Harmony and Duality

An introduction to music theory

Maksim Lipyanskiy

§ INTRODUCTION

“Writing about music is like dancing about architecture”

We develop aspects of music theory related to harmony, such as scales, chord formation and improvisation from a combinatorial perspective.  The goal is to provide a foundation for this subject by deriving the basic structure from a few assumptions, rather than writing down long lists of chords/scales to memorize without an underlying principle.  While learning to read lead sheets and studying jazz improvisation were the initial motivation for this approach, it is equally applicable to music arrangement and composition in a much broader context.

A more ambitious long-term goal of this project is to provide an alternative approach for learning to play the piano.  Traditionally, one starts by learning to read music notation and then progresses linearly through pieces of increasing difficulty.  We want to advocate for an alternative approach.  The idea is to practice basic harmonic structures and start improvising and experimenting with harmony from the beginning.  Much like a beginning fine art student, who does not spend all of their time painting by numbers or copying the works of the masters (though that too can be useful), we would like to advocate for a more personalized approach to learning music where the student is the creator from day one.  Our view is that this approach is complementary to the traditional piano method because it focuses on building a set of skills quite different from sight-reading or finger dexterity.  As we will demonstrate, armed with a few basic rules, even a beginner can start creating their own arrangements and become a part of the creative process. For a simple illustration on how to use the concepts in this work to create an arrangement of a song, see section <ref>.

Here are the key concepts used in our approach to harmony:

* Constraints: The study of harmony involves several voices moving simultaneously and a choice of a scale dictates which tones are available to use in an arrangement. A simple constraint is to restrict to scales that avoid voice “collisions”.  For example, we can impose the constraint that two voices cannot be only a semitone apart as this is too dissonant.  We can then study scales that do not contain notes that are a semitone apart.  Furthermore, a more refined constraint would avoid three voices “colliding” by studying scales that do not have three notes separated only by semitones.

* Completeness: Imposing constraints as above limits the possible scales we may consider. Additionally, we require that our scales are complete, which roughly means that they are the maximal sets of tones that satisfy such constraints.  In other words, we cannot add further tunes to such scales without violating the constraints.  As it turns out, completeness as applied to these simple two/three voice constraints characterizes the types of scales that are commonly used in music composition.

* Duality: Surprisingly, there is a correspondence between scales subject to the two voice constraint and those subject to the three voice constraint.  We formulate this correspondence as a duality statement that provides a way to understand scales subject to one type of constraint in terms of scales subject to the other.

It will take us the majority of this work to fully explore these key ideas.  Unpacking these concepts will lead to a full classification of harmony and chords as well as provide a principled approach to elements of music theory.

Outline of Contents: This work is divided into several parts.  The first part (sections <ref> to <ref>) is introductory and focuses on major and pentatonic scales.  The next part (sections <ref> to <ref>) generalizes these basic ideas and contains a discussion of the relation between harmony and packings.  Sections <ref> to <ref> are a culmination of our approach and contain a classification of voicings for various harmonies using the ideas of the previous sections.  We relate our approach to harmony with the standard “lead sheet” notation in sections <ref> and <ref>.  Finally, the work ends with several appendices that review basic concepts as well as contain topics that are tangential to the main discussion.

Prerequisites:  We aim to make this work accessible to beginners, such as students familiar with initial chapters of Alfred's Basic Adult All-in-One Course (Palmer, Manus and Lethco).  Specifically, we assume that the reader is familiar with sheet music notation and has some basic exposure to the piano.  Some familiarity with chord notation would also be valuable.  While we summarize some of the basic concepts in the appendices, we wanted to focus on material that is not available elsewhere and thus this work does not provide a comprehensive introduction to all aspects of music theory.

Math and Music: While this work does not assume a strong background in mathematics,  we use language borrowed from math to explain the major ideas of this work.  For us, this is just the simplest way we know how to explain these ideas and not an attempt to make music theory into a branch of mathematics.  At the end of the day, we are interested in concrete applications for creating musical arrangements and composition.  On the other hand, to a willing music student, perhaps this work can serve an introduction to rather sophisticated mathematical concepts such as classification, representation, completeness, reducibility and duality.

What this work is not about:  We try to be universal in our approach and avoid context-specific notions like “happy”/“sad" chords, and generally try not to tie our foundation to a particular music style.  Our hope is that reader can apply these basic notions to the particular context they are interested in.  As the history of music shows, each generation tends to reevaluate existing conventions and develop new rules.  In general terms, we focus on the parts of harmony that can be formalized and leave the art to the practitioner. Using an analogy to writing, we focus on building a student's vocabulary and grammar and not on writing beautiful paragraphs or entire works in a particular style.  Armed with the basic knowledge, the student can go on to study chord progressions/voice leading in the particular context they are interested in exploring.

Future Work:  While we aim to present all the major concepts, at the moment, a lot more experience with examples is needed to appreciate how to use this theory in practice.  Ideally, one could develop a course/textbook based on this approach that would include many more examples and exercises to help the student apply these concepts.  We hope to return to this sometime in the future.  For now, we do offer some concrete suggestions of what to practice in section <ref> as well as section <ref>.  There is enough practice to material there to keep a beginner busy for months.

Acknowledgement: The combinatorial approach to defining harmony is due to the work of the pianist, composer and the author's teacher, Gustavo Casenave.  Specifically, he introduced the notion of a (semitone) cell and went on to classify harmony/modes using this notion.  We take his foundational insight and develop the framework in a different direction by focusing on harmony with zero cells while giving a dual description via our notion of “blocks” and “packings”.   A key motif throughout our work is to associate to a harmony a simpler “shadow" object that we call a packing. Mastering these packings will then in turn help us understand harmony since, as we will show, harmony is constructed from these simpler objects.  We will then put all these concepts together to develop a theory of chords/voicings. To our knowledge, this is the first time the relationship between harmony and packings is discussed and used to explore music theory.

§ OVERTURE: THE PIANO LAYOUT AND THE CIRCLE OF FIFTHS

We begin by justifying the piano key layout.  While this is typically taken for granted, we will argue below that one can, in fact, motivate the piano layout and in the process introduce the main constructions used in the remainder of the work.

Looking at the piano as if for the first time, we notice a few things right away.  Namely, the pattern of black and white is periodic and repeats every 12 keys.  See figure <ref>.  First, let's focus on this periodicity.  Any key on the piano corresponds to a vibrating string of some length.  If we half this length, or equivalently double the frequency, we obtain a key that is an octave higher.  Tones that are an octave apart are so consonant that we tend not to distinguish between them harmonically.  This is the reason behind the periodicity of the piano layout.  Effectively, it reduces the harmonic possibilities to the range inside a single octave.

Piano keys.

We now justify why we have 12 tones in an octave.  To do so, we start with a single tone vibrating with some base frequency $f$.  As discussed, to obtain the octave we double the base frequency to obtain $2f$.  If we keep doubling the frequency ($4f$, $8f$, …), we simply get higher multiples of the octave.  To get something new, we may instead triple the fundamental frequency to $3f$. This gives us a tone higher than an octave, which ranges between $f$ and $2f$.  Since we identify tones that differ by an octave,  we may halve this to $\frac{3f}{2}$ obtaining a tone inside the octave. This interval, known as a perfect fifth, is the most consonant interval inside an octave and has a ratio of 3:2 to the fundamental frequency.  It gives us our first nontrivial subdivision of the octave.  For example, if we started the fundamental frequency with the note F, we obtain the note C as the perfect fifth above F.

At this point, there are two ways to proceed.  We may consider higher harmonics given by higher multiples of the fundamental frequency $f$.  These will give rise to highly consonant intervals and can be used to further subdivide the octave.  However, this will by construction tie us to the fundamental frequency $f$ and will create a highly non-symmetric arrangement of tones.  This “harmonic” tuning is further discussed in appendix D.  The alternative, known as Pythagorean tuning, is to continue using the perfect fifth ratio to obtain new notes.  Thus, if we started with F to obtain C, we now apply the same construction to C and obtain the next note that is a perfect fifth from C, which is G.  This puts each note on the same footing as the original.  By iterating the procedure of taking perfect fifths, we obtain:

\begin{equation}
\text{F, C, G, D, A, E, B, F{\sharp}, C{\sharp}, G{\sharp}, D{\sharp}, A{\sharp}, F }\dots
\end{equation}

Notice that we cycle through all 12 notes of the piano before we come back to the initial F.  This is known as the circle of fifths. It gives us a natural ordering of the notes by consonance and explains why we need 12 tones in our scale.

There is a slight wrinkle in this description.  Namely, if we actually tuned our keys to follow the 3:2 ratio, we would not exactly come back to F after 12 iterations.  However, the error would be much smaller than a semitone and the modern piano is tuned (equal temperament tuning) in such a way that the actual ratio we use is slightly less than a perfect fifth ($2^{\frac{7}{12}}\approx$ 1.4982..). It is defined so that after 12 steps you exactly come back to the initial frequency.   The fact that, after using a perfect fifth for 12 steps, we are almost back to the initial note explains why we actually have 12 notes in an octave.  This is not a perfect system but is favored because it puts all the tones on the same footing.  See appendix B and C for an in depth discussion of Pythagorean tuning and equal temperament.

We have now addressed both the periodicity as well as why we have 12 tones in the scale.  The last point to explain is why the notes are divided into black and white keys in a specific arrangement.  In particular, this singles out the C major scale as composed of all the white keys.  Note that this is not simply a convenience to make the piano keys easier to navigate; rather, it lies at the foundation of Western music. Indeed, music notation for any instrument takes as its basis a major scale (C major or any of its shifts).  The remaining notes are considered “accidental”.  We now motivate this division of tones into black and white keys and in the course of doing so introduce concepts that are central to our treatment of harmony.

In practice, we tend not to use all 12 notes at a given time in a composition.  In fact, the whole subject of harmony and chords is focused on picking out interesting subsets of these 12 notes to use.  A general heuristic is that we want to stop before we get too much dissonance due to too many notes that are close to each other.  With that in mind, let us follow the circle of fifths to find reasonable stopping points.  The first one occurs after 5 notes and gives us the F pentatonic scale:

\begin{equation}
\text{F, C, G, D, A}
\label{eq:penta}
\end{equation}

To see why this is a reasonable stopping point, we observe that the next note E is the first time we see two notes separated by a semitone (E and F in this case).  Furthermore, observe that all subsequent notes (E, …A${\sharp}$) will be a semitone away from a note in the pentatonic scale (F, C, G, D, A). See figure <ref> (a).

We formalize this by calling two notes separated by a semitone a semitone block or simply a block.  Here, we do not distinguish notes that are an octave apart; such a block could consist of tones separated by many octaves, as long as we can shift them to be a semitone apart.  Notice that the pentatonic scale contains no blocks and furthermore we cannot add any other notes without creating a block (check this!).  Harmonically, we think of the pentatonic as the largest set of notes that avoids a form of dissonance created by blocks. In fact, we believe that the prevalence of pentatonic scales in world music is related to this harmonic characterization.

(a) F pentatonic scale     (b) semitone block

First stopping point.

Another harmonically significant stopping point occurs at

\begin{equation}
\text{F, C, G, D, A, E, B, G}
\label{eq:major}
\end{equation}

when we have exhausted all the white keys.  Here, a different form of dissonance becomes relevant.  Namely, observe that once we add F${\sharp}$, we create for the first time 3 notes that are a semitone apart: E,F,F$\sharp$.  We call a sequence of 3 notes that are a semitone apart a dissonant cell, as shown in figure <ref> (b).  Just like with blocks, all the remaining notes {F${\sharp}$, C$\sharp$, G${\sharp}$, D${\sharp}$, A${\sharp}$} will create a cell when added to the white keys (check this).  Thus, the collection of white keys contains no cells and we cannot add further notes without creating cells. For us, this is the fundamental justification behind the C major scale: it is a largest set of tones that avoids cells.  Harmonically, cells create a higher form of dissonance (as opposed to blocks) and should generally be avoided when creating chords.  This observation, first made by Gustavo Casenave, helps motivate the prevalence of major scales in Western music.

(a) C major scale     (b) semitone cell

Second stopping point.

In summary, we have defined two notions of dissonance, one based on a 2-note block and the other based on 3-note cell.  A pentatonic scale avoids blocks and is maximal with respect to that property: we cannot add more notes without introducing a block.  Similarly, the major scale avoids cells and is maximal with respect to that property.  These simple criteria will be the basis of our combinatorial approach to music theory.  Namely, blocks will lead us to the notion of a packing (pentatonic scale is an example) while cells will lead us to the definition of harmony (major scale is an example).  One of the main goals of the subsequent sections is to define and explore these concepts in general.  Furthermore, a combination of these ideas will be the foundation of our theory of chords and voicings.

Finally, let us note that it is a surprising fact that the black keys (which are the complement of the white keys) form a pentatonic scale!  This is unexpected since the stopping criteria used to define the white keys (and hence their complement) is based on cells and has little to do with blocks.  Thus, at least for this example, we see that the white keys which are an example of a harmony are complementary to black keys that are an example of a packing.  By symmetry, this is true for all pentatonic and all major scales. As we will explain in later sections, this leads to a dual relationship between scales and packings or, equivalently, between the two notions of dissonance introduced in this section.

C major scale (red) and its pentatonic complement (green).

Remark: The reader will note that we started our circle of fifths on F as opposed to C.  This is dictated by the fact that we wanted to highlight the C major scale (white keys).  The reader can check that starting at C would highlight the G major scale instead.  Indeed, more than just a convenience, this suggests that the most natural starting point for the C major scale is F, as opposed to C.  In general,  the major scale has seven starting points which are called modes.  Starting at F is known as the Lydian mode or the F Lydian scale.  Qualitatively, it is the brightest of all modes and has led to the so-called Lydian perspective on tonal organization as coined/explored by George Russell. We will have more to say about modes in a later section.

§ BLACK KEYS

§.§ Getting Started

We begin our study of harmony in the simplest setting by focusing on the pentatonic scale.  This is the simplest nontrivial scale one may consider and is ideally suited to introduce all the main concepts discussed in this work. Typically, students start with the white keys (C major scale) as this is visually easiest to see on the piano and music notation is designed to make white keys easiest to sight-read. Instead, we focus on the black keys which form a simpler harmonic structure and, in our view, provide a better entry point for learning harmony, improvisation and even composition.  In fact, anyone who played around with the piano as a child may have noticed that playing black keys randomly already sounds melodic.  This is due to the fact that the pentatonic scale avoids dissonant semitone blocks which makes melodic lines sound smooth while containing some degree of complexity.

Thus, in this section, we would like to advocate for the black keys as a natural starting point for learning the piano with an eye towards harmony, improvisation and composition.  Visually, black keys are as prominent as the white keys and the only drawback is that music notation is not best suited for them.  Indeed, we have to introduce 6 flats to the notation to use the F${\sharp}$ pentatonic scale (black keys).  However, we feel that the slightly awkward notation is a small price to pay for the conceptual and technical benefits.  Finally, as we will see in the next section, the black keys are not merely a toy example of harmony useful as an introduction, but form the building blocks of major harmony.

The pentatonic scale is rich enough to form basic melodies.  The song Amazing Grace by John Newton written at the end of the 18th century is a great example.[see <https://en.wikipedia.org/wiki/Amazing_Grace> for more information] Figure <ref> shows a simple arrangement.

Amazing Grace in  F${\sharp}$.

In this section we will introduce basic harmony and improvisation for this tune.  In fact, we will use this piece as the main example to illustrate our ideas throughout the text.

§.§ Chords

We now discuss how to form various chords using the pentatonic scale.  Generally speaking, restricting ourselves to this scale implies that we avoid having too many notes clustered together so in reality almost any combination of notes would sound reasonable.  This gives the performer freedom to experiment freely in this restricted setting.  However, we want to be systematic with an eye toward generalization and thus will now present a more principled approach to chord formation.

On one extreme, we may want to avoid notes that are at most a tone apart.  Subject to this condition, for our maximal chords we get the two options in figure <ref>, which are a F$\sharp$ major triad (a) and a D$\sharp$ minor triad (b). Triad chords are the basis of classical harmony and offer a basic way of creating chords for our scale.

(a) F$\sharp$ major triad (inverted)     (b) D$\sharp$ minor triad

Triad options.

On the other extreme, we may use all the five notes, creating a 5-chord (or a pentachord).  As shown in figure <ref> (a), this might sound too clustered if we keep the notes next to each other.  However, if we spread them as in (b) we can use both hands to create a better sounding voicing.  Harmonically, we do not distinguish between these alternatives but, practically, we still need to keep in mind that we are free to move any note of a chord by octaves in either direction.  This 5-chord is in fact used in practice, for instance in the jazz piano playing of Bill Evans and McCoy Tyner.

(a) clustered 5-chord     (b) spread out 5-chord

A few 5-chord inversions.

We propose an approach somewhere between these extremes by forming chords that avoid having three notes in a row that are at most a tone apart.  Figure <ref> shows the three possible maximal chords that can be formed this way using the pentatonic scale.  These chord shapes will be the basic building blocks of our approach to chord formation.  Namely, we view these shapes as fundamental and other chords as derived from these shapes.  Specifically, in practice we may:

* Invert the chord by moving a tone up/down by octaves.  One can “spread” the notes of the chord to be played with both hands. See the examples in figure <ref>.

* Omit any number of tones from the chord.  For example, we can view the triad chords as “derived” from the 4-chords by omitting notes.

* Add the missing note to get the composite 5-chord.  Once again, we view the 5-chord as derived from the fundamental 4-chord construction.

(a)      (b)     (c)

4-chord variants.

Remark:  The 4-chords we obtained in figure <ref> are known as the F$\sharp$maj6 (a), D$\sharp$min11 (b) and A$\sharp$min11 (c).  These standard naming conventions are meant to reflect that we add tones to basic minor/major triads.  For instance, F$\sharp$maj6 is obtained by adding the 6th degree of the F$\sharp$ major scale to the F$\sharp$ major triad.  Concretely, this adds the D$\sharp$ to (F$\sharp$, A$\sharp$, C$\sharp$).  For D$\sharp$min11, we add the 4th degree to the standard 7th chord tones.  This adds the G$\sharp$ to (D$\sharp$, F$\sharp$, C$\sharp$). We will review basic chord notation in appendix A.

Remark: We will generally not distinguish between chords that are related by an inversion.  For instance, if we start with D$\sharp$, then the inversion of F$\sharp$maj6 is also D$\sharp$min7.  It may seem odd that inversions have such different names and we will clarify this when we discuss modes in section <ref>.

(a)      (b)

(c)      (d)

Basic chord inversions.

Let's apply this idea to the tune.  Our goal is to replace the single melodic line with an entire chord that has the melodic line as its top note.  The way to proceed is a combination of structure (pick the 4-chord shape you want to use) and randomization (omit some of the chord tones or add the missing note to get a 5-chord).   Figure <ref> provides an example of creating chords to harmonize the melody using these 4-chords.  One variant of the spreading technique that is worth pointing out is called “drop 2” where we remove the second highest note from the right hand and move it down to the left hand.  This way, the melody note which is on top has a larger separation from the rest of the chord helping make the melody stand out.  See the F$\sharp$ in the second bar for an example.  Additionally, note that in bar 3 we use only two notes of a 4-chord to harmonize the G$\sharp$.  Finally, in bar 15 we use the “derived” 5-chord to harmonize the A$\sharp$.

Amazing Grace with chords in F${\sharp}$.

As we see with this simple example, we have quite a lot of freedom to create harmony even when restricted to a pentatonic scale.  From our perspective, having concrete chord shapes to keep in mind can be quite liberating to the beginning performer as it allows them to experiment with harmony from the start without worrying too much about creating dissonance.  On the other hand, even in this simple setting, the number of alternatives is already large so there is plenty of room for experiments.

§.§ Basic Improvisation

Improvisation is a large and complex subject that can be quite intimidating for beginners.  One of the main issues is the large number of degrees of freedom that can be paralyzing on one hand and can make it difficult to create interesting musical phrases on the other.  Using auxiliary structures, such as pentatonic scales, can help simplify this process for the beginner and aid the development of improvisational skills.

Much like with chord formation, restricting ourselves to a pentatonic scale makes virtually any melodic lines sound reasonable.  Let's illustrate this on our basic example.  Aside from restricting ourselves to the pentatonic scale, conceptually we think of the improvisational lines as either adding small variants to the melody and trying to end these lines with the same tones as the original melody, or using the scale to fill gaps in the original melodic lines.  Figure <ref> gives a simple illustration.  Of course, improvisation and composition are as much an art as a science and at the end of the day we are limited only by our imagination!

Amazing Grace with basic improvisation in F${\sharp}$.

§.§ Playing Outside

To simplify the concepts as much as possible, we have so far worked exclusively with the black keys.  Our aim was to illustrate that there is just sufficient complexity to create interesting melodic development as well as to introduce our basic viewpoint on chord formation.  We view such pentatonic scales as the most elemental building blocks of music and one can spend a great deal of time exploring these scales without running out of new territory to master.

On the other hand, our restriction to the black keys is artificial.  Ultimately, we want to be able to explore the entire keyboard in music and in this section we take our first steps in that direction.  Recall that the pentatonic scale does not contain any dissonant semitone blocks and is maximal with respect to that property: any notes not in the scale will be a semitone away from a note in the scale.

In the case of black keys, we may view this as saying that any white key sounds “dissonant” with respect to the black keys and has one or two resolutions to a black key.  Figure <ref> illustrates this basic point.  In (a) view the note B (marked in green) as creating a dissonant block with A$\sharp$ and we can resolve B to an A$\sharp$.  On the other hand, (b) adds the note A that can resolve either to G$\sharp$ or to A$\sharp$.  Adding A to the scale creates what is known as a “blues” scale with A acting as the external “blue” note. The pentatonic perspective on these additional notes is that they are outside the harmony and can be used to create additional tension/complexity in the music but should be used somewhat sparingly as a sort of seasoning added to the dish.

(a) B with one resolution     (b) “blue” note A with two resolutions

Adding white notes.

We illustrate the use of these additional tones in two examples.  First, in figure <ref>, we write a basic improvisation using just the single “blue” note A mentioned above.  Not only does this illustrate the concept in the simplest setting of one additional tone, the resulting “blues” scale is very common in jazz/blues and is often the main component of building professional sounding improvisational lines.  See Oscar Peterson's rendition of the song “Georgia On My Mind” for a great example.

Amazing Grace with a “blue” note.

In our second example, we try to incorporate all extraneous white notes into our pentatonic scale.  See figure <ref>. Once again, these are used as a sort of embellishment to add complexity to the melody without deviating too much from the original idea.

Amazing Grace with many “outside” tones.

Remark: Notice that some of the white keys have a single black note that is a semitone away (like B) while others have two such notes (like A).  When we discuss harmony, we will show that the notes with a single resolution can be included in a harmony such as a major scale while the others cannot be included into a harmonic structure.  Using the language from section <ref>, notes with two resolutions would form a semitone cell if added to the scale.  See section <ref>.

For simplicity,  we have mostly focused on improvisation in the right hand.  Naturally, there is nothing preventing us from applying these ideas to the left hand or, more precisely, the lower part of the piano range.  One small caveat is that forming complex chords below the middle range creates a muddled sound and is generally avoided.  Note that harmonically, we only care about tones up to an octave so such restrictions are not strictly part of the main discussion.  We saw a similar phenomenon with the 5-chord where it was more desirable to spread out the tones across a larger range to get a clearer sound.  Thus, when using the lower range of the piano for improvisation, we tend to restrict to fewer voices moving simultaneously.

§.§ Harmony, Melody and Microtones

We round out this discussion by mentioning an important idea implicit in these simple examples of playing outside the basic pentatonic scale.  Namely, we view these variants as melodic rather than harmonic.  This means that we are not intending to change the chord structure or thus the underlying harmony of the piece.  Instead, these outside tones simply enhance the underlying fixed structure.  To take this idea to the extreme, there is in fact no reason to even stick to the 12-tone system when creating these variants.  In fact, for many instruments including the human voice, we can easily interpolate between tones (or “bend” notes) and create “microtonal” melodic elements.  For example, when going up from a starting note in a scale to an ending target note, we can not only use the chromatic scale to approach this note, but may in fact continuously increase the frequency until we end up on the target.  Another example is given by the blue note mentioned above.  It has been argued that the true blue note is in fact a slightly flattened version of the A note above and this falls outside of the 12-tone equal temperament system.  In fact, from the perspective of the pentatonic scale, the white keys are themselves already microtonal since they lie between the black key tones!

Our perspective on these microtonal phenomena is that they are fundamentally melodic rather than  harmonic in nature.  In other words, we use them to enhance the melodic structure rather than to alter the underlying chords.  On some level, the distinction between melody and harmony is a theoretical construct rather than an underlying truth since one can certainly think of examples where the distinction between the two is blurred. However, it is safe to say that a serious study of microtonal harmony (scales, modes and chords) lies beyond the scope of the current work and is quite close to the limits of the human aural ability.

§.§ Exercises

Our work is mostly focused on providing a theoretical framework for harmony with an eye towards composition and improvisation.  Although much of this is accessible to the beginner, we have not devoted enough effort to building a method book full of examples and exercises.  In this section we wanted to mention a few basic exercises that further explore pentatonic scales:

* Practice chord shapes on the black keys.  Get comfortable finding the three basic 4-chords as well as their inversions.

* Try basic improvisation on the black keys.  You can start by modifying the tune as we did above or simply get comfortable moving in 8th notes in one hand while play a baseline or chords in the other.

* Incorporate some white notes into your improvisation and repeat the previous exercise.

* Once comfortable with the black keys, do this for the other 11 pentatonic scales.  It's a good idea to at first pick one scale for each practice session until you are comfortable recognizing that scale.

* Try basic improvisation where you change your pentatonic scale every measure or two.  You can move along the circle of 5ths, chromatically or randomly.

All these exercises are a direct application of the basic concepts introduced in this section.  However, they will take months (if not more!) to master and will provide a challenge to even an advanced pianist new to improvisation.  We will see that similar exercises can be applied to other packings such as the dominant pentatonic or the diminished 7th chord.  They provide the basic building blocks of harmony and mastering these scales will allow us to better navigate the world of harmony.

§ WHITE KEYS

§.§ The Major Harmony/Pentatonic Connection

Now that we have spent some time exploring pentatonic scales, we shift our focus to major harmony, which is usually considered the starting point for music study.  The simplest major scale is C major, defined by the white keys.  On the surface, our exploration of black keys seems antithetical to the study of white keys.  However, black keys give us the simplest example of a pentatonic scale and major harmony is in fact built from such scales.  Indeed, as shown in figure <ref>, the F (blue), G (green), and C (red) pentatonic scales cover the C scale.  One can therefore view these pentatonic scales as elementary building blocks of major harmony!

Pentatonics in major harmony.

With this in mind, we can apply all that we've learned in the previous section to major harmony.  The difference is that now we have three different pentatonics to choose from when constructing improvisation or chords.  Of course, we do not have to restrict to a single pentatonic during composition/improvisation but can freely switch these pentatonics at will.  Thus, the pentatonic decomposition provides an organizational tool for major harmony, which can guide one's thinking without necessarily being explicitly stated.

Let's apply this concept to create 4-chords in major harmony using pentatonics.  Recall from the previous section that a pentatonic gives rise to three 4-chord shapes up to inversion, obtained by omitting a note from that scale.  This means that each of the three pentatonic scales contributes three chord shapes.  However, the same 4-chord can come from different pentatonics.  Figure <ref> plots these chord shapes with the colors indicating the pentatonic source.

(a) Fmaj6 (blue)     (b) Dmin11 (blue)     (c) Amin11 (blue)

Gmaj6 (green), Cmaj6 (red)     Emin11 (green), Amin11 (red)     Bmin11 (green), Emin11 (red)

4-chords from pentatonics.

We see that there are in fact only seven distinct chords:

{Fmaj6, Gmaj6, Cmaj6, Amin11, Dmin11, Emin11, Bmin11}

Let's now discuss these pentatonics from a musical perspective.  For this, we need to select a bass note in major harmony.  Relative to this choice, the pentatonics will have a different feel.  For example, a common choice of bass note is F since the C major scale starting in F sounds the brightest.  If we hold down F in the lower part of the piano range and play one of the pentatonics, it will have a distinctive feel.  The F pentatonic will sound the most consonant and calm with F in the bass.  The C pentatonic will have the recognizable “jazzy” sound that is a bit more dreamy and remote.  Finally, the G pentatonic will sound even more “ethereal”.  It is not easy to use words here so we encourage the reader to try this for themselves.  Similarly, the chords formed from these pentatonics will have these characteristics.  This most strongly applies to the maj6 chord, while the min11 chord is shared across scales as we have discussed above.  We will return to the relation between chords and choice of bass note when we discuss modes in section <ref>.

In summary, the pentatonic scale has helped unlock some of the hidden structure inside major harmony.  We observed that there are three distinct embeddings of pentatonics into major harmony.  Furthermore, these pentatonics give rise to 4-chords within major harmony.  We have a maj6 chord at C, F, G in this case and a min11 chord starting at D, E, B and A.  Formally, we may view this as two chord shapes where the 6th chord has 3 representations and the min11 chord has 4 representations in major harmony.

§.§ Example: Amazing Grace

We now apply these concepts to once again harmonize the tune Amazing Grace.  We switch from the F$\sharp$ to the F pentatonic scale which we view as embedded in the the C major scale.  Therefore, we can use all the white notes to create chords for the tune.  We proceed much like before, though instead of having three basic 4-chords to use, we now have seven 4-chords at our disposal.  Thus, as we shift perspective from a pentatonic scale to major harmony we allow more possibilities for creating chords which allows for richer harmony.  Figure <ref> gives an illustration of the result, with chords that we used to harmonize annotated at the top.  Note that the pentatonic scales are not explicit in the tune.  However, having the three pentatonics in mind when creating an arrangement or improvisation is a helpful conceptual tool.

Amazing Grace chord arrangement.

§.§ Major Scale as an Extension of a Pentatonic

In this section, we observed that one can view major harmony as comprised of simpler pentatonic elements.  We would like to offer an alternative viewpoint that considers major harmony as an extension of the pentatonic scale.  Namely, if we consider the F pentatonic inside the C major scale, we see that the tones missing from the pentatonic are B and E.  In both cases, there is a unique tone in the F pentatonic that is only a semitone away from the note:  for B it is C and for E it is F.  We can view this as stating that very note in the C major scale has a unique “projection” to the underlying F pentatonic.  For the tones that are already in the pentatonic the project is the tone itself.  For the remaining tones B and E, the projections are C and F respectively.  This means that, we can always take a melodic line in C major and project it in a unique way to a pentatonic scale, which will typically remove some of the tension and simplify the melody.  We view this as reducing the more complex major harmony to its pentatonic shadow.  Finally, observe that all the remaining tones not present in the C major scale will not possess a unique projection.  For instance, F$\sharp$ projects either to F or to G which introduces ambiguity.  Thus, we may view major harmony as a maximal set of tones you can add to a pentatonic scale, which still have a unique projection down to the scale.

This completes our introduction to music theory via pentatonics and major harmony.  In the next section we take a step back to formalize these notions and produce more examples of this phenomena.  While the discussion is more theoretical, it's important to keep in mind that, at the end of the day, we can apply the same ideas we used for pentatonic/major harmony in this broader context.

§ HARMONY

§.§ Harmony Basics

Our definition of harmony is closely related to the notion of a scale. For the purposes of harmony, we do not distinguish notes that are an octave (12 tones) apart.  Thus, one may consider any subset of the twelve tones to be a scale.  A useful way to organize scales is by the number of dissonant semitone cells they contain.  By definition, a semitone cell in a scale consists of three semitones in a row.  See figure <ref> (b) for an illustration.

Figure <ref> gives examples of scales: (a) is the C major scale that has no cells (b) is a C minor blues scale that has one cell {F, F$\sharp$, G} and (c) is a double harmonic C major scale that has one cell {B, C, D$\flat$}.  An extreme example is a chromatic scale that encompasses all 12 tones and thus has 12 cells.

(a) C major     (b) C minor blues     (c) C major double harmonic

Scale examples.

While any set of tones can be regarded as a scale, it is useful to introduce constraints to restrict the types of scales we consider.  In particular, for the purposes of having multiple voices moving simultaneously, we want to avoid scales that contain cells.  On the other hand, we can always add more tones to a scale so long as we don't introduce additional cells to that scale.  This motivates the following definition:

Definition. A scale is (semitone cell) complete if the addition of any tone not in the scale increases the number of dissonant semitone cells.

In the examples above, C major and double harmonic scales are complete while the blues scale is not.  For example, we can add D to the blues scale without increasing the number of cells.  We are now ready to define the central concept of this document:

Definition. A harmony is a scale that is complete and has no cells.

Hence, in the examples above, C major is the only harmony, as the blues scale is incomplete while the double harmonic scale contains a cell.  Keeping track of cells in scales gives a effective way to classify them. Namely, as we can always add extra tones to a scale until it is complete, we restrict our attention to complete scales.

(Pedantic) Remark:  When we talk about major harmony, we actually mean any of the 12 major scales.  Thus, harmony really refers to an equivalence class of scales, where we identify scales that shift all the notes by some fixed amount.  For instance, the C major scale and the G major scale are examples of major harmony as we can shift all the notes of the C major scale by a perfect fifth to obtain the G major scale.  To formalize this, we can refer to an abstract harmony as defined by such an equivalence class while a particular scale is a representation of this abstract harmony.  We hope this subtle distinction will be clear from context.

§.§ Classification of Harmony

We now present the classification of harmony, following the work of Gustavo Casenave. The classification is achieved through a brute force approach, possibly with the assistance of a computer:

Classification of Harmony:  There are a total of seven (abstract) harmonies.  They are:

* Whole Tone (WTONE): This is the most symmetric harmony consisting of 6 notes all a whole tone apart.  There are only two such scales together they make up the 12 tones.  Thus, this harmony provides a way of dividing the 12 tones into two coherent complementary harmonies.  This harmony occurs both in classical music and jazz.  In classical, Claude Debussy made use of it in many compositions (perhaps most famously in the beginning of “Prelude to the Afternoon of a Faun").

* Major Harmony (MAJ):  Most common harmony in music.  There are 12 representations of this harmony, one for every tone.  Note that Western musical notation is built around major harmony.  Thus, even if we switch to an instrument that does not have the black-key/white-key division, musical notation still favors the use of such harmony.  In fact, all notes outside of the white keys are viewed as “accidental".  The inclusion of any of the black keys would introduce a dissonant cell to the scale and this observation lies at the heart of our approach to harmony.

* Melodic Minor (MEL):  Probably the most common scale after major harmony.  It has 12 representations and is related to major harmony by flatting the major third to a minor third.

* Diminished Harmony (DIM): One can obtain this scale by starting at any note and alternating between one semitone and two semitone steps. From this description one can see that there are only 3 distinct representations of this 8-note harmony.  Alternatively, this harmony is related to a diminished 4-chord.  Up to ordering, there are only three diminished 4-chords: (B, D, F, A♭),  (G, B♭, D♭, E), and (C, E♭, G♭, A).  Note that a such chord cannot fit into major harmony as they always contain a black key.  One can think of diminished harmony as existing to accommodate such chords.  Indeed, if we take any two of these 4-chords, we get a diminished scale!  Thus, once again we see that there are 3 representations of diminished harmony.

* Symmetric Augmented (AUG): This harmony is similar to diminished but built on augmented chords, as opposed to diminished chords.  There are only 4 augmented chords: (C, E, G♯), (B, D♯, G), (A♯, D, F♯) and (A, C♯, F). We can form an augmented scale by taking two augmented chords that are a semitone apart.  Alternatively, one can obtain this scale by starting at any note and alternating between one semitone and three semitone steps.  There are 4 representations of this harmony.  This harmony is used less commonly in composition.  One example use is in Liszt's “Faust Symphony”.

* Harmonic Minor (HMIN): Obtained from melodic minor by flatting the 6th note.  There are 12 representations of this harmony.

* Harmonic Major (HMAJ): Obtained from major harmony by flatting the 6th note.  There are 12 representations of this harmony.  This harmony is used quite rarely in composition.

Figure <ref> shows representatives of each of these harmonies.

(a) C Whole Tone (WTONE)    (b) C Melodic Minor (MEL)    (c) C Diminished (DIM)

(d) C Symmetric Augmented (AUG)    (e) C Harmonic Minor (HMIN)    (f) C Harmonic Major (HMAJ)

Harmony examples.

In a sense, the content of the current work is devoted to studying the properties of these seven harmonies.  Namely, we devote majority of the text to studying chord formation (voicings) as well as melodic/improvisational aspects within these harmonies. Furthermore, we will explain how these harmonies are implicitly used in music composition.

Harmony and Notation: Note that there is no standard notation for non-major harmony.  For some scales, such as the melodic minor, we can introduce such notation by combining sharps and flats on the music staff.  See figure <ref>.  For others, like whole-tone harmony, incorporating them into standard notation is not straightforward.  Jazz lead sheets address this issue by focusing on complex chord notation, which implicitly indicates an underlying harmony for each measure.

(a) G Melodic Minor     (b)  G Harmonic Minor     (c)  G Harmonic Major

Staff notation for non major harmony.

The brief description of the harmony above is just a peek at the full structure we wish to explore.  To that end, we conclude this section by mentioning an invariant that can be used to tell these harmonies apart.  Specifically, the number of consecutive notes a harmony shares with a whole tone scale can be used as an easy way to differentiate among the possible scales, without needing to memorize all the different representations of these harmonies.  Table <ref> displays the overlap for all the harmonies. We see that MEL has the most in common with WTONE while AUG has the least in common.  The only wrinkle is that both of the harmonic scales have 3 notes in common with WTONE.  From a more pragmatic perspective, we are beginning to address the question of “how can we hear/see the underlying harmony?".   Well, if you see/hear a fragment of music with say four sequential whole tones the only possible harmonies this could belong to are WTONE and MEL.  This is perhaps the simplest example of how our ears/eyes learn to recognize the harmonic structure in a musical piece.

harmony     WTONE     MEL     MAJ     HMIN     HMAJ     DIM     AUG

overlap with WTONE      6      5      4      3      3      2      1

Harmony by overlap with the whole tone scale.

§ PACKINGS

§.§ Initial Classification

We now turn our attention to dissonant blocks instead of cells and develop a classification that parallels that of harmony in the previous section.  Recall from section <ref>, that a dissonant semitone block is given by two notes that are a semitone away, up to an octave.  Conceptually, our aim is to distinguish between sets of notes related to dissonant cells, which we call scales and sets of notes related to dissonant blocks which we call packings.  Of course, at the end of the day, we are still talking about subsets of the 12 tones so the distinction is purely for organization. We have:

Definition. A packing is (semitone block) complete if adding any tone not in the packing increases the number of dissonant (semitone) blocks.

This definition helps explain our use of the term packing:  we want to pack as many notes as we can, while ensuring that two adjacent notes don't form a dissonant block.  In analogy with harmony, we can now classify complete packings that have no dissonant blocks:

Classification of complete packings with no blocks:  Up to shifts, there are a total of four complete packings that have no blocks.  They are:

* Pentatonic packing (penta): This is the pentatonic scale that we have encountered before and discussed at length in the section <ref>. See figure <ref> (a).

* Dominant pentatonic packing (dpenta):  This packing is obtained from penta by raising the 5th degree of the scale.  The crucial difference is that the packing now contains a tritone interval between the 3rd and the (raised) 5th degree.  This is particularly useful for dominant chords. See figure <ref> (b).

* Diminished packing (dim):  This packing is simply the diminished 4-chord.  Note that there are only 4 such chords, up to inversion. See figure <ref> (c).

* Whole tone packing (wtone):  This is the whole tone scale, now viewed as a packing instead of a scale.  This is a rare example where a packing and a scale coincide.  See figure <ref> (d).

(a) F pentatonic  (penta)     (b) G dominant pentatonic  (dpenta)

(c) C diminished  (dim)    (d) C whole tone  (wtone)

Complete packings with no dissonant blocks.

§.§ Irreducible Packings

While the classification of complete, block-free packings is fairly straightforward, we now turn our attention to complete packings that have some dissonance.  Indeed, many complex chords (for example, the major 7th chord) contain dissonant blocks and thus if we want to allow a more general setting for constructing chords as in section <ref>, we need to allow some amount of dissonance.  If we look at all complete packings, there are in fact 132 such packings and this is quite unwieldy.  We now specify a much smaller set that in a sense “generates" the other packings.

To motivate the definition, consider a complete packing $P$. We can always take away tones from $P$ to get a possibly incomplete packing and this justifies why we restrict to complete packings in the first place: other packings are obtained by taking away arbitrary number of tones.  We can in fact go in the other direction.  Namely, given a complete packing we add arbitrary number of missing tones to get new complete packings.  We call packings that are derived from existing complete packings reducible.  On the other hand, there are complete packings that do not arise from such a process:

Definition:  A complete packing is irreducible if no subset of it is complete.

In other words, irreducible packings are complete packings that are not derived from smaller ones.  In this sense, we see that we can restrict our attention to irreducible packings as they generate the rest of the packings.  Note that the packings in the classification from the last section are all automatically irreducible.  Indeed, consider a complete block-free packing $P$.  If there was a smaller packing $P' \subset P$ that was complete that would imply that if we add the missing tones to get back to $P$ we would introduce a block for each missing tone.  This would violate the assumption that $P$ is block-free.

We now classify all irreducible packings:

Classification of Irreducible Packings:  There are a total of seven irreducible packings, up to shifts.  They are given by the 4 block-free packings in figure <ref> and the 3 packings with blocks in figure <ref>.

(a) C augmented (aug)     (b) F harmonic major  (hmaj)     (c) D harmonic minor  (hmin)

Irreducible packings with dissonant blocks.

Note that the harmonic major (minor) packing is obtained from a pentatonic scale by raising (lowering) the second degree. Additionally, the augmented packing (a) contains 3 semitone blocks while the harmonic major/min have one block each. Moreover, note that the augmented packing is identical to the augmented scale AUG we encountered before.  Thus, along with $\textbf{wtone}$, there are two examples where a harmony and a irreducible packing coincide.  Much like the black keys of section <ref>, we can use all these irreducible packings to create improvisational lines and form chords.  In the next section, we turn to the fundamental relation between packings and harmony.

Remark:  We have introduced the concept of irreducibility, as applied to packings.  One can formulate as similar notion for scales and classify irreducible scales that may have dissonant cells.  We explore this in appendix F.

§ THE HARMONY-PACKING DUALITY

§.§ Statement of the Correspondence

As suggested in the previous sections, there is a correspondence between harmony and packings.   We now make this explicit:

Harmony-Packing Duality (or Block-Cell Duality):  There is a one-to-one correspondence between the seven harmonies and the seven irreducible packings:

MAJ $\Leftrightarrow$  penta

MEL $\Leftrightarrow$  dpenta

WTONE $\Leftrightarrow$  wtone

DIM $\Leftrightarrow$  dim

AUG $\Leftrightarrow$  aug

HMIN $\Leftrightarrow$  hmin

HMAJ $\Leftrightarrow$  hmaj

Moreover, given a harmony or a packing, the corresponding dual object is obtained by taking the complementary tones.

As we can see, this is a generalization of the black-key/white-key duality that we observed in section <ref>.  Namely, we observed that the major scale that is defined by a dissonant semitone cell condition has as its complement a pentatonic scale (black keys) that is defined by a dissonant semitone block condition.  We have used cells to identify seven harmonies and we have used blocks to identify seven irreducible packings.  The duality result states that these conditions are complementary to each other. Finally, observe that there are two exceptional cases of the correspondence given by AUG/aug and WTONE/wtone.  These are complementary to themselves.  Such objects are called self-dual.

§.§ Proof of Duality

Given the classification of both harmony and packings in previous sections, verifying the duality is simply a matter of checking the complement for all seven cases.  However, this does not give a complete conceptual picture since it is still rather mysterious why these two seemly different types of constraints are related.  In this section, we give a simple mathematical argument to explain the correspondence a priori, without referring to the explicit classification of either structures.  As this is of more theoretical interest, the reader can safely skip this section since it will not be used explicitly anywhere else in the text.

To prove the correspondence, we will prove an auxillary statement that will allow us to explicitly see the relation between blocks and cells:

Auxillary Result: By taking complements, there is a one-to-one correspondence between scales with cells and incomplete packings.

To prove this result let $S$ be a scale and $S'$ be its complementary packing.  We now show that if $S$ has a cell then $S'$ incomplete.  For this, take the cell in $S$ and for argument's sake let's assume it consists of the tones E, F, and F$\sharp$.  See figure <ref>.  In particular, none of these tones are in the complement $S'$.  This means that we can add F to $S'$ without creating an new block, since the neighboring tones E and F$\sharp$ are not in $S'$.  This implies that $S'$ is incomplete as desired.

We can similarly show that if $S'$ is incomplete, then $S$ must contain a cell.  By definition, $S'$ is incomplete when we can add a tone without creating new blocks.  For concreteness, take F to be such a tone.  Then it follows that E and F$\sharp$ are also not in $S'$ since the presence of either of these tones would create a dissonant block with F.  Thus, $S$ contains the cell given by E, F and F$\sharp$ as desired.  This finishes the proof of the auxillary result.

We can now use this to prove the Harmony-Packing duality.  First, note that another way to rephrase the auxillary result is to say that scales with no cells correspond to packings that are complete.  This is simply the contrapositive of the auxillary result.  For if the complement is not complete there would be a cell in the scale by the result above.  We now wish to show that if a scale $S$ is a harmony then the complement $S'$ is irreducible.  Since $S$ is a harmony, it has no cells and this means that the complement $S'$ is a complete packing. We want to show that $S'$ is not only complete but also irreducible.  Recall, this means that any smaller packing $T' \subset S'$ is not complete.  Well, the complement $T \supset S$ is larger than $S$.  Since we assume that $S$ is a harmony, this means that adding any extra tones to it will create cells.  Using the auxillary result once more, we see that $T'$ is thus not complete because its complement has cells.   This implies that $S'$ is irreducible as desired.

This establishes that the complement of a harmony is an irreducible packing.  Finally, given an irreducible packing $S'$, let us show that its complement $S$ is a harmony.  First, since $S'$ is irreducible it is complete which by our auxillary result implies that $S$ has no cells.  We must still show that $S$ is complete: if we add any tone to $S$ we will create a cell.  Take any tone and add it to $S$ creating a bigger scale $T$.  This means that its complement $T'$ is smaller than $S'$.  Since, $S'$ is irreducible, by definition this means that $T'$ is not complete.  By the auxillary result, this implies that $T$ must have a cell concluding the proof.

Remark:  Given the auxillary result, the remainder of the argument boils down to the fact that harmonies are maximal among scales that have no cells and irreducible packings are minimal among packings that are complete.  Since taking complements reverses maximal and minimal objects, the remainder of the argument follows from this observation.

§.§ Relating Harmony and Packings

In the previous section, we discussed the duality between harmony and packings. From a theoretical standpoint, this reveals a hidden correspondence of harmonic structures.  However, what is the practical use of these findings?  Hopefully, the reader has already anticipated the answer to this question.   Namely, we started this work by pointing out the duality between black keys and white keys, and we spent time developing chords and improvisation based on the black keys.  We then saw that we can apply all these techniques to major harmony since three copies of the pentatonic scale are contained within the major scale.  Thus, somewhat paradoxically, the complement of a harmony is key to unlocking the structure inside that harmony.

Something analogous happens with other harmonies and packings.  In this section, we discuss how these packings fit inside various harmonies.  Similar to the pentatonic scale, we can then use these packings to build chords and improvisational lines inside the corresponding scales. To this end, table <ref> summarizes the number of embeddings of a given packing inside various harmonies.  For concreteness, we take the various harmonies starting in C and indicate in which packings embed in a given harmony by listing the starting tones of those packings.  For instance, as discussed in section <ref>, there are three embeddings of $\textbf{penta}$ into C $\textbf{MAJ}$.  These are indicated by the tones C,F and G in the table.

wtone      aug     penta     dpenta     dim     hmin     hmaj

C WTONE     C

C AUG          C

C MAJ              C,F,G     G

C MEL              F     F,G

C DIM                      C,D     D,F,A$\flat$,B     D,F,A$\flat$,B

C HMIN                      D          A$\flat$

C HMAJ                      D     G

Embedding packings into harmonies starting in C.

The general trend we observe in the table is that typically, it has has nonzero diagonal elements.  This means that the packing and its dual harmony are closely related, despite being complementary.  For instance, there are 3 ways to embed the pentatonic packing into major harmony and only one way to do so for $\textbf{MEL}$. See figure <ref> and <ref> (a). Furthermore, there are no other embeddings of this packing into the remaining harmonies.  This illustrates the close relationship between the pentatonic scale and major harmony.  It also suggests that $\textbf{MEL}$ and $\textbf{MAJ}$ are related since they both contain an embedding of penta.

Similarly, we see in figure <ref> (b) that there are two embeddings of $\textbf{dpenta}$ into $\textbf{MEL}$ which in fact cover the whole scale.  Finally, there is one embedding of $\textbf{dpenta}$ into $\textbf{MAJ}$, as shown in figure <ref> (c).  This once more highlights the close relation between $\textbf{dpenta}$ and its dual $\textbf{MEL}$.  Once again $\textbf{MEL}$ and $\textbf{MAJ}$ are seen to be related as they both contain an embedding of $\textbf{dpenta}$.

(a) penta (green)     (b) dpenta (green and blue)      (c) dpenta (green)

inside MEL            covering MEL         inside MAJ

Embedding penta and dpenta in harmony.

Moving on, the packing/harmony $\textbf{aug}$ and $\textbf{wtone}$ are self-dual.  In particular, the packing perfectly covers the harmony and does not embed into any other harmony.  On the other hand, the diminished packing $\textbf{dim}$ has two embeddings into the harmony $\textbf{DIM}$ that cover the scale.  Indeed, as we discussed above,  $\textbf{DIM}$ is made of two adjacent diminished chords.  See figure <ref> (a).  Additionally, $\textbf{dim}$ has a unique embedding into  $\textbf{HMIN}$ and  $\textbf{HMAJ}$, as shown in (b) and (c) of the figure.

(a) dim (green and blue)     (b)  dim (green)     (c) dim (green)

covering C DIM     inside C HMIN     inside C HMAJ

Embedding dim in harmony.

The case of $\textbf{hmin}$/$\textbf{hmaj}$ is exceptional because the embeddings switch:  $\textbf{hmin}$ embeds into $\textbf{HMAJ}$ while $\textbf{hmaj}$ embeds into $\textbf{HMIN}$.  See figure <ref> for an illustration. Finally, both $\textbf{hmin}$ and $\textbf{hmaj}$  embed inside $\textbf{DIM}$.  Due to the symmetry of that harmony, we can always shift by a minor third and still land in the scale.  For this reason, there are in fact 4 distinct embeddings of $\textbf{hmin}$ and $\textbf{hmaj}$ inside $\textbf{DIM}$. See figure <ref> for an illustration of the packings starting at B.  There are three more such embeddings starting at D, F and A$\flat$.

(a) G hmin (green) inside C HMAJ     (b) A$\flat$ hmaj (green) inside C HMIN

Embedding hmin/hmaj in harmony.

(a) B hmin (green) inside C DIM     (b) B hmaj (green) inside C DIM

Embedding hmin/hmaj in harmony (cont).

In all cases except $\textbf{HMIN}$/$\textbf{HMAJ}$, there are enough embeddings of the dual packing to fully cover the corresponding harmony.  The case of $\textbf{HMIN}$/$\textbf{HMAJ}$ is exceptional since the dual does not embed in the harmony but the embeddings switch as discussed above.  Furthermore, even if we add the embedding of $\textbf{dim}$, we will still be missing a tone in the scale.  We will come back to this point when we discuss irreducible chords in section <ref>.

Overall, the table demonstrates the close relationship between a harmony and its dual packing.  Learning to find these packings inside the harmony helps unlock interesting improvisational structures inside the harmony as well as to build chords, as we demonstrated in section <ref> and section <ref>.  Furthermore, the existence of off-diagonal terms in the table points to interesting embeddings in which the packing is dual to a different harmony creating a more exotic structure inside the scale.  Exploiting all this gives us a large collection of harmonic structures to experiment with in our musical work.

§.§ Projecting Harmony to a Packing

Our initial approach to packings was to view them as complementary/dual to harmony.  We then made the practical observation that in fact there are many embeddings of packings inside harmony, providing simpler structures for forming chords and improvisation.  We can take the latter viewpoint further by viewing harmony as an extension of a packing “shadow” contained inside the harmony.  For instance, we observed in section <ref> that major harmony extends a pentatonic inside it by noting that each tone in major harmony has a unique projection to a given embedded pentatonic.  See discussion at the end of section <ref>.  This generalizes to any packing inside any harmony:

Projecting a Harmony to a Packing:  Given a harmony $H$ and a packing $P\subset H$ contained in that harmony, there is a unique projection of every tone in $H$ to $P$.  In other words, every tone in $H$ is either already in $P$ or has a unique neighbor a semitone away that is in $P$.

To see this, given a packing $P$ inside a harmony $H$, take any note (say F) in $H$.  If F is already in $P$, then there is nothing to project.  If not, then either its left neighbor (E) or its right neighbor (F$\sharp$) must be in $P$.  Otherwise, we would have the semitone cell {E,F,F$\sharp$} centered at F that is disjoint from $P$.  This would violate the fact that $P$ is complete as we can add F to $P$ without creating (semitone) blocks.  Furthermore, only one neighbor is in $P$; otherwise we would have the cell contained in the harmony $H$.  Thus, the note has a unique neighbor in the packing that is a semitone away.

A consequence of this observation is that any melodic line in $H$ has a unique projection to $P$.  This provides a simplification of the melodic line by creating a “skeletal” version that removes some of the tension.  This can be used in helping create a harmonic profile of a piece of music.

§.§ Harmony and Piano Layouts

(a) Whole Tone     (b) Melodic Minor

(c) Diminished     (d) Augmented

(e) Harmonic Minor     (f) Harmonic Major

Keyboard layouts for non major harmonies.

We round out the general discussion of harmony by returning to the basic question we asked at the beginning of this work: why are the piano keys arranged that way?  As discussed, the arrangement highlights major harmony by making the white keys the C major scale.  Dually, we also noted that the black keys highlight the pentatonic scale (we call this a packing).  As discussed in the text, there are actually seven distinct harmonies and their corresponding packings.  It is instructive to imagine the piano layout if it were designed to highlight each of these other harmonies.  Figure <ref> plots the imaginary layouts corresponding to each of the other six harmonies where the white keys correspond to the harmony and the black keys are the dual packing.  We see, for instance, the symmetry of whole tone harmony, as well as that of diminished harmony.  Additionally, we observe that harmonic minor/major are mirror images of each other, which helps explain their close relationship.  All this is hidden in the standard piano layout. Of course, since the major scale is the most commonly used in music, we are not advocating for actually building such keyboards and merely want to visually highlight the symmetries in the corresponding harmony and packing.

Remark (Duality in Mathematics):  It is worth noting that some of the ideas in this section fit into a general set of ideas in mathematics known as duality. In the words of Sir Michael Atiyah, “duality in mathematics is not a theorem, but a principle”.[see M Atiyah, “Duality in Mathematics and Physics” lecture notes from the Institut de Matematica de la Universitat de Barcelona (IMUB).]
Broadly speaking, duality relates two types of structures in a one-to-one fashion.  Duality appears in many important mathematical areas, from set theory, algebra, optimization, geometry and analysis to number theory and string theory.  In some situations, the dual structure is more tractable and can be used to gain insight into the original object.  We saw a version of this in our context where the dual object to a harmony is a packing and provides a way to understand the original harmony.  For a general overview of duality, see the Wikipedia entry on duality (<https://en.wikipedia.org/wiki/Duality_(mathematics)>).

§ INTERLUDE: INTRODUCTION TO MODES

§.§ Modes of Major Harmony

We now introduce the concept of a mode that is central to many forms of Western and Eastern musical traditions.  While most of this material is well known, the topic can be a bit confusing and we would like to clarify the concepts that are most relevant to our treatment of harmony.

Let's take the example of major harmony as represented by the C major scale.  On its own, the scale does not sound “happy” or “sad” and we must introduce a tonal center to give it that quality.  We may think of modes of the major scale as the seven choices of tonal centers that are in the scale.  How do we actually hear the tonal center?  For this, we go back to our discussion of tuning and fix a bass note with some fundamental frequency $f$.  Recall, that we think of other tones as derived from overtones, which are multiples of this $f$.  Smaller multiples of $f$ typically sound more consonant with the fundamental frequency.  We can think of a mode as a choice of a bass note frequency $f$ and then hearing the remaining notes in reference to this bass.  In music, the mode is typically established by playing a low bass note at the beginning of a measure.

Modes of major harmony are closely connected to the circle of fifths:

\begin{equation}
\text{F, C, G, D, A, E, B, F{\sharp}, C{\sharp}, G{\sharp}, D{\sharp}, A{\sharp}, F }\dots
\end{equation}

or, equivalently

\begin{equation}
\text{F, C, G, D, A, E, B, G{\flat}, D{\flat}, A{\flat}, E{\flat}, B{\flat}, F}\dots
\end{equation}

By symmetry, a major scale is given by taking any seven consecutive tones in this sequence. Suppose we want our scale to contain a given bass note, say F.  As discussed, the default option is to take {F, C, G, D, A, E, B} as these notes are the closest to F in the overtone series.  We can then shift the seven note window one at a time to obtain:

* {F, C, G, D, A, E, B} (F Lydian)

* {B$\flat$, F, C, G, D, A, E} (F Ionian)

* {E$\flat$, B$\flat$, F, C, G, D, A} (F Mixolydian)

* {A$\flat$, E$\flat$, B$\flat$, F, C, G, D} (F Dorian)

* {D$\flat$, A$\flat$, E$\flat$, B$\flat$, F, C, G} (F Aeolian)

* {G$\flat$, D$\flat$, A$\flat$, E$\flat$, B$\flat$, F, C} (F Phrygian)

* {B, G$\flat$, D$\flat$, A$\flat$, E$\flat$, B$\flat$, F} (F Locrian)

We have labeled the modes by their traditional Greek names and ordered them from brightest to darkest.  Note that each change replaces one tone by another tone that is further away in the overtone series from F.  This is the physical reason modes get darker as we alter more notes.  Note that the F major scale is given by the Ionian mode though, as we observed before, the Lydian mode is actually brighter.

Presenting modes by fixing a bass note and following the circle of fifths is perhaps practically most useful.  It illustrates that modes have to do with alteration of the scale, one note at a time.  Thus, we can think of the modes are labelling different scales for a fixed tonal center.  See figure <ref> for a visual illustration.  We start with the C major scale (F Lydian) and go counterclockwise until we get to the G$\flat$ major scale (F Locrian). Beyond this point we would lose the F note from the scale.

Circle of fifths.

An alternative way to view modes is to fix the scale and alter the bass.  For example, if we fix the C major scale we list the corresponding modes with their bass notes as:

* C Ionian {C, D, E, F, G, A, B}

* D Dorian {D, E, F, G, A, B, C}

* E Phrygian {E, F, G, A, B, C, D}

* F Lydian {F, G, A, B, C, D, E}

* G Mixolydian {G, A, B, C, D, E, F}

* A Aeolian {A, B, C, D, E, F, G}

* B Locrian {B, C, D, E, F, G, A}

Both perspectives are useful in their own right since they highlight complementary aspects of the modes.  For instance, suppose you wanted to find the A Dorian mode.  From the list above, we see that the Dorian mode is given by the second degree of a major scale.  This means that to find A Dorian we need to find a scale where A is the second degree.  This is given by the G major scale as G is one whole tone below A.  The reader is encouraged to try playing the C major scale randomly while holding down a fixed bass note.  Depending on the bass note, the scale will sound rather different reflecting how bright or dark the given mode is. For instance, if we have an F in the bass we will get a bright sound corresponding to the F Lydian mode, while holding a B in the bass will give the very dark Locrian mode.

As discussed, for a given bass note, we can order the modes from brightest to darkest.  We can view this progression of modes from the pentatonic perspective.  Namely, recall that the major scale can be decomposed into three (overlapping) pentatonics.  For example, C major is made of the F, C  and G pentatonics.  If we view this as the F Lydian mode, the next mode is F Ionian which decomposes as the F, C, and B$\flat$ pentatonics.  Thus, we swapped the G pentatonic for a B$\flat$ pentatonic.  Similarly, the next mode is the F Mixolydian which is composed of the F, B$\flat$ and E$\flat$ scales.  Thus, each time we pass to the next mode we swap out one of the pentatonics for a new one, following in reverse the circle of fifths as we proceed.

§.§ Modes of Other Harmonies

The concept of mode is equally applicable to other harmonies.  Effectively, it amounts to picking a bass tone and viewing the remaining notes of the scale in reference to this tone.  To give a concrete example, let's take whole tone harmony.  This is a 6-note scale, so as a first guess we might think that there are 6 distinct modes.  However, since all the notes are one whole tone apart, all these six modes are equivalent up to a shift.  Thus, whole tone harmony only has one distinct mode.  In general, we have:

* Whole Tone Harmony: one mode

* Augmented Harmony: two modes

* Diminished Harmony: two modes

* Major, Melodic Minor, Harmonic Major/Minor: seven modes each

We will not bother naming all the modes and will simply refer to them by number.  For example, let us understand the 4th mode of melodic minor (MEL).  As our scale we can take the C major scale with a flat 3rd degree.  The 4th mode starts with the 4th note in this scale which is F.  We claim this is the same as the Mixolydian scale except it has a sharp 11th.  Well, how do we find the Mixolydian scale?  From the discussion above, we see that it is the scale associated with the 5th mode of the major scale.  For F to be the fifth note we must take the B$\flat$ scale.  Thus, the F Mixolydian scale is the B$\flat$ scale which is the same as the C melodic minor except that the B$\flat$ is changed to a B.  This is exactly what we meant by the sharp 11th.  In fact, this scale is sufficiently common to have its own name: the Lydian Dominant scale.

Another important example from melodic minor is given by the 7th mode.  For the C melodic minor scale, this is the mode that starts with B.  Note that this mode contains the major third (B to E♭) as well as the minor 7th (B to A).  Thus, one can build a basic dominant chord for B using this scale.  However, this scale is very different from the standard scale associated with the dominant B chord.  In fact, one can check that all the remaining tones of this scale are altered from the standard B mixolydian scale.  For this reason, this is known as the altered scale.  Note that is the same scale as the Lydian Dominant scale for the bass that is a tritone away from B (which is F in this case).

As discussed above, modes play a significant role in music, since fixing a bass gives a “feel” to the underlying scale.  Modes also play a key role in lead sheet/chord notation as the letter of the chord denotes the bass note of the mode.  As we will discuss in section <ref>, we typically think of modes as associated to a with chord symbol.  For example, a G$^7$ chord by default denotes the G Mixolydian mode while a Cmin$^7$ denotes the C Dorian mode.  Similarly, the Lydian Dominant mode is associated with the F$^{7(\sharp 11)}$ while the altered scale is associated with the B$^{\text{7alt}}$. We will come back to the relation between modes and chord notation in section <ref>.

As a final observation, note that it is possible to order modes according to their brightness for other harmonies.  Let us for example consider $\textbf{DIM}$.  If we fix a diminished chord, say (C, E$\flat$, G$\flat$, A), we can either add an “upper” chord (G, E, D$\flat$, B$\flat$) or a “lower” chord (D, B, A$\flat$, F) to obtain a diminished scale.  Relative to C, this choice will determine which of the two modes we are in. Comparing these two diminished chords presented this way, we see that the second choice is further away  on the circle of fifths starting at C than the first.  For instance, G comes before D and E comes before B.  Thus, the mode that contains the (G, E, D$\flat$, B$\flat$) chord will be more consonant and hence brighter.

Similar remarks apply to the two modes of $\textbf{AUG}$. Here, starting with (C, E, A$\flat$), we can either add (G, B, E$\flat$) or (A, D$\flat$, F) to create an augmented scale.  Once again, (G, B, E$\flat$) is closer to C on the circle of fifths than the other choice.  Hence, the mode that contains (G, B, E$\flat$) will be more consonant.

Remark: We will return to the connection between the brightness of a mode and chords when we discuss lead sheets in section <ref>.  As a preview, note that when we labeled the $\textbf{DIM}$ and $\textbf{AUG}$ scales we chose the mode that is the darker of the two.  See figure <ref> (c) and (d) for the modes starting in C.  As we will discuss later in the text, the brighter mode is associated with a standard jazz chord (dom7$\flat$9 for $\textbf{DIM}$ and maj7 for $\textbf{AUG}$).  This leaves remaining modes of each scale to be associated with the dim7 chord for $\textbf{DIM}$ and the aug chord for $\textbf{AUG}$.  This way, the C $\textbf{DIM}$ scale is associated with the Cdim7 chord and the C $\textbf{AUG}$ scale with the Caug chord.

§ COMBINATORIAL APPROACH TO CHORDS AND VOICINGS

§.§ Overview of Classification

We now assemble the ideas introduced in this work to give a classification of chords.  The basic approach is similar to what we used to classify packings/harmony.  We introduce a certain dissonance criteria (related to blocks and cells) and use this to construct all the irreducible chords.  The choice of criteria is less canonical than that used for harmony/packings, so let us say a few words on the general goal of such a classification:

* The list of irreducible chords should be small, so that any other chord is obtained by either removing notes (forming incomplete chords) or adding notes (forming reducible chords).  Thus, knowing this small list of options will allow the musician to produce many different arrangements, as we already saw in section <ref> and section <ref>.

* Irreducible chords should include most of the commonly used chords.

* The irreducibility criterion is related to some simple property of tones, as opposed to an arbitrary combinatorial rule.

These goals do not single out one irreducibility criterion and slightly different classification schemes are possible.  However, at the end of the day, they will produce similar sets of irreducible chords which is the real goal of this classification.

Our irreducibility criterion will relate to both blocks and cells.   Recall, when we discussed chord shapes in section <ref>, we decided to avoid having three notes each a tone apart.  This is an analogue of a semitone cell, except now we require notes to be at most a tone apart:

Definition:  A dissonant tone cell consists of three distinct notes in a row which are at most a tone apart, up to shifts by an octave.

See figure <ref> for possible tone cell configurations with middle note F.

Possible dissonant tone cell configurations with middle note F.

Our completeness criteria for chords will combine semitone blocks and tone cells:

Definition:  A chord is complete if adding further notes will increase either the number of semitone blocks or the number of tone cells (or both).

Definition:  A chord is irreducible if it is complete, but no subset of the chord is complete.

We can now determine all the possible irreducible chords.  It turns out there are eleven of them.  In the following subsections, we go through all these chord shapes and comment on their embeddings into harmony and packings as well as their use to voice modes.  In most cases, these chords are well known and we will use standard notation to denote them.

Remark (Triads):  We are focused mainly on 4-note chords, which are more common in jazz harmony than classical harmony that focuses on triads.  We will explain how triads fit into our framework in appendix E.

Remark (Inversions):  As discussed from the very beginning, for the purposes of harmony, we do not distinguish tones that are an octave apart.  For chords, this implies that we identify chords that are related by an inversion.  For example, the A minor 7th chord (Amin7) and C major 6th (Cmaj6) are related by moving the root note of A up an octave.  Ultimately, we resolve this ambiguity by using the concept of modes.  Thus, the choice of a bass note, typically in the lower range of the piano, determines the quality of the chord.  See section <ref> for the discussion of how this relates to standard chord notation.

Remark (Guide to Reader):  The rest of this section is list of all possible irreducible voicings as well as their embeddings into harmony.  As such, it is recommend that the reader views the rest of this section as a reference to be digested slowly.  To aid the reader, we introduce the most common voicings and embeddings first.

§.§ min7 (maj6)

We have already encountered the minor 7th chord when we looked at the black keys in section <ref>.  An inversion of it gives us the major 6th chord so either naming convention is reasonable as long as we understand that we are studying these chords up such inversions.  As we already discussed, min7 occurs in the pentatonic packing penta and in major harmony MAJ.  As discussed in section <ref> there are three embeddings of min7 into major harmony.  Since there is a unique embedding of penta into MEL, the chord also has one embedding in MEL.  Additionally, the chord appears in the packings hmin and hmaj so there is a unique embedding into HMIN and HMAJ.  See figure <ref> (a), (b) and (c), where the chord is indicated in red while the scale is green.

Finally, there are four embeddings of this chord into DIM, as illustrated in (d) of the figure.  These correspond to the embeddings of hmin into DIM.  Note that in the figure, we only plot Dmin7 which is only one of four possible chords.  Due to the symmetry of the DIM scale, whenever we have a chord in the scale, we have the other three chord embeddings that make up the diminished 7th chord.  For instance, for C DIM, we can embed Dmin7, Fmin7, A$\flat$min7 and Bmin7 since these bass notes make up a diminished 4-chord.  Observe that Dmin7 is also Fmaj6 so that the scale also contains Dmaj6, Fmaj6, A$\flat$maj6 and Bmaj6.  This is a rare instance of a scale containing both the major and minor 4-chords with the same bass!

(a) Dmin7 in C MEL     (b) Fmin7 in C HMIN

(c) Emin7 in C HMAJ     (d) Dmin7 in C DIM

{F,A$\flat$,B}min7 not shown

Possible embeddings of min7.

This chord is most commonly used to voice the Lydian mode of major harmony.  Concretely, when we have C in the bass, we use Amin7/C and Emin7/C as in section <ref>.  In fact, using Emin7 over a C in the bass is the standard “rootless” jazz voicing of the Lydian mode.  It is called rootless since the chord Emin7 is missing the root note which is C.  The motivation here is that since we already have C in the bass, there is no need to repeat this note in the chord. Instead, we add the extra 9th note (D in this case) to create a more complex voicing.  Thus, since Emin7 consists of (E, G, B, D) we are voicing the 3rd, 5th, 7th and 9ths tones of the C Lydian scale.  Of course, we may use a min7 over any mode of any harmony where there is an embedding and this will produce a plethora of more exotic voicings for us to experiment with.

Remark:  We wanted to take a moment to clarify the relation to standard chord notation that we will discuss in section <ref>.  Namely, when you see a Cmaj$^7$ chord symbol in sheet music, this suggests that the underlying harmony/mode is the C Lydian mode.  We have the C in the bass and we can certainly use the Cmaj$^7$ chord to voice this harmony.  However, this would be a bit of a waste since we already have C in the bass.  Thus, if we want to form a 4-note chord for the mode, we use the Emin7 chord over the C in the bass.  This adds the extra 9th tone (D in this case).  This is bit confusing for the beginner since we are effectively saying to use a minor 7th chord whenever we see a major 7th chord in the notation.  However, this is standard practice and we hope by using accurate notation (Emin7/C) we can make the voicing clear.  See section <ref> for further discussion.

§.§ maj7

The major 7th chord consists of a major triad with a major 7th added.  Following the discussion above, the maj7 chord is commonly used to voice the Dorian mode (which is confusingly associated with the min7 chord symbol!).  For example, for D Dorian, which is the second mode of C major, we would use Fmaj7/D to voice this mode.

Despite its pervasive use, the maj7 chord can only be found in the aug packing!  See figure <ref> (a) for an embedding.  As far as harmony, maj7 occurs twice in MAJ, and once in HMIN and HMAJ.  See figure <ref> (b)-(d). Since AUG harmony is the same as the packing (we call AUG self-dual) we of course have the three embeddings of maj7 into this harmony. This is an interesting case of a chord shape “borrowed” from one packing to use in a harmony (MAJ) quite different from the original AUG.

(a) Cmaj7 in B $\textbf{aug}$/$\textbf{AUG}$     (b) Cmaj7 (green) and Fmaj7 (blue) in C MAJ

(c) A$\flat$maj7 in C HMIN     (d) Cmaj7 in C HMAJ

Embedding maj7 into harmony.

Another common use of this chord is to give a “suspended” voicing for the Mixolydian mode.  For example, we can use Fmaj7 over G to get a suspended sound for G Mixolydian. We will discuss additional voicings of this mode below.

§.§ dom7

This is a standard dominant chord obtained by adding a minor 7th to a major triad.  As such, it contains the tritone interval and is used to build tension before resolution to a more consonant chord.  It is contained in two packings: dpenta and hmin. See figure <ref>.

(a) Gdom7 in G dpenta     (b) Gdom7 in B$\flat$ hmin

Packing embeddings of dom7.

This chord has one embedding in major harmony where it is commonly used in classical music to voice the Mixolydian mode.  For example, in C major it is given by G, B, D, F and denoted Gdom7 or just G7.  See figure <ref> (a).  Note that since the chord already contains the bass note G of the Mixolydian mode (recall that the Mixolydian is the 5th mode which starts at G for C major), it is less complex as a voicing than say our voicing of the Lydian mode by Emin7/C.  Thus, for a more complex sound, other chords are often used to voice the Mixolydian mode in jazz. We will discuss these below.  Finally, note that dom7 occurs in MEL and HMAJ twice, in DIM four times and in HMIN once.  See the figures below for an illustration of this.

(a) Gdom7 in C MAJ     (b) Gdom7 in C HMIN

Embedding dom7 into harmony.

(a) Gdom7 (green) and Fdom7 (blue)      (b) Gdom7 (green) and Edom7 (blue)     (c) Gdom7 in D DIM

in C MEL     in C HMAJ     {B$\flat$,D$\flat$,E}dom7 not shown

Embedding dom7 into harmony (cont.)

§.§ dim7

The diminished 7th chord shape is also the diminished packing dim and most commonly associated with diminished harmony DIM.  As discussed in section <ref>, diminished harmony is constructed from two dim7 chords that are a semitone apart. See figure <ref> (a). Additionally, there are unique embeddings of the diminished chord into HMIN and HMAJ.  See figure <ref> (b) and (c).

(a) Cdim7 (green) and Ddim7 (blue)      (b) Ddim7 in C HMIN     (c) Ddim7 in C HMAJ

in C DIM

Embedding dim7 into harmony.

The diminished chord is typically used to voice the modes of diminished scale.  Recall that this scale has two distinct modes.  By convention the “upper” green chord of figure <ref> (a) is used to voice both of these modes.  Concretely, we could use Cdim7 over both C and F for this scale.

§.§ min7$\flat$5

This chord is obtained from a standard min7 by flattening the 5th degree.  Alternatively, it is called the half-diminished chord since it contains a diminished triad and denoted by the symbol Ø.  It is contained in both the dpenta and hmaj packings, as shown in figure <ref>.

(a) Bmin7$\flat$5 in G dpenta     (b) Bmin7$\flat$5 in D hmaj

Embedding min$\flat$5 into packings.

As far as embeddings into harmony, there are unique embeddings into MAJ and HMAJ, as well as two embeddings into MEL and HMIN.  Finally, there are four embeddings into DIM.  See figure <ref> and figure <ref>.

(a) Bmin7$\flat$5 in C MAJ     (b) Dmin7$\flat$5 in C HMAJ

Embedding min$\flat$5 into harmony.

(a) Bmin7$\flat$5 (green)       (b) Dmin7$\flat$5 (green) and      (c) Bmin7$\flat$5 in C DIM

and Amin7$\flat$5 (blue) in C MEL     Fmin7$\flat$5 (blue) in C HMIN      ({D,F,A$\flat$}min7$\flat$5 not shown)

Embedding min$\flat$5 into harmony (cont).

This chord is commonly used to voice the Locrian mode of major harmony, as well as the second mode of harmonic minor.  For example, the B Locrian mode (which uses the C major scale) would start at B and the chord is Bmin7$\flat$5/B.  The same chord is used for the second mode of harmonic minor, which for A HMIN would be Bmin7$\flat$5/B as well.

§.§ maj7$\flat$5

This chord is obtained from a standard maj7 by flattening the 5th.  It provides our first example of a chord that does not come from any packing.  Since it contains a block and a tritone interval, it is fairly dissonant.  It has a unique embedding into MAJ, MEL, HMIN and HMAJ and no other embeddings.  See figure <ref>. This chord is commonly used in jazz to give a more complex voicing to the Mixolydian mode.  For example, the chord Fmaj7$\flat$5/G can be used to voice the G Mixolydian mode.

(a) Fmaj7$\flat$5 in C MAJ     (b) E$\flat$maj7$\flat$5 in C MEL

(c) A$\flat$maj7$\flat$5 in C HMIN     (d) A$\flat$maj7$\flat$5 in C HMAJ

Embedding maj7$\flat$5 into harmony.

§.§ aug

The augmented triad is the only irreducible chord with 3 notes and is constructed by stacking 2 major thirds.  It occurs twice in the augmented harmony/packing and twice in whole tone harmony/packing, as shown in figure <ref>.  Furthermore, there is a unique embedding of this chord in MEL, HMIN and HMAJ as shown in figure <ref>.  It is commonly used to voice the modes of melodic major.  For example, for the C melodic minor scale, the chord Gaug/C is known as the Cmin(maj7) chord.

(a) Caug (green) and Faug (blue)    (b) Caug (green) and Daug (blue)

covering C AUG       covering C WTONE

Packings/harmony that is covered by aug.

(a) Gaug in C MEL     (b) Gaug in C HMIN     (c) Caug in C HMAJ

Other harmony embeddings of aug.

§.§ min11

We have already encountered the minor 11th chord when we looked at the black keys in section <ref>.  As we already discussed, min11 occurs in the pentatonic packing penta.  As discussed in section <ref>, there are four embeddings of min11 into major harmony.  Additionally, there is a unique embedding into HMIN and HMAJ and two embeddings into  MEL.  See figure <ref> (a), (b) and (c).

(a) Dmin11 in C HMIN     (b) Dmin11 in C HMAJ     (c) Dmin11 (green) and Amin11 (blue)

in C MEL

Embeddings of min11.

This chord is most commonly used to voice the Lydian mode of major harmony.

§.§ dom$\sharp$11

This is a dominant 7th chord, where instead of a 5th degree we have a sharp 11th.  It consists of two tritones that are placed a whole tone apart.  It occurs only in the whole tone packing and there are three distinct embeddings of it.   As far as harmony, it has one embedding in MEL and two embeddings in DIM.  See figure <ref> for the examples.

This chord is useful as a voicing of the dom$\sharp$11 chord, associated with the 4th mode of MEL.  For example, F is the 4th mode of C MEL and we would voice this as  Fdom$\sharp$11/F.  Finally, this chord can also be used to voice the 5th mode of MEL which is the same as Mixolydian but with a $\flat 13$ (or $\flat$6).  For C MEL, it would be Fdom$\sharp$11/G.

(a) F$\sharp$11 in F WTONE     (b) F$\sharp$11 in C MEL     (c) F$\sharp$11 in C DIM

({G,A}$\sharp$11 not shown)                                                              (D$\sharp$11 not shown)

Embedding dom$\sharp$11 into harmony.

§.§ dom11

The dominant 11th chord is another example of a irreducible chord that is not derived from any packing.  It is formed from a regular dom7 chord by replacing the 5th with a 4th. This chord has a unique embedding in MAJ, MEL, HMIN and HMAJ harmony and no other harmony.  See figure <ref>.

(a) Gdom11 in C MAJ     (b) Gdom11 in C MEL

(c) Gdom11 in C HMIN     (d) Gdom11 in C HMAJ

Embedding dom11 into harmony.

A possible use is to voice the Mixolydian mode of  MAJ.  For instance, for G Mixolydian,  if our top melody note is a G we may use the inversion (F, B, C, G) as a more complex variant of the vanilla (F, B, D, G) voicing of that mode.  The chord Gdom11 can be abbreviated to G11.

§.§ dem

This chord shape is constructed from two tritone intervals that are a semitone apart.  Due to the fact that it contains two semitone blocks, it is fairly dissonant and thus rarely used in practice.  The chord shape does not occur in any packing, has two embeddings in diminished harmony and no additional embeddings in other harmonies.   Therefore, it can be used to voice diminished harmony in a way that distinguishes it from all other harmonies. See figure <ref>.

(a) Adem     (b) Cdem

Embedding dem into C DIM.

The author is not aware of a standard name for this chord and, since we already have “dim” and “dom” chords, we decided to use “dem” for the name.  In the 19th century, the tritone was associated with the devil for its harsh sound and since this chord shape contains two tritones we may think of “dem” as short for demon.

One potential use of this chord is to voice the dominant chord that embeds inside diminished harmony.  Figure <ref> displays a  jazz “251” progression with a standard dominant chord and then shows a variant that uses the dem chord to voice an alteration of the dominant chord that lives inside DIM.[see appendix G for more on the 251 progression and substitutions]  While fairly dissonant, this voicing singles out the diminished scale as the only scale that supports this chord. The figure also includes the “tritone” substitute progression where we replace G by D$\flat$, which is a tritone away. This type of substitution is common since G and D$\flat$ dominant chords have the same tritone where the 3rd and 7th degrees are switched.  What makes the dem voicing interesting is that it is symmetric with respect to this substitution: the $\sharp$9th and 13th for G become 13th and $\sharp$9th for D$\flat$.  A symbolic expression of this is
$$\text{Fdem = Bdem}$$
where as usual we only care about the chord shapes up to inversion.

251 progressions in C.

This concludes our discussion of irreducible chord shapes and their embeddings.  Notice that, for the most part, these chord shapes do come from packings, with the exception of dom11, maj7$\flat$5 and dem.  From a our perspective, understanding these chord shapes and their embeddings lies at the heart of the study of harmony.  These chords provide us with the landscape of harmonic combinations that are possible in our framework.

§.§ Reducible Voicings and Completeness Results

We conclude the discussion by mentioning some common reducible voicings and placing our classification of chords into a more general context.  So far, we focused our attention on irreducible chords, arguing that these are the building blocks for all chords.  Now, we will elaborate on how this holds true in practice.

Let's consider complete chords. By construction, these are either irreducible and already appear on our list above, or are obtained by adding extra notes to irreducible chords.  Naturally, there are many more such combinations[265 to be exact] than the 11 irreducible chords we discussed.  However, most of them contain too much dissonance in the form of blocks and tone cells to be useful in most standard musical styles.  Let us look at what is perhaps the simplest nontrivial family of complete chords: those that contain zero tone cells and at most one (semitone) block.  We have the following result:

Characterization of chords with one block:  Consider the set of chords that are (chord) complete, contain at most one semitone block and zero tone cells.  These chords are either irreducible as above or one of the four reducible chords in figure <ref>.

(a) Caug$^{(\text{add 13})}$     (b) Caug$^{(\text{add 5})}$

(c) Dmin7$^{(\text{add $\flat$11})}$ (hmin)     (d) Dmin7$^{(\text{add $\sharp$11})}$  (hmaj)

Reducible chords with one semitone block and zero tone cells.

Thus, there are only four examples of reducible chords of this type.  The first two (a), (b) are obtained by adding one extra note to the augmented chord and we highlighted the additional tone in green.  Furthermore, the notation Caug$^{(\text{add 5})}$ is intended to reflect that we have added extra tones to an irreducible chord shape.  These chords can be used to voice modes of MEL.  Chords in (c) and (d) of the figure are simply the hmin/hmaj packings.

Another simple case of reducible chords is given by complete chords with no blocks and at most one tone cell.  Since such chords have no blocks, we can embed them inside a block-free packing, which are penta, dpenta, wtone and dim.  We have:

Characterization of chords with one cell:  Consider the set of chords that are (chord) complete, contain at most one tone cell and zero (semitone) blocks.  These chords are either irreducible as above or one of the two reducible chords in figure <ref>.

(a) Caug$^{(\text{add min7})}$ (Cdom$\flat$13 or Cdom7$\sharp5$)     (b) Cmaj6$^{(\text{add 9})}$ (Cmaj$^{6/9}$)

Reducible chords with zero semitone blocks and one tone cell.

Since these chords are relatively common, in addition to their notation in terms of smaller irreducible chords, we have included their standard lead sheet notation. We will discuss lead sheet notation in more detail in the following section.   The chord in figure <ref> (a) can be used to voice the 5th mode of melodic minor which is just the Mixolydian mode with a $\flat$13.  The chord in figure <ref> (b) is used to create pentatonic 5-note voicings.  In fact, we already saw an example of this in section <ref> at the penultimate bar of figure <ref>.  Since this chord includes the entire pentatonic scale, there are three embeddings of this in $\textbf{MAJ}$, one in $\textbf{MEL}$ and no other embeddings.  As discussed in section <ref>, this chord should be spread across the two hands to get a less cluttered sound.

We end this section by returning to one of the first applications of chords we saw in section <ref>.  Namely, given a melodic line, we want to form a chord that will have the melody note as its top tone.  In section <ref> and <ref>, we used irreducible chord shapes min7 and min11 to harmonize a melody.  There are sufficiently many irreducible chords to do this in general:

Harmonizing tones:  For any harmony and any tone inside that harmony, there is an irreducible chord in that harmony that contains the tone as its top note.

We can verify this proposition by directly checking all the seven harmonies using the chord shapes above.  For most cases, we observe that the dual packings cover the harmony so we can use chords from that packing for every note of the scale.  For example,  MEL is covered by two dpenta and we can use dom7, and min7$\flat$5 to harmonize all the tones.  This approach works for all harmonies except HMIN and HMAJ.  Here, even if we use all the packings that embed into the harmony we will still have tones that are not covered.  For example, hmaj and dim embed into HMIN but we will still be missing the 5th tone of the scale.  Fortunately, we can use a min7$\flat$5 and an aug chord to cover the scale.  See figure <ref>.

(a) A$\flat$ hmaj (blue) and D dim (green)     (b) E$\flat$aug (green) and Dmin7$\flat$5 (blue)

in C HMIN      covering  C HMIN

Covering HMIN by irreducible chords.

We will refine this discussion further when we will go over concrete voicings for the most common modes in section <ref>.  This will provide examples of explicit chords that harmonize every tone in a given scale.

§.§ Summary of Harmony-Chord Embeddings

We devoted this section to a rather detailed description of all irreducible chord shapes and their embeddings into harmony.  For convenience, we now present this information from a complementary perspective, starting with a harmony, and noting all the chord embeddings that fit into this harmony.  This way of presenting the information is useful in practice as we typically start with a harmony and then try to find chord shapes that fit inside this harmony, as opposed to starting out with a abstract chord shape.  Table <ref> summarizes all the chord embeddings for harmony that starts in C.  For example,  if we want to look up the irreducible chords for the C WTONE scale, we would look at the letters in the C WTONE row.  The possible chords are Caug, Daug, Cdom$\sharp$11, Ddom$\sharp$11 and Edom$\sharp$11.  While this table does not give an indication on when to use these chords in practice, it maps out the possible harmonic options one can potentially consider for composition and arrangement.  We will have more to say about the common use of these chords when we discuss lead sheets in section <ref>.

C harmony          min7     maj7     dom7     dim7     min7$\flat$5      maj7$\flat$5     aug     min11     dom$\sharp$11     dom11     dem

WTONE                                                                            C,D                 C,D

MAJ       C,F     C,F      G                 B               F                         D,E                     G

G                                                                            A,B

MEL        F                 G,F               B,A              E$\flat$                  G       D,A        F                G

HMIN       A$\flat$        A$\flat$          G          D       D,F        A$\flat$       G       D                 G

HMAJ     G     C     G,E     D     D      A$\flat$     C     D          G

DIM     D,F                 D,F            C,D     D,F                            D,F          D,B

B,A$\flat$          B,A$\flat$             B,A$\flat$

AUG          D$\flat$,F                       C,D$\flat$

Embedding irreducible chords into harmony starting at C.

The table above gives a concise description of all the embeddings of irreducible chords.  While it is useful to know the set of all possibilities, from a practical perspective, it is desirable to have a short list of chords that are particularly common to a given harmony/mode.  We will go over one possible list of chords for a few of the most common modes in section <ref>.

§ LEAD SHEETS AND HARMONY

§.§ Relating Chord Notation and Modes

In this work, we have given a definition of harmony and packings and classified the resulting objects.  Furthermore, we combined ideas from these constructions to define and classify irreducible chord shapes.  Finally, we explained how these chord shapes fit into various harmonies.  This is by no means the standard way to think about harmonic constructions in practice.  In this section, we would like to bridge the gap to the more conventional representation of this circle of ideas by discussing lead sheet notation and how it fits into our story.

Lead sheets provide a concise and powerful way to use chord notation to denote harmonic structure in practice.[see appendix A for a review of standard chord notation.]  We can view harmony as a three stage process:

HARMONY $\Rightarrow$ MODE $\Rightarrow$ VOICING

The most important thing to understand is that lead sheet notation gives you partial information about all three aspects of harmony.  While there are typically standard choices of chords to go with a chord symbol, the reality is that the notation gives the performer the flexibility to form their own arrangement of the tune.  Here, we view this through the lens of harmony/voicings and apply the techniques discussed in this work to build an arrangement/improvisation for the piece.

Let us start with the symbol for the major 7th chord, say Fmaj$^7$, also written as F$^{\Delta 7}$.  The most basic information is the letter F which denotes the bass note of the mode.  Now, we have to figure out the harmony and mode associated with this symbol.  By definition, in addition to the bass, a major 7th chord contains the major 3rd and 7th of the chord scale.  Theoretically, we can use any harmony that contains these notes as long as the melody fits into that harmony.  We follow the following principle:

Chords to Modes Correspondence Principle:  Given the chord tones associated to a chord symbol, find the brightest mode that contains these chord tones.

Thus, for a major 7th chord we need to find a mode that has a major 3rd and a major 7th.  F Lydian is the default choice as it is the brightest of all major modes and contains the desired chord tones. We can make other choices of mode.  For example, the F Ionian mode (which has a B$\flat$ instead of a B) also contains the chord tones but is less bright than the Lydian mode.  See our discussion of modes in section <ref>.  Of course, if the melody contains additional tones and thus provides further information, we should use the mode that fits these tones.

Having established the harmony and mode we discuss the choice of voicing.  The choice of voicing at the end of the day matters most as it is what is actually played by the musician.  Compare the three voicings in figure <ref>.  By definition, (a) represents the F major 7th chord.  However, one almost never sees this in a professional arrangement of a tune that contains the Fmaj$^7$ symbol.  The reason is that the 4 notes are very close together and in particular the bass note F is not low enough to stand out from the rest.  If we play the chord an octave lower we will get an even more muddy sound that in (a).  A better option is represented by (b) where the bass is moved down (or even played by different instrument).  Finally, (c) is the so-called standard “rootless” voicing of the major 7th chord. It includes the extra tone G which is the 9th note of the F Lydian scale.  This extra note is optional since it is not included in the notation.  However, it adds more color/complexity to the voicing.  If one wanted to be explicit about including the 9th, one can denote the chord by Fmaj$^9$ instead.  In practice, when we see the Fmaj$^7$ chord on a lead sheet, we use the Amin7/F as the standard voicing.

(a) Fmaj7     (b) Amin/F     (c) Amin7/F

Different voicings of Fmaj$^7$ chord.

As another example, let us take the minor 7th chord Dmin$^7$.  Once again we know that D is the bass note of the mode. We look for the brightest mode that contains a minor 3rd and a minor 7th.  This is given by the Dorian mode, which in this case is the C major scale starting in D. As before, we want to separate the bass note from the rest of the chord so a possible voicing is Fmaj/D.  A standard jazz voicing that, once more, will highlight the 9th scale tone is given by Fmaj7/D.

Our final example compares two chord symbols: G$^7$ and G$^{7(sus4)}$.  Both of these chords are associated with the G Mixolydian mode as it is the brightest mode with a minor 7th and a major 3rd.  However, the choice of voicing will differ.  For G$^7$ a basic voicing would be Bdim/G, though once more the extended voicing like Fmaj7$\sharp$11/G is a standard jazz voicing.  On the other hand, for G$^{7(sus4)}$ we want to avoid the major 3rd by convention.  We can use the voicing Fmaj$^7$/G as a standard choice as this avoids B which is the major 3rd of G. This example illustrates that while both chords are associated with the same harmony/mode (C major scale starting at G), we use different voicings to highlight different tones of the mode.

These examples illustrate some of the expressive power as well as subtleties of lead sheet notation.  The notation does not actually specify a chord but rather gives information on what chord tones the harmony should include.  Ultimately, it frees up the performer to experiment and create something of their own.  We hope that the classification of voicings we discussed in the text will give a systematic road map on how to create your own arrangements from scratch.

Lead Sheet vs. Voicing Notation: Throughout this text, we used chord notation to denote concrete chords, specified up to inversion.  As discussed above, lead sheets use this notation rather differently.  The chord symbol on a lead sheet is meant to suggest an underlying harmony or at least some voicing and not to specify an exact chord.  To aid the reader, we use slightly different notation to distinguish a lead sheet chord and a voicing.  For example, for lead sheets, we use Cmaj$^7$ while for voicing we use Cmaj7, typically specified over a bass.

§.§ Classification of Dominant Chords

Our perspective on chord notation is to view a chord as implicitly embedded in harmony, which gives us a choice of scale to improvise over the chord.  For a given chord, there are typically many choices of such embeddings and this gives us various possibilities for scales.  We now use this correspondence to classify all dominant chords.  By definition, a dominant chord contains a root note, together with a major third and a minor seventh.  The remaining tones can be altered to give variants of the basic dominant chord.  Our goal is to explore what alterations are, in principle, possible.  To understand this, we study all the possible embeddings of a basic dominant chord into the seven harmonies and look at the resulting tones.  We summarize this information in table <ref>.  Here, we list the mode which embeds the basic dominant chord and then list the alteration to the remaining tones that this embedding implies.  For example, let's take a basic F dominant chord.  We can check that this embeds as the 4th mode of C $\textbf{MEL}$.  To see the alterations, we compare this to the F Mixolydian mode.  Recall that, to find the F Mixolydian mode, we need to find the scale that has F as its 5th tone.  Thus, the scale of this mode is given by the B$\flat$ major scale.  Comparing this to C $\textbf{MEL}$, we note that the only difference is that the B$\flat$ is altered to B.  From the base point at F, this alters the 4th (or 11th) tone to be sharp.  This explains the alteration in the second row of the table.

mode     alterations

Mixolydian (5th mode of MAJ)     none

4th mode of MEL     $\sharp$11

5th mode of MEL     $\flat$13

7th mode of MEL     $\flat$9, $\sharp$9, no 5th, $\sharp$11, $\flat$13

5th mode of HMIN     $\flat$9, $\flat$13

5th mode of HMAJ     $\flat$9

3rd mode of HMAJ     $\flat$9, $\sharp$9, no 11th, $\flat$13

2nd mode of DIM     $\flat$9, $\sharp$9, $\sharp$11

1nd mode of WTONE     $\sharp$11, no 5th, $\flat$13

Possible Dominant Chords.

We see that there are eight alterations of the basic dominant chord.  We generally want to choose voicings that reflects these alterations when using these chords in practice.  Some common voicings are provided below.

§.§ Summary of Chords/Modes Correspondence

We end this brief overview of lead sheet notation by providing a table of common chord symbols and their associated modes and voicings.  If there is more than one standard option, we provide the alternatives in parenthesis.  See table <ref>.

chord symbol     mode(s)     voicings (all /C)

Cmaj$^7$ (C$\Delta^7$,  CM$^7$)    C Lydian (or C Ionian), 4th mode of G MAJ    Emin7 (Cmaj6)

Cmin$^7$ (C$-^7$,  Cm$^7$)    C Dorian, 2nd mode of B$\flat$ MAJ     E$\flat$maj7

C$^7$ (Cdom$^7$)    C Mixolydian, 5th mode of F MAJ     B♭maj7$\flat$5 (Emin7$\flat$5)

C$^{7(sus4)}$     C Mixolydian, 5th mode of F MAJ     B♭maj7

C$^{7(\sharp 11)}$     4th mode of G MEL     Cdom$\sharp$11

C$^{7(\flat 13)}$     5th mode of F MEL     B$\flat$dom$\sharp$11

C$^{7(\flat 9, \flat 13)}$     5th mode of F HMIN     B$\flat$min7$\flat$5

C$^{\text{7alt}}$ (C$^{7(\sharp 9)}$)     7th mode of D$\flat$ MEL     Emaj7$\flat$5

C$^{7(\flat 9)}$      2nd mode of B$\flat$ DIM     F$\sharp$dom7 (Gdim7)

C$^{13(\sharp 9)}$      2nd mode of B$\flat$ DIM      B$\flat$dem

Cmin$^{maj 7}$      1st mode of C MEL     E$\flat$aug

Cdim$^7$ (C$^{\circ}$)      1st mode of C DIM     Bdom7 (Cdim7)

Cmin$^{7(\flat 5)}$ (C$^{\text{\O}}$)      2nd mode of B$\flat$ HMIN (C Locrian)     Cmin7$\flat$5

Chord Notation/Mode/Voicing Correspondence.

Remark:  Chord notation, while powerful, can be used rather imprecisely at times.  A common mistake is denoting a $\sharp 11$ by a $\flat 5$.  Technically, these are the same note but what is implicit here is that $\sharp 11$ indicates that a perfect 5th is still present while $\flat 5$ should mean that there is no perfect 5th in the harmony.  Similarly, $\sharp 5$ is often used to denote $\flat 13$ even when the implicit scale still has a perfect 5th.

§ LEAD SHEET FOR AMAZING GRACE

We now illustrate the lead sheet/voicing correspondence for the tune Amazing Grace.  A starting point could be a simple arrangement of the tune, as shown in figure <ref>.  We have included the basic chord notation on top which helps us figure out the underlying harmony/modes and ultimately helps create more sophisticated voicings of the piece.  This chord notation (called a lead sheet for the piece) is a compact and effective way of representing the underlying harmony, as we discussed in the previous section.  However, at times this is not provided (it is almost never provided for classical sheet music) we want to explain briefly how one can infer this from the sheet music.

For this, we have to figure out the chord associated with each measure.  Notice that, as this is a simple arrangement, we only have triads and simple dominant chords in the progression.  Typically, the lower note gives us the bass note of the underlying mode.  Thus, the triads along with the bass notes help label the chords of the music.  One exception occurs in the last bar of the first line where we have a B$\flat$ in the bass.  The B$\flat$ with the E form the tritone of the dominant C$^7$ chord and we infer that the true bass note (which is C) is just not included.  This is simply because for a simple piece with straightforward harmonic structure the 5-1 progression (C$^7$ to F) is common.  If this were a more complex work, there could be some ambiguity in the bass note of the mode.  In any case, one can usually find the lead sheet for most pieces without doing such detective work.

Basic arrangement of Amazing Grace with chord notation.

Armed with a basic lead sheet, we can now add more complex chord structure and rewrite it with jazz chords as shown in figure <ref>.  The basic idea is to simply replace the major (resp. minor) chords with major 7th (resp. minor 7th) chords. Here, we simply replaced the major triads with the corresponding major 7th chords.  As discussed in the previous section, presenting the music this way outlines the basic harmonic structure without providing a specific arrangement.  Indeed, using the chords/harmony correspondence of the last section, we can make the harmonic structure explicit.  The advantage here is that now one can create one's own arrangement or provide accompaniment for a vocalist.  Once the harmonic structure is explicit, one can start improvising over the score as well.

Lead sheet for  Amazing Grace.

To be more explicit, we can use the notation developed in this work to assign concrete voicings to the lead sheet arrangement.  For this, we use table <ref> to create specific voicings associated with the standard chord notation.  Figure <ref> displays the results, along with an actual arrangement based on these voicings.  We have already seen an example of this in section <ref>.  We use inversions of the voicings we wrote down, possibly dropping some tones and spreading the chords across the two hands at will.

We now comment on some of the choices of voicing we made in this*/ arrangement.  In the second bar we use F$^6$/F to voice the Fmaj$^7$ chord.  This is because the standard jazz voicing (Amin$^{7}$/F) does not have an F in the voicing and our melody note is an F.  For a similar reason, we use C$^7$ as opposed to B$\flat$maj$^{7(\sharp 11)}$ to voice the C$^7$ chord in the last bar of the first line.  Finally, the penultimate bar voices the C$^7$ chord and we decided to use the diminished 7th chord G$^{\text{\O}}$ to do so.  This chord is not part of the Mixolydian mode which is associated with the C$^7$ chord.  In fact, this is a simple example of a reharmonization where we replaced the vanilla C$^7$  chord with the C$^{7(\flat 9)}$ chord that is associated with the first mode of diminished harmony.  Typically, any reharmonization that fits the melody is permitted and swapping a dominant chord for another dominant chord with the same tritone works particularly well.

Jazzy arrangement of Amazing Grace with voicing chord notation.

We hope this example illustrates how we can develop a lead sheet notation for a score, and then use the chord shapes in this text to create your own arrangement/improvisation of this piece.  While there is a lot of ambiguity and flexibility on how to do this, we hope that the approach presented in this work provides a structured way of going about this task.

§ FINALE: VOICINGS THE MODES

In a sense, knowing all the irreducible chords and their embeddings into various harmonies gives a satisfactory combinatorial picture of our approach to music theory.  However, this is perhaps too much information to use in practice, particularly when trying to create standard arrangements of pieces.  For a given mode, there is typically a standard voicing that is used for that mode, and we presented this information in section <ref>.  Here, we take this a step further by associating a standard voicing to each tone of a given mode.  This gives a concrete way to practice not just scales but chord progressions associated with a given mode.  Mastering these progressions will allow the performer to form complex voicings of melodic lines on the spot.

There is a bit of ambiguity in the choice of chord associated with a given tone of a mode.  Indeed, there are typically several choices of chord to use for a given tone, as we summarized in table <ref>.  However, there is usually a standard convention that can be used to create many of the required voicings.  For example, consider the voicing of F Lydian in figure <ref>.  The chord names on top are all to be played over F in the bass.

Voicing the tones of F Lydian mode (over F).

With each tone in the scale, we associate an irreducible chord that has the tone as its top note.  According to table <ref>, Amin7/F is the standard voicing for this mode and indeed we see this chord (with its inversions) occurs four times in the scale.  The remaining chords are the Fmaj6/F and the Emin7/F which also embed into the harmony.  These are not the only options but they represent “standard” choices for the Lydian mode.  Learning this chord progression in all keys provides a systematic way to voice the Lydian mode and add chords to any melody.  In practice, as discussed in section <ref>, one is not expected the play all the notes with the right hand but either move some to the left hand or even drop some of the chord tones entirely.  The chord shape is there as a conceptual reference point for starting to create an arrangement.

Keeping the C major scale, we can look at voicing the D Dorian mode.  Here, the most common chord is Fmaj7/D. Figure <ref> constructs a chord for every tone in this scale.  Note that while the scale is the same, we want to emphasize the minor 3rd and 7th of the mode and thus the chords we choose are rather different.

Voicing the tones of D Dorian mode (over D).

We can use a similar idea to construct chords for the G Mixolydian mode, which once again utilizes the C major scale. Here, we aim to emphasize the tritone (formed by F and B) and, thus, our standard chord options include Fmaj7$\flat$5/G or G7/G. In this case, we provide alternative voicings for some of the tones, resulting in a slightly more dissonant quality.

One of the most common chord progressions in music is the “5-1” progression, involving the dominant-to-tonic resolution. In the key of C, this progression is represented by the chords G7 to Cmaj. The Mixolydian mode is typically associated with the G7 chord, creating an expectation of more dissonance before its resolution to Cmaj. These alternative voicings can be employed to introduce additional tension, as desired. See figure <ref> for the chord progression.

Voicing the tones of G Mixolydian mode (over G).

Generally speaking, we choose the chords so that the top tone is the melody note and there is no chord tone a semitone down from the melody, as this would sound rather dissonant and would not highlight the melody strongly.

Even if we fix a mode, we can choose voicings that emphasize different tones.  For example, figure <ref> uses the chords from figure <ref> but places them over G.  This is a different voicing of the G Mixolydian mode that “suspends” the tension by not including the tritone.  This example shows that by fixing the chords/scale and changing the bass tone, we can create new voicings of modes.

Suspended (Sus) voicing the tones of G Mixolydian mode (over G).

We end this section by creating voicings for some other common modes used in jazz harmony.  As before, there is some ambiguity in these choices and we try to pick chords that best represent the mode by picking the chords that are most commonly associated with this mode.  Figure <ref>, picks chords for the Lydian Dominant mode, which is the 4th mode of melodic minor.  This mode is similar to the Mixolydian but has a $\sharp$11 in the scale.  Thus, the corresponding chords are chosen to include this $\sharp$11 when possible.

Voicing the G Lydian Dominant mode (4th mode of MEL) over G.

Figure <ref> lists chords for the 7th mode of melodic minor.  This is known as the altered scale as it alters all the tones except for the 3rd and (minor) 7th.  A quick way to remember this scale is by taking the Lydian Dominant scale for a bass that is a tritone away from the root.  For example, C$\sharp$ altered scale is the G Lydian Dominant scale. To voice this mode, we use the standard G Mixolydian voicing, except when the top note is the $\sharp$11 in which case we use the voicing from the G Lydian Dominant mode.

Voicing the tones of C$\sharp$ altered mode (7th mode of MEL) over C$\sharp$.

Next, we have the second and fifth modes of harmonic minor, which are the most frequently used of all its modes.  Once again, we try to use the chord most commonly associated with the mode from table <ref> and fill the rest with chords from the embeddings table <ref>.  We pick chords that are more dissonant for the 5th mode as it is associated with a dominant chord.  Note the use in figure <ref> of the composite chord Gdom$\flat$13 which is obtained from Gaug by adding a minor 7th.

Voicing the 2nd mode of MEL (over D).

Voicing the 5th mode of MEL (over G).

Now, we provide a voicing of the 1st mode of melodic minor.  This differs from the Ionian mode (major scale) by having a minor 3rd and is often used as way of ending a piece with a some suspence hanging in the air.  Consulting table <ref>, we have a few options for voicing.  We would like highight the presence of the aug chord that is missing from major harmony.  Therefore, we make heavy use of it when possible.  See figure <ref>.  In some cases, this amounts to using a composite chord, such as Gdom7$\sharp$5 and Gaug$^{(\text{add 5})}$.  See the discussion in section <ref>.

Voicing the 1st mode of MEL (over C).

Finally, we turn to diminished harmony and discuss options for voicing the tones.  Recall, that this harmony is composed of two diminished 7th chords that are a semitone apart.  For instance, for C $\textbf{DIM}$ these are the “upper” chord Cdim7 and the “lower” chord Bdim7.  This harmony is typically associated with the either a dominant chord or a diminished chord.  For C $\textbf{DIM}$, these could be an alteration of Fdom7, or just Cdim7.  In either case, we would want to highlight the upper chord Cdim7 since this contains the key tritone intervals characteristic of both Fdom7 and Cdim7.  We can use the Cdim7 chord to voice the upper tones of the C $\textbf{DIM}$ scale.  To voice the lower tones (contained in Bdim7) we simply move the top note of (an inversion of) Cdim7 down a semitone.  For example, for (C, E$\flat$, G$\flat$, A) we would obtain (C, E$\flat$, G$\flat$, A$\flat$), which is an inversion of A$\flat$dom7.  Combining these, gives us a basic voicing of all the tones in C $\textbf{DIM}$.  See figure <ref>.

Basic voicing of  $\textbf{DIM}$ (over any bass).

Practicing this chord progression for all 3 diminished scales will help learning how to use diminished chords in practice.  We also create more complex voicings of $\textbf{DIM}$ by using table <ref>.  For the upper tones, using Cdim7 may be a bit boring so we are free to move one of the tones down a semitone without leaving the scale.  Note that we want to fix the top tone of the chord as this is the tone we are trying to voice.  For example, for (C,E$\flat$, G$\flat$, A) we can alter either C, E$\flat$ or G$\flat$ obtaining an inversion of Bdom7, Ddom7 or Fdom7 respectively.  We can also use a more exotic voicing for the “lower” chord tones by using dem chord shapes.  For instance, Adem consist of (A, D, E$\flat$, A$\flat$) which gives us a way to voice the A$\flat$ tone.  Figure <ref> shows an example chord progression using these more complex voicings.

Complex voicing of  $\textbf{DIM}$ (over any bass).