Hellenic Music Notation by Pablo Bellinghausen

Introduction

Although invented in the 16th Centur y, equal temperament has only been accepted as the sole musical tuning system for the last hundred years or so, superseding various others such as meantone and well temperaments. The whole of Western musical theory is in fact based on the assumption that the mathematical proportion between semitones is dependent on their key; in other words, in meantone and well temperaments tones, the intervals of an A major chord have a different mathematical proportion to those on a B major, and so on. This is not the case in equal temperament, now ubiquitous in both classical and popular music, so those precisions are thus now mostly redundant. However, that distinction has profoundly shaped the way music theorists perceive, view, and notate music.

The use of non-equal temperament systems has several consequences in musical notation: there is for example a need to differentiate sharps and flats (since, for example, C flat would be a slightly different frequency than B). Additionally, a key change becomes more complex just than changing the root note; either the interval relationships or the actual note pitches themselves have to be changed during modulation, limiting the number of good-sounding keys (although arguably enriching the musical vocabulary by giving rise to key colours, a concept not applicable in equal temperament), or introducing added levels of complexity for the player (e.g. key pedals in some organs).

Most musicians are nowadays completely unaware of this problem. We have grown used to the slightly out-of-tune major and minor thirds of equal temperament, and in general see little need to go back; nevertheless, the musical community has carried on with an age-old notation system, with all its redundancies and unnecessary complexity. Keyboards have black and white keys, reflecting the arbitrary importance of the C major key and its associated modes, but in a world where the proportions between all semitones are identical, the C major scale should get no preference, whether in note naming, in keyboard note position, chord notation, or staff position; writing on the key of F# on a traditional staff is by design more cumbersome than doing so in C, but the underlying reason is now gone. Some now-forgotten keyboards have been invented with equal temperament in mind in order to be equally easy to play in all keys. Arguably the most accomplished design, the Jankó keyboard (admired by Liszt and Rubinstein at the time of its creation for its ease of use and possibilities in composition, making modulations and transpositions as simple as moving the hands further up and down the keyboard) was ultimately unable to succeed in a world where all professional notation follows the structural idiosyncrasies of pianos around the world.

On the other hand, string instrument players are naturally in touch with the chromatic structure of modern music scales; the preference for C major in music theory often tends to baffle those who started playing fretted string instruments before or without learning how to

read traditional music. They have to put up with the eccentricities of the system, or else use tablature, limited in comparison, and unable to transcend the instrument and tuning for which the part is composed. In any case, as acquainted to chord shapes and note relationships a musician can be, there is a certain psychological veil between our current notation systems and the actual notes we hear, distorting and shaping the composition process.

When an individual hears a B major chord, their mind doesn’t understand it as the separate notes B, D# and F# named as such, unless it has been conditioned for years to do so. For someone without absolute pitch who is nonetheless used to the concept of semitones, the natural way of hearing a major chord is more like {0,4,7} – a better representation of the semitone distance between the notes in the chord. Melody and chord recognition are relative to a moveable tonic, not necessarily to C, but the traditional notation system does not reflect this at all, tying itself into a knot through layers of semantic substitution on most keys and scales.

The best way to look for a solution for the problem is to root notation in a fully chromatic theoretical system, in which each note has only one specific frequency, name, and position in a staff. Doing so has many advantages over its traditional counterpart: it is more consistent and coherent, makes transposition trivial, and lays out the chromatic structure of chords and harmony in a way more related to human musical perception. A good chromatic system also has the potential to be way more convenient and easy to learn.

Two Systems As discussed above, several alternatives have been proposed in the last century, particularly by serial music composers, but they don’t seem to have been easy enough to use and read, or perhaps haven’t been designed to offer any substantial advantages to composers already trained in traditional notation. A new system should be simple to understand for both classically trained and self-taught musicians, as well as easy to implement with current materials.

Such a reform should deal with three separate things, which could be implemented or adopted separately if necessary. The two main reforms I hereby propose have to do with the naming of the notes, and with their graphical notation. They are as follows:

– The Hellenic note system – Standard Symmetric notation

They combine into what we could call Hellenic notation, or the Hellenic system. Their extensive implementation is, of course, unlikely, but their use can prove convenient even if used in isolation. There is no need to buy or print special paper, or use invented symbols; it can all be done on traditional staff paper and there is software that, if properly configured, easily allows the musician to write and read any musical piece in Hellenic notation.

1. The Hellenic Note System Equal temperament has given us a set of pitches with a single measurable frequency we can use as a basis for a new set of notes. But their current names are based on a single diatonic scale, and thus inadequate for a fully chromatic one, so a new naming convention is required. Using the Roman alphabet runs the risk of confusing matters, since it’s already used for the traditional system. Several naming conventions already exist (notably Chinese and Indian), as well as the moveable do solfège that do account for semitones, but they either keep referring to the same old note privilege system or relate to tunings other than equal temperament. The best way to rename notes is to take a series of twelve memorable symbols, one for every semitone or pitch class, that have names allowing for solmisation (a recognisable syllable per note for singing purposes) and that can be written on computers without the need for new typography. The Roman alphabet, although utilitarian looking, was satisfactory, but not the only option.

After having tried several different versions with different symbols, names and solmisation options, I propose to use the Greek alphabet, just as we have used the Roman one for so long. It passes the requirements of memorability, distinctiveness, ease of use, simplicity of use with computers, and cultural inclusiveness, paying homage to the significant advances the Greeks made in early musical theory. It also has an inbuilt sequence, so it’s easy to remember adjacent notes. Since several upper case letters are identical to the Roman alphabet, using the lower case alphabet greatly clarifies matters, as well as being arguably more aesthetic.

If we determine the letter alpha (α) as the first note, so that α4 is equivalent to C4, the whole thing unfolds seamlessly. Solmisation can be derived from the first syllable of each letter’s name, with a couple of easy-to-remember adjustments. The rather convoluted zeta symbol can be replaced with the roman letter z for ease of writing whenever needed. We therefore end up with the following table:

Other areas that are deeply affected by our current naming system are intervals and chords, the names of which at present have little consistency. Their reliance on a diatonic scale can render chord names often unclear and inaccurate, and their symbols are sometimes cumbersome – especially for extended or non-standard chords. New names and symbols for chords should therefore be used alongside our new set of chromatic note names, although since

these will always be preceded by a Greek letter, some of the old denominations can be borrowed wherever it makes sense to do so without making things confusing. I also propose to use the term “strict” to indicate that the following interval is in the number of semitones, instead of scale degrees on a diatonic scale. Here is a list of those terms, which can be used for and clearly define both intervals and related chords:

Old Name Pitch Class New Symbol New Name Solmisation (IPA phonetics in grey)

C 0 α Alpha Al |al|

C# 1 β Beta Be |biː|

D 2 γ Gamma Ga |ga|

D# 3 δ Delta Del |del|

E 4 ε Epsilon Ep |ep|

F 5 ζ / z Zeta Ze |ziː|

F# 6 η Eta He |hiː|

G 7 θ Theta The |θiː|

G# 8 ι Iota Io |yoʊ|

A 9 κ Kappa Ka |ka|

A# 10 λ Lambda Lam |lam|

B 11 μ Mu Mu |mjuː|

Accidentals disappear both in notation and in naming. Key names can be used, but they no longer alter any notes themselves. A “major” interval is always four semitones, unlike traditional thirds which can be three or four semitones depending on key.

Extended intervals and added chords (above an octave) are written after an apostrophe, which can be omitted if the relative pitch class is lower than the preceding one. Thus, a “minor λ chord with added flat” {0,3,7,22} includes the apostrophe λm’f but a “major λ chord with added second” {0,4,7,14} is λMs instead.

Inversions are also spelt out; the bass note is just written in subscript (or after a dot if subscripts are unavailable) after any added notes. For example, “the second inversion of C minor with added ninth” {-5,0,3,7,14} is an “alpha minor with added second and dominant bass” and written αms^ instead of Cm2add9.

One could even abbreviate by using solmisation, saying “add” plus the relative pitch class of the added interval, as well as bass notes, by saying “dot” plus the relative pitch class. The aforementioned chord αms^ would be called “Al minor add two dot seven”.

Traditional extended chords lose their particular name; Hellenic notation just specifies the type of leading-tone through f or d, still shorter than ext, and removes any guesswork in traditional notation as to whether any mediant and leading notes are major or minor.

A hybrid chord such as a “minor with major seventh” {0,3,7,11} is a “minor divergent”. Constructions using flats and sharps are similarly simplified; for example, G7♭5♭9 is written as θ*fI instead.

Since leading-tones are specified, a “dominant ninth” {0,4,7,10,14} is a “major flat second”, or a “major flat two”; C9 is written as αMfs

instead.

An “added ninth” {0,4,7,14} is a “major chord with added second”, or “major two” (since the second is a lower pitch class than the major the “added” is implied), and the symbol Cadd9 is αMs instead.

All in all, the Hellenic note system is more straightforward, flexible, and logical, and although superseding the traditional system is likely unrealistic, its ease of use can ultimately benefit those who make the choice to learn and use it.

Interval in Semitones

Chord in Semitones

Interval & Chord Name

Strict Interval Name

Interval & Chord Symbol (using α as example)

{-1,0} — divergent bass strict eleventh bass αd

{0,0} — unison unison α=

{0,1} {0,1,7} first strict first αI

{0,2} {0,2,7} second strict second αs

{0,3} {0,3,7} minor strict third αm

{0,4} {0,4,7} major strict fourth αM

{0,5} {0,5,7} suspended strict fifth α∨

{0,6} {0,3,6} diminished / tritone strict sixth α*

{0,7} {0,7,12} dominant / power strict seventh α∧

{0,8} {0,4,8} augmented strict eighth α+

{0,9} {0,7,9} convergent strict ninth αc

{0,10} {0,7,10} flat strict tenth αf

{0,11} {0,7,11} divergent strict eleventh αd

{0,12} {0,12} octave octave αx

{0,13} {0,7,13} added first strict added first α’I

3. Standard Symmetric Notation Now armed with a new set of notes without arbitrarily-set scales, we it is important to find a new chromatic notation that would improve not only on readability and efficiency, but also hopefully preserve the aesthetics and elegance of the current system in both printed and manuscript forms. The traditional notation system attempts to optimise readability by graphically omitting five of the notes and using accidentals instead, reducing the list to a more manageable seven. But if we’re to create a truly chromatic notation, we require something that allows for a kind of quick visual interval discrimination that doesn’t vary with the chosen key.

The ideal would be a system to make it easy to visually recognise different intervals when reading. For that effect, there are already a number of hidden patterns in the standard staff that are not being exploited by current notation but that we can use ourselves. The usual grand staff looks like this:

There are several axis of symmetry that can be exploited on it. We can for example divide the top and bottom staves down the middle ledger “middle C” and further split them in half, leaving us with four 3-line blocks. It is strange to think that such a remarkable property is ignored by the traditional notation system; even though middle C does split the grand staff in two, the non-symmetrical inner structure of the diatonic scale means that the bass and treble clefs don’t have the notes in the same place at all, forcing the musician to learn several note positions per clef on top of dealing with keys and accidentals. How about we use those axis of symmetry so that notes of the same pitch class always fall on the same position?

That would mean we would place α at the centre of both the staff and the double staff. We therefore have three lines per octave we can play with. A single line must then harbour 12/3=4 separate notes.

The traditional note shape is an oval, which in handwriting can go from almost circular to almost a line. It’s easy to write and to read. Squares and more complex polygons are harder because in a lot of handwriting they look pretty similar to a circle, but after several tests using different pens and small music paper, I’ve decided that great results can also be had with a triangular note head.

A particularly useful characteristic of triangles is that they have an up and a down shape, meaning we can use 2 distinguishable shapes on the same line, and thus make the note head bigger. If we combine circles and triangles, we end up with a system that can host four notes per line, as follows:

The staff ends up fitting two octaves of notes and looking like this:

This is both a lot cleaner and easier to read than the traditional notation system with all of its accidentals. There are even more benefits to this notation system however. First of all, the system is more compressed without sacrificing note size, which means fewer ledger lines are required (we’ll be able to almost get rid of them almost completely through another writing device explained later on). Also, interval and chord recognition become both easier to read and independent on key. For example, a major interval (four semitones, or major third in traditional theory) can be recognised by having both notes of the same shape and position one on top of each other.

A suspended interval, or five semitones, will always look like one of these four shapes:

In musicology and serial music, pitch class set transformations are common, and this system allows those to be made geometrically. For example, an inversion can be done by finding the geometrical reflection using an α as the centre. Thus, the reflection of, say, a κ (pitch class 9), will be δ (pitch class 3), which is indeed the correct class inversion. This property is the reason for the term “symmetric” in the name.

The notes can be filled or hollow, with any length value graphical cues remaining the same: rests, stems, flags, and hooks can remain identical to the traditional notation system.

The combination of Standard Symmetric notation with the Hellenic note system is a clear improvement in readability, creative possibilities and consistency. It also uses a standard staff (hence the “standard” in the name), which is a major advantage to other notation systems – standard music notebooks and paper can be reused with this system without modification.

Clefs Clefs are necessary in traditional notation to indicate note position, but they are unnecessarily complex to learn and use. Notes in the G clef are in different positions to those in the C or F clef, 8va transformations are not just geometrical translations, and no further staves can be added at the top or bottom, which in music using notes in several octaves involves the use of an unreasonable number of ledger lines. Standard Symmetric notation does away with these problems, removing the need for clefs entirely, making transposition exceedingly simple, and giving the composer the possibility of adding staves above or below the first one and still have the octaves of the same note in the same relative position.

Although clefs as note position indicators are no longer necessary, the C clef (or α clef ) works perfectly in standard symmetric notation: the centre of this conveniently symmetric clef symbol is the centre of our new staff, and since staves themselves are symmetrical, a grand staff becomes a double Alpha staff. The only thing that needs to be established is the octave at the centre of the staff. After attempting many different ways of doing so, like using accidentals to denote different octaves similarly to the way Helmholtz pitch notation uses commas and apostrophes, I think I have found a method that is clearer and more aesthetic, as well as more adaptable in real-world situations.

Keeping in line with the concept of joining the old and the new together, this method uses Roman numerals to denote the scientific pitch notation of the clef ’s Alpha, by writing it at the centre of the staff. Taking into account middle C is an α4, an α clef to supersede the standard G would have a Roman five next to it to indicate the middle of the staff is a C5. We can of course do the same for the bass clef which becomes an α3, and we could add further staves up and down at will, having triple staves if we so wish, without having to learn any new note positions. Transposing for different instruments becomes a thing of the past, and orchestral compositions are thus made easier to manage; a violin player can as a result read parts written for the viola and vice versa without a problem.

Roman numerals have another nice touch to them when dealing with manuscript composition. All too often we start in one octave to realise later on that it should have been an octave higher or lower. But a roman V is simple to transpose, by adding an I before or after it to neatly transpose the whole project up and down. It can even be done two octaves up or down, through the unorthodox but still clear use of IIV as III). Traditional 8vb and 8va, although less practical, can still be used for software notation that does not allow for customised staves.

When the specific octave hasn’t been decided yet, relative octave modifiers can be employed in much the same way: using the old ♭ and ♯ accidental symbols for upper or lower octaves, and the natural symbol ♮ to go back to the first one. This method actually works so well it can even be used in the middle of a bar, eliminating the need for more than a few, very occasional ledger lines. This quickly becomes so practical ledger lines themselves start to feel rather primitive and cumbersome.

Conclusion There are two ideological characteristics about this staff system which I find rather important. The first is that for the first time there is a simple mathematical and physical correlation between a note in a page and a frequency, without any transposing and juggling. A symbol on the page is clearly and precisely a signifiant for a specific pitch, bringing sound and music together, without making any sacrifices on intelligibility. The other is that here is finally one good-looking, simple and flexible staff for every instrument that is both familiar-looking and innovative. Piano, guitar, cello, violin players can all lift a piece of paper and know exactly what they are reading, without the need to mentally transpose two or three times every time there is a note on the piece of paper.

This is arguably a much-needed evolution in music, and in this digital day and age, in which most musicians and most music listeners know how to use an equaliser and note choice feels as free as ever, bringing everything we know about sound into one single piece of knowledge is a commendable goal. If by doing this we improve a critically important musical tool which is currently sadly underused by self-taught musicians, incidentally the one tool which according to many historians is the very thing that separates popular from art music, then that can only be a good thing.

Appendix № 1

This is an excerpt of Franz List’s Hungarian Rhapsody № 2, a playful piece that is particularly cumbersome to write in traditional notation due to its choice of key, as well as its chromatic nature. Note the frankly preposterous number of ledger lines, clef symbols, sharp and natural intonation markers, and the cumbersome nature of octave transposition.

Appendix № 2

This is the same passage written in Hellenic notation. This image was created in Finale, which has a few limitations with regards to transposition: the Roman numerals denoting the octave are impossible to write after the clef, and there is no way to change this per bar, which forces the software to use more ledger lines than strictly required. It is however obvious that the notation has been greatly simplified, making the whole thing is easier to parse, and ultimately far simpler to read.