A LIGHTWEIGHT NOTATION for PLAYING PIANO for FUN by R.J.A

Home , Augmented fifth, Eleventh, Major scale, Minor scale, Musical prefix, Tritone

A LIGHTWEIGHT NOTATION FOR PLAYING PIANO FOR FUN by R.J.A. Buhr 1. INTRODUCTION Music notation gets in the way of the joy of making music on the piano that can be had without first spending thousands of hours at the keyboard to develop piano “wizardry,” meaning a combination of keyboard skill and mastery of music notation. The message of this document is the joy can be had, without any loss of musical substance, with no more wizardry than is required to read and play single melody lines. The vehicle is a lightweight notation I call PKP (for Picturing Keyboard Patterns) for annotating the core flow of harmony in interval terms above the staff. I developed the notation as an adult beginner with no training in music theory after becoming frustrated with the way music notation obscures the keyboard simplicity of harmony. That the notation has substance in spite of this humble origin is indicated by the following opinion of PKP provided by a music theorist specializing in improvisational music (see supplementary pages): “The hook ... , at least in my opinion, is that it's possible to attain a deep understanding of chords (and their constituent intervals) without recourse to Western notation. This has direct consequences for physical patterning, fingerings, etc. Essentially, your method combines the utility of a play-by-ear approach with the depth of a mathematically-informed theory of music.” PKP follows from two observations that anyone can make. One is the piano keyboard supplies only 12 piano keys to play all the notes of an octave that music notation describes in many more than 12 ways by key signatures and accidentals (sharps, flats, naturals). This presents pianists with a 12-half- tone chromatic scale in which the intervals that provide what music is to the ears are directly visibly to the eyes as musical objects defined by size measured in half tones instead of by pairs of note symbols expressed in key-signature notation. The other observation is chord symbols describe chords without reference to context (melody line, other chords) in a way that obscures the often beautifully integrated flow of melody and harmony.1 A sufficient representation that captures this flow is a harmonic core formed of a sequence of two kinds of building blocks, which are keyboard intervals (measured in half tones) chosen to represent chords in context. Building blocks are intervals that split octaves in half in two ways. The first kind consists of fifths (7 half tones) and fourths (5 half tones) that split octaves into harmonious pitch halves. The second kind consists of tritones (6 half tones) that split octaves into dissonant keyboard halves midway in size between fifths and fourths. The closeness in size of these building blocks means that sequences of them establish smooth flow that cues what comes next. This works because chords described by chord symbols are formed either of single building blocks with a middle note (triad chords) or overlapped combinations of 2 or 3 building blocks (seventh or sixth chords). A melody line plus a harmonic core consisting of a sequence of primary building blocks captures the essence of the most sophisticated features of the harmony. It also identifies context in the form of tonic scales in play, enabling missing details to be filled in by eye and ear. The PKP notation for annotating this above the staff is easy to extract from music notation and is helpful for learning new pieces, all the way to guiding improvisations based on them.

1 The following ugly chord progression obscures the beautifully integrated ﬂow of melody & harmony of a haunting minor blues with a simple, mainly-black-key melody line (see Chapter 3): E♭7♯9—B9(13)—EM9—A7♯11—D♭9sus—B9(13)—D♭7sus—E♭7—A♭m11— B7(13)—Fm7♭5—B♭7♯5♯9—C13♯11—F7(13)—B7—EM7—A7(13)—A♭7—B♭7—D♭7—E♭7♯9—B7—EM7♯11—A7♯11. www.pianotheoryman.com

2. CONCEPTS & NOTATION I was puzzled as a youth, when I picked up playing the trumpet in school bands, by the heavyweight notation required for music with tonic changes that could be effortlessly handled by ear by anyone when humming, whistling or singing a tune. I wondered why music could not be written down as simply as it is heard, but accepted that it could not as one of the mysteries of life and got on with the business of learning to read the notation as best I could. The puzzlement resurfaced when I took up the piano as an adult beginner. This time I decided not to ignore it. Music notation is here to stay and must be dealt with as it is by anyone aspiring to be a piano wizard but I wasn’t aspiring to that — I only wanted to play for fun — and I was not convinced that such heavyweight notation was necessary for the purpose. I was motivated to play harmonically sophisticated pieces from the “jazz and standards” repertoire that are available in numerous fake books. I had in mind pieces by the likes of, in no particular order, Bill Evans, Duke Ellington, Billy Strayhorn, Michel Legrand, Charles Mingus, George Gershwin, Thelonious Monk, John Coltrane and many more. Beyond specific pieces, I also wanted to learn how to play blues, which is often complex in music notation because its scales and chords tend to be strongly chromatic (an example is footnoted in the previous chapter). I once heard a jazz musician say in a radio interview that learning to play the blues before learning music notation had prepared him for anything music notation later threw at him. I wanted to focus practicing on pieces I loved instead of on developing general sight-reading and pianistic skills needed to become a piano wizard. I thought that a suitably lightweight notation to guide such practicing might be possible in terms of intervals instead of the notes of key-signature notation, and so it turned out to be. The concepts are very simple but be warned that they go strongly against conventional wisdom. They are simple if you shut music notation out of the mind until they are grasped. This is not to suggest discarding music notation. PKP is an aid to understanding it, not a replacement for it. No one is better qualified to shut music notation out of the mind than a skeptical adult beginner not yet versed in it. Skepticism followed from the visible mismatch between chord-progression-ugliness of the kind footnoted in the opening chapter and the simple beauty of the music it represents. Looking for a lightweight notation became an absorbing retirement hobby. A LIGHTWEIGHT NOTATION BASED ON INTERVALS Key signatures are the basis of written music, so a lightweight notation based on intervals must start with the scales of key signatures if it is to be helpful in understanding written music. The chromatic scale of a tonic octave on the piano provides the basis for this. Chromatic Scale Any tonic scale on the piano can be understood in terms of the intervals of the chromatic scale of the tonic octave represented by a line marked off in equal intervals representing half tones, as shown next. Every proper tonic scale is framed by the tonic notes an octave apart and the pitch center of the octave, a fifth above the bottom (a fourth below the top).

pitch tonic chromatic scale center tonic

@ $ @ <== tonic scale "frame"

An unadorned divided line of the form shown above, displayed horizontally or vertically on the

PKP 3/9/16 —2— ©copyright R.J.A. Buhr www.pianotheoryman.com page will serve from now on as a representation of the chromatic scale of a designated tonic octave on the piano, understanding that the thicker markers identify the asymmetric divisions of the frame. Intervals have the same number of half tones whether they are understood in terms of this picture of the piano’s chromatic scale, or in terms of pairs of notes of music notation.2 The difference with music notation is not in the number of half tones but in their individual sizes in pitch terms. The half tones of music notation are nominally slightly different pitch sizes for overlapped octaves. Piano keys would need adjustable pitches to play the differences, and they don’t have them. One octave of the piano keyboard can be tuned to the musical perfection implied by music notation but this would make octaves overlapping it sound different. Instead, equal temperament tuning is used to make overlapping octaves sound the same, making them all nominally slightly out of tune. The book How Equal Temperament Tuning Ruined Harmony strongly expresses the opinion of its title, and the book Lies My Music Teacher Told Me explains that singers in choirs trained to sing the slightly different pitches by ear can find singing with the piano slightly uncomfortable. Opinions and comfort aside, the piano is the way it is and delivers intervals that are musically accurate in terms of sizes measured in half tones. This enables PKP to dispense with sharps and flats. Seasoned pianists tend to think they are needed as indicators of the musical functions of notes because they use them for that, but the fact is that the positions of notes relative to tonics are sufficient. Why not take advantage of this simplicity? The very thought goes strongly against musical tradition and conventional wisdom. I have been accused by piano wizards of being a musical know-nothing for even suggesting it. Read on and judge for yourself. Scales of Key Signatures The scales of key signatures are defined in interval terms, independently of music notation, by classical modes (scales in which interval sequences are defined by rotations of a single scale to start on different notes are called modes). The rotations are relative modes (same notes, different tonics). PKP is based on parallel modes (same tonics) with the same interval sequences as the relative modes. The prominent parallel modes of key signatures are the Ionian and Ionian modes that define the major and natural minor scales of key signatures.

tritone fifth major scale (Ionian mode) "I" core anchors "A" natural minor scale (Aeolian mode) fourth tritone

Each mode has a pair of core building blocks of the two different kinds that share a common anchor note. Knowing this anchor means knowing the mode by construction in interval terms on the keyboard. This is different in kind from knowing it by enumeration of the notes of a key signature. The

2 For example, the following are different representations in music notation of a fifth of size 7 half tones between the same two piano keys: (B-F♯), (C♭-F♯), (B-G♭), (C♭-G♭), (B ♯♮-F♯), (B ♯♮-G♭), (B♭♮-F♯) or (B♭♮-G♭); and this doesn’t even include double sharps or flats.

PKP 3/9/16 —3— ©copyright R.J.A. Buhr www.pianotheoryman.com by-construction identification is as follows. The scale frame provides the tonic and pitch center, the anchor identifies the core pair of building blocks, the pair identifies a half tone that must be balanced by a symmetrically disposed half tone relative to the tritone (both half tones are visibly inside or outside the tritone) and what’s left are obvious whole tones. The PKP Alphabet The letters A and I identifying the anchors are from the alphabet PADMIL. The alphabet is formed of the first letters of the names of the classical modes Phrygian, Aeolian, Dorian, Mixolydian, Ionian and Lydian, arranged in order of anchor position going up the keyboard, from a half tone above the tonic to a half tone below the pitch center. The letters are in a special boldfaced font to distinguish them from note symbols (for example, “A”) or RN symbols for chord roots (for example, “I”). There is never any possibility of confusion because the notations are never used together but the special font provides a helpful reminder. The same purpose is achieved in handwritten annotations above the staff by outlining the letters by boxes for fifos (fifths or fourths) and circles for tritones. A seventh classical mode, Locrian, is not represented because it is not needed (its tritone/fifo pair is the inversion of the Lydian pair) and it is different in kind because it cannot be a proper tonic scale (it does not include the pitch center of its scale octave). The PAD and MIL modes have, respectively, minor and major tonality.

pitch tonic chromatic scale center tonic

@ P A D M I L $ @ ⇐ frame alphabet

Identifying building blocks by anchors associated with classical modes does not restrict the building blocks to the modes, which enables tonic scales of any kind to be identified by the alphabet letters and words. Outside of classical modes, the only anchors with specific tonality are visibly D (minor) and M (major). The tritone is unique to a each mode but the fifos are shared with other modes to different degrees. The mode fifos are identified by anchor letters, as illustrated next for the Ionian mode. The PDL set of anchors is not in this mode but the AMI set is: I anchors the core tritone/fifo combination and A and M anchor only fifos. The scale has a total of 6 fifos, which means 4 fifths and 2 fourths within the tonic octave.

frame fifo building blocks 4 of the Ionian fifos mode are covered core by the universal frame plus 3 anchors

@ A M I $ @

A Dramatic Simplification The identification of building blocks by anchor letters from 6-letter alphabet provides a dramatic simplification of music without losing any substance.

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A single alphabet letter identifies a classical mode relative to a tonic as definitely as a key signature and a tonic identify it in music notation. The singularity of core tritone/fifo combinations with same anchor in classical modes (identified by single letters) transforms into the singularity of core tritone combinations with different anchors (identified by multi-letter words) when classical modes are altered by adding, sharping or flatting notes. A small number of words are ambiguous by themselves but the ambiguity is always resolved by knowing whether tonality is major or minor. In all cases, the difference is between construction relative to a tonic on the keyboard and enumeration of all notes in music notation. This difference dramatically simplifies dealing with tonic changes, which amount to repositioning a conceptual tonic pointer within the chromatic scale of the main tonic of a piece. The building blocks remain the same on the keyboard, only their functions change relative to different tonics. Pentatonic scales that are the foundation of much simple music worldwide are identified in a negative sense by the same single letters as classical modes — they are classical modes without their half tones (which means without their tritones). They are covered in the next section. Building-block composition of chords is identified independently of inversions, dramatically simplifying the representation of chords. Inversions are left unspecified, to be decided at the keyboard from local context. When there are only two kinds of building blocks identified by anchors and one is the same size in either inversion, this is very simple. The restriction to building blocks of only two different kinds is valid because almost all chords from any tonic scale are formed either by adding a middle note to a building block (triad chords) or combining two or three building blocks to form 4-part, 5-part or 6-part voicings of seventh or sixth chords (5-part voicings follow from fifos of different sizes sharing a note). Combinations introduce smaller intervals between chord notes (or their larger inversions) but the substance is combinations of building blocks. Anchors are needed for building blocks only to specify harmonic cores, which always include all the tritones in the harmony because of their importance for tonic scale identification. Enrichment of harmonic cores is provided by fifos that are relatively freely adjustable without altering harmonic substance. The lack of anchors in the upper pitch half of the tonic octave may seem awkward but is completely natural on the keyboard, as illustrated below by a progression of fifths going up by scale steps in a way that seems to, but does not, require anchors in the upper pitch half. In an anchor line, the symbol > or a sequence of them following an anchor means slide the anchored interval to different positions, as appropriate for the context (the meaning here is slide it to two successive Ionian scale positions).

notated as I > > understood in terms of treble notes of fifths @ I $ @ A M $ moving between anchors in the next octave

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CAREFREE FUN WITH THE PIANO Pentatonic Scales The most carefree fun with the piano can be had with music from simple pentatonic scales, exemplified by the black keys on the piano. Anything played on the black keys is fundamentally harmonious because the scales contain no half tones or tritones. Musical lines from these scales are easy for anyone with a musical ear to hum, whistle or sing. This includes not only melody lines but also harmony lines. Pentatonic major scales played on the black keys start on the first black key of the cluster of 3 black keys; relative pentatonic minor scales (same black keys) start one black key down from this. The intervals between successive scale notes are whole tones (2 half tones) or minor thirds (3 half tones). Although pentatonic sales contain no half tones, their intervals are measured in half tones for two reasons: one is the fact that a minor third is one and half times larger than a whole tone, requiring a common unit that is half the size of a whole tone; the other is tonic changes or parallel major/minor changes (same tonic) require half tones. Parallel pentatonic major scales are shown below (transposing the minor scale down a minor third yields a relative scale with the same notes).

pentatonic major scale

parallel pentatonic minor scale

These are visibly the Ionian and Aeolian modes without their tritone notes. This leaves pentatonic scale with only 4 building blocks, all of which are fifos.

4 fifo building blocks

pentatonic major scale

A M

Anyone Can Play Basic Blues Music from straight pentatonic scales lacks the tension between dissonance and consonance that makes much music interesting. However, it leads directly to harmonically interesting basic blues in which melody is mainly straight pentatonic that anyone can play, with simple 3-chord harmony based on two tritones that result from combining the pentatonic scales (shown below) plus the Ionian tritone that comes with the major scale of any key signature. This is simple but can quickly go deep into strongly chromatic territory in which music notation tends to become complex (for example, the footnoted chord progression of the the opening chapter). Think of the result shown below as following from singers sometimes “bending” major notes downwards to minor notes to get a characteristic happy-sad sound (equating major with “happy” and minor with “sad”). The resulting scale has 4 half tones, the notes of which provide 2 harmony tritones anchored at the circled chromatic scale positions. The mixed minor-major character of the blues is visibly obvious from the tritone anchors, which are the fundamental notes that determine minor (D) and major (M) tonality.

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added notes form tritones

8-note blues combination pentatonic minor scale pentatonic major scale

D M

Shown below is the 12-bar form of a simple, 3-chord blues piece called Backstreet Blues that uses this scale and chords with a couple of added wrinkles. I learned it in a blues workshop course as typical of half the blues pieces played by pop and jazz musicians. In it, mip indicates a descending melody line in the minor pentatonic scale and I and IV are the roots of the dominant-7 chords based on the two tritones of the combined scale. The wrinkles are an added tritone anchored by I in bar 9 (and an associated dominant-7 chord on root V), and an added passing note in bar 10 that changes the minor pentatonic scale into a minor blues scale indicated by mip+.

chord roots I IV I I IV IV I I V IV I I anchor line Ⓜ Ⓓ Ⓜ Ⓜ Ⓓ Ⓓ Ⓜ Ⓜ Ⓘ Ⓓ Ⓜ Ⓜ

melody | mip | mip | — | — | mip | mip | — | — | mip | mip+ | — | — |

The details are shown below. This picture is presented only for explanation, not to suggest such pictures have to be drawn to use PKP (they do not). There are many important things to take away from this picture, outlined following it.

o melody line is pentatonic minor, except bar 10 o o o o o - melody o - tritone o o o o oo o o o - fifo o L o o o o o o o o o o I I I I I o I I o o o o o o o o o o o o o o o o o o o o o o o o o o o V o o o o IV IV IV o o o IV o o o o o o o o o o o o I o o o M M o o o M D D D M M D M M

The picture is musically accurate in interval terms both on the keyboard and in relation to music notation. Neither staff lines and spaces nor sharps and flats of music notation are needed for musical accuracy in terms of intervals, or for understanding context (positions of notes relative to tonics are sufficient for that). Blues is harmonically rich but simple on the keyboard. The simplicity follows from the simplicity of pentatonic scales and tritones on the keyboard. The mostly pentatonic melody lines are

PKP 3/9/16 —7— ©copyright R.J.A. Buhr www.pianotheoryman.com simple in any notation. The harmonic core consists of tritones sliding by half tones or holding position from bar to bar, with the sole exception of one slide by a whole tone from bars 9-10. The tritones emphasize the mixed major-minor nature of blues. The chord roots and the harmonic core form 3-note voicings of the dominant-7 chords that are easily completed from the simple scales. The simplicity of simple blues in music notation is deceptive. There are only three chords and all are of the same kind, namely dominant-7. The V chord is from the major scale of the written key signature and so comes into play naturally. The I and IV chords are outside the written key signature but their origin in the combined pentatonic major and minor scales makes them easy to understand and remember. The dominant-7 chord on root V is general purpose chord in many kinds of music that signals upcoming melodic resolution. The I-IV-I-I, IV-IV-I-I and V-IV-I-I root sequences are simple — each opens with a different root and closes with the same two roots. The simplicity in music notation is deceptive because the effective blues scale is a 10-note scale identified by the word DMIL. This opens the possibility of strongly chromatic melody and harmony that is fully exploited by many blues pieces. Chord symbols become complex because they have to be adjusted for different chord scales, which is like “banging square pegs into round holes” — the results are messy (for example, the footnoted chord progression in the opening chapter). Opposite inversions of building blocks are harmonically equivalent relative to an established tonic, therefore any chord progression may be transformed by selective inversions into a harmonically equivalent version of it with all its building blocks inside the octave of the main tonic of a piece (as shown above). Given that this inversion is harmonically equivalent, it is arguably a more fundamental view of the chord progression than is provided by the chord symbols. The PKP concept is that a chord is the set of building blocks of which it is composed, and inversions are different voicings of it. The visible structure of chords as combinations of building blocks is general. These chords are combinations of two building blocks of different kinds but they illustrate the nature of the general case.3 The essence of the flow of harmony within a tonic octave is captured by a harmonic core in which building blocks slide or morph into other building blocks. In a core consisting only of tritones, the tritones slide into each other, but morphing enters the scene when the core includes fifos. A harmonic core is summarized by an anchor line that replaces a chord root line as the fundamental defining line of harmony. The example core consists only of tritones, which an anchor line identifies unambiguously, but anchor lines in general may contain fifo anchors, which are ambiguous by themselves but generally unambiguous in context of a melody line and the rest of the harmony. Chord roots are visibly secondary elements in the flow of the music because inversions move them into the body of the chord, where they may be omitted if they are implied by context (for example, the V root is often omitted because it is the pitch center of the tonic octave). Tritones of strongly chromatic harmony such as the blues provide so much harmonic variety that enrichment requires only adding depth. Depth is easily added by doubling the treble line of the

3 Here are a few examples of chords. Two fifos form major-7, minor-7 or major-6 chords. A fifo and a tritone form dominant-7, half-diminished-7 (a.k.a. minor-7-♭5) or minor-6 chords). Three fifos or two fifos and a tritone form 9th, 1lth and 13th extensions of these chords. When tonic scales depart from classical modes, the same building blocks are available from the scales but the number of tritones is now more than one. Two tritones form diminished-7 or dominant-7-♭5 chords. Two tritones and one fifo (that may share a note with a tritone) form variations of other chord types such as dominant-7- ♭9 . This is only the tip of the chord iceberg (see Appendix B for more)

PKP 3/9/16 —8— ©copyright R.J.A. Buhr www.pianotheoryman.com harmonic core an octave down, or the bass line an octave up, whichever suits the piece. The octaves provided by the doubling moderate the tritone dissonance. If the core includes fifos, the doubling also applies to them. The resulting 3-note shapes are very easy for anyone to play. They provide an initial pass at harmony that is easily modified by adjusting the doubled line by eye and ear to nearby scale notes. For example, the bass line provided by doubling the core treble line in the above picture an octave down could be morphed into a line that holds the tonic for all bars except bar 9, in which it would be raised a scale step. No more piano wizardry is required to have fun playing the piano than the ability to sight read melody lines in music notation. Anyone with eyes to see intervals on the keyboard can infer harmonic cores from anchor lines annotated above the staff. Anchors are derived by a simple 2-pass analysis process of a piece of written music: the first pass finds tritone anchors from chord symbols using a table in Appendix B; the second pass fills in core fifo anchors from tonic scales understood from tritone content. Chord voicings with 3 different notes follow by adding roots of written chords to the core or by the doubling-and-adjusting process just described. More depth is easily added by obvious scale notes within the core building blocks. The deep context provided by a melody line plus anchor line provides many benefits. It guides learning new pieces, remembering pieces not played for a while, recovering from mistakes, tracking harmonic changes, distinguishing momentary ornamentation from substantive changes, and playing variations and improvisations. The context is understood independently of music notation. The notation is independent of music notation; it does not conflict with it, add to it, or replace it; it translates directly and simply into it, thus helping to learn it. REVISITING CLASSICAL MODES Ironically, dealing with harmony from classical modes in terms of anchors looks more difficult than for the simple 3-chord blues just described because of the need to deal with fifo anchor ambiguity for many possible fifos. It is not more difficult, but understanding this requires getting used to the idea that fifos to enrich harmonic cores are relatively freely interchangeable to fit context, without reference to overly precise chord symbols. The examples shown below give a sense of this.

chord progression for assigned root line is chord progression for assigned root line IIm7 - V7 - IM7 is IVM7 - VIIm7♭5 - IIIm7 o o o one of VIIo o many o core o possible o o V o o enrichment o o o variations IVo o anchor line o for the IIIo same core o II o o o o bass line 2 scale I treble line 2 scale steps up steps down o shown an octave down

In these examples, the core is a downward-trending fifth-tritone-fifth sequence in which the opening fifth and tritone are the core combination of this mode and the closing fifth could also be a fourth, depending on the melody line. The number and variety of misleadingly different chord progressions that can be formed around this core is startling. The differences amount to small adjustments of fifos chosen to enrich the core.

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The illustrated shapes begin with an enrichment line two scale steps below the core (on the left) or above it (on the right). These are good choices because two scale steps is a basic chord interval. The enrichment lines suggest scale notes to add within the core to provide well balanced shapes. On the left, the shapes are visibly well balanced. On the right they are well balanced for the enrichment line as a treble line. The root lines are assigned after the shapes are determined. The chord progression on root line II-V-I on the left is one of the most fundamental chord progressions of music. Harmonically equivalent variations (same core) can look very different, as illustrated on the right. These are far from the only possible choices of root lines for these examples (Appendix B provides more on this subject). ALPHABET WORDS AS GENERAL SCALE IDENTIFIERS I started out to find a notation to annotate harmonic cores above the staff and ended by fortunate happenstance with a notation that identifies all the tonic scales that the piano can play in very simple terms, summarized by the following table.

master words identified scales parallel words | PADMIL 12-note chromatic P.DM.L, PA.MI, AD.IL 8-note diminished (atonal) | P.D.I, A.M.L 6-note whole tone (atonal) ————————————————————————————————————————————————— DMIL, ADM.L 10-note blues | DM.L 9-note generic blues | AD.I, A.MI 8-note generic minor/major | D.I 7-note melodic minor P.D, A.M, D.I, M.L, I.P, L.A * | A..I 7-note harmonic minor/major P..M, A..I, D..L ** | I 7-note Ionian mode P, A, D, M, i, L * - 5-note pentatonic

** identify 4 scales each with tonics a minor third apart * identify 2 scales each with tonics a tritone apart

Alphabet words describe tonic scales by construction on the keyboard, without reference to note symbols in any notation. The vertical bars on the left identify the main words to remember because the others are related to them in obvious ways. The (optional) dots in the multi-letter words indicate skipped alphabet letters that bring the form of the words forward to the eye at a glance. The meaning of “parallel words” on the right is words of the same form in different alphabet positions that identify parallel modes, which are transposed scales that include the main tonic, rotated to start on it. Parallel words of more than two letters are not shown because their scales tend to appear only as actual tonic changes and not as parallel modes. The table covers scales with tritone content ranging from nil (pentatonic scales) to six (the full chromatic scale). Identifying scales from tritone content is simpler higher up in the table because tritones are unambiguously identified by anchors and more tritones provide more scale notes. For scales with more than one tritone, the tritone plus scale frame provide from 6 to 10 notes. The notes cover the irregular parts of scales, leaving at most a whole tone or two to complete a scale. At the bottom end of the table, pentatonic scales are identified by tritone content in a negative sense — they are classical modes without their tritone notes.

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Not all the possible scales that may be identified by alphabet words are covered by this table, but the coverage is very wide and the identification by words is open ended. The table covers all the usual scales I have encountered in scale dictionaries such as The Source, piano theory books such as The Jazz Theory Book and Modalogy, and a wide variety of pieces of music. Many of the scales have finicky detail that is difficult to keep straight, and exotically complex names that are often ambiguous and generally difficult to remember. Knowing them by alphabet words is simple and unambiguous (see Appendix C for more on scales). The wide coverage of such a simple table seems implausible when stated baldly by itself. The idea of identifying scales this way would be unlikely to occur to anyone versed in music notation because the multiplicity of ways of representing the same interval get in the way. It is an easily observable fact if you stand back from music notation to look at how things actually are in interval terms on the keyboard. The words by themselves mostly self identify major vs. minor tonality because anchors D and M are definitive minor and major notes. DM self identifies blues as of mixed tonality, which is the source of its characteristic happy-sad sound. D self identifies the minor tonality of the 8-note generic minor scale (which is known in jazz by the awkward name “bebop melodic minor”) and the 7-note melodic minor scale. There is nothing inherently major or minor about P, A, I, or L by themselves because any of them can be in a scale of either tonality. The means that A..I identifies a harmonic minor scale or major scale, depending on context. The companion major scale of the 8-note generic minor scale is a harmonic major scale with an added major note in its minor-third interval (known in jazz as “bebop major”); the added note is harmonically neutral in a major context, so for all practical purposes, the harmonic major scale is the bebop major scale (the extra note can be added or not without any substantive effect on the nature of the harmony). For classical modes, D and M visibly identify minor tonality for the Dorian mode and major tonality for the Mixolydian mode, but otherwise minor vs. major tonality emerges by construction. The same words that identify the scales provide building blocks directly without having to go to the enumerated scales. TONIC CHANGES A powerful feature of PKP’s way of knowing tonic scales is the simplicity with which it represents tonic changes, illustrated below for the most common same-mode tonic changes, namely by fifths or a whole tones. The tonic octaves are shown offset for clarity but their overlapped parts actually coincide on the keyboard. A same-mode tonic change boils down to a tritone slide that changes to a parallel mode combined with a tonic pointer change that alters no notes. Any tritone on the keyboard is compatible with two modes of the same kind with tonics tritone apart. An established mode has an established tritone but a tritone slide opens up two possible destination modes. The most common choice is the one that alters the fewest notes. For half tone tritone slides, the possible tonic changes are a half tone or a it plus a tritone (a fifth, which inverts into a fourth) and a fifth alters the fewest notes. For whole-tone tritone slides, the possible tonic changes are a whole tone or it plus a tritone (an augmented fifth, which inverts into a major third), and a whole tone alters the fewest notes. In the fifth case, the tritone provides the single altered note. Tonic scales offset by a fifth are nearby scales in the sense of having almost the same notes (the opposite case is tonic scales offset by a half

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@ I $ @

fifth tritone fifth tonic change (alters 1 note) @ M I $ @ half tone tritone slide

@ I $ @

whole tone tonic change (alters 2 notes) @ D I $ @ whole tone tritone slide tone, which are distant scales). In the whole-tone case, the tritone slide is composed of two successive half-tone slides each of which alters one note, for a total of two altered notes. This is a deeply simple picture of things that tend to be complex in music notation across the board. Anyone who thinks tritones are not fundamental to music might want to rethink this in light of this simplicity. Parallel mode changes for the main tonic of a piece, and same-mode tonic changes are two sides of the same coin, enabling tonic changes to be understood without making alphabet changes. KNOWING TONIC SCALES “BY CONSTRUCTION” VS. “BY ENUMERATION” We have come a long way without note symbols. However, there is a need for a notation for notes of the chromatic scale of a tonic octave for a variety of purposes, of which identifying tonic changes is the most important one in the actual use of PKP. Such a notation may also be used to show tonic scales by enumeration that are understood by construction, but keep in mind that this is a result not a starting point. A very simple numeric chromatic scale notation does the job. Numbers 1-2-3-4-5-6-7 identify notes of the Ionian major scale and prefixed numbers p2-p3-p5-p6-p7 identify the five in-between notes. The prefix p stands for “phlat,” with an obvious meaning (the note in the whole tone gap below the numbered position) but with a different symbol and spelling to emphasize that it is a position indicator for five notes and not a conventional flat. This is a 12-symbol notation with no sharps or flats. Here is an example of using the notation to describe a scale by enumeration that is understood by construction. The Aeolian mode identified by letter A is formed of the frame 1-5-1, plus the core tritone 2-p6 and fourth 2-5, plus a half tone 2-p3 that provides a companion half tone to 5-p6 inside the tritone, plus two whole tones to complete the seven notes. The result is 1-2-p3-?-5-p6-?-1 where the question marks can only be 4 and p7 because they provide whole tones going up in the scale. Although the tritone is neither major nor minor by itself, the scale is minor by construction. The AD.I and A.MI scales and sub-scales provide a second example. The 3-letter words provide a visibly simple way of knowing the suite of scales they imply by construction. The harmonic minor and major scales are identical except for the different in tonality provided by p3 vs. 3. Unlike the other scales, which are fully defined by tritone content, these scales require knowing tonality, which amounts to a choice between frame 1-p3-5-1 and 1-3-5-1.

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8-note AD.I scale = frame 1-5-1 plus tritones 1-2-p3-4-5-p6-6-7-1 7-note D.I scale (ionian-♭3, a.k.a. melodic or jazz minor) omit p6 7-note A..I scale (harmonic minor) omit 6 ———————————————————————————————————————— 8-note A.MI scale = frame 1-5-1 plus tritones 1-2-3-4-5-p6-p7-7-1 7-note A.M scale (mixolodyian-♭6, a.k.a. jazz mixolydian) omit 7 7-note A..I scale (harmonic major) omit p7

The melodic minor, harmonic minor and harmonic major scales provide the starting point for parallel modes formed by transposing the alphabet words and rotating the result to start on the original tonic, if the transposition contains it. They harmonic scales have a minor-third gap that can be left open or filled by note 6 or p7 to create a corresponding 8-note minor or major scale that defines a whole suite of scales. Filling the gap in the minor scale with note 6 or in the major scale with note p7 adds a tritone but doing it the opposite way does not. For example, an important jazz scale results from filling the gap in the harmonic major scale with note 6. The result is the AD.I scale made into a major scale by replacing p3 by 3. The closeness between these scales is reflected by their jazz names (“bebop melodic minor” for one and “bebop major” for the other). Knowing a scale by enumeration is not needed to know the chords it can supply. For example, knowing that tritones 2-p6 and and 4-7 of any of these scales combine into a diminished seventh chord on any of these notes as a root, or into a rootless voicing of a dominant-7-♭9 chord on root V, does not require spelling out the scale. The only required use of this notation in PKP is to identify tonic changes. Tonic Changes Refer back to the earlier picture that illustrated Ionian-mode tonic changes by a fifth and a whole tone. The different tonics of this example may be understood as @1 (main tonic, understood), @4 (tonic a fifth down) and @p7 (tonic a whole tone down). The way these changes would be represented in an anchor line is shown below. In the first example, M is M@1 on the keyboard but I@4 functionally. In the second example, D is D@1 on the keyboard but I@p7 functionally. In either case, a return later to the original tonic would be identified by the appearance of @1 in the anchor line.

tonic change a fifth down from @1 I - @4 - M tonic change a whole tone down from @1 I - @p7 - D

“Rhythm Changes” is the name given by jazz musicians to the widely copied chord changes of Gershwin’s I Got Rhythm. As shown below, the underlying mechanism is half-tone tritone slides. The first example is called tonicization because it suggests tonics going down by fifths to the ear without actually modulating to them. The second example is actual modulation down by fifths (tonic sequence 6-2-5-1). The tritones are main-tonic tritones interpreted relative to the tonic change annotated before them. For example, @5 - L means the tritone is read as L@1 on the keyboard but interpreted as I@5 functionally. Variations may skip steps to move the tonic by whole tones.

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tonicization I - M - D - A - I modulation @6 - A - @2 - P - @5 - L - @1 - I

Multi-tritone variations use harmonic major or minor double tritones (minor third offsets) instead of single tritones. The word table shows that only three such double tritones exist, namely P..M, A..I and D..L, so a 3-step slide returns to itself. As above, slides identify tonic changes going down by fifths or up by half tones. Interesting harmony results from transitioning to the next tonic by holding the double tritone, adding the new tonic frame and then sliding the tritones. An example is coming up in the next chapter (All The Things You Are). Parallel Multi-Tritone Modes The tonics that provide the parallel multi-tritone modes in the word table are summarized next.

Melodic Minor Harmonic Minor Harmonic Major P.D @p7 P..M @4 P..M @4 A.M @4 A..I @1 @p6 D.I @1 @6 A..I @1 M.L @p2 D..L @p2 D..L @p2 @5 @3 @3 L.A @p3 @5 @5 @6 @p7 @p7

The concept is that the notes of the mode are provided by the tritones plus a frame determined by the tonic (for example the basic frame of the top left mode is p7-4-p7, which implies the inclusion of tonic 1 because melodic minor scales start with a whole tone). In actual use, tonic 1 plus the tritone notes provide the basics and the details falls into place from context. For this particular example, the mode is 1-p2-p3-4-5-6-p7-1. Examples are provided in the next chapter. Appendix C provides a full summary of all the modes, for reference and for comparison with the same modes described in a much more complex way by the book Modalogy. Musical Lines In ordinary use of PKP, melody lines are left to music notation, with numeric scale symbols optionally annotated next to pivotal melody notes. A small extension to the numeric-chromatic-scale notation provides a shorthand notation for melody lines that can be used with the anchor line notation to provide outline sketches that are independent of music notation. The extension replaces dashes that indicate up the keyboard by diagonal down arrows to indicate down. For example, 1-p3-4-5-p7-1 is the pentatonic minor scale going up the keyboard and 1➘p7-5➘4➘p3➘1 is a melody line in the scale that goes down a whole tone from the tonic, then up a major sixth to the pitch center, and then continues down from there to the tonic. Up arrows are needed only for notes going up an octave (for example, 5-5 means play the same note twice and 5➚5 means go up an octave). I tried using arrows for both up and down throughout but the effect is cluttered and difficult to read. With this notation, 1- p7-5-4-p3-1 means go up the keyboard by large jumps. Timing is left to music notation or memory. The anchor notation is not changed for different tonics, only interpreted differently for them; and

PKP 3/9/16 —14— ©copyright R.J.A. Buhr www.pianotheoryman.com the same goes for the melody notation. Changing notations for every tonic change is confusing. The notation may be used for any musical line in which ups and downs are important, such as walking bass lines. Scale Shapes Before going down the path that led to the PKP alphabet, I consulted piano teachers and references to try to find an existing music notation that would do the job and did not find one. The nearest I could find is “figured bass notation,” which expresses chords in terms of the numbers of scale steps between successive notes going up from a bass note. A stack of these numbers is written next to a bass note on a staff. For example, a seventh chord consisting of four notes two scale steps apart would be identified by the stack 2+2+2 (this is a linear textual representation of what would be a literal stack of numbers one above the other on the staff). I came to think of such stacks of scale-step counts as scale shapes. Any chord from a tonic scale may be represented as a scale shape. This is, to my mind, a better way of knowing chords than chord symbols. It provides a direct picture of what chords actually look like on the keyboard in any inversion, with or without their roots. It is dependent only on the tonic scale, not the chord scale (which is no more than a tonic scale rotated to start on a different note). It provides a direct picture of movements of a selected scale shape to different positions in the same tonic scale (for example, a progression of seventh chords moves the scale shape 2+2+2 to different bass notes in a scale). I learned from jazz pianist Taylor Eigsti, while discussing my ideas with him at various times, that he advises his piano students to practice playing harmony in terms of scale shapes of all kinds moving in and between different classical modes across the board (he uses the term “keyboard shapes” but I reserve that for shapes in which intervals are measured in half tones, not scale steps) This helps develop the instinctive moves required of jazz pianists without being distracted by chord notation. There are two problems with scale shapes. They require knowing the scale that provides the shapes in advance, and they become unhelpful for the more variable scales that come into play when music departs from the classical modes of written key signatures (the number of scale notes and the keyboard sizes of inter-note intervals are both variable across the range of possible scales). The advantage of seeing chords in terms of building blocks instead of scale shapes is it does not require knowing the scales in advance. Quite the contrary — it enables scales to be inferred from building blocks. That said, knowing the scale shapes of basic chords in classical modes is helpful (see Appendix B for more on this). AN OBSERVATION I have been accused of finding a hammer (tritones) and seeing everything as a nail. A hammer is the right tool when everything is a nail, and in a sense everything is a nail in the keyboard view. The facts that tritones are dissonant intervals, complex in music notation, and historically shunned do not invalidate this. Tritone dissonance is muted in chords, complexity in music notation is misleading, and historical shunning is historical. Tritones on the keyboard are, visibly, pivotal elements in chord progressions and their harmonic cores. The simplest harmony in PKP is in the upper part of the above table for the same reason that it is the most complex part in music notation, namely high tritone content.

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3. EXAMPLES The ideas are simple but convincing others of their worth has not been because they go strongly against the grain of conventional wisdom. Piano wizards tend to think them naive, invalid, or complex, which is fair enough reaction to unconventional ideas put forward in a strange notation by a musical amateur. The ideas are naive in an “emperor has no clothes” sense — the emperor being the piano keyboard and the clothes being music notation — but this does not mean they are invalid. They are only complex if combined with key-signature notation into a super notation that covers both notes and intervals, which is not the concept of PKP. PKP is independent of music notation: it doesn’t replace it or alter it, but gives a different perspective on it that helps to deal with its intricacies. A sense of the breadth and depth of PKP may be obtained by looking at integrated melody and harmony cores of a few widely different pieces of music. When a piece has been learned based only on a melody line and the harmonic core implied by an anchor line, the harmonic core tends to provide a sufficient future road map for the piece that cues a familiar melody line and the harmony that goes with it. However, it is not useful to look at anchor lines in isolation from melody lines because the two form an integrated whole. Keep in mind that the PKP contribution is the core notation, not the melody line notation — the latter is mainly for explanation independently of music notation. STRAIGHT MAJOR The example is Happy Birthday. The notes for the six syllables of the opening phrase ha-ppy-birthday-to-you are 5-5-6➘5-1➘7. An outline sketch of the entire piece is developed in stages below. The anchor line that identifies the core tritone/fifo pair of the Ionian (major scale) comes first because it establishes the basis for choosing fifo anchors at the question marks. The scale of this piece is visibly major when you scan the melody line by eye, but the repeated tritones anchored by I assert the scale locally at important points as the piece moves forward. These points are just before the melody line resolves to the tonic and at the the ends of melody phrases that don’t resolve, to indicate the kind of resolution that is expected. These points are so obvious for this simple piece that chords are hardly needed to get started

core | ? I | Ⓘ | Ⓘ Ⓘ | ? | ? I | Ⓘ | I Ⓘ | ? |

melody 5-5 | 6➘5-1 |➘7➘5-5 | 6➘5-2 |➘1➘5-5 |➚5 ➘3 ➘1 |➘7➘6-4-4 |➘3 ➘1-2 |➘1 |

The obvious first choices for the question marks are shown next. The fifo anchored by M is a fourth, not a fifth because it would provide a half tone dissonance with the melody line where it resolves to the tonic. The chord roots shown across the top are typical for this piece.

example roots VI II V V V I VI II V II V I core | M I | Ⓘ | Ⓘ Ⓘ | M | M I | Ⓘ | I Ⓘ| M |

melody 5-5 | 6➘5-1 |➘7➘5-5 | 6➘5-2 |➘1➘5-5 |➚5 ➘3 ➘1 |➘7➘6-4-4 |➘3 ➘1-2 |➘1 |

The core is very easy to play because its bass line is either held or moves by a half tone. Octave shapes follow easily. With these “in the fingers” from practicing, a bass line to voice the indicated

PKP 3/9/16 —16— ©copyright R.J.A. Buhr www.pianotheoryman.com chord progression follows by adjustments of the bass line of the octave shapes. For example, the final appearance of the tritone may be played as octave shape 7-4-7 under melody note 2. Adjusting the bass note upwards to be a basic chord interval of two scale steps below note 4 yields the shape 2-4-7. Adding a note a fourth up from the new bass bass note yields the well balanced shape 2-4-5-7 which inverts into 5-7-2-4, a dominant-7 chord on root V. The chord intervals are all thirds (major third with two minor thirds on top). In scale-shape terms, this is 2+2+2. The PKP process yields scale shapes as a side effect of bass line decisions. Put another way, knowing the scale shapes of basic chords guides bass line decisions. The chords roots shown are examples not prescriptions. The RN root symbols are from a fundamental chord progression of major harmony called the diatonic cycle: IV-VII-III-VI-II-V-I, of which the above root line uses only the last four roots. This is one of the few root lines of harmony that is standard enough to reliably imply chord types. That said, a startlingly large number and wide variety of chord interpretations that look very different follow from assigning different root lines in the major scale to the same core. ENRICHED MINOR Minor harmony must go beyond the Aeolian mode because its harmony cannot be distinguished from that of the relative Ionian mode with the same notes. The example is Summertime. The core illustrates the notation for double tritones from chords containing them: the first anchor in alphabet position is shown normally and the second is shown as kind of superscript it to indicate it overlaps the first tritone from above. The offset avoids the possibility of interpreting two separate anchors close Ⓘ together as a double anchor (Ⓓ is a double tritone and Ⓓ Ⓘ is two separate tritones). Chord symbols are shown to make this concrete but are not needed to understand or play the piece.

V-9♭13 V-9♭13 V-9♭13 Ⓘ Ⓘ Ⓘ Ⓓ >>> Ⓓ Ⓓ … melodic minor

5➘p3 | 5 | — 5➘p3-4-5➘p3 |➘1➘5

I-7♯9 II-m7♭5 V-7♯9 Ⓜ Ⓐ Ⓘ … segue to Aeolian & harmonic minor

| — ➚5➘p3 | 4-4 | — ➘p3➘1-p3➘1-p3 |➘2 — | — 5➘p3

V-9♭13 ♭VII-9 Ⓘ Ⓓ >>> Ⓐ

| 5 | — 5➘p3-4-5➘p3 | 1➘5 | — 5 … all the elements of AD.I minor

II-m7♭5 I Ⓐ D

| p7➘5-p7-1-p3 | 5➘4➘3 | ➘1 — | … Aeolian resolution

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There are some sophisticated wrinkles here that I learned in a piano comping course given by Susan Muscarella at the then Jazz School in Berkeley (now the Jazz Institute), and separately from pianist Taylor Eigsti who was an artist in residence there at the time. These wrinkles are worth presenting because they illustrate some fundamental points about PKP. The notation “>>>” indicates a kind of ornamental sequence I learned from Taylor Eigsti, in which the scale shape defined by a voicing (the double tritone D.I here) is moved ornamentally by scale steps up or down in the current scale (melodic minor here), without thinking in terms of any notation. For example, if this double tritone is voiced as 4-7-p3, its scale shape is 3+3. This particular shape is “all fourths” — an augmented fourth (tritone) with an ordinary fourth on top. All-fourths shapes are a sophisticated form of harmony that fall into the category known to jazz musicians as “hip voicings.” Moving this particular all-fourths shape up a scale step yields 5-1-4 — a pair of ordinary fourths. Moving it down a scale step yields a p3-6-p2 — a different tritone with an ordinary fourth on top. The movements yield shapes that sound right because they are all the same scale shape in an established scale. Spelling out these shapes in note symbols is only for explanation; they are played without thinking about note symbols. The M tritone cues a segue between the melodic minor scale identified by D.I and the Aeolian scale that follows it. The segue is via a tonic scale that seemed arbitrary when I first saw it, namely 1-p2-p3-3-p5-p6-p7-1 called “dominant seven sharp nine.” The scale visibly contains both tritone M and note p3 that provides ♯9 of the written chord, and visibly provides a segue to the the Aeolian mode via notes 1, p3, p6 and p7. But why pick this particular scale out of a hat? The answer is, it is a parallel mode of the melodic minor scale D.I@1 defined by transposing it up a half tone to M.L@p2 and rotating the result to start on 1 (the L tritone includes 1). This is the concept but actual transposition and rotation of an enumerated scale are not needed because M.L@p2 provides everything needed: the tritone is given and the p2 tonic tells us the scale contains p3 and four whole tones going up from a half tone above it. This example illustrates how PKP may identify a context (melodic minor here) in which scale shapes may be used to provide pleasing sounds within it (all fourths here). Scale shapes are not useful across the board because of the variability of scales across the board, but they work nicely for classical modes and highly regular scales such as the melodic minor scale. A MINOR VARIATION OF HAPPY BIRTHDAY Reharmonizing Happy Birthday as a minor piece is instructive because it is major tonality is indicated by only two appearances of note 3 near the end, which only need altering to p3 to make the piece minor (giving a happy piece an ironically sad twist). The following harmonic core uses the AD.I minor scale to harmonize this minor variation. The resolutions are in the harmonic minor sub-scale. The resolution sequences may be played as the double tritone going to the minor triad of the tonic inverted on D (an alternative that provides a strong mournful sound goes to the augmented fifth that outlines the D.I double tritone of the composite scale).

VI II V V V I VI II V VI II V I Ⓘ Ⓘ Ⓘ Ⓘ | Ⓓ Ⓐ |Ⓐ |Ⓘ Ⓐ | D | Ⓓ Ⓐ | Ⓐ | Ⓓ Ⓐ Ⓐ | D

5-5 | 6➘5-1 |➘7➘5-5 | 6➘5-2 |➘1➘5-5 |➚5 ➘p3 ➘1 |➘7➘6-4-4 |➘p3 ➘1-2 |➘1

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The same chord roots as before may be used but the chords are different in kind and more diverse. Working backwards from the end, the I chord is a minor triad, the V chord is dominant-7-♭9 and the II chord is minor-7-♭5. The only new thing before that is the VI chord, which is also minor-7-♭5. “RHYTHM CHANGES” The example is the first section of Over the Rainbow. Harmonizing major melody lines with more sophisticated harmony than the major scale can provide is common. Such harmonizations are often variations of the original “Rhythm Changes” in which, recall, IMDA provides tonicization and APLI provides modulation. However, there is nothing to stop using the linear combination IMDAPLI or any extension of it to provide only tonicization. Start this off with L and you have the reversed alphabet LIMDAP with a tag LI at the end to bring it to a major resolution. This sequence is the core of the harmony shown next (tritones anchors shown in red). The tritones are positioned to be consonant with melody notes.

| ? Ⓛ | Ⓘ ? Ⓜ| ? Ⓓ | ? | ? Ⓐ | ? ℗ | Ⓛ ? Ⓘ | ? |

| 1 ➚1 |➘7➘5 - 6 - 7 - 1|➘1 - 6 |➘5 |➘6 - 4 | ➘3➘1 - 2 - 3 - 4 |➘2➘7 - 1 - 2 - 3 |➘1 |

The questions marks are filled in with visibly obvious fifo anchors (black text) plus additional ornamental tritones that fit the flow (blue text). This harmony exploits the full chromatic scale in depth and yet is visibly simple. This is only the opening section of the piece but gives the general idea.

Ⓜ Ⓘ Ⓜ | M Ⓛ | Ⓘ M Ⓜ| M Ⓓ | ℗ | A Ⓐ | Ⓜ ℗ | Ⓛ I Ⓘ | M |

| 1 ➚1 |➘7➘5 - 6 - 7 - 1|➘1 - 6 |➘5 |➘6 - 4 | ➘3➘1 - 2 - 3 - 4 |➘2➘7 - 1 - 2 - 3 |➘1 |

BASIC BLUES Here follows a textual outline sketch of Backstreet Blues seen earlier.

Ⓜ Ⓓ Ⓜ Ⓜ

|5➘4➘p3➘1|5➘4➘p3➘1|1 — | —

Ⓓ Ⓓ Ⓜ Ⓜ

|5➘4➘p3➘1|5➘4➘p3➘1|1 — | — 5-p7➘5|

Ⓘ Ⓓ Ⓜ Ⓜ

|p7-p7 — ➘5 |1➘p7➘5➘p5➘4➘p3➘1|1 — | — |

In the original, the chords are dominant-7 chord on roots I (tritone M), IV (tritone D) and V (tritone

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I) but blues chords are all over the map. Three-chord blues using these chords is deceptively simple. Except in bar 10, the melody is all in the pentatonic minor scale 1-p3-4-5-p7-1. The scale in bar 10 is a 6-note minor blues scale formed by adding p5 to the pentatonic minor scale. A companion 6- note major blues scale not directly used in the written melody of this piece is formed by adding p3 to the pentatonic major scale (the addition is minor but the basic scale is major). These scales are shown next (the parentheses outline half tone sequences centered on the tritone anchors that help in remembering them).

6-note minor blues scale identified by tritone L: 1-p3-(4-p5-5)-p7-1 6-note major blues scale identified by tritone D: 1-(2-p3-3)-5-6-1

Between harmony shown earlier and these melody scales, this simple piece uses the 10-note DMIL blues scale. This scale is more of a toolkit than a proper scale. It provides the harmony tritones, the two 6-note blues scales just identified, a third 6-note blues scale containing tritone M that is a hybrid of them, and a number of other scales to boot (for example, major, melodic minor). The harmony tritones may be taken as cues for blues scales to use for improvisation in the different bars. Tritone I in bar 9 an outlier because it doesn’t suggest a 6-note blues scale and its note 7 is strongly dissonant with the melody note p7 that opens the bar. The dissonance can be avoided by playing one note on an upbeat and the other on a downbeat, or finessed by playing the tritone as 7-4 under p7. The resulting shape is an example of an “all-fourths” shape (augmented fourth 7-4 with fourth 4-p7 on top). As said earlier, all-fourths shapes are harmonically sophisticated; they are part of a repertoire of chord voicings jazz musicians refer to as “hip.” SOPHISTICATED TONIC/MODE CHANGES The example is All The Things You Are, which makes a sequence of tricky tonic and mode changes that may be understood as a double-tritone version of “Rhythm Changes.” Instead of half-tone slides of single tritones, it is based on half-tone slides of the harmonic major double tritone A..I (which is also the bebop major double tritone). As shown in the word table, there are only three different words of this form in the alphabet, so the slide goes through them and then back to itself.

@1 - A..I - @3 - D..L - @7 - P..M - ( @1 - P..M ) - @1 - A..I

The unusual tonic changes are are shown in different colors to bring them forward to the eye (the sequence of major tonics 1-3-7-1 is Ab-C-G-Ab, and parallel minor tonics are visited for each major tonic). The opening is in straight major but thinking of it as in harmonic or bebop major yields a usefully uniform picture. The parenthesized segment transitions back to main tonic by holding the double tritone and raising the tonic and then holding the tonic and raising the double tritone. The transition is simple in these terms, and easy to play, but is misleadingly complex in music notation because it goes from multiple sharps to multiple flats. Here follows an outline sketch of the main part of the piece, showing only the tritone core. Passages of different tonics are daisy-chained together via shared notes without one passage necessarily resolving to its tonic before going to the next one. Tonic A♭goes to tonic C via shared note F and tonic And tonic G goes back to tonic A♭ via note B that is a major for G but minor for A♭. The segue

PKP 3/9/16 —20— ©copyright R.J.A. Buhr www.pianotheoryman.com back to the main tonic is accomplished via the scales shown below. The frame 1-5-p6-1 stays fixed, the tritones go up a half tone, and the other notes fall into place from context.

parallel mode of P..M 1-p2- p3-3- 5-p6- p7- 1 harmonic minor of A..I 1- 2- p3- 4- 5-p6- 7- 1 harmonic major of A..I 1- 2- 3-4- 5-p6- 7- 1

Here is the outline sketch, showing only the tritones that determine the tonic changes.

@1 (A♭) Ⓘ | 1 |4➘1|➘7-7-7-7| 7-3➘7|➘6-6-6-6 @3 (C) Ⓓ Ⓛ |6-p3➘6|➘p6 — | — |➘5|1➘5|➘p5-p5-p5-p5|p5-7➘p5 |3-3-3-3

Ⓜ @7 (G) ℗ ℗ Ⓛ Ⓜ

|3-p5-5➘p5➘3|➘p3 — | — ➘p5-7-p5 |p5➘3-3 | — ➘5-p6-3| ➘p3 — | — ➘p5-7-p3

@1 (A♭) Ⓜ ℗

|p3➘p2-p2 | — ➘2-p3-p2 | ➘1 —

@1 (A♭) Ⓐ Ⓘ | — | 1 | 4➘1|➘7-7-7-7 …

The written chords are shown below with the core building blocks within the tonic octave shown below them. The vertical gray bars are fifos, anchors for which would appear in the full core.

Eb7 Am7b5 E7#9 D7 F#m7b5B7 Fm7 Eb7 Fm7 Bbm7 AbM7DbM7G7 CM7 Cm7 Fm7 Bb7 EbM7AbM7 D7b9 GM7 Am7 GM7 CM7 EM7 C7#5 Bbm7 AbM7 @ Ab

$ Eb

@ Ab

RAPID CHANGES BETWEEN DISTANT TONIC SCALES The example is John Coltrane’s Giant Steps, which takes rapid “giant steps” between distant key signatures of 5 sharps (B major), 3 flats (E♭ major), and 1 sharp (G major). However, it is actually easy to play when you see it in the following terms. An outline sketch of the first seven bars is shown

PKP 3/9/16 —21— ©copyright R.J.A. Buhr www.pianotheoryman.com next (nothing harmonically new happens after this). The note on which the melody line comes to rest at the end of the piece and in several places in between (including the end of bar 7 shown here) is F♯, designated as 1. The chord progression is a mix of V-I (mostly) and II-V-I (a few) sequences of basic seventh chords for three major tonics identified above. The core identifies the tonic scales but the melody line contains only fragments of the tonic scales and does not actually resolve to these tonics anywhere, not even where it comes to rest in bars 3 and 7 (note 3 is A♭, the pitch center of the E♭ octave, and note

1 is F♯ , the pitch center of the B octave). The over-lines identify harmonic resolutions and the under- lines identify melody segments of the same form in interval terms.

======core | M Ⓛ | I Ⓐ | P | Ⓛ | I Ⓐ | P Ⓜ | M

melody | 1 ➘p6 | ➘4 ➘p2 - 3 | 3 … | 4 ➘p3 | p6 ➘3 | ➘p2 ➘ 6 - 1 | 1 … ======

Two striking things leap off the page from this outline sketch. One is the identification by the tritone content (word A.M.L) of a framing whole tone scale 1-2-3-p5-p6-p7-1 (F♯-G♯-A#-C-D-E-F♯) that provides many of the notes of the changes. The other striking thing is the absence of the pitch center of the tonic octave (note 5, which is C♯) from the melody line, which is a cue that the melody line is not from any ordinary scale of tonic 1. The piece is very easy to play as shown without thinking of the changes. The two under-lined melody segments are all thirds, alternating between major and minor, with the first three going down and the last one going up. The harmony is so varied that playing octave shapes for the building blocks is sufficient, at least to get started. The harmonic changes are highlighted by the over-lines: Ⓛ-I (tonic G), Ⓐ-P (tonic E♭), and Ⓜ-M (tonic B) (the latter holds the bass note instead of dropping it because the tritone is oppositely inverted in the B and F♯ octaves). The major scales they establish provide notes that will sound right for the brief period of their establishment but this is not particularly helpful because the changes are too rapid. The piece provides many opportunities for different interpretations. For example, the melody line could be interpreted as from an altered Mixolydian scale of tonic 1 identified by the word A.M (the scale is 1-2-3-4-5-p6-p7-1, tritone content in red). Melody lines in this scale with harmony from it and the whole-tone scale will sound right in relation to the original. The original harmony tonics G (p2) and E♭(6) become passing notes not in this scale, but then they are no more than that in the original because the melody line never resolves to them. The appearance of outside note p3 in the transition bar 4 suggests a possible blues interpretation.

PKP 3/9/16 —22— ©copyright R.J.A. Buhr www.pianotheoryman.com

A HARMONICALLY SOPHISTICATED JAZZ BALLAD The example is Strayhorn’s Lush Life. This beautiful piece, said to have been written by Strayhorn when he was a teenager, is harmonically rich, and challenging to play as written because there are often two or more chords per bar, many of them chromatic relative to the written key signature of five flats. The piece is a bit of harmonic puzzle because the key signature and most of the melody is consistent with parallel major or minor scales of Db (the major tonic of the key signature) but the melody line never comes to rest on Db, and its final resting note in the final bar is F (note 3 relative to Db as note 1). This is a product of daisy-chaining brief modulations at the ends of passages back to the main tonic via a shared note. The modulations are so brief that the effect is of an impressionistic flow of tonicizations rather than of modulations. The main tonic of Db suffices as a base for understanding the flow. The melody line sounds good in its own terms and the flow of harmony sounds good with it. The key to understanding the flow is the tritone content of the harmony. This is adventurous music by any measure, and correspondingly difficult to remember and play in conventional terms. The only way I have been able to get on top of it is via the following outline sketches. In them, fifo anchors are mostly omitted to focus on flow. The melody of the verse and bridge are essentially major (p3 appears only once in the verse as a passing note and not at all in the bridge), and of the chorus moves between parallel major and minor. Verse

1. 2. @Db ======@ Ⓐ M Ⓐ M Ⓐ M > > > > Ⓘ A Ⓘ M A Ⓓ 5|1..1–2..2|3..3➘2-2 |3..3➘2-2|3-3-4-4-5-5-p6-p6|p7-p7-p7➘p6-p7➘p6|5..5-p7➘p3-p3|➘2-2➘5 |2.. -7|

The first and second endings are each three bars long in the written piece but the difference in the first two bars of each ending is only a small melody-line wiggle so I show the two endings as one bar and leave the melody-wiggle difference to the written music. Notations of the form 1..1 cue repeated melody notes, and of the form M >>>> cue building blocks moving to successive scale positions. The joint melody/harmony scale is A.MI, which supplies the mixolydian-♭6 scale identified by A.M for the melody (1-2-3-4-5-p6-p7-1) and tritones A and I for the harmony. The notation >>>> in bars 4-5 means play the same scale shape tracking the melody line. Tritone D in the second ending cues a change to Ionian of F (the tritone is the Ionian tritone of F). Bridge

Ⓛ Ⓛ ℗ Ⓓ ℗ Ⓓ Ⓘ

|➘7-p2 |3-5 |7-p2➘6-7 |7➘7|7-p2 |3-5 |7-1➘6-7 |➘6-7➘5-6-7➘3 |

Ⓜ Ⓜ Ⓛ Ⓘ Ⓐ ℗ Ⓓ Ⓐ

| 3 — | — 5➘4➘3 |➘2 | — 2-2-3➘1 |5 — | — |

PKP 3/9/16 —23— ©copyright R.J.A. Buhr www.pianotheoryman.com

The lead-in from the verse cues tonic 3 (F) of which the D tritone is Ionian and the P tritone is Dorian tritone. P.D identifies melodic minor of F. That said, it is simpler to think of the tritones as ornamental and note p2 as a passing note because the bridge quickly reverts to tonic Db with the reappearance of tritone I. Chorus

Ⓘ Ⓘ Ⓛ Ⓘ Ⓘ Ⓓ Ⓐ Ⓜ Ⓘ Ⓘ

| 5-6-p7-p3➘p6|➘5-6-7-p3➘p7|➘6➘5-1-3 -4 -5 |➘p7 ➘p6 |

Ⓘ Ⓘ Ⓜ Ⓓ ℗ Ⓐ

|➘5-6-p7-p3➘p6|➘5-6-7-p3➘p7|➘6➘5-1-3 -5 -p6 |➘7 — 1 |

Ⓛ Ⓐ Ⓛ Ⓓ Ⓜ Ⓐ ℗ Ⓛ Ⓘ

|2-3-4-4 — ➘p3 |➘2-3-4-4➘p3 |➘p6➚p7➚7-7 — ➘p7 |➘p6-p7 -7 -7➘6 |

Ⓘ Ⓘ Ⓘ Ⓘ Ⓓ Ⓐ ℗

|➘5-6➚p7-p3-p6 |➘5-6-7-p3➘p7|➘5-1-3-4 -5 | 6 ➘3 |

Ⓛ Ⓐ Ⓛ Ⓘ Ⓓ ℗

| 5➘4-5-5➘4 |➘3➘2-3➘7|2➘1-2-2➘1|1➘p7-6 — ➘3|

Ⓐ Ⓛ Ⓘ Ⓘ P > > Ⓜ A > M

| 5➘4-5-5➘4 |➘3-3➘2-3➘5 |p7-7-1-p2 |2-p3-3|

It’s possible to parse this in terms of tonic changes but there are too many of them and they are too transient to make it worth the trouble. The melody line visibly moves between major and minor as identified by appearances of p3 and 3 and the harmony goes with the flow. The signature half-tone run up of the melody line in the final two bars is accompanied by a scale shape run up.

A “POSTER CHILD” FOR CHORD COMPLEXITY This closes the loop on the footnoted example in the opening chapter. The piece is the haunting E- flat minor blues Goodbye Porkpie Hat written by Mingus in memory of Lester Young. The chord root line by a creative bassist is responsible for much of the chord complexity; that and the inclusion of blues melody notes in some chord symbols. The melody line is simple minor blues melody with a few forays into parallel minor. The written chord progression is grotesquely complex (next page) but the integrated melody plus harmonic core is simple and easy to play. The opening five bars and closing two bars are straight DM.L blues with a few minor touches. Between them are minor-ish variations bookended by an out-of-context (in blues) double tritone. The core bounded by the tonic octave, using the tonic notes as voicing notes wherever they fit, is very easy to play and so harmonically rich that it provides sufficient variety. As is often the case for blues, resolution is left to the melody line. I have combined tritones from single chords into double tritones in bars 6, 7 and 10 because they provide 4-part harmony all by themselves, sound good, cue melody notes, and “bookend” the variations.

variations ======Ⓛ ⓛ Ⓓ ⓛ Ⓛ |Ⓜ Ⓛ|L Ⓜ |I Ⓛ |I Ⓜ |I Ⓛ |Ⓐ |℗ |Ⓛ L |Ⓜ Ⓓ |Ⓐ |Ⓜ Ⓛ |℗ Ⓜ | The full whack is shown below, including melody line, chords and anchor line above the staff (using a different but visibly equivalent notation for the double tritones) and expanding the core harmony below the staff. The expansion illustrates the smooth flow provided by tritone slides within the tonic octave plus a few morphs involving fifos. The melody line is primarily in the 6-note minor blues scale of tonic Eb, which is the all-black-key pentatonic scale with one white key added (the flatted 5th). Knowing the simple melody line enables all of the following to be cued by the above anchor line.

bars 1-4 @Eb M L L M.L I L I M

1— @ - p7— - - - 5— $ p5— L L L L L 4— I I I M M M M p3— D minor blues minor A P 1— @ bars 5-8 I P.D L A..I L L

3 circled p2 7 p6 notes are 1— @ "outside" - p7— - - - 5— $ p5— L L L L 4— I I I M p3— D D A A P P 1— @ bars 9-12

M D A..I M L P M.L

1— @ - p7— - - - 5— $ p5— L L L 4— I I M M M M p3— D D A A P P 1— @

4. OBSERVATIONS & CONCLUSIONS PKP’s way of knowing tonic scales is lightweight, demonstrating the principle that less is sometimes more. A small number of master words expands by transposition within the alphabet into a much larger number of words that defines an even larger number of parallel modes that do not have to be remembered in detail because the way of knowing them by construction is very simple. The words provide shapes from the scales that are or voice chords from the scales directly without reference to the actual scales. Understanding things this way helps in learning new pieces, remembering pieces not played for a while, recovering from mistakes, tracking context changes, distinguishing momentary ornamentation from substantive changes, and guiding experimenting with variations and improvisations. There is an analogy between the PKP alphabet and biological DNA. In both cases, words from a small alphabet identify deep structure. Words of biological DNA are “expressed” as proteins, the building blocks of life; PKP words are expressed as building blocks of music. Biological DNA is extracted from samples and interpreted by sophisticated machines called sequencers; PKP words are extracted from written music by sophisticated sequencers called human eyes. Biological DNA is inherited from parents; PKP patterns are inherited from other pieces. The independence of PKP of music notation makes time spent disassembling and reassembling particular pieces of music in particular keys in PKP terms generally rewarding across the range of all possible keys, because what’s learned is generally applicable across the board. From doing this myself across a range of pieces in different keys, I have formed the impression that simple PKP pictures of harmony are the norm, no matter how complex the written music looks, because composers and arrangers try to make flow patterns of music smooth and well balanced in interval terms. The translation into music notation only records the results. Music on the grand staff captures the flow patterns but in a very detailed and misleadingly complex way that overwhelms all but experts. Chord symbols obscure the flow patterns because they represent each chord independently of context in chord scale terms, and flow patterns are all about context. Key signatures are sometimes unhelpful for sophisticated harmony because they sometimes seem to be chosen to minimize accidentals across a range of key changes rather than to reveal harmonic fundamentals (which key signatures are incapable of doing in any case for pieces with many context changes, which, remember, include other than just key changes). I see PKP as enabling anyone to understand harmonically sophisticated piano music well enough to tackle playing it for fun, independently of developing “chops.” That said, nothing gets in the way of developing chops and much can help. Harmonic flow seen in PKP terms is so natural to the ear and eye that the notation quickly vanishes from the conscious mind, until it is needed for remembering a piece not played for a while, or to guide experimenting with variations and improvisations. Spelling everything out in the notation is often not even necessary. Knowing that simplicity lies underneath written complexity can be sufficient to cue the underlying simplicity while playing, assuming knowledge only of the general nature of context changes in a pieee (tonics, major tonality vs. minor tonality vs. blues, and so on). I have personally encountered no limits to PKP’s usefulness within the domain of tonal music.

APPENDIX A: UNCONVENTIONAL TERMINOLOGY & NOTATION anchor: identifies a building block by position of its bottom note in the chromatic scale anchor line: anchor sequence written above the staff (outlined by circles for tritones & boxes for fifos) alphabet: PADMIL identifies tritone anchors by the first letters of the names of 6 classical modes building block: tritone or fifo (fifth or fourth) chromatic scale: 1-p2-2-p3-3-4-p5-5-p6-6-p7-7-1 context: tonic scale identified by an alphabet word core: sequence of building blocks of harmony identified by an anchor line fifo: fifth or fourth that are opposite inversions (add up to an octave) frame: defined by the tonic and pitch center of the tonic octave, sometimes augmented by another note pattern: an organized arrangement of intervals on the keyboard or over time pitch center: the note identified by a fifth/fourth octave split, symbolized by $ morph: transforming a building block into a different one by altering one note while holding the other outside: not in an identified scale (as distinct from “chromatic” meaning not in a key-signature scale) shape: combination of two or three building blocks that form or voice a chord keyboard shape: shape in which intervals are counted in half tones from a bass note as “0” scale shape: shape in which intervals are counted in scale steps from a bass note as “0” slide: size-preserving movement of a building block to a different keyboard position p: prefix standing for “phlat” identifying chromatic-scale notes in the whole tone gaps of the major scale toolkit: a tonic scale identified by multi-letter word that supplies sub-scales and building blocks tonic: bottom note of the chromatic scale of the main tonic of a piece, symbolized by @ tonic pointer: symbol of form @tonic appearing as a separate symbol or as a suffix on an anchor symbol word: formed of one or more alphabet letters (with optional dots indicating skipped letters)

APPENDIX B: CHORDS TABLE OF TRITONE CHORDS Chords are basically simple things, described in a notation that often gives individual chords and progressions of them a misleadingly complex aspect (not surprisingly, this is particularly so for tritone chords). The workhorse chords of classical modes are seventh chords formed of two overlapping fifths or an overlapping fifth and tritone. For basic chords, the overlap is two scale steps. Structurally, sixth chords are inversions of seventh chords and triad chords (as distinct from tritone chords) are either of these with the top note omitted; in both cases, harmonic function is determined by context. Multi- tritone chords originate in multi-tritone scales. The scales may have more than two tritones but chords with more than two tritones effectively do not exist as notated chords (for example, “splashes” of notes created by a hand or arm on the keyboard). Tritones are extracted from notated tritone chords using the following table. offset of lowest suffixes on root symbols, identifying tritone-bearing chords tritone notes up from root 7 7#9(13) 9b13 7b9 7b5 dim7 m7b5 m6 M7(11) m7b9 M7#11 (or #5) (or #4 or #11) (or o7)

fourth x

major third x x x x x

minor third x x x

whole tone x

half tone x x 0 x x x x

Chord roots are references for defining notes not reliable indicators of harmonic function (e.g., a chord on root II may be a V chord of a different tonic). In PKP, the indicators of harmonic function are tritones relative to a current tonic. Tritones are found from written chords and assigned anchor letters relative to tonics understood as just described. Chord symbols can be complex but understanding the details is not required, only recognizing tritone-bearing chords, which stand out as different in kind (this table). Mostly but not entirely, the tritones found from these varieties of chords are the same as the guide intervals used in standard chord-voicing practice. The difference is significant because standard chord-voicing practice does not include all tritones, and PKP requires all tritones. Many more combinations of suffixes than these are possible but these are the ones that have tritone implications — for example G(9)13 has the same tritone as G7 but G9b13 (or G9#5) has a second tritone. Sixth chords are renamed inversions of seventh chords and may have corresponding extensions. For example, Dm6 is a renamed inversion of Bm7b5, and Dm6(9) is a renamed inversion of Bm7b5(11). Triad chords are formed of a fifth with a middle note a major or minor third above the root. Structurally, they are seventh or sixth chords with the top note omitted. In chord notation, an unadorned root implies a major triad and a root with suffix m implies a minor triad. Although triad chords formed of a tritone with a middle note are possible, I am not aware of any chord symbol for them (perhaps this

PKP 3/9/16 —29— ©copyright R.J.A. Buhr www.pianotheoryman.com is because they are generally understood as functioning harmonically as seventh or sixth chords). The PKP way of understanding chord progressions makes the difference between chord symbols such as G7b5 and G7#11 unimportant because tonic scales of origin of chords are understood independently. The intent of notations such as #11 is avoiding implying that b5 replaces a scale note a half tone up. When the scales are independently known, this is not an issue. For example, in a C melodic minor scale, b5 of G7b5 is Eb and note E a half tone up is not in the scale. But in a D melodic minor scale, b5 of G7b5 is C# and note D a half tone up is in the scale. CHORDS WITHOUT CHORD SYMBOLS Chords are very simple objects in PKP. Chords with 4 or more notes are shapes defined by combinations of 2 or 3 building blocks. Building blocks are identified by anchors. Two anchors means 2 tritones, 2 fifos, or one of each. Three anchors never means 3 tritones because chords formed of 3 tritones don’t exist for all practical purposes — it means 2 tritones and a fifo, 2 fifos and a tritone or 3 fifos. In the latter two cases, one of the fifos can generally omitted as implied by context. Any combination of inversions of the building blocks going up or down from the anchors is harmonically the same chord. Triad chords are two-anchor shapes in which one anchor is left as a note. The large number and wide variety of chord symbols that can appear in written music is a result of notating chords in terms of a bass note and inter-note intervals going up from it. In PKP, both are results, not parts of the definition. In PKP, a chord root line is no more than bass line underneath the sequence of shapes formed by the building blocks (it may be the bass line of the shapes, include elements of it, or be entirely separate). In anchor lines, chords are represented by one anchor for all-fifo chords and one or two anchors for tritone chords. Single anchors in anchor lines are turned into 4-note shapes by adding a secondary fifo, which amounts to adding a bass note because the other note is generally implied by context. Double anchors in anchor lines (always tritone anchors) are left alone because they are already 4-note shapes. Shapes with more than 4 notes are left to be implied by context (the extra notes are generally either carried over from the previous shape or anticipate the next shape). “Left to context” means leaving open the option of playing or not playing them. Some examples of tritone chords of the Ionian mode are illustrated next. In spite of the large number of very different looking chord symbols along the bottom, there are only two fundamentally different chords here, namely AⒾ and MⒾ. The difference between the two sizes of A in AⒾ is the difference between including the pitch center or a note a whole tone above it in the Ionian scale in the chord. This difference is likely to be dictated by the flow of the music rather than any difference in the sound of the stand-alone chord. The 5-note versions of AⒾ can easily dispense with the pitch center because it is fundamental part of the context. The representation of 4-part harmony by double anchors is the general concept but anchor lines include double anchors only for double-tritone chords. Completion fifos are so easy to add from context that they are not worth representing by anchors. The messiness of the chord symbols is in stark contrast to the simplicity of the keyboard shapes that voice them. And this is all within a single classical mode. It gets much worse beyond classical modes. Suffixes b5, #5, b9, #9, #11 and b13 come into play, singly or in combination. The complex chord symbols are deeply misleading because the chords remain combinations of two kinds of building blocks. The sharped and flatted suffixes generally mean no more than an offset or morph of a building

PKP 3/9/16 —30— ©copyright R.J.A. Buhr www.pianotheoryman.com block by a half tone to fit the chord into a tonic scale. The phrase “pounding square pegs into round holes” comes to mind.

C @ 2+1+2 2+1+1+1 2+2+1 1+1+1+1+1 1+2+1 B x x x x x VII - A x x x x - G $ x x x V - F Ⓘ Ⓘ Ⓘ Ⓘ Ⓘ IV E M M - D A A A A II -

C-major, for example C @ VII: Bmb5#5 VII: Bm7b5#5 VII: Bm7b5 VII: Bm7b5#5(11) VII: Bm7b5(11) V: G7 V: G9 V: rootless G9 V: G13 V: rootless G13 "usual" chords IV: F6(9)b5 IV: F6(9)b5 IV:F6b5 IV: F6(9)b5 IV: FM7b5 in red text II: Dm6(11) II: Dm6(11) II: Dm6 II: Dm6(9)(11) II: Dm6(9) - the number sequences that look like arithmetic are "scale shapes" (the numbers are scale-step counts) - dashed lines indicate candidates for omission from voicings

The above chord examples bring forward to the eye two fundamentally different kinds of harmonic shapes, namely all-thirds and all-fourths. All-thirds shapes are the basic shapes of chords from classical modes. They are exemplified by the 4-note shape that results from inverting both building blocks of the AⒾ chord on the left (5-7-2-4), in which all inter-note intervals are minor or major thirds. In classical modes, the major and minor thirds of all-thirds shapes are always two scale steps, which leads to the way of notating them with strings of scale-step counts separated by plus signs (an adaption of “figured bass notation”). The basic all-thirds shape from a classical mode is 2+2+2, defining a seventh chord. Thinking in these terms helps in recognizing inversions. For example, 2+2+2 on note 7 inverts into 2+2+1 on note 2 and (a minor-6 chord), and 2+2+2 on note 5 inverts into 2+1+2 on note 2 (a variation of a minor-6 chord). All-fourths shapes are harmonically more sophisticated. They are exemplified by the 3-note shape that results from inverting the most elemental form of the MⒾ chord on the right to put the tritone on the bottom (an augmented fourth) with the M fifth inverted into a fourth stacked on top (notes 4-7-3). The shape has fewer notes than the corresponding all-thirds shape but the notes are spread out more and provide a sound that is different in kind, automatically suggesting a different chord interpretation. The basic all-fourths shape from a classical mode is 3+3, defining no specific kind of chord — as illustrated above, it voices many varieties of extended and altered chords on different roots. This is so simple conceptually, and actually on the keyboard, that it is hard to believe it is one of the most advanced harmonic shapes in music. I recently heard classical pianist Daniel Glover speak of it, at a concert of Scriabin music he was presenting, as invented by Scriabin and later picked up by jazz pianists, among them Bill Evans. The voicings of chords it produces are sometimes called “hip” by jazz musicians. Extended Chords Scale shapes provide a useful way of understanding extensions of seventh chords. This is for classical modes. The +4 on top of two of the extended chords is a consequence of avoiding 1+1+1

PKP 3/9/16 —31— ©copyright R.J.A. Buhr www.pianotheoryman.com sequences in the in-place shapes. The corresponding chord symbols have degree-number suffixes 6, 9, 11 or 13 added to basic symbols to identify the variations (13 implies 6 and 9, 6 is just 6)

chord type split (2) in place extended ninth (2)+2+2 1+1+2+2 2+2+2+2 eleventh (seventh+11th) 2+(2)+2 2+1+1+2 2+2+2+2+4 thirteenth (ninth+13th) (2)+2+(2) 1+1+2+1+1 2+2+2+2+2+4

Diatonic Cycle The anchor line, bass line and chord root line shown next define a model chord progression of major harmony known as the diatonic cycle.

anchor line Ⓘ M M | I Ⓘ M | M anchor wave tri fi fo | fi tri fi | fo 4-7 3-7 3-6 | 4-1 4-7 3-7 | 3-6 bass line 2 2 1 | 2| 2 1 | 1 secondary fifos fo fi fo | fo fi fo | fi 2-6 2-5 1-5 | 2-6 2-5 1-5 | 1-4 chord root line VII III VI | II V I | IV chords 7-2-4-6 3-5-7-2 6-1-3-5 | 2-4-6-1 5-7-2-4 1-3-5-7 | 4-6-1-3 chord type m7b5 m7 m7 | m7 7 M7 | M7

The anchor line is Ⓘ-M-M repeated twice and joined by I. The resolution sequence is I-Ⓘ-M. The repetition of M implies a chord root line ending in IV that completes a down-by-fifths pattern (IV-VII is down a tritone, I-IV is down a fifth). The bass line implies secondary fifos that provide all-thirds harmony shapes. All-Thirds & All-Fourths Shapes All-thirds shape go naturally with resolution anchor lines of the form shown below left (meaning all-thirds after inverting the middle shape) and all-fourths shapes with the form shown below right. On the left, an obvious bass line 2-2-1 two scale steps below the anchor line provides all thirds between the two lines. The anchor-line fifhs may be split into thirds to make all-thirds chords going up from the bass line. The secondary fifths 2-6 and 1-5 make the splits that provide the chords (minor-7 chord 2-4-6-1 and major-7 chord 1-3-5-7). The same cannot be done with the tritone in this inversion because it is all whole tones in this part of the scale, but it can be done with its opposite inversion. The fourth 2-5 follows by carrying over bass note 2. This provides full chords. The anchor-line building blocks plus bass line 2-2-1 provide standard 3-note voicing of the chords that are very easy to play and, once learned, to invert into the more difficult to play all-thirds form if desired (the inversion amounts to transposing two notes by an octave). The difference is between a flow-based style that requires minimal, small finger movements and a shape-based style that requires lifting and moving all the fingers by large intervals. This is only one example of winkling all-thirds shapes out of this anchor line. Another one is suggested by a bass line 3-2-2 that is one scale step below anchor line 4-4-3. This bass line transposes

PKP 3/9/16 —32— ©copyright R.J.A. Buhr www.pianotheoryman.com up an octave into a treble line that it is all thirds above the same anchor wave. The chords that result are ninth and thirteeenth variations of the same chords, found without reference to chord symbols. On the right, all-fourths shapes follow from simply stacking fourths on top of the anchor wave implied by anchor line 5-4-3. The anchor line suggests this wave because the opening building block is the fourth 5-1 and the tritone is an augmented fourth. The final fourth of follows naturally. Stacking fourths on top yields very simple all-fourths shapes that can voice a variety of altered or extended chords on different root lines. All-fourths shapes also follow from extending fourths down from the anchor line.

all-thirds shaped formed 4 all-fourths shapes formed around anchor line 3 around anchor line I-Ⓘ-M $-Ⓘ-M 2 inverts upward 1 into all-thirds 1 1 7 7 7 7 primary 6 building 6 6 blocks 5 5 5 5

4 4 4 4 3 3 3 secondary 2 2 fifos 2

1 1 7

Building harmony shapes based on anchor lines to voice different kinds of chord progressions amounts to bottom-up chord substitution without reference to chord symbols. A huge difference between chord symbols may be implied by small differences in notes.

APPENDIX C: SCALES & MODES PKP is not a replacement for music notation, only a different way of looking at its harmonic aspects. Music notation is here to stay if for no other reason than the large legacy of music written in it. The 12-note chromatic scale on which PKP is based is not a fundamental musical scale historically and tritones from it are not fundamental intervals to human ears. The replacement of key signatures by tritones to define scales is way outside the comfort zone of anyone trained in music notation. Music notation makes it almost impossible even to comprehend this as a possibility. The concept of primary intervals of chords being their 3rd-7th intervals goes out the window. The concept is based on chord root lines being the primary defining lines of harmony, which isn’t true even in conventional chord-voicing practice because of chord substitution. In PKP, tritones are always primary intervals of harmony because of their scale-identifying ability, independently of their position in chords. This is realized in PKP by representing all the tritones of harmony in anchor lines. Fifos in anchor lines complete an anchor wave without regard to their position in whatever chords they may voice. Chord symbols tear apart the symbiotic relationship between melody and harmony that enables a few notes to imply many notes. Chord symbols specify harmony notes as if context did not exist, thus over-specifying harmony and adding misleading complexity. PKP restores the relationship by making building blocks fundamental instead of whole chords. Chord symbols are results not primary definers of harmony. PKP provides a simple mental model of the flow of intervals that music notation does not provide On the staff, accidentals obscure it. In chord progressions above the staff, chord symbols obscure it. Mastery of the piano through practice requires pianists to form personal mental models of the meaning of written music that can be very different. Classical pianists may see a mental image of the score scrolling by the mind’s eye. Jazz pianists may have different chords mentally lined up, or see harmonic movement in terms of shapes that preserve intervals as they move. When I saw the first glimmerings of the PKP view, I thought that it might be the view expert pianists form from practicing but discovered, in many discussions, that this is not so. Music notation makes it almost impossible to form this view based on music-notation-driven practicing. I was only able to form it because I resisted such practicing. Music notation with its sharps, flats and key signatures evolved to represent a view of musical perfection for classical modes and the pentatonic scales that are their sub-scales, which the piano does not support across the board. Any single octave may be tuned to support it but this leaves overlapped octaves sounding off relative to it. The musical perfection represented by sharps and flats can only be achieved independently of the piano by, for example, choirs. The piano’s chromatic scale covers all possibilities for tonic scales and intervals that can actually be played on the piano. The notes may be spread out by transpositions up or down one or more octaves but this introduces no different notes or intervals. Using this chromatic scale as a starting point for for understanding piano music may be unconventional but it is not wrong. The difference is a PKP provides context that enables a few notes to imply many notes. KEY-SIGNATURE SCALES (CLASSICAL MODES) The following summary of key signature scales is a helpful reference. It shows the sharped notes of sharp scales in red and the flatted notes of flat scales in blue. Although there are only five black piano keys, scales with six flats or sharps exist because of the scale-spelling rule that the same letter note cannot appear on both sides of a half-tone interval. For example, note B is C♭ in a 6-flats scale.

major rel. minor key sig. major scale (Ionian) (rel. Aeolian) C A empty C-D-E-F-G-A-B-C F D 1♭ F-G-A-B♭-C-D-E-F B♭ G 2♭ B♭-C-D-E♭-F-G-A-B♭ E♭ C 3♭ E♭-F-G-A♭-B♭-C-D-E♭ A♭ F 4♭ A♭-B♭-C-D♭-E♭-F-G-A♭ D♭ B♭ 5♭ D♭-E♭-F-G♭-A♭-B♭-C-D♭ ——————————————————————————————————— G♭ E♭ 6♭ G♭-A♭-B♭-C♭-D♭-E♭-F-G♭ F ♯ D ♯ 6 ♯ F♯-G♯-A♯-B-C♯-D♯-E♯-F♯ —same notes ——————————————————————————————————— B G ♯ 5 ♯ B-C♯-D♯-E-F♯-G♯-A♯-B E C ♯ 4 ♯ E-F♯-G♯-A-B-C♯-D♯-E A F ♯ 3 ♯ A-B-C♯-D-E-F♯-G♯-A D B 2 ♯ D-E-F♯-G-A-B-C♯-D G E 1 ♯ G-A-B-C-D-E-F♯-G C A empty C-D-E-F-G-A-B-C

A useful picture of all possible classical modes in terms of sequences of whole tones (W) and half tones (h) is shown next. The striking diagonal pattern in which half tones are highlighted in red text brings forward the fact that the scale half tones identify the scale tritone in a systematic way that enables the combination of the tritone and the half tones to define each scale by construction, knowing only the position of the anchor letter in the alphabet (“by construction” means without using the letter as a cue to recall the scale from personal memory). The half tones alternate between inside and outside the tritone, so knowing the Ionian position (outside) means knowing all the positions. The rest of the scale is whole tones.

Mode Scale Pattern (not all aligned vertically) Lydian WWW……………hWWh Ionian (major) WW…………..hWWWh Mixolydian WW…………hWWh…………W Dorian W…………hWWWh…………W Aeolian (natural minor) W………hWWh……………WW Phrygian hWWWh……………WW ————————————————————————————————— Locrian hWWh………………WWW

In numeric chromatic scale notation, the following relationships exist between relative Ionian

(major) and Aeolian (minor) modes. tonic scale relative scale (same symbols) relative scale (own symbols) 1-2-3-4-5-6-7-1 6-7-1-2-3-4-5-6 1-2-p3-4-5-p6-p7-1 1-2-p3-4-5-p6-p7-1 p3-4-5-p6-p7-1-2-p3 1-2-3-4-5-6-7-1 The tritones are highlighted to bring forward their notes as defining notes of these scales, determined by positions relative to the tonic. Tritone 2-p6 at top right is a minor third closer to the tonic than tritone 4-7 at top left. Tritone 4-7 at bottom right is a minor third farther away from the tonic than tritone 2-p6 at bottom left. These tritone positions define the modes whether they are relative or parallel. The middle column makes clear that the tritones in the relative modes are the same tritones in opposite inversions. Things are equally simple beyond classical modes, where scales acquire multiple tritones but this simplicity is obscured by sharps and flats to the extent that no one notices it. Sharps and Flats The sharps and flats of music notation follow from the fact that half tones are not equal pitch intervals but progressively increase in pitch size within an octave to make the top pitch double the bottom one. If the pitch increases are uniform for overlapped octaves, the pitches will be slightly misaligned. This misalignment is actual musical perfection in terms of the pitches of notes. The piano does not support the misalignment. A single piano octave can be tuned to perfection but this leaves the overlapped octaves sounding different. Equal temperament tuning makes them sound the same, meaning all equally imperfect. That the piano can get away with this is proof that the imperfection is a tiny nuance that tends to get lost in the flow of music. Musicians with trained musical ears in ensembles such as choirs or string quartets can learn the tiny nuances by ear for sustained notes, but only independently of the piano. Pianists cannot play them. This means that sharps and flats are not central to the conceptualization of music, only to tiny nuances of musical perfection that can be viewed as optional add-ons. MULTI-TRITONE MODES The book Modalogy provides a comprehensive summary of modes of the melodic minor, harmonic minor and harmonic major in terms that are directly comparable to PKP. Its scale notation uses RN symbols I-VII instead of plain numbers 1-7 for the major scale but this is purely cosmetic. The substantive difference is it uses the full range of sharps and flats of music notation freely in combination with these symbols. For example, the tritone anchored by D that is a component of many of the minor modes is understood in PKP as p3-6 in the chromatic scale of a tonic octave, and that’s it. Inversions in different places in harmonic cores are left to context. In Modalogy p3 could be ♭III or ♯

II and 6 could be VI or �VII in different scales, and different inversions are always explicitly notated. This mirrors music notation and therefore retains its misleading complexity. In Modalogy, seeing the possibility that the tritone represented in these different ways might be fundamental scale-defining object is effectively impossible. This is demonstrated by a discussion in the book of defining and non-defining notes of the many and various modes of double-tritone scales that never mentions the possibility that tritones might provide defining notes. Here follows a summary that demonstrates PKP coverage of the modes described in Modalogy.

Melodic Minor Word Tonic 1 in Tonic Frame Tritones plus Frame Mode (Modalogy) P.D @p7 frame p7-1-4-p7 1-p2-p3-4-5-6-p7-1 Jazz Phrygian A.M @4 frame 4-5-1-4 1-2-3-4-5-p6-p7-1 Jazz Mixolydian D.I @1 frame 1-2-5-1 1-2-p3-4-5-6-7-1 Melodic (or Jazz) Minor M.L @p2 tritone p2-p3-p6-p2 1-p2-p3-3-p5-p6-p7-1 Jazz Altered @5 tritone 5-6-2-5 1-2-3-p5-5-6-p7-1 Lydian Dominant I.P nil L.A @p3 tritone p3-4-p7-p3 1-2-p3-4-p5-p6-p7-1 Aeolian Diminished @6 tritone 6-7-3-1 1-2-3-p5-p6-6-7-1 Lydian Augmented

Harmonic Minor P..M @4 frame 4-p6-1-4 1-p2-3-4-5-p6-p7-1 Phrygian Dominant A..I @1 frame 1-p3-5-1 1-2-p3-4-5-p6-7-1 Harmonic Minor, Aeolian ♮7, Jazz Minor ♭6, Mohammedan @6 frame 6-1-3-1 1-2-3-4-p6-6-7-1 Ionian Augmented D..L @p2 tritone p2-3-p6-p2 1-p2-p3-3-p5-p6-6-1 Leading Tone Minor Diminished, Super Locrian �7 @3 tritone 3-5-7-3 1-p3-3-p5-5-6-7-1 Lydian Blues Major, Lydian ♯2 @5 tritone 5-p7-2-5 1-2-p3-p5-5-6-p7-1 Romanian, Dorian ♯4, Mishebarakh @p7 tritone p7-p2-4-p7 1-p2 -p3-4-p5-6-p7-1 Jazz Phrygian Diminished, Locrian ♮6 Harmonic Major P..M @4 frame 4-6-1-4 1-p2-3-4-5-6-p7-1 Jazz Phrygian Dominant, Mixolydian ♭2 @p6 frame p6-1-p3-p6 1-p2-p3-3-5-p6-p7-1 Altered Phrygian Dominant, Phrygian ♭4, Superlocrian ♮5, Superphrygian A..I @1 frame 1-3-5-1 D..L @p2 tritone p2-4-p6-p2 1-p2-p3-4-p5-p6-6-1 Leading Tone Major Diminished Locrian �7, Locrian Diminished-7 @3 tritone 3-p6-7-3 1-p3-3-p5-p6-6-7-1 Lydian Blues Augmented Lydian Augmented ♯2 @5 tritone 5-7–2-5 1-2-p3-p5-5-6-7-1 Lydian Melodic Minor, Lydian ♭3

@p7 tritone p7-2-4-p7 1-2-p3-4-p5-6-p7-1 Jazz Minor ♯4, Lydian Diminished

REFERENCES 1. Barta, The Source: The Dictionary of Contemporary and Traditional Scales, Hal Leonard (1995), for helping me to be sure I was not missing important scales. 2. Mingus Fakebook, Hal Leonard (1991) for the poster-child-for-misleading-complexity chord progression of Goodbye Porkpie Hat. 3. The Real Book, Sixth Edition (Hal Leonard) for Giant Steps. 4. The Ultimate Jazz Fakebook, Wong, Hal Leonard (1988) for All the Things You Are. 5. The Standards Real Book, Sher Music (2000) for I Got Rhythm. 6. Mehegan, Jazz Improvisation 1: Tonal and Rhythmic Principles, Watson-Guptil (1984), for Roman-numeral chord notation, and for teaching me (unintentionally) that it is not a solution for complex chromatic chord progressions, but a problem if pushed beyond its basic function of specifying chord root lines. 7. Wikipedia definition of Figured Bass Notation (FBN). 8. Eskelin, Lies My Music Teacher Told Me, Stage Three Publishing (1994) for insight into the nature of scales and musical “perfection” and for encouraging me to think outside the box. 9. Dmitri Tymoczko, A Geometry of Music (2011) for stimulating discussions of how to think about music from different angles. 10.Mark Levine, The Jazz Theory Book, Sher Music Co. (1995) for providing examples of well known jazz scales and harmonic forms using them in conventional notation against which to verify PKP coverage. 11.George Russell, The Lydian Chromatic Concept of Tonal Organization, http:// www.georgerussell.com/lc.html, for making me aware that PKP covers the concept, because nothing is changed by replacing the Ionian mode by the Lydian mode as the default reference major mode for any piece of music. 12.Edward Frenkel, Love and Math: The Heart of Hidden Reality, Perseus (2013) (on Kindle), for insight into symmetry and symmetry breaking that helped me keep PKP simple. 13. Jeff Brent with Schell Barkley, Modalogy — Scales, Modes & Chords: the Primordial Building Blocks of Music, Hal Leonard (2011), for the most comprehensive treatment I have found of the conceptual essence of scales, modes and chords as determined by music notation, presented at a higher level of abstraction than note symbols. It brings forward to the eye, stripped of the details of key-signature notation, the emisleading complexity of this conceptual essence. 14. Ross W. Duffin, How Equal Temperament Tuning Ruined Harmony (and Why You Should Care), W.W. Norton (2007), for an understanding of the nature of ET tuning.

ACKNOWLEDGMENTS I did not take this musical journey alone. I received comments and help from many people over the decade or so the ideas were germinating and consolidating. Music theorist Paul Steinbeck encouraged me to continue writing at a time when I was becoming discouraged about finding a way of bringing my ideas before the music community. Jazz pianist, teacher and composer Taylor Eigsti has been an inspiration to me. Although I have never been a piano student of his, I have learned much from him in sporadic discussions in person and by email. I am deeply grateful for his willingness to take time away from a busy schedule to engage in these discussions. A short series of piano lessons from SF jazz pianist Michael Parsons helped me to see more clearly the relationship between my ideas and standard jazz-piano practice. Thanks to SMT (Society for Music Theory) members Charise Hastings, Peter Shultz and Neil Newton for insightful email comments and encouragement following announcements of a website on an SMT mailing list. Thanks to music professor Robert Rawlins for email encouragement, and for helpful examples and comments. Aaron Blumenfeld and Susan Muscarella gave helpful courses at the Jazz School in Berkeley that provided many examples to chew on (including, from Susan Muscarella’s course, some interesting variations on Summertime in D minor that helped me understand ornamental scales). Amateur pianist and Jazz School Board Chair Susan Brand, and concert pianist and music entrepeneur Robert Taub, provided early encouragement. My piano teacher in San Francisco for several years, Ken Fishler, provided inspiration and how-to information on chord voicings, while patiently tolerating and responding to my question-everything approach. My first piano teacher in Ottawa, Canada, Sally Robinson, started me off right as an adult beginner by helping me to learn favorite harmonically sophisticated pieces by following her fingers on the keyboard without understanding anything about what I was doing except that it sounded right — being able to play these pieces, even if clumsily and by rote, enabled understanding to seep in gradually. I think that without this particular way of starting out — fingers on the keyboard first, written music later — I would not have started thinking about keyboard harmony in the way I did. It forced to my attention the large gap between simplicity on the keyboard and complexity of the full notation that represents it. I wore out the patience of many music professionals by bombarding them with unsolicited email requests for comments on my ideas. I would like to thank four in particular, who politely responded to numerous emails in spite of being uncomfortable with my ideas: Harry Likas, Dmitri Tymoczko. Jeff Brent and Daniel Glover. Their criticisms helped me change my explanations without changing my mind. Thanks to friends Marva Black, Mike Budde, Peter Marchant and Selinda Spugies for helpful insights on aspects of this material that helped me to understand how to get the ideas across better. My wife Sheila brings a spark to our life together, and an accurate musical ear that helps me know what does and does not sound good on the piano. My grandsons Joshua and Ethan Feiber provided encouragement and comments; Joshua set up the website www.pianotheoryman.com.

SOME COMMENTS FROM READERS OF EARLIER VERSIONS These comments provide a kind of history of the development of the PKP method (the dates on the left identify when the commenters first read earlier drafts of this material). The unconventionality of putting intervals on center stage independently of the notes of which they are formed has tended to get in the way of expert musicians accepting the ideas at all, let alone seeing them as simple, which has not been helped by the fact that it took me a long time to find the simple way of explaining the ideas that appears in this document. Taylor Eigsti’s comment shows I was a long way away in 2008; I include it because it shows he saw substantive content in the ideas, and I exercised due diligence in seeking out expert advice and paying attention to it. Later commenters saw improved versions in which I sought to remove the complexity he saw. However, it must be said that full fledged pianists continue to resist these ideas. It seems that once you know how to do it, a simpler way of thinking about it becomes unimportant. I continue to think the ideas are important for learning the piano in a less restrictive manner than is conventional but have not yet been able to find an audience. At the very least, the more recent comments show that the PKP method is more than just a fantasy of an over-enthusiastic amateur. (2011) Paul Steinbeck. Assistant Professor of Music Theory; Washington University, St. Louis “The hook ... , at least in my opinion, is that it's possible to attain a deep understanding of chords (and their constituent intervals) without recourse to Western notation. This has direct consequences for physical patterning, fingerings, etc. Essentially, your method combines the utility of a play-by-ear approach with the depth of a mathematically-informed theory of music.” (2009) Robert Rawlins. University Music Department Chair (Rowan University); jazz musician; teacher; author of several books on jazz “I became aware of Raymond Buhr's novel method for analyzing and voicing chromatic chord progressions in 2008 through a draft of a paper he wrote on the subject. I have kept up to date on developments of the method and we have had many email exchanges discussing issues of interpretation and application. I am a member of his intended target audience—a jazz musician who is not a pianist who needs to work out harmonic patterns on the piano from time to time. I am also a music teacher who has actually tried out aspects of his method on students. I can vouch from personal experience for his method's helpfulness in dealing with complex chromatic chord progressions.” (2009) Susan Brand. Board Chair, The Jazz School in Berkeley; amateur pianist “When Raymond Buhr consulted me about his theory of chromatic chord progressions, I was immediately struck by his ability to analyze and attempt to simplify this complex subject. Mr. Buhr brings a unique perspective and a great deal of enthusiasm, depth of understanding and originality of viewpoint to the subject. Over the years I have watched the continuous work that he has put into editing and rethinking his work. He has had ongoing consultations with many knowledgeable musicians/ teachers/editors and all have contributed to the development of the method described in this book. His ideas offer a way of understanding musical theory that will add greatly to the field and will be extremely helpful to musicians and music educators.” (2008) Taylor Eigsti. Jazz pianist, composer, teacher, former Artist in Residence at The Jazz School in Berkeley "Through the brilliant lens of an engineer, Raymond Buhr has laid out an analysis of harmony that is a unique and complex look at the right-brain from the left-brain's perspective.”