Music theory, medieval & modern
Jun. 17th, 2024 07:54 am"Nice-sounding", or "pure", is shorthand for "small-integer frequency ratios", which the human ear seems to like. A unison is two exactly-equal frequencies; if you play two notes that are almost in tune, say 440 Hz and 441 Hz, you can hear a "wow-wow-wow" at the frequency of the difference, in this case 1 beat per second; guitar players routinely use this to confirm that two strings are really in tune with one another.
A pure octave is a 2:1 frequency ratio, e.g. 440 Hz and 880 Hz. A pure fifth is ideally a 3:2 frequency ratio, a pure fourth a 4:3 frequency ratio, a pure major third a 5:4 frequency ratio, and a pure minor third a 6:5 frequency ratio.
Pythagorean tuning, which as its name suggests is thousands of years old, is based on fifths and octaves: go up a pure fifth, then another pure fifth, and you're a little over an octave from where you started, so go down a pure octave, and repeat. Unfortunately, if you tune everything by pure fifths and octaves, you end up almost but not quite where you started: 12 pure fifths is a frequency ratio of 129.746, while 7 pure octaves takes you back down by a ratio of 128, leaving you at a frequency 1.36% (the "Pythagorean comma") higher than where you started. This is enough to be audible, even to the un-trained musician: all the intervals sound beautifully in tune except this one real clunker. Which means when you tune an instrument, you have to decide in advance which intervals you want to sound good, and which one can be horribly out of tune; in practice, you decide what keys you're most likely to play in, and pick an interval that's unlikely to show up in those keys.
Another problem with Pythagorean tuning is that getting all the fifths right gives you terrible thirds: a Pythagorean major third is a ratio of 1.266, over a percent "wider" than the ideal 1.25, while a Pythagorean minor third is 1.185, over a percent "narrower" than the ideal 1.2.
People have tried various ways to solve the problem: split the comma in thirds, so there are a bunch of beautiful intervals and three less-bad ones ("third-comma meantone"), or split it in fourths ("quarter-comma meantone"), or split it in sixths ("sixth-comma meantone"), or split it in twelfths so all the intervals are equally slightly out of tune. That's "equal temperament", the way most pianos are tuned today. In equal temperament, a fifth is a frequency ratio of 1.4983, about 0.1% narrower than a Pythagorean fifth; a fourth is likewise about 0.1% wider than a Pythagorean fourth; a major third is 1.26, almost a percent too wide; and a minor third is 1.19, almost a percent too narrow.
In the 15th-17th centuries there was a different approach, called "just tuning": rather than insisting on fifths über alles, maybe we should aim for getting thirds in tune. Combining a pure major third (5:4) and a pure minor third (6:5) actually gives you a 3:2 pure fifth, so you haven't lost that after all, and the thirds sound much better than in Pythagorean tuning. But tuning the thirds still doesn't help you get all the way around the scale and end up back where you started; some intervals will be beautifully in tune, while others are horrible, and you just try to place the horrible ones in out-of-the-way corners of the keys you usually play in.
By the way, modern singers, and players of un-fretted stringed instruments like violins, are taught to use just intonation, adjusting vocal pitch or finger position slightly so that each third, fourth, or fifth is in tune when you get to it, even though it may not be in tune with where the piece started. Since most musical pieces don't modulate very far from their original key, you can usually get away with this and end up in the same key you started in, but theoretically if you played on the violin a long piece that modulated all the way around the circle of fifths, you'd end up off by a Pythagorean comma from where you started; in particular, the notes you played on open strings at the beginning would have to be fretted at the end.
I was in an early-music workshop on Friday, and the teacher mentioned the "Berkeley manuscript", a 1370's-vintage music theory text currently owned by UC Berkeley. We had a copy (in facing-page translation) at home, so I've been reading through it in the last few days. In the first section the author describes the hexachord (or deducciones -- the word "hexachord" was invented retroactively by 19th-century musicologists) system, how the C, G, and F hexachords overlap, how and why one switches from one hexachord to another, how intervals that should be a whole tone in the gamut can be narrowed to a semitone by explicitly marking one note "mi" or "fa" (what we would now call "sharp" or "flat"), etc.
But in the fifth section of the treatise (which I haven't gotten to yet; this is what the teacher and the editor say) the author says a mi-fa semitone really shouldn't be half a tone, but two-thirds of a tone (a "major semitone"; the remaining third is a "minor semitone"). If each of the five whole tones in an octave is three units, and each of the two semitones is two units, that makes... 19!
So how well does this actually work, using modern mathematical techniques (roots and exponents)?
- A major third, ut-mi or fa-la, is two whole tones, or 6/19 of an octave, which is a ratio of 1.24469, less than half a percent off from a just major third at 1.25 -- about equally bad as a third-comma-meantone major third, although narrow where that was wide.
- A minor third, re-fa or mi-sol, is a whole tone and a major semitone, or 5/19 of an octave, which is a ratio of 1/2001, less than a hundredth of a percent off from a just minor third.
- A fourth, ut-fa or re-sol or mi-la, is two whole tones and a major semitone, 8/19 of an octave, a ratio of 1.3389, about half a percent off from a Pythagorean fourth.
- Similarly, a fifth, ut-sol or re-la, is three whole tones and a major semitone, or 11/19 of an octave, a ratio of 1.49376, again about half a percent off. (This had to be true, since the perfect fourth and perfect fifth are one another's inverses: if one was half a percent off, the other would have to be likewise.)
So this system has reasonably-good fifths, fourths, and major thirds, and very good minor thirds, and it has the conceptual advantage of equal temperament that you don't need to decide which key you want to sound best: any piece of music can be transposed into any other key and its intervals will sound the same as they did before.
If you wanted to build a keyboard in this system, it would probably have the usual seven white keys (A, B, ... G), with two black keys between A and B, C and D, D and E, F and G, and G and A, and one black key between B and C and between E and F.
Dream journal
Nov. 12th, 2023 07:36 amI have taught a two-hour class on the theoretical side of "what's a mode", and taken a week-long hour-a-day hands-on course about what each mode "feels" like and what's distinctive about each one, and I really didn't want to get into that much detail while whispering in a library, not to mention I had my own stuff to do there. But I started on the few-sentence explanation, involving playing only white keys on the piano.
At which point a female friend of the nebbishy guy (slender, probably in her 40's or 50's) walked over and said "And why can't he find any books about sets?"
Umm... there are LOTS of books about sets, and one can spend semesters or years of one's life studying them, but I wasn't about to get into that. So I said "Well, it helps if you text-search" [I mimed typing on a keyboard] "rather than asking aloud, or people will think you're looking for something else." About which there are even more books available.
Probably inspired variously by attending my friend Alec's "medieval music jam" last Thursday, at which he taught a little bit of "what's a mode", and by my visit to the farmers' market yesterday where I asked "What kinds of apples do you have?" and the young guy standing next to me said "There are different kinds of apples? I thought they were all just apples."
minority voting rights
Jun. 9th, 2023 07:19 amJustice Clarence Thomas, in dissent, complains that the Court cannot force a state to use race as a criterion in drawing district boundaries; racism is racism, even in the interest of a minority. Of course, his criticism on this point is one-sided and hypocritical: he seldom objects when a state uses race as a criterion in favor of a white majority, as long as they're not so foolish as to put it in writing. But what if the state didn't use race as a criterion in favor of a white majority? Would he have a legitimate point?
What would a truly race-blind redistricting system look like? It could use not geography, but something with no correlation with race or political party, such as your Social Security number or your birthday (as a number from 1-366). If your Social Security number, divided by 7, has a remainder of 1, you're in district 1; a remainder of 2, district 2; and so on. Absolutely and utterly race-blind, and disastrous for minority representation, because every district in the state would be 27% black; considering the importance of race in Alabama politics, it would be rare to ever see a black representative elected at all.
In Justice Thomas's dream world, the end of overt racism by government would lead to race becoming irrelevant to politics, people would win elections on individual merit, unaffected by race, and in the long run the percentage of black elected officials might resemble the percentage of blacks in the population. But in the real world, that could be a very long run indeed, and it's not clear that Alabama society would actually move in that direction at all; in the interim, black Alabamans would have even less representation than they have now.
The fundamental problem actually isn't about race; it applies to race, political party, sexual orientation and identification, and any other characteristic (call it X) that voters see as politically important. Single-seat, winner-take-all elections amplify majorities. If 60% of voters have characteristic X, an X-blind electoral map will generally give at least 60%, but usually far more than 60%, of the seats to candidates with characteristic X.
You can try to fix this with X-based redistricting, ensuring that only about 60% of the districts have pro-X majorities, but that has several problems. First, it leads to weirdly shaped, unnatural districts, such as the one that gave "gerrymandering" its name. Second, it requires government to explicitly consider X, and to discriminate on the basis of X, in its official proceedings, and that feels wrong, even if saying so puts me in agreement with Clarence Thomas (which also feels wrong). Third, it intentionally involves creating districts in which the general election is a foregone conclusion, and all the real action happens in primaries, which are dominated by more-extreme members of the respective parties. And fourth, what if political decisions depend not only on X, but on unrelated characteristic Y? Do we need to draw districts representing every possible combination of X and Y values?
Some democracies have tried to resolve this using proportional representation by party: if Party Z gets 30% of the votes, Party Z gets roughly 30% of the seats in a legislature (or one house of a legislature, or something). Unfortunately, this (a) fails to recognize any politically-important characteristic that isn't (yet) associated with a political party, and (b) gives political parties an official status in the (state or Federal) Constitution, whereas I would prefer to burn political parties to the ground.
A better answer is a voting system that tends to produce proportional representation (in a state's Congressional delegation or its state legislature), not only by race or party but by whatever characteristic voters feel is important. This cannot happen as long as each district elects a single winner independently; you need either a way for votes in one district to affect outcomes in another, or (more plausibly) multi-seat districts. And you need a way for each voter to express a preference, not only for a single candidate, but for a whole class of candidates who have characteristics that voter considers important (race, party, sexual orientation/identity, musical talent, business acumen, etc; the voter doesn't need to say or even consciously know what the characteristics are). That could in principle mean voters rank the candidates from first to last, or assign each candidate an independent numeric score from 1 to 10, or check off all the candidates they "approve" of, but you definitely need more than a single name per voter.
There are a lot of voting systems out there, each designed to solve a different problem and placing different priorities on the desirable properties of a voting system. I know of one designed specifically to achieve proportional representation, without pre-judging what characteristics voters will find important: Single Transferrable Vote, which is reminiscent of Instant Runoff but applied to multi-seat districts. Each voter ranks all the candidates in the district, and the system first looks at all the first-place votes. Any candidate with at least a threshold number of first-place votes is elected, and the ballots that listed that candidate in first place are removed from the pool, those voters having had their say. (There are variants involving redistributing the "surplus votes" over the threshold.) Of the remaining candidates, some with especially small numbers of first-place votes are eliminated, and we look at the second-place candidates on the ballots that chose them first. This continues in multiple rounds until all the seats are filled.
The system has many of the same problems as IRV: it pays more attention to whom you vote for than whom you vote against, thus favoring divisive candidates over consensus candidates; it's sensitive to "butterfly effects", in that a handful of votes changing hands between minor candidates can have a cascading effect on major candidates; and it's non-parallelizable, in that you can't do any substantial part of the counting work at the precinct level, you have to do everything globally. And it's even more complicated, difficult to explain, and difficult to justify than IRV. Among other things, there's no single obviously-right way to choose the thresholds for "elected" and "eliminated", nor any single obviously-right way to decide which surplus votes to redistribute. But it does a very good job of producing proportional representation, if there are enough seats per district (ideally, a single district for the whole state).
Is there a way to achieve something like proportional representation, but with a simpler, easier-to-understand, easier-to-justify voting system? I'm a big fan of the Borda count for single-seat elections; maybe that could be adapted? In the Borda system, if you vote for 7 candidates, your first choice gets 7 points, your second 6 points, and so on. So if there are 7 seats to fill, it seems natural to give them to the candidates with the top 7 point totals. The result won't be as nearly proportional as STV would produce, but you would expect a coherent, disciplined minority party to be able to elect a few top candidates over lower-ranked candidates of the majority party. But I don't think that works. I'm still working through the algebra, but it looks as though a majority party can fill all the seats, no matter what minority parties do, simply by telling its voters to rank the majority-party candidates randomly and ahead of all the other-party candidates. Furthermore, the majority party's advantage increases with the addition of minor, long-shot candidates -- not because they take votes away from either "real" party, but because they decrease the point ratio between a party's top-ranked and last-ranked candidates, and whenever this ratio is less than the ratio in party allegiance among voters, the majority party wins.
You could perhaps fix the above problem by doing a Borda count in which the number of candidates you can vote for (and thus the maximum number of points you give any one of them) is substantially less than the number of seats to fill. But that defeats one of the appeals of the Borda system: the fact that you can meaningfully downvote a candidate by ranking it last, and downvotes matter just as much as upvotes. I think Borda just doesn't work for multi-seat elections.
Re-reading George Boole
Mar. 13th, 2023 07:34 amThe first half of the book introduces a new way of writing logical statements and arguments based on the familiar notation of algebra (hence the modern term "Boolean algebra"), while the second half addresses the theory of probability, largely by generalizing from logic's two values (true=1 and false=0) to numeric values in between. I'm still in the first half on this re-read, so I won't say anything more about the probability stuff now.
When Boole wrote, not much had changed in logic instruction since Aristotle. The essential components of logic were statements of the form "All X are Y", "Some X are Y", "No X are Y", "individual x is Y", and "X means the same thing as Y", which can be combined in various ways called "syllogisms". Students memorized which of these syllogisms constituted valid inferences (e.g. "All men are mortal", "Socrates is a man", therefore "Socrates is mortal") and which invalid (e.g. "All men are mortal", "All fish are mortal", therefore "All men are fish"). Boole observed that most of this was really about classes and categories of objects, and he proposed to represent each class of objects with a letter, e.g. x, y, z, as in algebra. Putting two letters next to one another, the same notation as multiplication in traditional algebra, would mean the class of objects that have both properties, what we would now call the "intersection" of sets. Putting two letters together with a "+" sign in between would mean the class of objects that have one property or the other; a "-" sign would mean the class of objects that have the former property but not the latter; and so on. He's careful to point out that although the symbols are familiar from their use with numbers, and they obey some of the same rules as numbers, these are not numbers, and we can't assume that they obey all the same rules. For example, from xy = xz one cannot validly conclude y = z. But then, one couldn't conclude that in numbers either unless one knew that x ≠ 0.
The first place he makes what would now be considered a mistake is when he defines "+" to only be meaningful between two classes that are known to be mutually exclusive, e.g. even numbers and odd numbers, or men and women (remember, it's 1854). He's aware of the issue, and points out that some people might do things differently, but settles on this. As a result, he can say with confidence that x + y - y = x, which wouldn't be true using an inclusive definition of "+". On the down side, if I read him correctly, it means one can't tell whether a particular expression is meaningful or not without knowing about the meanings of the individual symbols. If I write something about the class of "sentient beings and fish", secure in the knowledge that there are no sentient fish, and then a sentient fish is discovered, I have to go back and re-write all my algebra.
Likewise, he defines "-", as in x - y, to be meaningful only when y is contained in x. As a result, (x + y) - y is meaningful whenever y is disjoint from x, while (x - y) + y is meaningful whenever y is contained in x; it's quite possible for either one to be meaningful without the other, it's impossible for both to be meaningful at once unless y is empty, and this doesn't seem to bother him. Indeed, he asserts that any sequence of additions and subtractions can be rearranged freely, for example x - y = (-y) + x, despite having never defined what "-y" means in its own right. This seems to me a serious problem, since there is no possible interpretation of "-y" that makes it behave as he wants it to behave: there is no class of objects which when "added" to x yields a result properly contained in x. But this doesn't bother him: as he warned earlier, there doesn't have to be an interpretation of every step in a chain of reasoning, as long as the starting and ending points are interpretable.
Anyway, from his notational definitions and straightforward observations about how logical statements behave in the real world, he concludes that xy = yx regardless of what classes x and y represent (analogous to the familiar "commutative" law of multiplication), x + y = y + x (analogous to the commutative law of addition), x(y + z) = xy + xz (analogous to the distributive law of multiplication over addition), and xx = x (which is decidedly not true of numbers, unless you restrict them to the values 0 and 1). Extending the analogy, he observes that familiar numeric notation has two special values 0 and 1, distinguished by 0x = 0 and 1x = x regardless of x. In the logical setting, the same is true if 0 represents "Nothing" (or in modern terminology "the empty set") and 1 "Universe" or "Everything". He then writes 1 - x, reasonably enough, for "all the things that are not in class x. Since he has already observed that xx = x, and the former can reasonably be notated x2, he has x2 = x, hence x - x2 = 0, hence x(1-x) = 0, whose natural interpretation is that no object can both have and not-have the same property -- a standard law of logic, but not one he has previously assumed, and he's managed to derive it from the idempotence of "multiplication". Cute.
Revisiting the question of inclusivity, he then translates "all the things with property x or y or both" as x + y(1-x), i.e. things that are either x or y and not x. Which works, but it strikes me with my modern-logic training as overly cumbersome. Extending it to three variables, you get things like x + y(1-x) + z(1-x)(1-y), and that way madness lies. Likewise, he translates "all the things with property x or y but not both" as x(1-y) + y(1-x).
Since classical logic is all about syllogisms, he then shows how to translate all the statements of classical syllogistic logic. "X means Y" becomes x = y. "All X are Y" is a little trickier, since he doesn't have a notation for "subset": he writes x = vy where v is a new class about which nothing is known except that it has elements in common with y (a stipulation that I suspect will get him in trouble later). Once he's got that trick, he can similarly write "No X are Y", or equivalently "All X are not Y", as x = v(1-y) and "some X are Y" as vx = vy (again, v is assumed to have elements in common with the thing it's in front of; he's not clear on whether the two v's in this equation are supposed to be equal to one another).
I had originally written, above, that he translates "No X are Y" as xy = 0, but on re-reading I realize that he doesn't give that translation. Which is a pity, since it fits cleanly into his notation and isn't semantically problematic.
[Update, 24 March:]
I've read a bit farther, and I think I have a better idea why he insists on defining + and - with semantic restrictions: because he really wants to use familiar algebraic manipulations as though logical statements literally were numbers. He writes all sorts of algebraic expressions, like (x-y)/(y-2z) + 3z2, evaluates them with specified values for the variables, and uses the results even when they involve 1/0 or 0/0. He wants to be able to do anything with logical variables that he can do with numeric variables, subject only to the constraint that every variable can take on only the values 0 and 1. So a definition that makes
x + y - y ≠ x ≠ x - y + y
would really mess up his plans, even though it has other philosophical benefits such as "if A and B are both meaningful expressions, then A + B and A - B are meaningful expressions regardless of the meanings of A and B." Indeed, he says very explicitly that he's OK with apparently meaningless expressions creeping in along the way as long as they disappear by the final conclusion. To me, this calls into question the validity of his inferences (if they don't even preserve meaningfulness, how can they possibly preserve truth?), but he's not trying to prove that his inferences are valid: he knows they're valid, because they're the familiar algebraic operations on numbers.
More later.
Home from Pennsic
Aug. 13th, 2022 05:47 pmI managed to get all the way through Pennsic without sunburn, but driving home (even though I was in the driver's seat most of the way, and therefore on the north side of the car) sunburned my hands to the "itchy" level.
At some point during Pennsic, D. picked up a class handout somebody had left behind, on Ptolemaic mathematics, so I read that when I needed a break from unloading and unpacking. Some of it is basic stuff about converting between decimal and sexagesimal, some is standard 9th-grade geometry involving similar triangles, and then they get into spherical geometry, which is much trickier, particularly when trigonometry hasn’t been invented yet. The workhorse tool seems to be something called Menelaus’s Theorem, which specifies a proportion among several arcs in a figure formed by the intersection of four great circles… except it isn’t really a proportion among the arcs, but rather among their chords. And even that isn’t quite right: it’s actually the chords of the double angles of each of these arcs. How did anybody come up with that? I guess the “double angles” part is a transformation between an angle whose vertex is on the opposite side of the sphere and one whose vertex is the center of the sphere, and the “chords” part allows us to work with nice simple planar triangles. Anyway, one can use this theorem to calculate the angle between equator and ecliptic at various times of year (at the equinoxes it’s zero), and thence the positions of sun, moon, and planets.
Then I picked up a book that’s been sitting on the coffee table since Christmas: Seb Falk’s The Light Ages: the Surprising Story of Medieval Science. And a good deal of the first chapter is about the same sort of astronomy: if you live in one place for years, you can observe the northernmost and southernmost places the sun rises and sets, observe the lengths of noonday shadows at various times of the year, and thence derive things like your latitude and the Earth’s axial tilt, which go into the aforementioned Ptolemaic calculations. Of course, everything is much easier if you assume the Earth is a sphere, and its orbit and those of the moon and all the planets are circles. I wonder how much more difficult it would be on a planet with a significantly eccentric orbit. Among other things, the warmest and coldest parts of the year might not correlate closely with the highest and lowest noonday suns, which would change the societal motivation for doing astronomy. And the obvious existence of a non-perfect circle in Creation would have interesting theological implications.
Last Monday we gave a concert in the performing arts tent, entitled “An Evening at the Salle des Ardents: Medieval Smoky Jazz”. All circa-1400 music. There’s a recording, which I need to retrieve from the recorder and edit on the desktop machine before posting it for public consumption. The opening number was a little chaotic, so I may substitute in a recording of it from a rehearsal, but most of the pieces went well and I think are suitable for posting. [Edit: see here. There are a number of wince-worthy moments in the concert, but a bunch of good moments too.]
To do before next Pennsic:
- Buy and/or build a new tent.
Roof panels are largely sewn together, but that's where they've been for several years; no significant progress in Covidtime. Then need to cut and sew together wall pieces, and build or commission poles, and make a bunch of rope attachment points and stake loops, and so on.
shalmestere points out that we're much more likely to have a new tent by next Pennsic if we order it from a tentsmith.
- Finish building new "birdcage" music stand.
Parts are mostly cut out, but they need to be stuck together, and we need an upright and a foot.
- Rebuild cooler chest.
The ends are fine, but the lid and sides are permanently warped, so it never actually closes, which defeats its purpose of hiding a styrofoam cooler; in addition, one of the lid's reinforcing battens which had been loose for years finally fell off completely this Pennsic. Also want to attach handles to the ends, which would make it enormously easier to carry.
- Lose enough belly to fit into my red linen pourpoint.
shalmestere has a photo of me in it from eleven years ago today, but when I tried it on at Pennsic, it was nowhere near closing.
- Make more braes and shirts
- Repair existing braes and shirts
- Make more points and aiglets
To do in the next week or two:
- Repair the drinking jug I knocked off the top of the piano while putting away krummhorns from Pennsic ✓
- Wash and put away clothes and dishes from Pennsic ✓
- Put away stuff that was left lying around from Pennsic packing ✓
- Replenish groceries ✓
- Cook normal food ✓
- Edit and post concert audio ✓
- Hem, and attach a linen collar-lining to, the gown that
shalmestere made me two weeks ago from one of her old dresses
- Add eyelets to the tailed hose
shalmestere made me a few weeks ago
- Finish building the fleece harp-case we started several months ago
- Contact harp-maker to say "yes, we're home from vacation now; feel free to ship the double-strung harp we commissioned two years ago" ✓
- Get Rid Of Stuff
I'd like to set us the challenge that every day we throw/give away one item that's been in our possession, but unused, for years. I figure after a month or two of that, we might notice a difference.
On Things Separate but Bound Together
Feb. 17th, 2022 12:52 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png)
But first, consider how sexual reproduction works in our world, in our species. I'm only talking about biological/chromosomal sex here, not anatomical or social or legal, and for simplicity I'm assuming that every individual is of one unambiguous biological/chromosomal sex -- no XXY or XYY trisomies. A man and a woman can produce offspring of either sex; both sons and daughters are genetically close to both of their parents (although sons are slightly closer to their fathers because that's where they got their unpaired Y chromosome). The result is basically a single gene pool for the whole species (except for mitochondrial DNA and a few Y-linked genes).
Now imagine that each human carried only one sex chromosome, rather than a pair, and that men and women were both capable of homosexual reproduction: two women could produce a (necessarily female) child, and two men could produce a (necessarily male) child. If this happened only occasionally, it wouldn't have much effect on the population. But if homosexual reproduction were universal, there would be genetic divergence, with males and females gradually becoming two mostly-separate gene pools and then two separate species. If there were an occasional instance of heterosexual reproduction against a mostly-homosexual background, the offspring would have genetic material from both pools, and thus (if the offspring reproduced) pull the two pools back together.
So, back to the sparrows. White-throated sparrows have two genetically-determined color morphs: brown-streaked and white-streaked, each forming roughly half of the population. If brown-streaked sparrows preferred to mate with other brown-streaked sparrows, and white-streaked with white-streaked, they would diverge genetically over time and we'd eventually have two separate species: brown-streaked and white-streaked. But in fact, females (both brown- and white-streaked) prefer brown-streaked males, while males (both brown- and white-streaked) prefer white-streaked females. As a result, white-streaked females and brown-streaked males get their pick of mates, which is each other, leaving the remaining brown-streaked females and white-streaked males to make do with one another. As a further result, essentially all mated pairs are either white-streaked-female to brown-streaked-male or brown-streaked-female to white-streaked male. Both sorts of pairs can produce any combination of male, female, brown-streaked, and white-streaked, so there's no genetic isolation and no speciation event: there are effectively four "sexes", but they're all still bound together into a single genetic population.
(Another point in the article: brown-streaked sparrows of both sexes tend to be more nurturing, while white-streaked sparrows of both sexes are more aggressive. Which has interesting implications of its own, but it's peripheral to the point I'm making here.)
By amazing coincidence, the same day that I read that article, I read Ursula LeGuin's short story "The Wild Girls", about a human society with three castes named Crown (nobility), Root (merchant class), and Dirt (slaves). In our world, wherever a caste system has arisen, cross-caste marriage is strongly discouraged if not forbidden, which (given enough time, and strict enough enforcement) would be expected to produce genetic divergence and speciation. But in "The Wild Girls", cross-caste marriage is mandatory: Crown men are only allowed to marry Dirt women, Dirt men with Root women, and Root men with Crown women. It's not entirely clear in the story, but I think a child's caste is always the same as its father's. In any case, there are effectively six "sexes", and they're all bound together stably into a single genetic population by the cyclic marriage rules. And although you can tell someone's caste by clothing, there's no genetic difference among castes and thus probably no way to tell the caste of a naked person. (Which raises plot ideas....)
And then I thought about fingerloop braiding. Consider a simple two-loop "braid", as discussed here. There are two interesting operations you can do: you can pass one loop through the other, and you can twist a loop on its own axis ("taking thy bowes reversed", as the middle English source says). If you just twist each loop on its own axis, never passing one through the other, you end up with two independent two-ply twisted cords, connected only at the anchor. If you only pass one through the other, never twisting either one, you again end up with two two-ply twisted cords, connected at both ends but otherwise independent. But if you pass one through the other, then twist one, in alternation, the two operations lock one another in place and you end up with a single four-strand braid. Likewise if you have three, or four, or five loops: if you only twist each one, you get five independent two-ply cords; if you only pass them through one another, but always "taking thy bowes unreversed", you get two 3-ply, 4-ply, or 5-ply braids connected to one another only at the ends. But if you pass loops through one another and reverse them, alternating operations reasonably often, you get a single bound-together braid of 6, 8, or 10 strands. If there's an occasional reversal in a mostly-unreversed braid, you get a cord with large "eyes", holes where the two halves run parallel but independent between one linkage and the next.
Do with that what you will.
A difficult book
Feb. 12th, 2022 09:30 amI dove into Anathem, read several chapters, and got bogged down. Came back to it a few months later, read a few more chapters, and got bogged down. Came back to it a year later, read a few more chapters, and got bogged down. Came back to it a year later, starting from the beginning to get a running start, and got bogged down. The issue seemed to be that Stephenson had spent an enormous amount of effort on world-building and forgotten about plot: nothing ever happened. And there was a remarkable amount of philosophical argument.
For whatever reason, last month I came back to it, started from the beginning to get a running start, and read a few pages whenever I was waiting for a long compile during my work day. And actually finished it this time. In fact, a great deal happens in the last, say, 200 pages, which feel sorta like an ordinary SF novel in their own right, but nothing that happens in those last 200 pages would make sense without the world-building and philosophical argument.
So let's see what I can summarize without excessive spoilerization. The book is set in a world quite similar to ours, and from the first page you can tell that people speak and think in a Romance-based language that's not quite any of the ones we know. It's an old culture: it has written history going back at least 5000 years, at a level of detail comparable to what we have for the past 500 years.
And it's a culture in which academics have consciously (and not entirely voluntarily) divorced themselves from popular culture, living sequestered in what are effectively monasteries. To maintain their perspective and avoid being polluted and confused by transient fads, many of the (co-ed) monastic/academics live under a rule that they can only leave the monastery, or even hear news from outside the monastery, once a year. To maintain even more perspective, other monastics live under a stricter ten-year rule: no information from the outside can get in except at ten-year intervals. An even more elite group live under a hundred-year rule, and the really hard-core ones live under a thousand-year rule; they've only opened their gates twice since the monastic rules were set up 3000 years ago. (The monastics aren't celibate, but they're sterile due to their diet; an unanswered mystery, which bothers the protagonist from time to time, is how the Hundreders and Thousanders stay populated.) There are even rumors of a sect living under a 10,000-year rule, which has another 7,000 years to go before they'll open their gates.
So some of the book explores the resulting sense of detachment. For example, the monastics and the outside world experience linguistic drift, so the longer they've been apart, the less they can understand one another. The monastics of different sects can mostly understand one another because they're all trained from the same books in Classical Orth, the analogue of Classical Latin. And when they do observe the outside world, it's all "the city had gone through cycles of growth and shrinkage over the millennia, and was currently in a shrinking phase, as shown by such-and-such characteristics typical of shrinking cities," or "the current governmental institution was dominated by a religious belief system with properties X and Y, but not Z," all properties identified in other religious belief systems over the millennia. The monastics have observed how the outside world thinks about them, and have categorized such beliefs into a dozen or more named Iconographies, each of which recurs every few hundred years (e.g. the monastics are a bunch of mostly-harmless mental patients, the monastics are secretly developing technology that will bring about a Golden Age when they finally share it with the rest of us, the monastics are a bunch of malevolent magicians scheming to take over the world, the monastics are a bunch of malevolent magicians who have already taken over the world but are keeping it secret, etc.) In short, the monastics view the current outside world as anthropologists might view an isolated tribe, comparing and contrasting it with other isolated tribes while trying to avoid getting thrown in a stew-pot.
Another thread of the book starts with a philosophical question which (in our world) is called Mathematical Realism. Mathematicians studying a topic like (say) prime numbers generally feel as though the concept of prime numbers was inevitable -- that any culture, or even any species, with sufficient cognitive powers would eventually come up with the same concept, and prove the same theorems about them. Indeed, a great deal of mathematics has this feel of inevitability to it: we're not inventing new things, but discovering things that in some sense were already out there waiting to be discovered; when we prove a theorem, we now know not only that it's true, but that it couldn't have been false in any conceivable world. A mathematics developed by small furry creature from the planet Zrax would have isomorphic conceptual structures to our own, differing only in names. But where is "out there"? Is it a world of Platonic ideals, made up of isosceles triangles and automorphism groups in the same way our world is made up of photons, electrons, protons, etc.? And how do we know this stuff is inevitable, as opposed to being artifacts of our own culture or historical accident?
This doesn't happen only in number theory or group theory. In theoretical computer science, people point out that the problem classes Rec [recursive], r.e. [recursively enumerable], and P [polynomial-time-computable] can each be defined in at least half a dozen simple, elegant, apparently unrelated ways that all turn out to be exactly equivalent, and conclude from this robustness that they're "natural" or "real" classes. Discoveries, not inventions. Which makes people more motivated to study them: it feels more rewarding to study something that "really exists" than to study a historical contingency that might have worked out any of a hundred slightly different ways instead.
An alternative view could be called Mathematical Formalism: this dispenses with the question of whether these mathematical concepts are "real", and says "we've agreed to manipulate symbols according to such-and-such rules; what results do we get by following those rules? What if we changed the rules slightly; what results would we get then?" This approach is summarized in the Bertrand Russell quote "Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." We "don't know what we are talking about" because there is no "about": we're just shuffling symbols according to arbitrary rules. And "we don't know whether it's true" because a statement's "truth" is a property not of the symbols but of a particular world (while in a different world, the same statement isn't true).
In our world, mostly starting in the 20th century, logicians have made serious progress reconciling these views. We can study a set of axioms and their implications under certain syntactical proof rules (we call this "proof theory"), and (following Tarski et al) we can also talk about concrete worlds that "model" those axioms and proof rules in the sense that, when you interpret each formal symbol as a particular object in or assertion about the world, the axioms all turn out to be true and the proof rules preserve truth ("model theory"). Logicians explored the difference between formal provability and semantic validity, showing that in certain simple systems they can be made to coincide perfectly ("Completeness"), while for complex systems of arithmetic they can't (to oversimplify Gödel's famous Incompleteness Theorems). For example, one of the tasks my thesis advisor set me (and at which I never entirely succeeded) was to take a particular result that a colleague had proven using model theory, and see if I could reach the same or a similar conclusion using proof theory instead. There's a branch of logic called "modal logic" which talks about not whether things are true but whether they could be or must be true; logicians for the past few centuries had dismissed this as meaningless mysticism, but it was given a precise proof system in the 1930's, and a precise semantics in the 1960's; see this overview article.
In the world of Anathem, the realist and formalist schools have been feuding for thousands of years, the formalists accusing the realists of quasi-religious mysticism and the realists accusing the formalists of pointless mental masturbation. The realists hypothesize that yes, there really is a world (or "causal domain") of Platonic ideals from which we get all our mathematical ideas, which means that information somehow flows from that abstract world to our concrete one. But once you've postulated two worlds with a unidirectional information flow, a natural question (to any mathematician) is "why only two?" Could there be several "concrete" worlds, independent of one another but all getting their ideas from the same abstract world? Could there be several abstract worlds, independent of one another but all contributing ideas to our world? Could there be an even-more-abstract world from which the Platonic world gets its "ideas"? Could there be an even-more-concrete world that gets its "ideas" from ours? Could there be an arbitrary DAG (directed acyclic graph) of worlds connected by information flow? And why does it have to be acyclic: could there be information flow in both directions? (Although if there's information flow in both directions between two worlds, they've effectively collapsed into one causal domain.) The formalists, of course, say this is all mystical nonsense. Until they meet visitors from one of these other worlds... but that's getting into spoiler territory.
A seemingly-unrelated philosophical thread in the book is about epistemology within a single concrete world. When you see a frying pan, or a friend, how do you know it's "the same" frying pan or friend as it was yesterday, or as it was when you looked at it from the other side? There's an argument from simplicity -- Occam's Razor (or "Gardan's Steelyard" in the world of Anathem) -- that it's simpler to hypothesize one persistent object than to hypothesize distinct objects that coincidentally have a lot of observable properties in common. But those observable properties aren't identical; they vary, but only in certain predictable ways. If I move clockwise around the frying pan, I expect to stop seeing what was on its right side, and see additional things adjacent to what was on its left side, and things fall down but not up, and people get older but not younger, and so on. All this requires building a complex model of how the physical world works, which implies a lot of conclusions about "what would happen if ...": if I looked at the frying pan from the other side, if the table the pan is on weren't present, if my friend were wearing different clothes, etc. In other words, you can't make sense of even a single concrete world without a lot of counterfactual reasoning -- or in other words, reasoning about worlds that are extremely similar to this one, but differ in one particular way. The realists of Anathem further hypothesize that this ability of the human mind is implemented using quantum computation: that rather than one classical brain exploring all these counterfactual worlds classically, it actually explores all of them at once in a superposition of quantum states, or in parallel universes, or something. And we're back to talking about multiple worlds -- except that now the information flow among them (via quantum interference), while limited, is symmetrical. Each version of your brain directly perceives the universe that it's in, as well as picking up some quantum interference from versions of itself in other nearby universes, and uses this to form counterfactual models and thus to reason about its own world.
So now that I've finally finished the book, I have some things to think about. And some logic to reread.
Dog leashes, braids, and groups
Nov. 21st, 2021 10:15 pmOf course, I have two dogs, whom I frequently walk together. Which means not only can each leash get twisted, but they can get twisted around one another, and you can't untwist either of them individually until you've undone the twist-around-one-another.
This sounds like a job for... Math! (Into a nearby phone booth, a quick clothes change, and out comes Algebraman!)
I'm sure that better mathematicians than myself have analyzed cord-twisting and braiding from a group-theory perspective, but (a) I want to try developing it from first principles myself, and (b) most of those mathematicians don't know about fingerloop braiding, so I might have something to add.
First, I should give a roadmap to what I'm doing and why. Much of what mathematicians do is "find a concrete phenomenon, erase as many details as possible, and prove things about the resulting abstract concept." If you can prove something about the abstract concept, then since the original concrete phenomenon was an example of the abstract concept, whatever you proved must be true of it. Furthermore, by erasing the details, you may find lots of other concrete phenomena, apparently completely unrelated, that turn out to be examples of the same abstract concept, so the same must be true of them too. So today's voyage is inspired by the concrete phenomena of twisting dog leashes and fingerloop braids, but it'll have things to say about lots of other concrete phenomena like piles of paper in my office and infinite checkerboards.
( Cut a lot of math-for-the-layman content )
Things get much more interesting when you have three or more bowes. But I'd better wrap this up and prepare for bed.
group theory and pop culture
Feb. 8th, 2021 07:35 amOne episode stands out in my mind because the resolution to the plot problems rests on a theorem of permutation theory. The professor has built a mind-swapping device, and two characters swap minds, then have second thoughts and want to swap back, but the way the device works, once a pair of minds have been swapped, that exact pair can never be swapped again. But all is not lost: they can swap through a third party: (a,b) = (a,c)(c,b)(a,c). No, wait: that repeats the (a,c) pair; maybe a fourth party? Before they can explore that possibility, the third party runs off in mid-sequence, and various other swaps happen for other motivations... eventually (IIRC) eight different characters are all in the wrong bodies, and would all like to get back home. Is this even possible?
In the penultimate scene, two black NBA players (why? because it's Futurama, of course) show up, do a bunch of figuring on a blackboard (which they still have in Futurama), and announce that no matter how screwed-up the permutation of minds and bodies, it's possible to get back to the original state with at most two "extra players". Which they do, in a dizzying montage of swaps that leaves everybody in the right body, and they all live happily ever after.
OK, so the theorem is "for any sequence of swaps on a set of n objects, there's an additional sequence of swaps (on the original set plus at most two additional objects) that inverts the original sequence, without the same swap ever appearing more than once in the combined sequence." Why is this true, and why two additional objects -- one would have expected either one or infinitely many? And how does the number of swaps grow? I guess it can't grow more than quadratically in n because there are only C(n+2,2) = (n+2)(n+1)/2 possible swaps on n+2 objects without repeating.
The n=2 case from above: (a,b) can be inverted by (a,c)(b,d)(a,d)(b,c) without violating the no-repeats rule. And it really can't be done with only one additional "player", as one can see by exhaustive search: there are only 3 possible swaps on {a,b,c}, and neither (a,c), (b,c), (a,c)(b,c), nor (b,c)(a,c) is equivalent to (a,b). (Side question: how big is this "exhaustive search"? There are only C(n+1,2) = n(n+1)/2 possible swaps, but then you have to look at all permutations of at most this many swaps... how many permutations are there of at most m objects, where m=C(n+1,2)? Let F(m) = 1 + m + m(m-1) + m(m-1)(m-2) + ... + m!; then F(0)=1, F(1)=2, F(2)=5, F(3)=16, F(4)=65, F(5)=306.... nothing obvious except that it grows at least as fast as m!. I should know this, or be able to solve it; I've taken combinatorics, thirty years ago. I'll get back to this.)
It'll be difficult to do the underlying theorem by induction because the no-repeats rule imposes a global constraint that doesn't lend itself to self-reduction. But maybe it's simpler than that: just grab the last swap in the sequence that needs to be reversed, label it (a,b), label the "extra players" c and d, do (a,c)(b,d)(a,d)(b,c) as above, and iterate. We know that nothing involving c or d was in the original sequence, so that doesn't violate the no-repeats rule, but it's not obvious that nothing involving a or b has already been used in the "undoing" sequence. We know that (a,b) doesn't appear twice in the original sequence, and if that's the only occurrence of either a or b in the original sequence, the above works, but it's likely that (a,*) and (b,*) each appears more than once in the original sequence, which could use up our allotted (a,c), (a,d), (b,c), and (b,d) swaps. I'll get back to this.
Who knew he had it in him?
Feb. 8th, 2017 07:43 amPresident Trump seems to be on his way after only two weeks, with an illuminating example of the Alternative Facts approach to mathematical proof.
dream journal
Dec. 16th, 2016 06:50 amWhy I'm having dreams about a math colloquium in the first place, much less one about this particular class of matrices, I don't know....
on voting systems
Oct. 30th, 2016 07:46 amI find this last argument peculiar because it's the opposite of an argument I was going to make. Voters whose first choice got knocked out early do have their votes counted in the contest between the final two. The problem is voters whose first choice is one of the final two: nobody ever even looks at their preferences other than the top. Consider an election among Alice Awesome, Peter Prettygood, Oliver OK, and Dr. Evil. If Awesome and Evil are the final two, and Alice is your first choice, it doesn't matter whether you put Dr. Evil in second, third, or fourth place because nobody will ever look at that part of your ballot. IRV pays attention to whom you like, not whom you dislike.
Related to this is another feature of IRV: it tends to favor divisive, polarizing candidates over broadly-acceptable, unifying candidates. Suppose, for example, in the above election Peter Prettygood was the second choice of every single voter in the country, although they divided bitterly between Alice Awesome and Dr. Evil for first choice. The argument could be made that Peter Prettygood is the best candidate to lead the country, because everybody finds him acceptable -- yet Peter Prettygood is the first one knocked out of the race because he's not anybody's first choice.
None of the readers seemed to question the equation of "ranked voting" with "instant runoff voting". In fact, "ranked voting" is about how you cast your vote, while "instant runoff" is one of several possible ways to count ranked votes, the other two leading ones being Borda and Condorcet. Condorcet says "for each pair (A,B) of candidates, how many people prefer A over B?" and declares the election for whichever candidate is preferred over the largest number of other candidates. Borda says "your first-choice candidate gets 4 points, second choice 3 points, third choice 2 points, and fourth choice 1 point" and declares the election for whichever candidate gets the most points. Both of these systems pay attention to all of your ballot, not just your first choice, and both of them would be likely to elect Peter Prettygood in the above scenario.
Of course, both Borda and Condorcet have better mathematical properties, e.g. "if your preferences are the exact opposite of mine, then your vote and mine exactly cancel one another out", which isn't true of IRV or single-vote plurality. But that's perhaps of more interest to mathematicians than the general public.
Not quite a book review
Jan. 10th, 2016 09:24 pm![[livejournal.com profile]](https://www.dreamwidth.org/img/external/lj-userinfo.gif)
Which is almost certainly true. I've actually had occasion to use a little bit of calculus on the job (in doing analysis of algorithms), and in answering idle-curiosity physics problems, and I've used trigonometry to design tents, and I've used both trigonometry and linear algebra to write graphics programs, but 99.99% of the U.S. population will never need to do any of those things, either on the job or in private life. Most people need to be able to do arithmetic (with the aid of a calculator, but they need enough of a feel for numbers that they can tell whether the answers are at all plausible), and read a graph, and it would be nice if they knew that correlation isn't causation, and what statistical significance means. As a logician, I'd like it if ordinary citizens knew that "not all cats are grey" is equivalent to "at least one cat is not grey", and that "if that's a duck, then I'm Henry Ford" is not equivalent to "if that's not a duck, then I'm not Henry Ford".
Anyway, the author documents vast numbers of students whose only academic problem is an inability to pass such irrelevant math classes (middle school, high school, or college), but who are denied the opportunity to study Shakespeare or Swahili or spot-welding. He points out that most of the doomsaying about an imminent shortage of STEM-qualified workers comes from employers who have a strong vested interest in creating an overabundance of such workers. And he likes to illustrate things with sample test questions.
Here's a question he likes:
A rectangular-shaped fuel tank measures 27-1/2 inches in length, 3/4 of a foot in width, and 8-1/4 inches in depth. How many gallons will the tank contain? (231 cubic inches = 1 gallon)
(a) 7.366 gallons
(b) 8.839 gallons
(c) 170,156 gallons
He likes this because it tests "did you read the question carefully?" -- specifically, did you convert 3/4 of a foot into 9 inches -- and do you know what needs to be multiplied and what divided? He says if you failed to notice the "feet", you would get the incorrect answer (a) (in fact, he's misplaced a decimal point: you would get .7366 gallons, which you should also be able to rule out through common sense because a tank that big has got to hold more than a gallon!)
Here's a question he doesn't like:
Two charges (+q and -q) each with mass 9.11 x 1031 kg, are place 0.5 m apart and the gravitational force (Fg) and electric force (Fe) are measured. If the ratio Fg/Fe is 1.12 x 10-77, what is the new ratio if the distance between the charges is halved?
(a) 2.24 x 10-77
(b) 1.12 x 10-77
(c) 5.6 x 10-78
(d) 2.8 x 10-78
I have to confess I do like this question, because it doesn't require doing any arithmetic at all, only remembering that both gravitational and electrical forces follow an inverse-square law. Although if you have two masses of almost 1032 kg half a meter apart from one another, they're both black holes and you have bigger things to worry about than measuring the forces between them.
However, I don't see a lot of benefit in asking this question on an MCAT (which is where it allegedly came from). Yes, identifying the relevant and irrelevant features of a problem is important to a doctor, but physics isn't.
For a certain board game, two dice are thrown to determine the number of spaces to move. One player throws the two dice and the same number comes up on each of the dice. What is the probability that the sum of the two numbers is 9?
(a) 0
(b) 1/6
(c) 2/9
(d) 1/2
(e) 1/3
Again, he doesn't like this question, and I do: it requires no arithmetic, no probability, no combinatorics, only the ability to see past the irrelevant stuff to what matters.
Is this what they call a "trick" question? One that requires common-sense reasoning, not just the application of a memorized procedure? If so, I'm all for them.
Anyway, I've only read a quarter of the book; we'll see what else he has to say.
Nothing new under the sun
May. 20th, 2015 02:46 pmDecades later, I read in Isidore's Etymologies:
Numbers are divided into even and odd numbers. Even numbers are subdivided into these categories: evenly even, evenly odd, and oddly even.... An evenly even number is one that is divided equally into even numbers until it reaches the indivisible unity, as, for example, 64 has 32 at its midpoint; 32 has 16, 16 has 8, 8 has 4, 4 has 2, 2, has 1, which is an indivisible singularity. An evenly odd number is one that can undergo a division into equal parts, but then its parts cannot immediately be evenly dissected, like 6, 10, 38, 50. As soon as you divide this kind of number, you run into a number that you cannot cut evenly. An oddly even number is one whose parts can be divided equally, but the division does not go to the point of one, like 24.
(From The Etymologies of Isidore of Seville, transl. by Stephen A. Barney, W. J. Lewis, J. A. Beach, and Oliver Berghof, ISBN 978-0-521-83749-1, Cambridge University Press 2006.)
OK, so Isidore beat me out by 1350 years....