1. Pappas and Szpiro
2. Fibonacci Music
3. Benchmark Test Song -- Conversion
4. Benchmark Test Song -- Coding
5. Packet Song -- Conversion
6. Packet Song -- Coding
7. Do-Ru-Ma -- Conversion
8. Do-Ru-Ma -- Coding
9. What About 11-/13-Limit Music?
10. Conclusion
Pappas and Szpiro
This is what Theoni Pappas writes on page 24 of her Magic of Mathematics:
"Not quite sure what to expect, I rang the doorbell. A voice asked me to please push the first five terms of the Fibonacci sequence."
This is the first page of the section "A Mathematical Visit." It's a story where the narrator meets a (fictional) renowned mathematician, Selath, and his unusual house.
Here are some excerpts from this story:
"I pushed 1, 1, 2, 3, 5 and the door slowly opened. As I passed through the doorway, I was struck by the catenary stone shaped archway independently suspended at the entrance. In the kitchen we came to a peculiar table with many legs. Selath pulled an equally unusual bottle from the refrigerator. 'As you noticed, this tabel and bottle are not your everyday accessories.'"
The only picture on this page is of Selath's unusual table and bottle.
Chapter 7 of George Szpiro's Poincare's Prize is called "What the Conjecture Is Really All About." I give its opening here:
"Toward the end of the nineteenth century the main thrust of the then-still-young discipline of topology was the classification of bodies and spaces."
As the title implies, in this chapter we finally get to see Poincare's Conjecture itself. Szpiro tells us that the conjecture is a bit tricky for laypeople to understand, and so he spends this chapter giving a few analogies (in lower dimensions) and simple explanations.
So we open the chapter by viewing topology as an attempt to classify different shapes. But how exactly can we do this? Szpiro writes:
"Clearly a single number, such as the number of the body's corners, would not do since a ball and a bagel have zero corners and would thus be classed in the same group even though they are topologically different, while a cube would be classed as different from a pyramid, even though one can be morphed into the other."
Poincare writes a paper about this, and here he makes a bold claim:
"In modern language, this statement would read, 'A three-dimensional manifold that has the same homology groups as the three-dimensional sphere is homeomorphic to it.'"
Szpiro ultimately rephrases this as, Any body that contains no holes and is not twisted can be morphed into a sphere -- though he warns us that the morphing of the 3D body could actually take place in 4D space. But the author compares Poincare's claim to that of Fermat when he wrote about his famous Last Theorem:
"Like Fermat's claim, Poincare's announcement was premature, Unlike Fermat's claim, it was wrong. The somewhat cocky Poincare apparently thought he had a proof in his pocket, but as on numerous occasions, he was mistaken."
His claim is false because he finds a counterexample -- but it's a 3D manifold that is twisted up so much that it can only exist in 4D. As Szpiro explains:
"Even the Klein bottle, which is, after all, nothing but a two-dimensional surface, must be embedded in four-dimensional space so that its features can be fully exhibited."
Hold on a minute here -- let's jump back to Pappas for a second. We've just read page 24, where we see something about a strange bottle on the mathematician Selath's desk. Pappas explains what the bottle is on page 25, but we're skipping it because it's blocked by the non-posting weekend. Well, let's end the suspense and find out what Selath's bottle really is:
'
"'The water container is known as a Klein bottle -- its inside and outside are one.'"
And there is a picture of a Klein bottle on page 25. According to Pappas, it's unusual in that its inside and outside are not distinct. And here Szpiro gives even more info about what it is -- it's simply a 2D manifold, but the twisting must occur in 4D in order to unite its inside and outside. A mere 3D twisting can't produce a Klein bottle.
Since you can't see the picture of the Klein bottle, let's link to one right here:
http://www.kleinbottle.com/
Apparently, you can actually order Klein bottles from the above website.
OK, let's get back to Szpiro and Poincare. Notice that the Klein bottle is not the counterexample to Poincare's claim above, since the Klein bottle is a 2D manifold and the claim is on 3D manifolds. At this point, the author continues to describe the difficulties in mentally picturing objects in 4D:
"Some people, such as the physicist Roger Penrose, claim they can think in four dimensions, but most people cannot. However, one can gain an understanding by appealing to analogy: The three-dimensional sphere is to the ball what the surface of the ball is to a disk."
Max Dehn, whom we mentioned in yesterday's post, tries to visualize our 3D counterexample in 4D:
"He suggested extracting a pretzel-like object from the three-dimensional sphere and sewing it back differently."
Another way to obtain this manifold, akin to fellow German Hellmuth Kneser, is to begin with a dodecahedron and glue two of its opposite faces together.
"Now do the same thing with the other five pairs of pentagons, and voila, you have the homology sphere."
Let's return now to Poincare's research:
"Having found a counterexample to his initial claim, Poincare realized that even a series of numbers, such as the torsion coefficients and the Betti numbers, would be unable to furnish a classification of spaces and bodies."
And in his next attempt, Poincare considers sliding rubber bands around a manifold. For example, a bagel might have a band around the outside and another band that loops through the hole:
"These two rubber bands can also be slid about the surface of the bagel, but they will never become aligned next to each other."
At this point, Szpiro suddenly starts writing about group theory. I've mentioned group theory in several posts here on the blog, most recently on October 30th when discussing John Conway, a group theorist who advocates the inclusive definition of trapezoid. (I've also mentioned group theory in connection with other mathematicians like Evariste Galois and Eugenia Cheng.) Szpiro reminds us:
"But not every collection of objects together with an operation represents a mathematical group. For a collection of objects together with an operation to qualify as a group, four conditions, let's call them the groupie requirements, must be satisfied."
Since we've discussed group theory in previous posts, let's just skip to statement of the famous Poincare conjecture:
"Is it possible that the fundamental group of a manifold be trivial and yet the manifold not be homeomorphic to the sphere?"
Poincare's conjecture is that the answer is "no." Here "the fundamental group" refers to the group of all "rubber bands," and this group is trivial if all bands can shrink to the sphere. As we've seen earlier, the fundamental group of a bagel is not trivial. And here "sphere" refers to the 3D sphere, which is a manifold in 4D space.
Of course, Poincare dies without ever proving his conjecture:
"All that was now needed was a proof. And for a hundred years, mathematicians from all over the world searched...."
Fibonacci Music
OK, let's get back to music. The Fibonacci sequence is on my mind today since it's the code to enter Selath's house. So we will start by composing music based on this sequence.
We begin with the 3-limit Waldorf scale. Here's a link to the emulator again:
http://www.haplessgenius.com/mocha/
Type in the following program:
10 DIM S(6)
20 FOR X=0 TO 6
30 READ S(X)
40 NEXT X
50 DATA 72,64,54,48
60 DATA 108,96,81
70 N=1
80 A=1
90 B=1
100 SOUND 261-N*S(A),4
110 T=A+B
120 IF T>6 THEN T=T-7
130 X=Y
140 Y=T
150 GOTO 100
Remember to click the "Sound" box before typing RUN.
Just like the previous versions of the song, this program reduces the Fibonacci numbers mod 7 and then maps the numbers to the scale, except this time it's the Waldorf scale. Notice that even though there are DATA lines to set up the scale, the song itself doesn't use DATA. Instead, the computer actually calculates the Fibonacci numbers. We see how line 110 adds up two numbers, and line 120 performs the reduction mod 7.
As we know, the Fibonacci numbers mod 7 repeat every 16 terms:
1-1-2-3-5-1-6-0
6-6-5-4-2-6-1-0
Notice how when we write the numbers in two rows of eight, the numbers in each column add up to seven, except the two zeros in the last column. Since the Waldorf scale is symmetrical around its center note (A in the original scale, D on the emulator), we take advantage of this by assigning 0 to the center note:
G-A-C-D-E-G-A
4- 5- 6- 0- 1- 2- 3
Then the second half of the song is the mirror image of the first half. This actually reminds me of Lesson 4-8 of the U of Chicago text, Third Edition. This lesson is called "Transformations in Music," which demonstrates how the Common Core transformations can be applied to music. The text refers to this "horizontal reflection" of sorts as an inversion.
It can be proved that in any Fibonacci song (that is, where the numbers are reduced mod m for any m and mapped to notes), the second half is an inversion of the first half, with zero as the line of symmetry. Since songs in the Waldorf scale are supposed to be symmetrical around its central note, we assign zero to it and there we have it -- a true Waldorf-style song:
E-E-G-A(high)-A(low)-E-C-D
C-C-A-G(low)-G(high)-C-E-D
This program has an infinite loop, so try pressing "Esc" (for "Break") to end the program. As usual, we can switch octaves:
70 N=2
Let's try playing Fibonacci in the major and minor scales. For this, we'll use all ten notes, 1-9 and 0 (to represent 10). To make it easier, change only the following lines:
10 DIM S(9)
20 FOR X=0 TO 9
50 DATA 72,180,160,144,135
60 DATA 120,108,96,90,80
70 N=1
120 IF T>9 THEN T=T-10
Here Note 0 is a perfect tenth, which is D (in the Bb+ major scale). Because the song is symmetrical in Notes 1-9, it might be good to make Note 0 a rest (as some Pi Day musicians do). Then Note 5 also becomes a line of symmetry:
145 IF A=0 THEN FOR I=1 TO 460:NEXT I:GOTO 110
This line, numbered 145 so we can enter it before line 150, starts a small loop where the computer counts from 1 to 460. This takes about one second (or one beat) to accomplish, and so it serves as a quarter rest. As it turns out, the Fibonacci numbers mod 10 take longer to repeat. It takes 60 terms -- in other words, the Pisano period mod 10 is 60.
For the minor scale, make the following changes to the major song:
50 DATA 30,72,64,60,54
60 DATA 48,45,40,36,32
Again, the minor scale can be played in one of three octaves: N=1 (high D minor), N=2 (low D minor), and N=3 (low G minor).
Finally, let's play it in the New 7-Limit Scale:
50 DATA 54,105,96,90,84
60 DATA 81,72,70,63,60
There are two possible octaves, N=1 and N=2.
For this scale, I don't like simply making 0 a rest, since we've chosen a special high G note for 0. So instead, I want to erase only the 60th note -- the final 0 before it repeats 0-1-1. This avoids the dissonant semidiminished octave between Notes 0 and 1. Meanwhile, I also want to avoid the sequence 1-6-7 (which does occur in Fibonacci since 1+6=7). This sequence has a dissonant semidiminished fifth between Notes 1 and 6. Indeed, when I hear this sequence, my ear mistakes 1-6 for a perfect fifth, and then when the true perfect fifth 7 is played, it sounds out of place:
145 IF (A=0 OR A=6) AND B=A+1 THEN FOR I=1 TO 460:NEXT I:GOTO 110
This means that if the current note (variable A) is Note 0 or 6 and the next note (variable B) is one more than the current note (that is, Note 1 or 7), then play the quarter rest instead. It turns out that the sequence 1-rest-7 occurs just a few notes before the 1-rest-1 sequence.
By the way, there's another way to refer to the two G's besides G and G+7. The first G is called "white G," while G+7 is called "greenish" G. This is Kite's color notation:
http://xenharmonic.wikispaces.com/7-limit+interval+names
The New 7-Limit Scale, starting on white G:
Benchmark Test Song -- Conversion
For the rest of this post, I wish to convert some of my old songs from last year to the new scales. As it is still November, I choose to convert the songs I sang last November (my original songs, not the ones that come from Square One TV).
At the end of the first trimester, I sang the Benchmark Test song:
Benchmark Tests -- by Mr. Walker
Verse 2:
Why do we take Benchmark Tests?
The first trimester is done so let's
See how much we know, know know!
It's some new stuff on Benchmark Tests.
If we don't know it, we take a guess.
'Cause there's still time to grow, grow, grow!
The teacher sees our Benchmark Tests,
Knows what to teach more or less.
That's the way to go, go, go!
When I first created this song, it was in the C major scale, excluding the note B. After I sang it a few times, I instinctively dropped the note F as well, to make the song pentatonic. This suggests that the Waldorf 3-limit scale is a good scale for this song.
Benchmark Test Song -- Coding
Let's start a NEW program:
10 DIM S(7)
15 FOR V=1 TO 3
20 FOR X=1 TO 7
30 READ S(X)
40 NEXT X
50 DATA 108,96,81
60 DATA 72,64,54,48
70 N=1
80 FOR X=1 TO 4
90 FOR Y=1 TO 7
100 READ A
110 SOUND 261-N*S(A),4
120 NEXT Y
130 FOR I=1 TO 460
140 NEXT I,X
150 RESTORE
160 DATA 3,4,3,4,3,3,3
170 DATA 4,7,3,7,6,5,4
180 DATA 5,4,5,4,5,5,5
190 DATA 4,1,5,1,2,3,4
200 NEXT V
In this song, I return to labeling the notes 1-7, with the center D as Note 4 rather than Note 0. But the song still follows the Waldorf inversion pattern. Because of this, it's better to repeat lines, as follows:
Why do we take Benchmark Tests?
See how much we know, know know!
The first trimester is done so let's
See how much we know, know know!
It's some new stuff on Benchmark Tests.
'Cause there's still time to grow, grow, grow!
If we don't know it, we take a guess.
'Cause there's still time to grow, grow, grow!
The teacher sees our Benchmark Tests,
That's the way to go, go, go!
Knows what to teach more or less.
That's the way to go, go, go!
Each line is played as seven quarter notes followed by a quarter rest. Each stanza consists of four lines, and then the song repeats for all three stanzas. To accomplish this, we insert a FOR V loop from line 15 to line 200. The DATA lines don't automatically repeat (not even if we place them inside a FOR loop), and so we must RESTORE the data. But RESTORE automatically restores all the data, including lines 50-60 which are used to set up the scale. So we must repeat those lines -- but we can't repeat the DIM line, which is why the outer loop begins at line 15.
Packet Song -- Conversion
The next song we will convert is the Packet Song. Right after Thanksgiving break, the other middle school teachers and I came up with the idea of including all homework and classwork for the week in a packet. This idea fell apart once the administrators insisted that I assign Illinois State homework on the computer instead.
Nonetheless, here is the song:
When I first came up with this song, I played it in G major, just like many of my other songs. But in hindsight, I think it's better to play the song in a minor key. Minor keys express sadness -- and of course students are sad that they are assigned so much work in a packet.
So far in my class, the only original songs I played in minor keys are the Mousetrap Song and the first day of school Dren Song. I also wrote the Dren Song so that the second half is an inversion of the first half.
And so let's use our 5-limit just minor scale to create the song.
Packet Song -- Coding
This is a NEW program:
10 DIM S(9)
15 FOR V=1 TO 3
20 FOR X=0 TO 9
30 READ S(X)
40 NEXT X
50 DATA 30,72,64,60,54
60 DATA 48,45,40,36,32
70 N=1
80 FOR X=1 TO 4
90 FOR Y=1 TO 12
100 READ A
110 SOUND 261-N*S(A),2
120 NEXT Y
130 READ A
140 SOUND 261-N*S(A),4
150 FOR I=1 TO 460
160 NEXT I,X
170 RESTORE
180 DATA 4,1,4,5,1,1,2,1,3,5,7,8,7
190 DATA 3,3,4,1,5,8,4,7,3,4,7,6,4
200 DATA 4,1,4,5,1,1,2,1,3,5,7,8,7
210 DATA 5,6,3,5,2,3,5,6,6,3,1,3,1
220 NEXT V
In this song, most notes are eighth notes. A longer quarter note appears at the end of every other line ("packet" and its rhymes).
Do-Ru-Ma -- Conversion
There was one more song that I tried to sing last November. It was a special election song, for the day of the presidential election:
VOTE
If I could, I surely would,
Vote in this election.
Make my choice for president,
Choose the leaders of our land.
Have you registered to vote? No!
Then let's register today.
Soon we'll finally have our say.
So let's all go out and vote, vote, vote!
This is actually a parody of the song "Do-Re-Mi" from Sound of Music. This isn't an original song, but it might be interesting to convert "Do-Re-Mi" to the New 7-Limit Scale.
Hey, that reminds me -- "Do-Re-Mi" is just solfege, and we need to come up with solfege for the New 7-Limit Scale.
Musician Andrew Heathwaite has already invented solfege for many alternate scales. It's helpful to look at EDO scales which represent the 7-limit well, such as 19EDO, or even better, 31EDO:
http://xenharmonic.wikispaces.com/31edo+solfege
Heathwaite explains that some of the note names are "grandfathered in" -- that is, they are the old names from the just major scale, such as "do," "re," and "mi." The other names start with the same letters as the grandfathered names but change the vowels. Since the major second "re" starts with R, all seconds start with R. Likewise all thirds start with M, and so on:
(In the Extended Scale, Eb+ is "le" and F#7 is "to.") But for our song, we will start it on greenish G, and so we start on "do" and finish on "da."
OK, let's plan the lyrics now for our new song, which will be called "Do-Ru-Ma," the first three notes of the New 7-Limit Scale. We'll keep the original song lines for grandfathered notes:
DO-RU-MA
Do, a deer, a female deer,
Ru, ____________________
Ma, ____________________
Mi, a name I call myself,
Fe, ____________________
Su, ____________________
Sol, a needle pulling thread,
La, a note to follow so,
Ta, ____________________
That will take us up to da, da, da!
OK, we want to enter lines that rhyme -- although not necessarily the rhyme scheme of the original song (as there are now ten lines instead of eight). First of all, the last three notes la, ta, da already rhyme, and so "Ta, a note to follow la," will allow the last two lines to rhyme.
For "fe," I was thinking about fae, or fairies. The line "Fe, a flying little elf," fits, and notice that "elf" happens to rhyme with "myself" in the "mi" line. So now we have:
Do, a deer, a female deer,
Ru, ____________________
Ma, ____________________
Mi, a name I call myself,
Fe, a flying little elf
Su, ____________________
Sol, a needle pulling thread,
La, a note to follow so,
Ta, a note to follow la,
That will take us up to da, da, da!
I'm inspired by the rhyme scheme of yesterday's mouse poem to rhyme the third and sixth lines. For "ma," this sounds like a mother, while "sue" is what you do in court. Let's see:
Do, a deer, a female deer,
Ru, the day that I will fear,
Ma, to mom the babies say.
Mi, a name I call myself,
Fe, a flying little elf
Su, the way I make you pay,
Sol, a needle pulling thread,
La, a note to follow so,
Ta, a note to follow la,
Take us up to da, da, da!
Here "ru" refers to "rue" -- as in "rue the day" that something terrible happens.
Do-Ru-Ma -- Coding
This is a NEW program:
10 DIM S(9)
15 FOR V=1 TO 2
20 FOR X=0 TO 9
30 READ S(X)
40 NEXT X
50 DATA 54,105,96,90,84
60 DATA 81,72,70,63,60
70 N=1
80 FOR X=1 TO 70
85 READ A,T
90 SOUND 261-N*S(A),T
95 NEXT X
100 RESTORE
110 DATA 1,6,2,2,3,6,1,2,3,4,1,4,3,8
120 DATA 2,6,3,2,4,6,2,2,4,4,2,4,4,8
As the title implies, in this chapter we finally get to see Poincare's Conjecture itself. Szpiro tells us that the conjecture is a bit tricky for laypeople to understand, and so he spends this chapter giving a few analogies (in lower dimensions) and simple explanations.
So we open the chapter by viewing topology as an attempt to classify different shapes. But how exactly can we do this? Szpiro writes:
"Clearly a single number, such as the number of the body's corners, would not do since a ball and a bagel have zero corners and would thus be classed in the same group even though they are topologically different, while a cube would be classed as different from a pyramid, even though one can be morphed into the other."
Poincare writes a paper about this, and here he makes a bold claim:
"In modern language, this statement would read, 'A three-dimensional manifold that has the same homology groups as the three-dimensional sphere is homeomorphic to it.'"
Szpiro ultimately rephrases this as, Any body that contains no holes and is not twisted can be morphed into a sphere -- though he warns us that the morphing of the 3D body could actually take place in 4D space. But the author compares Poincare's claim to that of Fermat when he wrote about his famous Last Theorem:
"Like Fermat's claim, Poincare's announcement was premature, Unlike Fermat's claim, it was wrong. The somewhat cocky Poincare apparently thought he had a proof in his pocket, but as on numerous occasions, he was mistaken."
His claim is false because he finds a counterexample -- but it's a 3D manifold that is twisted up so much that it can only exist in 4D. As Szpiro explains:
"Even the Klein bottle, which is, after all, nothing but a two-dimensional surface, must be embedded in four-dimensional space so that its features can be fully exhibited."
Hold on a minute here -- let's jump back to Pappas for a second. We've just read page 24, where we see something about a strange bottle on the mathematician Selath's desk. Pappas explains what the bottle is on page 25, but we're skipping it because it's blocked by the non-posting weekend. Well, let's end the suspense and find out what Selath's bottle really is:
'
"'The water container is known as a Klein bottle -- its inside and outside are one.'"
And there is a picture of a Klein bottle on page 25. According to Pappas, it's unusual in that its inside and outside are not distinct. And here Szpiro gives even more info about what it is -- it's simply a 2D manifold, but the twisting must occur in 4D in order to unite its inside and outside. A mere 3D twisting can't produce a Klein bottle.
Since you can't see the picture of the Klein bottle, let's link to one right here:
http://www.kleinbottle.com/
Apparently, you can actually order Klein bottles from the above website.
OK, let's get back to Szpiro and Poincare. Notice that the Klein bottle is not the counterexample to Poincare's claim above, since the Klein bottle is a 2D manifold and the claim is on 3D manifolds. At this point, the author continues to describe the difficulties in mentally picturing objects in 4D:
"Some people, such as the physicist Roger Penrose, claim they can think in four dimensions, but most people cannot. However, one can gain an understanding by appealing to analogy: The three-dimensional sphere is to the ball what the surface of the ball is to a disk."
Max Dehn, whom we mentioned in yesterday's post, tries to visualize our 3D counterexample in 4D:
"He suggested extracting a pretzel-like object from the three-dimensional sphere and sewing it back differently."
Another way to obtain this manifold, akin to fellow German Hellmuth Kneser, is to begin with a dodecahedron and glue two of its opposite faces together.
"Now do the same thing with the other five pairs of pentagons, and voila, you have the homology sphere."
Let's return now to Poincare's research:
"Having found a counterexample to his initial claim, Poincare realized that even a series of numbers, such as the torsion coefficients and the Betti numbers, would be unable to furnish a classification of spaces and bodies."
And in his next attempt, Poincare considers sliding rubber bands around a manifold. For example, a bagel might have a band around the outside and another band that loops through the hole:
"These two rubber bands can also be slid about the surface of the bagel, but they will never become aligned next to each other."
At this point, Szpiro suddenly starts writing about group theory. I've mentioned group theory in several posts here on the blog, most recently on October 30th when discussing John Conway, a group theorist who advocates the inclusive definition of trapezoid. (I've also mentioned group theory in connection with other mathematicians like Evariste Galois and Eugenia Cheng.) Szpiro reminds us:
"But not every collection of objects together with an operation represents a mathematical group. For a collection of objects together with an operation to qualify as a group, four conditions, let's call them the groupie requirements, must be satisfied."
Since we've discussed group theory in previous posts, let's just skip to statement of the famous Poincare conjecture:
"Is it possible that the fundamental group of a manifold be trivial and yet the manifold not be homeomorphic to the sphere?"
Poincare's conjecture is that the answer is "no." Here "the fundamental group" refers to the group of all "rubber bands," and this group is trivial if all bands can shrink to the sphere. As we've seen earlier, the fundamental group of a bagel is not trivial. And here "sphere" refers to the 3D sphere, which is a manifold in 4D space.
Of course, Poincare dies without ever proving his conjecture:
"All that was now needed was a proof. And for a hundred years, mathematicians from all over the world searched...."
Fibonacci Music
OK, let's get back to music. The Fibonacci sequence is on my mind today since it's the code to enter Selath's house. So we will start by composing music based on this sequence.
We begin with the 3-limit Waldorf scale. Here's a link to the emulator again:
http://www.haplessgenius.com/mocha/
Type in the following program:
10 DIM S(6)
20 FOR X=0 TO 6
30 READ S(X)
40 NEXT X
50 DATA 72,64,54,48
60 DATA 108,96,81
70 N=1
80 A=1
90 B=1
100 SOUND 261-N*S(A),4
110 T=A+B
120 IF T>6 THEN T=T-7
130 X=Y
140 Y=T
150 GOTO 100
Remember to click the "Sound" box before typing RUN.
Just like the previous versions of the song, this program reduces the Fibonacci numbers mod 7 and then maps the numbers to the scale, except this time it's the Waldorf scale. Notice that even though there are DATA lines to set up the scale, the song itself doesn't use DATA. Instead, the computer actually calculates the Fibonacci numbers. We see how line 110 adds up two numbers, and line 120 performs the reduction mod 7.
As we know, the Fibonacci numbers mod 7 repeat every 16 terms:
1-1-2-3-5-1-6-0
6-6-5-4-2-6-1-0
Notice how when we write the numbers in two rows of eight, the numbers in each column add up to seven, except the two zeros in the last column. Since the Waldorf scale is symmetrical around its center note (A in the original scale, D on the emulator), we take advantage of this by assigning 0 to the center note:
G-A-C-D-E-G-A
4- 5- 6- 0- 1- 2- 3
Then the second half of the song is the mirror image of the first half. This actually reminds me of Lesson 4-8 of the U of Chicago text, Third Edition. This lesson is called "Transformations in Music," which demonstrates how the Common Core transformations can be applied to music. The text refers to this "horizontal reflection" of sorts as an inversion.
It can be proved that in any Fibonacci song (that is, where the numbers are reduced mod m for any m and mapped to notes), the second half is an inversion of the first half, with zero as the line of symmetry. Since songs in the Waldorf scale are supposed to be symmetrical around its central note, we assign zero to it and there we have it -- a true Waldorf-style song:
E-E-G-A(high)-A(low)-E-C-D
C-C-A-G(low)-G(high)-C-E-D
This program has an infinite loop, so try pressing "Esc" (for "Break") to end the program. As usual, we can switch octaves:
70 N=2
Let's try playing Fibonacci in the major and minor scales. For this, we'll use all ten notes, 1-9 and 0 (to represent 10). To make it easier, change only the following lines:
10 DIM S(9)
20 FOR X=0 TO 9
50 DATA 72,180,160,144,135
60 DATA 120,108,96,90,80
70 N=1
120 IF T>9 THEN T=T-10
Here Note 0 is a perfect tenth, which is D (in the Bb+ major scale). Because the song is symmetrical in Notes 1-9, it might be good to make Note 0 a rest (as some Pi Day musicians do). Then Note 5 also becomes a line of symmetry:
145 IF A=0 THEN FOR I=1 TO 460:NEXT I:GOTO 110
This line, numbered 145 so we can enter it before line 150, starts a small loop where the computer counts from 1 to 460. This takes about one second (or one beat) to accomplish, and so it serves as a quarter rest. As it turns out, the Fibonacci numbers mod 10 take longer to repeat. It takes 60 terms -- in other words, the Pisano period mod 10 is 60.
For the minor scale, make the following changes to the major song:
50 DATA 30,72,64,60,54
60 DATA 48,45,40,36,32
Again, the minor scale can be played in one of three octaves: N=1 (high D minor), N=2 (low D minor), and N=3 (low G minor).
Finally, let's play it in the New 7-Limit Scale:
50 DATA 54,105,96,90,84
60 DATA 81,72,70,63,60
There are two possible octaves, N=1 and N=2.
For this scale, I don't like simply making 0 a rest, since we've chosen a special high G note for 0. So instead, I want to erase only the 60th note -- the final 0 before it repeats 0-1-1. This avoids the dissonant semidiminished octave between Notes 0 and 1. Meanwhile, I also want to avoid the sequence 1-6-7 (which does occur in Fibonacci since 1+6=7). This sequence has a dissonant semidiminished fifth between Notes 1 and 6. Indeed, when I hear this sequence, my ear mistakes 1-6 for a perfect fifth, and then when the true perfect fifth 7 is played, it sounds out of place:
145 IF (A=0 OR A=6) AND B=A+1 THEN FOR I=1 TO 460:NEXT I:GOTO 110
This means that if the current note (variable A) is Note 0 or 6 and the next note (variable B) is one more than the current note (that is, Note 1 or 7), then play the quarter rest instead. It turns out that the sequence 1-rest-7 occurs just a few notes before the 1-rest-1 sequence.
By the way, there's another way to refer to the two G's besides G and G+7. The first G is called "white G," while G+7 is called "greenish" G. This is Kite's color notation:
http://xenharmonic.wikispaces.com/7-limit+interval+names
Key Sound Degree Note Ratio Color
0. 45 216 G 1/1 white G
1. 51 210 G+7 36/35 greenish G
2. 69 192 A 9/8 white A
3. 81 180 Bb+ 6/5 green Bb
4. 93 168 B7 9/7 red B
5. 99 162 C 4/3 white C
6. 117 144 D 3/2 white D
7. 121 140 D+7 54/35 greenish D
8. 135 126 E7 12/7 red E
9. 141 120 F+ 9/5 green F
0. 153 108 G 2/1 white G
In this scale, "white" means Pythagorean, "green" means raised a syntonic comma, and "red" means raised a septimal comma. "Green" and "red" combine to form "greenish." According to Kite, there are also colors "blue" and "yellow," but these are otonal colors -- and as we already know, the EDL computer scale is utonal. So we're stuck with "green" and "red." (How festive -- today is Black Friday which leads to the Christmas season.)
Of course, it's possible to combine green and red in different ways. "Deep" means that there are two factors of the given prime, 5 or 7:
Sound Degree Note Color
36 225 Gb++ deep green Gb
65 196 G#77 deep red G#
86 175 Bb++7 greenish green Bb
16 245 E+77 greenish red E
11 250 Fb+++ triple green Fb
So we can restate our rule to avoid dissonant fifths as, "Avoid playing a white G or D adjacent to a greenish G or D." We also want to avoid dissonant fourths, so we could also say "Avoid playing a greenish G adjacent to white C, or white A adjacent to greenish D." Our Fibonacci song still has a few dissonant fourths, but these are harder to avoid, and so I kept them in the song. On the other hand, all seconds are melodically consonant, and so "greenish G to white A" and "white C to greenish D" don't need to be avoided.
Of course, it's possible to combine green and red in different ways. "Deep" means that there are two factors of the given prime, 5 or 7:
Sound Degree Note Color
36 225 Gb++ deep green Gb
65 196 G#77 deep red G#
86 175 Bb++7 greenish green Bb
16 245 E+77 greenish red E
11 250 Fb+++ triple green Fb
So we can restate our rule to avoid dissonant fifths as, "Avoid playing a white G or D adjacent to a greenish G or D." We also want to avoid dissonant fourths, so we could also say "Avoid playing a greenish G adjacent to white C, or white A adjacent to greenish D." Our Fibonacci song still has a few dissonant fourths, but these are harder to avoid, and so I kept them in the song. On the other hand, all seconds are melodically consonant, and so "greenish G to white A" and "white C to greenish D" don't need to be avoided.
Benchmark Test Song -- Conversion
For the rest of this post, I wish to convert some of my old songs from last year to the new scales. As it is still November, I choose to convert the songs I sang last November (my original songs, not the ones that come from Square One TV).
At the end of the first trimester, I sang the Benchmark Test song:
Benchmark Tests -- by Mr. Walker
Verse 2:
Why do we take Benchmark Tests?
The first trimester is done so let's
See how much we know, know know!
It's some new stuff on Benchmark Tests.
If we don't know it, we take a guess.
'Cause there's still time to grow, grow, grow!
The teacher sees our Benchmark Tests,
Knows what to teach more or less.
That's the way to go, go, go!
When I first created this song, it was in the C major scale, excluding the note B. After I sang it a few times, I instinctively dropped the note F as well, to make the song pentatonic. This suggests that the Waldorf 3-limit scale is a good scale for this song.
Benchmark Test Song -- Coding
Let's start a NEW program:
10 DIM S(7)
15 FOR V=1 TO 3
20 FOR X=1 TO 7
30 READ S(X)
40 NEXT X
50 DATA 108,96,81
60 DATA 72,64,54,48
70 N=1
80 FOR X=1 TO 4
90 FOR Y=1 TO 7
100 READ A
110 SOUND 261-N*S(A),4
120 NEXT Y
130 FOR I=1 TO 460
140 NEXT I,X
150 RESTORE
160 DATA 3,4,3,4,3,3,3
170 DATA 4,7,3,7,6,5,4
180 DATA 5,4,5,4,5,5,5
190 DATA 4,1,5,1,2,3,4
200 NEXT V
In this song, I return to labeling the notes 1-7, with the center D as Note 4 rather than Note 0. But the song still follows the Waldorf inversion pattern. Because of this, it's better to repeat lines, as follows:
Why do we take Benchmark Tests?
See how much we know, know know!
The first trimester is done so let's
See how much we know, know know!
It's some new stuff on Benchmark Tests.
'Cause there's still time to grow, grow, grow!
If we don't know it, we take a guess.
'Cause there's still time to grow, grow, grow!
The teacher sees our Benchmark Tests,
That's the way to go, go, go!
Knows what to teach more or less.
That's the way to go, go, go!
Packet Song -- Conversion
The next song we will convert is the Packet Song. Right after Thanksgiving break, the other middle school teachers and I came up with the idea of including all homework and classwork for the week in a packet. This idea fell apart once the administrators insisted that I assign Illinois State homework on the computer instead.
Nonetheless, here is the song:
THE PACKET SONG
1st Verse:
Your folks came up to us and said,
"Hey teachers, you can't hack it!
You make the claim you give our kids
Much work, but you can't back it."
So we got together and said,
"Let's stop all this racket!
We'll staple all our good worksheets
To make a ten-page packet."
2nd Verse:
Don't leave your packet behind,
Make sure that you backpack it.
Or if it starts to rain then
Hide it underneath your jacket.
Keep it in a folder so that
You can always track it,
Make sure that you never ever
Leave your ten-page packet.
3rd Verse:
When it's homework time then
Take it out and just attack it.
When there's extra time in class
Then you need to unpack it.
When it's time to turn it in
Make sure that you don't lack it.
To each and every teacher just
Turn in your ten-page packet.
When I first came up with this song, I played it in G major, just like many of my other songs. But in hindsight, I think it's better to play the song in a minor key. Minor keys express sadness -- and of course students are sad that they are assigned so much work in a packet.
So far in my class, the only original songs I played in minor keys are the Mousetrap Song and the first day of school Dren Song. I also wrote the Dren Song so that the second half is an inversion of the first half.
And so let's use our 5-limit just minor scale to create the song.
Packet Song -- Coding
This is a NEW program:
10 DIM S(9)
15 FOR V=1 TO 3
20 FOR X=0 TO 9
30 READ S(X)
40 NEXT X
50 DATA 30,72,64,60,54
60 DATA 48,45,40,36,32
70 N=1
80 FOR X=1 TO 4
90 FOR Y=1 TO 12
100 READ A
110 SOUND 261-N*S(A),2
120 NEXT Y
130 READ A
140 SOUND 261-N*S(A),4
150 FOR I=1 TO 460
160 NEXT I,X
170 RESTORE
180 DATA 4,1,4,5,1,1,2,1,3,5,7,8,7
190 DATA 3,3,4,1,5,8,4,7,3,4,7,6,4
200 DATA 4,1,4,5,1,1,2,1,3,5,7,8,7
210 DATA 5,6,3,5,2,3,5,6,6,3,1,3,1
220 NEXT V
In this song, most notes are eighth notes. A longer quarter note appears at the end of every other line ("packet" and its rhymes).
Do-Ru-Ma -- Conversion
There was one more song that I tried to sing last November. It was a special election song, for the day of the presidential election:
VOTE
If I could, I surely would,
Vote in this election.
Make my choice for president,
Choose the leaders of our land.
Have you registered to vote? No!
Then let's register today.
Soon we'll finally have our say.
So let's all go out and vote, vote, vote!
This is actually a parody of the song "Do-Re-Mi" from Sound of Music. This isn't an original song, but it might be interesting to convert "Do-Re-Mi" to the New 7-Limit Scale.
Hey, that reminds me -- "Do-Re-Mi" is just solfege, and we need to come up with solfege for the New 7-Limit Scale.
Musician Andrew Heathwaite has already invented solfege for many alternate scales. It's helpful to look at EDO scales which represent the 7-limit well, such as 19EDO, or even better, 31EDO:
http://xenharmonic.wikispaces.com/31edo+solfege
Heathwaite explains that some of the note names are "grandfathered in" -- that is, they are the old names from the just major scale, such as "do," "re," and "mi." The other names start with the same letters as the grandfathered names but change the vowels. Since the major second "re" starts with R, all seconds start with R. Likewise all thirds start with M, and so on:
Key Sound Degree Note Ratio Solfege Interval (from root note)
1. 51 210 G+7 1/1 do tonic
2. 69 192 A 35/32 ru neutral tone
3. 81 180 Bb+ 7/6 ma subminor third
4. 93 168 B7 5/4 mi major third
5. 99 162 C 35/27 fe semidiminished fourth
6. 117 144 D 35/24 su semidiminished fifth
7. 121 140 D+7 3/2 sol perfect fifth
8. 135 126 E7 5/3 la major sixth
9. 141 120 F+ 7/4 ta harmonic seventh
0. 153 108 G 35/18 da semidiminished octave
The last note can't be "do" because that's reserved for the perfect octave. Instead, Heathwaite uses "da" to represent a slightly flattened octave. (If we use the Extended 7-Limit Scale, then Eb+ becomes "lo" and F#7 is a just major seventh, hence it's grandfathered "ti.")
It's also possible to begin the scale on white G. Then greenish G becomes "di," officially a slightly sharpened unison:
The last note can't be "do" because that's reserved for the perfect octave. Instead, Heathwaite uses "da" to represent a slightly flattened octave. (If we use the Extended 7-Limit Scale, then Eb+ becomes "lo" and F#7 is a just major seventh, hence it's grandfathered "ti.")
It's also possible to begin the scale on white G. Then greenish G becomes "di," officially a slightly sharpened unison:
The New 7-Limit Scale, starting on white G:
Key Sound Degree Note Ratio Solfege Interval (from root note G)
0. 45 216 G 1/1 do tonic
1. 51 210 G+7 36/35 di quarter tone
2. 69 192 A 9/8 re major tone
3. 81 180 Bb+ 6/5 me minor third
4. 93 168 B7 9/7 mo supermajor third
5. 99 162 C 4/3 fa perfect fourth
6. 117 144 D 3/2 sol perfect fifth
7. 121 140 D+7 54/35 si semiaugmented fifth
8. 135 126 E7 12/7 li supermajor sixth
9. 141 120 F+ 9/5 te minor seventh
0. 153 108 G 2/1 do perfect octave
(In the Extended Scale, Eb+ is "le" and F#7 is "to.") But for our song, we will start it on greenish G, and so we start on "do" and finish on "da."
OK, let's plan the lyrics now for our new song, which will be called "Do-Ru-Ma," the first three notes of the New 7-Limit Scale. We'll keep the original song lines for grandfathered notes:
DO-RU-MA
Do, a deer, a female deer,
Ru, ____________________
Ma, ____________________
Mi, a name I call myself,
Fe, ____________________
Su, ____________________
Sol, a needle pulling thread,
La, a note to follow so,
Ta, ____________________
That will take us up to da, da, da!
OK, we want to enter lines that rhyme -- although not necessarily the rhyme scheme of the original song (as there are now ten lines instead of eight). First of all, the last three notes la, ta, da already rhyme, and so "Ta, a note to follow la," will allow the last two lines to rhyme.
For "fe," I was thinking about fae, or fairies. The line "Fe, a flying little elf," fits, and notice that "elf" happens to rhyme with "myself" in the "mi" line. So now we have:
Do, a deer, a female deer,
Ru, ____________________
Ma, ____________________
Mi, a name I call myself,
Fe, a flying little elf
Su, ____________________
Sol, a needle pulling thread,
La, a note to follow so,
Ta, a note to follow la,
That will take us up to da, da, da!
I'm inspired by the rhyme scheme of yesterday's mouse poem to rhyme the third and sixth lines. For "ma," this sounds like a mother, while "sue" is what you do in court. Let's see:
Do, a deer, a female deer,
Ru, the day that I will fear,
Ma, to mom the babies say.
Mi, a name I call myself,
Fe, a flying little elf
Su, the way I make you pay,
Sol, a needle pulling thread,
La, a note to follow so,
Ta, a note to follow la,
Take us up to da, da, da!
Here "ru" refers to "rue" -- as in "rue the day" that something terrible happens.
Do-Ru-Ma -- Coding
This is a NEW program:
10 DIM S(9)
15 FOR V=1 TO 2
20 FOR X=0 TO 9
30 READ S(X)
40 NEXT X
50 DATA 54,105,96,90,84
60 DATA 81,72,70,63,60
70 N=1
80 FOR X=1 TO 70
85 READ A,T
90 SOUND 261-N*S(A),T
95 NEXT X
100 RESTORE
110 DATA 1,6,2,2,3,6,1,2,3,4,1,4,3,8
120 DATA 2,6,3,2,4,6,2,2,4,4,2,4,4,8
130 DATA 3,6,4,2,5,2,5,2,4,2,3,2,5,16
140 DATA 4,6,5,2,6,6,4,2,6,4,4,4,6,8
150 DATA 5,6,6,2,7,6,5,2,7,4,5,4,7,8
160 DATA 6,6,7,2,8,2,8,2,7,2,6,2,8,16
170 DATA 7,6,3,2,4,2,5,2,6,2,7,2,8,16
180 DATA 8,6,4,2,5,2,6,2,7,2,8,2,9,16
190 DATA 9,6,5,2,6,2,7,2,8,2,9,2,0,16
200 DATA 0,4,8,4,9,4,7,4,0,4,4,4,1,8
210 NEXT V
Yes, I know this song, but I hope it's worth it.
What About 11-/13-Limit Music?
So far, we haven't written much about the 11- and 13-limits. But some musicians do like to use the 11th and 13th harmonic in music:
- The undecimal quarter tone is 33/32. When we lower Degree 32 (Sound 229) E by this quarter tone, we obtain Degree 33, Ed (where "d" looks like a half-flat symbol). In Kite's color notation, this is called "amber E."
- The tridecimal third tone is 27/26. When we raise Degree 27 (Sound 234) G by this third tone, we obtain Degree 26, Gh ("h" is as close I can get in ASCII to this symbol). In Kite's color notation, this is called "ocher G." (Why is it "ocher" and not "orange"? Ask Kite!)
We can keep going with higher and higher primes -- 17, 19, and so on. The largest prime we can play on the computer is the Fermat prime 257. Degree 257 (Sound 4) is E played a small subcomma, 257/256, below white E.
I may consider playing some 11- and 13-limit music. I might consider converting the Dren Song, which is already an inversion song, to 13-limit.
Conclusion
There's so much more I can do with the New 7-Limit scale. One thing about the Extended 7-Limit Scale with Eb+ and F#7 is that these two notes are very close to a just 6/5 minor third apart. (The difference is called a "subcomma," or 225/224.) This allows us to play part of the Extended Scale as a pseudo-harmonic minor scale on red E (where Eb+ sounds like the leading tone D#.)
As Christmas is coming, some holiday songs can be converted to the 7-Limit Scale, especially songs where the song is in G major, but without F# (unless we use the Extended Scale with F#7). For example, on the radio I often here Mannheim Steamroller's Deck the Halls played in the key of F, except sometimes Eb is played instead of E. This version may be converted to the 7-limit scale.
Another possible holiday song to convert is Rise Up Shepherd and Follow, which is lesser known but appears in my Christmas music book. The melody of this song is fully in F Mixolydian, where the note Eb always appears instead of E. Notice that this song is an African-American bluesy spiritual song, and bluesy songs often employ the septimal intervals found in the New 7-Limit Scale.
Oops -- I mentioned race in this post, but hey, this is the bottom of a lengthy holiday post again. Let's put up the "traditionalists" label on this post just to be safe.
And as this is a traditionalists post, let me point out that traditionalist Barry Garelick mentions race in one of his recent posts as well:
Yes, God forbid we should study one subject at a time so we can eventually apply it to other disciplines. Just meld it into one big coloring book activity for teachers to facilitate. And of course, it is understood that mathematics is just white privilege but I’m stepping into other territory so I guess it’s time to stop.
This concludes my post. We'll return to our regular posting schedule on Monday.
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