- #1

KiwiJosh

- 6

- 5

So I've been digging into music theory and want to understand the basic reasons for how it is constructed.

I've come across a particular relationship but I can't understand the reason it exists.

A quick musical lesson:

Imagine you've got a pure musical note: let's say C. From a low C up to a middle C (the next C up) is called an octave. From one C (or any note, for that matter) to the next octave up there are 12 notes:

**C, C#, D, D#, E, F, F#, G, G#, A, A#, B**. Notice that there are two sharps (#s) missing from here - that's just the way music is structured. For the purpose of my question, that doesn't matter, the only point is that there are

*twelve*notes in an octave. So don't get confused by the Letter names of the notes, which could lead you to believe there are only

*seven*distinct notes (A through G). The 12 notes are in fact pretty

*close to*(more on that soon) equidistant in pitch, meaning that mathematically it would be more useful to label them A through L, dispensing with the need for sharps, to avoid confusion. There are other musical reasons this is not done, but that's another story.

Halfway between any two Cs lies a particular note that sounds especially good with C: it's G. This relationship is known as a Fifth Interval (because, well, another story). Notice that G is not five but seven notes up from the root (

**C**, C#, D, D#, E, F, F#,

**G**)... as I said, not going to explain the reason here for naming it a Fifth. Anyway, it's halfway in between the two Cs when you're measuring pitch(aka frequency). But it's the seventh note of twelve, you say! That's not half! Except that, the twelve notes are not actually perfectly equidistant, for more reasons I won't go into here. Some gaps between notes are larger than other gaps. Suffice to say that what we now call a Fifth Interval is indeed halfway between the root and its octave,

*when measuring the frequency*. In fact, no matter which note you start from, if you go "up a Fifth" (seven notes up) then you get the same relationship: a note that sounds especially good alongside the root note.

Another particularly pleasing Interval is a Fourth Interval, which is actually five notes up from the root. So a Fourth up from C is F.

These two intervals are directly related, because seven notes + five notes = 12 notes. Meaning if you go up a Fifth from C you get G; but if you go

*down*a

*Fourth*from C you also get G. Just basic counting here, in two directions. So far so good.

So, a musical note can be expressed as a pure waveform. It can be measured by its wavelength in cm.

The sound of that note, or rather the frequency - how high or low it sounds - can be measured in Hz - oscillations per second.

So there are the two ways to measure a note.

I decided to look into what makes the Fourth and Fifth intervals so special.

Here is a chart containing the Wavelengths (in cm) and the frequencies (in Hz) of many notes: Frequency/Wavelength Chart

The story goes that our ancestors took a musical string and plucked it, obtaining a musical note. Then they took the string (which we can think of as a wavelength) and halved the length, noticing that the pitch (frequency) of the note was now twice as high - the same note but a higher version (what we now call an octave higher). This relationship can be expressed as a ratio of inverse-fractions, where WL = Wavelength and FR = Frequency:

½WL → ²⁄₁FR

(I'm using an arrow instead of "=" to express this relationship, as the values are not equal. I'm not sure of the proper symbol to use here but the arrow will do. Take it to mean "This

*leads to*That". Maybe a colon would be more appropriate?)

So then our ancestors said, how else can we split this string up? How about instead of halves, we split it into thirds?

So, they took the original string, cut a third of the length off it, and got a pitch which was 50% higher than the original. Let's say they took a string that made the note C, then cut it down to two-thirds of the length. They got the note G, which is actually halfway (in frequency) between the original C and the next C up (or, the original C + 50%).

This relationship can be expressed through another inverse fraction ratio:

⅔WL → ³⁄₂FR

Then they went on to base our whole system of Western Musical Scales on this "perfect fifth", as they called it (not to mention the ordering of the days of the week... but that is such another story, forget I even mentioned it).

Now, remember we discovered that the Fourth Interval is really just the other side of the coin to a Fifth, since going up a Fifth will give you the same note as going

*down*a

*Fourth*(albeit an octave higher). You can also say that a Fifth + a Fourth = an Octave (7 notes + 5 notes = 12 notes).

So, let's see how our WL → FR looks with a Fourth Interval (again, don't get distracted by the names Fourth and Fifth - those numbers are irrelevant to this working and can confuse you. We'd be better off calling them Bob and Jane...)

To achieve this, we cut a quarter of the length off the original string, and get a pitch one-third higher than the original. So:

¾WL → ⁴⁄₃FR

Another inverse-fraction.

Here's a quick example using the chart linked above.

We'll go with C

_{4}, also known as middle C, halfway up a piano keyboard.

Wavelength: 131.87cm; Frequency: 261.63Hz.

Make the wavelength a third shorter and we get G(88.01cm), which has a frequency 50% higher (392.00Hz). This is a Fifth Interval up from C.

Or we could go a Fourth Interval up from C, cutting only a quarter off the original wavelength, and we get F(98.79cm) which has a frequency one-third higher(349.23Hz).

These both confirm the inverse-fraction WL → FR relationships stated above.

So, this finally brings me to my question: why do these relationships exist? Is there a basic physical reason why taking a third off the wavelength gives a 50% increase in frequency, while taking a quarter off a wavelength gives a one-third increase in frequency?

It seems arbitrary until you express it as:

**⅔**WL →

**³⁄₂**FR (Fifth Interval)

**¾**WL →

**⁴⁄₃**FR (Fourth Interval)

not to mention

**½**WL →

**²⁄₁**FR (Octave Interval).

Expressed like this it shows this seemingly-divine symmetry of

*inverse-fractions in each relationship*.

Then add to it the fact that all three relationships are directly related: A Fifth + a Fourth = an Octave.

To me now it seems like there must be some underlying formula that justifies these relationships. Something basic at the level of math and/or physics.

But my father, an ex Math teacher, seems to think it is just a set of coincidentally pleasing relationships which were observed by our ancestors and seized upon for their seeming perfection. Is he right? I mean, I know that the concepts of Intervals, and even Notes, are human constructs. But the fundamental relationship between a wavelength and its frequency?

I'm struggling to see how these relationships could co-exist without some underlying reason.

Please help!

*Now, you'll have to bear in mind you're delivering answers to someone with not-far-above

*junior-high school*level physics understanding :/ sorry guys..

One possibility in my head is that the answers can be found in studying Waveforms. Now I did study calculus in Senior Highschool, and back then I could work with Sin, Cos and Tan. These days I really can't remember any of that. If they form part of your answer, I may be asking for some serious reschooling for clarification..!

Ok thanks. Peace

Josh