## EDOs with multiple prime mappings

Dave Keenan
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### Re: EDOs with multiple prime mappings

I feel we are converging on a mutual understanding. As the Stephen-Hawking-voiced Pink Floyd lyrics say, "All we need to do is make sure we keep talking".
cryptic.ruse wrote:Does this forum support pdf file uploads?
Yes, it does. When composing a reply, click on the "Attachments" tab below the "Preview" and "Submit" buttons. The size limit is 5 MiB per attachment.
David Keenan wrote:But we _are_ doing that whenever we can. The constraints are:
(a) that there is a one-to-one relationship between symbols and numbers of steps in any given EDO, and
(b) we use a consistent subgroup of the obvious mapping (patent val) or we notate as a subset of 2n-edo or 3n-edo, and
(c) symbols for lower primes have higher priority.
This does not change the fact that, in an effort to keep symbols down, information about primes is omitted. It is simply not the case that "when good, low-prime-above-3, JI approximations do exist we want to make that information available directly in the notation."
Yes, information about primes is omitted. But short of secretly probing our brains while we're asleep, I'm not sure how you could, even in principle, falsify my claim regarding what we "want to" do. You might be polite and just take my word for that. But I assume you don't really mean that, and simply missed the words "want to". We do it whenever we can within the constraints I mentioned -- constraints which, it seems to me, any good EDO notation requires, whether based on fractional 3-commas, or whole >=3-commas.
For that to be true the notation would have to reflect the presence of all the primes you consider well-approximated in each EDO; it's a goal that is exactly opposite to the goal of keeping symbol counts down.
I have no idea how any sensible and usable EDO notation could actually reflect the presence of _all_ the well-approximated primes. Please explain. I note that the complete Sagittal system has symbols, not only for primes up to 37, but also for many ratios of primes, products of primes and powers of primes. We have so many that we can divide the Pythagorean apotome into 233 approximately-equal parts.
At most you use as few of the best primes as possible in order to notate all possible steps of the ET. How can you possibly keep maintaining the position that whenever good low-prime-above-3 approximations exist in an EDO, the notation makes that information available, when I can point out numerous EDOs where good and consistently-mapped primes exist, yet aren't indicated in the notation?
There are clearly some unstated assumptions here, on both our parts, that are not shared by the other. We really need to get to the bottom of what they are. I will try to guess what mine might be and explain them, and will try to ask questions that might reveal yours.

I don't believe I have ever claimed that whenever good low-prime-above-3 approximations exist in an EDO, the notation makes that information available. I have only claimed that we "want to" do it, and that we do it "whenever we can" within the constraint that each step count can have at most one symbol (or one string of symbols consisting of a sagittal and a conventional sharp, flat or double). If the mapping from EDO step count to symbol string (and back again), is not one-to-one and onto, i.e. is not a bijection, then we do not consider that we have standardised an EDO notation, as we explain in the last paragraph on page 13 of http://sagittal.org/sagittal.pdf.
My argument is that you are okay with leaving that information out sometimes, in the service of a particular goal (minimizing the total number of symbols). My proposal takes this same rationale and applies it to all the EDOs in whatever range we might wish to define.
Sure. We are OK with leaving that information out in service of the goal of standardising a notation for a given EDO. i.e. we stop throwing it away when we get down to one symbol-string per step-count. But just because we're okay with throwing some of it away some of the time, doesn't mean we'd be happy to throw it all away all of the time.

If you're imagining that we begin with a list of every Sagittal symbol, including accented ones, as given in http://sagittal.org/sag_ji4.par, but expanded to include combinations with sharps, flats and their doubles, and calculate how many steps of the given EDO each one corresponds to, using the obvious mapping (patent val), then it must seem like we are throwing away an enormous amount of information. And then it would seem that your proposal to throw away everything but the 3-limit is only throwing away a tiny bit of extra information.

But not all symbols carry the same amount of information, in the sense that not very many people care about how an EDO maps prime 37, but many do care about how it maps prime 5. A rough quantification of this effect is given by these statistics obtained from the Scala scale archive. viewtopic.php?p=258#p258. So, crudely speaking, throwing away everything beyond the 7-limit would still leave you with 65% of the information. Throwing away everything beyond the 3-limit only leaves you with 26%.

But you've already said you're not interested in statistics, so here's another argument. We don't start with the full list and cut it down, instead we start with the 3-limit and work up the primes until we have a comma for each step-count up to half an apotome (actually about 0.6 of an apotome but no need to go into that here). Even if you assume the same informational value for every prime, using symbols for 5 and 7 doubles the amount of information relative to stopping at the 3-limit.
It's incontrovertible that using a variety of symbols reflecting a variety of higher-prime commas will result in a lower or equal sum-total of symbols across a given range of ETs compared to using a symbol set that reflects only relationships on the 3-limit.
I suspect you meant to write "higher or equal", in which case I would agree.
In 19edo you can represent the whole gamut with only 3-limit information, and as you'll see when I finally get around to posting my enharmonics, you can do this in all the ETs.
Of course you can. That doesn't mean it is always desirable to do so.
It only makes sense to NOT do this when higher-prime information is considered integral to the composition, ...
So you're saying it makes no sense to include any symbols for primes higher than 3 in any EDO notation unless the higher prime information is considered integral to the composition?

The best I can do is to meet you part way and agree that it makes no sense to include any symbols for primes higher than 3 in an EDO notation unless most composers are likely to make use of that information. One of the things I realised while making that chart of mixed-Sagittal EDO notations is that most of our subset notations are complete crap. And so are most of the notations we give for EDOs whose best fifth is outside the range of about 692c to 712c (i.e. more than a 10 cent error in the fifth). The largest such EDO Is 47. In order of fifth-size these extreme-fifth EDOs are:
Narrow fifths: 11 9 16 23 7 14 21 28 35 47 40 33,
Wide fifths: 32 37 42 5 10 15 20 25 30 18 13 8 6.

I agree that most composers using these EDOs are unlikely to be interested in their JI approximations, at least not in relation to their chain of best fifths. That list also includes many of the EDOs below 48 for which we have been forced to go outside of the Spartan set of symbols. So these are definitely candidates for having their existing standard notations replaced with ones that use your way of notating EDOs.

The whole point of designing a standard like Sagittal is to get away from the pre-existing situation where we had one notation (or more) per composer per EDO. If it can change with every composition, it isn't a standard. We'd like to have the situation where a performer can learn to read one 22-edo Sagittal notation and can then play any 22-edo piece by any composer. If one composer says, "I'm not interested in the 5-limit implications of a single step of 22, so I'm going to use the half-apotome [Edit: 1/3-apotome] symbol instead of the 5-comma symbol", then he shouldn't be surprised if the performer says, "What's this? I thought you said it was in 22-edo."

So what we're doing is asking that composer, for the good of the team, to ignore the 5-limit implications of the symbol in this case, and just think of it as the half-apotome [Edit: 1/3-apotome] symbol for 22-edo, because most composers _are_ interested in the 5-limit (and indeed the 7-limit) implications of 22-edo. Of course he's always free to use a non-standard notation while composing, and then translate it to the standard notation when it's done.
... in which case Full Sagittal will provide a better option than my proposal OR Standard Sagittal.
I don't know what you mean by "Full Sagittal". To me, the alternative to a standard Sagittal notation for some EDO, within the Sagittal system, is any number of non-standard Sagittal notations for that EDO. I have no idea what a "Full" Sagittal notation for an EDO would be. Please give an example. How about for 19-edo?

And please give your own notation for 19-edo if this is different from the standard Sagittal notation.
I only say "probably", because in some cases (such as the 7-comma in 19-edo) the comma's symbol is unused because it is the same size as the apotome (whose symbol has priority because it represents a lower prime) and in a few pathological cases the comma's symbol is unused because the comma is negative rather than merely zero.
Right, that is one of my complaints. In Standard Sagittal there exists no single inviolable rule for when symbols representing a given prime will be used or not. You have a multitude of different considerations and priorities that lead to exceptional cases and debatable choices.
That is true for those extreme-fifth EDOs listed above, but not for most of the others.

I think you are overstating the difference between our approaches here. Surely in your approach, you also do not use a symbol if it maps to zero or negative steps? And surely you have some priority ordering to resolve collisions, e.g. do not use a limma-fraction symbol if an apotome-fraction symbol fills the slot?
A good "standard" will be clear, concise, and consistent.
There are many aspects whose standardisation you might be talking about here.
The EDO-independent meaning of every symbol is clear, concise and consistent, as encoded in http://sagittal.org/sag_ji4.par.
Every standard EDO notation is likewise completely clear, concise and consistent, since it consists simply of a list of symbols for consecutive steps of the EDO -- one symbol per step-count as encoded in http://sagittal.org/sag_et.par.

You are correct if you mean that the rules for deriving the standard notation for any EDO are not fully standardised. Most users are not interested in the details of this process, and simply want a notation that everyone can agree upon, to facilitate communication.

However the derivation does begin with a concise algorithm that I have already described several times in several different ways. And many EDOs are successfully notated by that algorithm on its first pass. But some are not, either because some slots remain unfilled, or because some slots are filled only by symbols for ratios whose complexity is out of all proportion to the accuracy of their approximation. There is another algorithm that checks the result of the first, for consistency of flag (i.e. sub-symbol) arithmetic, which may then require the first algorithm to be rerun with some symbols eliminated. And there are other checks for other undesirable properties. And where the subjectivity enters is when several undesirable properties must be weighed against each other.

I am interested in adopting your algorithms in precisely those cases where the existing JI-based algorithm fails on its first pass.
If the rules of Full Sagittal are simpler than the rules of Standard Sagittal, that should give pause as to whether the standardization method is really meeting its goal.
I think I have already conceded, several times, that it is not meeting its goal in the case of certain EDOs. But again I am mystified by this supposed comparision with "Full Sagittal". What do you see as the differences between "Full Sagittal" and "Standard Sagittal"?
And I would argue that the rules of Full Sagittal are simpler than Standard, when it comes to ETs--the only rule determining the presence or absence of a given symbol is that one always chooses the symbol that best represents the approximated JI identity. Thus a single glossary of available symbols and an understanding of the ET's mapping is all that is needed for the JI-savvy composer working in any ET.
This gives me some glimmerings of what the term "Full Sagittal" means to you. It narrows things down a little. But I'm still not sure I understand.

Are you saying that, when using what you call Full Sagittal, you might use one accidental symbol to represent an alteration of say one step of the EDO in one part of a composition, then use a completely different symbol to represent the same alteration in another part of the same composition? As I say, that's fine for your own private use, but it won't be of much use in communicating with a performer, or even with other composers using that EDO, unless they happen to think about it in the same way you do.

And why would _you_ want to do that, given that you are designing notations where (I assume) there is only one string of symbols to represent each step-count of a given EDO, or at least only 3-limit identities can be represented?
I am not missing that fact, I am harping on it! If your standard was clear, concise, and consistent, there would not be a need to debate and the best notation would be generally uncontroversial. And "obvious" is a rather subjective quality, don't you think?
In the case of "obvious prime mapping", that's synonymous with "patent val" which is not subjective. By "obvious best subset", I mean not having two or more subgroups with approximately equal errors. Haven't we already used Graham's temperament finder as the (non-subjective) arbiter of that?
but your formulae do not include the psycho_visual_ issues of notation, and you sometimes seem to be considering only the composer and not the performer.
If my notation ends up following the visual rules of Sagittal, with appropriately-chosen Sagittal symbols, to what extent is it inferior to Standard Sagittal in a performer-oriented/psycho-visual sense?
None whatsoever. But when we get to that stage, you will then be forced to consider some of the messy psychological issues, which you currently consider a limitation of Sagittal.
And in any case, I'm not married to any particular visual implementation, nor even suggesting one at present; the ASCII symbols I use are more or less placeholders.
Yes, I fully understood that.
I wouldn't even begin to argue that I have a solid understanding of those issues, as even standard 12-EDO staff notation is a challenge to me.
So how about cutting us some slack in regard to these issues?
But as far as considering the performer, I think you've got it backwards. A performer may know little to nothing about a given ET and how it handles JI, and the less information that need be understood for the score to be interpreted, the better (in that case). From the standpoint of a naive performer, what do you suppose would be easier to learn: more symbols, more rules, more exceptions, or fewer symbols, fewer rules, and fewer exceptions? A simple single rule for symbol generation that applies universally, or a handful of subjective practical considerations that are carefully rebalanced on a case-by-case basis?
I agree with your priorities here. But it's a serious misconception about Sagittal if you think a performer, or even a composer, is expected to apply a bunch of rules to generate a notation, for any EDO they want to use. That's what the list of standard EDO notations is for -- so they don't have to.
Why limit it to those cases, though, when my approach works just as well for the "good" ones? Presuming that we can preserve all the visual benefits of Sagittal, what really is lost in my approach that could not be recovered with a small amount of meta-notational information?
What is lost is the connection to the JI Notation. Given any pre-existing JI notation, it would be an exercise in willful ignorance not to notate 72-edo using symbols for 81/80, 64/63 and 33/32, for example.

But I agree that somewhere along the path from 72-edo down to smaller EDOs with worse fifth-sizes, the need for such a connection to JI fades and is gone.
You think these meta-notational considerations are not present in Standard Sagittal?
Of course I don't think that.
I will concede they may be slightly reduced in Standard Sagittal vs. my proposal in the case of some ETs, but the amount by which they are reduced is minuscule.
I agree it is miniscule for some EDOs, but I consider it to be of major significance for others.
The host of intervalic equivalencies within various prime limits is tremendous, and not trivial to deduce from a given set of accidentals--especially when an ET is notated as a subset of a larger one that may not even be familiar to composer or performer.
For many EDOs I agree. But 72-edo is an example where a host of 11-limit equivalences _are_ trivial to deduce from a 5, 7, 11 prime notation for it. It's the perfect JI-based EDO notation.
On the flip side, it may not even be the case that the accidentals of the higher-prime commas convey information the composer finds relevant.
Sure. And I agree that's harmful if it causes the number of unique symbols used across all EDOs less than say 72 to increase. But if it was just re-using symbols used in 72-edo, for lower EDOs, where would be the harm?
It's always debatable which prime approximations are more important than others, ...
No. That isn't _always_ debatable. The most important prime approximations in 72-edo are not debatable.
.. but the 3-limit (since it is the basis for the nominals) is more or less of incontrovertible importance.
Agreed.
Last edited by Dave Keenan on Sun Oct 09, 2016 9:11 pm, edited 1 time in total.
Reason: Added a correction where I mistakenly equated the 5-comma and the half-apotome in 22-edo. It's really 1/3-apotome.

Dave Keenan
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### Re: EDOs with multiple prime mappings

One of the things I'm struggling with here, in regard to revising some of the standard notations to use 3-limit fractional comma notations, is how and where to make the transition from the JI-based notations (primes greater than 3 allowed, fractional commas not allowed) to the apotome-fraction notations.

The further transition from apotome-fractions to limma-fractions seems obvious enough: Only use limma fraction symbols for EDOs where the apotome is zero or negative (6, 11, 9, 16, 23, 7, 14, 21, 28, 35).

Ideally such a transition (JI to fractional-3) would somehow be gradual and seamless. It's difficult to imagine how that could happen, but it could be that the maximum prime allowed could be stepped down gradually. In any case it's worth keeping the idea of a smooth transition in the back of our minds in case something occurs to us.

But whether the change is gradual or abrupt, it's not totally clear to me what property of the EDO should determine the JI-ness or otherwise of its notation. Our first pass at this was that it would be determined by the size of the EDO -- large EDOs have JI Notations, small EDOs have fractional-3-limit notations. But that doesn't work because I think 22-edo and 29-edo should definitely keep their existing 5-limit notation, while 42-edo and 47-edo don't deserve anything above the 3-limit.

People may of course choose to use a fractional 3-limit notation for 22 and 29 -- they both divide the apotome into 3 parts -- but this would not be their standard notation.

So my current suggestion is to base it on the error in the fifth. Error greater than 10 cents gets a fractional-3-limit notation. The 10 cent cutoff seems somewhat arbitrary. Reducing it to 9 cents would add 26 and 52 on the narrow side and 27, 54 and 59 on the wide side, as EDOs requiring fractional-3-limit notations. With a 10 cent cutoff, 47-edo has the greatest number of steps to the apotome (7 steps). With a 9 cent cutoff, 59-edo has 9 steps to the apotome, requiring more fractional-apotome symbols. One might argue that 26-edo is a meantone and so deserves a JI-based notation, however, like 12 and 19, it only has 1 step to the apotome, so its standard notation is the same whether JI-based or fractional-apotome.

I toyed briefly with the idea of making the notation fractional-3-limit whenever the EDO was not 1,3,9-consistent, i.e. whenever 2 fifths are not the best approximation to 4:9. But that occurs whenever the error in the fifth is more than 1/4 of a step, and so it does not have any upper limit in ET numbers, and therefore no upper limit on the number of steps per apotome requiring symbols. So I currently prefer a fifth-error cutoff, and 10 cents seems right to me.

Dave Keenan
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### Re: EDOs with multiple prime mappings

I'm now favouring a 7.5 cent error in the fifth as the transition from JI-based to fractional-apotome for the standard EDO notations. That means fractional-apotome notations for these EDOs:
Narrow fifths: 47 40 33, 26, 52, 45, 64
Wide fifths: 71, 49, 27, 54, 59, 32 37 42 5 10 15 20 25 30 18 13 8 6

And fractional limma notations for these:
Very narrow fifths: 11 9 16 23 7 14 21 28 35
(In previous posts I mistakenly included 6-edo among the fractional-limma notations.)

The reason for the change is that I learned that we have suitable Sagittal symbols to notate up to 10 steps in the apotome (as required by 71-edo). See viewtopic.php?p=417#p417

Even if we had the symbols to do so, I wouldn't want to extend the apotome-fraction notation to EDOs with fifth-error any less than 7.5 c, because the next to be included would be 19-edo and its multiples (on the narrow fifths side) and and 22-edo and its multiples (on the wide fifths side). I feel strongly that they should keep their JI-based notations as their standard notation.

cryptic.ruse
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### Re: EDOs with multiple prime mappings

I have yet to catch up on your latest replies in this thread, as I've been traveling for business this weekend and have elected to focus on actually finishing the list of enharmonic equivalents in my proposal. I have finally completed it, although I confess much of it was typed out on airplanes, in the backseats of cars, and after long days. As such, errors are likely, as are minor inconsistencies in formatting. Hopefully they are few enough that they do not completely wreck the entire edifice. It was quite a slog!

I'll try it posted here enclosed in [ pre ] tags; if that does not display correctly, I'll render a .pdf and upload it.

Note that I include a symbol glossary for each ET, listing every symbol and the number of steps it maps to. For a select few ETs, I included an "extra" mapping that violates the formula, but which may be useful under certain circumstances nevertheless (usually for backwards-compatibility with smaller ETs). Nevertheless, here is every ET from 5 through 72, notated with five symbol pairs in total (counting the Apotome):
Apotome fractions:
Whole: # b
1/2: ^ v
1/4: / \
Limma fractions:
1/2: < >
1/4: ( )

I should note that this is not an exhaustive list of all enharmonic equivalents; whenever there are multiple accidentals, there are multiple ways to specify a single pitch with a single nominal. This makes it difficult to count all the combined accidentals that may crop up, since I have not exhaustively searched for circumstances when one spelling may be more "correct" than another (for instance, when one might wish to use F^^^ in place of F# in 22edo). I am not sure if you've spent much time exhausting the different ways Standard Sagittal accidentals may or may not combine in each ET.

Anyway, here it is!
5edo: # = +1, b = -1
D	D#	Dx
Eb	E	E#
Fb	F	F#

6edo: x = +1, bb = -1 (subset of 12edo)
D	Dx	Dxx
Ebb	E	Ex
Fbbb	Fb	F#

7edo: # b = 0
D 	E 	F
D#	E#	F#
Db	Eb	Fb

8edo: x = -1, bb = +1 (subset of 16edo)

D	Dbb
Ex	E	Ebb
Fx#	F#

9edo: # = -1, b = +1
D	Db	Dbb
E#	E	Eb	Ebb
Fx	F#	F

10edo: # = +2, b = -2, ^ = +1, v = -1
D	D^	D#	D#^	Dx
Eb	Ev	E	E^	E#
Fb	Fv	F	F^	F#

11edo, version 1: # = -1, b = +1 (3-limit patent val, 3/2 = 6 steps)
D	Db	Dbb
E#	E	Eb	Ebb
Fx	F#	F

11edo, version 2: ^^ = +1, vv = -1; #v = +2, b^ = -2; x = +3, bb = -3 (subset of 22edo)
D	D^^	D#v	Dx
Eb^	Evv	E	E^^
Fb	Fv	F^

13edo: x = +1, bb = -1 (subset of 26edo)
D	Dx	Dxx
Ebb	E	Ex	Exx
Fbb	Fb	F#
13edo, version 2: # = -1, b = +1 (2nd-best 3-limit mapping, 3/2 = 7 steps)
D	Db	Dbb
E#	E	Eb	Ebb
Fx	F#	F

14edo: > = +1, < = -1
D	D>	D>>
E<<	E<	E	E>	E>>
F<<	F<	F

15edo: # = +3, b = -3; ^ = +1, v = -1
D	D^	D#v	D#	D#^	Dxv	Dx
Eb	Eb^	Ev	E	E^	E#v	E#
Fb	Fb^	Fv	F	F^	F#v	F#
Gbb	Gbb^	Gbv	Gb	Gb^	Gv	G

16edo: # = -1, b = +1
D	Db	Dbb
E#	E	Eb	Ebb
Fx#	Fx	F#	F

17edo: # = +2, b = -2; ^ = +1, v = -1
D	D^	D#	D#^	Dx
Ebv	Eb	Eb^	E	E^	E#
Fbb	Fbv	Fb	Fv	F	F^

18edo, version 1: ^^ = +1, vv = -1; x = +3, bb = -3; bb^^ = -2, xvv = +2 (subset of 36edo)
D	D^^	Dxvv	Dx
Ebb	Ebb^^	Evv	E	E^^	Exvv	Ex
Fbvv	Fb	Fb^^	F#vv	F#

18edo, version 2: # = -2, b = +2, ^ = -1, v = +1 (superset of 9edo)
D	Dv	Db	Dbv	Dbb
E#	E^	E	Ev	Eb	Ebv	Ebb
Fx^	Fx	F#v	F#	F^	F

19edo: # = +1, b = -1; # = >, b = <
D	D#	Dx	Dx#
Ebbb	Ebb	Eb	E	E#	Ex
Fbb	Fbb	Fb	F

20edo: # = +4, b = -4, ^ = +2, v = -2, / = +1, \ = -1
D	D/	D^	D#\	D#	D#/	D#^	Dx\	Dx
Eb	Eb/	Ev	E\	E	E/	E^	E#\	E#
Fb	Fb/	Fv	F\	F	F/	F^	F#\	F#

21edo: > = +1, < = -1
D	D>	D>>	D>>>
E<<<	E<<	E<	E	E>	E>>	E>>>
F<<<	F<<	F<	F

22edo: ^ = +1,v = -1, # = +3, b = -3
D	D^	D#v	D#	D#^	Dxv	Dx
Eb	Eb^	Ev	E	E^	E#v	E#
Fb	Fb^	Fv	F	F^	F#v

23edo: # = -1, b = +1
D	Db	Dbb	Dbbb
E	Ex	E#	E 	Eb	Ebb	Ebbb
Fxx	Fx#	Fx	F#	F

24edo: # = +2, b = -2, ^ = +1, v = -1
D	D^	D#	D#^	Dx
Ebv	Eb	Ev	E	E^	E#	E#^
Fbb	Fbv	Fb	Fv	F	F^

25edo: # = +5, b = -5; ^ = +2, v = -2; / = +1, \ = -1
D	D/	D^	D#v	D#\	D#	D#/	D#^	Dxv	Dx\	Dx
Eb	Eb/	Eb^	Ev	E\	E	E/	E^	E#v	E#\	E#
Fb	Fb/	Fb^	Fv	F\	F	F/	F^	F#v	F#\	F#

26edo: # = +1, b = -1
D	D#	Dx	Dx#	Dxx
Ebbb	Ebb	Eb	E	E#	Ex	Ex#
Fbbb	Fbb	Fb	F

27edo: # = +4, b = -4, ^ = +2, v = -2, / = +1, \ = -1
D	D/	D^	D#\	D#	D#/	D#^	Dx\	Dx
Eb	Eb/	Ev	E\	E	E/	E^	E#\	E#
Fb	Fb/	Fv	F\	F	F/	F^	F#\

28edo: > = +2, < = -2, ) = +1, ( = -1
D	D)	D>	D>)	D>>	D>>)
E<<	E<(	E<	E(	E	E)	E>	E>)	E>>
F<<(	F<<	F(	F<	F)	F

29edo: # = +3, b = -3, ^ = +1, v = -1
D	D^	D#v	D#	D#^ 	Dxv	Dx
Ebv	Eb	Eb^	Ev	E	E^	E#v
Fbv	Fb	Fb^	Fv	F

30edo, version 1: # = +6, b = -6; ^ = +3, v = -3; / = +1, v = -1 (superset of 10edo)
D	D/	D^\	D^	D^/	D#\	D#	D#/	D#^\	D#^	D#^/	Dx\	Dx
Eb	Eb/	Ev\	Ev	Ev/	E\	E	E/	E^\	E^	E^/	E#\	E#
Fb	Fb/	Fv\	Fv	Fv/	F\	F	F/	F^\	F^	F^/	F#\	F#

30edo, version 2: # = -1, b = +1, > = +2, < = -2 (2nd-best 3/2, mavila-style)
D	Db	D>	Db>	D>>	Db>>	D>>>
E#<	E<	E#	E	Eb	E>	Eb>	E>>	Eb>>
F#<<	F<<	F#<	F<	F#	F

31edo: # = +2, b = -2, ^ = +1, v = -1
D	D^	D#	D#^	Dx	Dx^
Ebb	Ebv	Eb	Ev	E	E^	E#	E#^
Fbbv	Fbb	Fbv	Fb	Fv	F

32edo: # = +5, b = -5; ^ = +2, v = -2, / = +1, \ = -1
D	D/	D^	D#v	D#\	D#	D#/	D#^	Dxv	Dx\	Dx
Eb	Eb/	Eb^	Ev	E\	E	E/	E^	E#v	E#\	E#
Fb	Fb/	Fb^	Fv	F\	F	F/	F^	F#v	F#\

33edo: # = +1, b = -1; > = +2, < = -2
D	D#	D>	D#>	D>>	D#>>	D>>>
Eb<<	E<<	Eb<	E<	Eb	E	E#	E>	E#>	E>>
F<<<	Fb<<	F<<	Fb<	F<	Fb	F

34edo: # = +4, b = -4; ^ = +2, v = -2; / = +1, \ = -1
D	D/	D^	D#\	D#	D#/	D#^	Dx\	Dx	Dx/	Dx^
Ebv	Eb\	Eb	Eb/	Eb^	E\	E	E/	E^	E#\	E#
Fbb	Fbb/	Fbv	Fb\	Fb	Fb/	Fv	F\	F	F/	F^

35edo: > = +3, < = -3, ) = +1, ( = -1
D	D)	D))	D>	D>)	D>))	D>>
E<((	E<(	E<	E((	E(	E	E)	E))	E>	E>)	E>))
F<<	F<((	F<(	F<	F((	F(	F

36edo: # = +3, b = -3; ^ = +1, v = -1
D	D^	D#v	D#	D#^	Dx	Dx
Ebb	Ebb^	Ebv	Eb	Eb^	Ev	E	E^	E#v	E#
Fbbb	Fbbb^	Fbbv	Fbb	Fbb^	Fbv	Fb	Fb^	Fv	F

37edo: # = +6, b = -6; ^ = +3, v = -3; / = +1, \ = -1
D	D/	D^\	D^	D^/	D#\	D#	D#/	D#^\
Eb	Eb/	Ev\	Ev	Ev/	E\	E	E/
Fb	Fb/	Fv\	Fv	Fv/	F\	F

38edo: # = +2, b = -2; ^ = +1, v = -1
D	D^	D#	D#^	Dx	Dx^	Dx#
Ebbb	Ebbv	Ebb	Ebv	Eb	Ev	E	E^	E#	E#^	Ex
Fbbb	Fbbv	Fbb	Fbv	Fb	Fv	F

39edo: # = +5, b = -5; ^ = +2, v = -2; / = +1, \ = -1
D	D/	D^	D#v	D#\	D#	D#/	D#^	Dxv	Dx\
Ebv	Eb\	Eb	Eb/	Eb^	Ev	E\	E	E/	E^
Fbb/	Fbb^	Fbv	Fb\	Fb	Fb/	Fb^	Fv	F\	F

40edo: # = +1, b = -1; > = +2, < = -2
D	D#	D>	D#>	D>>	D#>>	D>>>
E<<<	Eb<<<	E<<	Eb<	E<	Eb	E	E#	E>	E#>	E>>	E#>>
F<<<	Fb<<	F<<	Fb<	F<	Fb	F

41edo: # = +4, b = -4; ^ = +2, v = -2; / = +1, \ = -1
D	D/	D^	D#\	D#	D#/	D#^	Dx\	Dx
Ebv	Eb\	Eb	Eb/	Ev	E\	E	E/	E^	E#\
Fbbb	Fbb/	Fbv	Fb\	Fb	Fb/	Fv	F\	F

42edo: # = +7, b = -7; ^ = +3, v = -3; / = +1, \ = -1
D	D/	D^\	D^	D#v	D#v/	D#\	D#	D#/	D#^\
Eb	Eb/	Eb^\	Eb^	Ev	Ev/	E\	E	E^
Fb	Fb/	Fb^\	Fb^	Fv	Fv/	F\	F

43edo: # = +3, b = -3; ^ = +1, v = -1
D	D^	D#v	D#	D#^	Dxv	Dx	Dx^
Ebb^	Ebv	Eb	Eb^	Ev	E	E^	E#v	E#	E#^
Fbbb	Fbb^	Fbv	Fb	Fb^	Fv	F

44edo: # = + 6, b = -6; ^ = +2, v = -2; / = +1, \ = -1
D	D/	D^	D^/	D#v	D#\	D#	D#/	D#^	D#^/	Dxv
Ebv	Eb\	Eb	Eb/	Eb^	Ev\	Ev	E\	E	E/	E^
Fb\	Fb	Fb/	Fb^	Fv\	Fv	F\	F

45edo: # = +2, b = -2; ^ = +1, v = -1
D	D^	D#	D#^	Dx	Dx^	Dx#	Dx#^	Dxx
Ebbbv	Ebbb	Ebbv	Ebb	Ebv	Eb	Ev	E	E^	E#	E#^	Ex	Ex^
Fbbbv	Fbbb	Fbbv	Fbb	Fbv	Fb	Fv	F

46edo: # = +5, b = -5, ^ = +2, v = -2, / = +1, \ = -1
D	D/	D^	D#v	D#\	D#	D#/	D#^	Dxv	Dx\	Dx
Ebv	Eb\	Eb	Eb/	Eb^	Ev	E\	E	E/	E^	E#v
Fbb	Fbb/	Fbb^	Fbv	Fb\	Fb	Fb\	Fb^	Fv	F\	F

47edo: # = +1, b = -1; > = +3, < = -3
D	D#	Dx	D>	D#>	Dx>	D>>	D#>>	Dx>>	D>>>
Ebbb	Ebb<	Eb<	E<	Ebb	Eb	E	E#	Ex	E>	E#>	Ex>	E>>
F<<<	Fbb<<	Fb<<	F<<	Fbb<	Fb<	F<	Fbb	Fb	F

48edo: # = +4, b = -4; ^ = +2, v = -2; / = +1, \ = -1
D	D/	D^	D#\	D#	D#/	D#^	Dx\	Dx	Dx/	Dx^	Dxx\	Dxx
Ebb	Ebb/	Ebv	Eb\	Eb	Eb/	Ev	E\	E	E/	E^	E#\	E#
Fbbb	Fbbb/	Fbbv	Fbb\	Fbb	Fbb/	Fbv	Fb\	Fb	Fb/	Fv	F\	F
49edo: # = +7, b = -7, ^ = +3, v = -3; / = +1, \ = -1
D	D/	D^\	D^	D#v	D#v/	D#\	D#	D#/	D#^\	D#^	Dxv
Ebv/	Eb\	Eb	Eb/	Eb^\	Eb^	Ev	Ev/	E\	E	E/	E^\
Fbb^	Fbv	Fbv/	Fb\	Fb	Fb\	Fb^\	Fb^	Fv	Fv/	F\	F

50edo: # = +3, b = -3; ^ = +1, v = -1
D	D^	D^^	D#	D#^	D#^^	Dx	Dx^	Dx^^	Dx#	Dxx
Ebb	Ebvv	Ebv	Eb	Evv	Ev	E	E^	E^^	E#	E#^	E#^^
Fbbb	Fbbvv	Fbbv	Fbb	Fbvv	Fbv	Fb	Fvv	Fv	F

51edo: # = +6, b = -6; ^ = +3, v = -3; / = +1, \ = -1
D	D/	D^\	D^	D^/	D#\	D#	D#/	D#^\	D#^	D#^/	Dx\	Dx
Ebv	Ebv/	Eb\	Eb	Eb/	Ev\	Ev	Ev/	E\	E	E/	E^\	E^
Fbb	Fbb/	Fbv\	Fbv	Fbv/	Fb\	Fb	Fb/	Fv\	Fv	Fv/	F\	F

52edo:	# = +2, b = -2; ^ = +1, v = -1
D	D^	D#	D#^	Dx	Dx^	Dx#	Dx#^	Dxx	Dxx^
Ebbbv	Ebbb	Ebbv	Ebb	Ebv	Eb	Ev	E	E^	E#	E#^	Ex	Ex^	Ex#
Fbbbv	Fbbb	Fbbv	Fbb	Fbv	Fb	Fv	F

53edo: # = +5, b = -5; ^ = +2, v = -2; / = +1, \ = -1
D	D/	D^	D#v	D#\	D#	D#/	D#^	Dxv	Dx\	Dx
Ebb/	Ebb^	Ebv	Eb\	Eb	Eb/	Eb^	Ev	E\	E	E/	E^	E#v	E#/
Fbbv	Fbb\	Fbb	Fbb/	Fbb^	Fbv	Fb\	Fb	Fb/	Fb^	Fv	F\	F

54edo: # = +8, b = -8; ^ = +4, v = -4; / = +2, \ = -2; > = +1, < = -1
D	D>	D/	D/>	D^	D^>	D^/	D#<	D#	D#>	D#/	D#/>	D#^
Eb\	Eb<	Eb	Eb>	Eb/	Ev<	Ev	E\<	E\	E<	E	E>	E/
Fbv	Fb\<	Fb\	Fb<	Fb	Fb>	Fb/	Fv<	Fv	F\<	F\	F<	F

55edo: # = +4, b = -4; ^ = +2, v = -2; / = +1, \ = -1
D	D/	D^	D#\	D#	D#/	D#^	Dx\	Dx	Dx/	Dx^	Dx#\	Dx#	Dx#/	Dx#^
Ebb	Ebb/	Ebv	Eb\	Eb	Eb/	Ev	E\	E	E/	E^	E#\	E#	E#/
Fbbb	Fbbb/	Fbbv	Fbb\	Fbb	Fbb/	Fbv	Fb\	Fb	Fb/	Fv	F\	F

56edo: # = +7, b = -7; ^ = +3, v = -3; / = +1, \ = -1
D	D/	D^\	D^	D#v	D#v/	D#\	D#	D#/	D#^\	D#^	Dxv	Dxv/	Dx\
Ebv	Ebv/	Eb\	Eb	Eb/	Eb^\	Eb^	Ev	Ev/	E\	E	E/	E^\	E^
Fbb/	Fbb^\	Fbb^	Fbv	Fbv/	Fb\	Fb	Fb/	Fb^\	Fb^	Fv	Fv/	F\	F

57edo: # = +3, b = -3; ^ = +1, v = -1
D	D^ 	D#v	D#	D#^	Dxv	Dx	Dx^	Dx#v	Dx#
Ebbb	Ebbb^	Ebbv	Ebb	Ebb^	Ebv	Eb	Eb^	Ev	E	E^	E#v	E#	E#^	Exv	Ex
Fbbbv	Fbbb	Fbbb^	Fbbv	Fbb	Fbb^	Fbv	Fb	Fb^	Fv	F

58edo, version 1: # = +6, b = -6; ^ = +3, v = -3, / = +1, \ = -1
D	D/	D^\	D^	D^/	D#\	D#	D#/	D#^\	D#^	D#^/	Dx\	Dx
Ebv	Ebv/	Eb\	Eb	Eb/	Ev\	Ev	Ev/	E\	E	E/	E^\	E^
Fbb	Fbb/	Fbv\	Fbv	Fbv/	Fb\	Fb	Fb/	Fv\	Fv	Fv/	F\	F

58edo, version 2: # = +6, b = -6; ^ = +2, v = -2, / = +1, \ = -1 (superset of 29edo)
D	D/	D^	D^/	D#v	D#\	D#	D#/	D#^	D#^/	Dxv	Dx\	Dx
Ebv\	Ebv	Eb\	Eb	Eb/	Eb^	Ev\	Ev	E\	E	E/	E^	E^/	E#v
Fbb	Fbb/	Fbb^	Fbv\	Fbv	Fb\	Fb	Fb/	Fb^	Fv\	Fv	F\	F

59edo: # = +9, b = -9; ^ = +4, v = -4; / = +2, \ = -2; > = +1, < = -1
D	D>	D/	D/>	D^	D#v	D#\<	D#\	D#<	D#	D#>	D#/	D#/>	D#^
Eb	Eb>	Eb/	Eb/>	Eb^	Ev	E\<	E\	E<	E	E>	E/
Fb	Fb>	Fb/	Fb/>	Fb^	Fv	F\<	F\	F<	F

60edo: # = +5, b = -5; ^ = +2, v = -2; / = +1, \ = -1
D	D/	D^	D#v	D#\	D#	D#/	D#^	Dxv	Dx\	Dx	Dx/	Dx^	Dx#v	Dx#\	Dx#
Ebb	Ebb/	Ebb^	Ebv	Eb\	Eb	Eb/	Eb^	Ev	E\	E	E/	E^	E#v	E#\	E#
Fbbb	Fbbb/	Fbbb^	Fbbv	Fbb\	Fbb	Fbb/	Fbb^	Fbv	Fb\	Fb	Fb/	Fb^	Fv	F\	F

61edo: # = +8, b = -8; ^ = +4,v = -4; / = +2, \ = -2; > = +1, < = -1
D	D>	D/	D/>	D^	D^>	D^/	D#<	D#	D#>	D#/	D#/>	D#^	D#^>
Eb\	Eb<	Eb	Eb>	Eb/	Ev<	Ev	E\<	E\	E<	E	E>	E/	E/>
Fbv	Fb\<	Fb\	Fb<	Fb	Fb>	Fb/	Fv<	Fv	F\<	F\	F<	F

62edo: # = +4, b = -4, ^ = +2, v = -2; / = +1, \ = -1
D	D/	D^	D^/	D#	D#/	D#^	D#^/	Dx	Dx/	Dx^	Dx^/	Dx#
Ebbv	Ebb\	Ebb	Ebv\	Ebv	Eb\	Eb	Ev\	Ev	E\	E	E/	E^	E^/	E#	E#/	E#^
Fbbb	Fbbv\	Fbbv	Fbb\	Fbb	Fbv\	Fbv	Fb\	Fb	Fv\	Fv	F\	F

63edo: # = +7, b = -7; ^ = +3, v = -3; / = +1, \ = -1
D	D/	D^\	D^	D#v	D#v/	D#\	D#	D#/	D#^\	D#^	Dxv	Dxv/	Dx\
Ebv	Ebv/	Eb\	Eb	Eb/	Eb^\	Eb^	Ev	Ev/	E\	E	E/	E^\	E^
Fbb/	Fbb^\	Fbb^	Fbv	Fbv/	Fb\	Fb	Fb/	Fb^\	Fb^	Fv	Fv/	F\	F

64edo: # = +3, b = -3; ^ = +1, v = -1
D	D^ 	D#v	D#	D#^	Dxv	Dx	Dx^	Dx#v	Dx#	Dx#^
Ebbb	Ebbb^	Ebbv	Ebb	Ebb^	Ebv	Eb	Eb^	Ev	E	E^	E#v	E#	E#^	Exv	Ex	Ex^
Fbbbv	Fbbb	Fbbb^	Fbbv	Fbb	Fbb^	Fbv	Fb	Fb^	Fv	F

65edo: # = +6, b = -6; ^ = +3, v = -3; / = +1, \ = -1
D	D/	D^\	D^	D^/	D#\	D#	D#/	D#^\	D#^	D#^/	Dx\	Dx	Dx/	Dx^\	Dx^
Ebv	Ebv/	Eb\	Eb	Eb/	Ev\	Ev	Ev/	E\	E	E/	E^\	E^	E^/	E#\
Fbb	Fbb/	Fbv\	Fbv	Fbv/	Fb\	Fb	Fb/	Fv\	Fv	Fv/	F\	F

66edo version 1: # = +9, b = -9; ^ = +4, v = -4; / = +2, \ = -2; > = +1, < = -1
D	D>	D/	D/>	D^	D#v	D#\<	D#\	D#<	D#	D#>	D#/	D#/>	D#^
Eb\	Eb<	Eb	Eb>	Eb/	Eb/>	Eb^	Ev	E\<	E\	E<	E	E>	E/	E/>
Fb\<	Fb\	Fb<	Fb	Fb>	Fb/	Fb/>	Fb^	Fv	F\<	F\	F<	F

66edo version 2: # = +9, b = -9; ^ = +3, v = -3; / = +1, \ = -1 (22edo backwards-compatible)
D	D/	D^\	D^	D^/	D#v\	D#v	D#v/	D#\	D#	D#/	D#^\	D#^	D#^/
Ebv	Ebv/	Eb\	Eb	Eb/	Eb^\	Eb^	Eb^/	Ev\	Ev	Ev/	E\	E	E/	E^\	F^
Fbv	Fbv/	Fb\	Fb	Fb/	Fb^	Fb^\	Fb^	Fb^/	Fv\	Fv	Fv/	F\	F

67edo: # = +5, b = -5; ^ = +2, v = -2, / = +1, \ = -1
D	D/	D^	D#v	D#\	D#	D#/	D#^	Dxv	Dx\	Dx	Dx/	Dx^	Dx#v	Dx#\	Dx#
Ebb	Ebb/	Ebb^	Ebv	Eb\	Eb	Eb/	Eb^	Ev	E\	E	E/	E^	E#v	E#\	E#
Fbbb	Fbbb/	Fbbb^	Fbbv	Fbb\	Fbb	Fbb/	Fbb^	Fbv	Fb\	Fb	Fb/	Fb^	Fv	F\	F

68edo: # = +8, b = -8, ^ = +4, v = -4; / = +2, \ = -2; ) = +1, ( = -1
D	D)	D/	D/)	D^	D^)	D#\	D#(	D#	D#)	D#/	D#^(	D#^	D#^)	Dx\	Dx(	Dx
Ebv	Ebv)	Eb\	Eb(	Eb	Eb)	Eb/	Eb^(	Eb^	E\(	E\	E(	E	E)	E/	E^(	E^
Fbb	Fbb)	Fbb/	Fbv(	Fbv	Fbv)	Fb\	Fb(	Fb	Fb)	Fb/	Fv(	Fv	Fv)	F\	F(	F

69edo: # = +4, b = -4, ^ = +2, v = -2; / = +1, \ = -1
D	D/	D^	D^/	D#	D#/	D#^	D#^/	Dx	Dx/	Dx^	Dx^/	Dx#
Ebbv	Ebb\	Ebb	Ebv\	Ebv	Eb\	Eb	Ev\	Ev	E\	E	E/	E^	E^/	E#	E#/	E#^
Fbbb	Fbbv\	Fbbv	Fbb\	Fbb	Fbv\	Fbv	Fb\	Fb	Fv\	Fv	F\	F

70edo: # = +7, b = -7; ^ = +3, v = -3; / = +1, \ = -1
D	D/	D^\	D^	D#v	D#v/	D#\	D#	D#/	D#^\	D#^	Dxv	Dxv/	Dx\
Ebv	Ebv/	Eb\	Eb	Eb/	Eb^\	Eb^	Ev	Ev/	E\	E	E/	E^\	E^
Fbb/	Fbb^\	Fbb^	Fbv	Fbv/	Fb\	Fb	Fb/	Fb^\	Fb^	Fv	Fv/	F\	F

71edo: # = +10, b = -10; ^ = +5, v = -5; / = +2, \ = -2; > = +1, < = -1
D	D>	D/	D/>	D^<	D^	D^>	D^/	D#\	D#<	D#	D#>	D#/	D#/>	D#^<	D#^
Eb	Eb>	Eb/	Eb/>	Ev\	Ev<	Ev	Ev>	E\<	E\	E<	E	E>	E/	E/>
Fb	Fb>	Fb/	Fb/>	Fv\	Fv<	Fv	Fv>	F\<	F\	F<	F

72edo: # = +6, b = -6; ^ = +3, v = -3; / = +1, \ = -1
D	D/	D^\	D^	D^/	D#\	D#	D#/	D#^\	D#^	D#^/	Dx\	Dx	Dx/	Dx^\	Dx^
Ebv	Ebv/	Eb\	Eb	Eb/	Ev\	Ev	Ev/	E\	E	E/	E^\	E^	E^/	E#\
Fbb	Fbb/	Fbv\	Fbv	Fbv/	Fb\	Fb	Fb/	Fv\	Fv	Fv/	F\	F

Dave Keenan
Posts: 896
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: EDOs with multiple prime mappings

cryptic.ruse wrote:I have yet to catch up on your latest replies in this thread, as I've been traveling for business this weekend and have elected to focus on actually finishing the list of enharmonic equivalents in my proposal. I have finally completed it, although I confess much of it was typed out on airplanes, in the backseats of cars, and after long days. As such, errors are likely, as are minor inconsistencies in formatting. Hopefully they are few enough that they do not completely wreck the entire edifice. It was quite a slog!
I really appreciate your effort. I have only made a random sampling so far and will have more to say after I have worked through it systematically. I look forward to your response to my latest replies when you do have a chance to catch up. I am particularly interested in your response to
viewtopic.php?p=417#p417
I should note that this is not an exhaustive list of all enharmonic equivalents; whenever there are multiple accidentals, there are multiple ways to specify a single pitch with a single nominal. This makes it difficult to count all the combined accidentals that may crop up, since I have not exhaustively searched for circumstances when one spelling may be more "correct" than another (for instance, when one might wish to use F^^^ in place of F# in 22edo). I am not sure if you've spent much time exhausting the different ways Standard Sagittal accidentals may or may not combine in each ET.
I'm not sure how you could have failed to understand, or notice, that we do not combine Sagittal accidentals (except in the one-symbol-per-prime JI notation). That's why, instead of a glossary giving the number of steps for each symbol, we just list the symbols for consecutive numbers of steps from natural to apotome. So in our case, the number of symbols is also the number of combinations.

But don't worry about counting your combinations, because, as you'll see in the above-linked post, I show that you don't actually need to use any combinations of your apotome and limma fraction symbols. However, to pull off that trick all the way up to 71-edo, you do need one extra symbol pair, for 1/8-apotome, to deal with the handful of EDOs that have 8, 9 or 10 steps to the apotome.

cryptic.ruse
Posts: 23
Joined: Tue Sep 22, 2015 4:38 am

### Re: EDOs with multiple prime mappings

Okay, going to try to work my way through your replies.
I have no idea how any sensible and usable EDO notation could actually reflect the presence of _all_ the well-approximated primes. Please explain.
If by "all", you mean "to the infinite limit" then of course, it would be absurd to attempt to notate that. What I meant was "all the primes you care about, which you consider well-approximated". The full set of Sagittal symbols would undoubtedly be enough for 99.999% of use-cases, and anything beyond it would be better served by something like AFMM notation, which simply appends cents offsets to the note heads (since memorizing the lexicon of 53-limit commas and assigning them symbols that are sufficiently distinguishable would more or less be impossible for all but the most utterly dedicated musician).
I don't believe I have ever claimed that whenever good low-prime-above-3 approximations exist in an EDO, the notation makes that information available.
Ok, you're right, there are some unstated assumptions. I assumed that, since you and George designed the notation, if you wanted it to do something, it would do it. But you clearly explain it's a balancing act, and I see that.
I don't know what you mean by "Full Sagittal". To me, the alternative to a standard Sagittal notation for some EDO, within the Sagittal system, is any number of non-standard Sagittal notations for that EDO.
By "Full Sagittal", I mean that the composer has access to the entire range of Sagittal symbols, and chooses whichever ones best reflect the rational identities they are trying to convey in the score. This may involve ignoring that certain commas vanish, and using them anyway, simply to indicate that a specific ratio is intended. Or, in other words, notating ETs as if they were pure JI.
So you're saying it makes no sense to include any symbols for primes higher than 3 in any EDO notation unless the higher prime information is considered integral to the composition?
Yes, that is precisely what I am saying. The most basic notation doesn't need to tell you anything beyond what pitches to play--something like guitar tablature, where you get fret numbers and nothing else. A step up from that is a notation that tells you something about the scalar relationships between pitches, how they fit into some kind of scalar framework or relate to some tonic note, or what kind of chord something is, and so on. This is what you get with common-practice staff notation--we can see relationships by fifths, and by proxy, relationships by diatonic scale degrees; spellings matter insofar as scalar/tonal relationships matter. My proposed notation is designed to translate this approach to broadly cover a range of manageably-sized ETs, without adding any additional information.

Full Sagittal is, in my opinion, another level of information, communicating not simply scalar relationships but also harmonic series ones. We gain a layer of intonational detail, telling us for example not simply that an interval is an augmented 2nd, but that it also represents a particular ratio (or ratios), indicating its relationship to other pitches along a whole new set of dimensions.

In my estimation, Standard Sagittal for ETs appears like an attempt to reduce Full Sagittal so that, in a given ET, it will behave more like traditional notation, i.e. not explicitly spell out all the important harmonic series relationships, and in fact be useable in a context where harmonic series relationships can be more or less ignored. But it doesn't _really_ do that, because you are still trying to hold on to some of those harmonic series relationships when you think they're important enough AND you have room for them in the notation. My entire thesis here is that that approach is not optimal for those who want a notation similar to common-practice notation (i.e. without the extra harmonic series information), nor is it optimal for those who want a notation that DOES communicate that extra layer of harmonic series information. It's a compromise between the two, and a very specific one that reflects your judgment, and George's judgment, and the judgment of anyone else whose input was considered in the standardization. As such, it will only be optimal for those who share the same priorities; for anyone else, it will either be overloaded or insufficient. If I didn't personally find it overloaded for my uses, we wouldn't be having this conversation and I would not have spent time developing the approach I propose.
In Standard Sagittal there exists no single inviolable rule for when symbols representing a given prime will be used or not. You have a multitude of different considerations and priorities that lead to exceptional cases and debatable choices.
That is true for those extreme-fifth EDOs listed above, but not for most of the others.
I do not know how to make it more clear that a rule holding for "most" cases is not the same as a rule holding for "all" cases, or that 25 exceptional cases is, in my estimation, quite a lot, and quite sufficient to say that the standard is NOT consistent. And I'm assuming those 25 "extreme fifths" cases do not include the ETs that are notated as subsets, either...if we include those too, what does the number grow to? I suppose I'll have to count them in your chart.
So what we're doing is asking that composer, for the good of the team, to ignore the 5-limit implications of the symbol in this case, and just think of it as the half-apotome symbol for 22-edo, because most composers _are_ interested in the 5-limit (and indeed the 7-limit) implications of 22-edo.
I believe this is asking more than is necessary, and is therefore less universal. When we look beyond the use-cases of each individual ETs, and instead look at ETs as a whole, we will be drastically forced to reduce our assumptions about what sort of harmonic relationships composers may or may not be interested in. Rather than making a patchwork notation based on assumptions of compositional interests for each isolated ET, I advocate for making a uniform notation based on as few assumptions as possible. I am not looking at individual ETs, I am looking at a range of them, and trying to throw out as much as possible that is not common between the individual use-cases.

Someone using 22edo along with 23edo, 25edo, and 28edo may have a different approach to 22edo than the person using 22edo along with 19edo, 31edo, and 53edo (for example). I personally would argue that the 5-limit is one of the least important aspects of 22edo, because there are plenty of other good 5-limit temperaments. I would use 22edo over (say) 19edo if I wanted better ratios of 7 and 11, so to me, the way 22edo implements 11-limit harmony is what is important about it, and would prefer an accidental that reflected that. Why should we ask some composers to sacrifice more "for the good of the team", when we can ask instead for all to make an equal sacrifice? That seems more fair, and thus more desirable, more universal, and more likely to "catch on".
I agree with your priorities here. But it's a serious misconception about Sagittal if you think a performer, or even a composer, is expected to apply a bunch of rules to generate a notation, for any EDO they want to use. That's what the list of standard EDO notations is for -- so they don't have to.
Here I see that we have different ideas of how to make something easy to learn. In my view, it is easier to learn a single rule than to memorize a table, though undoubtedly some memorization will take place as well. From a rule, one can re-generate information as necessary when memory fails, or when the chart is not handy. And one can count on certain symbols behaving certain ways. When I wrote out my chart of enharmonics, I did not do it from memory, nor did I do it in reference to a chart I had worked out previously. I found the three most important intervals—limma, apotome, and whole tone—and could generate all the enharmonic equivalents on the fly, because I knew the symbol set was consistent and could derive the ET-specific meaning of each symbol from the relationship between those three intervals. To be able to do that seems like a plus, to me--it provides a framework by which to understand the specifics of a given ET's notation, it's not blind memorization.
Given any pre-existing JI notation, it would be an exercise in willful ignorance not to notate 72-edo using symbols for 81/80, 64/63 and 33/32, for example.
From what I hear about the Boston Microtonal Society, that willful ignorance with respect to 72-edo may be more common than you think. Julian Carrillo comes to mind as well. It is easy to forget that many--perhaps most!--who care about microtones do not necessarily care about JI. We hear little from them in the online communities, but we ignore their existence at our own peril. And in any case, a meta-notational note would be sufficient to convert 3-limit symbols into their higher-prime synonyms within a given ET.

I submit that it is easier overall to add back in the "missing information" meta-notationally than it is to insist that all composers learn a more complex notation that may not reflect their interests and then ignore whatever information they deem unnecessary. Especially when we expect composers and performers to consult charts in learning the notation in the first place.
But if it was just re-using symbols used in 72-edo, for lower EDOs, where would be the harm?
The harm would be in assuming that everyone cares enough about 72edo to be familiar with all of its accidentals. As I've shown, I can completely notate 72edo by combining 3 pairs of symbols (including the apotome), all of which appear together first in 20edo. It's actually quite an easy ET to notate, 69edo and 71edo require more accidentals. So perhaps 72edo is not a good ET to use for a benchmark. In any case, the larger ETs get built up from the smaller ones for the most part, in my approach; they do not get dissected out of the larger ones. Only the handful of pathological ones (6, 8, 13, 18) where the limma is negative necessarily get treated as subsets, and even then in most cases they don't have to be--with the exception of 8edo, the 2nd-best 3/2 restores the limma to being positive, and thus the normal approach works.
No. That isn't _always_ debatable. The most important prime approximations in 72-edo are not debatable.
How are they not? The fact that 72edo approximates any primes at all is of debatable importance. Julian Carrillo cared not at all about its rational approximations. Even if one does insist its rational approximations are important, of the myriad primes it approximates well, it is entirely subjective as to which ones a composer may care about. I may only care about its 5-limit approximations and have no interest in how it handles primes 7, 11, or 13; or I may care not at all for its approximations to the ratios of 5, and primarily concern myself with its 2.3.7.13 approximations. It is not necessary for anyone to care about any particular prime approximations in order to use an ET.

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### Re: EDOs with multiple prime mappings

cryptic.ruse wrote:I assumed that, since you and George designed the notation, if you wanted it to do something, it would do it. But you clearly explain it's a balancing act, and I see that.
It makes me happy that you see that. Thanks for letting me know. And I hope you understand by now, that I have come a long way towards your position since you first announced your draft paper on Facebook. You may also recall that I encouraged you, some months prior to that, to undertake this investigation, in the hope that it might lead to improvements in Sagittal.
By "Full Sagittal", I mean that the composer has access to the entire range of Sagittal symbols, and chooses whichever ones best reflect the rational identities they are trying to convey in the score. This may involve ignoring that certain commas vanish, and using them anyway, simply to indicate that a specific ratio is intended. Or, in other words, notating ETs as if they were pure JI.
OK. At last I understand. Yes, I would simply refer to this as notating ETs using JI notation. So your "full Sagittal" is synonymous with "Sagittal JI notation". But there are several precision levels of JI Notation available in Sagittal, so I guess you mean the lowest level sufficient for the primes you care about.
Full Sagittal [aka Sagittal JI Notation] is, in my opinion, another level of information, communicating not simply scalar relationships but also harmonic series ones. We gain a layer of intonational detail, telling us for example not simply that an interval is an augmented 2nd, but that it also represents a particular ratio (or ratios), indicating its relationship to other pitches along a whole new set of dimensions.
Agreed.
In my estimation, Standard Sagittal for ETs appears like an attempt to reduce Full Sagittal [aka the JI notation] so that, in a given ET, it will behave more like traditional notation, i.e. not explicitly spell out all the important harmonic series relationships, and in fact be useable in a context where harmonic series relationships can be more or less ignored. But it doesn't _really_ do that, because you are still trying to hold on to some of those harmonic series relationships when you think they're important enough AND you have room for them in the notation.
I would say you were exactly right if you had left out the phrase "But it doesn't _really_ do that, because". Harmonic series relationships _can_ be more or less ignored, despite holding on to the symbols for some of them when we think they're important enough AND we have room for them in the notation.
My entire thesis here is that that approach is not optimal for those who want a notation similar to common-practice notation (i.e. without the extra harmonic series information), nor is it optimal for those who want a notation that DOES communicate that extra layer of harmonic series information. It's a compromise between the two, and a very specific one that reflects your judgment, and George's judgment, and the judgment of anyone else whose input was considered in the standardization.
That's exactly right. That's what it takes to produce a standard. The purpose of a standard is to allow people to communicate with each other while minimising the total effort required.
As such, it will only be optimal for those who share the same priorities; for anyone else, it will either be overloaded or insufficient. If I didn't personally find it overloaded for my uses, we wouldn't be having this conversation and I would not have spent time developing the approach I propose.
Of course. And the only reason I am still listening, is not because I think everyone should use the notation that is optimal for them -- if we all did that there would be no standard and no communication -- but because you have convinced me that your notational semantics, or something very like it, is a better compromise, for a certain class of EDOs, than what we currently have.
So what we're doing is asking that composer, for the good of the team, to ignore the 5-limit implications of the symbol in this case, and just think of it as the half-apotome symbol for 22-edo, because most composers _are_ interested in the 5-limit (and indeed the 7-limit) implications of 22-edo.
I believe this is asking more than is necessary, and is therefore less universal. When we look beyond the use-cases of each individual ETs, and instead look at ETs as a whole, we will be drastically forced to reduce our assumptions about what sort of harmonic relationships composers may or may not be interested in. Rather than making a patchwork notation based on assumptions of compositional interests for each isolated ET, I advocate for making a uniform notation based on as few assumptions as possible. I am not looking at individual ETs, I am looking at a range of them, and trying to throw out as much as possible that is not common between the individual use-cases.
You are not looking at EDOs as a whole, and I am not making a patchwork notation for isolated EDOs. We are both looking at a range of EDOs, and trying to generate good notations for them using a simple 3-prime-limit algorithm. You are looking at the range of EDOs whose cardinality is less than or equal to 72. I am looking at the range of EDOs whose fifth-error is greater than 7.5 cents (the largest of which is 71-edo).
Someone using 22edo along with 23edo, 25edo, and 28edo may have a different approach to 22edo than the person using 22edo along with 19edo, 31edo, and 53edo (for example). I personally would argue that the 5-limit is one of the least important aspects of 22edo, because there are plenty of other good 5-limit temperaments. I would use 22edo over (say) 19edo if I wanted better ratios of 7 and 11, so to me, the way 22edo implements 11-limit harmony is what is important about it, and would prefer an accidental that reflected that. Why should we ask some composers to sacrifice more "for the good of the team", when we can ask instead for all to make an equal sacrifice? That seems more fair, and thus more desirable, more universal, and more likely to "catch on".
It seems we agree that (at least for EDOs with good fifths, like 72-edo and 22-edo) there is a one dimensional tradeoff available here along a spectrum from no higher primes and one symbol-combination per step, thru the present scheme of some higher primes and one symbol-combination per step, to all-significant primes and multiple symbol-combinations per step. If for all three types of notation there exist composers for whom that type is optimal, then there is no choice of standard notation for which all composers would make an equal sacrifice (and what about the performers?). I'm saying we'll minimise the total effort over all users by choosing the type in the middle to be the standard. Even if we assume that ignoring meanings is just as difficult as learning new ones (which I dispute), the only way that choosing one of the extremes as the standard, could minimise the total effort, is if more than 50% of users find that extreme to be optimal. I can accept that that may be the case for EDOs with bad fifths, but I think it extremely unlikely for those with good fifths.
Here I see that we have different ideas of how to make something easy to learn. In my view, it is easier to learn a single rule than to memorize a table, though undoubtedly some memorization will take place as well. From a rule, one can re-generate information as necessary when memory fails, or when the chart is not handy. And one can count on certain symbols behaving certain ways.
It may be easy to learn a rule, but difficult to apply it without using a computer or calculator, even when one is familiar with the relevant mathematics.
When I wrote out my chart of enharmonics, I did not do it from memory, nor did I do it in reference to a chart I had worked out previously. I found the three most important intervals—limma, apotome, and whole tone—and could generate all the enharmonic equivalents on the fly, because I knew the symbol set was consistent and could derive the ET-specific meaning of each symbol from the relationship between those three intervals. To be able to do that seems like a plus, to me--it provides a framework by which to understand the specifics of a given ET's notation, it's not blind memorization.
Of course, George and I can do the same thing with the JI-based notations for the ETs with good fifths. But I'm guessing we all have a level of mathematical sophistication that the average musician should not be assumed to possess.
From what I hear about the Boston Microtonal Society, that willful ignorance with respect to 72-edo may be more common than you think. Julian Carrillo comes to mind as well. It is easy to forget that many--perhaps most!--who care about microtones do not necessarily care about JI. We hear little from them in the online communities, but we ignore their existence at our own peril.
I had the BMS firmly in mind as I wrote that. I don't usually think of Carrillo as using 72-edo, since he favoured 24, 48 and 96, but sure, throw him and his students in too. I still think it's a minority.
And in any case, a meta-notational note would be sufficient to convert 3-limit symbols into their higher-prime synonyms within a given ET.
Sure. But to ignore the higher prime meanings of JI-based EDO symbols doesn't even require an extra-notational note. Such composers would never have learned the JI meanings of those symbols in the first place. Apologies for the pedantry, but I prefer the term "extra-notational" for this, as "meta-notation" is notation used for describing other notations.
I submit that it is easier overall to add back in the "missing information" meta-notationally than it is to insist that all composers learn a more complex notation that may not reflect their interests and then ignore whatever information they deem unnecessary. Especially when we expect composers and performers to consult charts in learning the notation in the first place.
For those EDOs whose fifths are within 7.5 cents of just, their existing standard Sagittal notations are not more complex than your proposed notations. But for most of the other EDOs, I agree that some version of your notation will be simpler, and I am willing to adopt it.
But if it was just re-using symbols used in 72-edo, for lower EDOs, where would be the harm?
The harm would be in assuming that everyone cares enough about 72edo to be familiar with all of its accidentals. As I've shown, I can completely notate 72edo by combining 3 pairs of symbols (including the apotome), all of which appear together first in 20edo.
One doesn't have to care about 72-edo, in order to use the same 4 pairs of symbols (including apotome) for other EDOs. Sure, you use only 3 pairs, but only by combining a +3 symbol and a -1 symbol to make a +2 symbol (why not +1 and +1?). That's another kind of complexity in addition to the symbol-count complexity.
Only the handful of pathological ones (6, 8, 13, 18) where the limma is negative necessarily get treated as subsets, and even then in most cases they don't have to be--with the exception of 8edo, the 2nd-best 3/2 restores the limma to being positive, and thus the normal approach works.
I don't understand why you have to use a subset or a second-best fifth for these. They all have a positive apotome, using their best fifth. So why not notate them as apotome-fractions?
How are they not [debatable]? The fact that 72edo approximates any primes at all is of debatable importance. Julian Carrillo cared not at all about its rational approximations. Even if one does insist its rational approximations are important, of the myriad primes it approximates well, it is entirely subjective as to which ones a composer may care about. I may only care about its 5-limit approximations and have no interest in how it handles primes 7, 11, or 13; or I may care not at all for its approximations to the ratios of 5, and primarily concern myself with its 2.3.7.13 approximations. It is not necessary for anyone to care about any particular prime approximations in order to use an ET.
I agree with all that you say here. But when speaking of the "importance" of prime approximations in deciding a standard notation for an EDO, I am concerned with the sum over all users. And I don't think there is much doubt what the outcome would be if you could take a vote of all present and future users of 72-edo, even if you allowed the BMS's and Carrillos to give negative votes.

One should also consider the possibility that some composers may make use of 72-edo's JI approximations, say for shades of consonance and dissonance, without knowing anything of their mathematical basis.

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### Re: EDOs with multiple prime mappings

Some questions after having perused your notations:

1. (Mentioned above) When the limma is zero or negative, why not notate using apotome fractions only, instead of resorting to second-best fifths or subset notations?

2. When the apotome is zero or negative, why not notate using limma fractions only, instead of allowing to lower pitch and to raise it? That way madness lies.

3. Why use multiple symbols when a single symbol will do?

I expect your answer to all of these is that there aren't enough symbols, and you don't want to add any more, because the whole point was to minimise the symbol count.

But all three can be achieved without adding any more symbols, except in the case of six largish EDOs: 54, 59, 61, 66, 68, 71 (those with more than 7 steps to the apotome). We'll ignore those for the moment.

The algorithm works by first dividing EDOs into two classes -- those that have a positive apotome and those that do not. The non-positive-apotome class consists of the nine EDOs: 7, 9, 11, 14, 16, 21, 23, 28, 35. We notate positive-apotome EDOs using only apotome fractions and we notate non-positive-apotome EDOs using only limma fractions.

I expect you would have noticed that in Pythagorean tuning (3-limit JI), a limma is very close to 4/5ths of an apotome, which means a 1/2-limma is 2/5-apotome and 1/4-limma is 1/5-apotome. So when we're notating positive-apotome EDOs we repurpose the limma-fraction symbols as apotome-fraction symbols (using your ASCII placeholders):

)  (	1/5, 1/6, 1/7 apotome
/  \	1/4, 1/3, 2/7 apotome
>  <	2/5, 3/7 apotome
^  v	1/2 apotome
#< b>	3/5, 4/7 apotome
#\ b/	3/4, 2/3, 5/7 apotome
#( b)	4/5, 5/6, 6/7 apotome
#  b	1 apotome
e.g.
70edo: # = +7, b = -7; > = +3, < = -3; / = +2, \ = -2; ) = +1, ( = -1									:70edo
D	D)	D/	D>	D#<	D#\	D#(	D#	D#)	D#/	D#>	Dx<	Dx\	Dx(	Dx	Dx)	Dx/	Dx>
Ebb/	Ebb>	Eb<	Eb\	Eb(	Eb	Eb)	Eb/	Eb>	E<	E\	E(	E	E)	E/	E>	E#<	E#\
Fbb<	Fbb\	Fbb(	Fbb	Fbb)	Fbb/	Fbb>	Fb<	Fb\	Fb(	Fb	Fb)	Fb/	Fb>	F<	F\	F(	F
[Use the horizontal scroll bar at the bottom of the post to see the rest of 70-edo -->]

And when we're notating non-positive-apotome EDOs we repurpose the apotome-fraction symbols as limma-fraction symbols:

)  (	1/4, 1/5 limma
/  \	1/3, 2/5 limma
>  <	1/2 limma
^  v	2/3, 3/5 limma
e.g.
35edo: ^ = +3, v = -3; / = +2, \ = -2; ) = +1, ( = -1
D	D)	D/	D^
Ev	E\	E(	E	E)	E/	E^
Fv	F\	F(	F
Those with more than 7 steps to the apotome require one more symbol-pair -- call it the 1/8-apotome pair. I'll use curly braces as placeholders for them, } up, { down.

}  {	1/8, 1/9, 1/10 apotome
)  (	1/5, 1/6, 1/7, 2/9 apotome
/  \	1/4, 1/3, 2/7, 3/10 apotome
<  >	2/5, 3/7, 3/8, 4/9 apotome
^  v	1/2 apotome
#< b>	3/5, 4/7, 5/8, 5/9 apotome
#\ b/	3/4, 2/3, 5/7, 7/10 apotome
#( b)	4/5, 5/6, 6/7, 7/9 apotome
#{ b}	7/8, 8/9, 9/10 apotome
#  b	1 apotome
e.g.
71edo: # = +10, b = -10; ^ = +5, v = -5; < = +4, > = -4; / = +3, \ = -3; ) = +2, ( = -2; } = +1, { = -1				:71edo
D	D}	D)	D/	D>	D^	D#<	D#\	D#(	D#{	D#	D#}	D#)	D#/	D#>	D#^	Dx<
Eb\	Eb(	Eb{	Eb	Eb}	Eb)	Eb/	Eb>	Ev	E<	E\	E(	E{	E	E}	E)	E/
Fbb>	Fbv	Fb<	Fb\	Fb(	Fb{	Fb	Fb}	Fb)	Fb/	Fb>	Fv	F<	F\	F(	F{	F
[Use the horizontal scroll bar below to see the rest of 71-edo -->]

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### Re: EDOs with multiple prime mappings

Hi Cryptic. I expect you have a table like this that you have referred to while constructing these notations. This one shows EDOs from 5 to 100 arranged according to their steps per apotome (# or b) and steps per limma (B:C or E:F). Steps per whole tone (C:D, D:E, F:G, G:A, A:B) can be obtained by adding the preceding two figures.

I've coloured it to shown my proposed classes of EDO notation.
EdoNotationClasses.gif
[Edit: 66-edo should be blue on the diagram]
Last edited by Dave Keenan on Thu Oct 13, 2016 2:34 pm, edited 1 time in total.
Reason: Added a note that 66-edo should be blue on the diagram.

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### Re: EDOs with multiple prime mappings

I see I'm in trouble, with purely apotome-fraction notations for 40 and 47. Not really a problem for mixed Sagittal, since I can use triple sharps and flats, but there are no equivalent symbols in pure-Sagittal. So a limma-fraction notation might be preferable.

Apotome fraction notations
40edo: # = +1, b = -1
D	D#	Dx	Dx#
Ebbb	Ebb	Eb	E	E#	Ex	Ex#
Fbbb	Fbb	Fb	F

47edo: # = +1, b = -1
D	D#	Dx	Dx#
Ebbb	Ebb	Eb	E	E#	Ex	Ex#
Fbbb	Fbb	Fb	F
Limma fraction notations

For 47-edo, I need to add 1/6-limma as one of the allowed meanings for the 1/4-limma symbol.

)  (	1/4, 1/5, 1/6 limma
/  \	1/3, 2/5 limma
>  <	1/2 limma
^  v	2/3, 3/5 limma
40edo: ^ = +3, v = -3; / = +2, \ = -2; ) = +1, ( = -1
D	D)	D/	D^
Ev	E\	E(	E	E)	E/	E^
Fv	F\	F(	F

47edo: ^ = +4, v = -4; > = +3, < = -3; / = +2, \ = -2; ) = +1, ( = -1
D	D)	D/	D>	D^
Ev	E<	E\	E(	E	E)	E/	E>	E>
Fv	F<	F\	F(	F
Perhaps all the EDOs with fifths narrower than meantone should have limma-fraction notations. So all the orange EDOs on the upper-right of my diagram would be coloured red, like those above them. Let's see if we can do 64-edo, which has 7 steps to the limma and 10 steps to the whole-tone. To do that, I need to repurpose the curly-brace 1/8-apotome symbols } { for 1/7-limma.

}  {	1/7 limma
)  (	1/4, 1/5, 1/6, 2/7 limma
/  \	1/3, 2/5, 3/7 limma
>  <	1/2 limma
^  v	2/3, 3/5, 4/7 limma
64edo: ^ = +4, v = -4; / = +3, \ = -3; ) = +2, ( = -2; } = +1, { = -1								:64edo
D	D}	D)	D/	Dv
Ev	E\	E(	E{	E	E}	E)	E/	E^
Fv	F\	F(	F{	F
[Use the horizontal scroll bar at the bottom of the post to see the rest of 64-edo -->]

That doesn't quite get us there. We can do it if we add a 3/4-limma symbol. I'll use square brackets, ] up, [ down.

}  {	1/7 limma
)  (	1/4, 1/5, 1/6, 2/7 limma
/  \	1/3, 2/5, 3/7 limma
>  <	1/2 limma
^  v	2/3, 3/5, 4/7 limma
]  [	3/4, 4/5, 5/6, 5/7 limma
64edo: ] = +5, [ = -5; ^ = +4, v = -4; / = +3, \ = -3; ) = +2, ( = -2; } = +1, { = -1						:64edo
D	D}	D)	D/	Dv	D]
E[	Ev	E\	E(	E{	E	E}	E)	E/	E^	E]
F[	Fv	F\	F(	F{	F
[Use the horizontal scroll bar at the bottom of the post to see the rest of 64-edo -->]

The 3/4-limma symbols ] and [ can also replace the combinations #< and b> as 3/5-apotome symbols.

}  {	1/8, 1/9, 1/10 apotome		1/7 limma
)  (	1/5, 1/6, 1/7, 2/9 apotome	1/4, 1/5, 1/6, 2/7 limma
/  \	1/4, 1/3, 2/7, 3/10 apotome	1/3, 2/5, 3/7 limma
<  >	2/5, 3/7, 3/8, 4/9 apotome	1/2 limma
^  v	1/2 apotome			2/3, 3/5, 4/7 limma
]  [	3/5, 4/7, 5/8, 5/9 apotome	3/4, 4/5, 5/6, 5/7 limma
#\ b/	3/4, 2/3, 5/7, 7/10 apotome
#( b)	4/5, 5/6, 6/7, 7/9 apotome
#{ b}	7/8, 8/9, 9/10 apotome
#  b	1 apotome