Minor Scale
Minor Scale was a large-scale conventional high-explosive test conducted by the United States Defense Nuclear Agency on June 27, 1985, at the Permanent High-Explosive Test Site on the White Sands Missile Range in New Mexico, involving the detonation of 4,744 tons of ANFO (ammonium nitrate-fuel oil) explosive, equivalent to approximately 4 kilotons of TNT.[1][2] The test simulated the blast and thermal effects of a low-yield nuclear weapon to assess the vulnerability and hardening of military equipment, structures, and instrumentation against such threats without the complications of radiation or fallout.[3] Sponsored as part of nuclear effects research programs, Minor Scale featured over 100 test items, including armored vehicles, bunkers like the Keyworker blast shelter exposed to peak overpressures around 75 psi, and various sensors to measure shock waves, ground motion, and fireballs reaching heights of thousands of feet.[4][1] It remains notable as one of the largest planned non-nuclear detonations in history, providing empirical data that validated computational models for nuclear blast simulations and informed defense strategies during the Cold War era.[5]Fundamentals
Definition and Characteristics
In Western music theory, the minor scale is a seven-note diatonic scale defined by a minor third (three semitones) between its tonic and third scale degree, contrasting with the major scale's major third (four semitones).[6] This structural feature establishes the minor tonality, which forms the basis for minor keys and distinguishes it from major tonality through altered scale degrees.[7] Key signatures for minor scales incorporate three additional flats or three fewer sharps compared to the parallel major scale sharing the same tonic, reflecting the lowered third, sixth, and seventh degrees in the natural form.[7] The foundational natural minor scale employs the interval pattern of whole step–half step–whole step–whole step–half step–whole step–whole step (W–H–W–W–H–W–W) ascending from the tonic.[8] For instance, the A natural minor scale comprises the pitches A–B–C–D–E–F–G–A, with the relative minor of a major key located a minor third below its tonic and sharing the same key signature.[8] This configuration yields a perfect fifth above the tonic, maintaining diatonic consonance while the flattened degrees introduce characteristic tensions resolvable within minor harmony.[6] Acoustically, the minor scale's intervals align with just intonation approximations in traditional tuning systems, where the minor third approximates a 6:5 ratio, contributing to its perceptual stability as a tonal framework despite lacking the leading tone of the major scale in its natural variant.[7] The scale supports modal interchange and serves as the sixth mode (Aeolian) of the major scale, enabling relative key relationships that underpin much of Western tonal composition.[6]Interval Structure and Acoustics
The natural minor scale consists of seven diatonic pitches arranged in the interval pattern of whole step, half step, whole step, whole step, half step, whole step, whole step, ascending from the tonic.[6] This yields semitone intervals of 2, 1, 2, 2, 1, 2, 2 relative to the tonic, distinguishing it from the major scale primarily through the minor third (3 semitones above the tonic) and minor sixth (8 semitones above the tonic).[9] These flattened scale degrees—relative to the major scale—create the characteristic minor tonality, with the minor third serving as the primary intervallic marker of the mode.[10] Acoustically, the minor scale's intervals derive consonance from simple frequency ratios in just intonation systems, where pitches align with low-integer harmonics from the overtone series. The minor third, central to the scale's identity, corresponds to a 6:5 frequency ratio (approximately 315.64 cents), promoting harmonic stability through overlapping partials that minimize dissonance via reduced beating.[11][12] In contrast, equal temperament approximates this at about 300 cents (exactly 6 equal semitones), introducing slight inharmonicity that can alter timbral perception but preserves functional intonation for Western instruments.[13] Other intervals, such as the perfect fifth (3:2 ratio, 702 cents), maintain cross-cultural consonance due to their prevalence in natural harmonics, underpinning the scale's structural integrity across tunings.[11] The perceptual "darker" quality of the minor scale arises from the minor third's ratio, which, compared to the major third's 5:4 (approximately 386 cents), results in fewer coincident overtones and a narrower bandwidth of harmonic reinforcement, influencing emotional response through auditory processing rather than inherent physics. Empirical studies of interval perception confirm that 6:5 yields high consonance ratings, with roughness minimized at ratios below 64:1, supporting its role in stable minor triads.[11][13] Deviations in historical tunings, like Pythagorean (32:27 for minor third, about 294 cents), introduce wolf intervals that challenge modulation but highlight the acoustic trade-offs in fixed-pitch systems.[12]Variants
Natural Minor Scale
The natural minor scale is a diatonic scale comprising seven notes arranged in the ascending interval pattern of whole step–half step–whole step–whole step–half step–whole step–whole step.[6][14] This pattern yields a tonality distinct from the major scale primarily due to the minor third interval from the tonic to the third degree, which produces a flattened third, along with flattened sixth and seventh degrees relative to the parallel major scale.[15][16] It corresponds to the Aeolian mode from the medieval church modes and serves as the foundational form of the minor scale before alterations in harmonic or melodic variants.[17] Historically, the natural minor emerged as the unaltered diatonic collection in minor keys during the common practice period, reflecting modal origins without the raised seventh tone introduced later for stronger resolution to the tonic in harmonic contexts.[18] The scale can be constructed from its parallel major by lowering the third, sixth, and seventh scale degrees by a half step each, or equivalently, by starting on the sixth degree of its relative major scale, which shares the same key signature and notes.[16] For instance, the A natural minor scale (with no sharps or flats) consists of the notes A–B–C–D–E–F–G–A, matching the notes of C major but tonicizing A.[19] In key signatures, this results in the same accidentals as the relative major; for example, F♯ minor natural uses the three sharps of its relative A major (F♯, C♯, G♯).[7]| Scale Degree | Interval from Tonic | Note in A Natural Minor |
|---|---|---|
| 1 (Tonic) | Unison | A |
| 2 | Whole step | B |
| ♭3 | Minor third | C |
| 4 | Perfect fourth | D |
| 5 | Perfect fifth | E |
| ♭6 | Minor sixth | F |
| ♭7 (Subtonic) | Minor seventh | G |
| 8 (Octave) | Octave | A |
Harmonic Minor Scale
The harmonic minor scale consists of seven diatonic pitches derived from the natural minor scale, with the seventh scale degree raised by a semitone to form a major second from the sixth degree to the seventh, creating an augmented second interval unique to this variant.[14] This adjustment introduces a leading tone that resolves strongly to the tonic, enhancing harmonic tension and resolution compared to the natural minor's subtonic seventh degree.[21] The scale pattern follows the sequence of intervals: whole step, half step, whole step, whole step, half step, augmented second (minor third), half step.[22] For example, the A harmonic minor scale comprises the notes A, B, C, D, E, F, G♯, A, with no sharps or flats in its key signature beyond the natural minor's conventions.[23] This structure yields half steps between the second and third degrees, fifth and sixth degrees, and seventh and eighth (tonic) degrees, while the augmented second between the sixth and seventh degrees produces a characteristic dissonant leap often evoking modal or exotic flavors in melodic lines.[24] The scale is symmetric in ascent and descent, unlike the melodic minor variant.[10] In harmonic practice, the raised seventh enables the construction of a major triad or dominant seventh chord on the fifth scale degree—for instance, E major (E-G♯-B) or E7 (E-G♯-B-D) in A harmonic minor—facilitating authentic cadences (V-i) that mirror major key resolutions.[17] This feature addresses the weaker pull of the natural minor's minor v chord, making the harmonic minor essential for tonal stability in Western classical composition from the Baroque era onward, as composers like Bach employed it to strengthen phrase endings in minor keys.[25] Its augmented second also appears in jazz improvisations over dominant chords and in genres like metal or flamenco, where the interval adds tension, though melodic lines may revert to natural minor for smoother contour.[26]| Scale Degree | Note in A Harmonic Minor | Interval to Next Degree |
|---|---|---|
| 1 (Tonic) | A | Whole step |
| 2 (Supertonic) | B | Half step |
| 3 (Mediant) | C | Whole step |
| 4 (Subdominant) | D | Whole step |
| 5 (Dominant) | E | Half step |
| 6 (Submediant) | F | Augmented second |
| 7 (Leading Tone) | G♯ | Half step |
| 8 (Octave) | A | - |
Melodic Minor Scale
The melodic minor scale is a diatonic scale derived from the natural minor scale by raising the sixth and seventh scale degrees in its ascending form, resulting in the interval pattern whole-half-whole-whole-whole-whole-half (W-H-W-W-W-W-H).[6] This adjustment introduces a major sixth (from the root) and a major seventh, providing a stronger leading tone to the tonic and avoiding the augmented second interval present between the raised seventh and natural sixth of the harmonic minor scale.[28] For example, the A melodic minor scale ascends as A-B-C-D-E-F♯-G♯-A, contrasting with the natural minor's A-B-C-D-E-F-G-A.[17] In descending form, the melodic minor scale conventionally reverts to the natural minor pattern (H-W-W-H-W-W-W), emphasizing the minor sixth and seventh for a more authentic Aeolian flavor, as the raised degrees serve primarily melodic purposes in ascent.[29] This bidirectional asymmetry arose during the early Baroque period (circa 1600-1700) to facilitate smoother voice leading in polyphonic music while preserving the minor mode's tonal character; composers like Claudio Monteverdi employed such alterations to resolve melodic tensions without disrupting harmonic progressions.[30] In classical composition, the ascending melodic minor enhances resolution toward the dominant or tonic, as seen in works by Bach and Mozart where it appears in melodic lines over minor-key harmonies to create tension-release via the leading tone.[31] Jazz musicians, however, often apply the raised sixth and seventh bidirectionally, treating the scale (1-2-♭3-4-5-6-7) as a static parent scale for its seven modes, including the Lydian dominant (mode IV, used over dominant seventh chords with ♯11 and ♭7) and altered scale (mode VII, for altered dominant chords with ♭9, ♯9, ♭5, etc.).[32] [33] This modal framework, popularized in mid-20th-century bebop and modal jazz by figures like John Coltrane, expands improvisational options beyond strict classical usage.[34]Related Modes and Extensions
The Aeolian mode, equivalent to the natural minor scale, shares its minor tonality with three other diatonic modes: Dorian, Phrygian, and Locrian. These modes all feature a minor third above the tonic but differ in other intervals, influencing their characteristic sounds. The Dorian mode raises the sixth degree relative to Aeolian (intervals: whole, half, whole, whole, half, whole, whole), producing a brighter, less somber quality often used in folk and jazz contexts.[35] The Phrygian mode lowers the second degree (whole, half, whole, whole, whole, half, whole), evoking an exotic, tense flavor common in flamenco and metal genres.[35] The Locrian mode, with a diminished fifth and lowered second (half, whole, whole, half, whole, whole, half), is the most unstable due to its tritone tonic chord, limiting its use as a primary mode but appearing in passing diminished harmonies.[36] Extensions of the minor scale arise from the harmonic and melodic variants, each generating seven modes that expand harmonic possibilities beyond the diatonic framework, particularly in jazz and contemporary music. The harmonic minor scale (intervals: whole, half, whole, whole, half, whole, half, with raised seventh) yields modes such as Phrygian dominant (fifth mode: half, whole, whole, half, whole, whole, half), valued for its leading tone and flat second over dominant chords, and Lydian sharp second (sixth mode), which adds tension via augmented intervals.[37]| Mode | Parent Scale Degree | Key Intervals (from tonic) | Common Applications |
|---|---|---|---|
| Harmonic Minor | 1st | 1, ♭3, 5, ♮7 | Minor keys with V-i cadence |
| Locrian ♮6 | 2nd | ♭2, ♭3, ♭5, ♮6 | Half-diminished chords |
| Ionian ♯5 | 3rd | ♯5, ♮7 | Augmented major triads |
| Dorian ♯4 | 4th | ♯4, ♮6 | Suspended minor harmonies |
| Phrygian Dominant | 5th | ♭2, 4, ♮7 | Dominant 7th with ♭9, ♭13 |
| Lydian ♯2 | 6th | ♯2, ♯4, ♮7 | Lydian tensions over majors |
| Altered ♭♭7 | 7th | ♭2, ♯2, ♭3, ♭4, ♭5, ♭6, ♭♭7 | Fully altered dominant 7ths |
| Mode | Parent Scale Degree | Key Intervals (from tonic) | Common Applications |
|---|---|---|---|
| Melodic Minor | 1st | 1, ♭3, ♮6, ♮7 | Minor-major seventh chords |
| Dorian ♭2 | 2nd | ♭2, ♭3, ♮6, ♮7 | Half-diminished with ♭9 |
| Lydian Augmented | 3rd | ♯4, ♯5, ♮7 | Augmented dominant tensions |
| Lydian Dominant | 4th | ♯4, ♭7 | Dominant 7♯11 chords |
| Mixolydian ♭6 | 5th | ♭6, ♭7 | Dominant with ♭13 |
| Aeolian ♭5 | 6th | ♭5, ♮6, ♮7 | Half-diminished variants |
| Altered (Superlocrian) | 7th | ♭2, ♯2, ♭3, ♭5, ♯5, ♭7 | Altered dominant 7ths |
Theoretical Framework
Scale Degrees and Notation
In the minor scale, scale degrees are numbered from the tonic (degree 1) to the octave (degree 8), with specific interval alterations distinguishing it from the major scale. The natural minor scale features a lowered third degree (minor third above the tonic), lowered sixth degree (minor sixth), and lowered seventh degree (minor seventh), resulting in the interval pattern whole-half-whole-whole-half-whole-whole.[14] These degrees retain standard nomenclature: degree 1 as tonic, 2 as supertonic, ♭3 as mediant, 4 as subdominant, 5 as dominant, ♭6 as submediant, ♭7 as subtonic (distinct from the major scale's leading tone), and 8 as octave.[40]| Scale Degree | Name | Interval from Tonic | Example in A Natural Minor |
|---|---|---|---|
| 1 | Tonic | Perfect unison | A |
| 2 | Supertonic | Major second | B |
| ♭3 | Mediant | Minor third | C |
| 4 | Subdominant | Perfect fourth | D |
| 5 | Dominant | Perfect fifth | E |
| ♭6 | Submediant | Minor sixth | F |
| ♭7 | Subtonic | Minor seventh | G |
| 8 | Octave | Perfect octave | A |
Key Relationships and Transposition
The relative minor key of a given major key shares the same key signature and pitches but begins on the sixth scale degree of the major scale, creating a tonal center a minor third below the major tonic. For instance, C major (no sharps or flats) has A minor as its relative minor, with the A minor scale comprising A-B-C-D-E-F-G. This relationship facilitates modal interchange in composition, allowing composers to borrow chords or notes between the pair without altering the key signature. Similarly, every minor key has a relative major, located a major third above its tonic; A minor's relative major is C major.[7][41] In contrast, the parallel minor of a major key maintains the same tonic but adopts the minor scale's interval structure, resulting in a key signature that typically includes three additional flats (or the enharmonic equivalent in sharps for keys with many sharps). C major (no accidentals) parallels C minor, which uses the signature of Eb major (three flats: Bb, Eb, Ab) to accommodate the lowered third, sixth, and seventh degrees relative to the major scale. This adjustment reflects the natural minor's lowered intervals, though harmonic and melodic variants introduce further accidentals like a raised seventh. Parallel relationships emphasize tonal contrast around a shared root, common in modulations where a piece shifts from major to minor without changing the fundamental pitch center.[7][42] Transposition of a minor scale involves shifting all pitches by a uniform interval while preserving the original pattern of whole and half steps specific to the variant (natural, harmonic, or melodic). For example, transposing the natural A minor scale (A-B-C-D-E-F-G) up a perfect fifth to E minor yields E-F♯-G-A-B-C-D, adjusting the key signature from none to one sharp (F♯) to match the new tonic's diatonic requirements. In practice, the key signature for the transposed minor key aligns with that of its relative major, with accidentals applied as needed for harmonic or melodic forms; transposing harmonic A minor (A-B-C-D-E-F-G♯) up a minor third to C minor becomes C-D-E♭-F-G-A♭-B, incorporating the relative major's signature (three flats) plus the raised seventh. This process maintains intervallic relationships, such as the minor third from tonic to third, ensuring the transposed scale retains its minor character.[43][14]| Major Key | Relative Minor | Key Signature (Sharps/Flats) | Parallel Minor Adjustment |
|---|---|---|---|
| C major | A minor | 0 | C minor: +3 flats (Eb, Ab, Bb) |
| G major | E minor | 1 (F♯) | G minor: +3 flats (Bb, Eb, Ab) |
| D major | B minor | 2 (F♯, C♯) | D minor: +3 flats (Bb, Eb, Ab) |
Harmonic and Compositional Applications
Diatonic Chords and Progressions
In the natural minor scale, diatonic triads are constructed by superimposing thirds starting from each scale degree, yielding seven distinct chord types: the tonic i (minor), supertonic ii° (diminished), mediant ♭III (major), subdominant iv (minor), dominant v (minor), submediant ♭VI (major), and subtonic ♭VII (major).[46][47] These qualities arise from the scale's interval pattern (whole, half, whole, whole, half, whole, whole), where the lowered third, sixth, and seventh degrees produce the characteristic minor sonorities and avoid a strong leading tone to the tonic.[14]| Scale Degree | Roman Numeral | Chord Quality | Example (A minor) |
|---|---|---|---|
| i | i | Minor | A–C–E |
| ii | ii° | Diminished | B–D–F |
| ♭III | ♭III | Major | C–E–G |
| iv | iv | Minor | D–F–A |
| v | v | Minor | E–G–B |
| ♭VI | ♭VI | Major | F–A–C |
| ♭VII | ♭VII | Major | G–B–D |