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.

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7 thoughts on “Musical Fractions That Make Sense

  1. Pingback: Musical Fractions That Make Sense — mr a music place | I Write The Music

  2. Great discussion of an arbitrary convention we’ve inherited from tradition.

    I often find it useful to think of the bar as the principal unit of time, the measure of duration. From this point of view, the first beat, the ONE of each measure, establishes a higher-level pulse that may be variously subdivided (into three, or four, or five, …, or n beats of equal duration). Series of measures joined together form beat cycles, song forms, passages or sections, what have you. The measure is a key to meter and form.

    Practically speaking it’s an arbitrary matter, given our convention, whether to call the beats in a bar “eighth notes” or “quarter notes”, so long as we follow through once we’ve selected a term that suits our purposes. Sometimes we do away with this terminological burden entirely, and speak of, say, 6 or 12 beats to a bar, and think of this as a whole measure, without worrying about what to call a quarter or an eighth. So long as a beat is sounded or held in mind, players may practice subdividing each beat into any number of tones of smaller duration, and practice combining any number of beats or beat-subdivisons into longer tones.

    Of course one way to develop musical skill and insight along these lines is to throw staff notation out the window, and express the desired rhythmic proportions graphically, geometrically, arithmetically, linguistically… or by direct demonstration as musical sound. If one has a desire to integrate mathematical and musical education, this is one way to go about it. For instance, let students use blocks or graph paper to depict a series of proportions of duration, and then sound out the durations depicted by clapping, or singing, or playing instruments together or in turns. One might devise simple arithmetical tasks based on the series of proportions. For instance… what fraction expresses the ratio of the sum of duration-units we sang before Sally’s turn and the sum of duration-units we sang from Sally to Paul?

    I suppose one theoretical benefit of tying the “whole note” to common time (or in principle, to any single meter) is that it establishes a single proportion of duration corresponding to each duration-symbol (crotchet, quaver, etc.) across all meters available in the system. Perhaps this is easier on the sight-reader in cases involving multiple meters in a single piece — say a transition from 8/8 to 7/8. The symbols marked within each bar retain their value from one bar to the next, from one meter to the next. Accordingly, the length of the bar varies along with the meter: The beat takes priority, and determines the length of the bar.

    We could do it the other way around, and give priority to the measure: Let the bar indicate a uniform duration across all meters, and the duration of beats (breves, crotchets, quavers…) within each bar be determined by the meter assigned to the bar. Would this really be more complicated for the practiced sight-reader — or is it only a bias of training and convention?

    In any case, I find it helpful to get my nose up from the beat and breathe a bit more deeply at the bar from time to time.

    • Modern time signatures are refinements of mensural notation which was used 1260-1600. The predesessors indicated one ictus and one division. A hold over is the C now used for “common” time.

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