Rhythm +
math= C by Donald
J. Smith, Ed.D.
Please refrain from fainting, slipping into a fit of rage and
tearing this page out, or worse-- burning the RIMER: What you are about to read sets off
violent reactions among many musicians.
As much as we might try to escape it, deny it and/or ignore it, music and
acoustics are grounded in mathematics.
While I’m on a roll here, I’ll pile on another hot tamale, we have been
under pressure to collaborate with other subjects and this is one of the more
natural collaborations and the one that is often done poorly. I believe that embracing the
relationship between music rhythm notation and the embedded mathematic formulae
and requisite problem solving skills, teaching the concept of rhythmic notation
through a systematic approach may be an enlightening experience for your
performance ensemble members. This
would also be a very good fit for National Standard #5,
Disclaimer: In no way am I suggesting that this is the best or only way to present this topic. However, I cordially invite e you “along for the ride” in the hope that an insight in how you present this material might be improved. While a wordy (sorry) explanation of a practical pedagogical strategy is not nearly as effective as seeing it in action in a classroom, release time to watch each other work is much more difficult to set up than snuggling with your RIMER during some quiet time at home.
Most students naturally try to use lower order thinking strategies to fulfill assignments, especially with written assignments during performance ensemble time. They naturally want to just play their instrument. Likewise, we should strive to maximize the time that they are performing as well. However, portions of lessons like these should lead to more efficient use of performance time and especially practice time as the students develop into independent learners and they are utilizing higher order thinking skills while playing. It can be argued that music students already use their higher order thinking skills better than other students while they are involved in the music program. I believe that to be one of the most powerful reasons that a growing body of research is confirming that students become smarter and better students while participating in performance ensembles. I also believe that it is our responsibility to shape the higher order thinking skills that our students use through the curriculum and strategies we present. Therefore, by just providing a few examples of the relationship between the note/rests they will see everyday with mathematical concepts we would be letting them down and dumbing-up our curriculum. In order to fully grasp a concept, students need to be stretched beyond the commonplace. I try to set that standard in the first lesson and almost always get a n enthusiastic response from students as the “light-bulbs” in their heads are magically switching on.
Step 1: The consistent and logical relationship between notes and rests. (5-10 minute daily presentations for each step
I begin this lesson with a blank chalkboard (dry erase now), plenty of space and the necessity of maintaining a blistering pace through all of this. I draw a whole note in the center and ask students to identify a whole note. I add a stem and repeat the question. I ask what is the next step in reducing the note’s value (guiding the response to fill it in). At this point I add two more quarter notes beneath the first note, but side by side (or is that side by each?). I dramatize the moment that from this next step on, all notes are identified by the addition of a flag or a beam-as I convert the quarters into eighth notes by adding a flag and adding a beam to the two quarter notes (± to Ä and ± ± to Ö µ ). This is where the light bulbs start igniting, because we start adding flags/beams and asking what notes they are now: 16th, 32nd, 64th, 128th, 256th, etc. I increase the pace as we move through that sequence. We then do the same thing with rests with the drama of the difference between whole and half, the uniqueness of the quarter rest and then the pace quickens as we hit that dramatic moment with the eighth rest and then 16th, 32nd, 64th, etc. While all of my students have been drilled in the relationships between notes and rests by a great team of general music teachers throughout the district, they are at first bored and then typically become excited with the concept of expanding beyond 16th notes/rests (as usual, pacing is the key to success).
Step 2: Rapidly remind the students about the 2 to 1 mathematical relationship between notes/rests by sketching (or pointing out the chart if it is already in your room) the expanding 1 whole note splitting into two half notes, that spit into two quarter notes each, that split into two eighth notes each, etc.
±
° ±
¬
etc.
° ±
±
It is critical to extend this query Make sure the equivalence of same-named-rests connection is made at this point. Quiz them on how many eighth notes are IN a whole rest? How many 32nd rests are in a half note, etc. Once again, the stretch beyond the usual values help to breed enthusiasm and build the conceptual understanding. Make sure to stretch into fractions such as; how many half notes are equal to an eighth note? (answer 1/4th ). I find that it is critical at this stage of concept building to carefully control the inclusion of several different levels of calculation. In other words, control the progression of questions considering how many levels of subdivision are required to resolve the problem. One level of query would be whole to half, quarter to eighth, 32nd to 64th, two levels of query would be whole to quarter, eighth to half, 64th to 16th, and three levels of query would be whole to eighth, eighth to 64th, etc. By proceeding through one level, then two, then three or more and then mixing the levels, a mastery of the concept is more likely to occur. A final addition to this level is being able to deal with the summation of multiple levels (how many 1/8th notes are in 2 quarter notes and a half?) as well as fractional relationships (how many half notes are in a dotted quarter note? [answer, ¾]).
Exercises (place the number in the blank that will result in the notes/rests have equivalent duration):
1. ¬
=_____ °
4. ±
=____ °
7. ±
=____ á
10. Ö - µ
=____Æ
2. ±
=____ à
5. ° °
=____±
8. à =____
¶ »
11. ¬
=____ Ä
3. Å =____Æ
6. °
=____ Ä
9. Ä =____
±
12. ä
=____ Å
Step 3: This is the step that requires the most higher order thinking/problem solving skills and the step that causes even professional musicians to hesitate at times; adding a dot (and then double dotted, etc.). I always add a dot to a whole note first and carefully ask how many quarter notes it is equal to (we haven’t covered beats yet—that’s next in the sequential order of presentation and essential to leave out of the vocabulary at this time). Most students know that adding a dot doesn’t just add a “beat” or one quarter note, but they pause until the definition of adding one-half the original value concept is hopefully suggested by one of the students. We then rapidly explore the new layer of calculations required to equate the value of dotted notes/rests. This should involve a reminder about basing the calculations on the least common denominator (the least common denominator for a dotted quarter note are eighth notes, for dotted eighth notes it would be sixteenth, etc.). Once again, make sure to stretch into summations and fractions; such as how many half notes are equal to a dotted eighth note? (answer 3/8th ).
Exercises (place the number in the blank that will result in the notes/rests have equivalent duration):
13. °.
=
____ ±
14. °. = ____ à
15. ¬ . = ____ Å
16. à. ____ Å
17. ±. ____ °
I am reminded of a technique I was first exposed to in choral music that can be adapted to instrumental music effectively. The chorus director would have us count everything in eighth notes or 16ths (which ever was the least common denominator) throughout the piece, instead for using the lyrics. For the first (very amateur) Methodist choir that I directed, this pulled all of their attention from the lyrics and made them actually look at the notes and rests just above. Eventually they became a pretty good reading choir. For band kids, counting aloud (with dynamics and phrasing and stylistic considerations) can be a huge help in reading ability and fun! However, having the section play the subdivisions is a great way to quickly add rhythmic integrity to a piece. Instead of the written: ±. Ä ° Ä. Å Ä. Å ° Have the students play: Ö - - µ Ö - - µ ¶ = = » ¶ = = » Ö - - µ Better yet, have half the students play it as written and the other half play the subdivisions—then switch roles (while the percussion section counts out-loud, supporting the change or gears required when the least common denominator changes from 8th to 16th and back to 8th). I think they call that collaborative/cooperative learning, best practice and a lot of other “edu-bable” that we’ve been using in music for years!
Step 4: The denominator of the time signature
This number is the most powerful number of the time
signature, yet it is often the least understood. I believe the cause of this lack of
understanding is that students rarely
see anything other than a 4, and when they do, it is an 8 that number is
so inherently confusing that many theorists advocate for the number to be
replaced with a ±. so that the time signature would
look like 6/±. 12/ ±.
Please note that this step
is an extension of the previous step if the connection is made with the students
that you are now asking the same question as above, you just use the denominator
of the time signature to calculate the equivalency of notes/rests. In other words, for the problem: 12/2 ¬ .
=____ you must first identify that the half note gets one beat and
then ask the question, how many half notes are their in the dotted whole
note? Of course the answer will be
3.
Stravinsky and other 20th Century composers pushed the envelop on time signatures. My students usually don’t look at the time signature (or key signature, but that’s another topic) carefully enough prior to beginning a piece. They use that old strategy: ready, fire AIM! too frequently when beginning a new piece. I believe it is for the same reason that they stop watching conductors—if there isn’t enough valuable information or variety in the data being collected, why pay attention. If they always play in Bb or F, if the conductor really is nothing more than a starter-metronome-ender, and if the time signature is rarely anything but 4/4, why pay attention? Occasionally stretching the time signature is healthy. It is very possible to ask the students to play a piece in 4/4 tapping the quarter notes the first time through, tapping half notes the second, whole notes the third and then work your way back to tapping eighths and sixteenths, etc.
Exercises (place the number of beats each note or group of notes would receive in this time signature):
1. 12/4 ° =____
4. 6/2 °. =____
7. 9/8 °. =___
10. 12/32 ±. =___
2. 14/8 ± =____
5. 10/4 Ä. =____
8. 4/8 Ö - é
» =____ 11. 4/1 ¬ .=____
3. 15/32 Å =____
6. 5/16 Ä. =____
9. 7/32Ä
=___
12. 4/4 Ö µ ¶ =
» =___
Step 5: The complete time signature
This step helps the students assimilate all of the previous steps by using both elements of the time signature as they construct/adjust measures. The numerator (top number) is the last mathematical element we get to in this sequence, but it is frequently the only element noticed by performers—who skip all or some of the previous steps, or we just assume that they have mastered them during previous musical study. Of course is reveals how many beats are in a measure. Some folks prefer to say that the 12/8 time signature is the equivalent of 12 eighth notes in each measure. How ever you say it, there are more than a few required mathematical processes required to arrive at a level of understanding that can be generalized throughout the literature. And, as dry as that last sentence may sound, kids reslly get fired up when they play new and unusual time signatures. I wrote a series of children’s choir pieces when I was required to lead the children’s choir along with my second Methodist choir directing gig. I wrote a series of 12 pieces for them to sing (they performed once per month) and by far their favorite one was the one in 7/8 (2 2 3). The joy they emitted while we sang and clapped and marched and danced while learning that piece was unmatched by any of the other typical pieces that I had introduced to them. Middle School and High School and yes even those jaded College Students return to a youthful exuberance when they confront fresh rhythmic meters!
Exercises (change one note or rest to make the measure fit the required number of beats):
First of all, if you’ve made it this far, thank you for
enduring the necessary tediousness of this article. I assure you that the presentation of
all of these steps is a lot of fun in a classroom with a bunch of live, high
energy learners. Even though they
will probably groan a bit at the beginning of some these experiences, winning
them over is likely if you keep the pace up, your spirits and expectations up,
and present material with the comical high drama that so many directors do so
well. As a reward for surviving the
article, I’ll tell you what the “C” stood for in my mind as I wrote the equation
in the title: Take your pick: perfect for Collaboration with math teachers,
Complete Conceptual understanding, and/or Confidence with rhythmic
reading/writing. Thank you to all
of the teachers I have had in the past that I borrowed some of these ideas
from. And especially thank you to
all of the students who had to endure me as I stumbled through the early stages
of developing this sequence, driven by the daily reminders of the necessity of
rhythmic understanding in the quest to develop truly independent music
learners. Please contact me with
any comments or questions at [email protected]
and try the really cool free downloadable notation program I used for this
article at www.music.qub.ac.uk/~tomita/bach_r4.html (or Google to “Bach font” and you’ll
find it-Hint #2: the instructions don’t tell you to first “UNZIP” the font file
and install both version 3 and 4 of the font before following their
instructions—it should work after
that).)