Keyboard design for 19-scale (from comp.music.research)

Hello!

I don't know what has been discussed in this list except
the last 10 digest's, but John Chalmers encouraged me to
re-post here what I wrote some days ago in comp.music.research.
(There seems to be no newsgroup on tuning - yet.)

What I am writing about is (a) my suggestion to keyboard design -
if this idea is not new, tell me!, (b) useful notation to be
used with the 19-scale and the keyboard (very straightforward),
and (c) some remarks on major and minor chords and the quint
circle.

If there are better sources on this topic, please tell me.

Jens Christian

-------
>Thierry Rochebois writes :
>The numerical coincidences that validate the 12 semitone scale
>also validate the
>19 "semitone" one..."

If you multiply 1.037155 up (instead of 1.0594631)
you recognize that almost all tones appear, however,
the c/d interval (and c/b flat therefore) and
the c/f# (square root of 2) are missing.
This is because 19/12 is approx. 3/2 and therefore
3 semi(third)tones are near to 2 halftones.
And that the tritonus is missing is in any scale that
has an even number of tones per octave.


>I have considered such a system, and have seen some experimental
>keyboards designed to support a 19 semitone scale. Successful
>micro-tonal instruments tend to be fixed-pitch instruments [...]

I would also suggest experimenting with wind instruments, e.g.
flutes/recorders. Perhaps one must wait for the player with
14 fingers...

>What keyboard layout would you envisage? Of course, if you wish
>to confine yourself to computer-driven music ...

There is one canonic way of designing a keyboard:
Look what tones are the same as before and insert the new ones!

--------
My solution is:

One possibility is to duplicate all black keys (5) and
to insert a key between e/f and b/c.

Then you have the scale

| | | | | | | | | | | | | | | | | | |
| | | | | | | | | | | | | | | | | | |
| | | | | | | | | | | | | | | | | | |
+-+-+ +-+-+ +-+ +-+-+ +-+-+ +-+-+ +++
| | | | | | |
| | | | | | |
| | | | | | |
f& c&
c# d& d# e& e# f# g& g# a& a# b& b#
c d e f g a b c

(Approximately) Unchanged are:

e& a& b&
c d e f g a c

(9 of 12). Lost are c#,f#

Completely new in pitch are:
f& c&
c# d& d# e# f# g& g# a# b#
b


(There is one small alternative, i.e. putting
b& on a white key. This means however that
neither notation nor naming of the tones can
be taken from the 12-key scale!-->DISCOURAGED)

As the interval c-a is very pure (1.666524)
one can tune with both a and c, that is,
if you tune a to 440, you have the same
c in 12- and 19-scale.

Many diatonic, e.g. early music can be played
on this scale...

You can use conventional notation; however
most enharmonics from the 12-scale like c#=d&
split up to 2 different notes;
and the two new semitones
f&=e# and c&=b# are introduced.

The advantage ist clearly that the mapping between
white notes on the keyboard and white notes on the
music sheet still remains.

That is, any keyboard player can read the notation above
and play it on this layout without further explanation!!!

However, harmonic properties become more difficult.
-------

C major = c e g
c# major = c# e# g#
d& major = d& f a&
d major = d f# a
(up to here everyone agrees...)
d# major = d# g& a# (!!!)
e& major = e& g b&
e major = e g# b

e#=f& major = e#/f& a& b# (!!!)

f major = f a c
f# major = f# a# c#
g& major = g& b& d&
g major = g b d

a major = a c# e
a# major = a# d& e#/f&

(and so on, in summary d#, a# and the
enharmonics' e#/f& keys become strange...)

---------
However, as the quint/quart and minor/major third
remain, the traditional harmonic approach
can be generalized.
---------
Minor intervals:

Differ from major ones just by changing by 1 semitone.

c minor = c e& g
c# minor = c# e g#
d& minor = d& e#/f& a&
..
g minor = g b& d
---------
quint circle in 19:

e#=f& b#=c& g& d& a& e& b& f (c) g d a e b f# c# g# d# a# e#=f&

That is the usual circle with the alteration that the first
enharmonic coincidence is not f#=g& (in 12scale) but e#=f&.

Therefore you have 19 major and minor keys!

Jens Christian Gruel

--
=Jens Christian Gruel (gruel@theo-physik.uni-kiel.de) 49-431-880-4096=
=http://www.theo-physik.uni-kiel.de/~gruel/ Fax 49-431-880-4094/4100=
=Institut fuer Theoretische Physik D-24098 Kiel-University, Germany=
=Private: Jens Christian Gruel and Marret Claussen =
= Wesselburener Str. 20, D-24106 Kiel-Wik, phone 49-431-35464=

= ceterum censeo mururoam non esse delendam =


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