Music Harmony and Dounuts

Dr. Paul Dysart

Home

A monograph on conventional triadic harmony and scales unique in its approach to the subject through its persistent use of graphical illustrations. The topics covered are unified throughout by their constant connection with the Tonnetz - a network of musical notes first described by the German mathematician Euler in 1739. Nearly all musical concepts are displayed graphically using the Tonnetz as a workspace.

Although the discussion might seem academic, it is instead just anotherway to think about music harmony and scales. There's really nothing new in substance, only in its presentation which relies entirely on pictures and graphs. The presentation is structured like many other self-instruction books on music harmony with chapters on intervals, scales, chords, and harmonic progression. However in this case, the Tonnetz is used to graphically explore these and other musical elements such as prime scales, tonic centers, and modes without the use of conventional music notation. The absence of any music notation is by design, since it is fundamentally an attempt to describe music harmony via a new - but very old - visual system.

Music theory and self-instruction books commonly use a graphical device known as the circle-of-fifths. The circle-of-fifths arranges the notes of the chromatic scale on a circle in a sequence of fifth intervals that repeats after 12 notes. The circle-of-fifths is widespread in its use as a graphical aid in the presentation of musical concepts, and for many the circle-of-fifths is particularly appealing due to its presentation in the form of a simple geometric shape.

The use of simple graphical objects such as lines and circles to illustrate theories and concepts is a common and effective teaching strategy. The circle-of-fifths has been used in this way to visualize many musical abstractions. However, the circle-of-fifths is only one of several possible slices through a 2-dimensional surface first described long ago in the music literature as the tonnetz. The underlying shape of the tonnetz as recognized later by others is a torus - the shape of a donut. Although it's true that a circle is a cross section of other surfaces such a sphere or a cone, it is the torus that contains the circle-of-fifths and all the other intervals too. This musical donut has been used over the years to support musical developments of the most abstract kind 1. However, as a framework for displaying shapes and patterns, it can also be used to illustrate many common musical concepts, and help one visualize relationships that are often difficult to see on the grand stave. In the following, the tonnetz and its variants are used to explore the geometry of music in a way that can help illuminate the space through which music travels.



The Traditional Tonnetz

The tonnetz, a German phrase meaning 'harmonic network', has appeared in many forms since it was first used by the eighteenth century German mathematician Leonard Euler 2. An early version of the tonnetz developed by Riemann is shown in Figure 1-1, and a similar representation appeared as a table in the work of Oettingen 3.

Music theory and self-instruction books commonly use a graphical device known as the circle-of-fifths. The circle-of-fifths is simply an arrangement of the notes of the chromatic scale on a circle in a sequence of fifth intervals that repeats after 12 notes. The circle-of-fifths is widespread in its use as a graphical aid in the presentation of musical concepts, and for many the circle-of-fifths is particularly appealing due to its presentation in the form of a simple geometric shape.

The traditional form of the tonnetz similar to the one show in Figure 1-1 can be assembled in just a few steps by arranging the 12 notes of the chromatic scale on a 2-dimensional grid of rows and columns beginning with the circle-of-fifths. This is illustrated in Figure 1-2 where the circle-of-fifths is first disconnected and laid out in a straight line. The first row of fifth intervals could have started with any note as long as all 12 notes are included 4. In the next step, the same row of fifths is shifted and placed below and between the notes in the first row so the notes are a minor third apart. This pattern is repeated again below the second and third rows. It soon becomes apparent that the columns are moving down by half steps, so the columns can be filled-in with the remaining notes, and the result is the traditional form of the tonnetz.

Figure 1-1 Early version of the tonnetz by Hugo Riemann, "Ideen zu einer 'Lehre von den Tonvorstellungen,'" Jahrbuch der Bibliothek Peters 21-22 (1914-1915): 20.  in Hyer, Brian. 1995. "Re-Imagining Riemann." Journal of Music Theory 39(1): 101-38.

Figure 1-2 Construction of the traditional tonnetz beginning with the circle-of-fifths.


Surface Form

The tonnetz was first seen as a flat plane that extended infinitely in all directions. However, it was Lubin that first recognized that this arrangement of notes could be formed into the surface of a torus 5. This can be seen in Figure 1-2

where the last note at the bottom of each column of half steps, like the disconnected row of fifths, picks-up where the first note at the top of the column left off. This suggested that the rows and the columns could be connected into circles.

To see this more clearly, imagine that the grid is drawn on a sheet of rubber that can be rolled and stretched as shown in Figure 1-3. By rolling the sheet into a cylindrical tube and connecting the columns, the half-step columns now go around the tube in continuous circles. The M3 and m3 intervals can also be traced around the tube, but they follow a spiral path that still runs off at the ends. The rows of fifth intervals also run off at the ends, so the final step shown in Figure 1-3 is to connect the ends of the tube together by stretching and bending the tube into the shape of a donut. This connects the rows of fifth intervals into continuous circles again, and also connects the M3 and m3 intervals into spirals that now trace a continuous path around the donut.

Figure 1-3 Steps in the transformation of the traditional tonnetz into a two-dimensional closed surface.


Volume Forms

More recently, music theorists have considered full 3-dimensional networks such as the one shown in Figure 1-4. For example, the network at the top of the figure developed by Gollin is an arrangement of the notes in a network of pyramids 6. Other 3-D forms can be constructed by adding another axis to the traditional tonnetz that extends into the third dimension. This results in cubes like the one shown at the bottom of the figure. Like the donut, these are infinite volumes that can be joined at their faces to form 4-dimensional objects called hyper-cubes7.

The solid forms rendered in perspective as they are in Figure 1-4 are generally difficult or impossible to use as graphical aids since they can't be viewed from the inside out, or from different sides. However, we will discover later that 4-dimensional extensions of the tonnetz are necessary to represent certain musical elements. Fortunately, these can be constructed as projections onto a flat plane as we will see in the next chapter. These planar forms are much more practical to work with as long as the underlying structure is kept in mind.

Figure 1-4 Three-dimensional versions of the tonnetz.


Whole-Step Axis

Figure 2-1 gives a close-up look at the traditional tonnetz, and the intervals captured by this arrangement of notes. The main axes in this version of the tonnetz are those that pass through intervals of a fifth and a half step, and another set of diagonal axes that pass through major and minor third intervals. A number of examples of single intervals are shown in the figure as short line segments connecting two neighboring notes. An interval can actually point in either direction, so an interval line represents both the interval and its inversion, e.g., a 5th in one direction and a 4th in the other. The set of intervals/inversions shown in the figure includes the major third M3/#5th, the minor third m3/6th, half-steps (b9th/7th), and 5th/4th intervals. There are some intervals that don't fit along the main axes, such as the whole-step interval which crosses between the 3rd axes or jumps a fifth, and the tritone interval, shown as a jump along the m3 axis. Together with the intervals along the main axes, this is the complete set of intervals for the even-tempered scale.

Also shown in Figure 2-1 are examples of major and minor triads formed by combining a M3 and m3 interval. Major triads, for example G-Major (G-B-D) always look like a triangles when all the notes are connected, and minor triads like the F-minor triad shown (F-Ab-C) all have the shape of an inverted triangle.

Whole-Step Network

Although some intervals that don't fit along the main axes in Figure 2-1, the traditional tonnetz can be drawn in other ways by rearranging the notes to produce a new set of main axes. Figure 2-2 shows another traditional rendering of the tonnetz where the whole-step interval appears as one of the main axes. Again, the shape and orientation of intervals do not change.

Figure 2-1 Intervals and triads on the traditional tonnetz

Figure 2-2 Intervals on a Whole-step tonnetz


Hyper-Tonnetz

The traditional forms of the tonnetz are effective workspaces for representing musical elements such as intervals and triads. However,they are less effective in representing more dynamic musical elements.  A more useful form of the tonnetz is one where a third dimension has been included in some way. The hyper-tonnetz shown in Figure 2-3 can actually be formed by simply overlaying 2 whole-step networks. However it is more interesting to see where this version of the tonnetz actually comes from by starting with the 3-D cube again in Figure 1-4

. Figures 2-4 show the construction of the hyper-tonnetz.

The main axes in this version alternate between two intervals - a HS and m3 along one axis, and #5 and WS along the other.  As shown in Figure 2-4,  the overlay is actually a projection of a 3-D tonnetz onto a flat plane. This is what you would see from the inside as you looked into the 3-D cube. If you can imagine yourself inside the 3-D volume, the highlighted notes on the hyper-tonnetz are where you can travel without passing through another note.

The two colors allow movement in all directions. Along either color individually, the axes move through intervals of a whole-step and a major 3rd.  By alternating colors, one can move through the volume along interval axes that alternate between a m6 and a tritone, or a half-step and a minor 3rd.

Figure 2-3 The hyper-tonnetz as an overlay of whole-step networks.

Figure 2-4 The hyper-tonnetz as a projection onto a flat plane.


Properties of the Tonnetz

It's worth mentioning some of the general properties of the tonnetz networks that make them so useful. The first of these properties is periodicity. Examples of this important property can be seen in the traditional tonnetz where the fifth and half-step intervals repeat after 12 notes. Likewise, intervals along the other axes also form periods or cycles, like the M3 interval which repeats every 4 notes, and the m3 interval which repeats after three notes. The whole step interval repeats after 6 notes, and tritone interval repeats after only 2 notes. Even the alternating interval axes on the stair-step tonnetz repeat, although they can have cycles that are longer the length of the entire chromatic scale.

Probably the most important property of the tonnetz is that shapes like intervals, triads, and other can be moved around. This is called translational invariance and this property is demonstrated by the fact that the shape and orientation of an interval or a triad is the same wherever it is drawn.  In other words, they can be moved from place to place on the tonnetz without changing either the musical or physical distance between the notes.

One final property of the tonnetz is its mutability as we saw when the traditional tonnetz shaped into a donut. This means that the network can be rotated, stretched, and sheared like a woven elastic fabric to create new versions with the same general properties. Shapes are consistent or transposable on the mutated plane as you move them to other places on the tonnetz. In fact, any musical element that uses intervals as its building blocks can be represented on the tonnetz as a movable object that doesn't change in shape.


Prime Scales

Scales are another musical element that can be represented on the tonnetz as geometric shapes. As the fundamental source of melody and harmony in different musical genres and styles, it's not surprising that there is a long list of scales to choose from. However, with certain restrictions on their uniqueness and independence, there are really only five. These prime scales were originally introduced as a comprehensive basis for tonality 8, and as such they provide a useful foundation for the graphical development of scales.

Figure 3-1 shows a simple way to construct the prime scales graphically by connecting notes on the tonnetz. The derivation of the prime scales on the traditional tonnetz follows a few simple steps - find a path between any two occurrences of the same note. The only rules are that, when you're done, all the notes along the path are different and each is a note in a major or minor triad. It's easiest to see this on the traditional tonnetz where the notes can be identified as being on one of the points in a triangle. Figure 3-1 shows five prime paths all representing a different and unique sequence of major and minor 3rd intervals. The last path shown in the figure is somewhat different from the rest since a whole-step interval is used to complete its path.

Figure 3-1 Graphical derivation of five prime scales by connecting notes on the traditional tonnetz.

It turns out that the five paths shown are the only ones possible. To prove absolutely that there are only five, every possible path would have to be considered. Problems like these are actually easier to check than to solve, even though there are thousands of possible paths.  Let a computer do it for you and the proof is made in just a few seconds. The result is interesting in that all five scales are 7-note subdivisions of the octave interval without assuming anything more than membership in a major or minor triad.  In a sense this shows where 7-note scales come from.

Many useful insights into the nature of the prime scales and their relationship to each other can be gained by looking at these scales on the hyper-tonnetz where paths and connectors can actually move in three dimensions. In Figure 3-2 the notes in the five prime paths are connected on the hyper-tonnetz to create five basic patterns. The first pattern which looks like a double diamond is actually the common scale. The other prime scales, which also form interesting and unique patterns, are the melodic, harmonic, and double-harmonic scales. Note that there are 2 ways to draw all but the harmonic scales. It is also interesting to observe that all the scale patterns can be constructed with triangles.

Figure 3-2 Prime scales on the hyper-tonnetz


Prime Modes

Modes are usually defined as alternative versions of a scale that start in different places, but otherwise the same. Those familiar with modal scales might point out that there are seven modes in the common scale. However, modes defined this way don't really distinguish effective harmonic sources, or explain how modality relates to all the scales we have found. So how do you find the modes in a group of five scales? The answer is an intuitive and simple rearrangement of the scale patterns.

Since the prime paths, shown in Figure 3-1

and transferred onto the hyper-tonnetz in Figure 3-2 , were derived independently there is no apparent relationship among them. How can we organize into keys? This is illustrated in Figure 3-3 where all the patterns are simply attached to a single note, in this case Db. In other words, just pick a keynote and anchor the patterns on red and black occurrences of that note.

Let's consider the scale patterns in Figure 3-3 and make a few observations. First is the generally ordered appearance of the whole group. Second, the patterns are all similar except the harmonic scales which are different in several ways from the rest. Only three of the five prime scale patterns, the common, melodic, and double-harmonic scales, can be split into mirror images. The two harmonic scale patterns don't have this symmetry, which makes them unique. It is the asymmetric harmonic primes that are central and the only scales that can be altered by a single note to form all the other primes. Both these observations are based intuitively on the notion of symmetry 9.

Figure 3-3 The ordered primes starting on Db.

After the scales have been anchored to a single key note, we see that the only triad common to all black is Db and the only triad common to all red is Bb. This is shown in Figure 3-4 where the anchored scales are shown again as prime paths on the traditional tonnetz. As we might have expected, the common scale patterns for Db major and Bb minor are contained in exactly the same scale pattern. These we recognize as two modes or tonics for this scale - the Ionian mode or the major scale, and the Aeolian mode or natural minor. The major and minor modes of the melodic and double-harmonic scales are not contained in the same identical pattern as in the case of the common scale. However, these scales can be drawn in two different ways which identifies the major and minor modes for each.

Figure 3-4 also shows the union of all the primes for the major and minor modes as a region of overlapping black and red polygons. The whole region actually contains all the notes of the chromatic scale. We could continue to view the prime scales as groups of notes, but this would result in a system without much organization. However, the identification of chords in this region, and the harmonic connections between them will finally establish a practical framework for using the scales in composition and performance.

Figure 3-4 The union of prime scales and the two prime modes


Diatonic Triads

In Figure 2-1 we saw that all major triads on the traditional tonnetz appear as triangles, and all minor triads have the shape of an inverted triangle. However, the major and minor triads are only a subset of the many shapes that chords can take.

One important group of chords is the diatonic triads. These are the major and minor triads formed by grouping 3 consecutive notes along the prime path. Figure 4-1 shows examples of the diatonic major and minor triads appearing along the prime paths.

The prime paths in Figure 4-1 also show the different types of 7th chords there are. These 4-note diatonic chords can be constructed by simply adding the next note along the prime path to the diatonic triad such as the M7, m7, and Dim7.

Figure 4-1 The diatonic triads and 7th chords along the prime scale paths on the traditional tonnetz.


7th Chords

The context of 7th chords within the prime scale mix is best shown using the hyper-tonnetz where the primary 7th chords appear again as triangles as shown in Figure 4-2 10. The points intersected by the large triangles now identify the D7 and m7b5 chords. The m7b5 chord is probably better named as a m6 or neapolitan sixth chord which identifies the whole triad it contains. The Dim7 chord, which is contained in the major and minor harmonic scales, has the shape of a rhombus. Another shape is the cross which forms the augmented chord, and M7b5 chord which appeared in the double-harmonic scale. We include the augmented triad in the set of 7th chords since it is completes the cycle of major thirds with nowhere else to go.

Many of the 7th chords that we have identified within the prime mix are often referred to as substitute dominants and altered chords that are often considered non-diatonic. However, the prime scales from which they were derived are actually all in the same key with the same tonic centers and modes. Like the scales that contain them, the 7th chords are anchored in the hyper-space by attaching the shapes to either a red or black note indicating the tonic note in the chord. The M7 and m7 chords, which appear as another set of triangles, are most easily described as a combination of two diatonic triads 11. As a slash or polychord, the M7 or m7 can be read as the anchor note over or under the triad on the other two points of the triangle. A 7-note scale can be represented by a small number of chords

Figure 4-2 7th chords anchored in the hyper-tonnetz.


Progression

There are a great number of terms used to describe how chords progress from one to another. Terms such as voice leading, pedal point, inversion, modulation, etc. all relate in different ways to chord formation and how chords in the same key and in different keys move smoothly from one to another. For example, transitions between chords are made effective through the use of common tones, or the movement of defining tones by half-step or whole-step to defining tones in another chord. Chord inversion and pedal point are other mechanisms used in chord progressions to minimize or control the intervals between base movements. Modulation and cadence are devices used to accomplish others musical goals such as resolution, pause, or divergence. Then there are qualitative terms like tension, resolution, open and closed voicing, consonance, dissonance, and others are used to describe the sound in a useful but subjective way.

A cadence consists of just two chords in its strongest form from one of the 7th chords to a major or minor triad. To greatly simplify, we can summarize the cadential forms to be those that complete their resolution (full cadences), those which pause (half cadences), and finally all those cadences that go somewhere else (deceptive cadences).

Figure 4-3 7th chords cadences

On the hyper-tonnetz full and half cadences from and to 7th chords appear as the arrows shown in Figure 4-3. Other cadences are generally movements on the double diamonds and deceptive cadences. The most familiar cadences are contained in the familiar I-IV-V-I and I- II-V-I progressions and many others. It is these chords and cadences and simple progressions that are strung together to form music 12. However, the sense of resolution, divergence, and pause in longer compositions is largely a matter of context. Probably just as important as the cadence itself is how it is approached, the departure from it, and the return to it. This is in some ways the essence of modulation.

Voice leading, pedal point, inversion all relate to how notes are actually ordered and spaced on a particular instrument. And the qualitative measure like tension, resolution, open and closed voicing, consonance, dissonance have to be heard to be appreciated.


Visual Clues

The intent in this final chapter is to present a chord system that can be used to see and hear the chords and cadences shown on the tonnetz. At the same time, we can give the shapes on the tonnetz more meaning by matching them with real physical shapes on an instrument - in this case a keyboard.

In constructing a set of keyboard chord charts, the situation arises where musical intervals are not consistent with the physical distance between keys. On the tonnetz, where the 12 notes of the chromatic scale were freely arranged in two and three-dimensions, the physical shape of a particular scale or chord was always the same. On the keyboard and other instruments the same 12 notes are usually arranged along one dimension in a sequence of half-steps. As a result intervals on a musical instrument are not generally consistent with the physical distance between keys. The is show in Figure 5-1 where the frequency of sound is plotted as a function of distance along the keyboard. In simple terms, perfect symmetry is broken by the two half-step intervals B-C and E-F where the relationship between distance and sound becomes non-linear, and only D and Ab are left as points of symmetry.

Words like asymmetry and non-linearity might seem to imply imperfection, and the acoustic non-linearity and physical asymmetry of the keyboard is not something that can be easily changed. However, this imperfection produces a richness and diversity in both shape and sound that wouldn't be there otherwise. In any case, whether diversity or distraction, the keyboard retains enough symmetry to allow the shapes of chords to be grouped and organized in a practical way.

Figure 5-1 Keyboard non-linearity

  • Recognize groups of chords with the same shape that reduce the number of chords to be learned
  • Find whole-steps groupings
  • Find the base note cues in the notes being played
  • Use the black keys as reference points
  • Recognize visual symmetries that make the formation of chords easier
  • Relate cadences to physical motions of shift, expansion, and contraction

Diatonic Triad Shapes

We can begin to organize the triads in some useful way. Since these two are reflections of one another, a natural way to display the blocks is to place them on opposite sides of a page. The region contains all of the major and minor diatonic triads found along the prime paths, and shows them in all three positions.

Several examples of these are shown in Figure 5-2. For example, in Figure 5-3, the notes in a D major triad (D, Gb, A) reflected about D or Ab become (G, Bb, D). For example, an E major triad has the same shape on the keyboard as A major and D major. We can see now that E major, A major and D major are all mirror images of G minor, F minor, and C minor, as well as other triads named by their shapes on the torus. The reflected shape group also shows us that G minor, F minor, and C minor all have the same shape on the keyboard.

All 10th chords have same shape as 3rd position major and minor triads except for B which has same shape as 5th position triads.

Figure 5-2 Symmetries and anti-symmetries in the shape of major and minor triad groups on the keyboard.

Figure 5-3 The shape of the major chords in 10th position cued by the shape of root position major triads.

Figure 5-4 The shape of the B and Bb chords in 10th position cued by the shape of root position opposites.

Figure 5-5 The shape of the minor chords in 10th position cued by the shape of root position minor triads.


Upper/Lower Parts

In this version, each 5th position triad is repeated an octave higher on the left side of the map, and an octave lower on the right. there are three-note shapes shown in the boxes composed of a single note between an octave span. Although there are really only 2 notes in this shape, we'll call it an octave-triad. These are particularly useful triads since only 4 of these shapes account for the all the major and minor diatonic triads.

In terms of chord voicing, polychords are composed of upper and lower parts, where the note in the lowest part creates different inversions (i.e., root, 3rd, and 5th inversions). Physically can be played on the keyboard by simply copying a triad form with the right and left hands. Base Structures (octave triads, and 1st inversion 10th chords)

Figure 5-6 Upper and lower parts for major and minor chords played in root position and grouped by shape.

Figure 5-7 Upper and lower parts for m7 and M7 slash chords.

Figure 5-8 Using the reflective symmetries and black note cues to form Dim7, Aug, and M7b5 chords.


Whole-Step Clues

Figure 5-9 A voicing of D7 chords on the keyboard cued by the whole-step interval in the upper part and the shape of the 10th chord in the lower part.

Figure 5-10 A voicing of m6 (m7b5) chords on the keyboard cued by the whole-step interval in the upper part and the shape of the 10th chord in the lower part.


Basic Motions

In additional to the visual cues, there are two basic motions that can be applied to the upper and lower forms to transform chords in different ways. These basic motions are defined below and illustrated in Figure 5-11. These are Expand and Contract where the notes in the upper and lower parts generally moves in or out in opposite directions, and a Shift motion where the notes move generally in the same direction. Examples of cadences, short progressions, pedal point are shown.

Figure 5-11 Pedal Point example cued by whole step intervals.


Summary

For those interested in playing and understanding music, the tonnetz in its various forms is a medium for both performance and theory that relies largely on the ability to recognize shapes and patterns. Beyond the simple notion of cadence and progression is composition and melody which seem to elude a simple geometric expression. It might be said that these are a matter of invention, creation, and discovery. However, the tonnetz in its various forms appears to be a useful tool for illustrating musical elements and relationships that are sometimes difficult to see. With some effort, it helped to map these elements and relationships to an actual instrument. Needless to say, only some of patterns have been described, and only a few of the possibilities explored - What do you see?

What is background you see in the above header? Email me with your answers.


Copyright © 2017 Music Harmony and Donuts by Dr. Paul Dysart. All rights reserved