Johnston's notation defended (again)

Johnny Reinhard said (Sun, 27 Oct 1996):
> An example of Ben Johnston's "conflicting directions" in his notation
> would be an "F" with a combination of symbols in front of it: one symbol
> might be to substract 70 cents, another to add 22 cents...and there could
> even be a third. Unfortunately, I cannot send the "actual" notation by
> e-mail. Suffice it to say that muddling over how to calibrate a pitch
> through a succession of mathematical processes is unneccessary. Cents
> makes sense.

These cents approximations must be: 70 cents down -- a flat sign; 22 cents up
-- a plus sign. Since an Fb+ is rather out-of-the-way, I'll show one way to
arrive at an Ab+ instead, since all we're interested in is the combination of
signs which shift the pitch in different directions. Here's one path through
the 5-limit lattice from C to Ab+, letter-names on the left, corresponding
ratios on the right:

Ab+ C+ 81/50 81/80
F+ 27/20
Bb D 9/5 9/8
G 3/2
C 1/1

So, for instance, the player might be required to ascend a fifth (3/2) from C
to G, then a minor third (6/5) from G to Bb; this process is then repeated,
leading from Bb to F+ and finally to Ab+. Fifth, minor third, fifth, minor
third -- nothing more complicated. Where's an Ab without a plus sign then,
and what would that mean?

Ab C 8/5 1/1

The Ab without further inflections is the normal just minor sixth above the
C. An acquaintance with such lattices induces performers to think of the
different inflection signs as steps along different axes of the lattice and
not merely up or down a pitch continuum. Where are the conflicts then?
Nowhere, except in the eye of someone who takes pride in refusing to discover
how Ben Johnston's notation works. I'll assume that Johnny Reinhard is not
taking issue with the difficulty itself, but with what he sees as unnecessary
difficulty; but when a composer wishes that the performer should, as far as
possible, understand and hear the intervallic relationships in the
composition, then presenting the performer with cents deviations is no longer
an option (and deviations from what? -- who is able to intuit frequency
relationships based on 2^[1/12]?). Ben's notation is simply the one correct
notation for his own music; the application of this notational system outside
Ben's works can only be justified on the merits of individual cases. For
Partch, it is arguably successful, but there would be little point in
reworking, say, Xenakis's quarter-tones in any extended JI notation (in
general, building bridges between Ptolemaic and Aristoxenian territory is a
thankless, and usually pointless task).

Cents notation, in any case, erects an equal-tempered default, which is the
last thing any JI composer would wish to invoke. So is Ben Johnston's
notation the perfect language of tuning into which all systems can and should
be translated? Not in the least. Is it the best system for Ben Johnston's
music, with the potential for successful application in many other areas
because of its consistency and lucidity? Yes.

A parting comment on my own theoretical home ground: I've seen various
articles which purport to explain certain historic keyboard tunings; on
reading through them, I find that they never go beyond cents notation, which
cannot even begin to explain how the tunings were devised, which would
require a mixture of ratios and inflections by the roots of one or other
comma. The most cents notation can tell us in such circumstances is the
audible effect of the tuning, which is very well in itself, but empty of
explanatory content. Our lives would be very different today if the sciences
had remained on this level of incomprehending taxonomy.

(And by the way: yes, that message to Adam Silverman was not intended for the
list -- the first time I've made the classic e-mail list slip; on one list I
received a message from a subscriber to his sister, asking for her advice on
the proposal of marriage he was about to make. Mine was slightly less
embarrassing, but bad enough.)

--
Jonathan Walker
Queen's University Belfast
mailto:kollos@cavehill.dnet.co.uk
http://www.music.qub.ac.uk/~walker/

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