One Collection, Four Gravities: C–D–E–F–G–G♯–A–B as a Chromatic Harmonic Field

Abstract. This article examines the eight pitch classes C–D–E–F–G–G♯–A–B, with the octave C understood as a registral return rather than a ninth pitch class. Its central claim is deliberately conditional: the collection is most productive not as a scale awaiting a single proper name, but as a harmonic field whose identity is specified by centricity, spelling, harmonic rhythm, bass, registral emphasis, and voice leading. It contains C major plus G♯; the union of A natural and harmonic minor; the union of E Phrygian and E Phrygian dominant; and, after respelling G♯ as A♭, the union of F Lydian and George Russell’s F Lydian Diminished. These equivalences are structurally exact, but they are not analytically interchangeable until musical behavior establishes a center and a syntax.

Thesis. The strongest synthesis is therefore neither “an exotic C scale” nor a claim that classical tonality, jazz chord-scale practice, Russell’s theory, and flamenco are secretly the same system. It is an eight-note chromatic harmonic field: a fixed pitch inventory that supports several genuinely related seven-note strata and a network of exceptionally economical chord transformations. The relationships are real at the pitch-class and voice-leading levels; their functions remain context-dependent.

1. 1. Terms and method
2. 2. Pitch-class and interval analysis
3. 3. Multiple tonal centers
4. 4. Classical tonal interpretation
5. 5. Jazz harmonic interpretation
6. 6. George Russell and Lydian tonal gravity
7. 7. Flamenco and Andalusian-Phrygian hearing
8. 8. Context-dependent pitch identity
9. 9. Transformational and voice-leading analysis
10. 10. Comparative theoretical matrix
11. 11. Composition and improvisation pathways
12. 12. Conclusion
13. Bibliography

1. Terms and method

A pitch collection is an inventory: here, eight pitch classes without an intrinsic first note. A scale is an ordered presentation, normally octave-bounded and usually spelled to display intervallic or functional structure. A mode is more than a rotation of a scale: it entails a center and characteristic melodic or harmonic behaviors. A tonal center is the pitch or sonority heard as referential or stable; it may be established by bass, duration, metric placement, cadence, repetition, register, or formal return. A chord-scale is a local mapping between a chord and a larger pitch resource; it need not govern an entire phrase. A functional harmonic system adds directed syntax—predominant, dominant, tonic, applied functions, cadential norms, and hierarchical prolongation. Confusing these categories is the principal source of false certainty in naming this collection.

Enharmonic equivalence belongs to equal-tempered pitch class; enharmonic meaning belongs to notation, voice leading, and hearing. G♯ and A♭ occupy the same piano key, but G♯ normally rises to A as an altered fifth of C, leading tone of A, or third of E, whereas A♭ can descend to G, rise to A as a split third around F, or serve as the lowered sixth of C harmonic major. The analysis will therefore respell pitch class 8 whenever function changes. This is not cosmetic renaming: spelling records the directed interval and the scale-degree claim.

2. Pitch-class and interval analysis

With C = 0, the set is S = {0,2,4,5,7,8,9,11}. In the supplied C ordering its cyclic step pattern is 2–2–1–2–1–1–2–1 semitones. Transposition by five semitones maps it to the Forte prime form (0,1,2,4,5,7,9,10), hence set class 8-26, with interval-class vector ⟨4,5,6,5,6,2⟩. The vector’s six ic3 and six ic5 dyads help explain the wealth of tertian and quartal resources; four semitones and two tritones preserve strong chromatic pressure without making the set an octatonic collection. The classification and vector agree with the standard Forte-set table maintained by the Humdrum/CCARH pitch-class reference (Forte 1973; Straus 2016).

The inversional symmetry I₄ is musically suggestive. Its fixed pitch classes are D and G♯/A♭; C mirrors E, F mirrors B, and G mirrors A. The chromatic trichord G–G♯–A (set class 3-1) is itself centered registrally and inversionally on G♯. In F-spelling, G–A♭–A makes A♭ the central pitch between scale degrees 2 and 3, but the same sounding trichord does not acquire a single harmonic function from this geometry alone.

Subsets and embedded harmonic vocabularies

Two seven-note subsets are structurally privileged because they differ by a single pitch and have established theoretical identities. Removing G♯/A♭ yields C major = F Lydian = A natural minor = E Phrygian. Removing A yields C harmonic major = F Lydian Diminished = the fourth mode of C harmonic major. From A- and E-centered perspectives the analogous pairing is controlled by G/G♯: remove G♯ for A natural minor/E Phrygian; remove G for A harmonic minor/E Phrygian dominant. Thus the same eight-note set is the union of two different pairs of seven-note collections, depending on which enharmonic and centric partition matters.

The complement is unusually legible: {C♯,D♯,F♯,A♯}, reordered D♯–F♯–A♯–C♯, is D♯m7 (enharmonically E♭m7), Forte class 4-26. This complementary relation is exact and nontrivial, but it does not establish D♯ minor as a hidden key. Complementation is a set-theoretical relation; a tonic requires musical evidence.

Upper structures, polychords, and chromatic mediants

An E-major triad over C produces C–E–G♯–B, the essential Cmaj7♯5 sonority; retaining G natural as well creates a split-fifth C-major field. C major and E major are chromatic mediants sharing E; E major and G major are chromatic mediants sharing B; F minor and A minor are same-quality third-related triads sharing C. The augmented triad C–E–G♯ is their hinge. Other useful superpositions include G major over C (Cmaj9 without the third duplicated), D minor over G (a G9sus-type upper structure if C is added), and G♯°7 over E (rootless E7♭9). “Polychord” is appropriate when registral or timbral separation makes both components perceptible; otherwise a single extended-chord label is usually more economical.

3. Multiple tonal centers

C hearing is favored by a C bass, cadences G7–C, the tonic third E, the leading tone B, and repeated C-major closure; G♯ then reads most strongly as ♯5 or as a chromatic leading tone to A. F hearing is favored by an F pedal, B as ♯4/#11, E as major seventh, and F-major or F-minor-major tonic sonorities; pitch class 8 should then be written A♭ when it acts as ♭3. A hearing is favored by E7–Am, G♯–A melodic closure, and tonic A/C/E; G natural becomes ♭7 while G♯ is the leading tone. E hearing is favored by F–E, F/E–E, or recurrent E–F–E gestures, by a bass or final on E, and by B as the upper fifth; G and G♯ alternate as modal thirds rather than as the seventh degree of A.

These are not four “modes” in the strict sense merely because the same pitches can be reordered. Each reading requires different cadences and different hierarchies. A melody that simply traverses all eight notes evenly supplies a scale but may leave centricity undecided. Conversely, a three-note cadence F–E with G♯ in the tonic chord may establish E-centered flamenco-Phrygian behavior without presenting the complete collection as a scale at all.

4. Classical tonal interpretation

C major plus G♯

Within common-practice C major, G♯ is not diatonic and need not define a new scale. Its most immediate linear function is an ascending chromatic inflection of scale degree 5: G–G♯–A. Harmonically it is the third of E or E7, hence part of V/vi; it is the root of G♯° or G♯°7, hence vii°/vi or vii°7/vi; and it is the augmented fifth of C+, the harmonic form of the chromatic passing stage between C and Am. A descending line A–A♭–G would normally reverse the spelling and suggest mixture, C harmonic major, or a lowered sixth rather than a raised fifth.

G♯°7 = G♯–B–D–F resolves conventionally to Am: G♯→A, B→C, and F→E, while D may move to C or E. The same sounding four pitch classes can be respelled F–A♭–C♭–E𝄫 as a common-tone diminished seventh decorating F: F is retained, A♭→A, C♭→C, and E𝄫 may move to E or C. Enharmonic spelling here distinguishes applied leading-tone function from nineteenth-century common-tone embellishment. The collection does not force one resolution; the following chord does.

A minor: natural and harmonic forms in one inventory

Recentered on A, the collection is the exact union of A natural minor (A–B–C–D–E–F–G) and A harmonic minor (A–B–C–D–E–F–G♯). This is historically more orthodox than treating the union as an autonomous octatonic scale: tonal minor routinely varies scale degrees 6 and 7 according to line and function. Here only 7 varies. G supports v (Em), VII (G), and modal descent; G♯ supports V (E), V7 (E7), vii°7 (G♯°7), and the leading-tone ascent to A. C–E–G♯ is III⁺, the augmented triad generated by harmonic minor.

Thus Am–Dm–E7–Am is i–iv–V7–i, while Am–G–F–E7–Am is i–VII–VI–V7–i under an A-minor functional hearing. Mixture is not adequately summarized as choosing one fixed minor “scale”: melodic direction, applied function, and contrapuntal needs govern the mutable degrees. Nineteenth-century chromatic practice intensifies this mobility through augmented triads, common-tone diminished sevenths, and enharmonic reinterpretation, all of which this field supports locally. Yet adding the inventory’s pitches indiscriminately above every chord would erase the voice-leading distinctions on which that practice depends.

5. Jazz harmonic interpretation

Jazz chord-scale practice asks which subset articulates a current sonority and its available tensions. It does not require one global scale to fit every chord. The field is especially useful because it contains a full C-major diatonic cycle, an applied E dominant with multiple alterations, a complete diminished seventh, and two tonic-color axes: G/G♯/A and G/A♭/A. Modern pedagogy may describe the same event through modal interchange, secondary dominant, upper structure, chromatic approach, or tonic prolongation; the best label is the one that predicts voicing and resolution.

Containment does not imply that every standard chord-scale is contained. Cmaj7 may take the C-major seven-note subset; treating G♯ as a sustained tension changes the chord to Cmaj7♯5. The field lacks F♯, so it is not C Lydian or C Lydian augmented. G7 can draw on complete G Mixolydian—G–A–B–C–D–E–F—with A♭ added as ♭9; thus both ♭9 and natural 9 coexist. E7 is still richer: relative to E the field offers ♭9 (F), ♯9 (G, enharmonically F𝄪), 3 (G♯), 11 (A), 5 (B), ♭13 (C), and ♭7 (D). It is not the standard E altered scale, because it retains B natural and A while lacking B♭. The most accurate description is a composite E-dominant/Phrygian field, from which a player selects tensions according to line and voicing.

For diminished harmony, G♯°7 is exact, but neither complete octatonic diminished scale is identical to the total collection. Calling the entire field “diminished” would therefore confuse a contained chord with a governing scale. Likewise, G♯°7 may be a passing diminished sonority, a rootless E7♭9, or an applied leading-tone chord; those are functional readings of the same four pitch classes.

Two progressions in detail

Cmaj7 → Cmaj7♯5 → Am7. Hold C and E throughout. Move G→G♯ in the first change. At the arrival, let G♯→A and B→A, or—more clearly in four voices—voice C–E–G♯–B as C–E–G–A, sending G♯→G and B→A while A enters as the new chordal seventh/root-class. The former emphasizes the ascending 5–♯5–6 line; the latter emphasizes contrary contraction. Harmonic hearing can move from tonic, through tonic major-♯5 color, to vi7 without requiring a modulation.

Cmaj7 → E7 → Am7 → Dm7 → G7 → Cmaj7. From Cmaj7 to E7, retain E and B, raise G to G♯, and move C to D. E7 to Am7 has classical semitone attractions G♯→A and B→C, with E retained and D moving to C or E. Am7 to Dm7 retains A and C while E→F; Dm7 to G7 retains D and F while C→B; G7 to Cmaj7 retains G and B while F→E. This chain unites jazz seventh-chord vocabulary with a conventional I–V/vi–vi–ii–V–I syntax. Reharmonization occurs not by abandoning function but by inserting the applied dominant already latent in the inventory.

Useful upper structures include E major over C (Cmaj7♯5), G♯°7 over E (E7♭9), G major over C (Cmaj9 without obligatory third duplication), and D minor over G (a G9sus family when C is present). The distinction between “upper structure” and “polychord” is perceptual: close voicing favors a single extended chord; separated registers, timbres, or attacks can preserve two chordal identities.

6. George Russell and Lydian tonal gravity

Russell’s Lydian Chromatic Concept of Tonal Organization cannot be reduced to the slogan that Lydian is “brighter” than Ionian. His premise is an ordered theory of tonal gravity. A Lydian tonic is established as the lowest member of a ladder of six consecutive perfect fifths. On F, the fifth series F–C–G–D–A–E–B, compressed into an octave, yields F–G–A–B–C–D–E. Russell calls this seven-tone order the ingoing level and treats it as a unified tonal-gravity field without the “goal pressure” characteristic of horizontally resolving major-scale tonality. The official George Russell site’s Concept overview and sample chart documents the Lydian tonic, the ladder-of-fifths derivation, five tonal orders, and seven principal chord-producing scales (Russell 2001).

The twelve-tone Lydian Chromatic order for F is, in scale-degree terms, I–V–II–VI–III–VII–♯IV–♯V–♭III–♭VII–IV–♭II. Russell deliberately skips a fifth after the seventh member before continuing the ordering. The first seven tones form F Lydian. Subsequent characteristic tones support progressively less ingoing principal scales. This hierarchy—not a generic major-mode affect—is the theoretical core relevant here.

Michael McClimon’s historical reconstruction distinguishes Russell’s two major legacies: Lydian generation and chord-scale equivalence. His A Transformational Approach to Jazz Harmony, chapter 4, identifies Russell’s Lydian Diminished as the fourth mode of harmonic major and separates Russell’s principal scales from later Berklee-style chord-scale pedagogy (McClimon 2016, 99–145). The formula under discussion is therefore unambiguous: F–G–A♭–B–C–D–E is genuinely F Lydian Diminished in Russell’s terminology.

Outside the Concept, “Lydian ♭3” is the clearest formula name because it displays the two defining alterations relative to a major tonic: minor third and augmented fourth. “Fourth mode of C harmonic major” identifies a parent collection familiar to later scale taxonomies. “Melodic minor ♯4” is intervallic shorthand, not a parent-scale derivation. “Lydian diminished” is defensible when Russell or the harmonic-major modal tradition is explicit, but potentially confusing because jazz musicians also use “diminished scale” for the octatonic whole–half or half–whole collections. Some informal sources even reverse the words and assign “diminished Lydian” a lowered fifth; that is neither Russell’s formula nor a stable scholarly convention.

The eight-note union in Russell’s terms

F–G–A♭–A–B–C–D–E is the union of F Lydian and F Lydian Diminished. It co-presents the principal Lydian third A and the Lydian-Diminished third A♭. Yet it is not one of Russell’s seven named Principal Scales. Nor is it simply the cumulative first eight or nine tones of the F Lydian Chromatic order: the nine-tone order would include the eighth tone C♯ before the ninth-tone A♭, whereas the present collection contains A natural and A♭ but no C♯. Calling it a named “eight-note Lydian Chromatic scale” would therefore misstate Russell.

Five possible descriptions can now be ranked. (1) One synthetic eight-note scale is compositionally legitimate if a work orders and stabilizes it, but historically nonstandard. (2) Two overlapping seven-note scales is exact and closest to Russell’s documented scale inventory. (3) A chromatic Lydian field is useful only as clearly marked analytical language, not as Russell’s label. (4) A superposition of harmonic resources best describes passages that alternate F major and F minor-major tonic sonorities. (5) Most rigorously: it is an eight-note subset of the F Lydian Chromatic universe, organized in practice as the superposition of the F Lydian and F Lydian Diminished principal scales. The italicized synthesis is ours; the two seven-note names and their hierarchy are Russell’s.

The first three Russell scales around the fourth/fifth cycle

The following list answers a narrower compositional question: as Lydian, Lydian Augmented, and Lydian Diminished are transposed around the closed fourth-cycle F→C→G→D→A→E→B→F♯→C♯→A♭→E♭→B♭→F (read backward for the fifth-cycle), which scales hit pitch class G♯/A♭ without G, G without pitch class G♯/A♭, or both adjacent pitch classes? “Hit” means pitch-class membership, not tonal center. The roots remain in cycle order inside every cell.

The three requested profiles can be heard immediately from representative scales. C Lydian Augmented (C–D–E–F♯–G♯–A–B) hits G♯ and omits G. F Lydian (F–G–A–B–C–D–E) hits G and omits G♯/A♭. F Lydian Diminished (F–G–A♭–B–C–D–E) contains both literal G and A♭. The F-centered comparison is especially revealing: F Lydian and F Lydian Augmented both occupy the “G only” branch, whereas F Lydian Diminished occupies the adjacent-pitch “G + A♭” branch.

JolyMusic keyscales: seven notes per card

Each interactive JolyMusic keyscale below contains exactly seven distinct scale notes. The first three cards keep F as tonic and compare Russell’s first three Principal Scales directly; the fourth supplies the requested cycle representative that contains G♯ but not G. The eight-note field is created by comparing or superposing cards, never by placing eight notes inside a seven-note keyscale.

Enharmonic discipline still matters. The “both” column is a pitch-class test: C♯ Lydian, for example, spells the two relevant pitch classes F𝄪 and G♯, not G and A♭. A♭ Lydian, by contrast, literally contains A♭ and G. The table therefore maps cycle membership without pretending that every transposition gives the same notational or functional identity to pitch class 8.

7. Flamenco and Andalusian-Phrygian hearing

On E the field is E–F–G–G♯–A–B–C–D: the union of E Phrygian and E Phrygian dominant (the fifth mode of A harmonic minor). The crucial resource is not an abstract “Spanish scale” but the coexistence of G and G♯ above an E center. G supports Phrygian melody, G-major harmony, and modal descent; G♯ stabilizes an E-major tonic sonority and forms the augmented-second tetrachord E–F–G♯–A often compared with Hijaz. The pitch inventory models this duality unusually well.

Peter Manuel’s foundational study, “Evolution and Structure in Flamenco Harmony”, describes flamenco harmony as a syncretic modal/tonal practice inseparable from vocal melody, characteristic guitar voicings, and instrumental affordances. He notes the alternation of lowered and raised third degrees in Andalusian cantes, the fundamental F–E rather than B7–E cadence, and the special stability of Am within the Am–G–F–E schema (Manuel 1986, 46–57). John Moore’s practitioner-oriented “The Flamenco Key” likewise treats E–F–E and Am–G–F–E as core por-arriba patterns while warning that open-string voicings materially shape the idiom (Moore 2015).

“Phrygian dominant” is common and useful in jazz, rock, and guitar pedagogy for the seven-note formula 1–♭2–3–4–5–♭6–♭7. It is not a sufficient theory of flamenco. Many palos use major or minor tonality; Phrygian practices differ by palo, compás, melodic type, guitar position, and cadential convention; vocal inflection may not conform to twelve-tone equal temperament; and nontriadic open-string voicings are structural rather than decorative. Manuel’s more recent Flamenco Music: History, Forms, Culture places harmony within that wider historical and performative ecology (Manuel 2023).

The field therefore supports a responsible flamenco analogy, not a totalizing claim. E–F–G/G♯–A–B–C–D supplies the relevant scalar resources; a sounding sequence becomes flamenco-like only when rhythm, articulation, compás, melodic contour, voicing, and formal practice also participate. A jazz solo over E7♭9 using the same pitch classes is not thereby flamenco.

8. Context-dependent pitch identity

The line G→G♯→A has four different grammars. In C it is 5–♯5–6, often prolonging tonic before vi. In A it is ♭7–7–1, dramatizing the historical variability of the minor-mode seventh. In E it is ♭3–3–4, a modal-third mutation that can turn melodic Phrygian color into an E-major tonic. In F, respelled G→A♭→A, it is 2–♭3–3: not a scalar “leading tone” sequence in the common-practice sense, but a progression from Lydian-Diminished color to principal Lydian third.

This supports a broader theory of context-dependent pitch identity. A pitch class supplies acoustical location; a scale degree is a relation to a perceived center; a tendency tone adds directed motion; a chord member adds vertical function; and a historical style constrains which relations count as normative. Identity is therefore layered rather than arbitrary. The labels are not coincidental because they correspond to exact subset relations and parsimonious transformations, but neither can they be inferred from the unordered set alone.

9. Transformational and voice-leading analysis

The augmented triad deserves special emphasis. C+ is invariant under major-third transposition, so C+, E+, and G♯+ name the same pitch-class set. It lies one semitone from several triads in the field: lower G♯ to G for C major; lower C to B for E major; raise G♯ to A for A minor; raise E to F for F minor. Extended neo-Riemannian theory often treats augmented triads as pivots within hexatonic systems, but the present analysis need not claim that the whole eight-note collection is hexatonic. The justified statement is local: the augmented triad enables single-voice transformations among four salient triads.

Similarly, F major→E major is maximally smooth in total voice-leading distance—three semitones, one in each voice—but not parsimonious in the stricter “one voice moves, others remain” sense. It is best modeled as uniform transposition T₁₁ or as a Phrygian upper-neighbor resolution. G♯°7’s minor-third symmetry supports enharmonic reinterpretation, but the resolution pattern selects a root: as G♯–B–D–F it tends to Am; as F–A♭–C♭–E𝄫 it can embellish F. Transformational labels illuminate distances; they do not replace tonal or modal syntax (Cohn 2012; Rings 2011; McClimon 2017).

10. Comparative theoretical matrix

11. Composition and improvisation pathways

A composer can change the perceived system without changing the pitch inventory by reallocating duration, bass, cadence, and accent. The following pathways use only the eight pitch classes. Chord symbols describe practical voicings; Roman numerals are supplied only where their syntax is meaningful.

Interactive harmonic-center journey

The network below makes the pitch inventory primary. It queries the JolyMusic KeyScale catalog and admits only seven-note KeyScales whose pitch classes are wholly contained in C–D–E–F–G–G♯–A–B. It does not manufacture twelve transpositions for a preselected list of modes. A solid edge changes tonal gravity while preserving all seven tones; a dashed edge performs a one-tone substitution between two seven-of-eight subsets. Because every distinct seven-of-eight subset differs from every other by one tone, displaying the complete relation graph would produce a near-clique. The visualization therefore retains a minimum-cost connected gravity backbone plus each node’s nearest recentering and morph links, then chooses the weighted graph diameter as its initial farthest-to-farthest journey. At every node the inspector uses the catalog’s ordered and spelled scale tones to derive seven tertian triads and seven four-note chords. In chord views each enharmonically normalized root-and-quality chord appears once, with every KeyScale membership retained as an incident edge; clicking a tone or chord auditions it through the current JolyEngine route.

Three shorter compositional studies

Study A: the chromatic axis. C | C+ | Am | E7♭9 | Am. Sustain C–E in the upper voices through the first three chords and make G–G♯–A the soprano. In E7♭9, reinterpret G♯ as dominant third and F as ♭9; resolve G♯→A and F→E. Improvise C-major pentachords over C, C–D–E–G♯–B over Cmaj7♯5, and A harmonic minor over E7♭9–Am.

Study B: the split-third field. Fmaj13♯11 | Fm(maj13♯11) | E/F? | E. The first two sonorities differ only A/A♭. Avoid the ambiguous symbol E/F if it obscures the intended voicing; a more literal score can sustain F–A–C above E and resolve every upper voice downward to E–G♯–B. This moves from Russellian vertical F organization to the flamenco F→E cadence. Improvise F Lydian, then F Lydian Diminished, then E Phrygian with G♯ reserved for the E arrival.

Study C: dual reading of the Andalusian chain. Am | G | F | E | G♯°7 | Am | Dm7 | G7 | Cmaj7. If Am receives initial duration but E receives phrase-final stress, bars 1–4 are iv–III–II–I in E. The immediate G♯°7–Am can then rotate the syntax into A minor, and Dm7–G7–Cmaj7 completes the return to C. The same material demonstrates that reanalysis is temporal: the listener updates centricity when new cadences arrive.

12. Conclusion

The proposition survives critical testing with one important qualification. C–D–E–F–G–G♯–A–B is most rigorously understood as an eight-note, context-sensitive harmonic field rather than as a scale with one definitive historical name. Its set-theoretical identity is precise: 8-26, inversionally symmetric under I₄, complementary to a minor-seventh set, and rich in triadic, diminished-seventh, quartal, and extended-chord subsets. Its tonal decompositions are also exact: C major plus G♯; A natural plus harmonic minor; E Phrygian plus Phrygian dominant; and—after A♭ spelling—F Lydian plus F Lydian Diminished.

The F hypothesis is not a coincidence. Russell’s F Lydian Diminished genuinely is F–G–A♭–B–C–D–E, and its superposition with F Lydian genuinely yields the requested eight pitch classes. But Russell did not thereby authorize a named eight-note “Lydian Diminished” scale containing both thirds. The rigorous formulation is a superposition of two Russell principal-scale resources inside the F Lydian Chromatic universe, or—when speaking outside Russell’s system—an eight-note synthetic field with split third.

The four frameworks disclose different truths. Classical theory explains G♯ as directed alteration and applied function. Jazz theory operationalizes the field as chord-scale subsets, upper structures, and altered dominants. Russell explains why F Lydian and F Lydian Diminished belong to an ordered tonal-gravity system. Flamenco scholarship explains why E can be the true final of F–E and Am–G–F–E while G/G♯ remains variable. These are genuine structural relationships because they share exact subsets and economical voice leading; they become merely coincidental relabelings only when an analyst asserts a center without evidence from bass, cadence, duration, register, or style.

Final synthesis: call the total inventory the C–F–A–E chromatic harmonic field only as an explicit analytical or compositional term, not an established scale name. Describe its active seven-note stratum and center at each moment. Preserve G♯ when it rises, leads, or forms E/C-augmented harmony; write A♭ when it lowers the F-centered third or participates in C harmonic-major hearing. The collection’s unity lies not in a single mode but in a network of centricities linked by the semitone hinge G–G♯/A♭–A.

Bibliography

• Aldwell, Edward, Carl Schachter, and Allen Cadwallader. Harmony and Voice Leading. 4th ed. Boston: Schirmer/Cengage, 2011.
• Cohn, Richard. Audacious Euphony: Chromatic Harmony and the Triad’s Second Nature. New York: Oxford University Press, 2012.
• Forte, Allen. The Structure of Atonal Music. New Haven: Yale University Press, 1973. Set-class data cross-checked against the Humdrum pitch-class table.
• Levine, Mark. The Jazz Theory Book. Petaluma, CA: Sher Music, 1995.
• Manuel, Peter. “Evolution and Structure in Flamenco Harmony.” Current Musicology 42 (1986; published 1988): 46–57. doi:10.7916/D88051HJ.
• Manuel, Peter. Flamenco Music: History, Forms, Culture. Urbana: University of Illinois Press, 2023. doi:10.5622/illinois/9780252045332.001.0001.
• McClimon, Michael. “Reconceptualizing the Lydian Chromatic Concept: George Russell as Historical Theorist.” Paper presented at the Society for Music Theory annual meeting, St. Louis, 2015.
• McClimon, Michael. A Transformational Approach to Jazz Harmony. PhD diss., Indiana University, 2016, especially chapter 4.
• McClimon, Michael. “Transformations in Tonal Jazz: ii–V Space.” Music Theory Online 23, no. 1 (2017).
• Moore, John. “The Flamenco Key: Phrygian Mode, Andalucian Cadence, and Cambios.” University of California, San Diego teaching essay, 2015.
• Mulholland, Joe, and Tom Hojnacki. The Berklee Book of Jazz Harmony. Boston: Berklee Press, 2013.
• Rings, Steven. Tonality and Transformation. New York: Oxford University Press, 2011.
• Russell, George. George Russell’s Lydian Chromatic Concept of Tonal Organization: The Art and Science of Tonal Gravity. 4th ed., vol. 1. Brookline, MA: Concept Publishing, 2001.
• Straus, Joseph N. Introduction to Post-Tonal Theory. 4th ed. New York: W. W. Norton, 2016.
• Terefenko, Dariusz. Jazz Theory: From Basic to Advanced Study. 2nd ed. New York: Routledge, 2018.

Source-status note. Russell’s scale names, tonal orders, and hierarchy above are attributed to Russell and checked against his official Concept materials and McClimon’s historical reconstruction. “Fourth mode of harmonic major,” contemporary chord-scale equivalences, and the proposed eight-note “harmonic field” are later taxonomic or original analytical formulations. The flamenco section follows Manuel and Moore and does not claim that a pitch collection alone constitutes flamenco practice.