Math can be used in music in many different ways. Music theorists often use mathematics to understand music, and although music is evidently abstract in modern mathematics, mathematics is “the basis of sound” and sound itself “in its musical aspects exhibits a remarkable array of number properties. ” The average person lacking in great knowledge of math and musical theories would not categorize mathematics with music. In actuality, math and music are related, and we use this to describe and teach and learn music without even knowing it.
This research paper will e proving the relationship between math and music, and how one is used to assist the other. Music is read in ways very similar to how math symbols are read. “Music,” wrote the great 17th-century German mathematician Gottfried Leibniz, “is the sensation of counting without being aware you were counting. ” Musical pieces are divided into sections called measures, or bars. Each measure contains an equal amount of time. These measures are further divided into equal portions called beats. These are all “mathematical divisions of time”.
Furthermore, fractions are used in music to indicate he lengths of notes. The time signature is usually written as two integers, one above the other. In other words, these are fractions. The time signature indicates the rhythm of the piece. The number on the bottom, or the (denominator), tells the musician of the note with a single beat count. The top number, or the (numerator), tells the musician of the number of notes in each measure. Numbers are vital in musical pieces. But there is more to the mathematical and musical relationship than counting, and beats, and time signatures.
As the French baroque composer Rammer declared n 1722: “l must confess that only with the aid of mathematics did my ideas become clear. ” When we realize and further discover the mathematical foundations in music, we are able to gain much more in-depth knowledge on the structures of music and sound. An important factor in sound is frequency. Frequency is “the rate at which a vibration occurs that constitutes a wave, either in a material (as in sound waves), or in an electromagnetic field (as in radio waves and light), usually measured per second. The most common unit of frequency is the hertz (Hz’s). When we hear two notes an octave apart, we feel like we are hearing the same note. We even give them the same name. This is because the frequencies of these two notes are in an exact 1:2 ratio. It vibrations and weights. He was then able to discover that the pitch of a vibrating string is and can be controlled by its length. Ultimately, the shorter the string, the higher the pitch. This is why stringed instruments, such as the guitar or cello, are tuned tighter on certain strings, to create a higher or lower pitches.
Pythagoras also realized that the notes of some frequencies sound best when played together with overall other frequencies of the same note. For example, notes of GHz sound best with notes of GHz, and so on. From this, we can find harmony. Harmony occurs when two pitches/sounds vibrate at frequencies in small integer ratios. For example, the middle C and high C sound good together, because high C has twice the frequency of middle C. It is a 1:2 ratio, or the octave. Middle C and the G right above sound good together because the frequencies of G and C are in a 3:2 ratio. This is also called the Perfect Fifth.
Another ratio is the 4:3; Perfect Fourth, and the Pythagorean whole tone; 9:8. These pure integer ratios help us see deeper into the realms of harmony, and why certain notes sound good with others when heard by ear. The OCTAVE; 1:2 RATIO is the most basic ratio in music. It is basically the relationship of one thing vibrating twice as much as the other (hence the 1:2). If you play a string, then stop the string at half its length, it will sound exactly an octave higher. If you play a string, then play another string that is exactly twice its length, it will sound an octave lower.
The result is 2, and then the previous fixed number is added to the current sum. The sequence grows exponentially using the same pattern. Here is a formula that helps us easily put this sequence into practice. Using this formula, we can conclude that the first 6 numbers of the Sequence would So the question is, how is this sequence used MUSICALLY? The Fibonacci numbers and sequence are often found in the timing musical compositions. Music can be composed through three known methods using Fibonacci. There is the Binary Method, the Note to Number Method, and the Best Ratio Method. I will be expanding further on the Best Ratio Method.
This method involves the use of beats thin a musical time frame in order to achieve a golden ratio hierarchy through the Fibonacci Sequence. For example, the composer might choose to use “4/4 time”, meaning 4 beats per measure, to compose their piece. In relation to the length of a note or”beat”, the composer may have: 1 whole note per measure 2 half notes per measure 4 quarter notes per measure 8 eighth notes per measure 16 sixteenth notes per measure 32 thirty-second notes per measure 64 sixty-fourth notes per measure In the first bar or measure, you would have a single whole note which would mark 4 beats.
The next measure would incorporate two half notes marking the 4 beats. It is by the third measure that the golden ratio starts to form as a result of the sequence. In the third measure two quarter notes are used and one half note is used marking the 4 beats. The fourth measure will contain four eighth notes and one half note marking the 4 beats. The fifth will contain eight eighth notes marking the 4 beats. The next and final section will contain twelve sixteenth notes and one quarter note marking the 4 beats within a measure.
It can be noted that the number of notes laced within each measure thus far has incorporated a number in the Fibonacci Sequence. Measure 1 = 1 note Measure 2 = 2 notes Measure 3 = 3 notes Measure 4 = 5 notes Measure 5 = 8 notes Measure 6 = 13 notes reached and Fibonacci music is successfully composed! Although at first glance, we would not put math and music together, after a deeper scope into the matter, we can conclude that mathematics play a huge role in music. Music is actually extremely mathematical; every aspect of it ruled by a principle based on mathematics.
The discovery of frequency (by Pythagoras) gave us whole new outlook on sound and music. Beats being the mathematical divisions of time. Musical patterns based on mathematical sequences. These few factors prove to us that math is, in fact, needed to make music delightful. Math is needed in order to make a melody perfect to our ears. Essentially, math is needed in order for us to even listen to music. We use math to complete perfect harmonies, and to play beautiful music on our instruments. Math is all around us. Music is amazingly mathematical, and mathematics are amazingly musical.
