Tau Day Post (Peterik Chapter 4: Snagging Your Listeners with a Hook)
Table of Contents
1. Introduction: That Time of the Year Again
2. Tau Day and Summer School
Hmm, today's date is June 28th. And now I hear the sound of all of my readers double-checking the date....
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:
Definition:
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 fifteen or twenty 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 ten 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":
http://halftauday.com/
"The true circle constant is the ratio of a circle’s circumference to its radius, not to its diameter. This number, called
The author of this link is Michael Hartl. Here's a link to his 2022 "State of the Tau" address:
https://tauday.com/state-of-the-tau
According to this link, Hartl and his friends translated his Tau Manifesto this year into German. He also mentions a reference to tau in XKCD (a web comic about math -- in fact, some XKCD comics appeared in our Stats text this year) and Jarom Watts, who memorized 1000 digits of tau.
Oh, I notice that Hartl also has a "parable" where he compares using pi (instead of tau) to counting your age in "thines" (half-years, instead of years). Even though "thines" are weird, notice that "thine" is an Anglish word (in Old English it means "yours" -- Hartl chose it because "year" sounds like "ye are") that can be used to replace "semester" (the half-year of the school year).
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.)
This is my annual Tau Day post. So what exactly does this mean for our summer projects? Well, we will still do the next chapter in Peterik. Tau Day is a day for music, and so I'll also try to tie Tau Day songs to the current Peterik chapter.
Tau Day and Summer School
In past years, I've often pointed out that both Tau Day and Pi Approximation Day can be used as alternatives to Pi Day for summer school classes.
When many of the students have completed the test, music break would begin. On instruction days, the song I'd sing would be based on that day's lesson. But on special days, I could sing a song that's more fun instead.
On Tau Day, I would have played Vi Hart's Tau Day tune, "Song About a Circle Constant." These songs are fun to sing on special days -- and I could even generate a discussion about why we'd give 2pi a special name like tau. And in a summer Algebra I class, these songs about squares and circles can serve as a preview to the Geometry that they'll hopefully take in the fall (provided they pass the summer class, of course).
I can keep up the tau = 2pi idea and let the food prize be pie. Recall that back on Pi Day Eve, I purchased three full-sized pies (and two pizzas) for two classes. Notice that pi can be rounded down to 3.
So let's round tau down to six and buy personal pies for the top six students (in each of the two periods that summer teachers typically have). Actually, we could round tau up to seven, and then give the seventh pie to the most improved student -- that is, the student whose score rises the most from the first unit test (from last Thursday) to the second unit test.
This is actually something I regret not doing in my old middle school class at the charter school -- I awarded four Red Vines to the students who earned A's on my tests, but gave nothing to the most improved student. I can't help but think back to the day when the "special scholar" (from my Phi Approximation Day post) earned her first C on a test. I gave her one piece of candy that day anyway, but under this plan, the most improved student would have earned (almost) as much candy as the ones who got A's. Perhaps this would have helped me establish better rapport with her. Instead, she felt that she wasn't good enough unless she earned A's -- and then proceeded to cheat to get those A's.
By the way, since I didn't actually teach any summer school, what did I really eat today? Last year, I ate at Tau-co Bell, of course -- the Nachos Bell Grande Party Platter. (According to the advertisement, this platter contains two boxes of nachos -- just as tau is two times pi.) But this year they aren't having it, so instead I just celebrated Tau-co Tuesday at home and ate some homemade tau-cos. But I still had a bottle of Mountain Dew Baja Blast (also at Tau-co Bell).
(Today I also ate two personal 7-Eleven pies to celebrate the date -- since tau = 2pi. The flavor I chose is Key Lime, since Hartl mentions key lime in his 2017 manifesto.)
A Rapoport Math Problem
Today on her Daily Epsilon of Math, Rebecca Rapoport writes:
sqrt(sum _n=1 ^7 (n^3))
Of course, seven is small enough that we could just add up the first seven cubes without having to know any formula. But let's use the formula anyway:
sum _n=1 ^N (n^3) = N^2 (N + 1)^2 / 4
So the square root of this is just:
N(N + 1)/2 (which also happens to be sum _n=1 ^N (n), a coincidence).
Plugging in 7 gives us 7(7 + 1)/2 = 7(4) = 28. Therefore the desired answer is 28 -- and of course, today's date is June 28th, Tau Day.
Peterik Chapter 4: Snagging Your Listeners with a Hook
Chapter 4 of Jim Peterik's Songwriters for Dummies is called "Snagging Your Listeners with a Hook" -- and it serves as the start of Part II, "Unleashing the Lyricist in You." Here's how it begins:
"When a fisherman casts his lure, he waits -- sometimes all day -- for some unsuspecting fish to find the hook."
Likewise, songwriters wish to catch their listeners with the hook. In this chapter, we'll learn about five types of hooks -- melodic, lyrical, musical, rhythmic, and sound-effect:
"Sometimes you'll find more than one type of hook in a single song. In fact, most successful songs combine several different types of hooks."
Peterik begins with the melodic hook -- the hummable part that sticks out in a listener's mind. He points out that melodic hooks have been around for centuries:
"Because in Morse code, the letter V is 'short-short-short-long,' which just happens to be the melodic hook of Beethoven's Fifth. Though hundreds of years and musical light years away from Beethoven, 'The Way,' this 1999 pop gem performed by Fastball (written by Anthony Scalzo), is defined by a super-infectious melody that greets its listeners at the chorus."
The author moves on to the lyrical hook. As its name implies, this hook can be found in the lyrics of the song, and it often includes the title:
"Suffice it to say, you'll run into this terminology in the music business, so be prepared. Try setting up a groove on your rhythm machine."
An example of a song with a strong lyrical hook is "I Heard It Through the Grapevine" (written by Barrett Strong):
"The word grapevine in and of itself is a descriptive and colorful word that's unusual without being too obscure for the average listener. Try to make your title a kind of condensed version of the song itself."
Other examples of songs with strong lyrical hooks include "She Loves You" (The Beatles), "Just the Way You Are" (Billy Joel), and "Oops, I Did It Again" (written by Martin Sandberg).
Peterik proceeds with to the musical hook, which is easier to understand if I just quote him here:
"A musical hook can be a riff, like the guitar figure in the intro of 'Daytripper' (written by John Lennon and Paul McCartney, performed by The Beatles), or anything instrumental that cries for the ears' attention."
Another song known for its musical hook is "Someone to Call My Lover":
"This 2001 smash by the resilient Janet Jackson (written by Dewey Bunnell, James Harris, Janet Jackson, and Terry Lewis), samples (uses pieces of previously recorded snippets of music) the musical riff from the group America's 1972 hit 'Ventura Highway' (written by Dewey Bunnell) and adds to it a brand-new tune."
Other examples of songs with strong musical hooks include "Satisfaction" (Keith Richards) and "Vehicle" (written by our author Jim Peterik).
The author moves on to the rhythmic hook. The rhythm draws in listeners in such tunes as "Wipe Out" (The Surfaris):
"The crazed laugh at the intro was nice, the guitar work tasty, but everything else existed to set up the rhythm hook. As my high school band director used to say, 'First there was rhythm.'"
Other examples of songs with strong rhythmic hooks include "Hey, Bo Diddley" and "Stayin' Alive" (the Bee Gees).
Peterik finishes with the sound-effect hook. The Moog Synthesizer can be used to produce alluring sound effects, such as in "Reflections" (written by Lamont Dozier):
"The ethereal sound of the Moog becomes like another voice filling the song with bittersweet feelings. Although the electronic sounds of the Moog have been elevated to high art since then; in 'Reflections,' the sound was used more to create an ethereal mood."\
And the masters of the sound-effect hook are The Beatles:
"Their producer, George Martin, made available to them the vast sound effect library of the BBC (British Broadcasting Corporation) and its array of electronic devices for their experimentation and taught them techniques of slowing down and speeding up the tape machines to warp the tonality (or timbre) of the sound, and actually reversing the tape for even more other-worldly effects (like the backwards guitar solo in 'Taxman' by the Beatles)."
Other examples of songs with strong sound-effect hooks include "Let's Roll" (Neil Young), "Barbara Ann" (written by Fred Fassert), and "The Kiss Off (Good-Bye)" (written by our author Jim Peterik).
As the author reminds us, we don't need to stick to just one hook per tune:
"There are, of course, many examples where different types of hooks are used in one song. When you team them all together you get a pretty compelling package."
For example, "Music" (Madonna) starts with a lyrical hook, a musical hook, and a melodic hook:
"Now it's time for the main hook: 'Music makes the people come together....' It's a vaguely Eastern melodic motif combined with the powerful main message of the song.'"
But, as Peterik points out, we must be careful not to overdo the hook:
"If the lyrics have enough say, they'll build in power with repeated listening and give your song staying power. Not everyone will listen to every word of your lyrics the first few times they hear them, so make sure there are a couple of easy-to-digest, hook-like phrases that sum up the premise (or idea) of the song."
This takes us to the next "Practice Makes Perfect" section -- it's time for me to write a song now.
In our last post, we looked at what Math 8 Unit 2 would have looked like at the old charter school with these songs. Let's look at the ideal Unit 3, spanning Weeks 9-12 of school. The math standards to be covered are the EE standards on radicals and integer exponents
One thing about this part of the Math 8 curriculum is that it also corresponds to what I taught in 2020 during my long-term assignment. Because of this, some of the songs that I performed there might also fit into this unit. But our unit numbers, based on the Illinois State text, don't line up perfectly with that of the APEX curriculum that we used during the long-term.
In particular, our Units 1-2 correspond to Unit 1 of APEX, which wrapped up the day before my arrival at the long-term. APEX Unit 2 consists of 8.F standards -- this marks the main difference between the Illinois State and APEX orders. Then APEX Unit 3 corresponds roughly to 8.EE5-6 here, with the remaining EE standards making up APEX Unit 4.
Let's get back to the songs now, beginning with Week 9. On the original timeline, Week 9 was when I received our Bruin Corps volunteers for the first time. In their honor, I made a parody of the UCLA fight song -- based on the Grade 6-7 RP standards on unit rates. On the new timeline, the standards for all three grades, so a unit rate song might no longer match the Grade 6-7 standards. But fortunately, the new Grade 8 standards for this week are on unit rate and slope, so a unit rate song fits here after all.
Let's tie this to what we learned in today's Peterik chapter. The UCLA fight song has a obvious hook -- the part that goes "U-C-L-A Fight! Fight! Fight!" It counts as a rhythmic hook, since excited Bruin fans perform the eight-clap rhythm at this point. In my parody, the new hook is "U-N-I-T Rate! Rate! Rate!" and this part also draws in my students.
I perform this song on Week 9 Thursday-Friday, since Thursday is the Bruin volunteer's first day. I'm not quite sure what I would have sung on Tuesday of this week (and for this project, I won't attempt to fill in every song that I perform on this timeline).
For Week 10, there's a song from my long-term that fits here -- "The Slope Song." So let's assume that on this timeline, I would have composed it in 2016 instead of 2020 and performed it here. I'm not sure whether this song contains a true hook as written, though the line "Calculator!" could serve as one.
In Week 11, the students learn to solve equations, so an obvious song fits here, "Solve It." This is a cumulative song where the verses get longer as the students learn to solve more complex equations. But there is the last line of each verse that ends with "(That's all you have to do to) solve it!" My students always enjoy getting to this part, and so it counts as a hook -- a lyrical hook as it's also the title. I did ultimately perform this song at both the old charter and long-term schools.
On the original timeline, I perform "Ghost of a Chance" during Week 11-- this is a Square One TV song that I often play at Halloween. The 31st was a Monday that year, but I didn't sing on Mondays, so my last song day would have been Friday the 28th. Thus I could perform "Solve It" on Tuesday and "Ghost of a Chance" on Thursday and Friday. While "Ghost" has a chorus -- "Probability! Don't you mess with me" -- that's not necessarily the main hook. I try to simulate the song as posted on YouTube, with screaming, shrieking (during the rattlesnake verse), creeping around like the mummy (near the end), and switching my voice to represent both the delivery guy and the ghost. Thus all of these might count as sound-effect hooks.
That takes us to Week 12. I'm not quite sure what song I would have performed as it's rather tricky. It's possible that on Tuesday, November 1st, I would have performed "Solve It" again, except with two extra verses for combining like terms and using the distributive property. This is also around the time of first trimester Benchmarks -- while the last week of the trimester was Week 13, there were computer and written components of the tests, and so they ended taking over Week 12 as well. On the original timeline, I don't sing "Benchmark Tests" in Week 12, but perhaps I should have.
Before we compose our song, let's look at the science to be taught during the unit. The Illinois State science text includes two projects each for MS-PS1-4 and 1-5. For 1-4, the students learn about how temperature affects chemical changes -- both hot (dissolving baking soda in water) and cold (making ice cream). For 1-5, they get to see how mass is conserved even when performing chemical reactions that produce slime (using borax solution) and popcorn.
Of course, which projects we do depend on how easily I can obtain the needed materials. Other factors include the visit from Illinois State leaders on Week 10 Wednesday -- the observation was for Grade 7, so I likely try to find the simplest possible projects in Grades 6 and 8 in order to focus on seventh. I'm not sure how the Week 12 Benchmarks would affect science -- unlike a regular math test that's over in one day, the Benchmarks last several days and would take over the science blocks.)
On the original timeline, this is the time when the "special scholar" (mentioned earlier in this post) excelled on her math test -- it was Thursday Week 11, and we celebrated with a pizza party on Tuesday Week 12 (November 1st). But on the new timeline, the Week 11 assessment is now a science test (on Friday instead of Thursday), and the math test is replaced with the Benchmarks. So as heartwarming as the story about her success on this day is, it disappears on the new timeline. But I hope that with the interactive notebooks, the special scholar would have fared better during the entire year -- good enough that she never has to cheat.
OK, so let's get to the new song. Even though I'm labeling this as "Song for November," this tune will be for Week 10 (in mid-October) instead. That's because Week 12 (my original intention) doesn't really need a new song -- again, the extra verses for "Solve It" and "Benchmark Tests" can carry that week.
On the other hand, Week 10 already has a slope song -- but it can always use another slope song. Slope is one of the more notoriously difficult lessons for both Math 8 and even going into Algebra I. I didn't really teach slope at all during my time at the old charter school. I tried to teach it at the long-term school, but it wasn't successful. In my blogposts from 2020, I wrote how some of the eighth graders couldn't even identify the slope and y-intercept from an equation written in slope-intercept form!
And that's exactly what the hook of this song needs to be -- the slope-intercept equation. In other words, the hook is y = mx + b. (Yes, it appears in "Slope Song," but not as the hook.) That's the part that our students need to remember. On the original timeline, I sang "Diagrams" for Grades 6-7, but I think the younger students can figure these out without a song.
This song is written in 12EDL (same as "Slope Song"), in ABAB format, verse and chorus. The chorus contains repeated lines (170-180, again at 210-220) to represent the hook.
CCSS.MATH.CONTENT.8.EE.B.6
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
https://www.haplessgenius.com/mocha/
20 FOR V=1 TO 2
30 FOR X=1 TO 59
40 READ A,T
50 SOUND 261-N*A,T*2
60 NEXT X
70 RESTORE
As usual, click on Sound before you RUN the program.
Tau Day Links
Here are this year's Tau Day links:
1. Vi Hart:
Naturally, we begin with the three Vi Hart videos that we just discussed above. The third video is new this year -- it's not about tau, but the Supreme Court decisions this week. (Hart is politically left of center and has previously posted videos on issues.)
By the way, since one of my summer projects is to post song lyrics, I ought to post the lyrics for Vi Hart's Tau Day song:
A SONG ABOUT A CIRCLE CONSTANT by Vi Hart
First Verse:
When you want to make a circle, how is it done?
Well, you probably will start with the radius 1.
Then use a compass or a string, and a paper or the ground.
And if the radius is 1, how far did you go around?
It's tau, 6.28.
Yeah, it's tau, 6.28318530717958.
Second Verse:
If you pick a certain distance, and you pick a certain spot,
And you put the two together, then what have you got?
It makes a very special shape, and now you are the inventor,
If you take all the points a certain distance from your center.
And you've got a great collection, and it is a great invention.
And it makes a lovely circle, well depending on dimension.
Because it's 1, 2, 3, then there's 4 and even more.
But for a circle, how much circle's there if you take the distance and compare?
It's tau, 6.28.
Yeah, it's tau, 6.28318530717958647692528.
Third Verse:
I know what you are thinking, what about that other guy?
The one that's sometimes pronounced "pee," and it's sometimes pronounced "pie"?
I mean, it's fine if you are building, but does not belong in math.
All the equations make more sense when you use "taw" or you use "taff."
Well, you get the same answers no matter which way.
We get further from truth when we obscure what we say.
You know that math makes sense when it's beautiful and pure.
So please don't make it ugly with your bad notation and awful curriculum.
Use tau, 6.28.
Yeah, use tau, 6.283185307179586476925286766559005768394338798750211.
[2022 update: This song has a hook, "It's tau, 6.28." Vi Hart could have left this line out and just skipped to the lines with more digits of tau, but kept it in since it serves as a memorable hook.]
By the way, in the original video, Vi Hart sings the words "awful curriculum" and writes the phrases "Alternate Interior Angle Equality Theorem!" and "Reflexive Property of Congruence" as examples of an awful curriculum. This sounds like something that can be fixed in a Shapelore class.
In Shapelore, we can change AIA to "Otherside Inside Angle Worthlink Provedsaying," and the reflexive property to "Self Law of Sizeshapesameness." If Vi Hart's complaint is that the old names are too long (that is, with too many words), then unfortunately the new names are just as long. My idea is just to replace old words with new words that students understand, such as otherside for "alternate" and self for "reflexive."
[2022 update: The word "Shapelore" means simple-language Geometry, but I often use it to mean simple-language math. For example, "little numbers" for "exponents" in last post's song can count as "Shapelore."]
2. TAU: Pi Day Protest
In past years, I've posted another version of this video where these same students are blowing up a pie for Tau Day. But that video is no longer on YouTube, so I'm posting the old version instead. (And yes, I know that in this past month there have been more important things to protest than pi vs. tau -- again, see Vi Hart's chicken salad video above for more info.)
3. Numberphile:
His Tau vs. Pi Smackdown is a classic, and so I post this one every year.
4. Mathstreet Boys
This is "Larger Than Pi," a parody of the Backstreet Boys' "Larger Than Life." Michael Hartl mentioned it in last year's "State of the Tau" address.
[2022 update: The original Backstreet Boys song is written in ABCABCD format. The last line of the chorus, in both original and parody, is where the title is sung, so this could count as a hook.]
5.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.
6. Mathsmagik:
This is a new video for this year.
7. Jarom Watts:
This is the video that Michael Hartl mentions in his manifesto this year.
More Tau Day Music
But I can't help thinking about music again on Tau Day. Recall that on Phi, e, and Pi Days, I coded Mocha programs for songs based on their digits. Seeing the Vi Hart and Michael Blake songs again makes me want to code a tau song on Mocha.
And besides, last year I wrote about music on Tau Day. So it's logical to write a tau song in one of these new EDL scales, but which one should we choose?
[2022 update: I'm keeping some of this music discussion from last year. I was going to wait until we reach Peterik Chapters 9-10 on melodies and chords before bringing up EDL scales again, but I've already been writing in EDL scales, and I can retain this discussion from last Tau Day.]
Recall that Vi Hart uses a major tenth to represent 0. The ratio for a major tenth is 5/2 (found by multiplying the major third 5/4 by the octave 2/1). The ratio 5/2 is equivalent to 15/6, which is good because now the digits 2-9 fit between Degrees 15 and 6:
A Tau Day scale:
Digit Degree Ratio Note
1 15 1/1 tonic
2 14 15/14 septimal diatonic semitone
3 13 15/13 tridecimal ultramajor second (semifourth)
4 12 5/4 major third
5 11 15/11 undecimal augmented fourth
6 10 3/2 perfect fifth
7 9 5/3 major sixth
8 8 15/8 major seventh
9 7 15/7 septimal minor ninth
0 6 5/2 major tenth
- 5 3/1 tritave
This scale doesn't fit in an octave. Instead, we can think of this scale as fitting a tritave (3/1) instead, just like the Bohlen-Pierce scale. On Mocha, we can't actually play the full tritave since Degree 6, not 5, is the last degree playable in Mocha. But Degree 6 is the last degree needed for the tau song.
Indeed, this span occurs in real songs. Most real songs have a span of more than an octave -- we explored some holiday songs six months ago and found out how spans of a ninth or tenth appear to be fairly common. On the other hand, a twelfth, or tritave, is too wide. One famous song with a span of a tritave is our national anthem, "The Star-Spangled Banner." This song is considered one of the most difficult national anthems, and the reason is its tritave span. This tells us that while EDL scales based on an octave might be insufficient (as we'd want to venture a note or two beyond the octave), EDL's based on the tritave are more than enough.
Let's fill in Kite's color names for the EDL scale that we've written above, 15EDL. Notice that while octave EDL's must be even, tritave EDL's must be multiples of three, so they could be odd:
The 15EDL tritave scale:
Digit Degree Ratio Note
1 15 1/1 green F
2 14 15/14 red F#
3 13 15/13 thu G
4 12 5/4 white A
5 11 15/11 lavender B
6 10 3/2 green C
7 9 5/3 white D
8 8 15/8 white E
9 7 15/7 red F#
0 6 5/2 white A
- 5 3/1 green C
The names of some of these intervals are awkward. First of all, 15/14 is often called the "septimal diatonic semitone." But the name "diatonic semitone" ordinarily refers to a minor second, not an augmented unison -- even though the notes are spelled as the latter (gF-rF#). The reason that the name "diatonic" is used is that it's close to the diatonic semitone 16/15 (wE-gF) in cents. The name "septimal diatonic semitone" for 15/14 forces 15/7 (an octave higher) to be a "septimal minor ninth," even though it's spelled as an augmented octave.
Here we call 15/13 a "tridecimal ultramajor second." In terms of cents, it's even wider than the 8/7 supermajor second, hence the term "ultramajor." The alternative name "semifourth" for 15/13 is akin to the name "trienthird" for 14/13. Two "semifourths" (which would be 225/169) sounds very much like a perfect fourth (4/3 = 224/168).
Finally, here we call 15/11 an "undecimal augmented fourth." And indeed, we spell the interval as an augmented fourth (gF-luB). But it's actually narrower than 11/8, which we called a "semiaugmented fourth" (the difference is the comma 121/120). It might be better to call 15/11 a "subaugmented fourth" instead to emphasize that 15/11 is narrower than 11/8.
Here's what a Mocha program for the tau song might look like. (As usual, don't forget to click the Sound box to turn on the sound.)
NEW
10 N=16
20 FOR X=1 TO 52
30 READ A
40 SOUND 261-N*(16-A),4
50 NEXT X
60 DATA 6,2,8,3,1,8,4,3,10,7
70 DATA 1,7,9,5,8,6,4,7,6,9
80 DATA 2,5,2,8,6,7,6,6,5,5
90 DATA 9,10,10,5,7,6,8,3,9,4
100 DATA 3,3,8,7,9,8,7,5,10,2,1,1
This song ends at the same point as Vi Hart's song. If you want to add ten more notes to match Michael Blake's song, change 52 in line 20 to 62 and add digits 53-62:
110 DATA 6,4,1,9,4,9,8,8,9,1
Notice that unlike the songs for pi and other constants, I didn't need to set up an extra loop to code in the scale itself. Instead, the expression 16-A subtracts the digit from 16 to obtain a Degree in the range from 15 to 6 (though we had to represent 0 as 10 to make this formula work for zero).
Except for digits 1 and 0, the 15EDL scale doesn't exactly match the major scale used in the Hart and Blake videos. Notice that 15EDL that contains a just major triad on the root note -- the first even EDL containing the just major triad is 30EDL. But the just major triad for 15EDL is digits 1-4-6, while for Hart and Blake, 1-3-5 is a major triad. Moreover, in 15EDL, digits 7 and 8 are a major sixth and seventh respectively, as opposed to 6 and 7 in the Hart and Blake videos. Again, this reflects the nature of EDL scales, where the higher steps are wider than the lower steps.
This song uses a major tenth for 0, but suppose we wanted to follow Blake and rest on 0 instead. We might add the follow line:
35 IF A=10 THEN FOR Y=1 TO 400:NEXT Y,X
Making our song simulate Hart's is more complex -- that is, if we want to add the three verses of her song and all the other details. The easiest fix is to insert 4 in line 60, since Vi sings the note 4 to represent the decimal point. For Vi, 6-4-2 is a minor triad on ii, but for us, 6-4-2 corresponds to Degrees 10-12-14, which is a type of diminished triad (as I explained the day I posted about 14EDL).
Also, Hart doesn't sing all the notes as quarter notes, but varies the lengths -- so we'd need to add numbers representing lengths to the DATA lines. If at the very least, we want to sing the digits three times, with each verse containing more digits (just like Vi) then add the following lines:
15 FOR V=0 TO 2
20 FOR X=1 TO 9.5*V*V-.5*V+15
55 RESTORE: NEXT V
The strange formula here is an interpolating polynomial that passes through the points (0, 15), (1, 24), and (2, 52), to match the number of digits she sings in each verse.
Here are the roots of all the 15EDL scales available to us in Mocha:
210 rugu G
225 gugu Gb
240 green F
255 sugu E
Notice that Vi Hart plays her song in either A or G major, while Michael Blake's is in C major. The closest available key to Hart's A is thugu Ab (N=13 as thu notes are sharper than they look -- white A is at Degree 192) while the nearest key to Hart's G is rugu G (N=14). Meanwhile, the closest key to Blake's C is lugu C (N=11).
By the way, when I was looking for Tau Day videos I stumbled upon two more videos detailing the relationship between math and music. One is by 3Blue1Brown, and that video itself mentions another video. I post both of them here:
The Sweet Spot
The full Mocha Sound system starts at Sound 1 = Degree 260. Thus in a way, the Mocha Sound system can be described as a 260EDL scale. But this is a lot of notes, and so the EDL scales that I describe on the blog contain much fewer notes.
What exactly is the "sweet spot" of EDL scales? In other words, we seek out EDL's that contain enough -- but not too many -- notes to compose songs in.
In the past, I declared the sweet spot to be 12-22EDL. We started with 12EDL because the highest playable note in Mocha is Sound 255 = Degree 6, and one octave below this is Degree 12 -- thus 12EDL is the simplest fully playable EDL in Mocha. The next even EDL's also contain octaves, so these are 14EDL, 16EDL, 18EDL, 20EDL, and 22EDL. But then 24EDL contains 12EDL as a subset, since 24 has 12 as a factor. A song written in 24EDL is likely to emphasize the 12EDL subset, which is why I considered 22EDL to be the last EDL in the sweet spot.(Notice that last year, I never actually posted anything in 22EDL, and so 12-20EDL ended up being the sweet spot on the blog.)
Even though 24EDL might reduce to 12EDL, 26EDL doesn't reduce as simply. I was considering sneaking 26EDL into the sweet spot as well, if only because 260EDL -- the entire Mocha system -- has 26 as a factor. In other words, the range 12-26EDL is completely based on the highest and lowest playable notes in Mocha, our EDL instrument. This doesn't necessarily mean that this range makes the most musical sense.
In recent posts, I mentioned that the composer Sevish actually posted a song to YouTube that is written partly in 10EDL. Earlier, I considered 12EDL to be the simplest EDL in the sweet spot, but I can understand the allure of a scale like 10EDL. After all, we do have pentatonic scales and many songs written in them. (Of course, we also have a few songs with four notes, as well as the Google Fischinger player with four-note scales. But 8EDL doesn't really have the correct four notes.) Just as we did for the tritave-based 15EDL above, we'll have to cheat and end our scale on Degree 5, even though this last note isn't really playable in Mocha.
So we may want to include 10EDL in the sweet spot, since there is a real musician (Sevish) writing music in 10EDL. As far as I know, no one has written music in 20EDL, so perhaps this is a reason not to include 20EDL in the sweet spot. Meanwhile, I do see evidence for 18EDL being used as a scale in real music -- the interval 18/17, "the Arabic lute index finger." This name suggests that at one time, Arabic lutes (ouds) were fretted to divide the string in eighteenths for 18EDL.
The idea of 10-18EDL as the sweet spot also reminds me of one justification for bases 10-18 as the sweet spot for number bases (decimal through octodecimal). A few posters at the Dozenal Forum have mentioned the idea of "seven plus or minus two" (that is, the range 5-9) as the ideal length of lists that humans can handle. Thus bases 10-18 contain 5-9 pairs of digits, and the 10-18EDL scales contain 5-9 notes. Indeed, the most commonly played scales contain five (pentatonic) to nine (melodic minor) notes as well.
That settles it -- 10-18EDL is the sweet spot based on real music. Let's write out all of the scales in the sweet spot, using Kite's new color notation.
The 10EDL Octave:
Degree Ratio Cents Note
10 1/1 0 green C
9 10/9 182 white D
8 5/4 386 white E
7 10/7 617 red F#
6 5/3 884 white A
5 2/1 1200 green C
The 12EDL Octave:
Degree Ratio Cents Note
12 1/1 0 white A
11 12/11 151 lavender B
10 6/5 316 green C
9 4/3 498 white D
8 3/2 702 white E
7 12/7 933 red F#
6 2/1 1200 white A
The 14EDL Octave:
Degree Ratio Cents Note
14 1/1 0 red F#
13 14/13 128 thu G
12 7/6 267 white A
11 14/11 418 lavender B
10 7/5 583 green C
9 14/9 765 white D
8 7/4 969 white E
7 2/1 1200 red F#
The 16EDL Octave:
Degree Ratio Cents Note
16 1/1 0 white E
15 16/15 112 green F
14 8/7 231 red F#
13 16/13 359 thu G
12 4/3 498 white A
11 16/11 649 lavender B
10 8/5 814 green C
9 16/9 996 white D
8 2/1 1200 white E
The 18EDL Octave:
Degree Ratio Cents Note
18 1/1 0 white D
17 18/17 99 su D#
16 9/8 204 white E
15 6/5 316 green F
14 9/7 435 red F#
13 18/13 563 thu G
12 3/2 702 white A
11 18/11 853 lavender B
10 9/5 1018 green C
9 2/1 1200 white D
Actually, let's go ahead and sneak 20EDL and 22EDL into our sweet spot anyway (just as I wanted to sneak 24EDL and 26EDL back when 12-22EDL was our sweet spot). Here 20EDL and 22EDL may be useful only because they are the first EDL's with something resembling a "leading tone" -- the last ascending note that leads into the octave:
The 20EDL Octave:
Degree Ratio Cents Note
20 1/1 0 green C
19 20/19 89 inu C#
18 10/9 182 white D
17 20/17 281 su D#
16 5/4 386 white E
15 4/3 498 green F
14 10/7 617 red F#
13 20/13 746 thu G
12 5/3 884 white A
11 20/11 1035 lavender B
10 2/1 1200 green C
The 22EDL Octave:
Degree Ratio Cents Note
22 1/1 0 lavender B
21 22/21 81 red B
20 11/10 165 green C
19 22/19 254 inu C#
18 11/9 347 white D
17 22/17 446 su D#
16 11/8 551 white E
15 22/15 663 green F
14 11/7 782 red F#
13 22/13 911 thu G
12 11/6 1049 white A
11 2/1 1200 lavender B
Let's add two more tritave scales in this range -- since I already wrote 15EDL earlier in this post, let's add 18EDL and 21EDL:
The 18EDL Tritave:
Degree Ratio Cents Note
18 1/1 0 white D
17 18/17 99 su D#
16 9/8 204 white E
15 6/5 316 green F
14 9/7 435 red F#
13 18/13 563 thu G
12 3/2 702 white A
11 18/11 853 lavender B
10 9/5 1018 green C
9 2/1 1200 white D
8 9/4 1404 white E
7 18/7 1635 red F#
6 3/1 1902 white A
The 21EDL Tritave:
Degree Ratio Cents Note
21 1/1 0 red B
20 21/20 84 green C
19 21/19 173 inu C#
18 7/6 267 white D
17 21/17 366 su D#
16 21/16 471 white E
15 7/5 583 green F
14 3/2 702 red F#
13 21/13 830 thu G
12 7/4 969 white A
11 21/11 1119 lavender B
10 21/10 1284 green C
9 7/3 1467 white D
8 21/8 1671 white E
7 3/1 1902 red F#
[2021 update: Last year on Tau Day, I mentioned that I was considering writing one of my songs from the old charter school, "Roots," in 24EDL. Instead, I wrote it in 16EDL. Meanwhile, "Diagrams" was written in 20EDL.]
The Arabic Lute
Returning to EDL scales, I've mentioned how fascinated I am by the name "Arabic lute index finger" for the interval 18/17, and its suggestion that the oud must have been fretted to 18EDL.
The following YouTube video is all about refretting a guitar to experiment which tuning makes the song sound better. The piece, written by Cage -- I mean Bach -- is called "Air." (Hofstadter gives "Air on G's String" as the title of one of his dialogues, but I don't know whether it's related to the "Air" piece in the video.)
Of the four tunings, one is just intonation, one is standard 12EDO, and the others are compromises between JI and 12EDO, called "well temperament."
In the comments at YouTube, many people found JI to be the best-sounding near the beginning, where many major chords are played. JI is based on pure ratios, such as the 4:5:6 major triad. But near the end, the piece became more melodic than harmonic. At this point, the best-sounding tuning according to the commenters became 12EDO, whose equal step sizes make melodies sound nice. The two well-temperaments are intermediate in both the harmonic and melodic sections. (The first tuning is closer to JI and thus sounds better harmonically, while the last tuning is closer to 12EDO and thus sounds better melodically.)
Notice that EDL scales are based on ratios and thus are closely related to JI. It's a shame, though, that Mocha can only play one note at a time -- it's melodic rather than harmonic. (Last year on Tau Day, I did mention the Atari computer that could play EDL-based harmony .)
The fretting is quite complex for all of the tunings except 12EDO. Actually, a fretting based on EDL's (which our hypothetical oud has) would look even simpler. Like 12EDO, the frets would at the same position for all the strings. The only difference is that the frets would be equally spaced apart -- exactly 1/18 of the length of the whole string. (That's what 18EDL -- 18 equal divisions of length -- really means after all.)
Imagine if the guitars in the video were fretted to 18EDL. Let's keep the standard EADGBE tuning, except we assume that all of these are white notes (Kite colors). This means that the interval between consecutive strings (E-A, A-D, and so on) is the perfect fourth 4/3. Then all of the notes fretted at the first fret (by the index finger, of course) are colored 17u ("su"), Second fret notes become white, third fret notes are green, and fourth fret notes are red.
Now let's try playing some chords using this tuning. We start with an E major chord -- a basic open chord that beginning guitarists learn to play. This chord is played as:
EBEG#BE
wE-wB-wE-suG#-wB-wE
The JI 4:5:6 would require G# to be yellow rather than su, but yellow (an "over" or "otonal" color) isn't available in EDL (which is based on "under" or "utonal" colors). Fortunately, the su 3rd (about 393 cents) lies about halfway between the yellow 5/4 3rd and the 12EDO major 3rd. Thus this E major chord will probably sound like one of the well temperaments from the video.
EDL's are supposed to be better at playing under/utonal chords, which minor chords are. So let's try playing E minor rather than E major:
EBEGBE
wE-wB-wE-wG-wB-wE
Now all the notes end up white. Chords with all white notes are considered dissonant -- this chord is known as the Pythagorean minor chord.
Let's try some A chords now. We begin with A major:
xAEAC#E
wA-wE-wA-wC#-wE
This is another dissonant all-white chord -- the Pythagorean major chord. We move on to A minor:
xAEACE
wA-wE-wA-suB#-wE
The first fret on the B string isn't even C -- officially it's su B#. This note is 13 cents flatter than green C -- the note that belongs in an A minor chord. It probably won't sound terrible in a chord only because at 303 cents, the interval wA-suB# is only three cents wider than the 12EDO minor 3rd and sounds indistinguishable from it.
So far, we have the passable (or "well tempered") E major and A minor chords, and dissonant Pythagorean E minor and A major chords. Moving on to D major, an obvious problem arises:
xxDADF#
wD-wA-gD-wF#
Now the two D notes aren't even the same color, so now we have dissonant octaves. Changing this from D major to D minor doesn't eliminate the dissonant octave on D.
So why are we having so much trouble with these basic chords? If we return to A minor, we notice that the third fret on the open A string indeed plays the green C needed for A minor. But the A string can't be used for green C, because it's too busy sounding the white A! In other words, most of the time, two or three notes we need to make chords sound on the same string.
The following link describes how actual Arabic lutes may have been tuned:
https://larkinthemorning.com/blogs/articles/the-oud-the-arabic-lute
Notice that even on that blog, there's still no mention of the "Arabic lute index finger," the interval 18/17, or 18EDL fretting. But we do see that instead of EADGBE, traditional tunings for ouds include DGADGC, ADEADG, and EABEAD.
But unfortunately, even with EABEAD, some JI chords remain difficult to finger and play. One of these days I'd like to solve the mystery of the Arabic lute, its index finger, and 18EDL.
Conclusion
If I timed this properly, it should post at 3:18 Pacific Time. Happy Tau Day -- we'll get back to Chapter 5 of Peterik very soon!
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