cryptic.ruse wrote:I've been working on this post for 2 days, so apologies if you've already covered some stuff in replies during the time I've been writing this.

No worries.

The 26edo notation has the advantage of treating 13edo as a spiral of whole-tones, whereas the 39edo version is a bit more complicated. I have never heard a piece of music in 39edo, or heard of anyone interested in the tuning; 26edo seems quite a bit more popular, even if its fifths are a bit worse. I even know a couple people with 26edo guitars. Also, as someone who's spent a rather large amount of time in 13edo, I find the 26edo subset a lot more intuitive, as the scale generated by the 2-step interval in 13edo hits most of it's "consonances" faster than scales generated by other intervals (10/9 = +1, 16/13 = +2, 11/8 = +3; 9/5 = -1, 13/8 = -2, 16/11 = -3), and it feels very much like the whole-tone scale.

OK. I agree 13 should be notated as a subset of 26. But should 26 be notated with # and b or with 1/3 and 2/3-limma symbols? See

http://forum.sagittal.org/viewtopic.php?p=473#p473.

So I'm good with dropping 11, 13, and 18 from that approach and using them strictly as subsets.

Cool.

I'm with you there; however, his ET notation is completely orthogonal to the color notation, a totally separate entity.

Right. And it's completely orthogonal to any kind of JI notation. I prefer the way that Sagittal smoothly transitions from JI to large numbered EDOs, because at some point they are audibly indistinguishable.

It has never, and will never, be any kind of standard within Sagittal, because it does not have a one-to-one correspondence between pitch-alteration and symbol.

I didn't say it was a standard. Your quote (bolded in context below):

there is a one dimensional tradeoff available here along a spectrum from no higher primes and one symbol-combination per step, thru the present scheme of some higher primes and one symbol-combination per step, **to all-significant primes and multiple symbol-combinations per step**.

Is this 3rd option not what using full JI Sagittal notation for ETs would be? There's no way to cover all significant primes represented in an ET without simply treating it like JI, which prevents one-to-one mappings between symbols and step-counts unless the ET is large enough to represent all of the desired JI intervals distinctly. Whether or not you propose this as a method to notate ETs with Sagittal is beside the point; it exists as an option for those who insist on representing the entire spectrum of primes they find important (up to Sagittal's limitations). If this 3rd group of composers exists, they wouldn't need a standard, they would just need the Sagittal lexicon.

We do seem to be going around in circles on this one. I'm not sure I even remember the point of it, so I was tempted to just drop it. But I have a new idea of maybe why we're talking past each other on this. So I'll give it one more shot. (Yes, to your question)

We agree we're talking about communications standards for EDOs. Consider 72-EDO. It's pretty much where Sagittal started. Perhaps you think it's potentially possible for each of these 3 groups of composers to have their own "standard" notation for 72-EDO. One group using a 1/6-apotome (or 1/12-tone) notation, another using a 5, 7, 11 comma notation with a one-to-one mapping, and another using a JI Notation with symbols for primes 13, 17 and 19 as well as 5, 7 and 11 in a many-to-one mapping.

But to my way of thinking, 3 standards would be no standard at all.

If there are 3 different public formats for 72-edo, then to communicate, every group needs to translate 2 other formats. If there is only one public format (the standard format), then one group doesn't need to translate at all, and the other 2 groups only need to translate that one. And to minimise the total effort, it makes sense to use the 5, 7, 11 comma notation with the one-to-one mapping, because it is mid way between the other two and so requires little effort to translate to or from either of them.

Consistency is one of those concepts that the regular mapping paradigm has more or less dispensed with, by focusing instead on error.

I'm well aware of that. But that's acoustics. This is notation. We're using a notation whose backbone is 3-limit, and people have an expectation that when the best 2:3 can be spelled as C:G, G:D, D:A etc, the best 8:9 can be spelled as C:D, G:A etc. So it's important to know when this is not the case. And one shorthand way to describe this is "1:3:9-inconsistent". Do you have another way you prefer?

We could actually dispense with it here, too, and insist only that the *overall best* 9-limit mapping of the ET also be distinct. It seems impossible that ETs beyond a certain size will have their best mapping not represent the 9-odd-limit distinctly, so wouldn't that solve the problem of being non-limiting?

Yes there is a limit to the number of EDOs which are 9-limit-indistinct. But there is no limit to the number of EDOs that are 1:3:9-inconsistent. Half of all EDOs are 1:3:9 inconsistent.

The higher the limit at which an ET's best mapping distinctly represents JI, the narrower the gulf between "partial JI" and "full JI".

Sure.

In terms of the massive world of ETs, we're debating over a tiny percentage, considering that people are talking about ETs in the mid triple digits.

I think the vast majority of the action will always be in the double-digits.

The reason I don't care for your 7.5¢ 3-limit error window is because it includes tunings like 17edo, 19edo, 22edo, 24edo, 29edo, and 31edo, which really do not need more than one additional symbol (and thus gain little in terms of JI representation from your approach).

Fair enough.

Furthermore, there are many ETs in this window that aren't actually very good (relatively-speaking) at representing higher-limit JI. If your "max unweighted error < 0.5 steps" criterion is used, I find that in the 11-limit, this would exclude 17, 19, 24, 34, 36, 38, 39, 44, 48, 51, 55, 60, 61, 62, 65, 66, 67, 69, and 70 (conversely, it would not exclude 9edo, 16edo, 26edo, 35edo, 37edo, or 49edo, even though those ETs are not within your window).

But why apply 11-limit criteria to all of these? What's so special about 11-limit?

Taking subgroups of the 11-limit or 13-limit may resolve that, but if we're taking subgroups, that opens up a fresh can of worms in deciding which subgroups to consider. I'd rather we didn't go there.

OK. But that's exactly what we've done in designing the standard EDO Notations. We only use subgroups that are consistent in the obvious mapping.

Distinctness in general gives us a more holistic bounding with respect to JI approximation, but I agree it may be overly restrictive (or insufficiently restrictive if set too low). 5-odd-limit distinctness happens as early as 9edo; 7-odd-limit distinctness happens as early as 27edo; 9-odd-limit distinctness happens first at 41edo. Distinctness also doesn't necessarily have much to do with error.

Distinctness was worth considering. But I'm unconvinced of its usefulness here.

**However, it could be the case that we can have our cake and eat it, too. ** Rather than setting a hard boundary between our two approaches, we could find a way to smoothly transition, ...

Something I promoted earlier, but had no idea how to achieve.

... provided you're okay with slightly updating the Standard Sagittal.

That depends on which EDOs and in what ways.

Here's my idea: we find the ETs that are both small (i.e. require only one additional symbol beyond the apotome to fully notate) and are within your ±7.5¢ 3-limit error window. Then we see if we can find a single comma representing some additional prime that is mapped to a useful and common step-size in all of them. We take that symbol and use it for the 1/2-apotome symbol throughout the 3-limit-notated ETs as well, so we keep a layer of JI where it matters while still keeping the symbol counts down, and effectively create a smooth transition zone between 3-limit fractional and Standard Sagittal.

That is an excellent idea!

It looks like ETs that fit this bill would be: 17, 22, 24, 29, 31, 36, 38, 43, 50, and 57 (12 and 19 need zero additional symbols).

Since I want to minimise the changes to the existing notations for these EDOs, my approach is not to survey all the notatable commas, but to look at what symbols are already representing the 1/2-apotome in the EDOs in the blue area (<7.5¢ 3-limit error and 1:3:9-consistent). When I survey pages 16 thru 19 of [url]http://sagittal.org/sagittal.pdf[url], I find that

(32:33) represents the half-apotome in all the 2-step-per-apotome (17, 24, 31, 38), and in all the 4-step-per-apotome (34, 41, 48, 55 and also the non-blue 62, 69), and in nearly all the 6-step-per-apotome (58, 65, 72, 79 but not 51). And not in 68 or 75 in the 8-step-per-apotome row.

That makes

a pretty solid contender as a half-apotome symbol, but that's no surprise, as that was deliberately designed into Sagittal right from the beginning. But it's not universal, and that's one reason why I propose that

(left-scroll added) be used as the true half-apotome symbol, for non-blue EDOs below 72.

I understand that you include 22, 29, 36, 43, 50, 57 in the above list despite the fact that they have no 1/2-apotome, because you've been using the 1/2-apotome symbol also for 1/3-apotome, whereas I suggest repurposing the 1/2-limma symbol for this. So I will now survey the symbols used for 1/3-apotome in the existing notations for the blue EDOs.

(80:81) is 1/3-apotome in 22, 29, 51 (and the non-blue 15)

(63:64) is 1/3-apotome in 36, 43, 65, 72, 79

(35:36) is 1/3-apotome in 50 (and the non-blues 57, 64)

(54:55) or

(44:45) is 1/3-apotome in 58 (undecided)

It's not surprising that there can be no agreement here. Meantones can't use

and superpythagoreans can't use

. But

is used by the greatest number, and I suggest

(left-scroll added) be used as the true 1/3-apotome symbol, for non-blue EDOs below 72.

We could extend the transition zone to also include ETs that need two additional symbols. According to my system, this would be 34, 39, 41, 44, 46, 48, 51, 53, 55, 56, 58, 60, 62, 63, 65, 67, 69, 70, and 72. However, 44, 51, 56, 58, 63, 65, 70, and 72 all divide the apotome into at least 6 parts, so you might want them to have three additional symbols at least. If we leave those 8 ETs out, we're left with 34, 39, 41, 46, 48, 53, 55, 60, 67, and 69.

Yes. I will next look at 1/4-apotome symbols in the blue EDOs having 4 or 8 steps to the apotome.

(80:81) is 1/4-apotome in 34, 41, 68, 75 (and the non-blue 27)

(729:736) is 1/4-apotome in 48 (and 96)

(891:896) is 1/4-apotome in 55

Now 5 steps per apotome.

(80:81) is 1/5-apotome in 39, 46, 53

(7:11C) is 1/5-apotome in 60

(891:896) is 1/5-apotome in 67

(32:33) is 2/5-apotome in 39, 46

(80^2 : 81^2) is 2/5-apotome in 53

(45:46) is 2/5-apotome in 60

(35:36) is 2/5-apotome in 67

(704:729) is 3/5-apotome in 39, 46

is 3/5-apotome in 53

is 3/5-apotome in 60

(8192:8505) is 3/5-apotome in 67

Now 6 steps per apotome.

(63:64) is 1/6-apotome in 51

(80:81) is 1/6-apotome in 58, 65, 72 (and 79)

Now 7 steps per apotome.

(63:64) is 1/7-apotome in 56, 63

(80:81) is 1/7-apotome in 70 (and 77, 84)

Ran out of steam before surveying 2/7, 3/7, 4/7, 1/8, 3/8, 5/8 apotome symbols. Need to sleep.

Can we find a pair of commas that are such that the first comma maps to 1 step in the first set and 2 steps in the second set, and the 2nd comma maps to 1 step in the second set? More specifically, can we find a pair of commas that meet these criteria, and are also simple enough to be relevant to practice? If so, we may be able to put this debate to rest with a mutually-amenable compromise.

As you found out, this is very nearly true for the half-apotome, but not for any of the others, because they inevitably alternate between the 5-comma and the 7-comma between superpythagoreans and meantones.