For many centuries music has been captured in the Western world using the well-established European notation system that allows expressing musical concepts in written form. Among features such as duration and pitch, this system maps the 12 unique notes of the chromatic scale onto a staff developed for 7 unique note positions, i.e. the ones for the diatonic scale (the white keys on the piano); the remaining 5 chromatic notes (the black keys on the piano) have to be accommodated through symbols, e.g. the sharp sign (#), either in front of individual notes, or as keys at the beginning of a staff. Additional symbols, clefs, further alter the meaning of the position of all notes on the staff. All combined makes reading of sheet music hard and increases the cognitive load.
Perhaps true musicians get over these issues very quickly by learning to read sheet music in any key and clef; their brain maps the combination of position on a staff, key signature, clef and other symbols onto a 12-tone system in real time and without any difficulty. Amateurs struggle and consider these features rather as deficiencies. This unnecessary complexity constitutes both a barrier that prevents many people taking up music in the first place and an impediment to quick progression. The purpose of this story is to present the shortcomings of the European musical notation system in their physical and historic context as well as to show alternative but consistent systems.
Bijective Mapping of Tones
Human languages are considered to be “phonetic” if each sound in the spoken language (phoneme) corresponds to a symbol in the written language (grapheme), i.e. a bijective mapping exists. In music, a similar development occurred in the Western world as a set of tones practised by musicians was mapped to a written representation yielding the foundation of the current European music notation system. Other cultures developed their own musical tone and notation systems, notably the Ancient Greeks. In the following, however, the system pertaining to the established European notation system is considered, one based on Pythagorean Tuning and used initially in Gregorian chant.
Gregory the Great (pope from 590–604 CE) is understood to be the source of service music in the Roman church¹; he institutionalised a certain kind of singing psalms and hymns. However, there was no written form of the music; singing was purely an oral tradition. It took however a long time for monks to learn chants for all days of the year and its liturgy. Over time “signs” were introduced above the written text representing some aspects of the music. Initially derived from accents, these signs called neumes were mere visual aids that could help to reproduce chants but were not useful to learn new chants. Throughout Europe, many similar systems developed in the same period such as Frankish.
200 years after Gregory the Great, Charlemagne, Emperor of the Holy Roman Empire (from 800 CE), tried to unify the various ways of performing chant in his multi-lingual realms¹; to that end he set up schools and introduced modern writing (Caroline minuscule). A notation system was needed too in order to facilitate the “roll out” of one manner of chanting across the empire: the Roman rites were chosen harnessing the reputation of pope Gregory the Great. Consequently, neumes became unified but remained simple instructions with two essential features:
- differential, indicate the musical line, i.e. a tone relative to the previous tone akin to differential signalling in communications systems, and
- qualitative, indicate whether to stay at a tone (punctum), go up or down a bit (virga) in general terms but not by how much.
Similar to Human languages it is necessary to establish the “vocabulary” of the “spoken” language of music before devising a written representation. As for the spoken musical language, Gregorian chant appears to have used initially only 7 notes based on Pythagorean tuning, in essence the Pythagorean diatonic scale as a natural sequence of tones or ratios from a chosen base tone. The sequence of the first 6 Pythagorean “perfect fifths” can be projected to tones within an octave by starting at any base tone of any key in the European diatonic scale, e.g. F. The result is the sequence close to F, C, G, D, A, E, & B. A logarithmic plot shows the correlation of these 7 tones to the Western chromatic scale:
These seven tones correspond to the levers used to operate the hydraulis, the water organ. The associated seven physically equidistant levers for not quite sonically equidistant notes were the basis for the “white keys” of today’s keyboards. The distance or ratio between most notes (intervals) is 9/8, e.g. C to D or a whole step in our current thinking, and between some others about half of that, 256/243, e.g. between E and F, a semi-tone. This Pythagorean diatonic scale with its 7 tones (and equivalent tones in octaves above and below) constituted the tonal “vocabulary” of the musical language of that period. Note, however, that such a Pythagorean diatonic scale could be built on any base tone (or frequency).
Based on the tonal “vocabulary” practised up to the beginning of the 11th century, Guido of Arezzo introduced the first notation system that uses the notion of tone position. This was revolutionary as the system proposed more than indicative instructions but it was prescriptive. It allowed learning new music without having to have it heard before. Guido still uses neumes, but places them on 4 lines and their 3 spaces in between. Each of the 7 possible positions gets a name derived from the hymn Ut queant laxis: “ut–re–mi–fa–sol–la”. The last note “si” (or sometimes “ti”) was added a bit later to complete the diatonic scale.
Since the “ut–re–mi–fa–sol–la — si” scale can be built on any natural tone or audible frequency, this music notation can still be considered relative. There is still one degree of freedom left as the staff can be moved up and down in pitch as a whole, i.e. it can be transposed in order to adjust to a vocal range of performer or instrument. Therefore, some other device is needed to allow pinning written music onto a tonal reference frame.
In order to allow this “pinning” of a scale, Guido introduced what is now called a clef (French for key). A letter at the beginning of each staff and in front of a certain line, e.g. an F in front of the “ut” line, determines the absolute pitch by mapping that position on the staff to a note on the “ut-re-mi” scale and the chosen note, say, “ut”, to a reference pitch, say F, which would have to be a recognised tone at the time, e.g. from an organ.
In the following 200 years, Guido’s notation was further refined by various contributors, notably by Léonin and Pérotin from the School of Notre Dame¹, around the time of construction of the cathedral of Notre Dame de Paris. Five instead of four lines were used by then. Most distinctively, however, the square notation was created based on the new Gothic writing using certain types of quills; this is perhaps most recognisable.
The notation continued to be improved, e.g. by introducing the concept of rhythm through notes of different lengths by France of Cologne in his book “The Art of Measured Song”, written in the mid 13th century. During the 14th century Philippe de Vitry and Guillaume de Machaut extended the temporal aspect of music notation even further through additional refinement in time resolution¹. As a result, Guido’s music notation system was adopted throughout Europe and became:
- absolute, the combination of clef and position of a note on the staff determines the exact tone to be performed, and
- quantitative, the tonal space is quantized, and by virtue of a diatonic scale the intervals (or ratios) between each line are defined.
However, what is a virtue to some is a deficiency to others: introducing a clef adds complexity. A note’s position alone is not sufficient any more to determine its meaning; when sight reading, the performer has to keep in mind which clef (key) has to be used to literally “unlock” the note. As long as one remains in a single clef, there is little problem. Changing between clefs, however, adds unnecessary cognitive complexity.
Stretching the Concept
Towards the end of the Middle Ages, and in particular through the School of Notre Dame, some embellishments of the performance in chants were introduced in various ways¹: a) between sections of chants, but most importantly, b) synchronously with the chant in form of polyphony. At first, parallel voices were sung to the main chant, e.g. in fifths or even fourths. Later independent voices appeared on top of the traditional chant at the bottom, called Organum. Léonin produced a whole collection.
More Tones Needed
Polyphony comes with its own problems. Consider the simple case of embellishing a melody with a parallel motion in fifths, i.e. one singer sings the melody, another sings a fifth above. Most notes in the diatonic scale would have a corresponding fifth, e.g. the fifth from “ut” is “sol”. But the fifth on top of that last note “si” will land between “fa” (F) and “sol” (G), more about where F# can be found. A solution is simply to introduce more notes for more fifths such that for every note a pure interval to the next and previous note is available. The resulting ratios can be mapped into an octave starting at say C in the European diatonic scale. The result is a sequence close to the chromatic scale, which fills the gaps of the 7-tone Pythagorean scale, and yields more Pythagorean tones:
Allowing various fifths and fourths (fifth down from octave) results in two candidates for every “black key” on today’s keyboard. As it happens, these issues where already explored largely in the 16th century. Marin Mersenne proposed keyboards that are able to select 19 or even 31 tones per octave.
Settling for 12 Tones
During the Renaissance period musicians and scientists were searching for better tone systems²; eventually, the system settled for 12 tones per octave. A problem persisted: how should all individual tones be tuned? For instance, just intonation replaces some Pythagorean intervals with simpler but close to rational numbers, e.g. the major third (E to C) is set to 4/3 instead of 81/64 which has a positive side effect: a triad of say C-E-G would sound perfect. Conversely, a triad or a fifth built on top of E would sound terrible, producing so-called “wolf” sounds. Rolling Ball demonstrates the problem visually and sonically.
Fierce debates were carried out up to the Baroque era about the “perfect” tuning of 12-tone system², i.e. how to “temper” the Pythagorean or Just intonation, e.g. through mean-tone temperament and all its variation such as the Werckmeister temperament. Eventually, the Equal temperament was adopted, using equal ratios between all steps in the 12-tone scale. The celebration of this convergence towards a unified 12-tone scale is certainly Johann Sebastian Bach’s Well Tempered Clavier.
Mapping 12 Tones
A long evolution starting with Pythagorean fifths (ratio of 3/2) led to a system of 12 equally distributed tones in an octave for the European tone system. The main question remains unanswered: how and when was the medieval 5-line staff, that catered only for 7 tones, “extended” to allow for all 12 notes? It probably happened little by little from the early Renaissance to the Baroque. When B-flat or an F-sharp were added to the musical practise, the notation had to be extended somehow, e.g. by modifier symbols (sharp and flat signs). Also the concept of a “key” of a whole score was introduced at some point that makes a diatonic scale available at any new position of the 12-tone scale. The result was a notation system that is:
- absolute, the combination of clef, key (made up of tone modifiers) and additional symbols in front of the position on the staff determines the exact tone to be performed, and
- quantitative, the tonal space is quantized, and by virtue of a chromatic scale, the intervals (or ratios) between all notes are defined.
However, as before, what is a virtue to some is a deficiency to others: introducing modifier symbols (sharps and flats) as well as keys adds complexity. When reading music, the performer has to keep in mind which clef (key) and which key is used to literally “unlock” the real note, adding even more unnecessary cognitive complexity.
Towards a Consistent Notation
Unfortunately, at no point was the system completely re-designed, possibly because the changes happened gradually over many decades, even centuries, building up more and more legacy. Given that the Western tonal system has settled for 12 tones, however, it should be possible to compile a requirement specification for an alternative notation system based on common practise. The principal requirement is clear: 12 tones per octave need to be mapped to something where every note has its own place on a staff, unambiguously, and vice versa, to reduce cognitive complexity. Using “modifiers” in front of the original notes or at the beginning of a staff is a crutch from the past. In addition, the notation system has to cope for the entire range of notes in musical practise.
Alternative Notation Systems
There has been a stream of proposals³ over the past centuries, which in itself is a proof that there is something wrong with the established system. The Music Notation Project is a great resource for alternative notation systems as well as ChromaTone. Some of them use a similar approach with staves containing several lines but different note heads encoding whether a note is raised or lowered. Others are chromatic affording a position to every note on the 12-tone scale; some run the risk of becoming too large where it becomes difficult to make out any position at all.
The most promising alternative notation was Klavarskribo as it turned the notation by 90 degrees hence removing the mental burden from the conventional horizontal notation. Its inventor, Cornelis Pot, invested a lot of time and money in its development as well as in porting a lot of sheet music to the new notation. One downside is that it is difficult to use with lyrics. If my mind was a clean slate and if I had not been contaminated by any horizontal notation at all, I would go for Klavar, Mirck Version by Jean de Buur. It allows lyrics and the horizontal visualisation of the musical development while given every note its position.
Another alternative is Dodeka chromatic music notation system (and matching keyboard) with 4 lines per octave.
Impediments to Adoption
Most of the alternative notation system suffer from the problem that one could not approach them in an unbiased manner if one has seen and worked with the traditional notation system: certain positions became engrained in one’s brain, e.g. the C on bottom of the staff. Also, there is a large body of written music in use which constitutes inertia to change. The same reasons prevented perhaps the development of a new notation system once more notes had to be represented than Guido of Arezzo’s system allowed for: too much had been written in that system.
There is one argument counting towards the established system: the learning curve of traditional staff is not too steep. As one starts to learn reading music one would start with a simple C-major scale. Once comfortable, one can extend the scale as the musical ability increases, add B-flat, then F-sharp, in essence repeat the evolution of the current musical notation on an individual scale. If the notation system catered for the most complex case, it would have to be complex, too; consequently, the learning curve would be steep and become an impediment for adoption.
- Thomas Forrest Kelly, Capturing Music — The Story of Notation, W. W. Norton & Company (4 Nov. 2014)
- Stuart Isacoff, Temperament, Faber & Faber; Main edition (5 April 2007)
- Gardner Read, Source Book of Proposed Music Notation Reforms, Greenwood; Illustrated edition (3 April 1987)
- Evolution of European notation systems, from neumes to staves, https://chromatone.center/theory/notes/evolution/
(last visited January 16 2023)
- The Mathematical Problem with Music, and How to Solve It, https://www.youtube.com/watch?v=nK2jYk37Rlg,
(last visited January 18th 2023)