The 16EDL scale:
Degree Ratio Note
16 1/1 tonic
15 16/15 diatonic semitone
14 8/7 septimal whole tone
13 16/13 tridecimal neutral third
12 4/3 perfect fourth
11 16/11 undecimal semidiminished fifth
10 8/5 minor sixth
9 16/9 dominant seventh
8 2/1 octave
10 INPUT N
20 FOR D=8 TO 16
30 SOUND 261-N*D,4
40 NEXT D
This is a descending scale. To make the scale ascend, use:
20 FOR D=16 TO 8 STEP -1
This is our second 13-limit scale -- the other such scale is 14EDL. Which of the two 13-limit scales sounds inherently better? Well, it depends. The 14EDL scale contains seven notes, and so it reminds us more of the major and minor scales.
But 16EDL might be more musically useful. It contains a tridecimal neutral third ("ocher 3rd") on the root note (the first such EDL scale to do so) rather than the 14/13 "trienthird." This doesn't mean that it contains a full ocher triad, of course, since there's no perfect fifth. Instead, 16EDL contains a 16/11 "undecimal semidiminished fifth." This interval is also known as a "subfifth." I prefer to reserve the prefixes "sub-" and "super-" for 7-limit intervals, but "semidiminished" is such a mouthful -- and besides, there is no septimal subfifth. Thus I might give in and use the name "subfifth" anyway.
Once again, both 14EDL and 16-EDL are 13-limit, but only 14EDL is 13-odd limit, while 16EDL is a 15-odd limit scale. So 16EDL contains the entire 15-odd limit tonality diamond.
Using Kite's color notation, here is the scale again:
The 16EDL scale:
Degree Ratio Note
16 1/1 white E
15 16/15 green F
14 8/7 red F#
13 16/13 ocher G
12 4/3 white A
11 16/11 amber B
10 8/5 green C
9 16/9 white D
8 2/1 white E
Because the root note is white, we don't change the note colors to obtain the interval colors. Thus this scale contains both a "green 2nd" and a "red 2nd," as well as an "ocher 3rd," "white 4th," "amber 5th," "green 6th," and "white 7th."
The other interval names are more established. The diatonic semitone 16/15 between E-F is so-called because it's the same semitone that appears is in the diatonic (or just major) scale. I use the name "dominant seventh" to describe 16/9 between E-D since it appears as such in a just major scale. For example, in A major, the dominant seventh chord is E7. If the root A is white, then both D and E in the same scale are also white -- and so the interval between E-D is indeed 16/9.
The 16EDL scale is truly utonal as it's completely based on undertones -- all of the intervals above the root are ocher, amber, red, green, or fourthward white. One musician, Bart Hopkin, has written an article about this sort of scale:
Hopkin describes 16EDL as follows:
The subharmonic series is less familiar than the harmonic series, and for good reason: while the harmonic series appears frequently in nature and is integral to many musical sounds, the subharmonic series appears much more rarely. (It’s sometimes said that it does not occur in nature. I can think of an instance or two in which it does, but that’s a topic for another discussion.) The subharmonic series is the inversion of the harmonic series: while the harmonic series consists of integral multiples of a base frequency (f, 2f, 3f, etc.), the subharmonic series consists of f, f/2, f/3, etc. While the harmonic series is conceptualized as an upward progression of frequencies, the subharmonic series is usually thought of as a downward progression. And while the harmonic series starts with large intervals that get smaller as you ascend, the subharmonic series starts with large intervals which get smaller as you go down. As with the harmonics, with the subharmonics once again we get a sweet spot of scale-sized intervals between 8 and 16. This, then, is the second of the two complementary scales: subharmonics 16 through 8, repeating at the octave.
He tells us that he wishes to combine the otonal scale with the utonal scale. Using Kite's colors, The corresponding otonal scale contains intervals above the root that are emerald, jade, blue, yellow, and fifthward white. Some authors refer to this scale as 16ADO (arithmetic divisions of the octave).
Of course, I am not the first to suggest these two scales. I first ran into the idea in an essay and accompanying cassette called Sounds of Just Intonation written by Ralph David Hill in 1985. Prior to that, Harry Partch, in his seminal work Genesis of a Music, discussed the harmonic series and subharmonic series. He coined the terms otonality (from overtone) and utonality (from undertone) to refer to tonalities based in the two series respectively. I’ve been using the term over-under scales to refer to scale systems which bring the two tonal approaches together as complements to one another. By extension, I’m also using over-under a noun to refer to instruments built to play in such scales (of which, admittedly, there are very few at present). Example of the term used in a sentence: “Nice over-under you got there.”
And here's what he specifically says about 16EDL, the one that Mocha can play:
And what about the subharmonic scale? Perhaps less familiar, because the subharmonic series isn’t widely present in nature? No, to my ears the subharmonics 16-8 scale has that strange-yet-familiar feeling to an even greater degree. Don’t ask me why; it’s just my ears. But I take great pleasure in it.
So Hopkin decided to build an actual instrument that can play the 16ADO and 16EDL scales (which he calls the "over-under scale":
So I built the instrument! It consists of two sets of rectangular aluminum tubes tuned by length and played by percussion. This is a suitable material for such a project because the hollow aluminum bars can be tuned to pitch quite accurately and, once tuned, retain the tuning well (although they are subject to some variation with temperature). With percussion bars like this, there’s often a problem with unwanted inharmonic overtones within the bar tone coming through too strongly and dominating the fundamental, especially in the lower bars. In this instrument the problem is mitigated in those lower bars by air resonance which strengthens the fundamental but not the overtones. The air resonance is that of the air column contained within the rectangular tubes themselves: by placing flute-like toneholes in the wall of the tube, the resonance pitch of the air column can be tuned to match the fundamental of the percussion tone. Strike the bar, and this internal air resonance comes into play and reinforces the tone of the metal bar itself. This is like the idea of placing tuned air resonator tubes below a marimba bar, but in this case the resonator is not a separate tube, but rather is the hollow bar itself – a much more compact arrangement. The choice of rectangular bars rather than round tubes has to do with a certain detuning problem that arises in connection with the drilling of the toneholes. Without going into details, I can say that this problem is easier to manage in hollow rectangular bars than in cylindrical tubes.
And here are "a couple more observations about the over-under scale pair":
The harmonic and the subharmonic 8 -16 scales have a sort of natural integrity, but they each lack some basic intervals that one might like to have available. Most notable among these are the fourth for the harmonic scale, and the fifth for the subharmonic scale. But with the two scales paired and positioned adjacent to one another, these intervals become available as the player can borrow tones as needed between the two. In addition to occasional borrowing for essential intervals, less obvious tones borrowed between the scales can provide color and interest and complexity within the musical context. For that matter, one could create entirely new scales from various combinations of tones from the two.
In fooling around with these scales, I’ve found that particularly lovely tonalities can be found in the modes based on the 12th harmonic or subharmonic. (It’s an interesting coincidence that the mode based on the twelve turns out to be especially appealing for both scales.)
Notice that the mode based on the 12th subharmonic is essentially 12EDL. (It's possible that Hopkin's mode still contains 13 and 15, so it's not quite 12EDL, but closely related). Most likely, he finds 12EDL and 12ADO appealing because each contains a perfect fourth and fifth.
He tells us which note he uses as his tonic:
As the fundamental for my over-under scales I chose 62.4Hz, which corresponds to the note B natural plus 18.5 cents.
The 16EDL scale in Mocha is based on E, not B. This is because all of the notes whose Degrees are powers of two are E's, starting from Degree 8 (Sound 253), four octaves above middle C, all the way to Degree 256 (Sound 5), the E below middle C.
As we'll see later this week, the possible root notes for 16EDL scales are themselves members of the fundamental 16EDL scale. Thus there exist 16EDL scales with tonics at amber B (or B quarter-flat) and green C (C+ syntonic comma). It's too bad that we can't have a 16EDL based on Degree 21, which is red B (B> septimal comma of 27.3 cents, which is more than 18.5 but close). Unfortunately, 16 * 21 is 336, which takes us out of Mocha's range (or "compass," as Hopkin would put it).
Finally, here's a link to his instrument, the "tangular arc":
There is an audio clip based on "subharmonic 12" (12EDL, possibly with 13 and 15). If I had the score, then I could convert it into Mocha. Notice that 12 * 21 = 252 (Sound 9), and so we could even play the song in the same key (based on red E, the "12" of his scale where red B is 8 or 16) provided that no 13 or 15 appears. Otherwise, we must code the song in another key, such as amber B or, more likely, green C.
Our exploration of 16EDL will continue on Thursday.
Question 13 of the SBAC Practice Exam is on solving equations:
Emily is solving the equation 2(x + 9) = 4(x + 7) + 2. Her steps are shown.
Click on the first step in which Emily made an error.
Step 1: 2(x + 9) = 4(x + 7) + 2
Step 2: 2x + 18 = 4x + 28 + 2
Step 3: 2x + 18 = 4x + 26
Step 4: -8 = 2x
Step 5: -4 = x
Click on the solution to Emily's original equation.
-10.5 -6 -2 0 2 4.5 8
Ah, so this is definitely a strong first-semester Algebra I problem. We must check each step carefully to find an error. For Part A, we see that in Step 2, Emily must combine the like terms 28 and 2, but in Step 3, she writes the sum as 26. (Clearly, she was thinking of 28 - 2 instead of 28 + 2.) Therefore the first step that contains an error is Step 3.
For Part B, let's start from the last correct step (namely 2) and proceed with the sum 28 + 2 = 30:
Step 2: 2x + 18 = 4x + 28 + 2
Step 3': 2x + 18 = 4x + 30
Step 4': -12 = 2x
Step 5': -6 = x
So students must click on the value -6.
I think ahead to my summer Algebra I class. My students are likely to make mistakes like Emily's, and so I must be able to correct them. If Emily were a girl in my class, I could help her by praising her last correct step (namely 2) then and ask her, "What's 28 + 2?" Hopefully, she'll realize that it's 30, not 26, and then she can finish solving the equation.
Question 14 of the SBAC Practice Exam is on explicit and recursive functions:
Match each recursive function with the equivalent explicit function.
f (1) = 18; f (n) = f (n - 1) + 6; n > 2
f (1) = 18; f (n) = f (n - 1) + 12; n > 2
f (1) = 1; f (n) = 6f (n - 1); n > 2
f (1) = 1; f (n) = 12f (n - 1); n > 2
Possible explicit functions:
f (n) = 6^(n - 1); n > 1
f (n) = 12 + 6n; n > 1
f (n) = 12^(n - 1); n > 1
f (n) = 6 + 12n; n > 1
The first two recursive functions are arithmetic sequences, so they might conceivably appear in the first semester of an Algebra I class. The last two recursive functions are geometric sequences. These used to appear in Algebra II, but since advent of the Common Core, I've seen geometric sequences and exponential functions pushed down into the second semester of Algebra I.
Let's look at the two arithmetic sequences first. The first term of each sequence is 18, but in the first sequence the common difference is 6, while in the second sequence it is 12. The coefficient of n in the explicit function is the common difference, so we already know that the first sequence must match to the "+6n" function while the second matches to the "+12n" function.
Even though we don't need it, it's helpful to know how to determine the constant term in each of the explicit functions. Some texts tell the students that the explicit function is f (1) + (n - 1)d (where d is the common difference), and so they must expand (n - 1)d. The U of Chicago Algebra I text has the students work backwards to find a "zeroth term," and this is the constant. So in the first sequence, the zeroth term is 18 - 6 = 12, and in the second, it's 18 - 12 = 6. Therefore the first explicit function must be f (n) = 12 + 6n while the second is f (n) = 6 + 12n. How I plan on teaching this, by the way, depends on how it's presented in the text I'm likely to use this summer, the Glencoe text.
Now we move on to the geometric sequences. This time, the trick is that the common ratio must be the base of the exponential function -- that is, f (1)r^(n - 1). In each case the first term is 1, so all that remains is r^(n - 1). Therefore the third sequence is f (n) = 6^(n - 1) while the last is f (n) = 12^(n - 1).
SBAC Practice Exam Question 13
Common Core Standard:
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
SBAC Practice Exam Question 14
Common Core Standard:
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
Commentary: In the U of Chicago Algebra I text, the first lesson in which multi-step equations with both the Distributive Property and variables on both sides is Lesson 6-8, "Why the Distributive Property Is So Named." Arithmetic sequences appear in Lesson 6-4, "Repeated Addition and Subtraction," though they aren't called this. Meanwhile, although exponential functions are in Lesson 9-2, "Exponential Growth," geometric sequences don't appear at all.