![]() With this relationship, we can cancel out f₀ from both sides of the equation, leaving us with an equation that we can use to solve for the value of our constant c: Going back to Pythagoras equation, we can say that f₁₂ is also equal to f₀ times c¹² as expressed in this equation: For example, we can continue the above succession into the next octave:Ĭ, C#, D, D#, E, F, F#, G, G#, A, A#, B, C1, C1#, D1, D1#, E1, F1, F1#, G1, G1#, and so forth.Įxtending the observation that we can produce two similar sounds after 12 notes, but with a doubled frequency, we can say that: However, to make them distinct from each other, we can use numbers to do this. We can repeat these 12 notes also to represent the notes on the next octave. In the western chromatic scale, it has been the standard to use such notations to represent each note in the scale. To learn more about music note intervals, you can check our music interval calculator. It also means that the frequency of D# is equal to the frequency of D multiplied by "c." Theoretically speaking, we can consider a jump from C to D as one whole step with an increase in the frequency of c². D#, read as "d-sharp," is half a step up D. In this convention, we designate the notes in the scale as follows:Ĭ, C#, D, D#, E, F, F#, G, G#, A, A#, and B.Įach interval represents a half step up, following the increase of frequency at rate " c," as described above. An octave is a set of 12 tones determined by a scale convention we call as the chromatic scale. Our second equation comes from the world of music, which states that a frequency and double that frequency, produce two similar sounds, but at exactly one " octave" higher. ![]() Let's call this equation the " Pythagorean equation." This is our first equation. ![]() We can express f₃ in terms of f₀ as follows:Ĭontinuing this fashion, we can generalize this equation by:įₙ = cⁿ × f₀ where fₙ is the frequency of a note produced while pressing at the "nth" fret. The same goes when we substitute Equation 4 in Equation 3. Substituting f₁ in Equation 2 with the equivalent expression in Equation 1, we obtain: Rearranging these equations and following this trend, we can come up with a set of equations as follows: Where " c" is the said standard ratio, f₀ is the frequency of the open string, and f₁, f₂, and f₃ are the frequencies of the following notes while pressing on the 1st, 2nd, and 3rd frets, respectively. And since, according to Pythagoras, any two adjacent notes should have the same common ratio, we can express this in a series of equations:Īnd so on and so forth. Plucking the new shortened scale length produces a higher frequency compared to just plucking an open string. In a string instrument, pressing on the string at the first fret shortens the scale length of the string. ![]() Since we can distinguish sounds through their frequencies, a change in the pitch of a sound also means a change in its frequency - the higher the frequency of a sound is, the higher its pitch is.Īdjusting a sound's pitch to predetermined frequencies is how we produce notes. Pythagoras, the same person who discovered the Pythagorean theorem of a² + b² = c², found that for a string instrument to sound its "best", any two adjacent notes played on the same string should have the same standard ratio. Without further ado, here are the concepts: The fret placement formula is a combination of three equations, two of which are physics formulas, and the other is an assumption that we now consider standard. ![]()
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