In October I didn't do "A Day in the Life" for this class, but this time there are a few interesting things about the schedule to point out, so I will do "A Day in the Life" today. But of course I'll write about the music today. I mentioned this week how a few music posts made it into my top blog entries from the past twelve months, and so I have no problem with having another music post today.
8:00 -- Students arrive early to drop off instruments so they won't have to carry them around.
8:15 -- The day begins with homeroom, as usual. This is the same school as yesterday, and since the period rotation started with third yesterday, it starts with fourth period today.
8:25 -- ... at least, it would have started with fourth period had it not been for the SBAC. I mentioned how at a different middle school where I subbed, the state science test was on Tuesday. Today's school doesn't start the SBAC until next week -- and here the science test is last, not first. Both schools are in fact giving the English and math tests next week. (It remains, of course, that the SBAC Prep posts on the blog are timed perfectly with the middle school SBAC, not the high school exam.)
With today being the last day of school before the SBAC, a practice test is given. This school refers to this as a "dry run" of the state test. Students remain in their homerooms and spend twenty minutes each on the English and math SBAC Practice Exam. But specialist teachers like music (who have a homeroom yet no Chromebooks in that room) go to another room instead. Another regular teacher is there to make sure that the students get the correct online codes for the practice test.
By the way, of course I'll never post questions from the actual SBAC on the blog. But since this is a practice test, I see no harm in mentioning topic the first passage the seventh graders must read and answer questions on -- it's the story of Archimedes and "Eureka!"
9:10 -- The dry run ends and fourth period arrives. This is the first music appreciation class. The kids are assigned a video to watch, "Getting Started as a DJ: Mixing, Mashups, and Digital Turntables." I tell them to watch their own DJ video on an iPad and then answer a few questions.
9:55 -- Fourth period leaves for snack.
10:05 -- Fifth period arrives. This is a band class. The students must practice the song "Superheroes R Us," a medley of the Superman, Bond, and Batman themes. All classes that perform today must also record the class performance and email the video to the regular teacher.
10:50 -- Fifth period leaves and sixth period arrives. This is the second music appreciation class. The students here must take four lines of notes on two videos. The first is "The Transformative Power of Classical Music" and the second is the same video that fourth period watches.
11:35 -- Sixth period leaves. First period is the regular teacher's conference period, which leads directly into lunch. I begin by meeting yesterday's English teacher and telling her about the one student who was switching seats. She'll inform the assistant principal of the incident.
Like yesterday and everyday during Teacher Appreciation Week, food is served in the lounge. This time, the special food for us is cake -- five different kinds (including cupcakes).
1:05 -- Second period arrives. This is a full orchestra class. There is no silent reading after lunch due to the SBAC dry run schedule. Instead, the students are to perform -- and record -- the song "Bohemian Rhapsody."
1:55 -- Second period leaves and third period arrives -- or returns, since third period is always the same as homeroom at this school. I begin the class with a special song from Square One TV:
Did you get it? It's "Archimedes" -- the topic of the SBAC practice question! Skip to 1:07 where the singer retells the story of "Eureka!" and the king's crown.
2:00 -- The SBAC has altered the afternoon schedule as well. Now there's an assembly to encourage the students to excel on the standardized test. The theme of the assembly is "Crush the SBAC."
2:20 -- The assembly ends, and the seventh graders return to the class for strings. The students are to perform -- and record -- the songs "Gauntlet" and "El Toro."
2:55 -- Third period leaves -- but now I must go to afternoon yard duty.
3:05 -- My duty ends just in time to let three girls pick up their instruments after school.
3:10 -- My day finally ends.
Let me discuss some of the classroom management issues. If you refer to my October 22nd post, you'll notice that the tough classes are the music appreciation classes. Fourth period talks through the video and throws pencils at the ceiling. Many students are surprised when I try to collect their papers since they never hear me say that there's an assignment. Then they try to write something at random instead of watching the DJ video on the iPad.
Once again, I attempt to correct this during sixth period. I try to get the kids quiet at the start of class so that they're aware that there's an assignment to do during the videos. It works in that I'm able to cut the number of non-participating students in half from fourth period.
In October, I wrote that the percussion section was a problem. This time, the teacher allows the percussionists to use actual drums today. Someone accuses the girl of using a cell phone in class again -- another problem named in my October post. But I never catch her in the act.
Today's dry run reminds me of SBAC Prep time two years ago at the old charter school. At this school the dry run is just before the test. But at the charter, the administration wanted us to give several "dry runs" starting in February. In fact, this was a big issue, as I had trouble figuring out how to get the SBAC portal to work.
And of course, my problem was that the charter students simply didn't listen to me. Today's testing students were quiet -- but then again, it was the seventh grade strings class (and for some reason, strings players tend to be better behaved). In fact, I heard that one student in another class has to be kicked out from talking during the dry run. He's actually in the sixth period music class (and his absence is another reason why sixth period is so much quieter today).
Today, two students have trouble with their computers during the dry run. Some of my students had computer trouble two years ago as well -- but then I yelled at them. This was mainly because I was losing control of the situation. If I'd had better control -- where I could tell the students to do something and be reasonably sure that they'd do it -- then I wouldn't have needed to yell. But this was a management problem that started many months before SBAC Prep, much less the SBAC itself.
By the way, the "dry run" at the old charter school is the topic of the seventh resolution, which I rarely discuss, but is relevant today:
7. If there is an official assignment to review for state testing, then implement it fully.
OK, let's get to the music itself. I wish to use today's post to write about several musical topics.
We start with the music I played in class today. I already mentioned "Archimedes," but if you recall from October, I played some music directly from Mocha. And so I do the same again -- I create one song for the morning when the students drop off their instruments and another for the afternoon when they pick them up.
Which Mocha programs/scales did I use to create these two songs? Well, one of them is 18EDL, which is the scale I used back in October. That day, the song sounded Halloween-like to match that time of year. Today I want a happier song for the holiday coming up on Sunday. Of course, I'm talking about the birthday of Achilles -- actually, I'm not, since the Greek hero never did prove that May 12th was his birthday, didn't me. Actually, I'm talking about Mother's Day.
As you might recall, minor intervals sound sad and major intervals sound happy. The program I used in October emphasized Degree 17, since 18/17 is a sort of minor second. It's generally the quality of the third, not the second, that makes a scale minor. Yet Ben Zander (in the classical music YouTube video above) mentions the minor second when describing a Chopin piece (the composer followed the note B with C, since the C makes the B sound sadder).
Today, I changed the code from Degree 17 to 14. The interval 18/14 = 9/7 is a sort of major third -- the supermajor third. That's one thing about 18EDL -- it's convenient for both minor and major.
For the other song, I chose a different scale. Since today is SBAC Practice Test Questions 7-8, I went back to last year's post and noticed that I wrote about 14EDL that day. It was the second day that I blogged about 14EDL -- the first was the previous day (my most popular post of the past 12 months).
Let me reblog what I wrote about 14EDL in last year's (Questions 7-8) post, except that I'm replacing the old Kite colors with his new notation:
Today's task is to compose music in 14EDL. I'm seeking out music that is easy to play on both Mocha and the guitar.
The 14EDL scale:
Degree Ratio Note
14 1/1 red F#
13 14/13 thu G
12 7/6 white A
11 14/11 lavender B
10 7/5 green C
9 14/9 white D
8 7/4 white E
7 2/1 red F#
For 12EDL, we started with the main triad, which happens to be an A minor triad. The corresponding triad in 14EDL uses Degrees 14, 12, 10. This reduces to 7:6:5 and plays as an F# diminished triad.
But how would we finger F#dim on the guitar? The rule of thumb is that in order to have an open chord, at least two notes of the chord need to be open strings (E, A, D, G, B, E). The F#dim chord contains notes F#, A, C, and only C is open.
Well, the common trick is to add a seventh to this chord. This is called a half-diminished chord -- guitarists are more likely to write this chord as F#m7b5, "F# minor 7th with a flat 5." That this chord would appear makes sense, as the 14EDL scale is similar to the Locrian mode, and the Locrian mode is strongly associated with the m7b5 chord.
OK, so let's try to finger the F#m7b5 chord. The bottom string must play the root note, which is F# at the second fret. The next string can be open A. Then the next three strings are fretted as E, A, C, and then the top string is open E. Notice that this is simply an A minor chord played over an F# bass, and so the chord can also be written Am/F#. It's easiest to use the middle finger on the bottom string and then adjust the other fingers from Am accordingly.
This fingering is still awkward, so we might wonder whether a bar F#m7b5 is available here. The following link shows several chord shapes, but the one that feels the least awkward on my hand is the sixth one, where the bar is on the second fret and the other fingers are C (fifth string), F# (fourth string), and E (second string).
But now we want to play other chords. Starting from an F#m7b5 chord, the easiest chord to move to is Am -- just leave out the bottom string. But I'd like to play a chord that involves the three missing notes -- G, B, and D. (We ignore "thu" and "lavender," even though "thu G" is more like G# and "lavender B" is more like Bb.)
The notes G, B, and D obviously spell out a G major chord -- even though a true 13:11:9 chord sounds more like a minor chord than a major chord. Since we're ignoring the colors, I must play the chord as G major.
Now fortunately, G major is an easy open chord to play. The bad news is that it's not easy for me to move my hand from F#m7b5 to G. It's slightly easier to move from open F#m7b5 to open G if the index finger is not used for open G. Ironically, I find it easier to move from the bar F#m7b5 to open G with the usual fingering,
Don't forget that I'm an amateur musician. Professional guitarists can probably make all of these chord changes without thinking about it. But if I were a professional guitarist, I obviously wouldn't be a math teacher! I must focus on all of these chord changes deliberately to make sure that I compose something that I can actually play in the classroom.
Also, all of these chord changes are relevant to playing harmony (rhythm guitar). It's also possible to play a melody on the guitar (lead guitar). But I'm not talented enough to play lead guitar and sing the melody at the same time. My goal is to have Mocha play a melody while I sing along the lyrics, and then once I've memorized the lyrics, I play rhythm guitar to accompany my voice. (Once again, if I were more talented at guitar, I wouldn't need to teach math!)
At one of the links from yesterday, Seymour Duncan suggests playing the following progression in B Locrian mode: Bm7b5 - G - Am - F. In F# Locrian, this becomes F#m7b5 - D - Em - C. I tried playing this sequence on the guitar, and I think I can master it with some practice.
And so let's program Mocha to compose a random tune using this sequence:
10 CLS
20 N=16
30 FOR A=0 TO 7
40 B=4
50 X=A-INT(A/4)*4
60 IF X=0 THEN D=14
61 IF X=1 THEN D=9
62 IF X=2 THEN D=8
63 IF X=3 THEN D=10
70 PRINT D;
80 L=RND(B)
90 SOUND 261-N*D,4*L
100 IF L>1 THEN FOR I=1 TO L-1:PRINT " ";:NEXT I
110 B=B-L
120 IF B>0 THEN D=15-RND(8):GOTO 70
130 PRINT
140 NEXT A
150 PRINT 14
160 SOUND 261-N*14,16
I admit that Lines 60-63 are clumsy here, but I can't see an easier way to make the degree D have the correct value depending on the measure A. The song repeats F#m7b5 - D - Em - C twice until ending on a whole note F#. Notice that in true 14EDL, the C major chord sounds slightly like Caug, while the Em chord is completely off -- it sounds either like E major or Edim. The D chord becomes D supermajor, so it should still sound like a D major triad.
Returning to 2019, we wonder whether a song in 14EDL would sound happy or sad. After all, it contains the interval 14/12 = 7/6, a superminor third.
But as it turns out, 14EDL actually contains another "third," although it doesn't look like a third. The interval 14/11 is a strange one. It's spelled as a fourth, but it sounds more like a third. At 417.5 cents, 14/11 lies between the usual major third 5/4 and the supermajor third 9/7.
Since Mocha is melodic, it doesn't matter what the harmonic interval 14/11 sounds like. Melodically, 14/11 still sounds recognizably as a major third, and so using it in my song makes it sound happier for Mother's Day.
I still used the program above to create my 14EDL song, even though the chord progression that I mentioned last year (F#m7b5 - D - Em - C) doesn't mean a thing anymore. The home chord suggested by my notes is F# major, not F#m7b5.
Here is the code for the 18EDL and 14EDL that I created today. After I made the songs, I tweaked them a little so that they'd have the same number of notes in the DATA lines. Then all I needed to do is change the DATA lines rather than the values in the FOR loops:
18EDL:
http://www.haplessgenius.com/mocha/
10 CLS
20 PRINT "NOW PLAYING MOTHER'S DAY"
30 PRINT "MUSIC RANDOMLY GENERATED"
40 N=8
50 FOR V=1 TO 3
60 IF V=2 THEN Z=34 ELSE Z=15
70 FOR X=1 TO Z
80 READ D,T
90 SOUND 261-N*D,T
100 NEXT X
110 RESTORE
120 NEXT V
130 GOTO 50
140 DATA 18,16,14,8,13,8,18,8,11,4
150 DATA 10,4,14,8,16,8,18,12,16,4
160 DATA 14,8,9,8,18,8,17,8,18,16
170 DATA 18,8,13,8,14,4,11,8,12,4
180 DATA 18,8,17,4,11,4,14,12,16,4
190 DATA 18,4,18,12,14,8,14,4,15,4
200 DATA 14,8,12,4,17,4,16,16
Don't forget to click on the Sound tab before running it.
14EDL -- change the following lines:
140 DATA 14,8,9,4,11,4,9,16,8,12
150 DATA 13,4,10,8,11,8,14,12,14,4
160 DATA 9,8,14,8,8,16,10,12,11,4
170 DATA 14,12,11,4,9,4,7,8,8,4
180 DATA 8,12,9,4,10,12,13,4,14,4
190 DATA 14,4,10,8,9,12,12,4,8,8
200 DATA 8,4,7,4,10,12,11,4
Both songs focus on the major thirds in their respective scales (9/7 for 18EDL, 14/11 for 14EDL) -- the minor thirds (6/5 for 18EDL, 7/6 for 14EDL) appear only in the bridges. When I wrote the music on the board, I spelled Degree 11 as B in the 18EDL song and Bb in the 14EDL (in order to show the interval F#-Bb, technically a diminished fourth but sounds as a major third in standard 12EDO).
When I played the 14EDL song, I wonder whether it might sound better to switch the first four bars with the second four bars. Of course, it might make a difference if we were to add lyrics (where the lengths of the notes need to fir the words).
Hofstadter writes about computers as composers of music, in his first chapter:
"While [Charles] Babbage dreamt of creating a chess or tic-tac-toe automation, [Lady Ada Lovelace] suggested that his [Analytical] Engine, with pitches and harmonies coded into its spinning cylinders, 'might compose elaborate and scientific pieces of music of any degree of complexity or extent.'"
And here I'm using Mocha to fulfill her dream.
It's notable that when I first came up with the EDL scales, I wrote that the scale I'd write in the most would be 12EDL. But I keep going back to 18EDL, and when I finally switch to a different scale it ends up being 14EDL instead. Then again, my top posts by hit count include the entries where I discuss 14EDL, 18EDL, and 20EDL, but not 12EDL.
I usually consider 12EDL to be the simplest EDL scale, but what about 10EDL? Here's what the 10EDL scale would look like:
The 10EDL scale:
Degree Ratio Note
10 1/1 green C
9 10/9 white D
8 5/4 white E
7 10/7 red F#
6 5/3 white A
5 2/1 white C
I didn't use this scale because in Mocha, Degree 6 (Sound 255) is the highest note, so there is no Degree 5 available to complete the octave. But 10EDL resembles a major pentatonic scale -- the only difference is that 10EDL contains F# where C major pentatonic contains G instead. When I had a chance to convert a pentatonic song to Mocha a month and a half ago, I considered using 10EDL but instead I used 12EDL with Degree 10 dropped.
However, I wish to mention 10EDL because it's used in a song I find on YouTube:
The composer, Sevish, uses both 16ADO (16 otonal harmonics) and 10EDL in this song:
Two tunings were used, the harmonic series segment 8-16 and the subharmonic series segment 10-5. That subharmonic segment is a pentatonic scale that I love very much - one of its notes has a distinctly bluesy intonation.
Presumably, the note with the "bluesy intonation" is 10/7. Of course, we can always think of 10EDL as a subset of 20EDL, but this composer likes 10EDL as a pentatonic scale on its own.
By the way, here's another link I found on YouTube, since it's all about playing the music of Bach -- one-third of Hofstadter's trio -- in alternate scales:
Bach's music is played in four different scales. The second scale is just intonation where all of the major scales are a pure 4:5:6, while the third scale is the standard 12EDO scale. The first and last scales are called "well-tempered," or compromises between JI and 12EDO. The first scale is closer to JI while the second is closer to 12EDO.
In the comments, the consensus is that the first two scales (JI and near-JI) sound better for the harmonious first part of the Bach song, while the last two scales (12EDO and near-12EDO) sound better for the melodious last part of the song.
The creator of the video hides the frets for the first part so you can choose your favorite before the huge reveal. But the only fretting that's easy to tell just by looking at it is 12EDO -- the frets go straight across the fretboard.
It makes me wonder what an EDL-fretted guitar would look like. An EDL-fretted guitar would also have straight frets just like EDO. The difference is that EDL ("equal divisions of length") would actually have the distance between the frets be equal along the whole fretboard. For EDO-fretted guitars, the frets get closer to each other as we move away from the nut toward the bridge.
I've always been interested by the 18/17 minor second interval (that I used in the Halloween song that I mentioned earlier) and by its strange name, "Arabic lute index finger." An Arabic lute is also known as an "ud." But no website I can find, under "Arabic lute" or "ud," explains why the 18/17 interval is called "Arabic lute index finger."
My guess is that the old Arabic lute was actually EDL-fretted -- 18EDL, to be precise. Then the intervals can be named after fingers of the fretting hand:
18/17 = Arabic lute index finger
18/16 = Arabic lute middle finger
18/15 = Arabic lute ring finger
18/14 = Arabic lute little finger
But 18/16 (= 9/8, major tone), 18/15 (= 6/5, minor third), and 18/14 (=9/7, supermajor third) all already have established names, and while 18/13 would be a unique interval, we have already run out of fingers. Thus only 18/17 has a finger name.
I can only imagine what Bach would sound like on an EDL-fretted guitar. Suppose we had a guitar tuned to the standard EADGBE, with all of these notes white (Kite colors), but the frets followed 18EDL (just like our Arabic lute hypothesis). This would mean that all of the first fret notes would be 17u (or "su"), the second fret notes are white, the third fret notes green, and the fourth fret notes red.
At first, we might think that the chords would sound like just intonation, since intervals like 18/15 (or 6/5) are clearly JI. But if we try to play the standard fingered chords, none of them would sound right.
The best-sounding of the open chords would probably be E major. The low E, B, and high E strings would all be white, while the second and third fingers would fret another white B and E. The first finger would fret the note su G# -- but "su" is only seven cents away from "yellow," and so it would sound recognizably as a major chord. But to play an E minor chord, we'd play the open G, which becomes white. Thus we have E-G-B all white, a Pythagorean minor chord which doesn't sound as good as a just minor chord.
The other chords sound just as bad. The standard fingering of an A major chord would have all white notes, a Pythagorean major chord. And if we tried A minor, we'd notice that the first finger playing the first fret would officially play su B#, which isn't enharmonic to C at all. The D chords are even worse since the D played at the third fret would be green D, so it's dissonant when played over the open white D, and so on.
What works against us is that while it's easy in 18EDL to fret a white A string at the third fret so that it becomes green C, we can't play white A and green C in the same chord (an A minor chord), because they're on the same string! We might be able to tune an 18EDL guitar cleverly to sound perhaps like a well-tempered tuning, but probably not JI itself. This makes me wonder even more what our hypothetical EDL-fretted Arabic lute sounded like.
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
The area of this room's rug was enlarged 5%. How many square feet were added?
[Here's the given info from the diagram: the rug is a rectangle 3 1/3 yd. by 6 2/3 yd.]
Since we need an answer in square feet and the dimensions are given in yards, we must dimensional analysis to perform the conversion:
3 1/3 yd. * 3 ft./yd. = 10 ft.
6 2/3 yd. * 3 ft./yd. = 20 ft.
Therefore the area of the rug is 200 ft.^2. We know that 5% of 100 is 5, so 5% of 200 is 10. Therefore the area has been increased has 10 ft.^2 -- and of course, today's date is the tenth.
This is what I wrote last year about today's lesson:
Question 7 of the SBAC Practice Exam is on inequalities:
Which inequality represents all possible solutions of -6n < -12?
A) n < 72
B) n > 2
C) n < 2
D) n > 72
At last, we have a true first semester Algebra I problem. The answers are set up to test student knowledge of two important ideas. First to solve -6n < -12, instead of multiplying by -6, we should divide by -6. This eliminates choices A) and D). The other is that whenever we divide both sides of an inequality, we must flip the direction of the inequality. Thus the answer is B).
Both the girl and the guy from the Pre-Calc class correctly answer B) for this question.
Question 8 of the SBAC Practice Exam is on trig ratios:
Consider this right triangle.
[In Triangle ABC, C is a right angle, BC = 20, AC = 21, AB = 29.]
Enter the ratio equivalent to sin B.
This Geometry question is also straightforward. The sine ratio is opposite divided by hypotenuse. The side opposite B is AC = 21, while the hypotenuse is 29. Therefore sin B = 21/29.
The girl from the Pre-Calc class correctly answers 21/29 for this question. But the guy gives the answer sin(21/29) instead. He clearly forgets that once he finds 21/29 he's already found the sine, so he doesn't need to take the sine again. I wonder whether he would have made this mistake on the computer and typed in the letters "sin."
Today is an activity day. Once again, the activity days this year don't line up exactly with the activity days from last year, and so I must create new activity worksheets.
For both questions, I take the Exploration Questions found in the corresponding lessons of the U of Chicago text (either Algebra I or Geometry, as listed below in the Commentary.) It's coincidental that both of them involve squaring numbers on a calculator.
SBAC Practice Exam Question 7
Common Core Standard:
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
SBAC Practice Exam Question 8
Common Core Standard:
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
Commentary: Solving simple inequalities such as this one is covered in Lesson 4-6 of the U of Chicago Algebra I text. And the sine ratio, meanwhile, is defined in Lesson 14-4 of the U of Chicago Geometry text. Students should just be careful not to write sin() after already finding the sine.
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