A proposal to simplify the notation of EDOs with bad fifths

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

The following diagram is very useful for understanding EDO Notations and how they relate to each other. It assumes the nominals are in a chain of the EDO's best fifths. It was prompted by discussions with Cryptic Ruse aka Igliashon Jones.

An apotome is a chromatic semitone (Eb:E, Bb:B, F:F#, C:C#). A limma is a diatonic semitone (B:C, E:F). If you want to know how many steps there are in the major-whole-tone (A:B, C:D, D:E, F:G, G:A), just add together the steps per apotome and the steps per limma. The number of steps in the EDO's best fifth can be obtained by taking the EDO number itself and adding both its steps per apotome and its steps per limma and dividing by 2. e.g. (22+3+1)/2 = 13.

So this diagram tells us how the nominals, sharps and flats are spaced in a chain-of-fifths notation for a given EDO. If the steps-per-apotome are negative or zero, we don't use sharps or flats. If the steps-per-limma are negative or zero, we don't use the nominals B and F.

Of course this diagram alone, doesn't tell you what accidentals to use for each number of steps. And it only works for native-fifth notations, not subset notations (which are likely to be preferred for 6. 8, 13, 18 and 11 edos, as subsets of 12, 24, 26, 36 and 22 edos respectively).

The blue area shows EDOs whose best fifth has an error no greater than 7.5 cents. Meantone and superpythagorean temperaments are just inside those boundaries, on opposite sides. Pythagoreans, such as the bolded 41 and 53 are on a straight diagonal line that runs down the middle of the blue area, through 94.

Presently, all Sagittal EDO notations are JI-based, in the sense that a symbol always represents the tempered size of its comma role in just intonation. But this results in an explosion of obscure symbols being used for the EDOs in the red and amber areas, many of which are very simple and so don't deserve to have obscure symbols. Cryptic Ruse pointed out this problem, and showed that it could be solved by using fractional-3-limit notations, using only a small number of accidental symbols. He gave the semantics (in bold below) but left the symbols unassigned.

Here's my latest version of this proposal. I've only shown the upward symbols, but of course there are matching downward symbols.
[Edit: You can skip to a later version, that George Secor and I agree on, here: viewtopic.php?p=729#p729.]
```
Symbol	Pronunciation	Apotome fractions represented	Limma fractions represented
nai		1/10, 1/9, 1/8 apotome,		1/7 limma
pai		1/5, 1/6, 1/7, 2/9 apotome	1/6, 1/5, 1/4, 2/7 limma
tai		1/4, 2/7, 3/10, 1/3 apotome	1/3, 2/5, 3/7 limma
phai		3/8, 2/5, 3/7, 4/9 apotome	1/2 limma
vai		1/2 apotome			4/7, 3/5, 2/3 limma
= 	phao-sharp	5/9, 4/7, 3/5, 5/8 apotome	5/7, 3/4, 4/5, 5/6 limma
= 	tao-sharp	2/3, 7/10, 5/7, 3/4 apotome	6/7 limma
= 	pao-sharp	7/9, 6/7, 5/6, 4/5 apotome	1 limma
= 	nao-sharp	7/8, 8/9, 9/10 apotome
= 	sharp		1 apotome```
Notice how the width of the Sagittal symbols increases steadily with the size of their alteration. You will also notice that these 5 symbols and their apotome-complements are in the Spartan subset of Sagittal -- the simplest and most-commonly-used subset by far. My previous version of this proposal did not use Spartan symbols because it was thought that they should not be used when they do not represent the tempering of their defined 11-limit commas.

However, I think it is more important to limit the number of symbols that need to be learned. I think we should add to Sagittal a rule that says:
When the notational fifth is bad enough, the symbols cease to represent the tempered value of their higher-limit comma and revert to their untempered value, which is then treated as a fraction of an apotome or limma. If the fifth is narrower than that of 19-edo, they represent a fraction of a limma. If the fifth is wider than that of 22-edo, they represent a fraction of an apotome.

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

An immediate benefit of the above is that we obtain simple consistent notations for all 5n edos that have the same fifth size, and for all 7n edos that have the same fifth size, in a manner similar to the 12n stack shown in figure 10 on page 19 of http://sagittal.org/sagittal.pdf, as follows:
```5n-edos: A:B C:D D:E F:G G:A = 1\5, B:C E:F = 0, # b = 1\5 (Can use multishafts for >1/2-apotome instead of mixed Sagittal as shown.)
5-edo																	  5-edo
10-edo																	 10-edo
15-edo																	 15-edo
20-edo																	 20-edo
25-edo																	 25-edo
30-edo																	 30-edo
Apotome fractions:	1/6	1/5	1/4	1/3	2/5		1/2		3/5	2/3	3/4	4/5	5/6		1 apotome```
```7n-edos: A:B C:D D:E F:G G:A = 1\7, B:C E:F = 1\7, # b = 0 (Must use multi-shafts for >2/3-limma. Can't use # or b.)
7-edo													  7-edo
14-edo													 14-edo
21-edo													 21-edo
28-edo													 28-edo
35-edo													 35-edo
Limma fractions:	1/5	1/4	1/3	2/5	1/2	3/5	2/3	3/4	4/5		1 limma```
Use the horizontal scroll bar below to see the rest of the accidentals.

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

All suggestions, questions and criticism of the above will be gratefully received.

Here are the proposed limma-fraction notations for 9, 16 and 23-edo.
```7n+2-edos: A:B C:D D:E F:G G:A = 1\9, 2\16, 3\23; B:C E:F = 2\9, 3\16, 4\23; # b = -1
(Must use multi-shafts for >2/3-limma. Can't use # or b.)
9-edo
16-edo
23-edo
Limma fractions:		1/4	1/3		1/2		2/3	3/4			1```
I note that this does not constitute a single Mavila-temperament notation like the one given in figure 8 on page 16 of http://sagittal.org/sagittal.pdf.

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

Here are all of the proposed apotome-fraction notations. These are the EDOs whose best fifths are wider than those of 22-edo (> 709.5 c). 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. Mixed Sagittal is shown. The equivalent pure Sagittal symbols can also be used.
```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 narrower than those of 19-edo (< 694.5 c). 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.
```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

```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```

George Secor
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Location: Godfrey, Illinois, US

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

Dave Keenan wrote: Presently, all Sagittal EDO notations are JI-based, in the sense that a symbol always represents the tempered size of its comma role in just intonation. But this results in an explosion of obscure symbols being used for the EDOs in the red and amber areas, many of which are very simple and so don't deserve to have obscure symbols. Cryptic Ruse pointed out this problem, and showed that it could be solved by using fractional-3-limit notations, using only a small number of accidental symbols. He gave the semantics (in bold below) but left the symbols unassigned.

Here's my latest version of this proposal. I've only shown the upward symbols, but of course there are matching downward symbols.
```
Symbol	Pronunciation	Apotome fractions represented	Limma fractions represented
nai		1/8, 1/9, 1/10 apotome,		1/7 limma
pai		1/5, 1/6, 1/7, 2/9 apotome	1/4, 1/5, 1/6, 2/7 limma
tai		1/4, 1/3, 2/7, 3/10 apotome	1/3, 2/5, 3/7 limma
phai		2/5, 3/7, 3/8, 4/9 apotome	1/2 limma
vai		1/2 apotome			2/3, 3/5, 4/7 limma
= 	phao-sharp	3/5, 4/7, 5/8, 5/9 apotome	3/4, 4/5, 5/6, 5/7 limma
= 	tao-sharp	3/4, 2/3, 5/7, 7/10 apotome	6/7 limma
= 	pao-sharp	4/5, 5/6, 6/7, 7/9 apotome	1 limma
= 	nao-sharp	7/8, 8/9, 9/10 apotome
= 	sharp		1 apotome```
Notice how the width of the Sagittal symbols increases steadily with the size of their alteration. You will also notice that these 5 symbols and their apotome-complements are in the Spartan subset of Sagittal -- the simplest and most-commonly-used subset by far. My previous version of this proposal did not use Spartan symbols because it was thought that they should not be used when they do not represent the tempering of their defined 11-limit commas.

However, I think it is more important to limit the number of symbols that need to be learned. I think we should add to Sagittal a rule that says:
When the notational fifth is bad enough, the symbols cease to represent the tempered value of their higher-limit comma and revert to their untempered value, which is then treated as a fraction of an apotome or limma. If the fifth is narrower than that of 19-edo, they represent a fraction of a limma. If the fifth is wider than that of 22-edo, they represent a fraction of an apotome.
Although I agree with the need to unify (and thus simplify) the notation of "bad-fifth" EDOs, I have a problem with the particular symbols that are proposed to accomplish this. One of the foundational principles (and strengths) of Sagittal is that each symbol is assigned a unique definition that is independent of the tuning. Therefore, I would be very upset with a notation in which C - E, for example, did NOT represent the best 4:5 in nearly all of the bad-5th EDOs in this proposal, and likewise for as 7C and as 11M.

If , , or are going to be used, then why not use these important symbols in those divisions where they are actually valid? After all, the main purpose is to simplify the notation of bad-5th EDOs by reducing the total number of symbols, but why should this require using ALL of the most familiar symbols where they're not valid? Here's where those 3 symbols can be useful:
```
Symbol	Pronunciation	Apotome fractions represented		Limma fractions represented
pai		2/7, 3/10, 1/3, 3/8, 2/5 apotome	(none)
tai		(none)					1/4, 2/7, 1/3, 2/5 limma
vai		(none)					(none)```
Thus can be used as 5C in the following EDOs: 8, 15, 18, 25, 30, 32, 37, 54, 59, and 71, but would not be used in 6, 10, 13, 20, 27, or any of the limma-fraction EDOs.
And can be used as 7C in the following EDOs: 11, 16, 21, 23, 26, 28, 33, 35, 40, 45, 47, 52, and 64, but would not be used in 9, 14, or any of the apotome-fraction EDOs.
would not be used for any bad-5th EDOs, because: it's valid as 1/2-apotome in only 10-EDO (whereas as 13M is valid as 1/2 apotome in every single instance); and it's not very suitable for the limma-fraction EDOs (where is much more useful).

I further investigated which Sagittal symbols would be both valid and usable for the other fractions of the limma and apotome. The only complicated single-shaft symbol required was (slai, which is a "sly” way of using 143C or 11:35k to represent 1/6, 1/5, 2/9, and 1/4 apotome). There are a few instances where I was forced to use symbols that are not valid (for 1/10, 1/9, 8/9 and 9/10 apotome, and 1/6 and 5/6 limma), and those are followed by an asterisk (*):
```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```
One anomaly in the apotome-fraction hierarchy occurs with 71-EDO, where both 3/10 and 4/10 apotome are within the previously established range for (2/7 to 2/5 apotome), but is valid only for 3/10 apotome. Since another symbol, (valid as 7:13S), must then be used for 4/10 apotome, this breaks precedent with 18, 25, and 32-EDO, where is used for 2/5 apotome. The fact that C-E-G produces a good major triad in 71-EDO confirms that 3/10 apotome is the preferred choice.
```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```
Following is a clarification of some of the promethean-level symbols used in the limma-fraction notations:
``` 	11:13C
495C (limma less 55C)
7L (27:28; limma less 7C)
limma less 11:13C (99:104)
limma```
In summary, many of the symbols in this proposed notation are ones commonly used in Sagittal, and there are very few occurrences of invalid symbol usage. Having different symbol sets for the fractional-apotome and fractional-limma divisions helps to identify the category to which a given octave division belongs.
Last edited by Dave Keenan on Tue Jan 03, 2017 6:35 pm, edited 1 time in total.
Reason: Added a thin non-breaking space before :|(: in the last table so it would render correctly

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

That's great stuff George. Thanks. To help folks evaluate it (OK, to help me), I have a couple of requests.

1. Could you please give a table showing all the symbols you have used, in size order, and which apotome and limma fractions they correspond to. Similar to the one of mine that you quoted.

2. Could you please edit your existing tables to show where a symbol is not valid as the tempered version of its primary comma. Or putting it another way, to show where the comma you are using would have accent marks if it was notated in Olympian. Perhaps you could use asterisks for these and double-asterisks where you currently have asterisks.

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:That's great stuff George. Thanks. To help folks evaluate it (OK, to help me), I have a couple of requests.

1. Could you please give a table showing all the symbols you have used, in size order, and which apotome and limma fractions they correspond to. Similar to the one of mine that you quoted.

2. Could you please edit your existing tables to show where a symbol is not valid as the tempered version of its primary comma. Or putting it another way, to show where the comma you are using would have accent marks if it was notated in Olympian. Perhaps you could use asterisks for these and double-asterisks where you currently have asterisks.
Dave, below I have edited the two tables I previously provided only with the asterisk modifications you requested. I have also added two separate tables (one for apotome fractions, the other for limma fractions) showing the symbols in size order (also with asterisks); in those instances where promethean-level symbols are used, I have indicated the corresponding olympian-level symbols in the comments column.

Here is my proposal for bad-fifth native notation using Sagittal symbols to represent various fractions of an apotome (for very wide fifths) or fractions of a limma (for very narrow fifths). Some of the symbols are not used in their primary roles, and these are followed by an asterisk (*). They fall into two categories: 1) the symbol is being used in a secondary role, because its primary role is not valid; 2) each valid limma-complement ratio is notated using a promethean-level symbol, because its primary role requires an olympian-level symbol with diacritic(s) (accents). A few instances require the use of symbols that are not valid for a particular EDO, and these are followed by two asterisks (**)
```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

Symbol	Pronunciation	Apotome fractions represented	Comments
rai	19s	1/10**, 1/9**, 1/8, 1/7		Invalid for 59-EDO** and 71-EDO**
slai	143C	1/6, 1/5, 2/9, 1/4		11:35k* in 54-EDO* and 71-EDO*
pai	5C	2/7, 3/10, 1/3, 3/8, 2/5
*	janai	7:13S*	4/10*				For 71-EDO*; olympian
*	gai	13M*	3/7*, 4/9*, 1/2*		Olympian
*	dai	13L*	4/7*, 5/9*			Olympian
= #*	janao-sharp	6/10*				For 71-EDO*; olympian  = # (112:117)
= #	pao-sharp	3/5, 5/8, 2/3, 7/10, 5/7
= #	slao-sharp	3/4, 7/9, 4/5, 5/6		Apotome less 11:35k* in 54-EDO* and 71-EDO*
= #	rao-sharp	6/7, 7/8, 8/9**, 9/10**		Invalid for 59-EDO** and 71-EDO**
= 	sharp		1 apotome```
One anomaly in the apotome-fraction hierarchy occurs with 71-EDO, where both 3/10 and 4/10 apotome are within the previously established range for (2/7 to 2/5 apotome), but is valid only for 3/10 apotome. Since another symbol (valid as 7:13S) must then be used for 4/10 apotome, this breaks precedent with 18, 25, and 32-EDO, where is used for 2/5 apotome. The fact that C-E-G produces a good major triad in 71-EDO confirms that 3/10 apotome is the preferred choice.
```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

Symbol	Pronunciation	Limma fractions represented	Comments
*	nai	11:13k*	1/7*, 1/6**, 1/5*		Olympian 11:13k* is
tai	7C	1/4, 2/7, 1/3, 2/5
kai	55C	3/7, 1/2
*	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)```
All of the olympian symbols given in the commens column have as their primary roles the ratios indicated.

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

For the benefit of other readers, I note that if I used asterisks in the way George has done, in my proposal, I think every symbol would have an asterisk, most would have two.

I agree that where a spartan is valid in its primary comma role we should use it there, as you have done with pai and tai , so I no longer support my previous proposal. However, although your current proposal unifies very-narrow-fifth notations, and separately wide-fifth-notations, it doesn't really simplify them, as it requires the learning of 7 non-spartan single-shaft symbols, most of which aren't even athenian but are from the rarely-used promethean set.

I remind you of an earlier proposal of mine that used prometheans, but they were all recognisable as spartans with an added left scroll.

I see no point in using non-spartans for these bad-fifth EDOs if the non-spartan doesn't even represent its primary comma. But rai slai and kai may be worth using.

And I don't have a problem with using Spartans where they are not valid. I note that this is different from not using them where they are valid. I agree we should use them where they are valid, but shouldn't shy away from also using them where they are not, if it simplifies the notations.

So I need to go back to the drawing board and come up with a new proposal that hopefully preserves what I see as the best of both our proposals so far. Please feel free to do the same.

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

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 ```
?