I was showing someone how I approach programming. We can think of a problem from either the “top-down” or “bottom-up” perspective. In “bottom-up” we identify the building blocks needed to eventually build the solution — in programming, this translates into writing the low-level functions (or other abstractions) needed to solve the higher-level problem. In “top-down” we, instead, assume that we’ll fill in the low-level details as we go, and choose to focus on the “skeleton” of the solution first.

Bottom-up study of the atomic elements of musical improvisation is possible, and of course must be paired with “top-down” study of complete pieces of music. My own experience of bottom-up study is that it is always painful to start it, but that it always proves enormously fruitful musically.

Bottom-up study

Rhythm

As a method of studying rhythm, consider all the ways of accenting 16th notes in a quarter note.

number of accents 16th note accent pattern (A=accent, x=rest)
1 Axxx, xAxx, xxAx, xxxA
2 AAxx, AxAx, AxxA, xAAx, xAxA, xxAA
3 AAAx, AAxA, AxAA, xAAA

There are 14 permutations. These could be practiced as follows:

Each permutation can be practiced in isolation:

  • LH (left hand) quarter notes, RH accents
  • LH accents, RH quarter notes

Next step would be to alternate between various permutations:

  • Roles of hands remain consistent as the permutations alternate
  • Roles of hands also alternate, e.g. after cycling through a pair of permutations, switch the hand roles and use the same pair of permutations

The table above only shows the possible accent permutations for quarter note divisions of 4. Divisions of 3 should be practiced separately.

Finally, practice polyrhythms such as 3 against 4 and 4 against 3. Systematizing such study is beyond the present scope.

Harmony

The table below shows every permutation of triads and seventh chords by stacking thirds, as well as the function of each such chord in each of the 4 scales that, together, form the basis of the Western tonal system.

In the table columns below:

  • “3s” means how to construct the chord by stacking minor (m) or major (M) thirds.
  • The subsequent 4 columns express the function of a given chord in the major and each of the 3 minor scales
  • Note: all natural minor chords are also available in the descending melodic minor.
name 3s major natural harmonic melodic
diminished mm viiᵒ iiᵒ iiᵒ, viiᵒ viiᵒ
minor mM ii, iii, vi iv, v, i i, iv i, ii, vi
major Mm I, IV, V ♭III, ♭VI, ♭VII V, ♭VI IV, V
augmented MM     ♭III+ ♭III+
diminished 7 mmm     viiᵒ⁷ viiᵒ⁷
dominant 7 Mmm V⁷ ♭VII⁷ V⁷ IV⁷, V⁷
minor 7 mMm ii⁷, iii⁷, vi⁷ iv⁷, v⁷, i⁷ iv⁷ ii⁷
half-dim 7 mmM viiø iiø iiø viø, viiø
major 7 +5 MMm     ♭III△+5 ♭III△+5
major 7 MmM I△, IV△ ♭III△, ♭VI△ ♭VI△  
min +Maj7 mMM     i△ i△

Of note: minor keys use every available harmonic relationship; major keys do not.

There are 4 permutations of triads, each in 3 inversions.
There are 7 permutations of 7th chords (omitting the MMM construction), each in 4 inversions.
There are 12 keys (physically — 15 if we include the viable enharmonic equivalents).

There are (4 * 3) + (7 * 4) 40 “close-position voicings” available for each note.

Thus, there are (12 * 40) 480 physical “close-position voicings” available on the piano, composed of triads and 7th chords in all inversions across all keys.

That is a brute-force method of practicing all the physical “close-position voicings” that are available.

At this moment, I see the following ways of practicing. Cycle through 1:

  • Chord quality (across all keys and inversions) — probably start with the tough ones.
  • Inversion (across all chords and keys)
  • Key (across all chords and inversions)

Only practice can show which of these are useful. However, as the table indicates, each of these chords has various functions. The brute-force practicing misses the point of a tonal system, which is beyond the present scope of identifying what is possible. A system would describe how these chords work together, and enable practice thereof.

One perspective to consider would be modulation (across multiple keys, i.e. tonal centers). The chords that function in multiple keys (indicated by rows in the table that have multiple values in the last 4 columns) can serve as pivots between keys. Another level of analysis would be to catalog the common modulations and to practice those.

And then of course the complexity increases as we introduce non-closed position voicings — I do believe that a systematic “relationship” (at minimum, physically, at the piano) with these voicings — and free movement among them — can only be built on a solid foundation of closed position voicings.

The eventual aim of this effort was at one point expressed by Miles Davis to George Russell: “I want to learn all the chord changes. How can I go about doing this?” (Making of Kind of Blue: Miles Davis and His Masterpiece, Eric Nisenson)

Inspiration

Inspiration for the Rhythm section: Gabriel Cruser.
Inspiration for the Harmony section: The Harmony of Bill Evans, Volume 2 (chapters 1 and 2), by Jack Reilly.