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!