Palindrome Harmony

DISCLAIMER: Please note that this article is the personal essay of a single person, and not a broadly accepted term or theory. However, it bears on established ideas of 20th century harmony.

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Extension of a new theory of harmony building – by Hansen

Introduction

Palindrome Harmony, and more specifically, Isocord Theory, are basically new ways of chord building: Instead of tertian constructions, chords are built of interval series which are symmetrically ordered. This presentation as a YC Wiki article is an attempt to scheme approaches found as early as in Schoenberg's Harmonielehre (1911) and later in Persichetti's Twentieth-Century Harmony (1961).

Symbols used throughout this article:

Palindrome harmony consists of bi-symmetric chords, built of arbitrary intervals, superimposed in such a way that the interval series is the same in both directions, as seen from below and from top-down. Thus a bi-symmetric chord is a palindrome harmony (after the mathematical/linguistic term for numbers/words which are back and forth the same – Persichetti calls this "mirror writing", which can be done vertically or horizontally).

Example:

• C#-D-E-G-A-Bb, in semitone distances C#:1-2-3-2-1.

A special case of bi-symmetric chords are plain-symmetric chords of equal intervals (see the YC Wiki article Isocord Theory).

Palindrome Building

A palindrome harmony can be built either by transformation of isocords or by rules of thumb such as discussed by YC member DAI (see his respective post).

All in all, this results in another piano hands-on rule: Construct an arbitrary interval structure as your prime form. Then attach its retrograde, in between an axis of symmetry. For an even number of intervals, this axis is empty, otherwise it is any interval (including 0, which would be a prime, to be played by different instruments).

There is no restriction to include intervals which are bigger than an octave; however, their usability might be restricted when working with even bigger intervals (cp. the graphic of isocord harmony in the Isocord Theory article).

Operations with Palindrome Harmonies

(1) Transposition of Palindromes: Transpose all notes of a palindrome chord by an equal interval up or down.

Example:

• C#:1-2-3-2-1 –> F#:1-2-3-2-1 (= a p-4th up or a p-5th down)

(2) Inversion of Palindromes: While retaining the tones of a palindrome harmony, replace each interval by its complement interval.

Examples:

• i) 1-2-3-2-1 –> 11-10-9-10-11 (e.g. B-C-D-F-G-Ab –> Ab-G-F-D-C-B)
• ii) 2-3-2-3-2 –> 10-9-10-9-10 (e.g. C-D-F-G-Bb-C –> C-Bb-G-F-D-C)
• iii) 2-4-4-2 –> 10-8-8-10 (e.g. Ab-Bb-D-F#-G# –> G#-F#-D-Bb-Ab)

(3) Transformation of Palindromes: While retaining its tones, reorder a palindrome harmony into a new bi-symmetric series of intervals. Where applicable, tone doubling and octave-stretching of intervals is acceptable.

Examples:

• i) C:2-3-2 –> C:5-9-5, D:5-5-5, D:10-7-10, F:2-5-2, F:7-7-7, G:7-3-7, G:10-9-10
• ii) C:5-2-5 –> C:7-10-7, F:7-0-7, F:7-12-7, G:5-0-5, G:5-12-5
• iii) Ab:2-4-4-2 –> F#:2-6-6-2, F#:14-6-0-6-14, Bb:4-6-6-4, Bb:10-6-0-6-10, Ab:2-4-12-4-2, Ab:2-4-6-6-4-2

Re-Transformation

Any palindrome harmony can be traced back to an isocord by re-transformation – reordering its tones in such a way that an isocord becomes visible, even with holes, which could be filled in with the missing tones.

Examples (all the items from above):

• i) 1-2-3-2-1 –> 1-[1]-1-[1-1]-1-[1]-1-1 (e.g. the chromatic scale of 10 tones)
• ii) 2-3-2 –> 5-5-5 (e.g. C-D-F-G –> D-G-C-F quartal isocord)
• iii) 3-4-3 –> 5-10-5 –> 5-[5-5]-5 (e.g. D-F-A-C –> A-D-[G]-C-F quartal isocord)
• iv) 5-2-5 –> 5-[0]-5 (e.g. C-F-G-C –> G-C-[C]-F quartal isocord)
• v) 2-3-2-3-2 –> quartal isocord
• vi) 2-4-4-2 –> whole-tone isocord

(4) Transition of Palindromes: To proceed from a palindrome harmony to any non-symmetric chord – or optionally to another symmetric chord –, apply the minimal tone-steps rule.

Minimal tone-steps rule: Move on from a palindrome to another chord by minimal tone-steps of 0, 1, or 2 semitones in any direction. Two different sorts of transitions from a palindrome harmony are possible:

Concords: Consonant harmonies of major and minor chords (with or without their fifth), void harmonies (chords without the triad's critical third), and plain unisons (primes, octaves). Thus all consonant chords of traditional harmony theory are achievable in a one-step transition ("resolution").

Examples:

• i) 2-3-3-2 –> 0-3-4-5 (a minor harmony)
• ii) 2-3-3-2 –> 0-4-3-5 (a major harmony)
• iii) 2-3-3-2 –> 0-7-0-5 (a void harmony)

Discords: Any dissonant chord of common practice harmony and modern composition practice, including jazz harmony.

Examples:

• i) 2-3-3-2 –> 0-4-3-3 (a dominant 7th chord)
• ii) 2-3-3-2 –> 0-4-3-5 (a 4-voiced major chord)
• iii) 2-3-3-2 –> 0-4-4-3 (a jazz slash chord, e.g. E/C = C:4-4-3)

Note: The minimal tone-steps rule does not imply that progression of isocord harmony must follow this rule – progression of palindrome harmony is free of any restriction; however, it may be combined with isocord progression.

Some Interesting Special Cases

(1) Any interval repeated in its next octave register (up or down) results in a palindrome harmony – of 2 different tones only, however.

Examples (complete list):

• 0-12-0, 1-11-1, 2-10-2, 3-9-3, 4-8-4, 5-7-5, 6-6-6 (= tritone isocord), 7-5-7, 8-4-8, 9-3-9, 10-2-10, 11-1-11, 12-0-12

(2) Palindrome harmonies built of minor and major triads.

Examples:

• C:3-4-5-4-3 (= C-Eb-G-C-E-G, a bitonal palindrome)
• F:4-3-4-3-4 (= F-A-C-E-G-B, a bitonal or tritonal palindrome)

(3) Palindromes of all seventh chords proper to the diatonic major/minor scale.

Examples (in C minor/major):

• C-Eb-G-Bb, C-E-G-B, D-F-A-C, Eb-G-Bb-D, E-G-B-D, F-Ab-C-Eb, F-A-C-E, G-Bb-D-F, Ab-C-Eb-G, A-C-E-G, Bb-D-F-A, B-D-F-Ab

(4) Consonant palindromes (2 tones in 4 voices) – a subset of (1).

Examples (in C minor/major):

• C:12-0-12, Eb:9-3-9, E:8-4-8, F:7-5-7, G:5-7-5, Ab:4-8-4, A:3-9-3, C:0-12-0

(5) Empty strings palindromes.

Examples:

• Violin: G3:7-7-7, Viola: C3:7-7-7, Cello: C2:7-7-7

Discussion

The sonority of palindrome harmonies is more sophisticated than isocords. They are either bi-/polytonal or atonal – or, in the rare special case (4) above, tonally consonant.

In a body, isocord and palindrome harmony is basically a theory of sound by, as I would like to call it, vagrant harmony (to use Schoenberg's term in a somewhat varied sense). Thinking in sounds is therefore a seemingly natural way to compose soundscapes, so to speak, on the basis of the ever varying distinctiveness of isocords and palindromes with respect to the intervals used and the number of tones included.