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

Posted:

**Thu May 12, 2016 4:23 pm**[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 Xenharmonikôn 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:

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

Notice that the offset from 1/1 for each prime is the negative of the (bolded) 3-exponents for its comma above. That's because the power of 3 in the nominal plus sharp or flat must cancel out the power of 3 in the comma. Of course, no comma symbols are required for harmonics 1, 3 and 9.

For a different 1/1 nominal, just copy, paste and add or delete spaces to slide the bottom two rows right or left.

You will only see the symbols in the first two rows below if you have the Bravura or Bravura Text font installed on your computer. But the mina accents on the symbols for primes 41 and 43 will not display correctly until you have a version of Bravura or Bravura Text that implements SMuFL 1.4 or later (expected to be released in early 2021).

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 Xenharmonikôn 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 Accent Core to copy and set in Bravura font --------------------------------------------------------------------------------------------------------- pao 5 / \ 80/81 [ 4-41 ⟩ U+E303 tao 7 f t 63/64 [ -620 1 ⟩ U+E305 vai 11 ^ v 33/32 [ -510 0 1 ⟩ U+E30A dao 13 q d 26/27 [ 1-30..0 1 ⟩ U+E30F sanai 17 e o 4131/4096 [-1250..0 1 ⟩ U+E342 rai 19 ; r 513/512 [ -930..0 1 ⟩ U+E390 zai 23 ~ z 736/729 [ 5-60..0 1 ⟩ U+E370 jai 29 ? j 261/256 [ -820..0 1 ⟩ U+E346 jpao 31 31/32 [ -500..0 1 ⟩ U+E3A5 phrai 37 37/36 [ -2-20..0 1 ⟩ U+E3A0 mopai 41 82/81 [ 1-40..0 1 ⟩ U+E3F5 U+E302 momosanai 43 129/128 [ -710..0 1 ⟩ U+E3F7 U+E342 bijanao 47 47/48 [ -4-10..0 1 ⟩ U+E3F2 U+E349 or if a recent change to the Olympian boundaries for mina 75 was to be reversed: satao 47 47/48 [ -4-10..0 1 ⟩ U+E39D kao 53 53/54 [ -1-30..0 1 ⟩ U+E345 bodai 59 531/512 [ -920..0 1 ⟩ U+E3F3 U+E30E binai 61 244/243 [ 2-50..0 1 ⟩ U+E3F2 U+E300 ↑ Short ASCII representation

The symbols were obtained as follows: As we came to each new prime, we determined the Olympian symbol for its comma. Then if dropping all mina-accents didn't make the symbol the same as some smaller prime, the mina accents 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 1/5.**Edit (Mar 2021):**The following chain-of-fifths slide-rule shows, for 1/1 = G, which nominal plus sharp or flat to use along with the above Sagittals, for the first 23 odd harmonics:FbbCbbGbbDbbAbbEbbBbbFb 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 |--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--| offsets from 1/1 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 odd harmonics 17 19 7 11 1 3 9 13 5 15 23 21

Notice that the offset from 1/1 for each prime is the negative of the (bolded) 3-exponents for its comma above. That's because the power of 3 in the nominal plus sharp or flat must cancel out the power of 3 in the comma. Of course, no comma symbols are required for harmonics 1, 3 and 9.

For a different 1/1 nominal, just copy, paste and add or delete spaces to slide the bottom two rows right or left.

You will only see the symbols in the first two rows below if you have the Bravura or Bravura Text font installed on your computer. But the mina accents on the symbols for primes 41 and 43 will not display correctly until you have a version of Bravura or Bravura Text that implements SMuFL 1.4 or later (expected to be released in early 2021).