Comments on Brian's Post in #542.

I have two comments on Brian's post in Tuning Digest 542. As Bosanquet
defined them, positive systems are regular cyclic systems whose
fifths are greater than 700 cents, hence the Pythagorean tuning
is a positive system. Defining positive fifths as those sharper than
3/2 is a common error, which I have been guilty myself on occasion.

Secondly, Wuerschmidt defined constructing and defining intervals
somewhat differently than Brian alleges, though his definition is
very useful. According to Mandelbaum's dissertation (my principal
source for W), W's defining intervals are small intervals set to
zero size in defining an ET. In 12-tet, the syntonic comma and the
diesis are two such defining intervals. Constructing intervals,
on the other hand, are intervals set to 1 degree of the temperament.
Examples of such intervals in 12-tet are 16/15, 135/128, and 25/24.
Constructing intervals, therefore, are not the coordinate axes of
tonal lattices, but rather the melodic seconds of the JI scale
which is the acoustic justification for the ET. Different sets
of defining intervals lead to different sets of constructing
intervals and different scales to be mapped onto the ET
(or is it into?).

Prof. Fokker used a concept identical to W's defining intervals
in his derivation of ET's as periodicity blocks in the 3x5 and
3 x 5 x7 tonal lattices. The number of tones in the ET is equal
to the absolute value of the determinant of the square matrix whose
rows are vectors whose elements are the exponents of the defining
intervals in the lattice. Two by two matrices are used for 2-D
lattices with obvious extensions to more harmonic factors.
For the 2- and 3-D cases, the ET's are equal to the cross and box
products of the vectors.

--John

Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Sat, 28 Oct 1995 22:27 +0100
Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
for id GAA26264; Sat, 28 Oct 1995 06:27:50 -0700
Date: Sat, 28 Oct 1995 06:27:50 -0700
Message-Id:
Errors-To: madole@ella.mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu