The project
Paul Hindemith (1895–1963) sought a system that would explain any chord (even dissonant ones and chords not built in thirds) starting from acoustics, without giving up a tonal center. He set it out in Unterweisung im Tonsatz (1937; in English The Craft of Musical Composition). His ideas have not been adopted as “rules,” but they offer very useful tools for managing tension and clarity in non-traditional music.
Series 1: the kinship of the notes
Hindemith orders the 12 notes according to their kinship with a root (derived from harmonics and combination tones). Starting from C:
C – G – F – A – E – Eb – Ab – D – Bb – Db – B – F#
The first notes are the “closest” (G, F), the last the most distant (B, F#: the tritone is the farthest). It serves to choose the tonal centers of a piece and their order. In Ludus Tonalis (1942), a cycle of fugues and interludes for piano, the twelve fugues follow precisely this order of keys: C, G, F, A, E, Eb, Ab, D, Bb, Db, B, F#.
Series 2: the strength of the intervals
The intervals ordered from the most consonant and “harmonically strong” to the weakest:
octave – fifth – fourth – major third – minor sixth – minor third – major sixth – major second – minor seventh – minor second – major seventh – tritone
Each interval (except the tritone) has a root:
| Interval | Root | Inverted interval | Root |
|---|---|---|---|
| fifth | lower note | fourth | upper note |
| major third | lower | minor sixth | upper |
| minor third | lower | major sixth | upper |
| minor seventh | lower | major second | upper |
| major seventh | lower | minor second | upper |
| tritone | indeterminate |
Finding the root of any chord
- Look in the chord for the strongest interval according to Series 2 (first the fifth, then the fourth, then the major third…).
- The root of that interval is the root of the chord.
- If there are several equal intervals, the lowest one wins.
Example: C – F – Bb – Eb (a chord in fourths, from the bottom). It contains no perfect fifths going upward; the strongest interval is therefore the fourth. There are three (C–F, F–Bb, Bb–Eb): the lowest, C–F, wins, and its root is the upper note. The root of the chord is F (read as F7sus4/C without the third).
Chord groups
Hindemith classifies chords into two large groups:
- Group A (without tritone): from the most stable (major and minor triads) to those with seconds and sevenths, down to the indeterminate chords (e.g. in fourths, augmented).
- Group B (with tritone): from chords with a minor seventh/major second (like the dominant seventh) to those with minor seconds and major sevenths, down to the indeterminate ones (diminished).
Each group has numbered subclasses (I–VI) ordered by increasing tension.
Harmonic fluctuation
Harmonic fluctuation (harmonisches Gefälle) is the curve of tension over time, obtained by following the value of the chords according to the classification: a well-built phrase alternates growth and release, as in classical tonality, even if the chords are not triads. The root progression (Stufengang), evaluated with Series 1, guarantees the sense of center.
Why it matters to a producer
- It gives you a criterion for ordering complex chords (clusters, quartal chords, polychords) from the most tense to the most stable.
- It helps you choose a clear bass under ambiguous harmonies: put the Hindemith root in the bass.
- Series 1 is an excellent map for planning the keys of the sections of a long piece or a soundtrack.
Exercise
- Write 6 four-note chords that are not traditional (fourths, seconds, mixed).
- For each one find the root with Series 2 and put it in the bass.
- Order the chords by tension (Group A first, then B) and build a phrase that rises in tension and then falls back.
Related topics
Quartal and quintal harmony, clusters, the harmonic series, Schoenberg (introduction), polytonality.