Uniqueness

To evaluate a set, construct a vector of relations for each element of the set. The mathematical representation is

, where the subscript "i+j" is to be taken mod m. Then, a set satisfies Uniqueness if

Here is an example scale: {0, 2, 4, 5, 6, 9, 10}

And here are the vectors for each element:

   0:    {0, 2, 4, 5, 6, 9, 10}
*  2:    {0, 2, 3, 4, 7, 8, 11}
   4:    {0, 1, 2, 5, 6, 8, 10}
*  5:    {0, 1, 4, 5, 7, 9, 11}
   6:    {0, 3, 4, 6, 8,10, 11}
*  9:    {0, 1, 3, 5, 7, 8,  9}
  10:    {0, 2, 4, 6, 7, 8, 11}

One of the more interesting of Balzano's findings with Uniqueness is that symmetrical scales consistently fail it. It is, in fact, the asymmetry of the diatonic set that succeeds in this measure.

There are some caveats which may help to put this trait in perspective. As already stated, it is conducive to the emergence of a 'tonic' element. If that is not desired, a set which conforms to Balzano's Uniqueness criterion is not desired. Many common scales don't conform to it, like the whole-tone and octatonic scales. Then again, many scales that aren't used at all conform to it. As Balzano says, "... many more sets satisfy Uniqueness than fail it ... " (p. 326)

Balzano Scalestep-Semitone Coherence