Rothenberg questions

Regarding Rothenberg propriety, I asked three questions...

>>1. Can somebody show a method for finding which tunings will support a
>>given rank-order matrix? Preferably one that works on both just and
>>equal step scales?

Paul E. asked...

>What's a rank-order matrix?

I 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.
>
>To make a "rank-order matrix", take this diamond and replace each
interval's >logarithmic magnitude with an integer ranking its size relative
to all the >other intervals in the diamond.
>
>The idea is that several scales, each having a different interval matrix,
>may all be percieved as re-tunings of eachother if they share the same
rank->order matrix.

This first question remains unanswered. I asked two more...

>>2. Can a proper scale with one and only one ambiguous interval in each
>>mode exist?
>>
>>3. What about a proper scale with one and only one ambiguous interval
>>in each interval (steps) class?

Paul E. responded...

>An ambiguity occurs between two intervals, therefore it is hard to know
>what you mean by "one ambiguous interval."

I did phrase these last two questions in a confusing way. Perhaps I
should have asked: Can a proper (but not strictly proper) scale have its
ambiguous interval(s) distributed so that they occur only once in each mode
(question 2) or interval class (question 3)? Despite my confusion of
terminology, I think my subsequent post did make it clear what I was after...

>Let's take a look at its "interval matrix"...
>
> (C) (D) (E) (F#) (G#)
>2nds 2 2 2 2 *4
>3rds *4 *4 *4 *6 *6
>4ths *6 *6 *8 *8 *8
>5ths *8 10 10 10 10
>
>And rank-order matrix...
>
> (C) (D) (E) (F#) (G#)
>2nds 1 1 1 1 *2
>3rds *2 *2 *2 *3 *3
>4ths *3 *3 *4 *4 *4
>5ths *4 5 5 5 5
>
>Modes are the columns, interval classes the rows. Ambiguous intervals are
>marked by stars. My questions are then: 2. Can a scale exist that has
one >and only one star in each column? 3. Can a scale exist that has one
and >only one star in each row?

Since I asked them, I have figured answers to these last two questions.
The answer to question 3 is "no". If you insist on having at least one
ambiguous interval appearance in each interval class, then the least amount
you can get to appear is 1 in the smallest class, 1 in the largest class,
and 2 in each of the other classes (no matter how many there are).

The answer to question 2 is "yes". Scales can exist that have one and only
one ambiguous interval appearance in each mode. Take this generic 3 note
scale (rank order matrix)...

X Y Z
2nds 1 1 *2
3rds *2 *2 3
4ths 4 4 4

To see how this might look, let's tune it in 12tET. For simplicity let's
assume that the 4th is the interval of equivalence, and that it is 12 steps
of 12tET, and that the other intervals are scaled proportionally to their
rank. The interval matrix then looks like this...

X Y Z
2nds 3 3 *6
3rds *6 *6 9
4ths 12 12 12

In standard notation, we can notate this scale like this...

.C Eb F# C
..^ ^ ^
..3 3 6

In which case mode "X" starts on C, mode "Y" on Eb, and mode "Z" on F#.

Carl