Are there bricks sized between LEGO (classic) and DUPLO? I am wondering whether there is a brick-type construction block toys on the market that would be sized between the LEGO (classic/original) and LEGO DUPLO. More specifically I am looking for a 3/4 ratio of DUPLO size.

The fact is that DUPLO is 8x the volume of the "classic" LEGO bricks. Measured in 2D dimensions, 4x4 classical LEGO "units" are the same width as a single 2x2 DUPLO brick. Also DUPLO has double brick height than classic. I wonder whether there is a brick where a single 2x2 brick would have the size of 3x3 LEGO classic units and also would have 3/4 of DUPLO height.
One Single Stud Lego brick is 9.6mm H by 8mm W (technically 7.8mm).  So at the scale you are looking for we're searching for single stud bricks that are 28.8mm H by 24mm W.

After an extensive search among multiple Lego-style building block manufacturers, I can't find any that use the dimensions we're looking for.  Here is a list of companies I searched:

Best-Lock, Buggi, Built To Rule, Ciddi Toys, Hubelino, Intelliblox, Kre-O, Loc-Bloc, K'Nex, Mega Bloks, NanoBlock, Playbig Bloxx, Q-Bricks, Super Blox, Tente, Super Blocks, Unico

The Best-Lock Juniors were probably the closest I could find but the proportions are still off and I think 2010 was the last time the sold a Juniors set.

One of the biggest selling points most all of these companies made was that their blocks were compatible with other building systems (ie Legos) so chances are they wouldn't deviate from the standard sizes that Lego produces with their own special sizes.