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Most of us learned the basics of music in elementary school. The notes in music can be sung as: Do - re - mi - fa - so - la - ti - do There are even lyrics in The Sound of Music that most of us probably recognize: Doe, a deer, a female deer Ray, a drop of golden sun . . . . But none of this tells us the actual frequencies of the notes. To explain that, I'll go into a little more detail. The notes in do - re - mi etc. can also be expressed as |
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| C - D - E - F - G - A - B - C | ||||||||||||||||||||
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Where each note increases in pitch, until the second C, which is double the frequency of the first. The second C is one octave higher than the first. This is one definition of an octave – the same note, but double the frequency.
Now C - D - E etc. does not list all of the notes. There are also flats and sharps. So the complete list of notes is: |
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| C - C# - D - D# - E - F - F# - G - G# - A - A# - B - C | ||||||||||||||||||||
| Where # denotes sharp (so that C# is C sharp, D# is D sharp, A# is A sharp, and so on). The reason I haven't written down E# is that E# is the same as F. And B# is the same as C. I could just as well have written: | ||||||||||||||||||||
| C - Db - D - Eb - E - F - Gb - G - Ab - A - Bb - B - C | ||||||||||||||||||||
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Where b denotes flat (so that Eb is E flat and Bb is B flat, etc.) And just as B# is the same as C, Cb is the same as B.
Earlier I said that second C in our sequence of notes was twice the frequency of the first C. There is no reason to stop there. I could go up another octave from the second C, or go down another from the first. But before I do that, I can ask: If the second C is twice the frequency of the first, how are the frequencies of C and D (or C and C#, or C and E, or any two notes) related? |
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| TOP | ||||||||||||||||||||
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Let's try the following. I know C# has a higher frequency than C, but not double the frequency (since the second C in our list of notes is just double the frequency of our first C). Suppose there is some number N, that when I multiply the frequency of the first C by N, I get the frequency of C#. So, in terms of frequency: |
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| C# = N x C | ||||||||||||||||||||
| I don't know what N is, but I'm hoping to come up with some reasoning that will tell me what it is. Now suppose that the frequency D is related to that of C# in the same way: | ||||||||||||||||||||
| D = N x C# | I can combine (4) and (5): | |||||||||||||||||||
| D = N x C# = N x N x C = N2 x C | ||||||||||||||||||||
| Suppose that D# and E can be written in a similar way: | ||||||||||||||||||||
| D# = N x D = N2 X C# = N3 x C | and | |||||||||||||||||||
| E = N x D# = N2 x D = N3 x C# = N4 x C | ||||||||||||||||||||
| Now, finally, we can write: | ||||||||||||||||||||
| F = N x E F# = N x F | G = N x F# G# = N x G | |||||||||||||||||||
| A = N x G# A# = N x A | B = N x A# C2 = N x B | |||||||||||||||||||
| Here I have referred to the second C as C2, to distinguish it from the first C, which I will now call C1. I can now combine all of the expressions in (4) through (9): | ||||||||||||||||||||
| C2 = N x B = N2 x A# = N3 x A = N4 x G# = N5 x G = N6 x F# | ||||||||||||||||||||
| = N7 x F = N8 x E = N9 x D# = N10 x D = N11 x C# = N12 x C1 | ||||||||||||||||||||
| Since C2 = 2 x C1 the number N is the twelfth root of 2. I won't describe how to calculate this number, I'll just give its value: | ||||||||||||||||||||
| N = 1.059463094359 | If you wish, you can multiply it out twelve times and see that it gives 2. | |||||||||||||||||||
| What (11) means is that, given the frequency of any note on the guitar, I can figure out the frequency of all the others. And that's what I do next. | ||||||||||||||||||||
| TOP | ||||||||||||||||||||
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Recall that when I derived the value of N I referred to the first pitch of interest as C1, and the second pitch, an octave higher, as C2. I can define another set of notes, an octave higher, ending in C3. And I can define yet another set of notes, an octave higher, ending in C4. So the frequency of C2 is double that of C1; the frequency of C3 is double that of C2, and the frequency of C4 is double that of C3. To get the frequencies of notes in each octave I just use (5), (6), and (7). According to Scientific Pitch Notation, the note middle C is C4 and A4 is 440 Hertz (cycles per second). Since A3 is an octave below A4, and A2 is an octave below A3, this would make A2 110 Hertz. Note: This same reference states that A4 is the first A above C4, so that the notes in the fourth octave are: | | |||||||||||||||||||
| C4 - C#4 - D4 - D#4 - E4 - F4 - F#4 - G4 - G#4 - A4 - A#4 - B4 - C5 | ||||||||||||||||||||
I can find the frequency of C5 from the frequency of A4 by using:
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| C5 = N x B4 = N2 x A#4 = N3 x A4 | ||||||||||||||||||||
Multiplying A4 (which is 440 Hz) by N3, where N is given by (11), yields a frequency of 523.26 Hertz. This is C5; middle C is C4 which is half this or 261.63 Hertz.
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| TOP | ||||||||||||||||||||
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Table N contains the frequencies of all of the musical notes from C0 to B8, using (5), (6), and (7). The notes on the guitar range from a low of E2 (which is 82.41 Hertz) on the open sixth string, to (on my diagram, which is for an acoustic guitar) a high of G5, which is 784 Hertz. On an electric guitar you might be able to get to C5, which is 1046.51 Hertz. And on the guitar, middle C is obtained by holding down the string on the first fret of the second string. | | |||||||||||||||||||