A proposal to simplify the notation of EDOs with bad fifths

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

cmloegcmluin wrote: Wed Nov 18, 2020 11:38 am I didn't mean to imply that I thought groups ever had bounded the colors. My understanding was that before we had the noble mediants, we had nothing. We had just decided, up to 72-EDO, which ones were which color, and not worried about where the exact boundaries should be, until we wanted volleo6144 to build us a towering tornado of EDOs way past 72.
It depends what you mean by "exact". I defined tentative boundaries to the nearest 1/10th of a cent before the PT was even released. See: viewtopic.php?p=843#p843

These resulted from my attempts to find the minimum number of notation stacks that would cover the complete range of fifth sizes. You can see these attempts below.

Here are extracts from 5 emails I wrote to George Secor in late November 2019, about interpreting the JI-based EDO notations on the Periodic Table as also being apotome-fraction notations that operate over narrow ranges of fifth size.

The reason I worked on the periodic table first, instead of responding immediately to your 45 and 54 proposals, is that I wanted 45 and 54 to fit into an overall scheme where every EDO notation on the periodic table is consistent with some apotome-fraction notation that applies over a particular range of fifth sizes. In any case, all the EDOs using the same apotome-fraction (AF) notation would be shown with the same colour.

Two exceptions are:
(a) Those with fifth error < -10 cents (red) do not use any apotome-fraction notation, but instead use the following limma-fraction notation.

For EDOs with fifth errors less than -10c (red)
Symbol Limma fractions represented
|( 1/6, 1/5
|) 1/4, 1/3, 2/5
|\ 1/2
(|\ 3/5, 2/3, 3/4
||) 4/5, 5/6
||\ 1 limma

(b) 66-edo uses the AF notation for fifth error > +10 cents (amber) even though its error is only +7.1 cents (same as 22 and 44).

For EDOs with fifth errors greater than +10c (amber)
Symbol Apotome fractions represented
)| 1/9, 1/8, 1/7, 1/6
)~| 1/5, 2/9, 1/4
/| 2/7, 1/3, 3/8, 2/5
/|) 3/7, 4/9
(|\ 1/2, 4/7, 5/9

There are 11 colours on the table at present. They are only approximately assigned. I haven't yet checked for the existence of consistent AF notations corresponding to all of them. But we know amber, red and light blue are good. It might be possible to reduce it to 10 colours, by merging some on the left.

And thanks for correcting me regarding the bad-fifths cutoff. I agree it is not 10.0 cents, as we do want to include 59-edo in the bad-fifths category. Not sure how I got that wrong. I will refer to the bad-fifths cutoff as 9.8 cents from now on.

Regarding 66-edo, we never did figure out why your spreadsheet claims 143C )~| is valid as 2°66, while my spreadsheet shows that, in 66-edo, 143C is tempered to 26.3 cents. And to be valid as 2°66, it would need to be between 27.3 cents and 45.5 cents.

The following sets this out in more detail:
-------------------------------------------------------------------------------------------------
143C is 143:144 which is 12.1 cents untempered. It contains 3^2 on the large side of the ratio, so when prime 3 is enlarged by 7.1 cents, as it is in 66-edo, the comma grows by twice that amount and becomes 12.1 + 2×7.1 = 26.3 cents.

To be valid as 2°66 it would need to be between 1.5 and 2.5 degrees. That's
1.5/66 × 1200 = 27.3 cents
2.5/66 × 1200 = 45.5 cents
-------------------------------------------------------------------------------------------------

But none of this matters, because I'm happy to accept )~| as 2°66, since the alternative is to use ~|) 49S which we have not used for any other small EDO. I suspect we both agree that 66-edo is microtonal garbage heap material, although we're not allowed to say that publicly about any EDO.

So in that sense, I agree that the JI-based notation for 66 is the same as the bad-fifths apotome-fraction notation for it. And I take your point, that I don't need to colour it amber or consider it an exception to the 9.8 cent cutoff. Thanks for helping me see that.

I'd prefer if 66 had a notation that was consistent with 22 and 44, so it could be coloured the same on the Periodic Table. It took me most of today to realise that it already is consistent with 22 and 44. Duh!

I also thought it would be nice if 27 and 54 could be part of the same apotome-fraction notation as 22, 44 and 66, which should therefore also include 49 and 71, because their fifth sizes are intermediate between 27 and 22. It turns out it is also possible to include 61 and 83 in this notation. So this ends up being an AF notation for fifth errors from +6.0c to +9.8c.

But it requires changing the symbols for 1 degree of 49 and 61 from /| to )| . And changing 1°54 to )| . And changing 2°71 to )~| .

In the 49-edo notation which we agreed long enough ago (Figure 9 of the XH article), |\ isn't actually valid as 1°49. I guess we were swayed by the flag arithmetic.

So I'm proposing we change to the following notations:
49 )| /| /|\ (|)
54 )| /| /|\ (|\ (|)
61 )| /| /|\ (|\ (|)
71 |\ )~| /| /|\ (|\ (|)

And add this one (that won't be shown on the Periodic table):
83 |\ )~| /| /|\ /|) (|\ (|)

Or putting it another way, I propose we adopt the following apotome-fraction notation:

For fifth errors from +6.0c to +9.8c. (Light green on periodic table)
Comma Symbol  Apotome fractions represented
61
83   71   66  54  49  44  27  22
-----------------------------------------------
55C   |\      1/11 1/10
19s   )|                1/9 1/8 1/7 1/6
143C  )~|     2/11 2/10 2/9
5C    /|      3/11 3/10 3/9 2/8 2/7 2/6 1/4 1/3
11M   /|\     4/11 4/10     3/8 3/7
13M   /|)     5/11      4/9
13L   (|\     6/11 5/10 5/9 4/8     3/6 2/4
11L   (|)     7/11 6/10 6/9 5/8 4/7

You suggested using 11:13k for 1°74. This is also a valid choice, and it wouldn't introduce a new symbol. By the way, you gave its symbol as )|, but it's actually |( .

But when I designed the following AF notation to cover all meantones, which I define for this purpose as EDOs whose fifth errors are between -4.4 and -7.5 cents, I find that I want to use the same symbol for 1/5, 1/6, 1/7 and 1/8 apotome, and 11:13k |( is too large to be valid for any but 1/5, whereas 7:13S (|( is valid for all of those, and I use 11:13k |( as 2/7 and 2/8 apotomes in the notations for 105 and 117 edo.

The range of -4.4 and -7.5 cents just excludes 43-edo and 64-edo. It includes only the following EDOs:

Steps  Meantone  Notation
per    EDOs
apotome
1      19
2      31 38                        /|\
3      50 57                /|)             (|\
4      62 69 76         /|)         /|\         (|\
5      74 81 88       (|(      /|)       (|\
6      93 100       (|(     /|)    )/|\     (|\
7      105         (|(    |(   /|)       (|\
8      117        (|(    |(   /|)  )/|\     (|\

74  62
93  81  69  50  31
117 105 100  88  76  57  38
----------------------------------------------
7:13S   (|(   1/8 1/7 1/6 1/5
11:13k   |(   2/8 2/7
13M     /|)   3/8     2/6 2/5 1/4 1/3
5:49M  )/|\   4/8     3/6
11m     /|\                   2/4     1/2
13L     (|\   5/8     4/6 3/5 3/4 2/3

This doesn't require any change to any previously agreed notation. It's also not very consistent. e.g. 3/6 and 4/8 use a different symbol from 1/2 and 2/4. But what's worse, 1/4 and 2/8 use different symbols. This could be corrected by changing 1 degree of 62, 69 and 76 to |( .

Do you think it is worthwhile, having these apotome-fraction notations for various ranges?

Hi George,

Here's my next attempt at an apotome-fraction notation for a limited range of fifth sizes. We need a snappy name for those. This one is for errors between -3.2 and -4.4 cents. Dark green on the periodic table. And as with the meantones, it will cover many more edos than will be shown on the periodic table.

From -3.2 to -4.4 cents includes only the following EDOs:

Steps  EDOs            Notation
per
apotome
1
2
3      43                               |)            ||)
4          55                      )|(         /|\
5              67                )|(      /|)       (|\
6      86          79          |(     /|)     /|\     (|\
7        98                  ~~|   )|(    /|)       (|\
8         110             ~~|    )|(    /|)    /|\     (|\
9     129   122          ~~|  |(   )|(     |)       ||)
10     141    134       ~~|   |(   )|(   /|)   /|\   (|\
11      153     146    ~~|   |(   )|(  /|)   |)  ||)   (|\
12                158 ~~|   |(  )|(  /|)   |)  /|\  ||)  (|\

only these 3 edos
146  134 122          79  on periodic table
158  153  141 129 110  98  86  67  55  43
------------------------------------------------------
11:49C  ~~| 1/12 1/11 1/10 1/9 1/8 1/7
11:13k   |( 2/12 2/11 2/10 2/9         1/6
7:11k   )|( 3/12 3/11 3/10 3/9 2/8 2/7     1/5 1/4
13M     /|) 4/12 4/11 4/10     3/8     2/6 2/5
7C       |) 5/12 5/11      4/9     3/7             1/3
11M     /|\ 6/12      5/10     4/8     3/6     2/4
||) 7/12 6/11      5/9     4/7             2/3
13L     (|\ 8/12 7/11 6/10     5/8     4/6 3/5

This doesn't require any change to any previously agreed notation.

I suggest the existing apotome-fraction notation for the 12n-edos, that we call Trojan, might apply to fifth errors from -1.2 cents to -2.8 cents, i.e. 700c +-0.8c. The smallest non-12n edos that this would include are 127-edo and 137-edo. Those have 10 steps to the apotome and 12 steps to the apotome respectively and so would have the same notations as 120-edo and 144-edo. But we don't need to worry about those because they won't be on the periodic table.

Then we have the Pythagorean apotome fraction notation, whose finest divisions are 217 and 224-edo with 21 steps to the apotome. I suggest this covers fifth errors of -1.2 to +0.8 cents.

Here are suggested names for the 10 colours, and suggested boundaries in cents between them.

+98
+9.8
Green (Super pythagorean, 22n, 27n, 49, 61, 71)
+6.0
Blue (17n, 39, 46, 56, 63)
+2.0
Magenta (29n, 70)
+0.8
Grey (Pythagorean, 45, 53, 65)
+1.2
Orange or Tan? (12n, Trojan)
-2.8
-3.2
Yellow (43, 55, 67)
-4.4
Cyan (Meantone, 19n, 31n, 50, 69)
-7.5
Mauve (Sub meantone, 26n, 45, 64)
-9.8
Red or Rose? (Bad fifths limma fraction)
-48

cmloegcmluin