Keys and Scales are Sets

Summary: Lesson plan, designed for grades 6-8, for a math-music interdisciplinary activity that use musical keys and scales to give students a chance to practice identifying sets and intersections of sets.

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Here is a lesson plan for a math-music interdisciplinary activity that uses the basic concepts of math set theory to determine which musical keys are most closely related. The activity may be used in math class, as a real-world demonstration of sets that is quite different from most of the other examples they will see. It may also be part of a music theory class, to demonstrate the concept of related keys.

The key that a piece of music is in is simply the set of notes that are allowed and expected to be used. (See Key Signature or Major Keys for more on this.) A scale is a list of those notes, in other words, a list of the items in the set. This activity encourages students to explore the math concepts of sets and intersections of sets, using musical scales. The lesson will be most illuminating to students who are also familiar with musical scales and keys, but musical knowledge is not necessary to do the activity. Teachers who are not comfortable playing the demonstration tune in different keys can arrange to have someone else do so. (A student who has had some piano lessons, for example, will be able to do this easily.)

Goals and Assessment

Grade Level - 6-8, or any students who are learning about set theory (in math class), or about keys and scales (in music class).
Goals - The student will use musical keys and scales to explore the concepts of sets and of intersections of sets.
Student Prerequisites - The students should already be somewhat familiar with the math concepts of sets and intersections of sets.
Teacher Expertise - The teacher should be familiar and comfortable with major keys and scales, should understand the math concepts of sets and intersections of sets. The teacher should be able to play the demonstration tune in the various keys, or should arrange to have someone else do this during the class period.
Time Requirements - One (approximately 45-minute) class period
Objectives - The student will listen to and watch a tune (that uses all the notes of a major key) being played in four different keys. For each key, the student will construct a set of the notes used in that key. The student will then construct set intersections for each combination of two keys, and will draw conclusions about the relatedness of the keys, based on the number of notes in each set intersection.
Music Standards Addressed - National Standards for Music Education standards 8 (understanding relationships between music, the other arts, and disciplines outside the arts), and 5 (reading and notating music).
Other Subjects Addressed - The activity also addresses the following math skills: identifying sets, finding intersections of sets, and drawing conclusions about non-math systems using set theory.
Evaluation - Assess students by grading the worksheets.

Materials and Preparation

You will need at least one instrument for demonstration purposes. If you have enough that each student has an instrument to play on, that may be better for older students or students who are musically experienced. For younger or musically inexperienced students, simple demonstration may be better. Instruments that only play one note each, like hand bells, are ideal, but instruments like piano, xylophone, or electric keyboard, also work well. If the students must find the notes from a demonstration, the instrument should be one on which it is easy to see what notes are being played. If all of the students are playing their own instruments, and know the names of the notes they are playing, it does not matter what instruments are being played.
Have enough reproductions of the two worksheets for each student. Keep a copy of the answer keys for yourself. Here are PDF files of Worksheet 1, Answer Sheet 1, Worksheet 2, and Answer Sheet 2. In case you have any trouble with the PDF files, the worksheets and answer sheets are also included at the end of this module, but the PDF files will give you a nicer-looking worksheet.
Choose the song you are going to use for the demonstration. If you are not a musician yourself, the suggested song is "Happy Birthday to You". This song is very familiar to almost everyone in the U.S. If you are a musician, feel free to pick any tune that uses all the notes of the scale. If you are not a musician and live in a culture that is not familiar with the birthday song, ask a musician to help you pick out and prepare a tune that most people in your culture know. (If you email me through the Connexions Project, I will be happy to add your song to this lesson.)
Practice playing your chosen tune on your chosen instrument so that it will be easy to present it to your class. If you don't know a tune, look at the keyboard on the handout. You can even label your keyboard with letters on stickers or masking tape if necessary. Don't be embarassed about this; it will help the students, too. Now look at the lists of notes for "Happy Birthday to You" in the Procedure below. Play the notes listed, using the rhythm you know from singing the song.

Procedure

Explain that a scale is the set of all the notes in a key.
Play a chromatic scale for your students, that is, all the notes available from lowest to highest.
Explain that this chromatic scale is the "set of all notes available", and that most songs use a smaller set of notes, either a "major scale set" or a "minor scale set". You are only going to talk about major scales today. You may also need to clarify that, for the purposes of scales and sets, all C naturals are considered to be the same note, all C sharps are the same, all D naturals are the same, and so on. In other words, don't worry about whether a note is C or C' - in the set, they are the same thing.
Play your song in C Major. This is the key that uses only natural notes. In this key, "Happy Birthday to You" is:
G G A G C' B / G G A G D' C' / G G G' E' C' B A / F' F' E' C' D C' /
Position yourself so that the students can see what notes you are playing, or have them play it after you. If it helps the demonstration, physically block the notes you are not using, or, if they are separate bells, put the unused notes further away from you. Have them color or otherwise mark the notes used on the top keyboard on their worksheet. When they have all of them marked, have them write in all the names of the notes used in the set oval to the right of the keyboard.
Repeat steps 4 and 5 for the key of D major. "Happy Birthday to You" in D Major is:
A A B A D C#' / A A B A E' D' / A A A' F' D' C#' B / G' G' F#' D' E' D' /
Repeat steps 4 and 5 for the key of G major. "Happy Birthday to You" will be:
D D E D G F# / D D E D A G / D D D' B G F# E / C' C' B G A G /
Repeat steps 4 and 5 for the key of B major. "Happy Birthday to You" will be:
F# F# G# F# B A# / F# F# G# F# C#' B / F# F# F#' D#' B A# G# / E' E' D#' B C#' B /
They should now have completed the first worksheet. Now point out the intersecting set ovals on the second worksheet. Each is already marked to be filled in with two particular keys. Explain that if a note is in both keys, it should go in the area where the sets intersect. If not, it should go in the outer area of that set's oval. Go through one intersection with them as an example if necessary. Let them fill in all the set intersection ovals and answer the questions at the bottom of the worksheet.
When they are done, explain that related keys are keys that share a lot of notes. The more notes they share, the more closely related they are. Have them look at their intersecting sets and answer the following exercise questions.

Problem 1

Which of these keys is most closely related to C major?
Which of these keys is most distantly related to C major?
Which other key is closely related to G major?
Which of these keys is most closely related to B major?
Is D major more closely related to B major or to C major?

[ Click for Solution 1 ]

Solution 1

G major
B major
D major
D major
C major

[ Hide Solution 1 ]

If it is useful and you have time, ask other questions about the sets that are relevant to your studies. If your students are musically inclined and you have the time and patience, let them pick out other tunes they know on their intruments. See if they can find a way to play the tune in one of the keys studied here (i.e. using only that set of notes). If they are playing it in a different key, what key is it related to?