## 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:72 is as good a cutoff as any. As I said before, I don't think anyone is going to refret a guitar to any division this complex.
I agree. But I don't understand why this is relevant to whether or not a division should have a native fifth notation. There are many other things people may want to do with an EDO besides refret a guitar, and some of them might benefit from a native-fifth notation, if it is simple enough.

Subset notations are often more complex than native fifth notations because they have the same complexity as the superset notation. The exceptions are 11, 6, 8, 13 and 18. I think the SS notations are simpler than the NF notations for these. The NF notations are LF and AF notations in these cases, and the SS notations are of 22, 12, 24, 26 and 36.
There are two reasons for making [44] a subset of 176-EDO:
1) 176 is a much better division than 132; and
2) The notation for 88-EDO is as a subset of 176, so the tones common to 44 and 88 would be notated alike.
OK. I'm happy to leave 44's SS notation being of 176. [Edit: I change my mind below, but ultimately we agree on a native fifth notation and not to recommend any subset notation.]

I checked those possible 61-edo notations I flagged, and I find that the 54-edo AF notation is not valid for 61, because the 5-comma is 2 degrees, not 3. But the 68-edo notation is valid for 61. And yes, 61 is 1:3:5:11:13 consistent.
61: JB (same as 68)

66 is 1:3:17 consistent. It is not 1:3:p consistent for any other prime p up to 19 (no higher primes were checked). So no useful JI-based notation is possible for it. But the following apotome-fraction notation is acceptable. Although is not valid as 2 degrees, it's close.
66: AF (same as 59)

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

On second thoughts:
George Secor wrote:There are two reasons for making [44] a subset of 176-EDO:
1) 176 is a much better division than 132; and
2) The notation for 88-EDO is as a subset of 176, so the tones common to 44 and 88 would be notated alike.
When you say 176 is "better" than 132, I assume you're referring to its more accurate fifths and more accurate approximations of JI. But 132 is "better" because it requires fewer symbols. Surely one can go on forever, finding higher multiples of an EDO that have even more accurate fifths. I believe the recommended superset, for notational purposes, of a 1:3:9-inconsistent EDO, should be the smallest multiple that is not itself 1:3:9 inconsistent.

I don't feel that notating common tones alike, between 44 and 88 is sufficient reason to force a more complex notation on 44. FWIW I note that, if 44 is notated as a subset of 132, then 44 and 66 will have common tones notated alike.

This also means that 13 should be notated as a subset of 26, not 39. This thread is about simplifying notations.

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

Here's a diagram you can copy and make changes to (preferably highlighting the changes with some colour). It shows what notation types I think we should give (in sagittal.pdf and sag_et.par) for each EDO from 5 to 72, and the order in which I prefer the types.

We already have all these notations, of all these types, either in sagittal.pdf or earlier in this thread. It only remains for a few of the most recent to be agreed upon. And then we need to agree which is the preferred type of notation for each EDO.

Steps per limma B:C E:F (diatonic semitone)
-2	-1	0	1	2	3	4	5	6	7
Steps per apotome
-2  (chromatic semitone)	   Subset (SS)	11
notation	SS>LF
-1	steps per whole-tone		9	16	23	   non-preferred
C:D D:E F:G G:A A:B	      LF>MV>SS LF>MV>SS LF>MV>SS   Mavila (MV) notations

0		EDOs	0	7	14	21	28	35	Limma-fraction (LF)
LF=JB<SS LF=JB<SS	LF<SS	LF<SS	LF<SS	notations

1			5	12	19	26	33	40	47
AF=JB>SS	JB	JB     JB=NA>LF	LF>SS	LF>SS	LF>SS

2			10	17	24	31	38	45	52	 (NA) Narrow-apotome-
AF=JB>SS	JB	JB	JB	JB  JB=NA>LF>SS JB=NA>LF>SS   fraction  26 45
notations 52 64
3  Subset (SS)	8	15	22	29	36	43	50	57	64
notations	SS>AF  AF=JB>SS	JB	JB	JB	JB	JB	JB>SS	JB=NA>LF>SS

4	6	13	20	27	34	41	48	55	62	69
SS>AF	SS>AF	AF>SS	JB>AF	JB	JB	JB	JB	JB	JB

5		18	25	32	39	46	53	60	67
SS>AF	AF>SS	AF>SS	JB	JB	JB	JB	JB

6			30	37	44	51	58	65	72
AF>SS	AF>SS	JB=AF	JB	JB	JB	JB

7		 Apotome-	42	49	56	63	70	  JI-based (JB)
fraction (AF)	AF>SS  JB>AF>SS	JB	JB	JB	  notations
notations
8					54	61	68
JB>AF>SS	JB>SS	JB		Red means 3-error 7.5c to 10c

9					59	66			a>b means notation of type a
AF>SS	AF>SS			is preferred over type b

10						71			a=b means the notations of
JB>SS			types a and b are identical

The fact that 66 (709c) must be AF while 27 (711c) must be JB, eliminates any possibility of a simple criterion for when AF is to be preferred over JB—that is if 66 is to have any native-fifth notation at all.

The criterion for when a notation should have a SS notation could be: When it is 1:3:9 inconsistent or has a non-positive apotome or limma. The criterion for when the SS notation is preferred, could be: When it has a negative limma or -2 steps per apotome.

My next task will be to reproduce this diagram, but with the symbols of the preferred notation in place of the list of notation types. I will only give symbols up to the half-apotome because (a) the notation is completely determined by these, (b) it will reduce the freakout-factor, and (c) it will be equally meaningful to those using pure or mixed notations.

[Edit: Changed 44 from "JB=AF>SS" to "JB=AF" as per following post from George.]
[Edit: Changed 71 from "JB>AF>SS" to "JB>SS" as per the following post where the AF notation is simplified by excluding n/10-apotomes.]

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

Here is that diagram showing only my preferred notation types, with each type coloured differently.

Steps per limma B:C E:F (diatonic semitone)
-2	-1	0	1	2	3	4	5	6	7
Steps per apotome
-2  (chromatic semitone)		Subset	11
notation	SS
-1	steps per whole-tone		9	16	23
C:D D:E F:G G:A A:B		LF	LF	LF

0		EDOs	0	7	14	21	28	35	Limma-fraction
LF 	LF	LF	LF	LF	notations

1			5	12	19	26	33	40	47
AF	JB	JB	JB	LF	LF	LF

2			10	17	24	31	38	45	52
AF	JB	JB	JB	JB	JB	JB

3     Subset	8	15	22	29	36	43	50	57	64
notations	SS	AF	JB	JB	JB	JB	JB	JB	JB

4	6	13	20	27	34	41	48	55	62	69
SS	SS	AF	JB	JB	JB	JB	JB	JB	JB

5		18	25	32	39	46	53	60	67
SS	AF	AF	JB	JB	JB	JB	JB

6			30	37	44	51	58	65	72
AF	AF	JB	JB	JB	JB	JB

7		  Apotome-	42	49	56	63	70	  JI-based
fraction	AF	JB	JB	JB	JB	  notations
notations
8					54	61	68
JB	JB	JB

9					59	66
AF	AF

10						71
JB

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

Here is the diagram showing my preferred notations for EDOs from 5 to 72, with symbols up to the 1/2-apotome or 3/4-limma.

Steps per limma B:C E:F (diatonic semitone)
-2	-1	0	1	2	3	4	5	6	7
Steps per apotome
-2  (chromatic semitone)		Subset	11
notation	22ss
-1	steps per whole-tone		9	16	23
C:D D:E F:G G:A A:B
Limma-fraction
0		EDOs	0	7	14	21	28	35	notations
(no  or )

1			5	12	19	26	33	40	47

2			10	17	24	31	38	45	52

3     Subset	8	15	22	29	36	43	50	57	64
notations	24ss

4	6	13	20	27	34	41	48	55	62	69
12ss	26ss

5		18	25	32	39	46	53	60	67
36ss

6			30	37	44	51	58	65	72

7		  Apotome-	42	49	56	63	70	  JI-based
fraction	  	  	  	  	  	  notations
notations
8		(no B or F		54	61	68
for 5n-edos)

9					59	66

10						71


[Edit: Changed 30, 37 and 44 from to as per a following post from George.]
[Edit: Changed 58 from to as per a following post from me.]
[Edit: Corrected 42 from to as per the apotome fraction notation, as noted by George.]

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

Dave Keenan wrote:On second thoughts:
George Secor wrote:There are two reasons for making [44] a subset of 176-EDO:
1) 176 is a much better division than 132; and
2) The notation for 88-EDO is as a subset of 176, so the tones common to 44 and 88 would be notated alike.
When you say 176 is "better" than 132, I assume you're referring to its more accurate fifths and more accurate approximations of JI. But 132 is "better" because it requires fewer symbols. Surely one can go on forever, finding higher multiples of an EDO that have even more accurate fifths. I believe the recommended superset, for notational purposes, of a 1:3:9-inconsistent EDO, should be the smallest multiple that is not itself 1:3:9 inconsistent.

I don't feel that notating common tones alike, between 44 and 88 is sufficient reason to force a more complex notation on 44. FWIW I note that, if 44 is notated as a subset of 132, then 44 and 66 will have common tones notated alike.

This also means that 13 should be notated as a subset of 26, not 39. This thread is about simplifying notations.
I tried out both subset notations in Scala using the following commands:
equal 44
set nota sa132
(and played it on the chromatic clavier.)
set nota sa176
(and played it on the chromatic clavier.)

Then I tried it out using:
set nota e44
(and played it on the chromatic clavier.)

I concluded that a native notation is much better (and simpler) than a subset notation, so we should omit any mention of a subset notation, since the fifths are not really bad.

We came up with slightly different native notations:
44: (DK 7/20/18; 1:3:5:11:13-consistent)
44: (GS 7/20/18; 1:3:5:13:19-consistent)

The only difference is for 1 degree (and its apotome complement). 143C has a lower prime limit (13), but 19s is a member of our one-symbol-per-prime scheme. Which is simpler? As far as I'm concerned, it's not a big deal. I await further comments.

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

Dave Keenan wrote:I checked those possible 61-edo notations I flagged, and I find that the 54-edo AF notation is not valid for 61, because the 5-comma is 2 degrees, not 3. But the 68-edo notation is valid for 61. And yes, 61 is 1:3:5:11:13 consistent.
61: JB (same as 68)
I agree.
66 is 1:3:17 consistent. It is not 1:3:p consistent for any other prime p up to 19 (no higher primes were checked). So no useful JI-based notation is possible for it. But the following apotome-fraction notation is acceptable. Although is not valid as 2 degrees, it's close.
66: AF (same as 59)
I agree -- with one exception: according to my spreadsheet 143C is indeed valid as 2 degrees of 66-EDO!

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

George Secor wrote:I concluded that a native notation [for 44] is much better (and simpler) than a subset notation, so we should omit any mention of a subset notation, since the fifths are not really bad.

We came up with slightly different native notations:
44: (DK 7/20/18; 1:3:5:11:13-consistent)
44: (GS 7/20/18; 1:3:5:13:19-consistent)

The only difference is for 1 degree (and its apotome complement). 143C has a lower prime limit (13), but 19s is a member of our one-symbol-per-prime scheme. Which is simpler? As far as I'm concerned, it's not a big deal. I await further comments.
Thanks George. I agree that your 44 notation is better. is not only visually simpler than , and has a lower odd-limit, it also has a lower sum-of-primes, 19 versus 11+13=24. Sum-of-primes is a very useful figure because it strongly correlates with ratio popularity rank, based on the Scala archive statistics.

The only unfortunate side-effect might have been that 44's JI-based notation was no longer the same as its apotome-fraction notation, as shared with 37 and 30-edo. However I checked whether the AF notation could be changed to match 44's JB notation and I find that indeed it can! is valid as 1/6-apotome (i.e. 1 degree) for both 30 and 37, so we should simplify their notations in this manner too.

I suggest we change the AF notation from the earlier proposal:

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
to the simpler:

Symbol	Pronunciation	Apotome fractions represented
rai		1/9, 1/8, 1/7, 1/6
slai		1/5, 2/9, 1/4
pai		2/7, 1/3, 3/8, 2/5
gai		3/7, 4/9
dai		1/2, 4/7, 5/9
This incorporates the fact that we no longer need an AF notation for 71-edo, and we use 13L for the half apotome instead of 13M.

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

Dave Keenan wrote:I have responded to your recent proposals for new JI-based notations for some divisions—in most cases accepting them. I'd appreciate if you would address those I have not (yet) accepted, and address my recent JI notation proposals, including changing to everywhere that it is used as a half-apotome with a 13-limit meaning, including in the AF notation for wide fifths.
Dave, I informed you by email that I will be taking some time away from the internet for about a week, and at the moment I'm only responding to a few last-minute things.

Yes, I agree that where 13M is used as 1/2 apotome it can be replaced with 13L. However, in each instance we need to check that is actually valid as 13M (and not just 35M), so it would be good if each of us kept track of which divisions are changing so that we can compare notes. I have already done that, and these are the EDO notations that I have determined will change:
JI notations: 27, 37, 44, 45, 51, 68, 75, 78
Bad 5ths (apotome fraction) notations: 6, 10, 13, 20, 27, 54, 71

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

Thanks George. I really appreciating you working on tidying those loose ends. Particularly the following, that affects so many divisions.
George Secor wrote:Yes, I agree that where 13M is used as 1/2 apotome it can be replaced with 13L. However, in each instance we need to check that is actually valid as 13M (and not just 35M), so it would be good if each of us kept track of which divisions are changing so that we can compare notes. I have already done that, and these are the EDO notations that I have determined will change:
JI notations: 27, 37, 44, 45, 51, 68, 75, 78
Bad 5ths (apotome fraction) notations: 6, 10, 13, 20, 27, 54, 71
I wasn't aware that we had JI-based notations for 37 and 78 but I agree that 13M and 13L (and not 35M or 35L) are valid as the 1/2-apotome for those and so they should use . I didn't include 45-edo in my earlier list of affected EDOs, because I was thinking that should only be used for the 1/2-apotome when 13M is valid and 35M is not valid.

I still don't think should be used in 45-edo because 45 is highly inconsistent in regard to prime 13. It is not 1:3:13 or 1:5:13 or 1:7:13 consistent. But it is 1:3:5:7:35 consistent. [Edit: Oops. No it's not. It's not 5:7:35 consistent. Thanks George.] Also, it will simplify things overall, if 45-edo has the same notation as 52-edo. The half apotome is not valid as 13M or 13L in 52, only as 35M and 35L.

I agree re your list of AF notations. But I assume it was a simple oversight, due to being rushed, that you did not include 30 and 37 among the apotome fraction notations that should use 13L as the 1/2-apotome, since is valid as 13M (and not 35M) in those divisions.

So here are the EDO notations I think should change to
JI-based notations: 27, 37, 44, 51, 68, 75, 78 (your list minus 45)
Apotome fraction notations: 6, 10, 13, 20, 27, 30, 37, 54, 71 (your list plus 30 and 37)
Except I don't think 71 should have an AF notation at all.