If you thought music was not mathematical language, then think again. In fact, music and mathematics are very much intertwined, so much so that I think you might say could not live without the other. Here we explore the relationship clearly shows the strength of this tie. Let the music begin.
For those with a rudimentary knowledge of music, 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 points pattern wave physics and frequencies. The sine wave is one of the basic pattern surge in mathematics and is displayed well alternately Crest-lowest extent. Many physical phenomena and real-world can be explained by this basic wave pattern, including many of the fundamental tonic properties of music. Certain musical notes sound good together (musically this is called reconciliation or consonance ) due dots pattern waves strengthen each other at select intervals.
If you play the piano, then how each of the different notes sound to you depends on how your instrument is configured. There are different ways to tune instruments and these methods depend on mathematical principles. These artefacts are based on the whole frequency applied to a particular mind, and as such, this product whether groups of notes sound well together, but we say such comments are in harmony, or incompatible, but we say such remarks are out of harmony or dissonant.
When this product coming from the launch of criteria instruments manufacturer and today there are certain criteria that these fabricators follow. Still criteria Despite a multiple nature of mathematics. For example, in more advanced mathematics, students learn some numbers. A series is simply the pattern numbers are determined by some rule. One famous series is harmonic series . This includes the reciprocals of integers, it is 1/1, 1/2, 1/3, 1/4 … The harmonic series serves as one set of criteria for certain tunings, one in particular called Pythagorean intonation .
In Pythagorean intonation, comments are set according to “ rule that perfect fifth .” A perfect fifth consists of “ musical distance ” between the two tones, such as C and G. Again without trying to turn this article into a thesis on musical theory, are notes between C and GC #, D, D #, E, F, F #, and G. The “ from ” between each of these notes is called a half-step. Thus, the perfect fifth 7 half-step, CC #, C #, D, DD #, D # -E, EF, FF # and F # -G. When we consider the notes in a musical harmonic series, a number considered a C note and attributed to G mind will always be in the ratio of 2: 3 So the frequency of these notes will be adjusted to their ratios in accordance with the 2: 3 There is C- note the frequency will be 2/3 G-note frequency, or vice versa, G note frequency will be 3/2 C note frequency, the frequency is measured in cycles per second or Hertz.
Now, continue to adjust according to the perfect fifth, fifth above G is D. Use perfect fifth rate , D note will be tuned to the frequency of 3: 2 G frequency or watched from below, G note is 2/3 frequency D note. We can continue the same way until we complete what is called Ring of the fifth , bringing us back to the C note by applying a series of ratios 3/2 in the previous note in the cycle. This includes the Twelve Steps and when complete, the frequency of the second C or higher octave C note should be exactly twice frequency lower C note. This is a requirement of all octaves. But this does not happen by applying this ratio to 3/2.
Musicians have corrected this problem by resorting to none other than the field irrational number . Recall that the numbers are such that they can not give up the offense, that is, a decimal representation of them, like the number pi or the square root of two, providing not and do not repeat. Thus failure of a Pythagorean tuning method for producing perfect octaves, tuning methods have been developed to prevent this situation. Is called “ equal temperament ” Tuning, and this is standard procedure for most practical applications. Believe it or not, this tuning method involves the rational powers of the number two. That’s right: folded chose number two . So if you thought you were studying rational exponents for nothing in algebra class, here is one example of where such a debate is used in real life
Route equal temperament tuning works as follows :. Each note through its octave has a frequency multiplier repeated twelfth root of two to get to the next higher note. That is, if we start with the traditional note, which vibrates at 440 Hertz, let’s say, getting to # 440 multiplied this by 2 ^ (1/12). Because the twelfth root of two is equal to 1.05946 to five decimal places, A # was tuned to 440 * 1.05946 or 464.18 Hertz. And so continues tuning with the next Note B by taking 2 ^ (2/12) * 440. Note that with the increase of twelfth power 2 by 1 every time the power 2 of 1 / 12, 2/12, 3/12, etc.
What is nice about this process is exactness, unlike inexactness of Pythagorean intonation method discussed earlier. Thus when we arrived at the octave note, almost on top of the standard A, which should vibrate twice the original frequency of 440 Hertz A, we get A octave = 440 * 2 ^ (12/12) 440 * 2 = 880 Hertz, as it should be — exactly. As we noted earlier, when the set of the Pythagorean method, this does not happen because of repeated use rate 3/2, and the accommodation must be made to bring into line the inexactness of this approach. These great cause noticeable dissonance between certain notes in certain keys.
This tuning exercise shows that math and music are closely intertwined, and indeed it can be said that these two disciplines are inseparable. Music truly is mathematics and mathematics is well, yes music. Since many people think of musical talent coming from “ creative” species and math capabilities come from “ nerdy ” or non-creative types, this article is in some part help disabuse these same people of this idea. Yet the question is: If two ostensibly different fields like music and mathematics are happily married, how many other areas out there that at first seem to have nothing to do with mathematics, are just as intricately linked to this most fascinating material. Consider that for a while.