The Circle of 5ths and introduction

Hello all. I'm Dave, and this will be my first time posting to a blog. For those who know me, HI! For those who don't know me, you're probably not reading this blog. But in any case, I will share a bit about myself.

I have been trying my hand as a serious composer since the age of ~14, when I began teaching myself piano and learned to play/sing/write music by ear. I have written in myriad styles through the years, from the diversity of a video game soundtrack to attempted stage musicals to Choral music performed live by my college choir. I feel I've come very far and learned a lot in my time pursuing music, and I've got a long way to go for sure.

But I've decided I'd like to share some of what I learned, and here is where I'll do it.

Today's topic is the Circle of 5ths. If you're a music student, you've heard of this Circle before, and you probably know what a Perfect 5th is in some sense. In case you're not a music student, I will start from the beginning.

When most music teachers start from "the beginning" they say:
"This here's the circle of 5ths. It's the foundation of our system of music and shows how all 12 of our chromatic pitches are related. You must memorize this or you'll do poorly in my class."

Which is a horrid approach that completely misses the point. If you want your students to really learn what it is, the REAL beginning is the best place to start. If you approach it from there, you will learn some things that make the Circle of 5ths much more interesting.

For starters, it isn't really a circle.

Some of you may remember a man called Pythagoras. He lived in ancient Greece, and he was basically a wizard at everything. He (or his followers, whatever; I'll just call it Pythagoras) invented the Pythagorean Theorem familiar to anyone who's taken Math in high school. It stated that the relationship between the lengths of the three sides of a Right Triangle was a^2+b^2=c^2. This Theorem is so pervasive that everyone who hears the name Pythagoras thinks of it before anything else.

To me this is kind of a crime. Certainly the Theorem is of huge importance in mathematics, and we owe him a debt of gratitude for that alone. But another monumentally important mathematical principle helped Pythagoras to do something else extraordinary.

What Pythagoras did became the foundation of the system of Western music we use today. It's seriously that enduring; after thousands of years we still utilize the founding principles he established about music. And what was it he established? It begins with something called a Monochord.

A Monochord is a simple device: one very long string (as on a guitar) stretched across a single structure of wood. Pythagoras conducted an amazing experiment with it: when he plucked the string on its own, he could hear not only the tone the string was tuned to make, but also some higher tones that related to the string harmoniously. After noticing this, he experimented to find what those tones were, and how they related to the root note to which the string was tuned. Forgive me for what seems like a diversion, but now I must discuss something vital to the understanding of the rest of Pythagoras' story: The Harmonic Series.

Can you count? 1, 2, 3, 4, 5, 6, 7, 8 -- let's start with that. The harmonic series is as easy as counting; it establishes that most tones in nature have an acoustic relationship with other tones called harmonics. Perhaps that isn't the best way to phrase it, but bear with me. Here is a table of what the harmonics mean, musically:

1 = Root
2 = Octave
3 = Perfect Fifth
4 = Double Octave
5 = Major Third
6 = Octave-Fifth
7 = Harmonic Seventh
8 = Triple Octave

The Root is also called the Fundamental; it is the basis of a Chord. Chords are built out of tones that relate harmonically. All even-numbered harmonics are forms of the Root: 1, 2, 4, 8, 16, 32, etc. ; tones with this relationship will always be in perfect harmony with one another, because they're all powers of two. To make it more simple, if the harmonic "1" or Root is tuned to the note C, then every even-numbered harmonic will also be the note C.

On the other hand, odd-numbered harmonics will always be tones distinct from the fundamental tone. They provide us musically with unique pitches that still relate mathematically to the Root. With a combination of harmonics, be they odd or even, you create acoustic sounds that are mathematically, naturally, harmonious and pleasant to the ear.

A simple example would be to take the first three odd harmonics: 1, 3, and 5. Anybody remember Solfege? That system of giving each note in a scale a short name to use when you sing the scale? It goes like (duh): Do-Re-Mi-Fa-Sol-La-Ti-Do. These same scale degrees also have letter names, like C. If we made Do into C, the scale would become C-D-E-F-G-A-B-C. I'll tell you right now that the harmonics 1, 3, and 5 are the basis of what is called a Major chord. Most of you are probably familiar with the difference in sound between Major and Minor chords. If you're not, that's fine, just keep reading.

Let's say we want to make a Major chord out of notes in the scale C-D-E-F-G-A-B-C. Let's also say we want to use "C" as the Root of the chord. We need three pitches, and "C" is the Root, 1. What do you say we take the third, and 5th notes of the scale and add them above C. Does that make a Major chord?

C - E - G = Major chord

What do you know! C - E - G is a Major chord! And,

1 - 3 - 5 = Major chord

Brilliant! So that must mean that C - E - G is the same as 1 - 3 - 5!

C - E - G =/= 1 - 3 - 5

Except no, it isn't. Not really. The truth is, both are certainly Major chords in every sense. BUT, the problem is that E is not the third harmonic and G is not the 5th harmonic. It's the other way around.

The REAL way to spell C - E - G is with harmonics 4 - 5 - 6. This is because 4th harmonic is an Octave of the Root, 5th harmonic is the Major Third, and 6th harmonic is an Octave of the Perfect Fifth. Confused yet? Let me clear up something else important.

When we say "Major third" it doesn't refer to the 3rd harmonic. It refers to the third note in OUR musical scale, not harmonic order. And the third note in the Major scale is Mi; or, if we're using note names, when C is the first note of the scale the third note (the Major Third) is E. And when C is the first note, the fifth note of the scale (the Perfect Fifth) is the note G. So C - E - G is a Major chord spelled in a Triad (Triad meaning that the pitches are distances to each other by some form of Third, Major or Minor). And, 4 - 5 - 6 is the harmonic spelling of the Major Triad, with C - E - G being a C Major Triad.

Still, the most accurate representation of the Major chord, acoustically, utilizes its harmonics in the order they occur in series. This means that the first appearance of the third harmonic (the Perfect Fifth) must occur before/beneath the first appearance of the 5th harmonic (the Major Third) in order for a chord to be in true harmonic alignment and have a pure sound. For this reason 1 - 3 - 5 seems a viable candidate for the Major chord, the only issue being that the tones are actually pretty far apart acoustically.

I personally think that 1 - 2 - 3 - 5 - 8 is one of the finest Major chords you can make, but I'll save the reason why for another lecture.

So we see how a combination of different harmonics, like 4 - 5 - 6, can make a harmonious chord like a Major chord. By the same token, the harmonics 1, 3, 5, 7, 9 create what's called a Dominant 9 chord. Sadly, because of the way our instruments are tuned it's impossible to play most odd harmonics, like 11 and 13; we actually have approximations in our scale that can be used for 15, 17, and 19, but I'm getting ahead of myself.

Back to Pythagoras. He understood the mathematics of the harmonic series and wanted to harness it to create musical instruments. He experimented with his Monochord to try and capture the higher tones he heard when he plucked the open string. So he would place his finger on part of the string, pressing it down, and pluck the string at the length that wasn't pressed down. This changed the pitch, because shortening the length of the string increased the frequency of its vibrations. Pythagoras found that when the string was held down at certain points - point correlating to harmonics - he would hear the higher tones the string had been making on its own.

Specifically, when he held the string down at points correlating to Small Integer Ratios. A ratio is something like 1/2; one over two is one half or 0.5. But some specific ratios Pythagoras found important were these:

3/2, 5/4, 7/4

Because 3/2 is a Perfect Fifth, 5/4 is a Major Third, and 7/4 is a Harmonic Seventh. These are ratios that take the odd-numbered harmonics and reduce their distance from the Root (by dividing them by powers of two, like /2 and /4). With harmonic intervals using the right ratios, it would be possible to compress harmonics into the width of a single octave, creating a musical scale.

If you took only the ratios 4/4, 5/4, 6/4, and 7/4, this is the scale of notes you would get:

C - E - G - ~Bb (why the ~ is for another day, when I talk about Temperament)

This is a fine scale, but with only four notes it's very limiting. You can't make good melodies using only a scale with each note so acoustically distant from the others. So Pythagoras thought, "How can I make a scale with numerous notes, all of which have some harmonic relationships, which fits in the span of a single octave and would be practical for instruments to play?"

Which is a fine question! Eventually Pythagoras realized something. Apart from the Octave, the Third Harmonic/Perfect Fifth was the smallest integer ratio and therefore the most stable interval aside from the octave. If Pythagoras could devise a scale based around the interval of the Perfect Fifth, the musical possibilities would be grandiose and far-reaching.

But how to make such a scale? Hmmm...Eureka came around eventually, and he said, "I will start on one note, my Root, and find its third harmonic; then I will find the third harmonic of that third harmonic, and so on, and so forth, yea and verily, til death do us part!"

And so he did. I've already mentioned that the third harmonic is the numerical value of 3 (duhhhh), and that the Octave is the numerical value of 2. To create a scale made of P5s that fits into one Octave, the only thing to do is multiply 1.5 (3/2) by itself and multiply 2 by itself until their values intersect. As it happens,

1.5^12 = 129.746
2^7 = 128

Well there you have it! Those numbers are extremely similar, so we can make a scale based on Perfect 5ths. It turns out that the acoustic distance of seven octaves is only slightly less than the distance of 12 P5s.

...only slightly less...that should be fine...won't cause any problems that endure for centuries to come...

So if there are 12 P5s in an octave, that means there are 12 distinct notes before the scale repeats. This is known today as the Chromatic Scale, which contains these 12 pitches:

C - C# - D - D# - E - F - F# - G - G# - A - A# - B ... and repeats at C, going further up or down as you like. Here (click) is the Wiki article on the Circle of 5ths, so you can see a picture (if you've never seen one before) of the notes arranged in an actual visible circle, rather than a metaphorical acoustic one.

"So in conclusion, Pythagoras invented the Circle of Fifths that we use today in modern music."
I would say, if that was accurate. But here's the thing.

That "slight" difference between 12 P5s and 7 Octaves is significant enough that it causes a huge, huuuuge problem. That problem is called the Pythagorean Comma, 23.46 cents (I have not yet explained cents, but I will in a later entry). Here's how I normally explain it:

If you begin on A, and go up seven octaves, you will arrive on another pure A.
If you begin on A, and go up the entire circle of fifths, you will arrive on an overly sharp "A".
If you have a note that IS A, played against a note that is ALMOST A,
you get something acoustically disgusting and discordant.

Octaves must be played in tune; either note of an "octave" being too sharp or too flat creates a frequency ratio that is overly complex and thereby dissonant. Dissonance is not bad in and of itself, but if what you're attempting to create is consonant harmony (which was Pythagoras' intention with the scale) then you must tune every interval accurately. The Pythagorean Comma meant an octave would be detuned, if the entire Circle of 5ths were used on an instrument.

In essence, this means that the "Circle" of 5ths is actually a Spiral of 5ths. If you use true acoustic Perfect 5ths and continue moving up, you will absolutely never arrive on a pitch that is a pure octave of the note you started on. The 5ths may have been perfect, but the scale they created had its issues.

So the first response was, "Hey, let's just not use the entire Spiral." So the Diatonic scale was born, a seven-note scale we know as Do-Re-Mi-Fa-Sol-La-Ti-Do. In this scale, instead of using the overly sharp Do from the end of the Spiral, the second higher Do was a pure octave of the first. So that solved that problem. There was another concern, though, namely the tuning of "Mi." The Major chord as I mentioned before is Do-Mi-Sol, but the Pythagorean Major Third is way, waaay too sharp. How sharp exactly?

The Harmonic Major Third is the ratio of 5/4.
The Pythagorean Major Third is the ratio of 81/64.

Its ratio's complexity makes it grating on the ear, and more simply it's just too sharp to create a truly pleasant, consonant sound. However, it was close enough that this Diatonic tuning endured for a long time. There was a Pythagorean Chromatic Scale as well, that used all 12 of the pitches of the Spiral of 5ths. But this one had even more harmonic trouble in use due to both its complexity and the frequent frequential disagreement of many of the scale's intervals.

Well, Pythagoras eventually passed away, but his followers carried on his work and promulgated his discoveries. The Spiral of 5ths as a musical system became widespread and popular, due to its reliance on clear mathematics and the acoustic functions of the natural world--and because it sounded pretty. The question of how to deal with the Pythagorean Comma and the other oddities of the Spiral of 5ths baffled people for many centuries, and they never stopped trying to find a way to solve that riddle. There had to be some way to fix the Spiral. To close it.

Ladies and gentlemen, I'll see you next time. For next time we delve. Next time we discover.

Next time, we close the Circle.

-Dave

Stay tuned for the upcoming entry on musical Temperament!
PS: I typed 99% of this from memory, so if you notice any errors in my information please let me know and I will add your corrections to the document. :)

First edit: So I browsed this after reading,
http://en.wikipedia.org/wiki/Pythagoras#Musical_theories_and_investigations
And apparently though Pythagoras did connect musical notes to the natural harmonic series, he was not responsible for inventing the Circle of 5ths. However, there ARE musical tunings called Pythagorean and Pythagorean Chromatic, probably devised using the small integer ratios of the harmonic series.

This is described in the article on Pyth tuning, for anyone interested.
http://en.wikipedia.org/wiki/Pythagorean_tuning