41. The method of invariants (hidden simplicity) in improvisation and composition.

We must remind here the fundamental philosophy of musical composition and improvisation.

Musical improvisation is not a technical skill that one “learns to do.” It is a natural spontaneous process that occurs first in the imagination. It is often a natural language of the soul, as we have the language of words. But that is why it is understood by people that do not even speak the same language of words

The main goal of musical composition and improvisation is not the output musical piece, but the EXISTENTIAL FUNCTION of the process of creating and listening the musical piece. 

Here is an example of mediative improvisation by Estas Tonne

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

The basic triagle functioning here is the next

                                                1) Feelings identity of the music (poetical symbol: Waters)   

                                                    

    2) Mental identity of  the music                                3) Instrumental-fingers identity               (poetical symbol: Air)                                               of the music

                                                                                       (poetical symbol: Material Solidity)

1)-3) is mainly pactical improvisation

2) -1) is mainly composition 

2)-3) is mainly acquiring skills on the instrument.

We must remark here that the 2), the mental musical theoretical or harmonic identity of the music should be simpler than the 3) the instrumental-fingers identity of the music.

The improvisation should be 60%-80% due to feelings  15%-30% due to simple mental images about the music and 5%-10% only due to hands skills!

Each vertice of the triangle can act as catalyst in  helping for a friction-less flow of relation of the other two vertices but can also if it is not the appropriate it may block the relationship of the other two. This blogs with more abstract mathematical perceptions that are  in 1) may help for a better relation between 2) and 3).   
Classical jazz practitioners give emphasis to  3) , but sometimes lose 2). To find again a better 2) maybe a better 1) is needed. Classical musicians give emphasis to 1), but often miss the 2). To find a better relation between 1) and 2) free improvisational practise is needed, which is a better 3).  And so one.


IN MY APPROACH IN THIS BOOK I FAVOR MIXTURE OF AN IN ADVANCED COMPOSED MUSIC PIECE AND  A LATER IMPROVISATION OVER IT, RATHER THAN A 100% PRIMA-VISTA IMPROVISATION. THE REASON IS OBVIOUS. THERE ARE ADVANTAGES OF MUSICAL COMPOSITION THAT WILL TAKE MORE TIME THAN THE DURATION OF THE MUSICAL PIECE OVER A DIRECT IMPROVISATIONAL CREATION OF IT AS WE LISTEN TO IT. THE FORMER GIVES US THE OPPORTUNITY OF A BETTER QUALITY MUSICAL CREATION AND A BETTER BALANCE OF THE PREVIOUS TRIANGLE OF MUSICAL MENTAL IMAGES, SOUND FEELINGS AND FINGER ACTIONS WHEN WE IMPROVISE LATER ON THE ALREADY COMPOSED MUSICAL PIECE.

The various improvisations practises may have one of the more visible dimensions of knowing better,  the next 4 factors ,by relating them to the feelings.

1) The fretboard of the instrument in relation to the feelings. (Playing almost randomly on the fretboard, guided by the desired feeling of sounds)

2) The melodies in relation to the feelings. (Improvise melodies alone or with a chord progression in the background, or as chord-melodies)

3) The harmony in relation to the feelings. (Improvise with chord progressions)

4) The rhythm in relation to the feelings. (Improvise and learn celebrated or unknown rhythms) 

From all the above types of improvisation the chord-improvisation is the easiest, at least one the basic functions and relations of the chords are understood. The melody improvisation with an already determined  chord progression in  the background, is the second easiest, and the chord-melody improvisation is  the most difficult. 

So in learning improvisation, this  is I believe the simplest and easiest way, in 3 steps

1) Practice at first chord-progressions improvisations. At this stage, the complexity is only the number of chords in the chord progression , their shapes , positions, repetitions etc. Mainly we start with a predetermined chord progression and we play all possible different positions of the chord, and maybe also alternative permutaions of the chords of the chord progression,  keeping of course the basic harmonic structure invariant as this is that tells the main emotional story of the song. We utilize chord-harping or finger-picking with possible small melodic embellishments, with a free finger of the left hand or freeing a finger from the left hand, and around the chord shape. I consider  that using the 5 fingers of the right hand, by far a supperior and more subtle way to  control the musical sound, than using a single plastic pick. With the finger nails it is as if we have 4 picks! 

2) Then practice melody improvisation, while a chord-progression is played by another musician or the computer. Here we may apply the ways to distribute a melody among the strings as in the post 43. 

3) Finally practice chord-melody improvisation, that is played only by you. This would be  chord-harping simultaneous  with the solo, or alternating , only chord-harping (finger-picking) and then isolated melody playing or it would be genuine chord-melody with one composite chord for each note of the solo. 



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 and mutations of them can be created by keeping invariant an  initial pattern of  interval shifts of a note and initial pattern of sequence of melodic themes after  seeminly random pauses (omittings) of the parts of the fixed pattern



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

In order to create the melody over a chord progression we may proceed as follows.

1)We compose a simplicial sub-melodies one for each part of the song ,  with one note per chord, over the chord progression preferably at a chromatic sequence ascending and descending . 

2) We create moves or waves or cycles for each note of the simplicial sub-melody by sequencing during the chord with two types of notes a fast (usually outside the chord) and a slow of double duration on the notes of the chord again ascending or descending with smaller waves

3) We arrange a continuous sound instrument to play the simplicial sub-melody only and a discrete sound (guitar mandolin etc) to play the full waves melody.

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, (HARMONY) MELODIC MORPHOLOGY (VOICES), RHYTHM (ORDER AND DIMENSION OF RHYTHMS) ETC. 

The invariant is the hidden simplicity (as mental image also) that the improviser keeps within himself while conducting complicated musical actions and providing rather complex type of musical output. Simple invariants can be a single note, an interval, or a sub-chord triad , a scale , a theme of the melody or a chord progression .  But there  are also more abstract invariants like the musical type of a chord, chord relations, the pitch order pattern of a melody without pitch specifications, a rhythmic pattern etc. The theme of a melody is by far a more important invariant than the scale, because a melody must always have a  theme, but may not be inside a single scale always. The theme can  be inside a chord or over a chord transition .The improvisation in this technique makes a group of  transformations (called isomorphisms) in the concrete voicing and musical sounds leaving nevertheless always unchanged the invariant. So the balance of what  changes in almost unpredictable or  random way )  and what does not change (determinism)  , usually at different level of abstractness (changes ar at the concrete level, invariants at a more abstract level) creates a beautiful and interesting result. The invariants  allow the player or composer to proceed keeping in his imagination and feelings something simple (the invariant) while the pluralism is in seemingly randomness of the transformations of the invariant. We proceed with some examples. We may be also utilizing the chord-transitions of the chord progression as rules of transformation of the theme of the melody.

The concept of groups of transformations defined by some invariants that they leave unchanged ,  is very familiar thought abstract mathematics, and a powerful of abstract perception. In mathematics very often (e.g. in geometry but in algebra too) they define structures not through elements and their relations, but through a group of transformations that leave some properties  unchanged (these properties , structures or relations are called invariants of the group of transformations). From this point of view Harmony, either tonal or multi-tonal can be defined though musical transformations (e.g. improvisations) that leave invariant some properties. 

Here is a video where the improvisation is conceived with invariant a simple melodic pattern  (similar to the syntax of phrase with subject verb and object etc).. The  points he suggests are
1) Always use a finite set of melody motives , themes or moves. A theme may consists of the basic 4 melodic moves.  (The theme entity for melodies is so important as the chord in harmony. A theme can  be inside a chord or over a chord transition. Conversely any of the 3 types of chord transitions may define a theme for the melody, thus a finite set of themes for a chord progression)
2) Transform these melodic themes or moves  which will be the invariant of their transformations. The simpler the themes the easier the transformations.
3) Close  by returning to the initial theme. 
The ways to transform a theme are at least the next 5 and combinations of them
1) Translate it in different pitches (within a scale or not changing possibly the pitch distances )
2) Translate in time (repeat it)
3) Invert it in time or change its rhythm (if at the begging is slower and at the end faster it will be now the reverse etc)
4) Invert it or distort it in pitch (Create 1st 2nd 3rd or 4th voice versions, utilizing the chord progression as rules of transformation of the theme, or if it is ascending now it will be descending etc)

5) Change it as morphology  (from a non-waving ascending it may become waving ascending or iso-kratic) . We prefer spikes and scaling as  the main  morphological types, while the waving and isokratic as intermediate bridges. 

6) Increase or decrease the size of steps while ascending or descending (accelerate, decelerate not in time but in pitch movement)

https://www.youtube.com/watch?v=nMApv6DSv4w

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  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. 

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.

Some instructive remarks in the composition of the melody based on the chord progression

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.Similarly ascending melodic moves for  major chords.
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!

Let us see another  example. Le us see at first in the next video  how a chord progression can create a melody.

Here is  a very simpler but also beautiful way to create chord-melodies from chord progressions. The melody is a simple oscillation (waving or butterflying) around the highest note of the chord with intervals of 2nd or 3rd. We may enhance this method by  choosing different inversions of the same chord so that the desired note is always the highest in the chord. 

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

Now to add the concept of invariants as in the post 41. Let us say for convenience that we are doing musical composition rather than improvisation.

We lay down the chord progression R1, R2,...Rn, and we chose let us say one pitch order pattern and also rhythmic pattern P for the melody. Pitch order patterns aay which not is high from which but they do not specify how much and as what interval. So we set the pattern to fit in the first chord R1, e.g. by having only notes of the chord R1,  and symbolize it by P1, then similarly for the rest of the chords P2,P3,....Pn. We laso keep invariant the rhythmic pattern of the P1 etc.  We make sure that the notes of the melody P1,P2,....Pn are higher than the chords. We already have composed a song. We may chose also two or more different patterns , and do the same, by making also sure that the same pattern goes to similar according to some rule chords (e.g. minor or major etc). As patterns have the character of joy and sadness we may correspond the sad pattern to sad chords and the joyful pattern to joyful chords.

Concluding the invariant here is the pitch order pattern. But we may already have other harmonic invariants in the chord progression and the coupling with the appropriate rhythm.

The method of invariants works very well with the harmonic method of composition (see post 9).

TO BE MORE SYSTEMATIC WE MAY FOLLOW THE NEXT STEPS

1) Choice of the chord progression (the cycle of 24 chords may be used here, some rule of chord-relations invariants , together with joy-sadness, tension-release considerations)

2) Choice of an appropriate rhythm (the rhythm also must fit the joy-sadness  considerations)

3) Choosing the lyrics (the meaning of the lyrics must fit the joy-sadness, tension-release considerations)

4) Discovering  a  simplicial sub-melody from the chord progression. (Here improvising with an instrument, or even the computer ,with melodic lines after the chord progression may be helpful. In the simplicial sub-melody the chords enhance the simplicial sub-melody, they are not simplifications of it.)

5) Creating the final melody (that has to fit not only the chord progression but also the meaning of the words in the lyrics).

6) Writing it with appropriate software, and  produce the musical score.

( Optional 

7) create a midi file for this song

8) Create a video for this song )

I apply the method of pitch, rhytmic and harmonic invariants  together with the harmonic method of composition (see post 9). Here is a song composed by me in that way.

https://www.youtube.com/watch?v=vMcPrrxJ-d8

DEFAULT MELODIES  FOR A CHORD PROGRESSION.
Given a  chord progression it is direct how to create a melody that fits the chords, with the following rules
1) During  each chord, the entry note of the simplicial submelody , is the middle note of the chord.
2) During  each chord, the exit note of the simplicial submelody (two notes per chord here), for major chords (including 7nth chords and extensions) is the upper note of the chord, for minor, diminished and augmented chords it is the lower note of the chord.
3) During the chord the melody follows an harmonic theme in one or more octaves span, in other words from notes of the chords, and is walking the chord by a spike, straight scaling or waving (these are parameters for the composer or improviser to choose) from middle and down to up (joy) if the chord is major, or from middle and upper to down (sadness) if it is minor, diminished or augmented. If the chord is simply major or minor we may enhance its harmony by extending it with its upper and lower relatives thus  by an interval of 3rd at the highest note and up , or at the lowest note and lower (in normal position). In other words making it a chord with 6th and/or 7nth.
4) At chord transitions x->y , the melody utilizes a dense melodic move (anxiety), with steps from 1 or 2 semitones, and within a scale (including the chromatic 12-notes scale) from the exit note of x of to the entry note of y , of the simplicial submelody.
5) The harmonic move   lasts longer than the transitional dense melodic move , as the latter  takes less than 30% of the duration of x, and y.

From the rule of local fitness of a melody to a  chord  progression , such a default melody will fit the chord progression.