Notation for 22-tET etc. (was: 9-limit triangular lattices)

Durn it! I still got it wrong. Just change all the / and // to \ and \\ in
my last post. I dunno about anyone else but I pronounce these things as /
"up" and \ "down" and // "double-up" etc.

So here's a symmetric decatonic as chains of fifths.

A\ E\ B\ F#\ C#\
Eb Bb F C G

Here we extend it both ways to encompass all of 22-tET.

C\ G\ D\ A\ E\ B\ F#\ C#\ G#\ D#\ A#\
Gb Db Ab Eb Bb F C G D A E

Now we rearrange them in pitch order.

C Db C#\ D\ D Eb D#\ E\ E F Gb
F#\ G\ G Ab G#\ A\ A Bb A#\ B\ B

We see that the "tones" (A:B, C:D etc.) are 4 degrees, the "semitones"
(B:C, E:F) are 1 degree. A sharp or flat corresponds to a shift of 3
degrees, and \ means lower (and / means raise) by 1 degree.

These definitions let us keep the familiar note names for the fifths. The
4:7's come out ok as minor 7ths (two stacked fourths). But the \ and /
modifiers are introduced here to let us keep the familiar note names for
the thirds.

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Note that all the "useful" N-tETs can be expressed as N = 5t + 2s where t
and s are whole numbers and s <= t. t and s then correspond to the size of
the "tone" and "semitone" in scale degrees, and t-s the size of a sharp or
flat. These are just the ETs with usable fifths (but sometimes barely).

Heres the "periodic table".
t
1| 5 7
2| 10 12 14
3| 15 17 19 21
4| 20 22 24 26 28
5| 25 27 29 31 33 35
6| 30 32 34 36 38 40 42
7| 35 37 39 41 43 45 47 49
8| 40 42 44 46 48 50 52 54 56
9| 45 47 49 51 53 55 57 59 61 63
10| 50 52 54 56 58 60 62 64 66 68 70
11| 55 57 59 61 63 65 67 69 71 73 75 77
12| 60 62 64 66 68 70 72 74 76 78 80 82 84
13| 65 67 69 71 73 75 77 79 81 83 85 87 89 91
14| 70 72 74 76 78 80 82 84 86 88 90 91 92 94 96
...
--------------------------------------------
s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

If an ET appears more than once it means it has more than one fifth size in
the range. The one nearest the middle of a row is the best.

If you want more reasonable fifths you can restrict it to cases where s is
between t/9 and 6t/7 inclusive, when it looks like this:

t
1|
2| 12
3| 17 19
4| 22 24 26
5| 27 29 31 33
6| 32 34 36 38 40
7| 37 39 41 43 45 47
8| 42 44 46 48 50 52 (47 is marginal)
9| 47 49 51 53 55 57 59
10| 54 56 58 60 62 64 66
11| 59 61 63 65 67 69 71 73
12| 64 66 68 70 72 74 76 78 80
13| 69 71 73 75 77 79 81 83 85 87
14| 74 76 78 80 82 84 86 88 90 92 94
...
--------------------------------------------
s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Note that _every_ ET greater than or equal to 36 appears at least once.
Also note that some ETs in this table are still not "useful" because they
have such lousy thirds, e.g. 17-tET, 33-tET.

Regards,
-- Dave Keenan
http://dkeenan.com