Musical Fractions That Make Sense

Version 2The nomenclature we (except for those who use the quaver family of names) really is not very useful. Whole note, half note, quarter note, eighth note, and sixteenth notes are the terms by which teachers, both of music and of other subjects, connect music to fraction arithmetic. As far as it goes, they are correct. All of those names, that is, quarter, half and so forth are fractions, but fractions of what? They are fractions of a measure in common time. Because those names only apply to four-four meter, their meanings are irrelevant to every meter except four-four. What’s more, I’ve never, in 31 years of teaching, found a student for which knowing a quarter note was one quarter of a whole note really helped him or her perform the more accurately. Of much more value is that the quarter note is, at least frequently, equal to the pulse of the music; the ictus, the beat the coincides with what an ensemble conductor is conducting.

From there, it is helpful to know that eighth notes are two equal divisions of the quarter note pulse. It is not at all helpful to know that one eighth note is one eighth of a measure in four-four time, and, as I have pointed out, even less helpful if the child is not performing music in four-four meter. Eighth notes are the division of the quarter note pulse into two equal parts, or the dotted quarter note pulse into three equal parts. Stated as a fraction,  one eighth note is either one half or one third of an ictus beat. That is the sort of musical fraction that is helpful. There are two or three sounds of equal duration during each pulse. That is helpful, because musicians mostly divide beats, not measures. Quarter note pulses can be further divided into four equal durations, and dotted quarter note pulses can be further divided into six equal durations. When this occurs, those divisions of the beat are called sixteenth notes, and each sixteenth note is either one-fourth or one sixth of a beat. Even this fractionalizing is of limited value, because the performing musician is not measuring individual durations in relation to the beat, he or she is gauging how to evenly distribute a number of sounds over a single pulse. Nevertheless, fractions of a beat are more useful than fractions of a measure. Every note value less than the value of the ictus is a division of the beat, and the process by which beats are divided are an example of the mathematical operation of division, which includes fractions.

Notes that are longer than the duration of the ictus are elongations of the beat, and the process by which beats are elongated are an example of the mathematical operation of addition, including adding fractions. Once more, we don’t really care that a half note is half of a measure in four-four time, or that a whole note occupies the duration equal to an entire measure in four-four time. We do care that a half note is the duration of two ictus beats added together, and that a whole note is the duration of four ictus beats added together, though in both cases the quarter note must be the ictus for this to be true.

It is useful to think of note values as not only fractions of the ictus, but also as fractions of note_hierarchyeach other. Practicing sixteenth note passages while audiating an eighth note beat in a piece where the quarter note is the ictus, a practice referred to as subdividing, is used by many students and teachers as an effective way of achieving rhythmic evenness. Such thinking also facilitates shifting the ictus from, for example, the quarter note to the half note when the feel of the music suggests as much. This often happens when the composer transitions the music from a rhythmic section that is best understood in quarter notes, to a broad melody that comfortably soars above all in half notes. Understanding that those still present quarter notes are each half as long as the now predominant half notes makes the shift natural and enjoyable.

Thinking of note values as fractions of other note values also facilitates understanding rhythm when the ictus is not the quarter note. Knowing that  a quarter note is half of a half note, and that an eighth note is half of a quarter note makes dividing or elongating the half or eighth note ictus possible, and the concept of divisions and elongations of the beat transferable. Indeed, it is important for students to understand once they have begun to read music that any note value can be the ictus, and that it follows that any note value can be divided or elongated. Indeed, a whole measure can be the ictus, and a half note a division of the beat. It is also important to understand that before students begin to read divisions or elongations of the beat, they must learn them aurally, so that when they do read them, they have a sound to associate with what they see. Music Learning Theory (Gordon) and Conversational Solfege (Feierabend) both provide well researched and classroom tested procedures for doing this.

We cannot conclude our discussion without raising the issue of meter signatures. Though these look like fractions, and are frequently wrongly notated in texts as fractions and aurally referred to as fractions as in “three quarter time,” they are not fractions. Three-four meter does not indicate three fourths of anything. Instead, it is a convenient way of indicating that there is the equivalent of three quarter note durations in each measure of music. There is no way of knowing from a meter signature how the ictus is divided, whether into two or three divisions, nor is there any way of knowing what the ictus is. The bottom number of the meter signature may, and often does coincide with the ictus, but it frequently does not as well. That said, more often than not a meter signature with a 4 as the bottom number and a number evenly divisible by two but not three as the top number  has beats divided into two equal durations, and a meter signature with an 8 as the bottom number and a number evenly divisible by three as the top number has beats divided into three equal durations, though eight-eight meter is an exception to this (see Toccata by Frescobaldi).

Attempts to correlate music with Common Core Mathematics with fractions can be made with note value nomenclature, but such connections are not helpful or even confusing to music students. Connections between music and fractions are more advantageously made concerning fractions of ictus beats and fractions of other note values. While traditional nomenclature can and does continue to be used, the bases for names such as quarter and half notes is only relevant to four-four meter.

The Problem With Using Math to Teach Rhythm

2011 Symposium2

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 Musical-Balancestrong 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.”