Recently, I wrote a computer program to generate all the 2048 possible musical chords (from any given root after collapsing transpositions) in any twelve-note, equal-tempered octave system. Each chord's notes transpose into fair consonance by excluding major and minor seconds, minor ninths, and tritones. The above chords completely cover the inversions of the 351 chords, available after collapsing transpositions and inversions. For more background on why 351, see Wikipedia, Necklace (combinatorics).
I provide details on all the chord necklaces and their inversions, including note names. A readable yet concise and rigorous scheme (that I created) was followed to provide a chord name for each inversion. A goal was matching common practice as much as possible. This provides a useful, searchable inversion chart. One might read a chord's description and play it manually, for instance on a piano.
A provided MIDI file of these chords is more consonant when played on a capable electronic piano than by the usual FM (frequency modulation) synthesis: than for instance on a SoundBlaster computer sound card. RealPlayer shows elapsed time; for the QuickTime player, General MIDI synthesis (which may sound better, depending on your sound card) is available at least on PC's by selecting, 'Plug-in Settings/Audio/Default Music Synthesizer.' For listenability, the (MIDI) chord roots are made to descend, alternately by major and minor thirds.
This is better than all the lists previously available of pitch-class (PC) sets because the process of finding and sorting normal forms is more computer-oriented: it is simply the largest binary number of all the (necklace) rotations. Along with this way of numbering notes, leftward from G, it collects similar-sounding PC sets (especially in their '0-inversion' chords) because this collation first looks at sevenths (F# and F) rather than seconds (Ab and A).
Copyright (c) 2009 Mark D. Blackwell.