Just doing a bit of mental math tells me there should be some impact either on string diameter for a given tension, or on string tension for a given diameter...but then there's a little popping noise and I start looking at cartoons, so I haven't gotten all the way to the end of that train of thought.
"Long and short scale bass differences explained!
Q: "What's the difference between a long scale bass and a short scale bass?"
A: In the context of guitars, bass guitars and other string instruments, scale length refers to the relationship between the length and diameter ("gauge") of the strings and the pitches they produce. Short scale basses are generally defined as having scale lengths between 30" and 32". Long scale basses conventionally have a 34" string length. Remember that we are talking about string length - the distance between the bridge and the nut - not neck length, although one affects the other.
The gauges of different bass string sets vary but a common long scale set made by Fender includes (high to low) .040, .060, .080 and .100 millimeters. Short scale strings are often thicker (heavier gauge); a typical set has diameters of .60, .75, .90 and .115 millimeters. Sometimes the densities of short scale strings - a factor of the materials used and how tightly they are wound - are also greater.
The first and most obvious reason to use a short scale bass is physical size. With their shorter necks, less distance between frets and more compact general dimensions, short scale basses are a good choice for young players and anyone challenged by the extra reach a long scale instrument requires.
However, many studios pros have long known a secret about the sound of short scale basses. The shorter strings demand lower string tension to be properly tuned. This gives the strings a kind of soft and floppy feeling but it also creates fatter, "blooming" low notes and what musicians perceive as sweet upper notes.
In the 1960s, short scale basses were more popular, but many were generally cheap student models with narrow string spacing and poor tone. As a result, many bassists got a bad impression of them. Although many bassists find the closer spacing of the frets more comfortable to play, for various reasons (sound not the least of them), long scale basses have remained more popular since the introduction of the first Fender Precision Bass in 1951. With the exceptions of the Ampeg/Dan Armstrong "See-Thru" basses and a few special order Alembics, there aren't many professional-quality short scale basses on the market today."