A proposal to simplify the notation of EDOs with bad fifths

[Dave Keenan]

In pure Sagittal, the possible spellings of notes from D to F in 37-edo are:

37edo: DE = 7, EF = 1,  = 6,  = 5,  = 4,  = 3,  = 2,  = 1
D	D	D	D	D	D	D	D	D
E	E	E	E	E	E	E	E	E
F	F	F	F	F	F	F	F	F

I failed to mention, in the previous post, that the 1 degree symbol is not in the Spartan single-shaft section, but in the Promethean single-shaft section.

And of course, for the purely-Sagittal notation you'll also need the Spartan multi-shaft and Promethean multi-shaft sections.

[Dave Keenan]

I've been exploring the idea that we could have apotome-fraction notations that produce consistent notation stacks for other sets of EDOs which are multiples of each other (when they have the same best fifth), like 22, 44, 66, and 19, 38, 57, in the same way that we have long had a consistent notation stack for 12, 24, 36, ... , and now (thanks to this topic) we have them for 5, 10, 15, 20, 25, 30 and 7, 14, 21, 28, 35. Such apotome-fraction notations can operate over a small range of fifth sizes, 2 to 4 cents wide. For example an apotome fraction-notation for meantones would include multiples of 31 as well as multiples of 19, and would also include 50-edo whose fifth size is intermediate between those two.

I have found that only 10 such ranges are required, to cover all EDOs from 5 to 72-edo. I've given fifth errors in cents as the boundaries.

+98c
	Bad fifths apotome fraction notation, 5n, 32, 37, 42, 59 (6, 8, 13, 18 preferred as subsets of 12, 24, 26, 36)
+9.8c
	Super pythagorean notation, 22n, 27n, 49, 61, 71
+6.0c
	17n, 39, 46, 56, 63
+2.0c
	29n, 70
+0.8c
	Pythagorean notation (JI), 45, 53, 65
+1.2c
	Trojan notation, 12n
-2.8c
-3.2c
	43, 55, 67
-4.4c
	Meantone notation, 19n, 31n, 50, 69
-7.5c
	Sub meantone notation, 26n, 45, 64
-9.8c
	Bad fifths limma fraction notation, 7n, 9, 16, 23, 33, 40, 47 (11 preferred as subset of 22)
-48c

[Dave Keenan]

George, Thanks for agreeing (via email) that we could keep the existing 45-edo notation that uses for one degree.

I found that to make a consistent apotome-fraction notation for the Super-pythagoreans we need to change the symbol we use for 1 degree of 49, 54 and 61, and two degrees of 71, to the following:

49:
54:
61:
71:

In reviewing this topic, I noticed that for these "microtonal garbage heap" EDOs we were often undecided whether to use or since both are valid (as 19s and 143C respectively). So I can't imagine you having any problem with the above.

[Dave Keenan]

So here, at last, is the final result for EDOs from 5 to 72, with symbols up to the 1/2-apotome or 3/4-limma. Hoorah! And thanks to @cryptic.ruse (aka Igliashon Jones) who got us started on this 3 years ago.

 
	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
	Add them to get
-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
						    

[Dave Keenan]

I've been contacted by a Dmitri Mendeleev. He says he's from Elysian Fields. Studied in Saint Petersburg, Russia. Works for the Olympian Committee. He says the gods are pleased, and he thinks he can help us with the presentation of these EDO notations.

[Dave Keenan]

Woah! See https://www.facebook.com/groups/4971050 ... 508580246/

[Dave Keenan]

Yeah. OK. Dmitri Mendeleev is really me.

So it doesn't get lost in Facebook, here's the non-linear function of fifth size that is responsible for the compactness of the periodic table of small EDOs, and the linearity of steps per diatonic semitone on one side and steps per chromatic semitone on the other side.

The non-linear function is in two-pieces.
left = (fifthCents-700)/(100/7)
right = (700-fifthCents)/(100/5)
groupNumber
= IF(fifthCents>700,
10 - 72/7 * left/(left+1),
10 + 72/5 * right/(right+1))

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.

Group
number	EDOs
------------------------------
I	6
II	8
III	none
IV	5, 10, 15, 20, 25, 30
V	37
VI	22, 44, 66
VII	17, 34, 51, 68
VIII	29, 58
IX	65
X	12, 24, 36, 48, 60, 72
XI	67
XII	31, 62
XIII	19, 38, 57
XIV	26, 52
XV	47
XVI	7, 14, 21, 28, 35
XVII	none
XVIII	16
XIX	9
XX	11

[Dave Keenan]

Here's the periodic table:

Periodic table of small EDOs large.png (353.75 KiB) Viewed 94 times

[Dave Keenan]

Here's the reversed version for those who prefer the wide fifths on the right, and don't mind that the EDO numbers no longer increase left to right:

Periodic table of small EDOs reversed large.png (356.44 KiB) Viewed 94 times

[Dave Keenan]

This text-only version of the periodic table of EDOs is for Margo (@mschulter).

   I    II    III   IV     V    VI    VII  VIII   IX     X    XI    XII  XIII   XIV   XV    XVI  XVII  XVIII  XIX   XX
+98.0 +48.0 +28.5 +18.0 +11.6  +7.1  +3.9  +1.5  -0.4  -2.0  -3.4  -5.2  -7.2  -9.6 -12.6 -16.2 -20.9 -27.0 -35.3 -47.4
         +36.5              +9.2              +0.5              -4.3              -11.0             -23.7   
                                CG error (cents)
              #=1    5                                 Equal                                 7                 9    11
CD=1                                                 tempered                                    EF=1         |\   ss22

   6     8    #=2   10        Super-             P      12                       Sub-       14   EF=2   16
 ss12  ss24  CD=2   (|\     pythagorean          y                  Meantone   meantone     |\        |) (|\
              #=3                                t                                               EF=3
      #=4  13       15                17         h                        19                21       23  EF=4
CD=3      ss26      /|                /|\        a                                        |) (|\    |) |\ 
      #=5                                        g                                                   (|\
              18    20          22               o      24                      26          28      Negative #
CD=4         ss36 )~| (|\       /|               r      /|\                                |) |\
                                                 e                                          (|\  EF=5
                    25       27             29   a                  31             33       35
CD=5              )~| /|   /| (|\           /|   n                  /|\           |) |\    |( |)
              #=6                                                                   (|\      (|\   Don't use b or #
                    30     32         34                36                38         40     #=0   \!!/ or /||\ here
CD=6              )| /|  )~| /|     /| /|\              |)                /|\       |( |)
     Don't use B   /|)                                                               (|\  EF=6
     or F here    EF=0    37      39           41                43           45      47          Periodic Table
CD=7                     )| /|  /| /|\       /| /|\              |)           /|)    |( |)        --------------
                    #=7   (|\                                                       |\ (|\        of Small EDOs
                         42     44       46             48             50       52    #=1         -------------
CD=8                    )| /|  )| /|   /| /|\         |~ /|\           /|)      /|)            (Sagittal Notation)
                         /|)    (|\                                               .            -------------------
                       EF=1   49      51         53            55         57     #=2
CD=9                         )| /|   |) /|     /| //|        )|( /|\      /|)                  http://sagittal.org
                         #=8  /|\     (|\                                       EF=7 ---------------------------------
                             54    56       58          60          62       64   | Part- Nota- Name   Mnemonic       |
CD=10                     )| /|   |) /|    /| |\      (| /|~      /|) /|\    /|)  | ial   tion                        |
                         /|\ (|\   /|\      /|\                                .  |                                   |
                      #=9   59   61     63        65          67       69     #=3 | 13 (!/   O  E dao  rotated 1 and 3|
CD=11                   )| )~|  )| /|  |) /|     /| |)      )|( /|)  /|) /|\      | 11   /|\ O  C vai  sloping 1 and 1|
                        /| /|) /|\ (|\  /|\       /|\           .   .    .   EF=8 |  9       O  A                     |
                       EF=2     66    68     70         72       . .      . .     |  7  !)   O  F tao  7-color rainbow|
CD=12                       )| )~|   |\ /|  /| |\      /| |)      .       ..      |  6       O  D                     |
                       #=10 /| /|)  /|\ /|)  /|\        /|\     .  .    .  .      |  5    \! O  B pao  V as Roman 5   |
        BC=EF                  71      .   . .    .   .     . .     .  .  #=4     |  4       O  G                     |
CD=13   CD=DE=FG=GA=AB      |\ )~| /| .     .      . .      ..       .            |               Just Intonation     |
        CG=(edo+CD)/2        /|\ (|\  .    . .      .      .  .    .  .           |               ---------------     |
                             . #=10 . #=9 .  #=8  . #=7   .  #=6 .    #=5         |             (Sagittal Notation)   |
                            EF=3 EF=4   EF=5   EF=6    EF=7    EF=8                -----------------------------------