140th mina

Post by **cmloegcmluin** » Mon Sep 02, 2024 5:28 am

cmloegcmluin wrote: ↑Thu May 14, 2020 1:49 am
Dave Keenan wrote: ↑Tue Apr 07, 2020 12:32 pm I suspect the complete apotome-complement rules only exist in email so far, but here they are: In the pure notation, the apotome-complement of a core symbol is given by Figure 13 on page 24 of http://sagittal.org/sagittal.pdf, with the exceptions that the apotome-complement of $:)/|\:$ is $:)/|\:$  . The apotome complement of a diacritic'd symbol requires inverting all the diacritics in addition to taking the apotome complement of the core as above, except in the cases of
$:)/|\:$ $:)/|\:$ and $:)/|\:$  where the apotome-complement of $:)/|\:$ is $:)/|\:$  .
Whoa! Totally hadn't noticed that before.

In my "Anatomy of an Apotome" sheet I give $:)/|\:$ as the only one which is its own apotome complement. When you checked it before you didn't call that out as wrong. Since in that diagram I am using the Very High precision level to illustrate points about Mixed (Evo) vs. Pure (Revo) flavors and symbol element arithmetic and such, I believe that is still an accurate statement. It would only be upon reaching the Extreme precision level that it would cease to be its own apotome complement, because that's where it gets broken down into an even number of chunks.

Were it to get broken down into five chunks instead then we wouldn't have to make any exceptions (like $:)/|\:$  , $:)/|\:$  , $:)/|\:$  , $:)/|\:$  , and $:)/|\:$  ). Probably there was a reason that wasn't done though. I guess the problem is that $:)/|\:$ is already toward the bottom of that section. In that case I would actually recommend reducing from four chunks to three. Oh, but the problem with that is that the boundaries have to be symmetrical and $:)/|\:$ does not fall inside the middle of three chunks there. Well, you all probably already went through this before... alas...

Okay, so...

In the Medium Precision notation, there is no $:)/|\:$ at all:

half-apotome mirror
	╹
┌	╹	┐
( $:/\|\:$ )	╹	()
	╹

In the High and Ultra Precision notations (the latter introducing Herculean accents), $:)/|\:$ is its own apotome complement, straddling both sides of the half-apotome mirror:

half-apotome mirror
	╹
┌	╹	┐
( $:/\|\:$ )	$:)/\|\:$	()
	╹

In the Extreme Precision notation (introducing Olympian accents), this interesting issue arises where we have an asymmetry about the half-apotome mirror, mercifully quarantined to accents of $:)/|\:$  :

half-apotome mirror
			╹
┌	—	—	╹	—	—	┐
\|	┌	—	╹	—	┐	\|
\|	\|	┌	╹	┐	\|	\|
( $:/\|\:$ )	$:)/\|\:$	$:)/\|\:$	╹	$:)/\|\:$	$:)/\|\:$	()
			╹

So

$:)/|\:$ is used, but

$:)/|\:$ is not. Such imbalance is not unusual; a great many capture zones are split four ways in Extreme, such as

 ,

 , and $:/|\:$  . The weird part is just the apotome complement asymmetry.

And once placing $:)/|\:$ to one side of the mirror or the other, all further precision levels must follow suit, so the Insane Precision level (as well as all beyond Insanity precision levels!) will have this property too, where e.g. we might have

$:)/|\:$ but no

$:)/|\:$  . Though we likely won't define the Insane Precision level in our lifetimes (assuming we remain sane).

Post by **Dave Keenan** » Mon Sep 02, 2024 6:50 am

That's an excellent summary. Thanks.

I will add that this comes about because $:)/|\:$ is defined as the symbol for the 5/49-medium-diesis, 405/392, being the simplest and most popular comma near to its sum of flags.

The half-apotome $\sqrt{\frac{3^7}{2^{11}}}$ is 56.84 cents while $\frac{405}{392}$ is 56.48 cents, about a mina lower.

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140th mina

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Re: 140th mina