1. #51 Listening to Numbers Every instrument we hear, every note someone sings, every song on the radio has one basic idea in common; because of Equal- Temperament Tuning, the western music scale is constructed of 12 equal pitches. I am sure you are all wondering (as did we) where the number 12 comes from. Why not 5 or 6 or even 10, a number that many mathematical ideas are based off of. Interestingly enough, it all links back to the times of the Ancient Greeks. The Greeks were the first known humans to link mathematics with music. They found that every specific musical note has its own precise frequency. Think back to your 5th grade science classes; frequency describes the number of waves that pass a fixed place in a given amount of time. Musical instruments make sound waves which, in turn, have frequencies.

2. This picture shows the difference between a note that is high in pitch (or has a high frequency) and a pitch that is low in pitch (or has a low frequency). In music a certain pitch has a certain number attached to it, which is what we call it‘s frequency. The higher the frequency of the sound wave makes a larger number that is attached to it and, therefore, the higher-sounding the musical pitch is to our ear. The frequency of one of Alicia Keys’ high notes is much larger than the frequency of the country singer Trace Adkins’ lower notes.

3. The picture above is of a portion of a keyboard. The keys labeled C4, C5, and C6 all have the same frequency ratio, even though C5 has a higher pitch than C4 and C6 has a higher pitch than C5. It was also the Greeks who determined that the notes that have frequencies that are multiples of one another are harmonious, or in other words, sound the best together. This idea created the foundation of equal- temperament tuning, which in the most basic of terms implies that a musical scale is divided into 12 equal pitches that each have other pitches with the same frequency ratio. It is important later on that we remember each of these pitches can be represented as an irrational number.

4. Discovered by the ancient Greek mathematician, Pythagoras, was what we have dubbed the Golden Number: 1.5. Octaves: C to C, D to D, E to E, F to F, G to G, A to A, B to B Pythagoras found that two vibrating strings make a harmonious (good) sound together if the ratio of their lengths is a ratio of small whole numbers. In terms that we can use; two musical pitches make a harmonious sound together if their frequency ratio is of small whole numbers. The simplest whole-number ratio is 2:1 which is the ratio of pitches an octave apart. The second simplest ratio is 3:1, which is a ratio larger than the octave. In order to get the pitches within the octave (or between the numbers 1 and 2) you want to divide the ratio in half, a concept you will want to be familiar with because we will use it often. This makes the ratio 3:2 or reduced, 1.5:1. The number 1.5 represents an interval of a 5 th musically.

5. When building the scale, the idea is to make the pitches in the sequence the same ratio apart. Because the 5th is such a pleasing sound to the ear (remember it‘s ratio is 3:1, the second simplest ratio), we would like to be able to build a scale where the 5th of any pitch is present. So we begin with ratio 1: 1*1.5=1.5 then the 5th of this tone, 1.5*1.5=2.25 then the 5th of this tone, 2.25*1.5=3.375 then the 5th of this tone, 3.375*1.5=5.063 then the 5th of this tone, 5.063*1.5=7.595 …etc.

6. This process is naming frequencies that make up the western- music scale. In order to build the simplest scale, however, you must get the tone frequencies to be in the range of 1 to 2 because the ratio 2:1 is that of an octave. We can divide these numbers by 2 (I told you we would need this) until they reach a number that is between our goal range. Therefore: 2.25/2=1.125 3.375/2=1.688 5.063/2=2.532/2=1.266 7.595/2=3.798/2=1.899 …etc. This diagram shows the condensed pitch frequency matched with their letter name in the systematic order. It begins at the letter C and the number 1 and goes clockwise.

7. Now the question is: at what point does this process stop? Ideally, it would be when the condensed number 2 appears because this would mean we have reached the number that completes our simple ratio, 2:1. In the real world, however, nothing is ever that perfect. Instead, we must look for a number that is as close to 2 as possible. When using this process, the frequency number we get is about 2.028. It just so happens that when we finally reach the number closest to 2, we have come to our 13 th pitch. Since this 13 th pitch is considered to be the same as the one we began with, we are left with 12 pitches. Consequently, this justifies the Equal- Temperament Tuning implication that a musical scale is made up of 12 equal pitches.

8. It should be noted that there are many different ways of creating music scales and there is no consensus when it comes to determining which method is the correct method. In the U.S., however, we typically use the Equal-Temperament method and it is generally agreed that this is the most correct. As we have proved with our mathematic system above; music cannot be condensed into an exact science. The chart above displays the final frequency ratios in order as they appear in the western-music scale.

9. Music is everywhere and is treated as a necessity in our society. Without equal-temperament tuning the music we listen to in the United States would sound very different. You may not know it, but your ear has been tuned to the frequencies we have created with our system. It is what makes us cringe when someone sings off-pitch and what creates goose bumps on our arms when a soprano beautifully executes a high-note. Every musical piece is created with the same 12 base notes. Harvey Reid once said, “numbers are not some way to describe music – instead think of music as a way to listen to numbers, to bring them into the real world of our senses.” The Condensed idea…. from sound to sight.

10. Bibliography Assayag, G., H. G. Feichtinger, and J. F. Rodrigues, eds. Mathematics and Music. New York: Springer-Verlag Berlin Heidelberg, 2002. Print. Reid, Harvey. "Music + Mathematics." Woodpecker Records Home Page. N.p., Nov. 1995. Web. 07 Dec. 2010.. Rick. "Math Forum - Ask Dr. Math Archives: Music and Mathematics." The Math Forum. Drexel University, 28 Dec. 2000. Web. 01 Dec. 2010.. Rusin, Dave. NIU Math Department, 28 July 1993. Web. 01 Dec. 2010.. White, D. "Skytopia: Tuning & Music Scales Theory - Why Are There 12 Notes in Equal Temperament?" The 12 Golden Notes Is All It Takes. Skytopia, 2002. Web. 01 Dec. 2010.. By: Brittany Bertram and Kelsie Gudgeon