Beside binary floating point and decimal FP only IBM's sedecimal (hex) FP found wider usage. To a practical extend the base-16 FP is only a variation of binary.
So did any early computers use another base than binary, in order to improve the accuracy of rational arithmetic?
Decimal offers the advantage of producing exactly the same precision inherent to our human arithmetic. A consistent result with a known number of valid digits is in reality usually more relevant than a perfect one. For decimal FP after each calculation the number of valid (decimal) digits is well known and without any doubt. That's why even binary machines often implement BCD, as this, in addition with fixed point notation guarantees results following the well set rules for decimal numbers.
The mentioned base-16 is more of a sideline of binary FP, as it's precision wise comparable (*1). It was introduced due to the structure of the machine implemented for (*2). It had already a nibble based move hardware to handle BCD. Basically a barrel shifter working in multiple of 4. Using it to align FP values for computation and normalizing them later would result in optimal speed without the need for adding a bitwise barrel shifter (*3).
So using a particular radix solves the problem of representing fractions (*4) in a finite amount of space, but each radix does this only for certain fractions. [...]
Base-240 [...] The advantage [...] is that common fractions 1/3, 1/5, 1/7 etc can be represented exactly, which could lead to fewer rounding errors.
And any base used can only cover a very small number of common fractions. By looking close you'll notice, that it comes down to prime numbers (*5), and even base-240 (*6) only includes the first 3 primes: (1,) 2,3 and 5. Already number 7 you cite isn't in there, thus producing precision errors.
Only a use of variable base FP, one where every number is made to a best fitting base, would maybe lower precision errors. Much the same way a fraction based system does circumvent the problem by using an intermediate format. Except, a variable base format would not only increase calculation cost to a similar extreme, it will also work only with a subset, as only one fraction has to be used (and found first).
Last, but maybe even more important, when it comes to outputing the result, no matter what system with a single radix we use (except 10), conversion errors will again be introduced, as we surely prefer a base 10 output, don't we? And any meaningful statement about the precision (valid digits) will have to be made in that radix, not in base-10 as we would like it to have.
Conclusion: For human computer usage there are only two meaningful FP representations:
- Binary (Base-2) to create the most simple computer and
- Decimal (Base-10) to create one operating with the same precision errors we are used to/expect.
*1 - There is a precision 'wobble', as due the nibble (4 bit) nature the number of significant binary digits varies from n-3 to n with n being the number of bis assigned to the mantissa. Thus a 32 bit float with a 24 bit (6 nibble) mantissa has 21..24 bit precision. The price to be payed for the hardware shortcut. But not a big one either, as the default format wasn't 32 but 64 bit anyway.
*2 - IBM /360 if you have to ask :))
*3 - It's these details why I'm so impressed with Mr. Amdahl's work.
*4 - Looks like you're still toying with this idea, don't you? :))
*5 - Since any number can be factorized into primes, we only need to look at primes.
*6 - Using that example is quite nice, as it points to the blinds we carry without even noticing. We're so used to binary computers, that we try to fit everything in this format - even when asking for non binary solutions. A similar shortcut is the often used argument, that decimal would be spacewise inefficient as it needs four bits to store a digit. On a decimal computer storage wouldn't be binary - it would be of course decimal, with each decimal-bit holding 10 values (much like today's flash cells holding 8 or more values). A decimal-byte on such a machine may be made of 2 decimal-bits, quite sufficient to hold a character.
Thinking of it, a modern decimal computer would be most likely be base-5, with two bits per decimal digit and 4 per decimal-byte and 20 per decimal word. With the exception of the sense amplifiers, such a computer will way more compact as a binary.
In any case, such a machine would most definitely not waste space storing decimal numbers - rather the other way around when it comes to binary - but than again, who would care about binary anyway .))