Good Friday Post

Table of Contents

1. Introduction & Fawn Nguyen's Writing Prompts
2. Overcomer: Another Cross Country Movie
3. Calculating the Easter Date
4. The Easter Song in 12EDO
5. The Easter Song in 28EDO
6. More on the Theory of 28EDO
7. Coding the Easter Song in 28EDO
8. Traditionalists: Learning Styles & Productive Struggle
9. Cosmos Episode 9: "Magic Without Lies"
10. Cosmos Episode 10: "A Tale of Two Atoms"
11. Conclusion & Revisiting the Fourth Dimension

Introduction & Fawn Nguyen's Writing Prompts

Today is the Friday before Easter -- also known as Good Friday. Many schools that aren't closed the rest of Holy Week are nonetheless closed on Good Friday -- including my old district.

Indeed, on the original pre-coronavirus calendar, this is the start of a long weekend. This would have transformed yesterday's Lesson 14-5 into an activity day, forcing me to come up with a brand new worksheet for it. This is one of the few changes I was planning to make to worksheets on the blog from the 2018-19 school year to 2019-20. (The Exploration Question in the text is all about vectors and airplanes.)

But instead, the schools remain closed. This is my fifth spring break/coronavirus break post. Today, my old district announces a grading policy for online classes -- the semester grade must be at least the letter grade just below than the third quarter grade (so an A can go down no farther than a B, for example), a D grade can't go down at all, and pass/fail grades are allowed for both graduation and college admissions. (The LAUSD is more lenient -- third quarter grades can't go down at all.)

Oh, and Numberphile posted another coronavirus video today:

Only once have I ever blogged on Good Friday -- this was four years ago, the year before I worked at the old charter school. The blog calendar for that year only was based on another district that was still open on Good Friday. On that day, I posted an activity on the surface area of a sphere, briefly mentioned the traditionalists, and did side-along reading about the fourth dimension. Oh, and I also discussed how to calculate the Easter date.

The changing date of Easter has always fascinated me. And indeed, two years ago, I created a song based on the changing Easter dates, similar to songs based on the digits of pi.

For today's post, I want to write about how I came up with the idea for the Easter song. This includes how and why I chose a special scale in which to compose it.

But I'll be squeezing in lots of other topics as well. For starters, Fawn Nguyen posted yesterday:

http://fawnnguyen.com/one-word-writing-prompts/

Years ago at George Middle School (Portland, Oregon), the teachers were allowed to teach something we were passionate about. The class would be twice a week, right after lunch, for just 20 minutes. I was a science teacher at the school and asked if I may teach “writing for writing.” My principal reacted with slightly more enthusiasm than my [male] colleague’s “The Simpsons.”

Hmm, this sounds interesting. This reminds me of that so-called "Advisory" at my old charter school, which was on Wednesdays just before lunch. Indeed, this is something I could have done with that period, but the administrators never fully explained what "Advisory" was for, and it ended up being fully disorganized. It's interesting that Nguyen, a math teacher (and as she said above, a science teacher at the time) would choose writing.

Weary that a question prompt might not elicit interest or intrigue and having to hear them whine pitch perfect, “But I don’t know anything about that,” I give only one-word prompts.

• tiny
• red
• outside
• wax
• intelligence
• breakfast
• sand
• scream
• pale
• rain

My goal was writing for writing. I wanted the pen or pencil to move across the page for five minutes.

OK, then, let me start this blog post with one of these words  -- "outside." Outside is a place where we can't go much any more due to the coronavirus -- now we must stay inside. When I was in high school, one of my favorite things to do outside was run Cross Country and Track -- and I feel sorry for current high school athletes whose Track season has been completely cancelled.

I'll continue by reviewing a movie that I recently watched. Yes, I watched the movie "inside" my house, but the movie is about the "outside" activity that I mentioned above.

Overcomer: Another Cross Country Movie

Today the movie I'm reviewing here is one I checked out from the library -- all the way back on Pi Day, the last day before the libraries closed. I didn't watch it until this week, since I knew I had the luxury of time.

The film Overcomer was recommended to me because it's about high school running -- and I, as I've said on the blog before, used to run distances in high school. To my surprise, the movie's actually about Cross Country. The only other XC movie I know of is McFarland USA -- and when that film first came out, I blogged that it would be years before I'd see another XC movie again, if ever. Well, that day has now arrived, merely five years later.

There are a few similarities between McFarland USA and Overcomer. But one huge difference is that while McFarland USA is based on a true story, Overcomer is completely fictional.

Both McFarland and Overcomer feature coaches who originally lead other sports before taking on Cross Country. In Overcomer, John Harrison originally coaches basketball, but then most of his players transfer out when a local factory is shut down -- and the XC coach leaves the school as well, leaving Harrison to coach a new sport. Meanwhile, McFarland hires Jim White to coach the football team, until he realizes that his school has more talented runners than football players.

When the coaches in each movie hold tryouts, only a single athlete shows up. In McFarland, that lone runner, Johnny, helps Coach White find six more runners to make a team -- and the coach also blackmails some troublemakers into joining the team to avoid a suspension.

But in Overcomer, a girl, Hannah Scott, ends up being the only runner on the team. She's one of the few students who is transferring in to the school instead of out -- and since she ran XC at her old school as a freshman last year, she decides to join as a sophomore this year. Coach Harrison doesn't think much of Hannah, because she has asthma and needs and needs an inhaler to breathe.

Both movies feature a white coach and minority athletes. But while Hannah is black, race isn't really significant to this movie. In one scene, Harrison says that he is white -- but much more importantly, he is a Christian. Religion is very important in this movie -- indeed, Overcomer is produced by Christian filmmakers (and thus it's in the same vein as Unbroken, the Louis Zamperini story).

On the other hand, race matters much more in McFarland, where White's culture clashes with that of his Latino runners. Religion is only briefly mentioned there -- for example, when the coach wishes to paint over the image of the Virgin of Guadalupe in his new home.

I'm delighted that fifteen minutes into Overcomer, the distance "5K" is mentioned, and this is quickly clarified as being approximately three miles. At no point in McFarland is a distance ever stated.

Both films have the father of a runner appear unexpectedly -- and indeed, the relationship between father and child is significant in both cases. McFarland's top runner, Thomas Vallez, is surprised when his dad, who works in farm fields in a faraway town, shows up. Senor Vallez is angry when he learns that his daughter -- Tom's sister -- is pregnant. These family issues upset the star runner to the point where he contemplates suicide, until Coach White convinces him that life is worth living.

As for Hannah, she lives with her maternal grandmother because her parents had once been heavy drug users. Indeed, her mother is deceased -- and as far as she knew from her grandma, her dad is also dead. It's Coach Harrison who discovers her dad in the hospital, being treated for a heart attack and diabetes (leading to blindness) caused by his drug abuse. Harrison knows that Hannah's birthday is on Valentine's Day and the patient tells him that he has a long lost daughter born on that February holiday, allowing the coach to make the connection.

Harrison finally reunites Hannah with her father in the hospital. The patient asks his daughter for forgiveness, but she's reluctant to do so. It's actually the principal who reminds Hannah that while her dad isn't perfect, she also has a heavenly Father who is. She tells the girl that Jesus sacrificed himself on the cross. (Yes, this is a religious movie -- and you shouldn't be upset that I'm mentioning the cross in a post clearly labeled "Good Friday Post." If you wish, you can skip the rest of this section and proceed to the next section of my post -- it's also about Easter, but only the date, not its significance in religion.) And the principal -- of Brookshire Christian Academy -- convinces Hannah that just as the Lord was able to forgive sinners, she can forgive her dad. Indeed, she ultimately does reconcile with her father -- who knows that his days remaining in this world are numbered.

Unfortunately, Hannah's grandma isn't as ready to forgive the man, and she forbids Hannah from seeing him in the hospital. He is upset that the old lady is taking his daughter away from him -- but then again, the old lady feels that her son-in-law took her daughter (Hannah's mother) away from her 15 years ago (by introducing her to the drugs that killed her).

And both coaches must also repair the relationships with their own children. In McFarland, Coach White forgets his older daughter's fifteenth birthday and makes it up with a quinceanera. In Overcomer, Coach Harrison forgets that his older son is also worried about his chances of winning a scholarship when the school teams disappear. The two do finally shoot some hoops together.

The climax of each film takes place, as you'd expect, at the State Meet. McFarland concludes at the inaugural California State Meet. And this movie has an exciting finish as Danny Diaz -- the slowest runner on the team -- runs a gutsy race to become the fifth and final scorer as two of his teammates collapse before crossing the finish line.

Hannah also prepares to run at her State Meet. Since Brookshire is a fictional town, the name of the state is never mentioned. Maybe it's the same state as Springfield, where The Simpsons live. Actually, I prefer to believe that the state is Georgia, where most of the filming took place.

Anyway, Hannah arrives at her new school with no friends, but by the end she has many supporters at the final race (though sadly, never any teammates). And her father, still in the hospital (but a former runner himself), becomes her ultimate coach. He records messages about coaching and gives it to Harrison, who has Hannah play them as she is running.

Struggling to breathe due to her asthma, Hannah also collapses during her race -- but fortunately, she collapses right at the finish line, outleaning another girl to take the State Championship. The last thing she presents to her father is her gold medal. The next scene is at her father's funeral. She says that for the past six weeks, she had the best father in the world.

The final scene takes place six years later. It is Hannah's 21st birthday, and she is now a college athlete. As she goes on a practice run, she is playing another tape from her late father, just as she did on the day she won the State Meet. It turns out that on his deathbed, he had recorded a special message for her to play every year on her Valentine's Day birthday.

To me, this scene is especially poignant considering the recent passing of basketball star Kobe Bryant (mentioned previously on the blog). Notice that both Bryant and the father in the story coached their respective daughters until their very last day on this earth.

And I'm glad that I watched this movie this week, because it reminds me of how much I definitely miss being -- "outside."

Calculating The Easter Date

This is what I wrote four years ago about calculating the Easter date:

Today is Good Friday, two days before Easter Sunday. As many readers are aware, I'm often fascinated by calendars, and there's no holiday commonly observed in this country that's more fascinating than Easter. Many people wonder why Easter is late in some years and early in others. It's because of this wide variability that many schools have abandoned tying spring break to Easter. As I mentioned before, early Easters interfere with end-of-quarter exams, while late Easters interfere with the AP exam.

The following link discusses why the Easter date changes so much:

http://www.timeanddate.com/calendar/determining-easter-date.html

According to the Bible, Jesus’ death and resurrection occurred around the time of the Jewish Passover, which was celebrated on the first full moon following the vernal equinox.

This soon led to Christians celebrating Easter on different dates. At the end of the 2nd century, some churches celebrated Easter on the day of the Passover, while others celebrated it on the following Sunday.

So we see that the Christian Easter is tied to the Jewish Passover. We've mentioned before that the Hebrew calendar is a lunisolar calendar -- that is, it's tied to both the sun and the moon. And we pointed out during our description of the Chinese calendar (another lunisolar calendar) that the phases of the moon and the seasons of the year don't line up exactly. The solar year cannot be divided evenly into lunar months.

In 325CE the Council of Nicaea established that Easter would be held on the first Sunday after the first full moon occurring on or after the vernal equinox. From that point forward, the Easter date depended on the ecclesiastical approximation of March 21 for the vernal equinox.

Easter is delayed by 1 week if the full moon is on Sunday, which decreases the chances of it falling on the same day as the Jewish Passover. The council’s ruling is contrary to the Quartodecimans, a group of Christians who celebrated Easter on the day of the full moon, 14 days into the month.

So we infer that the Christians wanted an Easter date that is similar to -- yet independent of -- the Hebrew calculation. The date of the full moon was determined by looking it up on a table, rather than depend on the date of Passover. And so this complicated rule of determining Easter was devised.

The link gives a table of the earliest and latest Easters. The earliest Easters between the years 1753 and 2400 according to the table are:

March 22nd: 1761, 1818, 2285, 2353
March 23rd: 1788, 1845, 1856, 1913, 2008, 2160, 2228, 2380

The early Easter of 2008 is still fresh in my memory. A school that took off the week before Easter had March 14th as the last day before spring break, and school resumed on the 24th. But as early as that Easter was, we see that the earliest possible holiday is one day earlier. But Easter hasn't fallen on that date since 1818 -- long before any of us here were born -- and it won't fall on that day again until 2285 -- long after all of us here are dead. What makes March 22nd Easters so rare?

The problem is that there's only one way for Easter to fall on March 22nd -- and that's for there to be a full moon on Saturday, March 21st. If the full moon were a day later, on Sunday, March 22nd, then Easter wouldn't be until the 29th -- since, as the link points out, Easter is delayed by one week if the full moon is on Sunday. And if the full moon were a day earlier, on Friday, March 20th, then Easter wouldn't be until April 19th. This is because, as the link points out, March 21st is considered to be the ecclesiastical first day of spring. So March 20th would still be considered winter, and winter full moons don't count -- only spring full moons do. So we'd have to wait until Saturday, April 18th, the latest possible Paschal Full Moon, which would make the next day Easter.

So we see that if March 22nd were Easter, then the full moon must be exactly March 21st. But surely we shouldn't have to wait nearly 500 years (from 1818 to 2285) for March 21st to be the full moon!

The problem is that these full moons are determined by a table and aren't the dates of the actual full moon (unlike the Chinese calendar, which is based on astronomical dates). Now this table repeats every 19 years -- recall my mention of the Metonic 19-year cycle in earlier posts. So of the 29 dates from March 21st to April 18th, only 19 of those dates are found in the table. The Metonic cycle is not exact, and so the tables are adjusted every century.

What this means is that, in a given century, only 19 of the 29 dates from March 21st to April 18th can be possible full moon dates. If, in a given century, March 21st isn't one of the 19 chosen full moon dates, then March 22nd can't be Easter, since the 22nd isn't Easter unless the 21st is the full moon. As it turns out, the 20th, 21st, and 22nd centuries are all centuries for which March 21st isn't one of the 19 chosen dates. So March 22nd can't be Easter in any of them. And so 19th century was the last time that March 21st was the full moon, and it won't be full moon again that date until the 23rd century.

Now we look at the latest possible Easters:

April 24th: 1791, 1859, 2011, 2095, 2163, 2231, 2383
April 25th: 1886, 1943, 2038

Of course, the late Easter of 2011 is still in recent memory. A school that took off the week after Easter had April 22nd as the last day before spring break, and school resumed on May 2nd. But as late as that Easter was, we see that the latest possible holiday is one day later. This time, Easter will fall on April 25th, 2038. It's possible for me to be alive on that day, as I would be 57 years old -- as opposed to 2285 when I'd be over 300 years old.

Notice that there's also a column at the above link for "Julian calendar." Recall that our current calendar, the Gregorian calendar, was named after a pope -- so why would Eastern Orthodox Christians follow a pope's calendar? To this day, they still follow the predecessor calendar, the Julian calendar, as I explained back in my New Year's Eve post.

This has two affects on the Easter date. First, the equinox date of March 21st on the Julian calendar is actually what the Gregorian calendar calls April 3rd. So if the full moon is too early, it would still be considered winter on the Julian calendar, and Easter must wait until the next full moon. The other effect is that the full moon dates are based on the old table that isn't adjusted every century. So the Julian full moons are themselves later than the Gregorian. This year, the full moon on the Gregorian calendar was on April 7th, but the Julian full moon is a few days later. So the Julians must wait an extra week, until April 19th, to celebrate Easter.

Easter can fall in May on the Julian calendar. According to the link, the earliest Easter in the range of the chart was April 3rd, 1763. (Notice that 1753 was the year that the British converted from the Julian to the Gregorian calendar, which is why the chart begins in 1753.) In recent times, April 4th, 2010 is an early Easter. The latest Julian Easter during the 21st century will be on May 8th, 2078 -- which is Mother's Day in the USA!

In some years, both calendars have the same Easter date. Both calendars agreed that April 16th, 2017 was Easter Sunday. The above link mentions that some people want to reconcile the two Easters by using an astronomical rule, just like the Chinese calendar. In this calendar, not only are March 22nd Easters slightly more likely, but even March 21st becomes a possible Easter date. The following link (near the bottom of the page, with even more discussion on how to calculate Easter throughout the rest of the page) claims that with an astronomical calculation, March 21st, 2877 will be Easter:

http://www.webexhibits.org/calendars/calendar-christian-easter.html

The Easter Song in 12EDO

It was in 1999 when I first saw a table of Easter dates. Let's look at the dates of the first few Easters after I saw this chart (from that day up to the creation of this blog -- in other words, the same range I covered in my "21 Years a Bruin" post):

April 4th, 1999
April 23rd, 2000
April 15th, 2001
March 31st, 2002
April 20th, 2003
April 11th, 2004
March 27th, 2005
April 16th, 2006
April 8th, 2007
March 23rd, 2008
April 12th, 2009
April 4th, 2010
April 24th, 2011
April 8th, 2012
March 31st, 2013
April 20th, 2014

We notice that there's a nine-year stretch -- from 2006 to 2014 -- such that every April Easter in that range falls on a date that's a multiple of four (4th, 8th, 12th, 16th, 20th, 24th). And if you think about it, March 31st is essentially April 0th (with 0 also divisible by four), and March 27th and 23rd are four and eight days, respectively, before March 31st. Thus we have a full decade stretch, 2005-2014, where all the Easters are multiples of four days apart from each other.

Before that stretch, some of the Easters are one less than a multiple of four (April 11th, 15th, 23rd) while since that stretch, some of the Easters are one more than a multiple of four (April 5th in 2015, April 1st two years ago, April 21st last year). No Easter in recent memory is on a singly even April date in recent memory -- the holiday fell on April 6th in 1980, the year I was born, but then not again on that date until 2042.

The Webexhibits link above explains the reason for this:

Suppose you know the Easter date of the current year, can you easily find the Easter date in the next year? No, but you can make a qualified guess. If Easter Sunday in the current year falls on day X and the next year is not a leap year, Easter Sunday of next year will fall on one of the following days: X-15, X-8, X+13 (rare), or X+20. If Easter Sunday in the current year falls on day X and the next year is a leap year, Easter Sunday of next year will fall on one of the following days: X-16, X-9, X+12 (extremely rare), or X+19. (The jump X+12 occurs only once in the period 1800-2200, namely when going from 2075 to 2076.) If you combine this knowledge with the fact that Easter Sunday never falls before 22 March and never falls after 25 April, you can narrow the possibilities down to two or three dates.

Of those eight possibilities (X-15, X-8, X+13, X+20, X-16, X-9, X+12, X+19), we notice that half of them involve multiples of four. Two of them are one less than, and two of them are one more than, multiples of four. In past posts, I referred to the eight valid intervals as "Easter bunny hops."

This completely explains the mod 4 patterns that I observed earlier. The "one less than" and "one more than" multiples tend to cancel out, with slightly more "one more than" multiplies. And so there are long stretches where certain dates mod 4 are possible Easter dates, while other dates mod 4 are skipped over.

And this reminded me of musical scales. In the C major scale, all the white keys on a piano are played while all the black keys are skipped. Meanwhile, in the B major scale, all the black keys plus B and E are played while the other white keys are skipped.

Thus I imagined that there could be a song based on the Easter date pattern. My thought was that there might be a few bars where the song is in C major, then gradually it modulates. For example, the note F might disappear while the F# appears, indicating a change to G major. Then the note C might disappear while C# appears, for D major, and so on. Eventually every note in every key would be reached -- all I had to do is decide which Easter dates to map onto which notes.

Since it's been so long since I tried this, I don't recall the exact mappings I used. I suspect that since two of the possible Easter differences are -8 and -9 (that is, X-8 and X-9 -- from this point forward, let's drop the X and just use the integer difference), I tried to let 8 represent a minor third and 9 represent a major third (since these thirds are musically important). This means that two dates 17 days apart would be a perfect fifth (as m3 + M3 = P5):

3/22  G
3/23  G
3/24  G#
3/25  Ab
3/26  A
3/27  A
3/28  A
3/29  A#
3/30  Bb
3/31  B
4/1  B
4/2  C
4/3  C
4/4  C
4/5  C#
4/6  Db
4/7  D
4/8  D
4/9  D
4/10  D#
4/11  Eb
4/12  E
4/13  E
4/14  E
4/15  F
4/16  F
4/17  F#
4/18  Gb
4/19  G
4/20  G
4/21  G
4/22  G#
4/23  Ab
4/24  A
4/25  A

These notes correspondences were chosen in a way such that:

• differences of 8 days are m3, 9 days are M3, and 17 days are P5 as often as possible
• the ten Easters from 2005-2014 correspond to the C major scale

For the most part, enharmonic sharps and flats are interchangeable. In each case, I gave the first day the sharp name and the second day the flat name, but I can see already that there might be two Easters a few years apart with one sharp and one flat. (That can't correspond to a major scale.) We'll just see what happens as we go.

So now let's compose the music. We write out some Easters and see what note they correspond to, starting with 2005 since we know that part will be in C major:

3/27/05 = A

4/16/06 = F
4/8/07 = D
3/23/08 = G
4/12/09 = E
4/4/10 = C
4/24/11 = A
4/8/12 = D
3/31/13 = B
4/20/14 = G

Because of the thirds and fifths that I made correspond to common Easter intervals, there are plenty of recognizable triads. From 2005-2007 we see A-F-D (D minor), the next three years give G-E-C (C major), 2009-2011 give E-C-A (A minor), and the last three years give D-B-G (G major). Where three consecutive years don't form a triad, they are often three of the four notes of a seventh chord, such as 2007-2009, D-G-E (Em7) and the following three years, C-A-D (Dm7, or perhaps D7 since the third is missing).

OK, let's continue the song by calculating some more Easters. After 2014, sharps begin to appear:

4/5/15 = C#
3/27/16 = A
4/16/17 = F
4/1/18 = B
4/21/19 = G

And here's our first departure from a simple major scale -- in 2015 the note C# appears, but the F in 2017 is still F natural. No major scale has both C# and F natural. The resulting triad for the three years 2015-2017 is C#-A-F, which would be F augmented.

Let's continue with the decade of the 2020's:

4/12/20 = E
4/4/21 = C
4/17/22 = F#
4/9/23 = D
3/31/24 = B
4/20/25 = G
4/5/26 = C#
3/28/27 = A
4/16/28 = F
4/1/29 = B

Here we see F# in 2022 an C# in 2026, but then back to F natural in 2028. Recognizable triads during this stretch include Em (2018-2020), C (2019-2021), Bm (2022-2024), G (2023-2025), and once again, F augmented (2026-2028).

Let's do one last decade, the 2030's:

4/21/30 = G
4/13/31 = E
3/28/32 = A
4/17/33 = F#
4/9/34 = D
3/25/35 = G#/Ab
4/13/36 = E
4/5/37 = C#
4/25/38 = A
4/10/39 = D#
4/1/40 = B (Let's throw in one more year so that 2005-2040 = nine bars in 4/4, twelve bars in 3/4)

We're clearly seeing more sharps as we proceed through the circle of fifths. Starting from D major, we notice that 2035 produces either G# or Ab -- in the chart above, I labeled March 25th as Ab, but it's better to call it G# here since we're definitely in the sharp scales now. Both G# and the D# in 2039 indicate that we've now reached the E major scale. Recognizable triads here include Em (2029-2031), D (2032-2034), C#m (2035-2037), and finally A (2036-2038).

This is likely the best possible Easter song that can be produced using our 12EDO scale. It's much better than songs based on the digits of pi, where (assuming 1=C, 2=D, 3=E as we usually do) we must go three bars (4/4) before hitting any triads -- 3.14159265358... is E-C-F-C-G-D-D-A-G-E-G-C (last three notes spell out C major), and two more before the next triad (D minor). In the Easter song, triads appear in nearly every bar.

It was only in the last few years that I learned about other EDO's besides the standard 12EDO. (This was still before I found out that my old Mocha computer is based on EDL's, not EDO's). Since there are a few irregularities in the 12EDO version, I wondered whether any other EDO might produce a better Easter song.

The Easter Song in 28EDO

One thing about consonant thirds and fifths in 12EDO is that when inverted (turned upside-down), they produce fourths and sixths. In particular, m3 becomes M6, M3 becomes m6, and P5 becomes P4.

Suppose we wanted to choose a new EDO for our Easter song such that each day represents a different step of the scale, and inverting a valid Easter interval produces another valid interval. Let's look at the list of Easter bunny hops again:

X-8     X+20
X-9     X+19
X+12  X-16
X+13  X-15

Notice that we have four pairs that add up to 28 (or differ by 28 if we take sign into account). Thus if these were notes of 28EDO, then inverting an Easter interval indeed produces another valid interval, as desired. No other EDO does this -- for example, in 29EDO, -9 inverts to +20 and +13 inverts to -16, but -8 and +12 have no inversion partners.

And that made my decision obvious. The natural EDO for our Easter song will be 28EDO. We can probably figure out why 28 turned out to be the best scale. It's a multiple of seven, the number of days in the week (and Sundays matter to the Easter date), and indeed it's the multiple closest to the length of a month (since Easter has a lunar component).

Let's link to the Xenharmonic website to learn more about 28EDO:

https://en.xen.wiki/w/28edo

According to this link, seven of 28EDO's notes are called A-B-C-D-E-F-G just as in 12EDO. But these seven notes are equally spaced on the octave. Indeed, instead of the whole tones and half tones of 12EDO, the seven lettered notes form a 7EDO scale on their own.

The other notes are named using ups and downs. So ^D is the note right above D, while vD is the note right below D. The note halfway between G and A is labeled either ^^G or vvA.

There's a reason why we use ups and downs rather than sharps and flats, to be explained later. For now, let's just convert Easter dates to notes of 28EDO so that we can get a song. Again, we'll keep the natural notes on the Easters of 2005-2014, which all differ by multiples of four:

3/22  vG
3/23  G
3/24  ^G
3/25  ^^G/vvA
3/26  vA
3/27  A
3/28  ^A
3/29  ^^A/vvB
3/30  vB
3/31  B
4/1  ^B
4/2  ^^B/vvC
4/3  vC
4/4  C
4/5  ^C
4/6  ^^C/vvD
4/7  vD
4/8  D
4/9  ^D
4/10  ^^D/vvE
4/11  vE
4/12  E
4/13  ^E
4/14  ^^E/vvF
4/15  vF
4/16  F
4/17  ^F
4/18  ^^F/vvG
4/19  vG
4/20  G
4/21  ^G
4/22  ^^G/vvA
4/23  vA
4/24  A
4/25  ^A

So we know that the Easters of 2005-2014 match in 12EDO and 28EDO. Thus we'll write the Easter song for Easters in the 2015-2040 range:

4/5/15 = ^C
3/27/16 = A
4/16/17 = F
4/1/18 = ^B
4/21/19 = ^G
4/12/20 = E
4/4/21 = C
4/17/22 = ^F
4/9/23 = ^D
3/31/24 = B
4/20/25 = G
4/5/26 = ^C
3/28/27 = ^A
4/16/28 = F
4/1/29 = ^B
4/21/30 = ^G
4/13/31 = ^E
3/28/32 = ^A
4/17/33 = ^F
4/9/34 = ^D
3/25/35 = ^^G
4/13/36 = ^E
4/5/37 = ^C
4/25/38 = ^A
4/10/39 = ^^D
4/1/40 = ^B

The mod 4 pattern of the Easter dates becomes much more evident in 28EDO. We have only natural notes from 2005-2014, then a mixture of naturals and ups from 2015-2028, only ups from 2029-2034, and finally a mixture of ups and double-ups beginning in 2035.

More on the Theory of 28EDO

What exactly does 28EDO sound like, anyway? There are a couple of YouTube videos out there that play microtonal music, including 28EDO. The first one is "Happy Birthday" by Claudi Meneghin:

And the second one is from James Pulley, who comes up with his own notation for 28EDO, besides the one mentioned in this post:

Why does Pulley prefer a different notation for 28EDO, anyway? Actually, it has to do with what perfect fifths and major thirds sound like in 28EDO.

Recall from our discussion of EDL's that a perfect fifth is supposed to be the 3/2 ratio, while a major third is supposed to be the 5/4 ratio. No EDO can produce exact ratios, since it would violate the Fundamental Theorem of Arithmetic for any number of perfect fifths (3/2's) to combine (multiply) to equal a whole number of octaves (2/1's). Some EDO's are better at approximating certain intervals more than others.

It's traditional to use cents to demonstrate how accurate an EDO is. Each step of our standard 12EDO scale is defined to be 100 cents, so that the whole octave is 1200 cents. Since 28EDO has 28 steps, each step of 28EDO is 1200/28 = about 43 cents.

Now we use logarithms to determine how many cents a just 3/2 and 5/4 should be:

P5 = 1200log(3/2)/log(2) = about 702 cents
M3 = 1200log(5/4)/log(2) = about 386 cents

Notice that it doesn't matter whether these are common logs or natural logs, because we actually used the Change of Base Theorem. Indeed, we ought to use base 2 logs (often abbreviated lg), but there is no such button on our calculator.

In 12EDO, we have multiples of 100 cents available. Thus its perfect fifth is very accurate at only two cents flat, while the major third is less accurate at 14 cents sharp.

But 28EDO is the other way around. Nine steps of EDO are very, very close to a just major third -- indeed, the error is a mere 0.6 cents! And the perfect fifth is less accurate -- the best we can do is 16 steps, which give us 686 cents, about 16 cents flat.

As for minor thirds, the just minor 3rd at 6/5 needs to be 316 cents. Both 12EDO and 28EDO use the same flat minor third at 300 cents, one-fourth of each respective octave (that is, three steps of 12EDO and seven steps of 28EDO). Once again, m3 + M3 = P5 in all cases. In 12EDO, the flat minor third and sharp major third combine to form a near-just perfect fifth, but in 28EDO, the flat minor third and near-just major third combine to form a flat fifth.

A just major scale contains the following intervals:

1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1

The 12EDO major scale C-D-E-F-G-A-B-C represents the best approximations to these just ratios available in 12EDO. But there's one more thing here -- why is this mode (also called the Ionian mode) considered to be the major scale, as opposed to the Lydian (F-G-A-B-C-D-E-F) or Mixolydian (G-A-B-C-D-E-F-G) scales which also have a major triad on the root? It's because Ionian is the only mode that has a major triad on the three most important scale degrees -- I, IV, and V. On the other hand, Mixolydian has a minor triad on v, while Lydian has a diminished triad on iv.

This is what Pulley has in mind when he comes up with his notation. Our interval C-G in 28EDO is already 16 steps, and thus it's a perfect fifth. But our C-E is only eight steps. In order to make it a just major third, we must raise the E up to ^E. And if we want IV and V to be major, then we must also raise A up to ^A and B up to ^B.

So our major scale becomes C-D-^E-F-G-^A-^B-C. But this annoys Pulley, who wants his major scale to be C-D-E-F-G-A-B-C without annoying up signs. In addition, he prefers to use sharps and flats to up and down signs.

Thus Pulley redefines E, A, B to be one step higher than our E, A, B respectively. Other steps are redefined using sharps and flats, so one step below G is Gb, one step below C is Cb, two steps below C is Cbb or B#, and so on.

But I disagree with Pulley's notation and will stick to what I've been using in this post -- up-and-down symbols with C-D-E-F-G-A-B-C corresponding to 7EDO, not the major scale. And here's why -- I have no problem with Pulley wanting to favor the major scale over the 7EDO scale. My problem is that Pulley's system favors the C major scale over all other possible major scales.

For example, what is the G major scale in Pulley's notation? Since his C scale matches the 12EDO scale (C-D-E-F-G-A-B-C), we may expect his G major scale to match also, G-A-B-C-D-E-F#-G. But unfortunately, the G major scale in Pulley's notation is G-Ab-B-C-D-E-F#-G. We must make it Ab instead of A because Pulley's A, which is correct for C major, is one step too high for G major. And the D major scale is even worse -- D-Eb-F#-G-Ab-B-C#-D. If we go the opposite way on the circle of fifths and try a flat scale such as F, a sharp appears in the scale -- F-G-A-Bb-C-D#-E-F.

On the other hand, all of these scales look alike in our notation:

• C-D-^E-F-G-^A-^B-C
• G-A-^B-C-D-^E-^F-G
• D-E-^F-G-A-^B-^C-D
• F-G-^A-B-C-^D-^E-F

As for why we use up symbols instead of flats, it's because we use a definition of sharp and flat that's consistent for all EDO's. Here it is -- B-F# and Bb-F must both be perfect fifths in whatever EDO we've defined perfect fifth in. In 12EDO, these are both indeed perfect fifths. Technically, in Pulley's notation, these are both perfect fifths as well. (His problem is that D-A is no longer a perfect fifth -- instead, we must use D-Ab or D#-A.)

And so in our (not Pulley's) notation, we want B-F# to be a perfect fifth. But the note that's 16 steps above B already has a name -- it's just F (natural). Therefore F and F# are the same note -- just as F is enharmonic to E# in 12EDO, F is enharmonic to F# in 28EDO. We say that the difference between F and F# (called an "apotome") is "tempered out" in 28EDO. And this is why the note above F must be called ^F and not F# -- the interval B-^F is not a perfect fifth.

In our notation, the letter names strictly follow the circle of fifths. This means that in C-G-D-A-E, the difference between each interval must be a perfect fifth. The interval between the first and last notes in this sequence, C-E, may or may not be a major third. EDO's in which C-E (as defined by the circle of fifths) indeed gives a major third is called a meantone EDO. In particular, our standard 12EDO scale is considered to be meantone.

The name "meantone" comes from the fact that the difference between the 9/8 and 5/4 ratios in a just major scale needs to be 10/9, so we refer to 9/8 (204 cents) as a "major tone" and 10/9 (182 cents) as a "minor tone." In a scale like 12EDO, the C-D and D-E intervals are neither a major tone nor a minor tone but somewhere in between, a "meantone."

In our notation, C-D-E-F-G-A-B-C is a major scale only in meantone EDO's. But 28EDO clearly isn't meantone, since in C-G-D-A-E on the circle of fifths, the last E is not a 5/4 major third above C. And so we shouldn't expect C-D-E-F-G-A-B-C to be a major scale in non-meantone EDO's like 28EDO.

If we raise the note C by four perfect fifths, we obtain E at 81/16 -- and if we drop this note by two octaves, we reach 81/64. This differs from 5/4 by the syntonic comma, 81/80 -- and this is also the difference between the major tone 9/8 and the minor tone 10/9. In meantone EDO's like 12EDO, the syntonic comma is tempered out, but in non-meantone EDO's like 28EDO, it is not tempered out.

Indeed, we see that in the first three notes of the major scale C-D-^E, the interval D-^E is wider than C-D, even though 10/9 is narrower than 9/8. This means that the syntonic comma is mapped to -1 steps in 28EDO! Indeed, it's one of only fourteen EDO scales where the syntonic comma is mapped to a negative step. In 28EDO, the syntonic comma is negative because our fifths are so flat, so four perfect fifths end up being narrower than a major third.

Another flaw in Pulley's notation is that it doesn't work very well for minor scales. Suppose we want to write the C (natural) minor scale. We recall that a minor third in 28EDO is seven steps -- two steps less than a major third. Thus if C-D-E-F-G-A-B-C is a major scale in Pulley's notation, then we must lower E, A, B by two steps to get a minor scale: C-D-Ebb-F-G-Abb-Bbb-C. In our notation, we simply use downs -- C-D-vE-F-G-vA-vB-C.

And finally, as it turns out, not everyone uses C-D-^E-F-G-^A-^B-C as the major scale anyway. If we return to Claudi Meneghin's birthday video, we notice that while Meneghin, like Pulley, prefers sharps/flats to ups/downs, his major scale is different. But instead of C-D-E#-F-G-A#-B#-C as we would expect (using C-D-E-F-G-A-B-C as 7EDO, as I do), his major scale is C-D#-E#-F-G-A#-Bx-C (where Bx is B double-sharp). I can see why he might prefer D# to D -- C-D# is closer to a just 9/8 than C-D (214 cents vs. 171 cents), even though C-G-D consists of the two perfect fifths that should take us to 9/8. But I'm not sure why Meneghin would prefer Bx to B#, as C-B# is still closer to 15/8 than C-Bx is. Perhaps the "leading tone" effect (from the seventh to the octave) sounds better with Bx-C than B#-C does. Even so, it's easy to modulate C-^D-^E-F-G-^A-^^B-C to any other key.

One commenter tells Meneghin that 28EDO is the first EDO (other than 12) that sounds "normalish" to her. Meneghin replies that it's because 28EDO has an accurate "natural 3rd" (just major 3rd) -- and scales with accurate thirds, and fifths, or where four fifths equal a third (meantone), sound more normal to us. But he also points out that when there are more notes in the scale, it's easier to get closer to the third and fifth than scales with fewer notes.

 It also helps that Meneghin uses 28EDO to play a song in a major scale ("Happy Birthday"), and the four modulations he uses are by the same 300-cent minor 3rd as 12EDO (that is, from C to Eb, Fx, A#, back to C). If he used intervals that differ appreciably from 12EDO intervals (as our Easter song song), then the music will sound quite exotic.

Even though 28EDO has an excellent major third, its lack of an accurate perfect fifth is why even among microtonalists, 28EDO isn't as popular as other EDO's with better fifths. (Likewise, among EDL's I often find myself preferring 18EDL with its 3/2 fifth to 20EDL and its 5/4 major third.)

A few years ago, I made music posts on the blog about a few EDO's (in anticipation of finding the Mocha emulator, before I discovered that it's based on EDL's instead). Most of those EDO's were meantone EDO's, including 19EDO and 31EDO.

But I did mention one non-meantone EDO -- 22EDO. That day, I posted two ways of naming the notes, one where C-D-E-F-G-A-B-C is a major scale (as Pulley does for 28EDO) and another strictly based on the circle of fifths. In 22EDO, the syntonic comma is mapped to one positive step, so it's less confusing than 28EDO where the syntonic comma is one negative step.

Here is a complete list of EDO's categorized by the number of steps of the syntonic comma. There are 14 EDO's with syntonic comma = 1, 28 EDO's where comma = 0, and 42 EDO's where comma = +1:

Syntonic Comma = -1
EDO's: 2, 4, 9, 14, 16, 21, 23, 28, 33, 35, 40, 47, 52, 64

Syntonic Comma = 0 (meantone)
EDO's: 5, 7, 12, 19, 24, 26, 31, 36, 38, 43, 45, 50, 55, 57, 62, 67, 69, 74, 76, 81, 86, 88, 93, 98, 100, 105, 117, 129

Syntonic Comma = +1
EDO's: 3, 8, 10, 15, 17, 22, 27, 29, 34, 39, 41, 46, 48, 53, 58, 60, 65, 70, 72, 77, 79, 84, 89, 91, 96, 101, 103, 108, 110, 115, 120, 122, 127, 134, 139, 141, 146, 151, 153, 158, 170, 182

In many of the missing EDO's, the comma is mapped to +2 steps. There is a "tuning" problem here (sort of like the tuning problem for odd number bases, except it's musical tuning) -- every EDO below 100 where the comma is mapped to +2 is right next door to a meantone EDO (and in some cases, both neighboring EDO's are meantone). For example, 44EDO maps the the comma to two steps, and its neighbors 43EDO and 45EDO are both meantone.

If you want C-D-E-F-G-A-B-C to be a major scale, then stick to the meantone EDO's only. Using any non-meantone EDO's means that C-D-E-F-G-A-B-C is no longer a just major scale.

Coding the Easter Song in 28EDO

This is what I wrote two years ago about coding the Easter song in Mocha:

We know that music on Mocha is based on EDL (equal divisions of length) rather than EDO (equal division of the octave). Thus trying to play an EDO on Mocha is unnatural -- and indeed, we can only approximate 28EDO on the emulator.

I wrote earlier that the EDO's up to 12 (the "macrotonal EDO's") sound quite well on Mocha, but the accuracy drops off quickly past 12EDO. The multiples of four (16EDO, 20EDO, 24EDO, 28EDO) are slightly better than non-multiples of four. Once we reach 31EDO, the odd EDO's are marginally better than the even EDO's -- in reality, all of them are very inaccurate. (This is another situation where 16-bit Atari music shines -- although Atari music is also based on EDL, we'd be able to approximate EDO's better on Atari than on Mocha.)

As it turns out, Degree 210 (Sound 51) -- the root note of the New 7-Limit Scale -- is also a good root note for a 28EDO scale. To create the scale start with Degree 210 and divide by the 28th root of two for each step until we reach 105, one octave above Degree 210. Here's the resulting scale:

Step  Degree  Sound

0       210        51

1       205        56

2       200        61

3       195        66

4       190        71

5       186        75

6       181        80

7       177        84

8       172        89

9       168        93

10     164        97

11     160        101

12     156        105

13     152        109

14     148        113

15     145        116

16     141        120

17     138        123

18     134        127

19     131        130

20     128        133

21     125        136

22     122        139

23     119        142

24     116        145

25     113        148

26     110        151

27     108        153

28     105        156

Some of these are more accurate than others. These include:

Step  Degree  Sound

0       210        51

3       195        66

6       181        80

9       168        93

10     164        97

11     160        101

12     156        105

20     128        133

24     116        145

25     113        148

28     105        156

These can indicate how close some of the steps of 28EDO are to just intervals. For example, that Step 3 corresponds to Degree 195 tells us that this interval from Steps 0 to 3 represents the ratio 210/195, which reduces to 14/13 (a 13-limit interval).

One important interval is that from Steps 0 to 9. The ratio 210/168 reduces to 5/4, a major third. In fact, 28EDO approximates a just major third to within one cent -- better than any simpler EDO. And furthermore, -9 is one of the Easter bunny hops, so here I achieve my original goal of making the valid Easter jumps correspond to consonant intervals.

The reason that we see EDO's like 24 and 31 more often than 28 -- despite its accurate major third -- is that 28EDO perfect fifth is inaccurate (and fifths are more important than thirds). In fact, the perfect fifth of 28EDO is the same as that of 7EDO -- Step 16. This is confirmed by the presence of Degree 141 in the above chart for Step 16 -- had Step 16 been closer to a just 3/2, it would have been listed as Degree 140 (210/140 = 3/2), not Degree 141.

I've mentioned 7EDO in previous posts -- there was apparently an ancient Chinese scale (Qingyu) based on five notes of a 7EDO scale. We found out that this scale fails to distinguish between major and minor intervals.

In fact, the Xenharmonic website often gives 7EDO a special name -- whitewood. It refers to the idea of removing all of the black keys on a piano, leaving only the white keys. Without black keys, the interval from C-D is the same as that from E-F -- seven equal intervals add up to the octave.

https://en.xen.wiki/w/Apotome_family

Another name given at Xenharmonic for EDO's like 7 and 28 is "perfect EDO." This is because there are no "major" or "minor" intervals, only "perfect" intervals.

Of course, 28EDO does have a major third (Step 9) in addition to 7EDO's "perfect third." In other words, 28EDO is what we get if we cross a perfect EDO with a just major third.

This indicates what our Easter song will sound like. We'll hear essentially 7EDO music during the stretches when Easter falls on the same day of the four-week (like 2005-2014), then slowly more and more major thirds (and minor sixths) appear. Eventually we'll hit another stretch (like 2029-2034) with the same 7EDO scale transposed up a 28EDO step.

Let's program this song in Mocha. We'll begin by coding the 28EDO scale:

https://www.haplessgenius.com/mocha/

NEW

10 DIM S(35)

20 FOR X=1 TO 35

30 S(X)=INT(210/2^((X-4)/28)+.5)

40 NEXT X

Don't forget to use the up arrow for the exponentiation ^ symbol. This sets up a 28EDO scale, where Note 4 is Degree 210 and Note 28 is Degree 105. The idea is that Note 1 will correspond to March 22nd and Note 35 is April 25th, so that the earliest Easters play the lowest notes (which are the largest degrees). 

Now for the trickiest part -- calculating an Easter date. We begin by asking the user for a year:

50 INPUT Y

Notice the definition of the Remainder or mod function given:

Remainder(x|y) (or x mod y) means the remainder when you divide x by y. It is never negative, and is defined in terms of the [] operation as follows:

Remainder(x|y) = x - y[x/y]

In Mocha, [] is the INT operation, so we write:

R=X-Y*INT(X/Y)

This is actually the same remainder function we used in the repeating decimals song earlier. It would be easier if the mod function had a single symbol, like % in C and C++. But unfortunately, BASIC doesn't have such a symbol built in!

Let's proceed with the algorithm mentioned at the above link. For example, at the above link we see:

G = year mod 19

In the program, we write this as

60 G=Y-19*INT(Y/19)

Here is the rest of the algorithm:

70 C=INT(Y/100)

80 HH=C-INT(C/4)-INT((8*C-13)/25)+19*G+15

90 H=HH-30*INT(HH/30)

100 I=H-INT(H/28)*(1-INT(29/(H+1))*INT((21-G)/11))

110 JJ=Y+INT(Y/4)+I+2-C-INT(C/4)

120 J=JJ-7*INT(JJ/7)

130 L=I-J+7

You might notice that the link above gives L = I - J, not L = I - J + 7. But let's see why I included it:

L is the number of days from 21 March to the Sunday on or before the Paschal full moon (a number between -6 and 28)

In other words, L is the number of days from March 21st to Palm Sunday. We might as well add 7 to make this the number of days from March 21st to Easter Sunday. And then this becomes a number between 1 and 35, which we can enter directly into Mocha:

140 SOUND 261-S(L),4

The song is supposed to continue on to the following year to computer the next Easter:

150 Y=Y+1

160 GOTO 60

(As usual, click on "Sound" before you RUN the program.)

We can begin this song at any year, such as 2020, and hear a different part of the song. I suppose that purists should begin the song with 1583 -- the first spring of the Gregorian Calendar.

Just as with the repeating decimals, this song repeats -- but according to the link above, Easter repeats only after 5.7 million years!

The link also states that Julian (Orthodox) Easter repeats only after 532 years. The link doesn't state this, but the first lines (calculating I and J that we skipped over) are actually for Julian Easter. (By the way, this year Orthodox Easter is a week after Gregorian Easter on April 19th.) This we can code in Mocha by deleting some lines:

DEL 70-80

90 II=19*G+15

100 I=II-30*INT(II/30)

110 JJ=Y-INT(Y/4)+I

This song thus repeats after 532 notes. (Notice that 533 isn't a prime, much less a full reptend prime, but 541 is indeed a full reptend prime. So the Orthodox Easter song is similar in length to the song for the repeating decimal 1/541.)

Returning to the present, here are a few more things I want to say about the Easter song in 28EDO:

First, our program maps Day 4 (that is, March 25th) to Degree 210 -- in Kite's color notation, this note is rugu G. Earlier, we mapped March 25th to ^^G in 28EDO, which is good enough for our purposes since we never stated how to map 28EDO names to concert pitch. This works out better for certain notes more than others -- the middle C of 28EDO is only two Degrees below wa C, but wa E ends up becoming ^^E in 28EDO.

Even though Pulley makes a big deal about playing major triads and the just major scale in 28EDO, major triads don't actually appear in the Easter song. Yes, the major third of nine steps is a valid bunny hop, but the minor third of seven steps (which we need to place on top of a major third to make a major triad) isn't.

What triads do exist in the Easter song? The bunny hop of eight steps is very close to what we call a "neutral" third, which can be written as 11/9 (or sometimes 27/22). As it turns out, 28EDO is worse in the 7-, 13-, and 17-limits than it is in the 3-limit, but the 11-limit is fairly strong in 28EDO. But identifying eight steps as either 11/9 or 27/22 requires using either 5 steps for the harmonic 9, or 21 steps for the harmonic 27 (rather than 4 or 20 steps, respectively, as implied by the perfect fifth).

Thus the C perfect triad C-E-G becomes either 18:22:27 or 22:27:33. The other possible triads are called C-up-five (C-E-^G) and C-up-up-five (C-^E-^G). These involving combining the 5/4 major third with the 11/9 neutral third. (I won't write these as extended ratios because musicians typically write them otonally, as opposed to the preferred utonal notation for Mocha. For example, the major triad of 12EDO is usually written as 4:5:6 and the minor triad as 10:12:15, but we must reverse these for Mocha -- 6:5:4 is a minor triad and 15:12:10 is a major triad.)

There are other alternatives to 28EDO for the Easter song, but I won't post them here today. Instead of approximating EDO in Mocha's EDL system, we might just play the song in pure EDL. (We can't use 28EDL because that has only 14 notes in the octave from Degree 28 to Degree 14 -- instead, we'd need something like 56EDL.)

Another possibility is 28EDT, 28 equal divisions of the tritave (3/1 or octave + perfect fifth). This seems promising since eight steps here is either 11/8 or 15/11, hence producing a 8:11:15 triad.

Traditionalists: Learning Styles & Productive Struggle

Let's get back to math now. Even though today's Rebecca Rapoport question has nothing to do with Geometry, I like posting some of these questions anyway, since it's pretty much the only math I'm doing these days:

5^55 mod 17

Nearly all questions involving exponents and "mod" go back to Fermat's Little Theorem. According to this theorem:

5^55 == 5^x (mod 17)
55 == x (mod 16)

When we reduce 55 mod 16, we obtain 7:

55 == 7 (mod 16)
5^55 == 5^7 (mod 17)

So even though we were able to reduce 55 to 7, we must still reduce 5^7 mod 17, which we can only do by hand:

5^2 == 5 * 5 = 25 == 8 (mod 17)
5^3 == 8 * 5 = 40 == 6 (mod 17)
5^4 == 6 * 5 = 30 == 13 (mod 17)
5^7 == 6 * 13 = 78 == 10 (mod 17)

Therefore our desired answer is 10 -- and of course, today's date is the tenth. There are four Geometry problems coming up on the Rapoport calendar, so it's likely that my next post will land on one of those dates.

OK, let's get to our traditionalists. Our main traditionalist, Barry Garelick, posted this week:

https://traditionalmath.wordpress.com/2020/04/07/say-it-enough-times-and-it-becomes-the-truth-dept/

There are certain narratives in education that are repeated so many times, that they become viewed as the absolute truth.  The idea of “learning styles” comes to mind. Although articles and papers have been written debunking the idea of learning styles, the idea persists in ed schools and in other fora.  I hear it at my school when, for example, a teacher will say that some student is a “visual learner” and thus finds listening to the teacher on a video to be disconcerting.  This is finished up with “Too much ‘teacher talk’ and not enough guidance.”

I'm always surprised that traditionalists don't like the phrase "learning styles." To me, it should be obvious that different students learn differently. Yet Garelick uses the word "debunking" above when referring to learning styles -- so to him, the idea that learning styles exist not only isn't obvious, but has been proven wrong.

Garelick proceeds:

This last I can almost buy, because I believe students do need guidance, particularly in the form of direct and explicit instruction.  How you do this without talking, however, remains a mystery.

Oh, so this gives it away. To him and the other traditionalists, students don't have a variety of learning styles but only learn in only one style -- via traditionalist direct and explicit instruction.

Garelick refers to "visual learner" above. OK, Garelick, but there is a subject that's highly dependent on visual learning -- my favorite branch of math, Geometry. Almost every problem there involves some visual diagram -- indeed, of those four Geometry problems coming up next week, three of them require diagrams. Of course, it's hard for me to post those diagrams on the blog. How you teach Geometry (or graphing in Algebra) without visual diagrams, however, remains a mystery.

Furthermore, even Garelick knows that languages other than English exist -- and English learners who speak another language exist. When such students are in our classes, they might not understand the traditionalist direct instruction that the teacher is giving. How you teach English learners without visual diagrams, however, remains a mystery.

There's another phrase that gets Garelick's goat besides "learning styles":

One of the many other myths is the one about “productive struggle”.  I saw this in an article about the difficulties of distance learning.

I won't quote the article since the link is right there. It's about the distance learning that many of our students are forced to do since the coronavirus outbreak.

But I will quote Garelick's "friend." He's learning more about web programming (and as I mentioned in my last post, I might want to go back and learn about this myself):

Good teaching makes learning easier, not harder with more struggle. Students will struggle plenty with regular textbook problem sets even if you do everything possible to make learning easy.


So according to this friend (who sounds suspiciously like a traditionalist himself), students are learning when they find things easy and aren't struggling. To non-traditionalists, students are learning when they are struggling as opposed to leaving the worksheet blank. Traditionalists forget that students don't always do the work -- and that's even more likely these days with distance learning. I heard that even students who are hardworking when they are in brick-and-mortar classrooms are slacking off when it comes to distance learning.

Let's look at the comments. Garelick's "friend" outs himself -- I should have known all along that this friend is SteveH:

SteveH:
As the author of the second quote, it took me until the later years of college/grad school to fully realize that in most cases, it wasn’t me, but the teacher and the books. Since the time of that quote, I have come across a book that does a good job of explaining “Full Stack” software development. My life is easier and I don’t struggle as much. I’m learning and understanding faster.

Once again, we all try to analyze and carefully respond to half-baked and unproven edu-thought. It has no effect. These are the people who claim to teach “deep understanding.” We might as talk to a brick wall.

SteveH is a traditionalist who knows the value of working hard to make things easier, but most students sitting in our classrooms or visiting our distance learning websites don't. To them, making things easier means skipping the work altogether. Thus traditionalist learning that works for SteveH (or his son) won't necessarily work for our students. To summarize:

Traditionalist Perspective:

• students struggling = poor teaching, no learning
• students finding it easy = good teaching, lots of learning

Non-traditionalist Perspective:

• students struggling = students working, lots of learning
• students finding it easy = students leaving sheets blank/skipping online work, no learning

Another comment here, by Wayne Bishop, is short and sweet:

Wayne Bishop:
And if you don’t parrot the myth, you aren’t qualified for a teaching credential.

And this is the problem that traditionalists have with the system -- to them, credentialing programs are the ones that perpetuate "learning styles" and "productive struggle." Indeed, this echoes what Garelick writes earlier:

Although articles and papers have been written debunking the idea of learning styles, the idea persists in ed schools and in other fora.
[emphasis mine]

Oh, I sort of know what Garelick and Bishop are talking about here. When I was in a credentialing program, I recall a class where we learned how to write lesson plans for secondary math. I think mine was for a unit on graphing linear equations for Algebra I. When I received my grade for the assignment, some sections were marked down for being "too directed."

At the time, I knew only a little about the traditionalist debates through the old websites called "Mathematically Correct" (traditionalist) and "Mathematically Sane" (anti-traditionalist). But I can only assume today that the "too directed" lessons where more traditionalist (after all, "too directed" sounds like "direct instruction" mentioned by Garelick earlier).

By now, I know what the problem is with a lesson that's "too directed" -- direct instruction assumes that students actually do the work rather than leave the assignment blank. (By the way, I did hear some traditionalist voices during my teacher education -- my mentor during BTSA was an older teacher who was strongly traditionalist.)

We can guess what Garelick and Bishop would propose as a solution here -- either find a way for get teachers in the classroom without needing a credential, or find a way for teachers to get a credential without having to go through the "ed schools," or find a way for more "ed schools" to teach from a traditionalist perspective. (The "no credential" route is often recommended for second career teachers who already know what math is needed to operate in the real world. Traditionalists believe that if left to their own devices, they would teach math traditionally -- it's only the evil "ed schools" and the need to have a teaching credential in their way.)

I say, OK, traditionalists, let's try it your way. Set up an online/distance learning class based mostly on traditional lessons, and let's see what percentage of the students actually do the work.

Cosmos Episode 9: "Magic Without Lies"


Here is a summary of Cosmos Episode 9, "Magic Without Lies":

• Ever since the discovery of a paradox, the universes have never been the same.
• Isaac Newton, co-creator of Calculus, believed that light consists of particles, "photons."
• By sticking a needle in his eye, he discovered how white light contains colors of the rainbow.
• Dutch scientist Christian Huygens proposed the "magic lantern" centuries before modern film.
• Huygens, co-creator of probability theory, believed that light consists of waves, like sound.
• Thomas Young's double-slit experiment noticed a wavelike interference pattern.
• By the end of the 19th century, JJ Thomson discovered and observed the electron.
• When he observed the photons, there was no interference pattern and acted like particles.
• According to quantum mechanics, both Newton and Huygens were right on the nature of light.
• Quantum mechanics tells us that particles act differently when they are being observed.
• To understand, we consider a two-dimensional world, like Edwin Abbott's Flatland.
• Flatlanders know nothing about our third dimension until we show it to them.
• Zero-, one-, and two-dimensional worlds are easy to visualize as we're three-dimensional.
• Imagining quantum mechanics is like trying to visualize a four-dimensional hypercube.
• Albert Einstein devised modern physics yet had trouble with quantum entanglement.
• In the casino-like world of quantum mechanics, there is no objective reality.
• According to the Many Worlds Hypothesis, any that can happen does happen in some universe.
• In the world of quantum mechanics, which takes place inside all of us, we're all Flatlanders.

As I've written before, I never learned about waves as a young middle school student at all. But now waves are included as part of the middle school NGSS. The Illinois State text for physical science mentions waves at the end of the text, and it's apparently still part of the eighth grade curriculum under the Preferred Integrated model.

Most likely, I wouldn't have reached the end of the text the year I was at the old charter school. This would especially the case if the Green Team curriculum kicked in, as it should have. If the Green Team never got the ground, then there might be a little discussion of waves in May, just before the California Science Test is given.

Cosmos Episode 10: "A Tale of Two Atoms"

Here is a summary of Cosmos Episode 10, "A Tale of Two Atoms":

• When we burn a match, a chemical reaction takes place, and chemical energy is released.
• Since we're all made of atoms, the story of atoms is the story of all of us.
• Hydrogen/helium atoms in a gas cloud fuse to form a carbon atom (shown in blue) in a star.
• On the other side of the galaxy, a uranium atom (shown in red) is formed in a supernova.
• Both the carbon and uranium atoms eventually land on the ancient earth as it is formed.
• The carbon atom in Marie Curie's eye saw the uranium as it emitted powerful nuclear energy.
• HG Wells, in his World Set Free, predicted that nuclear war would be a reality by the 1950's.
• Hungarian physicist Leo Szilard read Wells and realized that his atomic bomb was possible.
• During the prehistoric Agricultural Revolution, people could kill someone far away with arrow.
• The kill radius has gradually increased, from the range of an arrow to our entire planet.
• German physicist Paul Harteck hoped to produce a nuclear bomb for Hitler in 1939.
• Hungarian Edward Teller informed Einstein about the producing an atom bomb in the US.
• Meanwhile, Russian Georgy Flyorov wanted to help Stalin create the bomb first.
• Robert Oppenheimer led the Manhattan Project that successfully built the first atom bomb.
• In Martinique in 1902, there was volcanic ash, yet the mayor wanted his Ascension Banquet.
• When the volcano erupted, only two out of 30,000 survived, including one prisoner.
• The uranium atom from our story eventually decays into thorium, which is lethal to our cells.
• The carbon atom from our story is inside one of us viewers.

Nuclear Chemistry really isn't part of the middle school curriculum, but plain atomic Chemistry is. In the Illinois State text, it's part of the physical science curriculum, so I should have taught it to eighth graders my first year. I believe Chemistry is part of the seventh grade curriculum under the Preferred Integrated model, and so I would have taught it to both Grades 7-8 in my second year.

Conclusion & Revisiting the Fourth Dimension

Since I'm reblogging my Good Friday post from four years ago, and since Neil deGrasse Tyson mentioned the fourth dimension today, this is what I wrote four years ago on Good Friday regarding Rudy Rucker and the Fourth Dimension:

Chapter 9 of Rudy Rucker's The Fourth Dimension: Toward a Geometry of Higher Reality is called "Spacetime Diary." It's the start of Part III, "How to Get There" (to the fourth dimension, of course).

In this chapter, Rucker describes the idea of time as the fourth dimension. This idea is a cornerstone of Einstein's Theory of Reality -- instead of the three dimensions of space, we must consider the four dimensions of spacetime, He writes:

Later. Do you hate time? Alarm clocks, sure. Changing the clocks for daylight-saving time [as we just did last week -- dw] is the worst. How can they just take away an hour like that? Remember in 1973 when Nixon took away two hours for the oil companies?

"The older I get, the faster time goes," my [Rucker's -- dw] mother told me."The years just fly by. Every time I turn around, it's Christmas or Thanksgiving."

Let's look at the four puzzles of this chapter:

Puzzle 9.1:
If we say that the fourth dimension is time, then it is possible to construct a hypersphere in space and time. How?

Answer 9.1:
Take a small spherical balloon. Blow it up and then let the air out. The entire spacetime trail of the balloon's surface and inside is a solid hypersphere. The trail of the surface alone is the hypersurface of the hypersphere.

Commentary: We may draw the temporal definition as if it were a spatial dimension. The resulting diagram is called a Minkowski diagram.

Puzzle 9.2:
What kind of ideas about the past and future are embodied in this picture, where one thinks of the spacetime solid like a block of ice that melts from the bottom up?

Answer 9.2:
"The melting future" world view corresponds to the notion that future events exist, stored up and waiting for us. A uniform "now" moves forward with the passage of time, and instant after instant is permanently used up. In this viewpoint, past events are totally nonexistent. It is not uncommon for people to feel this way about their lives. Life here becomes a scarce resource that is consumed, and once something is over it doesn't matter at all. This is probably the least rewarding way possible to think about spacetime, as can be seen by thinking about the kind of personal philosophies inherent in the other three world views shown in the figure [only now "exists," only the past and now "exist," and the past, now, and future all "exist" -- dw]. Whenever you cut yourself off from your past, you're in an extremely rootless and vulnerable position. But if you do throw out the past, you might as well throw out the future too, and get totally into the "now."

Commentary: We really get deep into Einstein's Theory of Relativity in this puzzle:

Puzzle 9.3:
"The relativity of simultaneity" says that differing moving observers will have different opinions about which events are simultaneous. In this problem, we will see how the relativity of simultaneity follows from the two basic assumptions: (1) that moving observers are free to think of themselves as being at rest, and (2) that light always travels at the same speed.

The situation is as follows. A rigid platform is moving to the right at about half the speed of light. On the left end stands Mr. Willy Lee, and on the right end stands Mr. Rye. Mr. Lee sends a flash of light down the platform toward Mr. Rye. Mr. Rye holds a mirror that bounces the light flash back toward Mr. Lee. Mr. Lee receives the return signal. Call these events A, B, and C, respectively. Mr. Lee notes the times of events A and C on his world line. After a little thought he decides which event X on his world line is simultaneous with B. Where does he put X, and why? (Hint: We would place X horizontally from B, but Mr. Lee will not. Simultaneity is relative!)

Answer 9.3:
Mr. Lee will put X halfway between A and C. The reason is that Lee will assume that it takes the light just as long to travel from the other end of the platform as it took it to get there from his end. It is natural for him to think this, in view of the assumptions (1) and (2) mentioned in the puzzle. We, of course, feel that it really takes the light longer to get from A to B than it takes it to get from B to C ... But Mr. Lee will say that we just think that because we're racing past him at half the speed of light!

Commentary: This next puzzle brings up the idea that space is cyclical -- that is, just a circle:

Puzzle 9.4:
In the figure, we drew a picture [a cylinder -- dw] of a circular space that remains the same size as time goes on. A widely held present-day view of the universe is that our space is an expanding hypersphere, which started out as point-sized about twelve billion years ago. Can you draw a picture of spacetime that represents our space as an expanding circle?

Answer 9.4:
The picture would be a sort of "conical" spacetime, as drawn here. The starting point is known as "the initial singularity," or as the Big Bang. Whether or not our space will eventually contract back to a point is unknown at present. Apparently it depends on how much mass is actually in our universe: if there is enough mass, then the gravitational forces will pull things back together.

Commentary: The idea that gravity will pull things back together is called the "Big Crunch." It is often believed that a Big Crunch will be followed by another Big Bang, and Rucker mentions the idea that time is cyclical -- that is, just a circle. This idea appears in yet another Futurama episode -- "The Late Philip J. Fry."