Ariel Kasler. A Thesis

Transcription

1 GIANT STEPS: CHORD SUBSTITUTIONS AND CHORD-SCALES FOR IMPROVISATION Ariel Kasler A Thesis Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of MASTER OF MUSIC May 2014 Committee: Chris Buzzelli, Advisor Nora Engebretsen, Committee Member

2 2014 Ariel Kasler All Rights Reserved

3 iii ABSTRACT Chris Buzzelli, Advisor This thesis examines harmonic possibilities inherent in John Coltrane s composition Giant Steps from the perspective of a jazz improviser. Chord-scales suggested by the chord progression and subsets of these chord-scales form a tool for analyzing segments of melodic improvisations over Giant Steps by the composer and by other significant jazz musicians. Common-tone possibilities created by the application of various chord-scales are explored theoretically and through analysis of performances. Additional chord substitutions that alter the harmonic rhythm are also studied. The chord substitution-based approach to improvisation and analysis presented here partially explains the continued interest in Giant Steps while also offering improvisers many ways to approach this composition.

4 iv ACKNOWLEDGEMENTS I would like to express my gratitude to the members of my committee, Prof. Chris Buzzelli and Dr. Nora Engebretsen, whose differing perspectives were essential in researching and writing a thesis that borrows from both jazz and traditional music theory. I could not have succeeded in this endeavor without their ongoing direction, feedback and advice. My thanks to Prof. David Bixler, who unintentionally inspired my inquiry of this topic, to Prof. Tad Weed for his encouragement and advice in pursuing this line of research, and to Prof. Jeff Halsey, Prof. Morgen Stiegler, Dr. Roger Schupp, and Dr. Ann Corrigan for sharing their knowledge and offering their support during my studies at Bowling Green State University. Thank you to all of my music teachers, past and present, especially Profs. Mick Goodrick and Hal Crook at Berklee College of Music, for shaping my conception of musical improvisation. I would like to thank my wife Bobbi Thompson, my parents Tamar Berkowitz and Dr. Jon Kasler, my sister Shira Kasler, and all of my extended Berkowitz-Kasler-Prinz family members not only for their love and support throughout my life, but also for instilling in me their belief in the value of higher education. Finally, I would like to extend a special thanks to my mother for her assistance and expertise in editing this document and to my wife for her love, caring, and insight that guides me through life.

5 v TABLE OF CONTENTS Page INTRODUCTION... 1 CHAPTER I. THE MAJOR SCALE AND ITS SUBSETS... 6 CHAPTER II. TONIC CHORD-SCALES AND THEIR SUBSETS CHAPTER III. DOMINANT AND IIm7 CHORD-SCALES AND THEIR SUBSETS CHAPTER IV. COMMON-TONE POSSIBILITIES The Augmented/Hexatonic Scale The Whole-Tone Scale and Its Fragments The Octatonic Scale and Its Fragments Common-Tone Sus4 Chords Pivot Chord Analysis and Its Implications CHAPTER V. HARMONIC RHYTHM POSSIBILITIES Delay/Anticipation Polyrhythmic Harmonic Motion Subtraction and Addition of Chords and 6-Measure cycles CHAPTER VI. SUMMARY AND APPLIED EXAMPLES REFERENCES APPENDIX A. CHORD-SCALES, THEIR SUBSETS, AND EXAMPLES FROM THE LITERATURE APPENDIX B. GLOSSARY... 86

6 vi LIST OF FIGURES Page 1.1. Harmonic analysis of Giant Steps The augmented/hexatonic scale as a chord-scale for the tonic and dominant chords in Giant Steps Kenny Garrett s application of the augmented/hexatonic scale Michael Brecker s application of the augmented/hexatonic scale Coltrane s application of the augmented scale in the first three measures of One Down, One Up Harmonization of a descending whole-tone scale Michael Brecker outlines a descending whole-tone progression The whole-tone scale and its subsets applied to Giant Steps Kenny Garrett s application of both whole-tone scales HW diminished/octatonic common-tone triads HW diminished/octatonic aggregates across modulations Three suspended 4 th chords over the changes Pivot chord analyses Pairing of major triads a minor 2 nd apart Kenny Garrett anticipates Excerpt from Flow by Omer Avital Dotted quarter polyrhythmic harmony The big V Adding chords to measures

7 vii measure cycle measure cycle Juxtaposing 3- and 6-measure cycles and 6-measure cycles in the chord progression Cycles reversed All minor 7 th chords All major 7 th chords All suspended 7 th chords Upper structure triads Minor 7 th, minor 6 th An alternate 3-measure cycle Static root motion Diminished 7 th approach... 37

8 1 INTRODUCTION The release of John Coltrane s album Giant Steps in 1960 is widely accepted as a milestone in jazz history. During the decades since, the title track has become a core part of the jazz repertoire, continually inspiring and challenging jazz musicians and audiences. This tune specifically, and the 3-tonic system, more generally, have held the attention of generations of jazz musicians and scholars. Giant Steps has been recorded again and again, and scholars continue to write articles and books about it and the 3-tonic system. According to David Ake, it is safe to say that no piece enjoys as much prestige or overall aura in all of jazz education as this one [ Giant Steps ] does. 1,2 While the 3-tonic system at the core of the tune Giant Steps has played a part in many of John Coltrane s improvisations and compositions before and after its release, the tune is unique two ways. First, it is Coltrane s only composition that is comprised entirely of the 3-tonic system rather than a re-harmonization of another tune from the jazz repertoire. Second, its compact 16-measure form and symmetry make it a perfect case study for investigating the possibilities inherent in the 3-tonic system. Like countless other jazz musicians, I have spent many hours practicing and studying Giant Steps, and since my first attempts at its performance while in high-school, have found that it has continued to offer me interest and challenge. That being said, the questions that led to the writing of this thesis began in 2012 when Professor David Bixler, Director of Jazz Activities at Bowling Green State University, showed me a chord substitution that George Coleman taught him. Coleman started the tune with a D# minor 7 th chord, followed by a C major 7 th chord. 1 David Ake, Jazz Cultures (Berkeley: University of California Press, 2002), Although Ake makes this claim in the context of a critique of jazz education s focus on nineteenth-century European aesthetics, this does not diminish the direct meaning of his statement.

9 2 This led me to wonder which other chord substitutions could be applied to Giant Steps. After some early exploration, I showed some of my ideas to Tad Weed, my piano teacher, who encouraged me to further pursue my inquiry, perhaps with the intention of publishing my findings in some form. This suggestion led me to seek a more serious and systematic approach to finding answers to my question. Around the same time, I was studying Set Theory with Professor Nora Engebretsen, and was thinking about the possible application of this approach to jazz improvisation. I began seeing parallels between Set Theory and the systematic methods used in books by my former guitar teacher, Mick Goodrick. 3 As a counterpart to the theoretical and abstract form of my inquiry, I also began listening to every recording of Giant Steps I could find and reading every relevant book I could get a hold of. In listening to recordings, I was trying to find out what pitch-class content significant jazz musicians used when playing Giant Steps. In the books, I hoped to find a discussion of the same issues. While many good and useful books have been written about Giant Steps and Coltrane s 3-tonic system, most turned out to be only marginally relevant to my specific interest in chord substitution. In Coltrane: A Player s Guide to His Harmony Walt Weiskopf and Ramon Ricker first discussed the origins and analyses of the 3-tonic system, and then presented several melodic patterns and etudes to be played over various related chord progressions. 4 In recent years, several books have been released specifically with guitarists in mind: 3 Mick Goodrick and Tim Miller, Creative Chordal Harmony for Guitar: Using Generic Modality Compression, ed. Jonathan Feist (Milwaukee, WI: Berklee Press, 2012), Walt Weiskopf and Ramon Ricker, Coltrane - a Player's Guide to His Harmony (New Albany, IN: Jamey Aebersold, 1991), 1-48.

10 3 In Coltrane Changes: Applications of Advanced Jazz Harmony for Guitar, Corey Christiansen presented an analysis, possible chord-voicings, patterns, re-harmonized tunes with written solos, and various segments and simplifications of the 3-tonic chord progression for practice. 5 In Giant Steps for Guitar: a Six-Stringer's Guide to Mastering Coltrane's Epic, Wolf Marshall suggested that the guitarist s approach to improvising over the changes to Giant Steps could be guided by chord voicing. He then discussed the construction of melodies, presented 49- patterns to be played over the first 4 measures, and discussed rhythmic approaches, ii-v-i patterns, and the construction of model solos. 6 In Giant Steps: An In-Depth Study of John Coltrane s Classic Joe Diorio included 17 composed solos over the changes to Giant Steps, each focusing on a specific concept, interval, or rhythmic figure. He also introduced several chord substitutions and re-harmonizations of the tune. Out of all the books mentioned so far, this one was the most relevant to the writing of this thesis. 7 While these books and other articles offer useful information and insight for any jazz musician, I have not found a systematic approach to chord substitutions for improvisation as it relates to the tune Giant Steps. In this thesis, I intend to offer such an approach. After consulting various books relating to jazz harmony, I decided not to base my approach on examples of chord substitutions from any 5 Corey Christiansen, Coltrane Changes: Applications of Advanced Jazz Harmony for Guitar (Pacific, MO: Mel Bay Publications, Inc., 2004), Wolf Marshall and John Coltrane, Giant Steps for Guitar: A Six-Stringer's Guide to Mastering Coltrane's Epic, Pap/Com ed. (Milwaukee, WI: Hal Leonard, 2009), Joe Diorio, Giant Steps: An In-Depth Study of John Coltrane s Classic (Van Nuys, CA: Alfred Music, 1998), 1-44.

11 4 specific source, but rather to derive the chord substitutions as subsets of chord-scales relating to each chord in Giant Steps. The most concise and relevant source I found for applying chord-scales to changes was not a book, but rather a page on Dan Haerle personal website. 8 Haerle s significance as a jazz educator is evident in the many pedagogical books he has published, through his decades of teaching at the University of North Texas, induction into the International Association of Jazz Education "Hall of Fame" and more. 9 On this page he organized the various scales by Scale Group and Chord Group. The chord group organization was the most valuable for my puposes, as it allowed me to list the various chord-scales that apply to each of the chords in Giant Steps. When listing subsets of various chord-scales, I did not exclude subsets that included avoid notes such as the 4 th degree of the major scale. Since there are many differing perspectives on this issue, I leave the reader the choice of excluding or not excluding these subsets rather than predetermining this decision. My study of improvisations over Giant Steps demonstrates that avoid notes are not always avoided, although there are always many possible interpretations of note choice. In Chapter I, I introduce the method used throughout the thesis, essentially listing collections (scales and chords) for improvisation that are familiar to jazz musicians using settheory, and focus on subsets of the diatonic major scale. In Chapter II, I survey alternate chordscales and their subsets as they relate to the tonic chords. In Chapter III, I do the same for the dominant chords and their related IIm7 chords. In Chapter IV, I address various collections that connect, rather than differentiate, the three tonics. Chapter V addresses chord substitution and 8 Dan Haerle, Scale Choices for Improvisation, January 7, 2014, accessed January 7, 2014, 9 Dan Haerle, Biographical Info, March 2, 2014, accessed March 2, 2014, Http: //danhaerle.com/bio.html.

12 5 possibilities that alter the harmonic rhythm of the progression, and therefore do not fall into any of the previous categories. In Chapter VI, I summarize my findings and apply some of them to the entire chord progression. The resulting compendium of chord substitutions over Giant Steps will serve several purposes. First, some chord substitutions make Giant Steps easier to play, increasing the possibilities and probabilities of successful and expressive improvisations. More significantly, each chord substitution creates a new progression, which can inspire an improviser to improvise different melodies, creating a wide palette of melodic possibilities otherwise unlikely to occur.

13 6 CHAPTER I. THE MAJOR SCALE AND ITS SUBSETS The art of improvising over chord changes has been approached with various methods. In this thesis I present an approach that focuses on the pitch-class content of the music. Treatment of other essential aspects of improvisation such as timbre, articulation and dynamics are beyond the scope of this thesis. Many issues that are closely related to pitch-class, but depend upon register (the realization of pitch-classes in specific octaves) issues such as contour, chromatic passing tones, and voicings are also outside the scope of this approach. The vital issue of rhythm will be dealt with only in relation to how it affects pitch-class content. Some improvisers and educators approach improvisation by using chord-scales, where the improviser selects notes from a chosen scale that fits a specific chord. According to Ake, The chord-scale method works perfectly on Giant Steps. 10 Others prefer to conceive of structures with fewer notes as their vehicle for improvisation. These structures can simply be the chord tones of the relevant chord, or they can comprise upper-structures triads or other collections that are derived at least in part from the extensions (9, 11, 13 and their alterations) of a chord. This thesis will consider both approaches. I will start with full chord-scales and then derive subsets of 3-6 notes from each scale. Those subsets that are easily labeled will then be used to form various chord substitutions. These chord substitutions could be used by a soloing performer as pitch-class material for improvisation, a comper as pitch-class material for voicings, an arranger for re-harmonization possibilities, a bass player as material for alternative chord roots, or an analyst as a means of explaining note choices in recordings and transcriptions. While 10 Ake, Jazz Cultures, 130.

14 7 this method could be applied to any chord progression, the changes to Giant Steps by John Coltrane are interesting in several ways and will be used here as a case study. 11 The chord changes in Giant Steps comprise three major tonal centers, separated by intervals of major thirds, dividing the octave into three equal parts. In the progression, every tonic chord is preceded by its own dominant chord. Some of the dominant chords are, in turn, preceded by a IIm7 chord, forming a cadence. This progression is typically analyzed as containing only direct modulations, with no pivot chords: 12 Figure 1.1. Harmonic analyses of Giant Steps 11 Unless otherwise noted, I use the term Giant Steps to refer to the tune by John Coltrane, not the entire album of which the tune is the title track. 12 Throughout this thesis I have used uppercase Roman numeral notation for easy conversion to chord symbol notation and in order to avoid ambiguity when discussing non-functional chords.

15 8 For the improvising musician, the tonic major scale is the default chord-scale choice when improvising over a major cadence. Jazz musicians tend to treat the 2-5 part of the progression in one of three ways. The first is to play a melodic line that clearly distinguishes the two different chords. The second is to ignore the presence of the IIm7 chord, and play melodic lines that outline the V7 chord. This second approach is especially common at fast tempi. The third approach is to ignore the V7 chord and outline only the IIm7 chord. This third method is especially associated with guitarist Pat Martino. 13 In light of these different approaches, for the purpose of this thesis I will treat the IIm7 and V7 chords as interchangeable. Transcriptions of John Coltrane s iconic solo on Giant Steps show that he used the major scale almost exclusively over tonic chords, and as the most frequent choice over the IIm7 and V7 chords. 14,15 Coltrane did include notes at intervals of b9 and #5 above the root of dominant chords, which will be discussed in Chapter III. Other non-diatonic notes function as chromatic passing tones, since they are preceded by a note one semi-tone lower and followed by a note one semi-tone higher, or vice-versa. As mentioned above, this thesis will not discuss the use of chromatic passing tones as approach notes or be-bop scales. Table A.1 in Appendix A is a list of easily labeled triads, seventh chords, pentatonic scales, and triad pairs. I have included a verbal description, a MOD12 set-class, and an example of a chord symbol (or other label) in C. These 3- to 6-note set-classes will form the basis for chord substitution possibilities presented in this thesis. Out of the many 3 to 6-note subsets of scales commonly used in the jazz idiom, relatively few have a common label that a jazz musician could instantly respond to as an improviser. Other 13 Pat Martino, Linear Expressions (Milwaukee, WI: Hal Leonard, 1983), Unless otherwise noted, John Coltrane s solo refers to the original master recording, or take 5 from the May 5, 1959 recording session. 15 David Demsey, John Coltrane Plays (Milwaukee, WI: Hal Leonard, 1996),

16 9 sets could certainly be useful in inspiring improvisation and are included in the tables, but this thesis will focus on the easily labeled (and therefore readily identifiable) sets. Common three-note sets include the four triads (major, minor, augmented, and diminished) and the suspended 4 th chord. Four-note sets include 7 th chords, suspended 7 th chords, and four specific triads with an added note, which have been included due to their prevalence in John Coltrane s Giant Steps solo and, consequently, in much of the jazz pedagogy related to Giant Steps. 16 Five-note sets include common 9 th chords and two distinct pentatonic collections. Six-note sets include different pairings of triads with no common-tones. Table A.2 lists all of the 3- to 6-note subsets of the major scale, attributing a chord symbol where appropriate. All uppercase roman numerals are used to allow for simple conversion to specific chord symbols. The MOD12 column shows which sets have an identical structure, differing only by transposition, while the MOD7 column shows similarity, allowing for variation of intervals between scale degrees. Under Roman Numeral Chord Symbols some chords a shaded grey. These are less common chords that nevertheless can be labeled with a chord symbol. In the remainder of this chapter, I provide one example from the literature of each scale or subset I could find. In analyzing these examples from the literature, I treated specific pitchclasses from any subset as identical regardless of order, octave, or rhythm. While I strived to find non-ambiguous examples, the presence of a melodic step can always be interpreted in more than one way. 16 1,2,3,5 and other instances of MOD7(0124), especially MOD12(0247) form alarge part of the vocabulary in Coltrane s solo. Many other soloists have adopted this vocabulary and a significant amount of jazz pedagogy is devoted to the subject. Vol. 1 of the Inside Improvisation Series by Jerry Bergonzi, for example, is entirely devoted to this topic.

17 10 When a specific subset is identified in an improvised solo, it is not necessary to claim that the performer was thinking in the terms presented here. Various conceptual and non-conceptual approaches could lead an improviser to play a certain collection of notes. In this thesis, I offer one possible interpretation of the performer s note choices. The examples in table A.9, as well as in tables A.10, A.11, and A.12 were taken from several recordings, published transcriptions and books by renowned jazz musicians. I started by analyzing a transcription of John Coltrane s original master take, and then moved on to analyze a transcription of Kenny Garrett s solo on his album triology. 17,18 I also included examples written by Joe Diorio in his book Giant Steps, which features predominantly suspended chords. 19 Transcribed segments of various other recordings are included. Segments from Bob Mintzer s recording from his album Twin Tenors featuring Michael Brecker, and Joel Frahm s solo performance on the CD included with The Jazz Musician s Guide to Creative Practicing, by David Berkman were of particular interest. 20,21 While some transcriptions for these recordings are available, their accuracy varies and I referred to the recording directly as the only reliable source. Examples were found in both table A.9 and table A.10 for the vast majority of the major scale subsets labeled with a Roman numeral chord symbol in table A.2. Considering the limited scope of my research, it is reasonable to assume that examples of the remaining subsets could also be found in the multitude of commercially available recordings. 17 David Demsey, John Coltrane Plays, Kenny Garrett, The Kenny Garrett Collection: Alto Saxophone Artist Transcriptions (Milwaukee, WI: Hal Leonard, 2004), Joe Diorio, Giant Steps: An In-Depth Study of John Coltrane's, John Coltrane, Giant Steps, performed by Bob Mintzer with Michael Brecker, RCA NJC , CD, David Berkman, The Jazz Musician's Guide to Creative Practicing (Petaluma, CA: Sher Music Company, 2007), included CD.

18 11 The notable exception to this generalization is the 6-note triad pair sets. While I found many excerpts that include six of the seven notes in a major scale over a chord, I only found one instance where these presented as two distinct triads, and have therefore included that example only. This may be expected during the first half of Giant Steps, since there are usually only two beats per chord, but I did find the absence of triad pairs during measures 9-16 somewhat surprising. While several examples in table A.9 include scale degree 4 (the avoid note), none of them are taken from Coltrane s solo. This distinction is not made in table A.10, partially due to the treatment of dominant and IIm7 chords in one group.

19 12 CHAPTER II. TONIC CHORD-SCALES AND THEIR SUBSETS Dan Haerle provided a lists of chord-scales appropriate for improvising over different chord types on his personal website. 22 Under Major Seventh Chords, he included Ionian, Lydian, 3 rd mode harmonic minor (major #5), 6 th mode harmonic minor (Lydian #9), Lydian-augmented (4 th mode melodic minor), augmented/hexatonic, major pentatonic (on 1,2 or 5), major blues, harmonic major, and 6 th mode harmonic major (Lydian-augmented #9). Due to the fast tempo at which Giant Steps is normally played, as well as its frequent modulations, scales other than major are used less commonly than in other settings. However, some examples do exist. This and the next chapter focus on additional chord-scales and their application to Giant Steps. Haerle s list of scales can be derived from the following shorter list as modes/rotations or subsets: major, melodic minor, harmonic minor, harmonic major, major blues, and augmented/hexatonic. As far as subsets for chord substitutions are concerned, the major blues scale will not offer any additional distinctive or useful subsets, since they are all either included in one of the major modes, or contain three consecutive semi-tones and therefore have no common label. Each of the remaining chord-scales is characterized by specific chord extensions and creates additional chord substitution possibilities. All of the following chord-scales include a root, a major third, and a major seventh. A perfect fifth is assumed unless noted otherwise. The Lydian chord-scale includes extensions natural 9 th, #11 th, and natural 13 th. The 3 rd mode of harmonic minor includes an augmented 5 th, and natural 9 th, 11 th, and 13 th. The 6 th mode of harmonic minor includes a #9 th, #11 th, and a natural 13 th. The Lydian-augmented chord-scale includes an augmented 5 th, a natural 9 th, #11 th, and natural 13 th. The augmented/hexatonic chord-scale, starting with a minor 3 rd interval includes 22 Dan Haerle, Scale Choices for Improvisation,

20 13 an augmented 5 th, a #9 th and a b13 th. Augmented/hexatonic is the only major 7 th chord-scale with only six notes. Harmonic major includes a natural 9 th and 11 th and a b13 th. Finally, the 6 th mode of harmonic major includes an augmented 5 th, a #9 th, #11 th and a natural 13 th. Since subsets by definition include only some of the notes in a given scale, the chordscale implications of each subset often allow for some ambiguity. For this reason, after each example in table A.11 I include a brief discussion of the chord-scale implications. As examples are scarcer, I will also include some that are not easily labeled with a chord symbol. Alternate major 7 th chord-scales are entirely avoided by Coltrane, but are clearly present in more recent improvisations. Research of a larger scope would undoubtedly find more examples; however, it seems likely that the above chord-scales and their subsets include much territory that has yet to be covered.

21 14 CHAPTER III. DOMINANT AND IIm7 CHORD-SCALES AND THEIR SUBSETS Returning to Dan Haerle s list of Scale Choices for Improvisation, under Dominant Seventh Chords, we find: Mixolydian, 5 th mode harmonic minor (Mixolydian b9,b13), Lydian, b7 (Lydian-dominant, 4 th mode melodic minor), Mixolydian b6 (5 th mode melodic minor), super Locrian (fully altered, 7 th mode melodic minor), whole tone, HW diminished (octatonic, starting with a half-step), major pentatonic (on 1 or b5), minor pentatonic (on 1), major blues, minor blues, 3 rd mode harmonic major, and 5 th mode harmonic major. These scales are derived as modes/rotations or subsets from this shorter list: major, harmonic minor, melodic minor, whole tone, octatonic, harmonic major, and major blues. Here too, the major and minor blues scales will not be analyzed for subsets. The augmented/hexatonic scale, starting with the interval a semi-tone, was surprisingly omitted from this otherwise comprehensive list. Due to its significance in relation to Giant Steps, it will be added to the analysis here. Under Minor Seventh Chords, Haerle includes: Dorian, Phrygian, Aeolian, harmonic minor, 4 th mode harmonic minor (Dorian #4), melodic minor, Dorian b2, minor pentatonic (on 1,2, or 5), minor blues, 3 rd mode harmonic major, and 4 th mode harmonic major. For the purpose of this thesis, V7 and IIm7 are treated as interchangeable, therefore, Dorian, melodic minor, Dorian b2, and minor pentatonic (on 1,2, or 5) become redundant. Last, scale choices for Half-Diminished Seventh Chords (IIm7b5) can also be considered as a substitution for IIm7, implying Vsus7(b9). Here Haerle included Locrian, 2 nd mode harmonic minor, Locrian #2 (6 th mode melodic minor), and the 2 nd mode harmonic major. The 2 nd modes of harmonic minor and harmonic major have been covered as the 5 th modes of these scales under Dominant Seventh Chords ; therefore I will not cover them again.

22 15 Each chord-scale is characterized by a specific combination of extension. All V7 chordscales (except for the augmented/hexatonic scale) include a root, major 3 rd, and b7 th, while the additional notes vary with each chord-scale. The 5 th mode harmonic minor scale includes a b9 th, natural 11 th, and b13 th of the V7 chord. Lydian b7 includes natural 9 th, #11 th, and natural 13 th. Mixolydian b6 includes a natural 9 th and 11 th and a b13 th. Super Locrian/altered includes b9 th, #9 th, #11 th /b5 th, and #5 th /b13 th, but no natural 5 th. The whole-tone chord-scale includes a natural 9 th, #11 th, and #5/b13 th, again with no natural 5. This chord-scale is also characterized by containing only six notes. HW diminished/octatonic scale includes a b9 th, #9 th, #11 th, and 13 th. Having 8 notes, this scale allows for many harmonic possibilities, as can be seen in Appendix A. The 3 rd mode of harmonic major includes a b9 th, #9 th, b13 th, and notably no 4 th /11 th. The 5 th mode of harmonic major includes a b9 th, and natural 11 th and 13 th. The augmented/hexatonic, which Haerle may have omitted from his list due to its lack of a b7 th, does contain a root, b9 th, major 3 rd, natural 11 th, #5, and natural 13 th. IIm7 chord-scales all include a root and a minor third, and all except for the harmonic minor include a minor 7 th. Phrygian mode includes a b9 th, natural 11 th, and b13 th. Aeolian includes a natural 9 th and 11 th, and a b13 th. Harmonic minor includes a natural 9 th and 11 th, and a b13 th. 4 th mode harmonic minor includes a natural 9 th, #11 th, and natural 13 th. Last, the IIm7b5 chord-scales all include a minor 3 rd, diminished 5 th and minor 7 th. Locrian includes a b9 th, natural 11 th, and b13 th, while Locrian #2 includes a natural 9 th and 11 th and a b13 th. Table A.12 presents examples from the literature of the above chord-scales and their subsets. Unlike alternate major 7 th chord-scales, which Coltrane avoided in his solo, a few

23 16 examples of the inclusion of the b9 th and the #5 th of the dominant chord are presented. As might be expected, the more recent recordings feature many more alternate dominant and IIm7 chordscales. That being said, the above list of chord-scales and the subsets that can be derived from the related tables in Appendix A contains so many subsets that exploring all options in performance would be a monumental task.

24 17 CHAPTER IV. COMMON-TONE POSSIBILITIES John Coltrane s approach to soloing over Giant-Steps focused on outlining the changes as clearly as possible. Generally, changes are clearly outlined by accentuating the notes that change at each modulation, in other words, the antithesis of the common-tone approach. Some more recent recordings have abandoned outlining every single chord, focusing instead on ways of connecting the changes. This chapter explores the use of common-tones in improvisation over Giant Steps. The Augmented/Hexatonic Scale The most well-known approach to common-tones over Giant Steps is the use of the B augmented/hexatonic Scale (same as Eb or G augmented/hexatonic Scale). This 6-note scale includes the notes of the 3 tonic major 7 th chords in Giant Steps, as well as augmented triads for the 3 dominant chords. Figure 4.1. The augmented/hexatonic scale as a chord-scale for the tonic and dominant chords in Giant Steps

25 18 This scale is often used as a single chord-scale that fits the entire form, much like a blues scale is applied over a blues form. Improvisers can either choose specific notes from the scale that outline the harmony or play melodic lines that may not fit the changes directly, but relate more generally to the harmonic landscape through pitch-class content. Kenny Garrett played the entire collection in the first phrase of his 10 th chorus (2:36): Figure 4.2. Kenny Garrett s application of the augmented/hexatonic scale Michael Brecker started his solo with this scale, maintaining a strict relationship to the chord of the moment: Figure 4.3. Michael Brecker s application of the augmented/hexatonic scale Somewhat ironically, it seems that John Coltrane, who played an important role in popularizing the use of the augmented/hexatonic scale in jazz, never applied it to his own composition, where it fits so perfectly.

26 19 Figure 4.4. Coltrane s application of the augmented scale in the first three measures of One Down, One Up 23 The Whole-Tone Scale and Its Fragments Another scale that is symmetrical at the interval of a major 3 rd is the whole-tone scale. One interpretation of the first half of Giant Steps is to view it as a harmonization of a descending whole-tone scale: Figure 4.5. Harmonization of a descending whole-tone scale A variation on this harmonization would substitute the 2 nd inversion V7 chords with a root position IImin7 chords, as outlined by Michael Brecker in the 3 rd chorus of his solo, bars 1-2 (0:54): 23 Walt Weiskopf and Ramon Ricker, The Augmented Scale in Jazz: A Player's Guide (New Albany, IN: Jamey Aebersold, 1993), 6-6.

27 20 Figure 4.6. Michael Brecker outlines a descending whole-tone progression As a chord-scale, WT0 (C,D,E,F#,G#,A#) is a fairly standard option over D7, Bb7, and F#7. The entire scale does not fit as neatly over major 7 th chords, but 4 out of the 6 notes in WT0 are present in the Eb, G, and B major chord-scales. Figure 4.7. The whole-tone scale and its subsets applied to Giant Steps Kenny Garrett offered an interesting and less strict example of applying both whole-tone scales to the changes of Giant Steps in the first half of the 5 th chorus of his solo (1:29). His use

28 21 of a rhythmic motive, resolution of dissonance, and confident delivery allowed him to include some uncommon extensions over the chord changes: Figure 4.8. Kenny Garrett s application of both whole-tone scales The Octatonic Scale and Its Fragments Unlike the previous two scales, the octatonic scale divides the octave into 4 equal parts rather than 3, and therefore may be considered a counter-intuitive choice. The reason for its inclusion lies in the ascending minor 3 rd motion from one tonic to the dominant of the key a major 3 rd lower, and this movement s diminished implications. During the ascending major-3 rd modulation, the Lydian and 6 th mode melodic minor chord-scales allow for additional common chord possibilities. In the following figure, a B major triad serves as a common chord in the first measure, as a subset of both B major chord-scales and D HW diminished/octatonic. D7 is a subset of both 6 th mode harmonic minor on Eb and D HW diminished/octatonic, while F7 is a subset of both Eb Lydian and D HW diminished/octatonic. I then transposed these relationships so that they apply to the entire progression:

29 22 Figure 4.9. HW diminished/octatonic common-tone triads A contrasting approach alternates two 4-note chords at the interval of a tritone, resulting in a complete HW diminished/octatonic scale. This application results in a common chord-scale, rather than a common chord.

30 23 Figure HW diminished/octatonic aggregates across modulations Common-Tone Sus4 Chords The major scale contains three major 3 rd intervals within it (scale degrees 1-3, 4-6, and 5-7). As a result, each of the major 3 rd modulations in Giant Steps maintains 3 common-tones. These tones form a sus4 chord: B and G Bsus4 G and Eb Gsus4 Eb and B Ebsus4 Since Bsus4 is a subset of both the B major scale and the G major scale, it can be played over 6 of the 9 chords in Giant Steps (Am7, D7, G, C#m7, F#7, B). In addition, Bsus4 is also a subset of the Bb altered scale, one of the chord-scales for Bb7. The same principle can be applied to the remaining suspended chords: D7 Ebsus4

31 24 F#7 Gsus4 Figure Three suspended 4 th chords over the changes

32 25 Pivot Chord Analysis and Its Implications The modulations in Giant Steps are typically analyzed as direct modulations, as discussed in Chapter I. Another interpretation of ascending major 3 rd modulations is possible: Figure Pivot chord analysis In this analysis, the tonic major chords also function as chords borrowed from the parallel minor of the key that follows. The result is a minor pre-dominant to dominant progression, resolving unexpectedly to a major tonic, which in turn also functions as a minor pre-dominant chord in a new key. A chord-scale that is often applied to minor pre-dominant and dominant chords is the tonic s harmonic minor. In this case, the 6 th and 5 th modes/rotations have been applied respectively. This interpretation displaces the point at which the pitch-content changes. A subset of this scale, the pairing of two major triads a minor 2 nd apart, can work nicely:

33 26 Figure Pairing of major triads a minor 2 nd apart The intensity created by the rapid modulations of Giant Steps is one of its main characteristics, and part of the composition s appeal. The slower harmonic movement in the second half of the tune offers some contrast to this intensity. The common-tone possibilities explored in this chapter enable a performer to increase this contrast further. Proving that something hasn t been done is a difficult task, but I have not found any explicit examples of most of the techniques discussed in this chapter.

34 27 CHAPTER V. HARMONIC RHYTHM POSSIBILITIES In the material introduced up to this point, I avoided altering the number of chords in the progression, their starting point, and their end point. In this chapter, I will examine several approaches to altering the harmonic rhythm of Giant Steps. Delay/Anticipation A tool commonly used by improvisers to create interest and tension is the delay and/or anticipation of a chord change. The duration of any chord can start or end earlier or later than notated, as in the following example from Kenny Garrett s solo, 4 th chorus, m (1:27): Figure 5.1. Kenny Garrett anticipates Polyrhythmic Harmonic Motion The length of each chord change can be altered in a consistent manner that implies a different meter than the notated meter. Flow, a composition by Omer Avital that uses the chord progression from Giant Steps, starts with two 5-beat phrases, creating conflict with the 4/4 meter:

35 28 Figure 5.2. Excerpt from Flow, by Omer Avital 24 This technique can be applied to the entire chord progression. The following example applies a consistent dotted quarter pulse to the chord changes: Figure 5.3. Dotted quarter polyrhythmic harmony Subtraction and Addition of Chords In addition to the common-tone examples presented in Chapter IV, another way of slowing down the harmonic rhythm is to simply ignore some of the chord changes by lengthening the duration of one chord. Since Coltrane first used major 3 rd modulations as an 24 Omer Avital, Flow, performed by OAM Trio, Fresh Sound Records, FSNT-136, CD, 2002.

36 29 expansion of a dominant-tonic progression, a reversal of this expansion would result in a return to such a progression, as Michael Wolff discussed in a YouTube video. 25 The following figure applies this idea to the first three bars of the progression: Figure 5.4. The big V Alternatively, chords may be added to the progression, especially in the second half of the progression, which features slower harmonic motion. Common additions include approaching a chord by its dominant chord, a leading-tone diminished chord, a tritone substitute dominant chord, common-tone diminished chord, a diatonic chord a step below or above the target chord, and a parallel chromatic approach chord from below or above: Figure 5.5. Adding chords to measures Michael Wolff, Giant Steps Improvisational Concepts, YouTube video, 17:11, posted May 23, 2013, accessed February 18, 2014,

37 30 3- and 6-Measure Cycles The chord-progression in Giant Steps can also be understood as being derived from two related chord cycles, a 3-measure cycle and a 6-measure cycle: Figure measure cycle Figure measure cycle When juxtaposed, every two-measure section of these progressions includes 4 beats of identical harmony and 4 beats of differing harmony: Figure 5.8. Juxtaposing 3- and 6-measure cycles

38 31 With the exception of measure 16, the entire chord progression of Giant Steps can be derived from alternating between these two cycles. While the following figure labels the cycles as alternating, there is always a four-beat overlap between cycles: Figure and 6-measure cycles in the chord progression Keeping this analysis in mind, a possible re-harmonization could feature chords from the cycle not chosen in the original progression: Figure Cycles reversed

39 32 Altering the harmonic rhythm creates countless possibilities. Each of the techniques presented in this chapter can be applied flexibly to any part of the form or to the entire progression. Much like the techniques presented in Chapter IV, these techniques can also be used to create relief from the intensity of rapid modulations. These techniques differ from those in Chapter IV in that they can also be used to increase the musical intensity. Delay/anticipation creates instability, especially when suggesting a conflicting meter. The addition of chords naturally increases the intensity of the piece, just as superimposing the 3-measure cycles over 6- measure cycle sections would. Examples of performers delaying and/or anticipating single chords are plentiful in recent recordings, as are cases of the inclusion of phrases that imply polyrhythms. Subtraction and addition of chords is also fairly common, but since there are so many chords that could be added, it is likely that not all possibilities have been covered. I have heard recordings that may suggest the substitution of 3- and 6-measure cycles, but never very explicitly.

40 33 CHAPTER VI. SUMMARY AND APPLIED EXAMPLES When combined, the chord substitution possibilities introduced in the preceding five chapters and in the tables in Appendix A allow for near-infinite variations on the chord changes to Giant Steps. Since chord changes do not determine an improviser s phrasing choices, each variation may inspire countless melodies. The scope of this thesis allowed for an analysis of segments of a small portion of commercially available recordings of Giant Steps. The analysis method used here could be applied to segments of any other solo over Giant Steps, especially those that do not feature primarily chromatic approach notes and their targets. It should be noted that chord substitutions do not inherently create additional difficulty for the improviser. Some substitutions actually result in a progression that is more manageable for a jazz student just learning the piece. The descending whole-tone scale harmonization in Figure 4.5 is one example of a progression that may be less difficult to master. Slowing down the harmonic motion of measures 1-8 through the application of The Big V (Figure 5.4) or the 6- measure cycle (Figure 5.4) are additional methods for simplifying the progression, thereby increasing the likelihood of a student s successful improvisation. From an artistic standpoint, the presence of choice is a precondition for creativity and expression. While the near-infinite possibilities for the improviser presented here may seem overwhelming, the intention is that an individual would explore some of the options and gravitate towards a few that he or she finds esthetically pleasing. As a conclusion, I offer the following applied examples of the entire progression that I personally find esthetically pleasing to both the mind and the ear:

41 34 Figure 6.1. All minor 7 th chords Figure 6.2. All major 7 th chords

42 35 Figure 6.3. All suspended 7 th chords Figure 6.4. Upper structure triads

43 36 Figure 6.5. Minor 7 th, minor 6 th Figure 6.6. An alternate 3-measure cycle

44 37 Figure 6.7. Static root motion Figure 6.8. Diminished 7 th approach

45 38 REFERENCES Ake, David. Jazz Cultures. Berkeley: University of California Press, Avital, Omer. Flow. Performed by OAM Trio. Fresh Sound Records FSNT-136. CD Berkman, David. The Jazz Musician's Guide to Creative Practicing: Notes on the Difficult, Humorous, Endless Path of Becoming a Better Improvising Musician. Petaluma, CA: Sher Music, Christiansen, Corey. Coltrane Changes: Applications of Advanced Jazz Harmony for Guitar. Pacific, MO: Mel Bay Publications, Inc., Coltrane, John. Giant Steps. Performed by Bob Mintzer with Michael Brecker. RCA Records NJC RCA. CD Coltrane, John. Giant Steps. Performed by John Coltrane. Rhino Records R CD Coltrane, John. Giant Steps. Performed by Kenny Garrett. Warner Bros CD Demsey. David. John Coltrane Plays Giant Steps: Tenor Saxophone (Artist Transcriptions). Milwaukee, WI: Hal Leonard, Diorio. Joe. Giant Steps: An In-Depth Study of John Coltrane s Classic. Van Nuys, CA: Alfred Music, Garrett, Kenny, and Timo Shanko. The Kenny Garrett Collection: 14 Authentic Transcriptions Including Countdown, Koranne Said, Night and Day, Wayne's Thang, and Wooden Steps. Artist Transcriptions. Milwaukee, Wisconsin: Hal Leonard, Haerle, Dan. Biographical Info. March 2, Accessed March 2, Haerle, Dan. Scale Choices for Improvisation. January 7, Accessed January 7,

46 39 Marshall, Wolf. Giant Steps for Guitar: A Six-Stringer's Guide to Mastering Coltrane's Epic. Pap/Com ed. Milwaukee, WI: Hal Leonard, Martino, Pat. Linear Expressions - Pat Martino. Milwaukee, WI: Hal Leonard, Weiskopf, Walt, and Ramon Ricker. The Augmented Scale in Jazz: A Player's Guide. New Albany, IN: Jamey Aebersold, Weiskopf, Walt and Ramon Ricker. Coltrane: A Player s Guide to His Harmony. New Albany, IN: Jamey Abersold, 1991.

47 40 APPENDIX A. CHORD-SCALES, THEIR SUBSETS, AND EXAMPLES FROM THE LITERATURE Table A.1. Choice of subsets Full Name Mod 12 Example of label with C as root 3-note (triads) Suspended 4 th triad (027) Csus4 Diminished triad (036) Cdim Minor triad (037)a Cmin Major triad (037)b C Augmented triad (048) Caug 4-note (7 th chords) Diminished (major 7 th ) (0147)a bvidim(maj7) Minor (major 7 th ) (0148)a Cmin(Maj7) Major 7 (#5) (0148)b CMaj7(#5) Major 7 (b5) (0157)a Cmaj7(b5) Major 7 (0158) Cmaj7 Minor add 2 (0237)a Cmin(add2) Major add 4 (0237)b C(add4) Major add 2 (0247)a C(add2) Minor add 4 (0247)b Cmin(add4) Dominant 7 (#5) (0248) C7(#5)

48 41 Full Name Mod 12 Example of label with C as root Dominant 7, suspended 4th (0257) C7sus4 Minor 6/minor 7, flat 5 or half-diminished (0258)a Cmin6 or Amin7(b5) Dominant 7 (0258)b C7 Dominant 7 (b5) (0268) C7(b5) or F#7(b5) Minor 7/major 6 (0358) Cmin7or Eb6 Diminished 7 th (0369) Cdim7 5-note (9 th chords) Minor 9, major 7 (01348) Cmin9(Maj7) Major 9 (01358)a CMaj9 Minor 9 (01358)b Cmin9 Dominant 7, flat 9 (01369)b C7(b9) Dominant 7, sharp 9 (01469)b C7(#9) Dominant 7, flat 9, flat 5 (02368)a C7(b5,b9) Minor 9 (b5)/ dominant 7, flat 9, sharp 5 (02458)a Cmin9(b5) or D7(b5,b9) Augmented, major 7, natural 9 (02458)b CMaj9(#5) Dominant 7, natural 9, flat 5/dominant 7, natural 9, (02468) C9(b5) or D9(#5) sharp 5, whole-tone segment Dominant 7, natural 9 (02469) C9 5-note pentatonic scales Major pentatonic or minor 7 pentatonic (02479) C pent or Amin pent. Minor 6 pentatonic (01368)b Cmin6 pent.

49 42 Full Name Mod 12 Example of label with C 6-note (triad pairs with no common-tones) as root Diminished triad paired with a diminished triad one (013467) Cdim and C#dim triads semitone higher Diminished triad paired with a minor triad one semitone (013468)a Cdim and Dbmin triads higher Major triad paired with a diminished triad one tone (013468)b C and Ddim triads higher Major triad paired with a diminished triad three (013469)a C and D#dim triads semitones higher Diminished triad paired with a minor triad one tone (013469)b Cdim and Dmin triads higher Major triad paired with a major triad one semitone (013478)b C and Db triads higher Diminished triad paired with a major triad one semitone (013568)a Cdim and Db triads higher Minor triad paired with a diminished triad one tone (013568)b Cmin and Ddim triads higher Diminished triad paired with an augmented triad one (013569)a Cdim and Dbaug triads semitone higher Minor triad paired with a major triad one semitone (013578) Cmin and Db triads higher

50 43 Full Name Mod 12 Example of label with C as root Augmented triad paired with a major triad one tone (013579)b Caug and D triads higher Minor triad paired with a minor triad a tritone apart Major triad paired with a major triad a tritone apart Augmented triad paired with a minor triad one tone (013679)a Cmin and F#min triads (013679)b C and F# triads (014579)b Caug and Dmin triads higher Minor triad paired with a major triad a tritone apart (014679) Cmin and F# triads Diminished triad paired with a diminished triad one tone (023568) Cdim and Ddim triads higher Major triad paired with a major triad one tone higher Minor triad paired with a minor triad one tone higher Minor triad paired with a major triad one tone higher (023579)b C and D triads (023579)a Cmin and Dmin triads (023679)a Cmin and D triads Major triad paired with a minor triad one tone higher (024579) C and Dmin triads Table A.2. Major Scale subsets Major Scale Degree MOD7 MOD12 Roman numeral chord symbol 3-note 1,2,3 (012) (024) 1,2,4 (013)a (025)a 1,2,5 (014) (027) Vsus4 1,2,6 (013)b (025)b

51 44 Major Scale Degree MOD7 MOD12 Roman numeral chord symbol 1,2,7 (012) (013)a 1,3,4 (013)b (015)b 1,3,5 (024) (037)b I 1,3,6 (024) (037)a VImin 1,3,7 (013)b (015)a 1,4,5 (014) (027) Isus4 1,4,6 (024) (037)b IV 1,4,7 (014) (016)a 1,5,6 (013)a (025)a 1,5,7 (013)b (015)b 1,6,7 (012) (013)b 2,3,4 (012) (013)b 2,3,5 (013)a (025)a 2,3,6 (014) (027) VIsus4 2,3,7 (013)b (025)b 2,4,5 (013)b (025)b 2,4,6 (024) (037)a IImin 2,4,7 (024) (036) VIIdim 2,5,6 (014) (027) IIsus4 2,5,7 (024) (037)b V 2,6,7 (013)a (025)a 3,4,5 (012) (013)a

52 45 Major Scale Degree MOD7 MOD12 Roman numeral chord symbol 3,4,6 (013)a (015)a 3,4,7 (014) (016)b 3,5,6 (013)b (025)b 3,5,7 (024) (037)a IIImin 3,6,7 (014)b (027) IIIsus4 4,5,6 (012) (024) 4,5,7 (013)a (025)a 4,6,7 (013)b (026)b 5,6,7 (012) (024) 4-note 1,2,3,4 (0123) (0135)b 1,2,3,5 (0124)a (0247)a I(add2) 1,2,3,6 (0124)b (0247)b VImin(add4) 1,2,3,7 (0123) (0135)a 1,2,4,5 (0134) (0257) V7sus4 1,2,4,6 (0135) (0358) IImin7 or IV6 1,2,4,7 (0124)a (0136)a 1,2,5,6 (0134) (0257) II7sus4 1,2,5,7 (0124)b (0237)b V(add4) 1,2,6,7 (0123) (0235) 1,3,4,5 (0124)b (0237)b I(add4) 1,3,4,6 (0135) (0158) IVmaj7

53 46 Major Scale Degree MOD7 MOD12 Roman numeral chord symbol 1,3,4,7 (0123) (0156) 1,3,5,6 (0135) (0358) VImin7 or I6 1,3,5,7 (0135) (0158) IMaj7 1,3,6,7 (0124)a (0237)a VImin(add2) 1,4,5,6 (0124)a (0247)a IV(add2) 1,4,5,7 (0134) (0157)b IMaj7(sus4) 1,4,6,7 (0124)b (0137)b IV(add#4) 1,5,6,7 (0123) (0135)b 2,3,4,5 (0123) (0235) 2,3,4,6 (0124)a (0237)a IImin(add2) 2,3,4,7 (0124)b (0136)b VIIdim(add4) 2,3,5,6 (0134) (0257) VI7sus4 2,3,5,7 (0135) (0358) IIIm7 or V6 2,3,6,7 (0134) (0257) III7sus4 2,4,5,6 (0124)b (0247)b IImin(add4) 2,4,5,7 (0135) (0258)b V7 2,4,6,7 (0135) (0258)a VIImin7(b5) 2,5,6,7 (0124)a (0247)a V(add2) 3,4,5,6 (0123) (0135)a 3,4,5,7 (0124)a (0137)a IIImin(addb2) 3,4,6,7 (0134) (0157)a IVMaj7b5 3,5,6,7 (0124)b (0247)b IIImin(add4)

54 47 Major Scale Degree MOD7 MOD12 Roman numeral chord symbol 4,5,6,7 (0123) (0246) 5-note 1,2,3,4,5 (01234) (02357)b 1,2,3,4,6 (01235) (01358)b IImin9 1,2,3,4,7 (01234) (01356) 1,2,3,5,6 (01245) (02479) I pent. Or VImin pent. 1,2,3,5,7 (01235) (01358)a IMaj9 1,2,3,6,7 (01234) (02357)a 1,2,4,5,6 (01245) (02479) IV pent. or IImin pent. 1,2,4,5,7 (01245) (01368)a 1,2,4,6,7 (01235) (02358)a 1,2,5,6,7 (01234) (02357)b 1,3,4,5,6 (01235) (01358)a IVMaj9 1,3,4,5,7 (01245) (01378)a 1,3,4,6,7 (01245) (02378)a 1,3,5,6,7 (01235) (01358)b VImin9 1,4,5,6,7 (01234) (01357)b 2,3,4,5,6 (01234) (02357)a 2,3,4,5,7 (01235) (02358)b 2,3,4,6,7 (01245) (01368)b IImin6 pent. 2,3,5,6,7 (01245) (02479) V pent or IIImin pent. 2,4,5,6,7 (01235) (02469) V9

55 48 Major Scale Degree MOD7 MOD12 Roman numeral chord symbol 3,4,5,6,7 (01234) (01357)a 6-note 1,2,3,4,5,6 (012345) (024579) I & IImin triads 1,2,3,4,5,7 (012345) (013568)a VIIdim and I triads 1,2,3,4,6,7 (012345) (013568)b VImin and VIIdim triads 1,2,3,5,6,7 (012345) (024579) V and VImin triads 1,2,4,5,6,7 (012345) (023579)b IV and V triads 1,3,4,5,6,7 (012345) (013578) IIImin and IV triads 2,3,4,5,6,7 (012345) (023579)a IImin and IIImin triads Table A.3. Subsets of the harmonic minor scale Harmonic Minor Scale Degree MOD7 MOD12 Roman Numeral Chord Symbols 3-note 1,2,3 (012) (013)b 1,2,4 (013)a (025)a 1,2,5 (014) (027) Vsus4 1,2,6 (013)b (026)b 1,2,7 (012) (013)a 1,3,4 (013)b (025)b 1,3,5 (024) (037)a Imin 1,3,6 (024) (037)b bvi 1,3,7 (013)b (014)

56 49 Harmonic Minor Scale Degree MOD7 MOD12 Roman Numeral Chord Symbols 1,4,5 (014) (027) Isus4 1,4,6 (024) (037)a IVmin 1,4,7 (014) (016)a 1,5,6 (013)a (015)a 1,5,7 (013)b (015)b 1,6,7 (012) (014)b 2,3,4 (012) (013)a 2,3,5 (013)a (015)a 2,3,6 (014) (016)b 2,3,7 (013)b (014)b 2,4,5 (013)b (025)b 2,4,6 (024) (036) IIdim 2,4,7 (024) (036) VIIdim 2,5,6 (014) (016)b 2,5,7 (024) (037)b V 2,6,7 (013)a (036) #Vdim 3,4,5 (012) (024) 3,4,6 (013)a (025)a 3,4,7 (014) (026)b 3,5,6 (013)b (015)b 3,5,7 (024) (048) biiiaug 3,6,7 (014)b (037)a bvimin

57 50 Harmonic Minor Scale Degree MOD7 MOD12 Roman Numeral Chord Symbols 4,5,6 (012) (013)b 4,5,7 (013)a (026)a 4,6,7 (013)b (036) IVdim 5,6,7 (012) (014)a 4-note 1,2,3,4 (0123) (0235) 1,2,3,5 (0124)a (0237)a Imin(add2) 1,2,3,6 (0124)b (0137)b bvi(add#4) 1,2,3,7 (0123) (0134) 1,2,4,5 (0134) (0257) V7sus4 1,2,4,6 (0135) (0258)a IImin7(b5) or IVmin6 1,2,4,7 (0124)a (0136)a 1,2,5,6 (0134) (0157)a bvimaj7(b5) 1,2,5,7 (0124)b (0237)b V(add4) 1,2,6,7 (0123) (0236)b 1,3,4,5 (0124)b (0247)b Imin(add4) 1,3,4,6 (0135) (0358) IVmin7 1,3,4,7 (0123) (0146)a 1,3,5,6 (0135) (0158) bvimaj7 1,3,5,7 (0135) (0148)a Imin(Maj7) 1,3,6,7 (0124)a (0347) 1,4,5,6 (0124)a (0237)a IVmin(add2)

58 51 Harmonic Minor Scale Degree MOD7 MOD12 Roman Numeral Chord Symbols 1,4,5,7 (0134) (0157)b 1,4,6,7 (0124)b (0147)b IVmin(add#4) 1,5,6,7 (0123) (0145) 2,3,4,5 (0123) (0135)a 2,3,4,6 (0124)a (0136)a IIdim(addb2) 2,3,4,7 (0124)b (0236)b 2,3,5,6 (0134) (0156) 2,3,5,7 (0135) (0148)b biiimaj7(#5) 2,3,6,7 (0134) (0147)b bvimin(add#4) 2,4,5,6 (0124)b (0136)b IIdim(add4) 2,4,5,7 (0135) (0258)b V7 2,4,6,7 (0135) (0369) VIIdim7 2,5,6,7 (0124)a (0147)a bvidim(maj7) 3,4,5,6 (0123) (0135)b 3,4,5,7 (0124)a (0248)a biiiaug(add2) 3,4,6,7 (0134) (0258)a IVmin7(b5) 3,5,6,7 (0124)b (0148) bvimin(maj7) 4,5,6,7 (0123) (0236)a IVdim(add2) 5-note 1,2,3,4,5 (01234) (02357)a 1,2,3,4,6 (01235) (02358)a 1,2,3,4,7 (01234) (01346)a

59 52 Harmonic Minor Scale Degree MOD7 MOD12 Roman Numeral Chord Symbols 1,2,3,5,6 (01245) (01568)b 1,2,3,5,7 (01235) (01348) Imin9(Maj7) 1,2,3,6,7 (01234) (01347)b 1,2,4,5,6 (01245) (01368)b IVmin6 pent. 1,2,4,5,7 (01245) (01368)a 1,2,4,6,7 (01235) (01369)a 1,2,5,6,7 (01234) (01457)a 1,3,4,5,6 (01235) (01358)b IVmin9 1,3,4,5,7 (01245) (01468)a 1,3,4,6,7 (01245) (01469)a 1,3,5,6,7 (01235) (01458)a 1,4,5,6,7 (01234) (01457)b 2,3,4,5,6 (01234) (01356) 2,3,4,5,7 (01235) (02458)b 2,3,4,6,7 (01245) (01369)a 2,3,5,6,7 (01245) (01478) 2,4,5,6,7 (01235) (01469)b V7(b9) 3,4,5,6,7 (01234) (02458)a 6-note 1,2,3,4,5,6 (012345) (013568)b Imin & IIdim triads 1,2,3,4,5,7 (012345) (013468)a VIIdim and Imin triads 1,2,3,4,6,7 (012345) (013469)a bvi and VIIdim triads

60 53 Harmonic Minor Scale Degree MOD7 MOD12 Roman Numeral Chord Symbols 1,2,3,5,6,7 (012345) (013478)b V and bvi triads 1,2,4,5,6,7 (012345) (023679)a IVmin and V triads 1,3,4,5,6,7 (012345) (014579)b biiiaug and IVmin triads 2,3,4,5,6,7 (012345) (013569)a IIdim and biiiaug triads Table A.4. Subsets of the melodic minor scale Melodic Minor Scale Degree MOD7 MOD12 Roman Numeral Chord Symbols 3-note 1,2,3 (012) (013)b 1,2,4 (013)a (025)a 1,2,5 (014) (027) Vsus4 1,2,6 (013)b (025)b 1,2,7 (012) (013)a 1,3,4 (013)b (025)b 1,3,5 (024) (037)a Imin 1,3,6 (024) (036) VIdim 1,3,7 (013)b (014) 1,4,5 (014) (027) Isus4 1,4,6 (024) (037)b IV 1,4,7 (014) (016)a 1,5,6 (013)a (025)a 1,5,7 (013)b (015)b

61 54 Melodic Minor Scale Degree MOD7 MOD12 Roman Numeral Chord Symbols 1,6,7 (012) (013)b 2,3,4 (012) (013)a 2,3,5 (013)a (015)a 2,3,6 (014) (016)b 2,3,7 (013)b (014)b 2,4,5 (013)b (025)b 2,4,6 (024) (037)a IImin 2,4,7 (024) (036) VIIdim 2,5,6 (014) (027) IIsus4 2,5,7 (024) (037)b V 2,6,7 (013)a (025)a 3,4,5 (012) (024) 3,4,6 (013)a (026)a 3,4,7 (014) (026)b 3,5,6 (013)b (026)b 3,5,7 (024) (048) biiiaug 3,6,7 (014)b (026)a 4,5,6 (012) (024) 4,5,7 (013)a (026)a 4,6,7 (013)b (026)b 5,6,7 (012) (024)

62 55 Melodic Minor Scale Degree MOD7 MOD12 Roman Numeral Chord Symbols 4-note 1,2,3,4 (0123) (0235) 1,2,3,5 (0124)a (0237)a Imin(add2) 1,2,3,6 (0124)b (0136)b VIdim(add4) 1,2,3,7 (0123) (0134) 1,2,4,5 (0134) (0257) V7sus4 1,2,4,6 (0135) (0358) IImin7 or IV6 1,2,4,7 (0124)a (0136)a 1,2,5,6 (0134) (0257) II7sus4 1,2,5,7 (0124)b (0237)b V(add4) 1,2,6,7 (0123) (0235) 1,3,4,5 (0124)b (0247)b Imin(add4) 1,3,4,6 (0135) (0258)b IV7 1,3,4,7 (0123) (0146)a 1,3,5,6 (0135) (0258)a Imin6 or VImin7(b5) 1,3,5,7 (0135) (0148)a Imin(Maj7) 1,3,6,7 (0124)a (0236)a VIdim(add2) 1,4,5,6 (0124)a (0247)a IV(add2) 1,4,5,7 (0134) (0157)b 1,4,6,7 (0124)b (0137)b IV(add#4) 1,5,6,7 (0123) (0135)b 2,3,4,5 (0123) (0135)a

63 56 Melodic Minor Scale Degree MOD7 MOD12 Roman Numeral Chord Symbols 2,3,4,6 (0124)a (0137)a IImin(addb2) 2,3,4,7 (0124)b (0236)b 2,3,5,6 (0134) (0157)a biiimaj7(b5) 2,3,5,7 (0135) (0148)b biiimaj7(#5) 2,3,6,7 (0134) (0146)b 2,4,5,6 (0124)b (0247)b IImin(add4) 2,4,5,7 (0135) (0258)b V7 2,4,6,7 (0135) (0258)a VIImin7(b5) 2,5,6,7 (0124)a (0247)a V(add2) 3,4,5,6 (0123) (0246) 3,4,5,7 (0124)a (0248)a biiiaug(add2) 3,4,6,7 (0134) (0268) IV7b5 3,5,6,7 (0124)b (0248)b biiiaug(add#4) 4,5,6,7 (0123) (0246) 5-note 1,2,3,4,5 (01234) (02357)a 1,2,3,4,6 (01235) (02358)b 1,2,3,4,7 (01234) (01346)a 1,2,3,5,6 (01245) (01368)b Imin6 pent. 1,2,3,5,7 (01235) (01348) Imin9(Maj7) 1,2,3,6,7 (01234) (01346)b 1,2,4,5,6 (01245) (02479) IV pent. or IImin pent.

64 57 Melodic Minor Scale Degree MOD7 MOD12 Roman Numeral Chord Symbols 1,2,4,5,7 (01245) (01368)a 1,2,4,6,7 (01235) (02358)a 1,2,5,6,7 (01234) (02357)b 1,3,4,5,6 (01235) (02469) IV9 1,3,4,5,7 (01245) (01468)a 1,3,4,6,7 (01245) (02368)a 1,3,5,6,7 (01235) (02458)a VImin9(b5) 1,4,5,6,7 (01234) (01357)b 2,3,4,5,6 (01234) (01357)a 2,3,4,5,7 (01235) (02458)b 2,3,4,6,7 (01245) (02368)b 2,3,5,6,7 (01245) (01468)b 2,4,5,6,7 (01235) (02469) V9 3,4,5,6,7 (01234) (02468) 6-note 1,2,3,4,5,6 (012345) (023579)a Imin & IImin triads 1,2,3,4,5,7 (012345) (013468)a VIIdim and Imin triads 1,2,3,4,6,7 (012345) (023568) VIdim and VIIdim triads 1,2,3,5,6,7 (012345) (013468)b V and VIdim triads 1,2,4,5,6,7 (012345) (023579)b IV and V triads 1,3,4,5,6,7 (012345) (013579)b biiiaug and IV triads 2,3,4,5,6,7 (012345) (013579)a IImin and biiiaug triads

65 58 Table A.5. Subsets of the harmonic major scale Harmonic Major Scale Degree MOD7 MOD12 Roman Numeral Chord Symbols 3-note 1,2,3 (012) (024) 1,2,4 (013)a (025)a 1,2,5 (014) (027) Vsus4 1,2,6 (013)b (026)b 1,2,7 (012) (013)a 1,3,4 (013)b (015) 1,3,5 (024) (037)b I 1,3,6 (024) (048) Iaug/IIIaug/VIbaug 1,3,7 (013)b (015) 1,4,5 (014) (027) Isus4 1,4,6 (024) (037)a IVmin 1,4,7 (014) (016)a 1,5,6 (013)a (015)a 1,5,7 (013)b (015)b 1,6,7 (012) (014)b 2,3,4 (012) (013)b 2,3,5 (013)a (025)a 2,3,6 (014) (026)a 2,3,7 (013)b (025)b 2,4,5 (013)b (025)b

66 59 Harmonic Major Scale Degree MOD7 MOD12 Roman Numeral Chord Symbols 2,4,6 (024) (036) IIdim 2,4,7 (024) (036) VIIdim 2,5,6 (014) (016)b 2,5,7 (024) (037)b V 2,6,7 (013)a (036) #Vdim 3,4,5 (012) (013)a 3,4,6 (013)a (014)a 3,4,7 (014) (016)b 3,5,6 (013)b (014)b 3,5,7 (024) (037)a IIImin 3,6,7 (014)b (037)b III 4,5,6 (012) (013)b 4,5,7 (013)a (026)a 4,6,7 (013)b (036) IVdim 5,6,7 (012) (014)a 4-note 1,2,3,4 (0123) (0135)b 1,2,3,5 (0124)a (0247)a I(add2) 1,2,3,6 (0124)b (0248) III7(#5) 1,2,3,7 (0123) (0135)a 1,2,4,5 (0134) (0257) V7sus4 1,2,4,6 (0135) (0258)a IImin7(b5) or IVmin6

67 60 Harmonic Major Scale Degree MOD7 MOD12 Roman Numeral Chord Symbols 1,2,4,7 (0124)a (0136)a VIIdim(addb2) 1,2,5,6 (0134) (0157)a bvimaj7(b5) 1,2,5,7 (0124)b (0237)b V(add4) 1,2,6,7 (0123) (0236)b 1,3,4,5 (0124)b (0237)b I(add4) 1,3,4,6 (0135) (0148)a IVmin(Maj7) 1,3,4,7 (0123) (0156) 1,3,5,6 (0135) (0148)b bvimaj7(#5) 1,3,5,7 (0135) (0158) IMaj7 1,3,6,7 (0124)a (0148)b Imaj7(#5) 1,4,5,6 (0124)a (0237)a IVmin(add2) 1,4,5,7 (0134) (0157)b 1,4,6,7 (0124)b (0147)b IVmin(add#4) 1,5,6,7 (0123) (0145) 2,3,4,5 (0123) (0235)a 2,3,4,6 (0124)a (0236)a IIdim(add2) 2,3,4,7 (0124)b (0136)b 2,3,5,6 (0134) (0146)b 2,3,5,7 (0135) (0358) IIIm7 2,3,6,7 (0134) (0258)b III7 2,4,5,6 (0124)b (0136)b IIdim(add4) 2,4,5,7 (0135) (0258)b V7

68 61 Harmonic Major Scale Degree MOD7 MOD12 Roman Numeral Chord Symbols 2,4,6,7 (0135) (0369) VIIdim7 2,5,6,7 (0124)a (0147)a bvidim(maj7) 3,4,5,6 (0123) (0134) 3,4,5,7 (0124)a (0137)a biiimin(addb2) 3,4,6,7 (0134) (0147)a IVdim(Maj7) 3,5,6,7 (0124)b (0347) III(add#2) 4,5,6,7 (0123) (0236)a IVdim(add2) 5-note 1,2,3,4,5 (01234) (02357)b 1,2,3,4,6 (01235) (01358)b 1,2,3,4,7 (01234) (01356) 1,2,3,5,6 (01245) (01468)b 1,2,3,5,7 (01235) (01358)a IMaj9 1,2,3,6,7 (01234) (02458)b 1,2,4,5,6 (01245) (01368)b IVmin6 pent. 1,2,4,5,7 (01245) (01368)a 1,2,4,6,7 (01235) (01369)a 1,2,5,6,7 (01234) (01457)a 1,3,4,5,6 (01235) (01348) IVmin9(Maj7) 1,3,4,5,7 (01245) (01378)a 1,3,4,6,7 (01245) (01478) 1,3,5,6,7 (01235) (01458)b

69 62 Harmonic Major Scale Degree MOD7 MOD12 Roman Numeral Chord Symbols 1,4,5,6,7 (01234) (01457)b 2,3,4,5,6 (01234) (01346)b 2,3,4,5,7 (01235) (02358)b 2,3,4,6,7 (01245) (01369)a 2,3,5,6,7 (01245) (01469)b 2,4,5,6,7 (01235) (02469) V7(b9) 3,4,5,6,7 (01234) (01347)a 6-note 1,2,3,4,5,6 (012345) (013468)b I & IIdim triads 1,2,3,4,5,7 (012345) (013568)a VIIdim and I triads 1,2,3,4,6,7 (012345) (013569)b bviaug and VIIdim triads 1,2,3,5,6,7 (012345) (014579)a V and bviaug triads 1,2,4,5,6,7 (012345) (023679)a IVmin and V triads 1,3,4,5,6,7 (012345) (013478)a IIImin and IVmin triads 2,3,4,5,6,7 (012345) (013469)b IIdim and IIImin triads Table A.6. Subsets of the augmented/hexatonic scale Augmented/Hexatonic Scale Degree MOD6 MOD12 Roman Numeral Chord Symbols 3-note 1,2,3 (012) (014)b 1,2,4 (013)a (037)a Imin 1,2,5 (013)b (037)b bvi

70 63 Augmented/Hexatonic Scale Degree MOD6 MOD12 Roman Numeral Chord Symbols 1,2,6 (012) (014)a 1,3,4 (013)b (037)b I 1,3,5 (024) (048) Iaug/IIIaug/bVIaug 1,3,6 (013)a (015)a 1,4,5 (013)a (015)a 1,4,6 (013)b (015)b 1,5,6 (012) (014)b 2,3,4 (012) (014)a 2,3,5 (013)a (015)a 2,3,6 (013)b (015)b 2,4,5 (013)a (015)b 2,4,6 (024) (048) biiiaug/vaug/viiaug 2,5,6 (013)a (037)a bvim 3,4,5 (012) (014)b 3,4,6 (013)a (037)a IIImin 3,5,6 (013)b (037)b III 4,5,6 (012) (014)a 4-note 1,2,3,4 (0123) (0347) 1,2,3,5 (0124) (0148)b IIIMaj7(#5) 1,2,3,6 (0123) (0145) 1,2,4,5 (0134) (0158) bvimaj7

71 64 Augmented/Hexatonic Scale Degree MOD6 MOD12 Roman Numeral Chord Symbols 1,2,4,6 (0124) (0148)a Imin(Maj7) 1,2,5,6 (0123) (0347) 1,3,4,5 (0124) (0148)b bvimaj7(#5) 1,3,4,6, (0134) (0158) IMaj7 1,3,5,6 (0124) (0148)b IMaj7(#5) 1,4,5,6 (0123) (0145) 2,3,4,5 (0123) (0145) 2,3,4,6 (0124) (0148)a IIImin(Maj7) 2,3,5,6 (0134) (0158) IIIMaj7 2,4,5,6 (0124) (0148)a bvimin(maj7) 3,4,5,6 (0123) (0347) 5-note 1,2,3,4,5 (01234) (01458)b 1,2,3,4,6 (01234) (01458)a 1,2,3,5,6 (01234) (01458)b 1,2,4,5,6 (01234) (01458)a 1,3,4,5,6 (01234) (01458)b 2,3,4,5,6 (01234) (01458)a

72 65 Table A.7. Subsets of the octatonic/hw diminished scale Octatonic/HW Scale Degree MOD8 MOD12 Roman Numeral Chord Symbol 3-note 1,2,3 (012) (013)a 1,2,4 (013)a (014)a 1,2,5 (014)a (016)a 1,2,6 (014)b (016)b 1,2,7 (013)b (014)b 1,2,8 (012) (013)b 1,3,4 (013)b (014)b 1,3,5 (024) (036) Idim 1,3,6 (025) (037)a Imin 1,3,7 (024) (036) VIdim 1,3,8 (013)a (025)a 1,4,5 (014)b (026)b I(b5) 1,4,6 (025)b (037)b I 1,4,7 (025) (037)a VImin 1,4,8 (015)a (026)a 1,5,6 (014)a (016)a 1,5,7 (024) (036) #IVdim 1,5,8 (014)b (026)b #IV(b5) 1,6,7 (013)a (025)a 1,6,8 (013)b (025)b

73 66 Octatonic/HW Scale Degree MOD8 MOD12 Roman Numeral Chord Symbol 1,7,8 (012) (013)a 2,3,4 (012) (013)b 2,3,5 (013)a (025)a 2,3,6 (014)a (026)a 2,3,7 (014)b (026)b VI(b5) 2,3,8 (013)b (025)b 2,4,5 (013)b (025)b 2,4,6 (024) (036) #Idim 2,4,7 (025) (037)b VI 2,4,8 (024) (036) bviidim 2,5,6 (014)b (016)b 2,5,7 (025)b (037)a #IVmin 2,5,8 (025) (037)b #IV 2,6,7 (014)a (026)a 2,6,8 (024) (036) Vdim 2,7,8 (013)a (014)a 3,4,5 (012) (013)a 3,4,6 (013)a (014)a 3,4,7 (014)a (016)a 3,4,8 (014)b (016)b 3,5,6 (013)b (014)b 3,5,7 (024) (036) #IIdim

74 67 Octatonic/HW Scale Degree MOD8 MOD12 Roman Numeral Chord Symbol 3,5,8 (025) (037)a biiimin 3,6,7 (014)b (026)b #II(b5) 3,6,8 (025) (037)b biii 3,7,8 (014)a (016)a 4,5,6 (012) (013)b 4,5,7 (013)a (025)a 4,5,8 (014)a (026)a 4,6,7 (013)b (025)b 4,6,8 (024) (036) IIIdim 4,7,8 (014)b (016)b 5,6,7 (012) (013)a 5,6,8 (013)a (014)a 5,7,8 (013)b (014)b 6,7,8 (012) (013)b 4-note 1,2,3,4 (0123) (0134) 1,2,3,5 (0124)a (0136)a Idim(addb2) 1,2,3,6 (0125) (0137)a Imin(addb2) 1,2,3,7 (0124)b (0236)b 1,2,3,8 (0123) (0235) 1,2,4,5 (0134) (0146)a 1,2,4,6 (0135)a (0147)a #Idim(Maj7)

75 68 Octatonic/HW Scale Degree MOD8 MOD12 Roman Numeral Chord Symbol 1,2,4,7 (0124)a (0347) VI(add#9) 1,2,4,8 (0124)a (0236)a 1,2,5,6 (0145) (0167) 1,2,5,7 (0135)b (0147)b bvmin(add#4) 1,2,5,8 (0125)b (0137)b bv(add#4) 1,2,6,7 (0134)b (0146)b 1,2,6,8 (0124)b (0136)b Vdim(add4) 1,2,7,8 (0123) (0134) 1,3,4,5 (0124)b (0236)b 1,3,4,6 (0235) (0347) I(add#9) 1,3,4,7 (0135)b (0147)b VImin(add#4) 1,3,4,8 (0134) (0146)b 1,3,5,6 (0135)b (0147)b Imin(add#4) 1,3,5,7 (0246) (0369) Idim7 or biiidim7 or #IVdim7 or VIdim7 1,3,5,8 (0135)a (0258)a Imin7(b5) or biiimin6 1,3,6,7 (0135)a (0258)a VImin7(b5) or Imin6 1,3,6,8 (0235) (0358) Im7 or biii6 1,3,7,8 (0124)a (0136)a VIdim(addb2) 1,4,5,6 (0125) (0137)b I(add#4) 1,4,5,7 (0135)a (0258)a #IVmin7(b5) or VImin6 1,4,5,8 (0145) (0268) I7(b5) or #IV7(b5) 1,4,6,7 (0235) (0358) VIm7 or I6

76 69 Octatonic/HW Scale Degree MOD8 MOD12 Roman Numeral Chord Symbol 1,4,6,8 (0135)a (0258)b I7 1,4,7,8 (0125) (0137)a VImin(addb2) 1,5,6,7 (0124)a (0136)a #IVdim(addb2) 1,5,6,8 (0134) (0146)a 1,5,7,8 (0124)b (0236)b 1,6,7,8 (0123) (0235) 2,3,4,5 (0123) (0235) 2,3,4,6 (0124)a (0236)a #Idim(add2) 2,3,4,7 (0124)b (0137)b VI(add#4) 2,3,4,8 (0124)b (0136)b 2,3,5,6 (0134) (0146)b 2,3,5,7 (0135)a (0258)a biiimin7(b5) or Gbmin6 2,3,5,8 (0235) (0358) biiim7 or bv6 2,3,6,7 (0145) (0268) VI7(b5) or III7(b5) 2,3,6,8 (0135)a (0258)b biii7 2,3,7,8 (0134) (0146)a 2,4,5,6 (0124)b (0136)b #Idim(add4) 2,4,5,7 (0235) (0358) #IVm7 or VI6 2,4,5,8 (0135)a (0258)b #IV7 2,4,6,7 (0135)a (0258)b VI7 2,4,6,8 (0246) (0369) #Idim7 or IIIdim7 or Vdim7 or bvii dim7 2,4,7,8 (0135)a (0147)a bviidim(maj7)

77 70 Octatonic/HW Scale Degree MOD8 MOD12 Roman Numeral Chord Symbol 2,5,6,7 (0125) (0137)a #IVmin(addb2) 2,5,6,8 (0135)a (0147)a Vdim(Maj7) 2,5,7,8 (0235) (0347) 2,6,7,8 (0124) (0236)a Vdim(add2) 3,4,5,6 (0123) (0134) 3,4,5,7 (0124)a (0136)a biiidim(addb2) 3,4,5,8 (0125) (0137)a biiimin(addb2) 3,4,6,7 (0134) (0146)a 3,4,6,8 (0135)a (0147)a IIIdim(Maj7) 3,4,7,8 (0145) (0167) 3,5,6,7 (0124)b (0236)b 3,5,6,8 (0235) (0347) 3,5,7,8 (0135)b (0147)b biiimin(add#4) 3,6,7,8 (0125) (0137)b biii(add#4) 4,5,6,7 (0123) (0235) 4,5,6,8 (0124)a (0236)a IIIdim(add2) 4,5,7,8 (0123) (0146)b 4,6,7,8 (0124)b (0136)b IIIdim(add4) 5,6,7,8 (0123) (0134) 5-note 1,2,3,4,5 (01234) (01346)a 1,2,3,4,6 (01235)a (01347)a

78 71 Octatonic/HW Scale Degree MOD8 MOD12 Roman Numeral Chord Symbol 1,2,3,4,7 (01235)b (01347)b 1,2,3,4,8 (01234) (01346)b 1,2,3,5,6 (01245)a (01367)a 1,2,3,5,7 (01246) (01369)a 1,2,3,5,8 (01235)a (02358)a 1,2,3,6,7 (01245)b (02368)b 1,2,3,6,8 (01235)b (02358)b 1,2,3,7,8 (01234) (01346)a 1,2,4,5,6 (01245)b (01367)b 1,2,4,5,7 (01346) (01469)a 1,2,4,5,8 (01245)a (02368)a I7(b5,b9) 1,2,4,6,7 (01346) (01469)b I7(#9) 1,2,4,6,8 (01246) (01369)b I7(b9) 1,2,4,7,8 (01235)a (01347)a 1,2,5,6,7 (01245)a (01367)a 1,2,5,6,8 (01245)b (01367)b 1,2,5,7,8 (01235)b (01347)b 1,2,6,7,8 (01234) (01346)b 1,3,4,5,6 (01235)b (01347)b 1,3,4,5,7 (01246) (01369)a 1,3,4,5,8 (01245)b (02368)b 1,3,4,6,7 (01346) (01469)a

79 72 Octatonic/HW Scale Degree MOD8 MOD12 Roman Numeral Chord Symbol 1,3,4,6,8 (01346) (01469)b I7(#9) 1,3,4,7,8 (01245)a (01367)a 1,3,5,6,7 (01246) (01369)a 1,3,5,6,8 (01346) (01469)a 1,3,5,7,8 (01246) (01369)a 1,3,6,7,8 (01235)a (02358)a 1,4,5,6,7 (01235)a (02358)a 1,4,5,6,8 (01245)a (02368)a #IV7(b5,b9) 1,4,5,7,8 (01245)b (02368)b 1,4,6,7,8 (01235)b (02358)b 1,5,6,7,8 (01234) (01346)a 2,3,4,5,6 (01234) (01346)b 2,3,4,5,7 (01235)a (02358)a 2,3,4,5,8 (01235)b (02358)b 2,3,4,6,7, (01245)a (02368)a biii7(b5,b9) 2,3,4,6,8 (01246) (01369)b biii7(b9) 2,3,4,7,8 (01245)b (01367)b 2,3,5,6,7 (01245)b (02368)b 2,3,5,6,8 (01346) (01469)b biii7(#9) 2,3,5,7,8 (01346) (01469)a 2,3,6,7,8 (01245)a (02368)a VI7(b5,b9) 2,4,5,6,7 (01235)b (02358)b

80 73 Octatonic/HW Scale Degree MOD8 MOD12 Roman Numeral Chord Symbol 2,4,5,6,8 (01246) (01369)b #IV7(b9) 2,4,5,7,8 (01346) (01469)b #IV7(#9) 2,4,6,7,8 (01246) (01369)b VI7(b9) 2,5,6,7,8 (01235)a (01347)a 3,4,5,6,7 (01234) (01346)a 3,4,5,6,8 (01235)a (01347)a 3,4,5,7,8 (01245)a (01367)a 3,4,6,7,8 (01245)b (01367)b 3,5,6,7,8 (01235)b (01347)b 4,5,6,7,8 (01234) (01346)b 6-note 1,2,3,4,5,6 (012345) (013467) Idim and #Idim triads 1,2,3,4,5,7 (012346) (013469)a VI and Idim triads 1,2,3,4,5,8 (012345) (023568) bviidim and Idim triads 1,2,3,4,6,7 (012356) (013479) VI and Imin triads 1,2,3,4,6,8 (012346) (013469)b bviidim and Imin triads 1,2,3,4,7,8 (012345) (013467) VIdim and #VIdim triads 1,2,3,5,6,7 (012456) (013679)a Imin and #IVmin triads 1,2,3,5,6,8 (012356) (014679) Imin and #IV triads 1,2,3,5,7,8 (012346) (013469)a #IV and VIdim triads 1,2,3,6,7,8 (012345) (023568) Vdim and VIdim triads 1,2,4,5,6,7 (012356) (014679) #IVmin and I triads

81 74 Octatonic/HW Scale Degree MOD8 MOD12 Roman Numeral Chord Symbol 1,2,4,5,6,8 (012456) (013679)b I and #IV triads 1,2,4,5,7,8 (012356) (013479) #IV and VImin triads 1,2,4,6,7,8 (012346) (013469)b Vdim and VImin triads 1,2,5,6,7,8 (012345) (013467) #IVdim and Vdim triads 1,3,4,5,6,7 (012346) (013469)a I and #IIdim triads 1,3,4,5,6,8 (012356) (013479) I and #IImin triads 1,3,4,5,7,8 (012456) (013679)a VImin and biiimin triads 1,3,4,6,7,8 (012356) (014679) VImin and biii triads 1,3,5,6,7,8 (012346) (013469)a biii and #IVdim triads 1,4,5,6,7,8 (012345) (023568) IIIdim and #IVdim triads 2,3,4,5,6,7 (012345) (023568) biidim and biiidim triads 2,3,4,5,6,8 (012346) (013469)b #Idim and biiimin triads 2,3,4,5,7,8 (012356) (014679) biiimin and VI triads 2,3,4,6,7,8 (012456) (013679)b biii and VI triads 2,3,5,6,7,8 (012356) (013479) biii and #IVmin triads 2,4,5,6,7,8 (012346) (013469)b IIIdim and #IVmin triads 3,4,5,6,7,8 (012345) (013467) #IIdim and IIIdim triads Table A.8. Subsets of the whole tone scale Whole Tone Scale Degree MOD6 MOD12 Roman Numeral Chord Symbol 3-note 1,2,3 (012) (024)

82 75 Whole Tone Scale Degree MOD6 MOD12 Roman Numeral Chord Symbol 1,2,4 (013)a (026)a 1,2,5 (013)b (026)b 1,2,6 (012) (024) 1,3,4 (013)b (026)b 1,3,5 (024) (048) Iaug 1,3,6 (013)a (026)a 1,4,5 (013)a (026)a 1,4,6 (013)b (026)b 1,5,6 (012) (024) 2,3,4 (012) (024) 2,3,5 (013)a (026)a 2,3,6 (013)b (026)b 2,4,5 (013)a (026)a 2,4,6 (024) (048) IIaug 2,5,6 (013)a (026)a 3,4,5 (012) (024) 3,4,6 (013)a (026)a 3,5,6 (013)b (026)b 4,5,6 (012) (024) 4-note 1,2,3,4 (0123) (0246) 1,2,3,5 (0124) (0248) III7(#5)

83 76 Whole Tone Scale Degree MOD6 MOD12 Roman Numeral Chord Symbol 1,2,3,6 (0123) (0246) 1,2,4,5, (0134) (0268) II7(b5) or bvi7(b5) 1,2,4,6 (0124) (0248) II7(#5) 1,2,5,6 (0123) (0246) 1,3,4,5 (0124) (0248) bvi7(#5) 1,3,4,6, (0134) (0268) I7(b5) or #IV7(b5) 1,3,5,6 (0124) (0248) I7(#5) 1,4,5,6 (0123) (0246) 2,3,4,5 (0123) (0246) 2,3,4,6 (0124) (0248) bv7(#5) 2,3,5,6 (0134) (0268) III7(b5) or bvii7(b5) 2,4,5,6 (0124) (0248) bvii7(#5) 3,4,5,6 (0123) (0246) 5-note 1,2,3,4,5 (01234) (02468) II9(b5) or III9(#5) 1,2,3,4,6 (01234) (02468) I9(b5) II9(#5) 1,2,3,5,6 (01234) (02468) bvii9(b5) or I9(#5) 1,2,4,5,6 (01234) (02468) #V9(b5) or bvii9(#5) 1,3,4,5,6 (01234) (02468) #IV9(b5) or bvi9(#5) 2,3,4,5,6 (01234) (02468) III9(b5) or #IV9(#5)

84 77 Table A.9. Examples from the literature of improvisation over the tonic chords Roman Numeral Chord Symbols Scale Degrees Location Entire collection 1,2,3,4,5,6,7 Coltrane, 1 st chorus, m. 3 I 1,3,5 Coltrane, 1 st chorus, m.1, beats 1-2 IIImin 3,5,7 Brecker, 3 rd chorus, m.13, beats 3-4, 1:04 V 2,5,7 Garrett, 6 th chorus, m.2, beats 1-2 VImin 1,3,6 Frahm, 4 th chorus, m.2, beats 1-2, 1:09 Isus4 1,4,5 Diorio, solo 1, m.7 IIsus4 2,5,6 Diorio, solo 1, m.9 IIIsus4 3,6,7 Garrett, 3 rd chorus, m.11 Vsus4 1,2,5 Coltrane, 1 st chorus, m.15 VIsus4 2,3,6 Diorio, solo 2, m.5, beats 1-2 I(add2) 1,2,3,5 Coltrane, 1 st chorus, m.6, beats 1-2 V(add2) 2,5,6,7 Coltrane, 2 nd chorus, m.11, beats 1-2 I(add4) 1,3,4,5 Brecker, 4 th chorus, m.7, beats 3-4 IIImin(add4) 3,5,6,7 Coltrane, 7 th chorus, m.11 V(add4) 1,2,5,7 Coltrane, 2 nd chorus, m.7, beats 3-4 VImin(add4) 1,2,3,6 Diorio, solo 11, m.7, beats 3-4 IMaj7 1,3,5,7 Coltrane, 3 rd chorus, m.13, beats 3-4 IImin7 1,2,4,6 Frahm, 3 rd chorus, m.3, beats 1-2, 0:57 IIImin7 2,3,5,7 Coltrane, 4 th chorus, m.13, beats 1-3 VImin7 1,3,5,6 Coltrane, 6 th chorus, m.1, beats 1-2 VIImin7(b5) 2,4,6,7 Brecker, 5 th chorus, m.9, beats 1-2, 1:26

85 78 Roman Numeral Chord Symbols Scale Degrees Location II7sus4 1,2,5,6 Diorio, solo 9, m.1, beats 1-2 III7sus4 2,3,6,7 Diorio, solo 14, m.15, beats 3-4 V7sus4 1,2,4,5 Diorio, solo 4, m.3, beats 3-4 VI7sus4 2,3,5,6 Diorio, solo 9, m.13, beats 1-2 I major pentatonic 1,2,3,5,6 Garrett, 4 th chorus, m.13 IV major pentatonic 1,2,4,5,6 Garrett, 12 th chorus, m.15 V major pentatonic 2,3,5,6,7 Coltrane, 3 rd chorus, m.7 IMaj9 1,2,3,5,7 Coltrane, 10 th chorus, m.13 Table A.10. Examples from the literature of improvisation over the dominant and IIm7 chords Roman Numeral Chord Symbols Scale Degrees Location Entire collection 1,2,3,4,5,6,7 Coltrane, 6th chorus, m.8 I 1,3,5 Coltrane, 14 th chorus, m.4 beats 3-4 IImin 2,4,6 Coltrane, 4 th chorus, m.10, beats 1-2 IIImin 3,5,7 Garrett, 6 th chorus, m.2, beats 3-4 IV 1,4,6 Garrett, 11 th chorus, m.8 V 2,5,7 Frahm, 1 st chorus, m.1, beats 3-4, 0:28 VImin 1,3,6 Coltrane, 8 th chorus, m.8, beats 1-2 VIIdim 2,4,7 Frahm, 4 th chorus, m.1, beats 3-4, 1:09

86 79 Roman Numeral Chord Symbols Scale Degrees Location Isus4 1,4,5 Diorio, solo 2, m.16 IIsus4 2,5,6 Diorio, solo 1, m.4, beats 3-4 IIIsus4 3,6,7 Garrett, 4 th chorus, m.14, beats 1-2 Vsus 1,2,5 Diorio, solo 4, m.1, beats 3-4 VIsus4 2,3,6 Diorio, solo 1, m.8, beats 1-2 I(add2) 1,2,3,5 Frahm, 3 rd chorus, m.16, beats 3-4, 1:08 IImin(add2) 2,3,4,6 Coltrane, 1 st chorus, m.2, beats 3-4 IV(add2) 1,4,5,6 Coltrane, 8 th chorus, m.6, beats 3-4 V(add2) 2,5,6,7 Coltrane, 1 st chorus, m.1, beats 3-4 VImin(add2) 1,3,6,7 Coltrane, 9 th chorus, m.8, beats 1-2 IIImin(add4) 3,5,6,7 Coltrane, 3 rd chorus, m.4, beats 3-4 V(add4) 1,2,5,7 Frahm, 1 st chorus, m.8, 0:34 IMaj7 1,3,5,7 Coltrane, 3 rd chorus, m.12, beats 1-2 IImin7 1,2,4,6 Coltrane, 1 st chorus, pick-up measure, beats 1-2 IVMaj7 1,3,4,6 Coltrane, 6 th chorus, m.16, beats 1-2 VImin7 1,3,5,6 Coltrane, 9 th chorus, m.16, beats 1-2 V7 2,4,5,7 Coltrane, 1 st chorus, m.6, beats 3-4 VIImin7b5 2,4,6,7 Coltrane, 1 st chorus, m.5, beats 3-4 II7sus4 1,2,5,6 Diorio, solo 9, m.4, beats 1-2 III7sus4 2,3,6,7 Coltrane, 2 nd chorus, beats 3-4

87 80 Roman Numeral Chord Symbols Scale Degrees Location V7sus4 1,2,4,5 Diorio, solo 11, m.4, beats 1-2 VI7sus4 2,3,5,6 Coltrane, 4 th chorus, m.4, beats 3-4 IV major pentatonic 1,2,4,5,6 Garrett, 9 th chorus, m.8 V major pentatonic 2,3,5,6,7 Coltrane, 2 nd chorus, m.8 beat 3 to m.9, beat 1 IImin9 1,2,3,4,6 Coltrane, 3 rd chorus, m.10 V and VImin 1,2,3,5,6,7 Frahm, 3 rd chorus, m.14, 1:06 Table A.11. Examples from the literature of improvisation over the tonic chords using alternate chord-scales Roman Numeral Scale Location Possible Chord-Scale Implications Chord Symbol Degrees Iaug 1,3,#5 Frahm, 1 st chorus, m.1, beats 1-2, 0:28 II 2,#4,6 Garrett, 8 th chorus, Implies Lydian augmented, augmented/hexatonic, or harmonic major Implies Lydian or Lydian augmented m.9, beats 2-4 IVdim 4,b6,7 Garrett, 13 th chorus, m.3 VIIsus4 3,#4,7 Diorio, solo 2, m.15 Implies harmonic major or 3 rd mode harmonic minor Implies Lydian, Lydian augmented, or 6 th mode harmonic minor

88 81 Roman Numeral Scale Location Possible Chord-Scale Implications Chord Symbol Degrees II7 1,2,#4,6 Frahm, 6 th chorus, Implies Lydian or Lydian augmented m.1, beats 1-2, 1:35 IIImaj7 #2,3,#5,7 Diorio, solo 13, m.15, beats 3-4 Implies Lydian-augmented or harmonic major VIIdim7 2,4,b6,7 Frahm, 2 nd chorus, m.5, beats 1-2 0:45 VIIm7 2,#4,6,7 Diorio, solo 5, Implies Lydian or Lydian augmented m.3, beats 3-4 VII7sus4 3,#4,6,7 Diorio, solo 2, m.9 Implies Lydian, Lydian augmented, or 6 th mode harmonic minor No chord symbol No chord symbol No chord symbol 2,3,#5 Garrett, 9 th chorus, m.7 #2,3,7 Garrett, 8 th chorus, m.3 1,2,3,#5 Garrett, 9 th chorus, m.13, beats 1-2 Implies Lydian augmented, harmonic major or 3 rd mode harmonic minor Implies 6 th mode harmonic minor or augmented/hexatonic scale Implies Lydian augmented, harmonic major or 3 rd mode harmonic minor No chord 2,3,5,b6 Garrett, 2 nd chorus, Implies harmonic major symbol m.13

89 82 Table A.12. Examples from the literature of improvisation over the dominant and IIm7 chords using alternate chord-scales Roman Numeral Scale Location Possible Chord-Scale Implications Chord Symbol Degrees Entire HW #1,2,3,4,5, Frahm, 5 th diminished b6,b7,7 chorus, m.14, collection 1:33 Vaug #2,5,7 Coltrane, 2 nd chorus, m.4 Implies 5 th mode harmonic minor, Mixolydian b6, super locrian/altered, whole tone, 3 rd mode harmonic major, or augmented/hexatonic V7 chord scales #Vdim 2,#5,7 Frahm, 1 st chorus, m.2, beats 3-4, 0:29 Implies 5 th mode harmonic minor, HW diminished, 3 rd mode harmonic major, or 5 th mode harmonic major V7 chordscales bvim(add2) b3,b6,b7,7 Frahm, 2 nd chorus, m.10, Implies altered or 3 rd mode harmonic major V7 chord-scales 0:49 bvi(add2) 1,b3,b6,b7 Frahm, 3 rd Implies IIm7(b5) Locrian chord-scale chorus, m.4, beats 1-2, 0:58

90 83 Roman Numeral Scale Location Possible Chord-Scale Implications Chord Symbol Degrees biim7 b2,3,b6,7 Diorio, solo 16, Implies HW diminished V7 chord-scale m.8, beats 3-4 bvidim(maj7) 2,5,b6,7 Frahm, 4 th chorus, m.12, beats 1-3, 1:18 bvimin(maj7) b3,5,b6,7 Frahm, 4 th chorus, m.8, beats 3-4, 1:15 bvimaj7 1,b3,5,b6 Frahm, 6 th chorus, m.4, beats 1-2, 1:38 VI7 #1,3,5,6 Frahm, 3 rd chorus, m.2, Implies 5 th mode harmonic minor, 5 th mode harmonic major, or HW diminished V7 chord-scales Implies 5 th mode harmonic minor, 3 rd mode harmonic major, altered or augmented/hexatonic V7 chord-scales Implies 5 th mode harmonic minor V7 chord-scale, or Locrian IIm7(b5) chordscale Implies HW diminished or 5 th mode harmonic major V7 chord-scales beats 3-4, 0:56 VIImaj7#11 b3,4,b7,7 Frahm, 2 nd Implies altered V7 chord-scale chorus, m.8, beats 3-4, 0:48 VII7#5 #2,5,6,7 Coltrane, 10 th chorus, m.8, Implies whole-tone or Mixolydian b6 V7 chord-scales beats 3-4

91 84 Roman Numeral Scale Location Possible Chord-Scale Implications Chord Symbol Degrees biii7sus b2,b3,b6,b7 Diorio, solo 11, Implies altered V7 chord-scale m.8, beats 3-4 IV7sus4 1,b3,4,b7 Diorio, solo 11, Implies Locrian IIm7b5 chord-scale m.14, beats 3-4 bvisus4 b2,b3,b6 Garrett, 12 th chorus, m.5, Implies super Locrian/altered V7 chordscale beats 3-4 biii major 1,b3,4,5,b7 Garrett, 8 th Implies a Phrygian IIm7 chord-scale or a pentatonic chorus, m.12, Locrian IIm7b5 chord-scale beats 1-3 with 8 th note pick-up No chord symbol b3, 4, 5 Garrett, 7 th chorus, m.12, beats 3-4 Implies 5 th mode harmonic minor, whole-tone, super Locrian/altered, Mixolydian b6, or 3 rd mode harmonic major V7 chord-scales No chord 1,4,5,b6 Garrett, 9 th Implies Mixolydian b6 or 5 th mode symbol chorus, beats 3- harmonic minor V7 chord-scale or 47 Phrygian IIm7 chord-scale or Locrian IIm7b5 chord-scale

92 85 Roman Numeral Scale Location Possible Chord-Scale Implications Chord Symbol Degrees No chord 4,5,b6,7 Coltrane, 1 st Implies 5 th mode harmonic minor, HW symbol chorus, m.4, diminished, altered, or 3 rd mode beats 3-4 harmonic major V7 chord-scales

93 86 APPENDIX B. GLOSSARY Augmented/hexatonic scale. Also known as the double augmented scale, this symmetrical 6- note scale alternates three-semitone and one-semitone intervals. Avoid notes. Notes from a chord-scale that are not chord tones and not considered available extensions to the chord (e.g. in the C major chord-scale, the note F is considered an avoid note). Be-bop scale. Generally, an 8-note scale that includes a passing chromatic note. Be-bop scales are typically played in such a way that a chord tone is played on every downbeat. Changes. In jazz jargon, synonymous with chord progression. Chord-scale. A scale that is applied to a specific chord, the notes of which can form a basis for melodic improvisation, composition and chord voicings for comping or arranging. Chord substitution. For the purpose of this thesis, I use the term to refer to replacing a given chord with another, related chord. This is different from re-harmonization, in which a new progression is related to a given melody, and does not necessarily relate to the original chord progression. Comper. In jazz jargon, a chordal accompanist. Harmonic major. A 7-note scale that includes the major 3 rd of the major scale and the minor 6 th of the harmonic minor scale. The harmonic major scale is similar to the major scale except for the lowered 6 th scale degree. HW diminished/octatonic scale. A symmetrical 8-note scale that alternates one-semitone and two-semitone intervals. Lydian-augmented. The 4 th mode/rotation of the melodic minor scale, which includes an augmented 4 th as in the Lydian mode and an augmented 5 th as in an augmented triad. Similar to the Lydian mode, except for the raised 5 th scale degree.

94 87 Melodic minor. Also called jazz minor or ascending melodic minor, this scale is similar to a major scale except for the lowered 3 rd scale degree. For the purpose of this thesis, the melodic minor scale s pitch content does not depend on whether the melody ascends or descends. MOD12. A method of assigning an integer from zero to eleven to each pitch-class in the chromatic scale (C=0, C#/Db=1, etc.). Expressing a chord or subset in mod12 notation makes it relatively easy for the reader to get a sense of the exact distances, in semitones, among its pitch classes. MOD7. A method of assigning an integer from zero to six to each scale-degree in a given 7-note scale (scale degree 1=0, scale degree 2=1, etc.). Expressing a set in mod7 notation allows the reader to see similarities between sets based on generic interval sizes, counted between scale degrees, rather than exact semitonal distances captured in mod12 notation. Mod7 notation captures the sense of a third spanning three scale degrees, whereas mod12 notation specifies a 3- or 4-semitone interval, a minor third or a major third. MOD6 and MOD8 are used for scales with six and eight notes, respectively. Mode/rotation. Maintaining the intervallic structure of a scale, while changing which of the scale degrees functions as a tonic, or focal pitch-class (e.g. the 4 th mode of C melodic minor includes the same pitches as C melodic minor, but its tonic is F, the 4 th note in the C melodic minor scale). Pitch class. Refers to a pitch and its transposition to any octave, regardless of enharmonic spelling or register. Set. A collection of pitch-classes.