Music: Chords, an introduction

Some time ago, I wrote some blogs about scales and intervals. We now come to the good stuff, to chords.

+ Show Spoiler [The blog series about intervals and sc…] +

Part 1: Introduction and the major scale
Part 2: More about intervals, and the minor scale
Part 3: The history of modal scales
Part 4: Twelve notes for all scales

What This Is About

I recommend to read the other blog series first, if you haven't done it yet. As the four blog posts in the spoiler cover the basics, I will now go quite deep. Chords are much more complex than single notes or intervals, but the foundation of harmony is simple. The first half of the blog will show how the harmonic series arises from nature and ultimately mathematics. Very simple mathematics.

Creating A Sound

Let us play just a single note. We begin with putting mechanical energy into the musical instrument. We strike a guitar string with our hand, or pressure an air column by blowing air out of the mouth into the recorder, or do something else: In the end, we transport energy from our body to the instrument.

At first, the energy will create random frequencies, so no clear note can be heard. But the instrument is configured to play a certain pitch. This is done by having a particular length of the air column, if we using a wind instrument, or by a certain string length and strain when we play a string instrument.

To keep the math clear, we assume the instrument will be configured to create a sound at 100 Hertz. Let us think of a string. It can vibrate, meaning its middle is swinging upwards and downwards while both ends are fixed of course. Depending on the material and tension, a particular frequency can be achieved, having both ends fixed and the middle vibrating. The vibration periodicity implies a certain wave cycle length: The string cannot vibrate in way that a quarter of a wave is at an end point, because the string is fixed there.

That means, all other frequencies beside the one resonance pitch will cancel each other out very quickly. Or will they?

Where Overtones Come From

The air which surrounds the string will be compressed and dilated, in our example 100 times a second. That means, in 1/100 of a second, the air gets compressed and dilated. Then the pattern of compressing and expanding air repeats. This wave covers a single cycle, which represents 1/100 second:

This is not a picture of the string shape, but of the waveform of the surrounding air density in generates in 1/100 second. The string could also vibrate with the doubled frequency, because the according waveform fully fits the original one. Lets superimpose both waveforms, each covering 1/100 second:

We also could add a pitch three times as high:

The string of our instrument actually does vibrate in those frequencies, too. This is because these pitches are also in resonance with the instrument configuration. Now what do we hear?

Sound Color

Let us remember the waveform of 100 Hertz, for 1/100 second:

Now a waveform of 200 Hertz, also over 1/100 second. It results in this shape:

The combined shape looks like this:

The shape is different, but still repeats every 1/100 second. This remains true if we add waves with pitches with 300 Hertz, 400, 500 and so on. All those additional pitches change the combined waveform and therefore sound color. But we keep the wave pattern frequency of 100 times per second = 100 Hertz. So the perceived pitch does not change when we add these so-called overtones.

Harmony Without Overtones

We now know where overtones come from: The resonance frequency allows to vibrate in integer multipliers of the resonance, too, and some energy which we put into the instrument to generate a sound gets into those overtones. As the waveform shape does change depending on the volume of each overtone pitch, the sound color changes. It can be bright, dull, mellow, shrieking and so on. This enables different instruments to sound different even when they play the same note.

We can also derive harmonic properties from the spectrum of overtones: An octave interval, which represents a doubled pitch, creates no new overtones, it just amplifies every second one of the lower note. Therefore an octave doesn't even create a new note.

The fifth interval, which represents a pitch 1.5 times as high as the lower note, creates as few new overtones as possible. So it does create another note, but a very fitting, a very harmonic one.

But would the fifth interval of factor 1.5 still be harmonic when we have tones which only consists of their base pitch and no overtones? Yes, because this pitch ratio provides the smallest possible combined wavelength pattern of two different pitches beside the octave. So even if we have instruments which do not generate any overtone spectrum at all, the rules of harmony still apply.

Harmony Is Simple

If we use the expression lower:higher pitch, the 1:2 ratio represents the the octave. This interval is so simple that it doesn't get us a new note, we get the same note in another octave. 2:3 is the next best harmonic interval, creating a new and very harmonic note. The interval is called fifth. Octave and fifth form the start of the harmonic series as 1:2:3. It continues of course with :4:5:6 and so on.

It is simple math, using ratios of small integer numbers. Numbers like pi or the golden ratio do not play a role in the foundations of musical harmony.

In reality, the overtones are not at exact integer multiples. The actual overtone spectrum looks much more complicated than in our idealized model. But to understand the basics of harmony, we can use our simplification.

Music Abstracts Overtones

A note with a pitch three times as high as another note is a different note, even though no new overtones are created as in the octave. This is because the aforementioned 1.5x pitch factor does create a new note, and 3x in pitch is 1.5x – creating a new note – and an octave of factor 2.

We cannot use overtone characteristics blindly to create a theory of music, we also have to pay attention how we actually experience music. An interval where the higher note is 3x as high in pitch as the lower still is extremely harmonic. But intervals that wide are used quite seldom, because to recognize intervals as such, the notes should not differ too much in pitch. Our brains assumes a connection based upon similarity. While playing two notes at the same time does hint to a causality, a pitch difference too high is in conflict with the assumption of a connection of two notes.

How Much Off Is Equal Temperament?

When we mention tuning ratios, we will use the just intonation ratio even though most instruments use a slightly different temperament, namely equal temperament. Now complicated math does come into play, but only to approximate simple – the just – intervals with twelve equal steps per octave.

The advantage is that each musical key can be played without having one sounding better, more pure, than the other. Composers can freely modulate through musical keys without having to worry to get from a good-sounding to a bad-sounding key. The disadvantage is that we have no just tuned interval beside the octave.

With equal temperament, the frequency is always less than 1% off compared to just intonation. This appears like a very good approximation, but has to be put in perspective: The difference of a semitone implies a pitch difference of roughly 6%. This means we are up to almost a 1/6th semitone off.

While that could be considered an issue at first, the fifth interval is almost exactly in tune. Only some other intervals like the thirds are more or less off. This is one of the reasons of the success of the equal temperament: The important fifths are for all practical reasons intoned as good as just.

The Major Triad

For the time being, all our chord consideration will be for the natural major scale. Any note example will be for C-major, as this scale can be played with white-key notes only.

To make a sound fuller compared to a single-note sound, one can play additional notes. If the pitches of the additional notes correspond to overtone pitches, we get a harmonic sound.

The major triad fits this approach especially well, as it is tuned as a consecutive part of the harmonic series, which is the same as the overtone series. A major triad is tuned as 4:5:6.

The Tonic Triad

Let's say we play a C-major triad in the octave number 4, which is in the middle of a piano keyboard. The triad would be C4-E4-G4. To play the entire harmonic series up to that point, meaning to play notes in the pitch ratio of 1:2:3:4:5:6, we would play the notes C2-C3-G3-C4-E4-G4.

We get two additional Cs and one additional G in lower octaves. That means, even though the part "1:2:3" is not played by the triad, only the following "4:5:6" part, each note of the first part is still present. We play each note only in the highest octave in which it appears, but we are not leaving any note out.

The tonic triad uses the tonic of the scale as the triad root. The tonic note, which is the first degree of a scale, represents the point of rest, where any tension is resolved. Is that true for the tonic triad, too?

Note-By-Note Analysis

E has a pitch 1/4 higher than C, G has a pitch 1/2 higher than C. Both notes are in an simple mathematical, therefore easily recognizable harmonic relationship to C and stabilize C as the root note. If we try to view E or G as root note, we get much more difficult relationships. With C as root, the relationship between all three pitches is so clear, that we can even put C in another octave and play E-G-C. We even can put E in another octave as well and play G-C-E. Despite the octave shift of some triad parts, we still realize that C is the root.

The overarching triad interval, the perfect fifth, plays a large role here. As it is an extreme harmonic interval, we recognize it instantly and consider the root of the fifth interval as the root note of the entire chord.

Before the major/minor dualism was established, both inverted triads with one or two notes shifted by an octave, were used as standalone chords. Since we are covering the basics only, we continue with our interpretation on major triads, though.

C is the tonic tone itself and creates no tension. E is quite close in pitch to C, creating a rather complex overtone spectrum. But the simple pitch relationship to C avoids dissonance, so we get a consonant color to the sound. G is in an even simpler interval to C. Conclusion: Within the context of the C-major scale, a C-major triad represents ease, rest and peace.

The Fifth Triad For The Tonic

The easiest triad for a scale is the tonic one, using the first degree of the scale as root of the triad.

The three notes of the tonic triad represent almost half of the seven-note scale. The major third interval within the major triad tells us, that the entire scale has a major characteristic.

Which other triad sounds good? As the fifth seems so important, let us play a triad which starts at the fifth of our scale. In C-major, it is the G-major triad. For this one chord, we will perform a very extensive analysis.

The Root Analysis

Let us consider the major triad just a more fancy way to play the root note, and look on the root notes only.

As G is the most harmonic note to C beside other Cs in another octaves, we should expect that a G triad sounds very harmonic to a C triad. Let us assume we don't have any further harmonic context, only these two notes. Because the fifth interval is so simple, we tend to assume that we hear the fifth and not an inverted fourth or another complex relationship of the notes.

That works even if the play the G note below C, so we have a real fourth interval between G and C. If we play C and the lower G in succession and end with G, we would confirm the interval of the two notes as being a fourth. But a fourth is not stable and not as harmonic as the fifth. If we however resolve with C, we are more satisfied as the interval now is confirmed to be a fifth, only inverted.

That means, it doesn't even matter which note is the bass. Provided we don't have any further context, a G note helps to clarify that C is the tonic. Therefore we can expect that a G triad should confirm that the C triad is the tonic.

Two Types Of Scale Analyses

If we use white-key notes only, as in C-major, but use G for our first degree, we will get the Mixolydian scale. This is a modal scale and a major variant. The G-major chord could be viewed as a representative of Mixolydian, a scale with a close relationship to major (or Ionian, which is another name for natural major) as it uses the same notes, only in a different order. Therefore we should expect the G chord quite fitting.

But today, we use major and minor almost exclusively. Let us interpret the G-major triad as a representative of the G-major scale, to avoid other scale modes beside natural major and minor. G-major uses one different note compared to C-major (F♯ instead of F.) So we move a bit away from C-major, but not very much. The one different note of the G-major scale doesn't even appear in the G-major triad. Viewed this way, the G-major chord should be heard as harmonically close to C-major because the corresponding scales are almost the same, they differ in only one note.

The Note-By-Note Analysis

The G-major triad is G-B-D. The note G, as the fifth to C, is the harmonically closest note to C because the fifth is the best harmonic interval. B on the other hand is extremely far from C in harmonic terms. If we count the number of fifths we need to get from C to B, we need more than for any other note within the C-major scale

Also the tuning ratio is the most complicated, no other note is tuned in a more difficult ratio to the first scale degree. But pitch-wise, B is just a semi-tone away from C, indicating a close relationship. In the end, we have a number of effects here: A narrow relationship due to similar pitch, and extreme tension to the tonic center due to the complicated tuning ratio. This makes our ears asking for a resolution to the tonic, where any tension is resolved.

D is a whole tone step away from C and creates a weaker tension to the tonic. D also is the fifth to G and therefore strongly indicates G as the root of the chord, and the root note is the one which carries the meaning of the entire triad. Therefore, D – as fifth of G – mainly it supports the role of G.

Conclusion

If we have established C as the tonal center, a G-major chord is recognized as harmonically close. But the chord also creates quite some tension to the tonal center. Even though much tension is created, the chord itself sounds consonant, because it is a major triad. The tension is not within the chord itself, but in our head as we remember C as the tonic.

It is common to begin the accompaniment of a song with the tonic chord. The dominant is then used to confirm this chord as the tonic.

The Dominant Triad

In C-major, a G-major triad is the dominant. The word can be used as a noun. The dominant has several uses. It makes clear that yes, the tonic is actually the tonic, and it also indicates that the tonic will appear rather soon.

The dominant can be played from the fifth scale degree, but is often played an octave lower. It is still a chord of the fifth scale degree! The dominant is often referred as V. That is a roman numeral, meaning five and indicates the degree.

The tonic triad of a scale is of course the chord on the first degree, and therefore the I.

The V-I chord progression, playing the dominant and then the tonic chord, is also labeled an authentic cadence because the harmonic chord is falling (falling in Latin: cadens) a fifth down to the tonic. There are other types of cadences, too, but not all are authentic.

Next blog: Another chord and its meaning

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