Extending Trojan notation to finer resolution

[cmloegcmluin]

Oh, wot? You thought you'd won the argument? Ha!

It's the journey that counts, not the destination!

And in the character map, use "12 Relative (Trojan)" and uncapitalise "monzo". You can uncapitalise all but the first word of all the titles if you want. Making the above "12 relative (Trojan)". Of course leave things that are normally capitalised, like the initialisms EDO and IPA.

Ah! I had promised the Monz I would uncapitalize monzo for him. I see that I did it in the source document (in Google Sheets) but failed to propagate it to the hosted file on sagittal.org yet.

I think we should keep names of notations capitalized as we previously discussed, though. And I think the hyphen is better (lest anyone, even for half a moment, think it's a "Relative Notation" with aspect of "12" rather than a "Notation" with aspect of "12-Relative").

[Dave Keenan]

I think we should keep names of notations capitalized as we previously discussed, though. And I think the hyphen is better (lest anyone, even for half a moment, think it's a "Relative Notation" with aspect of "12" rather than a "Notation" with aspect of "12-Relative").

Agreed.

[cmloegcmluin]

I'm still in suspense about my symbol choices for the actual Trinary 12-Relative notation, @Dave Keenan.

And I realized that I've just done the thing I bothered you about the other day: link from a public thread to one of our private ones (private because I have volunteered to do a bunch of work on Sagittal resources and I often need to ask embarrassing questions, not because we're trying to leave anyone out of something they'd be interested in). Here's the relevant info from the link, about capitalization:

But what about the different notations Sagittal offers? Should they all be capitalized? My vote is yes, as it calls attention to them as specific things to learn, differentiate, and be mindful of, rather than generic things which may or may not exist in other systems and for which we might expect prior external knowledge to have significant importance. But similar to how I thought for symbol sets, I don't think the word "notation" should be capitalized. So among our standard notations we would have Prime Factor notation, 108EDO notation, Extreme Precision JI notation, etc.

I agree with all your capitalisation preferences.

[Dave Keenan]

I like your suggestion of as its secondary comma 7:25k (224:225), instead of  , for 1°324 (using a 700 ¢ notational fifth), to eliminate a symbol size reversal. I confirm that it is a valid choice and has the tempered size you claim.

And I accept that we have no choice but to use the untempered sizes for the accents.

I don't see the need to use the secondary 3125C for  . Doesn't the primary 7C work just fine for that here? But apart from that, I think your trinary 12-relative notation is perfect. So that's a wrap. Well done.

[cmloegcmluin]

I don't see the need to use the secondary 3125C for  . Doesn't the primary 7C work just fine for that here?

The 7C does work just fine, yes.

I suppose I only suggested the 3125C because it was slightly closer to the 33.333 target in its tempered size (and its untempered size, for whatever that's worth).

Had I done this work today, I may not have suggested the 3125C at all. My intuition now says that the commas in this context are for justification purposes, and that a secondary comma would only ever be something to resort to when the primary comma did not suffice.

And I suppose if we wanted to be objective about what "sufficing" entailed we'd want the tempered value of the comma to be closer to 33.333 than the next value in the trinary notation — in terms of the equation c = 100×3⁻ⁿ — so, given that the 7C's value of ≈31.174¢ is well above 100*3⁻¹⋅⁵ ≈ 19.245¢, I should definitely have stuck with that.

Perhaps I was just excited at the time, having just learned about using secondary commas to expand possibilities when designing notations, and I overcompensated.

Later version:

• Re: the 3125C, I think I may have just overcompensated upon learned that we could use secondary commas. It seems clear to me now that we'd only want to do that as a last resort, when the tempered version of the primary comma didn't work at all. And since 100×3^-1.5 ≈ 19.245¢, which would be the midway point of sorts between 33.333 and 11.111, and the tempered 1/7C at 31.174 is well on the correct side of that bound, it should certainly win the day.

Here's the final chart then (now with directed comma names! and I guess "apotome" should be the "1A"):

cents		symbols	tempered size	untempered size	EDO step	comma	primary?
±100		 	100.000		113.685 	12		1A	yes
±33.33333333	 	31.174		27.264		36		1/7C	yes
±11.11111111	 	11.199		24.884		108		19/5C	yes
±3.703703704	 	3.802		7.712		324		25/7k	no
±1.234567901	 	-13.686		1.954		972		5s	yes
±0.4115226337	 	4.333		0.423		2916		1/455n	yes

But apart from that, I think your trinary 12-relative notation is perfect. So that's a wrap. Well done.

Thanks for indulging my curiosity! It may not be destined to be the most popular Sagittal notation, but I can certainly see it being the right notation for something I would write one day.

And of course thank you for your guidance in making it happen!

[Note from Dave: I accidentally hit the edit icon on this post instead of the quote icon — a hazard of having moderator powers. I have now restored it from my browser cache. The short post that I intended as a response is quoted in full in Cmloegcmluin's following post.]

[cmloegcmluin]

If we put a nickel in a jar every time one of us edited each other's post instead of quoting it, we'd be... well, we'd probably have about a 7C by now.

Maybe the worst part is that I don't even get notified when it happens! I just came back here to grab my final table for the docs, when I noticed it was gone.

I forget — do you have a better way of getting my previous post back besides using ChromeCacheView or the like?

I think the apotome should be called "3A". If is 1A, what is  ?

My first reaction was "ah, yes, great point... 1's reciprocal is itself, so we can't effectively indicate what its inversion would be; the 1/1A is the same thing, but 1/3A is different from the 3A."

But then I realized: wait, wouldn't just be a 1A down? Just as is a 1/5C up and is a 1/5C down?

[Dave Keenan]

What used to be the next 9 posts in this topic, have now been moved to a new topic: How best to use directed comma names in the "Developing Educational Resources" subforum.

[cmloegcmluin]

Cmloegcmluin Xenharmonic Feisbeuk
...the piece I'm working on is not JI. If I were to notate it with a JI notation system ... it would connote JI, which is not how I'm thinking about it, and I don't want the performers to interpret it that way.

...

I feel like Sagittal connotes JI, since it is based on comma alterations...

Looking back at these concerns I expressed just over a year ago, I now think they're absurd. I was concerned that Sagittal implied my tunings were based on JI when they were not (or in some cases that they were based on different JI ratios than they were really). Sagittal was designed for both EDOs and JI from the start, and so e.g.  's association with the syntonic comma as its default value, and thus always at least a suggestion, but certainly not its ultimate meaning, is critical. Sagittal shouldn't be thought of as implying that any pitch is the JI suggested by the Sagittal symbols' elements or primary commas; Sagittal should instead be thought of as merely indicating JI intervals which are likely to be sensed in the music.

Said another way: it's not the notation which connotes JI, but the music itself, whether I like it or not; the music can't help but sound that way given how humans hear harmony. Or, said yet another way: don't kill the messenger (the music is the message, Sagittal is the (winged) messenger).

And I should do with that information what I will. 99% of the time, a Sagittal user will be like, "thanks Sagittal, that is indeed what I meant". And for the 1% of weirdos like me out there, if my goal is not to evoke the interval my sagittal is pointing out it will evoke, then it has thus alerted me to rethink my composition.

Honestly I'm no longer sure what it was exactly I wanted back then, if it was not what Sagittal provides. Perhaps I wished for a set of symbols with some of Sagittal's good design properties — like sight-readability and size-to-pitch-alteration proportionality — but otherwise randomly shaped, without any musical logic? Such a system would provide a different service to the community than Sagittal does, sure, but its defining anarchic dissociation from psychoacoustic truth would be of so little value that I can't imagine it being worth the cost to develop and maintain alongside Sagittal. In other words: why scramble that channel of information about underlying JI harmonic structure, when at worst, you could just choose to ignore it, and pretend sagittals were random shapes?

I think that even if I was proactively seeking to avoid JI as much as possible, such as with metallic scales / "merciful intonation" — a purpose which seems fundamentally at odds with what individual sagittals are equipped to convey — the Sagittal community will someday find a way of applying Sagittal symbols which leverages their JI basis in service of this goal.

[Dave Keenan]

I totally agree. Couldn't have said it better. That's effectively what we say in the XH article when we introduce the Trojan notation: You can just treat them as random symbols (with a rough increase of symbol size with alteration size and some handy flag arithmetic in a few cases). Never mind about us obsessives who felt we had to tinker in the background to make them consistent with this weird numerological fetish of ours.