Harmonic Structures in a Diatonic Context
The main objective of this article is to present a symbolic system for expression of chord structures within diatonic context. The purpose of such a system is to be a framework for perceiving and understanding of harmonic features in musical pieces. As well some basic properties of these harmonic structures are mentioned here. But it is just for illustration and better understanding, not an exhaustive description of them .
Diatonic 7-tone scale lies on twelf-step equaly tempered scale and consists of half and whole-step intervals (semitones and tones) in a sequence: step, step, half-step, step, step, step, half-step (the name diatonic comes from the fact that it consists just of the two types of intervals).
Before we talk about diatonic scale(s) more detaily, we should clarify intervals - distances between two tones (all music rests on intervals. When more tones sound simultaneously, the impression we get from it primarily comes from their interferences/intervals). Within one octave we recognize following intervals:
Unison - 1 (note that it's size is 0, but naming convention is 1)
Minor second - m2
Major second - M2
Minor third - m3
Major third - M3
Perfect fourth - P4
Tritone - TT (T)
Perfect fifth - P5
Minor sixth - m6
Major sixth - M6
Minor seventh - m7
Major seventh - M7
Octave - 8
It is good to know that they form complementary pairs filling octave: 1-8, m2-M7, M2-m7, m3-M6, M3-m6, P4-P5, T-T.
Back to diatonic scale topic. We know that in C major (diatonic) scale the tones are C D E F G A B (C). As well we know there are twelve musicaly equivalent major scales (keys): C, G, D, A, E, B, F#(=Gb), F, Bb, Eb, Ab, Db. They all are instances of the same scale, they are just mutualy shifted/transposed. All that applies to one of them, applies to all others. It is both good and bad news. The good aspect is that theoretically it is a simplification, the bad side is that for us as pianists, each such scale is completely different for playing (consists of different set of white and black keys) what brings practical difficulties.
For each such a major scale there are 7 modes: ionian, dorian, phrygian, lydian, mixolydian, aeolian, locrian. They are just scales consisting of the same tones, but starting on different key / root. For C major scale Ionian mode starts on C (so is equivalent with C maj scale itself), dorian on D, phrygian on E etc. There is no reason to be concerned much about the modes. They naturaly apply in songs that are based on diatonic scales (more modes apply in one song in time) what makes such songs sound interesting although they are laid on one simple scale. Explicitly it is necessary to be aware just of two modes - ionian and aeolian. Ionian is equivalent to it's owner major scale, aeolian is it's related minor scale (we as players know, that C major and A minor scales have the same signature and the same set of tones, G maj and E min too etc., they just have different root - starting key).
Now let's start talking about harmony - simultaneously sounding tones. Harmony is usualy described by chords. But let us be aware, that chords are not a dogm. They are not immovable construction units. It is better to perceive them more like elementary musical structures through and around which the music flows.
Elementary harmony unit is build on thirds (major and minor) going through diatonic scale. Such a unit containing three tones is called triad. In each diatonic scale there are 7 (6+1) basic and most important triads - chords. Let's present them for C maj scale (what analogously applies to all the other 11 scales):
1 : C E G
2 : D F A
3 : E G B
4 : F A C
5 : G B D
6 : A C E
(7 : B D F)
(for each of 12 diatonic scales - each with 7 modes - there are just these 6(+1) spinal chords, that are the most basic, the most important and the most probable/frequent ones, no matter what mode the song is written in)
There are 3 major chords, 3 minor chords, chord 7 is diminished (see later). Chord 7 doesn't contain P5 interval (very important interval - see later in the article of mathematical rudiments of music), this chord is quite sound inert. Although it is a diatonic chord, it has no strong diatonic function and rarely is used in it's pure form. It rather appears with conjunction with 5-th chord what creates 5-th four-note dominant chord G B D F (what strengthens it's tendency to go to root-tonic) or with conjunction with 2-th chord (what creates inverted sixth chord - 26 - with tone 7 as a bass). For these reasons we will perceive this chord a bit aside of the other 6.
So we have 6 (main) chords. Let's look at them. 3 of them are major (consist of intervals M3+m3), 3 are minor (m3+M3). Chords 4 5 1 are major, 2 3 6 are minor. Notice that these two sets are positioned on the same intervals between them (their roots) - the M2 (4-5 and 2-3) and the P4 (1-4, 5-1 and 6-2, 3-6). We say that major scale chord 1 (called tonic) has it's related dominant (5) and predominant (4) chords as well as minor scale root chord 6 has it's dominant (3) and predominant (2).
(here appears an importancy of fourth interval upwards (see Intervals in Diatonic Scale). It is used a lot in music. Especially in relation with predominant and dominant there is function root->predominant as a step 'from home away'. Step dominant->root is usual 'returning home' pair)
To summarize this part, we have 7 basic diatonic chords/triads defined along with their symbols: 1, 2, 3, 4, 5, 6, 7.
We met triads, other important basic structures are four-note chords (seventh chords). Let's present them for C major scale:
1 : C E G B
2 : D F A C
3 : E G B D
4 : F A C E
5 : G B D F
6 : A C E G
(7 : B D F A - N/A - in fact is D6/B, in our notation 26/7 chord)
For four-note chords we will use symbols with added 7: 17, 27, 37, 47, 57, 67
It is important to realize and know that quality of each such chord in a diatonic scale differs:
1 - major chord
2 - minor chord
3 - minor chord
4 - major chord
5 - major chord
6 - minor chord
For seventh/four-note chords:
17 - major major (highest interval is M3)
27 - minor dominant (highest interval is m3)
37 - minor dominant
47 - major major
57 - major dominant (dominant seventh chord)
67 - minor dominant
It comes from the fact that these chords are natural diatonic chords (what makes them easy to identify and play for us as pianists).
As mentioned already, all these diatonic chords lie on one diatonic circle of thirds (major and minor):
Now more generaly again - we should be aware there are chord inversions - first, second (and third for seventh chords). Important is to know that chords appear not only in their basic form (close harmony) - like CEG for major triad, but in open position too - CGE. The quality of the chord is given by the set of the keys in it, not by their order. The inversion plays bigger role when the lowest chord tone is further from it's closest higher mate than in close harmony. Then such a bass tone sounds distinctly.
As well we should be aware that there are incomplete occurences of chords when only 2 of 3 keys or 3 of 4 are used. Even 2 of 4 variants can appear.
Up till now we have been talking exclusively about basic 7 diatonic tones. There 5 more. We will refer them as 2/3, 3/4, TT(or T), 5/6, 6/7. We know that in musical notation we get these tones when sharpening (#) or flattening (b) are applied. Because of this ambiguity, for individual tones (not for chords) we will use simplified symbolism - neutral slash x/(x+1) notation. TT is for tritone - tone positioned in the middle of octave - between 4 and 5 (name tritone comes from the fact that it is ending tone of three whole-tone steps from the root).
Chord notation: Chord symbols are commonly known and could be useful for explaining some harmonic features, but have drawbacks. First, they define chords as independent elements, not within wider/diatonic context. Second, their naming is not 100 % unique and unambiguous. And for more compliacated chords their name doesn't imply the quality of the chord quite intuitively. Definitely chord symbols are good for guitar players (and generaly are good for instant playing and improvising). But here we will use a different convention for identifying chords, primarily based on diatonic numbers:
1 - triad laid on root of diatonic scale - thus major triad
1(-3) - the same as above, but with flattened inner third (3 is relative to 1 here), thus minor triad
3 - triad laid on the third of diatonic scale - thus minor triad
3(+3) - triad laid on the third of diatonic scale with sharpened inner third - thus major triad
17 - same as 1, but with added 7-th note, thus Xmaj7 chord
17(-7) - same as previous, but with flattened 7-th, thus X7 (Xdom7) chord
14sus - triad with 4 instead of middle tone 3
12sus - dtto with 2 instead of 3
etc. (we will extend this list when needed)
Note, that in each chord first number is it's position within diatonic context (position of it's root). The other numbers within the chord symbol are relative positions to that root.
Regular scales and chords: All the chords mentioned are irregular/diatonic. They consist of intervals M3 and m3. As well diatonic scale itself is irregular, it consits of M2 and m2 intervals. But there are regular structures that are not used so often, but they appear in compositions, have specific character and meaning and thus are important too. Let's summarize them.
Chromatic scale is a scale of 12 m2 intervals. It contains all the existing keys.
Whole-tone scale is a scale of 6 M2 intervals. In fact there are just two 'disjunct' whole-tone scales.
Diminished chords are very useful and important structures, they consist purely of m3 intervals. If we consider them as scales, there are just three such 'disjunct' scales existing. Practically diminished chords appear in form of triads or four-note chords. For diminished chords we will use a specific and simple notation. The first number (only) will be diatonic position of first dim chord key. The rest tones implicitly follow m3 interval form:
1dim - is in fact diatonic 1(-3-5) (what would be too complicated enrollment)
17dim - in fact 16(-3-5) (7 here is the fourth note of dim chord, not 7 of a diatonic chord)
1dim(+1) - in fact 1(+1)
17dim(+1) - in fact 17(+1-7)
2dim - in fact 2(-5)
27dim - in fact 26(-5)
etc.
A bit more about diminished chords now. Technicaly diminished chords are chords composed of m3 intervals. Functionaly diminished chords are chords diminishing a diatonic chord (in a certain context, and thus create a chromatic step). These two definitions are not entirely inline. Here we prefer the functional perspective (and thus we don't consider chord 7 as a standard diminished chord). Diminished chords can't contain transitions on their 3 and 5 positions, it would change their diminished character and function, but on the 7th position, there can be a transition in a specific chromatic context - like e.g. 67dim(+7), although contains M3, still works as diminished chord (diminishes chord 6).
Augmented chords are triads based on M3 intervals. They occur in triad versions only (not in four-note form). We will use standard diatonic notation for them:
1(+5)
2(+3+5)
etc.
(but notation 1aug, 2aug is possible)
Tritone is last regular structure. We can't consider it neither scale nor chord. It is just an interval. There are just two tritone intervals to a scale. Tritone itself plays distinctive role in music too (e.g. dominant seventh chord contains charasteristic tritone interval).
Perfect fourth interval triad is not such an ideal regular structure as all the above because one octave is not divisible by P4 intervals. Even it is not being considered as a chord.
That is roughly all we need to know to start studying composition structures. But to make our know-how even more complete, we recommend to go through Intervals in Diatonic Scale mini-course.
Examples of analysed pieces can be found here: