In early 2016 I began creating a new music notation that would evolve over the following months into what is now called pitch bracket notation. I wanted a notation that visually captures the intricate mathematical processes and patterns at play in classical music.
I have always been fascinated by the complex patterns of pitch and rhythm found in classical music. I could hear these patterns in performances but I could not ‘see’ them in traditional music notation. So I decided to create a new music notation which visually communicates to me the beautiful sounds of classical music.
The documents reproduced here show various stages of development. I should point out that my early experiments lead to a great of deal of frustration and so many early sketches have been lost to the trash bin.
Music develops in the dimensions of time and space. In the context of music, space is pitch. So I chose a two dimensional array of symbols to represent music. All symbols in a given column occur concurrently whereas all symbols in a row occur sequentially. Each measure appears in a box. Within the boxes the top two rows represent two melodic voices with notes written in scale degrees. Chords are written ‘in-line’ with hyphens distinguishing them from a melodic sequence. The bottom, rhythm row represents note duration for all the notes appearing in that column. The letters, A through H, are assigned to note values 1, 1/2 , ¼, 1/8, etc. The meaning of upper case and lower had some particular meaning which escapes me now.
The next document show only a few changes. Each voice now has its own rhythm row. Notes may have a line above or below them to signify the note an octave above or below. The hyphens joining chords disappear and are inferred by the lack of a rhythm symbol in those columns. Small vertical lines appear on the bottom rhythm row to distinguish beats.
Here I began to expand the symbol vocabulary. Notes that are sharp or flat are written with uppercase or lowercase letters respectively, while non-modified notes are written with numbers 1 to 7. Lines above and below a note still raise or lower by an octave. At the bottom I am starting to replace the rhythm letters with non-alphanumeric symbols. This was necessary if the upper and lower case letters were allocated for sharps and flats.
I knew I wanted to be able to communicate relative pitch within a melody. To include this in my symbol arrays, I started writing absolute pitch with letters and relative pitch with numbers. I allocated letters A to U for the absolute notes A3 to G5, a three octave stretch. Then numbers following a letter represent offsets in scale degrees. Rhythm letters are replaced with non-alphanumeric characters.
I started using uppercase and lowercase letters to distinguish between something. I can’t seemed to decipher what my intentions were here. The rhythm symbols became more exotic. At this point, I realized how convoluted and unintelligible this was. I had to think in a new direction.
I drew lines with dots for notes. These horizontal lines are not staff lines. Rather, they are lines connecting the notes of a melody. Notes on the same line occur consecutively. In order to show the change in pitch from one note to the next I began writing short vertical lines. These are like staff lines turned 90 degrees. Multiple vertical lines changed pitch by multiple scale steps. At the bottom left is a table of intervals written in this scheme. Notes could have a vertical line or no vertical line which corresponds to a staff line or space. The rest of the documents shows my attempts at polyphony using this scheme. I also used unfilled dots to signify something which I don’t recall.
There were two important problems with these scheme; the pattern of vertical lines between two notes depends not only on there interval but also the starting note, whether on a slash or not. The second problem was that the direction of a melody could not be shown. The interval between two notes may be a 3rd, but is it a 3rd up or a 3rd down in pitch?
At this critical point, I began bending the vertical lines in the direction of increasing pitch. This made the direction of the melody obvious. From here on, I will refer to these vertical lines as pitch brackets. After many experiments with pitch brackets, I realized how tedious it is to write many of them between two notes. I first created what you might call an “extended pitch bracket” which you can see in the middle of this document. These simply multiply the number of bracket by the enclosed integer. I also continued experimenting with empty dots, in this case used to indicate a minimum or maximum pitch. At the bottom of this document you can the first b2 and b3 brackets.
By the time I created this document, I invented the complete set of pitch brackets. The table at the top right is pretty much complete. The rationale for this set of symbols was the following: parenthesis has one “side”, a bent line, the angle bracket has two sides, the square bracket has three sides. The bracket pairs with matching shapes, 1/6 and 2/5, and 3/4, add to 7 which is the octave. The larger brackets have a bold, filled-in appearance. The octave bracket, b7, resembles a combined b3/b4.
With this new set of brackets I began experimenting. I tried to write polyphonic music with vertical brackets making chords and horizontal brackets making melodies. To the right, I tried combining opening and closing bracket pairs into various closed shapes (circles, diamonds, etc). I also wrote sharps and flats with diagonal slashes through notes.
I continued experimenting with pitch brackets. This document shows my attempts at combining adjacent pitch brackets into various shapes that surround a note.
I began to recognize the value of matching pitch brackets, that is, pairs of opening and closing pitch brackets that construct a hierarchical pitch structure. I continued to experiment with polyphony, but I started focusing on single voice melodies. The experiments on the left are from Beethoven’s 5th symphony. An interesting side development at this point was the idea of “pitch shapes”. By connecting matching opening and closing pitch brackets, a nested sequence of shapes emerges. The example at the bottom is quite beautify although not as practical as I would have liked. The problem was that too much vertical space was taken by these shapes. The more complex the melody, the taller they grow.
This document shows my first experiments with pitch bracket algebra. I started with addition and later multiplication.
I hope the details I shared here encourage people to explore their own creative aspirations. While at times I was quite frustrated and hopeless during this process, my passion for music kept me going to reach what I believe is a beautiful new way of communicating and thinking about music.
Over the following months I wrote two papers with many further example. I will post these at a later time.