1st, 2nd 3rds CHROMATIC VECTORS IN MELODIC AND HARMONIC RELATION AS STRUCTURE OF THE THEMES OF A MELODY.
1st, 2nd 3rds CHROMATIC VECTORS (denoted by e1,e2,e3) are simply very short melodic themes that are achromatic move of a chromatic internal of 2nd and of 1 ot 2 or 3 notes either inside the 7-notes diatonic scale or inside the chromatic 12-notes scale.They have a fixed rhythmic pattern e.g. (a,a, g# a)
Such melodic themes are the smallest and shortest span in the scale. Each such triplet melodic theme can substitute a note of the simplicial sub-melody.
But there are 2 more categories as follows which are essentially arpeggios!.
1st, 2nd 3rds MELODIC VECTORS (denoted by m1,m2,m3) are simply very short melodic themes that are melodic move of an internal of 3rd or 6th and of 1 ot 2 or 3 notes either inside the 7-notes diatonic scale or inside the chromatic 12-notes scale.
1st, 2nd 3rds HARMONIC VECTORS (denoted by h1,h2,h3) are simply very short melodic themes that are melodic move of an internal of 4th or 5th or 8th and of 1 ot 2 or 3 notes either inside the 7-notes diatonic scale or inside the chromatic 12-notes scale.
Furthermore there are 3 more hybrid categories of up to 3-notes melodic themes which combine the chromatic with the melodic and harmonic
1st, 2nd 3rds CHROMATIC YODELING VECTORS (denoted by cy1,cy2,cy3) are simply very short melodic themes that are melodic move of an internal of 2nd and an interval 8th (octave) or higher and of 1 ot 2 or 3 notes either inside the 7-notes diatonic scale or inside the chromatic 12-notes scale.
More general we may have
1st, 2nd 3rds CHROMATIC-MELODIC OR CHROMATIC-HARMONIC OR CHROMATIC YODELING VECTORS (denoted by cm1,cm2,cm3 and ch1,ch2,ch3 and cy1,cy2,cy3) and are simply very short melodic themes that are chromatic-melodic moves or chromatic-harmonic moves or chromatic-yodeling of 1 or 2 or 3 notes where the chromatic part are either inside the 7-notes diatonic scale or inside the chromatic 12-notes scale.
The division of e1,e2,e3 versus (m1 m2 m3), (h1 h2 h3) is also known in melodies are tail-arpeggio division of a melodic them.
Each such triplet melodic theme can substitute a note of the simplicial sub-melody or the 2 notes in a 2 notes per chord simplicial sub-melody
There is also one more hybrid combination of melodic-harmonic which is essentially an arpeggio.
When the variation shift are melodic (intervals of 3rd or 6th) and harmonic (intervals of 4th or 5th) such shift-variations can be within a single chord or by chord transition. Thus we may have high density of melodic notes per chord or lowest density of melodic notes per chord. In other words the variational shifts that are optimal for more complicated melodies within a single chord is the melodic shift (by intervals of 3rds or 6ths) . In Greek folk Buzuki music this was repeatedly implemented by the Greek folk composer Zambetas.
Musical instruments like the hang and steel pan or the African Kora harp or the small Kalimba but also the harmonica and zamponia pan flute have such arrangements of the notes that favor triplet melodic themes as the latter CHROMATIC-MELODIC OR CHROMATIC-HARMONIC VECTORS cm1,cm2,cm3 and ch1,ch2,ch3 .
It seems that the optimal number of (melodic ) variations of the small melodic theme within a chord is 2- or 3.
This is a matter of combination of the combinatoric topology of a part of the melody with the combinatoric topology of its underlying chord. A very important concept in the composition of melodies and underlying harmony for which very little has been written!
We may make the metaphor that the melodic triplets (e1, e2, e3) (m1, m2, m3), (h1,h2,h3) , (cm1, cm2, cm3) , (ch1,ch2,ch3) are the electrons neutrons and protons or elements of a period system and that chromatic , melodic harmonic relations are the 3 bonds that make them in to "chemical compounds" or final complex melodic themes or melodies.
E.g. Here 2nds chromatic vectors
In C major (a,a g#,a)->(harmonic transition by 5th) (f,eeee) chord a-c-e
Then again (a,a g#,a)-> (melodic transition by 6th) (g,f fff) chord f-a-c
Or
In C major (a,a g#,a)->(harmonic transition by 5th) (f,eee) chord a-c-e
Then (b,b a#,b)-> (harmonic transition by 5th) (g,f fff) chord b-d-f
A 3rds interval chromatic vector can be half in the chromatic 12-notes scale an half in the diatonic scale and when shifted melodically or harmonically it can preserve its half-structure giving rise to simultaneous transposition and modulation. This adds blue-chromatic notes in the 7-notes diatonic scale.
Other types of melodic themes over 2 chords in the 15-syllables pattern e.g. G choprd 6 times d& 2 times and D7 6 times G 2 times
Are the next
1) Ascending or descending sequences with folding
2) Cycles of 4 or 5 notes part of the scale
3) Minimalist 2- o3 notes per chord in chromatic sequence
4) Minimalist Tail-arpeggio themes for ukulele, dobro guitar , trombone or overtone tuned bowed strings.
5) super fast and complicated sequences retarded at the end or start and change of the chord.
