69 The simplest hidden symmetry of the composition of beautiful melodies of songs

See also post 114.

From the morphological point of view the basic melodic patterns as  we have mention in earlier posts are the next 5

1) Straight scaling up or down (including spikes) in one or more of the melodic speeds (straight sadness or joy). Here the notes of the simplicial submelody are the starting and ending notes.
2) Ascending or descending waving (complex sadness or joy). Here the notes of the simplicial submelody are the starting and ending notes.
3) Flat equilibrium waving (serenity and equilibrium emotion).Here the notes of the simplicial submelody are the upper level and lower level ofthe flat channel.
4) Flat diminishing waving (serenity and diminishing emotions). Here the notes of the simplicial submelody are the starting upper or lower level and h ending note of the diminishing channel
5) Flat expanding waving resolving up or down  (serenity emotions exploding to either sadness or joy). Here the notes of the simplicial submelody are the starting note and the ending note at the upper or lower level of the expanding channel.

Therefore we may start by choosing the morphology of the melody as a  sequence of the above 5 patterns, that say an emotional story e.g.
1) serenity emotions exploding to  sadness (pattern 5)
2) serenity emotions exploding to  joy (pattern 5)
3) Serenity and diminishing emotions(pattern 4)


Another way to put it is the next
We remind at first that except the melody itself we may have   HIGHER ORDER SIMPLICIAL SUBMELODIES. In other words except the 1st simplification of the melody, which is the 1st order simplicial submelody, we may have the 2nd order simplicial submelody, the 3rd order simplicial submelody, each one simpler that its previous. A path of grids from the complexity to simplicity. One of them should correspond of course to the complexity of the chord-progression, that is have one note for each chord of the chord progression. E.g. the starting ending notes of the meloduc themes may be a simplicial submelody while the centers of the melody a higher order simplicial submelody.

Now the best simple  hidden symmetry and simplicity  of the melody that is the subject of this post, is that one of the higher order simplicial submelodies of the melody  has a single or simple sequence 2-3 only,  of the basic 5 melodic patterns! 


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 or themes (5 basic  melodic patterns)	Words
Chords	Sentences
Simplicial submelody	Subject-verb-object

These sequence of the 5 patterns are best realizable with a very simple simplicial sub-melody! For this it helps very much if the simplicial submelody has long lasting notes  (even for more than one chord!) so that each of these patterns is e.g. 3-4 notes of  the simplicial submelody. The first note of the pattern and the last note may serve as notes of the simplicial submelody. See also a more detailed correspondence below in this post

The simplicial submelody with these 5 patterns may serve as a pattern for self-similar (fractal) repetition with shorter duration notes , and shorter duration similar patterns when composing the full melody ( a  method sometimes utilized by Bach) 

Another characteristic of the happy and joyful melodies is to define two notes (or interval) for the simplicial sub-melody for each chord so that in aver all the melody is maximally harmonic (see post 40) and we  may use almost exclusively  the maximum large intervals (within a scale) that exist in the chords of the song. And this would be intervals of 8th, 6th (for triad-chords) , 5th and 4th. In other words we use almost exclusively the maximum harmonic melodic  speed that the chords allow (see post 68). 

This idea of maximum harmonic speed in melodies is also an idea that can give pretty directly improvisation melodies over a chord progression! This is good for happy melodies. It directly defines improvisational beautiful melodies from the chord progression,  because the maximum intervals of a chord are unique or very few for each chord! In fact a single large such interval from each chord can define the melodic-rhythmic pattern for each chord! 

The standard preference is to use 

a1) For  a major chord x1-x2-x3, the 1st x1-3rd x2   notes interval of pure 5th (7 semitones), or the 1st nx1-2nd x2 notes interval of major 3rd (4 semitones)

a2) For  a minor chord x1-x2-x3, the 1st x1-3rd x2   notes interval of pure 5th (7 semitones), or the 1st x1-2nd x2 notes interval of minor 3rd (3 semitones)

a3) For  a dominant 7th and major 7th chord x1-x2-x3-x4, the 1st x1-3rd x2   notes interval of pure 5th (7 semitones), or  the 1st x1-4th x4   notes interval of minor 7th (8 semitones), or of  major 7th (9 semitones). 
An interesting case of simplicial submelody is  the first choice  always  (interval of 5th or 4th). 

Or we may allow this interval of 4th or 5h of each chord sound 2/3 of the time of the chord sounding and 1/3 of the time the other middle x2 note for minor or major  , or 7th note of the 7th chords.
 Still another case is the minimal harmonic simplicial submelody (but always with notes of the chords) where we take always the 2nd choice (the x1-x2 interval of 3rd, or x1-x4 interval of 7th) where this sounds 2/3 of the time and 1/3 of the time the 3rd note of the chord. This simplicial submelody gives emphasis to the character of each chord, that is being minor , major or 7th etc. 

But another more maximal  harmonic  method is based on the next rules

b1) For each chord the simplicial submelody consists of at least two notes one entry and one exit (that may though coincide)

b2) Complementary chords (e.g. Cmajor, Dminor) can transition with intervals of 5 or 7 semitones (e.g. exit note of Cmajor is the c, and entry note of Dminor is the f).

b3) Successive chords in the cycle of 4ths or 5ths, and relative chords have common notes, this the exit note of the first chord and the entry note of the 2nd chord are identical.

b4) If the entry note of the a chord and its exit  note is an interval of minor 3rd  (3 semitones) we may add two more notes during the chord which is twice the 3rd note of the chord, but at one octave distance, and convert the minor 3rd interval to major 3rd (4 semitones) which has higher harmonic score (see post 40). E.g. G7-->C-->E7 , entry of C=g3, exit of C=e2, so we add c2, c3, and the simplicial submelody goes like this g3-c2-c3-e2, duringthe chord C. We converted the minor 3rd interval g-e, to a major 3rd c-e. 

b5) Itis prefered that intervals of 1,2,3,4 semitones are converted to their complemntary of 11,10,9,8 semitones, by changing octave.

The so derived simplicial submelody singles less melody than the chord progression itself!

E.g. forthe Chord progression Am->F->G7->C->G7->C->G7->C->E7->Am, the sumblicial submelody with these rules would be a3-a2a2-f2f2-g3g3-g3g3-g3g3-g3g3-g3g3-c2c3e2e2-e3e3-a3.

This simplicial submelody can be the centers of  full melody over this chord progression

Then correspond this emotional story with the emotional story of the chord progression
E.g. 1) is parallel to  cycle of minor chords like Em, Am, Dm  2) is parallel to a cycle of major chords like
G, C, F, and 3) parallel to  cycle of major chords like A, D, E. (major relative scale A major of A major that were the previous 2 cycles)

If we make sure that the parts with middle and high harmonic speed  of the melody last say more than 70% compared to the parts with chromatic and diatonic speed of the melody, then the correspondence of chords is almost unique and easy! Alternatively  any descending , ascending or waving sequence of notes at diatonic speed such that the odd or even number of them is exactly the notes of the chord (extended probably by 7nth or 6th) and these motes sound e.g. 3 times more than the notes of the est of the scaling is a melody that fits the particular chord! Irish melodies do it often. Notice that e.g. since for the C major scale , both the minor cycle (Em, Am, Dm) and the major cycle (C, G, F) cover all the notes of the scale, corresponding a pattern on the major or minor cycle , is simply the way we start to span in middle harmonic speed the scale. Taking the middle notes as roots of the minor chords in the sequence Dm->Am->Em we get the sequence of the major chords F->C->G , and both triads are in the middle harmonic melodic speed. 

Notice that although a chord may have duration  for many notes of the melody, the above 5 melodic patterns may have duration for many chords of the chord progression. Thus these 5 melodic patterns are intended to by more macroscopic than a chord. By utilizing the inversions of a chord, and shifts in higher or lower octaves, the chords themselves, if they have been determined before the above 5 melodic patterns in the harmonic method of composition, can serve as vectors of move at middle or high harmonic melodic speed, to shape on of the above 5 melodic patterns. 

Of course as the approach of composition in this book, favors the greater simplicity, it is desirable to start composing from the more macroscopic and abstract patterns first to the smaller and more detailed patterns afterwards. Thus we may very well have at first a composition of a sequence of the above 5 melodic patterns, then the chord progression, then the simplicial submelody and  finally the detailed full melody

As an example we may discover exactly such a design in the song tico-tico non fuba by  Zequinha de Abreu in 1917(see https://en.wikipedia.org/wiki/Tico-Tico_no_Fub%C3%A1  and https://www.youtube.com/watch?v=lsMNvmqRdC0 and https://www.youtube.com/watch?v=Vo-OpQS2zdQ) which had huge success internationally during the 20th century. The melody of the song is designed with
1) serenity emotions exploding to  sadness (pattern 5)
2) Serenity and equilibrium emotion (pattern 3)
2) Serenity and diminishing emotions(pattern 4).
The pattern 5 is created in 2-3 variations with combination of chromatic, diatonic and middle harmonic speed over the mainly minor (sadness) chords B7, E, Am, Dm . Then a variation of pattern 3 is repeated parallel to the major (joy) chords G, C, F. And finally, the pattern 4 is created in 1 or 2 variations over the major (joy) chords E, A, D.

Other examples are celebrated musical pieces of Paganini , who loves to design whole pieces over the patterns 2 and 3

The way that patterns 1,2,3,4,5 can be realized with different melodic speeds (see post 68), the appropriate rhythm,  "sketching" or variations of them and over different chord progressions is practically unlimited! But the hidden simplicity of the emotional story of the 5 patterns behind it may always be very simple!

THE BASIC CONCEPT OF MUSICALLY BEAUTIFUL  IN THIS BOOK IS THE CONCEPT OF HIDDEN SIMPLICITY OF  RULES OF PROPORTIONS OF 2 AND 3 , AND IN GENERAL OF VERY SMALL INTEGER NUMBERS IN FREQUENCIES, (INTERVAL HARMONY) MELODIC MORPHOLOGY (VOICES), RHYTHM (ORDER AND DIMENSION OF RHYTHMS) AND EVEN PERCENTAGE OF MAJOR (=2/3)-MINOR(=1/3) CHORDS ETC. 

THE 5 BASIC MELODIC PATTERNS ARE REDUCED TO THE ELEMENTARY 3 MELODIC PATTERS THROUGH THEIR CRITICAL MELODIC INTERVAL!

We explain: The basic melodic patterns are 5 as we mentioned at the beginning of this post. But elsewhere we have mentioned the elementary melodic patterns that re only 3 (the up the down , the isoskratic or plain). Now the way to derive an elementary pattern from a basic pattern is through their critical melodic interval. A melodic interval is of course always  up , down , or the isokratic  plain. The critical melodic interval of the basic melodic pattern is the most characteristic interval of the pattern, that subconsciously the subjective listening would simplify in it the basic pattern. Usually it is the first and last notes of the pattern but not always. As with the critical path in the flow charts of the projects scheduling, the critical interval is usually the longest in the pattern or the one that sounds more time.

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)

1/3-2/3 POETIC MELODIC PATTERNS

These are melodic simple patterns where at the 1/3 of their duration sound transient short notes and 2/3 of their duration a long lasting note. E.g. S1+L,  or S1+S2+L or S1+S2+S3+L where the duration of the short notes S1 or S1+S2 or S1+S2+S3 is 1/2 of the duration of the long note L. Then the short note serve also as transient notes, while the Long note is always a note of the underlying chord. In this way the harmonic part of the melody (L-notes) is always double of the transient-chromatic part of the melody. 
Such melodies are very common on Irish traditional music, but also in Greek Islands traditional music.

 In traditional  poetry the poetic measures can define such melodic patterns. 
The poetic measures e.g. in the ancient and modern Greek poetry are the next 4 
1) Iamviko      S+L   duration(S)=1/2duration(L)
2) Trochaiko  L+S  duration(S)=1/2duration(L)
3) Anapestiko  S1+S2+L   duration(S1+S2)=1/2duration(L)
4) Daktiliko     L+S1+S2  duration(S1+S2)=1/2duration(L)
5) Amphivrachi   S1+L+S2 duration(S1+S2)=1/2duration(L)