This is a scale length of 356mm (14in) to 405mm (16in) and these days usually about 380mm (15in)
1920's Martin S1 Taropatch
My 1928 Harmony "Johnny Marvin Professional Tenor" Today a Concert
My 2011 Ibanez UEW20SM
The scale length kind of existed since the start of the Ukulele as an instrument, (and possibly before as it is roughly the same scale length as a Mandolin), because this was the original scale length of the Taropatch Ukuleles, (as the 8 string, 4 course Ukulele variant it can't just be coincidence that it is bigger that the standard Ukulele but the same size as an 8 string, four course Mandolin though I have seen no firm evidence to say that the early makers set out to mane a Mandolin but then tuned it in 4ths rather than 5ths) It came about as a recognised separate scale in the 1920's because Martin was finding that the Taropatch it had in its catalogue was disliked because of the difficulty in keeping 8 strings in tune, but liked because its larger size gave it more acoustic volume so in 1925 they started to sell the 4 stringed version and called it Concert scale. Why this is referred to as Concert and not Alto when all of the others go with the musical sizes I don't know for sure?. The most plausible explanation I have seen is that the push for the larger 4 string instrument came for the professional and semi professional players, and as it was tuned the same as the standard, scale lengths musically didn't really come into it so initially it was Standard and Concert because Concert sounded more professional? It was only with the advent of the Tenor and the greater acceptance of the C tuning for the larger sizes over the D tuning for the standard that musically there became a difference. Concert then became the battleground with Standard, (now Soprano), tuned one way (D) and Tenor tuned another (C), but Concert going either way and there not really being a space for an intermediate tuning, (unless you used a C# tuning and that really wasn't friendly to any of the commonly use keys!).
The other argument that did rage far more publically at the time than the largely academic musical notation was the actual physical size of a Concert Ukulele. Martin had come up with a premium term for a Ukulele, (Concert was being sold as a premium instrument for professionals, as oppose to an ordinary Standard instrument), but Martin was the only one of the big Mainland makers who had been making Taropatches, so they were the only firm to have the equipment in place, (I don't know how long it takes to set up new moulds in a Ukulele factory and the like?) So in 1926 the other makers got a public proclamation from a body called "the National Association of Musical Instrument and Accessories Manufacturers" that set the "official" size of a Concert as 13¾in (347mm) to 14½in (368mm). This meant that the main Chicago and New York firms biggest offerings were now officially Concerts and Martins Concerts weren't. This may have fooled the ordinary punters but the professionals wanted the bigger size not a shift in goalposts, so some of the bigger firms had set up new moulds and were producing Ukulele of equivalent size to the Martin. To keep these legitimate in their official proclamation called them Tenors, (it said a Tenor had a string length - it never actually used the word "scale" - from 14½ (368mm) to 15¾in (400mm) and had no mention of anything bigger) which is why a lot of pre WWII US Ukuleles are called Tenor but are actually Concert size.
Martin ignored this proclamation especially since they brought out a 17in string length Ukulele with 2 years of it, and with the decline in the popularity, (and production), of the Ukulele in the 1930's it increasingly became less of an issue. After the war there was a resurgence in the popularity of the Ukulele in the US but the world was a very different place.
Now there were electric pickup and Amplifiers, and the Ukulele even with its new popularity wasn't really seen as a professional instrument, most punters just want something cheap and cheerful they could play at picnics or whenever; and even this kind of use was seriously eroded by the portable radio. The few firms that remained, (most had gone under either with the Wall St crash, the War or just the general decline in sales over the preceding decade), had cut back production to their most basic models even before the war, but that was OK because that was all most of the public wanted. The marketing had changes a bit now though and these basic models were all Sopranos, no longer standard as this made them sound more like serious instruments. Martin was pretty much the only firm to still make others sizes and with the rise of the Baritone, which needed the intermediate scale to justify its name, the fact it was 20 years after the initial arguments and that the other sizes had never been that common, it was Martin's idea of string length not the NAMIAM one that became the accepted norm, and it didn't matter much because no one but Martin made them - One thing I should say here is today there is no "official" definition of string/scale lengths the figures I use here are my best guesses having looked at the output of a lot of manufacturers post millenium, their sizes and what they choose to call their instruments. Martin are not, and have never claimed to be any authority on any Ukulele sizes other than their own; and every maker out there has exactly the same right
Elsewhere in the world the whole debate, including the NAMIAM definitions, was completely ignored. In Europe and the British/French Empires, (which at the time covered pretty much everywhere else in the world the Ukulele was thought of), there was only standard and long scale however there was no attempt at a size definition of standard and long scale only meant that the instrument had more than 12 frets to the neck or that there was a "Standard" model in the manufacturers catalogue that it was larger than. And of course Hawaii, which was the only other bastion of Ukulele, had given up trying to compete for the mainland mass market before 1925 so never really got involved in the debate. Hawaiian made was more synonymous with quality than Concert so they didn't need too. A few, more market oriented people, like the Summers Brothers, saw some potential in stocking larger size Ukuleles for the more discerning visitors in the 1930's but they also knew that the models to emulate were Martins so they followed that sizing, and Kamaka the only major manufacturer to have Ukuleles in continuous production since 1916 taking up these scale lengths for their post WWII production, (before WWII they only made Sopranos), this only added to the strength of argument for those definitions to become the norm.
One last thing to say about the evolution of the Concert, originally they still usually had 12 frets to the body. These days though some still do have 12 fret bodies it is more common to have a longer fret board, meeting the body at the 14th fret and going further down the body than the Soprano.