I think I once posted to the effect that this measure is simply the shortest
distance along the edges of the triangular lattice I have proposed.
But it appears I was wrong about that. For example, your expressibility
measure for 81:80 would be 4*log(3) while for 135:128 it would be 3*log(3) +
log(5). However, in the triangular lattice both intervals have length
3*log(3) + log(5).
Rectangular:
135/128
|
|
|
|
1/1-------3/2--------9/8--------27/16-------81/64
|
|
|
|
81/80
Triangular:
135/128
/
/
/
/
1/1-------3/2--------9/8--------27/16
\
\
\
\
81/80
So it appears I was wrong about the triangular lattice. Yet the triangular
lattice distance measure seems more intuitive to me as a "complexity
measure" when the only consonances are 3:2, 5:4, 6:5, and their inversions.
--------------------------
It would seem that your measure and mine are
very closely related. For example, here is the right half of the lattice of
all points which form intervals with 1/1 whose "expressibility" is below
log(100):
25/16-75/64
| \ | \
| \ | \
| \ | \
5/4--15/8--45/32
| \ | \ | \
| \ | \ | \
| \ | \ | \
1/1---3/2---9/8--27/16-81/64
| \ | \ | \ | \ |
| \ | \ | \ | \ |
| \ | \ | \ | \ |
8/5---6/5---9/5--27/20-81/80
| \ | \ | \ | \ |
| \ | \ | \ | \ |
| \ | \ | \ | \ |
32/25-48/25-36/25-27/25-81/50
while here is the lattice with "expressibility limit" 150:
125/64
| \
| \
| \
25/16-75/64
| \ | \
| \ | \
| \ | \
5/4--15/8--45/32-135/128
| \ | \ | \ |
| \ | \ | \ |
| \ | \ | \ |
1/1---3/2---9/8--27/16-81/64
| \ | \ | \ | \ |
| \ | \ | \ | \ |
| \ | \ | \ | \ |
8/5---6/5---9/5--27/20-81/80
| \ | \ | \ | \ |
| \ | \ | \ | \ |
| \ | \ | \ | \ |
32/25-48/25-36/25-27/25-81/50
| \ | \ | \ | \ |
| \ | \ | \ | \ |
| \ | \ | \ | \ |
128/--196/--144/--216/--142/
125 125 125 125 125
Meanwhile, if we consider the --- connections to have length log(3) and the
|
|
|
and
\
\
\
connections to have length log(5), the points within log(100) units along
the connections from 1/1 are:
25/16-75/64
| \ | \
| \ | \
| \ | \
5/4--15/8--45/32
| \ | \ | \
| \ | \ | \
| \ | \ | \
1/1---3/2---9/8--27/16-81/64
| \ | \ | \ |
| \ | \ | \ |
| \ | \ | \ |
8/5---6/5---9/5--27/20
| \ | \ | \ |
| \ | \ | \ |
| \ | \ | \ |
32/25-48/25-36/25-27/25
while the points within log(150) units are:
125/64
| \
| \
| \
25/16-75/64
| \ | \
| \ | \
| \ | \
5/4--15/8--45/32-135/128
| \ | \ | \ | \
| \ | \ | \ | \
| \ | \ | \ | \
1/1---3/2---9/8--27/16-81/64
| \ | \ | \ | \ |
| \ | \ | \ | \ |
| \ | \ | \ | \ |
8/5---6/5---9/5--27/20-81/80
| \ | \ | \ |
| \ | \ | \ |
| \ | \ | \ |
32/25-48/25-36/25-27/25
| \ | \ | \ |
| \ | \ | \ |
| \ | \ | \ |
128/--192/--144/--216/
125 125 125 125
So my intuition was only a little bit off when I thought they'd be the same!