1: The harmonic series.

Music is an art, but it isn’t artificial. The reason melodies and harmonies sound sweet or harsh is completely scientific. For this reason, different human cultures independently developed similar musical systems.

The basis of melody and harmony is the ‘harmonic series’. When a vibrating body with a musical tone (like the string on a guitar or the column of air in a trumpet) vibrates, it does so ‘harmonically’. It vibrates along its whole length, but it also divides and vibrates with harmonic overtones.

To use an analogy of a guitar string…. The most important harmonic is the ‘fundamental’, the vibration of the whole length of the string. The string also vibrates in two halves, this is called the ‘first harmonic’; and in three thirds, the ‘second harmonic’ and so on. These vibrations all occur simultaneously.

The first 6 tones – the fundamental and the first 5 harmonics – are the most important. And it’s from these tones that the notes and harmonies of Western music, and most of the world’s musical systems, are derived. You will see that (in the key of C, for example) there are 3 Cs, 2 Gs and 1 E.

The notes of the scale.

The harmonic series gives us the notes C (‘tonic’ or root note, which appears at the fundamental and octave), G (the ‘fifth’) and E (the major third).

Transposed down to within the range of a single octave, G (produced by the string vibrating in 3 parts) has the ratio of 3/2 and E the ratio of (the string vibrating in 5 parts) the ratio of 5/4.

Placed on a diagram showing the equal semi-tones of the western chromatic scale, you notice that the pure harmonically derived notes are not an exact match to their ‘equal tempered’ notes used in modern Western music.

The ‘Blue Note’.

I’ve also included the 6th harmonic (7/8 the string in 7 equal parts). This note is quite flat by Western standards, and wasn’t included in the theoretical systems of Europe, India or China, but that tone, and notes derived from it are used all the time in music – it’s the ‘blue note’, the very beautiful and rather flat minor seventh or minor third used in jazz and blues.

The ‘just’ scale.

Simple extrapolations from these primary tones give us all the notes of the major scale. A note is included midway between the tonic (C) and third (E): this is produced by playing a fifth on top of a fifth 3/2 x 3/2 = 9/4 (transposed down to 9/8). A fifth below the tonic gives us the fourth (4/3, or 3/2 inverted). A major third above the fourth gives us the major sixth (4/3 x 5/4 = 20/12 -> 5/3), and a major third above the fifth gives us the major seventh (3/2 x 5/4 = 15/8).

Now we have all the notes of the major scale, plus the three major chords of the scale (C, G and F), all derived simply from the mathematical ratios of the harmonics. It’s not surprising that the major cultures of the world all discovered this scale independently.

The major third sits naturally between the tonic and the fifth, if you invert this interval, and extend the major third below the fifth, you get the minor third (3/2 x 4/5 = 6/5), and extending this interval from the fourth and fifth gives the minor sixth (4/3 x 6/5 = 8/5) and minor seventh (3/2 x 6/5 = 9/5).

Here are some other intervals. The semitone, or minor second (an inversion of the major seventh, although there are other ways to derive this interval that are slightly different); the blue notes (the blue third is fifth below the pure blue seventh: 7/8 x 2/3 = 7/12), and the ‘pythagorian third’. The pythagorian third is interesting – it’s somewhat dissonant when played in a chord, but sounds sweet and bright when played alone in a melody, and is the addition of two major seconds (9/8 x 9/8 = 81/64).

Have a look at where the blue note is. It’s a very natural thing, and sounds sad and pretty, to play a minor third or minor seventh about a third of a semitone flat. Singers often do this naturally without realising they are singing outside the traditional scale.

The modes.

Although the minor scale can be derived harmonically as above, there’s an easier way to derive it, you simply play the major scale starting at a different note! For example, the most famous and popular of the modes, the Dorian mode or ‘D’ mode, is simply the same notes as the major scale, played from the second (or D in the key of C).

And by the same process, the other modes can be derived. Note that in modal music, the original names for the major and minor scales were the Ionian mode and the Aeolian mode.

I’ve highlighted the tonic and fifth in these diagrams to show how the tones and semitones relate to these, the most important, notes.

In theory, there’s also another mode you could play, starting on ‘B’. It does have a name, it’s called the ‘Locrian’ mode, but because it doesn’t have a natural fifth (instead it has a flattened or ‘diminished’ fifth), it lacks a strong tonal centre and is rarely used, so I didn’t include it.

  • Ionian Mode (C)

    The same as the major scale. In C, it uses the I, IV, V chords C, F, G

  • Dorian Mode (D)

    In medieval times, this was the most popular mode, and it is now very popular again, being the basis of the blues scale, and is the most-used scale in modal jazz and jazz-fusion. It uses the I, IV, V chords Dm, G and Am

  • Aeolian mode (A)

    The natural minor scale. With all minor chords, it uses the I, IV, V chords Am, Dm and Em.

  • Lydian mode (F)

    Hugely popular in Indian music, like the major scale but with a sharp IV, which acts as a ‘leading tone’ to the fifth which gives it a brightness and playfulness jazz musicians also love. It doesn’t have a IV chord, but uses the I, V chords C and G.

  • Phrygian mode (E)

    An all-minor mode with a flat second, this popular mode has an exotic Middle-Eastern or Eastern European sound to Western ears. The flat second interferes with the V chord, so it doesn’t have a simple V chord, but uses the I, IV chords Em and Am.

  • Mixolydian mode (G)

    Despite its odd name, this is a common mode in modern music. It’s like the major mode with a minor seventh and uses the I, IV, V chords G, C and Dm.

Counterpoint and Harmony

The tonic-fifth relationship is a naturally occurring one, the most important in all music and one which exists in all the musics of the world. All music to some degree makes use the feeling of psychological resolution of melody and harmony away from and then back to the tonic or ‘home’ key. A multitude of songs and pieces from all cultures and eras begin in their home key, then move around somewhat, before returning, with a feeling of finality and resolution, to their home key at the end. There’s nothing unusual about this.

What we call ‘harmony’, in the medieval modal period was referred as ‘polyphony’. They saw their music more as multiple melodies playing together and, while aware of the chords, were not as concerned with chordal-based harmony.

But the resolution or ‘cadence’ of certain chords was important; in particular the V-I or ‘perfect cadence’. In the most emphatic form of a perfect cadence, the fifth will fall to tonic (V-I or g -> c) and the major third of the V chord will rise a semitone to the tonic (b -> c). Composers came to feel these notes, in particular the major seventh (‘b’), which is not normally present in the minor modes, because it was so important in cadence, was a necessary part of minor scales.

The other common cadence is the IV-I or Plagal cadence. This is the ‘amen’ cadence in church music.

With this new emphasis on cadence, the medieval modes were replaced with the major and minor scales. In fact the major scale is simply the Ionian mode, but the minor scale became something very different.

The minor scale

The need to incorporate the major seventh into the minor scale led to a curious hybrid scale. In Classical music musicians practice two minor scales, the harmonic minor – the scale which outlines the harmonies of the minor scale with a V-major / V-7 chord, and the melodic minor, which has a different descending and ascending form.

With these departures, Western music was becoming less tied to a particular key. Composers were experimenting with using cadence to move or ‘modulate’ from one key to another (at first usually from I to V and back again). The scales of music were becoming more flexible and chromatic.

Resolution and Tonality

J.S. Bach was famous for his masterful modulations within small-scale pieces. However, Western music, in the time of C.P.E. Bach (one of sons of J.S. Bach), developed a new compositional technique based on moving a piece of music away from its home key and then resolving it back, in a very specific, formal way. C.P.E. Bach is actually credited with inventing this; it’s sometimes called ‘tonalism’ or ‘tonality’ or ‘Classical harmony’, and it underpins the Sonata and Symphony forms in Classical music.

These large-scale musical forms depend upon a sense of key (or ‘tonal centre). Compositions are given a strong sense of structure or ‘form’ (hence the term ‘Classical’ – like the structure of Classical Greek architecture), by moving themes away from the tonal centre and then back again. This is why parts of Sonatas and Symphonies are called ‘Movements’.

The use of tonality was hugely exciting development in Western music. It added a completely new dimension of semantic meaning to phrases and sections.

Naturally enough, Mozart was a wonderful exponent; and it isn’t just on a large scale – from movement to movement – that he makes use of it. On the smaller scale, of musical phrasing, he plays with our sense of key very subtly and expertly. In this way he adds a new level of context to his phrasing. Melodies are not just harmonised for colour, they take on a kind of abstract meaning depending on their relationship to the tonal centre. The effect is difficult to describe, but very powerful to the listener.

Someone might say of Mozart, ‘it feels so balanced’ – what they’re experiencing is the feeling of the balance and tensions of the melodic phrases moving around the tonal centre.

‘Two-Five-One’

The usual pattern in early Sonata form (prior to Beethoven who began an era of much more adventurous use of tonality) was to move from I to V and back to I again. The resolution from V to I is a natural one; but in order to rest on the V for a while, the music must first cadence or resolve to it. So cadence will begin on the fifth of the fifth, which is II, and cadence to the V, then from the V cadence to the I.

From that we get the II-V-I (‘two-five-one’) progression.

This became a very important progression in Jazz. If you have a look at the long chorus in the Jazz standard ‘That Old Devil Called Love’ you can see that, as well as a lot of chromaticism, it’s full of II-V-I progressions based on different tonal centres.

Fm-Bb-Eb (II-V-I to Eb) … Cm7-Fm-Bb (II-V-I to Bb) … D7-(Ab in passing)-Gm7-C7 (II-V-I to C) … C7-F9-Bb7 (II-V-I to Bb) …

The Circle of Fifths

It’s possible to extend the II-V-I further, starting at a fifth above each chord and resolving, so you you get VI-II-V-I, or III-VI-II-V-I … or further: in theory you can go all the way round the ‘circle of fifths’ and back to the starting point. Although few songs extend that far, Classic Jazz is full of such segments of the Circle of Fifths.

The cadencing of the resolving fifths gives a satisfying feeling of forward movement, and the changes in key (or ‘tonality’) give the player melodic material to work from.

Modern modalism

During the late 19th and early 20th Centuries, adventurous Classical composers felt they had extended tonalism as far as it would go; some began to use new techniques of ‘atonalism’, but others began to explore earlier music, including medieval modalism.

In avante-garde Jazz during the 60s, the II-V-I pattern common to Jazz standards started to many musicians to feel like a cliche – the same chord patterns again and again, the same repeating key changes like gear changes in a car. It was predictable and boring. It was probably in part the influence of 20th Century classical music, and also partly the influence of blues and rock, which have always naturally used a lot of Dorian mode; but Jazz musicians during this time also returned to modalism as a freer, simpler, spacier basis for their compositions.

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