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I understand that the first note of each octave is twice the frequency of the first note from the octave before, but how are the frequencies for each note chosen subsequently after that. So far, I’ve been able to gather that an equation using geometric progression is used and it has something to do with harmony. I’m having trouble wrapping my head around how it all works. How does geometric progression translate into emotional value in music?

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There are probably more versions of this than I am aware of but I'll cite 2 or 3.

In the just tuning system the other intervals were chosen to be a rational fraction of the lowest note in the scale (Do). For example the frequency of a fifth (Sol) is 3/2*(frequency of Do), and a second is 9/8*(Do), etc. You can look up the full chart on wikipedia. The rational for this is that many of these tones are natural harmonics of a linear vibrating system.

The harmonic sequence is

fn = n*f1

where n is an integer and f1 is the lowest note in the sequence. n = 2 is obviously an octave, n = 3 is even higher and not within a single octave but you can divide this frequency by 2 and get 3/2 which is a perfect 5th. It turns out not all of the notes in the major scale are perfectly represented in this sequence but enough are to generate a major triad out of harmonics. If this process is repeated starting a 5th up in the scale and again a 5th below (on the 4th) one will generate all the just intonation scale notes. So one can think of this for of tuning as being based on the physics of vibrating systems.

Later in history people wanted a system that had a fixed ratrio for the half step and devised equal tempered tuning. This is based on fixing 12 half steps in an octave and defines the half step as the 12th root of 2.

f(1/2) = 2^(1/12)*f

this formula defines the frequency of the half step above f. In this manner you can build the major scale completely from half steps. The 5th would be 7 half steps and,

f(5th) = 2^(7/12)*f approximately 1.4983 * f

as compared to 3/2 = 1.5.

This form of tuning uses an irrational number which can never be perfectly represented so there is some fudging in this system.

Beyond that there is a system based entirely on 5ths, and probably a few others.

In fact the human can hear with much better resolution than a half step and frequency truly forms a continuum.

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    @YoungCapone And it's not about emotion, it's abut physics and biology, because that's how we perceive pitch and music. What we call octave is just the frequency doubled, so that's our reference. From there you can put whatever number of notes you feel like in between the octaves. 12 made sense in the context of the first overtones of the harmonic series. After that it made sense to make the space between them equal. The equations are just a way to explain, visualize, and model the phenomenon, but the phenomenon doesn't need them to happen. – Von Huffman Mar 2 at 0:35
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    @YoungCapone, while I can understand your frustration if you are not math inclined your statements are quite harsh. "What is the emotional value of this math? Why value does using an equation like this actually bring to music?". These equations are the foundation of music whether you understand it or not. Even if primitive man chose slightly different frequencies they were close to these relations, and for several hundred or even thousand years people understood sympathetic resonance even without physics to explain it. – ggcg Mar 2 at 1:38
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    Perhaps your question is not fair. Perhaps choice of frequencies never amounts to emotional value, rather rhythm and phrasing, dynamics, etc all provide emotion. – ggcg Mar 2 at 1:40
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    Thank you everyone or the responses! You've helped tremendously... it seems what's important in the progression of the frequencies in notes is...the distance between the frequency of any two notes is not the same, but the ratio between any two notes of the same distance is the same. For example, the difference in frequency between C1 and D1 is not equal to the difference between C2 and D2… however, D1/C1 = D2/C2 D1 - C1 does not equal D2 - C2… however D1/C1 equals D2/C2. Does this seem accurate? My next question would be why is it important that the ratio's remain the same? – YoungCapone Mar 2 at 2:17
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    @YoungCapone Exactly! The distance between them is not the same, but the distance changes at the same ratio. You multiply your current frequency by 1.0594... and you get the frequency of the next note. – Von Huffman Mar 2 at 8:09

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