198edo

FloraC
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198edo

Post by FloraC »

Hello. I'm new here but you may have known about me. Straight to the topic, 198edo has become my latest mega-fav-edo and I'd like to work out a solution in sagittal notation.

Here's a brief intro to 198edo.
  • It's a nice 13-limit temperament yet to receive much attention.
  • It's distinctly consistent in the 15-odd-limit.
  • Its 13-limit TE error is lower than any previous edos.
  • Its 13-limit TE simple badness is only outperformed by two previous edos (72 and 130).
  • It's contorted in the 7-limit with the same mapping as 99edo. An interesting property therefore follows that all odd-ordered undecimal and tridecimal intervals are odd steps, whereas all 7-limit intervals are even steps.
I highly value the notable fact that it tempers out 5120/5103 and 352/351, since that gets rid of certain nuisances in ji (at least to me), and it's the optimal edo for that purpose. However, that also renders the sagittal notation particularly tricky.

I'll call a step of 198edo, ~6 cents, a quarter-comma, for the size of 4 steps is extremely close to the mean of 81/80 and 64/63. Note also 198b val supports meantone and is close to the quarter-comma tuning (slightly sharp than 31edo). 198bbdddd val on the other hand supports superpyth and is also close to the quarter-comma tuning (literally identical to 22edo for 198 = 22*9). It's a sharp-20 system; here are the steps:

1\198: 176/175, 385/384, 540/539, 896/891, 196/195, 325/324, 351/350, 364/363
2\198: schisma (way too large), 121/120, 243/242, 144/143, 169/168
3\198: 99/98, 100/99
4\198: 81/80 :/|: , 64/63, 66/65, 78/77
5\198: 55/54 :|\: , 56/55, 65/64
6\198: Pyth comma (too large), 49/48, 50/49
7\198: 45/44 :(|: , 40/39
8\198: (81/80)^2 :/ /|: , 36/35
9\198: 33/32 :/|\: , 1053/1024
10\198: half-apotome, 28/27
11\198: 729/704 :(|): , 26/25, 27/26
12\198: 25/24
etc.

Several problems can already be seen through this.
  • As mentioned above, both 7/5k :|(: and 13/11k :|(: are tempered out. Very unusual for edos of this size. Consequently 81/80 and 64/63 share a single degree :/|: , same for 33/32 and 1053/1024 :/|\: .
  • 4096/4095 is -1 step, so we see 36/35 is 8 steps whereas 1053/1024 is 9 steps. A possible solution is to assign :/ /|: to 8\ and :/|\: to 9\ – Unfortunately, (81/80)^2 is closer to 7\. We can safely assign :|\: to 5\ and :(|: to 7\.
  • We must fill in the blanks of 1\, 2\, 3\, 6\ and \10. Unfortunately, there're two possible mappings for 17. By the fact that :|(: is tempered out, 4131/4096 :~|(: and 2187/2176 :~|: should be mapped to the same degree by what I learned is known as flag arithmetics. Since :~|(: and :~|: make a pyth comma, both should be 3 steps. That leads to the 198 patent val in the 17-limit. Similarly, there're two possible mappings for 19. Adopting 896/891 :)|(: as 1\ indicates 513/512 :)|: is also 1\. That leads to 198 patent val in the 19-limit.
  • For 2\, we got 144/143 :)~|: , yet it contradicts the flag arithmetics of :~|: and :)|: established above.
  • I'm completely clueless about 6\ and 10\. I confess I'm not familiar with all the symbols.
I know 270edo can be notated without accents. Now 198 = 270 - 72 so I guess 198 should also work. Maybe my understanding of this notation system is still shallow, or this edo is really a hard one. Should this be solved, it might well apply to 205edo and 212edo with minor tweaks. Hope somebody would help.

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volleo6144
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Re: 198edo

Post by volleo6144 »

Welcome!! (...wait, did someone approve this post before I could see it? or was getting rid of the approval process part of what all this maintenance stuff was about?)
7/5k :|(:
This would be either "5:7k" (undirected, with the 5 first because it's smaller; one-sided commas don't have the "1:" written) or "5/7k" (directed, with the 5 in the numerator because it's in the 5120 and not the 5103). It's ... fine; everyone fumbles a little with these things when they start.
45/44 :(|:
44:45 is :(|~:. See above. (:(|: is the 7/11C of 45056:45927; this complicated ratio was chosen so that :(|): can become the apotome-complement of 11M :/|\: 32:33.)
I confess I'm not familiar with all the symbols.
The main website's SagittalJI.gif has the definitions of every unaccented (no :': or :`: or :,: or :.:) symbol as well as a few notable accented ones like :'::/|: (5s + 1/5C = Pythagorean comma). 48:49 specifically is :~|):, and 49:50 and the aforementioned Pythagorean comma are both accented symbols. (There's been some sort of in-joke here about the "Promethean" precision level—with all the unaccented symbols and no accented ones—being bad in some way, but I don't really understand why.)
27:28 (7L) is notated as 8192:8505 (35L) minus the 5s, and 8192:8505 :(|\: is relevant because it's an apotome minus 35:36 (1/35M) :/|):. 26:27 (1/13L) is notated as 8192:8505 plus 4095:4096 (1/455n) :`:, except that the :`: is often omitted.

I would also have no idea which of the symbols to use, especially since staying within the Athenian set won't be possible.

24:25 in JI would be notated as an apotome minus 6400:6561 (1/25S) :/ /|:, which (being a S-diesis and not an M-diesis) doesn't have a single-shaft apotome complement. Mixed/Evo notation would write it as just :\ \!::#:, while pure/Revo notation would write it as :)||(: (with nothing to do with :)|(: except that the apotome-complement of :)|(: is also :/ /||:).
Last edited by cmloegcmluin on Tue Nov 10, 2020 4:18 am, edited 1 time in total.
Reason: fixing a broken smiley code by replacing a pipe "|" with a bang "!". Dave sometimes advocates for "lazy" shafts where "|" can represent either an up or down shaft, but we don't happen to support smiley codes for tha
A random guy who sometimes doodles about Sagittal and microtonal music in general in his free time.

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Re: 198edo

Post by cmloegcmluin »

volleo6144 wrote: Tue Nov 10, 2020 12:51 am Welcome!! (...wait, did someone approve this post before I could see it? or was getting rid of the approval process part of what all this maintenance stuff was about?)
Sorry about the hiccup. Dave was trying to fix a server issue which I complained about, where uploads to the forum would fail sporadically. His initial approach to fixing this issue broke some PHP scripts involved in approving posts, though, which is why FloraC's post sat there for a few days and the forum got locked down just in case. Dave is certainly asleep right now, but it looks he's now managed to fix both issues. Thanks @Dave Keenan!

And welcome to the forum, @FloraC! Helluva first post. Determining a new EDO notation is just about the most all-encompassing and tricky thing one could attempt to do in Sagittal. And this is exactly the right place to do it.

I'll go ahead and lead with the bittersweet news that a notation has already been determined for 198-EDO by the co-creators of Sagittal, Dave and George Secor:

 1°198 :)|(:  
 2°198 :|~:
 3°198 :~~|:
 4°198 :/|:
 5°198 :|\:
 6°198 :~|):
 7°198 :(|(:
 8°198 :/ /|:
 9°198 :/|\:
10°198 :)/|\:
11°198 :(|):

By no means should you have been expected to be familiar with it! It's not one of the lucky EDO notations which can be found in the Xenharmonikôn article. In fact, there's only a single way you could have known about it, I think: if you had installed Scala and browsed around the available Sagittal notations Dave and George submitted to the programmer of Scala. The Sagittal configuration files for Scala are hosted on the main Sagittal site here (http://sagittal.org/sag_et.par) but you'd never find them because they're not surfaced on the main site as a link. That's partially because the notations there are not even written in the standard ASCII format used on the forum, Facebook, etc. because, you know, some character sequences don't work in certain codebases or whatnot. Dave has said they're "not for human consumption" or something like that. I sort of agree, but we both agreed that something would be better than nothing. But then I took a note to add those links and sadly I hadn't gotten to that task yet. I probably could have saved you a lot of effort, so I'm sorry about that! (I've gone ahead and done it now. You can find it on the front page of sagittal.org by searching "Here are links to some configuration files used for Sagittal notation in Scala".)

I'm new to the Sagittal forum myself, just joining earlier this year, though I have been really active since then. My name's Douglas and I've volunteering my time this year to work on Sagittal. I had known about Sagittal for 10+ years but had never thought about it much because I primarily write computer music. It wasn't until March when I first set my sights on a traditional live acoustic performance of some xenharmonic compositions of mine (just in time for a global pandemic!) that I got interested in microtonal notation, and quickly identified Sagittal as pretty much the only solution for composers such as myself who use a great variety of tuning systems (not only JI or only EDO). But I was frustrated with how difficult it was to figure out how to use Sagittal. I couldn't believe that for a notation as mature and popular as Sagittal, the only place some questions' answers existed was on this forum. But that is indeed still the case. Dave and I have identified that we could use a page with documentation, glossary, tutorials, examples, etc., so people don't have to plumb the forum to discover concepts such as flag arithmetic!

I'll leave giving a detailed response to your foray into EDO notation to the experts, but there are a couple things I'm confident enough to say. First of all, most of the symbols you proposed do agree with what Dave and George decided, so... great work!
By the fact that :|(: is tempered out, 4131/4096 :~|(: and 2187/2176 :~|: should be mapped to the same degree by what I learned is known as flag arithmetics.
Speaking of flag arithmetic, your basic understanding seems good: two symbols in a notation which visually combine to make a third symbol in the notation should also sum their step sizes to the step size of the third. We can see this in your

:/|: + :|\: = :/|\:
4\ + 5\ = 9\

which is shared by the notation developed by Dave and George. But flag arithmetic (or "element arithmetic" as we're starting to call it now, because it works on the accents too, and symbol "elements" encompasses both flags and accents... again, more stuff for the upcoming glossary...) is just one of many, many considerations that go into deciding an EDO notation. It's okay for it not to work out, but if you can make it work, you should.

Also worth saying that Dave and George's word is not law. They certainly had a solid idea what they were doing when they first designed Sagittal, but I've been here long enough to catch a few mistakes or spots where there is room for improvement. So we may well find there're some properties of your notation which are preferable, perhaps in harmonic contexts which — with your deep specific appreciation for 198-EDO — you've foreseen but we haven't yet. So let's keep the conversation going.
Should this be solved, it might well apply to 205edo and 212edo with minor tweaks.
Right. From my own notes: "EDOs that differ by 7 sometimes have the same notation, because they sometimes have the same number of steps per apotome (and having the same number of steps per apotome sometimes leads to them having the same notation)."

I'm completely clueless about 6\ and 10 ... symbols.

Here's what I would have done for 10\. You've identified that it's the half-apotome. Whenever there's a need for a symbol around there, halfway between the :/|\: and :(|): , a good choice is :)/|\: , as you can see on this diagram of the JI notation, which I can see @volleo6144 shared as well.
7/5k :|(:
This would be either "5:7k" (undirected, with the 5 first because it's smaller; one-sided commas don't have the "1:" written) or "5/7k" (directed, with the 5 in the numerator because it's in the 5120 and not the 5103). It's ... fine; everyone fumbles a little with these things when they start.
Exactly.
45/44 :(|:
44:45 is :(|~:. See above. (:(|: is the 7/11C of 45056:45927; this complicated ratio was chosen so that :(|): can become the apotome-complement of 11M :/|\: 32:33.)
Yes.
I confess I'm not familiar with all the symbols.
The main website's SagittalJI.gif has the definitions of every unaccented (no :': or :`: or :,: or :.:) symbol as well as a few notable accented ones like :'::/|: (5s + 1/5C = Pythagorean comma). 48:49 specifically is :~|):, and 49:50 and the aforementioned Pythagorean comma are both accented symbols.
Updating that chart is one of the things I've been working on. First of all, the accents on the right sides of symbols nowadays would be written furthest to the left, because we decided that it was best to have the simple rule that smaller pitch alterations are always furthest to the left. I plan to make the updated chart an interactive SVG, so that even if a symbol's primary comma doesn't make the cut to be displayed in the main view, it can be found by hovering.
(There's been some sort of in-joke here about the "Promethean" precision level—with all the unaccented symbols and no accented ones—being bad in some way, but I don't really understand why.)
Ah! That was probably me making some sarcastic comment about it being "everyone's favorite". I certainly don't mean to leave anyone out of any jokes here. And I'm barely in the joke myself. I'll have to dig up my notes about why the High* precision level is the problem child...

Here's a post where Dave explains. It's the part that starts with "George and I aren't so sure the Promethean level was a good idea..."

A bit further down that page in this post he has a few more relevant words, beginning with "Removing mina accents from Olympian..."

The gist is this. If you kind of squint your eyes while looking at the JI precision levels diagram and focus on the colors, you can see the Promethean is the odd man out when you look at vertical columns of color. (oops! when I first submitted this post, I forgot to finish this bit) Because each precision level tries to distribute its symbols evenly, the new symbols introduced in Promethean that weren't in Athenian squeeze in and reduce the size of the Athenians' capture zones. But then as soon as you go up to Herculean and have access to accents, the Athenian symbols reclaim most of their original territory, since those zones are more accurately notated by combinations of the new accents with the Athenians than they are with combinations of the accents with Prometheans. However, if you had never introduced the Prometheans, then by the time you got up to the Olympian notation, you just wouldn't have enough options of base symbols, or "cores", to cover all the territory. So I think Dave and George were lamenting that the High/Promethean precision level broke the continuity of the precision level notations and maybe we shouldn't promote it as much.

*That's another thing I'm planning to tweak with the new version of the diagram; in the current version, the size words "Medium", "High", etc. are smaller and lowercase while the Greek mythology words "Athenian", "Promethean", etc. are prominent, but technically speaking, the Greek mythology words refer to the symbol extensions (and resulting new subsets) which these notations are strongly correlated with, while the notations' actual names should be the size words, capitalized. There a couple exceptions in the correlation between the levels and the symbol extensions which I think makes it a tad confusing to refer to them by the name of the extension.
27:28 (7L) is notated as 8192:8505 (35L) minus the 5s, and 8192:8505 :(|\: is relevant because it's an apotome minus 35:36 (1/35M) :/|):. 26:27 (1/13L) is notated as 8192:8505 plus 4095:4096 (1/455n) :`:, except that the :`: is often omitted.
Right. So the 7L can be written as :.::(|\:, since the 5s down is :.::!: and the 35L is :(|\: . Re: that omitting process, I want to demystify it if there's any uncertainty about it. Accents like :.: and :`: can (and arguably should) be omitted when the pitch represented by the unaccented symbol is not also needed in the same work. In other words, accents should be used to distinguish pitches that need distinguishing. One of the most powerful features of Sagittal is how each symbol has a "capture zone" which in addition to containing its "primary comma", the default value, contains a number of useful "secondary commas". Often it may simply be clear from context which comma to interpret the symbol as. Other times it may be best to include some sort of key up top of your sheet music (we'd like to determine a standard format for such a key at some point).
24:25 in JI would be notated as an apotome minus 6400:6561 (1/25S) :/ /|:, which (being a S-diesis and not an M-diesis) doesn't have a single-shaft apotome complement. Mixed/Evo notation would write it as just :\ \!::#:, while pure/Revo notation would write it as :)||(: (with nothing to do with :)|(: except that the apotome-complement of :)|(: is also :/ /||:).
Yep. 24:25 is ~70.67¢, which is just beyond the reach of single-shaft symbols, the largest of which notates up to ~68.57¢, the bound between the L- and SS- size categories of commas in the comma naming scheme Sagittal uses (that happens to be half of an apotome plus a Pythagorean comma, 113.69 + 23.46 = 137.15, 137.15 / 2 = 68.57). So you don't need to find a symbol for 12°198; beyond 11°, you should use exactly what volleo6144 suggested. Just choose one flavor of Sagittal or another — Evo or Revo (formerly Mixed or Pure, respectively) — and stick with it, at least within a given piece of music.

FloraC
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Re: 198edo

Post by FloraC »

@cmloegcmluin Thanks! You answered a lot of my questions.

Dave has emailed me the solution in ascii which is really difficult to read. Nice to see the graphical form.

Yeah I checked the diagram, but some commas are too complex to approach, such as the :(|(: = 45927/45056. I didn't have any idea how I could find that ratio. Now I see it's the apotome-complement of 22/21. Plus, I thought :)/|\: was :/|\: with a schisma accent.

Still some questions:
  • Is it reasonable to use :)~|: instead of :|~: for 2\198?
  • Both 45927/45056 and 45/44 (they indeed differ by 5120/5103 :|(: ) map to 7\198. Should I still bother with :(|(: instead of simply :(|:?

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Re: 198edo

Post by cmloegcmluin »

FloraC wrote: Thu Nov 12, 2020 3:37 am @cmloegcmluin Thanks! You answered a lot of my questions.
Good to hear. :)
Dave has emailed me the solution in ascii which is really difficult to read. Nice to see the graphical form.
Yeah, the graphical form, or "smilies" as we call them, are a game-changer. That's why we're always trying to get people to move Sagittal-related conversations over from Facebook or wherever to the fourm.

I totally empathize with the ASCII form of the Sagittal symbols being just One More Thing to work through. Once you get used to them though, you start to like them. It's pretty remarkable just how well Sagittal symbols can be evoked using what anyone can type on a keyboard already (as opposed to some other microtonal notations). If you haven't found it yet, the Sagittal-SMuFL-Map is the canonical reference at this time for the ASCII form (and much more!).
Yeah I checked the diagram, but some commas are too complex to approach, such as the :(|(: = 45927/45056. I didn't have any idea how I could find that ratio. Now I see it's the apotome-complement of 22/21.
Correction: :(|(: is 45/44, which is far less intimidating. It's actually the :(|: which is 45927/45056. It's the difference of the presence or absence of a right scroll. In ASCII they're (|( and (| respectively.

But yeah, some of those ratios look surprisingly complex at first. The ratios for the Athenian symbols go:

 5120/5103
  896/891
 4131/4096
   81/80
   64/63
45927/45056
   45/44
 6561/6400
   36/35
   33/32

But if you remove all the 2's and 3's from them, they look less lopsided:

    7/5
   11/7
   17/1
    5/1
    7/1
   11/7
   11/5
   25/1
   35/1
   11/1

And for notation purposes, those're what matter more, and those are the quotients you'll find in the names of the commas, which are also in that chart. Let's go through a concrete example. I'm doing this as much for myself as I am doing it for you, honestly, since I was actually just yesterday resolving my own lack of clarity regarding this concept which is pretty fundamental to Sagittal.

So all that 45927/45056 really is is the pitch you'd get if you applied the :(|: symbol to the Pythagorean nominal (A,B,C,D,E,F,G) which was your 1/1. Let's go with D for this example. So :(|: D is 45927/45056. But that's only 1 out of 56 different ratios in the octave which this accent gives you the power to notate! Why 56? Let's break that number down.

We've got 35 different pitches we can get to already using only 2's and 3's: :bb: F through :x: B. and for each of those pitches we can apply the accent upwards :(|: or downwards :(!: , so that's 35×2 = 70 pitches. Well, it would be 70, were it not for the fact that Sagittal doesn't go beyond two sharps/flats/apotomes (chromatic semitones) away from the nominal. I've referred to it fondly as "the edge of the world". So actually 14 of those 35 you can only notate in one direction from, namely, back toward the natural, whichever direction that is (up for double-flats, down for double-sharps). So 70-14 = 56.

Okay. So let's look at some of the other 55 pitches besides the up-from-D one we already looked at. As you already pointed out, 45927/45056 is the apotome complement of 22/21, so that's another way of saying that if we went up an apotome (or sharp), it would take one move by 45927/45056 back down to get us to 22/21. So :(!::#: D is 22/21. And here're all the remaining pitches notated by :(|: I find which are simple enough such that no number in their quotient has more than 2 decimal digits:

:(!::#: G 88/63
:(!::#: A 11/7
:(!::#: E 33/28
:(!::#: B 99/56
:(|::b: A 63/44
:(|::b: D 21/11
:(|::b: G 14/11
:(|::b: C 56/33

There are still a bunch of really nasty ones in there. Take :(|::#: B which is 2711943423/1476395008. You're probably not going to need that one. Or :(!::b: F which is 2952790016/2711943423. I grabbed these two pitches in particular because it doesn't get any worse than them; they're the furthest pitches away from 1/1 on the chain of fifths which still apply the Sagittal alteration away from the natural in the same direction. It's a way to explain that "edge of the world": the next step away from D past :#: B is :x: F, and we say that it no longer makes sense to move :(|: past that point, because you'll only get worse junk than 2711943423/1476395008.

And to complete the point I'm making, if you use one of the simpler-looking symbols, like 64/63, it's not that the pitches it notates are overall simpler. It simply displaces which addresses in the chain of fifths are the ones with the big nasty looking ratios with all the 2's and 3's in them and which are the ones with the really useful pitches. It all just depends on how many each of the 2's and 3's are required to pull the part of the ratio which includes 5's and beyond back down to commatic size. At a certain point, you can find ratios which have the combination of 5's, 7's, 11's, etc. that you need but it just takes too many darn 2's and 3's to get it to be comma sized, and then it ceases to be of much use for notating, because nowhere even in that chain of 35 Pythagorean fifths (ranging from :bb: F [27 -17⟩ to :x: B [-26 17⟩ ) can you find any position where the final pitches you get are worth trying to play (or as you put it, can you "find" them). That's why when choosing ratios for Sagittal, Dave and George looked at the ATE, or absolute three-exponent, of the pitches, to determine — even if the 2,3-free pitch classes the commas notate are popular ones which many musicians would likely want exactly notated — is the comma actually notationally useful.

Anyway, I encourage you to do your own investigations to convince yourself how this works out, if my explanation doesn't do the trick. Hopefully it helps, though!
Plus, I thought :)/|\: was :/|\: with a schisma accent.
Whoops, nope. That little curl on the left is a "scroll", meant to be the same flag as you see on the :)|: symbol, or )| in ASCII.

Accents will never be contiguous, or attached, to the symbol core. :/|\: with a schisma accent looks like this: :'::/|\:, or '/|\ in ASCII.

The left scroll which represents the flag with the smallest pitch alteration value of any Sagittal flag. You can see that because it's the furthest symbol to the left in the JI precision levels diagram. If you go any smaller than :)|: you're in the realm of the accents. And since the flags were designed so that their visual size corresponds to the size of their pitch alteration, so it's the smallest flag in appearance too. So it's not surprising that if you were to confuse any flag with an accent, it'd be this left scroll.

Dave and George did their best to make the symbols as distinct as possible at a glance, for ease of sight-reading. Some of those symbols with left scrolls, though, do get a big tough, as Dave himself will attest. Worst offenders probably being the :)/|: vs. the :)~|:.
Is it reasonable to use :)~|: instead of :|~: for 2\198?
Short answer: even if it is, prefer using the standard notation, to reduce the burden on yourself and the community. Unless you have an important reason for your exception.

One consideration when deciding an EDO notation is accounting for the tempered fifth. Some symbols' values change a ton in terms of their position within the apotome when you vary the size of the fifth. We say they have high "apotome slope". This is another way in which symbols can be notationally unuseful.

The primary comma for :)~|: has two 3's in its monzo, [4 2 0 0 -1 -1⟩, and since 198-EDO's fifth at 703.030¢ (116°198) is sharp by 1.075¢, we have to adjust by 2×1.075¢. Since [4 2 0 0 -1 -1⟩ = 12.064¢, we end up with 14.215¢ (some rounding happened in there). And now how about :|~: ? Its monzo is [5 -6 0 0 0 0 0 0 1⟩. It's going to change more, by -6×1.075¢, from 16.544 to 10.093. What's the exact size of 2\198, then? 12.121. It's a close call. :)~|: is 2.094¢ off, while :|~: is 2.028¢ off. On this basis, :|~: would be slightly preferred. But I'm sure there are other probably more important reasons. I'll defer to @Dave Keenan for those. So is it reasonable? I certainly wouldn't say it's unreasonable. But maybe not recommended.
Both 45927/45056 and 45/44 (they indeed differ by 5120/5103 :|(: ) map to 7\198. Should I still bother with :(|(: instead of simply :(|:?
Let me know if this isn't addressed by the earlier part of my response here.

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Re: 198edo

Post by volleo6144 »

FloraC wrote: Thu Nov 12, 2020 3:37 am Yeah I checked the diagram, but some commas are too complex to approach, such as the :(|(: = 45927/45056. I didn't have any idea how I could find that ratio. Now I see it's the apotome-complement of 22/21. Plus, I thought :)/|\: was :/|\: with a schisma accent.
:(|(: is 5/11S (44:45). You meant :(|:.

I don't think 45056:45927 was chosen because of its apotome-complement :)||~: 21:22, really. I'm pretty sure it arised out of :/|: 5C, :|): 7C, :/|\: 11M already being chosen, and the way they wanted to have :/|): be the apotome-complement of :(|\: 35L (chosen because of its proximity to :`::(|\: 13L) and :/|\: be the apotome-complement of :(|):, which therefore must be 11L (704:729). Flag arithmetic on this and :|): 7C gives you 7:11C = 45056:45927 = (212.11):(38.7), which was the primary comma they assigned to :(|:. (You'll find that all the more complicated ratios are just more simple ratios with large powers of 2 and 3 added to make them actually be worth notating as accidentals.)

Medium-precision symbols (excluding M/L-dieses), and their apotome complements:
:|(: 5:7k (5103:5120); :/||): 5:7LS (10485760:11160261) - 5/7k is the difference between 1/5C and 1/7C
:)|(: 7:11k (891:896); :/ /||: 7:11LS (1835008:1948617)
:~|(: 17C (4096:4131); :(||(: 17MS (17:18)
:/|: 5C (80:81); :||\: 5MS (128:135)
:|): 7C (63:64); :||): 7MS (131072:137781)
:(|: 7:11C (45056:45927); :)||~: 7:11MS (21:22)
:(|(: 5:11C (44:45); :~||(: 5:11MS (2560:2573)
:/ /|: 25S (6400:6561 = 802:812); :)||(: 25SS (24:25)

Yeah, this seems mostly coincidental. Other commas notable enough to be on SagittalJI.gif that have "simpler" apotome-complements (that aren't also on the chart) include:
:'::|: 5s (32768:32805) and :.::/||\: 5LS (15:16)
:)|: 19s (512:513) and :(||~: 19LS (76:81)
:,::~|: 73k (1024:1029) and :`::)/ /||: 73LS (686:729)
:~|: 17k (2176:2187) and :)/ /||: 17LS (16:17)
:'::)|(: 11:35k (2816:2835) and :.:/ /||: 11:35LS (280:297)
:,,::|~: 7:25C (409600:413343) and :``::/|~: 7:25MS (189:200)
:)|~: 19C (19456:19683) and :(||: 19MS (18:19)
:'::/|: 3C (524288:531441) and :.::||\: 3MS (243:256) - Pythagorean comma and limma, tempered out in 12 and 5 respectively
:)/|: 5:19C (40960:41553) and :)||): 5:19MS (19:20)
:`::.::|): 77C (2048:2079) and :,::'::||): 77MS (77:81)
:'::|): 5:7C (3584:3645) and :.::||): 5:7MS (20:21)
:`::(|: 29S (256:261) and :,::)||~: 29SS (232:243)
11:23SS (22:23) is missing
:'::~|): 5.72S (524288:535815) and :.::~~||: 5.72SS (245:256)
:,,::(|(: 7:13S (1664:1701) and :``::~||(: 7:13SS (112:117)
:~|\: 23S (16384:16767) and :)~||: 23SS (23:24)

Also the chart implies that :``::)/ /|:* 37M (36:37) and :,,::)|\ \: 5:77L (77:80) are apotome-complements... and they're actually 693:740-complements.
Last edited by volleo6144 on Fri Nov 13, 2020 7:40 am, edited 2 times in total.
A random guy who sometimes doodles about Sagittal and microtonal music in general in his free time.

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cmloegcmluin
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Re: 198edo

Post by cmloegcmluin »

volleo6144 wrote: Thu Nov 12, 2020 6:26 am Also the chart implies that :``::)/|: 37M (36:37) and :,,::)|\ \: 5:77L (77:80) are apotome-complements... and they're actually 693:740-complements.
I think you meant to write :``::)/ /|: as the 37M. You're just missing one left barb.

That said, I think you've found a bug in that chart, @volleo6144 . Congratulations. @Dave Keenan , tell him what he's won!

I'd never noticed that. Yeah, I've never seen 80/77 anywhere else... the Scala config files, the spreadsheet calculator... weird! It's not even turning up in the results for when we found the best commas per semitina zone recently. Its 2,3-free class {77/5}₂,₃ is notated by :,,::)/|:, which represents the 77/5C (coincidentally, I think, the same core as you mistakenly labeled the 37M, but with the accents in the opposite direction). But that symbol is small enough that its apotome complement is not single-shafted.

That chart, as I mentioned a couple posts ago, needs to be updated already, for three other reasons, and now we have a fourth. Fortunately, that's my top priority.

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Re: 198edo

Post by Dave Keenan »

cmloegcmluin wrote: Thu Nov 12, 2020 8:40 am
volleo6144 wrote: Thu Nov 12, 2020 6:26 am Also the chart implies that :``::)/|: 37M (36:37) and :,,::)|\ \: 5:77L (77:80) are apotome-complements... and they're actually 693:740-complements.
I think you meant to write :``::)/ /|: as the 37M. You're just missing one left barb.

That said, I think you've found a bug in that chart, @volleo6144 . Congratulations. @Dave Keenan , tell him what he's won!
Congratulations @volleo6144, you have won ... my admiration (I'm holding my undying gratitude in reserve). ;)

Or perhaps it should be the inaugural

:star: George Secor memorial: So you found where I screwed up and Dave missed it too Award :star:

Well done. :)

While the file http://sagittal.org/sag_ji4.par correctly has :,,::)|\ \: as the apotome complement of 37/36, the diagram has it wrong.
I'd never noticed that. Yeah, I've never seen 80/77 anywhere else... the Scala config files, the spreadsheet calculator... weird! It's not even turning up in the results for when we found the best commas per semitina zone recently.
Why would 5/77L (80/77) have turned up in our "best comma" results? It's greater than the half-apotome. However its apotome-complement, 77/5M, [-15 7 -1 1 1⟩ certainly did turn up, as the lowest badness comma in the capture zone for :``::)/ /|: . This is one of the 20 zones flagged for review after we've finished the educational material.

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Re: 198edo

Post by Dave Keenan »

I have found some email exchanges between George and I, from 2006, in which we discussed the 198edo notation. I hope to eventually post them. One thing to note is that we thought it desirable that the 198edo notation should be a superset of the 99edo notation. But in http://sagittal.org/sag_et.par I see that this is not so, although it is very close. The 99edo notation is

99: :~|: :/|: :~|): :/ /|: :)/|:

Only the first symbol above differs from what it would be as every second degree of 198edo.

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Re: 198edo

Post by Dave Keenan »

In an email thread entitled "Re: XH18" (which refers to issue 18 of the journal Xenharmonikôn for which we were preparing the article on Sagittal), on 18-Feb-2006, I emailed George:
Dave Keenan wrote: At 04:26 AM 8/02/2006, you wrote:
> --- Dave Keenan <d.keenan@bigpond.net.au> wrote:
> > I'm happy to post something to the tuning group about it, unless you
> > tell me you'd rather do it.
> >
> > I would just tell people there's a new version and list the
> > significant changes.
>
> Yes, please do. I've barely been able to keep up with the tuning lists
> -- only time to scan the new message subjects and perhaps read a few
> each day while I eat lunch.


Please see
http://launch.groups.yahoo.com/group/tu ... sage/64269
and its replies.

So would you now go for this 99-ET notation?

~| 1
/| 2 |) equiv
~|) 3
//| 4
)/|\ 5

If so, feel free to tell Gene? You could show him a bitmap version.

Sorry I reverted about 5 years and thought ~| was athenian. Duh!

But its presence is still justified by the presence of ~|) another non-athenian.

-- Dave
George replied:
George Secor wrote: --- Dave Keenan <d.keenan@bigpond.net.au> wrote:

> At 04:26 AM 8/02/2006, you wrote:
>> --- Dave Keenan <d.keenan@bigpond.net.au> wrote:
>>> I'm happy to post something to the tuning group about it, unless
you
>>> tell me you'd rather do it.
>>>
>>> I would just tell people there's a new version and list the
>>> significant changes.
>>
>> Yes, please do. I've barely been able to keep up with the tuning
lists
>> -- only time to scan the new message subjects and perhaps read a few
>> each day while I eat lunch.
>
> Please see
> http://launch.groups.yahoo.com/group/tu ... sage/64269
> and its replies.
>
> So would you now go for this 99-ET notation?
>
> ~| 1
> /| 2 |) equiv
> ~|) 3
> //| 4
> )/|\ 5

Assuming that you would now agree with this,
198: )|( |~ ~~| /| |\ ~|) (|( //| /|\ )/|\ (|)
I would still have to say no. I think that the nuisance imposed by not
having 99 notated as a subset of 198 would outweigh the two advantages
of having a flag common to 1 and 3deg99, especially when one of those
advantages is not that strong in that it requires an intermediate step
of logic:

~| + |) = ~|)
|) is equivalent to /|
therefore ~| + /| = ~|)

I don't see flag commonality as an important factor when you have only
5 single-shaft symbols in the set, or, to frame it in your terms, I
don't see lack of flag commonality as a significant "complexification"
imposed on 99-ET by treating it as a subset of 198.

Instead, I see an advantage in favor of |~ in that the flags for 1 and
2deg99 are on opposite sides, making the symbols less liable to be
misread.

However, the strongest reason I find for using |~ for 245C (243:245)
concerns the semantics of the notation, which I think Gene would find
attractive. All pairs of symbols differing by 2deg99 differ in size by
a 5-comma:

deg99 symbol semantics
1 |~ 245C, i.e. 49C 5C
2 /| 5C
3 ~|) 49C
4 //| 25C, i.e. 5C + 5C
5 )/|\ 5:49M, i.e. 49C + 5C

Hmmm, this is disturbingly interesting. In light of the latest
(3-flag) developments, did we use the wrong combination of flags for
49C, seeing that /| + )|\ = )/|\? (I'm just thinking "out loud" here,
because this would create some problems, e.g., nearly 3 times the error
for SoF and bad flag arithmetic in 494)

Anyway, getting back to the subject at hand, I have given 3 reasons for
using |~ for 1deg99:
1) It agrees with the 198-ET set;
2) It is less likely to be misread as /|; and
3) Taken as 245C, it results in elegant notational semantics.

> If so, feel free to tell Gene? You could show him a bitmap version.
>
> Sorry I reverted about 5 years and thought ~| was athenian. Duh!

It happens to the best of us.

> But its presence is still justified by the presence of ~|) another
> non-athenian.

I'll consider that statement irrelevant, since both ~| and |~ are
non-athenian.

--George
I replied:
OK. You've convinced me.
George replied:
(Gasp! 8-()

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