Tonnetz Torus — Circles of Thirds

ABSTRACT: This post describes a diagram or structure that can be used to identify the notes of major and minor triads (and other thirds-based chords) and the notes of the diatonic major and minor scales. This structure is based on the Euler Tonnetz (tone network), which extends the concept of the Circle of Fifths to also include the major and minor thirds. I represent the tonnetz as sets of intersecting circles of major and minor thirds in a donut shaped diagram, or torus. I call this the Tonnetz Torus or Circles of Thirds. I have also built a conceptually equivalent three-dimensional version using triangles and squares, resembling a house, which I call the House of Thirds.

House of Thirds

  F C G
b       D
e       A
a       E
  d g B
The Circle of Fifths (shown above, where lower-case letters represent flats) is a well-known diagram for constructing key signatures and harmonic progressions. But there is a little-used diagram that is even more powerful: Euler’s Tonnetz (tone network). This diagram displays not only the fifths, but also the major and minor thirds. Because it displays the thirds in a certain order, it can also be used to identify the notes of the major, minor, augmented, and diminished triads and sevenths (or any chord formed from “stacked” major and minor thirds), and also the notes of the diatonic major and minor keys and the principal chords in those keys! The Euler tonnetz is a hexagonal pattern that displays not only the fifths (across), but also the major and minor thirds along the forward and backward diagonals, respectively.
e b F C G D A E B g d a e
 g d a e b F C G D A E B g
  A E B g d a e b F C G D A
   C G D A E B g d a e b F C
    e b F C G D A E B g d a e
In the equal-temperament system, if we equivalence all octaves of the same tone, the tonnetz diagram can conceptually be stretched and folded into a donut shape (torus) composed of four cycles of minor thirds intersecting with three cycles of major thirds. I call this the Tonnetz Torus or Circles of Thirds.
      B g
     G D A
    e b F C
     g d a e
      A E B
       C G
Notice that the upper left and lower right diagonals are identical, as are the lower left and upper right. Imagine rolling the figure into a tube to match up the identical sides, then stretch the tube and bend it into a donut, matching up the ends. You will then have four circular rows of three notes in one direction and three circular rows of four notes in the other. One can then follow a path along the circular arcs, switching from major to minor circles in a fixed pattern, to identify the notes of the major, minor, augmented, and diminished triads and sevenths (or any chord formed from “stacked” major and minor thirds), and also identify the notes of the diatonic major and minor keys and the principal chords in those keys! Of course, the circular representation of the cycles of thirds is purely an artifact. If the points are connected with straight instead of curved lines, one would have four triangles and three squares. These can be rendered as a three-dimensional structure somewhat resembling a house, the House of Thirds. I have constructed a simple model of this figure using gumdrops and colored toothpicks. (See photo near the beginning of this post.) I used red, yellow, green, and blue toothpicks for the sides of the triangles, and I used white, orange, and black gumdrops for the notes, with C=orange for consistency with Roy’s Pertchik’s tri-color keyboard coloring. Each of the four triangles has sides of the same color and three different colored nodes. Each of the four squares has nodes of the same color connected by natural color toothpicks. There is also a “direction” around the circles/triangles/squares. [If available, one could use coctail toothpicks with feathers or hilts on one end to suggest arrows to indicate the direction.] In my structure, I follow a “direction” through the cycle of colors, as follows: Along squares the direction is: blue -> green -> yellow -> red -> blue (reverse spectral hue order), along triangles the direction is: black -> orange -> white -> black (increasing brightness order). To label the notes, I pasted on typewritten letters. I used lower case letters for the black notes, understood as flats. The circle of fifths could be added to the structure, e.g. by threading a strand of yarn (or sticks of spaghetti) through the gumdrops in the appropriate sequence, major+minor (= minor+major): g d a e b F C G D A G B F#=g This structure could be constructed by children as a fun lesson in harmonic structure! (To preserve edibility when working with children, you might use cake decoration icing for labeling the notes.) — The notes of scales and chords can be determined by following a path in the direction of the cycles: A major triad (major minor) follows a path: triangle square. A minor triad (minor major) follows a path: square triangle. For a diatonic major scale (IV I V7), start with the subtonic, create the intervals: major minor major minor major minor minor: Trinagle square triangle square triangle square square. For example, for the C major scale, the principal chords are: IV (FAC), I (CEG), and V7 (GBDF). start on the subdominant note F (yellow triangle, black square) and follow the path described by the intervals above: follow the yellow triangle [major] to the orange square (A), then the orange square [minor] to the red triangle (C), then red triangle [major] to the white square (E), then the white square [minor] to the blue triangle (G), then the blue triangele [major] to the black square (B), then the black square [minor] to the green triangle (D), then the black square again [minor] back to the yellow triangle (F). Following the same path, but starting on D, gives the minor chords (ii vi iii7) and the relative minor scale. The minor scales (relative, harmonic, and melodic) also consist of a pattern of 3 major and 4 minor thirds, which line along other paths around the torus. For minor scales, start on the subtonic of the minor (ii of the relative major), e.g. start on D for A minor, the relative minor of C ). for relative minor (ii vi iii7) , use intervals: minor major minor major minor major minor (D F A C E G B D). [This is actually the same path as the major scale, but starting on D instead of F]. for the harmonic minor (ii vi III7): minor major minor major major minor minor (D F A C E G#=a B D ) for the melodic minor ascending (II vi III7): major minor minor major major minor minor (D F#=g A C E G#=a B D). Note that all scales, when constructed from thirds, contain three major and four minor intervals, forming a cycle of seven notes. Because of the uneven structure of the pattern, starting on a different note will always give a different set of diatonic scales, and corresponding principal chords, for each of twelve notes of the chromatic scale. Other chords can be formed in a similar way. Examples: Augmented: major major Diminished: minor minor Seventh: add a minor to any major or minor triad Dominant seventh: major minor minor Minor seventh: minor major minor Major seventh: major minor major Dim seventh: minor minor minor Min 7 flat 5: minor minor major All these can be read from the House of Thirds by starting on the root, then following the triangle edge for a major third or the square edge for a minor third.

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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