Thursday, June 28, 2018

Computer Music Post: Tau Day Edition

Table of Contents

1. Pappas Question of the Day
2. That Time of the Year Again
3. June Gloom
4. Ingenuity with Edgenuity
5. Tau Day Links
6. More Tau Day Music
7. Other Tritave-Based EDL's
8. Converting Srutis to EDL
9. The Atari 16-Bit EDL Scale
10. Conclusion

Pappas Question of the Day

Today on her Mathematics Calendar 2018, Theoni Pappas writes:

Find the volume of this cone to the nearest whole number.

(Givens from diagram of right cone: height = 3, slant height = 3sqrt(2), right angle at top of cone, and 45-degree angle at base of cone.)

The only value required by the right cone volume formula that isn't given is the radius. It's easy to see that the radius, height, and slant height form a 45-45-90 triangle -- as usual, Pappas likes to include special right triangles in her problems. Since the height (one leg of the 45-45-90 triangle) is 3, the radius (the other leg) is also 3.

We now plug these values into the cone volume formula, which is V = (1/6)tau r^2 h:

V = (1/6)tau r^2 h
V = (1/6)(6.28)(3^2)(3)
V = 28.26 cubic units

Rounding off to the nearest whole number, the volume is 28 cubic units -- and of course, today's date is the 28th.

That Time of the Year Again

Hmm, that doesn't look like the usual formula for the volume of a cone -- V = (1/3)pi r^2 h. Instead, we used V = (1/6)tau r^2 h, and we substituted in 6.28 for tau. 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:

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 seven 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 2018 "State of the Tau" address:

According to this link, Hartl updated his Tau Manifesto this year to incorporate the areas and volumes of hyperspheres -- that is, spheres in dimensions higher than three.

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.)

June Gloom

It's June, the beginning of summer. It's a time for students -- and many teachers -- to be happy. Yet I can't help but feel gloomy this summer -- and it's all because of the lost opportunity to earn some money by teaching the summer Algebra I class. And on no day do I feel sadder and gloomier about this lost opportunity than today, Tau Day.

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. Therefore, if enrollment hadn't dropped and I'd been allowed to teach summer school, today would have been a Tau Day party in my classes. And now you can see why I'm so sad -- I could have had a party with my students today!

At this point you might be wondering, weren't the summer school classes based on Edgenuity software, with a strict pacing plan? So how could there be time for a Tau Day party?

Well, let's think about the schedule again. The summer classes are three weeks long, Monday through Thursday, and today is the second Thursday. The agreement I made with the other two Algebra 1A teachers (that is, the only two who would actually teach Algebra 1A) was that the last day of each school week would be the unit test on Edgenuity. Today's test, the second such test, is on graphing linear functions and correspond to Chapters 3 and 4 in Glencoe. Students are granted the entire period (nearly two hours) to take the test.

The idea I had for my own classes was that Thursdays -- during the time left over if the students finished early -- would become somewhat relaxed. (Indeed, my policy on cell phones would have been that they're only allowed on Thursdays after the test is complete.)

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 Thursdays after the test, I could sing a song that's more fun instead. For example, last Thursday I would've played a song from Square One TV, such as "Square Song." I choose this song (really a rap) only because it was originally posted to YouTube eleven years ago on June 21st (as you know how I like to celebrate YouTube anniversaries).

Today I would have played Vi Hart's Tau Day tune, "Song About a Circle Constant." Notice that neither Square Song nor Song About a Circle Constant are directly related to Algebra I -- indeed, both would fit better in Geometry. But again, these songs are fun to sing after a long, hard test -- 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).

No party would be complete without food, of course. I could distribute snack to everyone in the class, or I could make the students earn the food. And considering that the students have just finished their second unit test -- one that's been graded instantly by Edgenuity. This means that I can immediately check the scores and award food to the students with the top scores.

I can keep up the tau = 2pi idea and let the food prize be pie. Recall that back on Pi Day, I purchased personal 7-Eleven pies for three students. There were three pies because I passed them out to the video news class I subbed for that day (which contained only three students, since the rest of them were with their teacher on a field trip to Tennessee), but notice that pi can be rounded down to 3.

So let's round tau down to six and buy pies for the top six students (in each of the two periods that I would have been assigned). 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.

Instead of pies, I could give out cupcakes instead -- just like the cupcakes mentioned in Michael Hartl's State of the Tau address. By the way, since I didn't actually teach any summer school today, what did I really eat today? Well, I ate at Tau-co Bell, of course. I probably should have ordered some "tau-cos," but instead I tried the new $5 Steak Nachos Box. According to the advertisement, this box contains two times as much steak as before -- just as tau is two times pi.

Ingenuity with Edgenuity

I can't help but be curious about how well this year's summer school students are doing in the Algebra I class that I almost, but didn't, get to teach. And then I suddenly remember -- I was granted access to Edgenuity (in anticipating that I'd teach the summer class), and I still have that access. This means that I can log in right now and see how well the students are faring.

And so I peek at one of the two remaining Algebra I teachers -- the one who student-taught earlier this year -- and find out her kids' scores on today's unit test. I'll point out the students who would have earned pies, if only these were my own classes.

As I wrote earlier, I'd give pies (or cupcakes) to the top six students plus the most improved student in each period. On Edgenuity, I can't distinguish between first and second period, and so I'll list the top 12 scores here. Also, on Edgenuity, there are three units that are tested today: "Introduction to Functions," "Linear Functions," and "Point-Slope Form and Linear Equations." Nearly every student completed the first unit test today, about half completed the second, and only one completed the third, therefore this list is based on only the first unit, "Introduction to Functions." There are 25 questions on the unit test, and so each question is worth four percentage points.

Score     # of Students
92          1
88          1
84          1
80          2
76          3
72          5

Well, that's actually 13 students -- so only one pie is left. (Most likely, it would have depended on whether it's first or second period in which the 72's are scored.)

Of course, this is a summer class, so it's expected that most of the scores are low. Only one student earns an A on the test. His overall grade is also the highest at 84% -- a middling B.

As for the pies awarded to students whose grades improved the most, I must admit that most scores dropped from last week's to this week's test. I suppose this is expected -- even as a student teacher, I noticed that graphing linear functions (Chapter 4 of Glencoe) is always more difficult that just solving linear equations (Chapter 2). And to think that Edgenuity actually expects kids to graph the linear equations before solving them! (That why we override the Edgenuity order.)

Actually, the student who earned 92% is tied for the most improved student -- on last week's test, his score was only 76%. (The student he's tied with improved from 60% to 76%, so both have an increase of 16 points.) Depending on which period the students are in (including the 13th student who earned a 72%), the 14th and final pie would go to either the student who improved from 60% to 68%, or the one who maintained a score of 60%. (Due to the difficulty of the second test, even maintaining the grade counts as a major accomplishment.)

Well, that's how the grades in the actual class went. In many ways, it's better to award food based on the number of pies I brought as opposed to saying in advance, "Everyone with an A or B gets a pie!" (which might result in my having too few or too many pies). You might argue that this is sort of like grading on a curve (except that the letter grades don't change based on how much food I bring).

Tau Day Links

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.

Oh, and by the way, Vi Hart adds a new Tau Day video for this year -- "Suspend Your Disbelief":

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. 3Blue1Brown:

Michael Hartl links to this video in his State of the Tau Address this year. The first mathematician to use many of our current math symbols -- including pi -- was Euler (yes, the same Euler who solved the Bridges of Konigsberg problem). But according to this video, Euler was considering defining the symbol pi to represent the number we now call tau.

5. The Coding Train:

Here Dan uses geometric probability to approximate the values of pi and tau. Here's a simple program that does the same on Mocha:

10 A=0:B=0
20 B=B+1
30 X=RND(0)
40 Y=RND(0)
50 IF X*X+Y*Y<1 THEN A=A+1
60 PRINT 4*A/B,8*A/B
70 GOTO 20

Estimates for pi appear in the left column and estimates for tau appear in the right column. This program has an infinite loop, so press Esc when you feel that the approximations are good enough.

6. Algy Cuber:

Here Algy Cuber uses Euler's identity to calculate the ith power and ith root of i. Cuber brands this as a Tau Day video -- but unfortunately, she actually uses pi in her calculations.

7. Samuel El Pesado:

Oh, this is a video that was first posted six years ago. But once I created this blog I couldn't find the video again until today. Here a group of high school students blow up a pi(e) for Tau Day! (Happy early Fourth of July!)

8. 280zman:

This is the Square Song from Square One TV that I mentioned earlier in this blog entry. I decided that I might as well post it today even though it has nothing to do with tau or circles.

More Tau Day Music

And now you're probably saying, here we go again! Even after my summer school class is cancelled, first I start whining about summer school again, and now I go right back to music -- a topic that I wouldn't discuss unless I had a class to sing songs in.

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. It was around this time last year when I first read the Pappas book about musical scales. On Tau Day, I wrote about 12EDO, our usual scale, and its relationship to 5-limit ratios. A few days later, I wrote about the Indian sruti scale and 22EDO, and then I kept writing about more EDO scales until I stumbled upon the Mocha computer emulator and realized that EDL scale, not EDO scales, fit Mocha's sound command.

This is also the first day devoted to a constant since I wrote about those EDL scales. So it's logical to write a tau song in one of these new EDL scales, but which one should we choose?

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    ocher G
4           12             5/4        white A
5           11             15/11    amber 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-aB). 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:

10 N=16
20 FOR X=1 TO 52
40 SOUND 261-N*(16-A),4
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

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:

Possible 15EDL root notes in Mocha:
Degree     Note
15            green F
30            green F
45            green Bb
60            green F
75            deep green Db
90            green Bb
105          greenish G
120          green F
135          green Eb
150          deep green Db
165          amber-green C
180          green Bb
195          ocher-green Ab
210          greenish G
225          deep green Gb
240          green F
255          umber-green 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 ocher-green Ab (N=13 as ocher notes are sharper than they look -- white A is at Degree 192) while the nearest key to Hart's G is greenish G (N=14). Meanwhile, the closest key to Blake's C is amber-green 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 (who posted the Euler video above), and that video itself mentions another video. I post both of them here:

Other Tritave-Based EDL's

The first true tritave-based EDL available in Mocha is 18EDL, since it ends on the last playable Degree (that is, Degree 6):

The 18EDL tritave scale:
Degree     Ratio     Note
18            1/1         white D
17            18/17     umber D#
16            9/8         white E
15            6/5         green F
14            9/7         red F#
13            18/13     ocher G
12            3/2         white A
11            18/11     amber B
10            9/5         green C
9              2/1         white D
8              9/4         white E
7              18/7       red F#
6              3/1         white A

And the first odd tritave-based EDL available is 21EDL. It's easy to argue that it isn't truly a tritave EDL unless it's an odd EDL, since the octave is available in even EDL's:

The 21EDL tritave scale:
Degree     Ratio     Note
21            1/1         red B
20            21/20     green C
19            21/19     khaki C#
18            7/6         white D
17            21/17     umber D#
16            21/16     white E
15            7/5         green F
14            3/2         red F#
13            21/13     ocher G
12            7/4         white A
11            21/11     amber B
10            21/10     green C
9              7/3         white D
8              21/8       white E
7              3/1         red F#

In all tritave EDL's, the perfect fifth is available. In fact, any note above the octave appears one octave lower (such as gC, wD, wE, rF# in 21EDL).

It's interesting to find EDL's that approximate Bohlen-Pierce, either the full 13-note scale (that is, 13EDT) or the nine-note mode known as Lambda. So far, all of our tritave EDL's have an even number of notes (since 3n-EDL contains 2n notes in a tritave). Thus the closest EDL's to Lambda are 12EDL (eight notes) and 15EDL (ten notes), while the closest EDL's to full BP are 18EDL (twelve notes) and 21EDL (fourteen notes).

The creators of the Bohlen-Pierce scale point out that the scale is based on odd harmonics. So we can argue that a true EDL akin to BP should contain only odd degrees. For example, we might consider 27EDL, with only the odd Degrees played from 27 to 9, as akin to Lambda. Notice that the first interval of this scale would be 27/25, which is part of BP but not Lambda mode -- this is similar to how 14EDL contains the same number of notes as a major scale but sounds like Locrian mode. The mode of BP closest to 27EDL is probably (Elaine) Walker A.

Similarly, 39EDL played on odd Degrees has the same number of notes as full BP. In a previous post, I wrote that an extremely accurate EDL for full BP is 63EDL. I found this EDL by running the Mocha program that converts EDO's to EDL's and changing it to tritaves (EDT's) instead.

Converting Srutis to EDL

Since I already mentioned the Indian sruti scale in this post, let's try converting it to EDL. Recall that this is a 22-note octave-based scale. The sruti scale contains one unison and one perfect fifth, but otherwise two versions of all the recognizable intervals of 12EDO (minor second, major second, minor third, and so on).

The EDL with the same number of notes is 44EDL. But this scale would sound nothing like the original sruti scale. The sruti scale contains many pairs of notes that are a syntonic comma (in other words, 81/80) apart. The smallest interval in 44EDL is 44/43, which is already larger than this comma (also known as the green comma -- the difference between white C and green C).

Of course, you might point out that one step of 22EDO is also larger than a syntonic comma, so EDO's are no better than EDL's in this regard. Still, the "patent val" of 22EDO (that is, the mappings of the primes 2, 3, 5 to notes of 22EDO) maps the green comma to a single step (hence it's not a meantone EDO), and all of the srutis are mapped to the corresponding note of 22EDO. On the other hand, EDL's don't have "patent vals" that map 5-limit intervals.

One way to find a suitable EDL for the sruti scale is to find an EDL that can play the major scale exactly (such as 180EDL) and then add in the missing notes. The minor third, minor seventh, and tritone (45/32) all appear exactly in 180EDL, leaving only the minor second and minor sixth that must be approximated.

Then we raise or lower each note by one or two Degrees to represent the syntonic comma. Notice that the green comma is exactly one step between Degrees 81 and 80, or exactly two steps between Degrees 162 and 160. So we use two steps to represent the comma for the lower part of the scale and one step for the higher part of the scale. The dividing line between a two-step comma and a one-step comma would be the perfect fifth (which has no comma).

The tonic for this proposed scale is Degree 180 (green Bb). This results in some of the notes having awkward names, such as Cb for the minor second. An alternative basis for a sruti scale would be Degree 256 (white E). This time, we begin with the umber/khaki E major scale as mentioned in a previous post and proceed from there. Moreover, two exact syntonic commas are already available for this scale -- 243/240 at the minor second (that is, white/green F) and 162/160 at the minor sixth (white/green C).

By the way, I was also recently thinking about 5-limit rhythms. After all, just intonation and Mocha music are all about introducing higher and higher primes to the pitch -- so that now we go beyond the 5-limit music of 12EDO or Indian sruti music to new primes as high as 17 and 19.

Yet with rhythms, we're basically stuck at the 3-limit. Musical notes are generally based on powers of two -- whole, half, quarter, eighth, and sixteenth notes. We occasionally see the prime 3 when we have dotted notes (as in the dotted half note of three beats) or triplets (as in the eighth note triplet -- three make up a quarter note, so these are basically twelfth notes). It's ironic that with pitch we're venturing all the way up to 19, but for rhythm we can't even reach 5.

A famous jazz song, "Take Five," is written in 5/4 time. Just as changing the line in the randomizer program I wrote a few weeks ago to B=3 creates a song in 3/4 time, changing that same line to B=5 makes a random song in 5/4 time.

On the other hand, fives in the denominator almost never appear in music. But just as we choose a green root in Mocha to play a 5-limit major triad or scale over that root, we can declare that a whole note to be of length 20, so that we can play both quarter notes of length 5 and "fifth notes" which would be of length 4. Moreover, the PLAY command is even more appropriate for introducing different rhythms, since we can use L5 for fifth notes, L7 for seventh notes, and so on. These rhythms don't appear in real music, but there's nothing stopping us from programming such songs in Mocha.

The Atari 16-Bit EDL Scale

As I mentioned last year, one EDL scale that piqued my interest is designed not for Mocha, but another 1980's-based computer, the Atari:

This is another cliffhanger -- I said that I'd write more about the Atari 16-bit scale in a later post, but then I never returned to it. The fact that we have access to the Mocha emulator but not an Atari emulator discourages me from looking too deeply into Atari music.

But still, I just can't get the idea that we could generate more accurate 16-bit music (as compared to the 8-bit music of Mocha) as well as play harmony (two 16-bit notes, or one 16-bit and two 8-bit notes) out of my mind completely.

Once again, I don't have an Atari emulator, so all I know about the music comes from this chart:

16-Bit and 8-Bit Note Table
C27357OCTAVE 1C84830OCTAVE 6
C13675OCTAVE 2C42114OCTAVE 7
C6834243OCTAVE 3C207OCTAVE 8
C3414121OCTAVE 4C100OCTAVE 9

The first thing we notice about this chart is that in both the 8-bit and 16-bit columns, the lower notes have larger values than the higher notes. This indicates that this chart uses the Degrees of EDL directly rather than subtracting the Degrees from Bridge 261 as in Mocha.

Let's look at the 8-bit column first as this is simpler. This is very similar to Mocha -- by convention, Middle C is in "Octave 4," and so one octave below Middle C and three octaves above Middle C are available in 8-bit.

As usual, if two notes are an octave apart, the lower note should have exactly twice the value of the higher note. Indeed, D#3 has value 204, and D#4 has value 102. Then C#3 has value 230, so that C#4 should have value 115 -- but 114 is listed in the table instead. So what gives?

Notice that if the low note has an odd value, then the value of higher note always has half the value of the lower note rounded down. Thus C3 is 243 and C4 is 121, not 122.

This actually implies that there is a Bridge corresponding to a string of no length -- except that the Bridge is -1, not 0. So C4 is 121 because 121 is exactly halfway between C3 (243) and Bridge -1, while even values have a 50-50 chance of being rounded up or down in calculating the mean of that value and -1.

The notes such that the number of steps from Bridge -1 is a power of two are 0, 1, 3, 7, 15, 31, 63, 127, and finally 255. The values 15 and 31 are given as B6 and B5 respectively. Therefore let's declare all such powers of two to be the note white B and then we proceed to name all of the notes just as in Mocha. (The values 15 and 31 are considered to be Degrees 16 and 32 -- that is, add one to the value to find the Degree. Just intervals are found by take the ratio of Degrees, not values.) We see that the Atari Degrees are a perfect fourth lower than the corresponding Mocha Degrees -- so just as Mocha Degree 256 is white E3, Atari Degree 256 (the lowest playable note in 8-bit) is white B2.

A similar analysis reveals that the Bridge for the 16-bit column is -7. Again, this is revealed when we notice that the values for C1 and C2 are 27357 and 13675. If we add seven to each of these values, then 27364/13682 is exactly 2. And if the value is even (so that adding 7 makes it odd), then there's again a 50-50 chance whether we must add or subtract 1.

The 16-bit notes with values -6, -5, -3, 1, 9, 25, 57, and so on (Degrees 1, 2, 4, 8, 16, 32, 64, namely the powers of two) appear to be A's, so we can label these as white A.

But let's make a quick comparison here -- white A4 in 16-bit would be 2041, slightly flatter than the 2027 value given in the table. Meanwhile, white A4 in 8-bit would be 71 (two tritaves below white B7 which is Degree 8), slightly sharper than the 72 value given in the table. (Once again, notice that in 8-bit, Degree 72 is value 71.) I presume that the A4 given in the table is Concert A4 (440 Hz), which is apparently between 8-bit white A and 16-bit white A.

The discrepancy between 8-bit white A and 16-bit white A would be insignificant (after all, we don't know how Mocha white A compares to any of these three notes) -- except for the fact that harmony is possible on the Atari. Since it's possible to play chords with one 16-bit and two 8-bit notes, we'd really need to know the difference between the two A's.

Here's how I'd find this difference. Let's make the ansatz that the value given for the A1 in the 16-bit table is exactly Concert A (since it's the most accurate after all). The chart gives us the value 16264, which corresponds to Degree 16271. We've defined 16-bit white A to be Degree 16384 (a power of two), and so we find that 16-bit white A is:

16384/16271 -- log(16384/16271)/log(2) * 1200 = 11.9816

or about twelve cents flatter than Concert A.

Now let's make the same calculation for 8-bit white A. The value for A3, the lowest A given on the chart, is 144 (that is, Degree 145), while 8-bit white A has been defined as Degree 144. So we find that 8-bit white A is:

145/144 -- log(145/144)/log(2) * 1200 = 11.9809

or about twelve cents sharper than Concert A. (That the number of cents is almost exactly the same both ways is a coincidence!)

And so the two A's are about 24 cents -- around the size of a Pythagorean comma -- apart. Since the values in the table aren't necessarily exact, I've obtained different estimates for the difference between 16-bit "white" and 8-bit "white" notes using notes other than A. All estimates lie somewhere between the size of the syntonic (green) comma of 21.5 cents and the septimal (red) comma of 27.3 cents.

If we assume that the difference is the syntonic comma, then this implies that 8-bit F4, 16-bit A4, and 8-bit C5 together form a just F major triad. On the other hand, if we assume that the difference is actually the septimal comma, then this is no longer a just major triad.

When there are two values -- an 8-bit and a 16-bit value -- for the same note, dividing the latter by the former seems to be close to 28. For example, for the note C3, we divide 6834/243 = 28.123. If we use Degrees instead of values, then the quotient is even closer to 28 -- 6841/244 = 28.037.

So let's conjecture that if the 16-bit and 8-bit Degrees are exactly in the ratio 28/1, then the two notes are in unison. The fact that 28 = 7 * 4 gives away that this has something to do with 7-limit or the septimal comma. We can verify this by noting that 28/1 is close to five octaves or 32/1 -- that is the fundamental notes (Degree 1) for 16-bit and 8-bit are about five octaves apart. It's short of five octaves by the value 32/28 = 8/7, a septimal whole tone. But we've labeled these notes "white A" and "white B," which are 9/8 apart. The difference between 8/7 and 9/8 is indeed the red comma 64/63.

Let's assume that the white notes for 8-bit are indeed white. Then the note we've been calling "white A" should instead be the note 8/7 below white B, which is "blue A." The otonal color blue appears because the 16-bit degrees are divided by 28. All other colors that appear in 16-bit are utonal.

Thus 8-bit F4, 16-bit A4, 8-bit C5 is not a just major triad (but it's close). Instead, 8-bit F4, 16-bit Ab4 (Degree 4374 or value 4367), 8-bit C5 form a subminor (blue) triad.

I'd prefer to have yellow notes over blue notes -- if there are going to be any otonal notes at all, let them be 5-limit, not 7-limit. But the appearance of the number 28 in the table gives it away that the 7-limit is more significant here. And of course, today's date is the 28th -- which takes us back full circle (Get it?) to 28, 6.28, and tau.


Last year I sneaked the traditionalists' debate into the Tau Day post. I won't do so this year -- luckily for you, the traditionalists have been inactive lately (except for Ze'ev Wurman making a second recent comment at the Joanne Jacobs website).

If I timed this correct, this post should appear on exactly 6/28 at 3:18 -- as in tau = 6.28318. In other words, Happy Tau Day everybody!

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