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.

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

Symbol Name Prime Comma SMuFL code points Characters Inverted characters

to copy/paste and set in the Bravura font

---------------------------------------------------------------------------------------------------------

pao 5 80/81 U+E303

tao 7 63/64 U+E305

vai 11 33/32 U+E30A

dao 13 26/27 U+E30F

sanai 17 4131/4096 U+E342

rai 19 513/512 U+E390

zai 23 736/729 U+E370

jai 29 261/256 U+E346

jpao 31 31/32 U+E3A5

phrai 37 37/36 U+E3A0

paimo 41 82/81 U+E302 U+E3F3

sanaibo 43 129/128 U+E342 U+E3F3 U+E3F3

satao 47 47/48 U+E39D

kao 53 53/54 U+E345

odai 59 531/512 U+E3F3 U+E30E

inai 61 244/243 U+E3F2 U+E300

The symbols were obtained as follows: As we came to each new prime, we determined its Olympian symbol. Then if dropping all right-accents didn't make the symbol the same as some smaller prime, the right accents stayed dropped. This first applied to prime 13, then 29. The first prime for which we need to

*keep*the right-accent 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