Dear reader: this writing is in tutor/student dialogue format for more entertaining discourse. Enjoy!
TUTOR: OK, student, are you up for the music thing?
STUDENT: Yes
TUTOR: I promise it will be awesome!
STUDENT: Ok! Now, from what I understand, you are going to argue that music is not entirely relativististic, is that right?
TUTOR: Yes, because there are music students who have been educated into stupidity, like a lot of liberal arts majors.
STUDENT: In what way do music profs indoctrinate their students into believing that music is absolutely relativistic?
TUTOR: Well, they argue that as you go from different culture to different culture, what is perceived in music in terms of partitioning of the music period and emotions evoked is different, meaning, you can go one place, and what sounds somewhat discordant to Western ears may be entirely in tune to that culture. Plus, what we regard as sad in music, they may regard as happy, and vice versa. Their explanation therefore, is that there is absolutely no law from nature as to how you should or should not partition the music period and what is really happy or sad. It is entirely relative to individual human experience. Meaning, there is no real objectivity to music.
STUDENT: Do you agree with this?
TUTOR: Well, I think there is some truth to it, but in the end, some objectivity simply does exist, whether these liberal relativists want to admit it or not.
STUDENT: OK, tell me first where you give them credit.
TUTOR: OK, I give them credit in that it is true that human beings can be conditioned, there is no doubt of this. After all, dictatorships and ideologies can condition people. Indeed, the fall itself testifies to relativism, meaning, the human creature is different from animals. Animals have instinct, and their instinct is objective. If the sheep hear the wolf coming, they run like hell. No sheep is really capable of considering a live, out-front wolf as something good. Similarly for prey of lions and such. They just run. They cannot consider the lion benevolent. But humans are capable of being deceived. For example, young people are capable of thinking that promiscuity is something good and healthy. They are able to walk away from their parents at college and reject their upbringing and embrace licentiousness. Too, entire societies might be promiscuous and not realize that it is bad. This is their relativism at work. They are doing something gravely harmful, like a sheep hugging a wolf, and not realize they are in serious rebellion from God. This can then explain why certain cultures are conditioned to hear something different than what is objectively in music.
STUDENT: Do you have another example?
TUTOR: Sure. There was a story of a girl who was kidnapped preteen years by a Jehovah's Witness couple and the father raped her for years in his backyard. The JW man constantly told the girl that his religion was correct. Now, consider this: what if all the time the man was raping the girl, he played vibrant, joyful classical music; then eventually, joyful classical music would invoke fear, horror and anguish in the girl every time she heard it, even though it is objectively happy. Whereas on the other hand, if we suppose that when the girl was with the woman alone, the woman was compassionate with her and told the girl that she wanted to help her escape but would be killed if she did, and that while they communed, she played somber funeral organ music, then eventually the girl would associate the somber funeral music not with sadness but with peace and joy, even though funeral music is objectively sad.
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!
TUTOR: You are absolutely right, and it only gets better from here! We are going to kick those relativists in the ***! So now, as we hinted at, if we depress the string at the 1/2 mark, we get the first octave. NOW, when speaking of "following the implications," what we do to logically try to develop further notes while still using the same fraction you already have, you basically compound the fraction.
STUDENT: What does that mean?
TUTOR: It means, when you have depressed the string at the fraction point, you then take the string between that point and the end as a new string of reference and apply the fraction to that!
STUDENT: I see, meaning, you take the one half of the half, then the half of that, and so forth, and derivatively, if you had one third, take the one third of the one third, etc.
TUTOR: Exactly. Hence, returning to one half, if we depress the string there and then use that latter half string as the new string, and then apply one half to that new string, you get the octave of that, which is the octave of the octave.
STUDENT: Got it, and yet, the octave of the octave is just the same note we ultimately started with, only much much higher. And I can see that if we continue, we will never get anything but higher octaves of the ultimate starting point.
TUTOR: Right, so that, in other words, the ultimate implications of ½ never gets us anything but one note, only at higher pitches. Hence, if music is ever to get any more complex, we must move on to the next more complex fraction, which is clearly 1/3.
STUDENT: Right, and you already said that 1/3 is step 5 in do re mi, or, that is, “so.”
TUTOR: Yep, and, in fact, if we keep taking the fifth of the fifth, of the… we end up generating all twelve notes of western music. At least effectively. So that western music is natural, it is based on harmonic law and not just arbitrarily placing our finger anywhere on the string. Because obviously, there are infinite possibilities of where to put your finger on a string, just as there an uncountable number of Real Numbers on any interval of numbers.
STUDENT: Can we go through the compounding of the 1/3 fraction?
TUTOR: Well, it is not necessary to work this out for our purposes, since that process is a little complex math and geometry proportion skills; suffice it to say, one can find that process online. But to whet one’s appetite, we can show that to simply get the fifth of the fifth is a little dirty.
STUDENT: Ok, hit me!!
TUTOR: Ok, you asked for it. Tighten your stomach muscles. Ok, so again, 1/3 from the top of the string gives you the fifth step of joy relative to the note that you are taking the 1/3 of, which in the do re mi scale is “so”. So from here, if we want the fifth of the fifth, we do like this:
We consider our new string to be from our 1/3 mark to the end of the string. Then we take a third of that.
Ok, well firstly, the new string length is what is left after 1/3 which is 2/3 [the sum of the fractions must equal 1, or 100%]. Hence, the new fifth is the distance from the original 1/3 mark to the new 1/3 mark, which is clearly 1/3 of the new string length, or 1/3 of 2/3, or 1/3 * 2/3 = 2/9.
But this is just the distance from the old 1/3 mark to the new one. We need the new 1/3 mark’s ultimate position on the string, which is 1/3 + 2/9, or 3/9 + 2/9 = 5/9. Soo, in other words, the new fifth is at the 5/9 overall mark on the string.
STUDENT: Are we done?
TUTOR: Unfortunately not. In fact, we have to use another proportion.
STUDENT: You have got to be kidding me.
TUTOR: Unfortunately for you, I am not.
STUDENT: Why?
TUTOR: Because 5/9 is not in the first octave.
STUDENT: Why?
TUTOR: Because the first octave, remember, is ½, and 5/9 > ½, which means, again, we are into the next octave section on the string. Obviously, we want to keep our notes in the same octave section if at all possible.
STUDENT: Why?
TUTOR: Because otherwise, we get higher and higher (“We’ll get higher and higher, straight up we’ll climb, so baby dry your eyes, savor all the tears----”) oops, got a little carried away; Sammy get out of my brain!
STUDENT: I didn’t appreciate the Van Hagar excursion.
TUTOR: Sorry bout that. You can hear about that later.
STUDENT: And I am not appreciating the DLR excursion.
TUTOR: What is your problem, man, don’t you like Van Halen?
STUDENT: I do but I want to get on with this. I have tickets to Pink Floyd tonight.
TUTOR: Ok, then, we better hurry up. So where was I? Oh yeah, so again, the 5/9 overall mark on the string is unfortunately beyond the ½ mark, meaning it is the next octave section. So what we need to do is measure the distance between ½ and 5/9 and consider its proportional distance relative to the second octave length and then take that proportion from the top of the string relative to the first octave. EVERYBODY GOT THAT?! GOOD!
STUDENT: Umm, can I interject. Basically, this sucks royally.
TUTOR: I know, it does, at least for non-math majors. I don’t have a problem with it, cuz I am a math nerd, but for people who have less than critical thinking skills, it sucks. And this is why I warned you that it is messy.
STUDENT: You got that that right, I should have listened. Anyway, since we came this far, might as well finish the blasted thing.
TUTOR: You got it. Actually, it isn’t too bad. For, note, once we get the position we have relative to the
second octave dimension on the string, the corresponding proportion in the first octave is just twice that.
STUDENT: This I think I can see: because the ½ mark is the octave relative to the string in question, the second octave is just ½ of the ½ or ¼. Hence, the second octave length is just ¼ of the whole string, and obviously the first octave we have is ½ the string. Hence, any relative second octave position is fulfilled as twice as much in the first octave.
TUTOR: Amen, you got it bro, there may be some hope for nerdiness for you. Hence, to complete our thing, the fifth of the fifth is the 5/9 overall string position, and so that is, relative to the second octave dimension, 5/9 – ½ = 10/18 – 9/18 = 1/18. Hence, the first octave equiv is 2 * 1/18 = 1/9.
Ok, now you understand why this sucks for non-math majors. I can handle it because I was a math major, but most people are not math majors, and probably because of things like this. So anyway, you can see it is a can of worms. Nevertheless, the point is this, sans TMI: if you are insane enough in your scientologist, hyper-narcissistic drive to find truth, and hence to continue to plow through this diabolical algorithm, it just so happens that at the twelfth fifth, with a horrifically complex rational number in your concoctions, a very bright light suddenly erupts into the cave, where there had henceforth never been any light. And “blinded by this light” that is 3cm from your pitiful face, is the ½ octave!!!
In other words, once we kidnap princess vespa and for her father, king roland, to give us the combination to the air shield, we will thereby DESTROY plane druidia, and SAVE planet spaceball! EVERY GOT THAT?! GOOD!
STUDENT: So, in other words, we are back where we started, only one octave higher, or almost.
TUTOR: Right, cuz our fraction is almost one half, but not quite, and is very messy. Toward that end, this discrepancy of micro-proportion, known as the pythagorean commas, was the subject of heaty debate in the Middle Ages, as to how this mysterious gap should be filled or divided. They almost killed one another over it, kind of how religious wars got started, know what I mean?
STUDENT: Yeah, so anyway, that was very much TMI. Point is, I can see that the twelve notes of Western Music are not relative at all, but follow from natural law. Ok, so then, you were going to tell me why in the heck we even went into this?
TUTOR: Yes, well, the issue is, later in our series, we are going to show that Western Music has some amazing analogies for our faith, entirely appropriate and doctrinal, and we are preemptively doing this so that if any wiseass comes along and says music is completely random, and that, therefore, our analogies are probably just a wild coincidence, we can point them to this article.
STUDENT: Good, I look forward to it.
TUTOR: Great! Class dismissed!