Dave Keenan wrote: ↑Mon Jun 15, 2020 7:32 pm
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.
- 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.]