IntroductionI have been learning music since I was 5 years old. I started playing the piano and developed a passion for music, then I moved on to playing the cello and viola. I played a variety of genres too, from romantic to solemn to inspirational and although the individual notes remain the same, the chords, which are a group of (typically three or more) notes sounded together, the basis of harmony, changed depending on the type of music. Chords are the foundation of music, found in all types. What I found interesting was that in some jazz music, or sad music, the chords sounded ugly, out of place. Chords have different effects in music, giving a happy ending to a piece or leaving it unfinished. As I progressed in my musical capabilities, I found pieces that sounded beautiful because of the use of chords, and others that I couldn’t bear to play because of how disconnected they sounded. I would wonder why a composer would try to make their music sound like that, and how they would decide which chords to play, or would they pick random notes and put them together? Then I encountered an article about Robert Schneider, a mathematician rockstar in the 90s who was in a band called The Apples (Hansen). He used math to help him tune his instruments and to come up with harmonies and pleasant sounding chords. My knowledge about logarithms, from class, helped me to understand how he was able to use math to tune his instruments. However, questions still remained about how he used frequencies and musical intervals to see whether a note would sound dissonant, which is notes that are unsuitable or unusual in combination (Lacking harmony), or harmonious. This lead to my research question ‘what makes a sound dissonant?’ Basic mathematical knowledge about modeling trigonometric equations, ratios and knowledge about basic functions are impertinent to understanding the relationship between musical intervals and frequencies. Although having learned about these components, using them in music poses a very different understanding of the concept. Throughout this exploration, I aim to:Model different frequencies (of octave and harmonic) waves using trigonometric equations Study the frequency of a single string, such as those in a violin, using geometric and arithmetic sequencesApply the understanding of frequencies to beat intervals by using ratios Explore just and equal temperament using ratios found previouslyBackground InformationTo be able to understand the concept of math within music, musical terms need to be defined:Frequency: the rate per second of a vibration constituting a wave, either in a material (as in sound waves), or in an electromagnetic field.Musical Intervals: In music theory, an interval is a difference between two pitches. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.Harmony: The combination of simultaneously sounded musical notes to produce a pleasing effect.Harmonic: An overtone accompanying a fundamental tone at a fixed interval, produced by the vibration of a string, column of air, etc. in an exact fraction of its length.Dissonant: Lacking harmony or unsuitable or unusual in combination that is clashing.Octave: a series of eight notes occupying the interval between (and including) two notes, one having twice or half the frequency of vibration of the other.Equal Temperament: the adjustment of intervals in tuning a piano or other musical instrument so as to fit the scale for use in different keys; in equal temperament, the octave consists of twelve equal semitones.All these concepts are some of the many musical terms that can be found and are detrimental to understanding how to analyze music. Each one is intertwined in the other and combined with math, it can help to understand how frequencies and harmonic intervals are used to cause a note to sound dissonant.The Basics of FrequencyA musical sound is caused by vibrations, modeled as a sine wave. The domain is time, which represents the x-axis, while the range is pressure, which represents the y-axis. P = Asin(2?ƒt)The variables used are P, which is the pressure (measured in decibels or Pascals), A,which is the amplitude (measured in decibels or Pascals), ƒ – which represents frequency (measured in hertz), t – which is time (measured in seconds), T – represents the period (T = 1/ƒ). This is modeled in Figure 1.Figure 1: Sine Wave of a VibrationSound generally has two components – pitch and volume.The pitch of a note played is associated with its frequency. For example, if a note has a higher frequency then it is a higher note and vice versa. Frequency is measured by the number of waves passed per second. It encompasses a range, depending on the instrument, but audible frequencies are from a range of 20 Hz to 20,000 Hz. A piano, for example, has a frequency of 261.6 Hz, at the note Middle C. If using this frequency, along with 65dB as the amplitude, for 0.01 seconds, the pressure would be as follows:P = Asin(2?ƒt)P= (65 dB)sin(2?(261.6 Hz)(0.01 s)P= 18.392254499 dBVolume (which is loudness), is relative to the pressure. If one goes to a screamo or rock concert, the music will be loud, causing large oscillations in the form of pressure that can be felt. Pressure is normally measured by force per unit area. Since amplitude and pressure are linked, their units must also correspond to each other. Pascals can be converted into decibels, which is useful so that softer sounds are spread out, allowing for a control on volume. PdB=20logPPa210-5Using the answer to the frequency of Middle C on a piano, the answer can be converted from Decibels to Pascals using this equation as follows:PdB=20logPPa210-518.4 = 20logPPa210-50.92 = logPPa210-50.92 = log Ppa- log (210-5)0.92 = log Ppa- (-4.6987)-3.77897 = log Ppa Ppa= 1.663510-4Hence, the Middle C on the piano has the pressure of 18.4 dB, or 1.663510-4 Pa.Frequencies of Octaves and HarmonicsCombinations of certain notes make a harmony, both other notes placed together they sound dissonant. To comprehend why, it is necessary to look at the most basic form of a frequency: a vibrating string. The formula for this is:Frequency = 12lengthtensionline densityFrequency is measured in Hertz, which is 1/seconds. Length is measured in Meters and Tension is measured as a force, which is Newtons (kgMetersseconds2). Line density is the thickness of the string and is measured in Kgmeters. Frequency and pitch can be changed in 3 main ways:By tightening the string, which results in an increase in tension and thus increases frequencyBy using a thicker string, which increases the string density and thus lowers frequencyBy using fingers on the frets, which decreases the length of the string and thus increases frequency. The length of the string is inversely proportional to the frequency of a string. Meaning that if a frequency is doubled, then the length of the string is halved, which also means that the frequency just became an octave higher.NoteFrequencyLow Low Low G49 HzLow Low G98 HzLow G196 HzMiddle G392 HzFigure 2. Table showing frequencies of different notesThese frequencies shows in Figure 2, form a geometric sequence. A geometric sequence is where each term after the first term can be found by multiplying the previous one by a common ratio. To find the common ratio for this sequence, the formula is: rn=anan-1 rn=9849 rn=2Thus, the common ratio for this sequence is 2. This means for the frequency of the same note in different octaves can be found by multiplying the previous frequency by 2, therefore, the sequence grows exponentially. This can be noticed as instruments that have lower frequencies are much larger in size; for example, a double bass has a much lower frequency than a violin or fiddle. Organ pipes are another example, as they get larger as their frequencies become lower, “This is why the organ pipes at the front of a church if arranged in descending order, approximate an exponential curve” (Wright). For frequencies relation to harmonics, they form a different type of sequence:NoteFrequencyHarmonicLow low low A55 HzFundamentalLow low A110 HzSecondLow E165 HzThirdLow A220 HzFourthMiddle C275 HzFifthMiddle E330 HzSixthMiddle G385 HzSeventhMiddle A440 HzEighthFigure 3. Table showing different notes’ frequencies and the corresponding harmonicThis sequence in Figure 3 is arithmetic, whether a constant is added to the previous term. The constant for this sequence can be found through the formula:an = a1 + (n – 1)d 110 = 55 + (2-1)d110 = 55 + dd=55From this we can see that the common difference is 55, meaning that as a harmonic increases, the frequency will increase by 55 Hz. To summarize, frequencies of octaves form a geometric sequence whereas frequencies of harmonics form an arithmetic sequence. Beats and IntervalsWhen two sounds are played with a similar frequency to each other, beats are formed. The first graph’s curves are sometimes aligned and sometimes are aligned contrastingly. In the figure demonstrating the summation of the above frequencies, the pressure is doubled when the two curves are aligned and when they are aligned contrastingly the pressure is canceled out. When the frequencies are separated by a half step and a minor third, the beats are the strongest. When they are separated by an interval smaller than this, the beats cannot be distinguished by the human ear as they are too slow. Contrastingly, if the beats are separated by a larger interval, they cannot be heard as they are too fast. The sounds that humans like to hear and do not like to hear are controlled by physiological reasons, and hence, beats create a dissonant sound to human ears. Thus, the assumption is made that if frequencies are similar to one another, will not be harmonious as they produce beats.In Figure 14, it shows that depending on how strong the beats are, it can represent the harmony between two frequencies. Using this graph, the conclusion can be made that beats that are dissonant are created by frequencies that are similar or close. However, to understand why some notes sound better together than others, intervals must be used. To examine how harmony and musical intervals are related, the keyboard below, Figure (PUT # HERE) helps us to do so. As most of the harmonics in the diminished fifths do not line up exactly but are close, they create beats which then create a more dissonant sound. Comparatively, the harmonics of the third and fifth notes line up exactly, or are too far away from each other, creating a less dissonant sound. In Figure 15, the graph is taking many intervals and recording its dissonance using the frequency ratio. Just and Equal TemperamentAs harmonics line up perfectly in major intervals, the frequency ratio equates an exact fraction.For example, the frequency of G is exactly 3/2 times that of C. Beginning with a frequency of 65.4 Hz for C, the major scale can be built using these ratios. When these intervals are played in sync, the sound is harmonious. This type of tuning is called just temperament. However, just temperament has limitations. The frequency ratios from AB, FG and CD amount to 1.125, however for GA and DE, it is 1.111. This means that if a clarinet was tuned in just temperament, then the harmonics only sync for one key. This means the instrument can only sound harmonious in one key and not the other keys. “Just temperament instruments were the standard until the 1700s. A flutist would have had to own several flutes each tuned to a different key. Likewise, harpsichords had to have several keyboards for different keys” (citation needed). Bach in the 18th century created a new tuning strategy called equal temperament, which allowed instruments to be played in all keys as they had the same ratio between each key. To this day, modern instruments all use equal temperament as a way to sound more harmonious. Since in the equal temperament, the frequency ratio is the same, it remains 1.122. Interestingly, this number is between 1.125 and 1.111, which were the ratios found in just temperament. The ratio 1.122 was found as follows:As there are 12 half steps in an octave, and the frequency of an octave is 2, the ratio of each half step in an octave is 122=1.059.An octave has 12 half steps, but 6 whole steps. So, the frequency ratio for whole steps in an octave is 62=1.1222, which is the equal temperament frequency ratio. ConclusionUsing the math in equal and just temperament, beats and intervals, frequencies of octaves and harmonics, it all intertwines to answer the question of what makes a sound dissonant? From finding a frequency of a note to tuning the note to make a harmonious noise, math can be a fundamental necessity in music and its notations.