Chord complexity: Difference between revisions

Given all of this, it may be useful to see some examples. Let's look at the set of all chords of size 1-5 notes, with coefficients of at most 5, just to see what we get. Higher values on the list are "stronger" or "less complex". This is with the Benedetti Height with {{nowrap|''s'' {{=}} 1}}:

'''Benedetti Height, s=1'''

1:2:3:4:5 - 0.52103

If we do this instead with the Weil height, we get something which looks slightly tidier at first glance, but which on further inspection has some strange features. Let's take a look:

'''Weil Height, s=1'''

1:2:3:4:5 - 1

We can always interpolate between the two as well; if we go with the Tenney-Weil height for {{nowrap|''k'' {{=}} 0.5}}, and again {{nowrap|''s'' {{=}} 1}}, we get something intermediate:

'''Benedetti-Weil Height, k=0.5, s=1'''

1:2:3:4:5 - 0.72183

Let's focus on just the Tenney height, and look at how the value of ''s'' changes things. If we instead set {{nowrap|''s'' {{=}} 0.5}}, we get the following:

'''Benedetti height, s=0.5'''

1:2:3:4:5 - 0.10421

Likewise, if we make s larger and move it towards infinity, the Tenney height tends toward just the geometric mean, without being divided by any normalizing term. Then we get this:

'''Benedetti height, s=Inf'''

1 - 1

The viewpoint of this author is that the most sensible results are when we have {{nowrap|''s'' {{=}} 1}}, with Tenney height preferred to Weil height. The all-around best seems to be the Tenney-Weil height with k somewhere near 0.5, although having {{nowrap|''k'' {{=}} 0}} is probably good enough and slightly simpler to work with, particularly since it reduces to an L1 norm for dyads. We will do an extended listing of both, this time with at-most tetrads and with coefficients up to 7. First the Tenney height:

'''Benedetti height, s=1, tetrads with max-coefficient=7'''

1:2:3:4 - 0.55334

The results seem reasonably sensible to me, although with a few little caveats here and there—we have 1:7 ranked above 4:5:6:7, partly because there is no notion of octave-equivalence involved, and partly because Tenney height may not be prioritizing small-span intervals quite enough. But this is at least ballpark-sensible. We can tweak it slightly by looking at the Tenney-Weil norm with {{nowrap|''k'' {{=}} 0.5}} and {{nowrap|''s'' {{=}} 1}}:

'''Benedetti-Weil height, s=1, k=0.5, tetrads with max-coefficient=7'''

1:2:3:4 - 0.74387