Music and Mathematics

If you thought music was not a mathematical language, then think again. Here we examine a relationship that clearly demonstrates the strength of this tie. Let the music begin.

For those with a rudimentary knowledge of music, the diatonic scale is something quite familiar. To understand why certain pairs of notes sound good together and others do not, you need to look into the sinusoidal wave patterns and the physics of frequencies. The sine wave is one of the most basic wave patterns in mathematics and is depicted by smoothly alternating crest-trough regularity. Many physical and real-world phenomena can be explained by this basic wave pattern, including many of the fundamental tonic properties of music. Certain musical notes sound well together (musically this is called harmony or consonance) because their sinusoidal wave patterns reinforce each other at select intervals.

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If you play the piano, then how each of the different notes sounds to you is dependent on how your instrument is tuned. There are different ways to tune instruments and these methods depend on mathematical principles. These tunings are based on multiples of frequencies applied to a given note, and as such, these multiples determine whether groups of notes sound well together, in which case we say such notes are in harmony, or poorly together, in which case we say such notes are out of harmony or dissonant.

Where these multiples come from depend on criteria set by the instrument maker and today there are certain standards that these fabricators follow. Yet criteria notwithstanding, the multiples are inherently mathematical. For example, in more advanced mathematics, students study series of numbers. A series is simply a pattern of numbers determined by some rule. One famous series is the harmonic series. This comprises the reciprocals of the whole numbers, that is 1/1, 1/2, 1/3, 1/4...The harmonic series serves as one set of criteria for certain tunings, one notably called Pythagorean Intonation.

In Pythagorean intonation, notes are tuned according to the "rule of the perfect fifth." A perfect fifth comprises the "musical distance" between two notes, such as C and G. The "distance" between each of these notes is called a half-step. When we number the notes in a musical harmonic series, the number ascribed to the C note and that ascribed to the G note will always be in the ratio of 2:3. Thus the frequencies of these notes will be tuned so that their ratios correspond to 2:3. That is the C-note frequency will be 2/3 the G-note frequency, or vice versa, the G note frequency will be 3/2 the C note frequency, in which frequency is measured in cycles per second or Hertz.

Now, continuing by tuning according to perfect fifths, the fifth above G is D. Applying the perfect fifth ratio, the D note will be tuned to a frequency which is 3:2 the G frequency, or looking at this from below, the G note is 2/3 the frequency of the D note. We can continue in like manner until we complete what is called the Circle of Fifths, bringing us back to a C note by applying successive ratios of 3/2 to the previous note in the cycle. This takes twelve steps and when complete, the frequency of the second C, or the higher octave C note should be exactly twice the frequency of the lower C note. This is a requirement of all octaves. Musicians have rectified this problem by resorting to none other than the field of irrational numbers. Recall that those numbers are such that they cannot be expressed as fractions, that is, their decimal representations, like the number pi or the square root of two, do not end and do not repeat. Thus as a result of the failure of the Pythagorean tuning method to produce perfect octaves, tuning methods have been developed to obviate this situation. One is called "equal temperament" tuning, and this is the standard method for most practical applications. Believe it or not, this tuning method incorporates rational powers of the number two. That is correct: fractional powers of the number two. So if you thought you were learning rational exponents for nothing in algebra class, here is one example of where such a topic is used in real life.

The way equal temperament tuning works is as follows: each note throughout its octave has its frequency multiplied by successive twelfth roots of two to get to the next higher note. Since the twelfth root of two is equal to 1.05946 to five decimal places, A# would be tuned to 440*1.05946 or 464.18 Hertz. And thus the tuning continues with the next note B obtained by taking 2^(2/12)*440. Note that we increment the twelfth power of 2 by 1 each time, obtaining powers of 2 which are 1/12, 2/12, 3/12, etc.

What is nice about this method is its exactness, unlike the inexactness of the Pythagorean intonation method discussed earlier. Thus when we arrive at the octave note, the next A above the standard A, which should vibrate at twice the frequency of the original 440 Hertz A, we get A octave = 440*2^(12/12) which is 440*2 = 880 Hertz, as it should be---exactly. As we stated earlier, when tuning by the Pythagorean method, this does not happen because of the repetitive use of the ratio 3/2, and therefore accommodations must be made to bring in line the inexactness of this approach. These accommodations result in perceptible dissonances between certain notes and in certain keys.

This tuning exercise demonstrates that mathematics and music are well intertwined, and indeed one could say that these two disciplines are inseparable. Music is truly mathematical and mathematics is, well, yes musical. Since many people think of musical talent coming from the "creative" types and mathematics ability coming from the "nerdy" or non-creative types, this article in some part helps disabuse these same people of this notion. Yet the question remains: If two ostensibly different fields as music and mathematics are happily married, how many other fields out there, which at first seem to have nothing to do with mathematics, are just as intricately linked to this most fascinating subject. Meditate on that for awhile.

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