Friday, January 26, 2018

80. The pitch translation homomorphism between melodic themes and underlying chords.

The pitch translation homomorphism between melodic themes and underlying chords.

When the melody is composed from little pieces called melodic themes M1, M2, M3 etc and each one of them or a small number of them (e.g. M1, M2)  , have the same underlying chord C1, then we have a particular simple and interesting relation between the chords C1, C2 , C3 and the melodic themes (M1, M2), (M3, M4) ,...etc. This is not the case when the melodic themes start at one chord and end to the next, that we usually call in the book, as "external melodic Bridges" . We are in the case of "internal melodic Bridges". This relation is based on the pitch translations of the melodic themes and of the chords. Actually this is also a scheme of composition of melodies based on small melodic themes (see post 9), when the chord progression is given or pre-determined.

So let us say that the melodic themes (M1, M2), have underlying chord C1. Then as we have said there are only 3 possible chord-transition relations in a chord progression (see post 30 ): C2 will be either in resolution relation with C1 which means that the rood of C2 is a 4th lower or higher relative to the root of C1, or the root of C2 is an interval of  3rd  away from the root of C1 , or and interval  of 2nd away from the root of C1. Let us symbolize by tr4(), tr3(), tr2() , where tr() is from the word translation, of these three pitch shifts. Then we may also translate the melodic themes similarly
tr4(M1), tr4(M2), or tr3(M1), tr3(M2), or tr2(M1), tr2(M2), Then automatically the new translated melodic themes will have as appropriate underlying chord the C2. Actually in the case of intervals of 3rd or 5th, the melodic themes tr3(M1), tr3(M2)  or tr5(M1), tr5(M2) may as well as appropriate underlying chord the C1 again as the 3rd and 5th of the chord is a pitch translation that leads to a note again inside the chord. This is the reason we called this relation homomorphism and not isomorphic. In mathematics , and correspondence H is call homomorphism relative to some relations R, if the the objects H(x1), H(x2) are in relation R , if the objects x1, x2 are in relation R. Here H(M1)=C1 and H(tr(M2))=C2 and C2=trn(C1) that is are in distance of interval of n (=2,3,4 etc) if tr(M1) and M1 are in distance of interval of n. It may happen that H(x1)=H(x2). But if when x1 is different from x2 then always also H(x1) is different from H(x2) we say that H is an isomorphism. Here because it may happen that H(M1)=C1, and H(tr(M1))=C1 again the correspondence of melodic themes and chord is not 1-1, that is H is an homomorphism not an isomorphism in general. 
By continuing in this way translating in pitch the initial melodic themes M1, M2 according to the interval shifts of the roots of the chord progression an remaining inside a diatonic scale , we compose a melody (or simplicial sub-melody too, see post 9). Of course in order for the melody not to be too monotonous we may vary also the melodic themes from ascending to descending etc.

Other translations of the melodic themes can be during the same underlying chord, and are obviously of an interval of 3rd.

Now even when we are at external melodic bridges e.g. M1 which starts at underlying chord C1 and ends in underlying chord C2, even then this homomorphism is of use! The way to make it work is to take the range of the melodic theme (usually starting and ending note as simplicial submelody) equal as interval to the interval of the roots of the underlying chords C1, C2. 

Chord progressions that two successive chords  are always either 1) an interval of  4th , that is successive n the wheel of 4ths 2) Relative chords where major turns to minor and vice-versa, thus roots-distance  an interval of 3rd 3) Chromatic relation , in other words the roots differ by a semitone
are best chord progressions for parallel translations of melodic themes by intervals of octave, 4th-5th, 3rd and semitone. 

Here is a video of jazz improvisation which uses this idea. The chord progression id C, F, G, and the of pitch translations of the initial melodic themes are always intervals of 4th or 5th.