Lowest common denominator
In mathematics, the phrase lowest common denominator (LCD) means (adapting from the OED’s definition):
the lowest number that is a multiple of the denominator of two or more fractions.
For example, the LCD of 1/3, 1/4, and 1/6 is 12. This means 1/3 can be expressed as 4/12, 1/4 as 3/12, and 1/6 as 2/12; no number lower than twelve can be used to thus rewrite these three fractions, hence 12 is the LCD. (It can also be done with any multiple of 12: 8/24, 6/24, 4/24.)
The phrase may come from mathematics, but it is perhaps more often used figuratively, such that “appealing to the lowest common denominator” means
appealing to the lowest thing common to or characteristic of a number of things, people, etc.
In other words: mindless humor, sex, and violence.
Did you notice that the implied metaphor makes no sense? It assumes that, if I have a large, diverse audience, the LCD will have considerably lower standards than any particular audience member. But mathematically, the opposite is true: the more fractions with different denominators I have, the higher will be the LCD. For example, LCD(1/3, 1/4, 1/6)=12, but LCD(1/3,1/4,1/5,1/6,1/7)=420.
The metaphor needed here is not lowest common denominator, but greatest common divisor (GCD):
the highest number that is a divisor of two or more numbers.
For example, the GCD of 8 and 12 is 4. The GCD of 8, 10, and 12 is smaller: 2. And the GCD of 8, 9, 10, 11, and 12 is only 1–the lowest possible GCD. (That’s what always happens when you take the GCD of three consecutive numbers.) This is exactly what we were looking for: an operation that moves from a set of numbers to a number such that the more numbers in the set, the lower the result will tend to be.
But of course, no one will ever start referring to mindless humor, sex, and violence as “appealing to the greatest common divisor.” We accusing a writer of “appealing to the lowest common denominator” as an insult, after all: we mean that he didn’t just make something bad but was trying to be bad. Sell-out! Evil capitalist pig! It would be harder to demonize him in this way if our language revealed his actual motive more clearly: he was, in fact, trying to be good, shooting for greatness; alas, the constraints he chose for himself ensured that the highest he could be was at the bottom.
“Appealing to the lowest common denominator,” conversely, would be better applied to those artists who achieve the most. The Iliad, one could argue, is the LCD of Western civilization: it is the least old thing, and the least great thing, in which all of us can recognize ourselves. (Although, if there were a greater, perhaps the Iliad would no longer be great enough.) But we do no like this either; we do not like calling great works of art “low.” Is this dislike anything but the Romantic cult of genius rearing its ugly head yet again? Of course great artists aim to make something as simple as they can manage; thankfully, the constraints they chose for themselves ensure that the lowest they can be is quite high indeed.
The constraints, note, are those of division and denomination. The hack writer fails, not because he aims low, but because he aims at our components, rather than at us; while the great artist succeeds, not because he aims high, but because he addresses what we are, not what we are made of.