Rachel N. Wallace
MATH 4161 B
Musical Math: Will the songs ever end?
In this activity students explore how people have not run out of songs to compose. We look at some basic music theory, including note names, sharps, flats, time signiture, staff, barline, rhythm (including the notation of notes and rests). The students work through probablity problems, determing how many combinations can be made from a certain number of notes. After investigating all these aspects of music the students create their own unique musical piece demonstrating how even they can create a musical composition all their own. The music theory in this activity can be difficult for some students to understand depending on the amount of background knowledge the students possess. The math involved is quiet simple. I believe that this is moderately difficult and is most appropriate for middle to high school grades.
Musical Math: Will the songs ever end?
There are many different genres of music available for people to listen to and enjoy. Whether you are a heavy metal rocker or a folk music kind of person there is always a multitude of songs for you to listen to. But have you ever wondered why in thousands of years that the world has existed we have not run out of songs to create? Songwriters are coming up with catchy new tunes everyday. Math is in every aspect of music. The answer to our question can be answered through a study of probability and making combinations. Before we begin you need a little background information about music.
How many notes are there?
In music there are eight basic tones: A, B, C, D, E, F, G. This is called the musical alphabet. You may also be familiar with the musical syllables: Do, Re, Mi, Fa, Sol, La, Ti, and Do. The first and last Do’s have the same sound but different pitches. Pitch is the highness or lowness of a tone, as determined by the number of vibrations in the sound. They sound the same but one is low in pitch (fewer vibrations) and one is high in pitch (more vibrations).
If the C major scale contains the notes C, D, E, F, G, A, B, C
· Which note would correspond to the syllable Sol? ______
· Which note would correspond to the syllable Re? ______
· Which note would correspond to the syllable Do? ______
What do notes look like?
If we wanted to write the C major scale down we would use a staff. A staff contains five lines and four spaces. A bar line divides the measures:
C D E F G A B C
Each note rests on a line or a space. In the treble, or G, clef, which most people are familiar with, the notes that land on spaces are F, A, C, and E and the notes that land on lines are E, G, B, D, and F.
Now that we see how notes are written on the staff we can begin to examine why there are so many songs still to be written. Consider these notes:
F, G, A or
How many combinations of these notes are possible? ______
List the combinations in the following table:
FFF / GGG / AAAThis can also be represented another way. Imagine three blank spaces (for the three notes):
______
3 3 3 or 3^3
______
Each space has the possibility of having one of three notes (F, G, or A). If we multiply 3x3x3, or 3^3 we get 27, showing us that at most we could create 27 combinations from these three notes alone!
How many combinations would there be for these four notes: C, D, E, F ? _____
______or ______
How many combinations would there be for these five notes: C, D, E, F, G? ______
______or _____
How many combinations would there be for these six notes: C, D, E, F, G, A? _____
______or _____
How many combinations would there be for these 7 notes: C, D, E, F, G, A, B? _____
______or _____
How many combinations would there be of these 8 notes: C, D, E, F, G, A, B, C,? ____
______or _____
As you can see from these examples many combinations can be made from the C major scale alone. The notes in the C major scale are natural tones. Sharps (#) or flats (ь) can be added to these natural notes. Sharps raise a note by ½ step and a flat lowers a note by ½ step. There are 7 sharp keys (G, D, A, E, B, F#, and C#) and 7 flat keys (F, Bь, Eь, Aь, Dь, Gь, and Cь). We already know how many combinations can be made using all eight tones in the C major scale.
If we added all the possible combinations of the sharp and flat keys to that of C major how many would there be in total? ______
Rhythm
Let’s add one more factor of how music can be combined. Music contains harmony, melody and rhythm. Notes are not just placed haphazardly on the staff. Music is measured in a specific number of beats. The music’s time signature tells how many beats are in each measure. A common time signature is:
4 – the top number tells you how many beats are in the measure
4 – the bottom number tells you what note receives one beat
Within the measure of a piece with a 4/4 time signature there are four beats and only four beats per measure. Please see the chart below for the explanation of how each note is notated and what it’s value is.
Name of Note / Value (in 4/4 time) / NotationWhole note / 4 beats
Half note / 2 beats
Quarter note / 1 beat
Eighth note / ½ beat
Sixteenth note / ¼ beat
Whole rest / 4 beats silence
Half rest / 2 beats silence
Quarter rest / 1 beat silence
Eighth note rest / ½ beat silence
Sixteenth note rest / ¼ beat silence
Now that we have investigated the musical notes, their placement on the staff, and rhythm it is your turn to create your own piece of music. From our calculations it is guaranteed that your composition will not be like anyone else’s in your class. Use the C major scale (C, D, E, F, G, A, B, C) and the different rhythms you have learned to create your song. Remember that there can be no more than 4 beats per measure.
Conclusion: So how many possible combinations are there of notes when composing music? As we have seen there are endless combinations of the major scales alone. When things such as sharps, flats, or rhythms are added to compositions the possibilities really do outstretch the imagination!
Answer Key
If the C major scale contains the notes C, D, E, F, G, A, B, C
· Which note would correspond to the syllable Sol? G
· Which note would correspond to the syllable Re? D
· Which note would correspond to the syllable Do? Low C or high C
How many combinations of these notes are possible? 27
List the combinations in the following table:
FFF / GGG / AAAFGF / GFF / AFF
FGG / GFG / AFA
FFG / GGF / AAF
FAA / GAA / AGA
FAF / GAG / AGG
FFA / GGA / AAG
FGA / FAG / GFA
AGF / GAF / AFG
How many combinations would there be for these four notes: C, D, E, F ? 256
4 4 4 4 or 4^4
How many combinations would there be for these five notes: C, D, E, F, G? 3125
5 5 5 5 5 or 5^5
How many combinations would there be for these six notes: C, D, E, F, G, A? 46656
6 6 6 6 6 6 or 6^6
How many combinations would there be for these 7 notes: C, D, E, F, G, A, B? 823543
7 7 7 7 7 7 7 or 7^7
How many combinations would there be of these 8 notes: C, D, E, F, G, A, B, C,?
16777216
8 8 8 8 8 8 8 8 or 8^8
If we added all the possible combinations of the sharp and flat keys to that of C major how many would there be in total? C major equals 16,777,216 combinations. 7 sharp keys X 16,777,216 equals 117,440,512. 7 flat keys X 16,777,216 equals 117,440,512. 117,440,512 + 117,440,512 + 16,777,216 = 251658240
The musical composition will vary for each student. The teacher should examine the composition to ensure that the student only put 4 beats per measure.