The Case for Numerical Music Notation. Part 5: A Guide to Modernized Diatonic Numerical Notation

In the first four articles of this series I have discussed the history of moveable-do numerical music notation, its strengths and weaknesses, and discussed its potential as both an educational aid and musician’s tool. The aim of these discussions is the development of a modernized system of numerical musical notation as a supplement to Standard Music Notation (SMN). To this end, a basic idea, dating back to the Middle Ages, having developed into a variety of notation systems and educational methods from the early 1700s onwards, has been subjected to an analysis on the strength of modern musical requirements, the issues they raise and challenges they bring. In the light of this analysis, I have concluded that numerical notation is well suited, and potentially highly beneficial, to notation assignments where the aim is to communicate relational insight or to sketch out musical ideas with freedom of interpretation in mind. Practically speaking, this puts numerical notation into the role of “top line and chords.” Thus, central aspects of the basic, linear notation of Rousseau and the closely related Chinese JianPu have been modified to take rhythm notation from SMN and furnished with Relative Figured Bass, a chord symbol system of annotated scale degrees, inspired by aspects of functional harmonic analysis and basso continuo (figured bass).

In this article I will outline and explain a modification of existing approaches to rhythm notation, after which I will sum up the structure and characteristics of the modernized diatonic notation system with the help of examples.

As covered earlier, Rousseau notation’s graphic approach to rhythm is impractical for the same reason that SMN’s graphic approach to pitch is: Since graphics rely on the spacial positioning of symbols(as opposed to the difference between symbols) in order to impart meaning, the parameter they show is by definition impossible to notate without both the symbols and the positional grid. Note heads alone (without the staff) can’t tell you anything about pitch. Numerals alone (without Rousseau’s system of spacing) can’t tell you anything about rhythm. This reliance on some kind of coordinating grid severely constrains the use of the graphic system with any third system, like song lyrics, for example. As any arranger who has tackled this problem knows, laying out SMN with song lyrics is very complex. Happily, software has considerably eased and speeded things up, but even for the most practised SMN writer, it is and will always be a long-handed drafting process, coming nowhere near the vicinity of a shorthand for quick or even real time use. I hasten to add that the impracticality of SMN is a trade off of its enormous notational power. Impracticalities of Rousseau’s graphic rhythm system are easily overcome by replacing it with a symbol system.

JianPu solves the practical problem by using symbols for both parameters (pitch and rhythm) in an integrated and elegant system, in which the symbols for pitch (numerals) in default (non annotated) form represent 1/4 notes (crotchets), whilst underlining or adding dashes subdivides or multiplies their length respectively. The same goes for rests, depicted by 0 (zero). Whilst I admire the economy and elegance of this, I would prefer a completely separate set of symbols for the rhythm and have therefore chosen to employ the rhythm notation of SMN, placed over the pitch symbols.

Fig. 1. In Modernized Diatonic Numerical Notation, note heads are replaced by the numbers 1–7 for the degrees of the diatonic scale. Octave change up or down is given by apostrophe (‘) and comma (,) respectively. Rhythm is notated with the same symbols as in standard notation. Single stems are 1/4 notes, halved to 1/8 notes or smaller by joining. 1/2 and whole notes have an open rectangular note head.

Modernized Diatonic Numerical Notation (MDNN) is not intended as a replacement for SMN, but as a supplement, primarily for use in music education. This is by virtue of the fact that it depicts the diatonic tonal relationships that are only implicit in SMN, yet crucial to reading and understanding it successfully. For both educational and practical purposes, it’s good to have a way of separating the two “axes” (pitch and time) of music notation from each other in the form of two independent symbol systems, so that they can be tackled in isolation then combined as required. Since the educational shortcomings of SMN lie within the realm of pitch, rather than rhythm, it is expedient to combine the best aspects of existing numerical notation with standard rhythm notation, thus making a pupil’s step from MDNN to SMN easier, i.e. without the need for learning a new rhythm notation.

Music is a complex phenomenon. Apart from the two axes of events and time for any one voice or part, it also builds on the complexity created by voices moving in conjunction with each other. This can be viewed from various perspectives, but has developed in Western popular art music to a question of melody against harmony, where a more or less fixed part (a melodic sequence) is underpinned and “framed” by other voices running in conjunction with it and matched to each other according to a set of underlying principles. The craft of the musician — if he is to be independent and not just the servant of the visions and skill of others — relies on understanding how harmony arises from and interacts with melody. To this end, MDNN breaks with previous paradigms of numerical notation (like Chinese JianPu) as a system of “top line and chords” rather than a “full score” notation, where each part is notated specifically. In MDNN, only the melody part is notated specifically (as steps in time), whilst harmony is notated as chord changes, with or without annotations for chord embellishments, modifications or moving harmony voices.

Fig. 2. Harmony is given by chord notation in the form of Relative Figured Bass, in which chords are defined by their bass root (relative to the key root, 1) in brackets (or circle) and embellished or modified as desired/necessary by scale degree annotations. Whilst (1), (5) and (4) in the first line are understood nominally as the diatonic major triads on these roots, the 1 and 3 over (5) in the third bar modify the chord effectively to a (1) chord on bass note 5. (6), (2) and (3) are nominally minor triads. #4 modifies (2) in the second line from minor to major, whilst the melodic context also makes it a dominant seventh, as is specifically annotated in the last bar of line 2 by the addition of 1. An important aspect of this harmony notation is that it’s fully context dependent, meaning that the melody note alters the chord, where appropriate. An example is (3) in the penultimate bar, which is not annotated, but which harmonizes the melody note 1, effectively changing it from minor chord (3) to major chord (1) on bass note 3.

One purpose of this method of harmony notation is to give broad scope for practical interpretation and voicing by leaving a great deal of embellishment and even modification optional. All cases of (5) in the above example (fig. 2.) could, for example, could be embellished with 4, making them dominant sevenths, but this is up to the player’s discretion. Minor chords on (6), (2) and (3) can, where appropriate, be embellished with 5, 1 and 2 respectively, rendering them minor sevenths. (3) in the second line is nominally a minor triad, but could appropriately be altered to (1) on bass note 3 if the player desires. In all cases, a minimum of information leaves room for freedom of interpretation.

Another purpose of this is educational: The student of harmony is required, by virtue of the notational form, to think contextually, rather than just read chords without insight into their aesthetic function. One aspect of this contextual reading is the emergence of patterns in time, as in the first two bars of the last line of the above example (fig. 2.), where the bass line descends chromatically, reharmonizing the thematic restatement and creating an effect that is applicable again and again in many other tunes, i.e. a useful harmonic “set piece” for the empowered musician’s toolkit. In absolute chord notation, and indeed harmonic analysis, the chords of this sequence could be named in various ways, depending on the chosen perspective, yet none of the choices would capture the essence of what’s going on contextually from one instant to the next. This is because these naming systems focus on what is present at any one instant, rather than how it comes about. By contrast, MDNN in conjunction with relative figured bass describes the contrapuntal movement of voices, requiring the reader to interpret what’s present (or could be present) as a consequence of a musical journey in time from somewhere to somewhere new, the present moment as a product of origin and destination.

Another aspect of contextual relevance is the acknowledgement of melody notes and the way they alter chords. As any arranger knows, absolute chord notation is apt to be read far more literally than intended, especially by inexperienced players on harmony instruments not concerning themselves with melody or its relation to the harmony they’re playing. This is probably a consequence of education that neglects the significance of melody for such instruments, simply because they don’t directly play it, meaning that the chords are read literally as chords rather than the context dependent harmonic functions they actually are.

Fig. 3.

The final bar of this example (fig. 3.) contains a situation where a harmony instrument like guitar is given a dominant (5) as a harmony for the key root, 1. This is a classic situation where the melody note alters the chord, as the nominal chord contains degree 7, which will create an unpleasant disharmony with the 1 if the chord is read literally as a dominant seventh chord. An alternative for the arranger is to assign a 7th with sus4 or an 11th at this point, yet at the risk of getting some construction fresh from the chord manual that may not be necessary or very appropriate, because what’s really needed is more like “some suitable notes here.” The relative figured bass notation doesn’t tell the player what to play, but gives the harmonic function in context, implying that an appropriate chord (of which many are possible) must be found by harmonic insight rather than the literal reading of symbols.

Fig. 4. Modulation signatures. Apart from its chromatic content, this tune also presents the notational challenge of modulating as part of its structure. The modulation signature [bbbb] at the end of line 3 indicates a movement of four flats on the circle of fifths (i.e. down four perfect fifths = up four perfect fourths). Unlike an equivalent change of key signature in standard music notation, which is placed at the beginning of a bar, this relative signature can be placed more freely and strategically, as here, in the middle of a bar, before the modulating dominant. The opposite change back after the middle 8, [####], is similarly placed before the modulating dominant.

Fig. 4., above, is an example of numerical notation pushed to the limits of its useful application, where one naturally speculates as to whether standard, absolute notation wouldn’t do a better job. This is dependent on what that job is deemed to be. If the purpose is to inform experienced musicians of the content of an arrangement, then numerical notation seems superfluous, because we could expect such players to read and interpret standard notation appropriately. The purpose of MDNN should never be to notate music at any cost, but only to bring a fresh perspective where this is relevant, and possible. One such perspective might arise in an educational setting, where this way of notating musical content forces the student of harmony to read contextually and interpretively rather than just read and obey.

One challenge is a high degree of chromaticism (deviation from the diatonic), both in the melody itself as well as the harmonization. A function like (b6) harmonizing scale degree b3 (occurring in all penultimate bars of the main theme) could be justified in harmonic analysis by some circuitous “explanation” (involving for example tritone substituted dominant dominant or modal transformation to the common root minor key). Our purpose with notation is not so much to justify as to show patterns of where we’re headed musically (coming from and going to). In this case, (b6) is surely best understood as a stepping stone on the way to the dominant, (5), but it’s not obvious what kind of a chord it should be. The melody note b3 indicates a perfect fifth to the chord root and in the first occurrence (first time bar), I’ve annotated “1,” completing the major triad on b6. It is probably most commonly embellished with #4 (making it a dominant seventh chord, which I specifically annotate in a separate occurrence), but could equally take 5 (making it a major seventh). In absolute notation, we would have specified which (eg. Ab7 or Abmaj7 in the key of C) for the integrity of the finished arrangement (and to avoid nasty clashes between harmony instruments), but in an educational context, the use of numerical notation serves the purpose of requiring the student to weigh up various possibilities and complete the harmonization for himself. To this end, tunes can be notated to varying degrees of completeness in order to leave a desired amount of space for interpretation.

Fig. 5. As an educational assignment, only the most basic harmony notation is given, leaving the student to figure out appropriate chords from the context. This is particularly relevant for modified or revoiced chords like (#1), (#4), but also applies to different instances of (2), (3) and (6), which are not necessarily uniform throughout the arrangement and can be modified and/or embellished as deemed appropriate.

Fig. 5 (above) is an example of brief or incomplete harmony notation, designed to challenge the student in his harmonic knowledge and interpretation. Here, all functions are given in basic form (as defined by their bass notes), requiring the student to modify, embellish and annotate as necessary in order to harmonize the melody. In fig. 6 (below), the assignment is a more advanced one.

Fig. 6. The Danish folk tune “I skovens dybe stille ro” is given only partial harmonic treatment in the form of some basic functions to get the student started in each section of the tune. Many different harmonic interpretations are possible (and indeed appropriate). Different approaches to functions (2) and (3) in the first line alone will have a bearing on the overall mood and subsequent direction of the harmonization.

A harmony assignment of this type (fig. 6.) is designed to bring all interpretive knowledge to bear by making a suggestion, then leaving the student to run with it. Depending on the student’s level of knowledge and experience, the assignment can be solved either purely diatonically or with varying degrees of chromaticism. Here, the broad suggestion (in the first two lines) is for scalar bass lines, clearly shown by the movement of the numerical notation. One challenge for the student is to weigh up the merits of different types of revoicing and chord modification and their impact on the overall harmonic structure. As an example, (3) in the first line can be understood as either the nominal function on 3 (with or without modification), or as function (1) over bass note 3, each of which will tend to send the harmonic interpretation in different directions, one being more “classical” in nature, whilst the other possibly opens the door to a more “jazzy” mood.

Many other types and levels of educational assignment are possible using Relative Figured Bass with melody lines. Even for accomplished readers of SMN, Relative Figured Bass is a useful tool that can be used in conjunction with “topline” in SMN to promote better understanding and application of harmony. As I have tried to emphasise throughout, notation of any kind should be applied sensibly to the material to be communicated and illuminated. It should never be an aim in itself to notate numerically when this would overstep the bounds of what the notation can reasonably do.

Thus, I present the complete system of Modernised Diatonic Music Notation (in which Relative Figured Bass is a component) as an educational notation technology, designed to empower the music pupil in his basic learning of musical theory, as well as to supplement and ease his learning of Standard Music Notation and Absolute Chord Notation.

Whilst developing and applying this system in educational situations, I have often been asked why two notational systems for music should be better than one. I feel that this question should be firmly answered now. Music can be understood in both absolute and relative terms. Both perspectives are important and have their merits and shortcomings. For whilst it’s beneficial to get an insight into the abstract relationships that music builds on, it’s also essential to know about absolute pitch, what absolute key you’re in, especially if instruments are to play together. By the same token, whilst the neutrality of absolute notation makes it applicable to anything within music (including experimental forms that depart from diatonism and even standard pitches), it’s also crucial to acquire an insight into the (by far) most prevalent tonal form, standard practice, and its building blocks. With this insight, the building blocks can be used actively and discerningly by musicians who master their art, rather than languishing in the pitfalls of “a little learning.”

One of these pitfalls is the illusory overconfidence (and poison gas to education) associated with an idea that “anything goes.” For whilst the range of what constitutes music and can be referred to as music is vast (and probably without limit) only a tiny subset of it will be deemed to have quality and be worthwhile for an audience. If music is to matter, something must be better than something else. The other pitfall is a lack of root knowledge and reflexive ability within the art, rendering one a servant to the ideas and groundwork of others and, at best, rote learning hereof.

It would quite probably be superfluous to supplement an absolute notation system with another absolute notation system, or indeed a relative system with another relative system (even though there are plenty to choose from). It is the covering of both essential perspectives that should surely be the dedicated teacher’s goal.