Notation of n-tet?

I'm new to this list, but I've been playing with alternative tunings for a
while. I started in the mid-80's, experimenting with 19-tet and 1/4-comma
meantone. (Here's a bit of trivia: when I wrote the AdLib sound drivers for
the games Ultima VI and the PC version of Times of Lore, I set the tuning
tables to play in 1/6-comma meantone, with up to seven sharps or flats.)
Although I haven't written much music in the last 10 years, I've recently
started writing again, and some of the more interesting results are up on
my web page (http://www.io.com/~hmiller/music/music.html).

I've been exploring different equal scales (in particular 15- and 22-tet),
different kinds of well-tempered tuning, and just intonation, but I didn't
even consider exploring some of the more exotic equal tunings such as 16-
and 20-tet until I heard Easley Blackwood's CD of microtonal etudes in all
the equal tunings from 13 to 24. I did try 13-tet once just to see if it
was really as bad as Wendy Carlos hinted on _Secrets of Synthesis_, and it
turned out to be not quite that bad if you can manage to avoid the
dissonant fifths. But now I'm interested in exploring all of these scales
for their exotic musical properties, at least up to 31-tet, although 41-tet
also looks nice, and even 37-tet looks somewhat interesting. So far, I've
written brief "experiments" in 16-tet and 26-tet (on my web page as MIDI
files), and I'm in the middle of writing one in 20-tet.

The one problem I've been having is that I can't figure out how to notate
all these exotic scales in a way that makes sense musically. The meantone
scales (12, 19, 26, 31, 38, 43, ...) are easy enough. The scales that
resemble just intonation in having two different sizes of whole steps (15,
22, 27, 29, 34, 39, 41, ...) are a little more complicated, but only an
extra pair of accidentals is needed to represent the difference between a
major third and two large whole steps (or four fifths up / two octaves
down).

But that still leaves quite a few scales that don't fit easily into one of
these two categories. 17-tet is intermediate between these categories in
some ways, and it isn't really a meantone scale, although that seems to be
the best way to notate it. 11-tet seems especially hard to notate, although
it's not as interesting as some of the other tunings. I eventually figured
out that I can notate 16-tet by *adding* a step to the major thirds, rather
than subtracting a step as in 15-tet. It's bizarre, but it works.

The scale that I'm especially interested in right now, 20-tet, is resisting
my efforts to notate it. It has a very flat "major" third (the closest
approximation to 5/4), which in conjunction with the very sharp fifth
sounds more like a neutral third than a major one. The next interval in the
scale is a very sharp third, close to 14/11 (33.7 cents sharp of 5/4), and
somewhat dissonant, but to me it sounds more like a major third in the
context of a 0-7-12 triad compared with the "neutral" 0-6-12 triad. I'd
also like to be able to recognize the familiar sounding diminished seventh
chords and the 5-tet pentatonic scale without too much difficulty.

If you have Real Audio (at least 3.0), you can hear a sample of music using
this scale (about 300K): http://www.io.com/~hmiller/music/20test.ra

Has anyone come up with a good system of notation for this or other equal
scales?

I would be curious too which notation Blackwood used for 20-tET, perhaps
Paul Hahn can find it somewhere.
Because four cycles of fifths (with only five notes) are needed to
cover the scale, any notation that uses the seven diatonic names will
be inconsistent. The Pythagorean semitone and whole tone are both 4
steps. A possible approach is to use only the first five names,
starting at F: F-C-G-D-A and not use E and B. If one can live with the
high A. Then use Fokker's accidentals the semisharp, sesquisharp,
semiflat and sesquiflat for 1, 3, -1 and -3 steps, and the regular
sharp and flat for 2 and -2 steps. This gives:
0: 1/1 C
1: 60.000 cents C| Db,
2: 120.000 cents C# Db
3: 180.000 cents C#| D,
4: 240.000 cents D
5: 300.000 cents D| Fb,
6: 360.000 cents D# Fb
7: 420.000 cents D#| F,
8: 480.000 cents F
9: 540.000 cents F| Gb,
10: 600.000 cents F# Gb
11: 660.000 cents F#| G,
12: 720.000 cents G
13: 780.000 cents G| Ab,
14: 840.000 cents G# Ab
15: 900.000 cents G#| A,
16: 960.000 cents A
17: 1020.000 cents A| Cb,
18: 1080.000 cents A# Cb
19: 1140.000 cents A#| C,
20: 2/1 C

There is another notation by Zweifel for ET's of the form n(n-1), so
12 (4x3), 20 (5x4), 30 (6x5), etc. which he calls generalised diatonic
scales. This is the one for 20-tET:
0: 1/1 C
1: 60.000 cents C# Db
2: 120.000 cents D
3: 180.000 cents D# Eb
4: 240.000 cents E
5: 300.000 cents E# Fb
6: 360.000 cents F
7: 420.000 cents F# Gb
8: 480.000 cents G Hb
9: 540.000 cents H G#
10: 600.000 cents H# Ib
11: 660.000 cents I
12: 720.000 cents I# Jb
13: 780.000 cents J
14: 840.000 cents J# Kb
15: 900.000 cents K
16: 960.000 cents K# Ab
17: 1020.000 cents A Bb
18: 1080.000 cents B A#
19: 1140.000 cents B# Cb
20: 2/1 C

Zweifel, Paul F. "Generalized Diatonic and Pentatonic Scales: A
Group-Theoretic Approach", _Perspectives of New Music_ vol. 39 no. 1,
1996, pp. 140-161.
Paul Rapoport has done much work on notation of equal temperaments.
The best articles to start with are
Rapoport, Paul. "The Notation of Equal Temperaments", 1991,
_Xenharmonikon_ vol. 16, 1995, pp. 61-84.
Rapoport, Paul. "The Structural Relationships of Fifths and Thirds in
Equal Temperaments", _Journal of Music Theory_ vol. 37 no. 2, pp.
351-389, fall 1993.
Rapoport, Paul. "(K)no(w) More Notation", _1/1_ vol. 9 no. 2, 1995, p. 4.

Manuel Op de Coul coul@ezh.nl

On Wed, 13 Jan 1999 11:59:58 +0100, manuel.op.de.coul@ezh.nl wrote:

> The Pythagorean semitone and whole tone are both 4
>steps. A possible approach is to use only the first five names,
>starting at F: F-C-G-D-A and not use E and B. If one can live with the
>high A. Then use Fokker's accidentals the semisharp, sesquisharp,
>semiflat and sesquiflat for 1, 3, -1 and -3 steps, and the regular
>sharp and flat for 2 and -2 steps. This gives:

I think with a few modifications this might satisfy my criteria. First, if
I allow E to be synonymous with F, and B with C, the diminished seventh
chord on C would be C - Eb, - F# - A, (or with the notation that I've been
using, C - Ebh - F# - Ah: "h" for half-flat and "z" for half-sharp). F| (or
Fz) is close to 11/8 in this system, which is the same way that I've been
notating 11/8 in 17-tet, 24-tet, and 31-tet. 7/4 could be written as Bbb.

I don't really like the C - Ebh for the minor third, but I could get used
to it. Using C - Eh for a wide major third is a little too strange, though.
Perhaps I could use the up and down arrows that I use for 15-tet and 22-tet
instead of the semi-flats and sharps.

Just for fun, here are my systems for notating some other equal scales:

15-tet: C Db^ Dv D Eb^ Ev F Gb^ F#v G Ab^ Av Bb Bb^ Bv C
16-tet: C C#>> Db< D> Eb< E> Fb<< F F#>> G G#>> Ab< A> Bb< B> Cb<< C
17-tet: C Cz C# D Eb Eh E F Fz F# G Ab Ah A Bb Bh B C
19-tet: C C# Db D D# Eb E Fb F F# Gb G G# Ab A A# Bb B Cb C
22-tet: C C#vv Db^ Dv D D#vv Eb^ Ev E F F#vv F#v Gb^^ G G#vv Ab^ Av A Bb
Bb^ Bv Cb^^ C
26-tet: C C# Cx Db D D# Ebb Eb E E# Fb F F# Fx Gb G G# Gx Ab A A# Bbb Bb B
B# Cb C

(Note that in my 16-tet notation, sharps and flats seem to work in reverse!
But this allows a consistent notation of the thirds, fourths, and fifths.)

Herman Miller wrote:

> Just for fun, here are my systems for notating some other equal scales:
> 15-tet: C Db^ Dv D Eb^ Ev F Gb^ F#v G Ab^ Av Bb Bb^ Bv C

This is almost the same as Blackwood's. He uses a kind of up and down
arrow. Only he takes F^ instead of Gb^.

> 16-tet: C C#>> Db< D> Eb< E> Fb<< F F#>> G G#>> Ab< A> Bb< B> Cb<< C

Blackwood's notation for 16-tET is
C C# Db D D# Dx E F F# Fx G G# A A# Bb B C
He didn't try to make it consistent and only mentioned that it worked
well for him.

> 17-tet: C Cz C# D Eb Eh E F Fz F# G Ab Ah A Bb Bh B C

This can be notated without special symbols. The apotome is 2 steps:
C Db C# D Eb D# E F Gb F# G Ab G# A Bb A# B C

> 22-tet: C C#vv Db^ Dv D D#vv Eb^ Ev E F F#vv F#v Gb^^ G G#vv Ab^ Av A Bb
> Bb^ Bv Cb^^ C

Can also be notated without special symbols like
C Db Ebb C# D Eb Cx D# E F Gb E# etc.
But one might prefer yours, or use the comma which is one step in 22-tET,
notation upwards:
C C/ C// C# D D/ D// D# E F F/ etc. and downwards:
C Db D\\ D\ D Eb E\\ E\ E F Gb etc.
The diesis is also one step in 22-tET. Rapoport's symbols are
) diesis up, ( diesis down.
Anyway I try to avoid combining accidentals which go in different
directions as much as possible. Only for higher ETs it may serve brevity.

Manuel Op de Coul coul@ezh.nl

It's also well worth looking at Erv Wilson's presentation of keyboards and
notations in the first three issues of Xenharmonikon.

Wilson prefers, in some systems, a set of twelve nominals, in fifth order:
Gamma - Delta - Alpha - Epsilon - Beta - F - C - G - D - A - E - B. In most
cases, with added accidentals, these can be mapped down to a set of seven
nominals. Likewise, while Wilson prefers a twelve nominal staff with
alternating groups of two and three lines,

C
B
Beta ------------------------------------------------
A
Alpha------------------------------------------------
G
Gamma------------------------------------------------
F
E
Epsilon----------------------------------------------
D
Delta------------------------------------------------
C
B

-- which has been previously proposed as a keyboard notation, and is quite
easy to read at a conventional keyboard -- these can also be mapped down to
the conventional staff.

Hi Herman et al,

You might want to check this out :
<https://www.mindeartheart.org/micro.html>

Quoting from Ted Mook's site :

MICRO 2� is a Postscript(c) font designed for the 1/12th-tone notation
system developed by Ezra Sims for his own music and now taught in the
microtone classes of New England Conservatory. It was created using
Altsys Fontographer, and is available (upon request) for Mac,
DOS (Windows), SUN and Next platforms. In addition, the font set
contains other commonly used microtonal diacritical marks
(backwards flat, Tartini sharps), including a set of symbols
used by the composer Mathew Rosenblum in his hybrid 19-note scale.
I have also included two alternate sets of 1/12-tones symbols with
varying degrees of sleekness, which might work better in certain
kinds of densely voiced situations.

Salut,
Drew

On Fri, 15 Jan 1999 14:41:11 +0100, manuel.op.de.coul@ezh.nl wrote:

>Herman Miller wrote:
>
>> Just for fun, here are my systems for notating some other equal scales:
>> 15-tet: C Db^ Dv D Eb^ Ev F Gb^ F#v G Ab^ Av Bb Bb^ Bv C
>
>This is almost the same as Blackwood's. He uses a kind of up and down
>arrow. Only he takes F^ instead of Gb^.

If it's part of a Bb^ triad or a D minor triad, I'd write it as F^. But
spelling it as Gb^ makes a nice 5x3 array of notes:

Dv Av Ev Bv F#v
Bb F C G D
Gb^ Db^ Ab^ Eb^ Bb^

>> 22-tet: C C#vv Db^ Dv D D#vv Eb^ Ev E F F#vv F#v Gb^^ G G#vv Ab^ Av A Bb
>> Bb^ Bv Cb^^ C
>
>Can also be notated without special symbols like
>C Db Ebb C# D Eb Cx D# E F Gb E# etc.
>But one might prefer yours, or use the comma which is one step in 22-tET,
>notation upwards:
>C C/ C// C# D D/ D// D# E F F/ etc. and downwards:
>C Db D\\ D\ D Eb E\\ E\ E F Gb etc.
>The diesis is also one step in 22-tET. Rapoport's symbols are
>) diesis up, ( diesis down.

That's a useful symbol to have, essentially a substitute for my #vv and
b^^. Then I could use )^ and (v in place of #v and b^ for the chromatic
semitones. It's an extra pair of symbols, but it's less cumbersome to use.

On Fri, 15 Jan 1999 14:41:11 +0100, manuel.op.de.coul@ezh.nl wrote:

>Blackwood's notation for 16-tET is
>C C# Db D D# Dx E F F# Fx G G# A A# Bb B C
>He didn't try to make it consistent and only mentioned that it worked
>well for him.

I tried this with the first two bars of my 16-tet piece, which in my
current notation looks like this (letting "b" stand for "b<<"):

G< F< G<------ A Bb C< Bb A | Bb Ab G<------ A G< A G< Eb
Eb D Eb------ F< G< A F Eb | G< F<<Eb D F<<D Eb--- C< Bb
Bb A G<------ Eb D C< | Bb C< Bb--- C< A Bb G< A G<
C<--- C<--- C<--- Db--- D---- | Eb--- Eb--- Eb--- Bb--- C<---

In Blackwood's notation it comes out like this, which is a bit easier to
read (mainly because C< minor is an awkward key for my current notation):

F# E F#------ G# A B A G# | A G F#------ G# F# G# F# D
D Cx D------- E F# G# F D | F# Eb D Cx Eb Cx D---- B A
A G# F#------ D Cx B | A B A---- B G# A F# G# F#
B---- B---- B---- C#--- Cx--- | D---- D---- D---- A---- B----

In fact, it's similar to my original 16-tet notation, which I abandoned
mainly because the usage of sharps and flats was inconsistent:

C C# Dh D D# E Ez F F# G Gz G# A A# Bh B C
(using h for the half-flat symbol and z for the half-sharp symbol)

Another system I've been considering is the following:

C C# Db D Eb E E# F F# G G# Ab A A B Cb C

The only problems with this system are that the note a fifth above G is Db
instead of D, a fifth above B is F instead of F#, and the interval B - D is
a major third instead of a minor third. But it's easier to read and write
than my current system. I had a similar system for 15-tet:

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

Again, it's the G - D, B - F#, and B - D intervals that have unusual
spellings. If this kind of notation works well for other scales that I
haven't tried yet, I might go back to it, at least for the scales that are
awkward in my current notation.

Herman Miller wrote:

> From: hmiller@io.com (Herman Miller)
>
> On Fri, 15 Jan 1999 14:41:11 +0100, manuel.op.de.coul@ezh.nl wrote:
>
> >Blackwood's notation for 16-tET is
> >C C# Db D D# Dx E F F# Fx G G# A A# Bb B C
> >He didn't try to make it consistent and only mentioned that it worked
> >well for him.
>
> I tried this with the first two bars of my 16-tet piece, which in my
> current notation looks like this (letting "b" stand for "b<<"):
>
> G< F< G<------ A Bb C< Bb A | Bb Ab G<------ A G< A G< Eb
> Eb D Eb------ F< G< A F Eb | G< F<<Eb D F<<D Eb--- C< Bb
> Bb A G<------ Eb D C< | Bb C< Bb--- C< A Bb G< A G<
> C<--- C<--- C<--- Db--- D---- | Eb--- Eb--- Eb--- Bb--- C<---
>
> In Blackwood's notation it comes out like this, which is a bit easier to
> read (mainly because C< minor is an awkward key for my current notation):
>
> F# E F#------ G# A B A G# | A G F#------ G# F# G# F# D
> D Cx D------- E F# G# F D | F# Eb D Cx Eb Cx D---- B A
> A G# F#------ D Cx B | A B A---- B G# A F# G# F#
> B---- B---- B---- C#--- Cx--- | D---- D---- D---- A---- B----
>
> In fact, it's similar to my original 16-tet notation, which I abandoned
> mainly because the usage of sharps and flats was inconsistent:
>
> C C# Dh D D# E Ez F F# G Gz G# A A# Bh B C
> (using h for the half-flat symbol and z for the half-sharp symbol)
>
> Another system I've been considering is the following:
>
> C C# Db D Eb E E# F F# G G# Ab A A B Cb C
>
> The only problems with this system are that the note a fifth above G is Db
> instead of D, a fifth above B is F instead of F#, and the interval B - D is
> a major third instead of a minor third. But it's easier to read and write
> than my current system. I had a similar system for 15-tet:
>
> C C# D D# Eb E F F# Gb G Ab A A# Bb B C
>
> Again, it's the G - D, B - F#, and B - D intervals that have unusual
> spellings. If this kind of notation works well for other scales that I
> haven't tried yet, I might go back to it, at least for the scales that are
> awkward in my current notation.

I have a 16 tone scale based on the Chopi in S.E. Africa. It is not an equal
scale but is close using higher harmonics. It was formed by just doing simple
modulations until it formed a cycle. The closest scale you would have to this
tuning would occur if you superimposed 7 units of your tuning. You would end
up with an MOS at 5,7,9,&16. On the keyboard I have for this tuning I have a
conventional keyboard that alternates 3 blacks and 4 blacks instead of 2 and
3. One person remarked that it made them feel like they are seeing double!
I've been using F F# Gb G G# Ab A Bb B C C# Db D D# Eb E. Which has problems
too but thought you might like the MOS subset!
-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com