Sunday, June 25, 2017

Pappas Music Post: Introduction

I have two things to say about my blogging schedule. First of all, my copy of Eugenia Cheng's latest book, Beyond Infinity, has arrived from the library. Over the next three weeks, I plan on going back to edit my spring break posts to accommodate Cheng's book.

And so here's the plan. Recall that the goal is for my Cheng posts to line up with the Theoni Pappas pages on infinity -- and the Pappas pages correspond to the day of the year (Julian day count). Let's look at a list of those pages:

Pappas Page     Date     Pappas Topic                                              Corresponding Cheng Chapter
106                   4/16      Cantor & the Infinite Cardinal Numbers   8. Comparing Infinities
113                   4/23      Cantor & the Uncountable Real Numbers 6. Counting Beyond Infinity

Notice that April 16th was Easter Sunday and the 23rd was the following Sunday. In between those two Sundays was spring break at my subbing school district. I made four posts during that stretch, and so I'll edit Cheng into those as well. Since I already have to break Cheng's chapter order (Chapter 8 on the 16th and Chapter 6 on the 23rd), I decided to choose random chapters to cover the weekdays of spring break.

Here are links to the posts where I'll be posting about Cheng's book. I haven't made the edits yet, so these are just placeholders until I complete those edits:

Chapter 8: Comparing Infinities (April 16th)
Chapter 11: Things That Are Nearly Infinity (April 18th???)
Chapter 3: What Infinity Is Not (April 19th)
Chapter 16: Weirdness (April 20th)
Chapter 12: Infinite Dimensions (April 21st)
Chapter 6: Some Things Are More Infinite than Others (April 23rd)

The reason for the question marks is that April 18th was my monthly "Day in the Life" post, and I might not want to violate a post that has already been submitted to the challenge. If I do choose to edit it, notice that I've already selected the Cheng chapter for that day.

The original Pappas topic for the spring break weekdays was "The Parable of Pi." For some reason, Pappas interrupts her infinity pages with a story about pi. But hey, at least that constant is in the title of Cheng's first book, How to Bake Pi.

Meanwhile, this is what Theoni Pappas writes on page 176 of her Magic of Mathematics:

"The composer creates music that fits so beautifully and effortlessly together in the rigid structure of the written score."

This is the start of Chapter 7 of the Pappas book, "Mathematics Plays Its Music." No sooner do we finish one of my favorite Pappas topics -- non-Euclidean geometry -- do we begin another one of my favorites -- music. Much of music is mathematical -- and I've alluded to this fact in several blog posts in the past few years. As Pappas herself writes farther down the page:

"Since ancient times, mathematics has been used to explain music. Today computer modeling and digitization, the quantization of music and sounds coupled with the study of acoustics and acoustical architecture are producing new sounds sensations."

And so I'll manipulate my blogging schedule again so that I can devote several posts to discussing math and music. Yes, this means that Glen Van Brummelen's Spherical Trigonometry has now been put on hold. I have to spend time reading Cheng since I got her book from the library (unlike Van Brummelen's which I purchased, so it has no due date), and I acquired the Pappas book before Van Brummelen's, so those topics have priority. Cheng and Pappas have taken over all of my other planned summer blog topics.

For today's music post, I refer to last week's Google Doodle to celebrate the 117th birthday of Oskar Fischinger, a German-American animator. He was known for "animating" music, and so Google created an interactive Doodle on which one can create some simple music.

Here's how it works. For the rhythm, there are 16 beats that repeat over and over. We can treat these as quarter notes, and so there are four bars in common (4/4) time. As for the pitch, here are the possible notes that can be played:

C, D, E, G, c, d, e', g', c', d', e'

Those who aren't familiar with music might not know what the notes C, D, and so on mean. Let's have Pappas explain how musical notes work:

"With ratios, the Pythagoreans (585-400 B.C.) were the first to associate music and mathematics. They also found that harmonious sounds were given off by equally taut strings whose lengths were in whole number ratios -- in fact every harmonious combination of plucked strings could be expressed as a ratio of whole numbers. For example, starting with a string that produces the note C, then 16/15 of C's length gives B, 6/5 of C's length gives A, 4/3 of C's gives G, 3/2 of C's gives F, 8/5 of C's gives E, 16/9 of C's gives D, and 2/1 of C's length gives low C."

Yes, this is the same Pythagoras who proved the theorem on right triangles. He is strongly associated with music as well as Geometry.

Pappas goes on to explain that the length of the string is inversely proportional to the frequency of the curve associated with the sound wave. Musicians who analyze notes therefore associate the ratio 2/1 with the high C an octave above the starting note, since it has twice the frequency -- and all the frequency ratios from D to B are found by dividing 2/1 by the string lengths. Because of this, the frequency ratios associated with the notes are:

Name     Solfege     Ratio     Harmonic     Cents
C            Do             1/1            24               0
D            Re             9/8            27               204
E            Mi             5/4            30               386
F             Fa             4/3             32              498
G            Sol            3/2             36              702
A            La             5/3             40               884
B            Ti              15/8           45              1088
c             Do             2/1            48               1200

This is known as a justly tuned major scale.

Returning to the Google Doodle, the notes that are visually the lowest are musically the highest, so they are positioned the opposite from how they would appear on the musical staff. Also, it appears that there's an error, as there are two notes that both sound like e'. Most likely, the middle octave should contain an e and a g, but instead they sound one octave too high. So the (musically) highest octave contains a g', but the middle octave is missing both e and g. Oh, let's call the highest octave the "treble octave" in order to avoid much confusion between the musically highest and visually highest octave. The musically lowest octave is the "bass octave."

There are not many songs we can play with just the notes C, D, E, and G. One song that I was able to play was "Mary Had a Little Lamb":


Normally, the notes before the skips would be half notes, but since these are unavailable, I use quarter notes and skips (quarter rests) instead. It's possible to add a bass line to this, and I chose:

C-(skip 3)-C-(skip 3)-G-(skip 3)-C-(skip 3)

Another simple playable song is "Au Clair de la Lune" (French for "by the light of the moon"):

Bass:      C-r-C- r-C- r-G-r-C-r-G-r- C- r3
Melody: c'-c'-c'-d'-e'-r-d'-r-c'-e'-d'-d'-c'-r3

Most of the songs I sang in class are not playable. One tune that comes close is "Row, Row, Row Your Boat," which I used for both "Mode, Mode, Mode the Most" and "Same Sign, Add and Keep." I need an F to make this song work.

But actually, "Row, Row, Row Your Boat" is usually written using quarter notes and triplets, which aren't available in the rhythm anyway. So I could skip the F notes and try something like:


I was forced to use the bass octave for the melody, since the middle octave is missing both e and g and the highest octave doesn't have the high c" note we need for "Merrily." If we add the following accompaniment to the treble octave, the missing notes in the melody become less noticeable:


If you click "Modify," you can change the tempo. I recommend a tempo of 60 or less for "Row, Row, Row Your Boat" to make it easier to sing along. (A tempo of 60 indicates 60 beats per minute, or one beat per second.) Changing the instrument also helps, as some instruments make the bass octave easier to hear.

I also notice that it's also possible to change the key from the default of C major. Unfortunately, changing this key outside of the range from C to E messes up the octaves. But for "Row, Row, Row Your Boat," the octave change is beneficial.

Let's change the key to A major. (Any key in the range from A-flat to B will work.) Then we can use the middle octave for the melody along with the bass octave, as follows:

(The notes C-D-E-G are now written as A-B-c#-e, where c# means c-sharp.)

Bass:      A-r-  A-r- A-r-    e-r-A-r- A-r-   e-r-  A-r
Melody: a-a-a-c#'-c#'-c#'-e'-r-a'-e'-c#'-a-e'-c#'-a-r

Play this with a 60 tempo, and this is the best-sounding version of "Row, Row, Row Your Boat."

By the way, I'm tagging this as traditionalists -- and you may wonder what the connection is between music and the traditionalists. Well, here it is: I was disappointed that the Fischinger player could only play four notes, C, D, E, and G. I realized that Google did this so that novice musicians can play notes that sound good together, which is easier when there are fewer notes. But of course, I'm not a novice, so I want to be able to play F's, A's, and B's.

Similarly, traditionalists are disappointed when Common Core encourages all eighth graders to take Common Core 8 when some students are not novices -- they could excel in Algebra I and beyond.

Anyway, back to posting schedules, I still haven't decided whether I'll post one more "Day in the Life" post for July (maybe even on the 18th) and call that my "summer post," or just declare the June 18th post to be my "summer post" and final submission to the challenge. Well, I have plenty of time to make that decision.

My next post will be soon, since I want to post during Pappas Chapter 7 as often as possible.

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