Prime-factor Sagittal JI notation (one symbol per prime)

[Dave Keenan]

[MODERATOR NOTE: This topic was previously called "Multi-Sagittal JI notation (one symbol per prime)".]

Juhani Nuorvala has made me realise that George and I had been far too dismissive of one-symbol-per-prime notations for JI. We made it clear in the Xenharmonikon article, that such a notation was possible using Sagittal, and we gave symbols for all primes up to 29, but we confusingly gave two symbols for most of them.

Dave Ryan recently raised the question of algorithms for choosing a single best comma for each prime. So I asked George to list his choices, as far up the list as he wished to go, and I did the same. We then exchanged lists and found that we disagreed only for primes 23 and 59. I immediately saw that I had been mistaken in my choice for 23 and George saw that he had been mistaken in his choice for 59. So we quickly agreed.

It turns out that we agree with a simple algorithm that looks only at commas whose exponent of the prime 3 is in the range -6 to +6 and whose size is smaller than 68.57 cents, and we choose the one whose 3-exponent has the smallest absolute value. This minimises the need for sharps and flats in the most common keys—the most common choices for 1/1. And when there are two commas with the same absolute 3-exponent, as in the case of the prime 23, we choose the smaller comma. The 68.57 cents [sqrt(3¹⁹/2³⁰)] is the upper limit for single-shaft symbols in Sagittal (about 60% of a sharp or flat).

The following diagram shows the only Sagittal symbols most JI composers will ever need, with some aids to remembering the primes they correspond to. The mnemonics are explained here.



Here's the full list, as far as we went:

 
Symbol	Name	Prime	Comma	  Prime exponent vector	SMuFL code points    Characters      Inverted
		  Up Dn		  (Monzo) 3-exponent			 to copy and set in Bravura font
--------------------------------------------------------------------------------------------------------
	pao	5  / \	80/81	    [  4 -4  1 >	U+E303					
	tao	7  f t	63/64	    [ -6  2  0  1 >	U+E305					
	vai	11 ^ v	33/32	    [ -5  1  0  0  1 >	U+E30A					
	dao	13 q d	26/27	    [  1 -3  0..0  1 >	U+E30F					
  	sanai	17 e o 	4131/4096   [-12  5  0..0  1 >	U+E342					
   	rai	19 ; r 	513/512	    [ -9  3  0..0  1 >	U+E390					
   	zai	23 ~ z 	736/729	    [  5 -6  0..0  1 >	U+E370					
   	jai	29 ? j 	261/256	    [ -8  2  0..0  1 >	U+E346					
	jpao	31 	31/32	    [ -5  0  0..0  1 >	U+E3A5					
	phrai	37 	37/36	    [ -2 -2  0..0  1 >	U+E3A0					
	mopai	41 	82/81	    [  1 -4  0..0  1 >	U+E302 U+E3F3				
    momosanai	43 	129/128	    [ -7  1  0..0  1 >	U+E342 U+E3F3 U+E3F3			
	satao	47 	47/48	    [ -4 -1  0..0  1 >	U+E39D					
	kao	53 	53/54	    [ -1 -3  0..0  1 >	U+E345					
	bodai	59 	531/512	    [ -9  2  0..0  1 >	U+E3F3 U+E30E				
	binai	61    	244/243	    [  2 -5  0..0  1 >	U+E3F2 U+E300				
		    ↑ Short ASCII representation

The symbols were obtained as follows: As we came to each new prime, we determined its Olympian symbol. Then if dropping all mina-diacritics didn't make the symbol the same as some smaller prime, the mina diacritics stayed dropped. This first applied to prime 13, then 29. The first prime for which we need to keep the mina diacritic is 41 to distinguish it from 5.

If you have the Bravura font installed you should see the symbols below.
                   
5 7 11 13 17 19 23 29 31 37 41 43  47 53  59  61

[cam.taylor]

Nice!!! This is great!!!
Cool that the single-symbol stuff has been worked out too, I probably wouldn't use it myself very much but I'm sure it will be quite helpful for some, and it has a much shorter learning curve. Awesome work!

So from a 1/1 of D, would this be correct?:

1 D
3 A
5 F
7 C
11 G
13 B
17 E
19 F
23 G
29 C
31 D
37 E
41 F
43 G
47 A
53 B
59 C
61 C

[Dave Keenan]

Yes. Absolutely correct. Thanks for that.

Making a font containing every combination of sharps and flats (and their doubles) with every combination of prime Sagittals that anyone is likely to want, is a very daunting prospect.

It's hard to understand why the developers of the various notation programs never considered that people might sometimes need more than one symbol per accidental. I understand that Lilypond allows it, with playback, but I also understand that Lilypond's user interface is not for everyone. I had hoped we'd have Steinberg's new notation software by now, which will also do this [wrong!]. But it's likely to be expensive.

So I guess there's nothing for it but for me to assemble a huge number of combinations, as single characters in a font. Finding the combination you want could be something of a nightmare. But I've been learning about Windows keyboard layouts and have purchased a brilliant program for creating them, called KbdEdit. [Edit: I'm now using WinCompose, which is even easier. See viewtopic.php?f=16&t=396.] It will let you create unlimited sequences of "chained dead keys". You may be familiar with the way dead keys are used to type letters with diacritics in some keyboard layouts. I could repurpose some rarely used key such as Insert or Scroll Lock or the right-hand Ctrl key, as a Sagittal key.

For example, this would allow us to type the sequence Sagittal 7 b Space to obtain the single character for the combination
Sagittal 7 / 5 Space for
Sagittal / 5 # Space for
Sagittal 5 7 b Space for
Sagittal 5 5 Space for
Sagittal 5 7 / Q b Space for

Typing Q for 11 comes from the idea of mapping the primes to the keyboard as follows:

 1 2 3 4 5 6 7 8 9
  Q W E R T Y U I O
   A S D F G H J K L  
    Z X C V B N M

becomes

         5   7    
 11  13      17  19
      23          29
   31          37

Note that the units digit remains constant down each slanted column, and the tens digit depends on the row.

Creating all the dead key sequences for such a keyboard layout is nearly as daunting as creating all the combination characters in the font. BTW, I hope you like the new [ kbd ] [ /kbd ] BBCode I made.

[cam.taylor]

Here are respectively a Prime-Factor-Sagittal 13-prime and 37-prime 45-limit diamond for anyone interested in what this stuff looks like.

One might get scared by the number of symbols behind each note, but one can see at a glance what the relations are to other pitches on the staff, within the chord, etc, like Johnston and other popular JI notations.

Enjoy!!!

[Dave Keenan]

Awesome work as usual, Cam. Thank you very much for these.

Could you please check to see if you have some Private Messages from me from a few days ago. Look in the grey bar near the top of the window. You may also want to turn on email notifications for these with your User Control Panel > Board Preferences > Edit Notification Options.

[Dave Keenan]

By the way, anyone can edit a copy of the Bravura font, to add whatever Sagittal combinations ("ligatures" in font jargon) you need, using the free cross-platform font-editing program called FontForge. FontForge is the work of one man, George Williams, who deserves some kind of award for it. Its graphical user interface is one of the most elegant and consistent I have ever used, and it is well documented. But even so, it does take some time to get up to speed with how fonts work.

All of SMuFL is in what is called the Unicode Private Use Area from U+E000 to U+F8FF because SMuFL has no official standing within Unicode, as yet. The Sagittals go from U+E300 to U+E41F (288 code-points, not all of them occupied). In Mus2 these appear in ranges 455 and 456. The SMuFL committee has set aside a sort of even-more-private-use area within the Private Use Area, for non-SMuFL ligatures and stylistic variations, from U+F400 to U+F8FF (1280 code-points). So that is where any Prime-Factor-Sagittal ligatures should go. The Bravura font already uses 405 of these, from U+F400 to U+F595, and may use more in future. So I suggest starting the prime-factor Sagittals at U+F5E0.

[Edit: I originally wrote: That will give us 800 code points for the following 80 combinations of primes above 3. We only need to consider primes above 3 because prime 2 is the octave, and prime 3 is taken care of by the nominals, sharps and flats. But we need 10 code-points for each combination of primes below, because each can appear with double-flat, flat, sharp and double-sharp as well as appearing alone (5 combinations) and each of these will be accompanied by its inverse.]

That will give us 800 code points. This allows for the following 80 combinations of primes above 3, and 20 more that might be found useful in future (100 combinations). We only need to consider primes above 3 because prime 2 is the octave, and prime 3 is taken care of by the nominals, sharps and flats. But we need 8 code-points for each combination of primes, because each can appear alone, subtracted from sharp, added to sharp and subtracted from double-sharp (4 combinations) and each of these will be accompanied by its inverse (8 combinations).

#rank (anchor)
A simple way of enumerating the combinations of primes, in such a way that popular combinations tend to come before less popular combinations, is to rank them according to a weighted sum of the absolute values of the exponents of the primes, where the weight for each prime-exponent is the prime itself. Or putting it another way: When the ratio is expressed as its prime factorisation (as they are below), replace all multiply and divide signs by plus-signs to obtain the rank. When several ratios have the same rank, list them in order of prime limit, and when they have the same prime limit list division before multiplication. It's remarkable how well this matches the ranking obtained from the Scala archive statistics.

Rank	Ratio
----------------
5	5
7	7
10	5x5
11	11
12	7/5
	7x5
13	13
14	7x7
15	5x5x5
16	11/5
	11x5
17	5x5/7
	5x5x7
	17
18	11/7
	11x7
	13/5
	13x5
19	7x7/5
	7x7x5
	19
20	5x5x5x5
	13/7
	13x7
21	7x7x7
	5x5/11
	5x5x11
22	5x5x5/7
	5x5x5x7
	11x11
	17/5
	17x5
23	5x7/11
	5x11/7
	7x11/5
	5x7x11
	5x5/13
	5x5x13
	23
24	7x7/5x5
	7x7x5x5
	13/11
	13x11
	17/7
	17x7
	19/5
	19x5
25	5x5x5x5x5
	5x7/13
	5x13/7
	7x13/5
	5x7x13
26	5x5x5/11
	5x5x5x11
	7x7x7/5
	7x7x7x5
	13x13
	19/7
	19x7
27	5x5x5x5/7
	5x5x5x5x7
	5x5/17
	5x5x17
28	7x7x7x7
	5x5x5/13
	5x5x5x13
	17/11
	17x11
29	29
30	5x5x5x5x5x5
	5x11/7x7
	5x7x7/11
	7x7x11/5
	5x11x7x7
	17/13
	17x13
	19/11
	19x11
31	31
37	37

The last few are incomplete in the sense that there may be other combinations with the same rank, however they are complete in the sense that all ratios which are more popular than prime 37, according to the Scala archive stats, are included.

[Dave Keenan]

Many thanks to Dave Ryan for pointing out that, in the Prime-factor Sagittal JI notation, the pythagorean part of the notation for any prime number (its nominal plus sharps or flats) can be determined simply by reducing the prime to the first octave, calculating its size in cents and looking it up in the following table of 12 pythagoreans.

Take the prime number 5 for example. Reducing it to the first octave gives 5/4. Converting it to cents is log2(5/4) * 1200 = 386.31 c. That's between the boundaries of 362.71 c and 429.47 c in the table below, so it will be notated as an altered E if 1/1 = C, or an altered B if 1/1 = G, etc.

Boundary	Notation for various			Fifths		Pythagorean
(cents)		choices of 1/1				offset		(cents)
0
		C	G	D	A		0		0
68.57
		D	A	E	B		-5		90.22
135.34
		D	A	E	B		2		203.91
272.48
		E	B	F	C		-3		294.13
362.71
		E	B	F	C		4		407.82
429.47
		F	C	G	D		-1		498.04
566.62
		F	C	G	D		6		611.73
633.38
		G	D	A	E		1		701.96
770.53
		A	E	B	F		-4		792.18 [Thanks to cmloegcmluin
837.29 										for this correction]
		A	E	B	F		3		905.87
927.52
		B	F	C	G		-2		996.09
1064.66
		B	F	C	G		5		1109.78
1131.43
		C	G	D	A		0		1200
1200

You can go further and determine the Sagittal accidental for it by subtracting the cents for the chosen pythagorean. For the prime number 5 that's 386.31 - 407.82 = -21.51 c. It's negative, so it will be the downward version of the symbol. We look up the Olympian symbol for 21.51 cents at the top of the diagram below. The symbol is so the prime 5 will be notated as E when 1/1 = C.

Click on the diagram to see it full size. For more precise boundaries between the Olympian symbols see George Secor's spreadsheet.



If the symbol has a right accent mark, you can omit it, provided the symbol is not then the same as the symbol for a lower prime.

[cmloegcmluin]

I'd like to confirm some information about the table of 12 Pythagoreans shared above. I apologize for the rushed nature of this post.

• Why stop at providing only columns for C, G, D, and A? I figure that there's nothing special about those particular nominals, and that four is just the count of examples y'all capped out at. But please let me know if the process for calculating a prime-factor-sagittal accidental would be different for other nominals. I believe I've been able to work out for myself what the proper sequences of sharped and flatted nominals should be for each assignment of a nominal to 1/1.
• I notice that every one of the examples uses a chain of 12 fifths from five steps down to six steps up. Is that a convention, for the chain of fifths to go from -5 fifths to +6 fifths? It would certainly be possible to select 12 notes by going from -8 fifths to +3 fifths instead, or -0 fifths to +11 fifths, etc. I'm new to notating JI music (I don't think this question matters much in standard tuning, where the notes are enharmonic), and I'm wondering if I select C as my 1/1 would people be surprised if I used a Gb instead of an F# (which would be the result of my chain of fifths going from -6 fifths to +5 fifths)?
• If -5 fifths to +6 fifths is indeed convention, is it considered bad form to set Fbb, Cbb, Gbb, Dbb, Abb, Cx, Gx, Dx, Ax, Ex, or Bx as your 1/1, because if you did, your -5 to +6 chain of fifths would extend beyond the space of double sharps or double flats? (Or is it more simply considered bad form to set any double sharped or double flatted nominal as your 1/1, because that's weird? Or is it even weird/bad form to set any single- sharped or flatted nominal at 1/1?)
• If any of these things are not convention in general for JI composition, are they in the Sagittal community? Or would people like them to be?
• When describing a JI scale, is there a convention around the choice of 1/1 for any reason? I feel like I often see it on either E, D, C, or A.
• After playing around with a bit, it seems like there is a benefit to choosing -5 fifths to +6 fifths over other options. In particular there is a benefit to having the number of fifths you go up by and the number you go down by be as close as possible, which maximizes the minimum of the two numbers. And the reason you want the minimum of the two numbers maximized is so that, if you set one of the nominals from the extreme ends of the cycle (i.e. F or B), you end up with as many of the other "naked" nominals selected in your set of 12 as possible ("naked" will be my word here for nominal without a sharp or flat; please let me know if there is an accepted term for this already). It is an inevitability that one of these two nominals' chain of 12 fifths will not include every naked nominal. With -5 fifths to +6 fifths as the convention, then B as 1/1 becomes our exception: since it's at the extreme upper end of the chain, then reaching only 5 fifths downwards only reaches C; in other words, a naked F does not appear in B's chain of 12 fifths, and instead it has an E#, while every other naked nominal as 1/1 includes all 7 naked nominals in its chain of 12 fifths. However if we instead used -6 fifths to +5 fifths, then we'd solve the problem for B but create the opposite problem for F: if you set F as 1/1 then you'd only reach 5 fifths upwards, so you would only reach to E; instead of a B in your chain of 12 fifths you'll have a Cb. (By the way, if the range went from -6 fifths to +5 fifths, the cents value for that position would be 588.27 instead of 611.73).
• I sought to test my understanding of this convention via the existing EHEJIPN calculator: https://www.plainsound.org/HEJI/ If you choose B as your 1/1 and enter 3/2 as your ratio, I was expecting to confirm that it gives you an E#, but instead it gives you an F with both a natural sign and a sharp sign. Is that even a thing, or is that a bug? If a bug, I'd like to report it.
• Anyway, so primarily then I'm seeking to confirm this understanding of the chain of fifths we should base prime-factor-sagittal notes on. Because it would be nice if we could assume this was common knowledge for our performers, rather than needing to be explicit about it on the page. I noticed that the JI calculator, for any input JI pitch, gives you as many options as it can, either 2 or 3 (if you imagine the seven nominals spaced out across the octave in a just diatonic scale, each of them emanating "coverage" upwards and downwards in pitch, 2 apotomes in either direction, the resultant "coverage map" results in some sections being covered by three different nominals and others only getting covered by two of them; please let me know if you'd like me to share the detailed analysis I conducted of this "apotome topology"). Anyway, the point here is, that the general, non-prime-factor-sagittal JI calculator provided by the creators of Sagittal just provides you as many options as it can, and makes no distinction between which is the "proper" sharp or flat to use in between C & D, D & E, F & G, G & A, and A & B. Perhaps it should provide that information additionally, as an extra help, just in case (I plan to augment it accordingly soon). But while that may be the case for non-prime-factor-sagittal, it seems like for prime-factor-sagittal, it's especially important to conform to this convention of going -5 to +6 fifths from your nominal that is 1/1.
• I think there is a mistake in the "Pythagorean (cents)" column above: in the row for a fifths offset of -4, the value should be 792.18, not 729.18.

I also have a question about the results in the initial post:
What nominal was used when calculating the multi-prime symbols above? This I haven't had time to delve into yet, but seems like it should matter which one you choose. For example, the symbol for 13 changes depending on what letter you use. And it might matter therefore which nominal you set as 1/1 when you're deciding on the prime-factor-sagittal set.

One last thing: in case it is of use to anyone else, I've gone through an example of following the above technique to derive the symbols for when your pitch has more than one prime in its JI composition (I couldn't simply rely on the tonality diamond shared above because both of my primes are in the same direction):

• 11: log2(33/32)*1200 = 53.27
• 13: log2(26/27)*1200 = -65.34
• 11*13=143, log2(143/128)*1200 = 191.85
• with C = 1/1 that's between 135.34 and 272.48 therefore it's a D
• D is 203.91
• so 191.85 - 203.91 = -12.06
• so it's going to be overall downward
• +53.27 + -65.34 = -12.06
• so that's easy, you just put both of the symbols on.

[Dave Keenan]

I'd like to confirm some information about the table of 12 Pythagoreans shared above. I apologize for the rushed nature of this post.

I see that you, like Blaise Pascal, did not have time to make it shorter.

• Why stop at providing only columns for C, G, D, and A? I figure that there's nothing special about those particular nominals, and that four is just the count of examples y'all capped out at. But please let me know if the process for calculating a prime-factor-sagittal accidental would be different for other nominals. I believe I've been able to work out for myself what the proper sequences of sharped and flatted nominals should be for each assignment of a nominal to 1/1.

Those four choices of nominal for 1/1 (C G D A) are indeed special. They are by far the most common choices. C gives a major heptatonic with no sharps or flats. A does the same for the natural minor, and to some it seems logical that the first letter of the alphabet should be 1/1. G gives a complete 13-limit otonality with no sharps or flats, and D gives symmetry, so that any chord and its inversion exchange sharps for flats. There is little incentive to use any other nominals for 1/1 in JI, as you can always just declare your A (or other nominal) to have a frequency other than 440 Hz, or to correspond to some other note in standard 12-equal tuning, similar to what happens with transposing instruments. Minimising accidentals is usually the name of the game.

But in the rare case where there is some reason to use a nominal other than C, G, D or A for 1/1, the process would not be any different. It would simply shift everything, in one direction or the other, along this chain:
Fbb Cbb Gbb Dbb Abb Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# Fx Cx Gx Dx Ax Ex Bx.

• I notice that every one of the examples uses a chain of 12 fifths from five steps down to six steps up. Is that a convention, for the chain of fifths to go from -5 fifths to +6 fifths? It would certainly be possible to select 12 notes by going from -8 fifths to +3 fifths instead, or -0 fifths to +11 fifths, etc. I'm new to notating JI music (I don't think this question matters much in standard tuning, where the notes are enharmonic), and I'm wondering if I select C as my 1/1 would people be surprised if I used a Gb instead of an F# (which would be the result of my chain of fifths going from -6 fifths to +5 fifths)?

We initially considered -6 to +6 but the first prime that had a single-shaft comma involving those two offsets was prime 23. It has it's smallest comma in the +6 case, not the -6. So that determined that we only needed -5 to +6.

It's not a convention to go from -5 to 6. It's not even a convention to limit it to 12 Pythagorean notes. It just seemed like a good idea to George and I. But I note that this is only the case for notating ratios that do not have any factors of 3. Factors of 3 shift us along the chain in the obvious manner, and so may take us outside of the -5 to +6 range from 1/1.

Something that I don't think has been mentioned in this thread so far, is that even the more-distant Pythagoreans in the chain, could be notated using only 12 central Pythagoreans, by using a Sagittal accidental for prime 3, namely , the accidental for the Pythagorean comma, equivalent to a shift of 12 positions on the chain — an enharmonic shift.

• If -5 fifths to +6 fifths is indeed convention, is it considered bad form to set Fbb, Cbb, Gbb, Dbb, Abb, Cx, Gx, Dx, Ax, Ex, or Bx as your 1/1, because if you did, your -5 to +6 chain of fifths would extend beyond the space of double sharps or double flats? (Or is it more simply considered bad form to set any double sharped or double flatted nominal as your 1/1, because that's weird? Or is it even weird/bad form to set any single-sharped or flatted nominal at 1/1?)

Yes. It would be weird to set any sharped or flatted nominal as 1/1. But a calculator might as well allow it if it's easy to do so. But no need to allow those double-flats or double-sharps, for the reason you gave.

• If any of these things are not convention in general for JI composition, are they in the Sagittal community? Or would people like them to be?

Like them to be a convention?

• When describing a JI scale, is there a convention around the choice of 1/1 for any reason? I feel like I often see it on either E, D, C, or A.

If I was to add a fifth column, it could be for 1/1 as E.

• After playing around with a bit, it seems like there is a benefit to choosing -5 fifths to +6 fifths over other options. In particular there is a benefit to having the number of fifths you go up by and the number you go down by be as close as possible, which maximizes the minimum of the two numbers.

Exactly our thinking.

• And the reason you want the minimum of the two numbers maximized is so that, if you set one of the nominals from the extreme ends of the cycle (i.e. F or B), you end up with as many of the other "naked" nominals selected in your set of 12 as possible ("naked" will be my word here for nominal without a sharp or flat; please let me know if there is an accepted term for this already).

"Naked" is OK. I call them "bare".

• It is an inevitability that one of these two nominals' chain of 12 fifths will not include every naked nominal. With -5 fifths to +6 fifths as the convention, then B as 1/1 becomes our exception: since it's at the extreme upper end of the chain, then reaching only 5 fifths downwards only reaches C; in other words, a naked F does not appear in B's chain of 12 fifths, and instead it has an E#, while every other naked nominal as 1/1 includes all 7 naked nominals in its chain of 12 fifths.

True. But factors of 3 may result in the other bare nominals being used.

• However if we instead used -6 fifths to +5 fifths, then we'd solve the problem for B but create the opposite problem for F: if you set F as 1/1 then you'd only reach 5 fifths upwards, so you would only reach to E; instead of a B in your chain of 12 fifths you'll have a Cb. (By the way, if the range went from -6 fifths to +5 fifths, the cents value for that position would be 588.27 instead of 611.73).

Right.

• I sought to test my understanding of this convention via the existing EHEJIPN calculator: https://www.plainsound.org/HEJI/ If you choose B as your 1/1 and enter 3/2 as your ratio, I was expecting to confirm that it gives you an E#, but instead it gives you an F with both a natural sign and a sharp sign. Is that even a thing, or is that a bug? If a bug, I'd like to report it.

With 1/1 as B, you'd need 729/512 to get E#. We expect F# for 3/2. I have no idea why it gives a natural as well as a sharp. That is a thing, in some systems, but I think it's only to cancel some other accidental that was previously applied to the same nominal. See viewtopic.php?p=464#p464.

• Anyway, so primarily then I'm seeking to confirm this understanding of the chain of fifths we should base prime-factor-sagittal notes on. Because it would be nice if we could assume this was common knowledge for our performers, rather than needing to be explicit about it on the page. I noticed that the JI calculator, for any input JI pitch, gives you as many options as it can, either 2 or 3 (if you imagine the seven nominals spaced out across the octave in a just diatonic scale, each of them emanating "coverage" upwards and downwards in pitch, 2 apotomes in either direction, the resultant "coverage map" results in some sections being covered by three different nominals and others only getting covered by two of them; please let me know if you'd like me to share the detailed analysis I conducted of this "apotome topology"). Anyway, the point here is, that the general, non-prime-factor-sagittal JI calculator provided by the creators of Sagittal just provides you as many options as it can, and makes no distinction between which is the "proper" sharp or flat to use in between C & D, D & E, F & G, G & A, and A & B. Perhaps it should provide that information additionally, as an extra help, just in case (I plan to augment it accordingly soon).

Good idea.

But while that may be the case for non-prime-factor-sagittal, it seems like for prime-factor-sagittal, it's especially important to conform to this convention of going -5 to +6 fifths from your nominal that is 1/1.[/list]

Right. But only when there are no factors of 3.

• I think there is a mistake in the "Pythagorean (cents)" column above: in the row for a fifths offset of -4, the value should be 792.18, not 729.18.

Fixed. Thanks.

I also have a question about the results in the initial post:
What nominal was used when calculating the multi-prime symbols above? This I haven't had time to delve into yet, but seems like it should matter which one you choose. For example, the symbol for 13 changes depending on what letter you use. And it might matter therefore which nominal you set as 1/1 when you're deciding on the prime-factor-sagittal set.

It doesn't matter. We're just minimising the absolute value of the offset from 1/1 along the chain of fifths. I don't understand why you say the symbol for 13 changes depending on the nominal. The point of this notation is to standardise on one symbol for one comma for each prime.

One last thing: in case it is of use to anyone else, I've gone through an example of following the above technique to derive the symbols for when your pitch has more than one prime in its JI composition (I couldn't simply rely on the tonality diamond shared above because both of my primes are in the same direction):

• 11: log2(33/32)*1200 = 53.27
• 13: log2(26/27)*1200 = -65.34
• 11*13=143, log2(143/128)*1200 = 191.85
• with C = 1/1 that's between 135.34 and 272.48 therefore it's a D
• D is 203.91
• so 191.85 - 203.91 = -12.06
• so it's going to be overall downward
• +53.27 + -65.34 = -12.06
• so that's easy, you just put both of the symbols on.

It's simpler than that. You know right from the start that you're going to put both symbols on, because that's the nature of this notation. It uses one sagittal symbol for every prime factor above 3.

• 11 = 3⁻¹
• 13 = 3³
• 11×13= 3²
• with C = 1/1, 3² is a D
• So it's D

[cmloegcmluin]

I see that you, like Blaise Pascal, did not have time to make it shorter.

Exactly... I've exposed you to dangerous levels of my writing-to-myself writing...

Those four choices of nominal for 1/1 (C G D A) are indeed special. They are by far the most common choices. C gives a major heptatonic with no sharps or flats. A does the same for the natural minor, and to some it seems logical that the first letter of the alphabet should be 1/1. G gives a complete 13-limit otonality with no sharps or flats, and D gives symmetry, so that any chord and its inversion exchange sharps for flats. There is little incentive to use any other nominals for 1/1 in JI, as you can always just declare your A (or other nominal) to have a frequency other than 440 Hz, or to correspond to some other note in standard 12-equal tuning, similar to what happens with transposing instruments. Minimising accidentals is usually the name of the game.

This chunk is a godsend for me. That's so cool that it works out that way for G (I worked it out and I assume what you mean is that none of 3/2, 5/4, 7/4, 9/8, 11/8, or 13/8 require accidentals when G is 1/1, but you wouldn't necessarily get that for other nominals as 1/1... you don't get it for the couple others I checked). Where can I learn more of this? I'm embarrassed to say I haven't read Doty's Just Intonation Primer yet.

I've heard about setting A to something else than 440 Hz, but usually it's pretty close to 440 Hz. Is the choice of letter really just all about the accidentals? Meaning I could set it to A if that gave the nicest accidentals, but if I actually wanted my piece to be in a different register than that I could just say that A is something crazy like 311.13 Hz, and most performers would be okay with that?

Something that I don't think has been mentioned in this thread so far, is that even the more-distant Pythagoreans in the chain, could be notated using only 12 central Pythagoreans, by using a Sagittal accidental for prime 3, namely , the accidental for the Pythagorean comma, equivalent to a shift of 12 positions on the chain — an enharmonic shift.

Cool idea. Why not just keep adding conventional sharp signs though? Wouldn't it be fairly intuitive to indicate a triple sharp like ? I feel like I'd rather see that than myself.

Yes. It would be weird to set any sharped or flatted nominal as 1/1. But a calculator might as well allow it if it's easy to do so. But no need to allow those double-flats or double-sharps, for the reason you gave.

The Sagittal JI calculator spreadsheet does allow setting the double-flats and double-sharps as 1/1, and seems to work alright. Is that something I should fix then, when I do the work to have it help you out with choosing the most conventional options it suggests? Or do you think I may as well leave it that way?

It doesn't matter. We're just minimising the absolute value of the offset from 1/1 along the chain of fifths. I don't understand why you say the symbol for 13 changes depending on the nominal. The point of this notation is to standardise on one symbol for one comma for each prime.

Sorry, I should have given an example of what I mean. is the prime-factor-sagittal for 13. If I go to the Sagittal JI Calculator and put in 13, leaving C as the default 1/1, indeed I get as my result (on an A, and technically it's but let's just ignore the diacritics for now). G and D as 1/1 also agree with . But if you use A as your 1/1, is not one of the options; the only option with a bare nominal is F, and the accidental is . E and B also give this result. (F agrees with C, G, and D). So it's split roughly evenly, 4 against 3. Was preferred because 4 > 3? Or is there something deeper I'm not getting yet?

I definitely don't understand the statement "We're just minimising the absolute value of the offset from 1/1 along the chain of fifths".

It's not a convention to go from -5 to 6. It's not even a convention to limit it to 12 Pythagorean notes. It just seemed like a good idea to George and I. But I note that this is only the case for notating ratios that do not have any factors of 3. Factors of 3 shift us along the chain in the obvious manner, and so may take us outside of the -5 to +6 range from 1/1.

True. But factors of 3 may result in the other bare nominals being used.

Right. But only when there are no factors of 3.

It's simpler than that. You know right from the start that you're going to put both symbols on, because that's the nature of this notation. It uses one sagittal symbol for every prime factor above 3.
11 = 3⁻¹
13 = 3³
11×13= 3²
with C = 1/1, 3² is a D
So it's D

So... I do think there may be some fundamental inner workings of Sagittal (and extended H-E systems in general) that I don't quite intuit yet. I'll try to take you through my experience so far. Keep in mind that I don't have a strong musical background -- I've been writing music as long as I can remember, but I've never mastered an instrument or studied it academically.

• Ah, interesting. Commatic alterations. That makes sense. Because each of the primes has a different deviation from standard tuning.
• Okay, each comma can have a bunch of powers of 2 in it. Fine, I understand octave equivalency.
• ...Wait, what...? These commas have a bunch of powers of 3 in them, too! I don't like that. I almost never use fifths. They also aren't pitch class equivalent, so now I'll be limited by that nature. And I'll probably have to memorize the circle of fifths, etc...
• (Ignores JI notation systems for many years, focusing on writing music for computers that don't need to deal with all this nonsense)
• (Decides he'd like to have some of his music performed by humans, starts trying to figure out JI notation systems again)
• I still don't really understand the powers of 3. They seem to magically work out a lot of the time

Part of the reason I included that snippet of me working out 11*13, if only semi-consciously, is that I expected you might say something like this: that I should have just known that right off the bat. But I don't think I get yet why those powers of 3 just work themselves out. I still feel like I have to do it manually. And that's why I'm so uncomfortable with the idea of anything other than -5 to +6 fifths and just following instructions someone who knows what they're doing has told me will work. Either that or I really do just need to learn this by working out enough examples until things click (or by implementing it in code, although I'd rather work out the bugs in my understanding before I try programming them... there'll be room enough for bugs later in just the implementation errors!).

I'm trying to figure out all the thoughts behind the sentences: "It's not a convention to go from -5 to 6. It's not even a convention to limit it to 12 Pythagorean notes. It just seemed like a good idea to George and I." The chart above seems to have been provided by a guy named Dave Ryan. Did I miss somewhere where this concept of -5 to +6 fifths is codified for Sagittal in general? It seems like it's only relevant to figuring out these prime-factor sagittals. Otherwise you're free to use whichever sharped and flatted nominals you want. My sense is that these prime-factor-sagittal might only "work out" with respect to the count of 3's in their monzos as long as you followed this exact -5 to +6 chain.

Where are you getting these counts of three? 3⁻¹ for 11, 3³ for 13? They don't seem to correspond to the monzos for or , which have powers of 1 and -5, respectively, not -1 and 3.

Again, I apologize for Just Intonation 101 level questions.