279. A CLASSICAL GROUP OF MELODIC THEME VARIATIONS FOR BEAUTIFUL DIATONIC TONAL MUSIC TRANSLATIONS EXPANSIONS,INVERSIONS MUTATIONS

 A  CLASSICAL GROUP OF MELODIC THEME VARIATIONS FOR BEAUTIFUL DIATONIC TONAL MUSIC. TRANSLATIONS , EXPANSIONS, INVERSIONS, MUTATIONS

THE KEY-WORD HERE IN THE 4TH GENERATION DIGITAL MUSIC FOR THE MUSICAL-THEORETIC IDEAS OF THIS   POST (AS FAR AS MORDEN SOFTWARE FOR MUSIC MAKING IS ) IS MELODY-SEQUENCERS 

THE TERM  SEQUENCER MEANS HERE A LOOP OR RHYTHMIC CYCLE OF   A  MELODIC THEME THAT IS VARIATED INTERACTIVELY BY THE USER  IN A MELODIC SEQUENCER.

THERE ARE MANY GOOD SOFTWARE PROGRAMS FOR THIS COMPOSITION AND IMPROVISATION LIKE FUGUE MACHINE, YAMAHA MOBILE SEQUENCER, THUMPJAM ETC

By studying e.g. the art of Vivaldi melodic themes variations (but other folk music too, e.g. Celtic and Irish music), we can directly get the idea of a nice finite group of melodic themes transformations. We use of course the term finite group in the standard mathematical algebraic meaning of a set of transformations closed to composition and inversion. The transformations  are most often  tonal translations (in other words shifting the melody inside the diatonic scale and preserving the nature of intervals as 2nds, 3rds, 4ths, 5ths etc). From the mathematical algebraic point of view such a finite group has a finite set of generators and a  finite set of defining relations for the presentation of the group (https://en.wikipedia.org/wiki/Presentation_of_a_group).

And the group of such transformations are of course 

1) Tonal translations by 3rds (melodic translation) , 5ths or 4ths (harmonic translation) , when we ae inside the accompanying of a single chord (but also in different chords related melodically or harmonically to the first) And by 2nds (chromatically) which requires usually changing the underlying chord too. This tranasformation includes the  Harmonic Complementation (Blue Shaffling). If the instrument is harmonica tuned as a Zamponia panflute (double row of odd-even notes of a diatonic scale, called also cyclic tuning, or melody king tuning) we have an additional variational transformation that we may call complementation (Blue shuffle) . E.g. if the melodic theme has notes 1,1 , 2, 3,3 a complementation of it would be to inverse even with odd numbers e.g. 2,2,3,4,4 etc

2) Tonal inversion . It is an inversion of the pitch order of the notes of the melodic theme which results in melodic theme again inside the initial diatonic scale. Besides the pitch-inversion there is of course also the time-inversion of the melodic theme, which is a different concept.

3) Tonal expansions-contractions: It is when we keep an initial part of the melodic theme fixed while expand or contract the rest of it , but always resulting again inside the initial diatonic scale. E.g. the initial melodic theme over chord A maybe within ione octave and when we translate it to fit the next chord B we also expand it to span 2 octaves. 

 The statistical frequency of such tonal translations is at least 2/3 of the times by intervals of 3rds and 5ths/4ths (melodic and harmonic tonal translations) and at most 1/3 of the time by intervals of 2nds (chromatic tonal translations). A more tolerating quantification would be to require that the statistical frequency of such tonal translations is at least 50% of the times by intervals of 3rds and 5ths/4ths (melodic and harmonic tonal translations) and at most 50% of the time by intervals of 2nds (chromatic tonal translations).

In the formation of such group of variations of melodic themes and in creating such nice melodies and music we utilize of course the closure properties of the diatonic scale in shifts by intervals of 3rds, and 5ths or 4ths. (Which is also a good opportunity for the reader to refresh and re-discover. E.g. For every note of the diatonic scale there is an interval of 3rd and of 5th or 4th such that up or down of the note and away by that interval it is again a note of the diatonic scale). So that such variations create cycle of melodic themes that repeat.

The above group of variations of the melodic theme can be both in the context of tonal music but also in the context of chromatic tonal music as in the post 263

As we mentioned in post 263  the order of the chromatic notes 5#, 4#, 1# , 2#, 6# would give the next order of altering majors to minors or vice versa

3 minor ->3 major or 4 major->4 minor (for 5#) 
2 minor-> 2 major (for 4#)
6 minor -> 6 major (for 1#)
1 major-> 1 minor (for 2#)
5 major-> 5 minor (for 6#)

Boldly speaking, the topological shape of the melodic themes are of three  kinds:
A) Oscillations (arpegios of a chord or vector-chord)
B) Up or down Continuous Moves 
C) Discontinuous  jump (Spike) by an interval of 5th/4th or larger.

THE DIATONIC-GUITAR OR HARMONICA-GUITAR  AS IN POST 90 HAS A DIRECT ADVANTAGE OF APPLYING THE CLASSICAL GROUP OF VARIATIONS OF MELODIC THEMES AS DESCRIBED IN THIS POST  (TONAL TRANSLATIONS BY 3RDS AND 5THS OR 4THS IN AT LEAST 2/3 OF THE CASES AND CHROMATICALL IN AT MOST 1/3 OF THE CASES). THE MELODIC THEME SUCH TONAL  TRANSLATION BY 3RDS OR 5THS/4THS IS SIMPLY SHIFTING THE MELODIC THEME FROM ONE STRING ON THE SAME FRET  VERTICALLY TO  THE ADJACENT STRING ( A 3RD) OR NEXT TO ADJACENT STRING (5TH OR 4TH). (SEE POST 90).

See also about chromatic inversion  (pitch inversion)

https://www.youtube.com/watch?v=0oI2iFrzA0o