From 22ee558393ede3eabaae6e3b0ee455d3cabde7f5 Mon Sep 17 00:00:00 2001 From: Trance-0 <60459821+Trance-0@users.noreply.github.com> Date: Thu, 30 Oct 2025 23:51:17 -0500 Subject: [PATCH] Create Music1260_M2.md --- content/Swap/Math4501/Music1260_M2.md | 255 ++++++++++++++++++++++++++ 1 file changed, 255 insertions(+) create mode 100644 content/Swap/Math4501/Music1260_M2.md diff --git a/content/Swap/Math4501/Music1260_M2.md b/content/Swap/Math4501/Music1260_M2.md new file mode 100644 index 0000000..7743155 --- /dev/null +++ b/content/Swap/Math4501/Music1260_M2.md @@ -0,0 +1,255 @@ +# Music 1260 Jazz Theory I + +> [!TIP] +> +> In most of the time, it is sufficient to get a 60% with minimal effort so that you can focus on the more important things. +> +> Especially, for a course that you currently don't have time to learn. + +We will try to debunk the entire jazz theory in this notes, used for self-learning. + +I will try to formulate the jazz theory in a way that **I** can understand. + +## Basics of 7th chords + +It is a enumeration over the 7 notes of the scale with step size 2 (tritone, the musician like to call it the 3rd but that don't make sense to me). + +You can pick any using this simple function: + +```python +scale = ['C', 'D', 'E', 'F', 'G', 'A', 'B'] +def get_7th_chord(root, scale): + return [scale[(root + i*2) % 7] for i in range(7)] +``` + +There are some stupid math going on here since the scale is not uniform in a mathematical sense, you need to know that `E-F` and `B-C` are 1 half step apart where others are 2 half steps (they call whole step) apart. + +This results in stupid math like this: + +```python +chromatic_scale = ['A', 'A#', 'B', 'C', 'C#', 'D', 'D#', 'E', 'F', 'F#', 'G', 'G#'] +note_value = {e:i for i, e in enumerate(chromatic_scale)} +def get_number_of_half_steps(note1, note2): + return min(abs(note_value[note1] - note_value[note2]), 12 - abs(note_value[note1] - note_value[note2])) +``` + +You can construct this table in exam for simple counting. + +| Note\Note | **A** | A# | **B** | **C** | C# | **D** | D# | **E** | **F** | F# | **G** | G# | +|-----------|---|----|---|---|-----|---|-----|---|----|----|---|-----| +| **A** | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10| 11 | +| A# | 1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | +| **B** | 2 | 1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | +| **C** | 3 | 2 | 1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | +| C# | 4 | 3 | 2 | 1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | +| **D** | 5 | 4 | 3 | 2 | 1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | +| D# | 6 | 5 | 4 | 3 | 2 | 1 | 0 | 1 | 2 | 3 | 4 | 5 | +| **E** | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | 1 | 2 | 3 | 4 | +| **F** | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | 1 | 2 | 3 | +| F# | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | 1 | 2 | +| **G** | 10| 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | 1 | +| G# | 11| 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | + +They use different names for interval types, resulting in a more consistent naming scheme for the chords types. + +| Interval | Number of Half Steps | Short Name | +|---------------|----------------------|------| +| Augumented | 5 | *+ | +| Major | 4 | *M | +| Minor | 3 | *m | +| Diminished | 2 | *dim | + +_the `*` denotes the root note_ + +Thanks god you already pass the first exam so you don't need to make distinction between this notation and the major minor stuff for describing the interval types. + +But you still need to take these names to consider: + +| Interval sequence | Name | Short Name | Even shorter| +|-------|-------|---------------|------| +| (4,4,3) | Major-Augmented 7th | Augmented 7th | *M7#5 | +| (4,3,4) | Major-Major 7th | Major 7th | *M7| +| (4,3,3) | Major-Minor 7th | Dominant 7th | *7| +| (3,4,4) | Minor-Major 7th | Minor-Major 7th | *mM7| +| (3,4,3) | Minor-Minor 7th | Minor 7th | *m7| +| (3,3,4) | Diminished-Minor 7th | Half Diminished 7th | *m7b5| +| (3,3,3) | Diminished-Diminished 7th | Fully Diminished 7th | *dim7| + +_the `*` denotes the root note_ + +So we can construct the basic 7th chords like this, take the simplest example, C major 7th chord: + +| Chord | Notes | Interval Type | Short Name | Even shorter| +|-------|-------|---------------|------|------| +| I | C, E, G, B | (4,3,4) Major-Major 7th | Major 7th | CM | +| ii | D, F, A, C | (3,4,3) Minor-Minor 7th | Minor 7th | Dm | +| iii | E, G, B, D | (3,4,3) Minor-Minor 7th | Minor 7th | Em | +| IV | F, A, C, E | (4,3,4) Major-Major 7th | Major 7th | FM | +| V | G, B, D, F | (4,3,3) Major-Minor 7th | Dominant 7th | G7 | +| vi | A, C, E, G | (3,4,3) Minor-Major 7th | Minor 7th | Am | +| $\text{vii}^\circ$ | B, D, F, A | (3,3,4) Diminished-Minor 7th | Diminished 7th | Bdim7 | + +## Primary Chords + +Given the home key, the primary chords are the chords that are built on the 1st, 4th, and 5th degrees of the scale. + +For example, in C major scale, the primary chords are: + +| Chord | Notes | Interval Type | Short Name | Even shorter| +|-------|-------|---------------|------|------| +| I | C, E, G, B | (4,3,4) Major-Major 7th | Major 7th | CM | +| IV | F, A, C, E | (4,3,4) Major-Major 7th | Major 7th | FM | +| V | G, B, D, F | (4,3,3) Major-Minor 7th | Dominant 7th | G7 | + +### Diatonic median substitution + +Diatonic median substitution is a technique that allows you to substitute a **primary chord** with a **secondary chord** or vice versa. + +Formula is given by belows: + +```python +def diatonic_median_substitution(primary_chord): + # if primary_chord.roman_numeral == 'I': + # return ['iii','vi'] # +2, -2 + # elif primary_chord.roman_numeral == 'IV': + # return ['vi','ii'] # +2, -2 + # elif primary_chord.roman_numeral == 'V': + # return ['vii','iii'] # +2, -2 + if primary_chord.is_primary_chord(): + # consider the addition is defined and modulo 7 (number of keys in scale) + return [primary_chord._chord_key_shift(note=2), primary_chord._chord_key_shift(note=-2)] + else: + raise ValueError(f"Invalid primary chord: {primary_chord}") +``` + +## Secondary Chords + +Given the home key, the secondary chords are the chords that are built on the 2nd, 3rd, and 6th degrees of the scale. + +For example, in C major scale, the secondary chords are: + +| Chord | Notes | Interval Type | Short Name | Even shorter| +|-------|-------|---------------|------|------| +| ii | D, F, A, C | (3,4,3) Minor-Minor 7th | Minor 7th | Dm | +| iii | E, G, B, D | (3,4,3) Minor-Minor 7th | Minor 7th | Em | +| vi | A, C, E, G | (3,4,3) Minor-Major 7th | Minor 7th | Am | + +## Basis of Tonality, or key center + +ii-V-I + +is a classical chord progression, we will play a lot with this mathematically nonsense later. + +## Secondary Dominant Chords + +Deduction of the secondary dominant chords from the primary chords. + +Every major minor chord has a secondary dominant chord. (that means $\text{vii}^\circ$ chord don't have a secondary dominant chord) + +We can construct the secondary dominant chords from the primary chords by subtracting 2 half steps from the root note, the do a dominant 7th chord based on that. + +Or equivalently, consider the root notes for the 7th chord, denote as `r`, and then pick the V chord based on major of `r`. + +```python +def secondary_dominant_chord(primary_chord): + return _construct_chord(root=primary_chord._chord_key_shift(half_steps=-2), interval_type=(4,3,3)) # Using major-minor (dominant) as interval type +``` + +List of secondary dominant chords: + +Assume we build chord on C major scale. + +| Chord | Name | Notes | Secondary Dominant Chord | Name | Notes | +|-------|-------|------|-------|-------|------| +| I | CM | C, E, G, B | V | G7 | G, B, D, F | +| ii | Dm | D, F, A, C | V/ii | A7 | A, C#, E, G | +| iii | Em | E, G, B, D | V/iii | B7 | B, D#, F#, A | +| IV | FM | F, A, C, E | V/IV | C7 | C, E, G, **Bb** (keep consistency with the interval) | +| V | G7 | G, B, D, F | V/V | D7 | D, F#, A, C | +| vi | Am | A, C, E, G | V/vi | E7 | E, G#, B, D | + +### Companion ii chord with secondary dominant chords + +#### Companion ii chord with secondary dominant chords for major quality chords + +IV and V chords have companion ii chords. + +Should always be the **minor quality ii chord**. + +For example, in C major scale, the companion ii chord for IV chord (FM) is Gm, and the companion ii chord for V chord (G7) is Am. + +| Chord | Name | Notes | Companion ii Chord | Name | Notes | +|-------|-------|------|-------|-------|------| +| IV | FM | F, A, C, E | ii/IV | Gm | G, Bb, D, F | +| V | G7 | G, B, D, F | ii/V | Am | A, C, E, G | + +#### Companion ii chord with secondary dominant chords for minor quality chords + +ii, iii, vi have companion ii chords. + +Should always be the **diminished-minor quality ii chord**. + +For example, in C major scale, the companion ii chord for ii chord (Dm) is Em7b5, and the companion ii chord for iii chord (Em) is Fm7b5. + +| Chord | Name | Notes | Companion ii Chord | Name | Notes | +|-------|-------|------|-------|-------|------| +| ii | Dm | D, F, A, C | ii/ii | Em7b5 | E, G, Bb, D | +| iii | Em | E, G, B, D | ii/iii | Fm7b5 | F, G#, C, D# | +| vi | Am | A, C, E, G | ii/vi | Bm7b5 | B, D, F, A | + +## Tritone Substitution + +A tritone is the M3 and m7 of a dominant 7th chord. (6 half steps, or 3 whole steps) + +Consider the dominant 7th chord, G7 (G, B, D, F), the tritone is B and F, so we can some how flip the order of the note to derive a new dominant 7th quality chord. + +To find such chord, we align the remaining note as desired. + +Example: G7 (G, **B**, D, **F**) -> Db7 (Db, **F**, Abb, **Cb**) (keep consistency with the interval (4,3,3)) + +| Primary Chord | Secondary Dominant Chord | Key for secondary dominant chord | Tritone Substituted Secondary Dominant Chord | Key for tritone substituted secondary dominant chord | +|-------|-------|-------|-------|-------| +| CM | G7 | G, **B**, D, **F** | Db7 | Db, **F**, Abb, **Cb** | +| Dm | A7 | A, **C#**, E, **G** | Eb7 | Eb, **G**, Bb, **Db** | +| Em | B7 | B, **D#**, F#, **A** | F7 | F, **A**, C, **D#** | +| FM | C7 | C, **E**, G, **Bb**| F#7 | F#, **A#**, C#, **E** | +| G7 | D7 | D, **F#**, A, **C** | G#7 | G#, **B**, D#, **F#** | +| Am | E7 | E, **G#**, B, **D** | A#7 | A#, **C#**, E#, **G#** | + +### Tritone Substitution for secondary dominant chords + +Using the same strategy, but carefully list the key for the secondary dominant chord. + +### Companion ii chord with tritone substituted secondary dominant chords + +The same, just consider the secondary dominant chord as the V chord and derive what is the companion ii chord for it. + +## Modal borrowing + +Use notes from parallel minor scale to construct the chords. + +|Major|C|D|E|F|G|A|B| +|-----|---|---|---|---|---|---|---| +|Parallel Natural Minor|C|D|Eb|F|G|Ab|Bb| +|Parallel Harmonic Minor|C|D|Eb|F|G|Ab|B| +|Parallel Melodic Minor|C|D|Eb|F|G|A|B| + +### From parallel natural minor + +#### Primary, secondary dominant chords in parallel natural minor + +Rewrite the key for chords in major to minor. + +#### Companion ii chord with tritone substituted secondary dominant chords in parallel natural minor + +### From parallel harmonic minor + +#### Primary, secondary dominant chords in parallel harmonic minor + +#### Companion ii chord with tritone substituted secondary dominant chords in parallel harmonic minor + +### From parallel melodic minor + +#### Primary, secondary dominant chords in parallel melodic minor + +#### Companion ii chord with tritone substituted secondary dominant chords in parallel melodic minor