## 270edo in athenian+trojan only

FloraC
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### 270edo in athenian+trojan only

I talked about the significance of 224edo and 270edo on this forum. Later I named "the five essential comma-size 13-limit edos" (41, 46, 53, 58 and 72), and "the four essential athena-size 13-limit edos" (183, 224, 270, and 311).

There are other candidates including 198, 212 and 217, all of which I really love, but the "four essential" set was carefully picked to represent the temperament of {2080/2079, 3025/3024, 4096/4095}. I hope to cover 183edo and 311edo later.

I'm showing 224edo notation first because we knew athenian is designed for it. Since 224edo is sharp-21 and 270edo is sharp-26, extrapolating 224edo notation to 270 will render three gaps. I'd like to see if they can be filled by trojan.

1\:
2\:
3\:
4\:
5\:
6\:
7\:
8\:
9\:
10\:

This uses every symbol from spartan in addition to from athenian, altho the harmonic 17 in 224edo is over 40% sharp. The only unused athenian symbol is , which is interchangeable with in 224, but we'll see they're distinct in 270. It turns out that we can map the symbols from 224edo to 270edo:

1\224 > 1\270
2\224 > 2\270
3\224 > 4\270
4\224 > 5\270
5\224 > 6\270
6\224 > 7\270
7\224 > 9\270
8\224 > 10\270
9\224 > 11\270
10\224 > 12\270

So the gaps are at 3\, 8\, and 13\. Now 8\ is of the size between 45/44 and 55/54 and is nicely covered by . For 3\ we may take from trojan.

Now, unfortunately, there's no symbol from athenian+trojan available for 13\, the semisharp step. But look, it's the semisharp! We could take the conventional semisharp for it. The result:

1\:
2\:
3\:
4\:
5\:
6\:
7\:
8\:
9\:
10\:
11\:
12\:
13\: +
where + denotes a conventional semisharp.

It's kinda surprising that 224edo uses all of athenian but either or whereas 270edo can't be accomplished by athenian+trojan without the conventional semisharp. It'd be nice if the conventional semisharp was never needed and could take another from trojan instead.

FloraC
Posts: 19
Joined: Sat Nov 07, 2020 4:19 am

### Re: 270edo in athenian+trojan only

I still find the system a little bit glitchy because when I try extending it to 494edo:

2\:
4\:
6\:
7\:
9\:
11\:
13\:
14\:
16\:
18\:
20\:
22\:

it doesn't work well with changing to 6\ and changing to 7\.

For 494edo, since
I think, only Athenian with schisma accents.
said in viewtopic.php?f=5&t=413, and for 270edo, the schisma accent is absolutely undesired, a preferable result would be to get an athenian+trojan system that maps consistently as:

(Category 1: these show up for all the edos in question)

1\224 ~ 1\270 ~ 2\494
2\224 ~ 2\270 ~ 4\494

3\224 ~ 4\270 ~ 7\494 (?)
4\224 ~ 5\270 ~ 9\494
5\224 ~ 6\270 ~ 11\494
6\224 ~ 7\270 ~ 13\494

7\224 ~ 9\270 ~ 16\494
8\224 ~ 10\270 ~ 18\494
9\224 ~ 11\270 ~ 20\494
10\224 ~ 12\270 ~ 22\494

(Category 2: hidden for 224edo, required for 270edo, optional for 494edo)

2\224 ~ 3\270 ~ 5\494
6\224 ~ 8\270 ~ 14\494
10\224 ~ 13\270 ~ 23\494

or

(Category 3: hidden for 224edo, required for 270edo, optional for 494edo)

3\224 ~ 3\270 ~ 6\494
7\224 ~ 8\270 ~ 15\494
11\224 ~ 13\270 ~ 24\494

There's clearly a MOS pattern implied, if either of category 2 or 3 is picked in entirety.

For the gap between 13\494 and 16\494, can only be 14\. Seems the one from category 2 has to be chosen. (That and are only 1.4 cents apart is seriously unsettling. ) 46/45 maps to 16\ and thus is useless unless redefined.

We should probably use the patent val of 494edo in 2.3.5.7.11.13.17.23, so will be 6\ and will be 7\ (we don't have anything for 5\). 224 & 270gi will give you 4131/4096 as 3\270 and 736/729 as 4\270. This option retains the relative sizes between them, but 270gi is a bad mapping. 224gi & 270 will give you 4131/4096 as 4\270 and 736/729 as 3\270 with sizes reversed, but the advantage is that it doesn't show up in 224edo so the bad mapping 224gi doesn't matter. A perhaps better alternative would be to redefine them in terms of 13-limit intervals.

So my result is now

1\224 ~ 1\270 ~ 2\494
2\224 ~ 2\270 ~ 4\494
(3\224) ~ 3\270 ~ 6\494
3\224 ~ 4\270 ~ 7\494
4\224 ~ 5\270 ~ 9\494
5\224 ~ 6\270 ~ 11\494
6\224 ~ 7\270 ~ 13\494
(6\224) ~ 8\270 ~ 14\494
7\224 ~ 9\270 ~ 16\494
8\224 ~ 10\270 ~ 18\494
9\224 ~ 11\270 ~ 20\494
10\224 ~ 12\270 ~ 22\494

13\270 lacks a symbol anyway, and the MOS pattern sadly broke by picking 6\ and 14\.

Dave Keenan
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### Re: 270edo in athenian+trojan only

Thanks for those. I haven't had a chance to think deeply about them yet. But here are a few shallow thoughts:

I note that the Stein semisharp is available as a forum smiley :>: and the narrow semiflat is :<: . When editing a post, if you click on "View more smilies" you'll see them near the end of the list. The smiley code :+: gives the Wilson plus symbol which is an alternative to as a 5-comma symbol.

As you may know, George and I agreed on the following notation for 183edo:
183:
as given in the Scala file https://sagittal.org/sag_et.par

This list of symbols in N2D3P9 order may be of interest.

As might this Euler diagram of the relationships between sagittal subsets. It shows that Trojan is not strictly a part of the hierarchy.

Unfortunately it doesn't name regions as "extensions", as in SMuFL. The Athenian extension is Athenian minus Spartan. The Trojan extension is Trojan minus Athenian (asymmetric set difference). The Promethean extension is Promethean minus the union of Trojan and Athenian. The Herculean extension is Herculean minus Promethean. etc.

An alternative to as the semisharp in 270-edo, and many large divisions, is the Promethean the symbol for the 5/49-medium-diesis 405/392.

Dave Keenan
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### Re: 270edo in athenian+trojan only

I found an email to George in 2014 in which I wrote:
"There are also several [EDO notations] that I believe we have agreed on that are not in the XH article or in Scala. These are 270, 282, 306, 311, 342, 388, 494, 612, 2460. These should be added to the Scala sag_et.par file."

But we never got to do this. I note that 494, 612 and 2460 were assumed to correspond to using most of the symbols from Promethean, Herculean and Olympian respectively. But we've since learned that the 47-EDA subset of Promethean has a slightly different mapping from 494-EDO (inconsistent re prime 19).

Further searching of email turned up this from George in 2012:
270:

And this from George in 2006:
311:

I didn't find any of the others from the list above.

FloraC
Posts: 19
Joined: Sat Nov 07, 2020 4:19 am

### Re: 270edo in athenian+trojan only

I note that the Stein semisharp is available as a forum smiley and the narrow semiflat is  . When editing a post, if you click on "View more smilies" you'll see them near the end of the list. The smiley code gives the Wilson plus symbol which is an alternative to as a 5-comma symbol.
Good to know.
As you may know, George and I agreed on the following notation for 183edo:
183:
as given in the Scala file https://sagittal.org/sag_et.par
Yup 183 should be ez.
This list of symbols in N2D3P9 order may be of interest.
Interestingly rates higher than , and it has lower fifth-slope too. Why isn't 2\183 instead?
As might this Euler diagram of the relationships between sagittal subsets. It shows that Trojan is not strictly a part of the hierarchy.
An alternative to as the semisharp in 270-edo, and many large divisions, is the Promethean the symbol for the 5/49-medium-diesis 405/392.
I still wish 270edo and 311edo could be notated without any promethean extensions. I think 270edo basically passes. I checked 311edo but it seems impossible. There are gaps at either 3\ or 5\, and 11\, unless

It does use exactly every symbol from athenian+trojan, but it's probably not a great notation.

@cmloegmcluin cuz we had a talk on discord.

FloraC
Posts: 19
Joined: Sat Nov 07, 2020 4:19 am

### Re: 270edo in athenian+trojan only

I viewed trojan extension as a library to auxilliarate 224edo so as to extend it to something a little bit larger but still athenian-like, such as 270 or 311. Maybe it's promethean that works in that role?

Dave Keenan
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### Re: 270edo in athenian+trojan only

FloraC wrote: Wed May 12, 2021 5:48 pm
As you may know, George and I agreed on the following notation for 183edo:
183:
as given in the Scala file https://sagittal.org/sag_et.par
Interestingly rates higher than , and it has lower fifth-slope too. Why isn't 2\183 instead?
Good question. The only reason I can find is that with , in the Revo flavour of Sagittal, the notation's sequence of double-shaft flag combinations, matches a part of its single-shaft sequence.

     1  2  3 4  5  6  7  8  9  10
183:

11 12 13 14 15 16 17

I note that this is true of George's 270edo notation too. But not his 311edo notation.

The full set of apotome-complements is given in this figure from page 24 of http://sagittal.org/sagittal.pdf

But you should feel free to use for 2\183, particularly if you are using the Evo flavour.
I still wish 270edo and 311edo could be notated without any promethean extensions. I think 270edo basically passes. I checked 311edo but it seems impossible. There are gaps at either 3\ or 5\, and 11\, unless
311:
1  2  3  4 5  6  7 8  9  10 11 12 13 14 15
It does use exactly every symbol from athenian+trojan, but it's probably not a great notation.
Yes, it has the problem that the symbols  ,  , and are not mapped according to their primary commas 17C, 19/5C, 23C and 23/5S. These map to degrees 4, 7, 4 and 10, not 3, 4, 5 and 11 as you have them above.

BTW,
183edo is {1, 3, 5, 7, 9, 11, 13, 15, 17, 23}-consistent (no 19).
217edo is 21-odd-limit consistent (no 23).
224edo is {1, 3, 5, 7, 9, 11, 13, 15, 23}-consistent (no 17 or 19).
270edo is {1, 3, 5, 7, 9, 11, 15, 17, 19, 21}-consistent (no 13 or 23). [Edit: Corrected previous claim that 23 could be combined with 17.]
270edo is also {1, 3, 5, 7, 9, 11, 13, 19, 23}-consistent (no 15, 17 or 21).
270edo is also {1, 3, 5, 7, 9, 11, 13, 15, 21}-consistent (no 17, 19 or 23).
311edo is fully 23-odd-limit-consistent (in fact it's 41-odd-limit consistent!).
494edo is only 17-odd-limit-consistent (no 19 or 23). [Edit: Corrected previous claim that 23 was included.]
FloraC wrote: Wed May 12, 2021 5:54 pm I viewed trojan extension as a library to auxilliarate 224edo so as to extend it to something a little bit larger but still athenian-like, such as 270 or 311. Maybe it's promethean that works in that role?
That's correct. The Trojan set is intended for notating (any tuning) relative to nominals, sharps and flats in a chain of 700 cent fifths (699.1831¢ to 700.7552¢). When notating EDOs using their best fifth, whose size is outside that range, then the Trojan and Promethean extensions should be treated equally.

Historically, Athenian and Promethean came first, then a subset was chosen from Promethean for notating relative to 12edo and called Trojan. When we're not notating relative to 12edo, we ignore the Trojan boundary and just treat the symbols in the Trojan extension as if they are members of the Promethean extension.
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cmloegmcluin
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Real Name: Douglas Blumeyer (he/him/his)
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### Re: 270edo in athenian+trojan only

Sorry just seeing this now. Your @ mention missed, because I've made my name too bizarre to spell easily. Sorry about that.
FloraC wrote: Wed May 12, 2021 5:48 pm Why isn't 2\183 instead?
I happen to know that Dave has got a deadline on a work project right now and I must protect him from procrastination But I'm not sure about this one. If it has anything to do with consistency with other related EDO's notations, I don't expect to find it. But I can look at it in and of itself.
• It's not because of a difference in flag arithmetic.
• They're both consistent with 2\183.
• Neither nor creates a symbol reversal.
• What about symbol flag recapitulation (in the double-shafts when using Revo flavor)? The apotome complement of is which is why that's 15\183. If it was then by apotome complements 15\183 would be instead. That in turn would call for 6\183 to change from to  . And if was 6\183, then its apotome complement would need to be 11\183, replacing  , which also checks out. How does fare as 6\183, compared with  ? Well, by flag arithmetic, doesn't have anything going for it, while is clearly + where is 3\183 so that's actually a net win. It's also consistent with 6\183 and doesn't create a symbol reversal.
• I'm putting this bit last because my notes might be wrong here. I had written that a consideration for sagittal assignment to an EDO step was the proximity of its primary comma's tempered value to the EDO step size, but that's a restatement or edge-case generalization of the consistency requirement, so maybe what I meant was its untempered value should be as close as possible to its tempered value. If that's true, then would have an edge over  , because 2\183 = 13.11¢, = 9.69¢, and = 14.73¢.
So, I suppose wouldn't be a bad choice for 2\183, depending on if I got that value proximity rule right or not, and how important it is.
FloraC wrote: Wed May 12, 2021 5:54 pm I viewed trojan extension as a library to auxilliarate 224edo so as to extend it to something a little bit larger but still athenian-like, such as 270 or 311. Maybe it's promethean that works in that role?
Short answer: promethian is the main extension to athenian, yes. For some reason I have in my notes that athenian is associated with 217-EDO. And I associate trojan with 240-EDO because it's got 20 symbols and 20×12=240. Promethian is associated with 494-EDO. But when I say this I don't mean it was designed to notate 494, so much as 494 was a convenient as a grid to quantize capture zone boundaries to for a sweet-spot subset of JI intervals for the JI notation. Happy to go into detail more on that if you're interested.

Re-reading what I said on Discord, I think I had gotten myself really confused, and passed it on to you. Sorry about that. When I said we had 26 symbols to work with for 25 slots, I was thinking we'd need 25 different symbols, when of course we only need half that (rounded up) (because the other half will be pointing downward versions of the same symbols). Maybe you gave me a pass on that one and didn't correct me, or I didn't make it clear that's what I was thinking. We only need 13 symbols, as you clearly knew already.

And when I said athenian & trojan symbols would just cover us, what I was really thinking was "all the single-shaft core symbols" i.e. all the ones with one shaft and no accents, which includes athenian, trojan, and promethian.

I dunno how I came up with that 26 number. I'm pretty sure there's 31 of them: 8 spartan, 5 athenian, 3 trojan. So if we got really lucky, maybe we could have even done it with only athenian because 8+5=13.