June Gloom Post: Crying Over Spilled Milk

Table of Contents

1. Introduction
2. Attempts to Find Information
3. My Most Important Class
4. What I Could Have Done Better: Math
5. What I Could Have Done Better: Science
6. What I Could Have Done Better: Classroom Management
7. A Day in the Life of My Ideal Seventh Grade Class
8. The Language Barrier
9. The Feynman Point
10. Conclusion

Introduction

This is a strange post, because here I'm going to make a strange announcement. Readers of this blog are aware that I was formerly a middle school teacher at a K-8 charter in Los Angeles. I was only there for less than a year, and since then I've made numerous blog posts about how I could have had a better experience there.

Every five years, charter schools must apply for a renewal of the charter -- the actual document that allows the school to operate. Renewal isn't automatic -- the school must demonstrate to the chartering district -- in this case the LAUSD -- how the school has been successful during the previous five years in order to be granted the renewal.

Anyway, my announcement is that a few months ago, the district denied the charter renewal, mainly for financial reasons. It's possible for the school to appeal to either the county or the state to accept the renewal -- and indeed, I've heard of several charter schools that are currently authorized through LA County or the State of California because their respective districts denied their renewals.

Thus it's possible that my charter school will live on. But all I know as of now is that today -- June 30th, 2019 -- is the day when the charter officially expires. That's why I choose today to post, so I can announce the charter expiration. The "old charter school" that I've mentioned in previous posts is in danger of being an old (as in former, no longer existing)

You might ask, why does the possible closure of my old charter school affect me when I'm no longer a teacher there? Once again, recall that I currently a substitute teacher in search of a full-time job as a regular teacher. Most job applications require prospective teachers to list their former employers along with their contact information.

So I need to know what's going on with my old charter in order for the information on my applications to be current and complete. But while the current students, teachers, and other employers are aware of what's going on, there's no way for that information to reach me. I fear that this will affect my ability to fill out applications and be hired at a school next year.

I'm sorry to mix metaphors here, but this post is "June gloom" and "spilled milk" rolled into one. This post is about all I know about the situation, and what it means for my future as a teacher.

Attempts to Find Information

From time to time, I drive past my old charter school. I often make these visits during weeks of major events -- such as field trips or Pi Day -- and then report it on the blog. But those visits were both before the denial of the renewal petition.

I always knew that the year I taught at the school was the third year of the current five-year charter, and so I was aware that today would be the last day of the charter. But I had always taken it for granted that the charter would be renewed this year, both before and after I left the school.

The school has both Facebook and Twitter pages. But while these were regularly updated during the month before the petition denial, neither has been updated since. I suppose I can't blame the school leaders from being too subdued to want to post anything -- just consider my own posting habits two years ago, right after I left the school. I deleted posts and skipping blogging on many days.

I assume that the school leaders are giving teachers, students, and parents information about the renewal situation face-to-face, or perhaps by sending letters home. But once again, it's frustrating because there's no way for me to access that information.

Indeed, it was just recently that I grew curious about the renewal situation and decided to perform a Google search for my school. I then saw the minutes from the LAUSD board meeting a few months earlier during which the petition was denied. That explained the sudden lack of posts on Facebook and Twitter.

Last week, I decided to visit my school one last time to try to find some information. Of course, by now it's summer vacation and so there are no students there, but I was hoping that some adult affiliated with the school would be there. The back gate was open, and a group of men and boys were playing basketball. But it appears that none of them have anything to do with the school -- they likely live across the street and just played there because the court was accessible. The boys appeared to be upper elementary or middle school students, but they probably attend a nearby public school.

I walked around to the front of the school. There were still signs indicating that students could enroll at the charter school. Yet the front door -- where prospective parents would presumably enter the office to enroll their children -- was locked, and there was no sign of anyone there.

At 2:00, the players left the court. Shortly after, I saw someone enter her car to leave, though it wasn't anyone I recognized from my time at the school. I was hoping that maybe she worked in the office, and thus she would lock the back gate on her way out. That would have indicated that she indeed worked at the school, and so I could ask her about the renewal situation. But she drove off without locking the gate.

Recall that during the year that I worked there, our charter was co-located with a district school. Then the year after I left, the charter moved to its current location -- thus I wasn't familiar with any of the buildings there. I did notice that there was a church very close to the school. So it's possible that the person I saw worked for the church, not the school. If she did work for the school, she was mostly likely a one-woman skeleton crew left behind just to answer the phone until the end of the month.

The signs announcing that students could enroll at the school contained a phone number. So the next day at around noon, I tried calling the number. As it turned out, it was an old landline number from back when the school was still co-located with LAUSD (that is, when I was still there). A tape recorded message announced the school's current numbers, So I called the new number, but there was no answer -- instead, a message directed me to wait until someone picked up the phone. But two minutes later, my call was automatically disconnected.

I tried calling again after hours, hoping for a tape recorded message announcing the current status of the renewal and the availability of student enrollment. Instead, the message announced only that the office was closed (for the day, that is).

I was also provided with the "current" number of our sister campus. But this was even worse -- there was only a message from the phone company that the number was no longer in service.

Therefore after all of this research, I still don't know whether my school will open in the fall or not. I looked back at my job applications to see which phone number I'd provided for previous employer contact info. I'd already known that the numbers from the co-located days were invalid. It turns out that the number I provided is for our sister campus -- a number that I know doesn't work now. But there's no point in switching it to the number for our own campus unless I'm sure that someone will be there to answer the phone.

By now, the only way to find out what's going on -- short of someone actually updating Facebook and Twitter -- is to wait until the fall to see whether the school is open or not. It's possible that another charter might take over the building -- or perhaps the church might even open a private school. It means that this will be a lost summer for my job applications -- prospective employers might raise their eyebrows when they see the invalid phone number on my application.

I will point out that another charter school in the district, upon learning that its own renewal petition was denied, just closed down completely -- the first week in May, shortly after the denial. Thus this school didn't even attempt to appeal to the county or state. When I walked around my old school, I saw, through a window, the date "May 31st" written on a classroom whiteboard. This suggests that my school at least stayed open through the end of May, and probably the end of the school year. So it's likely that my school at least attempted to get a charter renewal with the county or state.

My Most Important Class

Once again, the main reason for the petition denial is financial -- the district did not agree with the charter school leaders regarding how money was spent. But at that fateful board meeting, it was also mentioned that standardized test scores weren't as high as they could have been.

Let me warn you that I'm about to mention race in this post, so if you wish to avoid race or politics, you should stop reading this post now. (But this is a vacation post, and I'll add the "traditionalists" label as I usually do for race-based posts.)

Let's analyze the test scores during the four years of the current charter (since the fifth year state tests, taken a few weeks ago, obviously haven't been scored yet). We see that the scores were highest in the second year -- the year before my arrival -- and have been declining since.

But it's important to look at the breakdown by race. As I wrote on the blog about a year and a half ago, our sister charter is almost 100% African-American, while our own school has both black and Hispanic students. Thus even though the score reports combine the two campuses, we can use "Hispanic" as a proxy for "was taught at our school, not the sister charter."

Furthermore, scores are also broken down by grade level. In five of the six testing grades (3rd-8th), there were ten or fewer Hispanics who took the test, and thus no score is reported. The one class with enough Hispanics to get a report score was in seventh grade during the year I was there.

Thus, of all the scores visible on the report, those for Hispanic seventh grade math are the scores for which I was the most personally responsible. No one else at my school, nor anyone at the sister charter, was more responsible for Hispanic Grade 7 math than I was. And our middle school English teacher had the most control over the Hispanic Grade 7 ELA scores.

The conclusion is inescapable -- during the year I was there, the seventh grade class was by far the most important class I taught. We can argue about whether it's OK to judge students and teachers by test scores, or whether it makes sense to break them down by race, until the cows come home. But the fact of the matter is, I was judged by my Hispanic Grade 7 math scores. And so I should have worked harder to make sure that my Hispanic seventh graders were succeeding.

When that school year first started, it was obvious that there were only two Hispanic students in the eighth grade class, with the rest black. I knew that there were more in the sixth and seventh grade classes, but I didn't count them. Only when it's time for the field trip to the Hidden Figures movie -- a film whose themes include race and gender -- did I actually count the number of students by race.

Not only should I have counted the students by race at the start of the year, but then my focus should have been on either the most diverse class, or the class whose racial majority isn't the same as the whole school's racial majority. Either one of these would have identified seventh grade as the focus class for both our English teacher and me.

It's sometimes tough having three preps, since a teacher must devote time to preparing for each of the different classes. What I'm saying here is that if I had only a limited time to prepare for class and must make a choice between setting up seventh grade and setting up another grade, then I should have chosen seventh grade. And this includes the time I spent on this blog.

You might recall that on the blog, my focus class was eighth grade. I created a "Math 8" label and then described that class more than the others. I did so because this is a Common Core Geometry blog, and Grade 8 is a key year for the geometry (G) standards under the Core. But once again, my needs in the classroom should have taken precedence. If it made more sense for me to focus on seventh grade, then I should have created the "Math 7" label instead.

Indeed, there were so few eighth graders that, even though they were more than ten, just enough were absent on testing day for them not to receive a math score in the report -- enough were present for ELA, but not math. This meant that seventh grade turned out to be even more important for me than for the English teacher (but there was no way to know this at the time).

Then again, the English teacher was better set up for seventh grade success than I was. Of the three grades, seventh was the grade I saw the least. Both sixth and eighth grades saw me for IXL after lunch, but not seventh grade. On Wednesdays, they didn't have me at all. Even when IXL time was replaced with SBAC Prep, I saw the sixth graders for an hour, but seventh grade for only half an hour.

In the end -- and I mentioned this a year and a half ago on the blog -- the ELA scores for Hispanic seventh graders in my year were much better than their math scores. So our English teacher had something she could be proud of in the score report, but not me.

As teachers, we should care about our students' needs, not my needs or the school's needs. But the truth is, if I'm no longer an employee at the school, or if the school is shut down, then there's no way for me to help students learn and satisfy their needs. Thus what I needed to think about the whole time I was there is keeping my job and keeping the school open -- and the way to accomplish both was to focus on Hispanic seventh grade math SBAC scores.

In this post, I wish to write only about those factors I had control over. I had no control over the bell schedule and the fact that seventh grade had much more time in English than in math. And I had no control over how the school leaders spent money (that is, I could have raised every single student's math score by one full level and they might have nonetheless closed the school over finances). The goal in this post is to describe what I could have done to improve the situation. What could I have done to improve Hispanic seventh grade math scores? What could I have done so that I wouldn't have needed to leave my job? What could I have done to get the renewal petition renewed?

What I Could Have Done Better: Math

In many previous posts, I described two weekly plans with regards to how to schedule all parts of the Illinois State project-based curriculum. Both take into account that the coding teacher visited my classroom every Monday:

Plan I:
Monday: Coding Monday
Tuesday: STEM Project
Wednesday: Traditional Lesson
Thursday: Learning Centers
Friday: Weekly Assessment

Plan II:
Monday: Coding Monday
Tuesday: Traditional Lesson
Wednesday: Learning Centers
Thursday: STEM Project
Friday: Weekly Assessment

The first plan is the one suggested by Illinois State itself. But this one is not suitable for seventh grade, because there was no class on Wednesdays for the traditional lesson -- and the students wouldn't have been able to do any projects without getting the traditional lesson. The second plan was created with the seventh grade schedule in mind. Under this plan, seventh graders would miss Learning Centers, which sixth and eighth graders would get. Yet, as you'll see later in this post, there might have been a way for seventh grade to get Learning Centers anyway -- at least sometimes.

Illinois State recommends that one standard be taught each week. If we look at the Common Core Math 7 standards, we see the following schedule:

Weeks 1-6: RP (1, 2abcd, 3)
Weeks 7-15: NS (1abcd, 2abcd, 3)
Weeks 16-20: EE (1, 2, 3, 4ab)
Weeks 21-26: G (1, 2, 3, 4, 5, 6)
Weeks 27-37: SP (1, 2, 3, 4, 5, 6, 7ab, 8abc)

In reality, there's no way we could get through all of the SP (stats & prob) standards, but we should at least start them before the SBAC. My rule of thumb is to assume ten solid weeks of teaching during each trimester. This means that second trimester starts with NS2 (number sense, mostly integer ops), and the third trimester starts with G (geometry).

Notice that the first Tuesday of the third trimester that year was Pi Day. Technically, pi is taught as part of Standard G4, but the third trimester starts with G1. In this case, it isn't too bad to start with G4 instead, so that the students could have had a happy Pi Day. Once again, recall from the blog that even though I was no longer teaching at the school by Pi Day, I did buy my old class a pizza. It turned out to be for sixth graders, only because their class was the one that met before lunch (and after the pizza place opened) due to the special Parent Conferences schedule at the start of the trimester.

What I Could Have Done Better: Science

I mentioned recently (after subbing for a seventh grade science class) that my seventh graders really drew the short end of the stick when it came to science. I didn't teach much science at all to any of the three grades -- and now we combine that with the lack of class time I had with seventh grade.

I've devoted many posts to how I could have taught science better. I explained that the three-week cycles that I'd envisioned for math tests actually should have applied to science instead.

In fact, notice that there were actually four types of assessments that I wanted to give my students -- Dren Quizzes, (math) Regular Quizzes, (math) tests, and science tests. Thus, if there was to be any sort of cycle, it should be four weeks, not three.

In fact, let's divide up the entire year into four-week cycles, as follows:

Weeks 1-4: Science Unit 1
Weeks 5-8: Science Unit 2
Weeks 9-12: Science Unit 3
Week 13: End of Trimester 1
Weeks 14-17: Science Unit 4
Weeks 18-21: Science Unit 5
Weeks 22-25: Science Unit 6
Week 26: End of Trimester 2
Weeks 27-30: Science Unit 7
Weeks 31-34: Science Unit 8
Weeks 35-38: Science Unit 9

These weeks don't really line up with the weeks I listed for math standards above, since this now shows the entire year, including special weeks when there might not be time for a math standard.

Now within each unit, the four assessments are given in the following order:

First Week: Dren Quiz
Second Week: Math Quiz
Third Week: Science Test
Fourth Week: Math Test

Science is to be taught on Thursdays. This means that the "STEM project" listed above will usually be a science project. In fact, the idea is to have traditional lessons during the first and third weeks of each cycle, and projects during the second and fourth weeks. This allows me to fulfill the Illinois State requirement of submitting feedback from science projects every two weeks.

Most assessments are given on Fridays. This includes the science test, which is given after the traditional lesson on Thursday.

There are many special weeks where this pattern isn't used -- but even during special weeks, we try to follow this pattern as closely as possible. During the first week of school, the traditional science lesson is on science safety, and the Dren Quiz is on 10's. The second week of school, we give the mousetrap project to all grades (the first project of the year and the only one that's not science), and that week's Benchmark Tests take the place of the Math Quiz (and is given earlier that week). The third week of school, the science test on Thursday is actually the Study Island pretest. (There was no school on Friday due to the four-day Labor Day weekend.) Then the first test that actually counts in student grades is the Math Test the fourth week, which covers the earliest math standards (mostly RP1, with possibly some RP2a). The first real science project is also given the fourth week.

The fifth week starts Unit 2 -- the first real science unit. A short traditional lesson can be given on Thursday, followed immediately by the Dren Quiz on 2's. This is because the field trip to the LA County Fair is on Friday (where a real science lesson -- observing farm animals -- can occur).

Now what science would I teach the seventh graders that year? I wrote in earlier posts that we should consider seventh grade to be the last cohort using the old California Standards (with life science in seventh grade and physical science in eighth), with that year's sixth graders to be the first cohort using the Preferred Integrated curriculum of the NGSS.

Thus seventh graders get life science. I use the online version of the Illinois State science text and then simply cover all of the projects in the order they are given. The traditional lessons and assessments thus correspond to the projects that I'm giving. (I wish I still had access to that online text so that I can post what these fascinating lessons actually are.)

There's no real reason for me to number the units here, except to note that the Dren Quiz that I give during each unit matches the unit number (except that I give the 10's Dren Quiz during Unit 1). The unit numbers are actually more significant for sixth grade, where I teach the Preferred Integrated model, following not the Illinois State order, but rather the Study Island order. That curriculum actually does number the units (with Unit 1 indeed being the pretest as stated above).

What I Could Have Done Better: Classroom Management

Of course, many of my biggest problems that year involve classroom management. Here I will discuss some of the things I could have done to manage the classroom much better than I actually did.

Recall that my class had a support aide -- and a huge problem was that many of the students respected her much more than they respected me. So we can look at what my schedule was that year and the times when the aide was available. Then we can figure out how I could have stepped up my management game during the times when she wasn't there.

Since seventh grade had coding on Mondays and music on Wednesdays, the three days that are significant to management are Tuesday, Thursday, and Friday. On Tuesdays and Fridays, seventh grade was the second class of the day, and my aide had usually arrived by then. But on Thursdays, seventh grade met before my aide arrived. Thus I especially needed to be careful about classroom management on Thursday mornings.

I've explained in previous posts why my management went awry that year. One problem was that I entered the classroom expecting my students not to listen to me, and so I avoided telling them to do things as much as possible. For example, I knew that students would not want to sit in their assigned seats and try to move next to their friends. Thus I wanted to let them choose their own assigned seats until I at least knew their names. Once I learned their names, then I could move them, since now I could catch anyone who tried to switch seats.

Anyway, the history teacher -- who had taught these students as sixth graders the previous year -- informed me of the four boys who were likely to cause trouble. Since I'm discussing race in this post, I'll mention that of this quartet, two were Latino and two were black. The history teacher then recommended that I seat these students in four corners of the room, so that they would be separated as far as possible. But I didn't, since I'd already planned on letting the students choose their seats.

This, of course, was the beginning of all my troubles. The four guys were loud and tried to take over the class. The other seventh graders saw them and decided that they had the right to be loud too. And then the sixth and eighth graders heard my seventh graders when they were in English room -- and they decided that they had the right to be loud in my classroom too.

Instead, I should have followed my colleague's advice and seated the four boys in the corners. It's likely that if I did so, one of them would have made a snide remark such as, "Hey, this is the same as the history class. Can't you come up with an original seating chart?" My response should have been something like "No, I can't." If I haven't learned all the students' names yet, I should at least learn the names of the four corner students first, so that I can catch them when they inevitably try to switch to a different seat.

This should have been my general approach to classroom management. Sometimes that year, I felt as if I was more afraid of snide remarks than of bad behavior -- if I told a student to stop misbehaving, they'd reply that I was mean or unfair, or simply "I wasn't talking." Instead, I should just ignore such remarks if I know that I'm right.

Instead of avoiding telling students not to do things that they might find unpleasant, I should intentionally tell them to do so -- but sure I know in advance what to do if they don't obey me.

For example, I considered using some sort of binder or interactive notebook system. This would have helped the students immensely as they tried to learn math and science. But I was afraid that many students would refuse to purchase the notebooks, or buy them and then forget to bring them more than half the time.

As small as my classes were, I can buy enough notebooks for all the students. Then I can come up with punishments for students who don't bring them -- this includes a parent phone call for those who go multiple days without bringing a notebook, and perhaps lesser punishments for those who go one or two days without a notebook.

Then to get the class quiet, I tell the students, "Open your notebooks and close your mouths." This is to establish a routine -- when the lesson begins, it's time to stop talking.

One major problem I had with the seventh grade was that when there were only a few kids talking (such as the four troublemakers), I didn't saying anything. I was still thinking like a sub -- if I'm subbing and there are only four students talking, I considered it a miracle!

But as a regular teacher, when I let four students talk and only confronted students when there were more talking, the newer students genuinely thought that I was being unfair. I still remember two black girls who thought I was being sexist when I told them to stop talking -- because I had let the four guys talk throughout the class period. One of these girls had received the top score on the Benchmark Tests at the start of the year, but she no longer respected me because of this incident. Her grades dropped the whole time that I was in that classroom.

There's one more thing I could have done with this group of seventh graders. At the end of the Benchmark Tests at the start of the year, I introduced a game of "Fraction Fever." My plan was that this game was something to play after Benchmarks, and so I didn't play it again until November, at the end of the first trimester.

What I didn't expect was how much the seventh graders in particular enjoyed Fraction Fever -- both the song and the game. When I saw this, I should have found a way to use the Fraction Fever song and game to my advantage.

On Wednesday afternoons, there was often extra time left between music class and lunch. This would have been a good time to play the Fraction Fever song. The class is divided into two groups, and if one group gets in trouble, I just drop that group one level in the game. And the next day -- which as you might notice, is Thursday morning, the key day (since both my aide hasn't arrived yet, and it's the science day) -- I make the class be quiet before handing out Fraction Fever prizes. This helps me out with classroom management on those critical Thursday mornings.

This is how effective classroom managers do what they do -- they look for something that works and then do it. I instead, did the opposite -- I made plans (for both the seating chart and Fraction Fever) and stuck to them no matter what. If I had been more adaptable, I could have taken advantage of what I had, and ended up with a better class.

Incidentally, once I move the four boys to the four corners, I should monitor their behavior. Chances are two of them will improve their behavior once the four are separated. I should figure out which two improve the most by, say, the end of the first trimester, and reward them by allowing them to sit with each other.

A Day in the Life of My Ideal Seventh Grade Class

I wish to describe what my ideal seventh grade class might have looked like if I had followed the principles mentioned earlier in this post (as well as the resolutions mentioned in other posts).

Let's start with a randomly chosen day from my seventh grade class that year -- January 10th. This was a Tuesday and the first day after winter break. As it turned out, this was one of my best days in the seventh grade class -- but by observing these principles, I could have made the class even better.

I wrote about that day on the blog, under the title "A California 'Snow' Day" (referring to the rain we had that day). Let's keep the "A Day in the Life" format and I'll describe what the ideal seventh grade class might have looked like.

9:45 -- This class officially starts at 9:45. But this is actually when the previous class of sixth graders leaves my classroom -- there are no bells and hence no official "passing period."

But if I'm an effective manager who enforces rules, there's nothing wrong with unofficially observing a five-minute passing period here. After five minutes, I can start enforcing rules -- most notably "no food in the classroom," since so many students want to eat between their English and math classes.

9:50 -- It's time for the Warm-Up. I'm required to give the Illinois State Daily Assessment, which is usually a quick review of the previous day's lesson. But this is the first day after a long vacation, and so no matter what problem I give, some students won't remember how to do it.

But in this class, the students have interactive notebooks. So I go up and down the rows and look for students who aren't doing the work, and ask them why. If they say "I don't remember how to do this," then I tell them which page of their notebook will be the most helpful.

9:55 -- One of my resolutions is for the Warm-Ups not to take too long. Illinois State recommends for the Warm-Up to take only five minutes. After five minutes, I go and stamp Warm-Up papers. I tell the students that they should be taking out their notebooks.

10:00 -- Since it's Tuesday, it's now time for the traditional lesson. As I wrote earlier in this post, I say "Notebooks open, mouths closed."

On the actual January 10th that year, my lesson was on angles (G5). But if we follow the pacing guide listed above, this week would probably be NS2d or NS3. But for the sake of this post, let's assume that I'm teaching G5 anyway, since I want to make a direct comparison to January 10th in the original (real) timeline.

There was only one real problem that I had on the real January 10th -- many students had trouble figuring out how to use the protractors. But by following the principles mentioned in this post, I could improve upon this lesson.

First of all, with interactive notebooks, I pass out worksheets for students to paste in the notebooks -- and these worksheets show angles for the kids to measure. And second, the line "Notebooks open, mouths closed" means that students should be quiet, so they are paying attention when I tell them how to use the protractors.

It's the second trimester, so by now two of the four troublemakers have been reformed. I remember that on the first day after winter break, one of the them (a black guy) tells me that he and his sister are transferring to another school. He wants to play competitive middle school sports, which our school doesn't offer. (Ironically, one of the things I do know about my old charter school in 2019 is that now our school does have sports teams. This, of course, was before the denial of the renewal petition.)

Let's assume that the two Latino boys are the ones still sitting in the corners. I remember that around this time in the original (real) timeline, one of their fathers requested that I indeed sit his son in the corner so that the kid can learn and be focused.

By the way, for the sake of my Hispanic students, I might try to help any English learners by defining words like "protractor," "acute," and so on in the notes. I actually don't believe that there were many English learners in the class. (Recall how well this cohort goes on to fare on the SBAC for ELA.)

10:25 -- It's time for music break -- but on this day, I have a very special music break. On this day, I sing Square One TV's "Angle Dance," since it fits the lesson perfectly. But in this song, the students should participate. It's based on an activity called "Human Protractor," which I learned about the previous day during a PD day -- "Responsible Classroom" training. (This is why the first day after winter break was Tuesday, since Monday was this PD day.)

On the actual January 10th, I was afraid that many students would refuse to do the Human Protractor, and just stay seated -- or stand yet do nothing. But the Human Protractor is an activity that would really help the students to learn the angles. Thus I should require students to participate.

Thus I threaten to punish students who aren't moving their arms for the angles. Since my aide should be in the classroom, she can help check to see who isn't participating. This is a proper use for my support aide -- she should help me with management, not be the main manager.

10:35 -- Only now should I have the students work in the "Student Journals" (which are Illinois State consumable texts). On the actual January 10th, I made the "Student Journals" the whole lesson, but now the students have notebooks to refer to for help.

11:00 -- During the last five minutes, I give an Exit Pass. I mentioned in old posts that Pappas-style questions where the answer is the date should be the Exit Pass, not the Warm-Up (which should be the Illinois State Daily Assessment).

On the original January 10th, I had the students answer a New Year's problem, "2 + 0 + 1 + 7 =." This is also based on another idea I've seen on the web -- make an entire activity where students do the "four 4's" problem but for the digits of the date, and give this activity on the first day after returning from winter break.

If I give this exit pass, I should make the students show work. If this were a typical problem (for example, draw a 10-degree angle) then there's no need to show work, but here I might require the students to write "Happy New Year!" to keep them in the habit of always showing their work on the Exit Passes.

11:05 -- The seventh graders leave after a fruitful 75 minutes of learning.

The Language Barrier

My goal in this post is to present an environment in which all my students can learn -- but especially the Hispanic students. One challenge in teaching Hispanic students is, of course, the language barrier.

Recently when I was subbing, I tried to tell a Latino boy to put his phone away. Instead, he started speaking Spanish and pretending that he couldn't understand me. But I believe that he was speaking English earlier in class, and so it was all just an act to avoid obeying my instructions.

Therefore I suspect that learning a few commands in Spanish won't help me get through to students such as these. He'd probably just act as if I wasn't pronouncing the Spanish properly -- I didn't roll my r's, so he still can't understand me. A student who really wants to follow directions would figure out what I'm saying (for example, asking another kid to translate) and just do so. A student who is determined to misbehave will find any excuse not to listen to me, regardless of his race. In this case, he just takes advantage of his race to pretend that he can't understand English.

Once again, when I was at the old charter school, most of the Hispanic seventh graders spoke English, and none of them tried to pretend that they didn't understand me. But when some students misbehaved and I threatened to call their parents, the kids told me that their parents didn't speak English, and so there was no point in calling home. The implication was that I might as well let the Hispanic students do whatever they wanted, since there was no way to contact their parents. Thus once again, students took advantage of the language barrier to disobey me.

This underlies a reason why I need to have a level of punishment between the initial warning and the parent phone call. There must be a way to punish students and redirect them to better behavior that doesn't involve any adult other than myself.

Once again, I should be able to use the seating chart as punishment. If two students are being disruptive, then I should separate them.

One of the worst-behaved students in the class was the Latino boy whose parents I never spoke to. I suspect that his parents speak no English, and thus calling his home was useless. But he had one weakness that I knew of -- he had a girlfriend, a girl in the class.

Thus I should have taken advantage of this fact. He should be seated in the corner -- definitely separated from his girlfriend. If he tries to switch seats to sit next to her, then I go ahead and punish both of them. Unlike his parents, her parents might speak English and be more responsive, so she might tell her boyfriend to return to his seat so she can avoid trouble. And the girlfriend should be surrounded by good students who don't want to break rules. Then most of the time, there isn't even an available seat for the boyfriend.

I wrote that the Angle Dance song engaged many students in the class -- the black girls. At the time, I wanted to engage this group because they were the target demographic of Hidden Figures, the upcoming movie field trip. I don't remember how many Hispanic students were engaged by the song.

I recall one Latino boy who wanted me to give an intro in Spanish to one of my songs -- open up with "uno, dos, tres, cuatro." I knew that there was one Square One TV song to which this line fits well -- "Sign of the Times," a song about multiplication. This song has a Latin beat and contains a line repeated in English and Spanish:

X, it's the sign of the times.
Equis es el simbolo de los tiempos.

I was thinking of singing this song and saving it for a very special day -- Cinco de Mayo. The problem, of course, is that I left my school and never made it to Pi Day, much less Cinco de Mayo.

I shouldn't wait until May to engage my Hispanic students. Since one guy wanted me to add some Spanish to my songs, I should have obliged much earlier. And as the song is about multiplication, the song fits best just before a Dren Quiz. I believe that the first Dren Quiz after this boy asked me to sing in Spanish was the 4's (as he transferred to our school just days before the 3's Dren Quiz), and so I should sing the song before the 4's quiz.

Notice that according to the schedule listed above, the 4's Dren Quiz should be Week 14. This is the first week of the second trimester and hence was Parent Conference week. Ordinarily I don't sing songs on minimum days such as Parent Conference days, but it's worth it to make an exception.

The Feynman Point

Two days ago was Tau Day. My tradition on the blog is to use tau, rather than pi, to solve problems involving circles -- from now until Pi Approximation Day.

Yet I mentioned pi and Pi Day several times in today's post. This is because I was referring to an actual seventh grade class -- and seventh graders learn about pi, not tau. I suppose I could have written "Half Tau Day" instead of "Pi Day" throughout this post, but that would have distracted readers from the points I was making about the seventh grade class.

There's one thing related to math and science that I actually did two days ago, on Tau Day. I was in a local comic book store, but there was one new comic book -- a graphic novel, actually -- that I found very unusual. The book is called Hawking -- as in Stephen Hawking (the subject of A Theory of Everything) -- and its author is Jim Ottaviani. He's also listed on the book as the author of another graphic novel, Feynman.

It turns out that Ottaviani has written several graphic novels about famous scientists -- and in fact, his first novel Feynman was published in 2011.

I didn't mention this at all in my Tau Day post, because I didn't think I'd be able to find it at my library this quickly. But the next day, yesterday, I had checked out a copy of Feynman.

This is our summer reading book -- you can consider this to be side-along reading if you wish. In my next post, I'll read and describe the first part of the book. For now, let me quote the front jacket:

"Richard O. Feynman (1918-1988) was one of the great minds -- and great personalities -- of the twentieth century. A Nobel-Prize winning physicist, he was also a safecracker, an adventurer, and a world-class raconteur.

"Feynman lived his life large and loud, becoming a public favorite with his autobiographical collections Surely You're Joking, Mr. Feynman! and What Do You Care What Other People Think? His innovations in the field of quantum electrodynamics profoundly changed the way we think about things as simple -- or complex -- as matter and light. He played a crucial role in uncovering the cause of the Space Shuttle Challenger disaster. His work on the Manhattan Project contributed to the creation of the atomic bomb, changing the course of world history.

"And he was really, really fun at parties."

The Feynman point of pi is named after Richard Feynman (although one of the first physicists fascinated with it was actually our last side-along author, Douglas Hofstadter). It refers to the six consecutive nines among the digits of pi, starting at the 762nd decimal place:

3.1415926535...1499999983...

Notice that tau must also have a Feynman point. If we imagine doubling the above number, we get:

6.2831853071...2999999967...

Doubling the 8 causes a carry in the next digit, forcing it to be odd, and then each of the 9's cause carries when doubling, so all of them remain 9's. The leftmost 9 causes a carrying when doubling, and this forces the 4, when doubled, to become 9 as well. Thus the Feynman point of tau contains seven 9's, one more than pi.

If we double this number again to 2tau = 4pi:

12.5663706143...5999999934...

this number again has seven 9's. But after one last doubling to 4tau = 8pi:

25.1327412287...1999999869...

then this number returns to having only six 9's, since the 3 after the last 9 doesn't cause a carry.

This year on Pi Day (uh, Half Tau Day), the students created a long colored chain based on the digits of pi. It's possible that it was long enough to have reached the Feynman point, but I don't know.

Returning to Ottaviani's book, I like the idea of depicting scientists as heroes, because they are. My students need to see that mathematicians and scientists are heroes, not nerds, and so they should strive to do well in my math and science classes. This includes my old Hispanic seventh graders (who by now have just completed their freshman year of high school).

Conclusion

Well, I don't know what to say. I'll continue to drive past my old charter school -- and perhaps even our sister school -- for any hints of what's happening in the fall. For the sake of my former coworkers, I hope that the school is able to get its charter renewed, whether at the district, county, or state level.

I'm glad that at least all of the middle school students that I taught have completed their time at the charter school. Thus none of them are affected by the current chaos. But unfortunately, I'm still definitely affected by the situation and how it will affect my job applications -- so I make make my return to full-time teaching.

Tau Day Post: A Return to Computer Music

Table of Contents

1. Pappas Question of the Day
2. That Time of the Year Again
3. Tau Day and Summer School
4. Ingenuity with Tau Day Pies
5. Tau Day Links
6. More Tau Day Music
7. The Sweet Spot
8. BACH and CAGE Again
9. The Arabic Lute
10. Conclusion

Pappas Question of the Day

It turns out to be bad timing for the Pappas calendar and blogging this week. This is the only post that I'm making this week, but as it turns out, there is Geometry on the Pappas calendar everyday this week -- except today. But I wish to post a Pappas problem today, so let me choose one of the problems from earlier this week.

This week on her Mathematics Calendar 2019, Theoni Pappas wrote:

A sector of a circle with an arc of pi/12 radians is 1/? of the circle.

Well, there are two ways to solve this problem. One is to recall that the entire circumference is 2pi radians, and so we can use dimensional analysis:

(pi/12)radians * 1 circumference/2pi radians = pi/(12 * 2pi) of the circle = 1/24 of the circle

Therefore the ? should be replaced by 24 -- and of course, this problem must have come from the 24th, last Monday.

The other method would be easier, if we could only read the answer directly from the fractional radian measure. The measure is pi/12 -- but no, this doesn't mean that it's 1/12 of the circle. Instead, since the circumference is 2pi radians, we multiply both the numerator and denominator by 2:

(pi/12)radians * 2/2 = (2pi/24)radians

Indeed, this problem would have been more straightforward if the 2pi were written with a single symbol, such as tau. Then the radian value would have been tau/24, and then we could read the 24 straight off.

That Time of the Year Again

Hmm, today's date isn't the 24th, but the 28th. And now I hear the sound of all of my readers double-checking the date....

You guessed it -- another Tau Day is upon us. This is what I wrote last year about tau:

But what exactly is this constant "tau," anyway? It is not defined in the U of Chicago text -- if it were, it would have appeared in Section 8-8, as follows:

Definition:
tau = C/r, where C is the circumference and r the radius of a circle.

Now the text tells us that C = 2pi * r, so that C/r = 2pi. Therefore, we conclude that tau = 2pi. The decimal value of this constant is approximately 6.283185307..., or 6.28 to two decimal places. Thus the day 6/28, or June 28th, is Tau Day.

It was about ten or fifteen years ago when I first read about Pi Day, and one old site (which I believe no longer exists) suggested a few alternate Pi Days, including Pi Approximation Day, July 22nd. One day that this page mentioned was Two Pi Day, or June 28th. But this page did not suggest giving the number 2pi the name tau or any another special name.

Now about eight years ago -- on another Pi Day, March 14th -- I decided to search Google for some Pi Day webpages. But one of the search results was strange -- this webpage referred to a "half tau day":

http://halftauday.com/

"The true circle constant is the ratio of a circle’s circumference to its radius, not to its diameter. This number, called τ (tau), is equal to 2π, so π is 12τ—and March 14 is thus Half Tau Day. (Of course, since τ=6.28…, June 28, or 6/28, is Tau Day itself.) Although it is of great historical importance, the mathematical significance of π is simply that it is one-half τ."

The author of this link is Michael Hartl. Here's a link to his 2019 "State of the Tau" address:

https://tauday.com/state-of-the-tau

According to this link, Hartl and his friend Juan Ferreiro translated his Tau Manifesto this year into Spanish. He also adds that just as Pi Day is Einstein's birthday, Tau Day is the birthday of both physicist Maria Mayer and rocket scientist Elon Musk.

Oh, by the way, here's one argument in favor of tau that I've rarely seen addressed by tauists -- that is, advocates of tau. It's pointed out that there's one formula in which pi shines:

A = pi r^2

This formula would be less elegant if it were written using tau:

A = (1/2) tau r^2

Tauists often respond by saying, "well, the 1/2 reminds us of the proof of the area formula" (for example, using the area of triangles) or "the 1/2 reminds us of other quadratic forms."

But actually there's a much stronger argument in favor of tau. The formula for the area of a sector of central angle theta is:

A = (1/2) theta r^2

Oh, so this formula contains a 1/2 no matter what! The full circle area is given when theta = tau. (I see that this formula actually is buried in the Tau Manifesto, but still the other arguments appear first in the Manifesto.)

Tau Day and Summer School

In past years, I've often pointed out that both Tau Day and Pi Approximation Day can be used as alternatives to Pi Day for summer school classes. Therefore, if I were teaching summer school, today would have been a Tau Day party in my classes.

Different districts have different schedules, so let's assume that this is a district with summer school on Fridays. The idea I had for my own classes was that Fridays -- during the time left over if the students finished early -- would become somewhat relaxed. (Indeed, my policy on cell phones would have been that they're only allowed on Fridays after the test is complete.)

When many of the students have completed the test, music break would begin. On instruction days, the song I'd sing would be based on that day's lesson. But on Fridays after the test, I could sing a song that's more fun instead. For example, last Friday I would've played a song from Square One TV, such as "Square Song." I choose this song (really a rap) only because it was originally posted to YouTube twelve years ago on June 21st (as you know how I like to celebrate YouTube anniversaries).

Today I would have played Vi Hart's Tau Day tune, "Song About a Circle Constant." These songs are fun to sing after a long, hard test -- and I could even generate a discussion about why we'd give 2pi a special name like tau. And in a summer Algebra I class, these songs about squares and circles can serve as a preview to the Geometry that they'll hopefully take in the fall (provided they pass the summer class, of course).

I can keep up the tau = 2pi idea and let the food prize be pie. Recall that back on Pi Day, I purchased three full-sized pies for two classes. Notice that pi can be rounded down to 3.

So let's round tau down to six and buy personal pies for the top six students (in each of the two periods that summer teachers typically have). Actually, we could round tau up to seven, and then give the seventh pie to the most improved student -- that is, the student whose score rises the most from the first unit test (from last Thursday) to the second unit test.

This is actually something I regret not doing in my old middle school class at the charter school -- I awarded four Red Vines to the students who earned A's on my tests, but gave nothing to the most improved student. I can't help but think back to the day when the "special scholar" (from my Phi Approximation Day post) earned her first C on a test. I gave her one piece of candy that day anyway, but under this plan, the most improved student would have earned (almost) as much candy as the ones who got A's. Perhaps this would have helped me establish better rapport with her. Instead, she felt that she wasn't good enough unless she earned A's -- and then proceeded to cheat to get those A's.

Instead of pies, I could give out cupcakes instead -- just like the cupcakes mentioned in Michael Hartl's State of the Tau address. By the way, since I didn't actually teach any summer school today, what did I really eat today? Well, I ate at Tau-co Bell, of course. I probably should have ordered some "tau-cos," but instead I tried the $5 Chicken Nachos Box. According to the advertisement, this box contains two times as much meat as before -- just as tau is two times pi.

(Today I also ate a personal 7-Eleven lemon pie to celebrate the date.)

Ingenuity with Tau Day Pies

Last year, I gave an example of a Tau Day pie distribution from an actual summer school class in one of my districts -- a class that met and took a test on Tau Day that year. (Don't worry about what class this is -- this is given only as an example of pie distribution.)

Score     # of Students
92          1
88          1
84          1
80          2
76          3
72          5

Well, that's actually 13 students -- so only one pie is left. (Most likely, it would have depended on whether it's first or second period in which the 72's are scored.)

Of course, this is a summer class, so it's expected that most of the scores are low. Only one student earns an A on the test. His overall grade is also the highest at 84% -- a middling B.

As for the pies awarded to students whose grades improved the most, I must admit that most scores dropped from last week's to this week's test. I suppose this is expected -- even as a student teacher, I noticed that graphing linear functions (Chapter 4 of Glencoe) is always more difficult that just solving linear equations (Chapter 2).

Actually, the student who earned 92% is tied for the most improved student -- on last week's test, his score was only 76%. (The student he's tied with improved from 60% to 76%, so both have an increase of 16 points.) Depending on which period the students are in (including the 13th student who earned a 72%), the 14th and final pie would go to either the student who improved from 60% to 68%, or the one who maintained a score of 60%. (Due to the difficulty of the second test, even maintaining the grade counts as a major accomplishment.)

Well, that's how the grades in the actual class went. In many ways, it's better to award food based on the number of pies I brought as opposed to saying in advance, "Everyone with an A or B gets a pie!" (which might result in my having too few or too many pies). You might argue that this is sort of like grading on a curve (except that the letter grades don't change based on how much food I bring).

Tau Day Links

Here are this year's Tau Day links:

1. Vi Hart:

Naturally, we begin with the two Vi Hart videos that we just discussed above.

Oh, and by the way, Vi Hart adds a new Tau Day video for this year -- "Suspend Your Disbelief":

2. Numberphile:

His Tau vs. Pi Smackdown is a classic, and so I post this one every year.

3.Michael Blake:

This is a perennial favorite. Blake uses a rest to represent 0. Like Vi Hart, he ends on the digit 1, but this is ten extra digits past Hart's final 11.

4. SciShow

This video explains some more uses for tau. Michael Hartl mentions it in this year's "State of the Tau" address, linked above.

5. The Coding Train:

Here Dan uses geometric probability to approximate the values of pi and tau. Here's a simple program that does the same on Mocha:

http://www.haplessgenius.com/mocha/

10 A=0:B=0
20 B=B+1
30 X=RND(0)
40 Y=RND(0)
50 IF X*X+Y*Y<1 THEN A=A+1
60 PRINT 4*A/B,8*A/B
70 GOTO 20
RUN

Estimates for pi appear in the left column and estimates for tau appear in the right column. This program has an infinite loop, so press Esc when you feel that the approximations are good enough.

6. MusiMasta

Here is a new song based on the digits of tau.

7. Samuel El Pesado:

Oh, this is a video that was first posted seven years ago. But once I created this blog I couldn't find the video again until last year. Here a group of high school students blow up a pi(e) for Tau Day! (Happy early Fourth of July!)

8. Arifmetix

Here's another quick tau song. Apparently, this song assigns other Greek letters to other even multiples of pi.

More Tau Day Music

And now you're probably saying, here we go again! Even after my summer school class is cancelled, first I start whining about summer school again, and now I go right back to music -- a topic that I wouldn't discuss unless I had a class to sing songs in.

But I can't help thinking about music again on Tau Day. Recall that on Phi, e, and Pi Days, I coded Mocha programs for songs based on their digits. Seeing the Vi Hart and Michael Blake songs again makes me want to code a tau song on Mocha.

And besides, last year I wrote about music on Tau Day. It was around this time last year when I first read the Pappas book about musical scales. On Tau Day, I wrote about 12EDO, our usual scale, and its relationship to 5-limit ratios. A few days later, I wrote about the Indian sruti scale and 22EDO, and then I kept writing about more EDO scales until I stumbled upon the Mocha computer emulator and realized that EDL scale, not EDO scales, fit Mocha's sound command.

This is also the first day devoted to a constant since I wrote about those EDL scales. So it's logical to write a tau song in one of these new EDL scales, but which one should we choose?

Recall that Vi Hart uses a major tenth to represent 0. The ratio for a major tenth is 5/2 (found by multiplying the major third 5/4 by the octave 2/1). The ratio 5/2 is equivalent to 15/6, which is good because now the digits 2-9 fit between Degrees 15 and 6:

A Tau Day scale:
Digit     Degree     Ratio     Note
1           15             1/1        tonic
2           14             15/14    septimal diatonic semitone
3           13             15/13    tridecimal ultramajor second (semifourth)
4           12             5/4        major third
5           11             15/11    undecimal augmented fourth
6           10             3/2        perfect fifth
7           9               5/3        major sixth
8           8               15/8      major seventh
9           7               15/7      septimal minor ninth
0           6               5/2        major tenth
-            5               3/1        tritave

This scale doesn't fit in an octave. Instead, we can think of this scale as fitting a tritave (3/1) instead, just like the Bohlen-Pierce scale. On Mocha, we can't actually play the full tritave since Degree 6, not 5, is the last degree playable in Mocha. But Degree 6 is the last degree needed for the tau song.

Indeed, this span occurs in real songs. Most real songs have a span of more than an octave -- we explored some holiday songs six months ago and found out how spans of a ninth or tenth appear to be fairly common. On the other hand, a twelfth, or tritave, is too wide. One famous song with a span of a tritave is our national anthem, "The Star-Spangled Banner." This song is considered one of the most difficult national anthems, and the reason is its tritave span. This tells us that while EDL scales based on an octave might be insufficient (as we'd want to venture a note or two beyond the octave), EDL's based on the tritave are more than enough.

Let's fill in Kite's color names for the EDL scale that we've written above, 15EDL. Notice that while octave EDL's must be even, tritave EDL's must be multiples of three, so they could be odd:

The 15EDL tritave scale:
Digit     Degree     Ratio     Note
1           15             1/1        green F
2           14             15/14    red F#
3           13             15/13    thu G
4           12             5/4        white A
5           11             15/11      lavender B
6           10             3/2        green C
7           9               5/3        white D
8           8               15/8      white E
9           7               15/7      red F#
0           6               5/2        white A
-            5               3/1        green C

The names of some of these intervals are awkward. First of all, 15/14 is often called the "septimal diatonic semitone." But the name "diatonic semitone" ordinarily refers to a minor second, not an augmented unison -- even though the notes are spelled as the latter (gF-rF#). The reason that the name "diatonic" is used is that it's close to the diatonic semitone 16/15 (wE-gF) in cents. The name "septimal diatonic semitone" for 15/14 forces 15/7 (an octave higher) to be a "septimal minor ninth," even though it's spelled as an augmented octave.

Here we call 15/13 a "tridecimal ultramajor second." In terms of cents, it's even wider than the 8/7 supermajor second, hence the term "ultramajor." The alternative name "semifourth" for 15/13 is akin to the name "trienthird" for 14/13. Two "semifourths" (which would be 225/169) sounds very much like a perfect fourth (4/3 = 224/168).

Finally, here we call 15/11 an "undecimal augmented fourth." And indeed, we spell the interval as an augmented fourth (gF-luB). But it's actually narrower than 11/8, which we called a "semiaugmented fourth" (the difference is the comma 121/120). It might be better to call 15/11 a "subaugmented fourth" instead to emphasize that 15/11 is narrower than 11/8.

Here's what a Mocha program for the tau song might look like. (As usual, don't forget to click the Sound box to turn on the sound.)

NEW
10 N=16
20 FOR X=1 TO 52
30 READ A
40 SOUND 261-N*(16-A),4
50 NEXT X
60 DATA 6,2,8,3,1,8,4,3,10,7
70 DATA 1,7,9,5,8,6,4,7,6,9
80 DATA 2,5,2,8,6,7,6,6,5,5
90 DATA 9,10,10,5,7,6,8,3,9,4
100 DATA 3,3,8,7,9,8,7,5,10,2,1,1

This song ends at the same point as Vi Hart's song. If you want to add ten more notes to match Michael Blake's song, change 52 in line 20 to 62 and add digits 53-62:

110 DATA 6,4,1,9,4,9,8,8,9,1

Notice that unlike the songs for pi and other constants, I didn't need to set up an extra loop to code in the scale itself. Instead, the expression 16-A subtracts the digit from 16 to obtain a Degree in the range from 15 to 6 (though we had to represent 0 as 10 to make this formula work for zero).

Except for digits 1 and 0, the 15EDL scale doesn't exactly match the major scale used in the Hart and Blake videos. Notice that 15EDL that contains a just major triad on the root note -- the first even EDL containing the just major triad is 30EDL. But the just major triad for 15EDL is digits 1-4-6, while for Hart and Blake, 1-3-5 is a major triad. Moreover, in 15EDL, digits 7 and 8 are a major sixth and seventh respectively, as opposed to 6 and 7 in the Hart and Blake videos. Again, this reflects the nature of EDL scales, where the higher steps are wider than the lower steps.

This song uses a major tenth for 0, but suppose we wanted to follow Blake and rest on 0 instead. We might add the follow line:

35 IF A=10 THEN FOR Y=1 TO 400:NEXT Y,X

Making our song simulate Hart's is more complex -- that is, if we want to add the three verses of her song and all the other details. The easiest fix is to insert 4 in line 60, since Vi sings the note 4 to represent the decimal point. For Vi, 6-4-2 is a minor triad on ii, but for us, 6-4-2 corresponds to Degrees 10-12-14, which is a type of diminished triad (as I explained the day I posted about 14EDL).

Also, Hart doesn't sing all the notes as quarter notes, but varies the lengths -- so we'd need to add numbers representing lengths to the DATA lines. If at the very least, we want to sing the digits three times, with each verse containing more digits (just like Vi) then add the following lines:

15 FOR V=0 TO 2
20 FOR X=1 TO 9.5*V*V-.5*V+15
55 RESTORE: NEXT V

The strange formula here is an interpolating polynomial that passes through the points (0, 15), (1, 24), and (2, 52), to match the number of digits she sings in each verse.

Here are the roots of all the 15EDL scales available to us in Mocha:

Possible 15EDL root notes in Mocha:

Degree     Note

15            green F

30            green F

45            green Bb

60            green F

75            gugu Db

90            green Bb

105          rugu G

120          green F

135          green Eb

150          gugu Db

165          lugu C

180          green Bb

195          thugu Ab
210          rugu G
225          gugu Gb
240          green F
255          sugu E

Notice that Vi Hart plays her song in either A or G major, while Michael Blake's is in C major. The closest available key to Hart's A is thugu Ab (N=13 as thu notes are sharper than they look -- white A is at Degree 192) while the nearest key to Hart's G is rugu G (N=14). Meanwhile, the closest key to Blake's C is lugu C (N=11).

By the way, when I was looking for Tau Day videos I stumbled upon two more videos detailing the relationship between math and music. One is by 3Blue1Brown, and that video itself mentions another video. I post both of them here:

The Sweet Spot

The full Mocha Sound system starts at Sound 1 = Degree 260. Thus in a way, the Mocha Sound system can be described as a 260EDL scale. But this is a lot of notes, and so the EDL scales that I describe on the blog contain much fewer notes.

What exactly is the "sweet spot" of EDL scales? In other words, we seek out EDL's that contain enough -- but not too many -- notes to compose songs in.

In the past, I declared the sweet spot to be 12-22EDL. We started with 12EDL because the highest playable note in Mocha is Sound 255 = Degree 6, and one octave below this is Degree 12 -- thus 12EDL is the simplest fully playable EDL in Mocha. The next even EDL's also contain octaves, so these are 14EDL, 16EDL, 18EDL, 20EDL, and 22EDL. But then 24EDL contains 12EDL as a subset, since 24 has 12 as a factor. A song written in 24EDL is likely to emphasize the 12EDL subset, which is why I considered 22EDL to be the last EDL in the sweet spot.(Notice that last year, I never actually posted anything in 22EDL, and so 12-20EDL ended up being the sweet spot on the blog.)

Even though 24EDL might reduce to 12EDL, 26EDL doesn't reduce as simply. I was considering sneaking 26EDL into the sweet spot as well, if only because 260EDL -- the entire Mocha system -- has 26 as a factor. In other words, the range 12-26EDL is completely based on the highest and lowest playable notes in Mocha, our EDL instrument. This doesn't necessarily mean that this range makes the most musical sense.

In recent posts, I mentioned that the composer Sevish actually posted a song to YouTube that is written partly in 10EDL. Earlier, I considered 12EDL to be the simplest EDL in the sweet spot, but I can understand the allure of a scale like 10EDL. After all, we do have pentatonic scales and many songs written in them. (Of course, we also have a few songs with four notes, as well as the Google Fischinger player with four-note scales. But 8EDL doesn't really have the correct four notes.) Just as we did for the tritave-based 15EDL above, we'll have to cheat and end our scale on Degree 5, even though this last note isn't really playable in Mocha.

So we may want to include 10EDL in the sweet spot, since there is a real musician (Sevish) writing music in 10EDL. As far as I know, no one has written music in 20EDL, so perhaps this is a reason not to include 20EDL in the sweet spot. Meanwhile, I do see evidence for 18EDL being used as a scale in real music -- the interval 18/17, "the Arabic lute index finger." This name suggests that at one time, Arabic lutes (ouds) were fretted to divide the string in eighteenths for 18EDL.

The idea of 10-18EDL as the sweet spot also reminds me of one justification for bases 10-18 as the sweet spot for number bases (decimal through octodecimal). A few posters at the Dozenal Forum have mentioned the idea of "seven plus or minus two" (that is, the range 5-9) as the ideal length of lists that humans can handle. Thus bases 10-18 contain 5-9 pairs of digits, and the 10-18EDL scales contain 5-9 notes. Indeed, the most commonly played scales contain five (pentatonic) to nine (melodic minor) notes as well.

That settles it -- 10-18EDL is the sweet spot based on real music. Let's write out all of the scales in the sweet spot, using Kite's new color notation (as Kite has changed it since last Tau Day).

The 10EDL Octave:
Degree     Ratio     Cents     Note
10            1/1         0            green C
9              10/9       182        white D
8              5/4         386        white E
7              10/7       617        red F#
6              5/3         884        white A
5              2/1         1200      green C

The 12EDL Octave:
Degree     Ratio     Cents     Note
12            1/1         0            white A
11            12/11     151        lavender B
10            6/5         316        green C
9              4/3         498        white D
8              3/2         702        white E
7              12/7       933        red F#
6              2/1         1200      white A

The 14EDL Octave:
Degree     Ratio     Cents     Note
14            1/1         0            red F#
13            14/13     128        thu G
12            7/6         267        white A
11            14/11     418        lavender B
10            7/5         583        green C
9              14/9       765        white D
8              7/4         969        white E
7              2/1         1200      red F#

The 16EDL Octave:
Degree     Ratio     Cents     Note
16            1/1         0            white E
15            16/15     112        green F
14            8/7         231        red F#
13            16/13     359        thu G
12            4/3         498        white A
11            16/11     649        lavender B
10            8/5         814        green C
9              16/9       996        white D
8              2/1         1200      white E

The 18EDL Octave:
Degree     Ratio     Cents     Note
18            1/1         0            white D
17            18/17     99          su D#
16            9/8         204        white E
15            6/5         316        green F
14            9/7         435        red F#
13            18/13     563        thu G
12            3/2         702        white A
11            18/11     853        lavender B
10            9/5         1018      green C
9              2/1         1200      white D

Actually, let's go ahead and sneak 20EDL and 22EDL into our sweet spot anyway (just as I wanted to sneak 24EDL and 26EDL back when 12-22EDL was our sweet spot). Here 20EDL and 22EDL may be useful only because they are the first EDL's with something resembling a "leading tone" -- the last ascending note that leads into the octave:

The 20EDL Octave:
Degree     Ratio     Cents     Note
20            1/1         0            green C
19            20/19     89          inu C#
18            10/9       182        white D
17            20/17     281        su D#
16            5/4         386        white E
15            4/3         498        green F
14            10/7       617        red F#
13            20/13     746        thu G
12            5/3         884        white A
11            20/11     1035      lavender B
10            2/1         1200      green C

The 22EDL Octave:
Degree     Ratio     Cents     Note
22            1/1         0            lavender B
21            22/21     81          red B
20            11/10     165        green C
19            22/19     254        inu C#
18            11/9       347        white D
17            22/17     446        su D#
16            11/8       551        white E
15            22/15     663        green F
14            11/7       782        red F#
13            22/13     911        thu G
12            11/6       1049      white A
11            2/1         1200      lavender B

Let's add two more tritave scales in this range -- since I already wrote 15EDL earlier in this post, let's add 18EDL and 21EDL:

The 18EDL Tritave:
Degree     Ratio     Cents     Note
18            1/1         0            white D
17            18/17     99          su D#
16            9/8         204        white E
15            6/5         316        green F
14            9/7         435        red F#
13            18/13     563        thu G
12            3/2         702        white A
11            18/11     853        lavender B
10            9/5         1018      green C
9              2/1         1200      white D
8              9/4         1404      white E
7              18/7       1635      red F#
6              3/1         1902      white A

The 21EDL Tritave:
Degree     Ratio     Cents     Note
21            1/1         0            red B
20            21/20     84          green C
19            21/19     173        inu C#
18            7/6         267        white D
17            21/17     366        su D#
16            21/16     471        white E
15            7/5         583        green F
14            3/2         702        red F#
13            21/13     830        thu G
12            7/4         969        white A
11            21/11     1119       lavender B
10            21/10     1284      green C
9              7/3         1467      white D
8              21/8       1671      white E
7              3/1         1902      red F#

END

BACH and CAGE Again

Yes, I'm aware that we're done with Hofstadter's book -- and I wrote that we wouldn't be reading his book past the summer solstice. But there's already music in the post, and Hofstadter mentions computer-generated music in his post. I can't help but wish to code several of the songs in his book into BASIC using Mocha.

I especially liked the author's jukebox that played BACH and CAGE -- that is, the notes B-A-C-H (in German -- same as Bb-A-C-B in American notation) and C-A-G-E. But one thing I found inelegant is the way he cavalierly interchanges the factors 3 and 3 1/3 to suit his needs. Let's recall that song:

Record B-1 (BACH): down 1, up 3, down 1
Record C-3 (CAGE): down 3, up 10, down 3
Record B-10 (BCAH): down 10, up 33, down 10

Suppose we had skipped directly from B-1 to B-10 -- it's suddenly not obvious why we multiply the downward intervals by 10 and the upward interval by 11. The real reason, of course, is that otherwise the notes wouldn't spell out BACH and CAGE. But what sequence of notes would play if we were to type in other letter-number combinations, such as B-4 or F-7? (Does the 4 in B-4 mean 4, or 4 1/4, or something else?) Again, the whole idea was just to spell BACH and CAGE, not create a full jukebox.

It would look much nicer if "3" could really mean three (rather than 3 1/3), and if there were no rounding needed. But we're limited by the number of semitones -- that is, degrees of our standard scale 12EDO -- between the notes of BACH and CAGE. This leads me to wonder -- is there another EDO, besides 12EDO, where this multiplication works out exactly?

Since 12EDO almost works, the correct EDO is likely to be one of 12EDO's neighbors, such as 11EDO or 13EDO, or maybe even 10EDO or 14EDO. We've previously written programs that convert EDO to EDL so that we can play these alternate EDO scales in Mocha.

Let's look at our goal again, except we write the indicated multiplication without rounding:

Record B-1 (BACH): down 1, up 3, down 1
Record C-3 (CAGE): down 3, up 9, down 3
Record B-10 (BCAH): down 10, up 30, down 10

The first thing we notice is that the notes A-C in BACH are the same as C-A in CAGE. Since the CAGE record is C-3 (multiplying by 3), this implies that A-C must be three steps. This is already the case in 12EDO.

But now we look at the C-A in BCAH. These two notes are just shy of three octaves apart -- so clearly if we add the A-C interval to this, we get three full octaves. Since C-A is 30 steps and A-C is three steps, we obtain 33 steps for three octaves. So each octave is 11 steps, indicating 11EDO.

Now we notice that the B-A in BACH is one step, while B-C in BACH is 10 steps. So these two intervals add up to 11 steps, or one octave. But recall that B is the German B (American Bb), so we really have Bb-A (minor 2nd) plus Bb-C (minor 7th) adding up to an octave. This is a problem, since the inversion of the minor 2nd is supposed to be the major 7th, not the minor 7th.

Of course, this is 11EDO, so one note is missing from from 12EDO. Who's to say that, for example, our scale from C-C skips the note B? A minor 7th (C-Bb) plus a minor 2nd would be an octave. The problem, though, is that the name BACH already distinguishes between notes Bb and B (that is, B and H in German). Unless we're prepared to change this to BACB or HACH, this is a no go.

We might notice that the interval A-G (in CAGE) ought to be a minor 7th, just as B-C (in German, Bb-C American) is also a minor 7th. But A-G is listed as nine steps, while Bb-C is ten steps.

There is actually a common solution to both of these problems. Instead of BCAH, we write the final sequence of notes as HCAB. This already looks nicer, since HCAB is a full reversal of BACH, while BCAH only reverses two of the letters:

Record B-1 (BACH): down 1, up 3, down 1
Record C-3 (CAGE): down 3, up 9, down 3
Record H-10 (HCAB): down 10, up 30, down 10

Now all the intervals work. A minor 2nd is one step, and its inversion the major 7th is 10 steps. A major 2nd is two steps and its inversion the minor 7th is nine steps. A minor 3rd is three steps and its inversion the major 6th is eight steps.

We might wonder what these 11EDO intervals actually sound like. Let's check out Xenharmonic:

https://en.xen.wiki/w/11edo

Compared to 12edo, the intervals of 11edo are stretched:

• The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
• The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
• The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
• The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
• The "perfect fourth," at 545.455 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

In other words, the 2nds and 7ths are nearest their 12EDO equivalents, the 4th and 5th are the farthest away from 12EDO, and the 3rd and 6th are somewhere in between. It's not mentioned here, but notice that a just minor 3rd (6/5) is 316 cents, so the minor 3rd is actually slightly more accurate in 11EDO than in 12EDO. On the other hand, as stated above, the major third is much worse in 11EDO than in 12EDO.

It's a good thing that no major thirds appear in the song (unless you count the C and E in CAGE, but those are nonconsecutive.) Indeed, notice that C-E is also three steps, just like A-C or E-G (which is forced since the original interval B-H is one step, just like A-B and H-C). But once again, neither 3\11 nor 4\11 is actually a major third (as a true 5/4 is almost exactly halfway in between).

When we run our EDO-to-EDL conversion program, we find that 162EDL is the best EDL in which to approximate 11EDO. A full octave of 11EDO falls on the following degrees:

162, 152, 143, 134, 126, 118, 111, 104, 98, 92, 86, 81

We see that Degree 162 is white middle C, and so this octave runs from C-C. Since C-E has been reduced to three steps, our 11EDO must omit one of the three notes Db, D, or D#. For this program, I decide to omit the note Db.

NEW
10 DIM A(8),L$(3),N(3)
20 FOR X=1 TO 8
30 READ A(X)
40 NEXT X
50 DATA 8,9,11,12,14,15,6,10
60 FOR X=1 TO 3
70 PRINT "LETTER (A-H) #";X
80 INPUT L$(X)
90 PRINT "NUMBER (1-13) #";X
100 INPUT N(X)
110 NEXT X
120 FOR X=1 TO 3
130 CLS
140 PRINT "PLAYING RECORD #";X
150 R=ASC(L$(X))-64
160 SOUND 261-INT(162/2^(A(R)/11)+.5),4
170 SOUND 261-INT(162/2^((A(R)-N(X))/11)+.5),4
180 SOUND 261-INT(162/2^((A(R)+2*N(X))/11)+.5),4
190 SOUND 261-INT(162/2^((A(R)+N(X))/11)+.5),4
200 NEXT X

Once again, use the up-arrow for ^ and the dollar sign $ for strings. As usual, don't forget to click the Sound box to turn on the sound.

In Hofstadter's story, the jukebox is three plays for a quarter (25 cents), so we must input three letters and three numbers. The INPUT command requires us to hit the Enter key after each one.

It's possible to code other songs from Hofstadter as well. The Crab Canon, where songs can go forward and backward, might be interesting. The easiest to code, of course, is Cage's 4'33":

10 CLS
20 PRINT "MOVEMENT #1"
30 FOR X=1 TO 18480
40 NEXT X
50 CLS
60 PRINT "MOVEMENT #2"
70 FOR X=1 TO 89600
80 NEXT X
90 CLS
100 PRINT "MOVEMENT #3"
110 FOR X=1 TO 44800
120 NEXT X

Trial and error suggests that an empty FOR loop runs about 560 times per second. This is based on the three movements at :33, 2:40, 1:20.

The Arabic Lute

Returning to EDL scales, I've mentioned how fascinated I am by the name "Arabic lute index finger" for the interval 18/17, and its suggestion that the oud must have been fretted to 18EDL.

The following YouTube video is all about refretting a guitar to experiment which tuning makes the song sound better. The piece, written by Cage -- I mean Bach -- is called "Air." (Hofstadter gives "Air on G's String" as the title of one of his dialogues, but I don't know whether it's related to the "Air" piece in the video.)

Of the four tunings, one is just intonation, one is standard 12EDO, and the others are compromises between JI and 12EDO, called "well temperament."

In the comments at YouTube, many people found JI to be the best-sounding near the beginning, where many major chords are played. JI is based on pure ratios, such as the 4:5:6 major triad. But near the end, the piece became more melodic than harmonic. At this point, the best-sounding tuning according to the commenters became 12EDO, whose equal step sizes make melodies sound nice. The two well-temperaments are intermediate in both the harmonic and melodic sections. (The first tuning is closer to JI and thus sounds better harmonically, while the last tuning is closer to 12EDO and thus sounds better melodically.)

Notice that EDL scales are based on ratios and thus are closely related to JI. It's a shame, though, that Mocha can only play one note at a time -- it's melodic rather than harmonic. (Last year on Tau Day, I did mention the Atari computer that could play EDL-based harmony .)

The fretting is quite complex for all of the tunings except 12EDO. Actually, a fretting based on EDL's (which our hypothetical oud has) would look even simpler. Like 12EDO, the frets would at the same position for all the strings. The only difference is that the frets would be equally spaced apart -- exactly 1/18 of the length of the whole string. (That's what 18EDL -- 18 equal divisions of length -- really means after all.)

Imagine if the guitars in the video were fretted to 18EDL. Let's keep the standard EADGBE tuning, except we assume that all of these are white notes (Kite colors). This means that the interval between consecutive strings (E-A, A-D, and so on) is the perfect fourth 4/3. Then all of the notes fretted at the first fret (by the index finger, of course) are colored 17u ("su"), Second fret notes become white, third fret notes are green, and fourth fret notes are red.

Now let's try playing some chords using this tuning. We start with an E major chord -- a basic open chord that beginning guitarists learn to play. This chord is played as:

EBEG#BE
wE-wB-wE-suG#-wB-wE

The JI 4:5:6 would require G# to be yellow rather than su, but yellow (an "over" or "otonal" color) isn't available in EDL (which is based on "under" or "utonal" colors). Fortunately, the su 3rd (about 393 cents) lies about halfway between the yellow 5/4 3rd and the 12EDO major 3rd. Thus this E major chord will probably sound like one of the well temperaments from the video.

EDL's are supposed to be better at playing under/utonal chords, which minor chords are. So let's try playing E minor rather than E major:

EBEGBE
wE-wB-wE-wG-wB-wE

Now all the notes end up white. Chords with all white notes are considered dissonant -- this chord is known as the Pythagorean minor chord.

Let's try some A chords now. We begin with A major:

xAEAC#E
wA-wE-wA-wC#-wE

This is another dissonant all-white chord -- the Pythagorean major chord. We move on to A minor:

xAEACE
wA-wE-wA-suB#-wE

The first fret on the B string isn't even C -- officially it's su B#. This note is 13 cents flatter than green C -- the note that belongs in an A minor chord. It probably won't sound terrible in a chord only because at 303 cents, the interval wA-suB# is only three cents wider than the 12EDO minor 3rd and sound indistinguishable from it.

So far, we have the passable (or "well tempered") E major and A minor chords, and dissonant Pythagorean E minor and A major chords. Moving on to D major, an obvious problem arises:

xxDADF#
wD-wA-gD-wF#

Now the two D notes aren't even the same color, so now we have dissonant octaves. Changing this from D major to D minor doesn't eliminate the dissonant octave on D.

So why are we having so much trouble with these basic chords? If we return to A minor, we notice that the third fret on the open A string indeed plays the green C needed for A minor. But the A string can't be used for green C, because it's too busy sounding the white A! In other words, most of the time, two or three notes we need to make chords sound on the same string.

The following link describes how actual Arabic lutes may have been tuned:

https://larkinthemorning.com/blogs/articles/the-oud-the-arabic-lute

Notice that even on that blog, there's still no mention of the "Arabic lute index finger," the interval 18/17, or 18EDL fretting. But we do see that instead of EADGBE, traditional tunings for ouds include DGADGC, ADEADG, and EABEAD.

But unfortunately, even with EABEAD, some JI chords remain difficult to finger and play. One of these days I'd like to solve the mystery of the Arabic lute, its index finger, and 18EDL.

Conclusion

Last year I sneaked the traditionalists' debate into the Tau Day post. I won't do so this year -- luckily for you, the traditionalists have been inactive lately (except for Ze'ev Wurman making a recent comment at the Joanne Jacobs website -- the topic was a slavery assignment in a middle or upper elementary history class).

If I timed this correct, this post should appear on exactly 6/28 at 3:18 -- as in tau = 6.28318. In other words, Happy Tau Day everybody!