As I write this, I’m looking at a page from a popular band method book. There is one of those boxes at the top of the page that directs students’ attention to an important concept or new learning. There is a pair of eighth notes followed by an equals sign followed by a quarter note. Next to these symbols is printed, “two eight notes equal one quarter note.” We know what is meant by this; that both two eighth notes and one quarter note occupy the same duration in time. But there are important ways in which two eighth notes do not equal one quarter note. With the two eighth notes, there are two sounds made within a single beat, whereas with one quarter note, there is only one sound made within a single beat. The duration of the beats is equal, but not the notes. One is a duration equal to the length of one beat, the other is a division of that beat into two equal parts. When someone looks at two eighth notes, they do not see something that looks like one quarter note. They look different, and when a person hears two eighth notes and then one quarter note, they sound different, so how can they be equal?
For those that use Curwin/Kodaly rhythm syllables, two eighth notes have a different name than one quarter–yet another way in which they are different. Two eighth notes are called ti-ti, and one quarter note is called ta. There’s no equality in that. So two eighth notes don’t look the same as one quarter note, they don’t sound the same, and they often aren’t given the same name, so how can we expect children to understand the statement “two eight notes equal one quarter note?”
Those of us who don’t use the British nomenclature of quaver, semi-quaver, and so forth, run into another problem when trying to use math to teach rhythm. We call a note in common time with a duration of 4 beats a whole note, a note with a duration of 2 beats a half note, of one beat a quarter note, and so forth. This makes perfect sense in only one meter signature–common time. In three-four time, a half note is two thirds of a measure, a quarter note is one third of a measure, and an eighth note is one sixth of a measure. Whole notes completely disappear–or should we call a dotted half note a whole note in three-four time? It a pretty sloppy thing to call something a half when it’s really two thirds.
Meter isn’t supposed to be about counting and fractions, it’s about recurring patterns of strong and weak beats. Our teaching must show students how the notion indicates the beats, beat elongations (durations longer than one beat) and beat divisions (durations shorter than one beat). We don’t make the first beat of each measure stronger than the others because that’s how it’s notated, we notate it that way because that’s what we hear when we listen to or audiate that music. Math is important to music because we naturally perceive patterns of strong and weak beats, beats, divisions and elongations. It is the beats that are equal, not the configuration of notes that occur within the beats.
I like to illustrate this to my students in this way. I have four (for common time) or three (for triple meter) students come to the front of the class and stand apart, with equal distances between them. These students represent beats. I then walk from in front of one student to the next, taking one step. My single step represents a quarter note, and the students represent beats. I have a time keeper clap each time I arrive at the next student. Then I go back and walk again, this time taking two equal steps to get from one student to the next. Now my steps represent eighth notes. The time keeper will report that I still got to the next student in one clap, and that he or she clapped at the same tempo as I moved across in front of the students. The class will observe that in order to make it to the next student in the same amount of time, I had to make my smaller steps faster. The steps were half as big so I had to walk twice as fast. I then have the class do a walk around the room. When I play quarter notes, they take single big steps. When I play eighth notes, they take half-sized but twice as fast steps. They listen to me play the rhythm first, then they step it. This teaches them that the beats are equal, but not the notes, and that every beat is the same, but that some beats have divisions within it and some beats don’t. So the next time you see or are ready to say “two eighth notes equal one quarter note,” remember, in the words of Ira Gershwin, “It ain’t necessarily so.”