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