ET, Limits

Gary: I think the context makes it clear that I was talking about equal
temperaments and contrasting closed cyclic systems to open, infinite ones.

Partch defined limit implicitly and used it in the titles of two chapters
(Chapter Seven, Analysis of the 5 Limit, and Chapter Six, Application
of the 11 limit). However, I can find no place where he explicitly
defines it, though it is clear from context (to me, at least) that the
Prime Limit of a tuning is determined by the highest prime number used to
define ratios in that tuning. Partch does use 9 and mentions 15 as odd
numbers
which define ratios, but does not use them to define Limits. He appears
to consider ratios such as 9/8, 81/64, 27/16 and their inversions to be
at the 3 limit.

One of the problems with HP's exposition is his technical vocabulary.
"Identity" as in the definitions of ratios, e.g., "Ratios of 9: those
ratios with identities no larger than 9, in which 9 is present: 9/8,
16/9, 9/5, 10/9, 9/7, 14/9" is not very meaningful as few people would
consider 9 as identical to 1 or any other number (except under the modulo
N operation). What he means is "odd multiple" and he carefully
distinguishes his concept of identity from that of partial or _"ingredient
of Harmonic Content"_.

Calling the identities "correlatives" does not make the concept
appreciably clearer, and he defines the correlatives as the
set 1, 3, 5, 7, 9, 11 ....

He further defines Odentity and Udentity according to whether the
odd number appears in the numerator (over numbers) or denominator.

At least in Genesis of a Music, HP seems to have restricted the term Limit
to prime numbers, though odd numbers serve as "Identities" in
defining ratios.

The definition of a Ratio of N is analogous to that of 9 given above.
Replace 9 by N in the first portion and list all ratios in the tuning
containing N, but no larger prime, as a factor. It is not necessary to
assume octave equivalence, though Partch does as he uses ratios both as
a labels for tones of his scale and for the relation to a 1/1. Two is thus
not considered a ratio defining number in his theory.

I can only say that with exposure, his technical vocabulary becomes
clearer.

Since the LCM is a function of all prime factors, including 2, and
their powers, the concept of LCM and Prime Limit are very different.
All inversions of both the major and minor triads are at the 5 Limit as
5 is the largest Prime Number that appears in any of them (1:3:5, 4:5:6,
5:6:8, 6:8:10, 10:12:15, 12:15:20, 15:20:24, /1:/3:/5 etc.).

--John

Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Sat, 12 Jul 1997 21:27 +0200
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA32325; Sat, 12 Jul 1997 21:28:24 +0200
Date: Sat, 12 Jul 1997 21:28:24 +0200
Received: from ella.mills.edu by ns (smtpxd); id XA32293
Received: (qmail 28887 invoked from network); 12 Jul 1997 19:28:18 -0000
Received: from localhost (HELO ella.mills.edu) (127.0.0.1)
by localhost with SMTP; 12 Jul 1997 19:28:18 -0000
Message-Id: <199707121525_MC2-1AB9-D028@compuserve.com>
Errors-To: madole@mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu

Ed Remler wrote:

<1) Broadwood's remarks relate only to ET's introduction to England. 1811
The problem with this remark is that it assumes a backward position for
England.
It is absolutely clear that piano manufacturing in the period in questionwas most
advanced in England (hence the demand for English instruments by all the
Viennese
masters), and it would be surprising to find that the tuning practice
would be behind the continent.

<2)Prior to the publication by Mersenne of the correct frequency ratios

The publication of the correct ratios is insufficient to produce a correct
tuning.
And accurate clock or metronome would also be neccesary and such
instruments
were not widely available until after Mozart's death. (The famous Paris
pendulum
measurements of tempi for Mozart are a good example of the latitude
tolerated,
which would certainly not be adequate to claim any tuning as ET).

The claims for the Jacobi-Kirche are anecdoctal and were - with the
available
technology - impossible to verify. In addition, I would be very skepticalabout the
influence of a single Evangelische Church organ in the Free and Hansa State
of Hamburg upon Catholic organs at general in the Austrian-Hungarian Monarchy.

(Those unfamiliar with German history or language should know that at thetime
in question, no large German state existed but many smaller ones, and that
the
divisions by religious confession and language/dialect were formidable. Acitizen
of Hamburg was usually Protestant and spoke a language that a Catholic from
Salzburg
would have understood only with great difficulty - or only through use ofan intermediate
language). Indeed, the musical demands upon Catholic Church organs duringthe
Josephian reform era were so circumscribed that the virtuoso organ
literature of
earlier eras would have been unusable in the Church - even if the composers
of
that repertoire had been Catholic. Meantone was entirely sufficient for the
repertoire
that was actually used in Austrian Catholic Churches. Kirnberg's tuning do indeed introduce sharp thirds but they are by no means
ETs.

<3) There is no doubt that all the great masters of the classical period
theory
that described unequal divisions of the whole tone. The Attwood-Studien are
certainly
the closest Mozart himself comes to describing an ideal, if he had one, and
that looks
much more like meantone than ET. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Sun, 13 Jul 1997 18:08 +0200
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA11716; Sun, 13 Jul 1997 18:09:03 +0200
Date: Sun, 13 Jul 1997 18:09:03 +0200
Received: from ella.mills.edu by ns (smtpxd); id XA11678
Received: (qmail 17277 invoked from network); 13 Jul 1997 16:09:00 -0000
Received: from localhost (HELO ella.mills.edu) (127.0.0.1)
by localhost with SMTP; 13 Jul 1997 16:09:00 -0000
Message-Id:
Errors-To: madole@mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu