**Table of Contents**

**1. The Pappas Book Ends -- The Borges Book Doesn't**

**2. Continued Fractions and Lunar Calendars**

**3. Continued Fractions and the Yerm Calendar**

**4. The Yerm Calendar and Blue Moons**

**5. The Yerm Calendar and Schools**

**6. The Yerm Calendar and Gubernatorial King Plan**

**7. Holidays on the Yerm Calendar**

**8. The Yerm Calendar and the QNTM List**

**9. Coding**

*Square One TV*'s "Angle Dance"**10. Conclusion: New Year's (New Yerm's?) Resolutions**

**The Pappas Book Ends -- The Borges Book Doesn't**

This is what Theoni Pappas writes on page 61 of her

*Magic of Mathematics*:

"In the 1900's infinity was featured in Jorge Luis Borges' book

*The Book of Sand*. Here the main character acquires a 'marvelous' book."

This is the last page of the section "Mathematical Worlds in Literature," which is the last section of Chapter 2, which is the last Pappas chapter that we're reading because today is the last day of 2017. I did say that our final Pappas post would be on December 31st, and voila!

Here are some more excerpts from this page:

*"'The number of pages in this book is no more or less than infinite. I don't know why they're numbered in this arbitrary way.'*This book adversely changes his life and his outlook on things, until he realizes he must find a way to dispose of it --

*'I thought of fire, but I feared that the burning of an infinite book might likewise prove infinite and suffocate the planet with smoke.'*You might want to read the book to find out how the hero resolved his dilemma.

"For example, in an episode of

*Star Trek -- The Next Generation*, the starship is being pulled by an 'invisible' force toward a black hole. Only when the ship's schematic monitor changes perspective does the crew realize the unknown force is a 2-dimensional world of minute life forms.

"Mathematics is full of ideas that make one's imagination churn and wonder --

*Are they real?*To mathematicians they are real. Mathematicians are familiar with the worlds in which these ideas reside -- perhaps not within our realm, but real in their own nonetheless!"

The Borges book may be infinite, but the Pappas book isn't. We have completed our reading of the Pappas book, which we began with Chapter 3 in March when I first bought the book, and then we finished today when we looped back to Chapter 2. Tomorrow I will return to

*The Mathematics Calendar 2018*by Theoni Pappas. Just as I did from 2014-2016, I'll only post questions from that calendar if they are related to Geometry, since that's the mathematical world we have always been explorers of on this blog.

**Continued Fractions and Lunar Calendars**

**It's still Calendar Reform week, so let's return to the use of continued fractions in calendars. So far, we've only looked at solar calendars, so let's see what happens if we try to make a lunar calendar.**

*Lunisolar calendars*use a lunar month as the Level-0 unit. The two major lunisolar calendars are the Chinese Calendar and the Hebrew Calendar. The following link provides us with the length of the lunar month:

https://mrob.com/pub/math/numbers-5.html

The length of the mean synodic month is 29.530588853 days. Since we already have a value of the tropical year as 365.242189 days, we divide the month into the year and enter this value into one of the continued fraction calculators:

12.36826637 = [12; 2, 1, 2, 1, 1, 17, 3]

0. The Level-0 cycle is the basic unit, the lunar month.

1. The first value in the CF is 12. The Level-1 cycle consists of 12 Level-0 cycles, or 12 months in a lunar year. This approximation is used in the Islamic Calendar, which is often characterized as a pure lunar calendar rather than lunisolar because it is so inaccurate. Because of this, Muslim holidays like Ramadan (and its conclusion, Eid al-Fitr) can fall in all four seasons during the course of a lifetime.

2. The second value in the CF is 2. The Level-2 cycle consists of 2 Level-1 cycles plus a Level-0 cycle, or two years with a Leap Month.

3. The third value in the CF is 1. The Level-3 cycle consists of one Level-2 cycle plus a Level-1 cycle, or three years with a Leap Month.

4. The fourth value in the CF is 2. The Level-4 cycle consists of 2 Level-3 cycles plus a Level-2 cycle, or eight years with three Leap Months. The mean year length is 12.375 months. This approximation is called an

*octaeteris*:

http://calendars.wikia.com/wiki/Octaeteris

The octateris was first used by the ancient Greeks. According to the link above, the octaeteris is about one or two days off after eight years, and so this calendar is comparable to a pure solar calendar with a simple approximation of 365 days -- this is also about two years off after eight years.

5. The fifth value in the CF is 1. The Level-5 cycle consists of one octaeteris cycle plus a Level-3 cycle, or 11 years with four Leap Months. The mean year length is 12.3636... months.

6. The sixth value in the CF is 1. The Level-6 cycle consists of one Level-5 cycle plus an octaeteris cycle, or 19 years with seven Leap Months. The mean year length is 12.368421 months. This approximation is called the Metonic Cycle:

http://calendars.wikia.com/wiki/Metonic_cycle

The Metonic cycle is named for the Greek astronomer Meton, but it may have been discovered many centuries earlier. Many lunisolar calendars, including the Hebrew Calendar, use the Metonic Cycle. It is an excellent approximation because the next number in the CF is the large 17.

7. The seventh value in the CF is 17. The Level-7 cycle consists of 17 Metonic Cycles plus a Level-5 cycle, or 334 years with 123 Leap Months. The mean year length is 12.368263 months. This approximation is used in the New Roman Lunisolar Calendar:

http://calendars.wikia.com/wiki/New_Roman_Lunisolar_Calendar

Because of its use in this calendar, we can call the 334-year cycle the "New Roman Cycle."

8. The eighth value in the CF is 3. The Level-8 cycle consists of 3 New Roman Cycles plus a Metonic Cycle, or 1021 years with 376 Leap Months. It is not used in any calendar I'm aware of, and since the next value in the CF is the huge 26, there's no point in going any further. All the CF values used in actual calendars also appear in the following list:

http://calendars.wikia.com/wiki/Lunisolar_calendar

But notice that even though these continued fractions tells us how many months are in a year, they don't tell us how many days there should be in each month. Recall that 29.530588853 days is the length of the lunar month.

The New Roman Lunisolar Calendar, for example, has ten months of exactly 30 days in length, even though this is slightly longer than the lunar month. (Because of symmetry, the two months near the middle of the year,

*Quintiliae and Sextiliae*, do start near new moons, but not the other months.) The other two months, called

*Brumias,*vary greatly in length. Then again, there's no claim that any month starts on the new moon except for New Year's Day, which is on the new moon if there's no Leap Month and the full moon if there is. Indeed, there is no Leap Month -- instead the

*Brumias*are longer in 123 of the 334 years. (On the other hand, the Chinese and Hebrew Calendars actually do start their months on new moons.)

**Continued Fractions and the Yerm Calendar**

**It is possible to enter 29.530588853 directly into the continued fraction calendar. This means that months can all start at the new moon. But the cycles that this CF produces won't necessarily line up with solar years at all, much less any of the longer cycles such as Metonic Cycle.**

The calendar produced by the continued fraction representation of 29.530588853 is called the

*Yerm Lunar Calendar*, and it was created by calendar expert Karl Palmen.

29.530588853 = [29; 1, 1, 7, 1, 2, 17]

0. The Level-0 cycle is the basic unit, the day. Actually, Palmen uses a "night" as the Level-0 cycle. I see the reasoning behind this -- this is a lunar calendar based on the moon, and the moon is the most visible at night. Nights on the Yerm Calendar run from noon to noon, just as days typically run from midnight to midnight.

1. The first value in the CF is 29. The Level-1 cycle consists of 29 Level-0 cycles, or 29 nights in a lunar month.

2. The second value in the CF is 1. The Level-2 cycle consists of one Level-1 cycle plus a Level-0 cycle, or 30 nights in a lunar month. (Apparently the New Roman Lunisolar Calendar uses Level-2 cycles for ten of its months.) Recall that whenever 1 appears in a CF, we can use the words "long" and "short" to describe them. Hence we have 29-night short months and 30-night long months.

3. The third value in the CF is 1. The Level-3 cycle consists of one Level-2 cycle plus a Level-1 cycle, or one long month and one short month. Let's call this the Long-Short Month Cycle. (This Level-3 cycle is the basic format of the Hebrew Calendar, plus adjustments.)

4. The fourth value in the CF is 7. The Level-4 cycle consists of 7 Level-3 cycles plus a Level-2 cycle, or eight long months and seven short months. This is a brand new 15-month cycle that Palmen calls a "yerm." One yerm is longer than one solar year. The mean month length is 29.533... nights.

5. The fifth value in the CF is 1. The Level-5 cycle consists of one Level-4 cycle plus a Level-3 cycle, or nine long months and eight short months. Since we have a CF value of 1, the words "short" and "long" apply again. Thus we have 15-month short yerms and 17-month long yerms. The mean month length is 29.529412 nights.

6. The sixth value in the CF is 2. The Level-6 cycle consists of 2 Level-5 cycles plus a Level-4 cycle, or two long yerms and one short yerm. Palmen calls this the "basic three-yerm cycle" and states its mean month length as 29.530612 nights.

7. The seventh value in the CF is 17. The Level-7 cycle consists of 17 Level-6 cycles plus a Level-5 cycle, or 17 basic three-yerm cycles and one long yerm. This is the full yerm cycle as used by Palmen, and the last one he considers, so we don't need to look at the eighth CF value at all. He states the mean month length as 29.5305882 nights -- more accurate than the Hebrew calendar.

According to Palmen, the current Level-7 cycle began at noon on November 11th, 1996 -- most likely the new moon closest to the day he created the calendar. Here is a link to the Yerm Calendar:

https://www.hermetic.ch/cal_stud/palmen/yerm1.htm

Yerms have nothing to do with solar years, since they have either 15 or 17 months. The first yerm started in November 1996, but the next yerm (the 17th) will start on January 17th, 2018. So not only do we have a new year coming up, but a new yerm as well. (Happy New Yerm!) And just as the Dee Calendar is the most accurate rule-based solar calendar (in keeping the spring equinox on a fixed date), the Yerm Calendar is the most accurate rule-based lunar calendar (in keeping the new moon on a fixed date). Of course, no rule-based calendar can ever be exact.

**The Yerm Calendar and Blue Moons**

**A blue moon occurs when there are two full moons in a calendar month. The last blue moon was in July 2015, and I mentioned it in my posts that month.**

Of course, there are no blue moons on the Yerm Calendar itself, in that every month starts on the new moon and contains exactly one full moon, around the 15th of the month. But we can use the Yerm Calendar to find full moons, and then we can determine the next Gregorian month that contains two of them.

According to the link above, the current month (the 17th and final month of the 16th yerm) began back on December 18th. This is the first night of the month, or the new moon, and so the full moon would be on the 15th, or 14 nights after the full moon. This works out to be January 1st -- in other words, tomorrow is a full moon.

Now the new yerm begins on January 17th, a new moon night. So we add 14 days to this to obtain the next full moon, January 31st.(On the Hebrew Calendar, this is Tu b'Shevat, the Jewish Arbor Day.)

And lo and behold -- January 1st and 31st are both full moon nights, and so we've already identified the next blue moon, January 31st. It's easy to see why blue moons exist in the Gregorian Calendar -- January has 31 days, which is longer than the 29.530588853 days that make up a lunar month. So if there's a full moon sufficiently early in the month (like the 1st or 2nd), then there could be a second full moon at the end of the month.

Even though we've already located the next blue moon on January 31st, let's find out the next full moon anyway. The second month of the yerm begins on February 16th. (This new moon corresponds to Chinese New Year). And so the next full moon night will be two weeks later, on March 2nd. (In the Chinese Calendar, this is the Lantern Festival.)

Hmmm, this seems strange -- there are full moons on January 31st and March 2nd, but no full moons in February at all. This is the

*opposite*of a blue moon -- a month without a full moon at all. And again, we can see why -- blue moons occur in months that are

*longer*than 29.530588853 days, and so there can be months without full moons in months that are

*shorter*than 29.530588853 days. In the Gregorian calendar, only February is short enough to lack a full moon.

And let's, for laughs, calculate one more full moon. The third month of the yerm is on March 17th, St. Patrick's Day. Two weeks later is the new moon, which we calculate as March 31st. (On the Hebrew calendar, this is the first day of Passover. It's also the Paschal Full Moon in the Easter calculation -- as this is a Saturday, Easter Sunday will be on April Fool's Day!)

And so March has two full moons, on the 2nd and the 31st -- hence March 31st is a blue moon. So you might have thought that "once in a blue moon" is rare, and yet 2018 will have two blue moons just two months apart.

Of course, the short February is the explanation for this. If January has a blue moon, then it would be late enough in January that the full moons skip February altogether. The full moons just

*barely*skip February though (since 28 is barely less than 29.530588853), and so the next full moon would be early in March -- early enough to allow for a second full moon that month.

How often do we expect blue moons to occur, anyway? We go back to lunisolar calendars and notice that if there is a Leap Month -- that is, 13 full moons in a solar year -- then since there are only 12 months on the Gregorian Calendar, one month must have two full moons. (This is the Pigeonhole Principle at work.) So we expect there to be as many blue moons as there are Leap Months:

3. Using the Level-3 cycle, there is one blue moon every three years.

4. Using the Level-4 cycle, there are three blue moons every eight years.

5. Using the Level-5 cycle, there are four blue moons every 11 years.

6. Using the Level-6 cycle, there are seven blue moons every 19 years (Metonic Cycle).

The Metonic Cycle is accurate enough for our purposes here. So there are seven blue moons every 19 years -- at least, there would be, if it wasn't for the short February problem.

Of the seven blue moon years, we expect one of them to be a January-March blue moon year. And indeed, the last such year (and the only other double blue moon in my lifetime) was in 1999, exactly one Metonic Cycle ago. (This can be confirmed using the Yerm Calendar for 1999, except that the first full moon is on January 2nd since it's a 29-day short month.) And so the final answer is that there are eight blue moons and one month without a full moon every 19 years.

Here's an interesting link about the lengths of lunar months:

http://earthsky.org/astronomy-essentials/lengths-of-lunar-months-in-2017

According to this link, the current lunar month (December 18th-January 17th) is in fact the longest lunar month of the

*century*(29.804722 days), because the new moon is farthest from the earth. (The corresponding full moon is at its

*closest*to the earth and is therefore a "supermoon.") It's interesting that the Yerm Calendar assigns 30 nights to the current month, since no rule-based calendar can predict this sort of variation in the lunar month length. (And the shortest lunation of the year is indeed assigned 29 nights on the Yerm Calendar.) In the Yerm Calendar, the months alternate 30 and 29 nights (Long-Short Month Cycle) with the first and last months of each yerm (15 or 17 months) having 30 nights -- so it has nothing to do with the actual variation of the lunation lengths.

It's interesting to see how blue moons work on other calendars. In the World Calendar, no month is shorter than 30 days, and so every month has a full moon. Blue moons occur about as often as predicted by the Metonic Cycle -- seven blue moons every 19 years.

On the International Fixed Calendar, every month is shorter than 29 days, and so no month has a blue moon (unless you count full moons on blank Leap Days as blue moons). With every month as short as February, months without full moons are more common. Since the Metonic Cycle has seven solar years with 13 full moons, it follows that there are 12 solar years with only 12 full moons. Hence months without a full moon occur 12 times every 19 years. (If we count December 1st-30th full moons as Leap Day blue moons, it's likely that neither November nor January has a full moon, but since not every year has a Leap Day, we expect this to be rarer than once per Metonic Cycle. We would have to go to the 334-year New Roman Cycle. My estimate is that Leap Day blue moons in the Int'l Fixed Calendar would occur once or twice in every New Roman Cycle.)

The Chinese Calendar also has a blue moon scenario -- except that this calendar counts new moons instead of full moons. As a lunisolar calendar, the Chinese Calendar has a lunar and a solar component -- the (mostly invisible) solar calendar divides the year into 12 solar terms (which line up with solstices, equinoxes, and astrological signs). Each lunar month is numbered for the solar term in which it begins, but if there's a "blue new moon" (a solar term with two new moons), the lunar month starting on the second new moon is officially the Leap Month. Hence Leap Months can occur at any time of the year (unlike the Hebrew Calendar, where they always occur near the Gregorian March).

As it turns out, the no-full-moon in February phenomenon can also occur in the Chinese Calendar. It happens that the solar terms aren't equal in length, but follow the elliptical orbit of the earth. (See my November 29th post for more info on the earth's orbit.) The shortest solar term is shorter than one lunar month (just as Gregorian February is), and so it could lack a new moon, with two new moons in each of the surrounding months. Unfortunately, the shorter solar terms (as we found out on November 29th) occur close to perihelion in January -- hence close to Chinese New Year. This results in a complication in the rule for Chinese New Year, when the New Year begins on the third new moon after winter solstice (instead of the usual second).

**The Yerm Calendar and Schools**

**This is an education blog, and so I often like looking at the school year on various calendars. So let's see what would happen if we change the school year so that it fits the Yerm Calendar.**

First of all, Karl Palmen describes a lunar week that fits with his Yerm Calendar:

The days of the month can be numbered from 1 to 29 or 30 as is conventional in most calendars. But I have in the first few months after inventing the calendar, used a lunar week whose weekends occur around the principal phases of the moon.

The 1st, 8th, 15th, 22nd and 29th 'nights' of the month are called 'Moonnight'. The first four of these form the start of a lunar week, whose 'nights' are

Moonnight, Tuesnight, Wensnight, Thursnight, Frinight, Saturnight and Soonnight (soon to be Moonnight).

For example, the 10th 'night' is referred to as the Second Wensnight.

This fixes the lunar week in relation to the standard 7 day week for the first four weeks of the month.

The 30th 'night' is a Soonnight, but the last night of any month can be referred to just as 'Lastnight' rather than 'Fifth Soonnight' or 'Fifth Moonnight'.

Short dates are ym(mm(w(d. So for example Fourth Frinight Month 2 Yerm 3 is 03(02(4(5.

I've also considered making the second lunar week have 8 days, but haven't decided how to name the 'nights'.

The name "Moonnight" refers to the phase of the moon -- new moons, first quarters, full moons, and third quarters are all supposed to fall on or near "Moonnight." But the similarity between the names "Moonnight" and "Monday" -- likewise "Tuesnight" and "Tuesday," and so on -- hints that the school week should last from Moonnight to Frinight, with Saturnight and Soonnight the weekend.

But there's a problem here -- the "nights" change at noon! This is convenient for observing the moon at night, but not for learning, since noon is in the middle of the school day.

I think the best plan is for the first school day to be when Moonnight changes to Tuesnight, and then the fifth school day is when Frinight changes to Saturnight. This has several advantages over the alternative of making Soonnight to Moonnight a school day:

- Just as New Year's Day is a holiday, presumably New Yerm's Day is a holiday. (Since all nights start at noon, so do yerms, and so "5, 4, 3, 2, 1, Happy New Yerm!" counts down to noon.) So the big party is at noon when Soonnight changes to Moonnight, and then the first school day of the yerm is the next day, as Moonnight becomes Tuesnight.
- Palmen writes, "...weekends occur around the principal phases of the moon." So Moonnight, when the principal phase of the moon occurs, is still part of the weekend. The school week begins with the moon longer visible -- the morning when Moonnight becomes Tuesnight.
- Palmen tells us that the 29th is always a Moonnight, so in short months, there are two Moonnights in a row. The 30th is always a Soonnight, and so in long months, the nights go Soonnight, Moonnight, Soonnight, Moonnight. It's convenient to have a three- or four-day weekend at the end of each month -- and this is justified because the school week doesn't begin until Moonnight changes to
*Tuesnight.*

It may be convenient to push all holidays to the end of the month, so that we have four full five-day weeks of school (or 20 days) per month, with a three- or four-day weekend at the end. This means that there is no "Big March" or long stretch without at least an extended weekend. (The Leap Week version of the Fixed Festivity Calendar pushes all holidays to a single week in each season. We might prefer having a full week off rather than just a three- or four- day weekend -- but the flip side is that we must go three straight months without a holiday before the week off. That's not a "Big March" -- that's a "Giant March"!)

OK, so now we've determined the school week, but what about the school year (yerm)? A yerm is much longer than a year -- yerms have either 15 or 17 months. We might enjoy having a standard nine months of school followed by six or eight months of vacation -- but then it will take 13 yerms (about 17 years) to finish grades K-12 rather than 13 years. High school grads would be old as college grads now are on the Gregorian Calendar!

So let's try something else. Palmen provides us with a rule of thumb here:

The construction of this table was greatly aided by the fact that 3 yerms is exactly 2 weeks less than 4 Julian years, except when the 3 yerms begin with the last yerm of a cycle.

So three yerms are approximately equal to four years -- that is, a yerm is about 1 1/3 years. So we figure that just as a year has 180 school days, a yerm should have a third again as many days -- that is, 240 school days. Each month has 20 school days, so there are 12 months in a school yerm. The remaining three or five months will be vacation months.

In the 15-month short yerm, let's try declaring months 5, 10, and 15 to be vacation months. We don't attempt to have a three-month "summer vacation" because there is no "summer" on the non-solar Yerm Calendar. With vacation months 5, 10, and 15, the 240-day school yerm has been divided into three 80-day terms.

What should we name these three terms? Since each is a third of a school yerm, we might refer to each as a "trimester," with progress reports given midway through the term, at the "hexter." But the name "trimester" may be a misnomer here, since it means three months, not "one-third year," since the root "mes" is Latin/Spanish for "month." (It's just a coincidence that one-third of the Gregorian school year really is about three months.) Our "trimesters" on the Yerm Calendar contain four months of school, so perhaps some form of "quad-mester" would be better. (Likewise, it's uncertain whether the "hex-" in hexter means one-sixth of a year or six weeks, because, once again, one-sixth of a Gregorian school year really is about six weeks.)

Maybe the best name for each term is "semester." (This is a misnomer whether or not you believe "semester" means six months or a "semi-year," since our semesters are neither six months nor half of a yerm.) This reflects the fact that our semesters contain 80 school days, which is not far from the nominal length of a Gregorian semester as 90 school days. (The first semester of the blog calendar contained only 83 days, and the LAUSD first semester is exactly 80 days.)

Each term would end with a final exam (for high school). Midway through each term ("hexter," or perhaps "quarter" since it's half of a semester) could be a midterm test. Actually, I'm considering making months 1, 6, and 11 the vacation months because then two of the three semesters have a long four-day weekend at the halfway point, rather than a three-day weekend.

By the way, what happens in 17-month short yerms? It's illogical to make months that don't appear in every yerm part of the school yerm (just as blank days in other versions of Calendar Reform are almost always holidays). And so months 16 and 17 can join with either month 15 or month 1 to form a three-month vacation, similar in length to the original summer vacation (though, once again, this can occur in any season).

All that's left is to determine how many yerms students must attend school for. According to Palmer's rule of thumb, four years are equivalent to three yerms. So let's convert Grades 1-4 to Yerms 1-3, Grades 5-8 to Yerms 4-6, and high school Grades 9-12 to Yerms 7-9. (The nominal four years it takes to earn a bachelor's degree also fits this pattern -- it's a three-term degree.) We might create new school standards using this pattern -- so Yerm 1 Standards consist of all Common Core 1 standards plus one-third of the Common Core 2 standards, then Yerm 2 is the rest of Common Core 2 plus about two-thirds of Common Core 3, then Yerm 3 is the rest of Common Core 3 plus all of the Common Core 4 standards, and so on.

As for high school math -- the focus of this blog -- the four years Algebra I, Geometry, Algebra II, and Precalculus must be converted into three yerms. Some high schools, in an attempt to get students to Calculus by senior year, compress four years of math into three years. These courses can be the basis of new high school courses -- which are no longer compressed, since now students have three yerms to complete them rather than three years.

Kindergarten doesn't quite fit into this pattern. But it's possible to combine California's "Transitional Kindergarten" with true kindergarten to form a Yerm 0.

**The Yerm Calendar and the Gubernatorial King Plan**

So far, all we've done is naively convert school years to school yerms. But why should we keep the same old education system? We should take advantage of the Yerm Calendar to make the education system into something better.

It's possible, for example, to make Classical Education fit yerms, since the four-year classical cycle fits the three-yerm cycle like a glove. Of course, there will now be only three divisions in each cycle rather than four (for example, ancient, medieval, modern history).

In the Gregorian Calendar, the school year starts in August or September, but the calendar year begins in January. I find this a bit annoying -- for example, when it's time to file for taxes and I've held different teaching jobs that follow the school years rather than the calendar year. And so in the Yerm Calendar, let's align the calendar and school yerms since they don't follow the seasons anyway.

Since three yerms are just about four years, students can enter Yerm 0 when they have reached the age of three yerms by the start of that yerm. But this means that a student who turns three on the second day of the yerm must wait an entire yerm before starting Yerm 0. So some students start Yerm 0 at the equivalent age of four years, while others must wait until they're almost 5 1/2 years old.

Instead, perhaps students may be allowed to start at the second or third "semesters" of the yerm. We should take advantage of the Yerm Calendar by allowing students to start at different times during the school yerm.

The Yerm Calendar is also a great fit for the Gubernatorial King Plan. Recall that two years ago, I suggested a radical school plan. (I never gave this plan a name, but here "Gubernatorial" refers to recommendations made by the California governor that are indirectly related to this plan, while "King" refers to the date I originally posted the plan, Martin Luther King Day.)

*For example, let's divide the K-12 span into ten levels. Ten makes sense -- after all, I pointed out how each level can be subdivided into ten units. Then each student can receive a three-digit score -- the first digit gives the level from Level 0 to Level 9. The second digit tells which unit a student is on, and the third digit tells where within each level a student is approximately.*

So these Levels 0 through 9 don't readily correspond to grades. I have no problem assigning the top four levels, Levels 6 through 9, to the traditional high school disciplines of Algebra I, Geometry, Algebra II, and Pre-Calculus, especially since these themselves don't correspond exactly to high school years. I still like the idea of having this testing plan max out at Pre-Calc, since there's already a separate test for Calculus students to take -- the AP exam.

This means that Levels 0 through 5 will take us approximately from grades K-8. With six levels to take us through elementary and middle school math, it's not as obvious which level, say, a fourth grader should be placed -- and this is the intent.

Of course, when I am creating the standards, I need to know how to convert from grade to level -- even though the schools shouldn't know this. Let's figure it out -- we need six levels to take us through the nine years K-8, so each level is approximately three semesters.

So these Levels 0 through 9 don't readily correspond to grades. I have no problem assigning the top four levels, Levels 6 through 9, to the traditional high school disciplines of Algebra I, Geometry, Algebra II, and Pre-Calculus, especially since these themselves don't correspond exactly to high school years. I still like the idea of having this testing plan max out at Pre-Calc, since there's already a separate test for Calculus students to take -- the AP exam.

This means that Levels 0 through 5 will take us approximately from grades K-8. With six levels to take us through elementary and middle school math, it's not as obvious which level, say, a fourth grader should be placed -- and this is the intent.

Of course, when I am creating the standards, I need to know how to convert from grade to level -- even though the schools shouldn't know this. Let's figure it out -- we need six levels to take us through the nine years K-8, so each level is approximately three semesters.

Returning to the present, I see a clear correspondence between my "levels" and yerms (even though the correspondence isn't exact), with about three semesters in each level/yerm. The main idea is that just as students can start Yerm 0 at different times through the yerm, they can progress through the other yerms at their own pace, too.

At the end of each semester, a computer-based standardized test can take place. I wrote that the computer can return a three-digit score -- the first digit for the level/yerm, and the other two digits for the score at a particular yerm level. I wrote this old post before I became familiar with IXL software, but we can plainly see that the IXL scoring system fits well here. A score of 100 indicates mastery -- that the student is ready for the next yerm. The normative time to complete a yerm is three semesters, so gifted students might need only two semesters and special ed students might need four.

Even though yerms aren't supposed to correspond to grade levels, we might at least wish to divide the yerms into elementary, middle, and high school spans. Two years ago I used Levels 3-5 for middle school, but using today's yerms it might be better to use Yerms 4-6 for middle school. Even now, Yerm 4 starts with fifth grade standards, which we might still wish to include with middle school.

I wrote that old post before my job at the charter school began. It's interesting to think about what my time would have looked like if our school used yerms instead of years. And yes, I know what you're thinking -- I didn't even finish the whole

*year*at my school, so how could I have lasted a*yerm*? But perhaps my time there would have been better if the schools used yerms instead of years.
The first night of the yerm was September 2nd, 2016 -- not that long, of course, after the start of the real school year. Of course, if as I suggested Month 1 of a yerm is a vacation month, then the school year would actually begin on October 2nd. (Rosh Hashanah is a new moon holiday that fell close to that date last year.)

Since our charter school is so small, it seems doubtful that our school would have allowed students to start the yerm at different semesters. (That's one problem with any plan like this -- it presumes that the school is large enough to accommodate different tracks.)

Now let's assume that middle school consists only of Yerms 5-6 (with Yerms 0-4 at the elementary school on our campus and Yerms 7-9 at a high school somewhere else). Then with only two cohorts, this would have been similar to the situation at our sister campus -- two cohorts and two teachers, with one assigned to teach English/history and the other (me) to teach math and science, with time explicitly allotted on the schedule for both subjects. Then I obviously would have taught science better than I did on the original calendar, since there is time reserved for science on the schedule, plus only four preps to teach instead of six (counting science).

Of course, the class sizes would have been larger as three cohorts are condensed to two. Perhaps the class sizes would have been large enough to justify hiring a third teacher and then having the third cohort be, say, Yerm 5 starting from the second or third semester. Three cohorts (say Yerm 5 Semester 1, Yerm 5 Semester 3, and Yerm 6 Semester 2) probably wouldn't have looked that much different from the situation on the original calendar.

If we assume that Yerm 4 is also middle school, then the fifth grade class (and fifth grade teacher) are suddenly included in the middle school. That class is large enough that most likely there would have been four cohorts in middle school (Yerms 4-6 starting at different semesters), with four teachers. So the fourth teacher could have been science, and then I wouldn't have had to teach science myself!

It's difficult to tell whether I would have fared better under the Yerm Calendar or not. (For example, if one of the cohorts is in Semester 3 and is ready to take a computerized test which they must pass to get to the next yerm, I'd need better classroom management to make sure that they're ready for the test, or else everyone might fail it!)

But anyway, some readers might like this idea so much that they might wish to implement yerms at the school level, even if the rest of society keeps the Gregorian Calendar. If we assume that elementary (excluding kindergarten) is Yerms 1-3, middle school Yerms 4-6, and high school Yerms 7-9, then each span would have only eight semesters, instead of nine on the pure Yerm Calendar. So this would have to be considered when setting up standards and tests, since we want the average student to progress three yerms every eight semesters. (Or perhaps trimesters work better here, since each span would now be twelve trimesters, or four trimesters per yerm.)

One idea I favor is -- since there are 15 or 17 months per yerm rather than 12 -- we take advantage of the extra months by placing holidays from different cultures on the calendar. Some holidays are already associated with the new moon, such as Chinese New Year, Hanukkah,and Diwali (India), so these readily fit on the long weekends anyway.

But many popular holidays -- the aforementioned Tu b'Shevat, Lantern Festival, and Easter -- are associated with full moons, not new moons. Yet the calendar is set up only for new moon holidays.

Earlier in this post, I quoted Palmen:

I've also considered making the second lunar week have 8 days, but haven't decided how to name the 'nights'.

This results in eliminating four-day weekends, but replaces them with more three-day weekends -- including full moon weekends. This allows us to place holidays at the full moons. Under this plan every month has a long full moon weekend, but only 30-day months (the odd months) would have long new moon weekends. Keep in mind that yerms have nothing to do with seasons, and so it should be

We might assume that we are converting to the yerm calendar in 2018, and so we choose months that match the 2018 calendar. (After 2018, of course, the holidays won't match up any more.)

So let's dive in. As a first attempt, we place Tu b'Shevat at the first full moon, and then Chinese New Year at the first new moon, since this matches 2018:

Tu b'Shevat (01(15

Lunar New Year (02(01

Lantern Festival (02(15

But this is awkward -- a "New Year" festival at the second month of a yerm. Let's break from 2018 and place it at the start of the yerm:

Lunar New Yerm (01(01

Lantern Festival (01(15

Of course, you might ask, why is Chinese New Year granted the privilege of being allowed to start the yerm, rather than the Rosh Hashanah or the new year on any other calendar? Easy -- it's because their new year just happened to fall the nearest to the start of the 2018 yerm. And besides -- the current phrase "Lunar New Year" is automatically assumed to refer to the East Asian year.

How about this -- let's balance the calendar by placing a Jewish full moon holiday, Purim, at the next full moon:

Lunar New Yerm (01(01

Lantern Festival (01(15

????? (02(01

Purim (02(15

A new moon holiday is needed at the start of Month 2 (since Month 1 has 30 days), but not Month 3 (since Month 2 has only 29 days). In other words, new moon holidays are needed at the

This year the full moon of Month 3 lines up with Easter, so let's add it in. This means that the new moon at the start of Month 2 should be like Carnival. This holiday usually occurs about a month and a half before Easter (hence near a new moon) and so it makes Lent the correct length:

Lunar New Yerm (01(01

Lantern Festival (01(15

Carnival (02(01

Purim (02(15

**Holidays on the Yerm Calendar****Palmen never defines any holidays on his Yerm Calendar. We presume that the extra days at the end of each month (the 29th and 30th) are used to set up three- and four-day weekends, so holidays can be placed near the ends of months.**

One idea I favor is -- since there are 15 or 17 months per yerm rather than 12 -- we take advantage of the extra months by placing holidays from different cultures on the calendar. Some holidays are already associated with the new moon, such as Chinese New Year, Hanukkah,and Diwali (India), so these readily fit on the long weekends anyway.

But many popular holidays -- the aforementioned Tu b'Shevat, Lantern Festival, and Easter -- are associated with full moons, not new moons. Yet the calendar is set up only for new moon holidays.

Earlier in this post, I quoted Palmen:

I've also considered making the second lunar week have 8 days, but haven't decided how to name the 'nights'.

This results in eliminating four-day weekends, but replaces them with more three-day weekends -- including full moon weekends. This allows us to place holidays at the full moons. Under this plan every month has a long full moon weekend, but only 30-day months (the odd months) would have long new moon weekends. Keep in mind that yerms have nothing to do with seasons, and so it should be

*moon phases*that determine the placement of holidays, not seasons.We might assume that we are converting to the yerm calendar in 2018, and so we choose months that match the 2018 calendar. (After 2018, of course, the holidays won't match up any more.)

So let's dive in. As a first attempt, we place Tu b'Shevat at the first full moon, and then Chinese New Year at the first new moon, since this matches 2018:

Tu b'Shevat (01(15

Lunar New Year (02(01

Lantern Festival (02(15

But this is awkward -- a "New Year" festival at the second month of a yerm. Let's break from 2018 and place it at the start of the yerm:

Lunar New Yerm (01(01

Lantern Festival (01(15

Of course, you might ask, why is Chinese New Year granted the privilege of being allowed to start the yerm, rather than the Rosh Hashanah or the new year on any other calendar? Easy -- it's because their new year just happened to fall the nearest to the start of the 2018 yerm. And besides -- the current phrase "Lunar New Year" is automatically assumed to refer to the East Asian year.

How about this -- let's balance the calendar by placing a Jewish full moon holiday, Purim, at the next full moon:

Lunar New Yerm (01(01

Lantern Festival (01(15

????? (02(01

Purim (02(15

A new moon holiday is needed at the start of Month 2 (since Month 1 has 30 days), but not Month 3 (since Month 2 has only 29 days). In other words, new moon holidays are needed at the

*end*of odd months, hence at the*start*of even months. (And of course, Month 1 starts with a holiday since all yerms end with an odd 30-day month.)This year the full moon of Month 3 lines up with Easter, so let's add it in. This means that the new moon at the start of Month 2 should be like Carnival. This holiday usually occurs about a month and a half before Easter (hence near a new moon) and so it makes Lent the correct length:

Lunar New Yerm (01(01

Lantern Festival (01(15

Carnival (02(01

Purim (02(15

Easter (03(15

I haven't completely decided yet how I'd like to do the rest of the yerm. Since the months of no school are Months 1, 6, and 11, it might better to put Easter on (06(15, so that the month off fits the old Easter vacation. Then again, maybe we should put Christmas (a non-lunar holiday) on (06(15 instead, so that it fits the old pattern of Christmas as the first long break and Easter as the second long break of the year.

At any rate, holidays that are longer than one day might belong in vacation months. I already did this by placing Chinese New Year in Month 1 (as it's considered to be 15 days of continuous celebration lasting all the way to Lantern Festival). Multi-day Jewish holidays such as Hanukkah, Passover, and Sukkot might belong in the vacation months. I'd like to add at list one Muslim holiday -- perhaps the Ramadan fast with its culmination in the Eid al-Fitr feast

Also, we must decide whether holidays such as Passover and Easter should continue to line up. Of course, they correspond for historical reasons (in the Bible, the Last Supper was a Passover seder), but with extra months, we might wish to separate them so we can celebrate them both. We also have tough decisions regarding Christmas, Hanukkah, and Kwanzaa, as well as with Purim and Holi (India) if we wish to include them both.

Months 16 and 17 should be devoid of holidays, unless there are holidays that we wish to celebrate in long yerms and skipped in short yerms.

**The Yerm Calendar and the QNTM List**

By the way, who remembers the infamous QNTM list that seeks to imply that all Calendar Reform is bad and that we should just keep the Gregorian Calendar?

Let's see which boxes are checked by the Yerm Calendar -- a very radical change that would require many boxes to be checked. The most obvious box is:

(x) solar years are real and the calendar year needs to sync with them

since there are no solar years in the Yerm Calendar. As it's a lunar calendar, some of the lunar boxes should be checked, such as:

(x) the lunar month cannot be evenly divided into solar days

On the other hand, I don't check the following box:

( ) having months of different lengths is irritating

since the months of the yerm vary

*less*than the Gregorian months. (The idea, remember, is to show why any proposed calendar is*worse*than the Gregorian calendar.)
I'd check the following box:

(x) your name for the thirteenth month is questionable

To me, this box is for

*any*calendar with 13 or more months, even if the months aren't actually named.
For the repetition of Moonnight and Soonnight in certain weeks, this might be captured by:

( ) every civilisation in the world is settled on a seven-day week

although I prefer this box for calendars that eliminate seven-day weeks completely. Still, the repetition of these two days should be represented somehow. (Maybe the "lunar month can't be evenly divided into seven-day weeks" is better, but I reserve this box for Leap Week Calendars.)

Yerms and their nights begin and end at noon. We have the perfect box for that:

(x) the day of the week shouldn't change in the middle of the solar day

The Yerm Calendar starts in November 1996, so we would have:

(x) nobody is about to renumber every event in history

And of course, all proposed calendars require the following auto-checked box:

(x) the history of calendar reform is insanely complicated and no amount of further calendar reform can make it simpler

Here are a few miscellaneous calendar comments. This year, Thanksgiving fell on Thursday, November 23rd -- which was

*not*the last Thursday of the month of November. Both 2017 and 2018 are "Franksgiving" years -- those in which Thanksgiving is observed according to FDR's rules, not the ones first proposed by Lincoln.
On the other hand, Advent (the Christian countdown to Christmas) began on its latest possible date, Sunday, December 3rd, with the fourth Sunday of Advent not until Christmas Eve. One thing interesting about Lincoln's original Thanksgiving date is that it was always three days before the first Sunday of Advent -- thereby linking the Christian countdown to Christmas with the secular countdown to the holidays. I once read someone suggest that if retailers wanted to start holiday sales earlier, FDR should have just declared

*Black Friday*to be a week earlier (the Friday before Turkey Day) and left Thanksgiving alone. The problem, of course, is that presidents give*Thanksgiving Proclamations*, not Black Friday proclamations.
This year in New York, the last day before winter break was Friday, December 22nd, and the first day back for students is Tuesday, January 2nd. New York schools close for Chinese New Year on February 16th, which extends right into the President's Day week break. Meanwhile, spring break in New York lines up with Passover, which starts on Saturday, March 31st and extends into the entire week after Easter -- but Good Friday, March 30th, is always a day off. Thus all three major breaks (winter, February, spring) are the same length in the Big Apple -- a week plus an extra day.

Here is another link about the upcoming supermoon and blue moon:

https://www.huffingtonpost.com/entry/double-newyear-supermoon_us_5a481a15e4b0b0e5a7a7011c?ncid=inblnkushpmg00000009

Notice that even though tomorrow's full moon is

Therefore January 31st is a supermoon, a blue moon, and a blood moon.

One commenter at the following link threw a wet blanket on the upcoming blue moon:

Dave Ralph:

Here is another link about the upcoming supermoon and blue moon:

https://www.huffingtonpost.com/entry/double-newyear-supermoon_us_5a481a15e4b0b0e5a7a7011c?ncid=inblnkushpmg00000009

Notice that even though tomorrow's full moon is

*the*supermoon, the preceding (December 3rd) and following (January 31st) full moons are often large enough to be considered supermoons as well. In addition, this is an eclipse season (having been about six months since the solar eclipse), and indeed the January 31st full moon is associated with a lunar eclipse that is visible in certain locations -- possibly including California.Therefore January 31st is a supermoon, a blue moon, and a blood moon.

One commenter at the following link threw a wet blanket on the upcoming blue moon:

Dave Ralph:

*There is no such thing as a "blue moon". It is a made-up term aimed to deceive the public. It merely means that there is a full moon on the first calendar day of the month. It has no scientific or observational significance whatsoever.*
Strictly speaking, Ralph is correct -- "blue moons" are an artifact of the Gregorian calendar. On other calendars, January 31st won't be a blue moon:

In class last year, I sang "Triangle Song" in December and "Angle Dance" in January, and I was hoping to code the songs in their corresponding months. But as it turns out, "Triangle Song" is much more challenging to code. I need more time to plan it correctly, and so I'll switch things up and code "Angle Dance" in this post.

Here's a first attempt at the coding:

10 DIM S(9)

15 FOR V=1 TO 3

20 FOR X=0 TO 9

30 READ S(X)

40 NEXT

45 DATA 54

50 DATA 105,96,90,84

60 DATA 81,72,70,63,60

70 N=1

80 FOR X=1 TO 68

90 READ A,T

100 SOUND 261-N*S(A),T

110 NEXT X

120 FOR Y=1 TO V

130 FOR X=1 TO 27

140 READ A,T

150 SOUND 261-N*S(A),T

160 NEXT X,Y

170 RESTORE

180 DATA 7,4,7,2,7,2,7,4,5,2,4,2,5,4,4,8

190 DATA 8,2,8,2,8,4,8,2,8,2,8,4,4,4,4,12

200 DATA 4,4,5,4,5,4,5,4,5,2,5,2,5,4,1,4,1,4

210 DATA 1,2,1,2,4,4,4,4,4,4,5,4,4,2,2,2,1,12

220 DATA 7,4,7,2,7,2,7,4,5,2,4,2,5,4,4,8

230 DATA 8,2,8,2,8,4,8,2,8,2,8,4,4,4,4,12

240 DATA 4,4,5,4,5,4,5,4,5,2,5,2,5,4,1,4,1,4

250 DATA 1,2,1,2,4,4,4,4,4,4,5,4,4,2,2,2,1,12

260 DATA 3,2,3,2,5,12,5,4,5,4,3,8

270 DATA 0,4,3,2,3,2,3,4,5,4,5,4,5,12

280 DATA 3,2,3,2,5,12,5,4,5,4,3,8

290 DATA 9,4,9,2,9,2,9,4,9,4,0,4,6,4,5,8

300 DATA 3,2,3,2,5,12,5,4,5,4,3,8

310 DATA 0,4,3,2,3,2,3,4,5,4,5,4,5,12

320 DATA 3,2,3,2,5,12,5,4,5,4,3,8

330 DATA 9,4,9,2,9,2,9,4,9,4,0,4,6,4,5,8

340 DATA 3,2,3,2,5,12,5,4,5,4,3,8

350 DATA 0,4,3,2,3,2,3,4,5,4,5,4,5,12

360 DATA 3,2,3,2,5,12,5,4,5,4,3,8

370 DATA 9,4,9,2,9,2,9,4,9,4,0,4,6,4,5,8

380 NEXT V

This song fits the New 7-Limit Scale because it switches from major to minor, and my New 7-Limit Scale combines major and minor notes. But the problem is that it switches from G major, not to G minor, but to

To include the Eb, let's lower it to octave N = 2:

45 DATA 67.5

70 N=2

With N = 1, this causes an error since there's no Degree 67.5, but when N = 2 there is no error. Of course, as far as which D to use, greenish D fits with (greenish) G major, while white D fits with (white) C minor.

One thing about this song on

But fortunately, we notice that not only does the high g' not appear in the song (which is why we changed S(0) to eb'), but Larry the baritone never sings f'. Instead, Reg E. the bass sings f. So I can avoid singing any note higher than e' by jumping down to a lower octave -- and it's justified because I'm jumping down to sing Reg E.'s bass part.

Notice line 120 FOR Y=1 TO V. This sets up the refrain to repeat once after the first verse, twice after the second verse, and thrice after the third verse -- just like the actual song on TV. But unfortunately, this forces us to repeat so many DATA lines. Again, since many songs repeat a lot, it would be so much more convenient if we could use Atari's RESTORE command, which would allow us to specify which line of DATA to RESTORE.

Oh, and speaking of which -- no, I didn't find an Atari emulator, so I won't compose 16-bit music any time soon. By the way, I keep calling it "the emulator," but emulators have proper names. The proper name of our emulator is "Mocha":

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

Here's a YouTube clip of the Angle Dance song, so you can compare the version we programmed on Mocha to the original song:

- On the Yerm Calendar and most lunar calendars, there are no blue moons because every month, by definition, has exactly one full moon.
- On the New Reform Calendar, months go 30-30-31, and so the full moon date shifts to February 1st, hence it's no longer a blue moon. March, meanwhile, keeps its blue moon
- On the Symmetry010 Calendar, months go 30-31-30, and so not only does the full moon shift to February 1st, but the
*next*new moon shifts to February 31st. Hence*February*now has a blue moon, while neither January nor March has a blue moon!

On the other hand, supermoons and lunar eclipses still exist regardless of the calendar. Hence Ralph respects supermoons and lunar eclipses more than "blue moons."

**Coding***Square One TV*'s "Angle Dance"**I want to code one song on our computer emulator in each vacation post -- in particular, I'd like to code songs I sang in class. Even though I didn't want to code**

*Square One TV*songs, two songs in particular -- "Triangle Song" and "Angle Dance" -- are well-suited for the New 7-Limit Scale.In class last year, I sang "Triangle Song" in December and "Angle Dance" in January, and I was hoping to code the songs in their corresponding months. But as it turns out, "Triangle Song" is much more challenging to code. I need more time to plan it correctly, and so I'll switch things up and code "Angle Dance" in this post.

Here's a first attempt at the coding:

10 DIM S(9)

15 FOR V=1 TO 3

20 FOR X=0 TO 9

30 READ S(X)

40 NEXT

45 DATA 54

50 DATA 105,96,90,84

60 DATA 81,72,70,63,60

70 N=1

80 FOR X=1 TO 68

90 READ A,T

100 SOUND 261-N*S(A),T

110 NEXT X

120 FOR Y=1 TO V

130 FOR X=1 TO 27

140 READ A,T

150 SOUND 261-N*S(A),T

160 NEXT X,Y

170 RESTORE

180 DATA 7,4,7,2,7,2,7,4,5,2,4,2,5,4,4,8

190 DATA 8,2,8,2,8,4,8,2,8,2,8,4,4,4,4,12

200 DATA 4,4,5,4,5,4,5,4,5,2,5,2,5,4,1,4,1,4

210 DATA 1,2,1,2,4,4,4,4,4,4,5,4,4,2,2,2,1,12

220 DATA 7,4,7,2,7,2,7,4,5,2,4,2,5,4,4,8

230 DATA 8,2,8,2,8,4,8,2,8,2,8,4,4,4,4,12

240 DATA 4,4,5,4,5,4,5,4,5,2,5,2,5,4,1,4,1,4

250 DATA 1,2,1,2,4,4,4,4,4,4,5,4,4,2,2,2,1,12

260 DATA 3,2,3,2,5,12,5,4,5,4,3,8

270 DATA 0,4,3,2,3,2,3,4,5,4,5,4,5,12

280 DATA 3,2,3,2,5,12,5,4,5,4,3,8

290 DATA 9,4,9,2,9,2,9,4,9,4,0,4,6,4,5,8

300 DATA 3,2,3,2,5,12,5,4,5,4,3,8

310 DATA 0,4,3,2,3,2,3,4,5,4,5,4,5,12

320 DATA 3,2,3,2,5,12,5,4,5,4,3,8

330 DATA 9,4,9,2,9,2,9,4,9,4,0,4,6,4,5,8

340 DATA 3,2,3,2,5,12,5,4,5,4,3,8

350 DATA 0,4,3,2,3,2,3,4,5,4,5,4,5,12

360 DATA 3,2,3,2,5,12,5,4,5,4,3,8

370 DATA 9,4,9,2,9,2,9,4,9,4,0,4,6,4,5,8

380 NEXT V

This song fits the New 7-Limit Scale because it switches from major to minor, and my New 7-Limit Scale combines major and minor notes. But the problem is that it switches from G major, not to G minor, but to

*C minor*(in the refrain). This C minor part does use the Bb in our scale, but it also uses the Eb that's available in the Extended Scale. The Eb is only in the lower octave as Sound 126 (Degree 135) -- as its degree is odd, it can't be halved to raise it an octave.To include the Eb, let's lower it to octave N = 2:

45 DATA 67.5

70 N=2

With N = 1, this causes an error since there's no Degree 67.5, but when N = 2 there is no error. Of course, as far as which D to use, greenish D fits with (greenish) G major, while white D fits with (white) C minor.

One thing about this song on

*Square One TV*is that its lead singer (Larry) is a baritone, while its backup singer (Reg E.) is a bass. I've mentioned before that while the key of G is convenient for the computer, it's not convenient for my singing range (G-g is a bit low, while g-g' is a bit high).But fortunately, we notice that not only does the high g' not appear in the song (which is why we changed S(0) to eb'), but Larry the baritone never sings f'. Instead, Reg E. the bass sings f. So I can avoid singing any note higher than e' by jumping down to a lower octave -- and it's justified because I'm jumping down to sing Reg E.'s bass part.

Notice line 120 FOR Y=1 TO V. This sets up the refrain to repeat once after the first verse, twice after the second verse, and thrice after the third verse -- just like the actual song on TV. But unfortunately, this forces us to repeat so many DATA lines. Again, since many songs repeat a lot, it would be so much more convenient if we could use Atari's RESTORE command, which would allow us to specify which line of DATA to RESTORE.

Oh, and speaking of which -- no, I didn't find an Atari emulator, so I won't compose 16-bit music any time soon. By the way, I keep calling it "the emulator," but emulators have proper names. The proper name of our emulator is "Mocha":

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

Here's a YouTube clip of the Angle Dance song, so you can compare the version we programmed on Mocha to the original song:

**Conclusion: New Year's (New Yerm's?) Resolutions**

Since it's New Year's Eve, let me declare my New Year's Resolutions. I will list them today and then explain the rationale for them later this week, in my final vacation post.

1. Implement a classroom management system based on how students actually think.

2. Keep a calm voice instead of yelling at students.

3. Move on from past incidents instead of bringing them up with students.

4. Begin the lesson quickly instead of having lengthy warm-ups

5. Engage the students in the learning process instead of lecturing excessively.

6. If there is a project-based curriculum such as Illinois State, then implement all components of it.

7. If there is an official assignment to review for state testing, then implement it fully.

Of course, I can't keep any of these resolutions unless I'm in a classroom -- and I don't know whether I'll be hired to teach in 2018. Therefore, I declare these to be

Yes, 2017 was a very tough year for me -- perhaps the toughest year of my life. If I am to have a better 2018 in the classroom, I must keep the above resolutions. And I

Of course, I can't keep any of these resolutions unless I'm in a classroom -- and I don't know whether I'll be hired to teach in 2018. Therefore, I declare these to be

*New Yerm's Resolutions*so that I'll have until the end of the yerm -- June 3rd, 2019 -- to keep them. In other words, I have the rest of this school year and all of the next.Yes, 2017 was a very tough year for me -- perhaps the toughest year of my life. If I am to have a better 2018 in the classroom, I must keep the above resolutions. And I

*will*keep them -- but only if I am given the chance.
Happy New Year -- or should I say, Happy New Yerm!