I think, we have at least two different questions mixed up here. The first is: why are ukes tuned higher than guitars? And the answer is obvious: they are smaller, and the smaller bodies don’t really resonate well with a lower DGBE tuning. And on closer inspection, you’ll see that DGBE and a (linear) GCEA tuning aren’t different at all: The fingerings/chord shapes are exactly the same, just the names of the chords that ring are different: a g-major chord here is a c-majore there and so on.
So although it is technically possible to tune a smaller uke in (lower) DGBE, i.e. a perfect fourth below our standard uke tuning, it doesn’t really work well: either your strings get thick and wobbly, and even if that isn’t the case, you’ll find that most ukes don’t really SOUND well in such a low register (I’m sure there are exceptions, but by and large, I am sure, you’ll find this to be the case). And as already noted: a guitar with a capo on the 5th fret will sound like a linear tuned uke, if played on the higher four strings, leaving out the two bass strings.
An second question then is, why are ukes tuned gCEA, re-entrant, that is? As already mentioned, it has to do with the fact, that with re-entrant tuning, all four notes played on the four strings of a uke are in the same octave, that means, they are as close as possible. In a way, you could say, that the ukulele re-entrant tuning is the perfect opposite to the mandoline’s fifth tuning, where you stretch the four notes you play on the four (double) strings of the instruments as wide as possible on a four-stringed instrument. Just imagine to play a ukulele voicing of a c-major chord on a piano, and compare that to the mandoline voicing of the same chord played on a piano. You can easily play the uke voicing with one hand, as the notes are as close as possible. In contrast, for the Mandoline voicing you would need both hands, as the four notes are stretched to nearly two octaves. Due to the fifth tuning, you always skip one chord note between two played notes.
So whatever the reason for the historic development for the re-entrant tuning, there is no doubt that is the distinguishing factor which makes the ukulele sound stand out between other stringed instruments (although if it’s NOT the only one using a re-entrant tuning).
I read Dirk Wormhout’s opinion on the different tunings of the ukes, and I think there are some points to it. It is certainly correct, that historically, all four types of ukes were tuned differently. Our today’s common tuning was originally the concert uke tuning, whereas soprano and tenor ukes were tuned differently. Sopranos were tuned a full step up, aDF#B and tenors lower than the concerts: Dirk says it was fBbDG, Lyle Ritz claims it was one more half step lower, EAC#F# - Lyle also tuned his tenor uke in linear tuning, way back in the late 50s when he recorded „Ukulele Jazz“.
Whatever the actual notes for the single strings were, I think it’s save to state, that there was quite a bit of flexibility to a ukulele’s tuning way back in the 20s and 30s, when the different ukulele sizes were invented (I know that baritones came later). As Dirk states referring to Mai Singhi Breen and you can confirm when looking at the original song sheets from that era, it was quite common to change a uke's tuning for as much as a third up or down, or using a capo for the same purpose.
As far as I know, the different tunings had to do with the fact that there was no material which would have allowed the same tuning on the different scales of soprano and tenor ukes. Things look different since the invention of synthetic materials like nylon and, a little later, fluorocarbon.
However, I don’t buy all his arguments about the standardization of the c- or concert tuning for all types of ukuleles. I’m sure there is something right about the point that the rise of ukulele clubs helped the c-tuning to become the standard tuning. Of course, it’s simpler for everyone if all the people play the same chords on their instruments, and they all sound the same, too.
However, I disagree with Dirk about the profitability for publishers: Dirk claims they could sell more books to the uke community with one common tuning on all uke types. I could just as well argue, they could have sold even more books if there still had been (or still were) three different tunings. For people owning a soprano uke and a method book for soprano ukes, might be willing to buy a second book for the tenor tuning after buying a tenor uke (before they realized they always play the same chords, and they just had to learn different names for these chords).
And I vehemently disagree with his arguments against a linear g-tuning on baritone ukes. He argues the body of a baritone uke was too small for a low d-string to resonate properly. On the two baritone ukes I owned (and a handful more I have played) so far, there was no issue at all with the low D tuning. The low D-string sounds and resonates just as full and loud as the other four strings. On the other hand, I had completely contrasting experiences with low g strings on other uke types, and for those body sizes, I’d agree with Dirk. On most soprano and even a lot of concert ukes, the low g-string does indeed doesn’t sound as loud or full as the other three strings. It’s no surprise, I guess, that low g is mostly used only tenor ukes, as the low g-string usually resonates better with the bigger body. Again, there are exceptions here, as the concert Fluke, for example, which basically has a tenor body. Or my concert Dupont (manouche style uke), which also has a tenor-size body, and really booms with a low g, it’s a real cannon. But many concerts sound better with a high g.
Just my two (or maybe seven or eight) cents...
