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

[cmloegcmluin]

In case this is handy, here are the primes thus far addressed, with their size in cents, and then the order they should appear in if based on this size.

5    -21.506   10
7    -27.264    9
11    53.273    4
13   -64.915    1
17    14.730   13
19     3.378   16
23    16.544   12
29    33.148    7
31   -54.528    3
37    46.394    5
41    21.084   11
43    13.795   14
47   -35.697    6
53   -31.767    8
59    64.519    2
61     6.479   15

[Dave Keenan]

I did mean to link to that HEWM page. Search on <+ within the page (the only example).

I have a Sagittal songbook. But I was thinking we needed to show symbols on staves more often in introductory and other explanatory material.

Thanks. I'll look into the CSS webfont thing when I've finished renewing my teaching quals.

I agree that sharps and flats should not be separated from their nominals by a space. And nor should a sagittal symbol when it is the only accidental.

[cmloegcmluin]

I have a Sagittal songbook. But I was thinking we needed to show symbols on staves more often in introductory and other explanatory material.

Will do.

Thanks. I'll look into the CSS webfont thing when I've finished renewing my teaching quals.

Good luck!

I agree that sharps and flats should not be separated from their nominals by a space. And nor should a sagittal symbol when it is the only accidental.

My suggestion was that the space should be there even when the sagittal is the only accidental, for consistency,
e.g. ~|( E instead of ~|(E.
But I suppose this is just coming down to subtle matters of style. We probably don't need to make a recommendation on this matter.

[Dave Keenan]

I assumed this was about what to output from the calculator rather than what to recommend to others. I don't have a strong feeling either way about that case. But let's be clear that we're only talking about including any spaces, when using the long ASCII form of pitches in prime factor notation. Long ASCII for monosagittal JI notations doesn't need any spaces within a pitch, so why have them when the prime factor notation happens to be the same as the mono? It seems simplest to use spaces within a pitch, only to separate sagittals (when in long ASCII form).

[cmloegcmluin]

Ha, I wasn't thinking about that, but yes, whatever the calculator spits out will certainly influence the community

You've got another good point: while I'd only been considering consistency within prime factor notation thus far, it's important to consider consistency across all Sagittal notations for the simplest cases.

From the first page of the Xenharmonikon article:

...the authors worked together to expand and refine these ideas into a notation system that would be both versatile and powerful, but for which the required complexity would not make it more difficult to do the simpler things.

This has stuck with me as an important principle to uphold. In this case, it convinces me to agree with you.

[Dave Keenan]

In case this is handy, here are the primes thus far addressed, with their size in cents, and then the order they should appear in if based on this size.

5    -21.506   10
7    -27.264    9
11    53.273    4
13   -64.915    1
17    14.730   13
19     3.378   16
23    16.544   12
...

When ordering prime-factor accidentals so that the largest alteration is closest to the notehead, it is convenient that, as you add primes up-to-and-including prime 19, no new prime symbol comes in between any two previous prime symbols. From 5 to 13 each new prime goes to the right of all previous, and from 17 to 19 each new prime goes to the left of all previous. And this ordering remains the same if the 17-kleisma replaces the 17-comma. This can be diagrammed as:

19 17 5 7 11 13 Notehead

Only with the addition of prime 23, which almost no-one uses, do these adjacency relationships change.

[cmloegcmluin]

Excellent observation. That should definitely be part of a Prime Factor notation how-to.

[Dave Keenan]

I have have edited the tables in the first post of this topic,
viewtopic.php?p=263#p263
to give the code points for the new left-side mina accents, as in the upcoming SMuFL 1.4 release.

Thanks to Douglas Blumeyer and James Ingram for reporting the problem.

[cmloegcmluin]

@Dave Keenan I noticed that in the first post of this topic, the diagram is different than the one that appears on the home page. For some reason the mnemonics and actual symbols are flipped. That's fine, just curious. Except that the mnemonic for pao is... well, just a pao, not the Roman numeral mnemonic. If you prefer the vectorized version I put together for the upcoming tutorial video, I can snapshot that instead.

Also, I thought that maybe until we have a full-fledged interactive calculator ready for this notation — like we have for the precision level notation, in spreadsheet form — we might want to add a note explaining what to do with the bolded 3-exponents. I had totally forgotten about those, and also the bolding isn't exactly enough to help them jump off the page atcha to remind you they're there.

[cmloegcmluin]

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 is one of those things that I have a lot of trouble getting my head around and retaining. Can anyone think of a neat way of visually conveying this information? Or know of one which already exists?