Indeed I listen to a lot of Zappa, plus I use Slonimsky’s thesaurus of scales and melodic patterns all the time, (the descending octave run in the beginning of the first link, which is used in several places throughout it, is from the Thesaurus) which Zappa borrowed a lot from as well. There’s a great many similar reference works I use, like this one, D. Creamer’s the “Hidden Symmetry of the 43 Octatonic Scales and Tetrachords”. These works contain not merely scales, but organizational systems for incorporating scales multi-modally, as well as skeletal melodies and patterns any composer can readily expand on into something new. The text on music theory I am writing is a work of the same sort; I came to make a note on some of the organizational principles in it I have utilized.
I’ve been absent on the forum because I’ve been busy. Writing both music and more text for my work on music theory. I wanted to make a note here on an alternate formulation of the L-C scale.
8-tone harmony (the 15th chord) as the unexceedable vertical limit; nonatonics or 9-tone systems as a special case of harmonic motion where modulations involving the 15th chord consolidate the remaining 4 notes of the 12-tone chromatic scale (by superimposing 3-tone and 6-tone patterns) to reach maximal harmonic complexity. (Schoenberg reached 12 tones by randomly including them through an algorithm, so I don’t count any of that as chords or harmony, it’s just random notes. It’s the musical version of deconstructionist politics. I’m looking to incorporate the 12 tones with an actual theory of harmony.)
In earlier posts in this thread I explained how the Lydian-Chromatic scale is constructed using Coleman’s symmetrical motion concept and eliminating duplicated notes. If the duplicates are admitted, a different kind of scale can be created that illustrates several useful points in understanding and putting all of this together, including how hexatonic superimposition works, (superimposing 3-tone and six-tone collections; nonatonics by extension) how the 15th chord is constructed with the 17th overtone, etc.
" You can also reconfigure this tone collection [the Lydian scale and it’s symmetrical mirror] as a meta-scale, (eg. a “non-octave equivalent scale) in the vein of meta-Lydian, where notes are allowed to double as long as they remain functionally independent and you accordingly reduplicate the intervallic patterns of the component tetrachords of the scale across the circle of fourths/fifths, as Slonimsky explains while detailing what he calls the disjunct Lydian polytetrachord. (More precisely, you extend their whole-tone/half-step patterns across the octave instead of identifying the tonic and wrapping the scale around at the root, progressing in this manner ad infinitum. All major scales can be interconnected through their tetrachords in this manner and interlinked as one greater “meta-scale” along the circle of fifths which, after moving across 12 octaves, returns to the initial tone after 84 functionally independent notes, while all minor scales can be similarly interlinked along the circle of fourths.) Doing this, we see that the Lydian-Chromatic scale, in its non-octave equivalent formulation, completes its ascent at C#. C# corresponds to the 17th overtone of the natural harmonic series. Interestingly, it is at the 17th overtone that the tonal system reaches its own final ascent as the microtonal gamut departs from the natural series. When this 17th overtone is conceived in a tertian chordal structure and superimposed on the 13th chord, it creates the 8-tone 15th chord of Ubieta’s system, (C# finds its place at the 15th scale degree of the Lydian-Chromatic scale) which would designate the limit of both the diatonic schema itself and all current theory of harmony, admitting the existence of nonatonics as more a theory of cyclic patterns useful in the construction of modulations involving the 15th chord than a theory of harmony itself,- a collection of modulations which, in sum, would consolidate the remaining 4 tones of the chromatic scale via harmonic motion. In fact, nonatonic cycles, (3-tone a la. the pattern used in Giant Steps, 6-tone a la. Scriabin’s Prometheus, 9-tone a la. Stravinsky, etc.) as expressions of a certain mystical ambitus latent in the diverse labors of the harmonic maximalists toward the achievement of 12-tone, total harmonic saturation, or what Scriabin called the Chord of the Pleroma, were used to traverse the chromatic spectrum long before Slonimsky’s systematic treatment of inter/infrapolation or Coltrane. For example, following the exegesis of the ‘gnostic impulse’ in “Music in the Early Twentieth Century”, if we analyze the three-note aggregate harmony that grounds the Rite of Spring, we find that it superimposes perfect fourths and tritones upward from a low-C; since a tritone equals a perfect fourth plus a semitone, when you extend this series over the octave, (just as the meta-scales are constructed over the octave by extending the step-wise patterns of their constituent tetrachords outside the respective key) as implied by Stravinsky’s utilization of the Rite-chord, you drive a fourths-fifths circular progression alternating by semitones, (the 12-tone chromatic scale) thereby running the gamut of both intervallic circles (the ascent by fifths corresponding, in Neo-Riemannian language, to major tonality, and, by fourths, to minor or “Plagal” tonality) and exhausting the chromatic spectrum, achieving the same 12-tone integration sought for by Schoenberg through a series of modulations without relying on mathematical processes to artificially expand through the 12 tones randomly, while for this reason also preserving the intrinsic ambiguity of the major-minor regions (telluric and universal gravity) which the 15th chord recapitulates as a distillation of musical impressionism and serialism rejects in its pathological deconstructionist trend toward a kind of musical antihumanism. Fittingly, the completion of this nonatonic cycle and the realization of its “unexceedable limit” sent Ives on his journey toward the microtones beyond the 17th degree, while, in contrast, Scriabin was sent more deeply inward in a journey toward the underlying organizational principle we have here indicated by the over-undertones, their spirognomic polarities, and the influence of a super-telluric gravity on the farthest vertical sonority available to us, (without resorting to microtones) that is, the most highly extended possible harmonic construct, that being the 15th chord. While Scriabin could not complete this later task, it should be clear that it is the one that has been taken up here as well. Perhaps microtonal music may be called for one day, but not until this later task has been accomplished.”