The Subjectivity Of Everything: Music And Math Edition

Recently from a combination of world-weariness, pandemic-weariness, and me-weariness, I was seized by a desire to think about something abstract, useless, inert, and intellectually challenging. "I know what I'll do," I thought. "I'll buy a book about math."

I went to a small bookstore and asked, "Do you have a math section?" No. But there among the science books I came upon Music By the Numbers by math historian Eli Maor. I thought it would be about music theory and the mathematical underpinnings of classical music structures -- something I vaguely remember being interested in when I read about it in Gödel, Escher, Bach as a teenager forty years ago. But it was more about the fundamental mechanics of western music's organization of sound. Where did all these notes come from?

I knew that with a string, half as long means an octave higher. What I learned first is that the Pythagoreans and their followers based an entire scale on their idea of pleasing fractional intervals. The full story is a bit complicated, but taking fourths and fifths as a starting point, you can create a scale with a version of the "whole tone" interval we're familiar with (C to D, for example) based on a 9/8 ratio of a note to its predecessor (and a half-tone with ratio 256/243).

Maor says that while this way of creating a scale is mathematically elegant, it is out of step with the pitches produced in harmonic overtones -- and thus with the fundamentals of acoustics itself. From a philosophical point of view, Maor argues that their obsession with formal beauty led Pythagorean followers into a self-circular mathematical maze: by insisting on mathematical simplicity, pleasing ratios, and no irrational numbers, they "subject[ed] the laws of nature to their ideals of beauty."

But the more amazing moment for me came later, when I learned about tempering. As I kid, I knew about Bach's Well-Tempered Clavier, and I used to enjoy imagining that the well-tempered clavier had replaced a sour and irritable keyboard instrument known as "the ill-tempered clavier." But in all these years, I never learned what tempering is.

The background for tempering in the Western context happened around 1550, when a new scale was created bringing in intervals of thirds and their inversions. This "just-intonation" scale has intervals close to the ones we'd find on a modern piano. Maor argues that as it is based on the first six members of the natural harmonic series, and has pitches corresponding to the natural harmonics of musical instruments, it is acoustically and empirically superior to the Pythagorean scale. You could say that it has formalism that is not so much mathematical but rather musical.

However: a crucial feature of the just-intonation scale is that not all the tone ratios are the same. For example, in a C-major scale, the ratio of C to D would be 9/8 and from D to E, 10/9.

From a musical point of view, Maor argues, this is as it should be. It's from the practical and social point of view that these differences became a problem. For instruments with keyboards and fixed holes, a just-intonation tuning in one key will have notes at slightly different pitches from those in another key. Increasingly, people wanted to play together, with multiple instruments all at once. What if a group wants to play one piece in one key and another in another? The workarounds were complicated: early harpsichords had multiple keyboards, each tuned to a different key, for pieces written in different key signatures.

"Tempering," then, is creating a scale with even divisions, all the same. You just divide the octave into 12 equal semitones. Now all the instruments can play together, and playing in C is the same set of notes as playing in any other key.

Math people may already see that, unlike the just-intonation scale, the ratio of a note to its predecessor in the tempered scale is based on irrational numbers -- numbers that cannot be expressed as fractions. Having even divisions requires dividing the scale into equal parts -- in this case, 12 equal half-tones -- and to be equal, each ratio between a note and its predecessor in the sale would have to be the twelfth root of 2 to 1.

Maor: "This irrational number would have been regarded with horror by the Pythagoreans, as it cannot be written as a ratio of integers." !!!

Leaving aside Pythagorean worries, Maor describes tempering as "an acceptable compromise" between "the dictates of musical harmony" and "the practicality of playing a piece on the keyboard." The difference between the tempered semi-tone and the just-intonation semi-tone is just barely within what human ears can discern, a difference "most musicians were willing to live with." Maor alludes to a suggestion that Bach's Well-Tempered Clavier was partly PR: written to convince his fellow musicians of the benefits of this sociable way of organizing everything.

I found myself astonished by idea that the structural simplicity I associate with piano keys is the result of an "acceptable compromise" to solve practical problems of musicians playing together.

From a personal point of view, I guess I thought that the Western division of the octave into 12 equal semitones was somehow linked to the fundamental structure of music and sound. Not that it was the only way to exploit that structure -- I've always known that non-Western music was organized differently -- but that it was one way. Now this book says not only is that not so, but it's even weirder: there is structure of music and sound, and those equal piano divisions are really, deeply, not it.

From a philosophical point of view, the analysis provides an interesting reminder about the subjectivity of concepts like simplicity and elegance. For the Pythagoreans, these concepts translated to rationality. For the just-intonation fans, they relate to harmonics and ratios. For the temperers, it's like brute force to make it work -- hey, make 12 that are exactly the same, whether they are rational or not.

From a mathematical point of view, it's striking to see how slippery the difference between natural numbers and other numbers can be. The nineteenth-century mathematician Leopold Kroeneker is famous for having said "God made the integers; all else is the work of man." But when you can describe the tempered scale in terms of "12 equal divisions" or "12 irrational tone ratios" -- well, things start to seem less clear. Even in math, simplicity and elegance can mean one thing in one context and another in another.

From a sociological point of view, I found myself wondering: did books like this used to be more common, or more talked about, or something? I feel like Gödel, Escher Bach was a bit splash of a book, right around the time of other math books like the Gleick book Chaos. But now my bookstore doesn't even have a math section. Did the world change, or is it something about me that's different?

Advice Of The Pandemic, Ranked!

Among the by-products of the pandemic has been a tsunami of advice, especially about living your best life in lockdown. Was it any good?

Advice: if you're working from home, keep your routines
Early on, the news reports were that people working from home were flailing: cleaning the kitchen when they should be working, bingeing shows when they should be eating, eating when they should be cleaning the kitchen. Was this ever true, or was it just irresistible to imagine people collectively unable to get their shit together?

The advice was predictable and perfectly capitalistic. Keep your routines, they said. If you usually dress up and wear full make-up to work, get up in time to dress up and do your make-up, they said. If you usually commute to work, walk around the apartment for 30 minutes before you sit down at your desk. Take a virtual coffee break with your co-workers.

I thought to myself: who would be so stupid as to add the old burdens of their old life to the new burdens of their new life? Me, I worked in my exercise clothes and took naps over lunch. Did this advice work for anyone?



Verdict: terrible.

Advice: go easy on yourself
This one is complicated because clearly, for some people, "go easy on yourself" was the perfect advice. I'm here to tell you about the rest of us.

As for so many people, in March 2020 my job became an endless series of Zoom-style meetings, unanticipated problems, and difficult conversations. Early on, I thought "I'll go easy on myself" -- instead of trying to do things during weekends or early evenings, I'd spend the time "resting" or "hanging out."

Three weeks of this and I was mired in despair. My always-present sense of my life drifting away and passing me by intensified; thoughts in the "what am I doing and what is the point anyway" category buzzed around my brain like a bunch of gnats. I felt the mold between the bathroom tiles mocking me. With extra free time, my irrational body self-criticism meter ramped up from the blue "Guarded" category into the orange "High Alert!" range.

The advice to go easy often reminds me of some advice my mother once gave me about cleaning. I was complaining that on the occasions when I was on my own and cooking, I hated having to cook *and* having to clean up, which does feel to me like an outrageous burden at the end of a long day. My mom -- whose motto in life was "If you don't want to, you don't have to" -- told me I should clean up the dishes in the morning, and that if I didn't want to wake up to a messy kitchen, I should just "put the dishes in the oven" after dinner. I'm sorry, but dirty dishes in the oven? What kind of a bizarre suggestion is this?

Anyway, to stay sane in the pandemic, I switched into "mental discipline" mode. I scrubbed the floors. I did a zillion live-stream workout, yoga, and ballet classes. I made an elaborate system of calendar alerts. I recommitted to Duolingo.

Obviously, for some people this particular challenge didn't arise, because they were working massive hours or juggling child care or whatever, and obviously "go easy on yourself" makes total sense in that context. For other people, "go easy on yourself" was just what they needed to hear for whatever reason. It just didn't work for me.  

I feel like there's a lot of pointless social aggression out there between the "go easy" people and the "mental discipline" people. Can we please have a détente on this issue?

Verdict: variable.

Advice: create a Zoom social spot
This is the idea that if your job is virtual meetings and your social life is virtual meetings, you should do those two things in two different physical spaces in your home.

I love this advice, because it is geared toward people with lizard brains and I am 100 percent a person with lizard brain.

Virtual interactions give me a special kind of social anxiety. I'm not sure why, but one possibility is that when I'm with people, I'm used to relying a lot on subtle physical and facial cues in figuring out how to interact. Also, I feel like in a room with people, I enjoy the small pauses and silences -- I've always been good at being together and being quiet -- but somehow on a screen, I feel like I have to fill the air with talk, like some kind of demented TV show host. Ugh.

Because I have lizard brain, the layers of Zoom-social-anxiety particular to work-Zoom are queued right up for me when I'm in my work spot -- the chair, the wall behind me appearing on my own screen, the light bouncing off the desk in just such a way. What a relief to take the laptop to the dinner table for the friend-Zoom!

Verdict: the best.
 

Why Do I Care About The Gender Of Elena Ferrante?

When I first encountered the kerfuffle around the true identity of the novelist Elena Ferrante, I had two immediate and diametrically opposed reactions. First, smugness: I never concern myself with the autobiographical details of authors I love. So who cares? Second, freak out: wait, "Elena Ferrante" might not be a woman?

Normally I use mental discipline to avoid finding out about fiction writers. If I love a novel, thinking about which parts might be based on real life leads me down a mental rabbit hole of pointless philosophical questions. "If that part is real, did it really happen like that? Or is the author changing the details to manipulate me into feeling one way or another about a thing?" Then I'm like "What do you mean, 'manipulating,' anyway"? Talk about pointless and dumb.

However, I have also been shaken to learn certain facts about books I love, especially books representing the experience of women, especially books representing the experience of women and sexuality.

As a young person, I fell in love with Colette, the early twentieth-century French writer whose work and life challenged gender norms and all other kinds of conventions. Colette is best known now for the book that inspired the movie Gigi, but if that's all you know about Colette then you have the wrong idea -- as so often happens, the novel is darker, weirder, and more interesting. Colette is also famous for her many short stories. But the books I feel hardest for were the Claudine series.  

Over the course of the four Claudine books, Claudine grows from a precocious and rambunctious fifteen-year-old into a comfortably bourgeois but also free-thinking and unconventional adult. The first book, Claudine at School, gives an amusing side-on look at French social life in the provinces and the complex emotions of mid-adolescent feminine life. Among other complexities, Claudine has a massive crush on her young school teacher Aimée; they have a budding sensual romance when Aimée throws her over for the head schoolmistress, mystifying Claudine and giving her minor heartbreak. Aimée's younger sister Luce develops an unreciprocated love for Claudine, who doesn't understand her own desire to hurt Luce and cause her pain.

The way Claudine has romantic feelings for women and men and the way at fifteen she is both highly sensual and also a child both profoundly resonate with me. So I was disturbed to learn that the first version of Claudine at School may have been a more chaste and childlike story, and that Claudine's husband -- the writer Willy, who first told her to write up her memories and initially published the Claudine books under his own name (!), may have told her to tart them up, to liven them up with sex. I don't have sophisticated thoughts about this, but the idea that my favorite things were in there because some guy wanted them in there made me feel sad and weird.

Then a similar thing happened with erotic novel Story of O. That's not a book I loved (and the kind of submission/dominance thing in that book is not my kind of thing), but I always appreciated Story of O as a frank and explicit presentation of a certain version of positive sexuality from a woman's point of view. How many stories depict a woman being sexually adventurous and just basically enjoying it and having a great time? It's not a lot.

So, again, I was unsettled to learn late in life that while the author is a woman, the book may have been originally written for a man. That is, the author was in a relationship with a man, and wanted to get his attention, to turn him on, and to flirt with him. Again, I don't have sophisticated thoughts about this, but I find it disappointing to think that this, too, like so many things depicting women's sexuality, was ultimately crafted to appeal to a man's taste.

I loved Ferrante's Neapolitan novels, and I found The Days of Abandonment conveyed vividly what if feels like to be trapped in your life and in despair. Like Claudine at School, The Lying Life of Adults shows a textured experience of girlhood-womanhood adolescence.

It's been suggested that Ferrante may be a woman, or may not be a woman, or may be a woman and a man writing together. I feel like there's some sense in which I shouldn't care: the books are the books whoever the author is, and Ferrante herself has suggested that it's misogyny and sexism that makes people want to know.

But I also can't shake the feeling that the thoughts and feelings of people who aren't women so often affect what gets presented as the truth about women. It happens in pornography, it happens in movies, and it happens in books; given the way social gender norms work, it probably happens whenever there is an audience. While the books are the books whoever the author is, I also don't want the books by Elena Ferrante to be another example where the tastes, attitudes, and feelings of people who are not women are shaping narratives of our woman- and girl-hood experience.

What Compelled Me to Reread Dracula During Lockdown?

I don't know how it started, but I think it was something in the NYT crossword, something to do with Keanu Reeves. My partner said, "Keanu Reeves was in Dracula." Wait, what? From the miracle of modern self-surveillance, I know that I looked it up and learned that Keanu Reeves did not play Dracula (thank god) but rather Jonathan Harker, the lawyer. Whether Gary Oldman is any less WTF as Dracula I leave as an exercise to the reader.

I have a long history with Dracula, a book that obsessed my father when I was a kid. My father was the kind of guy where a lot of things were like 80 percent jokes and 20 percent serious and others were 20 percent jokes and 80 percent serious and it was always a little murky, probably even in his own mind, where we stood on things. After his annual rereading, my father would sleep with garlic under his pillow out of fear. You might think that's the 80 percent joke, but this was a man whose nightmares tended to actual devils actually chasing him, so I'm not so sure.

Later, I had the widely shared adolescent girl experience of being "into" vampires, whatever that means. I watched the movie The Hunger over and over, and read a lot of Anne Rice. My first reading of Dracula, around that time, I had a vivid sense of the erotic in the vampires' ways -- you may not remember this, but these are literally described as "voluptuous" in the book. Poor Jonathan Harker, on meeting the women in the castle who want to drink his blood, describes their voluptuousness as "both thrilling and repulsive." I remembered the book as basically anti-vampire, but not in any particular way.

On this rereading, by contrast, I felt the full weight of the Christian anti-sex moralizing. As Lucy's appearance begins to shift toward the vampiric zone, her friends are disgusted by the new sensuality of her face; after they destroy her body to free her from Dracula's spell, they're thrilled to see her previous sweet, pure expression and physiognomy return. A "diabolical sweetness" allows vampires to express love and desire to seduce new recruits. I guess I'm as against killing people and sucking their blood as anyone else, but this framing struck me as depressing and dumb.

Rereading Dracula during lockdown, I couldn't help but notice that it's partly a travel book. The best part of the story is when Jonathan Harker first goes to Transylvania, ostensibly to help the Count with some clerical matters, and slowly gets caught up in Dracula's web. There are trains, and ships, and transfers to carriages, and rides on horses. I was like, "Oh yeah -- travel!" Of course, so much of that late nineteenth-century mode, of going to truly unknown places and being completely cut off from anything familiar, is totally lost to us now. These days, Jonathan Harker could read on Yelp, "Castle looks interesting but they will kill you and drink your blood. One star."

The most melancholy aspect of my reading experience was the way that it was stupidly mediated by all the ridiculous parodies, take-offs, and remakes of the Dracula story that I've encountered in my time. Chief among these was the 1979 film Love at First Bite, which as a kid I saw on TV multiple times and found hilarious and awesome. How could I not love a vampire movie that featured the classic disco song "I Love the Nightlife"? There's also Young Frankenstein, which takes place at a castle in Transylvania for who knows what reasons. Memories of those films made me see Renfield as an annoying twerp, the horses and wolves under Dracula's command as side-shows, and even the Count himself in the light of a ridiculous show-boater.

Overall, the whole thing was a sadder and less fun affair than I'd hoped it would be, though whether that's because I'm old, or because of lockdown, or because the book isn't really that good is totally unclear.