Theory of Music #3

While discussing chromatic symbols we omitted a crucial issue. We only considered the so-called accidentals, meaning the sharps and flats added as necessary whenever we needed to note a pitch located under a black key on the piano. There is, however, another class of chromatic symbols: the key signature. They are the same signs, but placed once – at the very beginning – and remain active throughout the whole piece.

Chromatic symbols in the key signature affect the chosen tones throughout the entire piece

“Not a big deal!” – you could think. – “The only difficulty is remembering this, uhhh... key signature, throughout the whole song!”. Right. But if you dedicate some time to further reading, you will see how big of a consequence the use of key signatures carries with it. You will understand why an equally tempered scale was an excellent idea, why the keyboard of a piano is built so asymmetrically, and you will find out what to do when a song does not match the scale of your voice.

Most songs use small subsets of “acceptable tones”. For example “Mary Had a Little Lamb” can easily be played on four white keys of the piano (try it: E-D-C-D-E-E-E-D-D-D-E-G-G-—E-D-C-D-E-E-E-E-D-D-E-D-C).

Now let's assume that you want to improvise around a theme of a chosen song. Which sounds can be safely used? It turns out that there is a simple rule to determine this. In the case of “Mary Had a Little Lamb”, the rule will claim that we can use all of the white keys on the piano. We can say that in this case the white keys create a scale, whose tones you will use in your improvisation.

How should we know which sounds can we use? The rule is as follows: first choose one basic pitch (we will call this pitch tonic). Usually it is the sound the melody of the song ends with. Further actions depend on whether the song is sad or happy.

If the song is happy (we will say: major), then your scale will composed of the following pitches:
I) the chosen basic pitch (tonic),
II) the pitch two semitones up from I),
III) the pitch two semitones up from II),
IV) the pitch one semitone up from III),
V) the pitch two semitones up from IV),
VI) the pitch two semitones up from V),
VII) the pitch two semitones up from VI),
VIII) the pitch one semitone up from VII).

Construction of the major scale. The distance of a single semitones occurs between the III and IV as well as the VII and VIII degree.

If the song is sad (we will say: minor), then the scale will be composed of:
I) the chosen basic pitch (tonic),
II) the pitch two semitones up from I),
III) the pitch one semitone up from II),
IV) the pitch two semitones up from III),
V) the pitch two semitones up from IV),
VI) the pitch one semitone up from V),
VII) the pitch two semitones up from VI),
VIII) the pitch two semitones up from VII).

Construction of the minor scale. The distance of a single semitone occurs between the II and III as well as V and VI degree.

Both constructions are abstractions of musical scales. In other words, they are recipes for building a set of pitches that can be safely used in one song.

Why does a song usually end with the tone which is the first degree of the scale? Why does a song sound happy or sad, depending on which scale it is written in? It may be a matter of habit, or the way our brains are structured. The topic is interesting, worth exploring, and... maybe too difficult to deliberate upon right now.

There are also scales other than the major and minor ones. There are scales with different (than eight) numbers of degrees. They have been defined and named, but they rarely appear in contemporary Western music.

Please notice that the white keys on the piano build up a major scale (with C as its tonic). This is exactly why there are no black keys between E and F as well as B and C: because the third and fourth as well as the seventh and eighth pitches of the major scale must be one semitone away from each other. Put differently, the piano was constructed in such a way, so that its white keys could be the tones of C major!

The major scale built starting from the C tone is going to be called C major. The minor scale built starting with the F tone is going to be F minor, etc.

Unordered set of tones that constitute the scale, with one (tonic) pitch set apart (i.e. the one that gives an impression of the ending of a piece) will be called key. Naming of the keys will be the same as naming of the scales, meaning D♯ major, G♭ minor, B major, etc. In the Continental European notation minor keys are marked with small letters and the major keys with capital letters: D♯ (major), g♭ (minor), H (major).

Let us now build the minor (sad) scale beginning with the A tone. We have to remember about the design of the minor scale, i.e. which degrees of the minor scale are separated by a semitone, and which by two semitones:

Construction of the minor scale from the A tone. Only white keys!

As you can see, the A minor scale is also composed of only the white keys of the piano. It's not a coincidence – we say that C major and A minor are relative (twin) keys. At the same time they are the most commonly used keys, which was probably imposed on us by pianists who preferred white keys!

C major and A minor are relative keys, meaning they consist of the same tones (although the first one has the tonic pitch C, while the second one: A). Each major key has its relative minor key and vice versa. The tonic pitch of the major key is three semitones higher than the tonic pitch of its twin minor key.

Now try to play “Mary Had a Little Lamb” entirely shifted down by a semitone (meaning: D♯-C♯-B-C♯-D♯-D♯-D♯-C♯-C♯-C♯...). It sounds good! Moving a melody to another range of pitches, called transposition, is a common trick that allows us to adjust the song to one's vocal abilities.

If the “distance” between each two closest “musical” pitches – meaning a semitone – wasn't invariant, transposition would ruin the melody. It's one of the main advantages of the equal temperament system: whole pieces can be raised and lowered without consequence!

We are reaching the core of the issue: how is all this related to the key signatures? Writing down the entire melody of “Mary Had...” after moving it a semitone down would be pretty inconvenient – we would have to add accidental chromatic signs to most notes:

“Mary Had a Little Lamb” in the C major key
“Mary Had a Little Lamb” lowered by a semitone

Notes will look cleaner if we use appropriate key signature:

“Mary Had a Little Lamb” lowered by a semitone. Key signature allows for a cleaner notation.

Observe where on the staff the sharp symbols are located – they are, from the left, the spots of F, C, G, D, A. Therefore our sharps have raised the tones F, C, G, D, A to F♯, C♯, G♯, D♯, A♯, respectively, for the entire song. It is worth mentioning here that sharps and minors cannot be mixed. It means that next to the clef there are either sharps or flats, what – as you'll notice – is not limiting whatsoever.

Why did we add sharps to those particular lines on the staff? Why in this order? And why do we need to raise the pitches of A and F (to A♯ and F♯), even though they don't appear in “Mary Had...”? Well, we have quietly passed over a certain aspect of key signatures: chromatic signs have to be added in a strictly defined order. You cannot just write one sharp at the height of D after the clef, even if you will often need D♯ in your song. The acceptable order is as follows:

  1. the first sharp in a key signature must affect the F notes (making them F♯'s)
  2. the second sharp in a key signature must affect the C notes (making them C♯'s)
  3. the third sharp in a key signature must affect the G notes (making them G♯'s)
  4. the fourth sharp in a key signature must affect the D notes (making them D♯'s)

A separate rule governs the usage of flats in key signatures:

  1. the first flat in a key signature must affect the B notes (making them B♭'s)
  2. the second flat in a key signature must affect the E notes (making them E♭'s)
  3. the third flat in a key signature must affect the A notes (making them A♭'s)
  4. the fourth flat in a key signature must affect the D notes (making them D♭'s)

We're not going to continue these enumerations – let's remember the rule instead:

The first sharp in a key signature raises the F note. Each following sharp in a key signature affects the tone which is seven semitones higher than the previous one.

F + (7 semitones) = C
C + (7 semitones) = G
G + (7 semitones) = D

The first flat in a key signature lowers the B note. Each next flat in a key signature affects the tone which is seven semitones lower than the previous one.

B – (7 semitones) = E
E – (7 semitones) = A
A – (7 semitones) = D

Why such weird rules? Why can't we just write a sharp at, say, D in the key signature? The answer is easy: by adding an isolated, song-wide sharp to D we would ruin the musical scale. We would create a set of pitches: {C, D♯, E, F, G, A, B}; which cannot be ordered into any major or minor scale!

The rules of adding chromatic signs to the key signatures, which we have just learned, are called the circle of fifths. You won't read about why it's “of fifths” until the next lesson, but in a second you will find out why it's a “circle”.

The circle of fifths protects us from building a set of pitches which doesn't constitute a correct major or minor scale. A melody played on such a set would sound awkward! Moreover, there is only one correct way of adding chromatic symbols to the key signature, that is one which doesn't ruin scales. On top of that, the circle of fifths is “complete”, meaning it lets us achieve any key, using a sufficient number of sharps or flats.

With our newly gained knowledge we can immediately tell which key a piece is written in. Example:
There are five sharps next to the clef, so in this song we are going to use the following set of pitches: {C♯, D♯, E, F♯, G♯, A♯, B}. The distance of one semitone occurs between D♯ and E and also between A♯ and B. Therefore our set of tones can comprise:

  1. a major scale : B, C♯, D♯, E, F♯, G♯, A♯, B
  2. a relative minor scale: G♯, A♯, B, C♯, D♯, E, F♯, G♯.
If the piece is “happy” and ends with the B note, it's a sign that it is written in the B major key. If a piece is sad and ends with G♯, it's a sign that it is written in the G♯ minor key.

Enough confusion, it's time to see the collective composition of key signatures and the keys resulting from them. Of course you know all the rules which will allow you to draw out the circle of fifths on a deserted island, but it's good to have a printed image on hand (and not have to count n times at seven semitones and adjust the scale to the achieved pitches).

The circle of fifths: the recipe for all keys. Take a really good look at it, or even better, print it and hang it above your piano.

Let's consider one more example: a key signature of seven flats. In accordance with the rule of order of adding flats it indicates the tones{C♭, D♭, E♭, F♭, G♭, A♭, B♭}. The single semitone only separates B♭ & C♭ and E♭ & F♭. Therefore this set can make up two scales:

  1. major (C-flat major): C♭, D♭, E♭, F♭, G♭, A♭, B, C♭
  2. minor (A-flat minor): A♭, B, C♭, D♭, E♭, F♭, G♭, A♭.
Now notice that C♭ = B, D♭ = C♯, E♭ = D♯, F♭ = E, G♭ = F♯, A♭ = G♯, B = A♯. So, scales with seven flats are created by exactly the same pitches which make up the B major and G♯ minor scales! This shouldn't surprise us, because C♭ is B (meaning C♭ major = B major), and A♭ is actually G♯ (meaning A♭ minor = G♯ minor). Therefore seven flat key signatures gives the same exact effect as five sharp key signatures. And the circle closes.

At Temptonik you can familiarize yourself with the A♭ minor key through the example of the song “Hit the road Jack”.

In our crude definition we omitted one significant point. Let us recall: we have assumed the the scale is an ordered set of pitches used in one song. Sometimes, however, there is a change of key in a song (which is a pretty easy way to build tension). This process is called modulation.

Modulation can be indicated in musical score with a new clef (with new key signature). Yes, a clef in the middle of the song. Though at Temptonik we prefer adding accidental chromatic signs.

Modulation can be observed in, for example, “My heart will go on” before the words “You're here, there's nothing I fear”. In the song “Belle” from the musical “Notre Dame de Paris” the change of key takes place as many as three times!

That's quite a lot for one lesson. We have learned the rules of adding key signature chromatic signs, which determine the key of a piece. We also know that in songs you cannot use pitches outside of the chosen key (though we will be gradually easing this rule). The situation is thus simplified, because we are managing to slowly limit the set of pitches under consideration. Nonetheless this knowledge is still not enough to create beautiful melodies. We invite you to read further; in the next steps you will find out how important pairs of pitches (intervals) and thirds of tones (chords) are, and you will learn to name and recognize them.

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