1. Pappas Question of the Day
2. Barry Garelick: Ratios and Subtraction
3. Summer School Final Grades
4. Graphing Linear Equations in the Common Core
5. The Importance of Similarity
6. Music: Tall Kite's New Color Notation
7. Music: From Xenharmonic to Tall Kite
8. Conclusion: Lunar Eclipse and Mars Opposition
Pappas Question of the Day
Today on her Mathematics Calendar 2018, Theoni Pappas writes:
What is the heading of vector v from the harbor?
I know what you're thinking -- finally Pappas has included spherical geometry on her calendar! The givens include the coordinates (longitude and latitude) of both the starting and destination harbors, and then we have to set up the Spherical Law of Sines or Napier's Laws to find the initial heading....
Uh, that's not exactly what this question is. Instead, the givens include only that the heading is 61 degrees, as measured counterclockwise from the positive x-axis (as we normally do in trig). But in navigation, headings are usually measured clockwise from north, not counterclockwise from east. So we calculate 90 - 61 = N 29 E, or 29 degrees east of north -- and of course, today's date is the 29th. I guess we should have known that Pappas would never really put spherical geometry on her calendar!
Barry Garelick: Ratios and Subtraction
This is a traditionalists post. Here's a link to what Barry Garelick posted yesterday:
https://traditionalmath.wordpress.com/2018/07/28/ratios-subtraction-and-standing-up-for-yourself/
Garelick flashes back to his days as a student teacher:
During a lesson on ratios for my seventh grade math class, I introduced the concept by saying how when we compare things we sometimes use subtraction, like comparing heights. That led into how ratios is a comparison by division. There are times when subtraction is appropriate and times when ratios are a better measure of comparison. This is a standard introduction in most math books–it certainly was in mine, and I’ve seen it in many.
My teacher criticized me for talking about subtraction in my lesson, feeling it detracted from (rather than setting up the discussion for) ratios. The retired math teacher agreed. I said nothing, but I felt and still feel that they were dead wrong.
In past posts, I've written about the concept of an "anticipatory set." Back during the 2016 Blogging Initiative (MTBoS), some other math teachers wrote about how they began their classes with an anticipatory set -- a introductory lesson that gets the students thinking before the main lesson. And anticipatory sets were emphasized during my own days as a student teacher (high school Algebra I).
Anyway, apparently Garelick wanted to mention subtraction as an anticipatory set to introduce the seventh grade ratios lesson. But his master teacher and supervising teacher didn't accept this, even though "it certainly was in" his textbook. He should have been protected by the presence of this anticipatory lesson in the text, and yet he wasn't. (Then again, he could mean that it's in the text he teaches with now, not the text he used when student teaching.)
I decided to compare this to the U of Chicago seventh grade text (Transition Mathematics). Lesson 11-5 of that text is called "The Ratio Comparison Model for Division," but there's no real discussion of why we'd want to divide rather than subtract.
Neither was there any discussion of subtraction vs. division in the Illinois State text that I used during my one year of teaching at a charter middle school. Notice that by the time I started teaching there, all thoughts of "anticipatory sets" have been long since forgotten, since the anticipatory activities for Illinois State are the projects. When rates and ratios are mentioned in the projects, students are simply asked to divide -- for example, consider the opening project on mousetrap cars (calculate the speed) or the first main seventh grade project "Orienteering" (including draw a map to scale). So at no point are the students directed to compare division to subtraction.
Technically, Warm-Ups are not anticipatory sets. But it's possible for me to make a Warm-Up into an anticipatory set -- for example, before the first ratio lesson, just ask a subtraction question. I was required to use Illinois State daily assessments as Warm-Ups (and there were no subtraction questions there), but I suppose I could have asked a subtraction question on the first day of the unit. (They can't answer the Illinois State ratio questions because they haven't learned ratios yet, so there's a little leeway there.)
Garelick continues:
I’ve taught several seventh grade classes at this point, and I never hesitate to start off with talking about subtraction as the springboard into a discussion on ratios. As a way of illustrating how the difference between what subtraction and ratios measure, (and the appropriateness of each) I show a video of the old comedy team Abbott and Costello which no student has ever heard of.
Well, let me actually embed the Abbott and Costello video to which he refers here, which is called "You're 40, she's 10":
I've heard of Abbott and Costello, because of the Square One TV parody "Cabot and Marshmallow":
There's also a parody of the famous "Who's on First" routine. Cabot has set Marshmallow up on a blind date with a girl. What is her name. (Notice the period there.)
Anyway, it might even be possible to create a project based on subtraction vs. division. For example, students might have to draw a picture of themselves -- but the human body can't fit on a page. So they can either subtract or divide to make themselves fit on the page. A student's head might be seven inches tall, and so if she divides by seven, then she can draw her head to be one inch on the page -- but if she subtracts seven, then her picture will have no head at all (while her legs still can't fit).
This project, while not an Illinois State project, fits Garelick's subtraction vs. division idea. But of course, Garelick is a traditionalist, so he's opposed to any projects at all in math class.
Neither SteveH nor any other traditionalist has commented on Garelick's latest post yet as of now. I see only one "comment" listed there -- a pingback to a "Nonpartisan Education Group." (I'm not quite sure how pingbacks work -- after all, I'm linking to Garelick's site right now, but this post will never be "pingbacked.")
Summer School Final Grades
Thursday was the last day of the B session, so let's check out those final Algebra 1B grades. Once again, these grades don't include a written district final exam -- and again, the final exam might be graded, but the results will never be posted to Edgenuity, so I will never know them.
- The top student from 1A has a final 1B grade of 91%.
- This isn't the top student in 1B -- he was beaten by two girls with grades of 95% and 92%.
- The bottom student from 1A has a final 1B grade of 48%.
- This isn't the bottom student from 1B -- one student earned a dismal 34%.
I decided to look at the bottom 1B student in more detail. He earned a 65 on his first test, Exponents and Radicals, but all his other test scores are in the 20's and 30's. He was not enrolled in either of the 1A classes, so presumably he passed 1A in the fall and failed only 1B in the spring.
Meanwhile, the 48% student earned D's on three of the first four tests and a 50 on the other. But he fell off a cliff on the last two tests, with scores of 16 and 20 on the two quadratic tests. Since this is the bottom student from 1A, I go back to check those scores as well. He earned a B on the first test on Solving Linear Equations (right in the middle of the pack with 80), but then failed all the other tests with scores of 40, 55, 50 -- when graphing linear functions begins.
Once again, the Algebra II class actually has the final exam grades posted on Edgenuity. One guy actually earns a perfect score on the final -- his overall 2B grade is 94%. The second highest score belongs to a girl who missed who question and earned a score of 96, but she has the highest overall grade in the 2B class with a grade of 97%.
On the low end, one student earns a 50 on the final. His 2B grade of 58% is the lowest among all students (excluding those who attended none or fewer than half of the class sessions). He passed only one test (Rational Functions, 68) and failed all the others with scores between 36 and 55.
This is a traditionalists post. I decided to include the summer school grades in this post since after all, this is what the traditionalists' debate is all about -- what's the best way to teach math?
Once again, the Algebra II class actually has the final exam grades posted on Edgenuity. One guy actually earns a perfect score on the final -- his overall 2B grade is 94%. The second highest score belongs to a girl who missed who question and earned a score of 96, but she has the highest overall grade in the 2B class with a grade of 97%.
On the low end, one student earns a 50 on the final. His 2B grade of 58% is the lowest among all students (excluding those who attended none or fewer than half of the class sessions). He passed only one test (Rational Functions, 68) and failed all the others with scores between 36 and 55.
This is a traditionalists post. I decided to include the summer school grades in this post since after all, this is what the traditionalists' debate is all about -- what's the best way to teach math?
Returning to Algebra I, consider how an Algebra I class is divided into semesters, 1A and 1B. We see that 1A is more or less the "linear semester" and 1B is the "quadratic semester." This year, we've seen slight differences among certain topics such as systems of (linear) equations -- the Glencoe text places them in Chapter 6, which is the end of the first semester (since the text has twelve chapters, the natural semester division is Chapters 1-6 and 7-12). On the other hand, systems of equations were taught during the B session this summer. Otherwise, the first/second semester division is consistent over all Algebra I classes.
It goes without saying that the quadratic 1B semester is harder than the linear 1A semester -- which is why there were more 1B classes than 1A this summer. But within 1A, we see that a unit students always seem to struggle on more than others is graphing linear equations -- and this is exactly what we observe by looking at the grades on Edgenuity.
Furthermore, consider the bottom 1A student who failed his graphing linear equations tests. In 1B, he did more or less OK on the first few tests, but failed miserably on the first quadratics test, which presumably contains graphing quadratic equations. We can't expect a student who doesn't know how to graph lines to be able suddenly to graph parabolas. When I was a student teacher covering the first semester of Algebra I, my students had trouble with graphing linear equations.
Much of the big traditionalists' debate concerns whether Common Core is a good idea -- that is, can the Common Core method actually raise achievement? It would be a huge triumph for the Common Core method if it can be used to make graphing linear equations easier for students to master.
Graphing Linear Equations in the Common Core
As I just wrote, we can cut an Algebra I text in half, with each half corresponding to one semester of the class. The first half is linear, while the second half is quadratic.
I mentioned in previous posts that this division roughly explains the algebra content of the first two years of Integrated Math. The linear half is taught in Integrated Math I, while the quadratic half is taught in Integrated Math II. (In Math II, we may go a step further and introduce complex numbers.)
It also explains the algebra content of Common Core Math 8, with the linear half of the Algebra I text covered in eighth grade. Systems of equations are definitely mentioned in the eighth grade standards, and so for the purposes of this discussion, systems are an Algebra 1A topic.
I've said before that while an Algebra I text can be cut cleanly into halves, with the first half equal to both Common Core 8 and Integrated Math I, the Geometry text can't be cut in half as cleanly. It's true that most of the eighth grade geometry content does appear in the first half of the U of Chicago Geometry text, one major topic (formal proofs) is in Geometry A but not Math 8, while another major topic (similarity) is in Math 8 but not Geometry A.
On the other hand, the Geometry A/B split does correspond to the geometry content of Integrated Math I/II respectively. This means that Common Core 8 and Integrated Math I aren't quite identical courses -- they agree on the algebra content but not the geometry content. Still, the fact that the courses are almost the same implies that a school district on the integrated pathway might attempt to accelerate students towards AP Calculus by letting them go directly from Common Core Math 8 to Integrated Math II as freshmen.
Anyway, similarity (the one topic in Math 8 but not Geometry A) is the key. The Common Core Math 8 Standards clearly connect similarity to graphing linear equations:
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.
But we ask, do most Common Core Math 8 texts really teach linear equations this way? I can only write about my own experience with the Illinois State eighth grade text. That traditional text was set up so that all standards are studied in naive order. This means that all EE (Expressions & Equations) standards are taught before any G (Geometry) standards are. Yet this standard listed above (8.EE6) is clearly intended to be taught after similarity (8.G4).
I wonder how many other eighth grade texts out there -- even newer Common Core-compliant texts -- teach all EE standards before any G standards. Such texts can't possibly follow the spirit of the Common Core Standards, where similarity is introduced in 8.G4 and used in 8.EE6.
And so it does mean that there is much potential for improvement here. If we could find a way to teach the eighth grade standards 8.G4 before 8.EE6, will this improve the students' understanding of linear equations -- so that by the time they reach Algebra I, we can make that grade dip during the graphing unit disappear?
By the way, I wonder whether it's possible to organize a high school Geometry class so that it can be divided elegantly into two halves -- with the first half matching the geometry content of Math 8, just like Algebra I. This itself may be a little tricky, since standard 8.G9, on the volumes of cones, cylinders, and spheres, doesn't appear until Chapter 10 of the U of Chicago text. A Geometry A course based on Math 8 would require us to cover all of the first four chapters, then parts of Chapter 5 (alternate interior angles), 6 (translations/rotations), 8 (Pythagoras), 10 (volume) -- and of course, Chapter 12 on similarity.
I probably wouldn't want to make this many changes, but at the very least we'd want to move similarity up into Geometry A if we want it to match Math 8.
The Importance of Similarity
With all of this discussion of similarity, we're now wondering, how easy is similarity a topic for students to grasp, compared to other Geometry topics? If students stumble with similarity, then having a linear functions unit based on similarity will definitely be unhelpful.
Well, let's check out the summer school grades for Geometry. First of all, I notice that to my surprise, Edgenuity actually includes similarity as part of Geometry A. Here is the list of Geometry A units:
- Foundations of Euclidean Geometry
- Geometric Transformations, Part I [reflections, translations -- dw]
- Geometric Transformations, Part II [rotations, compositions -- dw]
- Angles and Lines
- Triangles
- Triangle Congruence, Part I [SAS, ASA, AAS -- dw]
- Triangle Congruence, Part II [SSS, HL -- dw]
- Similarity Transformations
- Applying Similarity Concepts
- Quadrilaterals and Coordinate Algebra
With similarity included in this course, this seems like a strong framework for both Geometry A and Math 8 (with similarity used to prepare students for 8.EE6). But this is only if the grades are strong and students are actually learning similarity.
Let me post the scores of a few Geometry A students at random. We'll include the grades for some of the earlier units, and then the one on similarity transformations:
Student Overall FoEG GTII Triangles Similarity
#1 72 80 64 88 76
#2 77 72 76 60 68
#3 76 76 76 64 64
#4 74 72 68 92 52
#5 72 80 84 84 52
#6 85 72 96 96 72
#7 54 68 28 40 24
We see that Student #1 earned a higher grade on similarity (76) than the overall grade (72), Student #2 improved from the previous test to the similarity test (from 60 to 68), and Student #3 remained flat (both at 64). But for the other four students, there was a dramatic drop from the other tests to the similarity test. This includes Student #6, the top student in the class, who went an amazing 96 on the triangles test to a mere 72 on similarity. Student #7 was one of the lowest scorers, and that student dropped from an already dreadful 40 to an outright terrible 24.
When I see so many low scores on the similarity test, this makes me prefer to follow the U of Chicago Geometry text (and my first instincts) and keep similarity in the second semester. It surely doesn't inspire me to want to emphasize similarity during the linear functions unit in eighth grade. I must ask, how would it help students do well in a unit on which they struggle (linear functions) by emphasizing another unit on which they struggle (similarity)?
But let's return to the original topic of this post -- ratios in seventh grade math. There is, in fact, a Common Core Math 7 standard that relates ratios to both similarity and graphing linear functions:
CCSS.MATH.CONTENT.7.RP.A.2.D
Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
Hmm, so I wonder whether it's possible to emphasize this standard more strongly. We can have the students learn and master the graphs of y = rx, as mentioned in this above standard. Perhaps we can even sneak in some extra notation -- change the r to m without actually introducing slope. We might even have the students graph y = rx (or y = mx) for some negative values of r (or m).
Let's compromise between Garelick's traditionalism and project-based learning here -- we motivate ratios using subtraction, but it's up to the teacher to decide whether to do it his way (that is, without a project) or our way (using the drawing project mentioned above).
If it becomes necessary to tie this to similarity (for the eighth grade standard), then we can just draw in the right triangles with (0, 0) at one vertex. We can't escape dilations here -- but it's directly stated in the Common Core that a line passing through the center of the dilation is invariant. Since the line y = rx (or y = mx) passes through the origin, it is invariant.
Finally, one more transformation is needed. To get from y = mx to y = mx + b, a translation of b units upward (if b is positive) is used.
I can't guarantee that students will learn these tricky topics better and avoid those grade dips, but I hope that this can put them on the right track.
Music: Tall Kite's New Color Notation
In my Pi Approximation Day post, I mentioned that Kite's color notation is changing. I'd stumbled upon his new forum website that day, but my post was jam-packed with other information and I didn't have time to peruse the entire thread in which he introduces his new colors.
http://www.tallkite.com/forum/index.php?topic=86.60
First, the colors white, yellow, green, and red are not changing.
As for blue (7-limit otonal), Kite wants to keep this color, but there's just one problem with it. The color "blue" starts with the letter "b" -- but the letter "b" is also used to represent "flat" in ASCII. And even on the musical staff, the letter "b" for "blue" and the flat symbol look alike.
For example, notice that the link above doesn't take you to the first page of the thread -- instead, it's the fifth page. I chose this link because this page was the fourth result of a Google search for Kite's color notation. But notice that the first thing you see after clicking the link is a song written by Kite himself, "Without You" Piano. Midway through the second full measure, we see the the eighth note "blue D-flat." It's notated with the letter "b" for indicate "blue," followed by a flat symbol. In ASCII, we would write this note as blue Db, or even bDb.
In this song, all of the blue notes are also flat. But later in this song, there is a flat note that isn't blue in the last line. The song is in the key of B-flat major and thus all B's and E's are flat, but in the last line, a reddish E-natural appears. Later on in the same measure, the flat symbol restores the (white) E-flat -- but a musician might mistake this for a blue symbol and play blue E instead of white E-flat.
And therefore, because of possible confusion between "blue" and "flat," Kite is considering changing blue to a different color. One color under consideration is "azure," since it's another name for blue and contains the letter "z" -- the one letter that doesn't already have a meaning in this notation.
It's a shame, though, since Kite first selected the color blue because 7-limit otonal intervals often appear in blues music (the harmonic or Barbershop 7th). And besides, "flat" doesn't even start with the letter "b" -- it's just that the flat symbol looks like a "b." Since no one is going to change the flat symbol to "f" instead of "b," Kite's only other option is to change "blue" to "azure" and "z."
Notice that blue/azure is an otonal color. Mocha EDL music is based on utonal notes, and so most of the time, the blue/azure issue won't arise. But then again, even if we won't play absolute blue/azure notes, some scales might contain relative blue/azure intervals -- most notably 14EDL, which contains a blue/azure 3rd, 6th, and 7th. Also, we've seen that 7-limit otonal notation might be significant in Atari music, which will also force us to use blue/azure.
More significant for Mocha will be Kite's changes to the higher-limit colors. All of our Mocha scales contain 11, and many include 13, 17, and 19 as well. I admit that it's often tricky to have both "amber" and "umber" as the colors for 11 and 17.
First of all, notice that Kite's original notation stopped at the prime 19. For other primes, he originally used the suffix "-ish" for otonal and "-esque" for utonal. Thus Degree 23 (Sound 228) in Mocha would be labeled "23esque A#." Anyway, Kite wants these names to be easier to remember, so he seeks to replace "-ish" and "-esque" with "-o" and "-u," for "otonal" and "utonal." Therefore the new name for Degree 23 will be "23u A#" or "twenty-thu A#."
Moreover, Kite wants to use colors only up to the 13-limit, since 17 and 19 are hard to remember. So the new names for "umber D#" and "khaki C#" are "17u D#" and "19u C#." The only problem is how to pronounce "17u" and "19u." On one hand, Kite wants to shorten 17 and 19 to "s-" and "n-" (as 7 already has its own colors and 9 isn't a prime, so there's no chance of confusion). But if we pronounce "17u" as "su" and "19u" as "nu," these sound like other words, "su" and "nu." Kite suggests saying "sunda" and "nunda" (as in utonal or "undertone") instead. Meanwhile I wonder why we can't just say "seventu" and "ninetu," since the original idea was to pronounce the prime with an "-o"/"-u" ending (just as with "twenty-thu").
So this leaves primes 11 and 13. Actually, Kite's vision is to use the "-o" and "-u" abbreviations for all primes, including 5 and 7. This allows us to avoid mixed colors like "greenish" (red-green), which we might struggle to remember ("greenish" = "green" plus what?), and say "rugu" instead (since both red and green are utonal).
But if we keep "amber" for utonal 11, then the abbreviation would be "au," which isn't easy to say. So the new colors would begin with consonants. For 11, Kite wants t- for otonal and f- for utonal. Then the syllables for 11 would be to- and fu- (as in meat substitute, "tofu"). We could keep the colors tan and fawn (formerly used for 17 and 19), but Kite also likes "topaz" and "fuchsia," with the advantage of containing the vowels "o" and "u" already.
Only the 13-limit remains. Kite likes the letters "j" and "k," for "jo" and "ku." He hasn't settled on any colors yet. Perhaps he could recast "jade" and "khaki" for the 13-limit. It's too bad that we don't know any colors starting with "jo-" or "ku-." Hmm, from a list of colors used by computers, I see "jonquil" (a shade of yellow) and "kumquat"(a shade of orange) as valid colors. Perhaps I'll use those names for 13, so that we can clearly distinguish between "khaki" (old 19) and "kumquat" (new 13).
So let's look at a list of old and new color names. For Mocha, the emphasis is on the utonal:
Sound Degree Old Name New Name
250 11 amber B fuchsia B
248 13 ocher G kumquat G
244 17 umber D# 17u D# (seventu D#)
242 19 khaki C# 19u C# (ninetu D#)
239 22 amber B fuchsia B
238 23 23esque A# 23u A# (twenty-thu A#)
236 25 deep green Ab gugu Ab
235 26 ocher G kumquat G
228 33 amber E fuchsia E
227 34 umber D# 17u D#
226 35 greenish D rugu D
...
90 171 khaki B 19u B
71 190 khaki-green A 19ugu-A (ninetu-gu-A)
66 195 ocher-green Ab kugu Ab
61 200 deep green Ab gugu Ab
53 208 ocher G kumquat G
31 230 23q-green F# 23ugu F# (twenty-thu-gu F#)
Kite is also concerned with what these names sound like in other languages. For example, we see that Sound 66 (Degree 195) is "kugu Ab," while Sound 61 (Degree 200) is "gugu Ab." In some languages, "gugu" and "kugu" might sound alike. And so he might change 13 to a different color, depending on what comments he receives from world language speakers. (So much for "kumquat" then!)
Not only were the color names for 11 and 13 debated (in two long threads, both at Kite's own website and on Facebook), but the issue over correct intervals for 11 and 13 also came up. Should Degree 11 be spelled "fuchsia B" or "fuchsia Bb?" Should Degree 13 be spelled "kumquat G" or "kumquat G#"?
The names that I've chosen (which are equivalent to "topaz 4th" = P4 and "jonquil 6th" = M6) result in simpler commas (33/32 between "fuchsia E" and "white E," also 27/26 between "white G" and "kumquat G" as well). But Kite himself points out that all spellings for 11 and 13 have advantages and disadvantages -- spelling t4 = P4 and j6 = m6 results in 13/11 (kG-fB) being spelled as a major 3rd, even though it's played as a minor 3rd.
Hold on a minute -- 33/32 and 27/26 are ratios, which is the main topic of this post! I've stated before that it's possible to motivate students to learn math using music. For example, if students ask where ratios are used in real life, music is a possible answer!
Let's consider a simple 6EDL "scale" -- which is really just a minor arpeggio:
Degree Note
6 white A
5 green C
4 white E
3 white A
We can play other 6EDL arpeggios using ratios -- for example:
Degree Note
216 white G
180 green Bb
144 white D
108 white G
Now is a good time to show the difference between subtraction (since "Degrees" are lengths of string, subtraction means cutting off the same length from each string) and division. If we divide each string above by the same number, then we get something that sounds like another minor arpeggio:
Division by 2:
Degree Note
108 white G
90 green Bb
72 white D
54 white G
Division by 3:
Degree Note
72 white D
60 green F
48 white A
36 white D
Division by 4:
Degree Note
54 white G
45 green Bb
36 white D
27 white G
But subtraction doesn't produce another minor arpeggio:
Subtraction of 100:
Degree Note
116 29u F#
80 green C
44 fuchsia B
8 white E
Here's a quick Mocha program that demonstrates this:
http://www.haplessgenius.com/mocha/
10 INPUT "ENTER 4 DEGREES";A,B,C,D
20 SOUND 261-A,4
30 SOUND 261-B,4
40 SOUND 261-C,4
50 SOUND 261-D,4
If we enter the first four combinations listed above (216, 180, 144, 108 down to 54, 45, 36, 27) then we hear a minor arpeggio, but the one we found by subtraction (116, 80, 44, 8) sounds awful.
Music: From Xenharmonic to Tall Kite
To me, it's amazing that so many posters are participating in the Kite and Facebook threads. After all, most people aren't musicians, and most musicians play in the standard 12EDO scale, so Kite's colors don't matter to them. Apparently, Kite's website allows users to download software called "alt-tuner" that allows them to play these exotic colorful scales. I'm also using software to play alternate scales, except our software is called "Mocha."
The Xenharmonic website is set to disappear in two days. I was considering making many more music posts throughout July ahead of the site disappearance, but I didn't. I've already devoted so many recent posts to scales anyway, and I didn't want to tie up even more posts with music. I wanted to cut-and-paste more information from Xenharmonic to preserve it, but that's also misleading -- most Xenharmonic pages are on EDO scales, while Mocha plays EDL scales, not EDO's.
My original idea was to follow the Pappas pattern -- whatever the date was, I'd cut-and-paste the corresponding EDO from Xenharmonic. But that wouldn't have worked anyway -- for example, my first post this month was on the Fourth of July. But 4EDO is just a subset of our usual 12EDO scale -- and besides, that day I wanted to post patriotic music (that's not written in 4EDO). And my third post this month was on the 12th. Why in the world would I post information on our usual 12EDO scale?
Anyway, I consider Tall Kite's website to be the successor website to Xenharmonic. So there's no need for me to preserve info from Xenharmonic anyway, as the site lives on with Kite.
Some of the Xenharmonic EDO pages mention Kite's colors (and I already cut-and-pasted some of those color charts). At first, Kite's colors should have nothing to do with EDO's, since Kite's colors are based on exact ratios (just intonation), which EDO's aren't. But sometimes Kite's colors can be used to distinguish between notes in an EDO.
These are the EDO's for which Kite's colors are used:
17, 19, 22, 24, 26, 27, 29, 31, 41, 46, 53, 72
For the lower EDO's, Kite's colors seem unnecessary to name the notes. In 17EDO and 19EDO, we just use standard note names, except that notes such as G# and Ab are no longer enharmonic. And the notes of 24EDO are readily named using quarter-sharps and quarter-flats.
But consider the highest EDO on this list, 72EDO. I already explained how Kite's colors can be used to name all 72 notes in this scale:
- First, the twelve notes 72EDO that are in 12EDO are all white. Here it doesn't matter whether we call a note "white G#" or "white Ab," since these are enharmonic in 72EDO. (In other words, 72EDO tempers out the Pythagorean comma.)
- One step above white is green, and one step below white is yellow. This gives the 5-limit.
- Two steps above white is red, and two steps below white is blue/azure. This gives the 7-limit.
- All that's left are the notes halfway between white notes. The 11-limit is sufficient to name these notes, though we must deal with the "topaz 4th = P4 or A4?" problem. If we follow our usual "topaz 4th = P4" notation, then the note halfway between "white E" and "white F" can be name either "topaz E" or "fuchsia F." (Using "topaz 4th = A4," these would have to be switched to either "fuchsia E" or "topaz F.") If we wish to avoid the "P4 or A4" problem, then we can just stick to the 7-limit and name this note "greenish (rugu) E" or "yellowish (zoyo) F." (But keeping 11 isn't as bad here, since the problems occur only when we use 11 and 13 together.)
Anyway, Mocha is designed to play EDL scales, not EDO scales. And higher EDO's are even more inaccurate than lower EDO's. Even though Mocha can play (more than) 72 notes in an octave, most of those notes don't correspond to 72EDO at all.
But for the lower EDO's, Kite's colors can help us approximate the EDO in Mocha. For example, let's look at the lowest EDO for which Kite's colors are given, 17EDO. From the old Xenharmonic site:
Combining ups and downs notation with color notation, qualities can be loosely associated with colors:
| quality | color | monzo format | examples |
|---|---|---|---|
| minor | blue | {a, b, 0, 1} | 7/6, 7/4 |
| " | fourthward white | {a, b}, b < -1 | 32/27, 16/9 |
| mid | jade | {a, b, 0, 0, 1} | 11/9, 11/6 |
| " | amber | {a, b, 0, 0, -1} | 12/11, 18/11 |
| major | fifthward white | {a, b}, b > 1 | 9/8, 27/16 |
| " | red | {a, b, 0, -1} | 9/7, 12/7 |
Chord Names
All 17edo chords can be named using ups and downs. Here are the blue, jade and red triads:
| color of the 3rd | JI chord | notes as edosteps | notes of C chord | written name | spoken name |
|---|---|---|---|---|---|
| blue | 6:7:9 | 0-4-10 | C Eb G | Cm | C minor |
| jade | 18:22:27 | 0-5-10 | C Ev G | C~ | C mid |
| red | 14:18:27 | 0-6-10 | C E G | C | C major or C |
Again, this is the old website, so we see "jade" and "amber" instead of "topaz" and "fuchsia." We also note that "yellow" and "green" are missing, since 5 is not consistently represented in 17EDO.
I once wrote a program in Mocha to calculate the best EDL to approximate an EDO. For 17EDO, the best scale is 140EDL. But Degree 140 is greenish (rugu) D. Kite's scales are usually based on having a white note as the tonic. And the chart above is based on C as the root. So instead of Degree 140, we can choose white C (Degree 162) as the tonic.
Starting from 162, we divide by the 17th root of two and round off to the nearest integer to find Degrees for the other notes:
162, 156, 149, 143, 138, 132, 127, 122, 117, 112, 108, 103, 99, 95, 92, 88, 84, 81
According to the chart above, blue and (fourthward) white are enharmonic in 17EDO, just as jade and amber are also enharmonic. Since Mocha is utonal, we focus only on red and amber (fuchsia), which corresponds to multiples of 7 and 11.
Some of the notes listed here are already red or amber (fuchsia). Step 5 is Degree 132, which is correctly labeled as fuschia E. Fuschia A (Degree 99) and fuschia B (Degree 88) are also properly included in the scale. Red notes already on the list are red F# (Degree 112) and red B (Degree 84), while in addition to white C, white G (Degree 108) is also listed.
But some notes can be adjusted. For example, step 3 is Degree 143, but white D corresponds to the next note, Degree 144. In our calculation, the Degree to two decimal places is 143.35, which is closer to 143 than 144. In other words, 162/143 is closer to three steps of 17EDO than 162/144 is, but 162/144 is exactly equal to the interval (9/8) which 17EDO is designed to approximate. So using Degree 144 instead of 143 results in more consonant music.
Another white note to include is Degree 96 (white A) instead of Degree 95. Finally, Degree 127 can be replaced with either red E (Degree 126) or white E (Degree 128). To me, red E is preferable, since there are already four white notes but only two red notes in the scale -- and besides, red E allows us to play the red triad mentioned in the chart.
All the other notes have no white, red, or fuchsia note available, so we keep them as they are. The resulting scale is:
162, 156, 149, 144, 138, 132, 126, 122, 117, 112, 108, 103, 99, 96, 92, 88, 84, 81
Let's try this for one more scale -- 29EDO (hey, today is the 29th after all). Here are the charts given at the old Xenharmonic site:
Combining ups and downs notation with color notation, qualities can be loosely associated with colors:
| quality | color | monzo format | examples |
|---|---|---|---|
| downminor | blue | {a, b, 0, 1} | 7/6, 7/4 |
| minor | fourthward white | {a, b}, b < -1 | 32/27, 16/9 |
| upminor | green | {a, b, -1} | 6/5, 9/5 |
| " | jade | {a, b, 0, 0, 1} | 11/9, 11/6 |
| downmajor | amber | {a, b, 0, 0, -1} | 12/11, 18/11 |
| " | yellow | {a, b, 1} | 5/4, 5/3 |
| major | fifthward white | {a, b}, b > 1 | 9/8, 27/16 |
| upmajor | red | {a, b, 0, -1} | 9/7, 12/7 |
All 29edo chords can be named using ups and downs. Here are the blue, green, yellow and red triads:
| color of the 3rd | JI chord | notes as edosteps | notes of C chord | written name | spoken name |
|---|---|---|---|---|---|
| blue | 6:7:9 | 0-6-17 | C Ebv G | C.vm | C downminor |
| green | 10:12:15 | 0-8-17 | C Eb^ G | C.^m | C upminor |
| yellow | 4:5:6 | 0-9-17 | C Ev G | C.v | C downmajor or C dot down |
| red | 14:18:27 | 0-11-17 | C E^ G | C.^ | C upmajor or C dot up |
The first chart mentions colors for 11 (jade and amber) but the second chart doesn't, which indicates that 11 isn't as well-represented in 29EDO as the 7-limit. Also, even though the second chart is based on C, there was another chart on the old site based on D. So we'll start this scale on white D (144):
144, 141, 137, 134, 131, 128, 125, 122, 119, 116, 113, 111, 108, 106, 103, 101, 98, 96, 94, 91, 89, 87, 85, 83, 81, 79, 77, 76, 74, 72
Because 29EDO has a very accurate perfect fifth, all of the white notes (E at 128, G at 108, A at 96, and C at 81) are already placed correctly on the scale. But a few changes must be made if we wish to add red or green notes:
Because 29EDO has a very accurate perfect fifth, all of the white notes (E at 128, G at 108, A at 96, and C at 81) are already placed correctly on the scale. But a few changes must be made if we wish to add red or green notes:
- Change 134 to 135 for green Eb.
- Change 125 to 126 for red E.
- Change 119 to 120 for green F.
- Change 111 to 112 for red F#.
- Change 89 to 90 for green Bb.
- Change 83 to 84 for red B.
- Change 79 to 80 for green C.
Here is the resulting scale:
144, 141, 137, 135, 131, 128, 126, 122, 120, 116, 113, 112, 108, 106, 103, 101, 98, 96, 94, 91, 90, 87, 85, 84, 81, 80, 77, 76, 74, 72
The chart for 41EDO contains the color red, blue, and gray for every note. Actually, "red major" means "red" and "blue minor" means "blue," but "blue major" means "yellow" and "red minor" means "green," and "gray" means "white." I don't repeat that chart, since 41EDO is probably too many notes to keep accurately even for this method of conversion to EDL.
Also, 29EDO's next-door neighbor 28EDO doesn't have Kite colors, because its fifth is inaccurate. I wrote about 28EDO for my Easter song, but I didn't use Kite colors that day. If we seek to add more just intervals to the 28EDO conversion, it would be better to base it on 5/4, a just major third (very accurate in 28EDO), instead of 3/2.
Conclusion: Lunar Eclipse and Mars Opposition
Let's conclude this post with a number of celestial events in the recent news. First of all, it's eclipse season, and on Friday, there was a lunar eclipse. Not only was it a blood moon, but it's said to be the longest lunar eclipse of the 21st century.
Lunar eclipses always occur at the moment of the full moon. Friday's full moon was at 1:20 PM Pacific Time, and so the eclipse was visible at any time zone where this is nighttime (which was nowhere in the Americas).
Recall that during every eclipse season, there's always at least one lunar and one solar eclipse. As it turns out, there are two new moons, and hence two solar eclipses, during this season. Both of these solar eclipses are partial. The July new moon occurred on the 12th at 7:47 PM Pacific Time, and the partial eclipse was visible in the ocean between Australia and Antarctica. The August new moon will occur on the 11th at 2:57 AM Pacific Time, and the partial eclipse will be visible in many of the same countries where the lunar eclipse was seen -- that is, Northern Eurasia, not the Americas.
Meanwhile, there is another event that is currently visible around the world -- Mars opposition. This means that the sun and the red planet are directly opposite each other in the sky, so that Mars is much closer to Earth than usual. Occurring almost once every two Earth years (or almost once every Martian year), the opposition occurred on July 27th, the same day as the full moon. Mars should be especially bright in the sky the morning of July 31st at around 12:50 AM Pacific Time.
A previous especially close approach of the red planet was in 2003, and the next such approach won't be until 2035. The 2035 opposition has often been suggested at the date when human beings will first arrive on Mars. The year 2035 is mentioned as a particularly elegant year for our arrival on Mars, since just as 66 years separated the Wright brothers' flight in 1903 to the moon landing in 1969, so should 66 years separate the 1969 moon landing from the 2035 Mars landing.
Each year in late December, I make my annual Calendar Reform posts. Even though Calendar Reform will make timekeeping easier everywhere on Earth, many people believe that it will never happen, since the Gregorian and other world calendars are so firmly entrenched in our minds. Thus Calendar Reform will never happen anywhere on Earth -- if we want a new calendar, we'd have to get ourselves another planet.
The basic unit of a Martian Calendar is the sol, or Mars day. A Mars sol is slightly longer than an Earth day -- about 24 hours 40 minutes. Meanwhile, a Martian year is about 687 Earth days, or 668.6 Mars sols. Thus two Earth years is about one Mars year (another ratio! -- 2/1).
Since a Mars year is about twice as long as an Earth year, most proposed Mars calendars are of one of two types. One, the "Stretched Gregorian," is so-called because it stretches the twelve Gregorian months so that they're about twice as long as their Earth counterparts. The other, the "Darian"-type calendar, has 24 months instead of 12, so that each Martian month could be about the same length as an Earth month.
When I was a young elementary school student and my school introduced the "Path Plan," I once joked to myself that "paths" are what a Martian school would call their grade levels, since the normative time spent in each path is two (Earth) years -- or about one Mars year.
Conclusion: Lunar Eclipse and Mars Opposition
Let's conclude this post with a number of celestial events in the recent news. First of all, it's eclipse season, and on Friday, there was a lunar eclipse. Not only was it a blood moon, but it's said to be the longest lunar eclipse of the 21st century.
Lunar eclipses always occur at the moment of the full moon. Friday's full moon was at 1:20 PM Pacific Time, and so the eclipse was visible at any time zone where this is nighttime (which was nowhere in the Americas).
Recall that during every eclipse season, there's always at least one lunar and one solar eclipse. As it turns out, there are two new moons, and hence two solar eclipses, during this season. Both of these solar eclipses are partial. The July new moon occurred on the 12th at 7:47 PM Pacific Time, and the partial eclipse was visible in the ocean between Australia and Antarctica. The August new moon will occur on the 11th at 2:57 AM Pacific Time, and the partial eclipse will be visible in many of the same countries where the lunar eclipse was seen -- that is, Northern Eurasia, not the Americas.
Meanwhile, there is another event that is currently visible around the world -- Mars opposition. This means that the sun and the red planet are directly opposite each other in the sky, so that Mars is much closer to Earth than usual. Occurring almost once every two Earth years (or almost once every Martian year), the opposition occurred on July 27th, the same day as the full moon. Mars should be especially bright in the sky the morning of July 31st at around 12:50 AM Pacific Time.
A previous especially close approach of the red planet was in 2003, and the next such approach won't be until 2035. The 2035 opposition has often been suggested at the date when human beings will first arrive on Mars. The year 2035 is mentioned as a particularly elegant year for our arrival on Mars, since just as 66 years separated the Wright brothers' flight in 1903 to the moon landing in 1969, so should 66 years separate the 1969 moon landing from the 2035 Mars landing.
Each year in late December, I make my annual Calendar Reform posts. Even though Calendar Reform will make timekeeping easier everywhere on Earth, many people believe that it will never happen, since the Gregorian and other world calendars are so firmly entrenched in our minds. Thus Calendar Reform will never happen anywhere on Earth -- if we want a new calendar, we'd have to get ourselves another planet.
The basic unit of a Martian Calendar is the sol, or Mars day. A Mars sol is slightly longer than an Earth day -- about 24 hours 40 minutes. Meanwhile, a Martian year is about 687 Earth days, or 668.6 Mars sols. Thus two Earth years is about one Mars year (another ratio! -- 2/1).
Since a Mars year is about twice as long as an Earth year, most proposed Mars calendars are of one of two types. One, the "Stretched Gregorian," is so-called because it stretches the twelve Gregorian months so that they're about twice as long as their Earth counterparts. The other, the "Darian"-type calendar, has 24 months instead of 12, so that each Martian month could be about the same length as an Earth month.
When I was a young elementary school student and my school introduced the "Path Plan," I once joked to myself that "paths" are what a Martian school would call their grade levels, since the normative time spent in each path is two (Earth) years -- or about one Mars year.
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