A proposal to simplify the notation of EDOs with bad fifths

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volleo6144
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Re: A proposal to simplify the notation of EDOs with bad fifths

Post by volleo6144 »

Dave Keenan wrote:
Mon Jul 13, 2020 9:56 am
But I have a better criterion. I realised that what makes a second-best fifth worth considering for notation, is when it is not much worse than the best fifth. A convenient cutoff is where the error in the second-best fifth is no more than 4/3 of the error in the best fifth.

Putting it another way: For a b-ET to be included, the fractional part of its number of EDO steps in a pure (2:3) fifth, frac(log2(3/2)×EDO), must be between 3/7 and 4/7.

That includes all those I listed earlier, except 37b, which is OK with me. I realise this gives an infinite number of b-ETs, but that's OK too. As you might guess, it adds on average one b-ET for every 7-EDOs. Up to 72-EDO it includes only 6b, 11b, 18b (9), 23b, 30b, 35b (5), 42b (7), 47b, 59b, 64b (32), 71b.
This is equivalent to setting the EDO × cents-error boundary at 686¢, the tempered perfect fifth of 7edo. (This will never result in a case that's exactly on the line, for the same reason there is no EDO that exactly represents the 2:3 perfect fifth.) This looks like a plausible answer to the question, and the best besides just listing some specific ones to include.

Honestly, "noble-mediant" sounds better anyway. My choice of "phidiant" was ... something I came up with for a reason I can't tell anymore.
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Re: A proposal to simplify the notation of EDOs with bad fifths

Post by volleo6144 »

There are only finitely many (non-"b") EDOs of each color except grey, with the last one of each being:
  • Black: 1
  • Gold: 42 (5n stack ends at 30/35)
  • Green: 83 (22n stack ends at 66/88; 27n stack ends at 54/81)
  • Blue: 259 (17n stack ends at 136/153)
  • Magenta: 724 (29n stack ends at 377/406)
  • Grey: ∞
  • Orange: 459 (12n stack ends at 300/312)
  • Pink: 194 (there are only five of these without "b"s; 194 = 91 + 103, and 103 = 206b has a worse fifth than 206)
  • Yellow: 170
  • Cyan: 105 (31n stack ends at 93/124; 19n stack ends at 76/95)
  • Purple: 64 (26n stack ends at 52/78)
  • Rose: 47 (7n stack ends at 35/42)
  • White: 4 (superset of 2)
I added some changes to the JS file that I used to create the abridged periodic tables (update the way the colors work, make the groups 4 boxes apart instead of 2, and go out to 764edo*), and got ... this:

*764 is the first zeta integral edo after the last magenta edo.
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Last edited by volleo6144 on Sat Jul 18, 2020 5:26 am, edited 2 times in total.
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Re: A proposal to simplify the notation of EDOs with bad fifths

Post by Dave Keenan »

volleo6144 wrote:
Mon Jul 13, 2020 1:09 pm
Dave Keenan wrote:
Mon Jul 13, 2020 9:56 am
For a b-ET to be included, the fractional part of its number of EDO steps in a pure (2:3) fifth, frac(log2(3/2)×EDO), must be between 3/7 and 4/7.
This is equivalent to setting the EDO × cents-error boundary at 686¢, the tempered perfect fifth of 7edo.
Ah! Thanks for pointing that out.
volleo6144 wrote:
Thu Jul 16, 2020 12:49 am
I added some changes to the JS file that I used to create the abridged periodic tables (update the way the colors work, make the groups 4 boxes apart instead of 2, and go out to 764edo*), and got ... this:
That's beautiful and very informative. Thank you.

I love the way you go to the trouble of hiliting the EDOs in your posts with the correct colours too. :)

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Re: A proposal to simplify the notation of EDOs with bad fifths

Post by volleo6144 »

Dave Keenan wrote:
Thu Jul 16, 2020 8:40 pm
I love the way you go to the trouble of hiliting the EDOs in your posts with the correct colours too. :)
Yeah.

It's especially nice that many of the colors are on vertices of the RGB cube (where all three values in the triplet are either 00 or FF), and most of the ones that aren't are the more common (bad-fifths or 12n) colors and so things that you memorize pretty quickly from typing them over and over:
Color.. = RR GG BB
Black.. = 00 00 00 // vertex
Gold... = CC A8 00 // bad-fifths (sharp); 5n
Green.. = 00 FF 00 // vertex; 22n & 27n
Blue... = 00 B6 FF // 17n; on an edge of the cube
Magenta = FF 73 FF // 29n
Grey... = AB AB AB // actually a shade of grey; specifically grey67 (as in X11's rgb.txt!)
Orange. = FF 8F 00 // 12n
Pink... = FF D4 D4 // rare (only five exist, such as 91 and 103)
Yellow. = FF FF 00 // vertex
Cyan... = 00 FF FF // vertex; 19n & 31n
Purple. = B3 9C FF // "B39CFF" just rolls off nicely? (there's only four of these); 26n
Rose... = FF 88 88 // bad-fifths (flat); 7n
White.. = FF FF FF // vertex
Last edited by volleo6144 on Thu Jul 23, 2020 2:21 am, edited 1 time in total.
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Re: A proposal to simplify the notation of EDOs with bad fifths

Post by Dave Keenan »

volleo6144 wrote:
Fri Jul 17, 2020 1:07 am
It's especially nice that many of the colors are on vertices of the RGB cube (where all three values in the triplet are either 00 or FF), and most of the ones that aren't are the more common (bad-fifths or 12n) colors and so things that you memorize pretty quickly from typing them over and over:
Actually, I found the magenta at the RGB cube vertex (#FF00FF) to be too dark. I determined that the ideal magenta for the periodic table would be #FF73FF. But when I typed its Red, Green and Blue values into Microsoft Word, in decimal, I accidentally typed 255, 73, 255. That is, I typed in the Green hex value as if it was decimal. So the periodic table on the web uses #FF49FF. I will change it to #FF73FF, along with a bunch of other updates, eventually.

So I'm sorry for making the magenta harder to remember. But let's make the pink easier to remember by changing it from #FFD2D6 to #FFD4D4, which is an almost imperceptible change.

If you wanted to use named colours instead of having to remember hex codes, here are the closest ones.

Periodic tableNearest named CSS3
colour(web) colour
#000000 black#000000 black
#CCA800 gold#DAA520 goldenrod
#00FF00 green#00FF00 lime
#00B6FF blue#00BFFF deepskyblue
#FF73FF magenta #EE82EE violet(poorly named) Violet is more like this:        
#ABABAB grey#A9A9A9 darkgrey(poorly named) Dark grey is more like this:        
#FF8F00 orange#FF8C00 darkorange
#FFD4D4 pink#FFC0CB pink
#FFFF00 yellow#FFFF00 yellow
#00FFFF cyan#00FFFF cyan
#B39CFF purple#E6E6FA lavender  is almost too light, but #9370DB mediumpurple is too dark
#FF8888 rose#FFA07A lightsalmon
#FFFFFF white#FFFFFF white

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Re: A proposal to simplify the notation of EDOs with bad fifths

Post by volleo6144 »

Dave Keenan wrote:
Sat Jul 18, 2020 1:09 am
#A9A9A9 darkgrey (poorly named) Dark grey is more like this: #373737
And this isn't even the weirdest part: grey #808080 is darker than darkgrey #A9A9A9.
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Re: A proposal to simplify the notation of EDOs with bad fifths

Post by Dave Keenan »

volleo6144 wrote:
Mon Jul 13, 2020 1:09 pm
Dave Keenan wrote:
Mon Jul 13, 2020 9:56 am
Up to 72-EDO it includes only 6b, 11b, 18b (9), 23b, 30b, 35b (5), 42b (7), 47b, 59b, 64b (32), 71b.
This is equivalent to setting the EDO × cents-error boundary at 686¢, the tempered perfect fifth of 7edo. (This will never result in a case that's exactly on the line, for the same reason there is no EDO that exactly represents the 2:3 perfect fifth.) This looks like a plausible answer to the question, and the best besides just listing some specific ones to include.
@volleo6144, would you please do one of your cool graphics to show the periodic table layout of all EDOs from 5 to 72 including the "b" EDOs listed above except for 6b. This should have groups one box apart.

And could you please list the group numbers (horizontal coordinates) of those "b" EDOs.

This would assist me in devising notations for them, and assist @cmloegcmluin in updating the version of the periodic table on the Sagittal home page.

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Re: A proposal to simplify the notation of EDOs with bad fifths

Post by Dave Keenan »

Here's a start on notations for them, based on your colours and parenthesised EDOs of the same fifth-size.
11b will be "ss22".
18b (9) will be "ss36".
23b can be the same as 30 and 37, i.e. :)|: :/|: :(|\: (but maybe it should be "ss46"?).
30b will be the same as 35 and 40, i.e. :|(: :|): :(|\: (a limma-fraction notation).
35b (5) will be the same as 42, i.e. :)|: :/|: :/|):
42b (7) will be the same as 47, i.e. :|(: :|): :|\: :(|\: (a limma-fraction notation).
47b :)|: :)~|: :/|: :(|\:
59b maybe same as 52? i.e. :/|):
64b (32) :)|: :)~|: :)|(: :/|: :(|\:
71b maybe same as 64? i.e. :/|):

3 degrees of 64b was a tough one. :)|(: is in a secondary role as 11:35k (2835/2816). Or putting it another way, it is really :'::)|(: with the accent dropped. This avoided introducing a new symbol for EDOs under 72, namely :(|(: 5:11S (45/44) which in any case looks too much bigger than :/|: to be used in this position.

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Re: A proposal to simplify the notation of EDOs with bad fifths

Post by volleo6144 »

Dave Keenan wrote:
Mon Sep 28, 2020 11:15 am
volleo6144 wrote:
Mon Jul 13, 2020 1:09 pm
Dave Keenan wrote:
Mon Jul 13, 2020 9:56 am
Up to 72-EDO it includes only 6b, 11b, 18b (9), 23b, 30b, 35b (5), 42b (7), 47b, 59b, 64b (32), 71b.
This is equivalent to setting the EDO × cents-error boundary at 686¢, the tempered perfect fifth of 7edo. (This will never result in a case that's exactly on the line, for the same reason there is no EDO that exactly represents the 2:3 perfect fifth.) This looks like a plausible answer to the question, and the best besides just listing some specific ones to include.
@volleo6144, would you please do one of your cool graphics to show the periodic table layout of all EDOs from 5 to 72 including the "b" EDOs listed above except for 6b. This should have groups one box apart.

And could you please list the group numbers (horizontal coordinates) of those "b" EDOs.

This would assist me in devising notations for them, and assist @cmloegcmluin in updating the version of the periodic table on the Sagittal home page.
Oh, this one's unfortunate—I got rid of the script for that some time aft...wait, I didn't? That feels ... uncharacteristic of me, but I'll take it. Done.
EDOGroup
11b1.6
18b19
23b3
30b17.2
35b4
42b16
47b4.75
59b14.28 = 14+2/7
64b5.2
71b13.75
...except with a lot of floating-point weirdness as well.

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Re: A proposal to simplify the notation of EDOs with bad fifths

Post by Dave Keenan »

That's awesome! Thanks volleo.

I took the liberty of editing your post so the image would show inline. The forum used to do that automatically, but we decided it was better to give the author the choice. Unfortunately the only way to give the author the choice was to have it default to not showing.

That shows up nicely, the cases where we will have to deal with overlaps. In the final product, I think we should always have the "b" EDO as the one that is partly obscured. And in general, the one that has the lower fifth error will be on top (like 61 on top of 59). And so we may have to use a smaller point size for the accidentals in the "b" box, and possibly wrap them over two lines. I'm pretty sure the only one that will require that treatment is 64b which has 5 symbols.

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