## EDOs with multiple prime mappings

Dave Keenan
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### EDOs with multiple prime mappings

Cryptic Ruse has raised in this facebook thread, the question of how best to notate those EDOs below 72 that have no single best prime mapping but instead have multiple mediocre mappings. To see a list of 13-prime-limit mappings for 7-edo, click on the following link. Change the number after "ets=" to see the corresponding edo.
http://x31eq.com/cgi-bin/rt.cgi?ets=7&limit=13 Thanks to Graham Breed for this.

Cryptic makes a good point that we should have notations for these that do not favour one mapping over another. He proposes to use a 3-limit notation having Sagittal accidentals for semi-apotome, quarter-apotome, semi-limma and quarter-limma. There is some discussion of what these accidentals might look like in this Sagittal forum thread.

I have worked through all the standard EDO notations up to 72 in http://sagittal.org/sagittal.pdf and I note the following cases where we have used a prime mapping that does not agree with _all_ the mappings given for that ET by Graham's app.
EDO  Sag uses    Maps agree only primes
21    7          2  3
27    5 13       2  3  5  7
33   11          2
34   11          2  3  5 13
38   11          2  3  5
39    5 11       2  3
45   13          2
47   various     2
48   11 23       2  3
51    5  7       2  3
55   11          2  3
60    7:11       2  3  5
64   13          2
65    5  7 11    2  3  5
67    7:11 13    2  3  7
68   55 11       2  3  5  7 13
69   11 13       2  3  5
70    5 11       2  3
I note that several of these have a problem only with the use of as the 11-M-diesis 32:33. But already has a secondary role in Sagittal as a semi-apotome symbol, and so, if it is read as such, the problem goes away. However, up 'til now, this symbol would only have been used if it is both the semi-apotome _and_ the 11-M-dieisis in the EDO's obvious mapping.

I note that the following EDOs have multiple mappings which can't even agree on their mapping of the prime number 3.
23, 25, 28, 30, 33, 35, 42, 45, 47, 52, 54, 59, 64, 66, 71.
That means that even a 3-limit fractional-comma notation will not be able to avoid favouring one mapping over another if it attempts a native-fifth notation for them.

The solution chosen for these, in the standard Sagittal EDO notations, is to notate them as a subset of the EDO that is 2 times (or sometimes 3 times) larger. Although in a few cases (23, 33, 45, 47, 64) we provide native fifth notations as well. In the case of 23, 33 and 47 these do not use any prime 3 accidental. i.e. they do not use sharps or flats.

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

I note that the following EDOs have multiple mappings which can't even agree on their mapping of the prime number 3.
23, 25, 28, 30, 33, 35, 42, 45, 47, 52, 54, 59, 64, 66, 71.
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. 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).

I also note that some of the problematic ones in your list (21, 34, 48, 55, 60) seem like reasonably important ETs.

To respond to your other comments from the facebook thread:
Firstly, I don't understand why you raise "not preserving JI identities across ETs" as an objection. Surely Sagittal does that better than your proposal, since your proposal deliberately refuses to recognise any JI identities beyond the 3-prime-limit.
I would debate that. 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. 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.

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. Smaller ETs equate intervals of lower complexity, which is more problematic for accidentals based on commas of higher-limit JI. Because my system does not make use of common vanishing commas for accidentals, and also has a simple universal rule that allows translating apotome fractions to limma fractions, there are very few circumstances where accidentals in an ET map to zero steps in a larger ET. In standard Sagittal, 22edo's 81/80 accidental maps to zero steps in all meantone temperaments, e.g. 26, 31, 33, 38, 40, 43, 45, 47, 50, 52, 55, 57, 62, 64, 67, and 69, as well as all 12n-ETs (24, 36, 48, 60, 72). 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; but in all of those, exchanging the fractional apotome accidental for the same fraction of the limma generally works fine without much further thought.
With existing Sagittal 22 and 24 ET notations. The fact that one has /| and the other /|\ does not mean that melodic features must be lost in translation. It simply means that a translation will involve a change of accidental and some thought will be required about how to do that. But even assuming that your proposal has the same accidental for 1 degree of each, there is still thought required because the translation will depend on which particular enharmonic spellings you choose.
Doesn't this defeat the purpose of signifying 81/80 with the 22edo accidental in the first place? My primary objection is that nothing accomplished with accidentals for multiple commas that function differently across ETs can't be accomplished just as well with fewer symbols and less work/thought with my proposed limit-agnostic accidentals. My system is simpler--fewer symbols, fewer mapping-related variables, fewer cases where symbols must be substituted to preserve melodic features, fewer special cases, fewer ETs notated as subsets of larger ETs, fewer rules for translating between different accidental symbols (e.g. the relationship between the limma and apotome in a given tuning is all one needs to know, as opposed to the relationship between various 13-limit commas). Within the range of small ETs, it is also does not lose any more "functionality" from a cross-ET JI perspective, despite its agnosticism on mappings, because similar tunings use accidental similarly and map the same identities to the same accidentals.
I also don't know whether your combinations may involve accidentals (other than sharp or flat) altering in opposite directions.
Only necessary for the purposes of keeping the total symbol count down on a given notehead. With one more pair of accidentals, I could eliminate these sorts of opposing accidentals across the entire range; I've not given much thought to this, because I don't consider 2 to 3 symbols to be excessive, but if standard Sagittal involves distinct individual symbols that represent combined accidentals, I could add these in to compare apples to apples.

What I think would be helpful to compare our approaches would be a table showing how the pitches between 1/1 and 9/8 are notated in ETs 5-72, as well as a count of total unique symbol pairs used to cover the range. I should expect that comparison would be sufficient to settle the matter, actually.

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### Re: EDOs with multiple prime mappings

First to correct a technical detail, 81/80 does not vanish in EDOs 48, 60 or 72, where it is positive, nor does it vanish in 33, 40, 47, 52 or 64, where it is negative. And consequently these are not meantones. There are several other EDOs you list where 81/80 does indeed vanish, but they are also not considered meantones because they have more than one parallel chain of fifths. These are 24, 36, 38, 57, and 62 (multiples of 12, 19 and 31). 12-edo itself is usually considered a meantone, albeit a borderline case. So the maximal list of meantone EDOs up to 72 is: 12, 19, 26, 31, 43, 45, 50, 55, 67, 69. But I should mention that some do not consider 26 and 45 to be meantones as their fifths are wastefully narrow, resulting in minor thirds that are wider than just (albeit only slightly so).

Edit: One could argue that 64-edo is a meantone despite 81/80 not vanishing, purely because it has a single chain of fifths of a size between those of 12-edo and 26-edo. But like 45 and 26 it is susceptible to the charge of harmonic waste. And strictly-speaking so is 19-edo, however this is by such a tiny fraction of a cent that it is considered equivalent to 1/3-comma meantone for all practical purposes.

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### Re: EDOs with multiple prime mappings

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.

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### Re: EDOs with multiple prime mappings

I hit "Submit" prematurely when I meant to hit "Preview" on my reply above. That's probably just as well. It was getting rather large.

I'm not sure why we're comparing cases of non-positive 5-commas with non-positive 1/2-apotomes. The 5-comma is more likely to be equated to a 1/4-apotome than a half. Even in the case of 22-edo, the single step is both the 5-comma and 1/3-apotome. And 1/3-apotome is closer to 1/4-apotome than it is to 1/2-apotome. 1/4-apotome is non-positive in all the EDOs listed above in which 1/2-apotome is non-positive, plus those in which the apotome divides into only 2 parts, namely EDOs 10, 17, 24, 31, 38, 45, 52.

So we have 5-comma unusable in
12, 19, 24, 26, 31, 33, 36, 38, 40, 43, 45, 47, 50, 52, 55, 57, 62, 64, 67, 69.
and quarter-apotome unusable in
5,  6,  7,  9,  10, 11, 12, 14, 16, 17, 19, 21, 23, 24, 26, 28, 31, 33, 35, 38, 40, 45, 47, 52.
Because my system does not make use of common vanishing commas for accidentals, and also has a simple universal rule that allows translating apotome fractions to limma fractions, there are very few circumstances where accidentals in an ET map to zero steps in a larger ET.
I'm not sure why it matters if accidentals in an EDO map to zero (or negative) steps in a larger EDO, but FWIW I think I have just shown that there are slightly more EDOs in which the quarter-apotome is non-positive than there are in which the 5-comma is non-positive.

The near-universal rule for translating a 5-comma accidental (in fact any accidental) in such a way as to preserve the 3-limit plus melody, is to note how many steps there are in the comma-accidental and the sharp in the original EDO and in the sharp in the final EDO. Then map the comma accidental to the accidental that represents the number of steps nearest to
final_sharp_steps * original_comma_accidental_steps / original_sharp_steps.
In other words, map it to the accidental that is closest to the same fraction of its apotome.
I say "near-universal" because it won't work if you're using a notation which does not have a sharp, for either the original or the final EDO. But for each such native fifth notation there is also a subset notation that does have a sharp.

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### Re: EDOs with multiple prime mappings

Doesn't this defeat the purpose of signifying 81/80 with the 22edo accidental in the first place?
I had to read this sentence several times because it didn't make any sense to me. I eventually realised it's because I had never seen it as "signifying 81/80 with the 22edo accidental" but as signifying 1 step of 22-edo with the 81/80 accidental. And the purpose of that is to indicate the best mapping of the next highest prime (since the mappings of primes 2 and 3 have already been indicated by the nominal plus sharps and flats). And I fail to see how this purpose is defeated if this accidental has to map to a different accidental, or no accidental, when translating to an EDO in which the 5-comma vanishes. It seems this is entirely what a user would expect to happen.
My primary objection is that nothing accomplished with accidentals for multiple commas that function differently across ETs can't be accomplished just as well with fewer symbols and less work/thought with my proposed limit-agnostic accidentals. My system is simpler--fewer symbols, fewer mapping-related variables, fewer cases where symbols must be substituted to preserve melodic features, fewer special cases, fewer ETs notated as subsets of larger ETs, fewer rules for translating between different accidental symbols (e.g. the relationship between the limma and apotome in a given tuning is all one needs to know, as opposed to the relationship between various 13-limit commas). Within the range of small ETs, it is also does not lose any more "functionality" from a cross-ET JI perspective, despite its agnosticism on mappings, because similar tunings use accidental similarly and map the same identities to the same accidentals.
That's a vast catalog of claims and you really need to be forthcoming with some proof of them, and you need to stop comparing against some imagined standard Sagittal and begin comparing against the notations actually given in the updated Sagittal Xenharmonikôn article at http://sagittal.org/sagittal.pdf. I fully expect that most of your claims will be shown to be true, particularly those about using fewer symbols, and yet they will still only show that your proposed notations are better for one very specific, and so far very unusual, requirement for translating between otherwise unrelated EDOs. That is except in the case of those 16 "difficult" EDOs listed above, where I think your notation may be of more general interest.
What I think would be helpful to compare our approaches would be a table showing how the pitches between 1/1 and 9/8 are notated in ETs 5-72, as well as a count of total unique symbol pairs used to cover the range. I should expect that comparison would be sufficient to settle the matter, actually.
I suggest notating each from D to F so there is a limma as well as a major-whole-tone, and an apotome or two. And I suggest you should count both the number of total unique symbol pairs and the number of total unique symbol-combination pairs. Where, in the case of Sagittal, the former would be compared against the number of total unique flag-pairs. I think we are using sharps and flats in the same way. If so, these could be ignored when making these counts. I also suggest showing enharmonic spellings in the manner of this example:
72-edo
D D D D
D D D D D D D
E E E E E E E
E E E E E E E
F F F F
I note that you have all the information you need to generate all the standard Sagittal notations, but I am still waiting for the updated draft of your paper.

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### Re: EDOs with multiple prime mappings

And I fail to see how this purpose is defeated if this accidental has to map to a different accidental, or no accidental, when translating to an EDO in which the 5-comma vanishes. It seems this is entirely what a user would expect to happen.
I say the purpose is defeated because any number of different accidental symbols could have been chosen to represent one step of 22edo. I have not internalized all of the intervals for which Sagittal has symbols, but in 22edo one step could be 21/20, 25/24, 28/27, 33/32, 36/35, 45/44, 49/48, etc. etc. Presumably 81/80 was selected over any of these for some specific reason? Is there an algorithm or rule for selecting accidental symbols for a given ET in standard Sagittal, or is it somewhat subjective?

In any case, the point still stands that when you use a symbol representing a comma in some ET, and it equates to multiple other commas in that ET, translating to another ET where the symbol's comma vanishes but other commas equated to it in the original ET do not, you either must re-analyze the piece and re-notate with distinct commas, or you lose information with respect to JI. So it is not the case that standard Sagittal is doing a better job of preserving JI approximations across tunings when you are restricting the symbol set.

This is why I suggested that a Sagittal user might prefer to notate 22edo using symbols for both 33/32 and 81/80 (or any others from that partial list I gave above)--so that when translating from 22edo to 24edo (for example) 81/80 does indeed vanish but 33/32 does not. This would be a purpose where full Sagittal excels over my proposed system, though I believe this is what you call "straw man Sagittal". In any case, I know that if my goal was to have notation reflect the most specific and important rational approximations of each sonority in an ET, I would want to use more symbols than those proposed for Standard Sagittal, particularly if I intended the composition to retain its sense in a different tuning. Otherwise, why would I care about using notation to reflect JI at all?

If you're not concerned with reflecting JI, and simply concerned with translating melodic information and perhaps perfect fourths and fifths, why would you use more symbols than necessary to cover the gamut of ETs of interest? If you're trying to keep the total symbol count down to the minimum necessary to completely notate each ET in your range of interest, it is reasonable to select accidentals based on intervals that do not commonly vanish in that range. When I began this project, I first attempted to find some set of higher-limit commas that never vanished across the range of small ETs, and also behaved in a consistent way in these ETs. What I found was that every comma vanishes somewhere in the small ETs—as do my fractions of the apotome and limma—but no set of higher-limit commas behaves as consistently as the fractional intervals. They represent the smallest set of intervals that can be consistently used across all the ETs in the range I give, at least to the extent I was able to investigate. It would be quite the exercise to examine an exhaustive list of all 11-limit commas that do not vanish in the majority of these ETs to see if they also stay in similar relationship to other accidentals between tunings.

A hypothetical case illustrating what I mean by "similar relationship": is 81/80 always smaller than or equal to 33/32 in tunings where they both appear? No, it is not--in 21edo, 81/80 is -1 step, while 33/32 is +1 step; in 16edo, 33/32 vanishes, while 81/80 is again -1 step; in one mapping of 20edo, 81/80 is two steps (360¢ 5/4) while 33/32 is one step (540¢ 11/8). This seems to be the case for any pair of independent commas. But 1/2 an apotome is always at most half the steps of the apotome, and always has the same sign as the apotome.
I fully expect that most of your claims will be shown to be true, particularly those about using fewer symbols, and yet they will still only show that your proposed notations are better for one very specific, and so far very unusual, requirement for translating between otherwise unrelated EDOs. That is except in the case of those 16 "difficult" EDOs listed above, where I think your notation may be of more general interest.
Having fewer symbols to learn for the most popular range of ETs seems like an improvement regardless of trying to translate between them. Having symbols whose selection is the result of a simple single formula based on the mapping of a single prime, and whose meaning varies the minimum amount possible among the ETs in that range also seems like an improvement. Having more ETs notated "natively" as opposed to as subsets also seems like an improvement, in as much as having notation reflect the logic of the tuning is desirable.
I note that you have all the information you need to generate all the standard Sagittal notations, but I am still waiting for the updated draft of your paper.
I do not have all the information necessary, or if I do I don't know where to find it. Specifically, in the lists I have seen of proposed Sagittal accidentals for various ETs, step counts assigned to each symbol do not seem to be apparent, and I also do not know how you want to handle the ETs where an alternative mapping conflicts with the Standard Sagittal notation. If there is some rule or algorithm for determining which Sagittal symbols to use, then a comprehensive list is not necessary.

For my system, there is a very simple algorithm: find the 3-limit patent val (which is given in terms of ET steps). Use it to determine the size of the apotome and limma in ET steps. If either is greater than 1, divide in half and round down to get the 1/2 apotome and/or 1/2 limma. If either of those are greater than 1, then divide and round down again to get the 1/4 apotome and/or 1/4 limma. I also posted a picture on the facebook group that shows how many ET steps these intervals are mapped to for the whole range we are considering, if you wish to skip the calculation. (I can also repost it here if this forum supports higher-resolution images.) Any of these intervals not mapped to zero are valid accidentals for the tuning, though some may be redundant and apotome-derived accidentals are favored when available.

It will take some time for me to revise the manuscript, but I also have most of the enharmonic equivalents generated in the range you're requesting (D-F). I believe that providing these will be the quickest way to settle the matter. I would suggest however that we supply enharmonic equivalents for every pitch, e.g. between D and E, every pitch should be notated as both an altered D and an altered E. Since I will be taking the time to supply this for my proposal, I would request that you return the courtesy and demonstrate the Standard Sagittal approach. It will make a handy reference table regardless of which approach proves more efficient.

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### Re: EDOs with multiple prime mappings

I'll work on it.

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### Re: EDOs with multiple prime mappings

Cryptic Ruse wrote:It would be quite the exercise to examine an exhaustive list of all 11-limit commas that do not vanish in the majority of these ETs to see if they also stay in similar relationship to other accidentals between tunings.
We did this. In fact up to the 23-prime-limit, and some up to 37. We defined a property we called the "slope" of the comma, which is simply the rate at which it changes its apotome-fraction as the notational fifth changes. We favoured commas with low slope so they would be more useful for notating EDOs, but many were too complex and so we chose to use many that did not have low slope. By definition, an apotome-fraction accidental would have zero slope.

The lowest slope rational comma I can find is the 13:35-comma, 105/104. It remains close to 0.146-apotome, approx 1/7-apotome, across a wide range of fifth sizes.
Likewise the 7:17-small-dieisis, 459/448, remains close to 0.37-apotome.
The 5:11-small-diesis, 45/44, remains close to 0.34-apotome.
The 5:49-medium-diesis, 405/392, remains close to 0.50-apotome.
The 17:23-comma, 69/68, remains close to 0.22-apotome.
The 343-kleisma, 1029/1024, remains close to 0.07-apotome.

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### Re: EDOs with multiple prime mappings

So I just spent an hour writing a reply to this and when I went to post it, I was asked to log in again, and then my reply disappeared. It seems I am automatically logged out after a short time. Bummer!

But anyway, to sum it up again quickly:
We did this. In fact up to the 23-prime-limit, and some up to 37. We defined a property we called the "slope" of the comma, which is simply the rate at which it changes its apotome-fraction as the notational fifth changes.
That makes little sense to me, because commas are not continuous functions of 3/2. 45/44, for instance, vanishes at various ETs between 9edo and 12edo (666-700¢) but not at others in that range. For example it vanishes at 33edo but not 40edo, 19edo but not 31edo, 26edo but not 45edo, 12edo but not 43edo, etc., at least not using their optimal 11-limit vals. Unless there's some way to take the slope of a discontinuous function that I'm not aware of, how can you say that 45/44 has a continuous relationship with the apotome when it does not have a continuous relationship with 3/2?