I'm keen to provide better Sagittal notations for these multi-mapping EDOs, and your approach sounds very promising. But I do want to establish exactly which existing Sagittal ET notations have problems in this regard, and which do not (and where any grey area may lie).
For example, 72-EDO has only one worthwhile mapping of the 11-limit, and it is such a good mapping, that to me it would be a dereliction of duty not to notate it with symbols for the usual 5, 7 and 11 commas. A user who is not interested in JI approximations is free to ignore these meanings and treat the accidentals purely as representing numbers of degrees (for EDOs 65, 72, 77 and 79 only) or 1/6, 1/3 and 1/2 of an apotome (for EDOs with best fifths of 700±1.3 cents only), since their relative sizes are indicated by the area within their arrowheads. But it doesn't work the other way. i.e. If it were notated with symbols whose only consistent meaning was as fractions of an apotome, a JI-phile user can ignore this, but there would be nothing to indicate their JI role in this tuning.
When you point out that 72 has subsets that admit of other mappings, this only says that we should be careful that the notations for those subsets do not favour one mapping over another, and indeed they do not.
And in the notation for 10-edo, to fail to indicate its astoundingly good approximation of 13, upon which both of its 13-limit mappings agree, would seem almost unfair to the user.
However I am still willing to consider that, in addition to providing better standard notations for the 16 EDOs you have helped me to identify, your method might also provide alternative non-standard 3-limit Sagittal notations for other EDOs.
cryptic.ruse wrote:This is only the case when looking at the full 13-limit. If you check them on the 3-limit only, that list shrinks down to 23, 35, 47, 59 and 71; otherwise one 3-limit mapping is clearly better than another.
OK. That's good. But what is the justification for not looking at mappings of 3 outside the 3-limit? I suppose it is that no accidentals are used which have meanings beyond the 3-limit. Fair enough. In that case, I have revisited my list above, and asked Graham's site only for mappings to the limit of the highest prime used in notating each ET. I find that 21-edo has only one worthwhile 7-limit mapping so the use of the 7-comma accidental isn't a problem. And I find that 55-edo has only two worthwhile 11-limit mappings which disagree only on the mapping of 5, and its notation doesn't use any accidental involving the prime 5. So 21-edo and 55-edo are no longer on the list of standard notations that favour one mapping over another.
And the major difference between my approach and your standard is that a disagreement on the 3-limit does not change which accidentals are used, only the number of steps subtending the symbols (with the exception of 35, in which either the apotome or limma vanishes, depending on the mapping).
Again you refer to a "standard" Sagittal approach that exists nowhere but in your own mind. As I explained in my previous post, and as you can see in http://sagittal.org/sagittal.pdf
, when there is a disagreement on the mapping of the 3-limit, our standard approach to these EDOs does not change which accidentals are used, for the simple reason that it does not provide alternative notations for the two mappings. It ignores both mappings and notates using a fifth of a size in between the two (a subset notation). And in a few cases (e.g. 23 and 47) it also takes the best mapping and notates it without using any 3-limit accidentals (a native-fifth notation).
I also note that some of the problematic ones in your list (21, 34, 48, 55, 60) seem like reasonably important ETs.
I totally agree. Although that set is now down to 34, 48 and 60.
No matter what accidentals you use, the nature of ETs means that any single-identity symbol is leaving out information about other identities the symbol may stand for. A 5-limit accidental will leave out 7-, 11-, and 13-limit information, for instance.
Sure. But in many cases it is obvious what is the most useful identity to symbolise for a given number of degrees. It may be a single-prime rather than a combination of primes. It may be the lowest prime, or it may be the most accurate prime, or as we have been discussing, it may be the prime that all worthwhile mappings agree upon.
For example, in 72-edo, once you know that 5/1 is -1 degree and 7/1 is -2 degrees and 11/1 is +3 degrees, then you can readily figure out that 1 degree is also 7/5 and 7*11, that 2 degrees is also 5*11 and 5*5, that 3 degrees is also 5*7, that 4 degrees is also 11/5 and so on.
Just because my approach is explicitly agnostic about the higher-limit implications of an accidental, does not mean that its meaning in higher limits cannot be extrapolated in a given temperament by applying the temperament's mapping in those limits, or that it necessarily loses more information than any other accidental would. Moreover, it does not impose a higher limit upon the music than what the composer might wish, as would be the case using 7-limit accidentals for 5-limit music in e.g. 21edo, or 13-limit accidentals for 7-limit music in e.g. 27edo.
Fair enough. But this requires knowledge of the mapping, and knowledge of the mapping will also tell you which other identities are hidden behind those that are explicitly symbolised. We use the 7-comma accidental in 21-edo in part because there is no single good mapping of 5. But the user uninterested in the 7-limit can still learn that C:E is one poor option for 4:5 and C:E
is the other. Neither is favoured by the use of a 5-comma accidental.
It will always be the case that intervals are equated across different prime dimensions, such that any accidental symbol in any ET reflects identities or fractions of identities in whatever limit the temperament is being conceived of.
I think I followed that, and agree, but don't see it as a problem.
Smaller ETs equate intervals of lower complexity, which is more problematic for accidentals based on commas of higher-limit JI.
Why is it more problematic? What is it that you are seeing as a problem here?
In standard Sagittal, 22edo's 81/80 accidental maps to zero steps in [DK's correction: 12, 19, 24, 26, 31, 36, 38, 43, 45, 50, 55, 57, 62, 67, and 69]. The only ETs in which the 1/2 apotome accidental maps to zero are those in which the apotome vanishes (7, 14, 21, 28, 35) or is not subdivided [DK's addition: 5, 12, 19, 26, 33, 40, 47]; but in all of those, exchanging the fractional apotome accidental for the same fraction of the limma generally works fine without much further thought.
I think you should have included cases where the accidental becomes negative, as well as those where it maps to zero, since it is unusable in both cases. That happens to the 5-comma in
12, 19, 24, 26, 31, 33, 36, 38, 40, 43, 45, 47, 50, 52, 55, 57, 62, 64, 67, 69.
It happens to the half-apotome in
5, 6, 7, 9, 11, 12, 14, 16, 19, 21, 23, 26, 28, 33, 35, 40, 47.