Saturday, July 22, 2017

Pi Approximation Day 2017

Today is Pi Approximation Day, so-named because July 22nd -- written internationally as 22/7 -- is approximately equal to pi.

It also marks three full years since my very first post here on the blog. Yes, I've declared Pi Day of the Century -- March 14th, 2015, to be the ceremonial start date of this blog. But the chronological start date was July 22nd, 2014. I like celebrating both Pi Day and Pi Approximation Day -- including eating pie both days -- and so I celebrate both of them as launch dates for the blog.

Last year, I used the anniversary to create a FAQ about my blog and myself. I do the same this year -- and this year's FAQ will reflect the changes that have happened to my career since last year. As usual, let me include a table of contents for this FAQ:

1. Who am I? Am I a math teacher?
2. Who is Theoni Pappas?
3. How did I approximate pi in my classroom?
4. What is the U of Chicago text?
5. Who is Fawn Nguyen?
6. Who are the traditionalists?
7. What are number bases?
8. Why wasn't my first year of teaching as successful as I'd like?
9. What did I do about cell phones in the classroom?
10. How should have I stated my most important classroom rule?

1. Who am I? Am I a math teacher?

I am David Walker. I'm trying to start a career as a math teacher, but I must admit that my career isn't off to as good a start as I'd like.

Last year was my first as a teacher at a charter middle school, but I left that classroom in March. And this year, I was just about to start at a charter high school, but I was told that enrollment at the school is declining, and so they decided not to hire me as a teacher.

By now, I fear that I won't be hired to teach this year at all. If I'm not hired, then I'll try to return to substitute teaching. But this will make the launch of my teaching career that much more difficult.

So the answer to this question is no, I'm not currently a math teacher, but I'm hoping that I'll be one again soon.

2. Who is Theoni Pappas?

Theoni Pappas is the author of The Mathematics Calendar. In most years, Pappas produces a calendar that provides a math problem each day. The answer to each question is the date. I enjoyed, but only rarely wrote of, the Pappas calendar during the first three years of this blog, 2014, 2015, and 2016.

Last year, as I was about to begin my first year of teaching, I came up with the idea of starting each class with my own Pappas question -- a Warm-Up question whose answer is the date. But this quickly fell apart when I found out that our school curriculum provided a "daily assessment" question that I was required to give as the Warm-Up. Of course, the answers to these questions weren't just the date.

I was disappointed when Pappas didn't publish her calendar for 2017. So in January and February, I had neither my own Pappas calendar in the classroom Warm-Up, nor any questions directly written by Pappas for my home calendar.

In March, my local library held its biannual book sale. For fifty cents, I found a book actually written by Pappas, The Magic of Mathematics, published in 1994. And so I decided to blog about one page from her book everyday that I post for the rest of this year, until 2018, when it appears that she will publish her actual Mathematics Calendar once again.

I wanted to retain the tradition of incorporating the date in choosing a page to blog about. And so I decided that the page number will be the same as the (Julian) day count of the year. For example, today is July 22nd, the 203rd day of the year, so I'll blog about page 203 today.

This is what Theoni Pappas writes on page 203 of her Magic of Mathematics:

"Ever wonder, irritatedly, how it is you receive so much junk mail -- catalogues you never requested, offers for 'super' buys? How were your name and address accessed? Welcome to the electronic age and the loss of privacy."

This page is in Chapter 8, "The Computer Revolution." It is the only page of the section titled "Cryptography: Anarchy, Cyberpunks, & Remailers."

Keep in mind that Pappas wrote the above in 1994, when the Internet was in its infancy. Imagine how much easier it is to access personal information now that the Internet is in its prime.

It also reminds me of the old show Square One TV. In a Mathnet segment produced a few years before Pappas wrote her book, the villain was a woman named I.O. Privacy. She used personal information on a computer to send mailers to those who were most likely to spend a weekend on vacation at Pocono's Paradise. When the victims were out of town, she would send her henchmen out to their homes and rob them. Oh, and the culprit's initials, I.O., stood for "Invasion Of."

So Pappas warns us that living in the digital age has its drawbacks as well as its advantages. Here's how she ends this section:

"Many feel that if governments can use methods and devices to encode important information, an individual also has a right to use similar methods to insure personal privacy."

3. How did I approximate pi in my classroom?

Since today is Pi Approximation Day, I should write something about approximating pi. I won't let my disappointment -- of thinking I'd be hired at a new job only to be denied -- get in the way of celebrating the special day.

The dominant approximation of pi in my classroom was, of course, 3.14. I didn't quite reach the unit on pi in my seventh grade class -- Grade 7 being the year that pi first appears in the current Common Core Standards. But I did reach a unit on the volume of cylinders, cones, and spheres in my eighth grade class.

The main driver of my use of 3.14 as the only approximation of pi was IXL. I used this software to review the volume problems with my eighth graders. The software requires students to use 3.14 as pi, even though I provided them with scientific calculators with a pi key. For example, the volume of a cylinder of radius and height both 2 is 8pi cubic units. IXL expects students to enter 8(3.14) = 25.12, even though 8pi rounded to the nearest hundredth is 25.13. IXL will charge the students with an incorrect answer if they enter 25.13 instead of 25.12. And even on written tests, I didn't want to confuse the students by telling to do something different for written problems, and so I only used 3.14 for pi throughout the class.

But there was another problem with our use of calculators to find volumes. Some of the calculators were in a mode to convert all decimals into fractions. Thus 25.12 appeared as 628/25 -- and I couldn't figure out how to put them in decimal mode. This actually allowed me to catch cheaters -- only one of the calculators was in mixed number mode. Here was one of the problems from that actual test, where students had to find the volume of a cylinder of radius and height both 3:

V = pi r^2 h
V = (3.14)(3)(3)(3)
V = 84 39/50

so it displayed 84 39/50 instead of 84.78 or 4239/50. So any student who wrote 84 39/50 on the test -- other than the one I knew had the mixed number calculator -- must have been cheating.

Here's an interesting question -- suppose instead of catching cheaters, my goal was to make it as easy as possible on my students as well as my support staff member, who was grading the tests. That is, let's say I want to choose radii and heights carefully so that, by using 3.14 for pi, the volume would work out to be a whole number (which appears the same on all calculators regardless of mode). How could have I gone about this?

The main approximation of pi, 3.14, converts to 314/100 = 157/50. It's fortunate that 314 is even, so that the denominator reduces to 50 rather than 100. So our goal now is to choose a radius and height so that we can cancel the 50 remaining in the denominator.

We see that 50 factors as 2 * 5^2. The volume of a cylinder is V = pi r^2 h -- and since r is squared in this equation, making it a multiple of five cancels 5^2. It only remains to make h even to cancel out the last factor of two in the denominator. Let's check to see whether this works for r = 5, h = 2:

V = pi r^2 h
V = (3.14)(5)^2(2)
V = 157 cubic units

which is a whole number, so it works.

If we had a cone rather than a cylinder, then there is a factor of 1/3 to deal with. We can resolve the extra three in the denominator by making h a multiple of six (since it's already even). Let's check that this works for r = 5, h = 6:

V = (1/3) pi r^2 h
V = (1/3)(3.14)(5)^2(6)
V = 157 cubic units

Spheres, though, are the trickiest to make come out to be a whole number. This is because spheres, with the volume formula V = (4/3) pi r^3, have only r with no h. Thus r must have all of the factors necessary to cancel the denominator. Fortunately, the sphere formula contains the factor 4/3, and the four in the numerator already takes care of the factor of two in the denominator 50. And so r only needs to carry a factor of five (only one factor is needed since r is cubed) and three (in order to take care of the denominator of 4/3). So r must be a mutliple of 15. Let's check r = 15:

V = (4/3) pi r^3
V = (4/3)(3.14)(15)^3
V = 14130 cubic units

Notice that the volume of a sphere of radius 15 is actually 14137 to the nearest cubic unit -- that's how large the error gets by using 3.14 for pi. Nonetheless, 15 is the smallest radius for which we can get a whole number as the volume by using pi = 3.14.

Now today, Pi Approximation Day, is all about the approximation 22/7 for pi. Notice that if we were to use 22/7 as pi instead of 157/50, obtaining a whole number for the volume would be easier.

For cylinders, the only factor to worry about in the denominator is seven. Either the radius or the height can carry this factor. So let's try r = 1, h = 7:

V = pi r^2 h
V = (22/7)(1^2)(7)
V = 22 cubic units

For cones, we also have the three in 1/3 to resolve. Let's be different and let the radius carry the factor of three this time. For r = 3, h = 7, we have:

V = (1/3) pi r^2 h
V = (1/3)(22/7)(3^2)(7)
V = 66 cubic units

For spheres, unfortunately our radius must carry both three (for 4/3) and seven, and so the smallest possible whole number radius is 21:

V = (4/3) pi r^3
V = (4/3)(22/7)(21)^3
V = 38808 cubic units

This is larger than the radius of 15 we used for pi = 3.14. But notice that there are many extra factors of two around in the numerator -- 4/3 has two factors and 22/7 has one. This means that we can cut our radius of 21 in half. Even though 21/2 is not a whole number, the volume is nonetheless whole:

V = (4/3) pi r^3
V = (4/3)(22/7)(21/2)^3
V = 4851 cubic units

To make it easier on the students, I could present the radius as 10.5 instead of 21/2. (Recall that the students can easily enter decimals on the calculator -- they just can't display them.) There is still some error associated with the approximation pi = 22/7, as the volume of a sphere of radius 10.5 is actually 4849 to the nearest cubic unit, not 4851. Still, 22/7 is a slightly better approximation than 157/50 is, with a much smaller denominator to boot.

With such possibilities for integer volumes, I could have made a test that is easy for my students to take and easy for the grader to grade. I avoided multiple choice on my original test since I didn't want to give decimal choices for students with calculators in fraction mode (or vice versa). With whole number answers, multiple choice becomes more feasible. Of course, the wrong choices would play to common errors (confusing radius with diameter, forgetting 1/3 for cones).

4. What is the U of Chicago text?

In 1991, the University of Chicago School Mathematics Project published a series of secondary mathematics texts. I was able to purchase both the Algebra I and Geometry texts for two dollars each at a local public library. One thing I discovered was that the U of Chicago Geometry text is based on the same transformations that appear in the Common Core Standards -- even though the U of Chicago published the text nearly two decades before the advent of the Core.

The common denominator is NCTM, the National Council of Teachers of Mathematics. This group was behind both the U of Chicago text and the Common Core Standards. Still, this means that one can satisfy most of the Common Core Standards by teaching out of the U of Chicago text.

There are newer versions of the Geometry text than the one that I own. I've read that while my old version introduces reflections in Chapter 4 and the other isometries in Chapter 6, in the new version all isometries appear in Chapter 4.

The U of Chicago text is not the only book that I mention here on the blog, not by a long shot. Many posts here refer to the Illinois State text. This is mainly because I was a math teacher, and I used the Illinois State text when teaching at my old school. 

To supplement the Illinois State text (mainly when creating homework packets), I used copies of a Saxon Algebra 1/2 text and a Saxon 65 text for fifth and sixth graders.

To learn more about some of these texts, here are some links to a webpage by Cathy Duffy. Duffy's goal is to help homeschooling parents choose a textbook for their children. Here is her review of the U of Chicago text:

and the Saxon series:

5. Who is Fawn Nguyen?

For years, Fawn Nguyen was the only blogger I knew who was a middle school math teacher. Back when I was a middle school teacher, I enjoyed reading Nguyen's blog, but now that I've left, it's not as important for me to focus on her blog over all others.

Nonetheless, let's take a look at Nguyen's blog today. She hasn't posted in two months, and so the following link is to her most recent post:

This is a computer program that takes two whole numbers as input. Then the computer draws a rectangle whose dimensions are these two numbers.

Afterward, the computer performs the following algorithm -- inside the rectangle, it draws the largest possible square that fits inside the rectangle. The side length of this square will equal the smaller dimension of the rectangle. If there is a rectangle left over after cutting out the square, then the computer performs the same algorithm on the remaining rectangle. The algorithm terminates when the final rectangle is itself a square.

The name "Euclid's Algorithm" reveals its origins -- the Greek mathematician proved that if the sides of the original rectangle are whole (or rational), then the algorithm terminates. Euclid referred to the side length of the final square as the "common measure" of the dimensions of the original rectangle, and nowadays we can think of this length as the GCF of the original whole number dimensions.

But not all lengths have a common measure. Some are incommensurate (irrational). The most famous example of a rectangle with incommensurate dimensions is the golden rectangle. If a square is taken away from a golden rectangle, the result is a rectangle that is similar to the original figure. And if a square is taken away from the remainder, the result is again a golden rectangle -- thus Euclid's algorithm fails to halt for this rectangle. The ratio of the dimensions of a rectangle is (1+sqrt(5))/2, which I mentioned in another post about a month ago as the golden ratio, Phi.

But today is Pi Approximation Day, not Phi Approximation Day, so let's try Euclid's algorithm on a rectangle with length pi. For simplicity, the width of the rectangle will just be 1. (By now we've left Nguyen's original post long behind, since her class only dealt with whole number lengths.) We can approximate pi by pretending that there is no remaining rectangle, which would then make the dimensions rational.

-- Our starting rectangle has dimensions (pi, 1).
-- The first iteration allows us to cut three unit squares from the rectangle. Ignoring the remainder would then imply that pi = 3 -- and actually, Nguyen does do the (3, 1) case. And indeed, this crude approximation pi = 3 is implied in the Bible.
-- The second iteration takes the remaining rectangle of dimensions (pi - 3, 1). Seven small squares can be cut from this rectangle. Ignoring the remainder would then imply that pi = 3 + 1/7 (since the seven small squares would then have the same length as one of the large squares). This, of course, is equal to 22/7, which makes today Pi Approximation Day. The Greek mathematician Archimedes was the first to use this approximation.
-- The third iteration takes the remaining rectangle and cuts out one more smaller square. Ignoring the remainder would then imply that pi = 3 + 1/8 = 25/8 (since it's now as if eight small squares can fit on the side of the unit square). The ancient Babylonians used this approximation.
-- The fourth iteration takes the remaining rectangle and cuts out 15 smaller squares. Ignoring the remainder would then imply that pi = 333/106. This is less obvious, but if we were to take Nguyen's program and enter (333/106, 1) -- actually (333, 106) has the same effect -- we would see three large squares, then seven medium squares, then one square of its own size, and finally fifteen of the smallest squares. There is a theory that this approximation also appears in the Bible -- here's a link to a Reddit post (written exactly six years ago on Pi Approximation Day):

-- The fifth iteration takes the remaining rectangle and cuts out one more smaller square. Ignoring the remainder would then imply that pi = 355/113. Again, we can enter (355, 113) in Nguyen's program and see that there is one more smallest square than there was for (333, 106). This approximation was known to the ancient Chinese.
-- The sixth iteration takes the remaining rectangle -- which is very thin, because 355/113 is such a great approximation. Indeed, the remainder is 292 times as wide as it is long. And so we normally cut off the approximation at 355/113.

The rational approximations 3, 22/7, 25/8, 333/106, and 355/113 are called "continued fractions." We can use continued fractions whenever we need to find a rational approximation. (Calendar reform and microtonal music are two topics mentioned on the blog that can make use of continued fractions, though I didn't explain how in any of those posts.)

I hope Fawn Nguyen posts again on her blog soon!

6. Who are the traditionalists?

I use the phrase "traditionalists" to refer to those who oppose Common Core for mathematical -- not political -- reasons. The traditionalists tend to believe that many of the topics in the Common Core Math Standards are taught too late.

My own relationship with the traditionalists is quite complex. I tend to agree with the traditionalists regarding elementary school math and disagree with them regarding high school math.

For example, they believe that the standards in Grades 4-6 that require the use of the standard algorithm for arithmetic should move down at least one grade level. I tend to agree with the traditionalists on this point here -- most of the reasons for opposing direct instruction, rote memorization, and drill don't apply to the youngest students. I remember once learning about two nonstandard algorithms for addition -- the Left-Right Method and the Plus-Minus Method -- on an old 15-minute math show for sixth graders, "Solve It," that used to air on our local PBS station back when I was that age. For mental math, these nonstandard algorithms are great, but for a young elementary student first learning to add two-digit numbers I prefer the standard algorithm.

On the other hand, the traditionalists also prefer that high school seniors take AP Calculus in order to open the door to college STEM majors. In particular, some STEM colleges here in Southern California, such as Cal Tech and Harvey Mudd, require calculus for admission. Even colleges that don't formally require calculus are likely to send students rejection letters if they apply without having calculus on the schedule, especially if they apply to STEM majors. Therefore they want the classes leading up to senior year to prepare students to take AP Calculus -- including Algebra I in the eighth grade. They oppose Common Core because its eighth grade math standards do not comprise a full Algebra I class. I only partially agree because I don't want seniors who can't handle AP Calculus or eighth graders who can't handle Algebra I to be labeled failures. Just as AP Calculus can open the door to STEM, it can close the door to a non-STEM career if students are forced to spend too much effort trying to pass their high school math classes.

OK, so I support the traditional pedagogy with regards to elementary math and oppose it with regards to high school math. But what's my opinion regarding traditionalism in middle school math -- you know, the grades that I'll actually be teaching? In middle school, I actually prefer a blend between the traditional and progressive philosophies. And the part of traditionalism that I agree with in middle school is making sure that the students do have the basic skills in arithmetic -- since, after all, that is elementary school math, and with elementary math I prefer traditionalism.

I've referred to many specific traditionalists during my three years of posting on the blog. The traditionalist who is currently the most active is Barry Garelick. Here is a link to his most recent post:

Garelick teaches middle school math right here in California. In this post, he criticizes the famous businessman Elon Musk for encouraging Project-Based Learning instead of traditional math lessons.

He highlights two commenters in this post. One of them is SteveH, a traditionalist in his own right. I consider SteveH to be a co-author of Garelick's blog since he posts there so often.

But today I wish to focus on the other commenter, Richard Phelps. Here's what he writes:

My high school physics class — in the early seventies — comprised project after project. Teacher called it “Harvard Physics.” Each class period was filled with setting up equipment and building contraptions, in small groups. Essentially, it was a “lab only” course. I learned less in that course than in any other over 4 years of high school, and I’m including gym in the comparison. Typically, an entire class period was devoted to delivering just one fact or concept — “authentically” — that could have been simply told to us in less than a minute or, with some discussion, in just a few minutes — un-authentically. Moreover, so much of our attention was focused on building the contraptions and getting them to work that, in most cases, the one factoid we were supposed to learn was lost in the morass of mostly irrelevant information.

Some traditionalists are satisfied by returning to traditional math while conceding that projects are at least useful in science. But apparently, to others like Phelps, even science should be project-free. To Phelps and other science traditionalists, students should spend most of science class either taking notes based on a teacher's lecture (which could include a lab demonstration, but only the teacher is performing the lab, not the students), or answering questions out of the text. Anything else, to the science traditionalists, is not learning.

Here's my problem with what Phelps is saying -- I don't know what state he is in, but I know that Garelick is a fellow Californian. Here in the Golden State, the two major university systems (UC and CSU) have what's known as the a-g requirements for college admission. Let's look at the a-g requirement for science:

Laboratory Science ("d")
Two units (equivalent to two years or four semesters) of laboratory science are required (three years are strongly recommended), providing fundamental knowledge in two of the following disciplines:
[emphasis mine]
-- Biology
-- Chemistry
-- Physics

So we see that the UC and CSU systems expect high school students to take laboratory science. A student who takes the lecture science courses that Phelps prefers would not be admitted to a UC, since the UC explicitly states that it expects incoming students to have had laboratory science.

Last year, I was hired as a math teacher at my middle school. But I suddenly found out that there was no science teacher, and I unexpectedly had to teach science as well as math.

Meanwhile, the Illinois State text that I mentioned above is strongly project-based, and I was required to have students complete a project once every two weeks. It is undoubtedly a curriculum of which the traditionalists would disapprove.

Now here's the thing -- one day, I tried to pass out a worksheet for science, and my eighth graders complained about it. I wrote about this in my January 27th post:

-- On [January] 13th, I decide to print the students a Study Island worksheet on Human Interactions. The idea is that this lesson would bridge the gap to the Green Team next month. But then partway through the lesson, the students complain that it was too boring and refused to work on it. One girl who had transferred in from another school told me that her old school had a real science teacher who gave interesting projects, such as an Edible Cell Model. I agree to try something similar in my class.

The girl told me that my lessons weren't "fun" enough. When I tried to explain that lessons should not be "fun," but instead should allow them to learn a lot (as Garelick, SteveH, and Phelps would argue), she countered that she'd learned much from her previous class, yet she nonetheless had fun then.

I suspect if I were to tell this story to a traditionalist, he'd tell me that this was insufficient evidence that science classes should be fun. "So," our hypothetical traditionalist would ask, "when you finally gave the project, the girl was engaged, right?"

Well, let's find out:

-- On [January] 17th (after the three-day weekend) I introduce an Edible Molecule Project to be completed at home, just like the Edible Cell Model from last year. But the students reject this as well, telling me that they can't do it since they haven't learned anything about molecules yet. This includes the new girl, who probably never wanted to do the project in the first place. She only brought it up in order to tell me how her old teacher was a much better science teacher than I am (which is true, since she was a genuine science teacher and I'm not).

"Oh," our hypothetical traditionalist responds, "there were other issues in the class, and so this doesn't show that science teachers should give more projects."

And part of the problem was that as a natural math teacher, I rarely covered science, and so it's understandable that the students would reject something that was unexpected. I suspect that if I had covered science the entire year (as I should have), my science lessons would have been accepted -- regardless of whether they were project-based or traditional.

I plan on writing about my eighth grade class -- indeed about one particular student. Earlier I wrote that I'd post it soon, and in fact I'd had nearly the entire post already prepared for posting. But that was the day I found out that I wasn't going to be hired at the new school. The major disappointment weakened my will to post, especially about teaching -- and besides, I would need to spend extra time applying for teaching jobs. Instead, I wrote about topics from books written by both Theoni Pappas and Eugenia Cheng.

Notice that the Illinois State curriculum covers both math and science. As I think about my science failure more and more, I now realize that I should have dropped the Illinois State math projects and covered only the science projects. Indeed, this is essentially what my counterpart did at our sister charter school. I could have met the two-week project requirement with only science projects and prepared the students for lab science "d" in high school to boot, and this would have provided me with a loophole to keep the math class more or less traditional.

By the way, I wonder when Phelps believes that students should have their first lab science class. I'm certain that he'd admit that Ph.D candidates need to do original research in labs. So it remains to be seen whether Phelps would accept lab science for lower division, upper division, and MS candidates.

7. What are number bases?

I write about number bases today because they are related to both the Pappas topic (on computers) and the traditionalists (and you'll find out why soon).

Pappas explains:

"The binary system (base two), which uses only 0s and 1s to write its numbers, held the key to communicating with electronic computers -- since 0s and 1s could indicate the 'off' and 'on' position of electricity."

Earlier in the book, Pappas counts from zero to twelve in binary:

0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, ...

As she explains, the first mathematician to work in binary was the famous Scottish mathematician John Napier. This was before computers, so he used a chessboard to display his binary numbers.

Pappas writes:

"For example, to add 74 + 99 + 46, each number is written out in a row of the chess board by placing markers in the appropriate squares of the row so that the sum of the markers' the number they represent. 74 has markers at 64, 8, and 2 since 64 + 8 + 2 = 74."

In modern binary notation, we would write 74 as 1001010. Pappas proceeds:

"After each number is expressed on the chess board, the numbers are added by gathering the markers vertically down on the bottom row. Two markers sharing the same square equal one marker to their immediate left. So two '2' markers produce one '4' marker. Working from right to left, any two markers sharing the same square are removed and replaced by one marker in the adjacent square at the left. At the end of this process, no square will have more than one marker. The sum of the values of the remaining markers represents the sum of the numbers."

Here's the connection to traditionalism -- notice that this is actually the standard algorithm for adding in base two. The line about "any two markers sharing the same square are removed and replaced by one marker in the adjacent square" corresponds to "carry the one" in the standard algorithm.

And so we see that by learning about other number bases, we can shed more light on how young students learn arithmetic in base ten.

There are other number bases besides base two, of course. There is, in fact, an entire message board devoted to base 12, or dozenal:

This website is the de facto home of number bases in general. Indeed, the members of this site come up with their own words to describe the properties of number bases. This actually deserves a post of its own, and indeed I'll be making such a post soon. I admit that lately on the blog, "soon" essentially means "whenever." Well, let's just say I'll post about number bases before we reach the last page of the Pappas chapter on computers (page 221, which corresponds to August 9th).

8. Why wasn't my first year of teaching as successful as I'd like?

I've been reflecting on this for the past four months. And I've realized something -- whenever I have a bad day in the classroom, whether it was this year or back when I was a sub, the reason for that bad day can be summarized in a single sentence:

The students were led to believe that they could do whatever they wanted, then they resisted when they found out that they couldn't.

I remember my first day of substitute teaching, years ago in an eighth grade math class. Several of the students refused to do any work, and I kept nagging at them to do it. The students eventually rebelled, and during one period, they pulled a classic sub trick -- they all jumped up at the same time. I asked myself, what was I thinking making the students do work when I'm just a sub? And so I thought it was better not to expect too much from students -- at least the students wouldn't rebel against me the way they did that first day. I was wrong to believe this -- yet it permeated my entire line of thinking when I was in the classroom, all the way until this year.

The real reason the students jumped up that was that I didn't make it clear from the start of the period that I expected them to be quiet and do work. When I was taking roll, the students continued to talk, yet I didn't interrupt roll to tell them to stop. From this, they concluded that they could do whatever they wanted, and so they were shocked when I tried to punish them for not doing their work. If I had made it clear that there was to be no talking even during the roll, they probably would not have jumped up when I told them to work.

I don't write about classroom management often on the blog. This year, however, I wrote extensively about management in several posts labeled MTBoS (Math Twitter Blogosphere).

In last year's FAQ, I wrote about E's, S's, and U's -- conduct grades used by the LAUSD. My charter school didn't use these conduct marks, yet I still mentally divide my students in into three groups corresponding to these conduct grades.

Also in that FAQ, I wrote about a participation points system that I would use this year. I was hoping to use this as my primary management system, but it failed. This is what I wrote:

At the start of each unit (that is, right after the test), each student has two Participation Points. I will award a point each time a student gives a correct answer, to a Warm-Up or any question that I ask during the main part of the lesson.But if a student fails to participate or otherwise misbehaves, I'll deduct a point. If the student's Participation Point total drops below zero, I will begin assigning consequences, beginning with a minute of detention for each additional point below zero. I expect these detentions to be the most effective in my eighth grade classes, because based on my school's block schedule, Math 8 always meets right before nutrition or lunch, so students will want to avoid those detentions. (Math 7 also meets before nutrition on certain days of the week.) Failure to show up to detention results in a doubling of the detention time, and further consequences occur when the total detention time exceeds a certain amount. 

The problem was that in all three classes (but especially sixth grade), there were many students who were smart, yet very talkative. They would run up their Participation Point total by answering questions, and so when they lost points, it was never enough to earn them a detention.

Since the students weren't punished, they thought that it was okay to talk. Soon it was similar to the worst days as a sub -- since they believed talking was OK, they were shocked when it was time for them to work, and I ended up yelling at them. My biggest fear from last year was realized -- all of the S-students became U-students.

When I came up with the point system, I wanted to make it as easy as possible. So I gave them a single score to represent both academics and behavior. This is what Eugenia Cheng warned about in her Beyond Infinity, when she wrote about using one variable to represent two dimensional data.

And so I should have kept track of academics and behavior separately. I could still have awarded points for correct answers. But I should never have deducted points for bad behavior. Behavior needed to be addressed with a discipline hierarchy separate from Participation Points.

9. What did I do about cell phones in the classroom?

I addressed this as part of last year's FAQ. I wrote:

And so, at the end of any detention earned due to cell phone use, I will require the student to say "Without math, there wouldn't be any cell phones" before releasing them. And if by chance I must confiscate a phone, I will require the student to say the same before returning the phone.

And yes, I really did make them say this last year. But I had a completely separate problem with cell phone use this year.

I would see a student apparently typing something on a cell phone. I would tell her to put the phone away, and then she'd claim that she didn't have a phone out. Then I'd say that I see her with a phone out -- only for her to reveal that she'd been pressing on her cell phone case, not the phone itself.

The idea here is that I can't reliably tell whether a student has a phone case or actual phone out, and so the only fair thing to do is let students use phones whenever they wanted. The argument was that it's far better to let ten students use phones with no punishment whatsoever than to punish even one innocent student for merely having a phone case out.

Of course, students aren't entertained by playing on phone cases -- they are entertained by playing on actual phones. The sole purpose of playing with the case was to "neuter" the no cell phone rule -- by exploiting a loophole to prove that the rule is "unfair."

In reality, a student playing on a phone case isn't innocent. Such a student is trying to create a classroom where she can use her phone with no punishment. Her goal obviously isn't to make herself smarter, learn a lot of math, or become a valedictorian.

Thus a student who plays with a cell phone case deserves punishment -- yet I was powerless to do so, since nowhere in my rules was it stated, "no cell phone cases."

This leads to our last FAQ item:

10. How should have I stated my most important classroom rule?

This is what I wrote in last year's FAQ:

Rule #3: Respect yourself and others.

Students respect themselves and each other as well as me, the teacher, by following all rules and allowing others to learn the material.

This rule worked for Fawn Nguyen, the teacher from whom I got this rule. But it didn't work in my classroom at all. Of course, a student playing with a phone case isn't respecting the teacher, but I needed a rule that would require the student to put the case away immediately.

Here's a much better rule:

Rule #1: Follow all adult directions.

And so if I say, "Put the phone case away," the student couldn't counter that phone cases were against the rules, because my direction was to put the case away.

If I ever find myself in the classroom again, this will definitely be my first and most important rule.

OK, here are some Pi Approximation Day video links:

1. Draw Curiosity

Notice that this video, from last year, actually acknowledges Pi Approximation Day.

2. Reed Reels

This video mentions that even Legendre -- the author of the geometry text we've been discussing in some of my recent posts -- contributed to the history of pi. In particular, Legendre proved that not only is pi irrational, but so is pi^2 (unlike sqrt(2), which is irrational yet has a rational square, namely 2).

This video also mentions a quote from the mathematician John von Neumann: “If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” We've seen Eugenia Cheng, in both of her books, say almost exactly the same thing.

3. Fact Retriever

This video gives ten amazing facts about pi.

4. Numberphile

No Pi (Approximation) Day is complete without a Numberphile video. In this video, we learn that pi is not just irrational, but is also transcendental -- that is, there doesn't even exist a polynomial f, with integer or rational coefficients, such that f (pi) = 0 (unlike sqrt(2), which is algebraic). This is even stronger than Legendre's result that pi^2 is irrational.

5. Sharon Serano

Well, I already gave ten facts about pi, and so this video is twice as good.

6. Vi Hart

No, I can't avoid another one of my favorite math YouTube posters, Vi Hart. Of course, in this video she argues that there's nothing special about pi at all

7. Danica McKellar

I've mentioned McKellar's video in my past Pi (Approximation) Day posts, and it's a favorite, so I'm posting it again this year.

8. A Song Scout

This is another pi song based on its digits. Unlike Michael Blake's song (listed below), it is in the key of A minor rather than C major.

9. Michael Blake

I like the idea of using music to teach math, so here's a favorite, Michael Blake's "What Pi Sounds Like." I tried to find a way to incorporate songs such as this one into the classroom.

10. The Ancient Melodies

Going back to number bases, this song uses the digits of pi in dozenal, or base 12. Dozenal was chosen because we commonly use the 12EDO scale in Western music.

Bonus. Radomir Nowotarksi

This song is actually based on the Fibonacci sequence, not pi. I mention it here because in a previous post, I wrote about a Numberphile video which featured a song where the notes (1 = C, 2 = D, 3 = E, up to 7 = B) were based on reducing the Fibonacci numbers mod 7. I never found that video, but here Nowotarski is doing the same thing.

Nowotarksi uses the Lydian mode in this song rather than the major scale. This means that note 4 corresponds to F# rather than F. Like Michael Blake, Nowotarksi places a chord on each note, but these are different chords due to the use of F# over F. Where Blake uses a D minor triad (D-F-A), Nowotarksi uses a D major triad (D-F#-A).

I've mentioned that many songs based on digits are manipulated so that they end on Note 1, which is the tonic of the scale. But Blake's song ends on Note 5, since five is the last digit before the first appearance of zero (which he was tried to avoid). Song Scout uses G# for both zero and seven -- and his song actually ends on G#. The dozenal song changes the last digit from Note 5 to Note 4 so it could end on Eb (which serves as a tonic in that section). The Fibonacci song repeats, and each section ends on Note 7.

And so I wish everyone a Happy Pi Approximation Day.

Thursday, July 6, 2017

Pappas Music Post: Conclusion

This is what Theoni Pappas writes on page 187 of her Magic of Mathematics:

"Points A and C are speaker locations. Point D is the location of the listener. B is the midpoint of segment AC. Distance BD must be greater than or equal to the distance AB. x is the speakers' distances from the walls -- a minimum of 3 feet."

Here Pappas is describing QSound -- a stereo that produces sound coming in from all directions (in three dimensions) towards the listener.

I knew that QSound is 1994 technology (as this is when Pappas wrote the book), so I'm curious as to whatever happened to QSound. Well, let's find out:

In the early 1990’s, QSound Labs brought effective, practical three-dimensional (3D) audio technology to music and film production studios in the form of a ¼-tonne digital audio processing station called the QSystem. Meanwhile, video gamers enjoyed ground-breaking QSound 3D audio on Capcom™ arcade machines.

Today, QSound technology is embedded in tiny devices such as cell phones and Bluetooth™ headsets, whereas the QSound digital audio algorithm suite has grown considerably.

Ok, so QSound still exists. By the way, notice how QSound works:

The basic QSound 3D building block is a proprietary audio process that creates two or more outputs from each input signal. These outputs incorporate subtle differences in level, frequency content, and timing. Played back on the specific delivery system (headphones or speaker arrangement) for which the algorithm is designed, the effect is to mimic directional cues that a listener would normally receive when listening to an actual sound source at a given location in 3D space.

Ah, so we see how all of the components of sound first discovered by Fourier -- pitch/frequency, loudness/amplitude, and quality -- are modified to make the sound seem to come from different locations in the room. This is similar to how perspective is used to make a 2D image appear 3D.

This is the final page of the "Mathematics Plays Its Music" chapter in Pappas. And so I want to use this post to wrap up what we've been discussing about musical scales.

In the book, Pappas makes a tongue-in-cheek reference to 300EDO as a scale in which each step is about four cents -- the just noticeable difference (JND). We aren't meant to consider 300EDO as an actual scale, but I notice that it has several interesting properties.

First of all, 300 is the LCM of 12 and 50. So 300EDO is the smallest scale that incorporates both our usual scale 12EDO and yesterday's scale 50EDO.

But with each step of 300EDO equal to the JND, we should be able to play any interval and make it sound like just intonation. So a 300EDO player would want to play the fifth of 12EDO at 700 cents and the major third of 50EDO at 384 cents. Notice that the difference between these, the minor third, is part of neither 12EDO nor 50EDO, but at 316 cents, the minor third is almost exactly just.

The 300EDO has a special property -- it's the last scale to temper out the Pythagorean comma. Recall that a scale tempers out the Pythagorean comma if it equates 12 fifths with seven octaves -- that is, if the perfect fifth is exactly 700 cents. Any EDO that's a multiple of 12 -- 24EDO, 36EDO, 48EDO, and so on -- has an interval of 700 cents.

But that 700-cent interval isn't always the perfect fifth. The just perfect fifth is 702 cents -- or to be more accurate, about 701.96 cents. In 300EDO, the next step after 700 cents is 704 cents, and so 701.96 is slightly closer to 700 than 704. But in the next multiple of 12, 312EDO, the next step after 700 cents is 703.85 cents, and 701.96 is a little closer to 703.85 than 700. Thus 703.85 cents is the perfect fifth, and far from tempering out the Pythagorean comma, 312EDO exaggerates it. And so 300EDO is the last EDO to temper out the famous comma.

It's interesting that Pappas cites 4 cents as the just noticeable difference. Some sources give 5-6 cents as the JDN. If we choose 6 cents as the JDN, then 200EDO with its 6-cent steps is interesting:

Notice that 702 cents is a multiple of six, and so 200EDO has a very accurate perfect fifth. In fact, the fifth of 200EDO is better than any lower EDO (from 2 up to 199). On the other hand, the third of 200EDO, like that of 300EDO, is inherited from 50EDO (at 384 cents).

Since 200EDO has such an accurate perfect fifth, we're inclined to use Pythagorean notation, where C-G-D-A-E are each a perfect fifth apart, and so C-E is the Pythagorean major third (81/16 or 408 cents) rather than the just major third (5/4 or 386 cents).

In 12EDO, E represents both 5/4 and 81/16, as 12EDO is a meantone tuning. On the other hand, 22EDO is not meantone, and so we must decide which third we mean by the note E.

To qualify as a meantone tuning, a scale must equate 10/9 and 9/8. It is not sufficient for a scale to have a tone between 10/9 and 9/8 to be considered meantone. For example, 48EDO has the same 200-cent 9/8 as 12EDO, but 10/9 is 175 cents (and 5/4 is 375 cents), and so 48EDO is not considered to be a meantone tuning. The "meantone" actually isn't the "mean" (or average) between the two tones, but rather is the average between the unison (1/1) and the major third (5/4). Only when 5/4 is divided into two equal tones does the scale qualify as meantone. Meantone scales always temper out the syntonic comma 81/80.

-- 12EDO tempers out both Pythagorean and syntonic commas. E, Fb, and major 3rd are enharmonic.
-- 300EDO tempers out the Pythagorean comma only. E, Fb are enharmonic (at 400 cents), but neither of these is the major third (at 384 cents).
--50EDO tempers out the syntonic comma only. E and major 3rd are enharmonic, but neither of these is the same as Fb (at 432 cents).

We see that 200EDO tempers out neither comma. Since the note E represents the Pythagorean major third and the syntonic comma isn't tempered out, E (at 408 cents) is not the major third. Meanwhile, E is not the same as Fb -- in a Pythagorean tuning, E and Fb differ by the Pythagorean comma.

And so an interesting question is, exactly which note is Fb in 200EDO? Since this is a flat note, we use a circle of fourths instead of fifths: C-F-Bb-Eb-Ab-Db-Gb-Cb-Fb.

So Fb is eight fourths away from C. Since each fifth is 702 cents, each fourth is 498 cents, so eight fourths is 3984 cents. Reducing this by three octaves (at 1200 cents each) gives us 384 cents, so we conclude that Fb is 384 cents.

Hey, that sounds familiar -- 384 cents is the major third in 200EDO. So we conclude that, while E is not the major third and E is not Fb, Fb really is the major third!

What's going on here? Notice that there are two intervals called "comma" here -- the Pythagorean and syntonic commas. The Pythagorean comma is about 24 cents and the syntonic comma is 22 cents -- so these two commas are about the same size. Some tunings, like 200EDO, take full advantage of the near equality of these two commas. The note E is a Pythagorean major third, but instead of lowering it by a syntonic comma, we lower it by a Pythagorean comma to obtain Fb.

And so in 200EDO, C-Fb-G is a major third. Also, C-D#-G is a minor third in 200EDO. Notice that this only happens in tunings that don't distinguish betwen 81/80 and the Pythagorean comma -- that is, they temper out the difference between the two commas (called a "schisma"). Both 12EDO and 200EDO temper out the schisma, and so C-Fb-G is major and C-D#-G is minor in both. On the other hand, 19EDO doesn't temper out the schisma, and so C-Fb-G isn't major (it's supermajor), and likewise C-D#-G isn't minor (it's subminor) in 19EDO.

Meanwhile, if you prefer five cents for the JDN, here's a link to 240EDO:

One way to generate music in alternate scales is to take 12EDO music and naively convert it to 17EDO, 19EDO, or a higher scale. After all, we've seen that the notes in these other scales are often named the same as in 12EDO, just with different enharmonicities.

If we start with a 12EDO song in the key of C major, it should be easy to convert this to 19EDO, since the C major scale is C-D-E-F-G-A-B-C in both 12EDO and 19EDO. But notice that most modern songs don't stick to a single scale. We often refer to this as "jazzing up" a song, but even songs in genres other than jazz tend away from a single major scale towards atonality.

For the recent Fourth of July holiday, I checked out a music book from the library: All American Patriotic Songbook by John L. Haag (dated 1996). Even if we look only at the songs written in the key of C major, we'll find plenty of sharps and flats in most songs.

The first song in the book is "Alexander's Ragtime Band," by Irving Berlin. Ragtime is a form of jazz, and so we expect plenty of accidentals. This song starts in the key of C major before switching to F major. But in the first line (which is in C major), we see the notes D#, F#, G#, and Bb. That's four out of the five black keys right there.

The next song is "Alabama Jubilee" by Jack Yellen. This song is in the key of C major, and yet it contains D#, F#, and G# in the first line. We skip to the next C major song, "America, I Love You," and this song contains the notes C#, D#, G#, and Bb in the first line. "Battle Hymn of the Republic" is the first song that actually sticks only to the notes of C major. Again, we think back to the Google Fischinger player that could only play C, D, E, and G. But even if we added F, A, and B, we still couldn't play most of the C major songs in the book.

The question I'm asking is, if we're converting this to 19EDO, which note should we play for the notes which aren't in C major? For example, G# and Ab are interchangeable in 12EDO but not in 19EDO, so it makes a difference which note we play. We could just convert the notes naively -- that is, play the note G# if it is written as G#, and Ab if it is written as Ab. But it could be the case that playing the opposite note might actually sound better in 19EDO.

In general, there are three reasons why black keys appear in a modern C major song:

-- as a secondary dominant. This was the main reason for non-diatonic notes in classical music -- the note F# may appear as part of a D7 chord, since D7 is the dominant of G, which is itself the dominant of C major.
-- as a secondary leading tone. The note right before a G might be F#, even without a D7 chord. In this case, the F# often appears as a sixteenth note, last note of a triplet, or other downbeat note.
-- as part of a symmetrical chord. In 12EDO, this could be an augmented chord, but more likely it would be a diminished seventh chord. So we might see F# as part of an F#dim7 chord.

One song that appears in my songbook is "Meet Me in St. Louis, Louis." Notice that I changed this to "Meet Me in Pomona, Mona" as part of music break in my class leading up to the field trip to the LA County Fair in Pomona.

In this book, the song is written in the key of C major. Yet there are plenty of accidentals, and for all three reasons listed above:

-- There are plenty of accidentals. The line "We will dance the 'Hoochee Koochee'" is played with a series of secondary dominants: E7-A7-D7-G7-C. The first three of these require sharps.
-- There are several leading tones as well. The line "So he said 'Where can Flossie be at?'" contains an A# between two B's in the melody: G-A-B-A#-B-c-B-A-d.
-- There are two diminished seventh chords: Ebdim7 and F#dim7.

Notice that Ebdim7 and F#dim7 are actually the same chord in 12EDO -- the difference is that the note in the name of the chord is the bass note. But F#dim7 is spelled out as F#-A-C-D#, while Ebdim7 is spelled out as Eb-F#-A-C. Neither of this is strictly correct -- the dim7 chord should only contain minor thirds, so F#dim7 should be F#-A-C-Eb, while Ebdim7 is Eb-Gb-Bbb-Dbb -- that is with two double flats. (I notice that there are guitar chords shown in the score, and both Ebdim7 and F#dim7 have the exact same fingering on the guitar.)

None of this makes any difference in 12EDO. The real problem is in trying to convert the song to 19EDO, where D# and Eb are not enharmonic and the dim7 chord is not symmetrical.

In fact, one interesting part of the song is at the beginning of the Hoochee Koochee dance. There is a leading tone D-D#-E sequence. But the chords at this point are awkward -- right between a G7 chord and the secondary dominant sequence E7-A7-D7-G7-C is an F7 chord. Most likely, the intent of the F7 is to bridge the gap from G7 to E7. But the D# note is being played over the F7 chord, which ought to be spelled F-A-C-Eb. If we change this to D#, then it becomes F-A-C-D# -- which is actually a F harmonic 7th chord in 19EDO! Still, having an F harmonic 7th chord between the G and E dominant 7th chords may be awkward.

Here's how I would convert "St. Louis" to 19EDO:

-- I'd leave the 7th chords as dominant 7ths. Yes, I know that the harmonic 7th is more consonant, but dom 7th aren't supposed to be consonant -- its dissonance allows the dominant 7th to resolve to the tonic, as in G7-C. If the G7 in G7-C is dominant, then all such 7ths should be. (Instead, the harmonic 7th should be reserved for jazz songs in which all the chords are harmonic 7ths, including the tonic.)
-- Leading tones should remain as diatonic semitones, as in B-C. One might argue that a true leading tone in 19EDO should be something like Cb-C. But if the leading tone in a C major scale is a diatonic semitone B-C, then all such leading tones should be. The first appearance of a leading tone in the main melody is A# (B-A#-B, listed above), so I'd play it as written. (In class, I often played my "Pomona" parody on my guitar in the key of G major, where this becomes F#-E#-F#. Of course on my 12EDO guitar I ended up playing F#-F-F#, but the leading tone to F# remains E# in 19EDO.)
-- Diminished 7ths are the trickiest to resolve. It depends on what role they are serving.

Sometimes diminished 7th chords act like dominant 7ths in certain situations. In this song, the Ebdim7 chord appears in the sequence C-Ebdim7-G. We notice that if we spelled Ebdim7 as D#dim7 instead, the resulting chord D#-F#-A-C has three notes in common with D7 (F#-A-C), which is the actual secondary dominant to G. So I'd play it as D#-F#-A-C in 19EDO.

Meanwhile, the F#dim7 chords in the song are part of the sequence F-F#dim7-C. If we spell F#dim7 correctly as F#-A-C-Eb, then it has two notes in common (A-C) with the preceding F chord. So I'd play it as F#-A-C-Eb in 19EDO.

The trickiest part for me to figure out is the D# played over the F7 chord. Using the leading tone rule it should remain D#, but using the dominant 7th rule it should be Eb -- it just so happens that D# and Eb are the same note in 12EDO, but not 19EDO. But notice that F7 isn't really a secondary dominant leading to its tonic (which would be Bb), but just a transition G7-F7-E7. I see no reason why we can't use E#7 instead, so that the dominant 7th would be D# (E#-Gx-B#-D#). In other scales such as 31EDO, the interval G-E (a descending minor 3rd) can be divided evenly in half, with the note halfway in between being F^ (F half-sharp). So we might use G7-F^7-E7 instead, and the dominant 7th over F^ would be Ev (E half-flat, as in F^-A^-C^-Ev).

There's also another point in the score where D# appears. At "it's too slow for me here," there is another leading tone sequence D-D#-E, but the chord sequence is F-Ab7-C. An argument could be made that we should play F#dim7 instead, since we've established the F-F#dim7-C riff earlier. We could also try play a chord like Ab-C-D#-F#, so that C and F# match F#dim7 from earlier but the D# fits the leading tone sequence D-D#-E. But this isn't any established chord.

The ultimate decision for this and all other chords is -- what sounds right? After all, music is meant to be heard. Perhaps future musicians accustomed to 19EDO already know which notes and chords to play in this situation.

This is the final music post. I've squeezed in so many music posts in order to follow the music chapter in Pappas -- more posts than I usually write during a week in summer.

And so once again I'm taking some time off from blogging -- as in a week or two. I'll continue to edit the spring break posts to accommodate Eugenia Cheng, but otherwise I'm not touching this blog.

Wednesday, July 5, 2017

Pappas Music Post: The 50-Note Scale

Note: So far, I have edited the April 16th post:

to incorporate Chapter 8 of Eugenia Cheng's Beyond Infinity.

This is what Theoni Pappas writes on page 186 of her Magic of Mathematics:

"Two people can carry on a normal conversation from the two focal points [of the Rotunda in the United States capitol building], undisturbed by the noise level of the hall."

Pappas writes that the ceiling at the House of Representatives is a parabola -- and based on the info provided at this YouTube link, it really is a parabola.

According to the link, President John Quincy Adams took full advantage of this whispering spot.

Let's continue with our musical scales. By now, you should have figured out the pattern:

-- Our next prime is 13, so it should incorporate the 13th harmonic.
-- Our last two scales were 19EDO and 31EDO, so we combine these to form a 50-note scale.

It would take too long to write out all the notes of 50EDO. But since an octave is 1200 cents, each step of 50EDO must be 1200/50 = 24 cents. So all the multiples of 24 cents are steps of 50EDO

Let's check out how accurate all the harmonics are in 50EDO:

-- 3/2: The perfect fifth is 702 cents, so we choose Note 29 (696 cents).
-- 5/4: The major third is 386 cents, so we choose Note 16 (384 cents).
-- 7/4: The harmonic seventh is 969 cents, so we choose Note 40 (960 cents).
-- 9/8: The meantone is Note 8 (192 cents).
-- 11/8: The harmonic eleventh, is 551 cents, so we choose Note 23 (552 cents).
-- 13/8: And the newest harmonic, the 13th, is 841 cents, so we choose Note 35 (840 cents).

As we can see, the 11th and 13th harmonics are very accurate. The seventh is the furthest away.

Here is the link to our Xenharmonic site with regards to 50EDO:

By the time we reach 50EDO, most musicians would say that there are too many notes. I couldn't find any YouTube links to 50EDO music. But I do see a poster on the dozenal (base 12) website, Leopold Plumtree, who is apparently a huge fan of 50EDO:

Plumtree writes:

Dividing the octave into a dozen parts gives us a 1:2 semitone:whole tone ratio (of their logs). Using the next pair of Fibonacci numbers (including 2 again) would be 2:3 which results in 19 divisions per octave (very close to 1/3 comma meantone) and next up would be 3:5 giving 31 divisions (with a fifth a tad wider than that of 1/4 comma meantone). [Keep on going and we approach the golden meantone].

But, if we go a little further and use 5:8 for the diatonic semitone:whole tone ratio, we get 50 divisions per octave, so the diatonic semitone is a tenth of an octave.

The 50-edo major scale in a decimal octave is...

0.00, 0.16, 0.32, 0.42, 0.58, 0.74, 0.90, 1.00

The major triad is 0.00, 0.32, <0.26>, 0.58, the 0.26 being the minor third between the major third and fifth.

Compare to the 5-limit Just major triad...

0.000, 0.322, <0.263>, 0.585

And we get a lovely fit for the thirds; the major third being out by about 2 millioctaves and the minor third by about 3 millioctaves.

As others have noted, 50-edo is also close to the least squares *meantone mapping of the major triad compared to its Just counterpart (or minor triad...same intervals in a different order), which results in a fifth of 0.580137 octave.

So unfortunately, we can't hear what 50EDO music actually sounds like -- nor for that matter what 13-limit music actually sounds like.

But one thing I've noticed is that, while the third harmonic is actually a (perfect) fifth and the fifth harmonic is actually a (major) third, the seventh harmonic really is a seventh. Likewise, the 11th harmonic actually is a sort of "eleventh" (octave+fourth) while the 13th harmonic actually is a sort of "thirteenth" (octave+sixth).

In fact, we actually have eleventh and thirteenth chords in addition to seventh and ninth chords. But one thing I notice about 11th chords in music (especially jazz) is that the 11th is often sharpened:


We already know that jazz musicians often play a harmonic seventh for the Bb. And so I often wonder whether the F# is intended to be a harmonic 11th, 11/8 -- since F# is the closest note to 11/8 in a 12EDO scale.

So perhaps in 31EDO and 50EDO, we could try to play a true harmonic 11th chord -- as well as a true harmonic 13th chord in 50EDO.

Monday, July 3, 2017

Pappas Music Post: The 31-Note Scale

This is what Theoni Pappas writes on page 184 of her Magic of Mathematics:

"Mathematical ideas have been twisting and turning music and sound waves for centuries. A walk around the interior of the dome in St. Peter's Cathedral in Rome will convince you that the curve of the dome's walls carries one's whispers to a listener on the opposite side."

This begins a new section in Pappas, "Mathematics & Sound." I've heard of the concept of a "whispering gallery" before. Pappas suggests that the whispering gallery at the Vatican consists of two parabolas. But actually, St. Peter's Basilica is an ellipse. Let's watch the following video:

A person who stands at one focus of the ellipse can listen to one who stands at the other focus. I assume that in this video, the camera or phone is at one focus, and what we are hearing is at the other.

I'm not quite sure why Pappas writes "parabola" instead of ellipse. At first, I thought it was because she fears that the readers are less familiar with ellipses than parabolas. But Pappas herself explains what an ellipse is on Pages 18-19, so that can't be the case.

Fortunately, parabolas do share with ellipses the whispering property. Ellipses have two foci but parabolas have only one focus, so, as Pappas writes, we need two parabolas to make it work.

Actually, further investigation reveals that there really is a parabola at St. Peter's. But it's not the whispering gallery. Apparently, Michelangelo designed the dome as a parabola. Here is a link to the interior of this parabola -- it's the last photo at the following link:

I hope this settles the debate about which conic sections are at St. Peter's Basilica.

Technically, we are done with musical scales in Pappas. But actually, we can go on forever with scales, since there's so much information from the Xenharmonic website.

The next limit to consider after the 7-limit isn't the 9-limit, since 9 is already included in 3 (so we already had 9/8 with Pythagoras). Instead, it's the 11-limit, as 11 is the next prime number.

Is the 11-limit worth playing in music? Well, we reduce our 11 by octaves to 11/2 and 11/4, and finally 11/8 is the interval to consider. This interval lies somewhere between the major third of 5/4 (or 10/8) and the perfect fifth of 3/2 (or 12/8). In fact it is 551 cents, and so in 12EDO, it's very slightly closer to the 600-cent tritone than the 500-cent perfect fourth.

Vi Hart -- yes, the same Hart who prefers tau to pi -- created a video about the harmonics. She intentionally stops at the 10th harmonic, since the 11th harmonic is essentially a tritone -- one of the two most dissonant intervals. (She skips the 17th harmonic for the same reason -- 17/16 is the other of our dissonant intervals, the semitone.)

But some musicians wish to include the 11th harmonic anyway. Intervals based on 11 are known as "undecimal," just as those based on 7 are "septimal."

Now we just need a scale to in which to play our undecimal intervals. As it turns out, some of our scales fit into a pattern: 5, 7, 12, and 19.

Moreover, in the 7EDO, we played a 5-note subset (the Chinese scale). In 12EDO, we can play both a 7-note subset (white keys, major scale) and a 5-note subset (black keys, pentatonic scale). A certain musician (Joseph Yasser) continued this pattern with 19EDO -- a 12-note subset (all the white keys and the flats, a chromatic scale) and a 7-note subset (all the sharps, a C# major scale).

And so the next scale in this pattern would be 31EDO, which would contain both a 19-note subset and a 7-note subset:

Degree      Cents
0               0
1               39
2               77
3               116
4               155
5               194
6               232
7               271
8               310
9               348
10             387
11             426
12             465
13             503
14             542
15             581
16             619
17             658
18             697
19             735
20             774
21             813
22             852
23             890
24             929
25             968
26             1006
27             1045
28             1084
29             1123
30             1161
31             1200

Let's find our harmonics:

-- 3/2: The perfect fifth is 702 cents, so we choose Note 18 (697 cents).
-- 5/4: The major third is 386 cents, so we choose Note 10 (387 cents).
-- 7/4: The harmonic seventh is 969 cents, so we choose Note 25 (968 cents).
-- 9/8: The meantone is Note 5 (194 cents).
-- 11/8: And the newest harmonic, the 11th, is 551 cents, so we choose Note 14 (542 cents).

All of these harmonics are improvements from 19EDO. In fact, these are all the best approximations for the harmonics in scales we've seen except for the perfect fifth (slightly better in 12EDO and 17EDO) and the 11th (slightly better in 22EDO, but that note was already assigned to 27/20 based on the Indian scale).

The Xenharmonic website is very detailed as to which ratios correspond to each note of 31EDO:

Degree      Ratio             Name
0               1/1                 Perfect unison
1               55/54, 33/32  Diesis
2               22/21, 25/24  (Undecimal) chromatic semitone
3               15/14, 16/15  Diatonic semitone
4               12/11, 35/32  (Undecimal) neutral 2nd
5               9/8, 10/9        Major 2nd
6               8/7                 Septimal supermajor 2nd
7               7/6                 Septimal subminor 3rd
8               6/5                 Minor 3rd
9               11/9               Undecimal neutral 3rd
10             5/4                 Major 3rd
11             14/11, 9/7      (Undecimal or septimal) supermajor 3rd
12             21/16             Septimal sub 4th
13             4/3                 Perfect 4th
14             11/8, 15/11    Undecimal super 4th
15             7/5                 Septimal diminished 5th
16             10/7               Septimal augmented 4th

The chart abruptly ends here, since each new interval is an octave inversion of a previous interval. So Note 17 is an undecimal sub 5th, or 16/11, since (11/8)(16/11) = 2/1, the octave. We can invert any interval be changing "major" to "minor" (and vice versa), "super" to "sub," "augmented" to "diminished," and keeping "perfect" and "neutral" the same. Oh, and 2nds become 7ths, 3rd become 6th, and 4ths become 5ths.

We notice that these new undecimal intervals tend to be "neutral." This is because the 11th harmonic is almost exactly halfway between a perfect fourth and a tritone. So neutral intervals tend to differ from Pythagorean intervals by 33/32, which is an undecimal "quarter tone."

Notice that if we scroll down to Note 13 (perfect 4th/meantone) we can see how all of the previous scales fit into the 31EDO -- three-, five-, seven-, 12-, and 19-note scales.

To name these notes, we see that there are too many notes for a simple sharp-flat system. We could use either double-sharps and -flats, or half-sharps and -flats. The link above uses half accidentals, which are symbolized using up (^) and down (v) symbols:

Degree      Cents     Solfege     Color
C               0            do             white unison
C^             39          di              jade quarter-tone
C#             77          ro              blue 2nd
Db             116        ra              green 2nd
Dv             155        ru             amber 2nd    
D               194        re              yellow 2nd
D^             232        ri               red 2nd
D#             271        ma            blue 3rd
Eb             310        me             green 3rd
Ev             348        mu            jade 3rd
E               387        mi             yellow 3rd
E^             426        mo             red 3rd
Fv             465        fe               blue 4th
F               503        fa               white 4th
F^             542        fu               jade 4th
F#             581        fi               blue 5th
Gb            619         se              red 4th
Gv            658         su              amber 5th
G              697         sol            white 5th
G^            735         si               red 5th
G#            774         lo              blue 6th
Ab            813         le              green 6th
Av            852         lu              amber 6th
A              890         la              yellow 6th
A^            929         li               red 6th
A#            968         ta              blue 7th
Bb            1006       te              green 7th
Bv            1045       tu              jade 7th
B              1084       ti               yellow 7th
B^             1123      to              red 7th        
Cv            1161       da             amber octave  
C              1200       do             white octave

The solfege system is explained at the following link:

The new undecimal intervals require two new colors, "amber" and "jade." The colors must appear in pairs since the inversion of an amber interval must be jade, just as the inversion of a yellow interval must be green. Notice that jade is visually similar to green -- and in fact, green intervals are converted to jade by raising them 55/54. Yellow intervals are converted to amber by lowering them 55/54.

We also see that all of the septimal intervals are distinguished in 31EDO. So the harmonic 7th is equal to neither the Pythagorean minor seventh nor the septimal supermajor sixth. But the harmonic 7th must still be played as A#, just as in 19EDO. The harmonic 11th is played as F^.

As usual, here is a song played in 31EDO: