What is the term for systems of measure whose units are not simple scalings by magnitude, but instead "arbitrary" units, that is, units unrelated by magnitude?
I'd regard metric lengths scaling simply by magnitude. For the purpose of this question, I'd regard imperial lengths (foot, yard, mile) and common time reckoning (hour, day, week, month, year) as arbitrary.
Mathematically speaking, the metric system uses a fixed (or standard, or ordinary) radix (or base), whereas the imperial system uses a "mixed radix" (or synonyms per the preceding).
From Wikipedia's article on mixed radix:
Mixed radix numeral systems are non-standard positional numeral systems in which the numerical base varies from position to position.
The most familiar example of mixed radix systems is in timekeeping and calendars. Western time radices include decimal centuries, decades and years as well as duodecimal months, trigesimal (and untrigesimal) days, overlapped with base 52 weeks and septenary days.
A second example of a mixed radix numeral system in current use is in the design and use of currency, where a limited set of denominations are printed or minted with the objective of being able to represent any monetary quantity; the amount of money is then represented by the number of coins or banknotes of each denomination. When deciding which denominations to create (and hence which radices to mix), a compromise is aimed for between a minimal number of different denominations, and a minimal number of individual pieces of coinage required to represent typical quantities. So, for example, in the UK, banknotes are printed for £50, £20, £10 and £5, and coins are minted for £2, £1, 50p, 20p, 10p, 5p, 2p and 1p—these follow the 1-2-5 series of preferred values.
Wolfram Mathworld goes a little deeper into to the construction of such systems:
In conventional positional notation systems, a numeral written as has the value where is called the radix or base of the number system. The multipliers for each digit thus proceed from right to left in geometric sequence and each is a constant multiplied by the multiplier of the digit to the right. This representation of numbers is often extremely convenient. There are instances, however, where it is useful to denote a numeric quantity where the ratio between the multiplier of a digit and the digit on its right is not necessarily a constant. Such representation systems are called mixed radix or mixed base number systems. This Demonstration shows how numbers represented in conventional positional notation systems can be represented as a mixed base form.
Alternatively, the 1860 paper "Walkingame's Arithmetic" uses the term "Compound Units".
