cryptic.ruse wrote:It all depends on how you quantify error, of course. Complex ratios are significantly less concordant than simple ratios (comparing, say, 17/12 to 7/5, or 14/11 to 5/4), so if you want error to reflect perceived reduction of concordance, then complex ratios are less sensitive and should be weighted accordingly. However, because what concordance they do have is primarily due to periodic relationships between higher harmonics, those effects will diminish more rapidly with mistuning, and if you want error to reflect that, then more weight should be given to higher harmonics.
Agreed. Which is why, in notation design, when I have no way of knowing the user's preferences, I consider only unweighted cents error, which steers a middle path between the two weightings you mention. And that is why I am very dubious of claims, based on Graham Breed's error measures, and assuming optimally-tempered prime 2, that such and such a small EDO, in some subgroup, is just as good (for notational purposes) as 12-edo in the 5-limit.
In any case, the smaller ETs are all subsets of larger ETs and tune certain subgroups identically. Regardless of how you weight error, if evaluated on the correct subgroup the error will never be more than the larger ET. In many cases it's an operation as simple as substituting 9 in place of 3, 15 in place of 5, etc. or picking a handful of ratios that are not integers but are co-prime with 2. I'd be happy to demonstrate how this works for any ETs of your choice, but that's outside this discussion. Suffice to say this has been my primary area of study for nearly a decade now.
I'm sorry I don't understand what you're claiming here. Of course n-edo is always a subset of 2n-edo, 3n-edo, 4n-edo, etc by definition. But almost always, n-edo has a different size of best fifth (prime 3) from the best fifths of 2n-edo, 3n-edo, 4n-edo etc, or a different size for the best major third (prime 5) or a different size for the best subminor seventh (prime 7). In that case, your "correct subgroup" would need to involve a very poor choice of mapping for the larger EDO, so I don't see how this is relevant to notation.
I don't understand what you're saying about substituting 9 in place of 3 etc, or why that would ever make sense. Although you say it's outside this discussion, I encourage you to try to explain it to me, as it may help me to understand where you're coming from, and therefore help indirectly with this discussion. [Edit: I see you have done so here: viewtopic.php?f=5&t=158
David Keenan wrote:I don't really understand this complaint. Yes, we're leaving out JI information, but that's because it's an EDO, it's tempered, it's not JI.
You said previously "But when good, low-prime-above-3, JI approximations do exist we want to make that information available directly in the notation." My complaint is that you're not doing that. You may indicate some of the higher primes, or you may not.
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.
You do so more in the larger ETs, less in the smaller, and sometimes not at all (as in the case of 19edo), and don't follow a clear pattern of what primes do or do not merit having information conveyed about them in the accidentals.
I think you're wrong about that. As far as the purely mathematical considerations go, we try to follow consistent rules, as above. I'm sorry if I haven't explained them clearly enough before now.
I note that when the user understands that we assign symbols in prime number order, the mere fact that the 5-comma symbol
is _not_ used in a notation, suggests that the best 4:5 is probably spelled C:E etc in that notation. Likewise if neither the 5-comma
nor the 7-comma
symbol is used, the best 4:7 is probably spelled C:Bb etc in that 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.
Proposals for such notations might go back and forth half a dozen times between George and I before agreement was reached.
So that's another complaint I have, then; the choice involves some subjective decision-making based on various parameters that may or may not reflect the needs or desires of any particular composer. Presumably if you had included a few other people in the bouncing back and forth, the agreement may have looked different or not been reached at all. At least in the case of my proposal, no consensus is necessary, because the process is entirely formulaic.
By taking this quote out of context, you may be missing the fact that I was referring only to notations for EDOs that do not have an obvious best subset of their obvious prime mapping. Otherwise the decision was almost entirely formulaic and there was little question of what the notation should be. As mentioned in the XH article, we did in fact discuss notations for the more popular EDOs with many others on the Yahoo groups "tuning" and "tuning-math". You can find the discussions, and the consensus reached, in the archives of those groups. And may I suggest you re-read pages 13 and 15 of http://sagittal.org/sagittal.pdf
And I believe that anyone who understands _all_ the issues would agree with the notations for many other EDOs. But few people have gone into it deeply enough. Only Graham Breed and Herman Miller come to mind. Even in your case, I feel you have a good understanding of the mathematical and psychoacoustic issues, 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.
We have to also consider whether a notation for a particular EDO contains symbols that look too much alike, e.g.
, or whose visual size order disagrees with their size order in EDO steps, such as
in the 12n-edo stack, or whether there is some symbol that looks as though it is the combination of two other symbols, like
, and yet this does not agree with the sum of their EDO steps.
We always held a purely algorithmic decision process to be our goal, and we nailed these down whenever we could. I'd be happy to send you our entire email exchange for the past 14 years so you could see that for yourself. But some of the psycho-physical issues are just too complex and so defeated us in that regard.
But I am quite willing to concede, that in the case of maybe a quarter of the EDOs below 72 (the "bad" ones), there would likely be little agreement, even among those who understand all the issues. And so I am very interested in considering your approach to notating them.
I can't agree with this. It seems to me that there is no more information contained in a fractional apotome symbol than there is in the apotome symbol. Once you know how many steps (n) is notated by the apotome symbol (the sharp or flat, which both our systems have) then you immediately know which number of steps corresponds to 1/n-apotome, 2/n-apotome etc. You don't need to see the additional fractional apotome symbols in order to know that.
So you are arguing that the user's ability to deduce information by translating from accidental->ET step->additional meaning is functionally the same as accidentals directly conveying that additional meaning. In other words that by giving higher-prime accidentals in addition to the apotome, you're not losing fractional-apotome information because users may perform a translation themselves by subdividing the apotome and then using the ET steps they get to apply the higher-prime accidental that maps to the same number of ET steps.
Yes, except that "functionally the same" is perhaps a little too strong. Surely you agree that they _can_ deduce this, using extremely simple mental logic, from the number of steps represented by the apotome symbol.
So you might argue that encoding only 3-limit information in the accidentals loses that higher-prime information because it can't be directly deduced from the information given in the initial specification.
That is indeed what I have argued on several occasions.
Yet you don't seem to have a problem with expecting users to deduce higher-prime approximations from explicitly 3-limit notation in, for example, 19edo, so you can't entirely believe that argument.
You are completely mistaken about that. I certainly do not expect the user to "deduce" the higher prime approximations from the purely 3-limit notation of 19-edo. I don't see how they could do that (at least not by simple mental logic). This information is (unfortunately) outside this particular notation and so must be calculated and/or learned separately.
The requirement for a performer to have a one-to-one correspondence between EDO steps and symbols seems so obvious as to not need discussion. And the fact that we favour prime 3 doesn't seem like something _you_ should be complaining about.
If it can be expected that users will deduce higher-prime approximations from 3-limit notation in some EDOs, why not all of them, at least up to some cut-off? In either case, the onus is on the user to determine the equivalences and unpack the meanings from symbols that do not directly give them all of the information.
Since the premise is false, the conclusion does not follow.
Furthermore, it wouldn't take much to include a specification of the important higher-prime intervals equated with the apotome or limma fractions in a given ET.
Sure, but I consider this information to be _outside_ the notation, since it consists of more than just
(a) what accidental is used for each number of steps of the EDO, and
(b) what each accidental means in non-EDO-specific terms, as one particular comma or comma-fraction.
It can be seen purely as information about what numbers of steps correspond to what commas in that EDO, without requiring any symbols to be used as intermediaries.