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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. http://forum.sagittal.org/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.