1/a^s + 1/b^s + 1/c^s + ... complexity

I've been exploring complexity measures of the form 1/a^s + 1/b^s +
1/c^s + ... recently as an alternative to Tenney height. This will end
up rating simpler chords as higher than more complex chords, so it
should be called a "simplicity measure" instead. I'll call the inverse
of this measure the "zeta complexity" of a chord for now, because the
complexity of 1:2:3:4:5:6:7:8:... is 1/zeta(s) depending on what value
of s you use.

The reason I originally liked this measure is because it lets us
compare chords like 4:5 and 4:5:6 and 4:5:6:7, with each one being
more concordant than the former. It also has a quasi-interpretation as
being the extent to which a chord activates a hypothetical harmonic
template in the brain, each corresponding to a virtual pitch, with a
1/N^s rolloff. The reason I say that this is a quasi-interpretation is
that in real life, there's huge difference between the chord
1:2:3:4:5:6:7:8:... played with sines with a 1/N rolloff, and the
chord 1:2:3:4:5:6:7:8:... played with sines with no rolloff at all.
The former sounds like a single note with a very clear pitch, whereas
the latter is called an "impulse train" and sounds extremely harsh -
see http://www.mikebattagliamusic.com/music/110.25HzBuzz.wav. And if
you want to consider complex timbres, then 1:2:3:4:5:6:7:8:... with
all notes equal volume and a complex timbre is even worse - you'll end
up getting a negative rolloff.

This measure doesn't model that behavior, and instead models
everything as a totally linear process. It just assumes that more
notes = stronger fundamental and models the Nth harmonic as
contributing 1/N^p of energy to the fundamental. It doesn't think
about volume at all. You can interpret it as naively assuming that all
notes are being played at their ideal volume for a perfect template
match. This limits the direct applicability of all this in modeling
something like a harmonic template, but it looks like it might be
somewhat useful anyway.

Even outside of all of any psychoacoustic interpretation it looks like
it might be somewhat useful. So let's see what happens for different
values of s.

Dyads for s=1
/tuning-math/files/MikeBattaglia/ZetaComplexity/zetas=1.txt

Note that in general, things /1 appear before things /2, with 15/1
even appearing before 3/2. Within our psychoacoustic interpretation,
this means that 15/1 generates a stronger VF than 3/2, because the VF
is explicitly right there in the dyad: it's x/1. By altering the value
for s, this behavior changes. Look at s=2:

Dyads for s=2
/tuning-math/files/MikeBattaglia/ZetaComplexity/zetas=2.txt

The behavior is more extreme this time, and is approaching ordering by
1/d. In fact, the more we increase s, the closer that the complexity
gets, for any given chord a:b:c:d:..., to ordering based on
min(a,b,c,d,...). We can lower s to attenuate this behavior

Dyads for s=0.5
/tuning-math/files/MikeBattaglia/ZetaComplexity/zetas=0.5.txt

Oddly, the complexity of 1:2:3:4:5:6:7:8:9:... for s=0.5 turns out to
be -1.460354509.

Note that 3/2 is moving up the rankings and is now between 12/1 and
13/1. We can see that the order is starting to reflect Tenney Height.
Finally, as s converges on 0, we end up getting an order which
perfectly reflects Tenney Height, which is a pattern I conjecture will
hold for any set of intervals (seems trivial to prove)

Dyads for s=0.00001
/tuning-math/files/MikeBattaglia/ZetaComplexity/zetas=2.txt

If two dyads have equal Tenney Height, the tie is broken by picking
the one with the lowest minimum component in a:b:c:d:e:f:g:....

There is one more interesting thing about all this, which is to see
how chords rank up with dyads. Here's what we get for different values
of s:

s = 1
4:5:6 - 0.616666666666667 - about as simple as 9/2
4:5:6:7 - 0.759523809523809 - between 5/2 and 3/2
4:5:6:7:9:11 - 0.961544011544012 - between 3/2 and 15/1

s = 2
4:5:6 - 0.130277777777778 - between 7/3 and 8/3
4:5:6:7 - 0.150685941043084 - about 5/3
4:5:6:7:9:11 - 0.171296082865347 - about 4/3

s = 0.5
4:5:6 - 1.355461885963821 - about 8/1
4:5:6:7 - 1.733426358973048 - between 2/1 and 1/1
4:5:6:7:9:11 - 2.368271036884146 - better than 1/1 (hmmmmmm…)

s = 0.00001
4:5:6 - 2.999952125468695 - close to 3
4:5:6:7 - 3.999932666556532 - close to 4
4:5:6:7:9:11 - 5.999886715886912 - close to 6

Note that in the last case, the zeta simplicity is very close to the
number of notes in the chord. Thus, I conjecture that as s -> 0, this
will lead to a total ordering on the powerset P(N) of natural numbers
such that sets of the same cardinality will be ordered by the inverse
of their Tenney Height, ties being broken by the chord with the
smallest integer component (with further ties being broken by the
second-smallest, and so on), and sets of different cardinality being
ordered by their cardinality.

Seems like it might come in handy. I'm especially digging that 0.5
ranking, personally.

-Mike