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STUDENT: I see what you mean. So in other words, if one is conditioned by a society and becomes accustomed to a certain environment, they can hear something in music that is not heard the same by another. Ok, but then how can objective music exist? How can there be a natural rigorous way to partition to the music period? And one that derives scales that are objectively sad or happy?

TUTOR: Good question. Here, we need to work through the laws of harmonics.

STUDENT: Ok, what is that?

TUTOR: Ok, it is a law that fractions on a string can end up developing the 12 distinct notes of Western music. Do you know any instruments, hopefully the piano?

STUDENT: A little, yeah.

TUTOR: Ok. Well, so you probably know this: regardless of instrument, you know that in Western Music, there are twelve total notes; that is it, presuming you are set in a tuning. You also know that any key is just a frame of reference. The base note of the key determines everything else.

STUDENT: Yep, got it. Yep. But I don't know each of the scales.

TUTOR: That is ok for now. The point for now, firstly. is... When you have any note to start with, from there, assuming that that is the base note of your key, the primary scale of joy follows the same proportions or partitioning points relative to our string. Meaning, for example, if we fasten a string at two ends and tighten it to a pitch, we can consider that note as the base key of joy. From there, the notes that make up the pure key of that will always fall in the same fractions of the string; The scale is simply this from the sound of music: do re mi fa so la ti do. From there, do re mi fa so la ti do is actually always the same proportional, or fractional points, on the string.

STUDENT: That makes sense, but, are not these partitionings of do re mi... just arbitrary points of partitioning, like how music is relative?

TUTOR: No! Here is the thing: from the scientific law of harmonics, do re mi is actually just following the law of harmonics to its logical path. More on that later. Which is also not to mention that it is common sense of developing.

STUDENT: What do you mean common sense?

TUTOR: It means that if we were to ask ourselves without any reference point, how to partition a string, we would obviously start with the simplest points of reference. Toward that end, for example, we would never use the reciprocal of square root two as a partition point because square root two is an infinitely complex number. Rather, we would start with rational numbers, and in fact, the simplest rational numbers. And what do you think those would be?

STUDENT: Well, obviously the calculus harmonics fractions: 1/2, 1/3, 1/4, and that is because the numerator and denominator should always be the simplest. Since one is the simplest and lowest number for numerator, this leaves the denominators to increase as whole numbers.

TUTOR: Exactly. Which is to say, the lower the numerator and denominator, the simpler the fraction. Hence, we would not use for example, 3/4 because 3 is too complex, rather 1/4. But there is more.

STUDENT: Ok, what?

TUTOR: Well, we should do what is called exhausting the full implications of a fraction before moving on to the next most complex fraction. What that means is we compound the fraction in question to derive further notes, and continue, and so forth, until, if it were possible, we return to the base key note where we started, as in the octave.

STUDENT: That makes sense, but how do we do it?

TUTOR: Well, let us work it out. Firstly, let us just point out the essential substance of harmonics: not only are the simplest rational fractions common sense starting points for partitioning the period, but they are actually natural harmonics.

STUDENT: Ok, what is a harmonic?

TUTOR: Yes, on a string, a harmonic occurs naturally if, at that very point on the string, when you lightly touch the string, pluck it, and then quickly let up on the string, a pretty, otherworldly ring emanates from the note in a more wonderful way than if you simply fully depressed the string at that point and plucked it. So that is what is incredible that God did for us: He took the most common sense starting points for string partitioning and made them natural harmonics.

STUDENT: Awesomeness! Wayne's World!

TUTOR: Yes! That being said, the first "do" is the base note, the full string open. The final do, the octave, is always at half the length. Meaning, 1/2 the string is always the octave, or open note but much higher pitch. That is, if we depress the string at the 1/2 mark, that is, 1/2 of the string's length, we get the first octave. Always! It is natural law! No relativism possible here! Similarly, if we depress the string at 1/3 length from the top, we get the "fifth." Which, in the "do" scale, ... is the fifth step. Regardless the key. Or "so". Always! A law! Meaning, regardless of how we tune the string, once we keep the tuning where it is at, the string depressed at the one third mark from the top is always "so"!

STUDENT: In some sense, it seems it would be that way but it is also kind of neat. It is like a transcendent law, that doesn't depend on how long the string is, or what it is tuned to, like the pythagorean theorem!

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