Music Trajectories

Trajectories are essential to the science of music performance, but music majors rarely study the physics of motion. Consider the following graph of a conductor beating in 4/4 time:

Discussion Questions

1. Draw a picture of the conductor from the camera's view. Label the points at which each of the beats are located. 
2. How is it possible on the graph to determine the change of direction of the baton? 
3. What is the form of mathematics which describes changes in motion? 
4. What is the maximum velocity of the baton? 
5. Is the acceleration of the baton 0 at any point in the graph? If so, what are the coordinates? 
6. Comparing the minimum and maximum points, does the baton move a greater absolute distance in the x or y direction? 
7. What is the total length of trajectory of the baton in the x direction? In the y direction? 
8. Calculate the difference between initial and final positions of the baton. 

Additional graphs of music performance can be located in Volume 3 of How Do You Play That Thingamabob? The Science of Music Performance from Schottenbauer Publishing. These graphs can be integrated into music education curricula from late elementary school through high school and college/university.


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Science of Music Memorabilia

Celebrate science of music with memorabilia from Zazzle! Colorful graphs from Schottenbauer Publishing are featured on these mugs, magnets, keychains, & postcards. Graphs include conducting, voice, piano, violin, recorder, flute, clarinet, saxophone, trumpet, trombone, drums, cymbal, and Tibetan bowl. A direct link is included below:

A variety of other STEM education collections are also available from Schottenbauer Publishing on Zazzle, which features regular sales on most items.  


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Acoustics Comparison: Carpet and Vinyl Flooring

What are the acoustic properties of flooring? The following sound samples from SoundCloud can be used to compare these two common flooring options!


Sound Samples

• Carpet Floor
• Vinyl Floor

Questions

1. Describe the sound in each sample.
2. On a scale of 1 (none) to 10 (highest), rate the amount of each of the following in each sample:
1. Resonance
2. Echo
3. Damping
4. Decay
3. Which of the following tone qualities applies most to sample 1? Sample 2?
1. Mellow
2. Harsh
3. Bold
4. Jazzy
5. Buzzy
6. Loud
7. Metallic
8. Edgy
9. Smooth
10. Tinny
4. Which flooring is better for recording music? Why?
5. Bonus Questions:
1. What is the instrument?
2. What is the name of the tune?

Learn more about the science of music with books and multimedia from Schottenbauer Publishing!

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Harmonic Overtone Series Lesson Plan

The harmonic overtone series is essentially related to music performance, but may easily be overlooked in music lessons or classes. Lesson plan material is available in several books on the science of music offered by Schottenbauer Publishing. Relevant information includes calculations, diagrams, graphs of real data from various musical instruments, and graphs calculated from mathematical formulas.

Consider the following pages, excerpted from Where Does Sound Come From? Volume 1 (Copyright 2014, All Rights Reserved). 

Discussion Questions

1. Identify the relationship between harmonics and overtones, using words.
2. Express the relationship between harmonics and overtones, using a mathematical formula.
3. Draw a three-octave keyboard on paper, using a light gray color for the black keys. Make 12 copies on a printer or copier. Label each page on the top with a different chromatic note (A, A#/Bb, C, etc.). Starting on each chromatic note, color the entire harmonic overtone series. List the names of each note on the keys.
4. Using a keyboard, any musical instrument, or voice, practice playing each harmonic overtone series. Begin on the lowest note possible, to ensure enough range to complete the task.

The resources below provide additional sources of information for learning about harmonics and the overtone series:

Auditory Demonstration of Harmonics

• YouTube Video
• Introduction to the Harmonic Overtone Series 
• Audio Recordings
• Multimedia Ear Training Program

Math of Harmonics & Tuning Systems

• Mathematical Formulas
• Appendices, Where Does Sound Come From? Volume 2
• Graphs
• Appendices, Where Does Sound Come From? Volumes 1, 3, & 4

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Decibels & Hearing

Decibels are the measurement unit for the level of sound. Noise level is measured on a logarithmic scale, with the threshold of hearing at 0 decibels. On this scale, a typical conversation is approximately at 60 decibels, a vacuum cleaner at 70 decibels, and a jackhammer at 100 decibels. 

According to OSHA, high sound level exposure is linked to hearing loss, particularly if the exposure to loud noises is prolonged over time. OSHA's permissible exposure limit is approximately 90 decibels.

Are musical instruments safe to perform? Consider the graph below, excerpted from Where Does Sound Come From? Volume 2, Revised Ed., from Schottenbauer Publishing:

Discussion Questions

1. Approximately what is the decibel level of the ambient sound in the room?
2. Approximately what is the decibel level of the trumpet?

Decibel readings are the theme of Where Does Sound Come From? Volume 2, Revised Ed., from Schottenbauer Publishing. Students can use these measurements to understand the nature of sound, by comparing and contrasting graphs from 27 musical instruments, including strings, woodwinds, brass, and percussion, plus voice.

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What is Sound? The Problem of Averaging a Messy Graph

Graphs provide a unique perspective into the nature of music and sound. Graph-reading skills can be integrated into music classes, supplementing traditional science and math curricula. 

For example, consider the following two graphs, excerpted from Where Does Sound Come From? Volume 1 from Schottenbauer Publishing. These graphs shed light on a perplexing problem pertaining to averages.

Discussion Questions

1. Does the graph suggest (a) the presence of wind, (b) random differences in gas pressure in the room, or (c) error of the measurement device?
2. Is it accurate to state that the gas pressure is: (a) 99.8, (b) 99.825, (c) 99.85, (d) constantly changing?

Discussion Questions

1. What are the initial maximum and minimum sound pressures in the graph?
2. By the end of the graph, what are the maximum and minimum sound pressures?
3. Using a solid line, trace the maximum values of the sound pressures.
4. Using a solid line, trace the minimum values of the sound pressures.
5. Is it appropriate to average the minimum and maximum values on this graph, in order to show the true sound pressure reading? Why or why not?

Additional graphs of sound pressure, gas pressure, wind speed, and physical vibrations (force) are available in the lab manual Where Does Sound Come From? Volume 1 from Schottenbauer Publishing. Students can use these measurements to understand the nature of sound, by comparing and contrasting graphs from 27 musical instruments, including strings, woodwinds, brass, and percussion, plus voice.

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The Science of Tone Quality

Tone quality is essential to all musical performance, but is not usually taught systematically in traditional music education. Rather than building an awareness of tone across instruments, individuals taking lesson in strings, woodwinds, brass may be offered individual suggestions for improving tone on their own instrument. Usually, their instrument is a student or intermediate model. If fortunate to take private lessons, students may also be provided exemplary tone quality examples from a specialist teacher performing on a professional instrument. In ensemble, students may be urged to "match tone quality" across their section, even when performing on different makes and models of instruments, which offer various tone quality. Options for learning about tone quality are even more limited on the piano, which is a common tool for music education.

How can educators and students learn about tone quality across the spectrum of band and orchestral instruments? A set of educational materials from Schottenbauer Publishing offer opportunities for learning multiple aspects of tone quality, including:

• Identifying tone quality in band and orchestral instruments
• Understanding the math and science of tone quality
• Linking instrumental skills to auditory and math/science skills

To meet the first goal, the student can obtain recordings of multiple instruments, including common examples of good and poor tone quality. By listening carefully to comparison and contrast examples, and reading descriptions of each sample in words, students can develop and hone skills for identifying good tone. The multimedia set Ear Training: Tone Quality (Level 1) from MusicaNeo contains over 250 computer audio files with performance samples from common musical instruments, including strings, woodwinds, brass, and percussion. The series is accompanied by a brief lead sheet with exercises for improving tone quality, as well as an audio file with a computer realization of the exercises.  

To meet the second goal, the student can obtain graphs of tone quality samples from multiple instruments, including common examples of good and poor tone quality. By comparing these tone samples visually, and by reading descriptions of each sample in words, students can develop a mathematical/graphical understanding of good tone. The book series Where Does Sound Come From? and the anthology The Science of Music provide graphs with performance samples from common musical instruments, including strings, woodwinds, brass, and percussion. The graph below provides one example of the contents of these book series.

To meet the third goal, students can record their own musical performance on a microphone at the computer, and analyze the sound wave using a free program such as Audacity. Students can follow procedures from the free blog article Easy Science of Music Experiment to analyze their tone quality.

These educational tools can be integrated into music education curricula from late elementary school through high school and college/university, in classrooms, after-school enrichment programs, individual lessons, independent study, and/or homeschool. Additional information is available on the Schottenbauer Publishing website, and from the links below. A free pamphlet from the publisher is also available on the website.

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Easy Science of Music Experiment

With the assistance of a computer, microphone, and free program download, anyone with a musical instrument can explore the science of music! Even with a plastic soprano recorder costing $5 or less at a local music store, students can gather meaningful data for a school project or summer academic enrichment activity. Simply follow the steps below, to begin to collect and analyze data.

Science of Music Activity

1. Download Audacity, a free internet program for recording and mixing sound.
2. Open the program, and use a microphone and a musical instrument or voice to record a few notes. You may also record music with other software, and then open the file in Audacity for editing. It may be necessary to convert the file first to a format read by Audacity. Converting files is easy with a program such as DVDVideoSoft's Free Audio Converter.
3. Explore the sound wave, using the zoom function (magnifying glass icon). What does the wave look like?
4. Use these steps to compare several sound waves. Try comparing different notes or different instruments. Or, try comparing an instrument to voice.

Example: Flute Solo

The following flute solo was recorded and opened in Audacity.

Using the zoom function several times provides more detail, but the sine waves are still obscured.

Further use of the zoom function reveals the sine wave.

Individuals interested in comparing the sound waves of musical instruments may benefit from the following publications from Schottenbauer Publishing, which, as a group, contain samples from typical Western strings, woodwinds, brass, percussion, and keyboard instruments, plus voice and conducting:

• Where Does Sound Come From? A multi-volume series with data on sound
• The Science of Music: An Anthology of Graphs A concise collection of graphs illustrating central concepts
• Ear Training: Tone Quality A series of multimedia containing sound samples labeled with verbal descriptions of tone quality, for training and/or testing

Other science of music publications from the same publisher include:

• How Do You Play That Thingamabob? The Science of Music Performance A multi-volume series on the forces and motion required for producing sound
• Bow Science & Exercises for the Violin & Viola A series of books containing practical science, applied directly to exercises

These educational tools can be integrated into music education curricula from late elementary school through high school and college/university. Additional information is available on the Schottenbauer Publishing website, and from the links below. A free pamphlet from the publisher is also available on the website.

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How to Use Graphs in Music Education

Graphs can be integrated into music education from approximately fourth grade through high school, as well as college and university classes. 

The following list describes possible learning activities centered on graphs:

Possible Lesson Plans

• Graph Reading 
• Identifying Minimum & Maximum 
• Determining Graph Shape 
• Redrawing the Graph on a Different Scale 
• Mathematical Modeling 
• Writing the Graph's Equation 
• Transforming the Graph's Equation 
• Theoretical Analysis 
• Describing the Scientific & Mathematical Concepts Related to Graph(s) 
• Describing the Scientific & Mathematical Theories Related to Graph(s) 
• Identifying the Theoretically Ideal Graph & Comparing to Individual Graph(s) 
• Understanding Measurement Error & Other Types of Error 
• Comparing & Contrasting Specific Graphs 
• Science Writing 
• Describing a Graph in Words 
• Writing a Comparison & Contrast of Graphs 
• Describing Scientific Concepts & Theories Relevant to Graphs 
• Developing Theories Based on Graphs 
• Re-writing Graph Titles in Scientific Format 
• Practicing Different Styles of Writing about Graphs (e.g., Journalistic, Scientific, Practical/Colloquial) 
• Exploratory Research 
• Data-Mining 
• Developing Theories Based on Graphs 
• Exploring Open-Ended Questions (e.g., Look at this book of graphs and write about a topic of interest.) 
• Estimation Skills 
• Understanding Range & Order of Magnitude 
• Developing Theoretical Comparisons from Commonly Available Data (e.g., Here is a graph of air flow while playing a note on a trombone mouthpiece. How much air would be required to play on a trumpet mouthpiece? On a tuba? Euphonium? Flugelhorn?) 

A full listing of books with graphs from Schottenbauer Publishing, indexed by book series and data type, are available in a Teacher Resource Guide from the publisher. Additional information about using graphs in class is available on the blog Graphs in Education.


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Understanding the Harmonic Overtone Series

The harmonic overtone series describes the science of the relationship between notes in the Western harmonic system. Harmonics are most obvious on string instruments, brass instruments, and the flute, which easily demonstrate the relationship between the lowest note, the octave above, and the perfect fifth above the octave, plus many of the other harmonic overtones in the series.

Did you know that harmonics are present in all naturally-produced notes, sounding softly in addition to the fundamental tone? It is not obvious to many listeners. A free YouTube video from the publisher, Introduction to the Harmonic Overtone Series demonstrates harmonic overtones on several instruments. A mathematical expression of the combination of harmonics is presented in graphical form in the appendices of Where Does Sound Come From? Volumes 1, 3, & 4. A mathematical analysis of several prominent tuning systems is included in the appendices of Where Does Sound Come From? Volume 2. 

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