Magic Notation (and Orwell)

Here are my thoughts on notating magic with 9 nominals.

The basis is an augmented triad. In magic, that's an MOS. In any temperament that tempers out 225:224 (which is a lot of them) you can write such a chord so that two of the thirds approximate 5:4 and the other one approximates 9:7.

The staff has three lines and each adjacent pair of lines are an approximate 5:4 apart. That means the distance between the top line of one staff and the bottom line of the next is an approximate 9:7. The distance between the staves helps you remember that this interval is slightly larger than the others.

The fourth note in a chain of magic thirds falls short of an octave by a small amount. Let's call this a "wand". A wand is the difference between an approximate 9:7 and 5:4, so it stands in for 36:35. It also stands in for a 25:24 in magic.

To extend the system we need a symbol for shifts by a wand. I'll use ^ for up and v for down today. Maybe this can be the sagittal quartertone symbol as used for 2 degrees of 41, although that does make it look a little small.

The next stage is to place a note on each space above and below a line, and make the difference between an adjacent line and space two wands. That means there are two spaces between each pair of lines, similar to Wilson's 12 note notation. The logic behind making each step 2 wands is that to get 9 nominals we need to divide each major third into 3 parts. In 19-equal, there are exactly 6 steps (equal to 6 wands) to a major third.

I'm numbering the staff positions from 1 to 9. So the steps in different magic ETs are:

1 2 3 4 5 6 7 8 9 1
2 2 2 2 2 2 2 2 3 /19
2 2 3 2 2 3 2 2 4 /22
4 4 5 4 4 5 4 4 7 /41

Notes 2, 5, and 8 are on a line. The step between staffs is the largest step in the nominals. The steps between a line and a space are smaller than the steps between adjacent spaces. If you prevent adjacent spaces from overlapping, the distance between notes on the staff reinforces the difference between their intervals.

To distinguish the two adjacent spaces, you can add a dotted line that's not a staff position. The result will be something like:

9
---------------------8---------------------------
7
- - - - - - - - - - - - - - - - -
6
------------5------------------------------------
4
- - - - - - - - - - - - - - - - -
3
---2--------------------------------------------
1

(Use a monospaced font to view that properly.) A lazy or "thrifty" composer may use standard manuscript paper and remember that two of the lines have a different status to the other three.

One feature of this system is that it has lots of enharmonic equivalences. Most of the steps between nominals are 2 wands and the accidentals are 1 wand. You could view this as being confusing or adding flexibility depending on taste.

Here's a 19 note MOS as an ascending scale that doesn't require naturals (except for the octave):

1 1^ 2 2^ 3 3^ 4 4^ 5 5^ 6 6^ 7 7^ 8 8^ 9 9^ 1v

So we only need two accidental symbols, and one of those only for one note. Because of the equivalences, we still need an extra symbol (w for 2 wands down) for the 22 note MOS (and a natural symbol, so consider it implied).

1 1^ 2 2^ 3 3^ 4v 4 4^ 5 5^ 6 6^ 7v 7 7^ 8 8^ 9 9^ 1w 1v

Some similar nominals for orwell (where there is a 9 note MOS) are

1 2 3 4 5 6 7 8 9 1
2 2 3 2 3 2 3 2 3 /22
3 3 4 3 4 3 4 3 4 /31
4 4 5 4 5 4 5 4 5 /53

This means note 6 is a degree of 22 lower than its counterpart on the magic staff and notes 8 and 9 are a degree higher. It looks like 5:4 major thirds are consistent between the two systems but 3:2 fifths can be different. 7-limit intervals follow 225:224 being tempered out in both cases.

Ennealimmal works a bit differently. The 9 nominals are equally spaced and the staff should probably make them look so. A 4 line staff would do this well. Also, 225:224 isn't tempered out.

Graham