Intervals, Complementary Inversions & Jazz ii–V–I Piano Voicings

Attendance & Class Context

Roll call confirmed: Cardi, Coltrane, Bella, Marcus, Finn (moved), Steve, Ezra, Selenk, Luke (moved), plus possible arrivals from Paul’s class.

Ear-training (recognising, singing & naming intervals). 
1. Piano/jazz theory (ii–V–I guide-tone voicings, adding extensions, arranging implications).

Core request when naming an interval: always give BOTH components → NUMBER + QUALITY (e.g. “Major 7th” not just “Seventh”).

Reverse them (sing up instead of down) if descending feels difficult. 
1. Locate lower note, then walk up a major scale until you recognise the distance. 
  1. Use associative melodies (mnemonics).

Mnemonic examples: • Descending Major 3rd ⇒ common “door-bell” chime. • Ascending Perfect 4th ⇒ “Here Comes the Bride”. • Ascending Minor 2nd ⇒ \text{Jaws} motif (half-step menace). • Descending Major 7th ⇒ opening of Cole Porter’s “I Love You”.
Comfort range: each student should be able to sing a complete major scale in at least one comfortable octave (practice at a piano until matching pitch is easy).
Whistling is acceptable when accurate; physical singing remains more connected for ear-training.

Term: Complementary (a.k.a. Inverse) Intervals ⇒ Two intervals that add up to an octave (=9) when number values are summed.
Rule set:
• If one interval is Major ⇒ complement is Minor. • Minor ⇒ complement Major. • Perfect ⇒ remains Perfect. • Augmented ⇒ complement is Diminished. • Diminished ⇒ complement Augmented.
Numeric relation: \text{Number}1+\text{Number}2=9. Example: Minor 6th ↔ Major 3rd ( 6+3=9 ), Augmented 5th ↔ Diminished 4th, etc.
Practical ear-training drill shown: 
1. Start on (C), sing a Minor 6th up ((C\rightarrow A\flat)). 
  1. Use rule to predict complement → Major 3rd ((A\flat\rightarrow C)). 
    1. Verify on piano.

Students practised a spectrum of intervals: Perfect 8ve, Minor 7th, Major 6th, Perfect 4th, Augmented 5th/Diminished 4th, etc.
Tips: • Large intervals (6ths, 7ths) are “between a 5th and an Octave”; use perfect 5th or octave as anchor. • Sevenths: identify whether it is a whole-step (Minor 7) or half-step (Major 7) from root.
Musical context demonstration: F♯ to B♭ (Aug. 5th / Dim. 4th) appears naturally as 7^{\text{th}} & 3^{\text{rd}} of G-melodic/harmonic-minor scale (Lydian-dominant scenario).

Most idiomatic harmony voices for a lead melody: THIRDS (or Sixths below). • Parallel 4ths/5ths create hollow sound; avoid unless stylistically desired. • Example: “Bye Bye Blackbird” — options: ◦ Below-melody minor 3rd ((C) under (E)). ◦ Jazzier choice: major 3rd above for colour (arranger’s discretion).
Concept of surrounding/neighbor tones (bebop & vocal harmonies): • Create tension by landing a half-step above/below target chord-tone, then resolving. • Bass line example: minor-ii–V motion in F{\text{min}} \rightarrow B\flat7 using a major 3rd over the minor-ii chord (chromatic approach).

Shape A – 7 on bottom, 3 on top.
• Dm7: C\,(7) \rightarrow F\,(3)
• G7 : F \rightarrow B
• Cmaj7: B \rightarrow E
Shape B – 3 on bottom, 7 on top.
• Dm7: F \rightarrow C
• G7 : B \rightarrow F
• Cmaj7: E \rightarrow B

Smooth voice-leading principle: retain common tones, move each voice by step where possible (sequence 7→3, 3→7, etc.).

Add a tone ABOVE guide-tones → choose from 5-family (ii & I) or 9-family (V).
5-family (extensions around the fifth): 5,\;4/11,\;\sharp4/\sharp11,\;6/13,\;\flat6/\flat13.
9-family (extensions around the ninth): 9,\;\flat9,\;\sharp9,\;1.
Vanilla example (C major):
• Dm7: C\,(7) – F\,(3) – A\,(5)
• G7 : B\,(3) – F\,(7) – A\,(9)
• Cmaj7: E\,(3) – B\,(7) – G\,(5)
Alternate colour (using 11 on minor): C–F–E for Dm11.
Resolution guideline: 5-family → tends to resolve to 9-family; 9-family → resolves back to 5-family.

Keep voicings within the acoustic ‘sweet-spot’: roughly C3 (bass clef middle) up to C5. Below ⇒ muddy with bass; above ⇒ clashes with melody/solo.
Left hand can play ROOTS (especially for practice); in ensemble context bass player covers roots.
Drill: play ii–V–I guide-tone cycles in every key until automatic; then add third voice; then experiment with extensions (#11, b13, #9 etc.).

A piano voicing = blueprint for section writing: assign each pitch to saxes/trombones/trumpets as desired.
Workflow recommendation for big-band charts: 
1. Map FORM first (melody, counter-melody, solo backings, shout chorus …). 
  1. Create CONDENSED SCORE of chord voicings & melodic lines. 
    1. Orchestrate (choose which instruments play which chord-tones).
Colour decisions (examples): • Put baritone sax on top note for bite. • Mix instrument families within a single voicing (e.g. Berry, Trumpet, Trombone, Alto) for unique timbre.
Pianist in big band often drops out when sections are holding pads; alternate role → sparse Count Basie-style “comp bumps”.

Diatonic vi chord is normally minor. Jazz practice often makes it Dominant 7 to FUNCTION as V of ii (secondary dominant). Example in C major: A7 \rightarrow Dm7 adds forward motion vs. diatonic Am7.

Half-step approaches create “gravity” → consonant once resolved. • Concept applies to bass lines, comping, and improvised melodies. • Surround-tone pattern: target note x can be preceded by x+\tfrac12, x-\tfrac12 or both (enclosure).

Daily ear session: sing/whistle all simple intervals ascending & descending. • Use reference tunes, but gradually wean off them – aim for internal recognition.
Piano lab: 
1. Two-note guide voicings → 12 keys. 
  1. Add 5-family tone. 
    1. Add 9-family tone. 
      1. Test altered dominants ((\sharp9,\;\flat13,\;\sharp11)). 
        Apply to real standards (read changes, comp while singing melody).
Arrangement analysis: pick favourite big-band chart; reduce to condensed score; observe how arranger distributed piano-style voicings.
Improvisation drill: compose lines that resolve guide-tones by step; incorporate neighbor tones and chromatic enclosures.

Complementary interval equation: n1+n2=9.
Perfect ⇒ stays perfect; Major↔Minor; Augmented↔Diminished on inversion.
ii–V–I skeleton: 7\rightarrow3,\;3\rightarrow7,\;7\rightarrow3 (smooth path). Example in C: C\,(Dm7) \rightarrow B\,(G7) \rightarrow E\,(Cmaj7).
Extensions shorthand: • 5-family = {11, \sharp11, 5, 13, \flat13}. • 9-family = {1, 9, \flat9, \sharp9}.
Range guide for comping: roughly C3\text{ to }C5.

Understanding theory is like mastering language syntax: same vocabulary, infinite creative output; each arranger/soloist writes a unique “book”.
Ear-training builds musicianship beyond notation – physical engagement (singing) deepens internalisation more than visual/paper methods.