Re: 22 tet notation

Alison wrote

> >I have read all I can on the net about
> >22 - tet and could find nothing on notation.

> Paul replied

> There've been many proposals, including mine, which I take it you did find
> on the net?

> Yes, sorry,but I found that a bit difficult to grasp. I'll try again though.

> >So I simply made a six line staff with your idea of the 1 3 5 7 9 1
> vertically
> >up the side. I take this to be the 'default' Standard Pentachordal
> >Major.
>
> Not quite -- it's a symmetrical, rather than pentachordal, decatonic mode.

I have put up a temporary page with the notation at
http://homepages.which.net/~alison.monteith3 Forgive me for not explaining
myself clearly. I take the 1 to 9 up the side to refer to the steps of the
Pentachordal Major. So line one is step 0 or 22, space one is step 2, etc. This
then is the default, no sharps no flats scale.

> >Maybe I'll just use the lowest 'e' on guitar
> >for now, though this is a problem that will need an answer if a
> conventional
> >notational system is adopted.
>
> The problem solved itself quite naturally to me -- since 1-5 and 6-0 are the
> only 9:7 major thirds in the "natural" scale, and since I use a guitar tuned
> in a 22-tET tuning very close standard tuning except the major third between
> the second and third strings is the 9:7 major third, my guitar strings are
> either tuned, low to high, 9-3-7-1-5-9, or 4-8-2-6-0-4, it doesn't matter
> which, all the key signatures and everything come out the same with either
> choice.

I'll need to look back to your paper to follow your numbers here, but I get the
gist. Out of lazy habit I've been tuning to a Boss tuner which I suppose gives
accurate cent measurements but have found the intonation to need adjustment.
Perhaps tuning strings to each other would be better.
The tonic question, however, still seems to me to be unresolved. All that this
notational system does is to tell an instrumentalist to press this key, or to
fret this note. If we had a 22- tet band with guitar, keyboard and tuned
percussion, say marimba , wouldn't we need to find a common our "zero" note in
order to play a piece together?

I always think of middle 'c' as the zero note of the 'default' scale in 12 -
tet because of the conventional piano. Poor brass and wind players have to
transpose. Going back to our fictitious band. If I was the guitarist I would
probably want to choose an open string, probably 'e', as the zero note,
certainly not the 'c' as it lies on the guitar. The tuned percussionist might
conceivably want the lowest note on the instrument. The digital keyboard
player would have no problem, but I'm imagining a fixed pitch harpsichord,
organ or piano that once tuned stays that way for a while. Agreement needs to
be be reached to avoid asking players to transpose at will.
Please correct me if I'm wrong, but at some point, were 22-tet to become
incredibly popular or considered for adoption as a new standard (who knows),
then an actual frequency would need to be chosen as our zero note.

I also haven't worked out the key signatures. I suppose some will be hybrid
sharp/flat signatures.

Best Wishes

I have put up a temporary page with the notation at
http://homepages.which.net/~alison.monteith3

Alison, I don't understand this:

>The flats and sharps are chosen according to the proximity of the pitch to
the natural. For >example, note 0 sharp is closer in cent values to note 0
than to note 1, hence 0 sharp rather than >1 flat.

Since we're talking about an equal temperament, 0 sharp is exactly the same
distance in cents from note 0 and note 1 (in your notation). Right?

Alison,

There is a reason for the numbering system I use and the fact that I use the
symmetrical instead of the pentachordal scale for the naturals. It comes
from the keyboard mapping I use, which is described in the paper. The "E"s
on the keyboard are left out of the mapping, and the black keys, starting
with c#, are numbered 1, 2, 3, . . . 0. The sharps and flats fall on the
white keys. The key signatures are also a lot less bizarre when the naturals
represent a symmetrical rather than a pentachordal scale. I tune the open
strings of the guitar to the black keys as described before. The notation is
perfectly consistent and middle C is taken to be the same on the remapped
keyboard as in standard 12-tET tuning. It's eminently logical and the guitar
is virtually tuned in standard pitch!

-Paul

"Paul H. Erlich" wrote:

> Alison,
>
> There is a reason for the numbering system I use and the fact that I use the
> symmetrical instead of the pentachordal scale for the naturals. It comes
> from the keyboard mapping I use, which is described in the paper. The "E"s
> on the keyboard are left out of the mapping, and the black keys, starting
> with c#, are numbered 1, 2, 3, . . . 0. The sharps and flats fall on the
> white keys. The key signatures are also a lot less bizarre when the naturals
> represent a symmetrical rather than a pentachordal scale. I tune the open
> strings of the guitar to the black keys as described before. The notation is
> perfectly consistent and middle C is taken to be the same on the remapped
> keyboard as in standard 12-tET tuning. It's eminently logical and the guitar
> is virtually tuned in standard pitch!
>
> -Paul

Thanks again for explaining this to me. The only valid test of any notational system is
whether it is practical and readable. I will get back to composing some studies and will try
to notate them using both systems for comparison. I'm not particularly attached to my
proposal, it just seems simple and logical and easy for the guitar on the face of it so I
thought I'd share it. Your system stems from the keyboard but as you say works well for the
guitar. I'm still not sure about the business of middle C. Finally, are there ane existing
scores written using your proposed system? If so I'd be grateful for the links.

I'd also be interested to hear your thoughts and anyone else's on the issue of a new tuning
standard as discussed in Patrick Ozzard-Low's paper on 21st century acoustic instruments.
(Excuse me if this has been discussed before my time on the list.) The argument for the
adoption of a new tuning standard seems to imply that the large symphony orchestra has a
future as a JI/Microtonal institution. The new music seems to me to be essentially written
for electronics, Partch type ensembles, and adapted individual instruments. I think this is
an important topic because in my experience composing for the symphony orchestra is still seen
as the pinnacle of achievement in many composition courses. I have absolutely no interest in
this as my interests lie in JI/Microtonal composition. Non-12 compositions for the orchestra
are pretty thin on the ground. Perhaps, as Sethares and others suggest the, modern orchestra
is timbrally optimised for 12-tet . So will I ever get to write JI/Microtonal music for
'major forces' and is this even valid? Or will we see 'schools' throughout the world with
their own instrument ensembles and tuning and notational systems? This would mean that
despite the ability to share ideas around the world virtually in real time we are moving away
from uniformity towards fragmentation. Fascinating.

--- In tuning@egroups.com, Alison Monteith wrote:
> http://www.egroups.com/message/tuning/13634>
>
> I'd also be interested to hear your thoughts and anyone else's
> on the issue of a new tuning standard as discussed in Patrick
> Ozzard-Low's paper on 21st century acoustic instruments.

Hello Alison. I'll give you my thoughts.

> (Excuse me if this has been discussed before my time on the
> list.)

No, no excuse... go back and check the archives.

(...relax... just kidding)

> The argument for the adoption of a new tuning standard seems
> to imply that the large symphony orchestra has a future as a
> JI/Microtonal institution. The new music seems to me to be
> essentially written for electronics, Partch type ensembles,
> and adapted individual instruments. I think this is an
> important topic because in my experience composing for the
> symphony orchestra is still seen as the pinnacle of achievement
> in many composition courses. I have absolutely no interest in
> this as my interests lie in JI/Microtonal composition.

Good for you! Follow your inclinations.

I hadn't really thought about it much, because I love symphonic
music so much and really wanted to be a symphonic composer when
I was young.... aw, OK, I *still* want to be one... anyway,
this got me thinking, and it really does seem that the symphony
orchestra is yet another institution that's outlived its
usefulness to a large extent.

> Non-12 compositions for the orchestra are pretty thin on the
> ground.

Yes, I think I'd have to agree with you there. Johnny Reinhard
turned me on to a fantastic microtonal piece for full orchestra:
_Trace_, by Manfred Stahnke (sp?). It's worth looking for.

> Perhaps, as Sethares and others suggest, the modern orchestra
> is timbrally optimised for 12-tet.

Don't forget also that the objective for instrument-builders
in Europe throughout the 1800s was 'perfect intonation',
meaning as close to exact 12-tET as possible.

So there's even more support for that suggestion; it's not
just timbral - it's to a great extent built into the
instrument's physical structure. I know this applies to
woodwinds, anyway. (Of course, good performers like Reinhard
can get any pitch they want out of their instrument, but
we're talking about 'optimisation'.)

> So will I ever get to write JI/Microtonal music for
> 'major forces' and is this even valid? Or will we see
> 'schools' throughout the world with their own instrument
> ensembles and tuning and notational systems?

We're already seeing a *lot* of this. I know personally of
two centers of microtonal music, San Diego and New York,
that have many disparate 'schools' of microtonal music in
one city!

And as Ozzard-Low will attest from what he saw in his global
travels while studying this, there are plenty of 'composers
seduced into carpentry' (and electronics and computers) all
over the world, building their own instruments and in some
cases (like Partch) whole ensembles. Bill Wesley is a
composer here in San Diego who has built an entire orchestra
of incredibly imaginative instruments.

> This would mean that despite the ability to share ideas
> around the world virtually in real time we are moving away
> from uniformity towards fragmentation. Fascinating.

I think this is very close to a post I just made the other day
about how this list is both bringing people together to collaborate
on projects and theoretical developments, and at the same time
fragmenting the microtonal community into diverse new factions
who go off and form separate lists with more restricted (but
more focussed) interests.

-monz
http://www.ixpres.com/interval/monzo/homepage.html

--- In tuning@egroups.com, Alison Monteith <alison.monteith3@w...>
http://www.egroups.com/message/tuning/13634

> I'd also be interested to hear your thoughts and anyone else's on
the issue of a new tuning standard as discussed in Patrick
Ozzard-Low's paper on 21st century acoustic instruments. (Excuse me
if this has been discussed before my time on the list.) The argument
for the adoption of a new tuning standard seems to imply that the
large symphony orchestra has a future as a JI/Microtonal institution.
The new music seems to me to be essentially written for
electronics,
Partch type ensembles, and adapted individual instruments. I think
this is an important topic because in my experience composing for the
symphony orchestra is still seen as the pinnacle of achievement in
many composition courses. I have absolutely no interest in this as my
interests lie in JI/Microtonal composition. Non-12 compositions for
the orchestra are pretty thin on the ground. Perhaps, as Sethares and
others suggest the, modern orchestra is timbrally optimised for
12-tet.
So will I ever get to write JI/Microtonal music for 'major forces'
and is this even valid? Or will we see 'schools' throughout the world
with their own instrument ensembles and tuning and notational
systems?

Our own AFMM leader, Johnny Reinhard, has conducted and produced (and
encouraged the composition of!) MANY microtonal works for full
orchestra in New York. I think Johnny has pretty much proven that
the large symphony orchestra is really no different that a small
ensemble in terms of the ability to produce microtonality... given
enough rehearsal time and the positive attitude of the musicians...
And, as for microtonal works for orchestra... there are oodles of
them! I'm sure Johnny could rattle off 20 from the top of his head,
without looking anything up!

The question, I believe, is more the PRACTICALITY of producing the
symphony orchestra for ANY new works, microtonal or 12-tET. Johnny
is a Hercules in this respect, but that's what it takes... since our
culture is not fostering... through grants anymore (NEA used to do a
little of it) the production of new pieces by the symphony
orchestra... The American Composers' Orchestra and the Manhattan
School of Music orchestras are about the only two that consistently
produce new works...

Insofar as considering the symphony orchestra the "ultimate" in
expression... well... that's "old hat." In fact, that attitude went
"out the window" with most professional composers in the 1960s...
although mostly due to the fact that new music became complex, and
more difficult to produce, so orchestras didn't want to devote the
rehearsal time to it. The orchestra is coming back in vogue, though,
with the "New Romanticists"... Most of the music is pretty "yucky,"
in my opinion, but the orchestra is coming back in some quarters.

As a related aside: I'm attending the Lou Harrison piano concerto by
the ACO in a couple of weeks... I don't know about anybody else...
____________ ___ __ __ _
Joseph Pehrson

Alison wrote,

>Finally, are there ane existing
>scores written using your proposed system?

Someday I'll get around to it -- the Decatonic Waltz will make a reasonable
notation project.

>I'd also be interested to hear your thoughts and anyone else's on the issue
of a new tuning
>standard as discussed in Patrick Ozzard-Low's paper on 21st century
acoustic instruments.

I've thrown around the idea of 31-tET as a "standard alternate tuning".

>Perhaps, as Sethares and others suggest the, modern orchestra
>is timbrally optimised for 12-tet.

But even more so for 31-tET! 12 beat out 31 in the 17th - early 18th century
only because of the convenience of 12 -- the sound of 31 was acknowledged as
better.

Joseph wrote

>
> Insofar as considering the symphony orchestra the "ultimate" in
> expression... well... that's "old hat." In fact, that attitude went
> "out the window" with most professional composers in the 1960s...

Not over here in the microtonal desert (the just desert?) of Edinburgh. Witness the rise of
James McMillan. Now, I think his music is quite lovely and well orchestrated, though not
memorable, but it is still seen as the pinnacle of achievement because it is for the biggest
and loudest and crowd-pulling-est ensemble and because bits sound like Shostakovitch.

Having said that I feel that the study of orchestration is essential. We will always need to
learn how to put timbres together and to be aware of registral differences and voicings.
Stravinsky wouldn't have been the man he was if it wasn't for the thousands of hours he put in
studying orchestration with the best. For new young composers, however, it is still the study
of the orchestration of instruments suited to 12-tet composition. If the JI/microtonal
community is moving away from standardised, conventional ensembles, new young composers who
wish to study JI composition and orchestration for the 'new' ensembles will be sent to the
'old' textbooks in the absence of an alternative. So will they initiate change and progress or
will they reckon that the years spent studying conventional orchestration ought not to be
wasted? This is what I feel Mr. Harrington was driving at in his frustration. Some of us just
don't have the energy, but I'm pleased to hear you tell me that others have.

In order to be able to score out pieces in 22 tet using conventional
notation I've tried to come up with a practical 5-line staff solution.
There's nothing fancy or groundbreaking herebut I would welcome
comments, suggested amendments and so on towards a working solution. The
up and down arrows might be replaced with half sharps and flats; this
would certainly be easier in my notation apllication, Sibelius. Thanks
in anticipation.

Best Wishes.

> From: Alison Monteith <alison.monteith3@which.net>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, July 07, 2001 4:15 AM
> Subject: [tuning] 22 tet notation
>
>
> In order to be able to score out pieces in 22 tet using conventional
> notation I've tried to come up with a practical 5-line staff solution.
> There's nothing fancy or groundbreaking herebut I would welcome
> comments, suggested amendments and so on towards a working solution. The
> up and down arrows might be replaced with half sharps and flats; this
> would certainly be easier in my notation apllication, Sibelius. Thanks
> in anticipation.

Hi Alison,

Your use of the words "conventional notation" leads me to believe
that you've already read Paul Erlich's paper "Tuning, Tonality,
and Twenty-Two-Tone Equal Temperament": (requires Adobe Acrobat Reader)
http://www-math.cudenver.edu/~jstarret/22ALL.pdf

But if you haven't, there it is. :)

The last page gives Paul's "decatonic" notation and key signatures.

On your page,
http://homepages.which.net/~alison.monteith3/22notation.gif

the ^ symbol appears *before* the noteheads on C, E, and F, and
*under* the noteheads on Eb, Ab, A, Bb, and B. Is there any
significance to this difference? (or did you just find yourself
having to cram all those notes in?)

I've made a pitch-height graph illustration your notation,
Alison, and put it at:
/tuning/files/monz/22edo-notation.gif

Feel free to add it to your webpage if you wish.

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

monz wrote:

>
> Hi Alison,
>
> Your use of the words "conventional notation" leads me to believe
> that you've already read Paul Erlich's paper "Tuning, Tonality,
> and Twenty-Two-Tone Equal Temperament": (requires Adobe Acrobat Reader)
> http://www-math.cudenver.edu/~jstarret/22ALL.pdf
>
> But if you haven't, there it is. :)
>
> The last page gives Paul's "decatonic" notation and key signatures.

Yes, thanks, I've read Paul's excellent paper and it has helped me a lot.

>
> On your page,
> http://homepages.which.net/~alison.monteith3/22notation.gif
>
> the ^ symbol appears *before* the noteheads on C, E, and F, and
> *under* the noteheads on Eb, Ab, A, Bb, and B. Is there any
> significance to this difference? (or did you just find yourself
> having to cram all those notes in?)

No significance. As I mentioned elsewhere, I might use half sharps and flats as Sibelius always
formats these properly.

> I've made a pitch-height graph illustration your notation,
> Alison, and put it at:
> /tuning/files/monz/22edo-notation.gif
>
> Feel free to add it to your webpage if you wish.

Many thanks. I'll get on the case.

Best Wishes

>

Alison Monteith wrote:

>In order to be able to score out pieces in 22 tet using conventional
>notation I've tried to come up with a practical 5-line staff solution.
>There's nothing fancy or groundbreaking herebut I would welcome
>comments, suggested amendments and so on towards a working solution. The
>up and down arrows might be replaced with half sharps and flats; this
>would certainly be easier in my notation apllication, Sibelius. Thanks
>in anticipation.
>
>Oh, and here's where to find the suggested notation.
>
>http://homepages.which.net/~alison.monteith3/22notation.gif

Hi Alison,

You've run into the basic problem of trying to notate a tuning where four
fifths does not give you a major third, using a diatonic-based notation.
There are two mutually incompatible systems that I know of, which I will
call Ben Johnston's and Adriaan Fokker's until someone corrects me.
Apparently you started off heading for Fokker's and then changed towards
Johnston's and ended up somewhere in between.

You implied that you wanted it to be based on 12-tET with half sharps and
flats, i.e. 24-EDO, so I've also given a scheme that uses the nearest such
24-EDO name when both systems (22 and 24) are centered on the C:G fifth
(i.e. centered on your implied lattice as shown below). Maybe this is what
you were aiming for.

I'll put my cards on the table right from the start: I prefer Fokker's
system. But I'll try to be as unbiased as I can in the following.

By the way, when I earlier suggested using "^" and "v" as accidentals in a
diaschismic notation, I was being facetious. These are already used to mean
so many different things it isn't funny. Below I'll use the original
author's + and - in the Johnston notation and / and \ in the Fokker and ^
and v in yours, but you can of course change them to whatever you like.

22-EDO and 34-EDO are both diaschismic, so the notations I gave previously,
on lattices, for 34-EDO, are just as valid for 22-EDO. These were all in
the Fokker style.

Fokker's is based on a Pythagorean scale, Johnston's on a JI-major scale.
Fokker's names a major third from C as E\ (as you did originally) while
Johnston's names it E (as you do currently). A minor third from C is named
Eb/ in Fokkers (as in yours) but Eb in Johnston's.

Fokker's can name all fifths and thirds in a uniform manner (all fifths are
the same as usual, all major thirds are lowered by \ from the usual, all
minor thirds are raised by / from the usual). Any system that gives you the
familiar A E B as major thirds, or Ab, Eb, Bb as minor thirds, from F C G,
must pay for it by having non-uniform naming elsewhere in the lattice. e.g.
A perfect fifth from D is A+ in Johnston's.

Of course none of these will give you a single fixed notation that will be
suitable for all uses of 22-EDO. These are notation _systems_. The
particular choice of note names will depend on what "key" a piece is in,
i.e. what part of the lattice it is centered on, and whether the
appropriate lattice is 5-limit or 7-limit. I've made the following
assumption about how to shape and center the (5-limit) lattice, based on
your proposed names:

12 3 16 7 20 11 2
14 5 18 9 0 13 4 17 8 21 22-EDO degrees
2 15 6 19 10 1

G\ D\ A\ E\ B\ F#\
Ab Eb Bb F C G D A E B Fokker Pythagorean based
Db/ Ab/ Eb/ Bb/ F/ C/

Gv Dv A E B F#
Ab Eb Bb F C G D A^ E^ B^ Proposed
Db Ab^ Eb^ Bb^ F^ C^

Gv Dv Av E B F#
Abv Ebv Bb F C G D A E^ B^ 12/24-tET based, C:G
centered
Db Ab Eb Bb^ F^ C^

G- D- A E B F#+
Ab- Eb- Bb- F C G D A+ E+ B+ Johnston JI-major based

Db- Ab Eb Bb F+ C+

Here we compare them in pitch order. I've put asterisks above those that
the different systems disagree on.

* * * * *
0 1 2 3 4 5 6 7 8 9 10 22-EDO degrees
C C/ Db/ D\ D Eb Eb/ E\ E F F/ Fokker Pythagorean based
C C^ Db Dv D Eb Eb^ E E^ F F^ Proposed
C C^ Db Dv D Ebv Eb E E^ F F^ 12/24-EDO based, C:G centers
aligned
C C+ Db- D- D Eb- Eb E E+ F F+ Johnston JI-major based

* * * * * * * * *
11 12 13 14 15 16 17 18 19 20 21 22-EDO degrees
F#\ G\ G Ab Ab/ A\ A Bb Bb/ B\ B Fokker Pythagorean based
F# Gv G Ab Ab^ A A^ Bb Bb^ B B^ Proposed
F# Gv G Abv Ab Av A Bb Bb^ B B^ 12/24-EDO based, C:G centers
aligned
F#+ G- G Ab- Ab A A+ Bb- Bb B B+ Johnston JI-major based

All the above systems except Fokker's, have the problem that they describe
centered universes, while 22-EDO of course has no fixed center. (To be fair
to Johnston, his notation is intended for JI, not temperaments.) While your
center agrees with theirs, things are relatively familiar. Once your center
moves away from theirs, it starts to get very messy very fast. With
Fokker's system you have to get used to a small unfamiliarity right from
the start, that major and minor thirds (and of course sixths) have
accidentals, but this remains constant no matter where your center moves to.

Also don't forget, if we pad the lattices out with duplicate notes (in
parenthesis) so we can see all the 5-limit intervals available in 22-EDO
(i.e. to show how the lattice wraps around), we get:

(6)(19)(10)(1) (14)(5) (18)
(8) (21) 12 3 16 7 20 11 2 (15)(6)
(1) 14 5 18 9 0 13 4 17 8 21 (12) 22-EDO degrees
(7) (20)(11) 2 15 6 19 10 1 (14)(5)
(17)(8) (21)(12)(3) (16)(7)

If you start using any intervals with the parenthesised notes you would
effectively be changing keys, and you might want to use a different name
for the parenthesised note, compared to how it is named near the center of
the lattice (unparenthesised) (e.g. exchanging a sharp for a flat or vice
versa). I can tell you how this works for Fokker's and Johnston's systems,
but not your proposal or the 12/24-based one.

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

Hello, there, everyone, and thank you, Alison and Herman, for sharing
notation systems showing some of the approaches to 22-tET.

As it happens, my own approach ties in with a "regularized" keyboard
designed for a neo-Gothic style, with regular whole-tones of four
steps and diatonic semitones of one step. Regular major thirds are
~436.36 cents, and minor thirds ~272.73 cents -- close to 9:7 and
7:6.

This is what I call an eventone tuning -- or more generally an
evenstep tuning -- with major seconds very close here to the mean
between 9:8 and 8:7. One might say that it's near a "1/4 septimal
comma eventone."

In addition to the usual sharps and flats, spelled just as in
Pythagorean or meantone (here sharps are higher than flats, of course,
as in Pythagorean), I've found a "+" symbol convenient to show a note
raised by 2/22 octave or ~109.09 cents. This is the distance between
my two regularized 12-note keyboards, each sharing the same pattern of
steps and intervals. Note that Eb/C#+ and Bb/G#+ appear on both
keyboards:

273 381 764 982 1091
C#+/Eb Eb+/D# F#+/Ab G#+/Bb Bb+/A#
C+ D+ E+/Gb F+ G+ A+ B+/Db C+
109 327 545 600 818 1036 1255 1309
-----------------------------------------------------------
164 273 655 873 982
C# Eb F# G# Bb
C D E F G A B C
0 218 436 491 709 927 1145 1200

In this system, a sharp or flat raises or lowers a note by three
steps, with diatonic semitones equal to one step or ~54.55 cents.

A neat feature of this tuning is that we get ~12:14:18:21 and
~14:18:21:24 sonorities on every step, using the regular spellings.
In neo-Gothic music, these are typically unstable sonorities resolving
to a stable 2:3:4 or the like, as in this example:

<http://value.net/~mschulter/22tei002.mid>

However, there's no reason why someone couldn't use 22-tET in this
arrangement for a style with these sonorities treated as stable
saturated tetrads. It might have a very interesting "21st-century"
sound.

One thing that may come out of this kind of exchange is to show how
different notations or keyboard arrangements might fit different
styles.

At the very least, I'd say, for example, that a system based on usual
intervals forming sonorities of 14:18:21:24 might call for a different
notation than one based on Paul Erlich's tetrads at 4:5:6:7. Here, as
the adage goes, form follows musical function.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:
> In order to be able to score out pieces in 22 tet using conventional
> notation I've tried to come up with a practical 5-line staff solution.
> There's nothing fancy or groundbreaking herebut I would welcome
> comments, suggested amendments and so on towards a working solution. The
> up and down arrows might be replaced with half sharps and flats; this
> would certainly be easier in my notation apllication, Sibelius. Thanks
> in anticipation.

Sounds good -- just remember that a notated major third (such as C-E) is 436 cents, not the
near-just 382-cent interval.

Dave K.,

I agree with everything you said, except that what you call the "Fokker" notation is not really
Fokker's notation -- he uses sharps and flats differently. We might call the one we both like best
the "Rapoport" notation instead, and I'm sure lots of other people proposed it too, but not
Fokker.

-Paul

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> Dave K.,
>
> I agree with everything you said, except that what you call the
"Fokker" notation is not really
> Fokker's notation -- he uses sharps and flats differently. We might
call the one we both like best
> the "Rapoport" notation instead, and I'm sure lots of other people
proposed it too, but not
> Fokker.
>
> -Paul

Ok. Thanks. I wasn't sure. Rapoport it is.

Can you briefly explain what Fokker does differently?

-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> Can you briefly explain what Fokker does differently?

I've posted the details before . . . for one example, the 5:4 above D is notated F# . . . he starts
with Pythagorean for the naturals but departs from it for the sharps and flats.