Ambiguous or Contradictory?

>>4. 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?
>
>What's a rank-order matrix? Can it apply to meantone-type or Keenan-type
>scales as well?

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 Lambdoma or Tonality
Diamond.

To make a "rank-order matrix", take this diamond and replace each
interval's logarithmic magnitude with a number ranking it's size relative
to all the other intervals in the diamond. Thus, any subset of an equal
temperament that uses all the intervals in the temperament will have
identical interval and rank-order matrices. Example: diatonic scale in 12tET.

This 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 works for meantone scales, so far as I can figure.
What's a Keenan-type scale?


>>2. Can a proper scale with one and only one ambiguous interval in each
>>mode exist?
>
>I don't know what modes have to do with it, as all modes of a scale have
>the same intervals, but isn't the usual diatonic scale in 12-tone equal
>temperament an example?

Isn't it amazing how many definitions the word "interval" has? Here, I
meant a (number of scale steps, acoustic magnitude) pair. So no, the usual
diatonic is not an example; the tritone is not a type of fifth in the
natural minor.


>>3. What about a proper scale with one and only one ambiguous interval
>>in each interval (steps) class?
>
>An ambiguity occurs between two intervals, therefore it is hard to know
>what you mean by "one ambiguous interval." How about C D E F# G# in
>12-tET?

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: 1. Can a scale exist that has one
and only one star in each column? 2. Can a scale exist that has one and
only one star in each row?

Aside from being fun, I think these are useful questions. For me,
ambiguous intervals are the common tones of a type of "melodic modulation".
You don't want too many or too few. In fact, I think I care more about
ambiguous intervals than I do contradictory ones.

Carl