EPTPart 1: Introduction and the major scale Please read this entry first if you haven't done it yet.
Let's recapitulate the first blog: Pitch differences of tones are called interval. An interval is expressed by a ratio or by a multiplier. Simple ratios lead a harmonic sound. The dominant harmonic interval is the fifth, as it is so simple. A musical scale spans an octave which represents a pitch multiplier of 2. To create a scale, we are stacking fifths and shift notes back into our octave if needed. The major scale can also be viewed as a series of three major triads.
Interval Names
The eighth note of a seven-note scale is the first note again, but with a pitch twice as high. That is why this interval is labelled an octave as the Latin word "octavus" means eighth.
We also learned about the interval which represets the pitch ratio of 2:3, factor 1.5 – as 3 is 1.5 times as high as 2. As it turns out, this interval is represented by the fifth note in a major scale, that is why the intervall is named the fifth. As mentioned in the last blog, the fifth interval is often expressed as 3:2, but we are using the expression of lower note : higher note.
How to play a fifth on a piano? If we use C-major, we only need the white keys. So we begin with C and count: D (second), E (third), F (fourth) and G. The fifth. C-G therefore represents a fifth.
The Brother Of The Fifth
The complementary interval of a fifth to an octave is the following: 2 (representing the octave factor) / 1.5 (the fifth) = 1.333... Written as ratio, it is 3:4. This interval has its own name. Because it is also the fourth step of a scale, the interval is labelled the perfect fourth.
This creates an interesting ambiguity. How does relate F to C? It depends: We just counted from C to the fifth which was G. F was the fourth. But if we start from F, then C is the fifth to F.
The Sound Of The Fourth
Within the fourth interval, the second note has a pitch 1/3 higher than the base note. Is this interval harmonic, or not?
Please bear with me now, because this is important. Lets imagine C-G, a fifth. But for our experiment, we move G one octave down, so we play G-C. What do we hear? From G to C we get a fourth: G, A (second), B (third), C (fourth.) We hear the interval of the fourth, but without knowing the context, we don't know if it actually should be a fifth, just moved one octave down. Since octaves and fifths are both so simple, this interpretation of the interval would make sense, too.
The fourth is an unstable interval. While closely related to the fifth as both its complement and the interval which follows the fifth in the harmonic series, it does not share the feature of the fifth to create stability.
Ancient World Views
This relationship between fourth, fifth and octave was already known by the ancient Greek mathematician and philosopher Pythagoras. All three intervals in relationship to the first note are represented by the beginning of the harmonic series, 1:2:3:4. 1:2 = octave, 2:3 = fifth, 3:4 = fourth.
The first four natural numbers played a large role in Pythagoras' world view. Let's not get too deep into his teachings, let's just marvel at the fact that 1+2+3+4 = 10, the number of digits we have. That is a mathematical accident with no meaning.
![[image loading]](http://i.imgur.com/IWlJAuE.png)
The Third Makes The Sound Fuller
If we play a note and its fifth, we get of course a very harmonic combined sound. It's so harmonic that is sounds quite empty, too. Nothing exciting happens when we play a note and another one with a pitch which is by 1/2 higher than the first note, meaning its fifth interval.
The interval third is called as such, because it is the third step the the scale. In a major scale, its pitch is 1/4 higher than the first note. The ratio is 4:5, as 5 is 1/4 higher than 4. The third interval is also is part of the harmonic series, at it expands the system of having 1:2, 2:3 and 3:4, by 4:5.
Even though this ratio is still quite simple – and therefore harmonic – it is also quite narrow. This results in a rather difficult combined overtone spectrum when the two notes are played together. That means we get some content into the sound. This interval is responsible for the tonality of major triads.
As most songs today are accompanied by chors which either are triads or are based on triads, we hear the interval of a third very often. We get a harmonic, saturated, bright sound.
Narrowing The Third
The interval 4:5, where the second note has its pitch 1/4 higher than the first one, has a follower in the harmonic series. It is 5:6. The second note now is only 1/5 higher than the first one. This interval is still labelled a third, but a minor third in contrast to the major third.
A minor triad still uses a perfect fifth as outer interval, but the middle note is a minor instead of a major third. The sound is flatter, duller, not as bright, not as clear. It still is a triad, but not the 'real' one – or is it?
The Minor Secret?
The major triad will trump the minor one, because it is more defined and natural due to its simple frequency ratio of 4:5:6. The tuning of a just minor triad is 10:12:15. Looking at this, we could hardly expect a harmonic sound, yet this triad does sound nice. We can use it even as the final chord.
An analysis of the overtones will get us not very far. The secret of the minor mode is its relation to the major one.
A String Of Minor Triads
Let us remember that the major scale can be considered a string of three triads with the tonic triad in the middle. With C as tonic center, we get the white-key scale, providing us major triads if we start with either C, the tonic, or F or G – which are just a fifth away from the tonic.
But if we start with D, E or A, and still use only white-key notes, we get minor triads. Those are: D-F-A, E-G-A and A-C-E. That means, the major scale offers as many minor triads as major ones, namely three of each type.
We can create an entire minor scale from minor instead of major triads: Beginning with D-minor, then A-minor in the middle, followed by E-minor. We now have a minor scale, still using only white-key notes, but instead of C, the note A is the tonic now.
As the major scale also contains three minor triads, the minor scale conversely contains thee major triads.
Relative Scales And Triads
C-major and A-minor are using the same notes, just in a different order. A-minor begins with A and then goes B-C-D-E-F-G, while C-major begins with C and then goes D-E-F-G-A-B. If you play a scale on an instrument, you will most probably play the eighth note, too, which is the first one but an octave higher.
Using the same notes for its scale, A-minor has a close relationship to C-major. Neither scale uses musical accidentals (sharps or flats), so they use the same key of no accidentals at all and can be played with using only white-key notes. Both keys are therefore relative keys.
If that is true for an entire scale, it should also be true for a triad. And it is. The A-minor triad is relative to the C-major triad, and vice versa. This works for other chords, too. We will have a closer look on it when we get to chords by scale degrees.
Major And Minor – The Dualism
A minor and and a major third complement each other to a perfect fifth, which is the outer interval of a triad anyway. That means, each triad includes both types of third intervals! But the inner interval determines the sound.
![[image loading]](http://i.imgur.com/dixqcTx.png)
A-minor and C-major have two notes in common, namely C and E. That is one reason of the similar sound of both chords. But what about E-minor? That triad – E-G-B – also shares two notes with the C-major chord.
Now the other reason for the relativitiy between triads comes into play: The E-minor scale is different compared to C-major. Since a scale consists of just seven notes, a triad with three notes covers almost half a scale. We perceive triads as smaller representatives of an entire scale. The E-minor triad is not as close to C-major as A-minor, because the E-minor scale is different. In contrast, A-minor uses the same notes as C-major, just in a different order.
In just tuning, A-minor uses slightly different pitches than C-major. To fully use the major/minor tonality dualism, we have to compromize interval purity. But first things first: Why do we only have only two scales, when we have seven notes?
The next blog goes into the history of modern scales and their tuning
