Note names in 22-tET

The discussion about note names in 22-tET led me to design another
notation system for it. The notation with diatonic names is not ideal
because for example the major third and seventh are very out of tune.
The new notation is based on Paul Erlich's decatonic scale
Standard Pentachordal Major (2 2 3 2 2 2 3 2 2 2).
The 10 natural names were chosen C H D E F J G A K B.
This scale does not consist of a single chain of fifths but a double
one. This introduces one inconsistency with respect to the traditional
names which is that D - A is not a 22-tET fifth (13 steps) but a
22-tET fifth flat by a syntonic comma (12 steps). So the two
fifth-chains are K F C G D - A E B J H
or, in a 2-dimensional lattice

A E B J H
K F C G D

This one inconsistency seems preferable above substituting A, E and B
(or C, G and D) with other letters. This way, C D E F G A B C is still
the best approximation to the major scale in 22-tET.

0: 1/1 C Hbb
1: 54.545 cents C# Hb
2: 109.091 cents H Dbb
3: 163.636 cents H# Db
4: 218.182 cents D
5: 272.727 cents D# Ebb
6: 327.273 cents Dx Eb
7: 381.818 cents E Fbb
8: 436.364 cents E# Fb
9: 490.909 cents F Jbb
10: 545.455 cents F# Jb
11: 600.000 cents J Gbb
12: 654.545 cents J# Gb
13: 709.091 cents G
14: 763.636 cents G# Abb
15: 818.182 cents Gx Ab
16: 872.727 cents A Kbb
17: 927.273 cents A# Kb
18: 981.818 cents K Bbb
19: 1036.364 cents K# Bb
20: 1090.909 cents B Cbb
21: 1145.455 cents B# Cb
22: 2/1 C Hbb

So the meaning of # and b is one scale step, and not the Pythagorean
apotome, which is a very extreme three steps in 22-tET.
The chords 4:5:6:7 in this scale are C-E-G-K, J-K-H-E and G-B-D-F.
An effect of the double fifth chain is that modulating by a fifth
does not introduce one sharp, but two:
C H# D# E F J G A K B C.
And by a fourth it gives two flats:
C H D E F J G Ab Kb B C.
For a moment I thought about choosing Fokker's semi-augmented and
semi-diminished signs instead of the normal sharp and flat since one
step is about a quarter tone, but discarded this possibility because
the minor third is better notated with C-Eb.
Paul's Standard Pentachordal Minor mode is notated as
C H D Eb F J G Ab Kb Bb C.

Manuel Op de Coul coul@ezh.nl

Manuel wrote,

>The new notation is based on Paul Erlich's decatonic scale
>Standard Pentachordal Major (2 2 3 2 2 2 3 2 2 2).

That's how I approached the notation problem in 1993, but decided shortly
thereafter that the Symmetrical decatonic scale would be a better basis,
since it makes all the key signatures easier (see last page of my paper) and
allows the black keys and white keys on a standard keyboard to function as
the "naturals" and "accidentals" respectively when a keyboard mapping that
omits key "B" is used.

Your system gives a Ben Johnston-style C major scale (D-A bad, as you note),
but at least that gives it some point of reference with "traditional"
notation.

On Fri, 11 Feb 2000 19:23:48 +0100, <manuel.op.de.coul@ezh.nl> wrote:

>The new notation is based on Paul Erlich's decatonic scale
>Standard Pentachordal Major (2 2 3 2 2 2 3 2 2 2).
>The 10 natural names were chosen C H D E F J G A K B.
>This scale does not consist of a single chain of fifths but a double
>one. This introduces one inconsistency with respect to the traditional
>names which is that D - A is not a 22-tET fifth (13 steps) but a
>22-tET fifth flat by a syntonic comma (12 steps). So the two
>fifth-chains are K F C G D - A E B J H
>or, in a 2-dimensional lattice
>
> A E B J H
> K F C G D

How about using the German names for B and H?

A E H J K
B F C G D

--
see my music page ---> +--<http://www.io.com/~hmiller/music/music.html>--
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Herman Miller:
> How about using the German names for B and H?

> A E H J K
> B F C G D

Nice idea. A small disadvantage is that the interleaved
alphabetical order is then lost.

Paul:
> That's how I approached the notation problem in 1993, but decided shortly
> thereafter that the Symmetrical decatonic scale would be a better basis,
> since it makes all the key signatures easier (see last page of my paper) and
> allows the black keys and white keys on a standard keyboard to function as
> the "naturals" and "accidentals" respectively when a keyboard mapping that
> omits key "B" is used.

So that's the Dynamic Symmetrical Major then (2 3 2 2 2 2 3 2 2 2).
There is no "D" so we need another letter, which is the price to pay to
avoid the bad D-A. If we use the German B and H, and reshuffle J, K and L into
alphabetical order we get
C J K E F L G A B H C.
In lattice form

A E H L J
K B F C G

Manuel Op de Coul coul@ezh.nl

I wrote,

>> That's how I approached the notation problem in 1993, but decided shortly
>> thereafter that the Symmetrical decatonic scale would be a better basis,
>> since it makes all the key signatures easier (see last page of my paper)
and
>> allows the black keys and white keys on a standard keyboard to function
as
>> the "naturals" and "accidentals" respectively when a keyboard mapping
that
>> omits key "B" is used.

Manuel wrote,

>So that's the Dynamic Symmetrical Major then (2 3 2 2 2 2 3 2 2 2).

That's what you get, starting on key "C#", if you omit "B" from the
keyboard. If you omit "E", you get 2 2 2 2 3 2 2 2 2 3, which seems a bit
easier to remember, especially if your keyboard is one that begins and ends
on "C".

>There is no "D" so we need another letter, which is the price to pay to
>avoid the bad D-A. If we use the German B and H, and reshuffle J, K and L
into
>alphabetical order we get
>C J K E F L G A B H C.

That's hard to remember. In the 1994 version of my paper, I omit "B" from
the keyboard and use

R S T U V W X Y P Q R

and in the 1998 version I omit "E" from the keyboard and use

1 2 3 4 5 6 7 8 9 0 1

though the latter system has made for some confusion in trying to
communicate with other musicians in a semi-improvisational context since
numbers are then used for both absolute pitch and relative scale degree.

Manuel, have you read my paper? I'm surprised you're just picking this up
now.

Also, the 22-tone "well-tempered" system I describe in my paper, assigned to
the standard keyboard (omitting "E" again) allows decatonic material using
mostly black keys, and Indian scales using mostly white keys. Sruti 0 falls
on "D". In fact, every other "E" can be put back in the mapping in order to
distinguish sruti 2 of ma-grama, 135/128 (or 45/32 relative to the ma-grama
tonic), from sruti 2 of sa-grama, 16/15 (which falls on "F").

Manuel wrote,

>In lattice form

> A E H L J
>K B F C G

In the 1994 version of my paper, I had a table of key signatures like the
central column of the one in the 1998 version. I labelled the key signature
with no accidentals (i.e., all black keys) "V or Q major; P or U minor". At
that time I was thinking of the Static Symmetrical decatonic scales as the
"natural" ones. The 5-limit lattice would run:

...
Y U Q W S
T P V R X
Y U Q W S
T P V R X
...

and a portion of the 7-limit lattice showing the 8 consonant tetrads:

T---------P---------V
/|\`. ,'/|\`. ,'/
/ | \ R / | \ X /
/ Y---------U---------Q
/,'/|\`.\|/,'/|\`.\|/,'/
Q--/-|-\--W--/-|-\--S /
/ T \ | / P \ | /
/,' `.\|/,' `.\|/
V---------R---------X

Paul wrote:
>Manuel, have you read my paper? I'm surprised you're just picking this up
>now.

Oh yes, I have the PDF file of the 1998 version but you use a number
system in it. That you presented a letter notation in the 1994 version
I must have forgotten completely. Enfin, thanks for the comments. Still
I think a mixture of traditional and new letters is a bit easier to
remember than a whole set of new letters but it may depend.

Manuel Op de Coul coul@ezh.nl