494EDO and Promethean accidentals

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Ash9903b4
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494EDO and Promethean accidentals

Post by Ash9903b4 »

I’ve been trying to represent 494EDO in Sagittal, but I don’t know how to make it work. I’ve read somewhere on this forum that the Promethean level was specifically designed for 494, but there are some inconsistencies I can’t figure out.

First of all, the Athenian subset works perfectly, with consistent flag arithmetic and everything, and with symbols spaced by either 2 or 3 degrees of 494:

:|(: 2, :)|(: 4, :~|(: 6, :/|: 9, :|): 11, :(|: 14, :(|(: 16, :/ /|: 18, :/|): 20, :/|\: 22, :(|): 25, :(|\: 27

But once extended to Promethean, there are a few problems. :~~|: (99/98) maps to 7 degrees, which doesn’t have an exact half, and :)|: (513/512) maps to 1 degree, which is inconsistent with :|(: and :)|(:. Do I need to start using schisma accents here, or is Promethean enough?

Ash9903b4
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Re: 494EDO and Promethean accidentals

Post by Ash9903b4 »

I did come up with a consistent mapping, but at the expense of breaking the Athenian symbol :)|(:, and switching its ordering with :~|:, so I’m not sure how useful this is.

:)|: = 1°
:|(: = 2°
:)|(: = 3° (2+1)
:~|: = 4°
:)~|: = 5° (1+4)
:~|(: = 6° (4+2)
:|~: = 7°
:~~|: = 8° (4+4), also :)|~: (1+7)
:/|: = 9°
:)/|: = 10° (1+9)
:|): = 11°
:)|): = 12° (1+11)
:|\: = 13°
:(|: = 14°
:~|): = 15° (4+11)
:(|(: = 16° (14+2), also :/|~: (9+7)
:~|\: = 17° (4+13)
:/ /|: = 18° (9+9)
:)/ /|: = 19° (1+9+9)
:/|): = 20° (9+11)
:(|~: = 21° (14+7)
:/|\: = 22° (9+13)
:(/|: = 23° (14+9), also :)/|\: (1+9+13)
:|\): = 24° (13+11)
:(|): = 25° (14+11)
:|\ \: = 26° (13+13)
:(|\: = 27° (14+13), also :)|\ \: (1+13+13)

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Dave Keenan
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Re: 494EDO and Promethean accidentals

Post by Dave Keenan »

Ash9903b4 wrote:
Sun Mar 15, 2020 11:39 am
I’ve been trying to represent 494EDO in Sagittal, but I don’t know how to make it work. I’ve read somewhere on this forum that the Promethean level was specifically designed for 494, but there are some inconsistencies I can’t figure out.
Hi Ash. Welcome to the forum. You're certainly on the bleeding edge in wanting to notate 494-edo. It isn't quite correct to say that Promethean was designed for 494-edo. It's more that 494-edo (really 47-eda [equal division of the apotome {chromatic semitone}]) was used to help decide where to close Promethean, i.e. where to stop generating what might otherwise have been an almost endless series of arrow-like symbols involving every possible combination of flags, many of which would have been very close in size to one another.

But that certainly does suggest that we ought to be able to make a notation for 494-edo using only Promethean symbols. However I could not find, in searching through emails, anywhere that George or I proposed such a notation, let alone agreed on it. Good on you for having a go.

For other readers, Promethean can be seen here: http://sagittal.org/SagittalJI.gif, where you'll notice that it actually notates 52 ratios to the apotome, because you can count 26 symbols to the half-apotome (56.8c). So there are 5 symbols (3 single-shaft) in Promethean that are outside any 47-eda set.
First of all, the Athenian subset works perfectly, with consistent flag arithmetic and everything, and with symbols spaced by either 2 or 3 degrees of 494:

:|(: 2, :)|(: 4, :~|(: 6, :/|: 9, :|): 11, :(|: 14, :(|(: 16, :/ /|: 18, :/|): 20, :/|\: 22, :(|): 25, :(|\: 27
I can confirm that the above are all correct, based on the primary comma role for every symbol.
But once extended to Promethean, there are a few problems. :~~|: (99/98) maps to 7 degrees, which doesn’t have an exact half, and :)|: (513/512) maps to 1 degree, which is inconsistent with :|(: and :)|(:. Do I need to start using schisma accents here, or is Promethean enough?
I did find email from George that mentioned that :~~|: and :)/|\: were additional to 47-eda, but are required to complete the Olympian level. Fortunately, we don't need to use :~~|: for 7 degrees because we have :|~: , the 23-comma (736/729), for 7 degrees. Also, both :(|(: and :/|~: map to 16 degrees, so if one of them breaks the flag arithmetic, we can ditch that one.

The other thing to be aware of is that while 494-edo is 17-odd-limit consistent (and possibly {1, 3, 5, 7, 9, 11, 13, 15, 17, 23}-consistent), it is not 19-odd-limit consistent, so you have two choices for how to map prime 19, which relates to the mapping of :)|: and other symbols containing that flag (left scroll or rai).

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Dave Keenan
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Re: 494EDO and Promethean accidentals

Post by Dave Keenan »

Good work on that proposed notation above.

I just figured out that 494-edo is not a very good approximation of 47-eda. Here's how Promethean notates 47-eda, from the file sag_ji2.par that we made for Scala. The offset symbols with the red text are not part of 47-edo. Note that we have allowed flag arithmetic to break rather than apply :)|(: to the wrong degree.

Symbol	Comma	Degree of 47-eda
--------------------------------
:)|:	513/512		1
:|(:	5120/5103	2
:~|:	2187/2176	3
:)|(:	896/891		4
:)~|:	144/143		5
:~|(:	4131/4096	6
:|~:	736/729		7
  :~~|:	99/98		7.5
:)|~:	19683/19456	8
:/|:	81/80		9
:)/|:	41553/40960	10
:|):	64/63		11
:)|):	57/56		12
:|\:	55/54		13
:(|:	45927/45056	14
:~|):	49/48		15
:/|~:	46/45		16
  :(|(:	45/44		16.3
:~|\:	16767/16384	17
:/ /|:	6561/6400	18
:)/ /|:	416/405		19
:/|):	36/35		20
:(|~:	176/171		21
:/|\:	33/32		22
:(/|:	4096/3969	23
  :)/|\:	405/392		23.5
:|\):	8680203/8388608	24
:(|):	729/704		25
:|\ \:	373977/360448	26
:(|\:	8505/8192	27
:)|\ \:	885735/851968	28

For 494-edo, it might be preferable to use only Athenian cores, with schisma accents. In this post I wrote:
George and I aren't so sure the Promethean [JI notation] level was a good idea (a bit like the Greek gods not thinking Prometheus' gift of fire to the mortals was a good idea ;)). You may find, somewhat paradoxically, that you actually need more core symbols in Promethean than you do in Olympian. That's because the most common ratios all have Olympian notations that use Athenian cores. I think that cryptic comment in George's brilliant JI notation spreadsheet, "The herculean & olympian levels are highly recommended", should actually read, "The promethean level is deprecated". After all, there's nothing wrong with Athenian. But the Promethean symbols have to exist, to be used for less common commas in Herculean and Olympian

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Dave Keenan
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Re: 494EDO and Promethean accidentals

Post by Dave Keenan »

If we simply adopt the above 47-eda notation as a notation for 494-edo, we could justify the use of :~|: for 3°494 by interpreting it as :.::~|: with its 5-schisma accent dropped. You can see in the JI Notation Spreadsheet, that :.::~|: represents the 5×17-kleisma (256/255).

To justify the use of :(/|: as 23°494, we have to interpret it as :n::(/|: with its mina accent dropped, where :n::(/|: represents the 31-medium-diesis (32/31). And similarly its apotome-complement :|\): can be 24°494 as the 31-large-diesis (67797/65536).

Ash9903b4
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Re: 494EDO and Promethean accidentals

Post by Ash9903b4 »

All of that makes sense, although I'd personally like to preserve flag arithmetic as much as possible.

I still have a few questions though.
Are 494edo and 47eda different enough for it to be useful to use different notations for them? In particular for 494, would you recommend using only Athenian symbols with schisma accents, the full Promethean notation, or some compromise? And if a compromise is feasible, are there tiers to the Promethean level, with some symbols being safer or more commonly used than others?

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Re: 494EDO and Promethean accidentals

Post by Dave Keenan »

Ash9903b4 wrote:
Wed Mar 18, 2020 2:44 pm
All of that makes sense, although I'd personally like to preserve flag arithmetic as much as possible.
Fair enough.
I still have a few questions though.
Are 494edo and 47eda different enough for it to be useful to use different notations for them?
Yes.
In particular for 494, would you recommend using only Athenian symbols with schisma accents, the full Promethean notation, or some compromise? And if a compromise is feasible, are there tiers to the Promethean level, with some symbols being safer or more commonly used than others?
I think, only Athenian with schisma accents. It would be nice if the unaccented symbols could be regularly-spaced, but since 47 is prime, that's impossible. But if you had a particular temperament, and hence a particular generator, in mind for producing useful subsets of 494-edo, that might inform which alterations should have unaccented symbols, which might require using some non-Athenian Prometheans.

A good rule of thumb is that you can determine if one symbol is more common than another by summing the prime factors above 3 in its comma ratio, irrespective of whether they are in the numerator or denominator. The symbol with the lowest sum will be more common. See the table at the end of this post, viewtopic.php?p=299#p299, and the paragraph preceding the table.

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