secondary comma for :(|: because it is used that way in the prime-factor JI notation.

29S does not appear in Olympian because it is competing with 13/17S for a 233eda capture zone. We preferred 13/17S despite the fact that 29 is slightly more popular than 17/13 according to both the Scala archive statistics and N2D3P9. I believe the reason is that we wanted to complete the 17-limit diamond. There is no other 17/13 comma in Olympian.

There are two commas for extremely unpopular ratios included in Olympian, simply because we couldn't find any better comma to fill their 233eda slots at the time. These two outliers have N2D3P9 values greater than 1000, and may be replaced in future, but all the other commas have N2D3P9 values less than 307.

Therefore I could not complain if you found a comma with N2D3P9 less than 307 that falls within the Olympian capture zone for :~|\: and maps to 14°400edo or 33°971edo.

The way that George and I found commas was to work our way down the list of 2,3-free ratios obtained from the Scala archive, combining them with powers of 3 from -14 to +14 (double-sharp to double-flat), and whatever power of 2 was required to bring their size into the range -600 ¢ to +600 ¢, and to look at only those whose absolute size was less than half an apotome.

A more modern approach would be to use a list of 2,3-free ratios in order of increasing N2D3P9 up to 307, as some of these may not appear in the Scala archive at all.

In searching, I was reminded that this has already been done, and the result is in the spreadsheet attached to this post: viewtopic.php?p=2636#p2636
A better description of the "LPEI-badness" used to compare candidate commas in that spreadsheet can be found here:
viewtopic.php?p=4451#p4451

So I can give you two secondary commas for :~|\: that map to both 14°400edo and 33°971edo and have N2D3P9 < 307.
 35/13S = [	-22	13	1	1	0	-1	⟩
175/19S = [	  0	-2	2	1	0	0	0	-1	⟩
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Re: Can all the promethean symbols be realized consistently in the same edo?

Post by רועיסיני »

Dave Keenan wrote: Wed Jun 07, 2023 1:27 pm So I can give you two secondary commas for :~|\: that map to both 14°400edo and 33°971edo and have N2D3P9 < 307.
 35/13S = [	-22	13	1	1	0	-1	⟩
175/19S = [	  0	-2	2	1	0	0	0	-1	⟩
What exactly do you mean by "map to"? 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.
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Re: Can all the promethean symbols be realized consistently in the same edo?

Post by רועיסיני »

Here is a revised accented Promethean notation for 971edo:
:``::|:	:,,::)|:	 :)|:	:,,::|(:	 :|(:	:,,::~|:	 :~|:
1	 2	 3	 4	 5	 6	 7
 :)|(:	:``::)|(:	 :)~|:	:,,::~|(:	 :~|(:	 :|~:	 :~~|:
 8	 9	10	11	12	13	14
:,,::)|~:	 :)|~:	 :/|:	:``::/|:	:,,::)/|:	 :)/|:	:,,::|):
15	16	17	18	19	20	21
 :|):	:``::|):	:,,::)|):	 :)|):	 :|\:	 :(|:	:``::(|:
22	23	24	25	26	27	28
 :~|):	 :/|~:	:,,::(|(:	 :(|(:	:``::(|(:	 ://|:	:``:://|:
 29	 30	 31	 32	 33	 34	 35
:,,::)//|:	 :)//|:	:,,::/|):	 :/|):	 :(|~:	:``::(|~:	:,,::/|\:
 36	 37	 38	 39	 40	 41	 42
 :/|\:	 :(/|:	:``::(/|:	 :)/|\:	:,,::|\):	 :|\):	 :(|):
 43	 44	 45	 46	 47	 48	 49
:``::(|):	:,,::|\\:	 :|\\:	 :(|\:	:``::(|\:	 :)|\\:	:``::)|\\:
 50	 51	 52	 53	 54	 55	 56
:,,::)||(:	 :)||(:	:,,::~||(:	 :~||(:	:``::~||(:	 :||~:	 :~~||:
 57	 58	 59	 60	 61	 62	 63
:,,::)||~:	 :)||~:	 :/||:	 :)/||:	:``::)/||:	:,,::||):	 :||):
 64	 65	 66	 67	 68	 69	 70
:``::||):	 :)||):	:``::)||):	:,,::||\:	 :||\:	 :(||:	:``::(||:
 71	 72	 73	 74	 75	 76	 77
 :~||):	 :/||~:	 :(||(:	:``::(||(:	 :~||\:	:,,:://||:	 ://||:
 78	 79	 80	 81	 82	 83	 84
 :)//||:	:``::)//||:	 :/||):	:``::/||):	 :(||~:	:``::(||~:	:,,::/||\:
 85	 86	 87	 88	 89	 90	 91
 :/||\:
 92
The double shaft cores recapitulate the single shaft cores except for the symbols for 59, 67, 69, 72, 74 and 82 degrees, for which I'd need either 10\971 to be notated as :.::~|(: or 33\971 to be notated as :~|\:, 18\971 to be notated as :.::)/|: and 23\971 to be notated as :.::)|):. The first is problematic because :~|\: is invalid for 33\971 and the other two became a problem because the "semi-complements" :)/|: and :)|): add up to 45\971, which is two steps more than :/|\: = 43\971.
I think that is the best trade-off, unless you prefer using :~|\: for 33 (justified by some comma we can both agree on), because I prefer to not introduce another accent to this notation, especially not on non-Athenian symbols.
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Re: Can all the promethean symbols be realized consistently in the same edo?

Post by רועיסיני »

Another option that uses less Prometheans and has the schisma but only on Athenian symbols while exhibiting perfect core recapitulation is:
:``::|:	:,,::)|:	 :)|:	:,,::|(:	 :|(:	:,,::~|:	 :~|:
1	 2	 3	 4	 5	 6	 7
 :)|(:	:``::)|(:	:.::~|(:	:,,::~|(:	 :~|(:	 :|~:	 :~~|:
 8	 9	10	11	12	13	14
:,,::)|~:	 :)|~:	 :/|:	:``::/|:	:'::/|:	:.::|):	:,,::|):
15	16	17	18	19	20	21
 :|):	:``::|):	:.::|\:	:,,::|\:	 :|\:	 :(|:	:``::(|:
22	23	24	25	26	27	28
 :~|):	 :/|~:	:,,::(|(:	 :(|(:	:``::(|(:	 ://|:	:``:://|:
 29	 30	 31	 32	 33	 34	 35
:,,::)//|:	 :)//|:	:,,::/|):	 :/|):	 :(|~:	:``::(|~:	:,,::/|\:
 36	 37	 38	 39	 40	 41	 42
 :/|\:	 :(/|:	:``::(/|:	 :)/|\:	:,,::|\):	 :|\):	 :(|):
 43	 44	 45	 46	 47	 48	 49
:``::(|):	:,,::|\\:	 :|\\:	 :(|\:	:``::(|\:	 :)|\\:	:``::)|\\:
 50	 51	 52	 53	 54	 55	 56
:,,::)||(:	 :)||(:	:,,::~||(:	 :~||(:	:``::~||(:	 :||~:	 :~~||:
 57	 58	 59	 60	 61	 62	 63
:,,::)||~:	 :)||~:	 :/||:	:``::/||:	:'::/||:	:,,::||):	 :||):
 64	 65	 66	 67	 68	 69	 70
:``::||):	:'::||):	:.::||\:	:,,::||\:	 :||\:	 :(||:	:``::(||:
 71	 72	 73	 74	 75	 76	 77
 :~||):	 :/||~:	 :(||(:	:``::(||(:	:'::(||(:	:,,:://||:	 ://||:
 78	 79	 80	 81	 82	 83	 84
 :)//||:	:``::)//||:	 :/||):	:``::/||):	 :(||~:	:``::(||~:	:,,::/||\:
 85	 86	 87	 88	 89	 90	 91
 :/||\:
 92
I'm not sure which one of them I like more, but I suspect you'll prefer this one.
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Dave Keenan
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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 1:59 am
Dave Keenan wrote: Wed Jun 07, 2023 1:27 pm So I can give you two secondary commas for :~|\: that map to both 14°400edo and 33°971edo and have N2D3P9 < 307.
 35/13S = [	-22	13	1	1	0	-1	⟩
175/19S = [	  0	-2	2	1	0	0	0	-1	⟩
What exactly do you mean by "map to"?
I took the matrix product of the patent map of the EDO with the vector of the comma.
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.

Of course when designing a notation like Trojan, intended to work for many EDOs, and even tunings that are not EDOs, you have no choice but to temper only prime 3.
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