Wednesday, December 28, 2005
Fake is better
Imagine somebody were to argue that Kool-Aid’s convenience outweighs the superior taste of real grape juice, and convinced society to switch en-masse to Kool-Aid. Imagine further that we grew so accustomed to Kool-Aid that grape juice began to taste somehow wrong, and we now preferred the taste of Kool-Aid. Sick, no? It happened, in music:
The twelve-tone equal-temperament (12-TET) system that we use today developed as an approximation to the first few elements of the harmonic series. The ratio of any two distinct tones in 12-TET is an irrational number, some power of the twelfth root of two (the half-step), and that irrationality creates dissonance. By contrast, scales whose frequencies are related by small whole-number fractions are minimally dissonant, because they synchronize – but we wouldn’t know, because we never hear such scales.
Fortunately, there now exists computer software that allows one to play in harmonically perfect scales from the computer keyboard. (“Harmonically perfect scales” = “just intonation” = “rational temperament” and other combinations thereof.)
I did a little experiment over Christmas, subjecting a few family members to blind taste tests of the two scales. I would play a major seventh chord (e.g. C, E, G, B) first in 12-TET, then in a scale for which the tones were perfect consonances (1, 5/4, 3/2, 15/8). Very consistently, everybody preferred the approximation to the real thing!
12-TET is a recent phenomenon, dating to around Bach, whose series of compositions, The Well-Tempered Clavier, was intended to showcase 12-TET’s capacity for modulation to any of twelve identical keys – all of them slightly wrong. Bach and others eventually convinced
But that is not what my family preferred about 12-TET, because I never changed key. They thought a single, static chord sounded better in 12-TET than with just intonation. Moreover, that reaction is typical of people who have just been exposed to harmonically perfect scales (according to I don’t remember who; sorry).
My uncle Steve, an occasionally-professional musician, said the 12-TET chord had more “soul”, while the other was “too consonant”. Too consonant to sound good. His comment, while provocative, is not a good defense of 12-TET: dissonance should be intentional, not ubiquitous.
Before 12-TET, composers had to be conscious of the frequency relationships between tones. Now they don’t – almost all of them identify the pitch spectrum with the piano keyboard, as if the latter begets the former. Composers have largely lost sight of the mathematical origins of the pitches they work with.
I think we have screwed ourselves out of a lot of good music. Can you imagine restricting dance to a lattice – and not just some abstruse branch of dance, but almost all the dance in the world? Or can you imagine visual art limited to 16 colors – and no longer because it is convenient, but because we have come to perceive colors from outside the grid as wrong? Can you imagine math restricted to some excruciatingly small subset of the real line? That last example actually happened, although we eventually saw our way past it. It is hard to imagine how we could see our way past 12-TET.
I realize I am being quixotic, but I really think this is sad.
Ways I might change my tune
I have so far complained about the near-total domination of 12-TET over more conscious tuning systems. Might there be a few silver linings?
(1) A few years back, I bought some music in just intonation, from the Just Intonation Network. It is indeed sublime. Perhaps exposure to such artistic heights, and frustration with my own instrumental resources, will conspire to drive me away from what has been a blood-sucking hobby.
But really, I’m not going to stop playing music, so that’s dumb.
(2) Maybe the 12-TET system prevents musicians from wasting time. There are, after all, a lot of other parameters musicians can work with – the words, the instrumentation, the cultural context – and the Pandora’s Box of true freedom might lead one to spend so much time on harmonic questions that the music is left too poor in orchestration, lyrical content, or social midwifery to accomplish anything. Indeed, much of the music I got from the Just Intonation Network could be accused of such faults.
Alas, argument (2) is also lame, because although it’s true that some people would be wasting there time working with just intonation (e.g. Bob Dylan), there will exist others whose time would be best spent there, given appropriate accomodations. Comparative advantages differ. I would like a world in which someone whose comparative advantage might lie in the exploration of rational scales would have access to keyboards made for its exploration. Presently, such people have to learn how to program, or else how to build instruments. That difficulty drives most of them to other pursuits.
Friday, December 23, 2005
Are marriage norms efficient?
Judge Richard Posner has hypothesized that the common law is economically efficient. He seems to be right that our intuitive notion of justice often aligns itself spookily well with the efficient rule that pops out of a cost / benefit analysis. On the other hand, there are many shocking examples (e.g. regulating the production of flu vaccines so tightly that the number of firms producing them drops from 30 to 3) of inefficiency in the common law.
I wonder whether cultural norms are similarly efficient. In this blurb I will consider two particular norms, regarding marriage: the expectation of an arranged marriage, and the taboo against divorce.
The qualities by which we would label a norm “good” or “bad” are the same ones by which we would judge a law: Law X is better than law Y if, on net, the total societal happiness under law X is greater than that under law Y. A norm against divorce, then, would seem to indicate that our society “believes” it will be happier if there are fewer divorces. (Whole societies don’t hold beliefs, but it can be useful to think of them that way.) This does not necessarily mean that society believes individual couples will be happier if they stay together (although that might be so); the existence of a norm against divorce suggests merely society’s belief that the couple, their children, their relatives, and everyone else dealing with the couple will on net be better off if there is no divorce.
I suspect the norm against divorce was efficient. It is weakening, and I think that is efficient, too. That is, I think it is more likely for the gains from a divorce to outweigh the losses today than it was a few centuries ago. Similarly, the expectation of an arranged marriage is less prevalent today than it was in the past, and I believe that is efficient as well – arranged marriages are less likely to work today than they were once.
The mass social phenomenon of falling in love is recent; one can watch it happening from the writings of Jean-Jacques Rousseau, who wrote from 18th-century
The question naturally arises whether an individual’s parents might be better suited to choosing that individual’s spouse.
For most of humanity’s history, people did not change social surroundings much. One generally inhabited either a town or a traveling tribe; either way, the social environment was stable. In such situations a family could gather all necessary information on a child’s marriageable peers. If so, and given that one’s ability to weigh future payoffs only fully develops in one’s mid to late twenties, a family might indeed be expected to consistently choose a better mate than the child him- or herself. Furthermore, in a town or tribe with little change in population over that child’s lifespan, the well-chosen mate selected by the parents early on is likely to continue to be the most appropriate mate years later. In these circumstances, a taboo against divorce would be helpful to the community.
Modern society is bigger, more complex, and subject to heavy traffic. Parents would have a difficult time surveying every candidate, because there are more qualities to consider (a point I will leave for a future post), more candidates, and more change in the pool of candidates. Further, certain kinds of information which are difficult for parents to judge have become more relevant.
For instance, one who expects to move a lot (what’s that statistic – the average American now changes jobs six times before retirement?) will care more about a spouse’s conversational potential than someone who can expect his buddies to live next door forever. For conversational potential to be high, mates’ interests must overlap. The range of potential interests today is wider than at any point in the history of our species, and the chance of overlap accordingly smaller. The search for a mate in a compatible social situation may be feasibly undertaken by one’s parents, but the search for a mate with compatible interests is difficult to proxy.
An individual’s mate may be optimal at one time, and become sub-optimal as other candidates are discovered. This can only happen in a society with significant human traffic, so it was less of a problem in pre-industrial societies.
Someone, for instance the Pope, who is categorically against divorce must argue that once one has committed to marriage, there can be no better use of resources than maintenance of the marriage. That is, the costs of divorce always outweigh any possible benefits. Aside from divine inspiration, what might lead someone to say that? Let’s consider the costs of divorce:
To divorce is to sacrifice a frighteningly large investment. Ceteris paribus, a nation full of investment-sacrificers will do worse than it would otherwise. Among those investments are some obvious ones (e.g. financial) and some less obvious ones (e.g. “firm-specific investments,” “on-the-job training” – the time spent learning how to make marriage with a particular individual go smoothly). Of course, the ceteris paribus condition does not hold; a nation that outlaws divorce creates a stronger incentive for people to choose mates that will last, and (arguing the other way) a nation full of divorcers may move from less efficient to more efficient uses of resources. (Imagine how efficient our labor force would be if employers were not allowed to fire and employees not allowed to quit.)
Another cost of divorce which former Popes might have had in mind when arguing that divorce is never worth it: Marriage was once largely a business decision. The families of spouses can expect to transact with each other regularly (not necessarily monetarily); they therefore have a strong incentive to pair offspring with other families that offer attractive transaction opportunities, and they usually succeeded. Thus, dissolution of a long-standing marriage could be highly disruptive to many people who were not direct parties to the marriage contract. Of course, this disruption-of-family-ties argument is weak today, because we are less reliant on family and more reliant on other institutions, such as the school, the workplace, the sports league, the interest group, etc – which seems to support the hypothesis that the weakening of the taboo against divorce is efficient.
The sacrifice of investments and the disruption of family ties are positive costs of divorce, but they are finite, and so (Papal admonitions notwithstanding) they could be outweighed by the potential gain from divorce.
The benefits of divorce are precisely the opportunity costs of marriage: the ability to do anything that marriage precludes, such as a different marriage, or life as an artist. When the life expectancy was 30, having children was something like one’s first and last accomplishment. Now, one may have time to do something else afterward, so it may be efficient to terminate the marriage contract once the project for which it existed is complete.
Monday, December 19, 2005
Gifts: good or bad?
Economists always knock the idea of giving gifts. The argument goes like this: If I give you a $20 pair of socks, they may be nice socks, and you may be happy to have them, but you’re not as happy to have them as you would have been to have $20, which you would have spent in some other way. Thus, purchasing a gift destroys wealth – at least if the gift is a nice pair of socks.
However, the same language and logic can provide an argument in favor of gift-giving, in certain circumstances. If I could find an object G such that the utility I predict you will receive from G is greater than the utility of anything you would have spent the money on yourself, then my giving G to you for the holidays is in fact efficient.
How often do circumstances favorable to gift-giving come about?
(If I’m right that you’ll like G more than anything else, then it would be equally efficient for me to simply convince you of that fact, leading you to spend your own money on G. I would have performed a valuable service, by delivering information that permits you to attain greater utility.)