Tuesday, May 8, 2018

Chapter 15 Review (Day 160)

Today is the second day of subbing in an eighth grade Algebra I class. It marks my first multi-day assignment in a math class since starting in the new district.

Here is the focus resolution for today:

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

That's because even though this teacher doesn't give warm-ups, the first thing I had to do today is go over last night's homework. This is another task that I don't want to take too much of the class time -- we don't want to take too much going over the previous lesson that we can't start the new lesson.

7:15 -- Zero period is the first Algebra I class. The new material is Lesson 10-2 of the Big Ideas text, on solving square root equations after graphing the radical functions yesterday. In this class, I believe that I took fifteen minutes going over last night's homework -- which might be too long. I resolve to shorten the length of the this opening task in subsequent classes. I consider this class to be the best behaved class of the day.

8:15 -- Zero period leaves and first period/homeroom begins. The length of the video for morning announcements today is only about half as long as the length of homeroom, and so I take advantage of the extra time I can spend on homework without taking away from the main lesson. In other words, I'm able to cover more examples without any actual improvement at shortening the time devoted to homework.

9:20 -- First period leaves and third period arrives. Since today is Tuesday, the official rotation for the day is (0-)1-3-4-5-6-2. This is not a math class, but ASB Leadership. The students create posters -- in addition to finishing up yesterday's posters, they are to design a new set of posters. These are to advertise the upcoming eighth grade trip to Boomers (a family entertainment center -- it has ten locations, including five here in Southern California). It's the second best behaved class of the day.

10:15 -- Third period leaves and it's time for snack, followed by fourth period conference. I go to the teachers lounge, where bagels are being served to celebrate National Teacher's Day.

Yes, I know that there was a Google Doodle for National Teacher's Day yesterday. But I made no mention of it because the actual Teacher's Day is today. The official definition of Teacher's Week is that it's the first full week in May (counting Sunday as the first day of the week), and Teacher's Day is the Tuesday of that week. Thus an equivalent definition of Teacher's Day is the Tuesday before Mother's Day (the second Sunday in May), and Teacher's Week ends as Mother's Day begins. That Teacher's Week leads into Mother's Day is logical, since mothers are often our first teachers.

It's possible that my old school celebrated Teacher's Week last year -- but I can't be sure, since I left before the calendar hit May. In the past few years, there's also been a National Charter Schools Week which appears to coincide with Teacher's Week. Since my school last year was a charter, it's possible that Charter Schools Week was emphasized last year rather than Teacher's Week.

11:25 -- Fifth period arrives. After yesterday's class was on the talkative side -- and in fact, today first period kids talk much more today than yesterday -- I put my foot down on classroom management. I start a list of misbehaving students on the board and continue to stop the lesson and grab a brown marker, threatening to write down names. (I don't know why I chose brown for my bad list -- maybe the song "Bad Bad Leroy Brown" is stuck in my head.) It works, as today's class is much quieter than yesterday's, and I don't need to write any names.

In fact, this is my best-paced class. I end up spending the first ten minutes of class on homework, five minutes on the first examples (the easier square root equations), and then about ten minutes each on the second through fourth examples (the last of which has extraneous roots). The students spend the remaining time starting the homework, while I do the fifth example (a word problem) on the board.

12:20 -- Fifth period ends and it's time for lunch.

1:05 -- Sixth period arrives. This is a Math Support class for students who need extra help. Just like yesterday, students work on ALEKS and Prodigy on Chromebooks. Unfortunately, the class doesn't go as smoothly as yesterday -- I must write the names of two boys who try to visit other websites (such as YouTube) rather than ALEKS or Prodigy.

2:00 -- Sixth period leaves and second period arrives. This is the last Algebra I class of the day. After today's first and sixth period mishaps, I treat this class like fifth period -- I threaten to write down names of talkative students.

2:55 -- Second period ends, but this teacher still has ten extra minutes of after-school yard duty.

3:05 -- Yard duty ends, and thus ends a long day of subbing.

There's one more management issue that I'd like to discuss here. Late yesterday, I notice that the teacher has used blue masking tape to number all of the desks (which are in rows of three). This allows me to use a random-number generator to call on the students -- just as I programmed my TI to select random students at random in my old charter school last year.

Apparently, some students object to this -- either smart students who volunteer to be called, or shy students who hope not to be called. And so they respond by removing the blue taped numbers. As of the conference period, only one band of tape has been removed, but by the end of the day, most of the numbers are gone. This means that either fifth or second period removes most of the tape (unless it's sixth period -- which would be strange as I don't use the numbers to call anyone in sixth). Notice that fifth and second period are both quiet today -- but they could be quietly removing the tape.

Overall, I'm glad that I'm able to work on some time management issues today. But there are still some classroom management issues that I'd like to fix before my summer school class begins.

Speaking of summer school, this will be my first music post as I prepare to compose new songs to perform during music break. Here's my plan -- each week, I play a new scale. I'll devote three posts a week to music -- on Tuesdays I'll introduce the scale, on Wednesdays I'll discuss properties of the new scale, and on Thursdays I'll attempt to compose music in the new scale.

Along the way, I'll (re-)define new terms associated with this new type of music. Today's post won't merely introduce the new scale -- I'll also (re-)introduce this type of computer music in general. I'll write this post in question-and-answer format.

What is Mocha?

Mocha is a computer emulator. It emulates the old computers that I used as a young child. Here is a link to Mocha:


What is the SOUND command in Mocha?

The SOUND command takes two parameters. These parameters range from 1 to 255 and are separated by a comma. An example of a typical SOUND command is:

SOUND 169,33

The second parameter represents the length of the note -- larger values are longer notes. For simplicity, we can take Length 1 to be a sixteenth note, so Length 2 is an eighth note, Length 3 is a dotted eighth note, Length 4 is a quarter note, and so on. Thus in the example above, Length 33 is (slightly more than) twice as long as a whole note (a double whole note or breve).

The first parameter represents the pitch of the note -- larger values are higher notes. But there isn't a simple relationship between Sound values and notes on, say, a piano. So Sound 169 in the example above is slightly higher than A440, the basis of the modern tuning system, but it's difficult to say exactly how much higher this note is,

By the way, I notice that on Google Chrome, where I'm typing this entry (since Blogger is also a Google product), the above sound link doesn't work. It does work on Microsoft Edge. I suspect that Chrome recently had an update that makes it impossible to listen to Mocha. The same thing happened at the old Bizzie Lizzie (Sailor Pi) site -- old music no longer played in new browsers. I fear that soon Edge will update as well, and then it will become impossible to hear Mocha at all.

What exactly is the relationship between Sounds and notes on a piano?

Back when I was young, I didn't pay attention to how Sounds correspond to notes. I knew that it took several Sound steps to add up to a semitone, but I never took the time to explore. A new version of my old computer had another command, PLAY, where the commands actually follow musical notation (where pitches are specified from A-G and lengths are given as whole, half, quarter). So if I wanted to play music, I just used PLAY and forgot all about SOUND.

It was in reading Pappas and other sources where I learned about microtonal music. It's not as if the note A, when gradually raised, suddenly becomes A# (A-sharp), with a discrete boundary between the notes A and A#. Instead, there are new notes, microtones, between the 12 notes we know. So I suddenly realized that for years, I had an instrument capable of playing microtonal music -- the computer I used many years ago.

And so when I found Mocha, I just had to use the SOUND command to figure out exactly what microtonal scale it played. When I started with Sound 1, the first note I heard was somewhere between E and Eb (E-flat), in the octave just below middle C. When I was approximately halfway to Sound 255, I had ascended an octave from my starting point -- and when I was about 3/4 of the way, I was two octaves above the starting point. This indicates that Mocha music is based on EDL.

What exactly is EDL?

"EDL" means "Equal Divisions of Length." It means that we take a string (such as a guitar string) and divide it into equal parts. It was the ancient mathematician Pythagoras who discovered that music is based on ratios of lengths of string. Cutting a string in half (or fretting/vibrating it at its midpoint) raises the pitch of the string by an octave, and cutting it in half again raises it another octave.

What is the difference between EDL and EDO?

"EDO" means "Equal Divisions of the Octave." The most commonly used musical scale is based on 12EDO, or twelve equal divisions of the octave.

The frets of a guitar are based on 12EDO. This means that the first fret shortens the vibrating string by a factor of 2^(1/12) (the twelfth root of two), the second fret by 2^(2/12), the third by 2^(3/12), the fourth by 2^(4/12), and so on. The twelfth fret shortens the vibrating string by a factor of 2^(12/12) or 2 -- that is, it halves the length.

But in EDL, the vibrating string is a rational fraction of the total length. If a guitar string were fretted to 12EDL rather than 12EDO, then the first fret would leave 11/12 of the string vibrating, the second fret would leave 5/6 of the the string, the third fret 3/4, the fourth fret 2/3, and so on. These rational lengths (just intonation) mean that more consonant (good-sounding) intervals are available in an EDL system than in an EDO system.

Exploring 2EDL

The simplest possible EDL scale is 2EDL -- the simple division of a string in half. It is the exact same trivial scale as 1EDO -- all we have is the octave.

We notice that the longer strings produce lower notes -- but with the SOUND command, increasing the first parameter produces higher notes. Therefore the Sound value doesn't represent the amount of string that is vibrating. Instead, it's the length of string that we cut away before letting the rest of the string vibrating. On the blog, I use the word "Degree" to indicate the length of vibrating string. For any note, its Sound value and Degree value add up to some constant to represents the total length of the string. Two notes are an octave apart if their Degrees (not their Sounds) are in the ratio 2:1.

It's tempting to assume that the total length of the string is 256. This would mean that the two highest playable notes, Sounds 255 and 254, are Degrees 1 and 2. But then these two notes should be exactly an octave apart since 2/1 = 2 -- and they clearly aren't (though it may be difficult to discern an octave between two very high notes). It's more obvious that 256 isn't the length of the string if we continue to play Sounds 252 (= 256-4), 248 (= 256-8), 240 (= 256-16), 224 (= 256-32), and so on -- these notes clearly aren't an octave apart.

Instead, the length of the string is 261. We prove this by running the following program in Mocha:

10 FOR X=3 TO 8
20 SOUND 261-2^X,4

Remember to press the up-arrow key for ^ (exponentiation) and to click the Sound box on the left side of the screen. You should now hear descending octaves. These notes correspond to E on a piano, and so I define these six notes to be E, the note on which Mocha music is based.

If we replace 261 with any other value -- including 260 or 262 -- the intervals no longer sound like true octaves. Even though the lowest notes sound like E's, the first note sounds like D if we begin with 260 and F# if we begin with 262. This tells us that the length of the string is 261. In some posts, I refer to this as Bridge 261. The bridge of a guitar is where the string ends, so Bridge 261 means that we're at Degree 0, the end of the string. The Sound and Degree of any note always add up to 261.

The lowest note is Sound 1 (Degree 260), and the highest note is Sound 255 (Degree 6). Since the lowest note is Degree 260, we could refer to the entire Mocha sound system as 260EDL.

Within this 260EDL system are many pairs of notes that are an octave apart. We can play these pairs by writing the following program:

20 FOR D=1 TO 2
30 SOUND 261-N*D,4

Here N is the Degree of the higher note. The smallest permissible value of N is 6 (since the highest playable note is Degree 6) and the largest permissible value is 130 (since the lower playable note is Degree 260, which is 2 * 130).

Exploring 4EDL, 6EDL, and 8EDL

We only consider EDL's that are even. This is because the octave is 2/1 -- odd EDL's won't have the octave available as one of the notes in the scale.

It's easy to modify the program to play the next three possible EDL's -- 4EDL, 6EDL, 8EDL. We only need to change the following line:

20 FOR D=2 TO 4

20 FOR D=3 TO 6

20 FOR D=4 TO 8

In each case, we start at the octave and descend to the base note. This is because as the Degree D increases, the pitch lowers. If you prefer ascending scales, we reverse the order of the notes:

20 FOR D=8 TO 4 STEP -1

Of course, the permissible values of N must change so that N * D is always between 6 and 260. We may write out all the possible scales:

The 4EDL scale:
Degree     Ratio     Interval
4              1/1         tonic
3              4/3         perfect fourth
2              2/1         octave

The 6EDL scale:
Degree     Ratio     Interval
6               1/1        tonic
5               6/5        minor third
4               3/2        perfect fifth
3               2/1        octave

The 8EDL scale:
Degree     Ratio     Interval
8               1/1        tonic
7               8/7        septimal whole tone
6               4/3        perfect fourth
5               8/5        minor sixth
4               2/1        octave

All of these scales are too simple to compose songs in. We see that 4EDL has only two notes -- the root and the perfect fourth. The three notes of 6EDL form a triad -- the minor triad -- but we need more than that to make a song. And 8EDL, with its four notes, reminds us of the Google Fischinger player from last summer. While the root and whole tone are two of the four Fischinger notes, the perfect fourth and minor sixth are not. Meanwhile, if we leave out the root and play the four notes starting with the "septimal whole tone" as a base, the resulting chord sounds half-diminished.

Notice that EDL's have half as many notes within a given octave as EDO's. Therefore 4EDL, 6EDL, and 8EDL have as many notes as 2EDO, 3EDO, and 4EDO. These three EDO's are all considered to be subscales of 12EDO -- and the three EDL's can also be estimated in 12EDO as well. But let's compare the three EDL scales to the corresponding EDO scales, all with C as the root note:

4EDL: C, F, C
2EDO: C, F#, C

6EDL: C, Eb, G, C
3EDO: C, E, G#, C

8EDL: C, D, F, Ab, C
4EDO: C, D#, F#, A, C

In each case, we notice that except for the root and octave C notes, all of the EDL notes appear to be about a semitone lower than the corresponding EDO notes. We notice that this phenomenon occurs in all EDL scales. If we compare 260EDL (the entire Mocha scale) to 130EDO (the EDO with the same number of notes), the notes near the beginning and end of the scale are nearly the same in both scales, but near the middle of the scales, the notes are about a semitone apart. Microtonal intervals are often measured in "cents," where 100 cents equals a semitone. So we can say that the notes near the middle of an EDL scale are approximately 100 cents lower than the notes in the corresponding EDO scale.

In fact, all the notes in an EDO scale are the same number of cents apart. So in 130EDO, the notes are all about 9.23 cents apart. In an EDL, the lower notes are closer in cents than the higher notes, so Sounds 1 and 2 (Degrees 260 and 259) are 6.67 cents apart, while Sounds 130 and 131 (Degrees 131 and 130) are about double that, 13.27 cents apart. In fact, the difference in cents continues to rise throughout the entire Mocha range. The highest two notes, Sounds 254 and 255 (Degrees 7 and 6) differ by the interval 7/6, the septimal minor third of 267 cents. The highest notes in Mocha differ by whole tones while the lowest notes differ by a few cents, the just noticeable difference.

Notice that even with these simple EDL's and EDO's, the EDL's are more consonant than the corresponding EDO scales. In 2EDO we have the tritone -- the devil in music -- while in 4EDL we have a perfect fourth. In 3EDO we have an augmented triad, which as I've mentioned before is so dissonant that train whistles play the augmented triad as a warning. In 6EDL we replace this with the more consonant minor triad.

Here's another key difference between EDL's and EDO's -- we know that one EDO contains another EDO if the former contains the latter as a factor. So 12EDO contains 2EDO, 3EDO, 4EDO as subsets, while 130EDO contains 5EDO, 10EDO, 13EDO, 26EDO, and 65EDO.

On the other hand, 260EDL contains every EDL that's less than 260EDL -- not just the EDL's that are factors of 260. In fact, we just played 6EDL and 8EDL on our 260EDL computer, even though neither six nor eight divides 260. Even 6EDL contains 4EDL as a subset (a 6EDL scale beginning on C contains 4EDL beginning on G), just as 8EDL (starting on C) contains 6EDL (starting on F).

Then again, factors of an EDL have the property that the smaller scale can be built on the base. So scales like 10EDL, 20EDL, 26EDL, and 52EDL can all be built on Sound 1 (Degree 260).

Exploring 10EDL

Our next EDL is 10EDL, which contains five notes. We know how popular pentatonic scales are -- and indeed, we often tell beginning pianists that they can play something resembling music merely by sticking to the black keys, since they form a pentatonic scale. So let's look at 10EDL:

The 10EDL scale:
Degree     Ratio     Interval
10             1/1        tonic
9               10/9      minor tone
8               5/4        major third
7               10/7      large septimal tritone
6               5/3        major sixth
5               2/1        octave

Notice that the so-called "minor tone" is actually a major second -- and so the major second, third, and sixth all fit the usual pentatonic scale. But the huge problem is the fifth -- we expect the pentatonic scale to contain a perfect fifth. Instead, 10EDL has a tritone. So we want our pentatonic scale to sound like C-D-E-G-A-C, but instead 10EDL gives us C-D-E-F#-A-C.

In a way, it's understandable that 10EDL is C-D-E-F#-A-C. We know that the step sizes, in cents, of an EDL scale increase as we ascend the scale. Our usual pentatonic scale contains three whole tones and two minor thirds. Thus the three whole tones are bunched together at the beginning of the scale, and the two minor thirds are at the end of the scale.

And so I don't really want to compose music in 10EDL either. That F# will always sound out of place because our ears are expecting a G there.

20 FOR D=5 TO 10

Exploring 12EDL

Our next EDL is 12EDL, which contains six notes. Here is this scale in more detail:

The 12EDL scale:
Degree     Ratio     Interval
12            1/1         tonic
11             12/11    small undecimal neutral second
10            6/5         minor third
9              4/3         perfect fourth
8              3/2         perfect fifth
7              12/7       supermajor sixth
6              2/1         octave

Since 12EDL contains 6EDL as a factor, we can play the minor triad of 6EDL on the root note. And it also contains three additional steps between the notes of the minor triad, in order to provide melody as well as harmony. It's the first EDL that contains both a perfect fourth and a perfect fifth.

For these reasons, I believe that 12EDL is the simplest EDL in which we can compose music. I'll devote the next two music posts to the 12EDL scale.

20 FOR D=6 TO 12

And notice that this is the first scale in which we can choose N = 1. And so 12EDL is, in a way, the simplest possible playable scale in Mocha, since we can play it using Degrees 6-12. Using Kite's color notation, we can notate this simple scale:

The 12EDL scale:
Degree     Ratio     Note
12            1/1         white A
11             12/11    amber B
10            6/5         green C
9              4/3         white D
8              3/2         white E
7              12/7       red F#
6              2/1         white A

This is what I wrote last year about today's lesson:

Notice that Chapter 15 contains nine sections, so normally we'd have to squeeze in the Chapter 15 test today (Day 160) so we could begin Lesson 16-1 tomorrow (which is Day 161). But notice that Chapter 15 is the last of the U of Chicago text. Since there is no Chapter 16, we can take our time and review today for the test tomorrow.

Last year I wrote a "Chapter 15 Test." But last year I didn't cover Chapter 15 as a whole -- instead it was a general "circles and spheres" test with questions from five different chapters. This year we covered Chapter 15 straight through, so I could post a genuine Chapter 15 test today.

I decided to take last year's test and create a new first page, since all of the Chapter 15 questions are included on the second page. My new first page includes two questions each from Lessons 15-3 and 15-4 and one each from 15-1, 15-5, 15-6, and 15-7.

Last year's Page 2 contains some questions from Lessons 15-3 and 15-8. But there are also some questions from different sections.

Questions 9 through 11 are on the equation of a circle, Questions 14 and 16 are on the volume/surface area of a sphere, Question 15 is on the Pythagorean Theorem, and Question 17 is on the relationship between radius and tangent.

The new version of the test still contains these questions from four other chapters, but now most of the questions are from Chapter 15.

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