Tuesday, May 27, 2014

Circle of Fifths

The circle of fifths is the first thing that we learned about in Physics of Music. It is something that musicians use to help represent the relationship of the 12 notes of the chromatic scale. It shows how closely the keys are related to each other. The circle of fifths also has a basis in science and the harmonic series, because the interval from the second harmonic to the third harmonic is a fifth as well (A to E).  It uses every single note in the 12-note scale once before coming full-circle. The dodecahedron was a 12 sided 3 dimensional object that we built during the second week of our class. It represents the circle of fifths and how it progresses from each note to the other.

The Harmonic Series

The harmonic series is the sequence of all multiples of a base frequency called the fundamental. During the harmonic series a string or a column of air divides itself endlessly and the frequency gets higher. Frequency and wave lengths are inverses. For example, when the waves length divides itself in half, the frequency multiplies itself by two. Many instruments use the harmonic series either directly or indirectly. A brass instrument uses the harmonic series directly and produces the different harmonics in order.

Frequency of Piano Keys

During the first few weeks of this class we studied frequency. Frequency is the rate at which vibrations occur in waves and are usually measured per second, which are called Hertz. If you have a string with a certain frequency and you cut it in half, the frequency would be multiplied by two and this will continue forever because as the string gets smaller by halves the frequency doubles.These are called octaves. When a string is divided into thirds, it forms a musical interval called a fifth.  A standard modern piano has 88 keys and one way of determining all their frequencies is to use octaves and fifths.  We used this method to calculate the frequencies for every note of the piano. Although this is the simplest method, it has some problems and musicians usually use a different method to tune a piano.  

Different Systems of Tuning

Pythagorean, equal temperament, and just intonation are 3 different systems of tuning. Pythagorean and just both are based on the naturally occurring harmonic series. Equal temperament is a compromise and can be derived mathematically. All produce scales with a different third. Pythagorean tuning uses only perfect fifths, just intonation uses perfect fifths and major thirds while equal temperament uses cents and 12 equal notes to an octave.

A Trip to the Organ

During Week 2, the class went on a trip to the St. Mary’s Church organ. This was a very interesting trip because the organ was big enough that we were able to crawl inside, and examine the vast number of pipes within the organ. We learned that the organ contains hundreds of pipes varying in shape and length. The largest pipe was 16 ft long and had a width that resembled that of a small tree, and the smallest pipe was about the size of a straw. Each pipe releases a completely different sound from one another. There are so many pipes that you need to use knobs (called drawbars or stops) and pedals to be able to access many sounds able to play in the organ’s vast range of musical sounds.   

Building Pipes Whistles

During the second week of class we built pipe whistles with pvc tubing. First, we picked one fundamental note for our whistle. Then we used the equation Velocity = Frequency x Wavelength to find the wavelength and then the pipe length for an open column of air. Then we cut the pvc pipes with the right measurements so they would produce a sound that was close to the note. For the whistle to work well, we had to make a sharp edge to split the air stream. When all of this was done, we tried to match the sound of the piano with the sound of the flutes to see if our calculations were correct. Finally we made more calculations to add the holes in the right place, so it could produce more notes.

Building the Guitar

In the second week we built a guitar. Dr. Fletcher built most of the body and our responsibility was to calculate where the frets would be placed, cut them and glue them in place. Unlike making the Pvc pipe whistles, where the length is fixed, we only had to worry about ratios because guitar strings can be tuned based on tension. We figured out each notes’ location by relating it to a fifth above it or below it. We put two strings on the guitar. Dr. Fletcher also explained to us that the placements of frets is not the only important part to have a well tuned guitar. You must also consider the string type, and how tightly the strings are pulled. We also had to create a box with a hole in it that allowed the vibrations to spread through the box. This amplifies the sound so we can hear it clearly.