Equal Tempered Tuning & Flaws in Just Tuning

Just tuning presents an interesting method of tuning the different notes of a scale based on specific frequency ratios, and ultimately ends up with twelve notes of unique frequency intervals that are perfectly harmonic. While this sounds like the best possible tuning system in theory, it runs into some substantial problems in practice. For example, when an instrument is tuned to C, the minor third of D-F ends up having a different frequency ratio than the normal minor third, C-Eb. There are many other cases in which these harmonic inconsistencies occur, such as changing the key of the song, and this leads to dissonance and a lack of flexibility in playing music.

Equal tempered tuning, also known as equal temperament tuning, aims to resolve the problems created by just tuning. It does so by making the twelve semitones of an octave equally spaced in terms of the relationship of their frequencies. This process adds consistency to the tuning process that just tuning lacks, but loses some of the harmonic purity of the fractional intervals. The relationship between the frequencies of notes in equal tempered tuning is given by the equation, frequency ratio = 2^(n/12), where n is the number of semitones (The Physics of Music and Color). To hear a comparison between equal temperament and just tuning, check out this video.

The general consensus is that the flexibility given by equal temperament tuning is essential and makes up for the lack of purity that is achieved through just tuning.

Works Cited:

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

Cents and Musical Intervals

While the western musical scale is broken into twelve notes, we often need more specific ways of describing the pitch of a sound, especially during the tuning process. For example, let's say that you are tuning one of your guitar's strings to the E note played by a piano. As you twist your tuning peg and strike the corresponding string, you get closer the frequency of the E and eventually reach the point where the frequency of your string is lower than F but higher than E. How can we quantify this difference? The piano player may tell you that you are 50 cents sharp of E, but what does this mean?

The chromatic scale is broken into twelve notes, but the cents system allows us to further break this up. The basic definition of this is that the interval between two semitones consists of 100 cents, evenly spaced frequency values between the two notes. Since an octave consists of twelve notes in a chromatic scale, we also know that an octave is made up of 1200 cents. Given this understanding, we now know when the pianist tells us that we are 50 cents sharp of E when tuning, our note is tuned exactly in between E and F. Though this frequency does not have a specific letter name, it can easily be quantified by using the cents system.

Many electronic tuning devices (or tuning apps) can help you tune your instrument to standard pitches. With remarkable precision, these devices are often able to show how many cents sharp or flat your detuned note is in order to help you reach the ideal frequency.

Works Cited:

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

The Just Chromatic Scale

The western musical scale consists of twelve notes, each a semitone (half step) apart, and the scale consisting of all twelve notes is called the chromatic scale. Tuning an instrument to be able to play all twelve half steps of the chromatic scale adds a bit more difficulty to the process of tuning a pentatonic or diatonic scale.

Starting with the eight frequencies of a just diatonic scale, major third intervals are used to calculate the needed sharp/flat notes to complete the chromatic scale. Both descending and ascending major thirds can be used to find the ratios of the new notes, and the process of reducing the octave, as mentioned in my previous post, can be used to ultimately end up with the twelve ascending ratios of a chromatic scale. The following graphic is very useful in understanding the just chromatic tuning process:

(The Physics of Music and Color)

Works Cited:

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

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.