Today I subbed in a high school music class. Again, there's no real "Day in the Life" because a student teacher took over the classes -- in fact, it's the same student teacher from my March 22nd and 23rd subbing in a middle school music class.
But of course, I can't help but think about music today after being surrounded by it. When I started my new side-along reading book yesterday, I wrote that it would mean less time for two topics I've overdone lately, traditionalists and music. So what happens? On the first day of reading, I wrote about traditionalists, and now on the second day of reading, I'm writing about music.
Second and third period classes today are both strings -- second period for freshmen, and third period for the older grades. (There is no first period, which really means zero period.) Both classes are learning the theme to the movie How to Train Your Dragon. Violinists actually move to a separate room to practice with a "coach," and so only violas, cellos, and basses remain.
Well, here's something indirectly related to traditionalism -- no, the high school classes didn't warm up with F-E-F (as the middle school kids did), but they did play G scales. The freshmen played a G major scale, while the older students played G minor. Both classes had additional exercises involving these scales -- the freshmen played "thirds" (G-B-A-C-B-D-C-E, etc.), while the older students played "modes." So they began with the G natural minor scale itself (that is, Aeolian mode), then A Locrian (the G minor scale but starting on A), followed by Bb major, and so on.
The theme from How to Train Your Dragon is fairly challenging for the students. First of all, the song is written using flats, while the easiest keys to play on string instruments are sharp keys. The song keeps changing back and forth between 4/4 and 3/4 time. And there's one more strange thing about the song's rhythm. Much of the melody is written with eighth note triplets -- but a common pattern is a triplet consisting of a dotted eighth note, a sixteenth note, and an eighth note.
I've spent so many Mocha posts writing about scales (7-limit, 13-limit). It's about time that I say a few things about rhythm in Mocha -- the second argument of the SOUND parameter.
First of all, what exactly is a "triplet" anyway? Well, an "eighth note triplet" consists of three "eighth" notes that are considered to equal one beat. Notice that twelve such notes add up to one whole note, or one whole measure (in 4/4 time). Thus "eighth note triplet" is really a misnomer -- they really should be called twelfth notes.
Thus a "triplet" note is 2/3 the length of the original note. Meanwhile, a dotted note is half again (that is, 3/2) the length of the original note. Hence a "dotted eighth note triplet" is actually the same as a regular eighth note! But in the actual score, it's written as a dotted eighth note triplet because all the other notes in the measure are triplets, so it's awkward to place a non-triplet eighth note there.
The section of the song where the dotted eighth note triplet first appears contains two flats in its key signature -- Bb and Eb. Yet the actual dotted eighth note triplet note is Ab -- and indeed, the number of flats in the signature keeps changing between one and four. (So now you can see why the students had trouble with flats today.) Let's try to program the first line in Mocha. Let's use PLAY first. To make it easier (since I want to focus on the rhythm, not the flats), we lower it by a semitone. This also allows us to avoid the major 2nd step from Bb to C, which Mocha considers to be in a new octave:
http://www.haplessgenius.com/mocha/
PLAY "L12;G;L6;AA;L12;E;L8;G;L24;A;L12;B;L6;A;L12;E"
This means that we start with a twelfth note (eighth note triplet) G, followed by two sixth notes (quarter note triplets) A, then a twelfth note (eighth note triplet) E. The G which follows is a dotted quarter note triplet, so in Mocha we write it as an ordinary eighth note. The next note A is a sixteenth note triplet, which we write as a 24th note. The next note is a twelfth note (eighth note triplet) B, and so on.
So far when we use the SOUND command, I declare the whole note to be of length 16, so that a half note is length 8, and so on down to the sixteenth note of length 1. If I wanted to play triplet music using SOUND, I'd just slow it down so that the whole note is of length 24 instead. This allows all the triplets to be played in the correct length ratio, all the way down to the sixteenth note triplet, which is now of length 1. By the way, some of the holidays songs I played in my Christmas Eve post required triplets, and so I wrote them using 24 as the whole note.
Let's try this same line using SOUND. This time, I'll keep the song in Bb (recall that green Bb is the only just major scale playable in Mocha). For Ab we'll use the deep green Ab (this note lies a just minor 3rd above green F) -- all the other notes in the songlet are green. (Here we just use Sounds rather than waste time with Degrees, since the focus here is on the note lengths.)
10 SOUND 61,2
20 SOUND 81,4
30 SOUND 81,4
40 SOUND 21,2
50 SOUND 61,3
60 SOUND 81,1
70 SOUND 101,2
80 SOUND 81,4
90 SOUND 21,2
If we use our usual formula for Sounds (Bridge 261 - N * Degree), it's possible to play these Sounds in one of three octaves.
The remainder of the classes are band instruments. In fourth period, freshmen play the song "Baba Yangu," whose title means "Our Father" (that is, the Lord's Prayer) in Swahili. (I explained what the Swahili language is back in my Kwanzaa post on December 28th). In fifth period, the older students play "Esprit de Corps," whose title means "Body Spirit" in French. The remaining class, in seventh period, is jazz band. These students play several songs, including "Act Your Age" and "Bewitched."
Only the freshmen play a scale before starting. It's Bb major, one of the easier scales to play on most band instruments. All of the band students struggle with the rhythm in these songs, since they all feature "syncopation," where notes begin on unstressed offbeats rather than stressed beats.
Before we leave Mocha, let me repost the code (now under the new "music" label) to play two of the scales we heard today, G minor and Bb major (as both just scales are playable in Mocha):
10 DIM S(9)
20 FOR X=0 TO 9
30 READ S(X)
40 NEXT X
50 DATA 30,72,64,60,54
60 DATA 48,45,40,36,32
70 N=380 A$=INKEY$
90 IF A$="" THEN 80
100 A=VAL(A$)
110 SOUND 261-N*S(A),2
120 IF A$>="0" AND A$<="9" THEN 80
As I explained in my Thanksgiving post, press keys 1-8 for G minor and 3-0 for Bb major. (Oh, and press keys 2-9 for A Locrian, one of the modes played today.) And as I also explained in that post, technically green Bb major uses a green C, while white G minor uses a white C. This program uses a white C, so that it's evident that Bb major is a mode of G minor (as played today in class).
By the way, I still think my attempt to make New 11- and 13-Limit Scales is awkward. In a later post, I'll make scales that work more naturally in Mocha.
Chapter 2 of Wickelgren is called "Problem Theory." Here's how he begins:
"To illustrate the concepts involved in the theory of problems described in this chapter, we will begin with four sample problems."
Well, here are the four problems Wickelgren describes here:
1. Instant Insanity
This problem is best described via a link, rather than my trying to type it all out. Keep in mind that the following link also gives the solution, so don't scroll down too far unless you want to be spoiled:
https://www.jaapsch.net/puzzles/insanity.htm
2. Chess Problem
From the board configuration shown in Figure 2-2 describe a sequence of moves such that white can achieve mate in five moves.
Of course I don't post Figure 2-2. Instead I'll just tell you where the pieces are:
White: King at f5, Rook at g7, Pawn at g2
Black: King at h5, Bishop at f2, Pawns at g3, h6
Find Problem from Mechanics
What constant force will cause a mass of 3 kilograms to achieve a speed of 30 meters per second in 6 seconds, starting from rest?
Proof Problem from Modern Algebra
You are given a mathematical system consisting of a set of elements (A, B, C) with two binary operations (call them addition and multiplication) that combine two elements to give a third element. The system has the following properties:
(1) Addition and multiplication are closed.
(2) Multiplication is commutative.
(3) Equals added to equals are equal.
(4) The left distributive law applies; that is, C(A + B) = CA + CB for all A, B, C in the set.
(5) The transitive law also applies.
From these given assumptions, you are to prove the right distributive law -- that is, that (A + B)C = AC + BC, for all A, B, C in the set.
For this last problem, I don't define the properties, since these are basically the postulates of algebra mentioned in Chapter 1 of the U of Chicago text -- and in fact, even today's Lesson 14-6 discusses how vector addition has some of these properties:
Properties of Vector Addition Theorem:
(1) Vector addition is commutative.
(2) Vector addition is associative.
(3) (0, 0) is an identity for vector addition.
(4) Every vector (a, b) has an additive inverse (-a, -b).
We see that vector addition satisfies (1), (3), and (5) as given by Wickelgren, but not the ones involving multiplication. And recall the big deal made about how according to the Common Core, polynomials are a set analogous to the integers -- that is, they satisfy rules (1)-(5) among others. So these properties, along with a few others, make the set of polynomials into a "ring."
Wicklegren tells us that the first thing to do in any problem is to identify the givens:
"This definition of givens encompasses expressions representing objects, things, pieces of material, and so on."
The next step is to determine the allowable operations:
"Operations refers to the actions you are allowed to perform on the givens or on expressions derived from the givens by some previous sequence of actions."
Wicklegren's clearest example of operations is in his chess problem:
"For instance, in chess a pawn first can be moved either one or two squares, but thereafter it can be moved ahead only one square at a time. Let us adopt the convention that an operation refers to a class of actions, with the actions being distinguished only by the operands -- expressions or objects -- to which the operation is applied."
And the last step in beginning a problem is to identify the goals:
"There may be more than one correct solution to a problem, but all formal problems discussed in this book have the property that a solution is either correct or incorrect, without ambiguity. One reason for discussing the completeness of specification of the goal is to clearly describe the nature of the difference between find and proof problems."
Of course, in Geometry we're fully aware of both find and proof problems. (Of course proofs have a given and a goal, and the operations are definitions, postulates, and previous theorems.) Wicklegren now gives us an example in Algebra I -- the equation problem 4x + 5 = 17:
"Equivalences like this obtain where operations are uniquely reversible (that is, where there exist inverse operations for all operations)."
On the other hand, some problems contain irreversible operations, which he calls "destructive." In the Algebra example, think about squaring both sides introducing extraneous solutions:
"Often problems with destructive operations are considered to have only a single expression representing their state at the current moment, with the operations being able to change that entire state into a new state."
In Geometry proofs, a reversible step would be a theorem whose converse is also a theorem -- that is, a biconditional theorem. All definitions are reversible.
But what exactly is a solution, anyway? Wickelgren returns to chess:
"In a chess problem, a solution consists of some given board configuration, followed by a sequence of board configurations, each of which is derived from the previous configuration by an allowable move, and ending with a checkmate configuration."
The author tells us that the set of problem states can be viewed as a tree. If I recall correctly, Brian Harvey uses a tree metaphor when discussing computer solutions to problems in Logo:
"That is, two or more action sequences, which result in two different nodes, may result in two identical problem states."
He continues:
"These possible differences from node to node do not alter the primary lesson to be learned from examining a state-action tree -- namely, how rapidly the number of possible nodes or action sequences increases in such a tree as a function of level, that is, the length of the prior action sequence."
Of course, it's much easier to verify a solution than to find one -- or, as the author puts it:
"In such problems, the main difficulty is to find the correct type of tree in a large forest; climbing the tree may pose only minor problem."
Wickelgren concludes the chapter by telling us that in some problems, it might be easier to work backwards and derive the given from the goal:
"One problem of this type was discussed earlier in the chapter -- namely, the equivalence of deriving x = 3 from 4x + 5 = 17 or vice versa."
This is what I wrote last year about today's lesson. Admittedly it wasn't much:
I post my originally planned lesson for Lesson 14-6, which contains many of those properties of vector addition from the Common Core Standards that I mentioned yesterday.
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