Starting at a given note, say \(C\), suppose we play \(D\#\), a note that is 3 half-steps higher, then \(F\#\) which is again 3 half-steps higher, then another 3 half-steps to \(A\), and finally after 3 more half-steps we end up at a \(C\) again. The topic of intervals in music leads to some interesting elementary number-theoretic questions. In addition, the complex version of the exponential function is useful for representing the sinusoidal functions and for understanding their basic properties, especially in light of one of the most beautiful and important equations in all of mathematics, namely, Euler’s identityĪn extraordinary equation that is simple and 5 of the most important constants in mathematics. For example, when we talk about loudness of a sound we use the decibels measure, whose understanding requires the logarithmic function. The process of going from an additive measure of length to a multiplicative one leads us to what is arguably the most important function in all of mathematics, the exponential function and its inverse the logarithm function.īeyond describing in a clear way how to transition from notes to frequencies, these functions are important in a variety of ways for dealing with sound. On the other hand, at the frequency level, this change is multiplicative. Importantly, the note scale is an additive one: we speak of moving up by a certain number of half-steps. So moving up a half-step corresponds to multiplying a note’s frequency by a factor of about 1.059463. Which implies that \(c\) is the 12\(^\) root of 2.which implies that \(c\times c \times \omega_0 = c^2 \times \omega_0.\)Ĭontinuing for an additional 10 half-steps, the frequency of the note we reach after 12 half-steps above the starting note will beīut this note is an octave above the first note, so we see that But then going up an additional half step the half-note increase corresponds multiplying the frequency by this same factor of \(c\), so this whole step would have a frequency of \(C\) to \(C\#\), and let \(\omega_0\) denote the frequency of the lower note, and \(\omega_1\) the frequency of the higher note, we haveįor some constant \(c\). If we pick two piano notes separated by a half-step, e.g. In mathematical terms, this is best explained by introducing exponents. So we see that if we view the piano notes as ticks on a ruler, the tick distance between notes, as measured by the number of half steps separating them) translates into fixed frequency ratios. Similarly, if we move from \(A\) to E, the physical length of our interval is still a fifth, and the frequency of that \(E\) note is the frequency of the \(A\) multiplied by about 3/2. Saving Earth Britannica Presents Earth’s To-Do List for the 21st Century.When we move from \(C\) to the next \(G\) we multiply the frequency by a factor of about 3/2, so the \(G\) above middle \(C\) has a frequency of 392.4 Hz.Britannica Beyond We’ve created a new place where questions are at the center of learning.100 Women Britannica celebrates the centennial of the Nineteenth Amendment, highlighting suffragists and history-making politicians.COVID-19 Portal While this global health crisis continues to evolve, it can be useful to look to past pandemics to better understand how to respond today.Student Portal Britannica is the ultimate student resource for key school subjects like history, government, literature, and more.From tech to household and wellness products. Britannica Explains In these videos, Britannica explains a variety of topics and answers frequently asked questions.This frequency of 8 Hz which is at the top end of the Theta brainwave state is the brainwave state that makes us feel relaxed but conscious and open to intuitive learning. This Time in History In these videos, find out what happened this month (or any month!) in history. So, when we tune an instrument to 432 Hz, we get a C note at 256 Hz, which, due to the sympathetic resonance of the note overtones, will produce another C at exactly 8 Hz.#WTFact Videos In #WTFact Britannica shares some of the most bizarre facts we can find.Demystified Videos In Demystified, Britannica has all the answers to your burning questions.
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