## EDOs with multiple prime mappings

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

I've started a thread on the timeout problem, in the Admin sub-forum, here.
cryptic.ruse wrote:
Dave Keenan wrote: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?
These relationships are only discontinuous when you restrict yourself to whole EDO steps. There is a level of description prior to rounding the comma to the nearest step of an EDO, where we can ask what its tempered size is in cents, given the size of the notational fifth, and the power of 3 that is contained in the comma. 45/44 contains 3^2, so it will increase in size by 2 cents for every cent that the fifth increases, while the apotome contains 3^7 so it will increase in size by 7 cents for every cent that the fifth increases. Now if the comma happened to be 2/7 of an apotome to begin with (when both comma and apotome are untempered, i.e. when fifths are Pythagorean) then it would stay 2/7 of an apotome no matter what size the fifth became. log(45/44, 2) * 1200 = 38.91 c. Apotome = 113.685 c. 38.91/113.685 = 0.34. Not far from 2/7 = 0.29.

105/104 has a 3-exponent of 1 and its untempered size is very close to 1/7-apotome, so it stays close to 1/7-apotome as the fifth is varied, except when the apotome itself approaches zero size. Any ultra-low-slope rational comma will approximate some whole number of sevenths of an apotome. The worst case is if we want to approximate (n+0.5)/7 of an apotome (e.g. 3.5/7 = 1/2-apotome). In that case, the absolute slope of the rational comma cannot be less than 0.5.

We defined comma_slope
= comma_3_exponent - apotome_3_exponent * untempered_comma_cents / untempered_apotome_cents
= comma_3_exponent - 7 * untempered_comma_cents / 113.685c

A low absolute-value of slope means a near-constant apotome fraction. Of course even an ultra-low-slope rational comma like 105/104 will vanish in a given EDO for exactly the same reason a true fractional-apotome will vanish, namely because it falls below half a step of the EDO.

We could call the above, the comma's apotome-slope, and could define its limma-slope as
= comma_3_exponent + 5 * untempered_comma_cents / 90.225c

And its whole-tone-slope as
= comma_3_exponent - 2 * untempered_comma_cents / 203.91c

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

There is a level of description prior to rounding the comma to the nearest step of an EDO, where we can ask what its tempered size is in cents, given the size of the notational fifth, and the power of 3 that is contained in the comma. 45/44 contains 3^2, so it will increase in size by 2 cents for every cent that the fifth increases, while the apotome contains 3^7 so it will increase in size by 7 cents for every cent that the fifth increases.
Oh, I get it. This is different than what I was originally talking about. That slope is only valid when the tuning of the other primes remains constant, which doesn't happen in ETs, so the slope doesn't give us a predictor of how a comma will behave in different ETs, and is quite far off in ETs that share a tuning of 3 but not of other primes (e.g. 7n's, 5n's, 12n's, 17n's, 19n's, etc. up to a certain n). Without knowing how a comma will behave across the range of ETs we're trying to notate, we can't really know if it's going to be a better or worse choice for a symbol if we are trying to minimize the total symbol count across all ETs.

So I have to ask: if the goal was to find commas with low slope, so that they behave as much like apotome fractions as possible, why not just go straight for the apotome fractions? It took me a while to even hit on the possibility of apotome fractions (or irrational accidentals in general), but it seems like you guys were aware of that from early on but decided to go for a larger set of rational accidentals instead. Is this because you were trying to avoid using different symbol sets between JI and the EDOs, and wanted to stick as close to simpler JI commas as possible (since they'll be more common in JI notation)?

If that's the case, it seems like the benefits of fractional apotome (and limma) accidentals were already clear to you guys, and that you also agree with my goal of keeping down the total symbol count people need to learn. But keeping the total symbol count down could be accomplished with exactly what I'm proposing--not simply using "low slope" accidentals, but actually repurposing existing Sagittal symbols to mean fractions of the apotome and limma when used in the context of EDOs. Best of both worlds, isn't it?

I guess then what we're debating here isn't really if fractional 3-limit accidentals do what I say they do, it's whether there's enough of a difference between the fractional 3-limit approach and the status quo to be worth changing the status quo. Guess I better get to work on those enharmonics so we can actually assess the discrepancies!

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

cryptic.ruse wrote:Oh, I get it. This is different than what I was originally talking about. That slope is only valid when the tuning of the other primes remains constant, which doesn't happen in ETs, so the slope doesn't give us a predictor of how a comma will behave in different ETs, and is quite far off in ETs that share a tuning of 3 but not of other primes (e.g. 7n's, 5n's, 12n's, 17n's, 19n's, etc. up to a certain n). Without knowing how a comma will behave across the range of ETs we're trying to notate, we can't really know if it's going to be a better or worse choice for a symbol if we are trying to minimize the total symbol count across all ETs.
Right. Let's call this the 3-tempered size of the comma. But we make sure the mapping of the other primes in the comma are such that they agree with simply rounding the 3-tempered size to the nearest whole number of steps of the EDO. We do this by ensuring that the set of commas used to notate an EDO, only contain primes that form a mutually-consistent set for the EDO's obvious mapping (or "patent val", a term that I feel just obscures things for newbies, and which I don't see the need for).
So I have to ask: if the goal was to find commas with low slope, so that they behave as much like apotome fractions as possible, why not just go straight for the apotome fractions? It took me a while to even hit on the possibility of apotome fractions (or irrational accidentals in general), but it seems like you guys were aware of that from early on but decided to go for a larger set of rational accidentals instead. Is this because you were trying to avoid using different symbol sets between JI and the EDOs, and wanted to stick as close to simpler JI commas as possible (since they'll be more common in JI notation)?
You have it exactly. Yes we knew about comma-fraction EDO notations from Paul Rappoport's paper. There is no sharp line between JI and EDOs. Some larger EDOs, e.g. 217-edo, are audibly indistinguishable from JI and we figured they should be notated accordingly. And no-one before now, has ever wanted 72-edo to be notated with anything other than 5, 7 and 11 comma symbols.

But I concede that some small EDOs just don't have any good consistent subset of primes in their obvious mapping, and so we should definitely look at using your method for those.
If that's the case, it seems like the benefits of fractional apotome (and limma) accidentals were already clear to you guys,

Yes. If you look at the Trojan notation, used for the 12n stack, but designed to also be used for other tunings, by people whose universe revolves around 12-edo. We fully expect people using that notation not to give a damn about prime mappings, so we have alternative names for its symbols, in terms like "1/4 tone" (really 1/2-apotome), "1/6-tone" (really 1/3-apotome) etc., which you can read in the 8th column of the character-map spreadsheet on the Sagittal website.
http://sagittal.org/Sagittal2_character_map.pdf

Remember you noted that 48-edo and 60-edo were two important EDOs that were on my list of 16 EDOs whose notational commas favour one mapping over others. That's because the commas were really chosen, not specifically for 48-edo or 60-edo, but as commas whose 3-tempered sizes were close to n/8 and n/10 apotomes in the specific case when the fifth is 700c (and maybe a cent or two either side of that).
and that you also agree with my goal of keeping down the total symbol count people need to learn.
Absolutely!
But keeping the total symbol count down could be accomplished with exactly what I'm proposing--not simply using "low slope" accidentals, but actually repurposing existing Sagittal symbols to mean fractions of the apotome and limma when used in the context of EDOs. Best of both worlds, isn't it?
It is, in the case of EDOs that don't have any good JI approximations. But when good, low-prime-above-3, JI approximations do exist we want to make that information available directly in the notation.
I guess then what we're debating here isn't really if fractional 3-limit accidentals do what I say they do, it's whether there's enough of a difference between the fractional 3-limit approach and the status quo to be worth changing the status quo.

Not so much that. I'm seeing your fractional 3-limit approach as something we might consider recommending as the standard for EDOs whose native fifth is outside certain bounds, and as a non-standard notation for others.

Then there are the details to be worked out such as which fractions of which commas to use, and what symbols to use for them.

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

You have it exactly. Yes we knew about comma-fraction EDO notations from Paul Rappoport's paper. There is no sharp line between JI and EDOs. Some larger EDOs, e.g. 217-edo, are audibly indistinguishable from JI and we figured they should be notated accordingly. And no-one before now, has ever wanted 72-edo to be notated with anything other than 5, 7 and 11 comma symbols.
Agreed that there is no sharp line between EDOs and JI, and particularly that as the EDO size pushes into the triple digits, the odds of someone using that EDO in such a way that simple JI commas vanish or are equated with each other also goes down, thus distinguishing between commas that are equated or vanish in smaller EDOs.
...I'm seeing your fractional 3-limit approach as something we might consider recommending as the standard for EDOs whose native fifth is outside certain bounds, and as a non-standard notation for others.
While there is no sharp line between EDOs and JI, there is also no sharp line between "good" and "bad" EDOs; the paradigm of subgroup temperaments has given us the tool to find ways in which even most of the "worst" EDOs can be used to sound as close to some type of JI as 12edo is to the 5-limit. I could probably rattle off a list of subgroups of the 13-limit on which every ET from 8 through 31 has damage at least as low as 12edo's 5-limit damage. There is a growing number of microtonalists who are interested in JI and particularly in using small EDOs to approximate it, in addition to using the more traditional JI-oriented EDOs. That's the main reason why I'm suggesting this notation that I propose is not limited to just the edge-case EDOs. Taking it out as far as 72edo may be over-kill, and I'll admit to suggesting that only as a proof of how widely-applicable it could be. But I would certainly advocate up to 43, at least as an auxiliary form of notation, because there are more EDOs in that range that are a bit "weird" than there are normal ones. People likely to use 72edo might occasionally find use for 43, 41, 36, 34, 31, 29, 26, 24, 22, or 19, but it's not terribly likely that they'll take 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 21, 23, 25, 27, 28, 30, 32, 33, 35, 37, 39, 40, or 42 very seriously. Meanwhile, anyone likely to spend time in the ETs below 19 or 17 is unlikely to spend much time above 41 (or 43 at most), let alone 218 or 581 or whatever.
It is, in the case of EDOs that don't have any good JI approximations. But when good, low-prime-above-3, JI approximations do exist we want to make that information available directly in the notation.
If that's the case, then your proposed Standard Sagittal is at odds with that for the smaller ETs (let's say below 43 at least). In Standard Sagittal you're still leaving out tons of JI information, because of how these ETs roll the different primes up together. 19edo is a decent 13-limit temperament, but (among other things) it equates 5/4 with 16/13, 11/8 with 7/5 and 18/13, 8/7 with 7/6 with 15/13, 10/9 with 9/8 with 11/10, etc. etc. If you want to indicate when a decently-approximated prime is being approximated, full Sagittal gives you the tools for that, but Standard does not. Standard Sagittal, in the context of the smaller ETs (below 41) satisfies neither the desire to completely convey salient JI information, nor the desire to keep the symbol set minimal in that range. It also does not consistently follow a pattern of which JI information will be explicitly included and which won't. In fact, as I have mentioned before, Standard Sagittal in general preserves no more JI information than the 3-limit system I propose; it just replaces 3-limit information with information about different primes in places--but sacrificed 3-limit information is still sacrificed information. It's robbing Peter to pay Paul. And when it doesn't do this, it agrees with my notational approach (e.g. the 12n stack).

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

cryptic.ruse wrote:While there is no sharp line between EDOs and JI, there is also no sharp line between "good" and "bad" EDOs;
Agreed.
the paradigm of subgroup temperaments has given us the tool to find ways in which even most of the "worst" EDOs can be used to sound as close to some type of JI as 12edo is to the 5-limit.
I find that hard to believe. As I said earlier, we use consistent _subgroups_ of the obvious mapping (not neccesarily limits), in notating EDOs. But even so, I seem to remember that the maximum absolute errors in many of them are close to half a step of the EDO, where 12-edo only has a 16 cent max-abs error in the 5-limit. Half a step is more than 16c for EDOs less than 38.

And the higher the prime, the smaller the error must be, for it to sound close to JI.
I could probably rattle off a list of subgroups of the 13-limit on which every ET from 8 through 31 has damage at least as low as 12edo's 5-limit damage.
Is that Paul Erlich's "damage", from his 'Middle Path' paper, that treats large errors in high primes as if they are no worse than small errors in low primes -- just the opposite of what I wrote above? You may have seen me write elsewhere, that while it is mathematically convenient, I don't think it accords with typical human psychoacoustics.

How do the measures used by Graham's temperament finder relate to this? I see TE Error in "cents per octave". That sounds like Paul's damage. But what is "Adjusted Error"? None of them seem to bear any resemblance to maximum absolute error, which is a measure that falls between "damage" and beat-rate in its weighting of primes.

And I note that Graham's error measures assume optimally stretched or compressed octaves, where Sagittal EDO notation does not. Few people optimally temper the octaves of their EDOs, and probably no-one has ever re-adjusted the stretch of their octaves when they change from using one mapping to another, in the same EDO, in the same piece of music.
There is a growing number of microtonalists who are interested in JI and particularly in using small EDOs to approximate it, in addition to using the more traditional JI-oriented EDOs. That's the main reason why I'm suggesting this notation that I propose is not limited to just the edge-case EDOs. Taking it out as far as 72edo may be over-kill, and I'll admit to suggesting that only as a proof of how widely-applicable it could be. But I would certainly advocate up to 43, at least as an auxiliary form of notation, because there are more EDOs in that range that are a bit "weird" than there are normal ones. People likely to use 72edo might occasionally find use for 43, 41, 36, 34, 31, 29, 26, 24, 22, or 19, but it's not terribly likely that they'll take 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 21, 23, 25, 27, 28, 30, 32, 33, 35, 37, 39, 40, or 42 very seriously. Meanwhile, anyone likely to spend time in the ETs below 19 or 17 is unlikely to spend much time above 41 (or 43 at most), let alone 218 or 581 or whatever.
I can go along with all that. Except that I would put 17-edo in the first group (after 22 and 19).
It is, in the case of EDOs that don't have any good JI approximations. But when good, low-prime-above-3, JI approximations do exist we want to make that information available directly in the notation.
If that's the case, then your proposed Standard Sagittal is at odds with that for the smaller ETs (let's say below 43 at least). In Standard Sagittal you're still leaving out tons of JI information, because of how these ETs roll the different primes up together. 19edo is a decent 13-limit temperament, but (among other things) it equates 5/4 with 16/13, 11/8 with 7/5 and 18/13, 8/7 with 7/6 with 15/13, 10/9 with 9/8 with 11/10, etc. etc. If you want to indicate when a decently-approximated prime is being approximated, full Sagittal gives you the tools for that, but Standard does not.
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. To me, an EDO notation should have one symbol for each number of steps that is notated. You seem to be saying that we should give a great list of symbols for each number of steps (the symbol for every comma that tempers to it). A composer is certainly free to do that themselves, but then she's really writing in JI, not the EDO. But whatever would a poor performer make of such notation when he's told the piece is in a particular EDO?

And in the specific case of 19-edo, no accidental other than the apotome is needed. I assumed your notation would be identical to ours for 19-edo.

You said you don't know where to find the information to notate every EDO between 5 and 72 in the standard Sagittal notation. I don't understand why you say that, because it is given on pages 16, 17 and 19 of http://sagittal.org/sagittal.pdf. Although I admit it can be difficult to find a particular EDO, as they are not in numerical order. You said there were no step numbers, which is true, but the symbols are always given in order of increasing step number from left to right. We always show the natural, then the symbol for 1 step, then the symbol for 2 steps, 3 steps etc. Every number of steps has its own symbol. One never has to use multiple Sagittal symbols per note.

It may be confusing that we give only the pure Sagittal notation. But the mixed notation can be obtained by replacing the Sagittal sharp symbol with a conventional sharp and working back from right to left replacing any double-shaft symbol with the appropriate combination of a conventional sharp and a downward single-shaft symbol. All the existing single-shaft symbols remain as they are.

You asked about the algorithm for choosing the symbol for a given EDO. It begins by working through a list of the commas for unaccented single-shaft symbols in popularity order, which agrees with a certain kind of complexity order. You can get an idea of this ordering here, although it only shows the 8 symbols of the Spartan set, and it lists the ratios notated, rather than the commas themselves (i.e. the commas have had their 2's and 3's removed).
viewtopic.php?p=258#p258

We work through the list and see, using a spreadsheet, how many steps of the ET each comma represents in the obvious mapping. Of course some are zero or negative, and therefore unusable. We prefer to use the comma closest to the top of the popularity list for each number of steps, but there are other considerations involved, such as the aforementioned consistency of the subgroup, and ensuring that the flags that make up the symbols can be assigned consistent values across the whole set. We codified a number of desirable properties like these and gave them shorthand names like "valid flag arithmetic". With the "good" EDOs these things usually pulled in the same direction and the choices were obvious, but with the "bad" ones there was a good deal of subjectivity in weighing these principles against each other. Proposals for such notations might go back and forth half a dozen times between George and I before agreement was reached.
Standard Sagittal, in the context of the smaller ETs (below 41) satisfies neither the desire to completely convey salient JI information, nor the desire to keep the symbol set minimal in that range. It also does not consistently follow a pattern of which JI information will be explicitly included and which won't.
I hope the above convinces you that. if not exactly a "pattern" there are at least guiding principles. The primary one being "Use the comma that notates the simplest ratio". Note it's not the comma ratio that is simplest, but the ratio being notated -- which is the comma with all its 2's and 3's removed, which reduces the apotome to 1/1, and therefore the simplest of all and so the first to be assigned, when it is not zero or negative.
In fact, as I have mentioned before, Standard Sagittal in general preserves no more JI information than the 3-limit system I propose; it just replaces 3-limit information with information about different primes in places--but sacrificed 3-limit information is still sacrificed information. It's robbing Peter to pay Paul. And when it doesn't do this, it agrees with my notational approach (e.g. the 12n stack).
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. But you don't yet know how many steps correspond to the 5-comma or the 7-comma or the 11-medium-diesis. If we are able to use the symbols for those in notating the EDO, then we have conveyed additional information.

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

I find that hard to believe. As I said earlier, we use consistent _subgroups_ of the obvious mapping (not neccesarily limits), in notating EDOs. But even so, I seem to remember that the maximum absolute errors in many of them are close to half a step of the EDO, where 12-edo only has a 16 cent max-abs error in the 5-limit. Half a step is more than 16c for EDOs less than 38.
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.

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 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. 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.
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.
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. 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. 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. 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.

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.

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

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. Thanks.]
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. and , or whose visual size order disagrees with their size order in EDO steps, such as and 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.

Dave Keenan
Posts: 882
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: EDOs with multiple prime mappings

cryptic.ruse wrote: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.
Do you mean like this? (Use the horizontal scroll bar, at the bottom of the post, to see the full width.)
72-edo
D	D	D	D	D	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	E	E	E	E	E
F	F	F	F	F	F	F	F	F	F	F	F	F	F	F	F
I note that, when used on the staff, the Sagittal symbol goes to the left of the conventional symbol(s), which go to the left of the notehead as usual.

To format this, I typed it in an off-forum text editor, so I could put tabs between the pitches. Then I pasted it into this forum post and enclosed it in [ pre ]...[ /pre ] tags (without the spaces).

I typed the smiley codes for the accidentals, like : /| : (without the spaces) for . But I understand that you will be using ASCII characters for your accidentals since we have not decided on suitable Sagittals as yet.

Dave Keenan
Posts: 882
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: EDOs with multiple prime mappings

That was a marathon effort, but I should have done it long ago. It's not quite finished, but I need to sleep. Apart from the fact that I haven't shown any of the notations that are subsets of EDOs beyond 72, it's complete to 46-edo, and it's only missing five EDOs between 46 and 72. However I'll need to add some more Sagittal smilies to the forum before 33 and 40 will display correctly. [Edit: Done]

You will need to use the horizontal scrollbar at the bottom of the post, to see the rightmost parts of notations beyond 48-edo.
5-edo:
D
E
F
G

5-edo as a subset of 50:
D
E	E
F	F
G

6-edo as a subset of 12:
D	D
E	E	E
F	F

7-edo:
D
E
F

7-edo as a subset of 56:
D	D	D
E	E	E
F	F	F

8-edo as a subset of 24 (2 x 12-edo):
D	D
E	E	E
F	F

9-edo:
D	D
E	E	E
F	F

9-edo as a subset of 36 (3 x 12-edo):
D	D
E	E	E	E
F	F	F

10-edo:
D	D	D	D	D
E	E	E	E	E
F	F	F	F	F
G	G	G	G	G

10-edo as a subset of 50:
D	D
E	E	E
F	F	F

11-edo as a subset of 22:
D	D	D	D
E	E	E	E
F	F	F	F

12-edo:
D	D	D
E	E	E	E
F	F	F

13-edo as a subset of 39:
D	D	D	D
E	E	E	E
F	F	F	F

14-edo as a subset of 56:
D	D	D	D	D
E	E	E	E	E
F	F	F	F	F

15-edo:
D	D	D	D	D	D	D
E	E	E	E	E	E	E
F	F	F	F	F	F	F
G	G	G	G	G	G	G

15-edo as a subset of 60 (5 x 12-edo):
D	D	D	D
E	E	E	E	E
F	F	F	F

16-edo:
D	D	D
E	E	E	E	E
F	F	F

16-edo as a subset of 48 (4 x 12-edo):
D	D	D	D
E	E	E	E	E
F	F	F	F

17-edo:
D	D	D	D	D
E	E	E	E	E
F	F	F	F	F

18-edo as a subset of 36 (3 x 12-edo):
D	D	D	D
E	E	E	E	E	E	E
F	F	F	F	F	F

19-edo:
D	D	D
E	E	E	E	E
F	F	F

20-edo as a subset of 60 (5 x 12-edo):
D	D	D	D	D
E	E	E	E	E	E
F	F	F	F	F

21-edo:
D	D
E	E	E
F	F

21-edo as a subset of 63:
D	D	D	D	D	D	D
E	E	E	E	E	E	E
F	F	F	F	F	F	F

22-edo:
D	D	D	D	D	D	D
E	E	E	E	E	E	E
F	F	F	F	F	F	F

23-edo:
D	D	D
E	E	E	E	E
F	F	F

23-edo as a subset of 46:
D	D	D	D	D	D	D
E	E	E	E	E	E	E
F	F	F	F	F	F	F

24-edo:
D	D	D	D	D	D
E	E	E	E	E	E	E
F	F	F	F	F	F

25-edo as a subset of 50:
D	D	D	D	D
E	E	E	E	E	E	E	E	E
F	F	F	F	F	F

26-edo:
D	D	D
E	E	E	E	E
F	F	F
27-edo:
D	D	D	D	D	D	D
E	E	E	E	E	E	E
F	F	F	F	F	F	F

28-edo as a subset of 56:
D	D	D	D	D	D	D	D	D	D
E	E	E	E	E	E	E	E	E	E
F	F	F	F	F	F	F	F	F	F

29-edo:
D	D	D	D	D	D	D	D
E	E	E	E	E	E	E	E	E
F	F	F	F	F	F	F	F	F

30-edo as a subset of 60 (5 x 12-edo):
D	D	D	D	D	D	D
E	E	E	E	E	E	E	E	E
F	F	F	F	F	F	F

31-edo:
D	D	D	D	D	D
E	E	E	E	E	E	E	E	E
F	F	F	F	F	F

32-edo can be notated as a subset of 96 (8 x 12-edo).

33-edo:
D	D	D	D	D
E	E	E	E	E	E	E	E	E
F	F	F	F	F

33 can also be notated as a subset of 99.

34-edo:
D	D	D	D	D	D	D	D	D
E	E	E	E	E	E	E	E	E
F	F	F	F	F	F	F	F	F

35-edo as a subset of 70:
D	D	D	D	D	D	D	D	D	D
E	E	E	E	E	E	E	E	E	E
F	F	F	F	F	F	F	F	F	F

36-edo:
D	D	D	D	D	D	D	D
E	E	E	E	E	E	E	E	E	E	E	E	E
F	F	F	F	F	F	F	F	F	F	F

37-edo can be notated as a subset of 111.

38-edo:
D	D	D	D	D	D
E	E	E	E	E	E	E	E	E	E
F	F	F	F	F	F

39-edo:
D	D	D	D	D	D	D	D	D	D
E	E	E	E	E	E	E	E	E	E
F	F	F	F	F	F	F	F	F	F

40-edo:
D	D	D	D	D
E	E	E	E	E	E	E	E	E
F	F	F	F	F
40 can also be notated as a subset of 80.

41-edo:
D	D	D	D	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	F	F	F	F

42-edo can be notated as a subset of 84 (7 x 12-edo).

43-edo:
D	D	D	D	D	D	D	D
E	E	E	E	E	E	E	E	E	E	E	E
F	F	F	F	F	F	F	F

44-edo can be notated as a subset of 176.

45-edo:
D	D	D	D	D	D
E	E	E	E	E	E	E	E	E	E	E
F	F	F	F	F	F

45-edo can also be notated as a subset of 180 (15 x 12-edo), or as a subset of a future 135.

46-edo:
D	D	D	D	D	D	D	D	D	D	D	D
E	E	E	E	E	E	E	E	E	E	E	E
F	F	F	F	F	F	F	F	F	F	F	F

47-edo:

47-edo can also be notated as a subset of 94.

48-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
F	F	F	F	F	F	F	F	F	F	F

49-edo:

49-edo can also be notated as a subset of 147.

50-edo:																				:50-edo
D	D	D	D	D	D	D	D	D
E	E	E	E	E	E	E	E	E	E	E	E	E	E	E	E	E
F	F	F	F	F	F	F	F	F	F	F	F	F	F	F	F
G	G	G	G	G	G	G	G

51-edo:																				:51-edo
D	D	D	D	D	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	E	E
F	F	F	F	F	F	F	F	F	F	F	F	F	F	F	F

52-edo can be notated as a subset of 156 (13 x 12-edo).

53-edo:																				:53-edo
D	D	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	F	F	F	F	F	F	F	F	F

54-edo can be notated as a subset of 108 (9 x 12-edo).

55-edo:																				:55-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	E
F	F	F	F	F	F	F	F	F	F	F

56-edo:																				:56-edo
D	D	D	D	D	D	D	D	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	E	E	E	E	E
F	F	F	F	F	F	F	F	F	F	F	F	F	F	F	F	F	F	F

57-edo:																				:57-edo
D	D	D	D	D	D	D	D	D
E	E	E	E	E	E	E	E	E	E	E	E	E	E	E	E	E
F	F	F	F	F	F	F	F	F	F	F
57-edo can also be notated as a subset of 171.

58-edo:

59-edo can be notated as a subset of 118.

60-edo:																				:60-edo
D	D	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	E	E
F	F	F	F	F	F	F	F	F	F	F	F	F

61-edo can be notated as a subset of 183.

62-edo:																				:62-edo
D	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	E	E	E
F	F	F	F	F	F	F	F	F	F	F	F

63-edo:																				:63-edo
D	D	D	D	D	D	D	D	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	E	E	E	E	E
F	F	F	F	F	F	F	F	F	F	F	F	F	F	F	F	F	F	F

64-edo:																				:64-edo
D	D	D	D	D	D	D	D	D
E	E	E	E	E	E	E	E	E	E	E	E	E	E	E	E	E
F	F	F	F	F	F	F	F	F	F
64-edo can also be notated as a subset of 128.

65-edo:																				:65-edo
D	D	D	D	D	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	E	E	E	E
F	F	F	F	F	F	F	F	F	F	F	F	F	F	F	F	F

66-edo can be notated as a subset of 132 (11 x 12-edo).

67-edo:

68-edo:

69-edo:																				:69-edo
D	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	E	E	E	E	E
F	F	F	F	F	F	F	F	F	F	F	F

70-edo:																				:70-edo
D	D	D	D	D	D	D	D	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	E	E	E	E	E
F	F	F	F	F	F	F	F	F	F	F	F	F	F	F	F	F	F	F

71-edo can be notated as a subset of 142.

72-edo:																				:72-edo
D	D	D	D	D	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	E	E	E	E	E	E
F	F	F	F	F	F	F	F	F	F	F	F	F	F	F	F	F
Don't forget to use the horizontal scroll bar below, to see more of the larger EDO notations.
Last edited by Dave Keenan on Fri Jan 06, 2017 11:26 pm, edited 2 times in total.

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

### Re: EDOs with multiple prime mappings

That was a marathon effort, but I should have done it long ago. It's not quite finished, but I need to sleep. Apart from the fact that I haven't shown any of the notations that are subsets of EDOs beyond 72, it's complete to 46-edo, and it's only missing five EDOs between 46 and 72.
I should have mine finished before the weekend. Does this forum support pdf file uploads? If not, I will try to follow your example.
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." 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. 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?
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.
Right, that does not invalidate my argument. 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. 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. 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. It only makes sense to NOT do this when higher-prime information is considered integral to the composition, in which case Full Sagittal will provide a better option than my proposal OR Standard Sagittal.
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. A good "standard" will be clear, concise, and consistent. 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. 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.
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.
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?
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? 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. 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.

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?
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.
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?
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.
You think these meta-notational considerations are not present in Standard Sagittal? 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. 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.

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. It's always debatable which prime approximations are more important than others, but the 3-limit (since it is the basis for the nominals) is more or less of incontrovertible importance.