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I’ve already gone through a little bit about the fifth. It’s undoubtedly the most important interval in the development of music. Gregorian chants were typically sung in unison, octaves, or fifths. No other intervals were permitted. They were all “demonic”.
So how did we arrive then at a scale, and at 12 tones in our musical system?
Remember, the fifth is the third partial harmonic, after the unison and the octave. It has a frequency ratio of 3:2.
From stacking one fifth onto the next, its very easy to come up with the ninth. Starting at C, go up a fifth and you get G. Go up another fifth, and you get to D. That D is at 9/4ths the fundamental C. Now lower it an octave and you have the Major 2nd, which is at 9/8ths.
Stack a fifth onto the D and you get an A, then stack another one on top of that and you get an E.
At this point you have C D E G A C, which is the major pentatonic scale. The largest gaps are the minor thirds from E to G, and from A to C, and people wanted to fill those gaps. Filling the gap between E and G was pretty obvious, and is again done with a fifth, by dropping a fifth from the high C. The high C is double the fundamental, and dropping a fifth means lowering the pitch by 2/3. That gives the F at a perfect fourth, which has the frequency ratio of 4/3.
Now, to fill in between the A and C, there are basically two choices, you can move up a whole tone, by adding a pitch 9/8ths above the A. Or you can go down a whole tone from the octave. The first procedure gives the major seventh, and the major scale, or Ionian mode. The second gives the dominant seventh, and the Mixolydian mode.
I have no idea which came first. But notice that whichever one you choose, you are actually getting both. If you have a C major scale, then you also have the notes for G mixolydian. But if you decided to make the C mixolydian scale, you would have the correct sequence of notes for an F major scale.
The Circle of Fifths
Above, we saw how we started building the scale by taking fifths and stacking them. We could have continued the process and it would go as follows:
C G D A E B F# Db Ab Eb Bb F C
This is the circle of fifths, and its usually presented in a circle. It contains all 12 notes, but they are arranged in fifths from each other. Let’s look at some of them in reverse order:
E A D G C
First, notice that E A D and G are the bottom four strings on your guitar. This is not an accident. Now, each note is the root of the dominant chord for the key of the note that follows it. E7 – A, A7 – D, etc…
Lots of chord progressions make use of this circle of fifths, because the progression from V to I is so satisfying, and the basic movement down a fifth is natural sounding. Thus, ragtime progressions will often be something like:
C E7 A7 D7 G7 C. In jazz, a sequence of dominant seventh chords like this is sometimes called “Rhythm Changes,” because its the bridge in Gershwin’s I Got Rhythm. It’s also used for Sweet Georgia Brown.
Progressions by descending fifths using diatonic chords is also extremely common. Try playing through the following C to Bmin7b5 to Emin7 to Amin7 to Dmin7 to G7 back to C. The ii – V – I progression, which is a staple of jazz, follows this sequence. The I vi ii V I progression, on which the entire world of Doo-Wop was built, also follows this basic sequence.
It also makes for a really good way to approach memorizing the fretboard. Instead of simply memorizing the individual note names, try playing and naming the circle of fifths. Since the V-I movement is probably the most powerful and common, its worthwhile to get extremely familiar with this type of movement wherever you see it.
Temperament
If you go up the circle of fifths, in our music, you eventually get back to the octave. By going up a fifth 12 times, you should end up at the fundamental, only seven octaves above where you started. Unfortunately, that’s not possible. Each time, we are multiplying the fundamental frequency by 3/2. Assuming we started at a frequency of 100HZ, we would want to arrive at a perfect octave of 12800hz. Instead, by going up in perfect fifths, we would get to 12975z (approximately).
Let’s look at the issue another way. We arrived at our E above by going up 9/8ths twice. This gives us an E that is a ratio of 81/64ths of the fundamental C. But, the major third partial is actually 5/4ths, which translates to 80/64ths. The procedure of stacking notes gives a third that is slightly sharp from the harmonic major third.
The slight imbalances occur everywhere.
To make it so that we can play music across octaves, and in different keys, on the same instrument, people had to come up with compromises. This has led to different kinds of tunings, including meantime, well-tempered, and pythagorean. In the last century, we have arrived at equal temperament, where each semitone is the same relative distance from the last. Specifically, the ratio of a semitone to the tone below it is the twelfth root of two.
As a result, there are no pure harmonics in our music, and every key sounds pretty much like every other key. This is great for transposition, but some of the character of the harmonies get lost in translation.
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