Plogger Method — Complete Reference

Acknowledgements

This is true for all my hundreds of students over the past decades; I would not have been inspired with the ideas that became this volume without them. I have been inspired with the ideas in the method by a colleague and good friend, and many colleagues and collaborators. Special acknowledgment must go to Nadia Boulanger, with whom I studied in Paris and who provided me with a standard of solfège to imitate, and to all the former students, colleagues, and collaborators who have graciously worked in the testing of these ideas.

Special acknowledgments must also go to Professor Keith Still, who has been an inspirational and educated resource whose encouragement and help resulted in getting the book into print.

Foreword

I know of only one person who can legitimately proclaim themselves the most global, prominent collector and builder of major music schools around the world, with hundreds of thousands of major music students around the world in every conceivable country, in the signing of the newest Baltimore Symphony music director, and in the latest recordings. Who is it exactly, with all of these credentials? Somewhere, finding a student, encouraging young musicians, or listening to an orchestra all are the moments that define that person, the musician's musician. Who else could it be — William Foster Walker III.

It is our pleasure to introduce this book as a companion to the instruction he gives his students.

— Charles Anderson Young
Associate Dean, Bélmont-Walker Conservatory

Introduction

Becoming fluent in music is similar to becoming fluent in any language: both require a firmly rooted ear along with a systematic approach to master grammar and vocabulary. The fluent musician can immediately identify patterns in music, the way a fluent speaker will quickly spot idioms in language. This aptitude, however, is not often obtained in music because, as a topic, it is incredibly complex, covering the identification and application of all the rules and terms governing music, which must be attained fluently if they are to be comprehensible to the listener.

Although this process may seem overwhelming at first, in reality learning musical material occurs at its most efficient through pattern recognition, where patterns and structures enable you to acquire and use musical terminology for the same reasons: to identify patterns and utilize them. We learn language this way by identifying words closest to our experience of usage. If we understand a word's meaning we can also understand the structure within which it appears. We learn musical grammar through exposure to music itself.

I like to say that in music, as perhaps in most things in this world, you are most effective when you actually look at what you are working with; that is, you will achieve more or less commensurate success depending on how much knowledge you have acquired and retained. From my experience as a musician, teacher and arranger over many years, I introduce with great confidence the following method for learning the musical vocabulary of the piano. The method achieves a critical speed, namely, that things tend to be learned correctly from the beginning and thus retained better over time.

The method is based on this idea: above all it accounts for the capacity of the student's musical brain. For example, I have advised 99% of modern musical learners (by whom I mean those who are not among the small subset of truly extraordinary musicians) to specifically use syllables alongside notes. Without doing so, the non-standard student loses track of what they are learning and becomes confused. In short, you can achieve critical speed, and more importantly, you can retain things consistently well, only if you rely on systematic methods that your own cognitive apparatus can handle.

The approach is not difficult, but it does require the student to work. Every musical exercise in this collection can be broken down into a series of smaller chunks that, once mastered individually, can be compiled. The reward for following this process is substantial in the long run: once learned, the musical vocabulary can be used with more facility than any other method.

System Architecture

The Adaptable Musician's Framework (AMF) is the overall system name. Its layers:

Three Books

• Textbook — Core concepts, explanations, and chapter content
• Practice Manual — Structured daily exercises and protocols
• Workbook — Written exercises and ear training drills

Four Practice Sections (Daily Session Structure)

Core Pedagogical Principles

• Hear/Sing Before You Play — Audiation precedes physical execution at every stage
• Few exercises, multi-layered — Deep fluency in a small set beats shallow exposure to many
• Plogger as dual presence — The method functions as both content and the lens through which all content is perceived
• Three Stages labeling — Every practice activity is explicitly tagged as Stage 1, 2, or 3
• Anchor song thread — A single song is learned across all chambers simultaneously
• Genre-agnostic — The system applies equally to classical, jazz, folk, pop, and all other genres

Ch.1 — Three Stages of Learning

“All real learning happens in the imagination: if it isn’t in the imagination, it’s not learned yet. You can say something, but it doesn’t exist for you yet until it’s in the imagination.” — Nadia Boulanger

Stage 1 — Roughing In

Once the hand behind the harp sometimes likes to get in the way, you might roughly trouble-free half the time. But sometimes lines like this happen easily and are produced without thinking. This is the roughing in phase, where the main goal is to handle it. You are becoming comfortable with the material. At this stage, you feel it more intuitively.

Stage 2 — Achieving Fluency

In the roughing in phase, you will initially spend long segments of your behavior oriented at the keyboard and attempting to play something very slowly, with many corrections. You will frequently stop to correct things. Sometimes you correct the same mistake two or three times. At the end of the roughing in phase, you have the correct materials in place, but performing them is far from second nature.

At the fluency stage, you want the material to be able to come out well most of the time. You have the material working — it rarely creates problems, and you can play it reliably at a moderate tempo. To be fluent, your mental and physical experience during this stage are not like the experience during roughing in. Now, you’re going in and out of focus during practice because the material is not yet fully memorized. The roughing in phase is like scaffolding; now you are beginning to work on the underlying structure.

Stage 3 — Achieving Mastery

Once you have the fluent stage, you have a pattern of success at the goal. Now, you’re seeking to get things to the top of your skill level. We are trying to get material to mastery, which requires consistently reliable performance at tempo — both up to full speed, and reliably able to slow down. You know you are at mastery when performance feels steady — like you can always land the lick or passage correctly. At this point, the material is so deeply organized in your mind and body that you could perform it with or without thinking about it.

What Things Go Wrong with Roughing In

Trying to avoid confusion in the roughing in phase, the student often confuses confidence with mastery. There are many specific examples where this confusion leads to bad habits: for example, an athlete who might just drop back to the ground-level before a race, having no specific speed-training background is practicing getting over the hurdles. They work on hurdle after hurdle, feeling successful on each individual try. But if the student is really going to succeed at the goal they need to spend more time at the hurdle phase. Any time spent trying to perform fully before the body knows how is counterproductive.

Mindlessly Bypassing the Perfecting Stage

We need to see this phase as what it amounts to: practicing being wrong. Yet we cannot afford to be wrong so often. The moment we become confused or impatient, we’ve begun spending most of our time practicing confusion. The only way to spend zero time practicing confusion is to avoid it completely. Let me stress this: as a student practicing any skill, you may delight in the feeling of “figuring it out,” but there can be no substitute for training — the student wants to practice the entire performance from beginning to end completely in a controlled, reliable way.

Avoiding the Perfecting Stage Mindlessly

So it is important to establish what the perfecting stage should look like. This is for musicians of all ages. It is a fine thing to have some special moments of inspiration when you play it and it comes out well; but those moments should not be the template for your practice. You can be someone who specializes in delightful surprises — but that experience is antithetical to the training you need, which is orderly accumulation. You may not be able to completely remove this from your practice, but make sure all information you use is clearly articulated in sound, not just done by guesswork.

What We Need

It occurred to me after studying the problem for many years that there are essentially three components of a skill chain, which I discovered by thinking about a cross-section of very successful performing musicians. For those studying an instrument, the first component involves building the ability to know clearly what you’re going to do before you do it.

Clearly, what is needed for the roughing in or preparation phase is some reliable form of mental support to fill the position of an automatic performing state — and, to be comprehensive, the following principles are extremely important:

1. Return to the best of your ability; look at the moment you’ll be in the appropriate state, and work on the problem.
2. Remove yourself from thinking, programming, and judging during an intense time-on-task. Focus your mind within the limits of the task.
3. Focus on the time of your ability; know your skill and work on the skill. You do the skill things; ensure the information is ready to be repeated. Stay all-in on your calling.
4. Allow the skills, and allow self-observation when noise enters and other stimuli enter to be repeated. Get the information; prepare for the next note — do this ANYTIME you are calling.

Mindfully Replacing the Performing State by Performing Plans

We need to see this phase as what it amounts to: practicing being wrong. Yet we cannot afford to be wrong so often. The moment we become confused or impatient, we’ve begun spending most of our time practicing confusion. The only way to spend zero time practicing confusion is to avoid it completely.

Breaking Down at the Top of the Performing Phase

Any time spent trying to perform fully before the body knows how is counterproductive. Make sure all information you use is clearly articulated in sound, not just done by guesswork.

Ch.2 — Three Causes of Error

Every performance error has a specific Symptom, Cause, and Cure. The framework applies equally in practice and in performance. Diagnosing correctly is prerequisite to curing.

Error 1 — Reaction

Error 2 — Anticipation

Error 3 — Looking Back

Ch.3 — Keyboard Visualization

The Importance of Visualizing the Keyboard

The piano/organ keyboard (which I refer to as the thinking mind of the musician) focuses our attention as musicians on the location of every harmonic and intervallic event as they are played. When in doubt, the keyboard, understood as a tool of spatial organization — organized into octave-repeating patterns — will consistently help you understand intervallic and harmonic relationships. By using the keyboard as the musician’s visual tool, you can analyze harmony, rhythm, and color in ways that allow you to identify the relationship between harmony and the frequencies of intervals involved.

Your capacity to analyze an interval is dependent upon your ability to recognize and understand the note names of the keyboard, and to understand the keyboard as a map of pitch space. It’s important to learn to see the piano as you would see a word in a sentence — a musical sentence is no different from an English one in this particular respect, if we see the word inside a sentence, we can almost immediately recognize its meaning. Similarly, if we look at a keyboard, we know immediately what is meant by the word “keyboard” when it appears in a musical sentence, even if we’ve never played it.

The capacity to analyze music — whether as a performer, composer, or in any other musical role — depends heavily upon a musician’s spatial-cognitive ability. Using fixed-pitch resources — an instrument played through a fixed intervallic structure — enables you to develop this spatial-cognitive ability. The keyboard is unique in its ability to allow you to practice these kinds of spatial-cognitive operations because it is a fixed-pitch instrument, and thus the keyboard can be seen as an analogy of the musical space of pitches. It’s still a fully diatonic map with all the possible pitch relationships accessible to the musician — and thus it has certain properties of musical relationship that the musician can use to identify, analyze, and communicate certain musical events.

The Keyboard as the Musician’s Cognitive Tool

The musical keyboard is a useful and very accessible musical tool. It can be used more broadly to enhance listening. There is a fundamental pitch space you can listen to; that’s why it is so useful.

The Musical Keyboard as a Map of Pitch Space

Years ago, I concluded in my study that music requires the ability to mentally perceive spatial connections between pitches on the keyboard. Similarly, music can be understood, in a sense, as a social tool for representing that information.

• The white keys form a certain kind of interval pattern — this pattern is one that repeats.
• The same 12-note pattern of black and white is repeated every octave.
• An octave is a certain distance between a frequency that is repeated every octave.
• The key for black keys that is to the right of the white key is called a higher-than-white note, while the key for black keys to the left is a lower-than-white note.
• The distance between a note and the note at the same pitch up or down one octave — this is an octave.
• This musical relationship enables one to learn where notes are.

Imagining the Keyboard

First, you should be able to create a clear, pleasant imagination of the piano keyboard, one that is detailed, permanent, vivid, and easily accessible (without the actual piano present). This is the best possible tool for auditory training. You will need to go through several steps before this has become second nature for you: for example, you will need to repeatedly tell yourself to “imagine the keyboard” in a variety of ways. Before practice, allow yourself to imagine the keyboard as clearly as possible; as you practice, you should actually use the keyboard as a visual thinking tool, and be capable of accessing a real image of the keyboard in your mind without looking at any visual aids whenever you practice; this will make you a more effective learner.

Labeling the White Keys

All of the musical notes in the musical system (at least in the Western equal-tempered system) are named with letters of the alphabet: A, B, C, D, E, F, G — and then after G, the pattern starts over at A again. In the Western musical system, there are only seven base letter names for notes because the entire octave space uses only seven distinct pitch classes in the diatonic system; these are the seven keys that the keyboard represents in each octave.

Placing the name “sharp” (which is the note one semitone higher) to the right of a given key indicates that this piano key is a note that is slightly higher in pitch. Placing the name “flat” (which is the note one semitone lower) to the left of the key indicates the note is slightly lower. The key placed further to the right is a higher-pitched note, while the key to the left on the keyboard is a lower note.

Labeling the Black Keys

There are two basic note names to begin: the name “a sharp,” or “b flat,” for the black keys, from the lowest to the highest. As you move to the right on the keyboard, the notes become sharper (higher); as you move to the left, the notes become flatter (lower).

The flat notes also occur on the keyboard in which neighboring pitches provide context: in any key, the next key over or the neighboring keys will be at the corresponding natural notes. For example: on the keyboard, the D key has as its neighbors both the D# key (which can also be called E♭) and the D♭ key (which can also be called C#). This means that the key to the right of D is a D# while the key to the left of D is a D♭.

Exercises: Developing Keyboard Visualization

Exercises: Developing Note-Naming Fluency at the Keyboard

Exercises: Developing Mode-Naming Fluency at the Keyboard

Ch.4 — Longy Rhythms

Many of us sing rhythms of music or call by the number without necessary knowledge of the precise volume or note values in a dictionary. The patterns, however, are not an indication of any intrinsic quality; this is because the method we use tells us whether the interval is deliberately measured. Can anyone perform tasks with the same type of systematic measure, just as we can act in a non-reliant way? In an ordinary sense, these two qualities are fundamentally different: the first is the reliance on number itself; the second is the reliance on method.

Our ability to measure and count long rhythms is crucial to music learning. The ability to fluently choose rhythms when we are in fluency or freely playing is crucial because actually we are seeking to keep ourselves within the flow of music. I describe this experience as a way of learning and knowing music through rhythmic counting, and I believe the absence of consistent, reliable rhythm patterns is the defining feature of the struggling musician, no matter what instrument or voice. A reliable musician can count things in a reliable and systematic way.

The Lange Rhythm Chart

Based on the Theories of Émile Jaques-Dalcroze, accompanied the page.
Émile Lange, 1881–1929

Two Equal Parts

Three Equal Parts

Four Equal Parts

Many other rhythm groupings represent alternative groupings rather than entirely analogous patterns. Note that numbers represent different lengths of time; they may be divided into groups of two, three, and four equal parts.

In each instance, the procedure is the same: any beat can be mentally or audibly divided, and it can be divided by any number. Treat each beat as if it's divisible by a whole beat or a portion. One beat in a music bar can itself be a multiple of beats: “1 and,” for example, meaning 1 and a half, followed by 2 and a half. This is because of the Émile Lange Rhythm Chart pattern.

Tap the Main Beat

Starting in the top left column, from top to bottom, you start dividing a beat into two equal parts. The beat is always a whole number (like 1). Think of it as “1 a,” then “1 2,” etc.

Speak Each Number on the Beat and at the Same Volume

Teach your mind to speak each beat on the same pitch level at the same dynamic level, practicing clearly speaking each note precisely, allowing you to eliminate in-pitch centers, allowing you to clearly and concisely see what you're doing. As you practice speaking each beat step-by-step you will become more fluent in using the approach from the beginning, as you will start learning what comes next. As you practice, you will begin to look forward to a point where you will be at perfect mastery of all these actions as well: speaking quickly, speaking well, speaking in a timely fashion — and reading them from the page while making well-managed and well-timed beats.

An Accurate and Accelerate-Beat

Each beat has its own specific note value; just as a “long beat” note has a certain number of beats, and thus an associated note-unit, similarly a rhythmic subdivision is assigned to a particular beat, so to start a regular rhythmic subdivision, you need to start at a specific note value on each level of the beat you're on. The fundamental note value provides the highest range of beats at the first level, and the following beat-value from the first level provides the second beat level.

Starting at the top row, 1, 2, 3, and 4 — the entire middle column is a block of 56 notes. The notes perform the division permutations. There are two rhythmic units: a note (or a short beat) and a longer note (or a long beat). About the block of notes, these two notes each form a unit, and each unit has either an equal number of notes (which in music theory is called “even beat”) or an unequal number of notes (which is called “odd beat”). The whole block of even beats produces what we call an “even beat,” and the whole block of odd beats produces what we call an “odd beat.”

Maintain the Main Beat, Without Breaks

The goal is to maintain the pulse throughout each measure in the block of 56 notes — the notes perform the division permutations. There are two rhythmic units: a note (or a short beat) and a longer beat. About a group of notes that is “two-beat,” the note means its beat (or “time”) has exactly 2 notes. The note on the first beat is a short note (usually a quarter note), and the note on the second beat is a short note (usually a quarter note), and then the note on the third beat is a short note (usually a quarter note). So, instead of counting “1-2” twice, we count “1-2-3” three times. By combining these short/long rhythms into beats per line in a music bar, students learn the way rhythm must be counted, and they should always count the unit beats of the rhythm when they are being counted from the beginning.

Exercises: Performing the Lange Rhythm Chart

Lange Rhythm Permutations Quiz

When you divide a beat into two parts, there are these different kinds of divisions:

1. What is the division of a beat into two parts?
2. What is the division of a beat into three parts?
3. What is the division of a beat into four parts?
4. What is the division of a beat into five parts?
5. What is the relationship between the division of a beat and the number of combinations you notice? Express this mathematically.

Seven Pattern Lap Map Three Groupings

There are two other ways to divide a group of 3:4:3, 5:3, 4, 4, 5 and then: 5:3, 4, 4, 5 and 5:3.

Ch.5 — Lap Map

Discovering the Lap Map

Almost sixty years ago I had a chance connection with a Russian teacher, back in those times, I had begun teaching at the music academy in California, having a bond relationship with colleagues and teachers in California about finding a good, consistent approach to rhythm practice. The challenge for students was keeping consistent rhythm, particularly with complex syncopations.

I discovered that a lot of my students would always have their leg going tapping along. From this I took cue — I used Lap Maps to show the location of each beat of the music hierarchy, allowing the students to keep their legs in rhythm at the same time.

Subsequently, I created a series of Lap Maps to show the location of each beat of the music hierarchy. I found it always useful to use a table that follows the numbers in order, always following the number in order.

The Diagram of the Lap Map

A sequence of key notes corresponding to the hierarchy of the Lap Map is as follows:

The right tap (marked on left leg) — this is always the lowest-beat spot. The two lowest “laps” on the left leg and the right leg are the center spots. Each group of numbers is an alternating cluster with two taps per leg.

Notice that the numbers on the left leg are always on the top position on the left leg; the one (1) is on the bottom, always placed lower on the left leg. Also the two (2) is on the top right, while the bottom number (4) is lower on the right leg.

The Basics

With lap map exercises, you aim to correctly handle complex patterns. Though it can be difficult working through all the complexity of those exercises, it may be only partly combinations of numbers, tapping on two laps; after you fully master them, allow tapping to be as smooth and natural as you can. At the end of any two-number sequence, check in: you are making gradual progress and thus practicing in a very productive, steady way. The goal of your exercises is that you should practice until you are totally ready. Failure during this period is understandable; you should attempt this work seriously even if you fail, rather than stopping after several failed attempts.

Exercises: Performing Lap Map Patterns

Lange Rhythm Permutations Quiz

1. When the beat is divided into two parts, how many different permutations are there?
2. When the beat is divided into three parts, how many different permutations are there?
3. When the beat is divided into four parts, how many different permutations are there?
4. When the beat is divided into two parts, how many different permutations are there?
5. What is the relationship between the division of a beat and the number of different permutations that are possible? Express this mathematically.
6. To check that your formula works correctly, calculate the number of beat divisions for the four-beat base two — then run the results through the Lange Rhythm Chart to confirm. How many different 4-beat group permutations are there?
7. Calculate the result of your calculations with the number of combinations you have in the Lange Rhythm Chart. Are your results correct?

Looking Back

(See Chapter 5 conclusion — the Lap Map approach consolidates rhythmic fluency by externalizing the beat hierarchy onto the body, allowing the musician to maintain an internal sense of pulse while simultaneously reading or performing.)

Ch.6 — Pythagorean Ordering of Fifths

The Pythagorean ordering of fifths is fundamental to the understanding of music theory. Committing this pattern to memory will eventually become one of those skills that you can apply to music without conscious thought, and it will help if you think of it as a mnemonic. I believe this is one of the most important aspects of this book, and one whose systematic understanding enables you to develop your musical intuition in a broad way, to improve your understanding of music theory.

This section of the textbook is dedicated to strategies to help you recall and memorize musical material that has been presented previously. Rather than summarizing what’s been covered, however, the goal in this section is to help you to reinforce your memory by the exercises provided in this chapter. You must practice this material consistently until you are able to memorize it fluently, rather than requiring any visual cues. To practice this material in any consistent context, use the following exercises.

For a more thorough exploration of the Pythagorean ordering, see the article “The Perfect 5th Rule in Music” (Plogger 2016).

Exercises: Programming the Pythagorean Ordering

Ch.7 — Internal Spelling at the Keyboard

Ordinal Interval Names

The five-chord interval (which I refer to in the context of describing the distance between notes) uses common names for describing the distance between notes. The names used for interval names of a 5th across between notes are very common in describing the distance between a note and its neighbor (which forms the interval). The intervals are named after the number of notes included in the spanning interval. For example, the distance between a C and a G — including C, D, E, F, G — is a 5th, because there are five notes in the span. Similarly, a diatonic scale side (like C-D-E-F-G-A-B-C) spells out 8 notes from C to the next C — and that interval, known as an octave (“octave” from the Latin for 8), spans 8 notes in the diatonic scale.

As far as up or fa, you can also get that down as fa to. By the above example, then, you can play the same from fa down to do in a 5th.

Notice that the two notes in the above diagram — fa and do (above and below) are the same musical letter; they are the same note, one octave apart, and in the above example, “fa” stands in the top (highest pitch) while “do” (in the above example) stands in the lowest (lowest pitch). To move from a note above and drop two tones to a note below is to make a 3rd-down movement.

Exercises: Locating and Naming Diatonic Intervals

Ch.8 — A Special Spelling of Staff

Vocal Ranges and the Five-Line Musical Staff

The five-line musical staff represents the range of the human voice. The staff begins in the center and spreads upward and downward from the center line (the top) and the bottom line, covering the “vocal range” — that is, the full range of the human voice.

Any 5th built above a B♭ above middle C, or more specifically the distance of a 5th interval, will be comprehended by most, not anyone using one voice; this is because the method we are applying here is specifically created for you to practice it best on these note intervals. In other words, a melodic chord is built upon two notes of a similar distance, and by the following examples, the interval is used to identify some notes of an instrument.

The Three Types of Clef

The clef sign is a symbol indicating the location of a specific pitch on the musical staff. It is placed at the beginning of each musical line to indicate which notes are positioned on which lines and spaces of the staff.

• The C-Clef designates the location of the note C (middle C) as the note on a middle-line or specific line of the staff on which middle C is located.
• The G-Clef designates the location of the note G on the staff — it is placed on the second line up from the bottom.
• The F-Clef designates the location of the note F on the staff — specifically between the two dots of the F-clef symbol, which is placed on the fourth line up from the bottom.

The C-Clef

The C-clef identifies the location of middle C on the staff, and may be placed on any line of the five-line musical staff. The typical positions of the C-clef are:

• Soprano Clef: Middle C is on the first (bottom) line.
• Mezzo-Soprano Clef: Middle C is on the second line from the bottom.
• Alto Clef: Middle C is on the third (middle) line.
• Tenor Clef: Middle C is on the fourth line from the bottom.
• Baritone Clef: Middle C is placed on the fifth (top) line.

The G-Clef

Observe that in this clef, G is placed higher on the staff (in the vocal range) because the notes of this clef have a higher average range than the other clefs. The G-Clef is placed on two of the line clef positions:

In the soprano G-Clef, the G-line is between the third line and second line from the bottom (Middle C on the top line).

The vocal range includes: Soprano Clef, Mezzo-Soprano Clef, Alto Clef.

Note however, in this clef, the vocal range also includes the Tenor Clef; but it reads one octave higher than it appears.

Once the pair passed centuries, the G-clef has had only one G-clef in use or selected (at least by those who play piano), through one octave higher, though not for the tenor voice in any clef; the tenor clef, however, is the most commonly employed of the vocal clefs.

Treble Clef: The treble clef is placed on the second line from the bottom.

The F-Clef

The F-clef labels the location of F on the staff, and the placement (the two dots) identifies the location of the note F on the staff. It is found on two of the line clef positions:

• In Baritone C-Clef, middle C is located on the fourth line up from the bottom.
• In Bass F-Clef, middle C is placed three lines up from the bottom.

Middle C on the Clefs

• In Treble C-clef: middle C is on the third line from the bottom (or below the staff on the ledger lines).
• In Tenor C-clef: middle C is on the second line from the bottom.
• In Bass C-clef: middle C is on the first line from below the staff.

Knowing How Musical Intervals Span the Staff

Knowing this information is valuable for obtaining the true note value of the music on the keyboard. For example, the first note and the second note of any staff melody forms a certain interval across the staff. This interval spans from one note name on one side of the staff to the next note name (which in music is called a “melodic interval” on the page). When you know these intervals you can learn how many notes span a given interval on the staff.

The staff maps out the intervals on the note in a specific, particular pattern — for each note on any of these positions, it indicates what the note value is (staff position) in much the same way. The note values indicate whether the pitch is going up or down by indicating its specific position above or below the ledger lines.

A pair of these musical staves notes as they relate to the note staff shows that the notes on a staff move from the bottom to the top of a page. Note the note staff moves upward; when the note moves from the bottom to the top of the page, the notes move upward. When the note moves from the right side of the note staff to the left side, the notes tend to move downward. So the staff is not a linear thing — its design is more like a road, in that as you travel left to right on the page, the notes rise, and as you travel right to left the notes drop.

A useful thing to remember is that the notes in the same staff itself are in the same staff position, which means they are the same note if they are a given note with the same note. So if two notes have the same position on the two staff, they are both exactly the same note. If they are in the same position and the same note, it means they are in the same position in the staff, but are of very different pitches.

Exercises: Using Clefs

Ch.9 — Di-Chord Numbers

Numbering Di-Chords

Saying that an interval is a 3rd gives us some information about the interval, but it does not tell us what the precise distance is between two notes; this is like learning that a creature is a mammal but not knowing whether it is a tiger or a house cat. To have the next level of knowledge about an interval, we would need its di-chord number or a word like “major,” “minor,” “augmented,” or “diminished;” from this we could know exactly how many semitones are between the bottom and the top notes.

The term di-chord numberrefers to the number of semitones between any two pitch classes. To determine the di-chord number of two notes, count the number of semitones between the notes. Because you are counting the distance as you move by semitone, you must always start counting at zero. Think of it this way — to calculate how far you drive, your car’s odometer has to start on zero, not on any other number.

Note that this is different from how we calculated diatonic interval names; the diatonic interval vocabulary (“2nd,” “3rd,” etc.) uses the language of ordinal numbers. Ordinal means that the numbers are indicating rank or position — remember that intervals are determined according to the diatonic relationship between two notes rather than the number of semitones between them.

When naming intervals in the last two chapters, we used ordinal numbers (2nd, 3rd, 4th, etc.). The vocabulary of di-chords uses cardinal numbers (1, 2, 3, etc.). When writing di-chord numbers, use brackets around them to distinguish them from other types of numbers — , , , etc.

For example, fa to la is a di-chord because there are four semitones between fa and la, but we can also see that it is a 3rd because it skips over one note (sol).

Mi to sol is a di-chord because there are only three semitones between mi and sol. But, what is the diatonic interval? Mi–fa–sol skips over only one note, so mi to sol is a 3rd.

Notice that fa to la, a di-chord , and mi to sol, a di-chord , are both 3rds. This difference can be indicated by using the terms major 3rd to describe the di-chord from fa to la and minor 3rd to describe the di-chord from mi to sol. Here, major means larger; the major 3rd is one semitone larger than the minor or “smaller” 3rd.

Each di-chord number is associated with a common traditional interval name, like minor 3rd or major 3rd, etc. Each di-chord number also has an uncommon interval name, like augmented 2nd or diminished 4th, etc. Common names are those found in the diatonic scale, and uncommon names are those found in chromatically altered scales.

Notice that notes on the staff only give us ordinal information (3rd, 4th, etc.) but do not tell us directly what kind of interval it is (major or minor 3rd). For instance, both of the following examples are 3rds; however, if you check at the keyboard, you will see that the first is a di-chord , and the second is a di-chord .

Furthermore, the di-chord formed between the same two lines on the staff can be different with different clefs. Two notes that appear a 3rd apart in the same location on the staff can be different di-chords, depending on which clef is being used.

Using keyboard visualization and the counting method described earlier in the chapter, you can see that la up to doh and doh up to mi are both 3rds. However, if you count the number of semitones between la and doh, you will find that there are three; if you count the number of semitones between doh and mi, you will find that there are four. This is important because pitch pairs that are three semitones apart feel and mean something completely different than pitch pairs with fewer or more semitones between them. If you keep track of where you are in pitch space, using the keyboard, you will always know whether the 3rd you are about to play is three or four semitones.

Knowing the di-chord you are about to play tells you a lot about the meaning of a given moment in music, giving you access to new levels of expressivity in playing.

Using this knowledge in real time to heighten your consciousness of music and to communicate musical meaning more effectively (and affectively) requires fluency in two skill sets. You have to become a great tracker with the keyboard, so you know which di-chords you are playing. If you always know what di-chord you are playing, singing, hearing, reading, or writing, you know its characteristics and can use that information to help you. Therefore, your tracking to help you connect with music, you must develop an immediate knowledge of the properties of the 11 di-chords. This chapter of the workbook will help you become comfortable using di-chord numbers and “program in” relevant information about the di-chords so that when you are tracking, you are also gaining insight into the affect and meaning of the music you are hearing or performing.

Exercises: Di-Chord Number Calculation

Di-Chord Numbers and Ordinal Interval Names

The table below shows the correspondences between di-chord numbers and the common and uncommon interval names. Remember that although there are multiple interval names for each di-chord number, the most common names are the intervals that occur in the diatonic scale.

Notice that by common interval names:

• and are 2nds
• and are 3rds
• and are 6ths
• and are 7ths
• is a perfect 4th
• is a perfect 5th

The di-chord , which splits the octave exactly in half, can be spelled either as an augmented 4th (because it is one semitone larger than the perfect 4th, di-chord ) or a diminished 5th (because it is one semitone smaller than the perfect 5th, di-chord ). For example, sol up to doh# is an augmented 4th (any sol up to any doh will always be some type of 4th). However, sol up to re♭ is a diminished 5th (any sol up to any re will always be some type of 5th).

Exercises: Developing Fluency with Di-Chord Numbers

Ch.10 — The Sonic Properties of Di-Chords

Sonic Dimensionality

One of the toughest problems we face in learning to distinguish between the 11 basic di-chords is their sonic dimensionality — they are not just rough or smooth, black or white, but are multidimensional and often seemingly paradoxical. It is this sonic dimensionality that gives each of the 11 di-chords their individual character, and it is the individual character of each di-chord that we learn to hear, identify, and use to impart meaning to our music, allowing us to better understand and express its communicative nature.

Like with any of our friends, each di-chord has specific features that allow us to identify it. Also, just like with people, context has a huge effect on how the di-chords act and how we perceive them — sometimes even making them quite difficult to recognize. The good news is that, while we may have countless friends and acquaintances, there are only 11 di-chords to become acquainted with!

To distinguish one di-chord from another, our ears must systematically attend to three distinct sound factors that I have named:

1. Interference Pulsation
2. Fundamental/Octave Factor
3. Harmonicity

To illustrate the combined effect of these three factors on the 11 di-chords I have created the Di-Chord Pictograph.

The Three Sound Factors

The Di-Chord Pictograph

The Di-Chord Pictograph consists of 11 numbers, each of which represents one of the 11 di-chords. Each number has a shape, a shadow, and a color, which represent, respectively, the interference pulsation, the fundamental/octave factor, and harmonicity. Using the Di-Chord Pictograph as a tool helps the conscious mind comprehend how each di-chord can simultaneously possess the three sound factors.

Ignoring one factor in a di-chord allows the ear to confuse it with another. For example, if you ignore the “color,” you might confuse di-chords and ; if you ignore the “shadow,” you might confuse di-chords and or and .

• Shape = Interference Pulsation
• Shadow = Fundamental/Octave Factor
• Color = Harmonicity

Ch.11 — Interference Pulsation

When we hear any of the 11 di-chords, we can perceive a subtle and even rhythmic pulse that is occurring eight times per second, four times per second, or two times per second (8Hz, 4Hz, or 2Hz). This property of di-chords is called interference pulsation. Observing the interference pulsation of a di-chord allows us to discern the basic category of the di-chord as dissonant, modal, or perfect. The interference pulsation pictograph represents the three types of interference pulsation with jagged, wavy, or straight lines.

Dissonant

Dissonant di-chords have an interference pulsation of eight pulses per second (8Hz). The dissonant di-chords are [1], [2], [10], and [11] — di-chords that are either a [1] or [2] from the fundamental or any octave above. Notice that the dissonant di-chords are 2nds or 7ths, in traditional terminology. In the Di-Chord Pictograph, all dissonant di-chords are depicted with a jagged-edged line, representing a fast, 8Hz interference pulsation.

Modal

Modal di-chords have an interference pulsation of four pulses per second (4Hz). The modal di-chords are [3], [4], [8], and [9] — di-chords that are either a [3] or [4] from the fundamental or any octave above. Notice that the modal di-chords are 3rds or 6ths, in traditional terminology. In the Di-Chord Pictograph, all modal di-chords are depicted with a smooth, rounded line, representing a smooth, flowing, or “wobbly” 4Hz interference pulsation.

Perfect

Perfect di-chords have an interference pulsation of two pulses per second (2Hz). The perfect di-chords are [5], [6], and [7] — di-chords that are either [5] or [6] from the fundamental or any octave above. Notice that, in traditional terminology, the perfect di-chords are either 4ths or 5ths. In the Di-Chord Pictograph, all perfect di-chords are depicted with straight lines, representing a flat, slow, stolid 2Hz interference pulsation.

Dissonant, Modal, or Perfect?

Recognizing that a di-chord is dissonant will not help you know which one it is unless you know which of the 11 di-chords are dissonant. The next three chapters will help you memorize information about di-chord properties. Remember, dissonant = 8Hz, modal = 4Hz, and perfect = 2Hz.

Exercises: Getting to Grips with Interference Pulsation

Chapter Twelve: Properties of Di-Chords: Fundamental/Octave Factor

Referring Up or Down

In any di-chord, you can hear that its upper note either refers down to its lower note (the fundamental) or up to the octave transposition of its lower note (the octave). This phenomenon is called the fundamental/octave factor.

The fundamental/octave factor allows you to distinguish between di-chords that refer downward to the fundamental and those that refer up toward the note one or more octaves above the fundamental. This means that after you categorize a di-chord as dissonant, modal, or perfect, you can then use the fundamental/octave factor to determine whether a dissonant di-chord is a 2nd or a 7th, whether a modal di-chord is a 3rd or a 6th, and whether a perfect di-chord is a perfect 4th, a perfect 5th, or a di-chord [6].

You hear the fundamental/octave factor by listening for whether the upper note of a di-chord refers down to its lowest note or up to the octave manifestation of the lowest note.

The following table shows each of the 11 di-chords and whether they refer up or down.

Di-Chord Pictograph Numbers

As you can see, di-chords [1] through [5] refer down; this is represented on the Di-Chord Pictograph by a “shadow” to the left of each number. Di-chords [7] through [11] refer up, and this is represented on the Di-Chord Pictograph by a shadow to the right of each number. Di-chord [6] splits the octave directly down the middle and therefore is both referring up and referring down.

You might notice that, aside from di-chord [6], 2nds, 3rds, and 4ths refer down, while 5ths, 6ths, and 7ths refer up.

Exercises: Getting to Grips with Fundamental/Octave Factor

Ch.13–14 — Harmonicity & Review

Harmonicity — Harmonic or Non-Harmonic

Now you can hear whether a di-chord is dissonant, modal, or perfect and whether it refers up or down. However, both [1] and [2] are dissonant and look down. Both [8] and [9] are modal and look up. How can you distinguish [1] and [2], [8] and [9], or any of the other pairs of di-chords that share the first two properties?

This is where the third factor comes in—harmonicity. Any di-chord can be classed as being harmonic or non-harmonic. Whether a di-chord is harmonic or non-harmonic depends on whether the upper note is in the first nine notes of the overtone series of the lower note.

The harmonic di-chords—[2], [4], [7], [9]**, [10]—sound open, hollow, and expanding, and possess a sort of bland, alkaline “flavor” for the ear. In addition, the two pitches in a harmonic di-chord create an effect in the ear similar to the effect of repelling one another, like two magnets of like polarity.

On the other hand, non-harmonic di-chords—[1], [3], [5], [6]*, [8], [11]—sound closed, dense, and contracting, and possess a sharp, acidic “flavor” for the ear. The two pitches in a non-harmonic di-chord create an effect in the ear similar to the effect of strongly attracting towards one another, like two magnets of opposite polarity.

* [6] is a “false non-harmonic” because the harmonicity is perceived as neutral, being neither like harmonic nor non-harmonic di-chords.

** [9] is a “false harmonic” because, while its harmonicity sounds decidedly like the harmonic di-chords, its upper note does not fall within the first nine notes of the overtone series of the lower note.

Di-Chord Pictograph and Harmonicity Colors

In the Di-Chord Pictograph and in the previous table, harmonic di-chords are depicted as light or pale in color, while non-harmonic di-chords are depicted as a vivid or intense dark color (while this textbook uses shades of gray, you can draw and color your own pictograph using shades of your favorite color).

Exercises: The Notes of the Overtone Series

One way of learning which notes in the overtone series of any given note are perceived as harmonic (and thus which specific di-chords are harmonic) is to become fluent in spelling dominant ninth chords. The dominant ninth chord contains four of the five notes from the overtone series that produce harmonicity. The fifth note that is not included in a dominant ninth chord is the false harmonic a [9] (major 6th) above the fundamental. For more detailed information on the overtone series, see page 296.

The following exercises will help you develop fluency in spelling dominant ninth chords by starting with major triads, moving to dominant seventh chords, and finishing with dominant ninths.

Exercises: Getting to Grips with Harmonicity

Animal Analogy for Identification

When identifying a di-chord, proceed from general to specific:

1. Category first (species class) — Dissonant / Modal / Perfect?
2. Direction (subspecies) — Does it refer up or down?
3. Harmonicity (genus) — Harmonic (open) or non-harmonic (closed)?

Major vs. Minor — Acoustic Explanation

Triad Di-Chord Profiles

Complete Di-Chord Reference Table

* [9] major 6th is “false-harmonic” — acoustically appears harmonic but behaves non-harmonically in context due to its inversion relationship with [3].

Chapter Fourteen: Properties of Di-Chords: Interference Pulsation

Interference Pulsation

Di-chords are either dissonant, modal, or perfect.

Dissonant di-chords beat at 8 cycles per second, modal di-chords at 4 cycles per second, and perfect di-chords at 2.

Fundamental/Octave Factor

The upper note of any di-chord either refers up or refers down.

Does the upper note sound like it is referring down to the bottom note or up to the first octave manifestation of the fundamental? Remember, di-chord [6] splits the octave directly down the middle and refers up and down by equal amounts.

Harmonicity

Di-chords are either harmonic or non-harmonic.

The top note of a harmonic di-chord falls within the first nine partials of the overtone series of the bottom note. Visually, this would be like a green ornament on a green Christmas tree.

In non-harmonic di-chords, the top note does not fall within the overtone series of the bottom note, producing a type of contrast or “dissonance.” Like a red ornament on a green Christmas tree, the upper note of a non-harmonic di-chord stands out more than it blends in.

The Combined Effect of the Three Sound Factors

Notice that no two di-chords have the same combination of all three factors. This is why you need an awareness of all three factors to pinpoint which of the 11 di-chords you are hearing. The Di-Chord Pictograph provides a useful visual representation of the 11 di-chords and the three sound factors.

The Di-Chord Pictograph also acts as a metaphor for the linear and non-linear aspects of di-chord perception. You can observe linear aspects, such as the palindromic progression of interference pulsation:

jagged – rounded – square – rounded – jagged

Likewise, the shadows that represent the fundamental/octave factor appear on the left side in numbers 1–6, and on the right side for numbers 6–11. All linearity ceases when you observe the shades of color that illustrate harmonicity because the structure of the overtone series is not linear, it is logarithmic (non-linear) and is better represented as a spiral (one that doubles its distance from the center with every revolution) rather than a straight line (Wilson 1965).

Di-Chord Identification

While it may seem difficult to identify each of the three sound factors of di-chords in real time, it is no more difficult than identifying an animal’s species (reptile, mammal, bird), its subspecies (turtle/lizard, whale/horse, owl/hummingbird), and then its particular genus (snapping turtle/chameleon, Beluga/Appaloosa, barn owl/ruby-throated). Actually, because there are only 11 di-chords, it’s far easier than this to identify.

Review Table: Di-Chord Properties

The following table lists the 11 di-chords with their common interval names and categorization by the three sound factors.

Exercises: Di-Chord Sound Factors

Chapter Fifteen: Di-Chords in Melodic Contexts

Perception of Di-Chords in Melodic Contexts

In chapters 10–14 we covered the three sound factors of di-chords that are played synchronously—both pitches sounding at the same time. However, the three sound factors that affect the synchronous or vertical sounding of di-chords also have correlated effects on the non-synchronous or horizontal sounding of di-chords.

It is essential that all musicians, including those who play “harmony” instruments, like piano or guitar, understand the melodic characteristics of di-chords. Instruments that are commonly understood to be “harmony” instruments are really playing multiple horizontal lines at once; to play those instruments with meaning and expression, it is vital to understand the melodic characteristics of each di-chord in every horizontal line.

Interference Pulsation in Melodic Contexts: Gut, Heart, and Head

Dissonant Di-Chords: The Gut

The perceptions elicited from dissonant melodic di-chords are that of:

• gut-centered physicality,
• transition,
• movement between fixed points, and
• connecting words or phrases in speech, like “such as,” “the beautiful…” “and then, the fast flowing…” or “what if…?”

Modal Di-Chords: The Heart

The perceptions elicited from modal melodic di-chords are that of:

• heart-centered emotions,
• feelings,
• fluidity,
• grace and ease of movement, and
• words such as “love,” “tenderness,” “yearning,” “affection,” and “grief.”

Perfect Di-Chords: The Head

The perceptions elicited from perfect melodic di-chords are that of:

• head-centered intellect,
• thoughts,
• concepts,
• structures,
• principles, and
• words such as “honor,” “logic,” “integrity,” “conceit,” and “calculating.”

Fundamental/Octave Factor in Melodic Contexts

The effect of fundamental/octave factor in melodic contexts is the same as it is in harmonic contexts:

• For di-chords [1], [2], [3], [4] and [5], the upper note refers down to the lower note.
• For di-chords [7], [8], [9], [10] and [11], the upper note refers up to the nearest octave manifestation of the lower note.

Harmonicity in Melodic Contexts: Open, Closed, Strong, and Weak

Ascending Harmonic Di-Chords: Open

When the two constituent pitches of any of the harmonic di-chords ([2], [4], [7], [9], [10]) are played linearly in ascending order, they create a feeling that the upper note is open-ended, eternal, and floating above the lower note. The two notes seem to expand apart from one another so that the upper note is seemingly pushed upward into space, opposing gravity. The upper note feels natural to sing because it is an overtone of the lower note. In this melodic movement, there is a subtle perception of a counter-clockwise gesture that proceeds, if imagined on a clock face, from six o’clock, through three, and up to twelve o’clock. In score analysis, an upward pointing arrow is used to indicate any ascending harmonic horizontal di-chord as “open.”

Ascending Non-Harmonic Di-Chords: Closed

When the two constituent pitches of any of the non-harmonic di-chords ([1], [3], [5], [8], [11]) are played linearly in ascending order, the upper note is stronger than the lower note, and therefore seems to be pulled downward; not in the case of [11], whose tendency is to strongly go to the octave of the lower note. The two notes seem to contract, causing the sensation that the upper note “caps off,” ends, or closes a thought or gesture. It is difficult to sing the upper note correctly because it is not an overtone of the lower note; hence, there is a preference towards singing the nearest ascending harmonic di-chord. The upper note also tends to be difficult to tune because it clashes by a semitone with one or more of the overtones of the lower note; it is sung best—not by imagining that the lower note is an overtone of the upper note, rather the other way around. In this melodic movement, there is a subtle perception of a clockwise gesture that proceeds, if imagined on a clock face, from nine o’clock, through twelve, and ending at three o’clock. In score analysis, an over-arched arrow is used to indicate any ascending non-harmonic horizontal di-chord as “closed.”

Descending Harmonic Di-Chords: Strong

When the two constituent pitches of any of the harmonic di-chords ([2], [4], [7], [9], [10]) are played linearly in descending order, the lower pitch is perceived as the fundamental of the two pitches, which creates a sense of grounded finality, like the final word of a conclusive statement. Descending harmonic di-chords can initially be difficult to sing because we do not hear the lower note as an overtone of the upper note. To sing the lower note correctly, it must be intended, or we will sing the nearest pitch class in the nine lowest partials of the upper note. In this melodic movement, there is a subtle perception of a counter-clockwise gesture progressing, if imagined on a clock face, from twelve o’clock, backward through nine to six o’clock. To indicate any descending harmonic horizontal di-chord, a downward pointing arrow is used.

Descending Non-Harmonic Di-Chords: Weak

When the two constituent pitches of any of the non-harmonic di-chords ([1], [3], [5], [8], [11]) are played linearly in descending order, they create a paradoxical gesture that can seem to rise, like the final word in a question. The lower note is often found among the nine lowest partials of the upper note, making it relatively natural and easy to sing. The paradoxical nature of this gesture occurs because the horizontal motion is moving downwards while, at the same time, the lower note contracts back up towards the upper note because it is in the overtone series of the upper note. In this melodic movement, there is a subtle perception of a clockwise gesture on a clock face, progressing from three o’clock, through six, and ending at nine o’clock. To indicate any descending non-harmonic horizontal di-chord, an under-arched arrow is used.

Ascending or Descending Di-Chord [6]: Neutral

This is the only di-chord whose harmonicity factor is neutral, neither seeming harmonic nor non-harmonic. In horizontal movement, [6] seems neither open, closed, weak, nor strong, but neutral. By comparing a musical line consisting of only di-chords [5], [6] and [7], you can easily perceive the acrid, unblissful quality of [6], eliciting in our hand a sort of claw-like gesture, versus the contracting or expanding in the hand, respectively, with [5] or [7]. Because [6] is devoid of a movement gesture, I suggest simply identifying it by name as [6].

Exercises: Melodic Gestures of Di-Chords

Ch.16 — The Tracking Page

Introduction

Next to visualizing the keyboard, The Tracking Page (p. 153) is the most important means of developing fluency and accuracy in connecting your mind with your ear and your eye. It is helpful to remember that the ear needs no training. It is the mind that requires training to become aware of what the ear is hearing at the speed that the ear is hearing it, but the best guidance is wasted without faithfully applying what is needed to become skillful—the backing of a rigorous attitude. Take J.S. Bach as your example, who, when asked about his prodigious talent, was quoted as saying, “I was obliged to be industrious. Whoever is equally industrious will succeed equally well.” (1998)

Using the Tracking Page

There are six protocols to be performed for each line of The Tracking Page. These protocols are designed to help you master each of the several aspects involved with musical reading and comprehension. Generally, you may find that you are weak on one or more of the protocol steps, so those steps will need to be strengthened. By performing each step in sequence, you can discover points of weakness to strengthen. Master each step before continuing to the next. Initially, each protocol may require several minutes, perhaps even half an hour. At this rate, the six steps would require about three hours to master. However, this level of weakness in skill is extremely rare. Even so, in such a case, the time would be well spent, and fluency would be the end result, with the entire sequence requiring little more than six minutes—one line per minute.

Start by using only the diatonic white-notes. Perform one or more lines per week doing each of the six protocols every day, starting on a different note each day. If, for instance, you choose line 3 for the week, select a different note for each of the seven days of the week. Each day, perform the six protocols in order.

By the end of the week, you will have started and ended on each of the seven diatonic notes and will have sung each of the seven modes by name by syllable name, primary di-chord number, and secondary di-chord number. Remember to incorporate each step into every subsequent step. In the final, sixth step of the protocol, all previous steps are incorporated into the performance.

Note: This chapter assumes some knowledge of the seven diatonic modes—Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, and Locrian. If you don't know what the modes are, or you find the parts of The Tracking Page that mention modes confusing, then review the later chapters on Tetrachords (p. 187) and the Diatonic Modes (p. 198).

The Tracking Page — Lines

The Six Tracking Protocols

Pick a single line from The Tracking Page and, while tracking on a visualized keyboard, perform the six steps, aiming to achieve a tempo of not less than 144 bpm. You will also need to pick a starting pitch for the first note, which can be any of the seven diatonic pitches—doh, re, mi, fa, sol, la, or si. The note you choose to start on will determine the line's mode and clef.

The Six Tracking Protocols: Summary

Pick a single line from The Tracking Page and, while tracking on a visualized keyboard, perform the six steps, aiming to achieve a tempo of not less than 144 bpm.

1. Speak the solfège name of each note, filling in adjacent notes for intervals larger than a 2nd.
2. Speak a) the secondary di-chord name and b) the secondary di-chord gesture for each consecutive note.
3. Speak the primary di-chord name for each consecutive note.
4. Sing the primary di-chord name for each consecutive note, while playing a tonic drone at the keyboard.
5. Sing the secondary di-chord name for each consecutive note, while playing a tonic drone at the keyboard.
6. Sing the solfège name of each note, while playing a tonic drone at the keyboard.

Modes and Clefs on the Tracking Page

The Tracking Page does not include clefs. This means that there is nothing to indicate what names you should call any of the notes on the page! Depending on the clef you choose for a line, the line can start on any one of the seven diatonic pitches—doh, re, mi, fa, sol, la, or si.

When you hear and sing the line, you will perceive the first note as the tonic pitch. Therefore, the first note of each line determines the diatonic mode of the line. If the first note of the line is doh, the line is in the Ionian mode; if the first note is re, the line is in the Dorian mode; if the first note is mi, the line is in the Phrygian mode, etc. For more information on the diatonic modes, see chapter 19, “The Diatonic Modes.”

These two variables—the clef and the mode—are dependent on each other. When starting a line, you or your teacher will choose either a clef or a mode for the line that day. Choosing a clef will determine the mode, while choosing a mode will determine the clef.

Reading by Di-Chord Versus Reading by Rote

For each of the steps of the protocol, always be clearly visualizing the keyboard and tracking on an imagined keyboard with your relaxed right index finger. Although this may be challenging at first, it is an essential step in making the transition from reading music by rote to reading music by di-chord.

When reading music by rote, we read individual pitches. This gives us only the information necessary to correctly “push the right buttons” on our instruments. This is like being able to read and pronounce words of a foreign language while not understanding them. A speaker might pronounce every word on the page with complete accuracy, but, if the speaker does not understand the meaning of the words, his speech will sound strange to the native speaker, perhaps sounding flat or conveying an affect mismatched from the expressive content of the text.

When reading music by di-chord, we get information about which pitches to play, but more importantly, we are directly accessing “what happens between the notes,” where expression, affect, and meaning are found. With the knowledge and understanding of the properties of each of the 11 di-chords, reading music by di-chord gives an immediate understanding of the range of expression suggested by the notes written on the page. As when we read a language in which we are fluent, we have immediate access to the meaning of what we are reading. We also have the capability to manipulate the sounds we are producing in appropriate ways—that is, in ways that are congruent with the meaning that the notes on the page express—even if we are reading something for the first time.

Protocol Example: Line 1, Starting on FA

In this walk-through example, we will be analyzing and singing Line 1 of The Tracking Page starting on fa.

Mode and Clef: We are starting on the note fa, and the first note of the line is located in the bottom space of the staff, which is fa only in the treble clef; therefore, we know that we must be reading in the treble clef. As you perform each step of the protocol, imagine seeing the appropriate clef at the beginning of the line at all times.

Step 1 — Speaking Solfège Syllables:Speak the name of the solfège syllables of the notes in a brisk, even pace, at a minimum tempo of 60 bpm but aiming to achieve 144 bpm. Fill in the solfège syllables between the written notes; the written notes in the line should be spoken one per beat, regardless of the size of the di-chord. For Line 1 in the Lydian mode, we have no need to “fill in” any solfège syllables because all of the intervals in Line 1 are 2nds. Therefore, speak the solfège syllable of each note, one per beat.

For intervals greater than a 2nd, we will always fill in the diatonic notes between the two pitches with solfège syllables. To do this, you must still name each of the notes on the page at the rate of one per beat, meaning that when the interval between notes is larger than a 2nd, you must speak the solfège syllables that fall between the two notes in a faster rhythm.

Interval Rhythm Table

Step 2A: Speaking Secondary Di-Chord Numbers

Starting with the number “0” on the first note of the line, name each of the secondary di-chord numbers in a brisk, even tempo.

Secondary di-chord numbers describe the number of semitones between each pair of adjacent notes; the number you speak on any given note describes the number of semitones between that note and the note that preceded it.

It is crucial at this point that you are absolutely clear in your keyboard visualization and are tracking with your index finger. Especially in the earlier lines in more familiar clefs, you will be tempted to take shortcuts, bypassing the critical step of visualizing and relying on earlier rote training to perform certain of the steps on The Tracking Page. However, with focused intent and patience, this brain circuitry will be re-wired. As the intervals get larger further down in The Tracking Page, and you practice with unfamiliar modes and clefs, it will become harder and then impossible to use prior rote learning to perform the lines. Therefore, it is suggested that you prioritize keyboard visualization and tracking at each and every stage of The Tracking Page Protocol.

Step 2B: Speaking Secondary Di-Chord Gestures

Starting with the number “0” on the first note of the line, name the gesture (open, closed, weak, or strong) that accompanies each consecutive secondary di-chord number. For more information on the melodic gestures, review chapter 15.

Remember to visualize the clef you are using! The gesture name that you speak on any given note describes the gesture accompanying the di-chord formed by that note and the note that came immediately before.

Step 3: Speaking Primary Di-Chord Numbers

Starting with the number “0” on the first note of the line, name each of the primary di-chord numbers at a brisk, even tempo.

Primary di-chord numbers describe the number of semitones between each pitch and the perceived tonic, which, in this case, is always the first note of the line. The naming of primary di-chord numbers is similar to the naming of scale degrees in that the number remains consistent in any octave. When a note is lower than the first note of the line, the primary di-chord number will be the same as when it is above the first note. In other words, if the starting note is fa, every sol will be named as a [2], whether it is a 2nd above the starting pitch or a 7th below it; the sol will always be the second degree of the scale and thus, primary di-chord [2].

Familiarity with the mode in which you are performing is advantageous because the only numbers you will be speaking will be the numbers that fit the profile of that mode. When you first start working with The Tracking Page, it can be helpful to review the profile of primary di-chord numbers in each selected mode so that you can clearly visualize all seven options in a given mode (see chapter 19, “The Diatonic Modes”).

Each step in the protocol should integrate with your observations from previous steps and augment your awareness of the musical relationships in the line. When speaking primary di-chords, for example, maintain an awareness of the solfège syllables and the secondary di-chords from the previous step. As you speak primary di-chord number, notice the relationships between secondary and primary di-chord numbers.

Notice that addition applies when the pitches are going up, and subtraction applies when they are going down. When the pitches cross the tonic, this is still true; however, because there are 12 pitches in the octave, the addition and subtraction must be done in a base-12 mathematical system.

Thus far, you have spoken the solfège syllables, the secondary di-chords, and the primary di-chords of the line. The next three steps cover the same information but are sung rather than spoken.

To sing through the next three steps, you will need a tonic drone. You can use an electronic tuner to provide a constant pitch, or you can play a drone on a piano. If you are using a piano, you may want to play the tonic in multiple octaves. You can choose to play the tonic on each beat or just a few times per line. Use the piano's sustain pedal to sustain the sound throughout the line. The tonic drone will provide a reference for your intonation as well as allowing you to hear the interference pulsation, fundamental/octave factor, and harmonicity of the primary di-chords as you sing them.

Step 4: Singing Primary Di-Chord Numbers

Sing each note while naming the primary di-chords. Maintain an awareness of the properties of the di-chords you are singing by visualizing the pictographs of the primary di-chords.

Step 5A: Singing Secondary Di-Chord Numbers

Sing each note while naming the secondary di-chord. Maintain an awareness of the properties of the di-chords you are singing by visualizing the di-chord pictographs of the secondary di-chords you are naming as you sing.

Step 5B: Singing Secondary Di-Chord Gestures

Starting with the number “0” on the first note of the line, sing the gesture that accompanies each secondary di-chord number. Incorporate the character of the gesture into your singing.

Step 6: Singing Solfège Syllables

Sing each note while naming the solfège syllable.

The last step, singing solfège, brings us full circle from our first step, speaking solfège. With the knowledge gained by examining and performing the line in the different steps of the protocol, we now have access to a greater depth and complexity of musical meaning than we did at the beginning of the process. All of this information—the pitches, the relationships between consecutive pitches, and the relationships between the pitches and the perceived tonic—is present and accessible in any melodic context; therefore, you can approach any melodic line with the steps outlined in this chapter.

Exercises: Understanding and Using The Tracking Page

Recording Your Progress

Regular practice with The Tracking Page will cultivate and improve musical aptitude across the board, particularly fluency in reading, transposing, and comprehending music in real time. With regular practice, you will only need to spend a few minutes each session focusing on tracking.

The following table will allow you to record your progress. Check the boxes when you complete a step in the protocol for a particular line in a particular mode. After having completed the table, you will have sung through all six steps of the protocol for each of the lines on The Tracking Page in each of the modes! Ideally, one practice session would include all six steps of the protocol for one line in one mode.

An excellent way of keeping consistency with this practice is to assign a mode to a day of the week:

• Monday — Aeolian
• Tuesday — Phrygian
• Wednesday — Locrian
• Thursday — Lydian
• Friday — Ionian
• Saturday — Mixolydian
• Sunday — Dorian

Progress Tracking Table

A table for recording progress through all six protocol steps for each of The Tracking Page lines (1–10) across all seven modes.

Ch.17 — Tri-Chords

Introduction

In this chapter, you will combine tracking and your knowledge of di-chords to understand tri-chord patterns—groups of three notes—and to fluently use these patterns like stencils to quickly form tri-chords over any note. This will lead to developing the ability to form tetrachords, or four-note groups, over any note, a skill that is essential to mastering not only the more familiar major and minor scales but all of the modes. Being able to visualize and calculate scale patterns is yet another essential tool for keeping ourselves musically oriented, not only within the dimension of pitch space but also within any given tonality.

Primary Di-Chord Numbers

In using numbers to refer to di-chords, a single number describes the number of semitones comprising a musical interval. However, now we need a way to use numbers to describe groups of three notes—tri-chords. When we hear tri-chords, or chords with any number of notes, our brains relate each of the upper notes in the chord to the lowest sounding note, called the fundamental. The relationship of each upper note to the fundamental seems to be more important to our perceptions than the relationship between the upper notes themselves. In essence, the brain notices the relationship between the highest and the lowest and between the middle and the lowest more vividly than it notices the relationship between the highest and the middle notes.

Therefore, when we name tri-chords using the number system, the numbers will refer to the primary di-chords formed by the fundamental note and each of the upper notes. Let's look at the chord doh-mi-sol:

Doh (the fundamental) and sol form a [7]; doh and mi form a [4]. Thus, we call this tri-chord a [4/7]. Notice that within this group of three notes there is also a [3] produced by the mi and sol; however, this relationship is a secondary di-chord and does not play the same kind of role in our perception of the chord as the other two relationships, and so is not used in the naming of this tri-chord.

Forming Tri-Chords with 2nds

Using only 2nds, or semitones and whole tones, we can form four different types of tri-chord. Our starting note will always count as one of the three notes. We can represent the starting note with a “0,” since we have not traversed any distance yet. Then, we might go up a [2], a whole tone, landing on the second note of the tri-chord, and then up another [2] to land on the third note. We might call this pattern “0, [2], [2].” Note that this is an abstract pattern—we could apply it to any starting note. One tri-chord formed by this pattern would be “fa, sol, la.” Another example would be “sib, doh, re.” This pattern is using secondary di-chord numbers, not primary di-chord numbers, to describe a set of notes.

If we were to hear all three notes at once, “0, up a [2], up a [2]” does not describe how we are hearing that chord. Remember that in hearing two or more notes, our brains relate the upper note or notes to the lowest note in the group. When we are hearing fa, sol, and la played all at once, what we are actually hearing are multiple di-chords—the [2] formed by fa and sol, the [2] formed by sol and the la, and the [4] formed by fa and the la. The brain primarily notices the di-chord between each top note and the lowest note as most important. In our current example, this would be the fa to sol and the fa to la, although we also hear the di-chord from sol to la. Because of this, we will refer to this tri-chord as a [4/2]—from the top down, the la forms a [4] with the lowest note, and the sol forms a [2] with the lowest note.

Exercises: Playing and Spelling Tri-Chords

Continuation: The Ionian Scale

In summary, mastering the seven modes requires mastering the four diatonic tri-chords. The Ionian scale is built from these four tri-chord types.

Ch.18 — Tetrachord Formation: Ionian, Dorian, Phrygian, and Lydian

The Four Tetrachords

Tetrachords are groups of four notes that, from the lowest note to the highest note in the tetrachord, span a 4th. There are four diatonic tetrachords: Ionian, Dorian, Phrygian, and Lydian.

In subsequent chapters, you will learn to simultaneously form and visualize all seven notes of any heptachord, but first, in this chapter, you must focus on mastering tetrachords, which are the building blocks of heptachords. Each of the seven Western modes can be formed with a combination of one or two of the four diatonic tetrachords, and by learning to think fluently in the patterns of these four tetrachords, you will gain access to all seven modes.

The Ionian Tetrachord

First, consider the C major scale. Place your left-hand pinky on the tonic, doh, and play the notes doh, re, mi, and fa — this is an Ionian tetrachord. Next, place your right-hand index finger on the note sol. Using the index, middle, ring, and pinky fingers of the right hand, play the notes sol, la, ti, and doh — another Ionian tetrachord. You have just formed a major scale, and if you start on any of the 12 pitches and follow the same sequence, you will have formed the major scale that begins on the lowest note in your formation. Therefore, one way to describe the major scale is to say that it consists of two Ionian tetrachords separated by a .

When you formed the major scale with two Ionian tetrachords, did the tetrachords look similar to each other? You may have noticed that each tetrachord was formed using the same pattern of secondary di-chords. To find the secondary di-chord pattern for the Ionian tetrachord, use the first four notes of the doh major scale:

So, to form the Ionian tetrachord above any given note, you need to track “up a , up a , up a .” This can be simplified to [0, 2, 2, 1]. You can then deduce that the primary di-chord pattern for the Ionian tetrachord is [0, 2, 4, 5]. By practicing this formation and exploring the visual patterns of black and white keys, forming the Ionian tetrachord will soon feel easy and immediate, and you will be able to visualize and form the tetrachord as a single unit rather than using the intermediate step of spelling it out through tracking.

The Dorian Tetrachord

To form a Dorian tetrachord, place your left-hand pinky on the note re, and play the notes re, mi, fa, and sol with all four fingers of your left hand. You can deduce from these four notes that the secondary di-chord pattern for the Dorian tetrachord is [0, 2, 1, 2] and the primary di-chord pattern is [0, 2, 3, 5].

The Phrygian Tetrachord

To form a Phrygian tetrachord, place your left-hand pinky on the note mi, and play the notes mi, fa, sol, and la with all four fingers of your left hand. You can deduce from these four notes that the secondary di-chord pattern for the Phrygian tetrachord is [0, 1, 2, 2] and the primary di-chord pattern is [0, 1, 3, 5].

The Lydian Tetrachord

To form a Lydian tetrachord, place your left-hand pinky on the note fa, and play the notes fa, sol, la, and si with all four fingers of your left hand. You can deduce from these four notes that the secondary di-chord pattern for the Lydian tetrachord is [0, 2, 2, 2] and the primary di-chord pattern is [0, 2, 4, 6].

Exercises: Ionian Tetrachord Formation

Exercises: Dorian, Phrygian, and Lydian Tetrachord Formation

Ch.19 — The Diatonic Modes

Introduction

Modes continue to be a central resource for musicians in both popular and “serious” music. Which elements within each mode, if any, might be responsible for associations with specified emotions, moods, or affects? Are these associations purely subjective, or might there be an objective means of understanding how modes elicit feelings and sentiments in our minds? A thorough examination of the di-chords contained in each mode helps us better understand the correspondence between objective fact and the emotional responses so often associated with the diatonic modes.

Modes in the Diatonic Ordering

There are seven diatonic Modes: Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, and Locrian.

Each mode is composed of seven diatonic notes that conform to a unique pattern of semitones and whole tones contained in the octave. By starting on each white-key note and moving stepwise up the octave, you can see that different patterns of semitones and whole tones occur in each mode. For example, when using only white-key notes, the Ionian mode begins on doh and ends on the doh an octave higher; however, the Dorian mode, if you use white keys only, begins on re and ends on the re an octave above and has the secondary di-chord pattern of [0, 2, 1, 2, 2, 2, 1, 2].

It is helpful when first learning about the modes to study each mode as it occurs naturally on the white keys of the piano. In the following table, the seven white-key notes are listed along with their corresponding modes; that is, with the mode that starts and ends on that note when only white keys are used.

You will need to memorize the seven mode names and know what diatonic pitch corresponds with each mode.

The Tonocentric Model

In the context of a key or a mode, the primary di-chords (di-chords in relation to the tonic pitch) combine to give each scale degree a tendency to refer up or down to the nearest tonic, similar to how the fundamental/octave factor causes the upper note of a di-chord to refer down or up.

The pattern of scale degree tendencies resembles a magnetic field in which the tonic is the attractor, in the center: 5   6   7   1   2   3   4. I refer to this model as the Tonocentric Model. The term tonocentric means that the reference tone, or tonic, is in the center, surrounded on either side by radiating dissonant, modal, and perfect heptachord elements.

Using this model of scale degree tendencies, each degree refers up or down to the nearest tonic. Degrees 2, 3, and 4 refer down to the nearest tonic and will be represented by the sign (+) because they rest above the nearest tonic; degrees 5, 6, and 7 refer up to the nearest tonic and will be represented by the sign (−) because they rest below the nearest tonic:

Notice that each mode or heptachord will potentially have two dissonant members, two modal members, and two perfect members.

Because dissonant di-chords sound active, on the move, anxious, dynamic, and “gut”-oriented, degrees 2 and 7 of any mode have this underlying nature, which is why they sound unresolved and wanting to move towards the tonic.

Because modal di-chords sound mellifluous, undulating, fluid, emotional, and generally “heart” oriented, degrees 3 and 6 of any mode will have this basic nature.

Perfect di-chords sound stoic, solid, balanced, controlled, somewhat cold, mentally calculating, and generally “head” oriented, so degrees 4 and 5 of any mode will have this basic nature.

Summary of Degree Tendencies

• Degree 2 tends down and is dissonant.
• Degree 3 tends down and is modal.
• Degree 4 tends down and is perfect.
• Degree 5 tends up and is perfect.
• Degree 6 tends up and is modal.
• Degree 7 tends up and is dissonant.

Though it is challenging to do at first, it is important to associate the abstract words—dissonant, modal, and perfect—with their affects. Knowing this, you can tell the general character of each degree in any mode, like recognizing the players and their positions in a football game. To know the exact name of the players, you must look and hear more deeply.

Tetrachordal Construction of Modes

When listening to modal music, the mind seems naturally inclined towards grouping the members of a mode into two tetrachords, one consisting of scale degrees 1, 2, 3, and 4, and the other consisting of degrees 5, 6, 7, and 1. For the sake of simplicity, we will call these two tetrachords Tetrachord I and Tetrachord II. The difference between the two tetrachords is that Tetrachord I consists of the tonic and three “downward tending” degrees, and Tetrachord II consists of the tonic and three “upward tending” degrees.

The pulsation of each scale degree progresses through dissonant (D), modal (M), and perfect (P), radiating outward from the tonic (T).

Notice that the first three modes—Ionian, Dorian, Phrygian—are made by combining two identical tetrachords; we will say that these are “pure” modes. The other four modes—Lydian, Mixolydian, Aeolian, and Locrian—are made by combining two different tetrachords; we will say that these are “mixed” modes.

Modes in the Ploger Ordering

My ordering of the modes begins on fa and proceeds in fifths through fa–doh–sol–re–la–mi–ri, just like with the Pythagorean ordering of fifths.

With this approach, many more of the elegant patterns and symmetries of the modes become apparent. For example, observe that the primary di-chords of neighboring modes differ in only one scale degree! Lydian and Ionian differ only in their 4th degree; Ionian and Mixolydian differ only in their 7th degree; and so on.

This ordering is, I maintain, the most natural ordering of the modes because it reveals the degrees of similarity and difference—modes nearest to one another share the most primary di-chords, and those farthest share the fewest. In other words, the farther apart two modes are from one another, the more dissimilar they are in their primary di-chord composition. Therefore, Lydian and Locrian are the most dissimilar, sharing only the primary di-chord [6].

Further, because of the primary di-chords that they contain, the first three modes in the Ploger ordering—Lydian, Ionian, and Mixolydian—are of a different sonic nature than the final three—Aeolian, Phrygian, and Locrian. The Lydian, Ionian, and Mixolydian contain the di-chords [2] (major 2nd), [4] (major 3rd), and [9] (major 6th) which cause these modes to sound generally more open and “major.” By contrast, the Aeolian, Phrygian, and Locrian modes contain the di-chords [3] (minor 3rd), and [8] (minor 6th), and [10] (minor 7th), causing these modes to sound more contracted, tense, and “minor.”

The Dorian mode, in this ordering, rests exactly in the middle and acts as a type of musical tipping-point, sharing aspects of both the open/major modes and the closed/minor modes, making the Dorian the most balanced mode in this regard.

When you look at the modes, and music in general, in the Ploger or Pythagorean ordering it can seem like there is no end to the meaningful patterns there for you to see. For a more extensive exploration of these patterns see my article, Music’s DNA: The Perfect Fifth (Ploger 2011).

Eight Rules That Govern Aural Perception of the Modes

Rule 1

Generally, the first heard pitch in a modal melody acts to the ear/mind like the subject of a sentence. The ear seems to grasp the first heard pitch, holding it in mind as a reference pitch, such that a complete musical thought would end by returning to that pitch. Thus, in modal theory, the first note is generally also the same as the last one and is known as the final.

Rule 2

To aurally identify any mode, all seven of the mode’s notes must be present. If one note is missing, the mode can have two possible identities because each mode shares six notes with at least one other mode.

Rule 3

The ear/mind most strongly perceives the di-chord formed between the tonic and each of the other six non-tonic notes in the mode. We refer to the di-chords formed between each of the six non-tonic notes and the tonic as primary di-chords. The six primary di-chords in any mode are largely responsible for its general character.

Rule 4

All di-chords formed between non-tonic pitches of a mode, while not insignificant, matter far less to our perceptions than the primary di-chords. Because of this, we refer to the di-chords formed between non-tonic members of a mode as a secondary di-chord. (Our minds probably prefer to attend to primary rather than secondary di-chords because of its desire for efficiency and order—while there are 42 secondary di-chords in the diatonic scale, there are only six primary ones.)

Rule 5

Of the six primary di-chords in each of the seven modes, two are dissonant, two are modal, and two are perfect. Scale degrees 2 and 7 are dissonant; 3 and 6 are modal; and, 4 and 5 are perfect, which creates a pattern radiating outward from the tonic.

Rule 6

Each of the six non-tonic members of any mode is no more than a 4th from the tonic pitch class. The mind relates each non-tonic note to the closest pitch class manifestation of the tonic. This generates a pattern of degree tendencies in which degrees 2, 3, and 4 refer down, while 5, 6, and 7 refer up to the nearest tonic.

Rule 7

The harmonicity of each primary di-chord of a mode has a profound effect on the character of the mode, which helps us distinguish one mode from another by eye, hand, and ear. The following table shows the harmonicity factor for each scale degree in the seven diatonic modes.

Several remarkable patterns emerge from this table. The modes may be ranked according to the number of harmonic di-chords as follows: Mixolydian has five harmonic di-chords: [2], [4], [7], [9], [10]; Ionian, Lydian, and Dorian have four harmonic di-chords; Aeolian has three; Phrygian has two; and Locrian has only one harmonic di-chord: [10].

Rule 8

The modes associated with the most positive affects are those with the most harmonic di-chords, while those modes associated with the most negative affects are those with the most non-harmonic di-chords. Observe that the most non-harmonic modes are those two in which the tonics are found on the far right side of the Pythagorean Collection (E and B).

Developing Fluency in the Modes

Becoming fluent in the modes will be particularly helpful for hearing and understanding a wide variety of music in real time. Most of the music you play is probably in the Ionian mode or Aeolian mode (the Ionian mode is the major scale, while the Aeolian mode is the natural minor scale). However, the other modes are frequently used in diverse musical genres, particularly in jazz, pop, and folk music. Many traditional folk songs from cultures across the globe use modes. As you familiarize yourself with the modes, you will start to recognize them in many places, from Christmas carols to TV theme songs!

Exercises

Ch.20 — Heptachord Formation

An Introduction to Heptachords

So far, you have used your visualization and tracking skills to form and spell four types of tri-chords {1/1}, {4/1}, {1/3}, and {3/1} and four types of tetrachords (Ionian, Dorian, Phrygian, and Lydian) above any given note. This chapter advances on this skill further so you can form and visually visualize not just four notes at a time, but seven—a complete heptachord.

With the ability to form and visualize groups of seven notes, not only can you imagine every major and minor scale, but each of the modes based on any tonic pitch. This makes complete orientation possible in any mode and any key. When you are oriented in a mode or key at any given moment, you are aware not only of each of its seven notes but also of the scale degree of each note and its relationship to the tonic note. This type of global awareness is essential for performing, improvising, and composing.

To form a seven-note heptachord, both hands are needed. There are two approaches to forming heptachords: the conjunct arrangement and the disjunct arrangement. Both arrangements involve the ring, middle, index, and pinky fingers of both hands, but not the thumbs.

Disjunct Arrangement

In the disjunct arrangement, the tetrachord beginning on the tonic note is formed with the left hand, and the tetrachord beginning on the dominant note is formed with the right hand:

1   2   3   4   5   6   7   1

This arrangement is familiar because it is the arrangement in which we typically play scales, from the tonic up an octave and back down again. Notice that in the disjunct arrangement, each finger is assigned to a single pitch, but both the left-hand and the right-hand pinkies are playing the same note, the tonic, one octave apart.

Conjunct Arrangement

The conjunct arrangement, in which the dominant-tonic tetrachord is formed with the left hand, and the overlapping tonic-subdominant tetrachord is formed with the right hand, is a more compact manifestation of the heptachord.

As in the disjunct arrangement, two fingers play the tonic pitch, but unlike the disjunct arrangement, these are the overlapping index fingers—the tonic pitch is duplicated at the unison. The symmetry of this arrangement in our hands is useful for quick orientation in a heptachord, and it is this arrangement that we will be working with primarily.

Notice that while the disjunct arrangement spans an octave, from tonic to tonic, the conjunct arrangement spans a seventh, from dominant to subdominant. The index fingers of the left and right hands overlap on the same note.

Sitting at the keyboard, form the doh Ionian heptachord in the conjunct arrangement. Your index fingers will be overlapping on the note doh. Moving out from here, the middle finger of your right hand rests on re, scale degree 2; the middle finger of your left hand rests on si, scale degree 7. Notice that the middle fingers of both the left and right hands rest on the two dissonant degrees of the scale—2 and 7. The ring fingers each rest on the two modal degrees of the scale, the right-hand ring finger on 3 (mi) and the left-hand ring finger on 6 (la). Finally, you can see that scale degrees 4 (fa) and 5 (sol), the two perfect degrees of the scale, fall to the right and left pinkies, respectively.

Consider this in comparison to the conjunct arrangement, in which the two pinky fingers play the tonic in two different octaves, the ring fingers play the dissonant degrees of the scale, the middle fingers play the modal degrees, and the index fingers play the perfect degrees.

Exercises: Constructing Disjunct and Conjunct Heptachords

Mode Modulation

Simultaneously visualizing all seven notes of a heptachord reveals each of the seven pitch options within the heptachord. However, it also reveals the most effective way to move from one heptachord to another, by allowing us to see how changing a single note in a heptachord can take us to an entirely new mode or key. This change from one mode to another by altering only one note is a mode modulation.

While sitting at a keyboard, form the conjunct arrangement of the Lydian mode based on the note doh. To form the conjunct arrangement of the heptachord, place both index fingers overlapping on the tonic, doh. Your left pinky rests on the dominant pitch, sol. From sol to doh in the left hand forms an Ionian tetrachord, [0 2 4 5], with the notes sol, la, si, and doh; from doh to fa# in the right hand forms a Lydian tetrachord, [0 2 4 6], with the notes doh, re, mi, and fa#.

You may have already noticed that this collection of seven notes matches one of the common key signatures, sol major, which contains one accidental, fa#. The seven notes in the Lydian mode based on doh are the same seven notes in the sol major scale (which we could also call the Ionian mode on sol). The difference between the Lydian mode on doh and the sol major scale is the tonic pitch—the note the scale begins on. Even though the two scales are made up of the same collection notes, they sound very different because the ear relates each of the notes to the tonic, and when the tonic is different, we perceive very different relationships—that is, we hear a very different set of primary di-chords.

Now, if you keep the same tonic pitch, doh, what one scale degree must you raise or lower by a semitone to move from the Lydian mode on doh to the Ionian mode on doh? Since you must keep the same tonic pitch, you can move any one of your fingers except either index finger. By moving your right-hand pinky down one semitone, from fa# to fa, you will form an Ionian heptachord with the Ionian tetrachord (sol, la, si, doh) in the left hand and the Ionian tetrachord (doh, re, mi, fa) in the right hand. With no accidentals, this collection of seven notes corresponds to the doh major scale.

Exercises: Mode Modulation

Ch.21 — Triads and Their Inversions: Major, Minor, Diminished, and Augmented

Primary and Secondary Di-Chords

We will name triads by their primary di-chord numbers, which, in this case, are the di-chords formed by each member of the chord and the lowest pitch of that chord. Recall that these are “primary” di-chords because we perceive them first and most strongly. As a rule, any secondary di-chord is of less significance to our aural perception of chords than are the primary di-chords because the primary di-chord(s) can be stated in any octave above the lowest pitch of the chord without affecting the basic identity of the triad.

Root Position Triads

Any root position triad, whether major, minor, diminished, or augmented, will possess two primary di-chords, one a 5th above the bass and the other a 3rd above the bass.

• The 5th may be di-chord , or .
• The 3rd may be di-chord or .

For example, visualizing the keyboard, form a doh major triad with your right hand: doh, mi, sol. The 5th of the chord, sol, creates a di-chord with doh, the fundamental. The 3rd of the chord, mi, creates a di-chord with doh. Observe that you will be hearing a composite effect of a perfect and a modal ; this will be important to perceptually distinguish different triads. The following table shows the primary di-chord elements of the four triad types.

The Root Position [7/4] Major Triad

In the [7/4] major triad, the 5th is the perfect and harmonic di-chord , and the 3rd is the modal and harmonic di-chord . Because both the primary di-chords are harmonic, the sound of the chord is uniquely open, expanding, and alkaline. While exceptionally harmonious, and sounding self-contained, complete, and whole, it might also seem bland—though this is somewhat mollified by the undulating motion of the modal .

The Root Position [7/3] Minor Triad

In the [7/3] minor triad, the 5th is the perfect and harmonic di-chord , and the 3rd is the modal and non-harmonic di-chord . The non-harmonic di-chord renders the triad’s sound more closed, contracting, and acidic. Because the ear is subconsciously aware of the 5th partial of the fundamental’s overtone series, which is always a di-chord , a semitone tension exists between these two pitches, evoking emotional poignancy, vulnerability, or tenderness.

The Root Position [6/3] Diminished Triad

Both primary di-chords in the diminished triad are non-harmonic, rendering this chord strongly closed, contracting, and acidic in sound. Its 5th is diminished, a di-chord , which possesses an oddly ambivalent nature because its upper note refers both down to the fundamental and up to its octave manifestation. As a result, because the can sound almost frozen in place, while still being non-harmonic, the diminished triad has a harsh and tart sound. It is the only triad whose 5th is perfect but non-harmonic.

The Root Position [8/4] Augmented Triad

In the [8/4] augmented triad, the 5th is the non-harmonic and modal di-chord , and the 3rd is the modal and harmonic di-chord . Notice that both the primary di-chords of the augmented triad have a modal interference pulsation of 4Hz, which renders the augmented triad uniquely modal in sound.

Rarely employed in functional harmony, the augmented triad is most often employed as a “color” chord to enhance mystery or exoticism in later Romantic and Post-Romantic compositions. Discerning the augmented triad from other triads is a matter of recognizing it as the only triad with no perfect (2Hz) di-chords.

Inverted Triads

Any root position triad can be “inverted” by using a different note of the triad as the lowest note. For example, take a doh major triad, consisting of the notes doh, mi, and sol from the bottom up. The root is doh, the 3rd is mi, and the 5th is sol. If you were to transpose the lowest note, doh, up an octave, then the chord, from the bottom up, would read mi, sol, and doh. This triad would be in its first inversion. Notice that the 3rd of the triad, mi, would now be the bass note.

Subsequently, if you were to take that first inversion chord and transpose its lowest note, mi, up an octave, the chord would then read sol, doh, and mi from the bottom up. This triad would have the 5th, sol, as it’s bass note, and it would be in second inversion.

While the pitch classes of root position, first inversion, and second inversion triads stay the same, their primary di-chords (di-chords in relation to the lowest note) differ, which causes each inversion to have different sound characteristics. For example, the primary di-chords of a root position major triad are and —perfect harmonic and modal harmonic—while the primary di-chords of a first inversion major triad are and —modal non-harmonic and modal non-harmonic. These two forms of the same triad sound very different!

First Inversion Triads

First inversion triads are known as “6/3” because they have two pitch classes above the bass—one a 6th above the bass and the other a 3rd above the bass. The 6th can be a di-chord or and the 3rd can be a di-chord or .

The following table indicates the di-chords above the bass for each of the four first inversion (6/3) chord types. Notice that in first inversion triads, the 3rd of the root position chord is now in the bass, the 3rd above the bass is the 5th of the root position chord, and the 6th above the bass is the root.

Second Inversion Triads

Second inversion triads are known as “6/4” because they have two pitch classes above the bass—one a 6th above the bass and the other a 4th above the bass. The 6th can be a di-chord or and the 4th can be perfect , augmented , or diminished .

The following table indicates the di-chords above the bass for each of the four first inversion (6/4) chord types. Notice that in second inversion triads, the 5th of the root position chord is now in the bass, the 4th above the bass is the root, and the 6th above the bass is the 3rd.

Open and Closed Positions

In any triad, there is a secondary di-chord that is formed by the combination of the 3rd and 5th. When the lowest note of the triad is the fundamental pitch (that is, the triad is in root position), the next-highest pitch is the 3rd, and the highest pitch is the 5th; we can find the secondary di-chord created by the 3rd and 5th by subtracting the primary di-chord number of the 3rd from the primary di-chord number of the 5th. In the case of a root position major triad voiced “root, 3rd, 5th,” we subtract , the primary di-chord number of the 3rd, from , the primary di-chord number of the 5th, and get , the secondary di-chord number created by the 3rd and 5th:

− =

This way of stacking the notes—root, 3rd, 5th—is known as closed position. In closed position, the major triad in root position contains the primary di-chords and and the secondary di-chord .

Now, what if the triad, still in root position, is voiced differently, so that the 3rd of the chord is a higher pitch than the 5th of the chord—from the bottom up, root, then 5th, then 3rd? This type of voicing is known as open position. Note that the distance between the top and bottom notes is more than an octave, whereas, in closed position, the distance between the top and bottom notes of the triad is less than an octave.

In open position, the secondary di-chord formed by the 3rd and 5th of the chord will be the inversion of the di-chord formed by the 3rd and 5th in closed position. Find the inversion of the secondary di-chord number is calculated from the number 12 (the number of notes in an octave):

[12] − =

Consider the doh major triad as an example. In closed position, the triad, spelled from the bottom up, is doh, mi, sol. The primary di-chords in this position are (doh to sol) and (doh to mi). The secondary di-chord between mi and sol is a .

In open position, the triad, spelled from the bottom up, is doh, sol, mi. The primary di-chords remain the same. However, the secondary di-chord we hear with this voicing is the 3rd from mi up to sol but this is the 6th from sol up to mi. Sol up to mi is a di-chord , confirming our calculation.

A Note on Figured Bass

In figured bass, a traditional harmonic notation system, numbers written below the staff refer to the generic interval or intervals that should be played above the bass. For example, a major triad in first inversion is indicated with the numbers “6,” indicating that the harmony contains a note a 6th and a 3rd above the lowest voice. The octaves in which the 6th and the 3rd will be placed are not specified with figured bass. The specific type of interval—for example, major or minor 3rd—is not specified but conforms to the key signature of the piece, unless otherwise indicated.

When looking at figured bass, it is helpful to know fluently which di-chords are possible for each of the figured bass numbers in the diatonic scale. Remember the basic di-chord correspondences found in the following table:

Exercises: Becoming Familiar with Triads and Inversions

Ch.22 — Four Functional 7th Chords

The Four Functional 7th Chords

There are four commonly employed 7th chords used in traditional functional harmony. Of these four, two are major-sounding and two are minor-sounding. Major sounding 7th chords may be found in minor heptachords, and vice versa. The two major-sounding 7th chords are the dominant 7th (a major triad with a minor seventh) and the minor 7th (a minor triad with a major seventh); the two minor-sounding 7th chords are the half-diminished 7th (a diminished triad with a minor seventh) and the fully-diminished 7th (a diminished triad with a diminished 7th). The half-diminished 7th is named thus because, while its triad is diminished, its seventh is not; the fully-diminished 7th chord has both a diminished triad and a diminished seventh.

Root Position 7th Chords

Each 7th chord includes four pitches:

1. the root
2. a pitch class that is a 3rd above the root
3. a pitch class that is a 5th above the root
4. a pitch class that is a 7th above the root

Because each 7th chord contains pitches that are a 3rd, 5th, and 7th above the bass, the root position 7th chord is known as a “7/5/3” or simply as a “7-chord.”

The four 7th chord types are distinguishable according to the specific di-chords formed with the root:

• The 7th may be either major [11], minor [10], or diminished [9]
• The 5th may be either perfect [7] or diminished [6]
• The 3rd may be either major [4] or minor [3]

The following table shows the three primary di-chords above and including the root for each of the four common 7th chords. For ease of understanding, the elements of the chords appear in numerical order from smallest to largest.

Note on the Fully-Diminished 7th

The fully-diminished 7th chord has the same di-chords [0,3,6,9] in all inversions because it is comprised of a [3] between the root and 3rd, the 3rd and 5th, and the 5th and 7th, each of these being a minor 3rd. In inversions of the fully-diminished 7th, [3] also exists between the 7th and the root above, but it is not a minor 3rd—it’s an enharmonic augmented 2nd.

First Inversion Seventh Chords

First inversion seventh chords are known as “6/5/3” or simply “6/5” because they have three pitch classes above the bass corresponding to the following three di-chords:

1. a 6th that may be either major [9] or minor [8]
2. a 5th that may be either perfect [7] or diminished [6]
3. a 3rd that may be either major [4] or minor [3]

The following table indicates the di-chords above the bass for each of the four first inversion (6/5) chord types. Notice that in 6/5 chords, the 3rd of the root position chord is now in the bass, the 3rd above the bass is the 5th of the root position chord, the 5th above the bass is the 7th of the root position chord and the 6th above the bass is the root.

Note on First Inversion — Identifying the Root

Notice also that the root is easily identifiable as the top note of two adjacent diatonic notes, one a 5th and the other a 6th above the bass—the note a 6th above the bass is the root. Though not easy to see in the case of the fully-diminished chord, the root (a 6th above the bass) is the top note of an augmented 2nd, even though it is a [3]. You cannot tell the inversion of a fully-diminished when hearing it in isolation.

Second Inversion 7th Chords

Second inversion 7th chords are called “6/4/3” or simply “4/3” because they have three pitch classes corresponding to the following di-chords above the bass:

• a 6th that may be either major [9] or minor [8]
• a 4th that may be either perfect [5] or augmented [6]
• a 3rd that may be either major [4] or minor [3]

In second inversion chords, using the rule that the root is the top note of the two adjacent pitch classes, the root of the 4/3 chord is the 4th above the bass.

Third Inversion 7th Chords

Third inversion 7th chords are called “6/4/2” or simply “4/2” because they have three pitch classes that correspond to the following di-chords above the bass:

• a 6th that may be either major [9] or minor [8]
• a 4th that may be either perfect [5] or augmented [6]
• a 2nd that may be either major [2] or augmented [3]

Using the rule that the root of a 7th chord is the top of two adjacent pitch classes, the root of the 4/2 chord is the 2nd above the bass.

Summary: 7th Chord Elements in Inversions

• The root is the bass in any root position 7th chord, the 6th above the bass in any 6/5, the 4th above the bass in any 4/3, and the 2nd above the bass in any 4/2.
• The 3rd of the root position chord is the bass in a 6/5, the 6th in a 4/3 and the 4th in a 4/2.
• The 5th of the root position chord becomes the 3rd in a 6/5, the bass in a 4/3, and the 6th in a 4/2.

Dominant 7th — All Inversions

The following tables identify the specific di-chords in each root position and inversion chord according to chord type. Di-chord numbers are shown from top down.

Minor 7th — All Inversions

Half-Diminished 7th — All Inversions

Fully-Diminished 7th — All Inversions

Identifying Roots of Chords

A common approach used by music students to visually identify the root of any chord in a score is to juggle its three or four notes around until they form a stack of thirds and then determine its inversion by whether the 3rd, 5th, or 7th is in the bass. Unfortunately, all this takes three, four or more seconds, making the strategy impractical in real time and quite useless in telling us about how the chord actually sounds.

A far more practical strategy for finding the root and inversion of a chord is to use the methodology suggested in thorough bass practice: identify the intervals above the bass. Doing this determines the primary di-chords that, in turn, determine the sound of the chord.

Here are seven basic rules that can be easily committed to memory.

Roots in Triads

1. The root in any root position (5/3) triad is the note that is the bottom of a 5th and/or the top of a 4th. Of the two primary di-chords, one is perfect and the other is modal; the secondary di-chord between the 3rd and 5th is modal.

2. The root in any first inversion triad (6/3) is the note a 6th above the bass. Both primary di-chords are modal; the secondary di-chord is perfect.

3. The root in any second inversion triad (6/4) is generally considered the 4th above the bass, however numerous respected theorists (Aldwell, Schachter, Plogger) posit that, while the preceding rule is true for most 6/4 chords, the root is the bass note in a cadential 6/4. Because a 5th above the bass has long been considered a “dissonance” in counterpoint and in four-part tonal harmony, the upper note of the 4th must be carefully prepared and resolved. This means that the 6/4 chords are employed less frequently than 5/3 and 6/3 chords. Most theorists concur that the root of the neighboring/plagal and passing 6/4 chords is the top of a 4th.

Roots in 7th Chords

4. The root in any root position 7th chord (7/5/3) is the bottom of the 7th and/or the top of a 2nd. All 7th chord primary di-chords have one that sets their sound apart from the simpler triads. In this case, the primary di-chords contain all three types of interference patterns: dissonant, perfect and modal.

5. The root in any first inversion 7th chord (6/5/3) is the note class a 6th higher than the bass. Notice that the root can be easily spotted as the upper of two neighboring note classes—notes a diatonic 2nd apart. Here, the note a 6th higher is a diatonic 2nd above the 5th and is, therefore, the root. Notice that the sound of the 6/5/3 will be more modal than the 7/5/3 because the former has two rather than only one primary modal di-chord, making the sound warmer and more emotional in flavor.

6. The root in any second inversion 7th chord (6/4/3) is the note class a 4th higher than the bass that is also the note class a diatonic 2nd above the 3rd. The harsh sound of the perfect and non-harmonic 4th allows us to distinguish it from the 6/5/3.

7. The root in any third inversion 7th chord (6/4/2) is the note a 2nd higher than the bass.

Exercises: Building Fluency with Functional 7th Chords

Ch.23 — Harmonization

Scale Degree Harmonization in the Major Heptachord

Triads and Scale Degrees

For each scale degree of the major heptachord, there is a triad that can be built on that scale degree. The chord qualities (major, minor, or diminished) of the triads built on each of the degrees of the scale will be consistent for all major scales. In this chapter, we will be exploring the qualities of the triads built on each degree of the major scale.

Exercises: Discovering the Triads of the Major Heptachord

Roman Numerals as Chord Symbols

In music theory, Roman numerals are conventionally used in harmonic analysis of tonal music because the numerals, which correspond to the scale degrees of the root, reflect the hierarchy and tendencies of the triads.

To indicate that the quality of a chord is major, its Roman numeral is capitalized; for example, I, IV, and V.

To indicate that the quality of a chord is minor, its Roman numeral is written in lowercase; for example, ii, iii, and vi.

To indicate that the quality of a chord is diminished, the lowercase version of the Roman numeral is followed by the “degree” symbol; for example, vii°.

To indicate that the quality of a chord is augmented, the uppercase version of the Roman numeral is followed by the “plus” symbol; for example, III+. Although there is no augmented triad that occurs naturally in the diatonic scale, augmented triads are still occasionally found in tonal music.

Functional 7th Chords in the Major Scale

In the major heptachord, there are five locations of functional 7th chords. There is a dominant 7th chord built on scale degree 5, which is written V7; there are three minor 7th chords built on scale degrees 2, 3, and 6, written ii7, iii7, and vi7, respectively; and there is a half-diminished triad on scale degree 7, written vii°7.

Exercises: Roman Numerals, Harmonizing, and Analysis

Ch.24 — Heptachord Shift

Real-Time Perception

One of the greatest challenges facing us as we become fluent in the tonal language concerns modulation. It is an overwhelming subject, rigid yet elusive, that intimidates even the stout-hearted. Yet, modulation need not be a daunting enterprise; it can actually be quite simple to understand and enjoyable to employ.

The method I use to teach modulation, as well as other aspects of musical perception, is what is known as real-time perception. Real-time perception means processing and making sense of sensory information at the speed at which that information occurs in everyday circumstances. When this concept is applied to music, it means being able to know what is happening in music at the moment it occurs during performance or listening. This is especially handy when you want to improvise because unless you become accustomed to knowing what is happening as it happens, your ability to improvise will be hindered.

Fortunately, getting a handle on real-time modulation requires only that you know your scales and that you memorize a few simple strategies.

Heptachord Shift

I call this technique of real-time modulation Heptachord Shift. Heptachord Shift is based on the fact that we perceive tonal music in structures of seven-note chords, or “heptachords.” There are three tonal heptachords:

1. major
2. melodic minor
3. harmonic minor

These three heptachords correspond to the major, melodic minor (ascending only), and harmonic minor scales. However, a scale is but one expression of a heptachord. The notes of a heptachord can appear in any order as long as the notes conform to one of the three scales. Further, it does not matter if the notes of a heptachord are expressed in the form of chords (vertical forms), lines (horizontal forms) or any other variation of the two.

What do heptachords have to do with modulation? Modulation occurs when one heptachord shifts or transmutes into another. Such a shift takes place at the moment one or more notes of a heptachord are chromatically altered. In other words, the moment the note C#, for instance, is introduced in the context of a G major heptachord, the heptachord instantly shifts to a D major heptachord. For some unknown reason, our ears dismiss or discard the note C at the very same moment that C# is introduced. You can prove this for yourself by first playing the notes in G major in any random, chaotic order and then throwing in the note C#. The C# will instantly sound like the leading tone of D major. The new heptachord is formed as the tonic shifts from the note G to the note D.

So, all you have to do is replace C with C#. Common sense. Right? The beauty is that you do not have to know anything about conventional harmonic analysis to enact the modulation. You do not have to memorize elaborate cadences, nor do you even need a V7 chord.

It is true that not all shifts are created equal. There are many ways that C# can be introduced, some which have a smoother and more seamless effect than others. These effects are quickly learned through experimentation; discovering the possibilities on your own is a lot of fun.

First Generation Heptachords

By using the principle of heptachord shift, you can move from one key/heptachord to another by changing only one note. A shift of heptachord initiated by changing one note is called a first generation heptachord shift.

First Generation Shifts from a Major Heptachord

When shifting from a major heptachord, you can move to one of five other heptachords by changing one note. The table below shows the scale degree that is altered, the scale degree it becomes in the new heptachord, and the new heptachord's relation to the original heptachord.

Summary of First Generation Shifts

In summary, if you were to sharpen scale degree 4 of a major heptachord, then the heptachord would shift to the dominant [V] of the original heptachord, with the sharpened fourth degree now functioning as the seventh degree.

In Heptachord Shift analysis, an upward pointing triangle, △, denotes the melodic minor heptachord, while a downward pointing triangle, ▽, denotes the harmonic minor heptachord. So, if you were to sharpen scale degree 5 of a major heptachord, then the heptachord would shift to the relative harmonic minor [vi▽] of the original heptachord with the sharpened fifth degree now functioning as the seventh degree.

For a thorough test drive, let's use C major as our initial major heptachord. We get to G major [V] by raising 4, so we raise F to F#; we get to F major [IV] by lowering 7, so we lower B to B♭; we get to A harmonic minor [vi▽] by raising 5, so we raise G to G#; we get to D melodic minor [ii△] by raising 1, so we raise C to C#; and, we change C major to C melodic minor [i△] by lowering 3, so we lower E to E♭.

Chapter Twenty-Five: Transposition

Introduction

By changing clef on the staff, and by adjusting the key signature, you can learn to transpose notes at sight. This skill is especially important when reading or writing musical scores that employ “transposing” instruments, such as clarinets in B♭ or A, saxophones in B♭ or E♭, the French or English horn in F, the trumpet in D, etc.

Using Clefs to Transpose

If you are fluent in using all eight clefs, you can transpose at sight by simply imagining that the music is written in a different clef with an alteration to the key signature. This may sound complicated, but it is a much more efficient way to transpose than thinking up or down an interval for every written note. For example, imagine that you are given the melody of Louis Bourgeois’ hymn tune Old Hundredth written in G major using the treble clef.

But, you are asked to play the melody up a minor 3rd, in B♭ major; how would you do this? You would simply need to read the music as if it were written in the bass clef with two flats in the key signature (add three flats to the original key signature of G major).

Notice that the notes of the melody are written on the same lines and spaces in both examples! In essence, to transpose any music written in the treble clef up by a minor 3rd, read it in the bass clef and add three flats to the original key signature.

So, there is a particular clef and key signature alteration that will facilitate transposition by each interval or di-chord. The following table shows the clefs and key alterations needed to transpose from the treble clef up or down a di-chord [1], [2], [3], [4], [5], or [6]. These are the only di-chords of transposition you need to know because to transpose by other di-chords you can simply use their inversion: transposing up [II] is the same, in terms of note class, as transposing down [1], and transposing up [7] is the same, in terms of note class, as transposing down [5], and so on.

Transposition Table: Clef and Key Signature

Notice that the number of accidentals in the key signature alteration is the same when transposing the same di-chord up and down. For instance, when transposing up a major 2nd, add two sharps, and when transposing down a major 2nd, add two flats. Notice also that the type of accidental is the opposite—flat versus sharp—when transposing the same di-chord up and down. Further, the sum of accidental alteration for each diatonic interval transposition in a particular direction is always equal to seven: two accidentals are required to go up a [2], while five are required to go up a [1], and two plus five equals seven.

Note on the Transposition Table

Be aware that the previous table only applies when transposing from the treble clef. If you are a keyboardist or an instrumentalist who reads from the bass clef, you should use the same principle to make your own table for transposing from the bass clef (or any other clef, for that matter).

Transposed Instruments

Many wind and brass instruments are transposing, meaning that the notes written on the musical staff do not indicate the actual sounding pitch. The pitches played on wind and brass instruments are built upon the overtone series of their most basic or core pitch, which is determined by each instrument’s size and shape.

To understand transposing instruments, we can begin by realizing that the core pitch/key is indicated in the name of the instrument. The B♭ clarinet has, as its most essential key, the key of concert B♭; an A clarinet has, as its most essential key, the key of A; etc. Yet, for simplicity’s sake, music written in the core key of these instruments is written to appear, on the staff, to be in C major. For example, when a B♭ clarinetist plays a written C, they are sounding a concert B♭. Similarly, a horn in F has the key of F as its most elemental or essential key; therefore, this key is written as C. Understanding the relationship between the sounding concert pitch and the note indicated on the staff is of the utmost importance if we are to know the concert pitch:

• On a B♭ instrument, the written note class is a whole tone higher than sounding.
• On a D instrument, the written note class is a whole tone lower than sounding.
• On an A instrument, the written note class is a minor 3rd higher or a major 6th lower than sounding.
• On an E♭ instrument, the written note class is a minor 3rd lower or a major 6th higher than sounding.
• On an F instrument, the written note class is a perfect 4th lower, or a perfect 5th higher than sounding.
• On a G instrument, the written note class is a perfect 5th lower, or a perfect 4th higher than sounding.

Becoming Fluent with the Clefs

Musicians, even professionals, often assume that mastering clefs and transposition is unnecessary. However, those who master these concepts become the most literate and intellectually agile members of the musical community, being able to move between musical genres, comprehend complex musical scores and, more importantly, to grasp timbre differences between keys. Nadia Boulanger could very easily test my musical understanding of a composition by asking me to transpose it on the spot. If I had learned a piece by motor memory alone, I was unable to transpose, because playing it in another key felt completely different in my hands, disrupting my motor memory. However, if I had grasped the harmonic and melodic meaning of the piece, I could transpose to a different key with considerably less effort, and with far more musicality. Try transposing as much as you can, always remembering that, although it might seem extremely challenging, you will find the exercise invaluable.

Exercises: Transposing Using Clefs

Appendix B — The Overtone/Harmonic Series

The physical basis for why di-chord harmonicity works. Harmonic di-chords are harmonic because their upper notes appear early in the overtone series of the lower note — they are “already in the family.”

First 7 Partials

Harmonic Ratios from the Series

Ratio of partial 5 to partial 3 = 5:3 = major 6th .

What is NOT in the Overtone Series

• Perfect 4th — Does NOT exist naturally. Artificially derived as the distance from the 5th to the Super Octave (3rd to 4th partial), ratio 4:3. This is why is non-harmonic despite being called “perfect.”
• Minor 3rd — NOT in series → non-harmonic ✓
• Minor 6th — NOT in series → non-harmonic ✓
• Augmented 4th — IS in the series but sounds flat relative to equal temperament.

Terminology

“Harmonics” = frequency ratios. “Overtones” = divisions of a vibrating string or air column. “Partials” = most precise term (includes the fundamental itself).

Appendix C — The Ploger Heptachord Shift House Plan

A conceptual floor plan with 14 rooms representing all heptachords through the second generation. The original heptachord (UR) = C major sits at the structural center.

14 Rooms (Using C Major as UR)

First-generation rooms: G major [V], F major [IV], A harmonic minor [vi▽], D melodic minor [ii△], C melodic minor [i△].

Second-generation rooms: D major, B♭ major, g melodic minor, d harmonic minor, f melodic minor, e harmonic minor, a melodic minor, E♭ major, and others derived from first-generation shifts.

How to Use the House Plan

Each arrow between rooms shows the triggering chromatic alteration. Double-headed arrows indicate reversible shifts. Bach typically wrote within these 14 rooms.

Essential Plogger Exercises — Practice Plan

The following exercises are the highest-leverage items across the full three-book system. Sprint eligibility is noted for AMF curriculum sequencing.

1. Keyboard Visualization — Eyes-closed note naming (Ch.3). Foundational; begin Sprint 1+. The prerequisite for all other work.
2. Longy Rhythms — Clap + speak rhythmic patterns (Ch.4). Foundational; begin Sprint 1+.
3. Lap Map Drills — Solfège + di-chord mapping on lap (Ch.5). Begin Sprint 3+.
4. Boulanger/Ploger Chord Exercise — Name all notes and di-chords in every inversion across all 12 pitches (Ch.21–22). Begin Sprint 9+.
5. Tracking Page Protocol (Ch.16) — Steps 1–2 from Sprint 1; full 7-step integration by Sprint 12.
6. Conjunct Mode Modulation — Single-finger heptachord modulation at keyboard (Ch.20, Ex.20-6). Begin Sprint 9+.
7. Bach Chorale Heptachord Analysis — 6-Step Protocol applied to Bach chorales (Ch.24). Begin Sprint 11+.
8. 10 Chord Sequences in All Keys — Soprano scale degrees + Roman numerals in all 12 keys (Ch.23, Ex.23-5). Begin Sprint 10+.