The Case for Numerical Music Notation. Part 2: Issues and Challenges

Numerical music notation has the potential to be a useful supplement to Standard Music Notation as well as an invaluable aid in the teaching of music skills. Yet because of its history and the way it has developed, it has found itself down a blind alley, particularly in the West, where it is little used. For all its faults and limitations, Standard Music Notation (SMN) is still the only available tool for certain types of notational role. This is fine for the relative few who actually master this tricky system, but limiting for those who require something more user friendly, or a supplement to help them properly learn music theory and SMN. Numerical notation is today limited by a number off issues and challenges. What I would classify as issues are questions arising from the use of the system that generally need addressing. Challenges I would see as more far reaching problems that require modifications and new protocols.

Simple chromaticism

One issue is the question of how chromaticism is dealt with. Simple and temporary deviations from the diatonic scale can be shown with accidentals as in SMN.

Fig. 1. Two examples of accidentals used in numerical notation. Whilst a) is a modal shift (to mixolydian), i.e. an altered scale with the same root, b) is a temporary modulation (new root). In either case, a change of key signature is not warranted in SMN and these accidentals play the same role and follow the same conventions as they would there.

It is fairly obvious that the sort of chromatic variations in the scale that would not warrant a new key signature in SMN don’t warrant any more than accidentals in numerical notation. Note that accidentals are relative, i.e. deviations from the default diatonic scale and therefore are not necessarily “black” notes, as they would tend to be in SMN. If example a) above were played in the key of D, the b7 (seventh scale degree flattened) would land on the note C (i.e. flattened from C#).

Minor mode

In Part 1, all of my examples of numerical notation were in major mode, but questions arise around the use of numerical notation in minor. There are two basic solutions for minor in moveable-do systems.

One way is to make no formal distinction between major and minor and simply notate everything from the perspective of major (ionian mode, centred on do). This is in keeping with SMN, where minor is a mood implied by context, rather than a restructuring of notation. If we regard minor in its basic form as aolian mode, the diatonic scale centered on la (step 6) in this approach, it is defined in numerical notation as the scale 6, 7, 1', 2, 3, 4, 5, 6. It can perhaps feel misleading to have a tonal centre on 6, but the question of tonal centre is by no means clear cut in minor, as evidenced by the frequency of major cadences (back to 1) in minor tunes. The minor key on 6 is not only related (parallel) to the major key on 1, but in some ways also to the major key on 6, making for a general prevalence of chromaticism in minor, emerging from this instability or short circuiting between tonal centres and modes. In any music from the common practice onwards, steps 4 and 5 (fa and sol) are often intermittently replaced by #4 and #5 so that melodically and harmonically, minor mode can use any and every combination of these four possible notes in conjunction with each other at any time. In short, we needn’t loose sleep over whether a tune is predominantly major or minor, but merely deal with chromatic aberrations as they arise with the help of accidentals.

Fig. 2. Treating minor not as an independent key/mode, but from a default perspective of major mode. This saves having to classify tunes as either major or minor, a question which is often unclear in light of the frequent commuting between major and minor tonal centres in modern music.

The other way to treat minor in relative notation is as an independent mode, changing the distribution of whole and half steps relative to the symbols. Steps 2 to 3 (re to mi) and 5 to 6 (sol to la) become the half steps (semitones) of the new scale, while the rest are whole steps. The typical chromatic characteristics of minor will now occur around 6 and 7 (la and ti), #6 and #7.

Fig. 3. Treating minor as modal scalar change. Note that the numerical notation has its own clear modal signature.

Although I would not rule out this modal scalar change solution for minor, it seems a rather technical/theoretical, rather than “organic” approach. I am inclined to stick to the first approach (6 as minor root) wherever possible, especially when numerical notation is intended as a teaching aid, in order to maintain a uniformity of interval patterns between symbols. I feel that these patterns should sit with the user/pupil at a subconscious, automotive level, as a routine connection between eye and ear, rather than requiring conscious calculation from scale to scale. In the end, it’s probably best for the user to do what feels fitting and natural, rather than constructed, depending on one’s own disposition and the given context.

Any change from the default scale of diatonic major (ionian) must of course be clearly indicated. In relative notation, where by definition, no absolute key is specified, a “key” signature denotes a modal change on the same root, rather than a new root. Thus, it is not a key signature at all, but a modal signature, indicating a change from the default mode (ionian) to another: 3 flats (bbb) for diatonic minor (aolian), one sharp (#) for lydian, one flat (b) for mixolydian, two flats (bb) for dorian. This all seems rather academic, as modern music rarely follows any such modes in a strict sense, i.e. without chromaticism and the need for accidentals. It also strikes me as being prone to misunderstanding (the mistake of reading the signature as absolute key rather than mode). Since there are also circumstances when absolute changes of key are required (moving the mode to another absolute tonal centre), any use of signatures in this position should be clearly marked as “mode.”

Structural chromatic modulation

Many tunes modulate as an inherent part of their structure. By this, I mean that integral parts of a whole repeated sequence (verses, bridges, choruses) are over different tonal centres. This would not have been a consideration in Rousseau’s late baroque/early classical musical world of 1742, where the idea of chromatic key was still relatively new (and the temperaments to play in all keys even newer). At that time, the most exploratory of compositions stayed close to “home,” venturing no further than subdominant, dominant and relative minor as temporary tonal centres. Modulation as a compositional device didn’t develop until further into the classical period. As with Bach’s Minuet (Fig.1. above), short sequences in closely related keys can be dealt with by accidentals. It is significant modulations over many bars, normally requiring a change of key signature in SMN, that require a corresponding treatment in numerical notation.

An example is “Dream a Little Dream” (Andre, Schwandt, Kahn), in which the bridge (middle 8) changes four flats away from the original root. This change could alternatively be expressed “down a major third” or “up a minor sixth,” but I prefer the cycle of fifths terminology that expresses key, leaving ideas of “up” and “down” in pitch optional. Thus the change, in numerical notation, can be given as [K: +4b] (Key: plus four flats). I can add “up” if I want the pitch to actually go up in the octave, or leave it up to the interpreter’s discretion. The opposite change back is given as [K: +4#] (Key: plus four sharps), but could equally be expressed as [K: -4b] (Key: minus four flats). These are not “key” signatures in the absolute sense, as they will not necessarily result in a particular number of flats or sharps, but simply add or subtract them from whatever absolute key one is playing in. A +4b change from the key of B major, for example, will land on the key of G (one sharp). To make this distinction, we should call them relative key signatures or, better yet, modulation signatures.

Fig. 4. “Dream a Little Dream of Me” (Andre, Schwandt, Kahn), melody only. The square brackets contain the modulations, relative to whatever home key is chosen. If the verse is in G, the middle 8 will be in Eb. If the verse is in E, the middle 8 will be in C. Interestingly, a famous version of this tune by The Mamas and The Papas raised the middle 8 one more semitone up from the home key, i.e. a major 6th (or minor 3rd down). Quiz: What would the modulation signature be for that version?

Another classic example of this kind of modulating tune is “Penny Lane” (Lennon, McCartney), in which the root moves two flats [K: +2b] from verse to chorus. In many ways, this modulation is more tricky to deal with, because of the logistics of where it occurs in terms of notes and bars and therefore, considerations of how it’s best communicated.

Fig. 5. “Penny Lane” (Lennon/McCartney) modulates +2b from verse to chorus, equal to a major second down or minor seventh up. Unlike the previous example, the modulations both to and from the new key are quite seamless, occurring in the midst of a continuous melodic sequence. For the first modulation, I have chosen a typical location for the signature, based on what would be conventional in SMN, but it is not quite satisfactory here, resulting in a step 4 becoming 5, which is misleading, as it’s the same note tied over. A better solution in this notation would have been a bar earlier, where there is more of a clean break. The return modulation is placed in such a way as to give ample warning and make the repeat resemble the lead-in bar.

The right logistical decisions can lessen the challenge of these structural modulations, but there’s no getting around the fact that they are a challenge and might possibly exceed the scope of what numerical notation can reasonably achieve. When faced with a challenging degree of chromaticism to notate, the primary question should not necessarily be, “How do we notate this numerically (at all costs)?” but rather “Who is the notation intended for?” Musicians advanced enough to handle chromatically complex material will reasonably be expected to read SMN. If not, then an insight into numerical notation, with or without the help of a teacher, should soon rectify the deficiency.

It’s a good idea for all users and students of theory to know the cycle of fifths, if not like the back of their hand, at least as a concept for quick reference. There are various rules of thumb: Fifths up give one more sharp each time, fourths up, one more flat, whilst adding a sharp (+#) is the same as taking away a flat (-b) and vice versa (so the key of Bb is -3b or +3# away from Db). A whole tone up is +2# (-2b), whilst a whole tone down is +2b (-2#). A minor third up is +3b (-3#), whilst a minor third down is +3# (-3b)

Fig. 6. By virtue of equal temperament, the keys of the 12-tone, chromatic octave meet in a cycle (or circle) of fifths, where the keys Gb and F# become the same key for practical purposes (which they aren’t in natural or just intonation). In this diagram, a modulation of three flats (+3b or -3#) is a movement anticlockwise, whatever the starting point, e.g. from E to G, Db to B, F to Eb. A modulation of five sharps (+5# or -5b) is a movement clockwise, e.g. from C to B, A to Ab, Gb to F.


In JianPu, polyphony has traditionally been given by parallel layers of voices, connected by common bar lines, just as parallel staves are joined by bar lines in SMN to indicate that they should sound simultaneously. Thus, piano, vocal or ensemble arrangements (for example) are often notated in much the same way as in classical SMN, where all voices are plotted in full.

Fig. 7. An excerpt of “Oh! Susanna” (Foster) with melody and a rudimentary bass line. This way of using numerical notation is widespread in Asia and in many ways mimicks the way SMN has been used in the West, particularly in the classical tradition, to impart specific arrangements for polyphonic instruments, groups of monophonic instruments or voices. The absolute key is even often specified.

The weakness of this type of arrangement (fig. 7.) is that it is specific about what each voice does, which is a classic trait (for better or worse) of SMN. Here, the bass is told exactly what to do, i.e. which notes in which direction up and down octaves. This very specific notational assignment (if necessary) is much better achieved in SMN in an absolute key, rather than a relative notation, where this exact bass line won’t have the same power or “instrument-friendliness” in every key. One workaround is to leave the octave change dots off of the bass part, allowing the bass player to use discretion. Another is to assign a specific key by writing “F=1” or “C=1”, but then we’re clearly in the realm of compromising numerical notation in favour of trying to make it do what SMN does better.

Fig. 8. An excerpt of “Nobody Knows You When You’re Down and Out” (Blom, Daams, Veen, Cox) arranged for two unspecified voices. Since the key and the voices are flexible, we could regard this as a rough sketch of an arrangement, possibly before committing to specific instruments (like winds or vocals) and (by necessity) an absolute key, then notating in SMN.

Fig. 8. shows a more legitimate use of polyphonic numerical notation, to sketch out an arrangement before specific instruments and absolute key are chosen. Any finished arrangement (with all the specifics decided) will be much better served by SMN, as in this version for two vocals with lyrics added:

Fig. 9. Excerpt from a finished arrangement of “Nobody Knows You When You’re Down and Out” for two vocals.

Yet this is not to negate an important role that numerical notation can play here. The upper harmony in this arrangement (upper stave, fig. 9.) is the sort of part that could be tricky to learn by heart, and diatonic notation can provide valuable support for singers who don’t (yet) read SMN. This could be done with diatonic numerals over the note heads in SMN or directly over the lyrics.

Fig. 10. the upper voice from the same arrangement as fig. 9. a) As SMN with numerical help. The octave change dotting is Rousseau method with comma and apostrophe instead of dots. b) As numerals over lyrics.

As a neutral plotting tool, a comprehensive guide to what note goes where, SMN is unparallelled. The strength of numerical notation lies not in recording every detail of a musical arrangement, but in the relational insight needed to figure out the ins and outs of music for oneself. To this end, numerical notation has lacked its own compatible chord symbol system.

In Part 3, I will take up this challenge by using various forms of harmonic notation as inspiration for a numerical system of chord symbols, suitable for use with numerical melody notation.

Quiz, fig. 4.: The modulation signature for moving up a major sixth/down a minor third is +3# (-3b).