Re: 15EDO

If you want to obtain a notation that corresponds in some way to our seven-line stave then the best approach is to determine the generators of 15EDO:

0 2 4 6 8 10 12 14 1 3 5 7 9 11 13 (0... [2]
0 4 8 12 1 5 9 13 2 6 10 14 3 7 11 (0... [4]

[I wish there was an easy way to draw ascii circles]

Now, choose a segment of the generator that contains as many notes as possible but not, when rearranged, contains groups of three adjacent pitch classes:

0 2 4 6 8 10 12 14
or
0 4 8 12

Next study the transposition of this group by the generator:

0 2 4 6 8 10 12 14 ---> 2 4 6 8 10 12 14 1
0 4 8 12 ---> 4 8 12 1

As can be seen one pitch is replaced by another, maximally adjacent, above it : 0 goes to 1

Keep repeating this until a cycle of 'keys' is formed.

Now, write the first key (the one starting with pitch-class 0) on a 'stave' such that each ascending note takes a line or a space:

0 ----------------(o----- This is the 'octave' above the lower 0 line.
14 o
12-------------o---------
10 o
8 ---------o-------------
6 o
4 -----o-----------------
2 o
0 -o---------------------

The same scale beginning on pitch-class 2 contains pitch-class 1, which is notated as

2 o
0 ----------------#o----- This is the 'octave' above the lower 0 line.
14 o
12-------------o---------
10 o
8 ---------o-------------
6 o
4 -----o-----------------
2 o
0 ----------------------

Thus you have a sequence of sharps in one direction and (when transposing downwards by the generator, 2) flats.
Obviously, 15EDO behaves like 12EDO in that 0# = 1 = 2b (if you'll pardon the orthography). This means that letter names can be used
A B C D E F G H, with A=0, say, so that the key of B is B C D E F G H A# B, and the key of Hb is: Hb A B C D E F G H A.
You will notice that the scale steps are as follows:

A 2 B 2 C 2 D 2 E 2 F 2 G 2 H 1 A.

For those of you more into scale theory, we could substitute 3 for 2 and 2 for 1 to end up with the 23EDO analogue to 15EDO (like 19EDO is to 12EDO):

A 3 B 3 C 3 D 3 E 3 F 3 G 3 H 2 A. Which of course would permit 'split keys' of a keyboard tuned in 23EDO. The next EDO up would be 38EDO: A 5 B 5 C 5 D 5 E 5 F 5 G 5 H 3 A.

As for the 0 4 8 12 segment, it is best seen as a subset of the 0 2 4 etc. segment.

I suppose this little bit of scale theory puts me in the notation per scale camp, rather than having a generalised notation, such as decimal or altered stave notation. I choose to devise a notation for each scale, with #s and bs as I think (IMHO) it leads to a consistent notation.

Mark

> Mark

I just wanted to acknowledge Mark for his fine contributions to this
thread and hope that next year, and those beyond, will lead to still
deeper comminglings between his ideas and those of the rest of
us . . .

-Paul