Tonnetz-Based 4×3 Notation

(Revised 2/25/2011 to correct errors in the Bach example.) (Revised 12/13/2010 to add Chromatic Scale example and correct an error in the Bach example, and other editorial revisions.) I’ve created a font to print music in a Tonnetz-based 4×3 notation. The notation is based on Euler’s Tonnetz (described in more detail below) which emphasizes the relationships of major and minor thirds.  Each note of the chromatic scale is represented by a notehead of a different shape/orientation and color.  Noteheads of the same shape belong to the same augmented chord (notes a major third apart) and noteheads of the same color belong to the same diminished seventh chord (notes a minor third apart).  The following illustrates the chromatic scale in this notation, on a special chromatic staff with staff lines vertical, a line for each “black” note and a space for each “white” note: The next illustration is Bach’s Prelude 1 from the Well-Tempered Clavichord.  
Bach Example

Bach WTC Prelude 1 in 4x3 Tonnetz

st7-5BachWTC1P1ref02250211 The first 33 measures are each condensed into a single 5-note chord.   This illustrates the benefit of the Tonnetz foundation, by which intervals of the chord can be discerned by matching shape and  color: major thirds have the same orientation, minor thirds have the same color. The Tonnetz (tone network) was developed by the mathematician Euler as a representation of the harmonic relationship of musical notes. The tonnetz is a hexagonal arrangement of the tones of the just tempered scale based on their frequency ratios, principally the perfect fifth (2:3), the major third (4:5) and the minor third (5:6). The same pattern can be reinterpreted in terms of the equal-tempered scale using the customary definitions of Perfect Fifth, Major third, and Minor Third as intervals of 7, 4, and 3 semitones, respectively. In the Tonnetz arrangement, the tones are arranged in a hexagonal honeycomb grid along three principal diagonals:  along one diagonal the tones are arranged in the sequence of the circle of fifths; the same sequence of tones lie along parallel rows, offset so that tones a major third apart lie on one diagonal and tones a minor third apart lie on the remaining diagonal. The three notes of a major or minor root-position triad are thus clustered in a triangle. One can imagine a musical notation based on the Tonnetz. Each of the twelve tones of the chromatic scale would be distinguished by a pair of characteristics, an “X” and a “Y”, the X’s in cycle of 4 and the Y’s in a cycle of three, such as: A1 B2 C3 D1 A2 B3 C1 D2 A3 B1 C2 D3 A1 … . Thus, all notes a Major 3rd apart are the same “X” (here, letter) and all notes a minor 3rd apart are the same “Y” (here, number). For example, if A1 is “do”, the next A is a major third up (A2=”mi”), the next 2 is a minor third up from that (D2=”so”). I use a triangle shaped notehead symbol cycling through 4 orientations (X=up, right, down, left) and 3 colors (Y=black, gray [crosshatch], white).  One orientation cycle = major third (4 semitones); one color cycle = minor third (3 semitones).   The noteheads thus provide a visual indication of the harmonic relationship among the notes of a melody or chord, without explicit  reference to pitch. I borrowed the 3-color idea from Roy Pertchik’s patented tri-chromatic keyboard system. Roy has also developed a note-naming system based on the 4×3 structure of the Tonnetz.  Enrique Prieto, John Keller and others have also suggested similar symbol systems.  For further discussion of these topics, refer to the forum of the Music Notation Project In this system, each pitch of the chromatic scale has a unique symbol.  The notehead system does not use staff position per se, so it can be adapted to any staff system.  In fact doesn’t even need a staff, if some other mechanism is used to indicate the octave. For illustration, I use a 7-5 “piano roll” staff that mimics the conventional piano keyboard. For convenience of editing/printing, I have used a “Klavar” style staff with the lines running vertical and pitches horizontal (discussed in prior posts), but the page can be rotated to read it horizontally if you prefer. Although the system uses color for pitch, it does not use stems or flags, so these could still be used to indicate timing and parts. This example was produced entirely with a spreadsheet, word processor, and home-made custom font as described in my earlier posts.  The font is available on as StaffTonnetz1b. To create sheet music, 12 staff positions are represented by  “line” characters: open bracket [ for gray line: Db Eb; close bracket ] for black line: Gb Ab Bb; and vertical bar | for “space”: ABCDEFG. A comma represents a 1/2 space between line characters.  To place a note on a line or space, just type the letter in place of the comma before the line or space on which it is to appear, using lower-case d e g a b for the flats and lower-case c f for sharps.  [d g a sharp and other accidentals must be entered as an enharmonic]. You could use some other staff-line pattern if you choose.

About DrTechDaddy

Dr Tech Daddy is a retired computer science professor with additional interests in music, robotics, STEM education, model railroading, mathematical physics, congenital heart disease and heart transplant, and Christian theology.
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