Isocord Theory

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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Introduction to a new theory of harmony building – by Hansen

Introduction

Isocord Theory, and in a wider sense, Palindrome Harmony, 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:

Isocord theory covers plain-symmetric chords, built of superimposed equal intervals. A plain-symmetric chord is an iso-interval chord, or isocord harmony (after the German term "Isointervallakkord", see Lexikon der Harmonielehre by Reinhard Amon, 2005). Note, that plain-symmetric chords are a special case of bi-symmetric chords (see the YC Wiki article Palindrome Harmony).

Example:

• F#-A-C-Eb, in semitone distances F#:3-3-3 (which is a diminshed 7th chord in traditional harmony theory).

An overview of isocord harmony is given in the following graphic.

Isocord building

An isocord is built by superimposing an interval i = 0–12 (measured in semitones) as often as the root tone X = A–G (note names, optionally signed #/b) does not recur. Note, that the term "root tone X" does not imply that X is, functionally seen, a fundamental of the respective chord – it is only the starting tone on which an interval series begins.

The following table shows all isocords of intervals i = 0–12.

The number of different tones n in an isocord is the number of included intervals i + 1 (i.e. n = i + 1). There is no restriction to build isocords with intervals bigger than an octave; however, their usability might be restricted when working with even bigger intervals (cp. the graphic of isocord harmony above).

The number of different isocords in each series is as follows:

An isocord harmony may consist either of all tones n of its respective series (= full isocord) or only of a sample m < n of its tones (= partial isocord). Where musically meaningful, augmenting isocords by doubling of tones is acceptable (i.e. in other registers).

Operations with Isocords

(1) Transposition of Isocords: Transpose all notes of an isocord by an equal interval up or down.

Example:

• C:2-2-2-2-2 –> G:2-2-2-2-2 (= a p-5th up or a p-4th down)

(2) Inversion of Isocords: While retaining its tones, replace each interval by its complement interval.

Example:

• F#:3-3-3 –> Eb:9-9-9 (m-3rds –> M-6ths)

Generally, the following equivalence relations hold for isocord inversion:

(3) Transformation of Isocords: While retaining the tones of an isocord, reorder them into a bi-symmetric series of arbitrary intervals (= palindrome harmony). In transforming an isocord, the original succession of intervals need not be complete (i.e. omitting tones of an isocord is feasible). Furthermore, optional doubling of tones is acceptable, where necessary.

A piano hands-on rule would be: With both hands, build a nested structure of the isocord's constituent interval, until an adequate palindrome harmony results (optional doubling of tones included).

Examples:

• i) 2-2-2 –> 4-6-4 (three M-2nds transformed into a M-3rd + a-4th + M-3rd series)
• ii) C-D-F#-A# –> Bb-C-D-F#-A#-C-D (incomplete M-2nds series gives 2-2-4-4-2-2)
• iii) D-G-C-F –> C-F-G-C-D-G (a C:5-2-5-2-5 palindrome by the hands-on rule)

(4) Transition of Isocords: To proceed from an isocord to any non-symmetric chord – or optionally to some other symmetric chord –, apply the minimal tone-steps rule.

Minimal tone-steps rule: Move on from an isocord to another chord by the minimal tone-steps of 0, 1, or 2 semitones in any direction. Two different sorts of transitions from an isocord 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) 2-2-2-2-2 –> 0-3-4-0-5 (a minor harmony, e.g. a 4-voiced C:3-4-5)
• ii) 2-2-2-2-2 –> 0-4-3-0-5 (a major harmony, e.g. a 4-voiced C:4-3-5)
• iii) 5-5 –> 4-8 (a major harmony without its fifth, e.g. C:4-8)
• iv) 5-5 –> 7-5 (a void harmony, e.g. C:7-5)

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

Examples:

• i) 2-2-2-2-2 –> 0-3-4-0-3 (a minor 7th chord, e.g. a 4-voiced Gm7:3-4-3, which is also a palindrome harmony)
• ii) 2-2-2-2-2 –> 0-4-3-0-3 (a dominant 7th chord, e.g. a 4-voiced G7:4-3-3)
• iii) 2-2-2-2-2 –> 0-4-0-4-3 (a jazz slash chord, e.g. E/C = C:4-4-3)
• iv) 2-2-2-2-2 –> 0-5-0-4-2 (a Messiaen mode-2 harmonization, e.g. G-C-E-F#)

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

Discussion

The symmetry property of isocord harmony is essential with respect to the sonority of a sound, depending on the kind of interval used and the number of different tones included. Insofar is isocord a generic term of the iso-interval structure of a sound as well as a synoptic term of discords (dissonant sounds) and concords (consonant sounds). Isocords built on primes and octaves (0 & 12) are consonant, built on minor seconds through major seventh (1–11) are dissonant, where the degree of dissonance varies with the density and ambit of isocord harmonies.

Taken together, isocord and palindrome harmony comprises a unified theory of building chords based on symmetry properties for musical sounds – instead of the property of fundamentals of traditional harmony theory.