chemistruck daniel thesis
TRANSCRIPT
Imposing Harmonic Restrictions on Symmetrical Scales: Creating a Tonal Center in the Half/Whole Octatonic Scale
Daniel Chemistruck
Undergraduate Thesis in partial completion of the
CCSU Honors Program
May, 2006
Advisor: Dr. Charles Menoche
Abstract
Octatonicism does not negate diatonicism. By imposing harmonic restrictions and relationships on chords and pitch class sets derived from the octatonic scale, it is possible to create a series of harmonic progressions that will establish a specific note as the tonic. This text presents a theoretical approach for incorporating references to common-practice period tonality, Jazz theory and Set theory to imply a tonal center—or at least centricity—in the octatonic scale; thus, overcoming the symmetrical nature of the scale.
Table of Contents
Introduction...................................................................................................................1 Chapter I: A Brief Overview of the Octatonic Scale ....................................................4 Chapter II: Imposing Harmonic Restrictions ............................................................14
Modal Relationships ................................................................................................14 Harmonic Relationships ..........................................................................................21
(1) Creating a Dominant to Tonic Relationship..................................................21 (2) Predominant Chords ......................................................................................22 (3) Chord Quality.................................................................................................24 (4) Root Progressions...........................................................................................25 (5) The Number of Vertical Structures in an Octatonic System ........................27
Vertical Spacing.......................................................................................................28 Intervallic Content...................................................................................................30 Voice Leading ..........................................................................................................32 Additional Considerations.......................................................................................35
(1) Subset Relations .............................................................................................35 (2) Reharmonization and Implied Harmonic Possibilities .................................36 (3) Metric and Rhythmic Considerations ...........................................................39
Chapter III: Creating a Tonal Octatonic System.......................................................41 Modal Relationships ................................................................................................43
(1) Creating a Dominant to Tonic Relationship..................................................44 (2) Predominant Chords ......................................................................................44 (3) Chord Quality.................................................................................................45 (4) Root Progressions...........................................................................................45 (5) The Number of Vertical Structures in an Octatonic System ........................46
Vertical Spacing.......................................................................................................46 Intervallic Content...................................................................................................47 Voice Leading ..........................................................................................................47
Conclusion ...................................................................................................................50 Bibliography ................................................................................................................50 Appendix A: Chord Lexicography .............................................................................53 Appendix B: Chord-Scale Relationships ....................................................................54 Appendix C: Modal Tetrachords Found in the Octatonic Scale ...............................60 Appendix D: Ron Miller’s Collated Order of All Constructed Modes......................61 Appendix E: Additional Subset Relations ..................................................................62 Appendix F: Additional Octatonic Systems ...............................................................63
1
-- Introduction --
Octatonicism does not negate diatonicism. A modified understanding of
functional harmony is needed in order to create a tonal center within the octatonic scale,
which will primarily be achieved by imposing harmonic restrictions and relationships on
existing chords and pitch class sets (pcs). Due to the symmetrical nature of the octatonic
scale, there is no way to ensure that certain scale degrees will maintain predominance
over others unless a set of guidelines is put in place to govern the construction of possible
vertical structures. Joseph Straus’s definition of traditional common-practice tonality (See
Figure i-1), the musical language of Western classical music from roughly the time of
Bach to Brahms, will serve as a reference for creating a tonal octatonic system.
1) Key. A particular note is defined as the tonic with the remaining notes defined in relation to it.
2) Key relations. Pieces modulate through a succession of keys, with the keynotes often related by perfect fifth, or by major or minor thirds. Pieces end in the key in which they begin.
3) Diatonic scales. The principal scales are the major and minor scales. 4) Triads. The basic harmonic structure is a major or minor triad. Seventh
chords play a secondary role. 5) Functional harmony. Harmonies generally have the function of a tonic
(arrival point), dominant (leading to tonic), or predominant (leading to dominant).
6) Voice leading. The voice leading follows certain traditional norms, including the avoidance of parallel perfect consonances and the resolution of intervals defined as dissonant to those defined as consonant.1
Figure i-1
When constructing a tonal octatonic system, adhering to or compensating for these six
attributes will provide the clearest relationship to tonality.
1 Joseph Straus, Introduction to Post-Tonal Theory 3rd ed. (NJ: Prentice Hall, 2005), 130.
Introduction 2
Chapter I will provide an introduction to the octatonic scale, describing the
octatonic scale’s construction and several symmetrical properties, along with an overview
of its use in musical contexts. Chapter II will provide the foundations for creating a tonal
octatonic system, while Chapter III will apply the concepts of Chapter II in the
explanation of a given octatonic system. Chapters II and III will be divided into the
following sub-sections:
• Modal Relationships will discuss the common subsets of the diatonic
collection, non-diatonic minor scales, and the octatonic collection.
• Harmonic Relationships will discuss creating the dominant to tonic relationship, predominant chords, chord qualities, and the total number of chords in an octatonic system.
• Vertical spacing will discuss the intervallic spacing and ordering of notes within a vertical structure to avoid harmonic ambiguity.
• Intervallic Content will discuss post-tonal relationships between chords.
• Voice Leading will discuss possibilities of creating new rules for harmonic progression.
• Additional Considerations will discuss additional subset relations, chord substitutions, and metric considerations.2
An explanation of chord symbols is given in Appendix A, with an explanation of
chord-scale relationships in Appendix B. Appendices C and D contain charts and
information from Ron Miller’s book Modal Jazz Composition and Harmony Volume 1,
which are referenced to throughout the text. Appendix E lists several subsets derived
from the octatonic scale, with Appendix F listing additional octatonic systems.
2 Additional Considerations is omitted from chapter III.
Introduction 3
Definition of Terms:
• Any reference to the “octatonic scale” refers to the half/whole ordering of the scale
unless otherwise noted. The preference of the half/whole ordering over whole/half is
due to the presence of the perfect fifth interval available from the tonic of the scale; a
more thorough discussion is presented in Chapter II.
• Any reference to the “octatonic collection” indicates that the ordering of the octatonic
scale is inconsequential, as the material being discussed is pertinent to both possible
orderings of the scale.
• All scales, melodic patterns, and harmonic progressions will be presented starting on
C or in relation to C unless otherwise noted.
• All discussions of scale degrees refer to the parent major scale of the given key. For
example, in the mode of C Lydian, a raised fourth scale degree indicates an F , even
though F is the naturally occurring fourth scale degree of the C Lydian mode.
• All instances referring to the “dominant” indicates the vertical structure built on the
note a perfect fifth away from the root of the scale; whereas all instances referring to
“dominant 7th” chords indicates a major-minor 7th chord built on a given note.
• The term “subdominant” refers to the vertical structure built on either the fourth or
raised fourth scale degree.
• The term “chord” is used to indicate a specific harmonic structure familiar to modern
Jazz harmony; whereas the term “vertical structure” indicates a collection of
simultaneously sounding pitches.
• The terms “scale” and “mode” are used interchangeably.
• All accidentals affect only the immediately following note.
4
-- Chapter I -- A Brief Overview of the Octatonic Scale
A brief overview of the octatonic scale is necessary to differentiate between the
concepts that will be presented herein and those of previous composers, although a
thorough discussion of the scale is beyond the scope of this text. Jazz improvisational
theory along with the works of Stravinsky and several other notable composers will serve
as the foundation for the historical use of the octatonic scale.
The octatonic scale is utilized prominently in Jazz improvisation over diminished
7th or dominant 7th chords, depending on the ordering of the scale. Due to the inherent
diminished 7th presence within the octatonic scale, since it is possible to extract two of
the three possible diminished 7th chords a minor second apart (see Figure 1-1), Jazz
terminology usually refers to the scale as the “diminished scale” in either “half/whole” or
“whole/half” ordering, referring to the order of alternating tones and semitones.3
Figure 1-14 The cycling of major and minor seconds facilitates the symmetry that is characteristic of
the octatonic scale, creating two possible orderings: alternating major and minor seconds
(whole/half), or alternating minor and major seconds (half/whole). Oliver Messiaen refers
3 Jim Hall, Exploring Jazz Guitar (Milwaukee, WI: Hal Leonard Publishing Corp., 1990), 18–21. 4 The numbers below the staff refer to the notes relationship to the parent major scale.
Chapter I – A Brief Overview of the Octatonic Scale 5
to the octatonic scale as the second mode of limited transposition because the octatonic
collection produces complete invariance at four levels of transposition and four levels of
inversion, allowing for a total of three octatonic collections, as listed in Figure 1-2.5
OCT0,1 (0134679T) OCT1,2 (124578TE) OCT2,3 (235689E0)
Figure 1-2
As a result of the symmetrical nature of the octatonic scale, Messiaen states that the
modes of limited transposition are “in the atmosphere of several tonalities at once,
without polytonality, the composer being free to give predominance to one of the
tonalities or to leave the tonal impression unsettled.”6 The predominance of a tonality
typically arises from notes that are stated frequently, sustained at length, placed in a
registral extreme, played loudly, and rhythmically or metrically stressed.7 Also,
associating a harmonically ambiguous vertical structure with a unique orchestrational
effect may serve to endow the vertical structure with an individuality that allows it to
function as a tonic sonority, at least to the extent of achieving a sense of return.8
However, each of these instances utilizes orchestrational techniques rather than
theoretical applications of harmonic progression to provide a tonal center. Chapter II
introduces several concepts on how to create a tonal center within the half/whole
octatonic scale through the use of voice leading and harmonic progression.
5 Oliver Messiaen, The Technique of My Musical Language (Paris: A. Leduc, 1966), 87. 6 Ibid., 96. 7 Straus, Introduction to Post-Tonal, 131. 8 Milton Babbitt, “The String Quartets of Bartok” Musical Quarterly 35, no. 3 (1949): 385.
Chapter I – A Brief Overview of the Octatonic Scale 6
Due to the symmetrical nature of the octatonic scale, its subset structure is
comparably restricted and redundant.9 Like the octatonic collection itself, many of its
subsets are inversionally and/or transpositionally symmetrical, providing the ambiguity of
tonality typically associated with the scale.10 Figure 1-3 shows several possible subsets of
the octatonic scale that are combined to form intervallic sequences, taken from Jim Hall’s
book Exploring Jazz Guitar.11
Perfect 5ths followed by augmented 5ths, or minor 6ths, depending on the spelling:
Figure 1-3(a)
Major 3rds followed by Perfect 4ths:
Figure 1-3(b)
Sequence of minor and major triads:
Figure 1-3(c)
In post-tonal music, and even in earlier music, octatonic collections frequently emerge as
by-products of transposing scale fragments around an interval cycle of minor thirds,
which can also be seen in Figure 1-3, as every two measures of each example is
successively transposed up a minor third.12 In jazz improvisation, and a certain amount of
Twentieth-century Classical music, many of the intervallic sequences derived from the 9 Straus, Introduction to Post-Tonal, 144. 10 Ibid., 144. 11 Hall, Exploring Jazz Guitar, 18–21. 12 Straus, Introduction to Post-Tonal, 147.
Chapter I – A Brief Overview of the Octatonic Scale 7
half/whole ordering of the octatonic scale are used in conjunction with dominant 7th
chords. This arises from the fact that the upper structure of a V7 9 chord in harmonic
minor is a diminished 7th chord (see Figure 1-4).
Figure 1-4
Substituting the diminished 7th sonority for the V7 9 chord provides an increased amount
of tension, increasing the chords tendency to resolve. If the root or additional upper tones
of the V7 9 chord were added to the diminished 7th structure, it would destroy the
equidistant minor-third relationship, causing the chord to relinquish its quality to that of
an altered dominant.13 Figure 1-5 shows several common jazz “licks” used over dominant
7th chords taken from Jerry Coker’s Patterns for Jazz.14
Brooker Ervin, “No Private Income Blues,” on Mingus In Wonderland, Mingus Group
Figure 1-5(a)
David Baker, “Honesty,” on Ezz-Thetics, George Russell Sextet
Charlie Mariano, “Deep River,” on Toshiko Mariano Quartet, Toshiko Mariano Quartet
Figure 1-5(b)
13 Ludmila Ulehla, Contemporary Harmony: Romanticism through the Twelve-Tone Row (NY: The Free Press, 1966), 126. 14 Jerry Coker, Patterns for Jazz (Miami, FL: Studio Publications Recordings, 1970), 115, 131.
Chapter I – A Brief Overview of the Octatonic Scale 8
John Coltrane, “Straight No Chaser,” on Milestones, Miles Davis Sextet
Figure 1-5(c)
Dan Chemistruck, “The Bombing of London,” on 5/2/06, CCSU Jazz Ensemble
Figure 1-5(d)
For more insight on the use of the octatonic scale in jazz improvisation, studying one or
more of the following books will be helpful:
• Jazz Improvisation by David Baker • The Complete Method for Improvisation by Jerry Coker • Patterns for Jazz by Jerry Coker • The Lydian Chromatic Concept by George Russell • Scales for Jazz Improvisation by Dan Haerle • Thesaurus of Scales and Melodic Patterns by Nicolas Slonimsky
A significant amount of Stravinsky’s “octatonic-diatonic interaction” adheres to
the practice of utilizing the half/whole octatonic scale over dominant chords. Stravinsky’s
most octatonic works include Petroushka, The Rite of Spring, and Symphony of Psalms.15
Each of these pieces uses symmetrical constructions within juxtaposed blocks, defying
internally motivated “development” along traditional tonal lines through the static
harmony present within each block. Pieter Van Den Toorn, a scholar of Stravinsky’s
octatonic works, states that “change, progress, renewal, or development is possible only
by abruptly cutting off the deadlock and juxtaposing it with something new in the
15 Pieter C. Van Den Toorn, Stravinsky and The Rite of Spring (Berkley/Los Angeles, CA: Univ. of CA Press, 1987), 125.
Chapter I – A Brief Overview of the Octatonic Scale 9
collectional reference.”16 The collectional reference typically consists of either the
octatonic or diatonic collections, although a significant amount of the work can also be
analyzed using the modes of non-diatonic minor scales.17 Regardless of analytical
approach, Stravinsky’s octatonic writing is characterized by his use of tonality by
assertion, along with creating forward motion by juxtaposing blocks of octatonic material
with blocks of non-octatonic material.
The techniques of superimposition, juxtaposition and repetition are essential to
Stravinsky’s art.18 Stravinsky assigned priority to certain pitch classes by means of
doubling, metric accentuation, and persistence.19 This practice is most vividly seen in
Petroushka, The Rite of Spring, and Symphony of Psalms. Figure 1-6 shows the chord
typically associated with each of the works.
Petroushka chord:
Rite chord:
Psalms chord:
Figure 1-6
The Petroushka and Rite chords persist throughout a significant portion of their
respective works as either source material for other melodic lines or as repeating ostinato
figures. The Psalms chord, while it is not overtly octatonic itself, acts as a punctuation
16 Pieter C. Van den Toorn, Music of Igor Stravinsky (New Haven, CT: Yale Univ. Press, 1983), 328. 17 Dmitri Tymoczko, “Octatonicism Reconsidered Again,” Music Theory Spectrum 25 (2003): 188. 18 Van Den Toorn, Stravinsky, 128. 19 Ibid., 144.
Chapter I – A Brief Overview of the Octatonic Scale 10
mark between blocks of interacting and interpenetrating diatonic and octatonic
collectional references, as it is a subset of both references.20
The Petroushka chord is a polychord, consisting of two major triads a tritone
apart, producing pitch class set (pcs) 6-30 (013679), which is a subset of the half/whole
ordering of the octatonic collection; however, the chord can also be analyzed as a
C7 9 11 chord, although it does not function as such. The Petroushka chord is then used
as the referential source for the piano arpeggios and orchestral interactions, functioning
as it its own independent block with no tendency to resolve, despite its dominant 7th
structure.
The Rite chord can also be viewed as a polychord, consisting of an E major triad
over an E 7 chord, producing pcs 7-32 (0134689); however, this pcs is not entirely
octatonic, as pitch class 8 is not a member of the half/whole octatonic scale. The chord
can be more thoroughly analyzed as the vertical manifestation of the E Mixolydian 2 6
scale, although it does not have any of the functional implications that are typically
associated with this mode.21 “Stravinsky claims that he was not aware of Phrygian
modes, Gregorian chants, Byzantinisms, or anything of the sort in relation to his octatonic
compositions. The works were conceived intervallically, not harmonically, as two minor
thirds joined by a major third (0134).”22 Despite the foreign note found in the Rite chord,
there is no harmonic motion presented or attempted by the chord, as it serves as a pitched
rhythmic figure. Van den Toorn states, “Tonally functional schemes of modulation or
definitions of key are irrelevant here. Dominant chords are likely to be heard and
understood with reference to these prior contexts as self-enclosed blocks, not as sustained
20 Van Den Toorn, Music of Igor, 346. 21 Mixolydian 2 6 is the fifth mode of harmonic minor. See Appendix B for more information. 22 Van Den Toorn, Music of Igor, 344.
Chapter I – A Brief Overview of the Octatonic Scale 11
dominants awaiting an impending resolution.”23 This is characteristic of the majority of
Stravinsky’s octatonic work, typically treating a block of octatonic material as one
section of a composition to be varied with contrasting blocks of non-octatonic material,
without any intentions of creating a sense of forward motion within the octatonic block
itself.
Van den Toorn asserts that “the dominants are, octatonically, incapable of
realizing the traditional tonal escape route: there can be no resolution to a tonic.” 24 This
statement is only true if the theoretical traditions of common-practice tonality are forced
onto the octatonic collection; however, there is no need to chain modern sounds into a
system for which it was never intended.25 While Van den Toorn’s statement is true, it
does not necessitate that dominant 7ths are incapable of any resolution within an octatonic
framework. Despite the lack of traditional leading-tone elements in the octatonic scale,
principals of modern contrapuntal writing and voice leading permit all intervallic leaps
and require no resolutions.26 This concept may be further refined by altering the
definition of resolution so that it includes a defined intervallic relationship established by
the composer, rather than limiting it to the traditional concept of stepwise or perfect fifth
motion; therefore, the effect of harmonic progression is possible analogically rather than
absolutely through the transposition of a vertical structure, where the harmonic
relationship associated with the interval of transposition defines the harmonic
relationship. This type of progression is one of tonal association rather than of tonal
23 Van Den Toorn, Music of Igor, 343. 24 Ibid., 326. 25 Ulehla, Contemporary Harmony, 264. 26 Ibid., 327.
Chapter I – A Brief Overview of the Octatonic Scale 12
function, as the only preserved relationship to traditional tonality is the root movement of
the vertical structures.27 This concept will be discussed in greater detail in Chapter II.
Stravinsky utilizes the Dorian tetrachord (0235) as a bridge between octatonicism
and various diatonic orderings.28 The octatonic-diatonic interaction that the Dorian
tetrachord allows blurs the distinction between the two systems, but once an octatonic
framework is brought more securely into play, the authenticity of the tonal reference
fades. “It becomes a semblance, a side effect; the triads of progressions acquiring a
different feel, a different identity, owing to the symmetry of which they are now felt to be
a part.”29 Although specific instances of Stravinsky’s vertical structures can be analyzed
as subsets of either the diatonic church modes or non-diatonic minor scales, when looking
at how the structures interact with each other, the octatonic relationships become
apparent as the chords are incapable of fulfilling their traditional tonal functions. When
enough alterations take place that the accustomed Classical progressions of the major and
minor tonalities are no longer present, and no single chord takes unquestioned status as a
tonic, a solid diatonic key center is lost and we must settle for a possible, or perhaps,
probable hypothesis.30 Stravinsky is able to avoid this ambiguity of tonal center through
his blocks of static harmony, creating a tonal center simply because there are no other
choices presented.
27 Babbitt, “The String Quartets of Bartok,” 380. 28 Pieter C. Van den Toorn, “Colloquy: Stravinsky and the Octatonic, The Sounds of Stravinsky” Music Theory Spectrum Vol.25, 2003: Pg. 168. 29 Van Den Toorn, Music of Igor, 327. 30 Ulehla, Contemporary Harmony, 183.
Chapter I – A Brief Overview of the Octatonic Scale 13
Suggestions for Further Research:
For additional insight and musical examples of Stravinsky’s octatonic music, see
any of the books or articles used for this text written by Pieter Van Den Toorn. An
analysis of Oliver Messiaen and Béla Bartók’s music will offer additional insight into
octatonic systems that are harmonically more active than Stravinsky’s, yet maintain use
of the scale’s symmetrical qualities. Further study into the octatonic scale’s use in Jazz
will yield primarily pattern-based melodies used for improvisation, although occasionally
composers will use diminished 7th chords in unorthodox circumstances. For example,
Duke Ellington will occasionally end a piece on a diminished 7th chord to create an
ambiguous sense of tonal center and completeness.
14
-- Chapter II -- Imposing Harmonic Restrictions
Modal Relationships:
Modal scales made a profound impact on harmony during the Impressionistic
period; however, the effect that modes had on harmonic progression is more pertinent to
creating an octatonic system.31 Ludmila Ulehla points out the following characteristics of
the changes that took place between the chromatic concept of harmony and the aspects of
Impressionistic harmonization: 32
1) Lack of leading-tone. 2) Triad qualities relating to a mode rather than the major or minor scale. 3) Root progressions utilizing the full scope of chromaticism. 4) Non-traditional resolution of seventh chords. 5) Vague sense of key due to the non-diatonic effects.
These concepts are essential to creating an octatonic system because they are inherently
present within the octatonic scale (with the exception of number three). The remainder of
this section will focus on characteristic number two (triad qualities relating to a mode
rather than the major or minor scale), discussing the common subsets of the diatonic
collection, non-diatonic minor scales, and the octatonic collection. The remaining topics
will be discussed in the following sections of the chapter.
The diatonic church modes are defined by characteristic notes of the scale, allowing
for incomplete harmonies (chords missing the root, third and/or seventh chord degrees) to
maintain a tonal relationship and function. The ability to imply modal relationships
without presenting the entire mode is a convenient way to maintain associations to
31 For a thorough discussion of the influence of modes on harmony, see Ludmila Ulehla’s Contemporary Harmony: Romanticism through the Twelve-Tone Row Chapter 9: The Influence of Modes on Harmony. 32 Ulehla, Contemporary Harmony, 171.
Chapter II – Imposing Harmonic Restrictions 15
tonality, as only portions of the diatonic church modes and other non-diatonic minor
scales are found in the octatonic scale. Modal relationships should be made wherever
possible, unless it is the composer’s specific intention to avoid these relationships. Figure
2-1 shows the characteristic scale degrees of each mode of the major scale (column one
being the most characteristic and column six being the least).
Diatonic Modes Priority Table: 1 2 3 4 5 6 Lydian 4 7 3 6 2 5 Ionian (1) 7 4 3 6 2 5 Ionian (2) 7 3 2 6 5 (no 4) Mixolydian (1) 7 4 3 6 2 5 Mixolydian (2) 7 3 2 6 5 (no 4) Dorian 6 3 7 2 5 4 Aeolian 6 2 5 3 7 4 Phrygian 2 5 4 7 3 6 Locrian 5 2 7 6 3 4
[Note: The order has been adjusted to conform to “common practice.”]33
Figure 2-134
With the root being the most important note of a chord, typically the third and seventh
chord degrees are the next most important notes in terms of function, the third chord
degree providing the quality of the chord—major or minor—and the seventh chord
degree providing direction. Although the third and seventh chord degrees are usually the
most important functional notes of a chord, it does not necessitate that they are also the
most characteristic notes of a chord, which is more apparent in the Aeolian, Phrygian and
Locrian modes. The first ordering of the Ionian and Mixolydian modes show the fourth
scale degree being more characteristic than the third scale degree of the mode; this is due
to the harmonic use of the modes and their relationship to the other modes. A vertical 33 See Appendix D for an explanation of “common practice.” 34 Ron Miller, Modal Jazz Composition & Harmony Volume 1 (Rottenburg, Germany: Advance Music, 1992), 20.
Chapter II – Imposing Harmonic Restrictions 16
structure consisting only of a major seventh interval can be interpreted as either the
Lydian or Ionian mode. Adding the third chord degree will do little in terms of aiding in
distinguishing between the two modes, as both modes contain a major third; however,
adding the fourth chord degree to the vertical structure will clearly indicate if the implied
mode is Lydian or Ionian. Unfortunately, this same logic cannot be applied to the first
ordering of the Mixolydian mode because the remaining diatonic church modes share the
first two characteristic notes of the Mixolydian mode. Figure 2-1 was taken from Ron
Miller’s book on modal Jazz composition, and it displays the book’s predisposition for
quartal voicings, as the characteristic notes of the Lydian mode and the first orderings of
the Ionian and Mixolydian modes (reversing the order of the first two characteristic notes
listed) ascend up a quartal structure (See Figure 2-2).
Figure 2-2
The second ordering of the Ionian and Mixolydian modes come from their melodic
function, aptly omitting the fourth scale degree of the two modes as it is usually
considered a melodic dissonance.35 The latter orderings are also based on tertiary
structures, with the perfect fifth being the least characteristic note of the vertical
structure, simply providing a clear root with little indication of intended mode.
The characteristic notes of a mode can be used to imply associations to traditional
Western modality. Using Figure 2-1 as a guide, it is possible to construct vertical 35 Mark Levine, The Jazz Theory Book (Petaluma, CA: Sher Music Company, 1995), 34.
Chapter II – Imposing Harmonic Restrictions 17
structures using the most characteristic notes of a diatonic mode that are also available in
the octatonic scale. Figure 2-3 lists all the possible notes for each mode that can also be
found in the octatonic collection, with the first variation (1) indicating the half/whole
ordering and the second variation (2) indicating the whole/half ordering of the octatonic
collection. Each mode can also be described by its Forte code (Name), pitch class set
(PCS), interval vector (Vector), and prime inversion (PI).36
Mode: Scale Degrees: Name: PCS: Vector: PI: Ionian (1) 1, 3, 5, 6 4-26 0358 012120 ---- Ionian (2) 1, 2, 4, 6, 7 5-25 02358 123121 03568 Dorian (1) 1, 3, 5, 6, 7 5-25 02358 123121 03568 Dorian (2) 1, 2, 3, 4, 6 5-25 02358 123121 03568 Phrygian (1) 1, 2, 3, 5, 7 5-25 02358 123121 03568 Phrygian (2) 1, 3, 4, 6 4-26 0358 012120 ---- Lydian (1) 1, 3, 4, 5, 6 5-25 02358 123121 03568 Lydian (2) 1, 2, 4, 6, 7 5-25 02358 123121 03568 Mixolydian (1) 1, 3, 5, 6, 7 5-25 02358 123121 03568 Mixolydian (2) 1, 2, 4, 6 4-26 0358 012120 ---- Aeolian (1) 1, 3, 5 7 4-26 0358 012120 ---- Aeolian (2) 1, 2, 3, 4, 6 5-25 02358 123121 03568 Locrian (1) 1, 2, 3, 5, 7 5-25 02358 123121 03568 Locrian (2) 1, 3, 4, 5, 6 5-25 02358 123121 03568
Figure 2-3
Notice that since each mode is derived from the same collection of pitches, the sets
consisting of either tetrachords or pentachords remain the same, with only the ordering of
each set changing, as seen in the Scale Degrees column.
Since the listed pentachords in Figure 2-3 are extremely redundant, limiting them
to tetrachord formations will create more variety among the pitch class sets derived from
the octatonic collection. A possible solution for choosing which tones to select is to start
with the root, third or seventh chord degrees of a mode before referring to Figure 2-1 to
36 The “----” in the PI column indicates that the given pcs is invariant under inversion.
Chapter II – Imposing Harmonic Restrictions 18
fill out the remainder of the tetrachord as shown in Figure 2-5(a). For example, the third
scale degree of the Ionian (2) mode is not present in the whole/half octatonic scale,
allowing for two more notes to be placed in the tetrachord. The next two most
characteristic notes available in the Ionian mode, as listed in Figure 2-1, are the fourth
and sixth scale degrees, both of which are available in the octatonic collection. Figure 2-4
shows the construction of the Ionian and Dorian tetrachords as listed in Figure 2-5(a).
Ionian (2) Tetrachord Derived From the Whole/Half Octatonic Scale:
Figure 2-4(a)
Dorian (1) Tetrachord Derived From the Half/Whole Octatonic Scale:
Figure 2-4(b)
The Dorian (1) mode provides an example that requires only one characteristic note to
complete the formation of a tetrachord, as both the minor third and minor seventh are
available in the octatonic scale. Figure 2-1 lists the major sixth as the next most
characteristic note for the Dorian mode. The Phrygian and Aeolian tetrachords listed in
Figure 2-5(a) are constructed in a similar manner; however, the Lydian, Mixolydian and
Locrian modes listed do not conform to this pattern. The Mixolydian mode uses the fifth
scale degree instead of the sixth scale degree that Figure 2-1 suggests, forming its tertiary
seventh chord, while the Locrian mode utilizes the fourth scale degree instead of the
lowered seventh scale degree to create a minor second interval between the fourth and
lowered fifth scale degrees, creating a dissonant cluster voicing. The Lydian mode uses
Chapter II – Imposing Harmonic Restrictions 19
the fifth scale degree instead of the sixth scale degree for the same reason as the Locrian
mode.
Mode: Scale Degrees: Name: PCS: Vector: PI: Ionian 1, 4, 6, 7 4-Z29 0137 111111 0467 Dorian 1, 3, 6, 7 4-13 0136 112011 0356 Phrygian 1, 2, 3, 7 4-10 0235 122010 ---- Lydian 1, 3, 4, 5 4-Z29 0137 111111 0467 Mixolydian 1, 3, 5, 7 4-27 0258 012111 0368 Aeolian 1, 2, 3, 6 4-Z29 0137 111111 0467 Locrian 1, 3, 4, 5 4-13 0136 112011 0356
Figure 2-5(a)
Figure 2-5(b)
Figure 2-5(b) shows all the tetrachords listed in Figure 2-5(a) that are available in the
half/whole octatonic scale. Notice how each chord is available four times, with the roots
of the four chords outlining a diminished 7th chord. This is fairly characteristic of the
octatonic scale due to the symmetrical nature of the scale. The octatonic scale also
Chapter II – Imposing Harmonic Restrictions 20
contains four major and four minor triads, along with eight diminished triads, all
behaving in similar fashions, as seen in Figure 2-9.
Disregarding Figure 2-1 completely, there is still another alternative to consider
when constructing chords. It is possible to use the tetrachords that form the scales upon
which the modal chords are based to create vertical structures.37 Each mode of the major
scale is created by combining two tetrachords, with at least one of the tetrachords
providing the unique color tones of the scale. The tetrachord formations are as follows:
Tetrachords of the Diatonic Church Modes: First Tetrachord: Second Tetrachord: Mode: Name: PCS: Vector: PI: Name: PCS: Vector: PI: Ionian 4-11 0135 121110 0245 4-11 0135 121110 0245 Dorian 4-10 0235 122010 ---- 4-10 0235 122010 ---- Phrygian 4-11 0135 121110 0245 4-11 0135 121110 0245 Lydian 4-21 0246 030201 ---- 4-11 0135 121110 0245 Mixolydian 4-11 0135 121110 0245 4-10 0235 122010 ---- Aeolian 4-10 0235 122010 ---- 4-11 0135 121110 0245 Locrian 4-11 0135 121110 0245 4-21 0246 030201 ----
Figure 2-6 The only shortcoming with using the tetrachords listed in Figure 2-6 is that the only listed
tetrachord that is also available in the octatonic scale is pcs 4-10 (0235), which is
typically referred to as the Dorian tetrachord. Therefore, using this method with only the
diatonic church modes will allow only one modal inference; however, it is also possible
to use the tetrachords shared by the non-diatonic minor scales and the octatonic scale,
which are listed in Appendix C.
37 See Appendix B for chord-scale relationships.
Chapter II – Imposing Harmonic Restrictions 21
Harmonic Relationships:
Due to the lack of leading-tone resolution in the octatonic scale, traditional
concepts of functional harmony cannot be realized. Although certain tonal root
progressions are still available in the octatonic scale, the vertical structures built over
these roots and their interactions with each other will be inherently different; thus, it is
still possible to maintain a tonal association through root progressions despite the lack of
tonal function. Before codifying a system of vertical structures to be used as an octatonic
system, harmonic relationships should be established by expanding upon common-
practice tonality. Without a tonal basis, the elements of chromatic harmony can
overwhelm the octatonic scale, producing confusion of key center and direction. Thus,
the creation of an octatonic system should attempt to preserve or compensate for as many
harmonic relationships found in traditional common-practice tonality as possible in order
to provide a clear tonal center. There are five main factors to consider when constructing
harmonic relationships: (1) creating a dominant to tonic relationship, (2) predominant
chords, (3) chord quality, (4) root progressions, and (5) the number of vertical structures
in the system.
(1) Creating a Dominant to Tonic Relationship:
The half/whole ordering of the octatonic scale was chosen over the whole/half
ordering because the former contains the common-practice period dominant (G) to tonic
(C) relationship. When constructing a dominant to tonic relationship, the two chords must
be given vertical structures and/or intervallic content that are unique from the other
vertical structures in the system. This will provide a distinct contrast between the tonic
Chapter II – Imposing Harmonic Restrictions 22
and dominant chords, and the other subordinate sonorities. The presence of additional
unique vertical structures in the system is inconsequential as the main purpose for
creating the disparity between the tonic and dominant from the remainder of the system is
to prevent the symmetrical structure of the octatonic scale from overwhelming all of its
vertical manifestations, as the octatonic scale contains numerous symmetrical
constructions as seen in Figure 2-5(b). Therefore, making the intervallic content of all the
vertical structures mutually exclusive within a given system will aid in the possibility of
inferring additional tonal centers within the system, whereas permitting a degree of
recursion among the subordinate sonorities will place a greater distinction on the tonic
and dominant chords.
(2) Predominant Chords:
In addition to establishing the dominant to tonic relationship, it is also important
to designate one or more chords as predominant chords. The predominant chords of
common-practice tonality typically consist of a vertical structure built on the
subdominant or supertonic; however, neither of these chords is present in the octatonic
scale. The most practical alternative for replacing the subdominant chord is a diminished
structure built on the raised fourth scale degree, functioning as a secondary leading-tone
chord in traditional harmony as seen in Figure 2-7(a). Just as practical a substitution for
the subdominant is the tritone dominant (a dominant chord built on the raised fourth, or
tritone, of the scale), allowing for the subdominant to also resolve to the tonic as seen in
Figure 2-7(b).38
38 Ulehla, Contemporary Harmony, 213.
Chapter II – Imposing Harmonic Restrictions 23
Secondary Leading-Tone: Tritone Dominant:
Figure 2-7(a) Figure 2-7(b) Another way to analyze the two possible chords presented in Figure 2-7 is to label the
tritone dominant as the subdominant of the octatonic system, with the diminished 7th
chord functioning as either a common-tone diminished 7th (the root of the chord
providing the common-tone) or as a secondary leading-tone chord, seen in Figure 2-8.
I IV13 VIIdim7/V Vdim addM7 I
Figure 2-8 This interpretation offers the most flexibility with chord progressions, as it allows both
dominant 7th and diminished 7th chords to be built on the raised fourth scale degree. Due
to the nature of the octatonic scale, it is possible to build a common-tone diminished 7th
chord from any note within the scale without adding foreign notes to the scale.
Chapter II – Imposing Harmonic Restrictions 24
(3) Chord Quality
When choosing the chord qualities for an octatonic system, it is important to
consider the structure and intervallic content of the other chords present in the system.
For example, maintaining the same triadic quality for the tonic, subdominant and
dominant harmonies will provide a distinguishable relationship to the major scale as each
of these harmonies consists of major triads in the major scale, although this is only
possible with the diminished triad in the octatonic scale. Despite the limited choices for
maintaining consistent triadic qualities of the tonic, subdominant and dominant chords, it
is possible to construct either a major, minor, or diminished triad on any of the notes
found in the C diminished 7th chord (See Figure 2-9). This characteristic of the octatonic
scale has potential for exploiting the various types of mediant relationships to allow a
sense of modulation to other tonal centers without introducing foreign notes to the scale.
Major Triads Available in the Octatonic Scale:
Minor Triads Available in the Octatonic Scale:
Diminished Triads Available in the Octatonic Scale:
Figure 2-9
Chapter II – Imposing Harmonic Restrictions 25
The chords presented thus far are largely based on tertiary triads, presenting only a
handful of the total possible trichords, tetrachords, etc that can be found in the octatonic
scale.39 Additional concerns regarding similarities between chords will be discussed in
the Intervallic Content section.
(4) Root Progressions
Root progressions in the octatonic scale are not limited to those arising from the
diatonic church modes. Although it is possible to reharmonize familiar tonal progressions
(see the Additional Considerations section), chromatic additions and mixture of modes
must prevail if a composition is not to become overwhelmed by its own unique
qualities.40 Chord progressions involving root movement of a minor second (leading-tone
resolution), major or minor third (mediant relationship), and perfect fourth or fifth
(cadential resolution) are all possible from several starting points of the octatonic scale,
as seen in Figure 2-10. Despite the lack of clear functional harmony, the root progression
accompanied by smooth voice leading provides forward motion until the progression
cadences on the C major triad.
I IIIdim VImin VIdim7 IV VImin7 5 III7 I
Figure 2-10
39 See Appendix E for a complete listing of additional subset relations. 40 Ulehla, Contemporary Harmony, 180.
Chapter II – Imposing Harmonic Restrictions 26
Figure 2-10 makes use of several inverted chords to provide a smooth bass line that
moves predominately by either semitone or perfect fifth motion. In addition to inversions,
it is also possible to use root progressions involving implied harmonies, but this will be
discussed in the Additional Considerations section.
A unique feature of the octatonic scale is the possibility of using interval cycles or
intervallic sequences. An interval cycle is created by starting on any pitch and moving
repeatedly by any interval.41 Although it is only possible to have an intervallic cycle
consisting of minor thirds or tritones in the octatonic scale, it is possible to create
intervallic sequences by cycling through a pattern of intervals. The first two staff systems
of Figure 1-3 show two basic intervallic sequences, the first consisting of up a perfect
fifth – down a tritone – up a minor sixth – down a tritone, and the second consisting of up
a major third – down a minor third – up a perfect fourth – down a minor third. Although
these sequences are presented in a melodic context, it is also possible to use them as root
progressions. There are also less complicated sequences that will go through a complete
sequence in fewer notes than the patterns listed in Figure 1-3. Figure 2-11 shows an
intervallic sequence consisting of up a perfect fifth and down a major third.
Figure 2-11 Likewise, it is also possible to create more intricate patterns than the ones shown here.
For an exhaustive listing of possible symmetrical progressions in the octatonic scale, see
41 Straus, Introduction to Post-Tonal, 154.
Chapter II – Imposing Harmonic Restrictions 27
Nicolas Slonimsky’s chapter on the Sesquitone Progression in his Thesaurus of Scales
and Melodic Patterns.42
(5) The Number of Vertical Structures in an Octatonic System:
The number of vertical structures that are present in a given system will affect the
total possible achievable tonal centers and will play a factor in the degree of relationships
occurring between each chord. Fewer chords will facilitate a higher degree of recursion
between vertical structures and will provide the clearest sense of tonal center due to the
limited amount of vertical structures being presented. Contrarily, additional chords will
increase the possibilities for creating alternative approach chords, tonal centers and
unique vertical structures, although it is still possible to have a high degree of recursion
due to the symmetrical nature of the octatonic scale’s subsets.
42 Slonimsky, Nicolas, Thesaurus of Scales and Melodic Patterns (NYC, NY: Schirmer Books, 1975), 51– 73.
Chapter II – Imposing Harmonic Restrictions 28
Vertical Spacing:
Equal in importance to choosing chord tensions is determining the vertical
spacing of the chord. The categories of chord spacing are:
1) Tertiary – The adjacent notes are of a major or minor third interval. 2) Cluster – The adjacent notes are of a major or minor second. 3) Quartal – The adjacent notes are of a perfect or augmented fourth. 4) Mixed – The adjacent notes are of a combination of seconds, thirds, and fourths.43
The most apparent relations to common-practice tonality will occur with the use of
tertiary structures, although the uses of the other possible spacings are equally practical.
Regardless of the chord spacing used, principles of orchestration will help clarify chord
qualities and intended roots. There are several concerns to consider avoiding any
additional harmonic ambiguity than is already present in the octatonic scale. According to
Ulehla, reasons for harmonic ambiguity that arise from poor orchestration and/or chord
voicings can generally be found in one or more of the following situations:
1) The lack of a prominent bass tone. 2) A melodic movement in the bass register. 3) Abnormal order of intervals in the vertical structure which negate the strength
of the overtone series. 4) A conspicuous use of the tritone, melodic or harmonic, in which either tone
may claim its right as a root. 5) An absence of the cadential root progression by which an association of tonal
movement could either anticipate a root or recognize it following the chord’s resolution.44
Adhering to these guidelines will provide a clear sense of tonal function. These rules are
essential to securing the tonal underpinnings of an octatonic framework due to the
octatonic scale’s inherent tendency to lend itself to a number of these situations.
43 Miller, Modal Jazz Composition, 20. 44 Ulehla, Contemporary Harmony, 219.
Chapter II – Imposing Harmonic Restrictions 29
Using inversions is possible, although they should be handled with care as the
symmetrical structure of the scale may lead to the confusion of the intended root. Figure
2-10 provides an example that successfully utilizes chord inversions. The G /B chord
can be analyzed as either a first inversion G Maj chord or as a B min 6 chord. The given
voicing doubles both the G and the B , but the interval of the perfect fifth between G
and D firmly establishes the G as the root of the chord. Additionally, the root of a
major triad sounds more clearly than the root of a minor triad because the overtone series
produces the major triad in lower partials.45 The chord must also be considered in its
contextual use. The preceding Adim7 chord can also be labeled as G dim7/A,
functioning as a common-tone diminished 7th chord to the G Maj chord. Figure 2-12
shows an example using the same chord with a different voicing and contextual use,
where labeling the chord G Maj would be wrong.
Figure 2-12
The root progression of B min 6 – E 7 functions as a IImin 6 – V7 in A minor, but is
deceptively resolved to Amin7/E. While the bass movement of the harmonic progression
follows the traditional deceptive cadence by having the dominant 7th chord resolve up a
semitone, the resolution chord is not the expected EMaj chord, but a second inversion
Amin7 chord. The Amin7 functions as an inverted chord because the only doubled note is
the root, with no functional choices for an alternative chord label. 45 Ulehla, Contemporary Harmony, 329.
Chapter II – Imposing Harmonic Restrictions 30
Intervallic Content:
In addition to the surface construction of the vertical structures, it is also
important to consider their intervallic content and pitch class relationships, as it will
allow additional relationships and distinctions to form between the various vertical
structures. Two intervallic relationships that will be discussed are the R relationships and
the Z relationship. There are many other possible relationships, but they are beyond the
scope of this text.
The R relationship exists in four forms: Rp, R0, R1, and R2. Allen Forte’s
description of each relationship is given in Figure 2-13.46
Relation: Interpreted As: Rp Maximum similarity with respect to pitch class. R0 Minimum Similarity with respect to interval class. R1 Maximum Similarity with respect to interval class.
Interchange feature. R2 Maximum Similarity with respect to interval class.
Without interchange feature.
Figure 2-13
To be in the relation Rp, two sets, S1 and S2, of cardinal number n must have at least one
common subset of cardinal number n-1.47 As Forte explains on page 47 of his book The
Structure of Atonal Music, this relationship is not especially distinctive since many sets
are related to a large number of other sets. For two sets to be in relation R0, two vectors
must have no entries the same. For two sets to be in relation R2, two vectors must have
four similar interval-class entries. Figure 2-14 compares the vectors of pcs 5-10 (01346)
and 5-Z12 (01356), indicating that entries for ic1, ic2, ic4, and ic6 correspond.
46 Allen Forte, The Structure of Atonal Music (New Haven, CT: Yale University Press, 1973), 49. 47 The “cardinal number” is the amount of different pitch classes in given a set.
Chapter II – Imposing Harmonic Restrictions 31
5-10 [ 2 2 3 1 1 1 ] 5-Z12 [ 2 2 2 1 2 1 ]
Figure 2-14
For two sets to be in relation R1, two vectors must contain the same digits and have four
corresponding interval-class entries, as seen in the comparison of pcs 4-2 (0124) with 4-3
(0134) shown in Figure 2-15.
4-2 [ 2 2 1 1 0 0 ] 4-3 [ 2 1 2 1 0 0 ]
Figure 2-15
As seen in Figure 2-15, the entries for ic2 and ic3 are interchangeable. In this sense, R1
provides a closer relation than R2 does.
Any two sets related by transposition or inversion must have the same interval-
class content; however, the converse is not true.48 Sets that have the same interval-class
content but are not related by transposition or inversion are called Z-related sets, and the
relationship between them is the Z-relation. Two prominent subsets of the octatonic
collection that are in the Z-relation are pcs 4-Z15 (0146) and pcs 4-Z29 (0137), which
also have the property of being the two “all-interval” tetrachords because their interval-
class vectors consist of 111111.
48 Straus, Introduction to Post-Tonal, 91.
Chapter II – Imposing Harmonic Restrictions 32
Voice Leading:
Due to the structure of the half/whole octatonic scale, it is not possible to strictly
adhere to common-practice tonality standards of voice leading. Therefore, certain rules
will have to be developed for each individual system with reference to concepts of
traditional voice leading, focusing primarily on stepwise motion and the cadential root
movement of down a perfect fifth. This paper focuses on the half/whole ordering of the
octatonic scale primarily because of the presence of the perfect fifth occurring between
the tonic and dominant of the scale, which is absent in the whole/half ordering of the
scale. The interval of a perfect fifth is essential to establishing a sense of tonality because
in harmonic contexts the perfect fifth establishes the lower tone as a root so strongly that
it will retain its tonal predominance through a significant amount of contrapuntal activity
until another combination of strong intervals seize harmonic priority.49 Specific
guidelines in relation to octatonic systems will be discussed in Chapter III. The remainder
of this section will focus on the limitations of the octatonic scale in relation to traditional
functional harmony.
The octatonic scale allows the construction of dominant 7th and secondary
leading-tone structures, but both are unable to realize their traditional tonal resolutions
without adding foreign notes to the octatonic scale. A significant amount of octatonic
literature is used in an octatonic-diatonic interaction because the octatonic scale adds
chromaticism to the dominant chord, increasing its tendency to resolve. For example,
Figure 2-7(b) shows that it is possible to construct an F 7 chord, but it cannot be resolved
down a perfect fifth to BMaj because the B is absent in the C half/whole octatonic scale;
49 Ulehla, Contemporary Harmony, 321.
Chapter II – Imposing Harmonic Restrictions 33
however, it is possible to imply this resolution because the upper structure of BMaj is
available in the octatonic scale. The concept of implied harmony will be discussed in
greater detail in the Additional Considerations section. The F 7 chord has two other
functional possibilities for resolution: CMaj and Gmin. The resolution of F 7 to CMaj is
shown in Figure 2-7(b) and is referred to as the tritone dominant. The second possible
resolution is up a semitone to Gmin, which will function as a deceptive cadence but the
fifth of the chord will have to be omitted, as D is not present in the C half/whole
octatonic scale and using the available fifth will produce a diminished triad, significantly
weakening the cadence. Figure 2-16 shows a possible voicing for this progression.
. Figure 2-16
Secondary leading-tone chords will function in a similar manner to the deceptive cadence
shown in Figure 2-16, as the only functional resolution available in the octatonic scale is
up a semitone to a minor triad with the fifth of the chord omitted to avoid creating a
diminished triad. Although neither of the choices provides as distinct a cadence as their
traditional tonal counterparts offer, each example resolves the dissonance of the given
vertical structure to a consonant tertiary sonority.
Following the guidelines of smooth voice leading, additional resolutions can be
justified by melodic movement. For example, Figure 2-10 shows an E 7 resolving down
a minor third to CMaj with all the notes of the chord either resolving through stepwise
Chapter II – Imposing Harmonic Restrictions 34
motion or sustaining, with the exception of the bass. To further simulate a cadential
progression, the E 7 could have been voiced in first inversion to create bass movement
from G to C to provide a dominant to tonic bass motion. If this harmonic motion, or any
other harmonic motion that the composer desires, is used consistently to designate phrase
endings within a piece, it will assume a cadential function as the piece progresses. It is
also important to remember that all intervals which have a closer placement to the bass or
lowest tone, have a greater harmonic contribution than do intervals rising above them;
therefore, it is necessary to maintain as strict a sense of harmonic continuity in the lowest
sounding register as the octatonic scale permits.50
When creating an octatonic system, the best results will occur by establishing
relations between a series of vertical structures that will be used consistently throughout a
piece. Whether this process occurs prior to the beginning of a composition or allowing
the relations to establish themselves as the piece develops is irrelevant, as long as there is
an underlying system of voice leading in place. While this text has a predisposition for
the use of smooth voice leading to provide possible alternatives for the lack of traditional
functional harmonic resolutions, more angular approaches towards voice leading are
equally as plausible, as long as the composer is consistent with the methods used.
50 Ulehla, Contemporary Harmony, 325.
Chapter II – Imposing Harmonic Restrictions 35
Additional Considerations:
Although the focus of this text is harmonic motion, it is not the only factor in
creating a tonal center. Several other musical aspects that have only been mentioned thus
far include (1) subset relations, (2) reharmonization and implied harmonic possibilities,
and (3) metric and rhythmic considerations. Each of these concepts is as potentially
useful as any of the other concepts presented in this chapter and will serve to further
solidify the intended tonal center.
(1) Subset Relations:
The Modal Relationships section showed that it is possible to imply a specific
vertical structure that is not entirely present in the octatonic scale by using the
characteristic notes of the related mode; however, it is also possible to imply additional
harmonies by using common subsets. Figure 2-17 displays a series of possible harmonic
implications using dyads over notes from the octatonic scale.51
Dyads:52 C C C C C C C C Dyads: D D E F G A B C
C7 9 Cmin CMaj C7 4 C5 C6 C7 C C Maj7 C Maj9 C mM7 C M7sus4 C M7 5 C M7 5 C Maj13 C Maj7 D 13 D 6 D 13 9 D dim7 D 13 D dim7 D 6 D 6 E13 5 EMaj7 5 E7 13 E9 13 Em7 6 E7 6sus4 E7Alt5 E7 13 F 7 11 F dim7 F m7 5 F 7 5 F 7 9 4 F dim F 7 5 F 7 4
Gdimsus4 G+sus4 G6sus4 GM7sus4 Gsus4 Gsus Gmin11 G7sus4A7 9 Adim Amin Adim7 Amin7 Amin Amin7 9 Amin
Implied Chords:
A min9 A sus4 A 9 5 A 9 13 A dim7 A Maj9 A 9 A 9
Figure 2-17
51 See Appendix D for Ron Miller’s Collated Order of All Constructed Modes. 52 The dyads are presented vertically, with the bass note from the octatonic scale and the implied chord label from the resulting trichord listed underneath each dyad in accordance to Ron Miller’s Collated Order of All Constructed Modes.
Chapter II – Imposing Harmonic Restrictions 36
Notice that the chords listed in Figure 2-17 are derived from the diatonic church modes or
non-diatonic minor modes.53 As all the dyads are formed with the note C as a bass, they
can be transposed to any of the other three notes of the Cdim7 chord to produce
additional harmonic implications. Although there is a large amount of harmony that can
be implied through dyads, if overused or similar harmonies are transposed symmetrically,
it will ultimately detract from a clear tonal center. Additional listings of harmonic
implications, including trichords and tetrachords, are given in Appendix E.
(2) Reharmonization and Implied Harmonic Possibilities:
Reharmonizing traditional tonal progressions using those available in the
octatonic scale will provide a way of working in and out of an octatonic system. It is not
necessary for an entire composition to strictly adhere to the guidelines and restrictions of
an octatonic system, as most of the octatonic literature to date consists of octatonic-
diatonic interaction. Figure 2-7(a) illustrates a IVdim7 – Vdim addM7 - I progression,
which can serve as a replacement for the traditional I – IV – V – I progression. If a
IVmin7 5 was used instead of the fully diminished 7th, it would serve the same purpose,
except that with the exception of the root, the remaining chord tones would be invariant
with the original IVMaj7 chord found in the major scale. As for the V dim addM7 chord,
as long as it is presented in root position it will function as a dominant structure because
the dominant to tonic root progression will remain intact as intervals that are placed in the
lowest register have a greater harmonic contribution than intervals rising above them.
Jazz theory refers to this practice as “modal interchange,” since the root of the chord
remains the same, while the chord-scale relationship changes. 53 See Appendix B for a list of chord-scale relationships.
Chapter II – Imposing Harmonic Restrictions 37
Implying harmony is another way to simulate functional harmony without
presenting a chord in its entirety; however, if done improperly, implied chords may lead
to the confusion of the intended chord and harmonic progression. Ulehla states, “Any
chord in which the root tone is omitted is less securely situated than if the root tone were
present and especially if located in the bass. Enharmonic changes may invite surprising
resolutions and thereby alter an analysis.”54 If an implied harmony does not use its
traditional tonal function, whether it is cadential resolution or simply a passing chord, the
implied harmony is likely to be seen with a different function and is more likely to be
mislabeled. Evaded progressions, or those involving root movements other than the
cadential dominant to tonic progression, are feasible but less convincing when using
implied roots.55 Figure 2-18 shows four different possible harmonizations of a 12 bar
blues. Figure 2-18(b) is an octatonic reharmonization of Figure 2-18(a) using implied
harmony on the subdominant chord in measures 2, 5, and 10 because the root F is not
present in the octatonic scale. Also, the dominant chord in measures 9 and 12 is
reharmonized as a diminished triad, since the typical V7 chord is not available in the
octatonic scale.
Figure 2-18(c) presents a more complex version of Figure 2-18(a), reharmonized
with a series of IImin7 - V7 progressions. This new progression is then reharmonized in
Figure 2-18(d) using only chords or implied harmony derived from the octatonic scale.
The only new implied harmony that is not present in Figure 2-18(b) is the D7 9 in
measures 9 and 12, serving as a secondary dominant chord to replace the Dmin7 in
Figure 2-18(c). However, the D7 9 is an example of an implied harmony that has several
54 Ulehla, Contemporary Harmony, 127. 55 Ibid., 129.
Chapter II – Imposing Harmonic Restrictions 38
12 Bar Blues:
Figure 2-18(a)
12 Bar Blues Reharmonized with the Octatonic Scale:
Figure 2-18(b)
12 Bar Bebop Blues:
Figure 2-18(c)
12 Bar Bebop Blues Reharmonized with the Octatonic Scale:
Figure 2-18(d)
Chapter II – Imposing Harmonic Restrictions 39
different ways of being analyzed. By omitting the root on the D7 9 chord, the four
remaining notes form an F dim7 chord, which can also be labeled as a secondary leading-
tone chord to Gdim addM7.56 Regardless of the label, the function of both chords is the
same; yet, if Figure 2-18(c) is played prior to Figure 2-18(d), the chord in question will
likely be heard, or at least understood, as a rootless D7 9 chord. Similar chord
substitutions are possible with any chord whose upper structure is found in the octatonic
scale, despite the lack of a root. In certain instances, simplifying a chord progression will
provide a clearer relation to the intended harmonies simply because there are fewer
harmonies to imply; this is seen as the basic progression of Figure 2-18(a) requires only
one reharmonization in Figure 2-18(b) compared to the more complex progression of
Figure 2-18(c) that requires four reharmonizations, seen in Figure 2-18(d). Since using
implied chords opens an analysis to several different interpretations, placing incomplete
chords on weak beats will help maintain their secondary function.
(3) Metric and Rhythmic Considerations:
Rhythmic and metric placement is an essential part of dictating harmonic
function, having an equally influential impact on a chord progression as voice leading or
harmonic relationships. Regardless of interval, rhythmic stress dictates the hearing of
harmonic versus non-harmonic tones.57 Ulehla makes the following statement about the
use of rhythm in non-diatonic settings,
Rhythmic stresses give attention to select groups of pitches. The pitch contour formed by the phrase produces some tones of prominence and others which serve in a supporting rhythmic capacity. Harmonies contain roots, without requiring a
56 In measure 12 of Figure 2-18(d), the Gdim addM7 chord is abbreviated as Go7 because any note a whole step above a chord tone in a diminished 7th chord is a possible tension. 57 Ulehla, Contemporary Harmony, 322.
Chapter II – Imposing Harmonic Restrictions 40
diatonic tertiary order. Those are the new developments recognized today. The tonality today is not one that necessarily centers on one central tonic for the entire composition. It shifts tonal centers at will. All twelve tones may take on the equivalent role of the former tonic. But they will always assume a position that governs twelve notes, each of which may hold reign above the others at any time. Tones which start a phrase, climax the contour of a phrase, become part of a cadence; all contribute towards the movement within the phrase. They lead somewhere. The recipient of that motion has more tonal power than the insignificant motivic assortment of tones which are heard ‘en route’. Tonality is fleeting, but it is there. It is not in the form of one key dominating all, but a transient assortment which may include all twelve tones in rotation.58
Despite the importance of rhythmic and metric placement, these considerations are listed
as a subsection of the Additional Considerations section rather than in their own section
because it is possible to establish a vertical structure as a tonic through sheer repetition,
rather than through the use of harmonic progression or other traditional tonal concepts,
effectively circumventing the focus of this text; therefore, only a limited discussion of
rhythmic and metric considerations is presented.
Similar to traditional common-practice tonality, placing the primary harmonies of
an octatonic system—typically the tonic, dominant, and possibly subdominant—on the
strong beats of a measure will help separate the primary chords of a system from the
subordinate sonorities, providing a clear sense of forward motion. Also, in terms of
melodic line, the tones which occupy the highest or lowest position of each small “peak”
are more prominent than the connecting tones between them.59 Therefore, another
possibility to assert harmonic importance on a vertical structure is to place it at the peak
of a melodic line. A vertical structure can also gain importance by sustaining longer than
the other structures surrounding it. For a more thorough discussion of rhythm in non-
diatonic settings, see the Linear Roots section of Ulehla’s Contemporary Harmony.
58 Ulehla, Contemporary Harmony, 322. 59 Ibid., 304.
41
-- Chapter III -- Creating a Tonal Octatonic System
This chapter will be limited to the naturally occurring tetrachords found in the
octatonic scale. There are twelve prime forms and an additional six prime inversions for a
total of eighteen separate tetrachords, which are listed in Figure 3-1.
Tetrachords in the Octatonic Scale: Name: PCS: Vector: PI:
4-3 0134 212100 ---- 4-9 0167 200022 ----
4-10 0235 122010 ---- 4-12 0236 112101 0346 4-13 0136 112011 0356
4-Z15 0146 111111 0256 4-17 0347 102210 ---- 4-18 0147 102111 0367 4-26 0358 012120 ---- 4-27 0258 012111 0368 4-28 0369 004002 ----
4-Z29 0137 111111 0467
Figure 3-1 The concepts discussed in Chapter II will help limit the eighteen choices of tetrachords to
eight or fewer, so that there is no more than one tetrachord per scale degree of the
octatonic scale. This chapter will only be divided into five of the six sections found in
Chapter II:
• Modal Relationships • Harmonic Relationships • Vertical Spacing • Intervallic Content • Voice Leading
The Additional Considerations section has been omitted from this chapter because it dealt
with more compositional concerns rather than specific theoretical applications.
Chapter III – Creating a Tonal Octatonic System 42
This chapter will relate the concepts presented in Chapter II to the octatonic
system presented in Figure 3-2(a), derived from System 2 of Appendix F. The octatonic
system is presented alongside the major scale to provide a comparison between the two
systems for later discussions.
Octatonic Scale:
[Note: The chord built on the fourth scale degree of the octatonic scale has been omitted.]60
Figure 3-2(a)
Major Scale:
Figure 3-2(b)
Figure 3-2(c)
60 See the Harmonic Relationships - subsection (5) of this chapter for an explanation.
Octatonic Scale: Major Scale: Chord: Name: PCS: Vector: Chord: Name: PCS: Vector:
I7 4-27 0258 012111 IMaj7 4-20 0158 101220 IIo7 4-28 0369 004002 IImin7 4-26 0358 012120 IIIMin7 4-26 0358 012120 IIImin7 4-26 0358 012120 IVMin7 5 4-27 0258 012111 IVMaj7 4-20 0158 101220
V dim addM7 4-18 0147 102111 V7 4-27 0258 012111 VImin7 4-26 0358 012120 VImin7 4-26 0358 012120 VIIo7 4-28 0369 004002 VIIMin7 5 4-27 0258 012111
Chapter III – Creating a Tonal Octatonic System 43
Modal Relationships: This system was created with the intention of forming relationships to the major
scale on the raised fourth and sixth scale degrees; several other modal inferences are
made on other scale degrees as well. A description of each relationship is given below.
• The tonic chord of the octatonic system is a dominant 7th structure, implying the
Mixolydian mode. The decision to use a dominant 7th structure as a tonic was derived
from a 12-bar blues progression—see Figure 2-18(a)—which also utilizes a dominant
7th as tonic. Also, the possibility of omitting the seventh chord degree will provide the
same tonic triad as the major scale.
• The lowered third chord of Figure 3-2(a) is completely foreign to the major scale
presented in Figure 3-2(b), yet the chord quality of the third chord in both systems is
the same, implying the Phrygian mode in the given context.
• With the exception of the root, the subdominant chord in both systems is invariant.
Using a rootless or implied harmony instead of the provided chord of the octatonic
system will maintain a closer relation to the major scale; however, the presence of the
root will provide the given chord with a higher degree of functionality.
• The dominant chord of the octatonic system has no direct correlation to a diatonic
church mode, as it was chosen for its intervallic qualities. Although it is also possible
to analyze the dominant chord as a first inversion G major triad over a G bass note,
the chord’s functional use within a composition will dictate the chord’s label.
• The VI chord in both systems is invariant.
• The remaining two chords of the octatonic system are diminished 7th structures,
functioning as subordinate sonorities due to their recursive qualities.
Chapter III – Creating a Tonal Octatonic System 44
Harmonic Relationships:
(1) Creating a Dominant to Tonic Relationship:
The dominant chords in both systems have unique chordal tensions allowing for
chromatic resolution to the tonic chord along with the root movement of down a perfect
fifth. In order to create a clear distinction in sonority, the dominant chord of the octatonic
system contains a unique pcs from the other chords present in the system. Similarly, the
tonic chord of the octatonic system has a unique chordal structure, although it shares its
pcs with one of the other vertical structures in the system: the subdominant.
(2) Predominant Chords:
This octatonic system attempts to preserve the similarities present between the
tonic and subdominant chords that are found in the major scale. Within each system, the
tonic and subdominant chords are derived from the same pcs, despite the differing
chordal structures in the octatonic system. Although it is possible to have dominant 7th
structures built on both the tonic and subdominant chords of the octatonic system, the
Modal Relationships section of this chapter states that the subdominant chord in the
octatonic system is trying to create maximum invariance with the subdominant of the
major scale.
To compensate for the lack of an F in the octatonic scale, the root of the
subdominant chord has been replaced with an F to create a secondary leading-tone chord
to the dominant to provide a suitable predominant chord. Of the four chords that are built
on diminished triads in the given octatonic system, the subdominant chord is the only one
capable of functioning in a traditional tonal setting by resolving up a half step.
Chapter III – Creating a Tonal Octatonic System 45
(3) Chord Quality:
The chord qualities of the given octatonic system have been chosen to provide a
significant amount of recursion without becoming overtly redundant, imitating the
structure of the major scale. The major scale has diatonic minor 7th chords built on the
second, third and sixth scale degrees; the octatonic system maintains a similar
consistency of tertiary structures on the lowered third and sixth scale degrees. Also, the
octatonic system contains diminished 7th structures built on the lowered second and
lowered seventh scale degrees.
(4) Root Progressions:
Since this system omits one of the possible chords in the octatonic system (see the
next subsection for a more thorough explanation), the possible root progressions will be
more restricted than normal. A prominent disadvantage to this system is not having the
perfect fifth root movement available between the third and sixth scale degrees, as the
chord built on the third scale degree has been omitted; however, through inversions it is
possible to utilize a bass progression of E to A, as seen in Figure 3-3.
Figure 3-3
Chapter III – Creating a Tonal Octatonic System 46
(5) The Number of Chords in an Octatonic System:
To imitate the major scale as closely as possible, this particular system consists of
seven chords rather than eight. Omitting the fourth chord of the octatonic system allows
the intervallic content to more accurately reflect that of the major scale by reducing the
total unique tetrachords in the given system to four, compared to the three unique
tetrachords found in the major scale. An alternate version of this system includes an
additional chord built on the fourth scale degree of the octatonic scale, labeled
IIIminMaj7 5. 61 This additional chord would introduce the previously foreign tetrachord
4-17 (0347) into the given octatonic system. Although the composer may desire a variety
of unique vertical structures, it will ultimately detract from isolating the dominant and
tonic relationship by placing an equal amount of importance on what was previously a
subordinate sonority.
Vertical Spacing:
This system makes use of tertiary harmony because it will provide the most
immediately recognizable foundation to traditional tonality. The tertiary underpinnings
will also aid in emphasizing the overtone series, providing a clearer sense of intended
harmonic structure and function. There are several untraditional occurrences of the tritone
that should be handled with care, particularly on the dominant chord to avoid the
analyzation of the chord as G Maj/G. For the best results, the root of the chord should
always be placed in the lowest voice, and when that is not possible, the root of the chord
should be doubled.
61 See System 2 of Appendix F for this alternative octatonic system.
Chapter III – Creating a Tonal Octatonic System 47
Intervallic Content:
In this particular octatonic system, the tonic, 4-27 (0258), and dominant, 4-18
(0147), chords are in the relationship R1 to form a stronger association with each other, as
seen in Figure 3-4.
4-27 [ 0 1 2 1 1 1 ] 4-18 [ 1 0 2 1 1 1 ]
Figure 3-4
Although this relationship is not present in the major scale, it is an attempt to compensate
for the lack of leading-tone resolution and other relationships that are incapable of being
reproduced in the octatonic scale without introducing foreign notes to the scale. Instances
of pcs transposition occur between the lowered third and sixth chords, the lowered second
and lowered seventh chords, and also the tonic and subdominant chords.
Voice Leading:
The half/whole ordering of the octatonic scale does not contain the leading-tone
found in the major scale, thus necessitating the need to redefine the voice leading rules
for the dominant to tonic resolution. Within the octatonic system, every note in the tonic
chord is approachable by half step, but two of the notes (G and B ) are already embodied
in the dominant chord, leaving the D to resolve down to C and the F to resolve either up
to G or down to E, as seen in Figure 3-5.
Chapter III – Creating a Tonal Octatonic System 48
Figure 3-5
Although there is a tritone present in the dominant chord, it exists between the root and
lowered fifth chord degrees rather than between the third and lowered seventh chord
degrees. For the resolution of this tritone, rather than having each note expand or contract
by half step—as is common-practice for traditional Western harmony—the lowered fifth
chord degree (D ) will resolve down a half step while the root (G) can either resolve
down a perfect fifth or sustain. Another possible resolution is to anticipate the D
resolution to C, creating another tritone between C and F and allowing the F to resolve
to G, seen in Figure 3-6.
Figure 3-6
Another possibility to consider is including both altered fifths (D and D ) in the
dominant chord, producing a pentachord. This will introduce additional voice leading
possibilities, as the D would resolve up by half step to the E of the tonic chord, as seen in
Figure 3-7, creating smoother resolution than the doubled F resolving to G and E in
Figure 3-6. Adding an additional note to the vertical structure also serves the purpose of
differentiating the sonority from the other vertical structures present in the system.
Chapter III – Creating a Tonal Octatonic System 49
Figure 3-7
An analogous process may be applied to the tonic chord by reducing the dominant 7th
structure to a major triad, providing the only traditional common-practice period
consonance of the system and establishing it as the tonic of the scale. Similar procedures
may be applied to other chords in the system, to create larger, more complex chord
progressions.
50
-- Conclusion --
Through the examination of harmonic restrictions and relationships on previously
existing vertical structures and pitch class sets found in the octatonic scale, this text
demonstrated that it is possible to create a series of harmonic progressions that will
establish a specific note as the tonic within the octatonic scale. Chapter I presented a brief
overview of the octatonic scale as used in both Jazz and Twentieth-century Classical
music. The intent of this overview was to establish a historical foundation of the
octatonic scale to be contrasted with the material presented in Chapters II and III. Chapter
II introduced theoretical applications for establishing a tonal center within the octatonic
scale, with Chapter III applying the concepts of Chapter II to a given octatonic system.
The main function of Chapter III is to provide the reader with an example of how theory
can become music. Breaking up chapters II and III into the six sub-sections—listed
below—aided in the organization and cohesion of the material presented in this text.
• Modal Relationships discussed common subsets of the diatonic collection, non-diatonic minor scales, and the octatonic collection.
• Harmonic Relationships discussed creating the (1) dominant to tonic relationship, (2) predominant chords, (3) chord qualities, and (4) the total number of chords in an octatonic system.
• Vertical spacing discussed the intervallic spacing and ordering of notes within a vertical structure to avoid harmonic ambiguity.
• Intervallic Content discussed post-tonal relationships between vertical structures.
• Voice Leading discussed the need for establishing new rules of harmonic progression.
• Additional Considerations discussed (1) additional subset relations, (2) chord substitutions, and (3) metric considerations.
Conclusion 51
Additional research possibilities include creating additional octatonic systems,
creating a more thorough subset relations chart (Appendix E), applying the concepts of
this text to other symmetrical scales (whole-tone, augmented/hexatonic, etc.), and
utilizing the presented guidelines in composition. Although it is not directly pertinent to
this text, establishing a standard for chord symbols and chord-scale relationships would
aid all Jazz literature, and would have facilitated the creation of the musical examples
used in this text.
Had more time been allotted for a composition to be completed utilizing principles
presented in this text, it would have aided in presenting the material of Chapter III by
drawing upon a single source rather than fragments of a composition that have not yet
been completely worked out. Additional examples of musical literature would have
improved the text as a whole; however, studying the footnoted references will provide a
wealth of musical examples and detailed analyses that will aid in understanding the
material presented, but are beyond the scope of this text. It is important to keep in mind
that this text would not have been possible without combining aspects of common-
practice period tonality, Jazz theory, and Set theory. As the distinctions between genres
of music continue to blur, it allows for new musical styles to emerge from the fusion of
previous styles; however, this fusion does not mean that one system must dominate over
the other. While this text does not explore the full range of possibilities of the octatonic
scale, I hope that it does convey at least one critical concept: octatonicism does not
negate diatonicism.
52
-- Bibliography --
Babbitt, Milton. “The String Quartets of Bartok.” Musical Quarterly 35, no. 3 (July 1949). Baker, David. How To Play Bebop. Bloomington, IN: Alfred Publishing Company, 1987. Coker, Jerry. Patterns for Jazz. Miami, FL: Studio Publications Recordings, 1970. Forte, Allen. The Structure of Atonal Music. New Haven: Yale University Press, 1973. Hall, Jim. Exploring Jazz Guitar. Milwaukee, WI: Hal Leonard Publishing Corp., 1990. Levine, Mark. The Jazz Piano Book. Petaluma, CA: Sher Music Company, 1989. Levine, Mark. The Jazz Theory Book. Petaluma, CA: Sher Music Company, 1995. Liebman, David. A Chromatic Approach to Jazz Harmony and Melody. Rottenburg, Germany: Advance Music, 1991. Messiaen, Oliver. The Technique of My Musical Language. Paris: A. Leduc, 1966. Miller, Ron. Modal Jazz Composition & Harmony Volume 1. Rottenburg, Germany: Advance Music, 1992. Slonimsky, Nicolas. Thesaurus of Scales and Melodic Patterns. NYC, NY: Schirmer Books, 1975. Straus, Joseph. Introduction to Post-Tonal Theory. 3rd ed. NJ: Prentice Hall, 2005. Tymoczko, Dmitri. “Octatonicism Reconsidered Again.” Music Theory Spectrum 25 (2003). Ulehla, Ludmila. Contemporary Harmony: Romanticism through the Twelve-Tone Row. NY: The Free Press, 1966. Van den Toorn, Pieter C. “Colloquy: Stravinsky and the Octatonic, The Sounds of Stravinsky.” Music Theory Spectrum 25 (2003). Van den Toorn, Pieter C. Music of Igor Stravinsky. New Haven: Yale University Press, 1983. Van Den Toorn, Pieter C. Stravinsky and The Rite of Spring. Berkley: University of California Press, 1987.
53
-- Appendix A -- Chord Lexicography
All chord labels adhere to the following rules:
• Roman numerals do not indicate major or minor triadic quality. Roman numerals only
indicate the root’s relation to the tonic chord of the parent major scale.
• A major triad is indicated by “Maj” or just by the chord’s root. (i.e., CMaj; C)
• A minor triad is indicated by “min”. (i.e., Cmin)
• A dominant 7th chord is notated by “7”. (i.e., C7)
• If both the third and seventh chord degrees are minor, it is notated “min7” or abbreviated as “m7”. (i.e., Cmin7; Cm7)
• If both the third and seventh chord degrees are major, it is notated “Maj7” or abbreviated as “M7”. (i.e., CMaj7; CM7)
• If the triad of a chord is minor and a major seventh interval is present, it is notated “minMaj7” or abbreviated “mM7”. (i.e., CminMaj7; CmM7)
• If a triad has the sixth chord degree present, while omitting the seventh of the chord, the label “6” replaces the position of the “7”. (i.e., C6 is a C major triad with an A above the root; Cmin6 is a C min triad with an A above the root; Cmin 6 is a C minor triad with an A above the root.)
• If higher scale degrees of a chord are present, they replace the position of the “7” in the chord label. (i.e., C13 indicates a C7 chord with an additional 13th; CMaj13 indicates a CMaj7 with an additional 13th)
• Additional alterations to a chord are listed after the first chord tension. (i.e., C7 9)
• Alterations to dominant 7th chords are listed as compound intervals, altering the 9th, 11th, and 13th chord degrees. Alterations to the 5th indicate a change in triad quality.
• The term “Alt” indicates that the 5th and 9th chord degrees are raised and lowered.
• The term “sus” indicates a tertiary structure that substitutes the third of the chord for the chord degree following the “sus”. (i.e., Csus2; C7sus4)
• The term “add” indicates additional tones are added to the preceding chord, without requiring any further alteration to the chord. (i.e., Cadd9 indicates the notes C, E, G, and D)
54
-- Appendix B -- Chord-Scale Relationships
Modes of the Major Scale:
Mode: Altered Scale Degrees: Tonic 7th Chord: Ionian: N/A Maj7
Dorian: 3, 7 min7
Phrygian: 2, 3, 6, 7 min7 9
Lydian: 4 Maj7 4
Mixolydian: 7 7
Aeolian: 3, 6, 7 min7 6
Locrian: 2, 3, 5, 6, 7 min7 5
Appendix B: Chord-Scale Relationships 55
Modes of the Harmonic Major Scale:
Mode: Altered Scale Degrees: Tonic 7th Chord:
Ionian 6: 6 Maj7, Maj7 6
Dorian 5: 3, 5, 7 min7 5
Phrygian 4: 2, 3 4, 6, 7 7Alt
Lydian 3: 3, 4 minMaj7
Mixolydian 2: 2, 7 7 9
Lydian 5 2: 2, 4, 5 Maj7 5, Aug
Locrian 7: 2, 3, 5, 6, 7 dim7
Appendix B: Chord-Scale Relationships 56
Modes of the Harmonic Minor Scale:
Mode: Altered Scale Degrees: Tonic 7th Chord: Aeloian 7: 3, 6 m7, mM7
Locrian 6: 2, 3, 5, 7 min7 5
Ionian 5: 5 Maj7 5, Aug
Dorian 4: 3, 4, 7 min7 4
Mixolydian 2 6: 2, 6, 7 7 9, 7 13
Lydian 2: 2, 4 Maj7 9
Altered 7: 2, 3, 4, 5, 6, 7 7Alt
Appendix B: Chord-Scale Relationships 57
Modes of the Ascending Melodic Minor Scale:
Mode: Altered Scale Degrees: Tonic 7th Chord:
Dorian 7: 3 m7, mM7
Phrygian 6: 2, 3, 7 sus 9
Lydian-augmented: 4, 5 Maj7 5
Lydian-dominant: 4, 7 7 11
Mixolydian 6: 6, 7 mM7 in 2nd inversion
Aeolian 5: 3, 5, 6, 7 min7 5
Altered: 2, 3, 4, 5, 6, 7 7Alt
Appendix B: Chord-Scale Relationships 58
Modes of the Ascending Melodic Minor 5 Scale:
Mode: Altered Scale Degrees: Tonic 7th Chord:
Melodic Minor 5: 3, 5 minMaj7
Phrygian 6 4: 2, 3, 4, 7 min7 5, dim7
Lydian 3 5: 3, 4, 5 Maj7, aug
Mixolydian 2 4: 2, 4, 7 7 11
Altered 6 7: 2, 3, 4, 5, 6, 7 dim7, 7Alt
Aeolian 5 7: 3, 5, 6 minMaj7
Altered 6: 2, 3, 4, 5, 7 7Alt
Appendix B: Chord-Scale Relationships 59
Diatonic Church Modes Priority Table:62 1 2 3 4 5 6 Lydian 4 7 3 6 2 5 Ionian (1) 7 4 3 6 2 5 Ionian (2) 7 3 2 6 5 (no 4) Mixolydian (1) 7 4 3 6 2 5 Mixolydian (2) 7 3 2 6 5 (no 4) Dorian 6 3 7 2 5 4 Aeolian 6 2 5 3 7 4 Phrygian 2 5 4 7 3 6 Locrian 5 2 7 6 3 4
Harmonic Minor Modes Priority Table: 63 1 2 3 4 5 6 Aeloian 7 6 7 2 3 5 4 Locrian 6 5 6 2 7 3 4 Ionian 5 4 5 7 3 2 6 Dorian 4 6 4 3 2 7 5 Mixolydian 2 6 2 3 5 7 6 4 Lydian 2 4 2 7 3 6 2 Altered 7 4 7 2 5 6 3
Melodic Minor Modes Priority Table: 64 1 2 3 4 5 6 Lydian-augmented 5 7 3 4 6 2 Lydian-dominant 4 7 3 6 2 5 Mixolydian 6 6 7 3 2 5 4 Dorian 7 7 3 6 2 5 4 Aeolian 5 5 3 7 6 2 4 Phrygian 6 6 2 4 7 3 5 Altered 4 7 6 3 5 2
[Note: At least two of the tones must be used to get sufficient modal definition in the non-diatonic minor modes.]
62 Miller, Modal Jazz Composition, 20. 63 Ibid., 90. 64 Ibid., 33.
60
-- Appendix C -- Modal Tetrachords Found in the Octatonic Scale
The following tetrachords are used in the construction the non-diatonic minor
scales and are also present in the octatonic scale. The tetrachords are taken from page 130 of Ron Miller’s book, Modal Jazz Composition & Harmony Volume 1. Each tetrachord is labeled with its name and intervallic construction.
Spanish (121): Dorian (212):
Hungarian Minor (213): Hungarian Major (312):
Hungarian Pentatonic (231): Hungarian Phrygian (123):
Hungarian Spanish (132): Blues (321):
61
-- Appendix D -- Ron Miller’s Collated Order of All Constructed Modes
Using the major scale as a reference, by considering the raising of a scale degree as a brightening and the lowering of a scale degree as a darkening, the resulting order of brightest to darkest is:
1. Lydian 5 3 2. Lydian 5 3. Lydian 2 4. Lydian 5. Lydian 3 6. Ionian 5 7. Ionian 8. Ionian 6 9. Mixolydian 2 4 10. Mixolydian 4 11. Mixolydian 6 12. Mixolydian 13. Mixolydian 2 14. Dorian 7 5 15. Dorian 7 16. Dorian 7 5 17. Dorian 4 18. Dorian 19. Aeolian 7 20. Aeolian 7 5 21. Aeolian 22. Aeolian 5 23. Phrygian 7 5 24. Phrygian 6 4 25. Phrygian 6 26. Phrygian 3 27. Phrygian 28. Locrian 6 29. Locrian 6 30. Locrian 7 31. Locrian 4 32. Locrian 33. Altered 6 34. Altered 7 35. Altered 6 7
62
-- Appendix E -- Additional Subset Relations
Trichords: Cdim Cmin Cmin6 Cmin7 C7Alt9 C7 9 C7+4 CMaj
C C C C C C C C E E E E E E E E G G A B D E G G
Trichords Continued:
C6 C7 C7 9 C7 9 C dim C min7 6 C dim7 C minMaj7 C C C C C C C C E E E E E E E E A B D D G A B C Trichords Continued: C min9 C min11 C C E E D F
Tetrachords: Cmin7 5 Cmin7 C7 5 C7 C7Alt9 5 C7Alt9 C7 9 5 C7 9
C C C C C C C C E E E E E E E E G G G G G G G G B B B B D D D D
Tetrachords Continued:
C7 9 5 C7 9 C7 9 4 C7 4 Cdim7 Cmin6 Amin6 C6 C C C C C C C C E E E E E E E E
G G G G G G G G D D F F A A A A
Tetrachords Continued: C dim7 C dim7 5 C mM7 5 C mM7 5 C min9 5 C m9 C m11 5 C m11 5
C C C C C C C C E E E E E E E E G A G A G A G A B B C C D D F F
63
-- Appendix F -- Additional Octatonic Systems
Note the following distinctions of the given octatonic systems:
• System 3 uses implied harmony on the D9 and F9 • System 4 uses quartal harmony • System 5 uses mixed voicings
1
2
3
4
5