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Lowest common denominator

January 18, 2014

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.

11 Comments leave one →
  1. January 19, 2014 6:58 am

    You’re right, of course, which is why it’s a cliche and not a metaphor…”lowest common denominator” /sounds/ more like what they mean (i.e. it’s “low” not “great”).

  2. Chris Wolfe permalink
    January 20, 2014 11:20 pm

    I’ve thought about this before, and you’re exactly right. But I love the phrase “appealing to the lowest common denominator” because it is a math pun

  3. January 21, 2014 12:52 am

    Yes, but it’s a math pun that appeals to the lowest common denominator of mathematical literacy.

  4. David Simmons permalink
    January 24, 2014 8:16 pm

    It seems the issue is that the mathematical situation the phrase is based on has no relation to reality. In mathematics, the thing that has the biggest impact is whether you are considering the set of divisors of your numbers, or the set of multiples of your numbers. Whether you choose the greatest or smallest number out of this set makes relatively little difference. But there seems to be no reasonable analogue of “the set of divisors” or “the set of multiples” in real life – both of these are captured by the same concept, namely the set of characteristics of a group of people. In which case, it makes a big difference whether you choose the least member of this set or the greatest member.

    Incidentally, appealing to the lowest common denominator is still better than appealing the lowest denominator, i.e. appealing to the worst instincts of the worst person. In which case, that part of the metaphor is correct: adding people can increase where the lowest common denominator is. On the other hand, the LCD would decrease it if you started with a person with good instincts who avoided media targeted at the “lowest common denominator”, and then started adding people. This somehow suggests that “the set of characteristics of a class of people” is really more of an averaging operator than anything else.

    So there’s a commonsense defense of the phrase “lowest common denominator”. Really, the only thing that doesn’t make sense about it is the word “denominator” – what the heck does it refer to? But idioms can’t be expected to completely make sense.

    Also I suspect that the word “denominator” is meant to reinforce the word “lowest” – everybody knows you put the denominator at the bottom of the fraction.

  5. January 24, 2014 10:34 pm

    I think you’re right that “denominator” is supposed to reinforce “lowest”… though it seems to me a bad pun.

    I’m not as convinced that people employ a distinction between “lowest common denominator” and “lowest denominator” the way you suggest. In my experience people use the phrases interchangeably. Though it’s true people sometimes mean that the movie fails because it’s trying to appeal to too many people (“when you try to entertain everyone, you entertain no one”), while the sometimes seem to be imagining a “worst person”–you know, the person who actually enjoys the LCD stuff, because of course WE don’t.

    The point I wanted to make, though, is that this “worst person” is something of a fantasy (though of course extremely crude people do exist). Whenever we watch a movie and enjoy it even while despising it for “appealing to the LCD,” *we’re still watching it and enjoying it*–and for some reason we feel compelled to insist we’re not the film’s real audience. This fact deserves closer attention. It suggests that what we like to pretend is going on, probably isn’t.

    It’s true not everyone enjoys films targeted at LCD(mass-market), and those people use “appealing to the LCD” as a way to insult films they don’t want to watch. But this is, in a way, an even more dangerous situation. People who go to see The Avengers know that the film will “appeal to the LCD” (though I’d argue it’s still a decent film, despite that), but the exact same thing happens with films meant to attract supposedly LCD-proof audiences. There’s such a thing as an ultra-hip indie film that appeals to the LCD of ultra-hip indie filmgoers–and such films can be just as bad as mass-market LCD films. They’re simply bad in different, more pretentious ways.

    We need to realize: When we say a film “appeals to the LCD,” we think we’re saying “this film is targeted at a bad audience”; *but the problem isn’t with the audience the film seeks; the problem is with the way the film seeks its audience.*

    This is where I tried to bring in LCD versus GCD. You say this difference doesn’t have a clear real-world analog, since both would have to refer to “the set of characteristics of a class of people,” which is “really more of an averaging operator.” But how do we average people together to find their common characteristics? I know how to average things I can quantify; I don’t know how to average things I can’t. If you assume it’s possible to average people together, you’ve already made an assumption that will determine more about the result of your average than anything in particular about the people in your set.

    One alternative to an “average” is a “norm.” Norms, unlike averages, are, well, normative: they require us to make judgments, not just calculate things. I can appeal to average(mass-market) just by doing a lot of polling and product testing (“what *will* be popular?”), but if I’m going to do anything involving norm(mass-market), I need to ask myself “what kind of movie *should* the common man watch?” The difference between these two is far more important than the difference between mass-market and indie.

    TL;DR: It’s absurd to be proud of yourself for belonging to one audience rather than another. Every audience can be pandered to, and, conversely, worthwhile works can be written for any audience. So enjoy pop culture for what it is, look for the good in everything, and remember that almost nothing is all that great.

    If you find something that is all that great, hold onto it–but don’t go begging for more.

  6. Chris Wolfe permalink
    January 25, 2014 12:08 am

    In my mind some fascinating cases of “appealing to the lowest common denominator” are Shakespeare’s jokes for the groundlings and Aristophanes’ bathroom-type humor. The interesting aspect of those authors’ appeals to the LCD is that Shakespeare and Aristophanes KNOW most of the audience will recognize an appeal to the LCD and do not consider themselves part of it. And then Shakespeare and Aristophanes often use one those same clowns to teach some of the more profound messages of the play.

    Shakespeare and Aristophanes have a strange way of having their plays “seek their audience,” as you put it Joseph. Maybe they really weren’t appeals to the LCD after all

  7. David Simmons permalink
    January 25, 2014 12:17 pm

    I think if you asked people, they would admit that there is something worse than the “lowest common denominator” – namely, trying to be bad for the sake of being bad, rather than for the sake of achieving an audience. For example, in my opinion Ayn Rand falls into this category.

    I agree that calling a film “lowest common denominator” while still watching it is hypocritical.

    Your comment regarding the LCD of ultra-hip indie filmgoers is too abstract for me to agree or disagree with it – I can’t think of any movies to which this would apply. I suppose I’m not an ultra-hip indie filmgoer though.

    Here’s how I think of the “averaging operator” of “the set of characteristics of a class of people”: Suppose I want to find out whether a given characteristic is part of that class. I can poll people to see how many agree with it. That’s quantitative. To be a “common characteristic”, some percentage of people would have to agree with it – what percentage exactly is a little fuzzy. But the point is that every individual person will have characteristics which are not “common”, and these get washed away when you average, leaving you with blandness. That I think is the real implicit criticism of the “lowest common denominator” – instead of having an artistic vision (individual to the artist) and attempting to achieve it, thereby at least satisfying those who can appreciate your vision, you are just averaging.

    Which is, in a sense, another way of stating your last paragraph – norms (the vision of a single artist) are better than averages (appealing to the lowest common denominator).

  8. January 25, 2014 5:55 pm

    Well, I mostly agree, but I think there’s an important nuance left out of your account of how to “average” people. In your second-to-last paragraph, you say: “Suppose I want to find out whether a given characteristic is part of that class. I can poll people to see how many agree with it.” That assumes something about what characteristics we’re interested in: namely, that they’re things we can ask about.

    Certain things are, e.g. “Does X appeal to you.” We can poll over all Xs and see which ones are most popular (this is basically what market research firms do); for a large and diverse enough audience, the winners will usually be “sex and violence.” But of course we can do this for more specialized and “sophisticated” audiences too. Maybe the answer there isn’t “sex and violence,” it’s “irony and black humor”; a movie can be constructed to appeal to that average as well.

    But there are some characteristics (I want to say) that can’t be averaged, things like “How would you evaluate movie X.” That’s it demands an act of conscious judgment on my part, not a mere unconscious reaction. (Importantly, I think, it’s not a simple yes/no question.) So it can’t be predicted–my answer is part of how I decide who I am, and so it has to actually be asked for there to be an answer. This is aesthetic freedom, parallel to ethical freedom. Everyone has an answer to “Is it OK to do X,” and I can make up an answer to “What would you do in situation X,” but both of these are pretty much irrelevant to my personal development. My character is shaped when I’m actually in situation X-, and actually have to do something.

    Since I want to talk about it in this way, I’m somewhat hesitant to say that the alternative to “appealing to the LCD” is “having an artistic vision (individual to the artist).” I don’t think good art comes from having a special “vision” (what’s that?) that makes you a good artist; rather, it comes from something like a combination of technical ability and moral insight–insight not into yourself, but into other people. Artists need to know what they should demand of their audience. When we complaint about the LCD, we’re really complaining that the work doesn’t demand anything of us, other than our money.

    If it’s unclear why I’m insisting on these distinctions, it’s probably because my thoughts on this issue are strongly influenced by my unarticulated quarrel with Romanticism.

  9. David Simmons permalink
    January 25, 2014 7:03 pm

    OK, I think I agree with everything you said there. I feel like a movie meant to appeal to the average of “irony” (I don’t know about “black humor”) has a good chance of being better than a movie meant to appeal to “sex and violence”, though. But you’re right, they can both just be appeals rather than works that are morally demanding in the sense you described.

    I think when I said “artistic vision” I was subconsciously thinking of something like this: Breaking Bad is a good show because the producers had a good idea of what they wanted to do with it from the beginning, whereas Lost is not because they just tried to make it “interesting” but had no idea where they were going. But maybe that’s not completely relevant to the discussion (or maybe it falls under the category of “technical ability”).

    I would still say that a good work of art should be distinctive to the artist.

  10. David Simmons permalink
    January 25, 2014 9:02 pm

    I have a question though: If a good work of art forces us to make a moral choice, does a better work of art make people more likely to make better choices? For example, there’s a large number of people who thought that Walter White was actually a good guy (see Is this a failure on the part of the show? Or does the fact that we have free will mean that the most the show can do is present us with the choice, and naturally some people will make the wrong one? Does the idea of a “right choice” and a “wrong choice” even make sense here?

    Sort of the aesthetical analogue of the “problem of evil”. Possibly I’m reading too much into your comment about what art demands of us. But it does seem to me that good art does tend to ask moral questions, without necessarily answering them, in order to make us think.


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