Reimagining Musical Harmony

The New Harken Algorithm for Arranging Chromatic Harmony

In the ever-evolving landscape of music theory, the quest for harmonious arrangements that resonate with both intellect and intuition remains a timeless pursuit. The Harken Music system introduces a groundbreaking new algorithm that redefines how we perceive and arrange musical scales and combinations. This algorithm leverages the Supremacy of the Tonic and a meticulously crafted Hierarchy of Intervals to automatically arrange any musical scale or combination, ensuring harmonic coherence and aesthetic balance.

Tonic Supremacy: The Foundation of Harmony

The principle of Tonic Supremacy is the foundation of this new algorithm. The Tonic—always denoted 0—is the fundamental pitch upon which all harmonic structures are built. In any arrangement, the Tonic must be included and is invariably positioned as the root or bass—the lowest frequency pitch. This anchoring of the Tonic provides a gravitational center, a sonic home base that players and listeners subconsciously return to throughout a piece.

While there are many instances when the Perfect 5th (7), Diminished 5th (6), or Augmented 5th (8), or other intervals, are inverted below the Tonic (0) for creative effect, these variations do not undercut its fundamental role. The algorithm accounts for these exceptions, ensuring that the core function of the Tonic remains unaltered.

Resolving Close Interval Conflicts: The Hierarchy of Intervals

A significant challenge in harmonic arrangement is the dissonance caused by close intervals, specifically adjacent pairs, or clusters. When semitone-adjacent notes sound together simultaneously, they produce tension that can alter or disrupt the harmonic balance. The Harken algorithm addresses this by implementing a hierarchy of intervals that dictates how to resolve these conflicts through octave displacement.

Key Principles of the Hierarchy:

• Octave Displacement: To mitigate for excessive dissonance, the lower pitch in a conflicting pair is raised by one or more octaves. This increases the interval between the two notes, reducing tension while maintaining harmonic function.

• Interval Prioritization: Certain intervals take precedence over others based on their harmonic relationships, where higher pitches are perceived as louder and therefore more prominent in the harmony. For example: when both Major 3rd (4) and Minor 3rd (3) are present in a combination, the higher pitch 4 takes precedence over the lower pitch 3. In this case, the Minor 3rd (3) is raised one octave to function as the ♯9 tension (or Augmented 2nd).

• Exceptions to the Rule: When the Tonic (0) and Minor 2nd (1) are both present in a combination, the Minor 2nd is routinely, but not always, raised one octave to function as the ♭9 tension.

Symmetry and Balance: Utilizing Symmetrical Note-Pairs

The Harken Music system also considers the concept of symmetry in harmonic structures. By analyzing symmetrical note-pairs, like 5, 7 or 10, 2, the algorithm ensures an even distribution of notes around the Tonic (0) within an arrangement. This balance is crucial for creating chords and scales that are both harmonically rich and aesthetically pleasing.

Possible Configurations for Symmetrical Pairs:

• Full Inclusion: Both notes of a symmetrical pair are included: 5, 7 or 10, 2.

• Partial Inclusion: One note is included, and the other is omitted: 5, x or x, 7.

• Omission: Both notes are omitted and considered null: x, x.

The Harken algorithm aims to create the most musically satisfying arrangement while adhering to the principles of the Harken Music system, while acknowledging Western arranging traditions.

Constructing Harmonic Order: Building from the Bass Up

Harmonic structures are constructed from the bottom up, starting with the Tonic. The algorithm adheres to the following systematic hierarchy.

1. Tonic (0): Establishes the foundation of the overall harmony.

2. Perfect 5th (7) and/or Perfect 4th (5): The Perfect 5th takes precedence over the Perfect 4th because of its close harmonic relationship to the Tonic. When both the Tritone (6) and Perfect 5th (7) are present, and the Perfect 4th (5) is absent, the Tritone 6 functions as the ♯11 tension (Augmented 4th) to avoid redundant “Fifths” within the same chord or scale.

3. Guide Tones (3rd and 7th): The Major 3rd (4) or Minor 3rd (3) and the Major 7th (11) or Minor 7th (10) determine the basic tetrachord quality. The harmonic order of the Third and Seventh are interchangeable.

4. Scale Extensions: The 2nd (Minor or Major), 4th (Perfect or Augmented), and 6th (Minor or Major) add color and harmonic richness to the basic underlying tetrachord.

5. Chromatic Tensions: Notes that are not diatonic to the scale (e.g., ♭9, ♯9, ♯11, ♭13, ♯13) are placed up at least one octave (or higher) to resolve potential conflicts with the underlying chord and scale tensions. If necessary, intervals may be raised two octaves to avoid ongoing conflicts in the higher registers, especially in dense harmonic structures like the chromatic scale. Some chromatic tensions can lowered one octave to back-fill any large interval gaps that may result.

6. Goal: The goal of this algorithm is to eliminate or minimize conflict, using octave displacement to separate adjacent conflicting semitones.

Step-by-Step: The Stages of Chromatic Harmony

1. Stage One: Tonic: 1 note (0) that determines the fundamental

2. Stage Two: Basic Chord: 3 notes that determine the basic chord quality

3. Stage Three: Scale Extensions: 3 notes that determine the heptatonic scale

4. Stage Four: Chromatic Tensions: 5 notes that complete the chromatic scale

• The Tonic (0): The fundamental is always present as the first and lowest frequency pitch in any combination.

• Basic Chord: Fifth, Third, and Seventh: These 3 basic intervals are interchangeable in their hierarchical order due to the invertible nature of basic tetrachords. Positioning the Fifth below the Third and Seventh provides a solid foundation for constructing larger chords while maximizing harmonic stability.

  • Special Conditions Concerning the Fifth:

    • If the Tritone (6) and Perfect 5th (7) are present, and the Perfect 4th (5) is absent or omitted, the Tritone functions as the ♯11 tension.

    • If the Tritone (6) and Perfect 4th (5) are present, and the Perfect 5th (7) is absent or omitted, the Tritone functions as the ♭5 or Diminished 5th.

    • If the Tritone (6) is present, and the Perfect 5th (7) and the Perfect 4th (5) are absent or omitted, the Tritone functions as the ♭5 or Diminished 5th.

• Scale Extensions: Second, Fourth, Sixth: These next three scale tones are generally, but not always, positioned one whole step above the Tonic, Third, and Fifth. If conflicts occur at this stage, one or more of the scale extensions may be raised one octave to resolve any conflicting semitones. As with the basic chord notes, scale extensions are interchangeable (invertible) in terms of their relative positions within the group.

• Chromatic Tensions: The remaining 5 notes, which form a pentatonic scale, are also interchangeable in terms of their relative positions in the group. But again, the principle goal of the algorithm is to eliminate or minimize adjacent semitones. Although some chromatic tensions can be mixed or interchanged with the scale extensions to back fill any large interval gaps, these tensions will generally remain in the highest registers of the arrangement to avoid conflicts.

• Example:

  • Tonic + Basic Chord + Scale Extensions + Chromatic Tensions

  • Mixolydian ♭6 Chromatic: [0] + [7, 4, 10] + [2, 5, 8] + [1, 3, 6, 9, 11]

    • More detailed examples provided below

Applying The Algorithm

1. Create or Select a Combination of Notes:

   Begin by choosing the set of notes you wish to arrange.

2. Analyze the Combination:

   Confirm the presence of the Tonic (0) and identify the presence or absence of key intervals: Perfect 5th (7), Perfect 4th (5), Tritone (6), Third (3 or 4) and Seventh (10 or 11).

3. Determine Symmetrical Pairs:

   Analyze the symmetrical pairs to identify whether one, both, or neither are present.

4. Resolve Conflicts:

   Use octave displacement to resolve dissonance caused by adjacent semitones. Apply the hierarchy of intervals to prioritize notes.

5. Sort Notes Following the Hierarchy of Intervals:

   Arrange the intervals in order of the hierarchy, from the bass upwards, following the Stages of Chromatic Harmony detailed above.

6. Output Result:

   Present the arranged scale or chord, in sequential order, always starting with the Tonic 0 and ensuring each note adheres to the Harken principles.

7. Apply Transformations (Optional):

   Utilize symmetry and transformations such as inversion and retrograde on each group—basic chord triad, scale-extension triad, and chromatic-tension pentatonic scale—separately before recombining them.

Example:

Mixolydian ♭6 Scale: 0, 2, 4, 5, 7, 8, 10

Chromatic Arrangement: 0, 7, 4, 10, 2, 5, 8, 1, 3, 6, 9, 11

• 0 Tonic remains fixed

• 7 Perfect 5th positioned above the Tonic

• 4 Major 3rd positioned above the Perfect 5th

• 10 Minor 7th positioned above the Major 3rd

• 2 Major 2nd raised one octave to scale extension ♮9th

• 5 Perfect 4th raised one octave to chromatic tension ♮11th

• 8 Minor 6th raised one octave to chromatic tension ♭13

• 1 Minor 2nd raised two octaves to chromatic tension ♭9

• 3 Minor 3rd raised two octaves to chromatic tension ♯9

• 6 Tritone raised two octave to scale extension ♯11

• 9 Major 6th raised two octave to scale extension ♮13th

• 11 Major 7th raised two octaves to chromatic tension or leading tone

Chord Quality: Mixolydian ♭6 Chromatic chord, Dominant 7th chord with 3 scale extensions and 5 chromatic tensions.

Transformations:

• Prime: 0, 7, 4, 10, 2, 5, 8, 1, 3, 6, 9, 11

• Inverse: 0, 5, 8, 2, 10, 7, 4, 11, 9, 6, 3, 1

More Examples:

1. Dorian Chromatic:

   • Prime: 0, 7, 3, 10, 2, 5, 9, 11, 1, 4, 6, 8

   • Inverse: 0, 5, 9, 2, 10, 7, 3, 1, 11, 8, 6, 4

2. Lydian Chromatic:

   • Prime: 0, 7, 4, 11, 2, 6, 9, 1, 3, 5, 8, 10

   • Inverse: 0, 5, 8, 1, 10, 6, 9, 11, 9, 7, 4, 2

3. Locrian Chromatic:

   • Prime: 0, 6, 3, 10, 1, 5, 8, 2, 4, 7, 9, 11

   • Inverse: 0, 6, 9, 2, 11, 7, 4, 10, 8, 5, 3, 1

4. Ionian Chromatic:

   • Prime: 0, 7, 4, 11, 2, 5, 9, 1, 3, 6, 8, 10

   • Inverse: 0, 5, 8, 1, 10, 7, 3, 11, 9, 6, 4, 2

5. Phrygian Chromatic:

   • Prime: 0, 7, 3, 10, 1, 5, 8, 11, 2, 4, 6, 9

   • Inverse: 0, 5, 9, 2, 11, 7, 4, 1, 10, 8, 6, 3

6. Mixolydian Chromatic:

   • Prime: 0, 7, 4, 10, 2, 5, 9, 11, 1, 3, 6, 8

   • Inverse: 0, 5, 8, 2, 10, 7, 3, 1, 11, 9, 6, 4

7. Aeolian Chromatic:

   • Prime: 0, 7, 3, 10, 2, 5, 8, 11, 1, 4, 6, 9

   • Inverse: 0, 5, 9, 2, 10, 7, 4, 1, 11, 8, 6, 3

More Examples (“Exotic” Scales)

Mixolydian ♭6 Chromatic:

• Prime: 0, 7, 4, 10, 2, 5, 8, 11, 1, 3, 6, 9

• Inverse: 0, 5, 8, 2, 10, 7, 4, 1, 11, 9, 6, 3

Lydian ♭7 Chromatic:

• Prime: 0, 7, 4, 10, 2, 6, 9, 1, 3, 5, 8, 11

• Inverse: 0, 5, 8, 2, 10, 6, 3, 11, 9, 7, 4, 1

Phrygian Harmonic Chromatic:

• Prime: 0, 7, 3, 11, 1, 5, 8, 10, 2, 4, 6, 9

• Inverse: 0, 5, 9, 1, 11, 7, 4, 2, 10, 8, 6, 3

Arabian Chromatic:

• Prime: 0, 7, 4, 11, 1, 5, 8, 10, 2, 3, 6, 9

• Inverse: 0, 5, 8, 1, 11, 7, 4, 2, 10, 9, 6, 3

Additional transformations are possible using rotation (modes) and retrograde (direction).

All of the above chromatic arrangements were calculated based on the Hierarchy of Intervals and Stages of Chromatic Harmony.

Exceptions and Contradictions

The goal of this algorithm is to minimize or eliminate conflict, when two or more adjacent notes create dense clusters or harsh dissonance. But as with all rules, certain circumstances may involve contradictions that require exceptions. One example, the Inverse of Prime above, includes 0 and 1 within the basic chord; in this case the Minor 2nd (1) is almost always going to be raised at least one octave to become the ♭9 tension. In general, the low interval limits of arranging dictate when octave displacement must be used to avoid excessive dissonance in the bass register.

Back Filling Large-Interval Gaps

With some chromatic arrangements, it may be desirable to modally rotate one or more of the upper stages in order to lower any chromatic tension to back-fill large-interval gaps that may occur in some cases.

Calculating All Possible Chromatic Arrangements

The total number of possible chromatic arrangements (A) for a given Tonic (0) is calculated as follows:

• 1! = 1 fixed Tonic (0)

• 3! = 6 possible arrangements of the three basic chord notes

• 3! = 6 possible arrangements of the three scale extensions

• 5! = 120 possible arrangements of the five chromatic tensions

A = 1! × 3! × 3! × 5! = 1 × 6 × 6 × 120 = 4,320 Possible Arrangements

The Harken Music System

The Harken Music system emphasizes the importance of mathematical foundations in music. By integrating numerical representations of pitches and intervals, it offers a quantifiable approach to harmony. This algorithm aligns seamlessly with the Harken principles by:

• Numerical Precision: Using numerical identifiers for pitches 0 to 11, it allows for precise manipulation and arrangement of notes.

• Mathematical Harmony: The hierarchy of intervals reflects the mathematical relationships between the Tonic (0) and each of 11 intervals (1-11).

• Innovative Conflict Resolution: Octave displacement and interval prioritization mirror the Harken system's focus on resolving dissonance through logical, mathematical methods.

By applying symmetry and transformations to each harmonic layer separately and then recombining them, the algorithm offers a structured yet flexible framework for harmonic arrangement. This method enhances complexity and creativity while maintaining harmonic coherence, reflecting the Harken Music system's dedication to blending mathematical rigor with musical artistry.

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Conclusion: A Harmonious Future

This algorithm represents a significant advancement in automated musical arrangement. By respecting the supremacy of the tonic and meticulously applying a hierarchy of intervals, composers and musicians can create harmonically rich and balanced scales and chords. The integration of the Harken Music system's principles further enhances the algorithm's effectiveness, bridging the gap between mathematical theory and musical artistry.

As music continues to evolve, tools like this algorithm empower creators to explore new harmonic landscapes while staying grounded in foundational principles. Whether you're a seasoned composer or an aspiring musician, embracing these concepts can unlock new dimensions of creativity and expression in your work.

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For a detailed explanation of the high-school-level math used in creating the Harken Music system, please read, “The Future of Music is Math” by Mitch Kahle.

See also: “Harken Music: A New Mathematical Framework for 12-Tone Harmony”, by ChatGPT 4o, and for an even deeper dive into the mathematics, read Factorials and Fractions “Navigating the 12-tone matrix,” by Mitch Kahle.

Please try our open-source proof-of-concept demo* at harkenmusic.com, where you can choose any combination size (from 1 to 12 notes) from the pull-down menu; then select any particular combination from the numbered list, to see and hear all the permutations, reflections (inversions), and rotations (transpositions).

*Note: a desktop or laptop computer and current web browser (Chrome, Safari, etc.) are required for using the demo; however there are no downloads, sign-up, or subscription requirements.

It cannot be overstated that the Harken Music system uses only mathematics and math related programming code. The programming does not include or use any databases, lookup tables, or other sources of data. Everything—nearly 1.2 billion musical combinations and permutations—is calculated on the fly.

This information and related software code are available on Harken Music and Github as open-source software under the standard MIT License set forth below.

Copyright 2024 Mitchell Kahle and Holly J. Huber

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

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