64. Symbolism of the melodic themes and their harmonic transformations structure in a melody. The algebra of melodic moves-themes. Melodic maths by Max Martin

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 MANY GOOD SOFTWARE PROGRAMS FOR THIS COMPOSITION AND IMPROVISATION LIKE FUGUE MACHINE, YAMAHA MOBILE SEQUENCER, THUMPJAM ETC. ALAO ARPIO  AND ARPEGGIONOME FOR GENERAL ARPEGGIOS ALTERNATED WITH MELODIC IMPROVISATIONS


MELODIC MATHS BY MAX MARTIN AND SYSTEMS OF CREATING MELODIC THEMES AND MUTATING MELODIC THEMES AND RHYTHMS

In the next videos one can see how melodic themes of notes  (but also of chords) and mutations of them plus repetitive combinations of them, can be created by keeping invariant an  initial germ-pattern or  melodic-seed of  interval shifts and pause (GERM PATTERN)  of a note (or chord) or  of  initial pattern of sequence of melodic themes of notes or chords after  seeminly random pauses (omittings) of the parts of the fixed pattern. 

Melodic themes of notes can be considered and created also as repettitive combinations of a small set of interval-steps (pitch transformations) in a scale plus a pause wchich may be called MELODIC GERM . A melodic germ  as basic invariant can give many melodic themes with an internal affinity which can  be considered a system of muttations of melodic themes

Comparing the melody with a speaking language suggests the next correspondence

Let us correspond to each vowel a number of steps inteval shift insidea scale

E.g.

empty space=pause
A=0 step
E= 1 steps
I= 2 steps
O=3 steps
OU=4 steps


Then the content of vowels of any phrase can be translated as a GERM-PATTERN for creating melodic themes as muttaions of this germ-pattern  (and latter also repettitive combinations of them)

https://www.youtube.com/watch?v=7HPkTMYoJnI


https://www.youtube.com/watch?v=sb3e4Mq6y3s


https://www.youtube.com/watch?v=w0-Ljf5gm4A


https://www.youtube.com/watch?v=Fc16Y1gKUDc



https://www.youtube.com/watch?v=w0-Ljf5gm4A

This post should be read together with post 106, about the Melodic Seed of a song, where starting from a small number of melodic themes that are independent (abstract algebraic dependence system or closure system) as far as variation transformations is concerned, from which the full melody of the song is derived by an algebra of transforming variations.

Here is a table of the analogy and correspondence of the levels of the musical language and Speaking languages

MUSICAL LANGUAGE	SPEAKING LANGUAGE
Note	Letter  of the alphabet
Interval (3 elementary melodic moves)	Syllables
Melodic moves and  themes (5 basic  melodic patterns, 4 basic transformations of melodic themes.)	Words (corresponding to the simple melodic moves) that make a simple proposition (subject verb object, that correspond to the melodic themes)
Chords duration may contain many musical themes	Sentences from a point to  a next point , that may contain many simple propositions

In this post we introduce special symbolism for the basic themes, their rhythm and their transformations structure in  melodies (Time and pitch inversions, time and pitch translations 1st 2nd 3rd 4th voices versions  of them inside or outside a scale , morphological type transformation, and harmonic relations) . This is very useful in musical composition and improvisation as introduces simplicity in the complexity of a melody. It is so important in improvising melodies as is important to write the chords in a  chord progression. The technique of this symbolism utilizes the more simple , mathematical exact and compact way of musical writing as presented  in the post 7.
As we have defined them in this book, the 3-elementary pitch moves are like the 3-elementary particles of electron , neutron and proton, the 4-basic melodic moves are like types of atoms, the melodic themes are like molecules, and a melody is like a material mixture of molecules. The symbolism we describe is not only like the chemical symbolism of long organic molecules. The only reason we introduce this symbolism is for purposes of simplifying composition and playing and  so as to have simplified perception of the structure of the melody in a stratified-structure and the 3 basic  harmonic transformations  of its themes (see post 30) . In this way we can easily visualize how the harmony of the chord progression of the song contributes in the structure of the melody.
The general shape of such structure is a follows. If a1, a2, a3, a4, are the themes of the melody, and f1,f2,f3,f4,f5 some of the transformations of the theme, then such a symbolism would have a shape for the melody as follows a1a2a4f1(a1)f2(a1)f3(a1)f4(a2)a3a1f3(a4) etc.........Nevertheless each of the  ai i=1,2,3,4 can be further decomposed in the way a theme is created from the 4 basic melodic moves  itch moves, andeach fi i=1,2,3,4 can be decomposed in the 5 basic theme transformations. In this way writing contrapuntal fugue, like those of J.S. Bach  becomes a lot easier.

The themes of a melody consist of a plot or sequence of the 4 basic moves (see post 59)  which by itself says an emotional story without the help of the harmony. If we have (as here we assume we do) an underlying chord progression, then utilizing almost all the notes of the chords and one theme for each of the 3-harmonic-types of chord transitions , we may define the set of themes of the melody in easy way. Alternatively we may define a theme for each type of emotion, sad, joy, anxiety or serenity, or a theme for each type of chord respectively minor (sad) major (happy), 7nth or diminished or augmented (anxiety) and r5 (serenity.) The chord progression serves as a way to transform and make variations of the themes. The notes of the simplicial submelody are the centers of the melody that sound longer and are usually the tops and bottoms of the 4 basic melodic moves that create the themes of the melody but also the notes of the underlying chord. 

Summarizing in  a simplistic way the correspondence of melodic pitch dynamics and the 4-basic emotions in music (joy, sadness, anxiety, serenity) we have 

1) Up pitch, moves correspond to joy

2) Down pitch moves to sadness

3) Small pitch intervals of 1 or 2 semitones (chromatic or interval of 2nd) correspond to anxiety

4) Large pitch intervals (e.g. 4th, 5th octave etc) correspond to harmony and serenity. 

Moving in the melody from intervals of 1,2 semitones to 7 (5th) , 12(octave) is moving from anxiety and stress to joy and serenity.

6) Ascending with larger steps that those of descending indicates favor of joy

7) Accelerating ascending indicates more joy, while decelerating ascending less joy. The converse with descending.

 A single melodic theme has a simple emotional meaning and this is a simple interplay or move inside the duality of emotions (positive-negative emotions).

A melodic  theme-move, can easily have three factors that characterize it

1) If it is sad (-) or joyful (+) (we may call it minor or major  melodic move, although its underground chords sometimes , rarely  may be a  major or a minor chord respectively).

2) Its melodic density (see the 4 melodic speeds or densities, chromatic, diatonic, middle harmonic and high harmonic in post 68)

4) Its range as an interval (this is related somehow by inequality to the density as in 2). melodic theme-moves that their range is more than one octave are special in stressing the nature of being sad or joyful. 

These three parameters still do not define the melodic move-theme even if we know its first note. As we see melodic theme-moves are much more complicated than 3 or 4 notes chords! When creating a melody through melodic theme-moves, ideas similar to those that structure a good chord progression may apply.

We may device a symbolism for a melodic theme move based on the above three factors as follows An1Bn2(-)(x) or An1Bn2(+)(x) where An1 is the first note and Bn2 the last note of the move (n1 n2 denote the piano octave of it) and a minus - or plus + sign if its is sad (minor)  or joyful (major) and (x)=1,2,3,4 denotes the dominating density of it is chromatic x=1, if it is diatonic x=2, if it is middle harmonic x=3 and high harmonic x=4  (see post 68)  e.g. G5A4(-)(2) . In this way we write the dynamics of he melody as a theme-progression ,much like a chord progression. 

The 4 basic transformations of them are

(we should remark that such transformations may be interpreted not only in one pitch dimension of the scale, but in modern digital musical instruments with 2-dimesional scales layouts (like Musix (see post 12 and post 312) e.g. horizontally by 2nds andvertically by 3rds) we may have 2-dimensional interpretaion of the transformations or melodic themes variations.  In 2-dimensional interprtation except of these 4 we may also have rotations! )

1) The translation (either with intervals of 2nd , (or diatonic density) or intervals of 3rd (or middle harmonic density) or  of intervals of 4th or 5th (or high harmonic density)). Translations with intervals of 3rd, may applied without changing the underlying chord, or changing it to a relative chord. Translations with intervals of 4th and 5ths, occur when the underlying chords are in resolution-relation that is successive chords on the wheel of 4ths. Translation by one semitone or chromatic translation may happen in the cases where the underlying chords are in resolution relation (successive chords on the wheel of 4ths) and the first is a dominant 7th chord, or when the underlying chords also have roots at  distance of one semitone. 

2) The melodic density change , density contraction or expansion (see post 68 and 78) . Often it is neither expansion neither contraction but rotation in the sense of stationary waving like an harping in a chord.

3) The inversion where the ascending pitch move becomes descending.

4) Rhythm transformation (which may vary)

More instructive remarks in creating the final melody based on the chords are the next.

1) In the part of the chord progression with minor chords, utilize descending melodic moves so that sadness from melody and sadness from harmony fit
2) In the sad melody parts of the melody (and minor chords) utilize rhythmic patterns that start with faster notes and end with slower notes, and the reverse for the happy part (and major chords).
3) In a triad or 7 nth 4-notes chord the most characteristic notes are the middle 2nd note (in 1-3-5 interval notation  is the 3) and the 7 nth (if it exists). So for the anxiety part of the melodic moves we may utilize 1-semitone trills around these two notes, or waving with 1 or 2 semitones steps and notes outside the chord in the interval of minor 3rd (3 semitones) of the chord. Alternatively instead of trill or small amplitude waves we may utilize chromatic monotone scaling by steps of 1 semitone , or scaling with steps by intervals of 2nd of the scale,  that go from these previous notes of the chord to the same such notes in the next octave. But always make sure that the notes of the chord sound in the average longer, than the notes of these anxiety transition moves with notes outside the chord. 
4) Alternate up (happy) and down (sad) pitch moves , or chromatic moves (anxiety), with harmonic (on chord notes) moves (serenity-harmony).
5) Utilize at least 2 octaves, or even 3 for the melodic moves repeating the notes of the underlying chord on the next octaves , so there is sufficient space for melodic moves, to express with sufficiency the emotions.
6) For the duality of emotions anxiety-serenity, it may be utilized also harmonic waves or monotone scaling over 2 octaves at least,  on the notes of the chord, but also chromatic trill wave over the notes of this wave or scaling (modulated wave on wave or move) and then return to the pure harmonic wave or scaling on the notes of the chord.

7) A chromatic wave by 1-semitones steps or all notes of the scale (steps by intervals of 2nd) that goes up and down at least 2 octaves, corresponds to a chord sub-progression of the song , of our choice that utilizes almost all the chords of the scale!

After the chord progression and simplicial submelody we chose, 
THE DEFINITION OF MELODIC BRIDGES THAN LINK TWO SUCCESSIVE CHORDS BETWEEN THEM AND START AND END AT THE NOTES OF  THE SIMPLICIAL SUBMELODY.

1) WHICH CHORD-TRANSITIONS (PAIRS OF CHORDS) WILL HAVE A MELODIC BRIDGE! (Usually the chord-trasnitions that are in resolutional relation, or resolutional-like relation)

2) THEN WHICH BRIDGES WILL BE ISOMORPHIC IN PITCH AND RHYTHMIC DYNAMIC SHAPE AND WHICH DIFFERENT, DEFINING THEREFORE A PARTITIONING IN THE BRIDGES.

3) THEN IF IN EACH EQUIVALENCE CLASS OF  ISOMORPHIC MELODIC BRIDGES IN THIS PARTITIONING, THE BRIDGES ARE  EVENTUALLY ASCENDING OR DESCENDING (This besides the emotional significance, determines also where to play the chord in one of the 3 neighborhoods of the fretboard)

4) FINALLY  HOW IN EACH EQUIVALENCE CLASS OF  ISOMORPHIC MELODIC BRIDGES IN THE PARTITIONING, THE COMPLICATED PITCH DYNAMIC SHAPE  OR WAVING AND RHYTHM WILL BE AS A REPETITION  OF SUCH PATTERNS OF PREVIOUS ISOMORPHIC MELODIC BRIDGES, OR VARIATION OF  SUCH PATTERNAS S SO NOT TO BE TOO BORING. (This pitch dynamic shape has again a significant emotional meaning)

(This post has not been written completely  yet)