Re Carl's Rothenberg Questions in TD#1

The formula to which Carl refers applies to Moment of Symmetry (MOS)
(aka Well-Formed Scales (WFS)), not scales in general. For MOS (WFS),
which by definition have two and only two sizes of step interval
(melodic seconds), if A and B are the two intervals, then propriety
demands that the ratio A/B (in cents or logs) be between 1/2 and 2.

There are two exceptions to this assertion. For MOS (WFS) of the form N
A's + 1 B, N => 2, and A >B, A may be arbitrarily larger than B.
Examples are 2 2 2 1, 9 9 9 9 2, etc. The other exception is the trivial
case of just two intervals A B where A and B may have any proportion.

I would like to know of any other exceptions, corrections, etc. I once
thought I had a semi-formal proof, but I can't reconstruct or find it
today.

The above assertion does not apply to scales which repeat within the
octave or other interval of equivalence, such as the modes of limited
transposition of Messiaen, even if they have only 2 sizes of scale step.
For example 5 1 5 1 in 12-tet is strictly proper. To compute propriety
of scales of this type, one can use only 1 "module" and replace the IOE
with the sum of the intervals of the module. The resulting difference
matrix is termed "reduced" by Rothenberg.

>What's a rank-order matrix?

CarI answered...

>A graphic method of determining propriety, no matter how many unique
>interior intervals a scale has (Chalmers' formula only works for two), is
>to construct its "interval matrix". This is just its tonality diamond.

The difference matrix is not the same as the Tonality Diamond. The
difference matrix of a scale or chord is an array in which each row
consists of the successive n-step intervals from each note. The first
row contains the melodic seconds, the 2nd, the thirds, etc. through all
scalar classes. The final row consists of the octave in whatever units
it is measured. In practice, only the first N*(N-1)/2 entries of the
first [N/2] rows need be considered.

For example, the Tonality Diamond of the major triad in 12 tet ( 0 4 7 )
is

0 3 7 where 0 is either 12 or 0. Note that the TD consists of
9 0 4 of major triads built on the octave inversion of the major
5 8 0 triad ( a minor triad on 5).

The Difference Matrix is

4 3 5 Note that the octave, the IOE, is added to the triad for
7 8 9 the purpose of computing intervals. The first row consists
12 12 12 of the successive intervals (major third, minor third, 4th).
The next row contains the sums of every two adjacent intervals, and the
last the sum of every 3 (which completes the octave here.) All summation
is done "end around" or toroidally.
>
>To make a "rank-order matrix", take this diamond (sic) and replace each
>interval's logarithmic magnitude with an integer ranking its size relative
>to all the >other intervals....

Correct if one is using the DM and working in JI. Otherwise, use the
tempered intervals directly. By ranking, assigning 1 to the smallest
interval, 2 to the next smallest, etc. is meant.
>
>The idea is that several scales, each having a different interval matrix,
>may all be perceived as re-tunings of each other if they share the same
rank->order matrix.

True. They are presumed to fall into the same "equivalence class" and
tend to be heard are different tunings of each other. However, this is
somewhat dependent upon the listener's discrimination and expectation.
While the major scale in
JI, Pythagorean tuning and 12-tet are in different EC's, many listeners
will
hear them as separate tunings if they ignore the commas and smaller
intervallic discrepancies. (in JI, the major mode is strictly proper; in
Pythag, improper; and in 12-tet, proper).

In other ET's, it depends on whether Pythagorean or JI is being
approximated.
The 7-tone fifth-generated MOS's in 17 and 22 are improper, the scale 4
3 2 4 3 4 2 in 22-tet is proper, and the scale 3 2 1 3 2 3 1 in 15-tet
is improper.

The rest looks pretty clear to me.

--John