The Case for Numerical Music Notation. Part 1: Introduction and History

This article is a presentation of the concept and history of notating diatonic music with the use of numerals, of which several systems exist. This concept has enormous potential, particularly in the field of music education, borne out by my own experiences teaching with it. However, I feel that existing systems of numerical notation lack some important developments in order to bring them up to date and optimize them to the task of notating modern music and teaching the skills required for playing it. I will therefore outline the potential I see in the concept, as well as tackle some problems that existing systems come up against. Finally, I will propose an optimized, modernized system for notating music numerically with suggestions on how it can be put to beneficial use.

In 1742, philosopher, writer, composer and music copyist Jean Jacques Rousseau presented a system of numerical music notation to the French Academy of Sciences, later published in his Dissertation on Modern Music (1743). He is by no means the only person to have thought of representing the notes of the diatonic scale with numerals, and it seems that many variations of this idea have cropped up more or less independently of each other since Rousseau (and possibly even before). Despite usage in Germany, France, and the Netherlands, numerical notation has never become widespread in the West, even though its central idea— numbers for degrees of the scale- is used as a sort of shorthand among jazz and country musicians to describe harmonic sequences as relative patterns, rather than absolute values.

Fig. 1. The opening 8 bars of “Look for The Silver Lining” (Kern/DeSylva), first in Ab, then F, then C. Standard Music Notation (SMN) centers on absolute pitch, so even though the three are exactly equivalent to each other — all internal relations the same— a new version is required for each new key. This is also true for absolute chord symbol notation. Working musicians who play in all keys as required would recognise the harmonic sequence of the first two bars as “1, 2, 5,” a common chord sequence occurring in numerous tunes. When (more rarely) chords are notated in numerical form, roman numerals are generally used. This is not only a shorthand — one version covers all 12 possible keys—but gives an insight into musical relationships and the equivalence of musical keys. This approach to reference (and notation) can accurately be described as relative (as opposed to absolute). It can also be thought of as an abstraction, showing the derivation of a single category from subordinate examples.

The concept of describing and notating musical notes relatively, rather than in absolute terms, has a long history. Chinese Gongche notation, in which syllables and their characters stand for degrees of the scale, dates back to the Tang Dynasty (AD 618 to 907). In Europe, the Aretinian syllables ut, re, mi, fa, sol, la, si denoting the seven steps of the diatonic scale (named after their inventor, music theorist Guido d’Arrezzo), date back to eleventh century Italy. This system has undergone development at various times and in various places and evolved into the system known variously as solfege, sol-fa, solfeo etc., in which do, re, mi, fa, sol, la and ti denote the degrees of the diatonic scale. There are two schools of thought regarding its use. In fixed-do, particularly in use in Latin speaking countries, these syllables are the names given to the absolute notes, known to us as C, D, E, F, G, A, and B respectively. It is of course the use of solfege as moveable-do, independent of absolute pitches, that is of interest here. It has particular relevance for singers — whose instrument is in its nature unconnected to absolute pitch- who can use it in conjunction with or without Standard Music Notation to pitch notes relative to a root note (do at a particular pitch) in order to read notation, sing, conceptualise and remember complex vocal parts. The use of solfege in this fashion owes much to the work of English music teachers Sarah Glover and John Curwen in the 19th century, as well as that of a French school consisting of Pierre Galin, Aimé Paris and Emile Chevé, whose Galin-Paris-Chevé method was arguably based on Rousseau’s ideas from 1742. A set of hand signs developed by Curwen, corresponding to the seven degrees of the diatonic scale (and popularised in Stephen Spielberg’s film Close Encounters of The Third Kind), was later adopted by Zoltán Kodály in the method he developed in the 20th century in Hungary.

Whilst intoned syllables and hand signs have communication value monophonically (a single voice of music at a time), the recording, communication and development of complex harmony (polyphony) requires written symbol systems. Standard Music Notation (SMN) is one such system, whose capacity for notating pretty much anything within music is unparalleled. However, in music education, SMN can be enormously misleading as a depiction of music, as well as requiring considerable skills and experience if one is to write in it. As an absolute pitch notation system and essentially a graph, SMN simply plots what happens where and when in music. As seen in fig. 1 above, it is not immediately apparent to the untrained eye that the three examples are equivalent (the same thing at different pitches), because neither SMN or the absolute chord notation says anything about the roles of the chords or their relationship to each other and back to the key they’re in. It is only in the numerical version that these relationships are described, because this version is an abstraction, based on movable symbols, variables. So whilst this neutrality makes SMN versatile in the hands of an expert, the abstract version provides a pattern recognition tool in conjunction with it, a supplement that can help the uninitiated reader to understand what’s going on musically.

Fig. 2. The diatonic scale represented as a tonal “staircase” with both the syllables of solfege and their corresponding numerals in a repeating pattern. Note the two different sizes of step. 3 to 4 (mi to fa) and 7 to 1 (ti to do) are half tone steps, whilst the others are whole tone steps. This assigning of numerals to this distinct pattern of tonal steps is the basis for numerical notation.

Despite never really catching on in the West, numerical musical notation is in widespread use in Asia, particularly in China. JianPu, a notation system closely related to Rousseau’s (and possibly indirectly derived from it via the Galin-Paris-Chevé method) was adopted around 1900 and is more entrenched in Chinese music culture than SMN. This is perhaps hardly surprising given the long legacy of Gongche,i.e. relative (as opposed to absolute) pitch, and symbol based (as opposed to graphic), notation.

Fig. 3. Two examples of monophonic JianPu notation, both of internationally known children’s songs. Time signature and bar lines are as in SMN. Free standing digits are 1/4 notes(crotchets). A single underscore gives 1/8 notes (quavers), whist a double underscore indicates 1/16 notes (semiquavers) and so forth. Rests are indicated by zeros in the same way. Dots after any of these values have the same meaning as in SMN. A dash lengthens the previous digit to a 1/2 note (minim), two dashes to a 3/4 note (dotted minim) and three dashes to a whole note (semibreve). A dot over or under a digit indicates a higher or lower octave respectively. Thus, the scale step over 7 is 1 with a dot over, whilst the step under 1 is 7 with a dot under. In JianPu, dots are assigned to specific octaves, so repeated dots indicate the present octave, as in bar 5 of example a), whist no dots is a default back to the “home” octave. Quiz: Can you name these two tunes?
Fig. 4. The same two examples in Rousseau notation. Although the two systems have the same numerical basis, they differ in two key aspects. In Rousseau, rhythm is notated not by symbols , but by the positioning of the numerals relative to each other, i.e. graphically. The dots denoting octave change are not assigned to specific octaves as in JianPu, but merely indicate an octave change upwards or downwards, after which the following notes (without dots) default to the present, rather than the “home” octave. In the last two bars of b) the change down from 1 requires a dot under the 5, after which the return up to 1 requires a dot over it. In bar 5 of a) a dot over the 1 changes the octave up, after which the next 1 (no dot) defaults to the same octave. The change back down requires a dot under 7, after which the rest of the steps (no dots) default to the present octave.

Whilst I in no way claim that numerical notation is better than SMN overall, it has certain advantages in various contexts. As symbols rather than note heads denote pitch, they are not only easy to write as sequences (rather than having to plot them on a graph) but can also be notated first, before rhythm notation is considered. This is ideal for a beginner, using his ear to pick out the notes of a tune and jotting them down for future reference. This kind of task, directly engaging the musical ear as a source of musical knowledge, is invaluable for musical development, but often neglected at beginner and even intermediate levels, precisely because it’s not supported by a user friendly notation, i.e. one for writing in, rather than just reading and blindly following. For a beginner, SMN can’t be anything other than an instruction manual, giving a misleading sense of where music comes from and “lives.”

For many pupils, such an assignment is greatly aided by the use of lyrics as a guide to work by, not least because lyrics can help to “frame” the rhythm, while identifying the syllables where note events occur. As numerical symbols, the notes can simply be assigned to syllables without the strict necessity of rhythmic markers, a task that SMN is quite unsuited to performing.

Fig. 5. In this example, note events are written over the syllables where they occur in a lyric. Because of its graphical nature, SMN can scarcely be used in this way, as lyrics have to be matched to existing music notation rather than vice versa. This flexibility of numerical notation allows for a sort of “half notation” in which rhythm in a strict sense is omitted, but is guided by the pupil’s knowledge of the song via the lyrical content. Jianpu rhythmic markers and bar lines can still be added if required in this setting. This one area where JianPu is superior to Rousseau, whose graphic rhythm notation doesn’t lend itself to lyrics.

Whilst I see numerical music notation as a valuable, but neglected, resource, I also feel that its own potential for further development has been neglected. My own work with the system as an educational aid has led to a preference for JianPu over Rousseau as a wholly symbol based system, free of the constraints of graphic notation. The rhythmic markers of JianPu mirror to a large extent those of SMN, a distinct advantage when using numerical notation as a learning aid and supplement to SMN. I do however find Rousseau’s octave change method (of defaulting to the present octave) superior and have adopted it in my own system of numerical notation.

In direct conjunction with SMN, numerical notation has various uses. One of these is close to the way solfege has been used to “translate” SMN into the diatonic language of the musical ear. As with the musicians’ harmonic (Roman numeral) shorthand, the relative notation shows equivalence between keys and common patterns in different examples and can be used as an aid to deciphering, learning and transposing SMN.

Fig. 6. Numerical notation is used to show equivalence of melodic sequence in two different keys. Used In conjunction with SMN, numerical notation can aid singers in learning a vocal part, or generally aid the reading and understanding of SMN. This “translation” into scale steps is also a way of easily transposing SMN. Octave change uses Rousseau method, but with apostrophe and comma replacing dots because of typeface availability.

This use of numerical notation, in direct conjunction with SMN, has been of considerable benefit to my own students, who have often written in the numerals as a help to reading and transposing SMN. It has also inspired me to make a number of modifications to numerical notation, which I will cover along the way.

It is, of course, not my intention to plead a case for numerical notation as anything more than a supplement to SMN, as for all its strengths as an educational aid, it is by its nature inherently limited to diatonic music. Also within this realm it has some problems that need addressing. These concern chromaticism, notation of tunes in minor mode and the quite substantial question of how polyphony (multiple simultaneous voices) can and should be tackled. I will take these questions up in the next article.

Quiz, fig.3: a) “Itsy Bitsy Spider.” b) “Frere Jacques”