Take a piece of string and tie it between two nails. Make it really tight. Now pluck it. It is going to fluctuate for a little while before it comes to a stop.
Depending on how the long the string is, it's going to fluctuate at different frequencies. Frequencies are measured in Hertz and they are defined as:
The hertz (symbol: Hz) is the derived unit of frequency in the International System of Units (SI) and is defined as one cycle per second.
Now place your finger on the middle of the string and push down firmly, you have essentially shorten the string by half and if you were to pluck it again it would now produce a frequency which is double that of the original one.
Now put this theory to practice and pluck the string below. If you pluck in on the left hand side of the divider it will be unshorten which produces a deep sound. Pluck it on the right side and you've done the equivalent of placing your finger in the middle of the string, shortening it by half and it now produces a hight sound. This would be similar to how a stringed instrument like a guitar or a cello works where the player uses their finger on their left hand to change the length of a string to produce different tones or notes.
Turns out that there is a special relationship between the sound that those two frequencies produce. In music, they are called an octave apart, and defined as:
...the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems".
If the string is divided in two parts again, we will end up with a quarter of the string's original length and four
times the frequency. There is exponential growth happening every time we half the string. If the original frequency
1, it will go:
[1, 2, 4, 8, 16...] every time it is divided.
Now we have a an instrument that can play the same note in different octaves, not that impressive (...and if you are wondering why it is called octave, meaning eight, well, we will get to that in a moment).
We need more notes, and for that we need to divide each of our octave range into some arbitrary number.
The western music system opted for twelve, but different system use different numbers of notes per octave. The modern Arab tone system is based upon the theoretical division of the octave into twenty-four equal divisions. Then there is the 17 tone equal temperament which dividing the octave into 17 equal steps. On the other end of this is the pentatonic scale which is a musical scale with five notes per octave.
Even though only one octave has been mapped out in this drawing, just remember that there are octaves above and below.
All the possible 12 notes are called the chromatic scale. The distance from one note to the other is called an
interval. It's like the ratio from one tone/note to the other. The distance from one note to itself is
distance from a note to its octave counterpart is
2:1, a note eight steps up from another has the ration of
to name a few.
But the thing with these ratios is that some are more pleasing to the ear than others when played in sequence. So a few hundred years ago, a few guys came together and picked out a few ratios that sounded good together and gave them names. (I don't think that some guys actually came together at a round table and decided on this, but it's a funny mental picture).
Let's have a look at one of the formulas they came up with, or scales as they are called. But first: a jump up one note is called a half step or a H and a jump up two notes is called a whole step or a W. The formula goes:
whole-step, whole-step, half-step, whole-step, whole-step, whole-step, half-step.
W W H W W W H
We end up with seven notes and the eight being and octave higher that the one we started with. That's why it's called an octave. A seven note scale is called diatonic.
Let's go ahead and label the notes with number from 1 to 7. From now on, we are going to be talking about the 3rd and the 7th and so on and when we do that, we are talking about these notes that we labeled. They are called scale-degrees (...And ohh, btw, this scale is called the major scale, but more on that later).
Putting the scale to good use, here is a little melody that you may or may not recognize. The important bits are that only the notes from the scale are used. Because of that, the melody sounds harmonious. Every once in a while there might be a small 8 that flashes above the note being played. That means that the note is an octave higher than transcribed. But as we have previously discussed, a note is always the same regardless of which octave it is in.
Let's move on to harmony
in music, the sound of two or more notes heard simultaneously. In practice, this broad definition can also include some instances of notes sounded one after the other. If the consecutively sounded notes call to mind the notes of a familiar chord (a group of notes sounded together),
Out of each of theses notes in our scale, we can build a chord or harmony by selecting other notes from the same scale by a pre-defined formula and then play them in unison. The most common one is the triad where we select every-other note. It's called a triad because we are only gonna have a group of three notes in the harmony.
Every other note will give us notes
First up is the One chord. It is sometimes called the Tonic. Count with me how many boxes are between
There are three boxes, right? That interval between
3 is called a major 3rd so this chord is a major
chord. The ratio between them is
5:4. Roman numerals are often used to label chord. Major chords get a capital letter.
This chord will be labeled I.
We consider our
1 note to be our root and we do our ratio calculations from the root.
How many boxes are between the
5? There are six. That interval is called a perfect fifth and has a
Next is the Two chord. Is is called the Supertonic. Again, let's count boxes. Between
3 are only two
boxes, while there were three in the previous chord. That interval is called a minor 3rd and has a ratio of
The Two chord is therefor a minor chord, which in labeled in roman numerals system with lower case letters.
This chord will be labeled ii.
There are still six boxes between
5 so this chord has a perfect fifth as well.
You might have noticed that the numbers got shifted up one spot. That is because we are now considering the first note in our chord to be our new root and we always count from our root.
On we go and the next one is the Three chord. It is called the Mediant It has two boxes between
so a minor chord. It has six boxes between
5 so also a perfect fifth. It will get the roman
The Four chord is called the Subdominant. It has three boxes between
3, so a major chord. It has
six boxes between
5 and therefor a perfect fifth. It will get the roman numerals IV.
You might have noticed that we have started counting from the left side, going in a loop. We could have quite as easily extended our octave to the right but it doesn't matter, a note is the same note regardless of which octave it is in.
The Five chord is the Dominant. Three boxes between
3 and a major chord. Six boxes between
5 and therefor a perfect fifth. It will get the roman numerals V.
The Six chord is the Submediant. Two boxes between
3 and minor chord. Six boxes between
5 and therefor a perfect fifth. It will get the roman numerals vi.
Lastly we have the Leading tone. Two boxes between
3 so therefor a minor chord. But it only
has five boxes between
5. That is a diminished fifth it will get a little degree symbol. The
roman numeral will be vii°
Summing it up into a table, it would look like this.
That was a lot of information. The central point is though. In each diatonic scale, there are seven notes and seven chord which are composed out of those same seven notes.
And finally here is that little melody from earlier but now with the added harmony or chords. Look out for the top left corner, it's going to display the diatonic chord degree.
If you have dabbled in music before, maybe play a little guitar now and then you might be thinking: What is all this numbers stuff. Aren't the notes supposed to be labeled with letters like C, D, E, F?
And you would be right in thinking that. I just wanted, before I got into that to really hammer in this point that the notes you play do not directly make up the melody. It's the ratio between the notes that make up the melody. We can move a melody up or down a key and it will still remain the same if the internal ratio or intervals between the notes are kept intact.
Let's revisit the chromatic scale, now with characters labels attached to them starting from the note C
All of the previous examples have been in the key of C Major. Figuring out which notes are in the diatonic C major scale is to apply the W W H W W W H rule from before. We will end up with
Which so happens, doesn't have any sharps or flats and are all the white keys on the piano.
Building up our first harmony or the Tonic from the scale by selecting the root, 3rd and 5th will give us
C E G
Does this look like something?
This is a C chord. Moving onto the Supertonic by selecting every other note starting at D will give us
D F A
Which is Dm. I'll leave the rest up for you to figure out (or you can just have a look at the table below).
Transposing to another key
Let's use D now to see how the name of the note change but the intervals between them don't. The chromic scale from the letter D looks like this and applying the W W H W W W H rule will give us.
You might have noticed that the letters did change between C to D, but the placements of colored and shaded boxes did not.
It is a fun exercise on a cold winter's night with pen and paper in hand to work out all of the notes and harmonies in a given key. But I'll be a nice guy and list them out for you :)
- A, B, C♯, D, E, F♯, G♯
- A, Bm, C♯m, D, E, F♯m, G♯°
- B, C♯, D♯, E, F♯, G♯, A♯
- B, C♯m, D♯m, E, F♯, G♯m, A♯°
- C, D, E, F, G, A, B
- C, Dm, Em, F, G, Am, B°
- D, E, F♯, G, A, B, C♯
- D, Em, F♯m, G, A, Bm, C♯°
- E, F♯, G♯, A, B, C♯, D♯
- E, F♯m, G♯m, A, B, C♯m, D♯°
- F, G, A, B♭, C, D, E
- F, Gm, Am, B♭, C, Dm, E°
- G, A, B, C, D, E, F♯
- G, Am, Bm, C, D, Em, F♯°
In a song.
Looking one more time at our little happy birthday melody. The chord progression went:
I · V · I · IV · I V I
Which in the key of C major, goes
C · G · C · F · C G C
Transposing it to the key of B major gives this chord progression.
B · F♯ · B · E · B F♯ B
It goes without saying that it is far easier to memorize the diatonic chord degrees of a song and be able to change its keys on the fly rather that memorizing the same song in twelve different keys.
A few shout outs
I nicked some code and ideas from these places, so ... thanks :)