Pythagorean Tuning & Just Tuning

One of the most basic and essential methods of tuning is called pythagorean tuning. It relies on a 3:2 frequency relationship between the fifth and the root of the scale. Starting with an initial frequency, for example middle C, the rest of the notes are tuned by multiplying the frequency of C by 3/2. This multiplication process is continued by treating the new note as the root note until each note has a specific frequency. This does create a problem, however, as the resulting notes do not fit into a scale within the same octave range. One essential relationship between notes is that the relationship between an octave is 2:1; this relationship is found in essential every method of western tuning. The octave relationship is applied to all of the highly tuned frequencies, and they are essentially halved until they form an ascending scale. This process is known as reducing the octave (The Physics of Music and Color).

Another important tuning method is called just tuning. Similarly to pythagorean tuning, just tuning relies on the 3:2 relationship between the fifth and the root of the scale. It also relies on a 5:4 relationship between the major third and the root of the scale. By applying these two relationships to the root note of a scale, and through the process of reducing the octave, a full diatonic scale can be defined through just tuning. After the process is completed, each note ends up with a fractional relationship to the root:

(The Physics of Music and Color)

Because of the reliance on the ratio of the third, just tuning creates a scale with relationships that differ from pythagorean tuning. 

Works Cited:

1.  Gunther, Leon. The Physics of Music and Color. New York, New York: Springer, 2012.

Musical Scales and Tuning

For the rest of my research this semester, I am going to be looking mostly at musical scales and the tuning of musical instruments as I prepare to construct a musical instrument.

Musical scales are sets of notes that instruments play centered around one note that acts as a resolution. For example, a C major scale includes eight notes, the white keys on a piano, and resolves on the note C. Another important scale is the chromatic scale, which includes every note in western music, or every white and black key on a piano. The relationships between the frequencies of notes in a scale can be quantified mathematically, and different scales have different physical relationships. As I continue with my research, I will spend a fair amount of time exploring these relationships within different types of scales.

Musical tuning is a more relative concept. For example, two instruments playing a C major scale can be playing notes of completely different frequencies, because they are tuned to a different frequency. One instrument can be tuned to A440 and the other could be tuned to A410, and they would be horribly out of tune. A440 is the conventional tuning method in today's instruments. This means that instruments are initially tuned by starting with an A note (the A just above middle C) at a frequency of 440 hertz. The remaining notes are tuned to this initial A note by using mathematical relationships. There are two main types of tuning relationships, just intonation and equal temperament. I will be exploring both throughout the next couple of weeks.

Works Cited:

1.  Gunther, Leon. The Physics of Music and Color. New York, New York: Springer, 2012.

Beats Experiment

Beats, or beat frequencies, are an acoustic phenomenon that occurs when sounds of two different frequencies overlap. As I have already learned through fourier's theorem, overlapping waves synthesize a wave of a new shape. This is fundamental in beats.

For my experiment, I first recorded the waveform of two tuning forks:

(C 256)

(G 384)

Both tuning forks appear to have a nearly pure waveform. As a recorded the ringing of two tuning forks simultaneously, I found a very different looking waveform:

The data pertaining to the graphs are summarized in the following table:

The frequency found when both tuning forks were played was found to be about 125 hertz, which equals the higher frequency minus the lower frequency. This difference is known as the beat frequency, and it appears because of the overlap of the two waves' different frequencies.

Beats play an essential role in the physics of music and harmony. Because the two tuning forks were in tune relative to each other, the beat frequency was also in tune and sounded pleasant. When instruments are properly in tune, beat frequencies are able to add to the harmonic richness of sound.

However, if two musical instruments are out of tune and are played together, their beat frequency is not in tune. When this happens, humans perceive the combination of sounds to be dissonant. In this sense, playing music is the act of creating air vibrations that act constructively with each other in order to synthesize something new. 

Works Cited:

 "Interference and Beats." The Physics Classroom. Web. 16 Nov. 2015. 

Simple Harmonic Oscillations and Hook's Law

As I mentioned in my first post, the sine wave is the basic building block of sound; this idea is developed through an understanding of Fourier's Theorem. A sine wave models simple harmonic motion, and because of this, understanding simple harmonic motion is critical to understanding sound. In this post, I am temporarily moving away from the direct physics of sound in order to focus on this foundational concept in another context.

The most traditional way of modeling the sine wave, through a simple harmonic oscillator, requires a simple setup that involves a spring and a mass. The spring hangs from a surface, and the mass is attached to the bottom. One then applies a force to the spring, by pushing it up or pulling it down. An interesting concept that occurs in this system is Hook's Law, which states that the displacement of the object on the spring is directly proportional to the force applied to the string. This is represented by the equation F = kX, where k is the spring constant, X is the displacement, and F is the force. The integral of this equation, or an equation derived with geometry, shows that the elastic energy of the spring is equal to x*x*k/2. The elastic energy stored in the spring causes the mass to bounce up and down in a sinusoidal manner. I was able to record the displacement over time of a mass on a spring:

The shape of the simple harmonic motion is clear in the spring system, and it is interesting to compare this graph to a musical wave. The following image is taken from my post about the sound of an ocarina and depicts the waveform of an ocarina through the relative air pressure over time: 

I would argue that based on this visual comparison, an ocarina acts as a better and more interesting simple harmonic oscillator based on the purity of the shape graph that it generated, but both systems model simple harmonic motion very well.

Works Cited:

1.  Gunther, Leon. The Physics of Music and Color. New York, New York: Springer, 2012.

Wave Velocity of String Vibrations

As I move away from the human perception of sound, I'm beginning to look at the physics of sound that are directly applied to musical instruments in order to prepare for my culminating project of the semester. The first thing I'm studying in this context is the wave velocity of strings, a topic essential to understanding how string instruments (piano, violin, guitar) work.

In a previous post, I established that the frequency of a string is proportional to the square of the tension divided by the mass. This makes sense at a basic level to anyone who is familiar with the tuning of a string instrument, such as a guitar. Increasing the tension of a string causes it to create a sound of higher pitch, and increasing the mass of a string causes it to create a sound of lower pitch. This ties into the idea of wave velocity, the speed at which a wave is able to travel on the string. A faster wave has a higher frequency and thus a higher pitch. The opposite is true for a slower wave.

I've established that the pitch of a string is dependent on its tension and its mass density. Linear mass density is defined as mass divided by length for a string. From this definition, we can see that changing mass of a string will affect its pitch.

With an understanding of wave velocity, an instrument creator has three ways to set the pitch of a string instrument. It should already be known that pitch is related to the length of a string, so one can change the length of the string. However, only being able to adjust string length does not grant much freedom for creativity in instrument design. Luckily, one can change the tension on a string and the linear mass density of the string in order to tune it. This is a fundamental idea in a guitar. While the strings of a guitar are similar in length, they clearly differ in thickness, which allows them to produce sounds of different pitches. A guitar player also knows that he or she can adjust the pitch of strings by making them more or less tense by turning the tuning keys.

Works Cited:

1.  Gunther, Leon. The Physics of Music and Color. New York, New York: Springer, 2012.