Engage
- Play the video . Engage students in a discussion. Ask: What do pianos and Phi have in common? Allow time for several student responses. Ask the class to pay particular attention to the music, listening for beat, rhythm, repeating musical phrases, and tones. Show student to see how the sequence is done with body percussion.
- Engage the class in a discussion about the videos, sharing what they noticed about the rhythm, tones, and musical phrases. Display the following photos and ask students to think about how art and nature connect to the videos they watched:
How did the music relate to the images they saw in the video? What do the subjects, images and sounds have in common? Is there a common thread between geometry, seashells, sunflowers, sounds, etc.? Could Phi or the Golden Ratio be the source of the beauty? If the Golden Ratio is a fundamental building block of beauty that exists in nature, can understanding and applying this ratio to human endeavors enable us to create something of beauty? Where else might you find evidence of the Fibonacci sequence of numbers? (Point out that there are many more instances of this ratio in our natural and human made world, i.e., the lengths of the bones in our fingers, the proportion of the human face, the shape of the human ear, musical instruments such as the violin, guitar, piano, famous works of music and art)
Possible observations:
- The repetition of musical phrases helped me to expect what was coming.
- The beat conveyed a sense of energy that related to the way each image morphed from one image into another.
- Yes, I think that the spiral is beautiful because it leads your eye around and around and makes you want to keep looking at it.
- I never thought about how all these things are connected before, but it is interesting, there must be some reason for it.
- It makes sense that if the Golden Ratio is a standard for beauty, then having it in your art will make it beautiful.
- I don’t think art has to be beautiful to be good.
- I think it’s spooky that all these things are sort of built the same way.
- It makes us wonder how this happened.
- Briefly review the origin of the Fibonacci sequence of numbers. Tell students Leonardo of Pisa brought Hindu-Arabic number system to Western Europe in 1202. He published his theory of how rabbits multiply in Liber Abaci and observed the pattern of numbers now called the Fibonacci sequence. This famous sequence of numbers is arrived at by adding consecutive numbers together to arrive at the next number in the pattern beginning as follows: 0, 1,1, 2, 3, 5, 8, 13, 21...etc. Growth rate of Fibonacci sequence can be expressed as: F(n+1)/F(n). The growth rate is the same as the Golden Ratio
Build
- Distribute the handout, . Students can work in small groups or independently to solve the problems. When students finish, have them check for accuracy and report their calculations to the rest of the class. Explain that this sequence can teach us more about patterns in math, recursion, Phi and the Golden Ratio, and music.
- Have students rejoin their groups to study the filled in Fibonacci sequence charts looking for whatever patterns they can find. Allow students sufficient time to contemplate the numbers or use the following observations to facilitate conversations:
- Observe the last digit of all the numbers. (pattern repeats after 60, every 5 number is a 5 or a 0)
- Look for patterns of odd/even numbers (odd, odd, even, odd, odd, even...)
- Look for patterns with prime numbers. (every prime number is a factor of some Fibonacci number)
- Look for patterns in numbers divisible by a constant. (Every 7 Fibonacci number is divisible by13)
- Look for Phi relationships. (The relationship between the numbers approaches Phi and is also the approximate conversion factor of miles to kilometers. For example, 3 and 5 are consecutive numbers in the sequence and 3 miles equals roughly 5 kilometers and so forth)
- Look for patterns in the products of Fibonacci numbers (The product of any 4 consecutive Fibonacci numbers is equal to the area of a Pythagorean triangle)
Student Challenge: If you extend the sequence, you will find that the last two digits repeat in 300, the last three digits repeat in 1500, the last four digits repeat in 15,000 and so forth.
- Tell students the Fibonacci sequence of numbers is an example recursion, a math concept, which is defining something in terms of itself. To demonstrate the concept of recursion, distribute two mirrors to each group. Direct the students to hold the mirrors parallel to and facing each other about 2 – 3 feet apart. The reflection from one mirror to the other produces an image that approaches infinity and is another example of recursion. Display an example of the to show several artist made images that illustrate this idea in another way.
- Display or share the . Have students share what they know about a piano keyboard. Record their responses on chart paper so they can use this information later to study and compose music. Ask the class if they can find examples of Fibonacci sequence numbers in the structure of the piano keyboard. Write down student responses. Possible responses:
- What is an octave and how many piano keys are in it? [13: 8 white and 5 black]
- What is a scale and how many notes are in it? [8]
- How many white keys are in an octave? [8]
- How many black keys are in an octave? [5]
- Black keys are split into groups of? [ 2 and 3]
- What is a pentatonic scale and how many tones are in it? [5]
- What is a chromatic scale and how many notes are in it? [13 – all the notes of an octave]
Note that all these numbers are in the Fibonacci sequence.
Apply
- Tell students they will use the math concepts they have learned and apply them to music. Have students listen to . Display the . Divide students into groups and have them identify the Fibonacci sequence in the excerpt. Once students have had time to identify the sequence, display the so they can check their answers.
- Share . Instruct students to look and listen for major musical events in the piece and take note of the measure in which they occur. For example, the opening phrase runs from the beginning through measure 8, the first major use of chromaticism is in measure 13, the piece’s climax is in measure 21, and the piece ends in measure 34 (8, 13, 21, and 34 are part of the Fibonacci sequence). Students can optionally listen to the piece as they analyze the musical events.
- Analyze the music for patterns or ratios that reflect the Fibonacci sequence of numbers. Students can analyze the following data:
- Count the total number of measures in a piece.
- Count the number of beats per measure.
- Consider the pattern of notes and their ratios to each other.
- Consider the time signature.
Student Challenge:
- Consider if the scales are chromatic (all 13 notes of an octave), pentatonic (five black keys with subsets of 2 and 3) or diatonic (all white keys in an octave).
- Consider where the buildup, climax and wind down are in the music, check the ratio between the segments for the Golden Ratio.
- Ask the class to develop a simple set of rules for writing music using the Fibonacci sequence. Model for students how this is done. Challenge student teams to come up with a “formula” for writing a jingle or a phrase of music. For example:
Fibonacci sequence
- Begin with the numbers 3, 5, 8, & 13.
- Use the 13 notes of a chromatic scale over 8 measures.
- Repeat a pattern of 3 beats plus 5 beats in each measure.
Golden Ratio
- Create a pattern of 8 notes going up, 5 notes going down, 8 notes up, 5 down.
- Alternatively: Employ 8 notes per measure in a 5 measure, 3 measure pattern repeat.
The idea is to experiment and explore what happens in the process. In this way of composing music, you are using both sides of your brain – the analytical (math concepts) and the artistic (musical sounds) strengthening and informing each.
- Distribute or the notation tool . Have students work collaboratively to compose and arrange an original musical phrase using the Fibonacci sequence of numbers, the concept of recursion, pattern, Phi, or the Golden Ratio as the foundation. Explain that a musical phrase is like a sentence in literature. Musical phrases follow each other and tell a musical story just like sentences in a paragraph tell a literary story. Consider the following to create your musical phrase.
- Have keyboards, bells, or xylophones available for student use. Allow sufficient time for students to experiment with the sound of their music after they have written it. Encourage students to listen to and revise their phrases until they create a sound they like.
Student Challenge: Create an adaptation or a harmony for their phrase.
Adaptations:
- Create musical phrases that are questions and answers. Questions end on a note that is not the key note. Answers end on a note that is the key note.
- Recreate the piece in another key signature.
- Creating a pattern of emphasis on some notes while playing others softer.
Harmonies:
- Harmonies can be written in intervals of thirds. Have students write the harmonies by placing the notes a third up or down from the melody.
- Remind students that when counting intervals, they must include the starting note in their count. For example, when creating a harmony that is a third up from C, students would count up three notes in the scale starting from C (CàDàE). Thus, students would harmonize a C with an E.
Reflect
- Give students an opportunity to play their piece for the class. Students can explain the “theory” of their music and perform it for the class.
- Ask the students to identify what characteristics their music may have from an artistic point of view. Consider factors such as rhythm, tempo, melody, harmony, whether the song has a “catchy” beat.
- Distribute the . Allow students time to have a “jam” session to inspire each other to create adaptations and collaborations of their work. Challenge the class to create a finished piece of music. Time permitting, students can add lyrics to the song.
Extend
- Teach the class about pentatonic and diatonic scales and chromatic scales. The pentatonic scale is a musical scale with 5 pitches per octave. (5 is a Fibonacci sequence number, the scale is played on the piano with the black keys which are subset in 2 plus 3; also, a Fibonacci sequence.) The major scale is C, D, E, G, A. The minor scale is D, Eb, F, G, Bb.
The diatonic scale is a musical scale with 8 pitches per octave. (8 is a Fibonacci sequence number) An example of keys for this scale are: C, D, E, F, G, A, B, C.
The chromatic scale is all 13 pitches of an octave. (13 is a Fibonacci sequence number) An example of keys of this scale is: D, D#, E, F, F#, G, G#, A, A#, B, C, C#, D.
- Lead the class in a choral singing activity. Sing a melody and harmony of a tune that features a pentatonic or diatonic scale or chromatic scale. Assign the melody and harmonies according to individual ability and preference. Practice the parts individually and then together as a group. Refer to music books in your school or public library. Consider the following songs or choose from favorites you may have:
Pentatonic Scale:
- The first two phrases of a melody from Stephen Foster’s “Oh Susanna”
- Opening bars of “My Girl” by the Temptations
- “Amazing Grace”
Diatonic Scale:
- Do- Re- Mi” from “The Sound of Music”
Student Challenge: Find a vocal piece that has a 13-pitch chromatic scale as a musical phrase in its arrangement.
- Begin with a breathing exercise. Encourage the students to practice good posture and breath control singing from their diaphragms and with expression. Demonstrate the following exercise or one of your own preferences.
Have students stand with their back straight, shoulders back and head held high.
Direct students to place one hand on their abdomen and one on their upper chest and breathe normally, observing the rise and fall of the chest with each breath.
Now ask the students to repeat the exercise but this time to pull the air into their abdomen as they inhale and push the air into their chest as they exhale, observing the rise and fall of each as they do.
Now ask the students to repeat this while singing a note.
Help students achieve technical accuracy through practice of proper use of tone and pitch by playing notes on a keyboard or xylophone for them to mimic. Experiment with changes of tempo, key, and meter. Record the singing and play it back for the class pointing out strengths and weaknesses and how they might improve the performance. Repeat the process. As the class improves in ability to create a technically proficient and artistically beautiful piece, acknowledge their progress and achievement.
Resources for this step:
Music paper
Simple piano sheet music
Sheet Music for “Often A Bird” from the Nature by the Numbers Video
Musical Phrase Examples
- Suggest that students continue to observe their culture and natural surroundings for evidence of the Fibonacci sequence of numbers pattern and share their findings with the class. Students can write poetry using the math concepts of Golden Ratio, pattern, and recursion as they did with music. They can set the poem to music, the cadence of stressed and unstressed syllables in Limericks is a Fibonacci sequence number pattern.