EPT![[image loading]](http://i.imgur.com/38RNIsQ.png)
The Acoustic Sensation
How we experience a piece of music depends on our personal experience which in turn is influenced by the culture in which we were raised.
The genetic code of all humans is very similar to each other. This means we can expect to have very similar brain 'hardware'. But there is more to it. The universe didn't have much choice because harmony depends on very simple concepts which are above physical laws.
Our human hearing sense has its root in fishes who already had a lateral line some umpteen million generations ago. Sensory cells with small hairs upon them – to amplify the sensation with the leverage effect – could sense pressure variations in the water.
As some fish developed lungs (a mutated part of the gut), getting a second source of oxygen, they were able to survive in ponds which dried out over the summer. Thick fins became primitive legs to crawl on the ground. While the gills were lost over the time, a part of the lateral line evolved into an organ we still have today – the ear.
Even the human ear still has sensory cells with small hairs upon them to amplify the vibrations. The faster they vibrate, the higher the pitch we perceive. The higher the amplitude in which they swing, the higher volume we perceive.
Natural Characteristics Of Sound
When a force strikes an item, it begins to vibrate. Depending on the material and shape of the items, it can generate an audible sound. This happens when the object transmits energy to the surrounding air, which in turn begins to vibrate. The density of the air increases and decreases, spreading sound energy to surrounding air and other objects nearby. We humans can perceive sound from about 20 of such vibrations per second to about 15000 of such vibrations. The physical unit is Hertz. 1 Hertz = 1 combined increase and decrease over one second.
The actual range of pitch we can hear depends on the age. We are losing sensitivity in the high pitch range first. Most humans quite sensitive at about 1000 Hertz regardless of age.
Almost any sound generated by a natural object consists of multiple pitches at the same time. Let's image a string which has a length and a strain applied on it to resonate with 100 Hertz when it is struck.
At first, the string needs some time to build up a clear sound. This can be described as the "attack phase", where the volume gets higher very quickly. At this time – often just a fraction of a second – no clear pitch can be discerned. But this phase has a great impact on how we perceive the overall sound.
However, for this blog we don't pay attention to the attack phase because we are interested in the sound once the string vibration built up.
We get a high volume sound at 100 Hertz, but also some volumes at integer multiples like 200 Hertz, 300, 400, 500, 600 and so on.
The Sound Processor In Our Head
In reality, the so called overtones are not exactly at those multiples, but often close enough. A certain stiffness of the string and other non-ideal circumstances cause deviations from the mathematical ideal. Still, those integer multipliers are the core of harmonics!
Even though the string propagates many different pitches, we just hear the base pitch of 100 Hertz. This is an advantage because all those tones are generated by just one object. Because of audio processing in our brain, which aligns those integer pitch multipliers to a single source, we perceive only one object emitting sound. The pattern of the volumes of those overtones in relation to the base pitch is perceived as tone color, as timbre.
A dull sound doesn't have high-frequency overtones with high volumes, whereas a screeching sound does. A common piano has more volume on even-number multipliers than on odd-number multipliers.
The perceived base pitch of a church bell is not even a part of the actual pitch spectrum, but reconstructed by our brain! Also due to its three-dimensional shape, which generates complex interactions, the overtone multipliers of a bell are far from being ideal. This is why the bell sound stands out compared to other, more naturally behaving musical instruments.
![[image loading]](http://i.imgur.com/Tvc0SEZ.png)
The Harmonic Series
What is true for a single sound, is also true for the musical scale of nature. Consider a column of air inside a flute or trumpet. You blow and apply pressure, causing the air to vibrate back and forth. Depending on the length of the instrument – the length of the air column – the inside air vibrates in resonance, giving you a particular pitch. Lets say you get 100 Hertz.
If you increase the pressure by blowing harder, you compress the waves until the entire column vibrates twice as fast. So you get 200 Hertz. Of course, each of those tones comes with its own overtones. But now we are exploring the base pitches.
The next higher tone you are able to play will be 300 Hertz, followed by a pitch of 400 Hertz and so on. The reason is simple: integer numbers of full wave lengths fit perfectly into the air column. As with the string, we are using an idealized model here. Actual brass instruments deviate more or less from the harmonic series.
Many brass instruments offer valves to change the lengths of the air column. This enables the musician to play more different notes using different natural tones combined with different valve positions.
Naming Notes: We Begin With The Letter A
Something is harmonic when it fits together well. If we play a note at a particular pitch, and another note with the pitch doubled, we also double the pitch of any overtone for that note. That means, we are not adding overtones to the lower-pitch sound! We just amplify every second one. That means we are just changing the tone color. These two notes match very well and are therefore very harmonic.
The interval of doubling or halving a pitch is called octave. Because setting a pitch into another octave doesn't change the harmonic relationship with other pitches very much, a musical scale needs to span just one octave and then can be repeated.
By an arbitrary definition, the note A in the octave number 4 is defined as a pitch of 440 Hertz. That means, A4 = 440 Hz. A5, one octave higher, has the frequency doubled, resulting in 880 Hz. A3, one octave lower, has the halved frequency, 220 Hertz.
Simple Intervals: Foundation of Harmony
The interval of factor 2 is too simple to generate a new note because we just change the octave of the same note. The next simplest ratio would be 2:3. That means, the second tone is by factor 1.5 higher than the first one. It is also common to express this ratio as 3:2, as 3/2 = 1.5, the factor we need. This blog however uses the order of "lower pitch : higher pitch", therefore 2:3.
This interval is called perfect fifth. The fifth is of great importance, because we can use it to calculate the pitch of the best fitting notes to any given note. Let's actually do it.
We have the note A4 and want to know the most harmonic pitch above A4. We calculate 440 Hertz * 1.5 = 660 Hertz. Now we want to calculate the most harmonic pitch below A4. We calculate 440 Hertz / 1.5 = 293.333... Hertz.
Both new notes (660 Hz or 293.333 Hz) have a strong relation to 440 Hz as both are just a fifth away.
Seven-Note Scales
Western music scales consists of seven notes and use the letters A through G. The history is quite complex, with the ancient Greeks using two symmetric 4-note-chords to form a 7-note scale. In the middle ages in Europe, the names of the Greek scales were mixed up. This blog will use the names assigned in the middle ages though, because they are used for the church modes and still used in jazz music today.
While the major and minor scales, which are widely used today, are not part of the authentic church modes, they fit into the same system. We will have a closer look at this later, once we know the major and minor scale.
The Starting Point For Our Major Scale
This blog will focus on the inner order of the major scale rather than its history. Also modern-named notes are used.
The major scale can be considered a series of perfect fifths. If we start with the first note and stack just four of the perfect fifths on top, we end up with a scale consisting of five notes. The difference between a string of fifths and a scale is that we push a note down an octave when required to fit all notes into the same octave.
That scale is called the pentatonic major scale, because it consist of five notes. Pentatonic scales are common throughout human history all over the world. The exact pitches differ, but the core idea is the stacking of four fifths on the first note – that interval is so harmonic, that it comes naturally to us. Of course, the very ancient Greeks used the five-note scale, too.
A Complete Major Scale
A seven-note scale requires some more thought and actual acoustic experiments, like some more recent ancient Greek mathematicians have done. That was roughly two and a half millenniums ago. They noticed the one-and-a-half-tone-steps between certain notes. The Greeks did it differently, but lets continue stacking fifths: From the last fifth we used to get the pentatonic scale, let's stack another fifth on top.
And from the from the first note – in our example, the note C – let's go a fifth backward. We now have seven notes in total. The white-key notes to be more precise, ordered by pitch: C-D-E-F-G-A-B.
If you want to do it on a piano, start with C and go five fifths upward, to G, D, A, E and B. Also one fifth downward from C, to F. You have now played all seven notes of the C-major scale.
Major Triads: Natural Sound
Remember the harmonic series? It begins with 1:2:3:4:5:6. Lets only consider the last part, 4:5:6. The factor 4 can replace the '1' and the '2' part of the series, because they represent the same note, just one or two octaves below. The factor 6 replaces the '3' part, because factor 6 is just one octave higher than 3. Remember: An octave shift means to multiply or divide by factor 2, and 6 = 3 * 2 (where '* 2' is the octave shift part of the caluclation.)
A chord with notes in the pitch ratio 4:5:6 is a major triad. Since it is also part of the harmonic series, any sound contains it own major triad. That is why a major triad sounds to natural, because it appears in (almost) any single sound itself!
The pitch between the first and the last note of a triad spans a perfect fifth. Since the perfect fifth delivers the most harmonic pitches to a given note, it should also provide the best fitting triads to a given chord.
A Triad Cadence
Back to our C-Major scale. The C-Major triad is C-E-G. The perfect fifth of C is G. So let's add a G-Major triad. It is G-B-D.
If we lower the pitch of C by a fifth, we get to F. F itself is also the fourth note in the major scale, so the interval from C up to F – instead of down to F – is called the perfect fourth. Let's see how the F-Major triad looks. It's F-A-C.
So all triads connected together gets us F-A-C-E-G-B-D. Again we have the full seven notes of the scale, but now explained with three triads. The root triad, the tonic triad, is in the middle. The two other triads are in strong relationship to the tonic center – because they are a fifth away from the tonic and therefore in a simple, easy to recognize relationship to the tonic.
To give us a complete picture of the C-Major scale, we can play an F-major chord, a G-major chord and a C-major chord. This chord progression is also called the authentic cadence. It has other names, too. The ability to play this authentic cadence with using only notes which are in the scale itself, was one of the reasons why the major scale went so successful.
Next blog: The minor chord and the minor scale.
Yea you are correct on that, D4 would be 293.332 and E5 would be 660 in JI.
