Wednesday, June 28, 2017

Pappas Music Post: Tau Day Edition

This is what Theoni Pappas writes on page 179 of her Magic of Mathematics:

"The speed of light, c, tau, e, phi, and Avogadro's number are all examples of constants of our universe. These are numbers which have vital roles in equations and formulas which define various objects of our world -- be they geometric, physical, chemical or commercial."

This begins a new section of Pappas, "Musical Scales and Mathematics." But before we begin discussing this section, I must point out that I slightly edited the Pappas quote.

In between the physical constant c and the mathematical constant e, Pappas actually mentioned another mathematical constant, "pi." But I changed that constant from "pi" to "tau."

You guessed it -- another Tau Day is upon us! This is what I wrote last year about tau:

But what exactly is this constant "tau," anyway? It is not defined in the U of Chicago text -- if it were, it would have appeared in Section 8-8, as follows:

tau = C/r, where C is the circumference and r the radius of a circle.

Now the text tells us that C = 2pi * r, so that C/r = 2pi. Therefore, we conclude that tau = 2pi. The decimal value of this constant is approximately 6.283185307..., or 6.28 to two decimal places. Thus the day 6/28, or June 28th, is Tau Day.

It was about ten or fifteen years ago when I first read about Pi Day, and one old site (which I believe no longer exists) suggested a few alternate Pi Days, including Pi Approximation Day, July 22nd. One day that this page mentioned was Two Pi Day, or June 28th. But this page did not suggest giving the number 2pi the name tau or any another special name.

Now about six years ago -- on another Pi Day, March 14th -- I decided to search Google for some Pi Day webpages. But one of the search results was strange -- this webpage referred to a "half tau day":

"The true circle constant is the ratio of a circle’s circumference to its radius, not to its diameter. This number, called τ (tau), is equal to 2π, so π is 12τ—and March 14 is thus Half Tau Day. (Of course, since τ=6.28, June 28, or 6/28, is Tau Day itself.) Although it is of great historical importance, the mathematical significance of π is simply that it is one-half τ."

The author of this link is Michael Hartl. Here's a link to his 2017 "State of the Tau" address:

According to this link, there's going to be a Tau Day party tonight right here in Southern California, but I probably won't be able to attend.

Oh, by the way, here's one argument in favor of tau that I've rarely seen addressed by tauists -- that is, advocates of tau. It's pointed out that there's one formula in which pi shines:

A = pi r^2

This formula would be less elegant if it were written using tau:

A = (1/2) tau r^2

Tauists often respond by saying, "well, the 1/2 reminds us of the proof of the area formula" (for example, using the area of triangles) or "the 1/2 reminds us of other quadratic forms."

But actually there's a much stronger argument in favor of tau. The formula for the area of a sector of central angle theta is:

A = (1/2) theta r^2

Oh, so this formula contains a 1/2 no matter what! The full circle area is given when theta = tau. (I see that this formula actually is buried in the Tau Manifesto, but still the other arguments appear first in the Manifesto.)

Let's go back to Pappas. This is a section ostensibly about music, so why is Pappas writing about c, e, and the circle constant anyway?

Well, Pappas writes:

"Among these famous constants, the concept of an octave should be included as a constant of a special nature."

Further down the page, she adds:

"Just as the ratio of a circle's circumference to its [radius] always produces the constant [tau -- dw], the ratio of the number of vibrations of a plucked string to a string half its length is the ratio 1/2."

So this "new constant" Pappas proposes is none other than 1/2. Hey, that means that the two circle constants fighting for supremacy -- pi and tau -- are also an "octave" apart.

In today's music post, we will extend music beyond yesterday's 5EDO and 7EDO scales. And we make this extension by approximating more just intervals by looking at the fifth harmonic.

So far, we have estimated the intervals 2/1 (octave), 3/2 (perfect fifth), and 4/3 (perfect fourth). So according to the pattern, we should look at the interval 5/4 next. This interval is called a major third, and the note a major third above C is the note E.

The next interval to consider is the interval 6/5. This interval is called a minor third, and the note a minor third above C is the note E-flat, or Eb.

Notice that this E and Eb are not the same as the Pythagorean E and Eb from yesterday. The thirds based on the factor 5 are associated with the mathematician Ptolemy -- yes, the same Ptolemy I associated in trig in my post from two weeks ago.

Pythagorean major third: 81/64
Ptolemaic major third: 5/4

Pythagorean minor third: 32/27
Ptolemaic minor third: 6/5

Due to their simpler ratios, the Ptolemaic thirds are more consonant than the Pythagorean thirds. We can use the Ptolemaic major third to form a major triad -- the ratio is 4:5:6, and the notes are C-E-G.

If we were to begin with the three most important Pythagorean intervals -- the unison (C), perfect fourth (F), and perfect fifth (G) -- and build a major triad on each, we obtain the following:

C major: C-E-G
F major: F-A-C
G major: G-B-D

If we combine all these notes, we obtain the just major scale that I mentioned in an earlier post:

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

We can do the same thing with the minor triads -- if we build a minor triad on the first, fourth, and fifth scale degrees, we can obtain a just minor scale:

C minor: C-Eb-G
F minor: F-Ab-C
G minor: G-Bb-D

Name     Solfege     Ratio
C            Do             1/1
D            Re             9/8
Eb           Me            6/5
F             Fa             4/3
G            Sol            3/2
Ab          Le             8/5
Bb          Te              9/5
c             Do             2/1

A just minor scale is often built on the note A instead of A minor. This way, there are no flats:

A minor: A-C-E
D minor: D-F-A
E minor: E-G-B

Music using only Pythagorean intervals is called 3-limit, since the largest prime that appears in the ratios is 3. Music using Ptolemaic intervals is called 5-limit.

As it turns out, the simplest EDO that incorporates the important 5-limit intervals is 12EDO. Let us look at the most consonant intervals in 12EDO:

Name     Interval      Notes of 12EDO
C            1/1 = 1        (2/1)^(0/12) = 1
Eb          6/5 = 1.2     (2/1)^(3/12) = 1.189
E            5/4 = 1.25   (2/1)^(4/12) = 1.26
F            4/3 = 1.333 (2/1)^(5/12) = 1.335
G           3/2 = 1.5      (2/1)^(7/12) = 1.498
Ab         8/5 = 1.6      (2/1)^(8/12) = 1.587
A           5/3 = 1.667  (2/1)^(9/12) = 1.681
c            2/1 = 2         (2/1)^(12/12) = 2

The perfect fourth and fifth are very close to just. The thirds and sixth are a little bit more off, but are still recognizable as thirds and sixths.

In 12EDO, we can use perfect fifths to generate all of the notes. This is the famous circle of fifths:


If we were to use fourths instead, we obtain:


Therefore F# and Gb are the same note -- they are enharmonic. Likewise C#=Db, G#=Ab, D#=Eb, and finally A#=Bb.

These sharps and flats are used to generate major scales on notes other than C -- after all, the purpose of using an equal division such as 12EDO is to aid transposing into different keys. Last year, I wrote a story about how I "discovered" the G-major scale:

I remember taking a piano class the summer after kindergarten. As it was only a beginners' course, the only major scale taught was the C major scale, which was played on the white keys. But I wondered to myself why one couldn't play a major scale beginning on notes other than C. The following December (either for my birthday or Christmas) I received a small electronic keyboard as a gift, and naturally I tried playing other scales. beginning on D, then E -- but none of them sounded like the proper major scale Do, Re, Mi, etc.

But one of these scales sounded almost right -- the scale beginning on G. The scale G, A, B, C, D, E, F, G sounded correct except for the last few notes. But I didn't know how to make the last part sound like the major scale. Disappointed, I started playing around with chromatic scales -- where I included the black keys as well as the white keys. But as I was still tantalized by the G scale, I sometimes started out by playing the part of the G scale that sounded right -- G, A, B, C, D, E -- and then switched to the chromatic scale. The black key between F and G is called F sharp, or F#, so what I played was G, A, B, C, D, E, F, F#, G, and I often played around with this "scale" for awhile.

Then one day, I accidentally skipped the F note, so what I ended up playing was G, A, B, C, D, E, F#, G. And what I played sounded exactly like the major scale that I had been seeking! And I still remember to this day how excited my seven-year-old self was to "discover" the G major scale! After this, I quickly realized that I could make all the other scales (the D scale, E scale, and so on) sound right by including some of the black keys as well as the white keys on my keyboard.

The theory of 12EDO isn't complete unless we mention the Pythagorean comma. Here is how I described the comma in that same post from last year:

Musicians are familiar with the circle of fifths. We begin at C and move up a perfect fifth to G. Then we move up another perfect fifth to D, and then to A, then to E, and so on. After twelve steps on the circle of fifths, we return back to C again -- or so we are told. That is, twelve perfect fifths are declared to be equal to a whole number of octaves -- seven, as it turns out.

But what exactly are seven octaves? Notice that one octave is 2:1, two octaves are 4:1, three octaves are 8:1, and so on. Since these are ratios, they must be multiplied, not added, just as performing a dilation with scale factor 2 thrice gives a figure that is eight times the original. So seven octaves are the ratio 128:1.

Now let's try twelve perfect fifths. Each fifth is 3:2, so twelve of them would be 531441:4096, which is not exactly 128:1. Cross multiplying these ratios would give 531441 = 524288, which is false. And we shouldn't be surprised that twelve fifths don't actually equal seven octaves. Since combining musical intervals amounts to multiplying, breaking them up amounts to factoring -- and we can't factor the same interval in two different ways (i.e., as seven octaves and twelve fifths) for the same reason that we can't factor the same number in two different ways -- this would violate the Fundamental Theorem of Arithmetic. The "equation" 531441 = 524288 states that a power of three equals a power of two, which is impossible.

But in music, we declare 531441 and 524288 to be equal. So the ratio 531441:524288 is considered to be equal to 1. In honor of the ancient Greek mathematician, we refer to the ratio 531441:524288 as the Pythagorean comma. And it's because it's the twelfth power of three that is approximately a power of two that our octaves consist of twelve notes -- and why it's the twelfth fret on the guitar that gives us the octave. (Notice that computer scientists have their own "comma." They declare 1024 = 1000 -- that is, a power of two equals a power of ten -- in order to justify the names "kilobyte," "megabyte," etc.)

Now let's actually use these notes in a song. Last year on Tau Day, I wrote about the "mathemusician" Vi Hart, who advocates using tau instead of pi. She actually posted two versions of her "Song About a Circle Constant" -- the first time in A major, the second time in G major. But to make it easier for us to analyze her song, we will convert it to C major.

Many pi enthusiasts create a song by using the digits of their favorite number. This is done by using a major scale: 1 is C, 2 is D, 3 is E, and so on. Hart does the same, except she uses tau instead of pi.

Let's look at the digits of tau in more detail:


Notice that there is a zero among the first ten digits of tau. Some "mathemusicians" use a rest for zero, while others use a low B just below the C for note 1. The digit zero doesn't appear among the first thirty digits of pi, and so pi musicians avoid the zero controversy by ending the song just before the first zero. Tauists, on the other hand, must face the zero problem right away due to the much earlier appearance of 0 in tau compared to pi.

Hart chooses to treat the digit 0 as a 10, and so she sings a high e (a major 10th above C). With this, we obtain the following notes:


Actually, Hart assigns a note to the decimal point as well -- the note F:


The reason she chooses F is so that the first three notes, A-F-D, form a D minor triad. (We could even include the fourth note, c, to make the chord D-F-A-c, a D minor seventh chord.)

Hart likes to end each verse with either the note 1 (the unison) or 8 (the octave). These notes are both known as the tonic, and most real songs end on the tonic. Her song includes 51 digits after the decimal point, since the 50th and 51st digits are both 1. On the other hand, pi songs often end on the dominant (the fifth), since the last digit before the 0 happens to be 5.

Don't forget that Hart doesn't sing her song in C major, but in A or G major instead. The original version of her song is in A major. This key requires three sharps: F#, C#, and G#:


Her new version is in G major, which as I "discovered" when I was seven, requires only F#:


Why did Hart change from A to G major? I suspect it's because of the guitar she's playing in the second version. Some chords are easier to play on the guitar than others. The opening chord (remember the decimal point?) in the key of C is D minor, the opening chord in the key of G is A minor, and the opening chord in the key of A is B minor. Now B minor is a harder chord to play on the guitar (though if we include the extra A to form B minor seventh, the Bm7 chord isn't so difficult as it only requires a slight finger change from an A major chord). On the other hand, it's much easier to play the Am and Am7 chords for G major.

Here are this year's Tau Day links:

1. Vi Hart:

Naturally, we begin with the two Vi Hart videos that we just discussed above.

2. Numberphile:

His Tau vs. Pi Smackdown is a classic, and so I post this one every year.

3.Michael Blake:

This is a perennial favorite. Blake uses a rest to represent 0. Like Vi Hart, he ends on the digit 1, but this is ten extra digits past Hart's final 11.

4. Math Babbler:

Math Babbler memorizes the digits of tau up to the first zero -- which, as I wrote earlier, is much easier than memorizing pi up to the first zero! He also mentions that he thinks of tau time as 6/28 at 3:18:53 -- which is unwittingly when I decided to play the video today!

5. Cody's Lab:

Apparently, Cody baked half a pie for Pi Day, and so today he bakes a whole pie for Tau Day. Notice that the digits of tau appear in the flour in the background.

6. SciShow:

In this video, Caitlin Hofmeister demonstrates many of the uses of tau. She says that she's staying neutral when it comes to pi vs. tau.

There are so many more videos that I could post, but many of them are longer. One of them is a pi vs. tau debate of about a half-hour, posted just before Election Day last year.

OK, I'll post another video because this one is so short. It's another song based on the digits of tau, but the digits are in dozenal, or base 12, so there's a digit of all the notes of 12EDO.

As I mentioned yesterday, using fewer notes (such as C-D-E-G or C-D-E-G-A) in a song based on random digits make it sound more consonant. With all 12 notes at our disposal, often the two sharp dissonances (tritones and semitones) appear in the song.

I was thinking about more songs that I can play based on digits. Many years ago, back before I knew that anyone was basing songs on the digits of pi, I considered a Fibonacci song. I was thinking about it because one of my favorite episodes of Mathnet (from Square One TV) was posted just before Election Day -- "The Case of the Willing Parrot," and Fibonacci plays a huge part in this episode. (I must point out that there really is a parrot in this episode, who repeats "1, 1, 2, 3, 5, Eureka!")

Actually, Numberphile has already created a Fibonacci song. It's another long video so I don't post it, but here's how the song works. Since Fibonacci is 1, 1, 2, 3, 5, 8, the song starts C-C-D-E-G. For eight, we know that due to octaves, 8 is the same as 1, another C. Actually, Numberphile reduces all of the notes mod 7 (the next Fibonacci number is 13, which reduces to 6 mod 7, or A), and so the following 16 notes end up repeating:


("The Case of the Willing Parrot" uses mod five instead of mod seven in one of its patterns.)

My old version of the Fibonacci song was different. For 8, I played a high c, and for 13, I treated it as 1 and 3 and played the notes C-E. The resulting song was:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89


Unlike the Numberphile version, this one is almost playable in the Google Fischinger playable, as it's spoiled only by the lone F. We can replace that F with a G without it being noticed. (Then maybe we can hide the treasure behind that G -- oops, I don't want to spoil the plot of "Willing Parrot"!) I also like the idea of ending the song with 55, with two G's and a rest in between them. On the other hand, we can't include the next Fibonacci number, since 144 would require two F's (and it would be too noticeable to change those F's to G's).

This song suffers the same problems as "Row, Row, Row Your Boat," with the higher c note. Again, it has the same solutions -- we either play the melody in the base octave, or switch to another key such as A major.

Oh, I see that I gave this the traditionalists label again. Well, let me repeat parts of a traditionalists post from Tau Day two years ago:

Aha -- if Michael Hartl himself, or any reader of his Tau Manifesto, really wanted to get a new generation of students to learn using tau instead of pi, he might have tried to convince David Coleman to incorporate tau in the Common Core Standards. Let's look at the relevant standard in the Core:

Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

We notice that this standard doesn't actually state any formula for the area or circumference -- neither the pi formula nor the tau formula. So technically, this standard alone doesn't rule out teaching the students C = tau * r instead of C = 2pi * r. But if we really want to enforce the new constant tau, let's explicitly require the students to learn it:

CCSS.MATH.CONTENT.6.G.B.4 (proposed)
Know that tau is defined as the ratio of the circumference of a circle to its radius; know the formulas for the area and circumference of a circle in terms of tau and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

Notice that I dropped this standard from seventh grade to sixth grade, since I already stated that the circle measurement formulas used to be part of the 6th grade standards in California. In fact, since we can't make the students learn about tau on Tau Day (unless we want to shorten summer vacation), why don't we teach tau to 6th graders 28% into the school year -- that is, at grade level 6.28 or tau.

On the surface, this proposal should work. Over the next 14 years, students would learn only about tau and not about pi -- and these would become our future mathematicians. With an entire generation of students now aware of tau, we should attain our goal and fulfill the Tau Manifesto by 2031.

But what would really happen if we included tau in the Common Core? This idea of mine is not a serious proposal at all -- instead I want to make about Common Core and why so many people are opposed to the standards.

For starters, there will be a backlash to including tau in the standards. There is already a Pi Manifesto written as a response to the Tau Manifesto:

But here's the point I'm trying to make here -- if tau were included in the Common Core Standards, then Common Core opponents would try as hard as they could to show that pi is better than tau -- and that includes arguments such as the one in Section 3.2. That is, if a math teacher first learns about tau by reading Hartl's Tau Manifesto, the teacher is more likely to agree with Hartl's position that if she first learns about tau by having it forced upon her in the Common Core Standards.

And of course, the biggest problem would be in the transition itself. Students who learned pi in middle school will be confused the first year of the new standard when they were supposed to be using tau instead. Critics would say that the new standard was designed to keep the top students down, since these students who already know about pi would be confused when they see tau on the test instead of pi. Sixth graders may ask their parents for help on their math homework -- and then the parents would be confused, since they don't know what tau means. Schools might even have to hold parent training nights so that the parents can learn about tau in order to help their children. I can almost see a parent's complaint now: "My child used to love math back when he learned about pi. But now they want to treat him like a guinea pig and force tau on him, and now my child hates math."

Of course, the Common Core doesn't really require students to learn about tau -- the above is simply a thought exercise. But what if we replace "tau" above with, say, "transformation geometry"? Many teachers, parents, and students feel the same way about transformation geometry in reality as they would about tau in our hypothetical scenario.

So let's try again. Suppose we want everyone to use tau instead of pi, and convince them that tau will help students learn math better than pi. We know now that simply including tau in Common Core won't convince anyone of anything. So how could we accomplish this goal?

Some people oppose not just Common Core, but public schools in general, due to capitalism. They believe that whoever pays for a school has the most power, so if all schools were private, the parents, not the government, have all the power over the children's education. Of course, then the problem is that not everyone can afford private school, and then the question is raised of who pays for the education of those who can't afford it, which quickly gets into politics.

Innovation in education is difficult because someone's children must be among the first to study under the new teaching method, and whoever's children they are will fear that they are being treating like "guinea pigs" because they are the first to learn it. Here the private school avoids the "guinea pig" problem because the parents knew that their students were learning tau before writing the tuition check in the first place. Once the private school demonstrates success, public schools, states, and the nation can adopt tau into their standards without making the children feel like guinea pigs, since tau would no longer be an unproven teaching method, but instead it would have been proven by the success of the private school.

Avoiding the transition problem is very tough. We already know that some schools grandfathered in the upperclassmen and teach Integrated Math only to the freshmen last year. Some people believe that the grandfathering should go all the way down to first grade -- in the first year of the new program, only kindergartners should be taught using the new program. But even that doesn't avoid the difficulty of having parents who learned the old way being unable to help their children learn the new way.

Interestingly enough, Dr. Hung-Hsi Wu, in his Common Core lessons, gave an argument which favors pi over tau. Dr. Wu defines pi as the area of the unit disk. This is mentioned in Section 3.1 of the Pi Manifesto -- we can't get an area of tau unless the radius is sqrt(2). Only an area of pi gives a simple radius of 1.

Happy Tau Day, everyone!

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