## 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

George Secor wrote:
Symbol	Pronunciation	Limma fractions represented	Comments
*	ktai	55L	4/7*				Olympian 55L is  (495:512, limma less 55C)
*	chai	7L	3/5*, 2/3*, 5/7*, 3/4*		Olympian 7L is  (27:28, limma less 7C)
= *	tao-sharp	4/5*, 5/6**, 6/7*		Olympian is  =  (99:104, limma less 11:13k)
= *	prao-sharp	1 limma*			Olympian limma is  =  (243:256)
Why not:

Symbol	Pronunciation	Limma fractions represented	Comments
*	wai	55L	4/7*				Olympian 55L is  (495:512, limma less 55C)
*	dai	7L	3/5*, 2/3*, 5/7*, 3/4*		Olympian 7L is  (27:28, limma less 7C)
= *	tao-sharp	4/5*, 5/6**, 6/7*		Olympian is  =  (99:104, limma less 11:13k)
= *	pao-sharp	1 limma*			Olympian limma is  =  (243:256)
?

So, including both the apotome-fraction and limma-fraction notations, I've reduced the number of non-spartan single-shaft symbol pairs from 7 to 3, and the last of those 3 is athenian. The three remaining non-spartans are rai 19s, slai 143C and kai 55C. All three represent their primary commas (except in 54, 59 and 71 edos). The other 4 non-spartans have been replaced with spartans in secondary comma roles, or equivalently with olympians in primary comma roles with their diacritics omitted, leaving only their spartan cores.

I don't consider it to be a problem that dai and pao-sharp appear in both apotome-fraction and limma-fraction notations. I think any advantage that might be gained by having completely disjoint symbol sets for the two cases would be outweighed by the disadvantage of having to learn two new non-spartan non-athenian symbol pairs.

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

George, I note that you have implicitly accepted, without discussion, my proposed cutoffs at fifth sizes narrower than 19-edo and wider than 22-edo. Some might consider that 26-edo should be notated as a meantone.

And you have implicitly accepted my suggestion that all such edos with narrow fifths should be notated as limma-fractions, despite the fact that some of them (including 26) have positive apotomes and so could be notated as apotome-fractions (in which case 26-edo would happen to also be notated as a meantone).

I just want to be sure you considered these problems or options.

And I want you to know that I did not simply accept your comma assignments to the various fractions of an apotome and limma. I derived them from scratch using my own spreadsheet and so I can confirm that you did a thorough job of finding the best. And what a lot of work it was! Well done.

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

Dave Keenan wrote:
George Secor wrote:
Symbol	Pronunciation	Apotome fractions represented	Comments
*	janai	7:13S*	4/10*				For 71-EDO*; olympian 
Why not
Symbol	Pronunciation	Apotome fractions represented	Comments
*	phai	7:17S*	4/10*				For 71-EDO*; olympian 
?
For that matter, why not
Symbol	Pronunciation	Apotome fractions represented	Comments
pakai	11M	4/10				For 71-EDO
...
jatai	11L	6/10				For 71-EDO
?

11M and 11L are both valid and spartan! Two possible objections are:

1) 11M, which represents 4/10 apotome, is larger than 13M, which respresents 1/2 apotome, so the size order is reversed. This is nothing unusual, however, since there are other divisions in which the symbol sizes are reversed: 51, 56, 63, 68, 75, 80, 162.

2) Since the apotome is an even number of steps (10), 11M should not be used if it's not valid as half that number (5). This is nothing new, since 68 also has the same situation:
68:
I submit that and are the best choices for 4/10 and 6/10 apotome, because they are very meaningful.

George Secor
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Joined: Tue Sep 01, 2015 11:36 pm
Location: Godfrey, Illinois, US

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

Dave Keenan wrote:
George Secor wrote:
Symbol	Pronunciation	Limma fractions represented	Comments
*	ktai	55L	4/7*				Olympian 55L is  (495:512, limma less 55C)
*	chai	7L	3/5*, 2/3*, 5/7*, 3/4*		Olympian 7L is  (27:28, limma less 7C)
= *	tao-sharp	4/5*, 5/6**, 6/7*		Olympian is  =  (99:104, limma less 11:13k)
= *	prao-sharp	1 limma*			Olympian limma is  =  (243:256)
Why not
Symbol	Pronunciation	Limma fractions represented	Comments
*	wai	55L	4/7*				Olympian 55L is  (495:512, limma less 55C)
*	dai	7L	3/5*, 2/3*, 5/7*, 3/4*		Olympian 7L is  (27:28, limma less 7C)
= *	tao-sharp	4/5*, 5/6**, 6/7*		Olympian is  =  (99:104, limma less 11:13k)
= *	pao-sharp	1 limma*			Olympian limma is  =  (243:256)
?

So, including both the apotome-fraction and limma-fraction notations, I've reduced the number of non-spartan single-shaft symbol pairs from 7 to 3, and the last of those 3 is athenian. The three remaining non-spartans are rai 19s, slai 143C and kai 55C. All three represent their primary commas (except in 54, 59 and 71 edos). The other 4 non-spartans have been replaced with spartans in secondary comma roles, or equivalently with olympians in primary comma roles with their diacritics omitted, leaving only their spartan cores.

I don't consider it to be a problem that dai and pao-sharp appear in both apotome-fraction and limma-fraction notations. I think any advantage that might be gained by having completely disjoint symbol sets for the two cases would be outweighed by the disadvantage of having to learn two new non-spartan non-athenian symbol pairs.
Okay, Dave, I agree! In fact, I thought of doing the same thing, but I decided to wait for your suggestions.

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

Dave Keenan wrote:George, I note that you have implicitly accepted, without discussion, my proposed cutoffs at fifth sizes narrower than 19-edo and wider than 22-edo. Some might consider that 26-edo should be notated as a meantone.

And you have implicitly accepted my suggestion that all such edos with narrow fifths should be notated as limma-fractions, despite the fact that some of them (including 26) have positive apotomes and so could be notated as apotome-fractions (in which case 26-edo would happen to also be notated as a meantone).

I just want to be sure you considered these problems or options.

And I want you to know that I did not simply accept your comma assignments to the various fractions of an apotome and limma. I derived them from scratch using my own spreadsheet and so I can confirm that you did a thorough job of finding the best. And what a lot of work it was! Well done.
Dave, I briefly considered the issue of cutoffs for fifth sizes, but I decided that it would be better to leave that for another time and just work on the complete set of EDOs you submitted so that, whatever we might decide later, our chosen apotome-fraction and limma-fraction symbols would take into account the complete range of EDOs for which they might be used.

And thank you for checking my work! It was indeed quite a bit of work, and I'm very happy that there was hardly anything about which we disagreed.

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

George Secor wrote:I submit that and are the best choices for 4/10 and 6/10 apotome, because they are very meaningful.
Sold! Now what? Sleep on it for a few months in case anyone comes up with any objections, before we update sagittal.pdf?

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

Here's a summary of the final version of this proposal, that George and I agree on.

For EDOs where the best fifth is more than 7.5 cents wider than just:

Symbol	Pronunciation	Apotome fractions represented	Comments
rai		1/10, 1/9, 1/8, 1/7
slai		1/6, 1/5, 2/9, 1/4
pai		2/7, 3/10, 1/3, 3/8, 2/5	Not for 2/5 in 71-EDO
vai		2/5				In 71-EDO only
gai		3/7 4/9 1/2
dai		4/7 5/9
wai		3/5				In 71-EDO only
= 	pao-sharp	3/5, 5/8, 2/3, 7/10, 5/7	Not for 3/5 in 71-EDO
= 	slao-sharp	3/4, 7/9, 4/5, 5/6
= 	rao-sharp	6/7, 7/8, 8/9, 9/10
= 	sharp		1 apotome
[Edit: See later version here]

For EDOs where the best fifth is more than 7.5 cents narrower than just:

Symbol	Pronunciation	Limma fractions represented
nai		1/7, 1/6, 1/5
tai		1/4, 2/7, 1/3, 2/5
kai		3/7, 1/2
wai		4/7
dai		3/5, 2/3, 5/7, 3/4
tao-sharp	4/5, 5/6, 6/7
pao-sharp	1 limma

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

Here's is the final version, which George and I agree on, of all the EDO native-fifth notations resulting from this proposal.

First we show all of the proposed apotome-fraction notations. These are for the EDOs whose best fifths are more than 7.5 c wider than just (> 709.5 c) (wider than those of 22-edo). Those on the same row have the same number of steps per apotome, and differ only in the spacing of their nominals. This spacing can be obtained from the chart at the start of this thread. Those with a zero or negative number of steps per limma should only use the 5 nominals ACDEG.

Proposed apotome-fraction notations (pure Sagittal):
Steps	0	1	2	3	4	5	6	7	8	9	10
5-edo
10-edo
8,15-edo
6,13,20,27-edo
18,25,32-edo
30,37-edo
42,49-edo
54-edo
59-edo
71-edo
Steps	0	1	2	3	4	5	6	7	8	9	10

Proposed apotome-fraction notations (mixed Sagittal):
Steps	0	1	2	3	4	5	6	7	8	9	10
5-edo
10-edo
8,15-edo
6,13,20,27-edo
18,25,32-edo
30,37-edo
42,49-edo
54-edo
59-edo
71-edo
Steps	0	1	2	3	4	5	6	7	8	9	10

Here are all of the proposed limma-fraction notations. These are the EDOs whose best fifths are more than 7.5 cents narrower than just (< 694.5 c) (narrower than those of 19-edo). Those on the same row have the same number of steps per limma, and differ only in the spacing of their 7 nominals. This spacing can be obtained from the chart at the start of this thread. Pure Sagittal is shown. The equivalent mixed Sagittal symbols cannot be used for these notations, as # and b have no meaning as limma-fractions. They are purely apotome symbols.

Proposed limma-fraction notations:
Steps	0	1	2	3	4	5	6	7
7-edo
9,14-edo
11,16,21,26-edo
23,28,33-edo
35,40,45-edo
47,52-edo
64-edo
Steps	0	1	2	3	4	5	6	7

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

Relative to the notations given in http://sagittal.org/sagittal.pdf at the time of writing, this proposal will add native-fifth notations for the following seventeen EDOs: 6, 8, 11, 13, 18, 20, 25, 28, 30, 32, 35, 37, 42, 52, 54, 59, 71.

And it will change the existing native-fifth notations for these eleven EDOs: 9, 16, 23, 26, 27, 33, 40, 45, 47, 49, 64.

And it justifies the existing notations for these six EDOs: 5, 10, 15; 7, 14, 21.

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

Well George, we've slept on it for 9 months now, and there has recently been a request on one of the facebook groups, for someone to add the corresponding Sagittal notation(s) to every EDO entry in the Xenharmonic Wiki. So we really ought to decide whether these will become the new standard notations for these poor-fifth EDOs, and update figures 8 and 9 on pages 16 and 17 of Sagittal.pdf (the updated Xenharmonikon journal article) accordingly.

Since no one else is arguing, I suggest that we both attempt to come up with reasons why the existing notations for these EDOs should not be changed, or should be changed in ways different from this proposal. i.e. play devil's advocate. For this purpose, it is useful to repost this diagram.

There are eleven existing native-fifth notations that would change under this proposal. These can be grouped as follows. You should locate each group on the above diagram.
Near-superpythagorean (amber): 27, 49 (also includes 54 (2x27) and 71, which don't presently have native fifth notations)
Near-meantone: (red) 26, 45, 64 (also includes 52 (2x26), which doesn't presently have a native fifth notation)
Narrow fifths with one step per apotome (red): 33, 40, 47
Mavila, -1 step per apotome (red): 9, 16, 23

I note that 27 is not simplified by this proposal, since 1\27 changes from the spartan to the non-spartan . Nor is 26 simplified, as it goes from being notated only with sharps and flats (apotomes), to requiring spartan symbols (for limma fractions).

One could argue that the blue area on the diagram (JI-based notations) should be expanded to include the first two categories above. This would change the boundaries, in fifth sizes, from +-7.5 c of just, to +-10 c of just. We might continue to show how apotome and limma fraction notations can be defined for those with fifth errors between 7.5 c and 10 c, but we need not list them as the standard native-fifth notations for those EDOs (the first two categories above).

People who think of 9, 16 and 23 in terms of the Mavila temperament, might be upset when we change their notations to use limma fractions, where the same number of degrees may have a different symbol in each of the three EDOs. A similar complaint could be leveled against changing the 33, 40, 47 group which presently have a sort of apotome-fraction notation.

The two middle groups above might all be coloured amber on the chart, because they all have a positive number of steps per apotome. But their apotome-fraction notation would need to be very different from that used in the amber region on the left of the above diagram.

I refer you back to these posts where I considered similar options.
viewtopic.php?p=416#p416
viewtopic.php?p=452#p452
viewtopic.php?p=457#p457

Edit: Here's a diagram that makes it easier to locate the 4 contentious groups of EDOs:
Small Edos.png