## A proposal to simplify the notation of EDOs with bad fifths

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

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

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.
Attachments
edos.png
2560×4154. Hope your screen is giant or your image viewer has zoom-in. (up to 764edo)
Last edited by volleo6144 on Sat Jul 18, 2020 5:26 am, edited 2 times in total.
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Dave Keenan
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### Re: A proposal to simplify the notation of EDOs with bad fifths

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.

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

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

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 table  Nearest 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

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

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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