MATH IN MUSIC:
Sine Waves, Ratios, and Exponents
Editor's Note: This submission is a piece from Lily C. ('15) that she did for her Precalculus class. She wrote
about writing sine graph equations for different musical pitches, and how math
was used in the development of different tuning systems in music.
Sine Waves
What we perceive as sound is vibration traveling through matter (most often, air). Through a complex system in our inner ear, this energy is received and turned into neural impulses that travel to our brain where we interpret them as different pitches. These vibrations cause rapid changes in the surrounding air pressure. Because these changes in pressure are cyclical, sound can be shown using a sine graph showing pressure vs. time.
For example, the note A4 (the A above middle C) is represented in the following graph.
What we perceive as sound is vibration traveling through matter (most often, air). Through a complex system in our inner ear, this energy is received and turned into neural impulses that travel to our brain where we interpret them as different pitches. These vibrations cause rapid changes in the surrounding air pressure. Because these changes in pressure are cyclical, sound can be shown using a sine graph showing pressure vs. time.
For example, the note A4 (the A above middle C) is represented in the following graph.
http://www.math.umn.edu/~rogness/math1155/soundwaves/
The different pitches that we perceive correspond to the different frequencies of these vibrations. Higher frequencies of vibrations correspond to higher pitches. The most commonly used unit of frequency is Hz (hertz), the number of cycles per second. When you know the frequency in Hz for a particular pitch, it can easily be translated into a sine curve.
The amplitude of the graph doesn’t matter in differentiating between pitches, because it tells us the volume of the sound. The important property in graphing pitches is the period of the sine wave.
In radians, the normal period of a sine wave is 2π. To modify the period, you put the reciprocal of the number you want to multiply the period by in front of the x. To change the sine graph to a period of 1, you divide the original period by 2π. The new equation would be y=sin(2πx). Now that the graph has a period of one rotation per second, because Hz is the number of cycles per second, you can get the graph of a specific pitched sound by putting its Hz frequency in front of the 2πx; the equation for the graph of a pitch would therefore be: y=sin((Hz)2πx). This is why A4, with a frequency of 440 Hz, is represented by the equation y=sin(880πx). (Shown in the graph above.)
Relative Pitch and Scales
Math is ubiquitous in music theory. In addition to sine waves, ratios and exponents play a less obvious but central role in music, particularly in tuning and the formation of scales.
Equal temperament is currently the most widely used system of tuning, in which there is an equal ratio between each adjacent note in an octave. When the frequency of any note is doubled, it becomes an octave higher. This being the case, the distances between the frequencies in equal temperament are exponential and based on ratios, and are not linear.
A unit called the cent (¢) is used to measure musical intervals. In a twelve-tone equal temperament (where each octave is split up into 12 notes) scale one octave is 1200¢. Each adjacent semitone has an interval of:
The amplitude of the graph doesn’t matter in differentiating between pitches, because it tells us the volume of the sound. The important property in graphing pitches is the period of the sine wave.
In radians, the normal period of a sine wave is 2π. To modify the period, you put the reciprocal of the number you want to multiply the period by in front of the x. To change the sine graph to a period of 1, you divide the original period by 2π. The new equation would be y=sin(2πx). Now that the graph has a period of one rotation per second, because Hz is the number of cycles per second, you can get the graph of a specific pitched sound by putting its Hz frequency in front of the 2πx; the equation for the graph of a pitch would therefore be: y=sin((Hz)2πx). This is why A4, with a frequency of 440 Hz, is represented by the equation y=sin(880πx). (Shown in the graph above.)
Relative Pitch and Scales
Math is ubiquitous in music theory. In addition to sine waves, ratios and exponents play a less obvious but central role in music, particularly in tuning and the formation of scales.
Equal temperament is currently the most widely used system of tuning, in which there is an equal ratio between each adjacent note in an octave. When the frequency of any note is doubled, it becomes an octave higher. This being the case, the distances between the frequencies in equal temperament are exponential and based on ratios, and are not linear.
A unit called the cent (¢) is used to measure musical intervals. In a twelve-tone equal temperament (where each octave is split up into 12 notes) scale one octave is 1200¢. Each adjacent semitone has an interval of:
hyperphysics.phy-astr.gsu.edu
The ratio of frequencies between two pitches can be found using exponents of 2 and the interval measured in cents:
2^(interval in ¢/¢ per octave) = ratio between two pitches. In 12 tone equal temperament, the specific formula would be 2^(interval in ¢/1200).
We know that 2 is the right base for this exponent formula because the ratio between the frequencies of two notes an octave apart is 2.
2^(1200/1200) = 21 = 2
Another example:
Frequency of C4= 261.63
Accepted Frequency of C#4= 277.18
The ratio of 277.18 to 261.63 is 1.059.
We can get this same ratio using the exponent formula. Because these two pitches are adjacent, the interval is 100¢.
2^ (100/1200)= 1.059
This concept was used to develop the 12 tone equal temperament scale of frequencies, which is widely used in western music. The system was developed by using the exponent formula to find the ratio between the frequency of one note and that of the next adjacent note. Using these ratios, the notes within the octaves can be spaced at exponentially equal intervals.
Much of modern western music is based on this tuning scale. There is some debate over whether this current system is better than other temperaments, such as Pythagorean tuning, which was commonly used in earlier music. However, it is no coincidence that equal temperament has become so popular. The system offers many advantages. For example, the consistent ratio intervals allow for pieces to be easily transposed from key to key. The same principles of twelve tone equal temperament have been applied to create other equal temperament systems, which are often used in non-Western music.
Even in ways that may not always seem obvious, mathematical concepts play a central role in basic pitch concepts and musical theory. While we may not always see it directly, our main system of tuning is based off of exponents, and understanding trigonometric functions is key (no pun intended) to understanding how pitch works.
2^(interval in ¢/¢ per octave) = ratio between two pitches. In 12 tone equal temperament, the specific formula would be 2^(interval in ¢/1200).
We know that 2 is the right base for this exponent formula because the ratio between the frequencies of two notes an octave apart is 2.
2^(1200/1200) = 21 = 2
Another example:
Frequency of C4= 261.63
Accepted Frequency of C#4= 277.18
The ratio of 277.18 to 261.63 is 1.059.
We can get this same ratio using the exponent formula. Because these two pitches are adjacent, the interval is 100¢.
2^ (100/1200)= 1.059
This concept was used to develop the 12 tone equal temperament scale of frequencies, which is widely used in western music. The system was developed by using the exponent formula to find the ratio between the frequency of one note and that of the next adjacent note. Using these ratios, the notes within the octaves can be spaced at exponentially equal intervals.
Much of modern western music is based on this tuning scale. There is some debate over whether this current system is better than other temperaments, such as Pythagorean tuning, which was commonly used in earlier music. However, it is no coincidence that equal temperament has become so popular. The system offers many advantages. For example, the consistent ratio intervals allow for pieces to be easily transposed from key to key. The same principles of twelve tone equal temperament have been applied to create other equal temperament systems, which are often used in non-Western music.
Even in ways that may not always seem obvious, mathematical concepts play a central role in basic pitch concepts and musical theory. While we may not always see it directly, our main system of tuning is based off of exponents, and understanding trigonometric functions is key (no pun intended) to understanding how pitch works.
Equal temperament. (n.d.). Retrieved from https://www.princeton.edu/~achaney/tmve/wiki100k/docs/Equal_temperament.html
Ideas associated with the equal tempered octave. (n.d.). Retrieved from http://hyperphysics.phy-astr.gsu.edu/hbase/music/etsum.html#c1
Soundwaves. (n.d.). Retrieved from http://www.math.umn.edu/~rogness/math1155/soundwaves/.
(n.d.). Retrieved from http://www.seventhstring.com/resources/notefrequencies.html
Ideas associated with the equal tempered octave. (n.d.). Retrieved from http://hyperphysics.phy-astr.gsu.edu/hbase/music/etsum.html#c1
Soundwaves. (n.d.). Retrieved from http://www.math.umn.edu/~rogness/math1155/soundwaves/.
(n.d.). Retrieved from http://www.seventhstring.com/resources/notefrequencies.html
