Music and Math: Three Concepts for Understanding Rhythm

Music and Math

My job as a music teacher is to help students understand the basic elements of music. Rhythm is the most difficult of the “right-or-wrong” music elements (the others are articulations and pitches.) Mastering rhythm often requires multiple approaches. Connecting music and math is one approach.

We all have different learning styles. One model of learning suggests that we have a preference for visual, auditory, or kinesthetic experiences. Understanding where your strengths are in the visual-auditory-kinesthetic spectrum can direct you to helpful systems of learning. Here’s a link to the  VAK Learning Styles Survey .

Some students resonate with a kinesthetic understanding of rhythm. They need to feel the beat in their bodies. In the studio, we may use marching, bouncing balls, and/or conducting the music to help students internalize the beat. For kinesthetic learners, Dalcroze Eurythmics is helpful.

Other students prefer to hear the rhythms. These are auditory learners. The blog post “Rhythm Silliness” talks about the words I like to use to help students learn rhythm by ear. We use words like “Mississippi,” “peanut butter” or “R2D2” for sixteenth notes. We do a lot of playing rhythms together or I’ll play one measure and the student will play it back.

Visual learners often prefer a more analytical approach to learning rhythm. Charts and visual aids are helpful to folks who have strong visual skills. If you are one of these people, read on! This blog post is for you.

This blog post will show three mathematical concepts and how they contribute to a deeper understanding of the division of the beat.

1. FRACTIONS

5th grade math and music

math workbook

My son brought home his fifth grade math book last night. This is from an introduction page to a unit on fractions. This page clearly shows the relationship between music and math. It even points out that the time signature looks like a fraction. By adding the note values at the bottom of the page, students can figure out where the barlines are supposed to be.

A word of caution: I have observed that younger students can be confused about fractions. They think that if a pie is divided into two parts, there must be more pie because two is more than one. For this reason, I don’t use fractions with young students until they have studied them in math at school. I have had the most success using rhythm words to teach rhythm to young students.

However, once a student understands fractions, the concept is easily applied to music and deepens the student’s understanding of how the beat is divided into equal parts. Notes that are tied together become easy math problems. (1/4+1/4=1/2) Students who understand fractions have an easier time subdividing the beat, as they have to do when the eighth note is the value that gets one beat.

2. RATIOS

I love to cook and sometimes I have to double of halve a recipe. This calls for an understanding of fractions and ratios. Take a look at my measuring cup:

music and math in the kitchen

measuring cup

If you look closely, you can see that there are lines for 1/4 cup, 1/3 cup, 1/2 cup, 2/3 cup, 3/4 cup, and the top is one whole cup. Imagine that this one cup of flour is a quarter note. That note can be divided in many different ways. Three 1/3 cups = 1 cup in the same was as a triplet will divide a quarter note into three equal short notes. I like to think of the image of this measuring cup when working on rhythms that switch between eighth notes and triplets. I have a clear mental image of how triplets in one instrument and sixteenth notes in another will line up.

Music and math makes great cookies!

3. MEASUREMENT

Daniel Buckvich, professor of music at University of Idaho has an innovative system for teaching rhythm, which he calls “Measured Music.” I became familiar with his ideas as an undergraduate student at The Ohio State University. It was one of those great “AHA!” moments when I opened the book and saw for the first time the visual representation of my mental concept of rhythm. Buckvich’s “Measured Music” system begins with a ruler, something with which we are all familiar. (I should note that this ruler is in inches, so if you are a metric person, this might not be as helpful.)

music and math

from Measured Music

A ruler has whole numbers (1, 2, 3, and so forth) that indicate inches. The inches have smaller divisions also. The longest line is at the half-inch mark. Shorter lines indicate the 1/4 and 3/4 inch. Finally, the smallest lines divide the 1/4 into 1/8 (and sometimes 1/16).

Buckvich’s system also uses small circles, like mini pies, to show the subdivision of the beat. Sometimes the smallest subdivision is 1/2, other times it is 1/3. In the above example, you will see that it is 1/4 or the sixteenth note. Below is a page from later in the book showing complex rhythms. I love how visual this music and math system is.

music and math

another page from Measured Music

If your brain isn’t wired for this kind of math, that’s totally fine. You likely need another system for internalizing the pulse and subdividing the beat. As I mentioned earlier, there are other systems for teaching rhythm that are good for kinesthetic and auditory learners.

If the above music and math concepts work for you, great! Use them to deepen your understanding of the division of the beat. Visually overlay a ruler, a measuring cup, an apple pie over difficult rhythms. Continue to explore how the fractions can be overlaid and be prepared for some “Aha!” moments of your own.

Music is a beautiful, mathematical marvel. If you liked this post, you may also be interested in learning about the math concepts behind the harmonic series, music theory, acoustics, and instrument design.

Practice Session, Part 3: Etudes

Etudes, or study pieces, are the third element of a productive practice session.

If you haven’t read the blog posts OVERVIEW, WARM UP, and SCALES, you may want to start with those before reading about part #3.

OK, so there are six parts to every good practice session:

  1. Warm up.
  2. Scales, short technical exercises.
  3. Etudes.
  4. Repertoire.
  5. Improvisation and revisiting old material.
  6. Review, warm-down, reflecting on goals and planning for the next practice session.

Our bodies and minds are warmed up and our fingers have worked on some fast, but short, technical exercises or scales. If our practice session is 30 minutes in length, you will want to spend about 6 minutes on etudes. If you’re watching the clock, the third part will be minutes 7-13. Remember, these are ballpark numbers and you may need to adjust your practice session if you need more work in one area than another.

By the way, the word etude comes from the French word “to study.” I think it is a lovely, elegant word.

practice "drawers"Continuing the drawer analogy I have used in the other posts, the third drawer (orange in the picture) is slightly larger than the second drawer. In this drawer we find study pieces. These etudes are longer than the two or three line exercises we worked on in the scale section. Depending on your level, the etudes could be anywhere from a half page to four pages in length. Etudes are different from repertoire pieces because they are learned in a shorter time frame, usually 1-4 weeks. These pieces are played at lessons but rarely in a performance. Etudes are often included in volumes with a variety of key signatures. Many etudes are written with a specific technical goal in mind, such as octave practice or double tonguing. Repertoire pieces, on the other hand, are long term projects and are often performed in recitals or contests. We’ll discuss repertoire pieces in the next part of our practice.

Here are some of the books in my etudes “drawer:”

etudes for the third part of a productive practice session

some of my etudes

Etudes bridge the work between scales and repertoire. Can I be honest with you? I hated etudes when I was in college. They felt like drudgery. I wanted to skip to the repertoire pieces that I loved. But old age brings wisdom and I now know that practicing etudes helps with the repertoire work. Etudes can be entertaining too! My younger flute students like using Irish music for their etudes. They are pleasant for everyone to listen to and because Irish music is very tonal, students can immediately know if there is a wrong note. An added bonus is that Irish tunes are a blast at fast speeds, providing motivation for working with the metronome. Every instrument has standard etudes. For flutists, it’s Anderson, Altes, and Berbiguier. Please leave a note in the comment section below with the best-loved (or hated!) etudes for your instrument.

And now, you’re ready for the largest chunk of time: REPERTOIRE.

Practice Session, part 2: Scales & Technique

Scales and Technique

The second of the six parts of a practice session.

In the first blog post of this seven part series, I gave a general overview the elements of typical practice session:practice "drawers"

  1. Warm up.
  2. Scales, short technical exercises.
  3. Etudes.
  4. Repertoire.
  5. Improvisation and revisiting old material.
  6. Review, warm-down, reflecting on goals and planning for the next practice session.

If we think of the practice session as a set of drawers, we start at the top and work our way down. Today, we will look in the second drawer: scales and technique exercises.

Opening the second drawer, we find lots of scale books, technique exercises, and things of that nature. This is minute 3-8 of our thirty minute practice session. In this part of the practice session, you will want to focus on short technical exercises. Ideally, you will choose exercises that strengthen your weaknesses and solve problems you are having in repertoire (element #4). When creating your plan for the practice session (element #1), try to pick exercises that will help you achieve your goals. These are a few of the books–and the scale cards–that I use during this portion of my practice:

scales and technique books for flute

scales and technique books for flute

Scales. There are many different kinds of scales– major, minor, chromatic, modes, whole tone, jazz, pentatonic, and so forth. Work on memorizing the scales. Check out the Scale Game I use with my students. Advanced players will also want to practice scales in thirds, octaves, full range, descending first (then ascending), and any other combination you can think of.

Intervals. Each instrument will have unique needs for interval practice, but we all benefit from working on difficult combinations of notes. I like to use an exercise of expanding intervals to increase flexibility. Lip slurs are great for brass players. All wind players can work on the harmonic/overtone series.

Articulations. Practice different kinds of accents, like marcato, staccato, legato. Work on the beginnings of notes. Don’t forget about articulations: single, double, and triple tonguing. Pianists  may use exercises to improve their repetition speed and pedaling. 

In a 30 minute practice session, you may want to spend about 5 minutes on scales and technical exercises. The technical exercises should be short. This will keep the brain entertained. Keep in mind the technical problems you have in your repertoire and pick scales and technical exercises that compliment that work. This is called “interleaving.”

Interleaving: That’s a cognitive science word, and it simply means mixing related but distinct material during study. –From How We Learn by Benedict Carey.

Choosing scales and technique exercises that are related to the repertoire you’re studying actually makes it easier. That’s really the whole point of scales and technical exercises anyway.

Longer exercises, or etudes, will be included in the next phase of practice. The two purposes of this phase of practice are

  1. strengthen skills needed in the solo repertoire
  2. prepare the body and mind for the longer, intense concentration needed for the longer pieces

Fingers flying, mind roaring, let’s move on to ETUDES.

Woodshedding: Grouping Game

GROUPING GAME for your Woodshedding arsenal.

Woodshedding is the process of isolating a difficult section of music and practicing it until the technical issues are resolved. In other blog posts, I have offered games for changing articulations and rhythms. (See posts Rhythm Spinner Game, Articulation Game.) Today’s blog post will feature another strategy: changing how the notes are grouped. This changes where the accents are placed.

Think about your phone number for a moment. If you live in the US, your number is in the form 614-555-1212. We chunk the number into two sets of three digits and a final set of four numbers. Now, group your number differently. Let’s say 61-455-512-12. Or 6145-551-212. The order of the numbers did not change but our perception (and perhaps how easily they can be remembered) did change significantly.

 

Music is often grouped also. Here’s an example of four groups of three notes from a Handel Sonata:

woodshedding example

Giga from Handel Sonata

In this example, there are 12 notes split by slurs into four groups of threes. By changing the groupings, we alter how the brain perceives the notes. The order and rhythm remain the same, but the accents are different. Try playing the same passage in groups of two, four, or six as shown below.

 

Groups of two:

woodshedding example by 2

same notes, now in groups of two

Groups of four:

woodshedding

groups of 4

 

Groups of six:

woodshedding

groups of 6

In this exercise, we are playing the notes in the same order with the same rhythms. Two elements are changing:

  1. Different notes receive a small accent because they are the first of each grouping.
  2. The brain chunks them together in a new way.

Just as phone numbers are perceived differently when we changed the groupings of numbers, so too your brain is challenged when we change the groups of notes.