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Mathematical Investigation Scott Marino II September 1 2013 An Investigation of Music & Math Introduction In my course of education the fields of Music and Math have fascinated me by their separation in description, math being considered a science and music a subject of art. However I now see them both as an art and a science. I am a cellist, pianist, and trumpeter and in my pursuit to master these instruments I began to blur the lines between math and music seeing music in a more mathematical light, recognizing the distances between notes on a cello’s finger board and opening the piano to see the harp and the length and thickness of stings; I started to think that math was in essence the basis of all music. For my mathematical investigation I wanted to see how exactly mathematical concepts existed in music and then to see if I could derive these concepts from my playing of cello. Mathematics is the fundamental building block of sound and in its musical form an even larger array of number phenomena exist. Although the mathematical properties of sound had been studied by the ancient Chinese, Mesopotamian, and Egyptian societies, the scholars of the Pythagorean School is ancient Greece were the first to investigate and formulate musical scales and tuning systems in terms of numerical ratios and functions. They had believed that all nature consists of harmony arising out of the small integers of 1, 2, 3, and 4; regarding them as the source of all perfection as did Confucius from China and Indian theorists. These numbers are crucial in the equations of musical scales and tuning systems. Tuning Systems Pythagorean Just Intonation The Pythagorean Scale is based on the ratio of 3:2 this ratio is found in the harmonic series and when multiplied by any frequency will produce a note known as the fifth in relation to it. Pythagoras had based his scale system upon this ratio because it is one of the most consonant or pleasing to the human ear since it uses what they referred to as pure harmonic integers or ratios using the numbers 1, 2, 3, and 4. Harmonic Series (Music): A harmonic is a wave that is an integer multiple of a fundamental frequency. It produces a pattern that can be formulated into an ( ) , where the frequency of a harmonic arithmetic sequence of is the fundamental frequency plus (n-1) multiples of the fundamental. The ratio of the fifth is obtained using the ratio (ascending 5th) or (descending 5th) where n correlates to the column of harmonics (figure 2). Harmonics 1 2 4 5 3 6 7 2 4 8 8 Note 16 32 64 128 Octave (1) 17 34 68 136 Minor second (2-) 9 18 19 36 38 72 76 144 152 Major second (2) Minor third (3-) 10 20 40 80 160 Major third (3) 11 21 22 42 44 84 88 168 176 Perfect fourth (4) Augmented fourth (4+) Diminished fifth (5-) 23 46 92 184 12 24 48 96 192 Perfect fifth (5) 13 25 26 50 52 100 104 200 208 Augmented fifth (5+) Minor sixth (6-) 27 54 108 216 Major sixth (6) 14 28 29 56 58 112 116 224 232 Augmented sixth (6+) Minor seventh (7-) 15 30 60 120 240 31 62 124 248 Major seventh (7) Augmented seventh (7+) 32 64 128 256 Octave (8) 16 The wavelength of a harmonic can be expressed by integrating the equation for frequency of a harmonic into the equation for wavelength and creating an explicit formula. where f is the frequency ratios of harmonics hence ( ( ) ) If the speed of sound was not known and the wavelength of the first harmonic was known, a arithmetic sequence could also be used to represent the wavelength of a harmonic Mathematics of Pythagorean Just Intonation The Pythagorean Tuning system uses the fifth ratio and powers of it along with powers of ratio 2:1, the ratio of the octave. An issue arose with this tuning system. When applied to extended fifths and creating a whole chromatic scale, no group of 3:2 notes (fifths) will fit evenly into a 1:2 ratio octave. Note Gb Interval from C Diminished fifth Db Formula Frequency Ratio 1.404663923 ( ) ( ) Minor second ( ) ( ) 1.05349742 Ab Minor sixth ( ) ( ) 1.580246714 Eb Minor third ( ) ( ) 1.185185185… Bb Minor seventh ( ) ( ) 1.777777… F Perfect fourth ( ) ( ) 1.33333… C Unison 1 G Perfect fifth 1.5 D Major second ( ) 1.125 A Major sixth ( ) 1.6875 E Major third ( ) ( ) 1.265625 B Major seventh ( ) ( ) 1.8984375 F# Augmented fourth ( ) ( ) 1.423828125 C Octave 2 In the formulas, the ratios 3:2 or 2:3 represent an ascending or descending perfect fifth, while 2:1 or 1:2 represent an ascending or descending octave. In order to create a 12-tone chromatic scale Pythagoras had to omit the Gb (which is supposed to be the same note as F#, however this tuning produces two different frequency rations for these notes) only using the 12 remaining notes to comprise a complete chromatic scale. The untrue fifth which does not justly adhere to the 3:2 ratios from F# to Db is left out of just intonation and is out of tune, being slightly smaller by what is known as a Pythagorean Comma (a quarter tone). This interval was given the name the wolf interval due to its displeasing nature. This mathematical phenomenon made it impossible for these two notes to be played simultaneously in the music. Based on which note a Pythagorean scale was formulated as its tonic, the wolf interval will shift. Due to the shifting ambiguity of the wolf interval, it made it impossible to play all keys in tune. F#-Gb A Solution to the Problem Pythagoras had proposed that instead of using both ascending and descending fifths to formulate a chromatic scale that only ascending fifths be used, which would, as he thought eliminate the existence of the wolf tone as well as the off pitch augmented fourth-diminished fifth twin tones (same tone). By doing so Pythagoras thought he would resolved the problem of not being able to transpose to other keys. Note C Interval from C Unison Formula or ( ) G Perfect fifth ( ) ( ) 1.5 D Major second ( ) ( ) 1.125 A Major sixth ( ) ( ) 1.6875 E Major third ( ) ( ) 1.265625 B Major seventh ( ) ( ) 1.8984375 F# Augmented fourth ( ) ( ) 1.423828125 Db Minor second ( ) ( ) 1.067871094 Ab Minor sixth ( ) ( ) 1.601806641 Eb Minor third ( ) ( ) 1.201354980 Bb Minor seventh ( ) ( ) 1.802032471 F Perfect fourth ( ) ( ) 1.351524353 C Octave ( ) ( ) 2.027286530 ( ) Frequency Ratio 1 Pythagoras had resolved the problem of the wolf tone and the frequency ratio discrepancy concerning the augmented fourth-diminished fifth. However he was not able to resolve the issue of playing in all keys in tune due to the fact that his new mathematics left the octave interval a Pythagorean comma (quarter tone) out of tune. Where the true just ratio of an octave is 2:1 while Pythagoras’ fifth (3:2) ratio based scale produces an octave that is 2.027286530:1, hence leaving the problem of transposition unresolved yet. This affirmed the inability to fit a series of 3:2 ratio fifths evenly among a 2:1 octave interval. The inability to achieve equal ratios prompted composers to consider what specific key they wanted to write in as depending on which note (frequency) they wanted to use as the tonal center, unique dissonances would ensure based on the shifting properties of the wolf tone. Composers often referred to a keys mood or colour as some would contain more dissonant or consonant intervals. Equal Temperament(12TET): Equal temperament is a tuning system which uses logarithms in order to have all pairs of adjacent notes be equidistant from another (same frequency ratio). The most common and relevant of the Equal Temperament tuning systems is the 12TET which has 12 tones within the distance of an octave. Other Equal Temperaments exist such as 5, 7, 19, 22, and 31TET however for the purpose of understanding the math behind this system 12TET is the best representation as it solved the issues of Just Intonation and is the most popular tuning system in modern time. History of 12TET The founding of 12TET is often attributed to Chinese scholar Zhu Zaiyu (1584) and Dutch mathematician Simon Stevin (1585); both of their studies were independent from one another and occurred concurrently Zhu Zaiyu Chinese culture had for centuries practiced a tuning system which was based on five equidistant notes where the first and last where at a ratio of 1:2 ; it was known as the Pentatonic scale. Zaiyu was a Music theorist and was interested in developing another equal tone scale. He accurately calculated the value of the 12th root of two using the properties of generalized continued fractions which states: Where; An is the numerator and Bn is the denominator, Zaiyu then applied the properties of generalized continued fractions to that of roots to the nth degree where can be restated as +y then the proceeding is true. √ √( ) Hence the 12th root of 2 is approximated by; √( ) ( ( ) = √ ( ( ( ) ) ( ( ) ( ) ( ) ) ( ( ( ) ) ) ) ( ( ) ( ) ) 1.059463028 *The actual value of the twelfth root of two has been more accurately calculated using logarithms to the value 1.059463094359295264561825. The Modern 12-TET Tuning System Interval Name Exact Value Decimal Value Unison 1.000000 ( √ ) Minor second 1.059463 ( √ ) Major second 1.122462 ( √ ) Minor third 1.189207 ( √ ) Major third 1.259921 ( √ ) Perfect fourth 1.334840 ( √ ) Augmented fourth 1.414214 ( √ ) Perfect fifth 1.498307 ( √ ) Minor sixth 1.587401 ( √ ) Major sixth 1.681793 ( √ ) Minor seventh 1.787797 ( √ ) Major seventh 1.887749 ( √ ) Perfect octave 2.000000 ( √ ) Application in Real World I have been instructed in the western style of music, my teachers had never told me explicitly that I was using the 12-TET tuning system. When practicing scales I would be told to play any of the major scales and although they had different names they all sounded the same. It hasn’t been until now that I understand the mathematics behind it. I wanted to see how accurately me playing a C-Major scale is to the frequency values calculated using 12-TET. I used sound analysis software that recorded data from wave patterns and displayed a spectrogram (A spectrogram, or sonogram, is a visual representation of the spectrum of frequencies in a sound). The frequency I will use will be an average of the two obtained from the spectrogram and wave pattern. Note C: ( ) ⁄ 263.72 *All frequency calculations follow suite Note: D ( ) Note E: ( ) Note F: ( ) Note G: ( ) Note A: ( ) Note B: ( ) Note C: ( ) Note Wave Data (Hz) 263.72 290.94 328.18 344.84 397.73 439.59 495.93 528.08 C D E F G A B C Spectrogram (Hz) 262.02 293.16 330.47 350.87 392.57 439.24 495.14 525.20 Average (Hz) 262.87 292.05 329.33 347.86 395.15 439.42 495.54 526.64 Calculated Values of Notes in a C-Major Scale To find the frequency of a note using the 12-TET tuning system the following equation can be used ( √ ) In this formula refers to the value of the desired pitch, refers to the value of a predetermined reference pitch (typically A440), and whole number values given to the desired and assigned pitch. These numbers refer to consecutive integer numbers assigned to the numbered key they are on a standard piano. Where ( ) and where if ( √ ) ( √ ) Calculations follow same process with respective Note C D E F G A B C values. Calculate Frequency 12-TET (Hz) 261.63 293.66 329.63 349.23 392.00 440.00 493.88 523.25 Comparison of Frequencies Slope of Experimental Data: 38.59 Slope of Calculated Data: 38.17 ( ) Conclusion In conclusion through my investigation I found a greater understanding of the tuning/scale systems that I use on a daily basis. I also determined that when practicing my scales in real life I do not play them perfectly in 12-TET but rather I have a percent error of 1.10%. Through more practice I hope to become mathematically accurate with my scales now that I know the history, mathematical concepts, and principles of determining the frequency ratios between notes in 12TET .