## 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:
Tue Dec 03, 2019 12:01 pm
I call it "group number" because many EDOs (but not all) have a whole-number value for this, like the element groups on the chemistry periodic table.
Here's all the fractions up to 72-edo:
Group
number	EDOs			Colour
-------------------------------------------------
0	1			Black
I	3, 6			Gold
II	8			Gold
2.5	13			Gold
2.8	18			Gold
III	23b			Gold - fifth = 14\23 - 13\23 is group 17.5
IV	5, 10, 15, 20, 25, 30	Gold
4.86	42			Gold - group = 4 + 6/7
V	37			Gold
5.2	32			Gold
5.33	59			Green - group = 5 + 1/3 - formerly Gold
5.5	27, 54			Green
5.71	49			Green - group = 5 + 5/7
5.8	71			Green
VI	22, 44, 66		Green
6.25	61			Green
6.4	39			Blue
6.57	56			Blue - group = 6 + 4/7
VII	17, 34, 51, 68		Blue
7.43	63			Blue - group = 7 + 3/7
7.6	46			Blue
VIII	29, 58			Magenta
8.29	70			Magenta - group = 8 + 2/7
8.5	41			Grey
8.8	53			Grey - actual 2:3 fifth is group X where X is the number of apotomes in a 9:16 minor seventh
IX	65			Grey
X	12, 24, 36, 48, 60, 72	Orange
XI	67			Yellow
11.2	55			Yellow
11.5	43			Yellow
XII	31, 62			Cyan
12.4	50			Cyan
12.57	69			Cyan - group = 12 + 4/7
XIII	19, 38, 57		Cyan
13.43	64			Purple - group = 13 + 3/7
13.6	45			Purple
XIV	26, 52			Purple
14.5	33			Rose
14.8	40			Rose
XV	47			Rose
XVI	7, 14, 21, 28, 35	Rose
XVII	37b			Rose - fifth = 21\37 - 22\37 is group V
17.5	23			Rose
XVIII	16			Rose
XIX	9			Rose
XX	11			Rose
XXI	17b			White - fifth = 9\17 - 10\17 is group VII
XXII	2, 4			White
XXIII	7bb			White - fifth = 3\7 - 4\7 is group XVI - 5\7 (7b) is group 4/7 = 0.57
XXIV	1b			White - 0¢ fifth

Most of the non-integer groups are in fifths or sevenths, which was kind of to be expected. (Halves, thirds, and quarters also show up occasionally.) The integer groups also often happen to fall on important EDOs—groups I through XX are all the fifths of these EDOs in order: 6, 8, 23b, 5, 37, 22, 17, 29, 65, 12, 67, 31, 19, 26, 47, 7, 37b, 16, 9, and 11.

The function's derivative is actually continuous at 700¢ (it's 0.72 groups per cent on both sides).

(also why is the fact that the exact color boundaries were never specified—even Trojan's was, after some searching—bothering me so much)

59 and 61 also overlap.
Last edited by volleo6144 on Thu Jul 16, 2020 3:32 am, edited 4 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

Hey random guy, I really appreciate your doodling. It was particularly fun to learn that groups III and XVII are not empty, but correspond to the second-best fifths of 23 and 37 EDOs. And from a purely mathematical point of view, its interesting that group 0 and groups XXI thru XXIV are also occupied. Perhaps group 0 should be Black and XXI thru XXIV should be White.

It was also fun to learn the piecewise-defined function is continuous in its first derivative.

I was confused at first, as to what you meant by
volleo6144 wrote:
Thu Jun 11, 2020 4:33 am
59 and 61 also overlap.
I eventually realised you meant their boxes overlap on the periodic table. Well-spotted. Of course, you could say that's just down to my choice to make the boxes exactly one group wide. But then, there were many advantages in doing so. But yes, in mathematical terms, they are the only pair of EDOs less than 73 having the same number of steps per 8:9 tone (and therefore being on the same row) whose group numbers differ by less than one. 6.25 - 5.33 = 0.92.

If the periodic table was to be extended, I think there would need to be a separate diagram which omits the gold and rose EDOs, and doubles the horizontal scale, since it would never go outside of the range from group 5.5 to group 14, and the boxes would then be only half a group wide (and so they would still be square). I wonder how far that would get us before a significant overlap occurred. For most EDOs beyond 72, we'd also need to have 3 lines of text in each box, to have enough room for all the symbols. So it would need to use 2/3 of the point size for the EDO number and sagittals. There would also be no expectation of providing notations for every EDO. In fact very few beyond 72 would have notations, mostly only those with very accurate fifths. It might also need a new colour for the range of fifth-errors between -2.8c and -3.2c.

(also why is the fact that the exact color boundaries were never specified—even Trojan's was, after some searching—bothering me so much)
Could you please explain. I assume you found this post: viewtopic.php?p=843#p843
But I agree the boundaries are not precise.

Or are you referring to the boundaries between symbols within a single notation colour? Which we really only have for orange/Trojan, as given here: viewtopic.php?p=858#p858

If you were interested in working out symbol boundaries for other colours (when this is possible), not as cents but as fractions of the EDOs' own tempered apotomes, that would be very useful. It would also be useful to see how far down the periodic table you can push such notations before they break (by running out of symbols for some degrees).

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

Dave Keenan wrote:
Thu Jun 11, 2020 10:25 pm
(also why is the fact that the exact color boundaries were never specified—even Trojan's was, after some searching—bothering me so much)
Could you please explain. I assume you found this post: viewtopic.php?p=843#p843
But I agree the boundaries are not precise.
Yup, I was referring to the boundaries between the colors here. Trojan's symbol boundaries took some searching to find (admittedly, I had totally forgot that the exact boundaries between the different comma-size classes were—wait, no, that was just a footnote in sagittal.pdf), but including it as a parenthetical was totally a bad idea on my part.

Reversing the group number calculation is actually kind of easy—about as easy to do by hand (I can't even decide which is easier) as calculating the group number for a fifth by hand in the first place:

3 = 10 - 7; 7 × 7/72 = 49/72; 72 - 49 = 23; (49/23) / 7 = 7/23; (7+7/23) × 100¢ = 730.4¢ = 14\23 (23b), and
17 = 10 + 7; 7 × 5/72 = 35/72; 72 - 35 = 37; (35/37) / 5 = 7/37; (7-7/37) × 100¢ = 681.1¢ = 21\37 (37b).
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volleo6144
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### Re: A proposal to simplify the notation of EDOs with bad fifths

Dave Keenan wrote:
Thu Jun 11, 2020 10:25 pm
If the periodic table was to be extended, I think there would need to be a separate diagram which omits the gold and rose EDOs, and doubles the horizontal scale, since it would never go outside of the range from group 5.5 to group 14, and the boxes would then be only half a group wide (and so they would still be square). I wonder how far that would get us before a significant overlap occurred. For most EDOs beyond 72, we'd also need to have 3 lines of text in each box, to have enough room for all the symbols. So it would need to use 2/3 of the point size for the EDO number and sagittals. There would also be no expectation of providing notations for every EDO. In fact very few beyond 72 would have notations, mostly only those with very accurate fifths. It might also need a new colour for the range of fifth-errors between -2.8c and -3.2c.
I did this. Sort of. It turns out that 165 (group = 136/19) and 167 (group = 23/3) just barely don't overlap—the gap between them (minus the 1/2-group width of the boxes) is 1/114 of a group.

The colors in the table below are a continuous rainbow from red on 1 to magenta on 23b to blue on 22 to cyan on 65 to green on 31 to yellow on 47 to red on 16 to magenta on 17b.

177 (group = 73/10) and 179 (group = 148/19) just barely do overlap—this time, they overlap by 1/95 of a group. I'm not sure what your definition of "significant" overlap is.

(The attachment is over 3000 pixels tall, and I don't know of any way to make it not embed itself at the bottom of the post other than by embedding it somewhere else in the post. Sorry.)
Attachments
edos.png
Particularly bare-bones, but it works. Maybe.
Last edited by volleo6144 on Tue Jul 14, 2020 6:29 am, edited 1 time in total.
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cmloegcmluin
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### Re: A proposal to simplify the notation of EDOs with bad fifths

I have to admit I'm still only barely understanding what y'all're talking about here, but that's a hella spiffy visual you've come up with, volleo6144!

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

cmloegcmluin wrote:
Tue Jun 16, 2020 9:48 am
I have to admit I'm still only barely understanding what y'all're talking about here, but that's a hella spiffy visual you've come up with, volleo6144!
I, uh... didn't actually come up with it, really. It's just the periodic table of small EDOs (shown on the previous page, too) with half-size boxes and continued until just a little beyond 300edo (just before 7\12 stops being the best fifth for the 12n edos, even though the boxes start to overlap—like 59 and 61 do in the original table—around 180edo), in response to this post from Dave about how the table should be extended.

And also I forgot that 5 was included in the original table, so it starts at 6 instead.

The discussion was mostly about that horizontal-placement function (Dave calls it the "group number" for the fifth, because the fifths of many important EDOs happen to land on integers) and the way the colors are assigned to the EDOs.
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cmloegcmluin
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### Re: A proposal to simplify the notation of EDOs with bad fifths

Oh! Yes I'm well aware of the original periodic table. Dave and George certainly did the heavy intellectual lifting on the idea. I was complimenting your spacing out of the tiles and extension beyond 72, which brings out the texture of this system in a nice new way.

I'm not sure if Dave will be a fan of the spectral color scheme. He had good reason for dividing the vertical columns with maximally-distinct colors as neighbors. I won't put words in his mouth though!

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

cmloegcmluin wrote:
Tue Jun 16, 2020 11:41 am
I'm not sure if Dave will be a fan of the spectral color scheme.
To be totally honest, I only used it because the code to generate it was something I wrote in about 15 minutes and I was feeling too lazy to actually use the original colors. And also because using it beyond the range where the boundaries are specified precisely enough (two of the EDOs I mentioned in the post with the actual diagram had a fifth in between the narrowest blue—46—and the widest magenta—29 and 58—below 72edo) felt wrong.
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Dave Keenan
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### Re: A proposal to simplify the notation of EDOs with bad fifths

Thanks @volleo6144, that is beautiful. I understood that the rainbow colour scheme was just an expedient way of getting some colour in there.

Would it be easy for you to eliminate the vertical white-space. i.e. eliminate the vertical gaps between rows?

I was deliberately vague about what would be a significant overlap. Initially I thought that since we have an 8% overlap in the existing table, that should be the limit for the table extension too. But if we don't actually have a notation for an overlapped EDO, then the amount of overlap doesn't matter. Overlap only becomes a problem if it prevents us from fitting a notation in a box.

A logical place to stop would be with the row containing 240-EDO as that's the largest we can notate with the Trojan set, and such a table would include the near-Pythagorean 224-EDO which is the largest to be notated with the Athenian set.

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

Done.
• Added 5 at the top.
• Removed vertical whitespace. (For some reason I was using the size of a group as the vertical spacing instead of the size of a box on the first one.)
• Stopped at 240 instead of 306. (237 [group 196/25] and 239 [group 33/4] are in the 41st row.)
Attachments
edos.png
Done.