Take for example C augmented. It sounds more consonant in the treble clef above middle C and more dissonant in the bass clef. However the proportions of the frequencies are exactly the same. This doesn't make any sense that it is more dissonant in the bass while the proportions are the same. Frequency proportions being the same should suggest the same amount of dissonance in every octave.

So why are dissonant intervals and chords more dissonant in the bass clef and more consonant in the treble clef? 


Question: Linear Independence of a set of Complex Vectors
Answer: Check the determinant just like you do for real numbers. If an n x n determinant A such that det(A) = 0, it's rank r &lt; n. This implies that at least one row or column is a linear combination of the rest.