notational inconsistency

I've been arguing that the inconsistency inherent in 24-eq
is not as important in terms of the notation as it is in terms
of the tuning itself. I thought it would be prudent to post
this message, to give a balanced view of my opinions on
notational inconsistency.

I've stated here before that I stumbled upon prime-factor
notation and lattices on my own, without knowing about the
work of Fokker, Wilson, and others previous to me.

My 'invention' of prime-factor notation stemmed from my
dissatisfaction with the inconsistency in Ben Johnston's
notation. I've always felt that Johnston's system is a good
compact and accurate JI notation, but his basic scale is
the 'usual' 5-limit JI 'major' scale, which uses two different
prime-factors (3 & 5, ignoring 2), and this is a problem, IMO.

As Daniel Wolf has pointed out (in private communication),
this causes anyone reading Johnston's notation to have to
perform a two-step mental process to decode the intervallic
relationships in the music.

This is avoided by using a notation based strictly on prime
factors, which is a one-step mental process. (Wolf has
designed one himself, published in a letter in _1/1_). [*]

The point, as far as this posting goes, is that altho I think
Johnston's notation has its merits, it also has a whole series
of inconsistencies (in the Erlichean sense) because of its
two-factor basis. So even tho each of the primes above 5 is
indicated consistently by its own symbol, the plus and minus
signs used for factor 5 itself vary, based on a particular
note's relationship to the reference 'major' scale.

For example,
if we're in the key of 'C', then F:C, C:G, and G:D all indicate
'perfect 5ths' of 3:2 ratio. However, because 'A' (without
any other 'accidentals') is designated as the 5:4 of 'F'
(i.e, 5:3), the notes D:A indicate not a 3:2 'perfect 5th',
but rather a 40:27 '5th' lowered by a comma. The 'perfect 5th'
from 'D' would be notated as D:A+, which I see as a distinct
disadvantage.

I point this out because, first of all, Johnston's notation
seems to become increasingly popular for JI scores and musical
examples in articles, and I think this is unfortunate - if
prime-factor the way I use it is too cumbersome, then certainly Wolf's
notation is far preferable; secondly, as I said above,
I wanted to give a balanced view of my own feelings on the issue
of notational inconsistency.

However, I still feel that it's not that much of an issue when
using an ET notation to convey a JI intention, probably
because ETs have their own consistencies apart from their
rational implications. Perhaps the reason it bothers me so much
in Johnston's case is simply because Johnston's notation *does*
give absolute rational accuracy.

-monz

[*] _1/1_, v 9 #3 (summer 1996), 'Correspondence', p 15.
Wolf uses pairs of symbols, similarly to Johnston, but with
the first and second symbol of each pair representing
p^1 and p^-1 respectively, where 'p' is a prime-factor
from 5 thru 23.

His symbols for 5, 7, and 11 are very much like both Johnston's
and my adaptation of Sims/Herf 72-eq, which makes all three
systems virtually synonomous for 11-limit music, excepting
the variant meanings of Johnston's +/- signs, as noted above.
Wolf's symbols for 13, 17, 19, and 23 are unique (as far as
I know).

Wolf told me he learned prime-factor notation from Erv Wilson,
but I'm only familiar with Wilson's keyboard 'modulus' notations
- I've never seen any of his prime-factor notations.
Perhaps a Wilsonite out there can give some info
(Kraig Grady? John Chalmers? Daniel Wolf himself?).

Joseph L. Monzo monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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Joe Monzo wrote,

>The point, as far as this posting goes, is that >altho I think
>Johnston's notation has its merits, it also has a >whole series
>of inconsistencies (in the Erlichean sense)

Not really; it's a different kind of inconsistency. My kind of inconsistency is one of approximations to JI by ETs and can never arise in Just Intonation itself.

But I agree that Wolf's notation is better than Johnston's. As I'm reading in Mandelbaum, both notations (in concept if not detail) have been around for centuries, the latter called "syntonic" notation by Mandelbaum.

Message text written by Paul Erlich
>
But I agree that Wolf's notation is better than Johnston's. As I'm reading
in Mandelbaum, both notations (in concept if not detail) have been around
for centuries, the latter called "syntonic" notation by Mandelbaum.
<

The idea is both so basic and so old that there's no point in attaching my
name to it! Perhaps 'pythagorean-commatic' would describe it well, in that
the notation is based upon known commas and comma-like deviations from
pitches in a basic pythagorean series.

As I have posted before, I have always used + and - signs for 81/80, 7 and
inverted-7 signs for 63/64 and up- and down-arrows for 33/32's. My own
use of signs for ratios of 13, 17, 19, 23 has not been consistent.

While the major virtue of such a 'commatic' notation is the consistant
spelling of intervals, this does comes at the price of some difficulty in
notating more complex ratios (i.e. 11/7) and the notation, being basically
pythagorean, does not always reflect melodic changes in pitch height, but
these difficulties are also encountered in the alternative notation.