Having defined a chord notation system for use with numerical notation, it is worth looking more closely at some aspects of how it is best used, as well as its potential for broader use than just in conjunction with numerical melody notation.
I have developed Relative Figured Bass (RFB) as a chord notation system for use in conjunction with numerical melody notation as I see “top line and chords” as a highly flexible approach to notation with many general applications and benefits for learning. Bringing this approach to numerical notation and vice versa is a way to achieve two goals: The first is to optimize numerical notation according to its strengths: brevity, looseness (inviting individual interpretation), relational transparency. The second is to bring more flexibility and educational value to “top line and chords” as a notational approach and form.
In the previous article, I covered the first goal. In this one, I will look at ways in which RFB can be used with a melody line — whether numerically or conventionally scored — as an asset to to teaching basic and advanced harmony, sketching out arrangements and experimenting with harmonic ideas.
The principle behind numerical notation is to notate pitches as scale degrees, i.e. as relationships according to an underlying pattern (the diatonic scale), rather than just as specific pitches. RFB extends this same principle from monophony to polyphony in the way it describes chords. The chord, described via its bass root in brackets or a circle, is understood nominally as a triad consisting of the diatonic scale degrees as they stack up in thirds.
Fig. 1. Which ever degree of the diatonic scale we start on, the basic chord type (triad) is built up of three alternate scale degrees, i.e. two third intervals stacked one on the other. Since any interval can be inverted or stretched without changing its harmonic significance (e.g. a minor sixth or major tenth is harmonically equivalent to a major third), any triad can be voiced in various pitch configurations (inversions).
Fig. 2. The diatonic scale is a pattern of scalar steps consisting of two sizes of basic interval: whole tones and semitones (half tones). The semitones are between degrees 3 ↔ 4 and 7↔1. A convenient way of seeing this pattern is to think of the diatonic scale as two equivalent halves or “tetrachords” (Greek for “four strings”) with a whole tone in between them. The sequence 1 to 4 is the same tonal pattern as 5 to 1'. This pattern of steps results in two different sizes of thirds consisting of either two whole tones or a semitone and a whole tone. This in turn results in different kinds of triads.
Fig. 3. The seven available thirds of the diatonic scale, identified as three major (large) and four minor (small) thirds. Major thirds consist of two whole tones, whilst minor thirds consist of a whole tone and a semitone, i.e. one and a half tones. The major thirds are on degrees 1, 4 and 5, whilst the minor thirds are on degrees 2, 3, 6 and 7. The distribution of these different thirds have significance for chord type. A triad with a minor third below and a major third on top (eg. 6–1–3 or 2–4–6) is a minor chord. A triad with a major third below and a minor third on top (e.g. 1–3–5 or 5–7–2) is a major chord. Thus, the diatonic major triads are those built on 1, 4 and 5, whilst the minors are those built on 2, 3 and 6. Each of these six chords, built from two different thirds (a major and a minor) is characterised by a perfect fifth between the outer degrees of the chord, e.g. 1–5, 2–6, 6–3. The chord on degree 7 is unique as it consists of two minor thirds. It is neither major nor minor and is known in harmonic theory as the “tritone,” a name describing the interval (three whole tones) between the chord’s outer degrees.
Fig. 4. A table of triads as harmonic functions in Relative Figured Bass notation. Reading each column upwards, the scale degrees rise by a diatonic third each time. A numeral, when circled, becomes the chord symbol for its triad, read three places (inclusive) up from the root. Thus, reading each column from the major row, we have the three major triads 1–3–5, 4–6–1 and 5–7–3. Reading each column from the minor row, we have the three minor triads 6–1–3, 2–4–6 and 3–5–7. The columns can be read to four places or more for optional embellishments, giving minor sevenths (e.g. 6–1–3–5), major sevenths (e.g. 1–3–5–7) or the dominant seventh 5–7–2–4. The columns are denoted according to Riemann’s theory of harmonic function: T=tonic, S=subdominant, D=dominant. As the the contents of each row are a diatonic third from each other, it will be noted that the triads of the major and minor rows are a third apart. This means, for example, that chords (6) and (1) not only share the common degrees 1 and 3 but are also inherently in harmony with each other, a relationship that also applies to (2) and (4) as well as (3) and (5). This relationship (in Riemann’s terminology, parallelism) between major and minor triads is one of a number of harmonic relationships giving rise to possibilities for chord substitution, whereby common chord notes allow alternative harmonization of a given melodic sequence.
As covered previously, numerical notation can be used for the type of educational exercise where the musical ear is used as a source of musical content, i.e. where the pupil searches out the notes of a familiar tune, rather than reading them in a score or otherwise being instructed in what to play. As a continuation of this exercise, a first, basic harmonization is attempted. Having successfully found and notated a melody, the pupil attempts to match bass notes with it. It is not an easy assignment for a pupil attempting it for the first time and facilitating the process demands extensive methodological resources of the teacher, but it can work very effectively as a first step to understanding harmony. In my current method, I give the assignment initially only with melodies in major mode and restrict the choices of bass note to the three major functions.
Fig. 5. A first exercise in harmony. An entire melodic sequence (major mode) is played over a pedal tone (drone) on the key root (scale degree 1). The pupil is asked to notice the changing tension (consonance and dissonance) between the melody notes and the drone as changing intervals are created. The pupil is encouraged to listen for instances where a change of bass note might be desirable. The craftsman’s sense of this tension and the way it is tempered by harmonic changes is a question of knowing what to focus one’s musical ear on and is taught by example rather than explanation.
Having experienced the melody with a drone on the key root (1), the pupil is trained to recognise instances where changes to the two other possible bass notes and back to the root could frame the melody in a beneficial way. With numerical notation as a support, all harmony is denoted as relationships rather than just seemingly arbitrary pitches. This allows patterns to become apparent, like the way that certain melody notes seem to attract certain bass notes. Recognition of such patterns is a precursor for insight into the structure of chords as in fig. 4., above. Knowledge of harmonic voicing, substitution and chromaticism build on this initial sense of harmonic function, constantly harking back to it as a fundamental tool. In this way, the immense complexity of the harmonic craft grows from simple ideas, regarded in a shifting perspective.
Note that chord symbols (in any “top line and chords” depiction) are only abstract “changes,” i.e. they don’t necessarily stipulate how a chord is voiced, sounded, or distributed in time, but merely how long it applies before giving way to the next.
Fig. 6. With the right guidance, the pupil learns to recognise musical tensions and the way in which the available harmonic functions diffuse or temper these. At this early stage, the three basic functions are still understood as bass notes rather than chords.The first line illustrates the “weight” of the three major functions, relative to each other. The key root (1) is clearly predominant, whilst (4) and (5) are merely visited briefly as temporary “havens” from the inevitability of the return “home” to (1). For the second line, the pupil is encouraged to reuse the same harmonic parings of melody and bass notes that occur in the first line. In this way, a two-part (melody + bass) version of the tune is created. At this stage, the idea of chords is introduced as the set of harmonic relationships depicted in the chord table, fig. 4.
This basic two-part harmonization between melody and bass is a good springboard for introducing extended harmony in the form of chords. It can be noted, for example, that parings like 3 over (1) or 6 over (4) — based on thirds — are fuller sounding than 5 over (1) or 2 over (5) — based on perfect intervals— whilst unisons like 1 over (1) and 4 over (4) are aesthetically even “drier” and more lacking in musical finish. This sets up the introduction of chord structure (fig. 4.) as a source of reference for a third (harmony) part, arising out of the question “what’s missing?” at any given point in the two-part harmonization.
Fig. 7. “Missing piece” strategy. With the help of the harmony table (fig. 4.), a third part is added as a harmonic line under the melody, i.e. in between the melody and bass lines. This harmonic line is found as a solution to the question of what is “missing” from the current chord function at any given point in the two-part version. This does not necessary complete the triad, but gives the pupil choices of which scale degree to add at any given point, illustrating by example that thirds are “fuller” and preferable to perfect intervals. The harmonic line created in this way is rather fragmented, as becomes clear when the three-part arrangement is played on a keyboard. However, since a requirement is that the melody line is pitched highest, this exercise introduces the idea of interval (and indeed chord) inversion, training the pupil to recognise and work with any one chord in a variety of possible voicings.
Fig. 8. With a few scale degree substitutions, the original harmony line (a) is reworked (b) to make it more practical by letting it move parallel with the melody line wherever possible, i.e. become scalar. With the exception of the unison at the beginning (for ease of playing), the melody and harmony lines now move in thirds in the first two bars, then sixths (i.e. inverted thirds) in the second two. Replacing 1 with 4 in the second bar under melody note 6 doubles the bass note (4), but has a negligible harmonic cost, whilst adding playability and elegance to the harmony line. In the last bar, replacing 7 with 4 under melody note 2 maintains the sequence of sixths and introduces the dominant seventh as a strong alternative to the third in the chord (5). Since the functions are still understood as bass notes at this early stage, the notation has almost taken the form of a literal mapping of all three voices (in terms of notes, if not their distribution in the bar). The important consideration from a developmental point of view is that the pupil, albeit with a good deal of guidance, has done the groundwork himself on the basis of his musical ear and access to a fledgling theory.
Fig. 9. This time, the bass line, rather than the harmony line, becomes scalar and is made to run parallel to the melody for the first three bars. Thus, it is freed from the constraints of the three major functions by two kinds of substitution. The first is a re-voicing of an existing function from the bass. In the first bar and again in the second, the bass moves from the root of (1) to a different note, 3, in the same triad. In both instances, the 1 over the bubble alters the triad, indicating that (3) is not the minor chord on 3, but (1) over its third. The other substitution is a parallel one, occurring in the third bar, where (4) is replaced by its parallel minor, (2). Note that the scalar bass line allows for a long stretch of static harmony line on 1, which both the melody and bass line move against. Again, this is both practical and elegant, serving not only technical playability, but also the aesthetic quality of the arrangement. Both this harmonization and fig. 8, b) are viable alternative strategies that, for the sake of variety and nuances of mood, can be interchanged and juxtaposed at the arranger’s discretion.
Thus, from a matching of bass notes to melody, followed by a “missing piece” strategy of assigning harmony notes, we move to various strategies for adjusting and trimming harmony parts in practical and aesthetic interests. The figured bass notation, because it uniformly relates back to the key root, illuminates and facilitates these strategies in a way that neither multi-part standard music notation, nor absolute chord notation can. Note how RFB supports the emphasis on the definition of chords by context, rather than just as autonomous blocks of notes, and chord sequences as voices moving against each other in time, rather than blocks of notes after each other. Although we can regard the events of the first beat of the second bar (fig. 9.) as a triad of degrees 4, 1 and 6 at once, we can also change focus and see any of these numerals as a captured moment in a single tonal part/voice. Both perspectives are equally important in harmonic theory.
RFB also has certain advantages in the study of advanced harmony, since (re-)harmonization strategies like revoicing, substitution, parallel harmony and counterpoint all have strong context relative significance, which the reader must be aware of in order to interpret absolute chord notation.
Fig. 10. An arrangement, notated as absolute (as opposed to relative) top line and chords, employing diverse strategies of revoicing, substitution and chromatic counterpoint. After the first instance of the main theme (from bar 1), each reiteration (from bars 5 and 13) gets a new, more complex harmonization of the same identical melodic phrase. For the initiated, this is easy enough to read, since the absolute notation meets the necessary context (i.e. key) relative knowledge. But without this advanced frame of reference, the chord symbols are viewed out of context and the growing complexity risks hampering insight into what’s going on.
Compare the absolute chord notation in the above example (fig. 10) with the same harmony, notated (fairly minimally) in RFB, below. Note that, unlike absolute chord notation, RFB takes account of notated melody notes in chord formation. This integration supports educational agenda by reflecting the close relationship — and indeed causality - between melody and harmony, not apparent in absolute notation, and consequently often overlooked by readers lacking the necessary implicit knowledge. It also means that a harmonic notation without the top line would require more annotations to take account of the missing melody notes.
Fig. 11. RFB used with conventionally notated top line. Each chord symbol is understood as a nominal function (triad), with or without embellishment (as desired), unless otherwise annotated. The first instance of the main theme uses only (1), (4) and (5) with no strict need for annotation. It is up to the interpreter’s discretion, for example, whether instances of (5) take a dominant seventh (scale degree 4), or whether (5) moves parallel with the melody in bar 4. The second time (bar 5) introduces alternative voicings as well as chromatic counterpoint. (3) in bar 5 is a tonic (1) chord over bass note 3, as indicated by the melody note 1 (F). In the same bar, a (2) is altered from its nominal minor (Gm) to dominant seventh (G7) by the annotations #4 and 1 (raising Bb to B natural and adding F). In bar 7, (4) becomes minor by annotation b6 (D becomes Db). In bar 15, the equivalent movement happens in the bass line. In bar 10, (4) is modified by the annotation 2, as well as melody note 7 (E) to become the dominant (5) over bass note 4. The same construction is achieved in bar 12 by the annotation 7, since the melody note is 5. In bar 13, (#4) with annotations b3 and 1 denotes a diminished on B natural. In the following bar, (#5) with annotation 2 is an ambiguous chord, related to both the diminished on #5 (C#dim) as well as the dominant seventh on 3 (A7#9). In bar 15, (5) with 3 and 1 indicates the tonic (1) on bass note 5 (F/c), which immediately gives way to (b7) with 3 and melody note 1, i.e. the same chord on its dominant seventh (F/eb).
The harmonic variations depicted here, albeit complex, can, for the most part, all be referenced back to the harmony table of diatonic relationships (fig. 4) and categorized either as revoicing (same chord, new root) or substitution (different chord, common harmony), with or without chromatic alteration. Exceptions, i.e. chords that defy diatonic/functional categorization (like diminished), can be identified with certain conventional positions like (#1), (b3) and (#4).
All in all, the purpose of this harmonic notation is to find a happy medium between a) indicating patterns of movement in time, and b) denoting harmonic concepts. Full multi-part standard notation (i.e. classical full score) will achieve a), but often at the cost of both brevity and interpretational freedom by neglecting b). Absolute chord notation can (with the right background knowledge) achieve b), but often at the cost of relational insight by neglecting a). As I have shown, the use of RFB is flexible, allowing for a certain “tweaking” between a) and b). For the beginner and intermediate level pupil, the chord symbols can be annotated as extensively as required (often as not by the pupil himself) in order to show patterns of movement. For the more advanced user, little or no annotation is needed over the chord symbols in conjunction with the top line, as the context will often suggest what is meant and/or invite alternative interpretations. The exercise of deciphering meaning from minimal relative notation is excellent training for the skill of reading absolute notation loosely, contextually and with freedom of interpretation.
In Part 5, I will sum up all relevant aspects of the modernized diatonic numerical notation, including approaches to rhythm notation, with extensive examples.
I would like to thank Jef Moine for his generosity and expertise in contributing a tech solution to creating graphic files in MDNN, of which fig. 5 to 9 are examples in this text. The solution is a modification of the popular ABC notation, and would need its own series of articles to do it justice.
