## EDOs with multiple prime mappings

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

Here's my latest proposal for standard Sagittal EDO notation classes. As well as notating all sub-meantones as limma-fractions, I've corrected where I mistakenly showed 66-edo notated as apotome-fractions.
EdoNotationClasses2.gif
The boundaries are at fifth errors of +-7.5 cents. Superpythagoreans and Meantones are just on the blue side of these boundaries. I suggest that if you want a fractional-3-limit notation for any of the EDOs in the blue area (a non-standard notation for them), they should be notated as apotome-fractions. i.e. You can imagine the amber underlaying the blue, like sand under water. But the red is fire, and stops at the water's edge.

Many thanks to Cryptic Ruse for promoting fractional-3-limit notations for EDOs below 72, and for showing that it is possible to do it with a small number of accidental symbols.

Here's my proposed mixed-Sagittal symbol set for fractional-3-limit notations (originally posted here: viewtopic.php?p=417#p417). I've only shown the upward symbols, but of course there are matching downward symbols too.

slai		1/8, 1/9, 1/10 apotome,		1/7 limma
prai		1/5, 1/6, 1/7, 2/9 apotome	1/4, 1/5, 1/6, 2/7 limma
ratai		1/4, 1/3, 2/7, 3/10 apotome	1/3, 2/5, 3/7 limma
phrai		2/5, 3/7, 3/8, 4/9 apotome	1/2 limma
vrai		1/2 apotome			2/3, 3/5, 4/7 limma
rachai		3/5, 4/7, 5/8, 5/9 apotome	3/4, 4/5, 5/6, 5/7 limma
ratao-sharp	3/4, 2/3, 5/7, 7/10 apotome
prao-sharp	4/5, 5/6, 6/7, 7/9 apotome
slao-sharp	7/8, 8/9, 9/10 apotome
sharp		1 apotome
I note that all of these symbols contain a left-scroll (to indicate that this is a fractional-3-limit notation and not a JI-based notation), and most of them are Spartan symbols with a left-scroll added. The left-scroll adds an "r" or "l" to the symbol's name.

Notice how the width of the Sagittals increases steadily with the size of their alteration.

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

### Re: EDOs with multiple prime mappings

I fear that in not replying to everything, I'm just burying certain points under continued discussion of others. So I'm going to endeavor to address all of your points I have not yet addressed in one marathon post. This discussion has spawned a lot of off-shoots!
1. (Mentioned above) When the limma is zero or negative, why not notate using apotome fractions only, instead of resorting to second-best fifths or subset notations?

2. When the apotome is zero or negative, why not notate as limma fractions only? Allowing to lower pitch and to raise it? That way madness lies.

3. Why use multiple symbols when a single symbol will do?
1. When the limma is negative, 7 nominals will not do; 8 are required, or alternatively, 5. I don't think either of those are good options. When it is zero, I do indeed only use apotome fractions.

2. When the apotome is negative, it still has a meaning with respect to the chain of fifths, but I agree that having it reverse meaning is problematic. In fact I seem to be the only person who isn't bothered by it. It had not occurred to me to simply use limma fractions instead...that is a splendid idea! That would essentially make 7edo the lower bound for tuning of the fifth where the apotome remains relevant. Of course, if one takes a meantone score and wants to play it in Mavila to hear all the interval classes invert qualities (major<->minor, augmented<->diminished, etc.), then there's no getting around # and b switching meaning as well...but for as many times as I've had this debate, I'd rather just go with something less contentious as the standard.

3. I think I misinterpreted this question at first, but reading further I see better what you mean. I hadn't thought of repurposing symbols. Without repurposing, I have already for the most part mixed limma and apotome symbols when they reduce the total number of combined symbols needed below 3 total. As you notice later on, limma fractions in positive-apotome tunings become increasingly useful as the fifth flattens and the ET cardinality increases. I'm not sure how I feel yet about repurposing symbols--I need to think about the ramifications a bit--but you make a powerful case for it with your demonstrations, and have convinced me that it absolutely warrants further contemplation on my part.
It makes me happy that you see that. Thanks for letting me know. And I hope you understand by now, that I have come a long way towards your position since you first announced your draft paper on Facebook. You may also recall that I encouraged you, some months prior to that, to undertake this investigation, in the hope that it might lead to improvements in Sagittal.
It makes me happy as well that you're having this debate with me in good faith. I don't have any ego attached to this; if at the end of the day the advantages of my proposed system are insufficient to make it a worthwhile alternative or adjunct, so much the better--one fewer approach to notation to muddy the waters. I've got a parallel debate going with Kite Giedratis via e-mail, as he's got a system even simpler than mine, where ^ and v always mean "one step of the ET" while # and b retain their usual meanings with respect to the circle of best fifths in a given ET. His notation agrees with mine on nearly half of the ETs up to 72, and is significantly simpler, but I dislike having to stack multiple symbols of the same meaning for consecutive steps, and also find it weird that the symbols bear no abstract relation to the apotome. Perhaps a modification of his to address these things would assuage my critiques and persuad me to join his camp. But ultimately I think what would be of most benefit is if the three of us can reach some sort of agreement on a standard, since I am sure we all agree about the general utility of standards per se.
I would say you were exactly right if you had left out the phrase "But it doesn't _really_ do that, because". Harmonic series relationships _can_ be more or less ignored, despite holding on to the symbols for some of them when we think they're important enough AND we have room for them in the notation.
Perhaps I was unclear. What I meant was, if the desire was to simply repurpose Sagittal symbols in such a way as to behave like traditional notation, *AND* therefore be agnostic about higher harmonic series relationships, it would almost certainly look different than the current Standard Sagittal that *WAS* derived to preserve some of the more obvious and pertinent higher harmonic series relationships.
That's exactly right. That's what it takes to produce a standard. The purpose of a standard is to allow people to communicate with each other while minimising the total effort required.
I could not agree more. The essence of what we are debating is what approach entails more effort on average, I think...and possibly the scope of information to be communicated in the first place.
You are looking at the range of EDOs whose cardinality is less than or equal to 72. I am looking at the range of EDOs whose fifth-error is greater than 7.5 cents (the largest of which is 71-edo).
Fair enough!
I'm saying we'll minimise the total effort over all users by choosing the type in the middle to be the standard.
I would not dispute that if it were not the case that your third type already exists and will continue to exist ("all-significant primes and multiple symbol-combinations per step"). The standard Sagittal is already auxiliary to what I've been calling full Sagittal, and thus the two together reflect a bias toward interest in higher primes. If the first case and third case are extremes, it seems more fair to split the middle rather than throw out one of the extremes entirely.
It may be easy to learn a rule, but difficult to apply it without using a computer or calculator, even when one is familiar with the relevant mathematics.
What is easier to memorize: a set of symbols that follows a pattern, or a set of symbols that doesn't? The rule is not a replacement for memorization, just an aid to it.
I had the BMS firmly in mind as I wrote that. I don't usually think of Carrillo as using 72-edo, since he favoured 24, 48 and 96, but sure, throw him and his students in too. I still think it's a minority.
A minority compared to what? Are you confident? I don't see a terribly large corpus of 72edo music in general, and those who use 72edo for its higher harmonic series resources have not really penetrated academia much (to my knowledge), nor is there an institution that comes to mind that treats 72edo as a temperament of extended JI. It's a popular subject of consideration in our particular esoteric theory circle, but if that has led to much music, then said music could use a bit of PR boost, as I'm not sure where to hear it. Regardless, microtonal music is not something that has become sufficiently widespread to permit generalization, at least in my opinion. Our sample population may not be terribly representative of the total population that could become interested in the subject given the right circumstances. Let's proceed cautiously with respect to generalizing from our own peculiarities, shall we?
One of the things I'm struggling with here, in regard to revising some of the standard notations to use 3-limit fractional comma notations, is how and where to make the transition from the JI-based notations (primes greater than 3 allowed, fractional commas not allowed) to the apotome-fraction notations.
That is certainly a good question. 72edo was a rather arbitrary cut-off for me, more a proof of concept than anything else, and I'd be perfectly happy to move the goal post around a bit.

However, I'm not a tremendous fan of going solely by the tuning of 3/2, since what we are interested is the higher primes. A reasonably-pure 3/2 would be a necessary condition, but perhaps not a sufficient one. Personally, I would advocate the shift to occur whenever the ET represents the 9-odd-limit consistently and distinctly. At that point distinctions between JI identities that most people consider consonant are matched by distinctions in ET intervals--i.e., the commas 16/15, 21/20, 25/24, 28/27, 36/35, 49/48, 50/49, 64/63, and 81/80 must all remain untempered. Off the top of my head, I'm not sure what ETs that includes in the range up to 72, though I believe it includes 41, 46, 53, 60, and 72 if I'm not mistaken. Considering that these commas are large and simple, they are less likely to vanish as ET size grows, and there is no doubt that they all have a largest ET after which they never vanish again under the best mapping (though of course a non-optimal mapping can always take them out again). This would be a sensible place to switch from one "extreme" to the other, I think, and if I'm not mistaken would also enforce a particular level of accuracy on the 3-limit, since we know the fifths will be neither flat enough to be meantone, nor sharp enough to be superpyth--so certainly within your ±7.5¢ boundary. However, if we drop the odd limit distinction down to 7, 81/80 and 64/63 can now vanish again--the reason I don't like this is because we get 5-limit accidentals when fifths are sharp of Just, but we can keep it to 3-limit when they are flat (within your 7.5¢ window)...that feels inconsistent to me. If one side of pure gets by with 3-limit only, then the other side should as well.

And now to throw a complete monkey-wrench into the whole edifice: I'm presently considering ditching the limma and apotome fractions in favor of fractions of 9/8. Whether that would be the best 9/8 or the one consistent with the best 3/2, I'm not sure; this is barely more than an inkling at this point. But the advantages I see would be in the fact that 9/8 is never tempered out nor sent negative, so there's never a matter of switching symbol sets, nor is there a reason to group ETs into families (which I do in the first draft of the manuscript). I'll dig in a little further and see what I find.

---

I think I have addressed most of the loose ends; feel free to remind me of something I may have missed!

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

cryptic.ruse wrote:I fear that in not replying to everything, I'm just burying certain points under continued discussion of others. So I'm going to endeavor to address all of your points I have not yet addressed in one marathon post.
Thanks. I really appreciate this.
1. When the limma is negative, 7 nominals will not do; 8 are required, or alternatively, 5. I don't think either of those are good options. When it is zero, I do indeed only use apotome fractions.
We have no qualms about recommending 5 nominals (C G D A E) for some EDOs in the Xenharmonikon article.

I must admit I didn't actually generate the apotome-fraction notations for 6, 8, 13 and 18 until now.
6edo: (3 nominals: G D A) tone = 2, apotome = 4, limma = -2
A	A/	A^		Av	A\	A
Dv	D\	D	D/	D^		Dv
G^		Gv	G\	G	G/	G^

8edo: (5 nominals C G D A E) tone = 2, apotome = 3, limma = -1
A	A/	A#\				Ab/	A\	A
C\	C	C/	C#\				Cb/	C\
Db/	D\	D	D/	D#\
Eb/	E\	E	E/	E#\
G#\				Gb/	G\	G	G/	G#\

13edo: (5 nominals C G D A E) tone = 3, apotome = 4, limma = -1
A	A/	A^	A#\	A#					Ab	Ab/	Av	A\	A
Cv	C\	C	C/	C^	C#\	C#					Cb	Cb/	Cv
Db	Db/	Dv	D\	D	D/	D^	D#\	D#
Eb	Eb/	Ev	E\	E	E/	E^	E#\	E#
G#\								Gv	G\	G	G/	G^	G#\

18edo: (5 nominals C G D A E) tone = 4, apotome = 5, limma = -1										:18edo
A	A)	A>	A]	A#(	A#								Ab	Ab)	A[	A<	A(	A
C[	C<	C(	C	C)	C>	C]	C#(	C#								Cb	Cb)	C[
Db	Db)	D[	D<	D(	D	D)	D>	D]	D#\	D#
Eb	Eb)	E[	E<	E(	E	E)	E>	E]	E#(	E#
G#(	G#								Gb	Gb)	G[	G<	G(	G	G)	G>	G]	G#(
[Use the horizontal scroll bar at the bottom of the post to see the rest of 18edo -->]

Well, yeah, it can be done. But I agree that most people are going to prefer them notated as subsets of something with a better fifth. I don't see the point of using their second-best fifth when their best fifth is already bad, and that's what's causing the problems with their notation. I don't think there's any question that 6 should be notated as a subset of 12, and 18 as a subset of 36. But it's arguable whether 8 should be notated as a subset of 16 or 24, and 13 as a subset of 26 or 39. 16 and 26 have their own problems, because their fifths aren't so great either. So I think I'd prefer to recommend 8 from 24 and 13 from 39, as we currently do in the Xenharmonikon article.
3. I think I misinterpreted this question at first, but reading further I see better what you mean.
Ah, yes. I see it was quite ambiguous, sorry. But you have correctly inferred that I meant "Why use multiple symbols per note (rather than per notation) when a single symbol will do?"
I've got a parallel debate going with Kite Giedratis via e-mail, as he's got a system even simpler than mine, where ^ and v always mean "one step of the ET" while # and b retain their usual meanings with respect to the circle of best fifths in a given ET. His notation agrees with mine on nearly half of the ETs up to 72, and is significantly simpler, but I dislike having to stack multiple symbols of the same meaning for consecutive steps, and also find it weird that the symbols bear no abstract relation to the apotome.
I'm with you there. And of course you lose the feature that the notation of a subset is a subset of the notation (when they have the same best fifth), e.g.
7, 14, 21, 28, 35
26, 52
19, 38, 57, 76
31, 62, 93
12, 24, 36, 48, 60, 72, ... 240
17, 34, 51, 68, ... 136
22, 44, 66
27, 54
5, 10, 15, 20, 25, 30
We currently don't manage to maintain that feature in all these cases in Sagittal, but we agree it's desirable, and by implementing your ideas we will have it in more of these cases.
Perhaps a modification of his to address these things would assuage my critiques and persuade me to join his camp. But ultimately I think what would be of most benefit is if the three of us can reach some sort of agreement on a standard, since I am sure we all agree about the general utility of standards per se.
It was good to read Kite's material. The more people there are thinking deeply about this stuff the better. But I think we're too far down our very different roads to ever have a meeting of the ways. The whole colour thing, in his JI notation, leaves me cold. We're designing a mapping from shapes-on-a-page to pitches (or perhaps harmonic numbers). I don't see the point of interposing another layer of indirection via colours. If the colours were tacked on the outside, like the pronunciations of the Sagittal symbols, I wouldn't mind. But they seem to be in the middle.
Perhaps I was unclear. What I meant was, if the desire was to simply repurpose Sagittal symbols in such a way as to behave like traditional notation, *AND* therefore be agnostic about higher harmonic series relationships, it would almost certainly look different than the current Standard Sagittal that *WAS* derived to preserve some of the more obvious and pertinent higher harmonic series relationships.
Oh, sure.
I could not agree more. The essence of what we are debating is what approach entails more effort on average, I think...and possibly the scope of information to be communicated in the first place.
Agreed.
I'm saying we'll minimise the total effort over all users by choosing the type in the middle to be the standard.
I would not dispute that if it were not the case that your third type already exists and will continue to exist ("all-significant primes and multiple symbol-combinations per step"). The standard Sagittal is already auxiliary to what I've been calling full Sagittal, and thus the two together reflect a bias toward interest in higher primes. If the first case and third case are extremes, it seems more fair to split the middle rather than throw out one of the extremes entirely.
No. The 3rd type does not exist. Or at least it didn't until you invented it. That's why I have been referring to it as a "straw-man" version of Sagittal. In other words it's a fictitious version of Sagittal EDO notation that you erected in order to argue against it. 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.
What is easier to memorize: a set of symbols that follows a pattern, or a set of symbols that doesn't? The rule is not a replacement for memorization, just an aid to it.
Of course a set that follows a pattern is easier to memorise, but only if that pattern is a simple and obvious one. In both the JI-based and the fractional-3-limit notations, the pattern is rarely obvious merely from looking at the symbols for any single EDO. It takes some mathematical explanation and a table showing what the consistent meaning of the symbols is across all EDOs to understand what the pattern is.
A minority compared to what?
Compared to the total number of people at least thinking about 72-edo.
Are you confident?
No. Not really. When I examine my own motivations, by posing myself the question: "What if someone could prove to you, right now, that 90% of people thinking about 72-edo didn't give a damn about any of its near-JI harmonies?" My answer is: I would still want its notation to relate to 11-limit JI notation because I would assume this was some kind of passing fad. It's just such basic psycho-acoustics -- the fact that sufficiently-close-approximations to sufficently-small whole-number ratios are maxima of consonance. I could easily accept that its ratios of 11 aren't accurate enough to worry about notating, but the Sagittal 11-diesis symbol has a secondary meaning as the half-apotome anyway.
I don't see a terribly large corpus of 72edo music in general, and those who use 72edo for its higher harmonic series resources have not really penetrated academia much (to my knowledge), nor is there an institution that comes to mind that treats 72edo as a temperament of extended JI. It's a popular subject of consideration in our particular esoteric theory circle, but if that has led to much music, then said music could use a bit of PR boost, as I'm not sure where to hear it. Regardless, microtonal music is not something that has become sufficiently widespread to permit generalization, at least in my opinion. Our sample population may not be terribly representative of the total population that could become interested in the subject given the right circumstances. Let's proceed cautiously with respect to generalizing from our own peculiarities, shall we?
Yes. You're right. But I note that 72-edo is quite large, so people tend to use carefully-chosen subsets, e.g. based on temperaments such as Miracle and Marvel. These should all count in favour of a JI-based notation for 72, as these subsets are usually chosen on the basis of their JI approximations.

There is however an interesting class of 72-edo subset (in fact usually also a 36-edo subset) that uses its close approximation to a frequency ratio of phi (the golden ratio) as a way of specifically _avoiding_ JI. But one could say that even the deliberate avoidance of JI is an acknowledgement of its salience.
However, I'm not a tremendous fan of going solely by the tuning of 3/2, since what we are interested is the higher primes. A reasonably-pure 3/2 would be a necessary condition, but perhaps not a sufficient one. Personally, I would advocate the shift to occur whenever the ET represents the 9-odd-limit consistently and distinctly.
I mentioned considering 1:3:9 consistency myself, but noted that such a cutoff places no limit on the number of steps per apotome or limma. 9-limit-consistency and distinctness would be similarly non-limiting. We could limit those by other means, e.g. by limiting to less than 72-edo, but I don't like having too many arbitrary parameters in the notation generating algorithm, and I know you don't either.
At that point distinctions between JI identities that most people consider consonant are matched by distinctions in ET intervals--i.e., the commas 16/15, 21/20, 25/24, 28/27, 36/35, 49/48, 50/49, 64/63, and 81/80 must all remain untempered. Off the top of my head, I'm not sure what ETs that includes in the range up to 72, though I believe it includes 41, 46, 53, 60, and 72 if I'm not mistaken. Considering that these commas are large and simple, they are less likely to vanish as ET size grows, and there is no doubt that they all have a largest ET after which they never vanish again under the best mapping (though of course a non-optimal mapping can always take them out again).
Too restrictive of JI-based notations I'm afraid. It's clear to me that meantones and superpythagoreans are most significant for their JI approximations, even if their mapping of 9 (as twice the steps in their 3) is not the best 9 they have.
This would be a sensible place to switch from one "extreme" to the other, ...
The extremes I spoke of most recently, in regard to EDO notation methods, were fractional-3-limit notation and notating EDOs as if they were JI. Neither I nor the Sagittal documentation has ever advocated the latter (certainly not as a standard). So the switching would be from one extreme (fractional-3-limit) to the middle (JI-based, one-symbol per step). I have been abbreviating the latter to "JI-based". Perhaps I should call it "Partial-JI-based" since you seem so intent on misunderstanding the intent of Sagittal in regard to EDO Notations. But I hesitate to do so, because it seems to lend legitimacy to your idea of "Full-JI-based" notations for EDOs, which neither George, nor I, nor anyone else that I am aware of, has ever advocated.
I think, and if I'm not mistaken would also enforce a particular level of accuracy on the 3-limit, since we know the fifths will be neither flat enough to be meantone, nor sharp enough to be superpyth--so certainly within your ±7.5¢ boundary. However, if we drop the odd limit distinction down to 7, 81/80 and 64/63 can now vanish again--the reason I don't like this is because we get 5-limit accidentals when fifths are sharp of Just, but we can keep it to 3-limit when they are flat (within your 7.5¢ window)...that feels inconsistent to me. If one side of pure gets by with 3-limit only, then the other side should as well.
I'm not sure I follow this. Meantones will not use the 5-comma symbol because they cannot. The 5-comma vanishes. But they may well use the 7-comma or 11-diesis symbols. Similarly, superpythagoreans will not use the 7-comma symbol because it vanishes, but may use the 5-comma or 11-diesis. Seems consistent to me.
And now to throw a complete monkey-wrench into the whole edifice: I'm presently considering ditching the limma and apotome fractions in favor of fractions of 9/8. Whether that would be the best 9/8 or the one consistent with the best 3/2, I'm not sure; this is barely more than an inkling at this point. But the advantages I see would be in the fact that 9/8 is never tempered out nor sent negative, so there's never a matter of switching symbol sets, nor is there a reason to group ETs into families (which I do in the first draft of the manuscript). I'll dig in a little further and see what I find.
That doesn't entirely surprise me. It occurred to me too, while typing up apotome and limma fraction notations. Also, 12-edo-centric microtonalists (like BMS) tend to talk in terms of tone-fractions -- twelfth-tone, sixth-tone, quarter-tone. I would be very grateful if you would investigate this option. I think, for notational purposes, it would have to be the 9/8 that's consistent with their best 3/2.
I think I have addressed most of the loose ends; feel free to remind me of something I may have missed!
You've done well. Thanks.

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

On seeing exactly which EDOs, having fifth-error less than 7.5 cents, fall outside the 1:3:9 consistent zone (see diagram below) I'm not so averse to them having fractional-3-limit notations as standard. But we'd be limited to those with no more than 10 steps per apotome and no more than 8 steps per limma. I suppose that would be OK.
EdoNotationClasses3.gif
In regard to reusing the same symbols for limma fractions and apotome fractions:
One might ask: "How does a user tell whether the symbols are representing apotome fractions or limma fractions in any given instance?"
1. Do they care? (OK, not really an answer. More an attempt to un-ask the question)
2. If there is any sharp or flat (alone or in combination with a Sagittal) then it's definitely an apotome-fraction notation. The equivalent in the pure Sagittal would be any symbol with more than one shaft. Otherwise it's probably limma-fraction.
3. If the fifths are narrower than 694.5 c then it's a limma-fraction notation.
4. Look it up in whatever the final version of the above diagram may be.

Kite has called for xenharmonic terminology that requires less memorization. His manifesto:
Things I refuse to memorize:
I refuse to memorize any ratio with numbers of more than 2 digits
I refuse to memorize any extended ratio with numbers higher than 20
I refuse to memorize the difference between a limna, a kleisma, a schisma, a diaschisma, and a diesis
I refuse to memorize the 62 sagittal accidentals
I refuse to memorize the 750 temperament names
I think that's all very reasonable. No one should have to memorise any of those things to make use of the standard pitch notation for any given EDO.

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

Having done apotome-fraction notations for EDOs with negative limmas, let's see what limma-fraction notations for EDOs with negative apotomes look like.
9edo: tone = 1, apotome = -1, limma = 2
D	D>
E<	E	E>
F<	F

11edo: tone = 1, apotome = -2, limma = 3
D	D/	D^
E\	E	E/	E^
Fv	F\	F

16edo: tone = 2, apotome = -1, limma = 3
D	D/	D^
E\	E	E/	E^
Fv	F\	F

23edo:  tone = 3, apotome = -1, limma = 4
D	D)	D>	D]
E[	E<	E(	E 	E)	E>	E]
F[	F<	F(	F
They look OK to me.

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

### Re: EDOs with multiple prime mappings

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.
. I don't think there's any question that 6 should be notated as a subset of 12, and 18 as a subset of 36. But it's arguable whether 8 should be notated as a subset of 16 or 24, and 13 as a subset of 26 or 39. 16 and 26 have their own problems, because their fifths aren't so great either. So I think I'd prefer to recommend 8 from 24 and 13 from 39, as we currently do in the Xenharmonikon article.
Well, in point of fact, these ETs *can* be notated as subsets of either, since...well, they're subsets of existing notations. But as far as picking a standard, let's see...8edo as a 24edo subset would be:
D	D#^	Dx
Ebb	Ev	E#
Fbbb	Fbv	F
Compared to a 16edo subset, with limma accidentals:
D	D>>	D>>>>
E<<	E	E>>
F<<<<<	F<<<	F<
Yeah, that's not even close...unless we add a 2/3 limma symbol, the 24edo notation is hands-down simpler. I'm good with that.

Now 13edo as a subset of 39edo:
D	D^/	D#/	Dx\	Dx^
Ebv	Evv	E\	E^	E#
Fbb/	Fb\	Fv\	F	F^/
Compared to as a subset of 26edo:
D	Dx	Dxx
Ebb	E	Ex	Exx
Fbbb	Fb	F#
I definitely struggled a bit working through the 39edo example. 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. I am thus not persuaded that 39edo is the better superset.
But I agree that most people are going to prefer them notated as subsets of something with a better fifth. I don't see the point of using their second-best fifth when their best fifth is already bad, and that's what's causing the problems with their notation.
Well, the point is having a native 7-nominal notation based on something fifth-like. If we use limma fractions as you suggest, we can use the same notational approach for 9, 11, 13, 16, 18, and 23. However, that really doesn't accomplish much, as I doubt the 2nd-best fifth in 13 and 18, or even the best fifth of them (and that of 11edo) will be of structural/organizational importance to anyone using those tunings. In 11 and 13, the ~650¢ intervals generate a rather uneven 7-note scale; and in 18edo, there's not much added to 9edo by the additional pitches, at least as far as consonances go--pretty much ratios of 17 only. The only way to use 18edo as any sort of consonant tuning that's not just contorted 9edo is to treat it as a superset of 6edo that adds the 21st harmonic, enabling chords like 8:10:18:21...and that makes more sense as a subset of 36 than anything else. So I'm good with dropping 11, 13, and 18 from that approach and using them strictly as subsets.
The whole colour thing, in his JI notation, leaves me cold.
I'm with you there; however, his ET notation is completely orthogonal to the color notation, a totally separate entity. I need to give it a closer inspection before I can say I prefer mine, yours, or the blend of the two that we seem to be converging upon.
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.
9-limit-consistency and distinctness would be similarly non-limiting.
Consistency is one of those concepts that the regular mapping paradigm has more or less dispensed with, by focusing instead on error. 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?
So the switching would be from one extreme (fractional-3-limit) to the middle (JI-based, one-symbol per step).
The higher the limit at which an ET's best mapping distinctly represents JI, the narrower the gulf between "partial JI" and "full JI". 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.

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). 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). 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.

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.

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

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).

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.

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.

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

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).
45/44 is the only comma I have found so far that works for these ETs; being the difference between 11/9 and 5/4 (or 9/4 and 11/5, or 15/11 and 4/3), I'd say it carries a good amount of relevant JI information pertinent to primes 3, 5, and 11. Mind you, this assumes we're using *optimal* mappings, not patent ones--17 and 38's 11-limit patent vals are not their lowest-error ones; 17c and 38d are optimal unweighted 11-odd-limit mappings as far as I can tell (interestingly, I am discovering that looking at odd limits instead of prime limits changes some of the optimal mappings, primarily because adding 9 to the mix doubles the error of 3 and can pull some of the primes in different directions).

As for the ETs that require more symbols, I give up. I've looked at all the simple commas of the 13-limit I can think of and none of them are 1 step in all of the 2nd-set ETs.

So, if we can use 45/44 for the 1/2 apotome symbol, that will cover the above ETs, which also happen to be the JI ETs up through the 3-steps-to-the-apotome region on your table. The rest of the blue region I am happy to acquiesce on, and leave the standard untouched, since those ETs are all near enough to JI and sufficiently large that I suspect most people using them will prefer some sort of JI-oriented notation.

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

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 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.

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

### Re: EDOs with multiple prime mappings

So, after spending some time on this, I have indeed discovered that by dividing 9/8 (the one consistent with the best 3/2, provided the best 3/2 is no sharper than 720¢) instead of the limma and apotome, I can accomplish exactly the same results as my original proposal with one fewer symbol pair, and in fact the result is even more effective: since we are deriving accidentals from divisions of 9/8, we can have meaningful single-symbol accidentals in 11edo, 13edo, and 18edo that exist independently of the apotome or limma (if we desire such things, anyway).

One thing I am currently trying to figure out (and you may already have figured it out, I need to re-read your replies about repurposing symbols), is what is the best way to repurpose symbols across ETs whose whole-tones do not share the same GCD. I.e., since I now only "need" three symbol pairs (plus the apotome), so if I am ok with having a total of six symbol pairs (including the apotome), what is the most effective way to assign multiple meanings to the five non-apotome symbol pairs to cut down the overall amount of symbols on the page?

I'm not entirely sure if this approach is more fruitful than the original premise of dividing the two pythagorean semitones, nor am I sure it wouldn't be better to divide the best 9/8 independent of 3/2 (incidentally, of ETs in this size range, almost exactly half are 1-3-9 inconsistent, in a sort of alternating pattern...I never realized that!). But I want to hack my way through this before moving onward.

I'll be back with a more in-depth reply in the next week, I think.

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

cryptic.ruse wrote:So, after spending some time on this, I have indeed discovered that by dividing 9/8 (the one consistent with the best 3/2, provided the best 3/2 is no sharper than 720¢) instead of the limma and apotome, I can accomplish exactly the same results as my original proposal with one fewer symbol pair, and in fact the result is even more effective: since we are deriving accidentals from divisions of 9/8, we can have meaningful single-symbol accidentals in 11edo, 13edo, and 18edo that exist independently of the apotome or limma (if we desire such things, anyway).
Interesting. Thanks. I assume you did not use the sharp or flat symbols? Or did you perhaps treat them as half-tone symbols rather than apotome symbols, in which case the reader can't assume Bb:F or B:F# are fifths?
One thing I am currently trying to figure out (and you may already have figured it out, I need to re-read your replies about repurposing symbols), is what is the best way to repurpose symbols across ETs whose whole-tones do not share the same GCD. I.e., since I now only "need" three symbol pairs (plus the apotome), so if I am ok with having a total of six symbol pairs (including the apotome), what is the most effective way to assign multiple meanings to the five non-apotome symbol pairs to cut down the overall amount of symbols on the page?
I didn't have any particularly clever way of deciding that. I listed all the fractions needed, converted them to decimal, sorted them and plotted them so I could see where the big gaps were. Then I did trial and error, moving the boundaries between symbols until I had a unique symbol for every degree of every division of the apotome and limma.
I'm not entirely sure if this approach is more fruitful than the original premise of dividing the two pythagorean semitones, nor am I sure it wouldn't be better to divide the best 9/8 independent of 3/2 (incidentally, of ETs in this size range, almost exactly half are 1-3-9 inconsistent, in a sort of alternating pattern...I never realized that!). But I want to hack my way through this before moving onward.

I'll be back with a more in-depth reply in the next week, I think.
I look forward to your further results