Kees-Euclidean interval complexity

The Kees expressibility of an interval is defined as the
larger of the numerator and denominator, once factors of
two have been stripped out. You can also think of it as
setting a fractional number of octaves to send the interval
to a unison. For example, the interval 5:4 is octave and
inversion equivalent to 8:5, 5:2, 16:5, 5:1, 10:1, and so
on. You can call it [-2.322, 0, 1> or 5:5 as an average of
these. The complexity of this impossible interval then
gives you the expressibility of the octave-equivalent
interval.

So, let's generalize this to a Euclidean metric. Take an
interval, strip out factors of 2, and measure its size in
octaves. Then set the negation of this as the factor of 2,
and calculate the TE complexity.

5:4 and friends still map to [-2.322, 0, 1>. The
complexity is sqrt(2.322^2 + 1) along with whatever
normalization you want. To get the TE complexity, multiply
by the square root of 2, to get 3.284. To normalize like
an RMS, divide by the square root of 3, to get 1.896.

6:5 and equivalents map to [-2.811, 1, 1>. The complexity
is 2.906 or RMS-like as 1.678.

I gave a formula for the inverse of Cangwu badness before.
This involves the TE complexity and a correction term
proportional to the square of the size of the interval.
The Kees-Euclidean complexity is the square root TE
complexity squared of the interval with octaves stripped
plus the size of the interval squared, when you measure in
octaves. That means the inverse Cangwu badness matches
Kees-Euclidean complexity when Ek=1/sqrt(d), where d is the
number of prime intervals.

Because octaves are defined to be zero for the intervals
you apply this to, you can use a smaller matrix to define
an octave-equivalent metric. For the 5-limit, it's
equivalent to taking straight-line distances on a
triangular lattice with 6:5 and 5:4 almost equal in length.

This happens not to be the same as the complexity of a rank
2 temperament class being the standard deviation of the
weighted octave-orthogonal mapping (including the octave as
zero). That happens to be the same as the relative TE
error (scalar badness) of the stupid 0 note equal
temperament defined by the octave-orthogonal mapping. As
such, it implies Ek=0.

I don't have temperamental Kees-Euclidean complexity worked
out yet. For rank 2 temperaments, any octave-equivalent
complexity counts generator steps. But for higher ranks,
it depends on which octave-orthogonal generators you
choose. What I think we want is a temperamental complexity
metric where the top-left (period-period) entry is 1 and
the other entries in the top row and left hand column are
all zeros. I believe there's a false mapping that gives
this, with the period mapping containing non-integers.
That's acceptable like the fractional octave exponents
above, because the point is to throw it away. Anyway, I
haven't found a recipe for finding this mapping. But once
you have it, you should be able to use a special case of
Cangwu interval badness to get an octave-equivalent
temperamental complexity. I think it'll still be
Ek=1/sqrt(d) with d counting primes.

This is all related to hobbits. I don't happen to think we
need it, though. You can always take the simplest interval
within an octave. You may think that 15:14 should be given
an extra (unnecessary as it happens) boost over 16:15,
though, because it becomes 15:7 instead of 32:15 in the
octave above. (This is balanced with 15:8 versus 28:15 of
course, but it's reasonable to ask for simple intervals
relative to the tonic.) In that case, all you have to do
is look at a different interval of intervals, for example
0.5 to 1.4999.... octaves.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
That means the inverse Cangwu badness matches
> Kees-Euclidean complexity when Ek=1/sqrt(d), where d is the
> number of prime intervals.

Interesting! I'll try to do more than skim your recent postings when I finish this composition project.

Until here I can follow.
Can you please explain in detail how you get the 2.811?

Thanks

Kees

On Fri, Dec 3, 2010 at 2:08 AM, Graham Breed <gbreed@gmail.com> wrote:

>
>
>
> 6:5 and equivalents map to [-2.811, 1, 1>. The complexity
> is 2.906 or RMS-like as 1.678.
>
>

On 4 December 2010 01:47, Kees van Prooijen <keesvp@gmail.com> wrote:
>
>
> Until here I can follow.
> Can you please explain in detail how you get the 2.811?

That's wrong. I was doing 15:8 instead of 6:5. So, taking 15:8, you
get 2.811 as sqrt[(log2(5))^2 + (log2(3))^2]. But that's a formula
for complexity, not size, so I shouldn't have left that number in
there at all.

I think it should have been [-0.737, -1, 1> where 0.737 is log2(5/3).

2.811 is the TE complexity of [0, -1, 1>, so the TE complexity of
[-0.737, -1, 1> will be sqrt(0.737^2 + 2.811^2) = 2.906, so that
number below is correct:

>> 6:5 and equivalents map to [-2.811, 1, 1>. The complexity
>> is 2.906 or RMS-like as 1.678.

I did the right calculation for 6:5 but forgot to change the ket.

The correct calculation for 15:8 should have given [-3.907, 1, 1>,
which will obviously give a much higher TE complexity than 6:5, as
we'd expect. The point of the octave equivalent metric is that
[0,1,1> and [0,1,-1> don't give the same result.

Graham