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Keys and Scales are Sets

Module by: Catherine Schmidt-Jones. E-mail the author

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.

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). The activity is adaptable, based on the students' ages and musical experience.
  • 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. Knowledge of music or music theory is not necessary.
  • Teacher Expertise - The teacher should be familiar and comfortable with major keys and scales and should understand the math concepts of sets and intersections of sets. The teacher need not be a trained musician or music teacher, but should have a good enough "ear" to recognize whether the chosen tune is being played correctly. 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 students will listen to and watch a tune (that uses all the notes of a major key) being played in four different keys (or play it themselves). For each key, the students will construct a set of the notes used in that key. The students 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; in this case, you may want to assign each student one note to play, and play the tune together,as a class, in each key. Instruments with a keyboard layout, like piano, xylophone, or electric organ, 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 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 below.
  • Choose the song you are going to use for the demonstration. Try to choose a tune that is familiar to you and to most of the students. Here are PDF files of three tunes that are in the public domain in the U.S.: God Save the Queen / My Country, Tis of Thee, Aloha Oe, and Three Blind Mice. (All are also included as figures below.) You may prefer a different song; any tune, or part of a tune, that contains all seven notes in the scale, and contains no accidentals, will work. Happy Birthday to You or Doe, a Deer may be a good choice for your class, for example. If your students know the Christmas carol Joy to the World, that might be particularly useful, as the first seven notes of that tune are the seven notes of the scale, in a row.
  • Based on the students' musical experience, decide on the procedure for "discovering" the notes of each set. Young, musically inexperienced students will need to have the tune played for them in each key. Students who have some musical experience and/or a "good ear" may take turns in class trying to play the tunes for each other in the different keys. Students who are already competent musicians may be able to list the sharps or flats in each key signature already, so you may be able to assign the "discovery" process to this group as homework.
  • Practice playing your chosen tune in all four keys on your chosen instrument so that it will be easy to present it to your class. If you have chosen a tune other than the ones provided, transpose the tune so that you know it in all four keys.
  • If you don't read music, or can't play a piano keyboard, study the handout for your chosen tune. You can even label your instrument with letters on stickers or masking tape if necessary. Don't be embarrassed about this; it will help the students, too. Now look at the lists of notes for your tune. Play the notes listed in each key, using the rhythm you know from singing the song.

Procedure

  1. Explain that a scale is the set of all the notes in a key.
  2. Play a chromatic scale for your students, that is, all the notes (sharp, flat, and natural) available from lowest to highest.
  3. 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.
  4. 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.
  5. Play your song in C Major, or have a student play it in C major.
  6. Have the students 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. Play the tune several times if necessary, making sure students that students have marked all the notes that they see being used, and also that they know what the tune is supposed to sound like.
  7. At this point, your procedure will depend on the students' musical competence. You may repeat steps 4 - 6 for the key of D major, reading from the lists of notes on the sheet, or transposing a common notation version, or playing the tune in the new key by ear. If students can play an available instrument, and if they can correct each other respectfully, in a spirit of exploration, rather than teasing or criticizing, let them volunteer to try to play the tune in the new key. Give them the starting note, and let them "discover" which notes must be used to play the tune correctly in the new key. If the students can't play the instrument, but do recognize when a wrong note is being played, you can turn this into a game in which you play the natural notes and let the students correct you when they hear a note that doesn't fit. If students have been learning their scales and key signatures on their instruments already, you can assign the "discovery" process as homework; give them a copy of the tune in C major and a sheet of music manuscript paper. Have them write out the tune in the other keys, and fill out the intersection worksheet, as homework. Have at least one class period before the homework is due, during which they can ask questions if they are having problems finding the tune in the other keys.
  8. Use the same procedure that you did in step 7, for the key of G major
  9. Use the same procedure that you did in step 7, for the key of B major
  10. Regardless of your chosen procedure, the students 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.
  11. 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.

Exercise 1

  1. Which of these keys is most closely related to C major?
  2. Which of these keys is most distantly related to C major?
  3. Which other key is closely related to G major?
  4. Which of these keys is most closely related to B major?
  5. Is D major more closely related to B major or to C major?

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?

Figure 1
Worksheet 1
Worksheet 1 (scalesets.png)
Figure 2
Worksheet 2
Worksheet 2 (intersectsets.png)
Figure 3
Worksheet 1 Answer Key
Worksheet 1 Answer Key (scalesetsA.png)
Figure 4
Worksheet 2 Answer Key
Worksheet 2 Answer Key (intersectsetsA.png)
Figure 5
Figure 5 (4KeyMyCountry.png)
Figure 6
Figure 6 (4KeyAloha.png)
Figure 7
Figure 7 (4Key3Blind.png)

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