581EDO

[cam.taylor]

I realise 581 is tricky, mostly because it looks really close to JI, but it's fifths are ever so slightly sharp as to make a very nearly pure 7:8 after 14 fifths, and virtually just 8:11 at 23 fifths, making it very practical to work with. Another reason 581 is tricky is because the schisma maps to 2\, even though it is around double the schisma's just size. In fact 2\ also represents 512:513 and 351:352, while it's difficult to find a simple interpretation for 1\ (as pointed out by George Secor in a recent discussion)

I've been thinking about this since late last year as I figured 581 is a great modular system to use to represent nearly anything we do to a reasonable accuracy, with 23- and 25-odd limit consistency and very low 23-limit damage, and nice little ~2c and ~4c lego blocks named "spooks" ($) by Scott Dakota, due to 581's "spooky accuracy"

I know we've had some discussions about 581 before, but could there be a solution to notating it? The 270edo notation posted in a recent thread might be helpful, as 581=270+311

270-edo:

So far I've got:

2$: schisma (one step too large), 512:513 , 351:352
4$: 891:896 , 207:208, 224:225
5$: 168:169
6$: 152:153, 135:136, 143:144
7$: 729:736, 120:121
8$: 4096:4131 (one step too large), 99:100
11$: 80:81
13$: Pyth comma (too large), 63:64
17$: 48:49, 49:50
24$: 35:36 , 1024:1053
26$: 32:33 , 2 Pyth commas, 2 Septimal commas


Help? Or is this going to get really messy?

[cam.taylor]

I'd like to emphasise the 23-prime limit, so not that keen to use higher primes unless it makes things a whole lot easier. Thanks.

[George Secor]

I have a solution for notating 1deg581: Define the symbol as 17:121n (the 17:121-schismina, 1088:1089). This is slightly larger than 5:19n (1215:1216, ~1.424c), which is notated and vanishes in 581-EDO.

This combination of opposite-altering diacritics (or accent marks) may then be added to various unaccented symbols to alter those by 1deg581. For example, is 11deg581, so may be 12deg581 and may be 10deg581. Since is 13deg581, might be used instead for 10deg581, so there would need to be a method for deciding between such alternatives (such as preferring augmentation of an unaccented symbol to its diminution).

Some combinations would need to be avoided, e.g., for 3deg581, since this symbol is already defined as 1225k (19600:19683) and would be 5deg581. Since is not presently defined, that could be used instead.

It's evident that 581 is going to take a bit of time and effort to get it done, but it's far better to have the problem of deciding between two alternatives for a given number of degrees than not being able to find anything that works.

[George Secor]

Some combinations would need to be avoided, e.g., for 3deg581, since this symbol is already defined as 1225k (19600:19683) and would be 5deg581. Since is not presently defined, that could be used instead.

I made a mistake in the foregoing. is 5deg581, so would be 4deg581, not 3deg. Instead, could be used for 3deg581, since it's not already defined, and can be 4deg581. So a possible symbol sequence might begin with:


One problem with the foregoing that might crop up later in the sequence is with symbol arithmetic involving , which is 2 degrees by itself (as 19s), but with and notating consecutive degrees, it's actually representing 1 degree.

I said that this was going to take some time and effort to get a working symbol set.

[cam.taylor]

Thanks for all this George, for taking the time to look into this so deeply. I wouldn't have thought it would be quite this tricky, since 581 is not too far from 23- and 25-limit JI, and with 270 looking not too radical, I thought 581 would be only a bit of gap-filling from there. Obviously, I was wrong. Does that mean a notation for 311 would also be rather dificult, as 581=270+311?

[George Secor]

Thanks for all this George, for taking the time to look into this so deeply. I wouldn't have thought it would be quite this tricky, since 581 is not too far from 23- and 25-limit JI, and with 270 looking not too radical, I thought 581 would be only a bit of gap-filling from there. Obviously, I was wrong. Does that mean a notation for 311 would also be rather dificult, as 581=270+311?

No, 311 is fairly straightforward:


There is a small problem with the flag arithmetic:
If = 1, = 1, = 6, = 7, = 8, and = 9,
then the number of degrees for is not the same for these four symbols , being either 2, 3, 2.5, or 3 degrees, respectively. I don't think this is serious enough to require replacing any of these with accented symbols.

[George Secor]

I posted the following, regarding notation of 581-EDO, on 9 April 2017 to a thread in the Facebook group Xenharmonic Alliance Mathematical Theory, and am reposting it here at Dave Keenan's request. Although most of this already appears in my previous postings here, my suggestion of the 85:121 schismina as a possible interpretation of is new:

<< Scott Dakota, do you have a ratio for 1\581? The principal problem for notating 581 with Sagittal is defining a symbol for the single degree. The 5-schisma (32768:32805) maps to 2\581, but is not particularly useful. The 17:121-schismina (1088:1089) maps to a single degree, but would be notated as -- rather clumsy, but the simplest ratio for that symbol (not defined for olympian level, hence theoretically in a "super-olympian" extension). The difference between these two small intervals, the 85:121 schismina (61952:61965), would then be 1\581, notated as -- rather convoluted in that it's valid only as a secondary ratio that's far down the list.

I previously had 581 in my notes, where I chose '|. as 17:121n for 1\581, with the first few members of the symbol sequence being:
581 ... >>

[Dave Keenan]

It seems wrong to me, for 581-EDO notation to use schismina accents. 612 and 624-EDO notations don't need them. However it seems unavoidable that 581 will need to use _some_ kind of accents. I believe 494-EDO is the largest that we can notate without accents. As with 612 and 624, I think we should use schisma accents for 581, but in the case of 581 we need to accept that they will not represent the 5-schisma.

[George Secor]

It seems wrong to me, for 581-EDO notation to use schismina accents. 612 and 624-EDO notations don’t need them. However it seems unavoidable that 581 will need to use _some_ kind of accents. I believe 494-EDO is the largest that we can notate without accents. As with 612 and 624, I think we should use schisma accents for 581, but in the case of 581 we need to accept that they will not represent the 5-schisma.

Okay, I agree. I checked several of the simplest alternatives to the 5-schisma, and the only one I found that's valid for a single degree of 581-EDO is the 17:121-schismina. I previously pointed out that 17:121n would define , which is not included in the olympian JI symbol set. Since we want to use only schisma (i.e., left) accents for 518-EDO, the schismina (right) accent can be dropped from , allowing to be used exclusively in a secondary role, 17:121n.

In order to avoid confusion, I think that it would be advisable, where possible, to avoid using herculean-level left-accented symbols in the 581-EDO symbol set, since the left accents in these symbols have already been defined with 5-schisma alterations. Among the most popular to avoid are:
1C, pythagorean comma
25C, diaschisma
7:25k
5:7C

Using the foregoing principles, I arrived at the following symbol set:

581:

Because all of the accented symbols (except the very first and last one) have non-athenian cores, there are only two herculean-level left-accented symbols in this set, for a single degree and its apotome-complement , and even these can be avoided using and instead. Normally we would try to use a few symbol cores as possible, but the advantage in maximizing the number of symbol cores (in addition to eliminating 5s-confusion) is that most of the accented symbols are very easy to distinguish from the unaccented ones.

[Dave Keenan]

That's quite a tour de force, George. I don't doubt that it is valid, however I find it unsatisfying. Of course it's easy to criticise, and much harder to come up with an alternative. It's unsatisfying that there is no regularity that might reduce the amount of information that needs to be remembered or looked up -- or at least none that I can discern. In particular, there is no consistent assignment of numbers-of-steps to all flags, and the double-shaft symbols do not repeat a flag sequence of the single-shaft symbols. And there are so many different core symbols.