43. The pitch geometry of the parallel strings , and alternative rules to distribute melodies among the strings .

We are used to the visible geometry of the fretboard. But what our imagination needs is a simpler pattern which is the pitch geometry of the parallel strings , so as to guide the fingers how to play, when the sound feeling of the fretboard is not yet adequate, or when it is adequate but we want an innovative kind of sound, that we have not practiced so far.

The pitch geometry of the parallel  strings is  created by taking an horizontal line of all pitches by semitones, and putting the strings as parallel straight line segments on this line and each one according to their pitch range. So in this way each of the pairs 6-5, 5-4 , 4-3 ,2-1 strings are positioned so that the second starts 5 semitone steps after the start of the first, while for the exception pair 3-2, the second starts 4 semitone steps after the start of the first 

In the pitch geometry, any scale has one only horizontal pattern according to their intervals, but different verical pattern according to the string transition rules or realizations among the strings. 

There are some useful observations about alternative rules to play melodies and improvisations on the parallel strings. Normally one has to play from 5 to 9 notes per string before moving to the next string, This comes by assuming that in the average ones utilizes 3.5 octaves that is 3.5*12 semitones=42 semitones, and then if we divide them by the 6 strings 42/6=7 semitones. IN this way the timbre or color of the sound changes from deep bass and hard on the lower strings and close to the upper end of the strings to soprano and vivid-soft of higher strings and in the middle of the fretboard.

Nevertheless of one is using in the melody say only 2.5 octaves, then this would be 2.5*12=30 semitones, and 30/6=5 semitones, thus 3-7 semitones per string. 

The reason one would prefer only 1-3 notes per string, which means fast way to change strings, is when this melody is anchored to one only position of  a chord (e.g. the particular inversion and position of the chord has a better voicing and sound) and thus the melody has to be also around the finger pattern of the chord. 

Here are some rules of distributing a melody or solo among strings together with their logic. Such rules  can be used also when in exercises  improvisation as in post 11, of the 1st  that is when we want to link the inner feeling of the sound of notes with frets of the guitar fretboard.

1) We play a solo on one only  particular string because 

a) we prefer the sound of the particular string among the other strings

b) We want to utilize another neighborhood  string as repeating root (isokratis)

c) It is easier to visualize the solo there, and its range is small covered by one string. e.g. we want to play on D minor scale, and it is convenient to do it on the string D, or on the E major scale, and it is convenient  to do it on the strings E etc. 
e) We want to move with sufficient speed from one position of a chord to another of the other 3 of its positions
f) We want to make waving od amplitude 1 or 2 semitones only

g) Other reasons

2) We play a solo on n only consecutive strings (e.g. n=2,3,4 and e.g. highest 4 strings 4th, 3rd 2nd 1st or lowest 4-strings 6th,5th,4th,3rd) . Possible reasons: 

a) The sound of the strings e.g. being highest or lowest 

b) being the highest 3, or 4 strings because the easier DAE pattern of the chords 

c) n=2 and because solo on one string is easier to visualize and the range of the melody is about 2 octaves requiring thus  only 2 strings. 

d) n=4 and it is the strings 6th 5th, 4th, 3rd because the mutual tuning is uniform (distance by intervals of 4th) which makes string transitions of the solo uniformly easy

e) Other reasons.

3) We play the solo on only  5 frets, with 1 or 2 or 3,rarely 4 notes per string , thus changing often the string, because 
a) we want to stay close to a single chord pattern and position, and not move the hand, that can control easily about 4 frets 

b) we want to stay on a particular area of the fretboard where the sound is soft (middle of the fretboard) or hard (beginning of the fretboard). See also the two almost mirror images of the fretboard each by about 5-frets in post 5.

c) We want to utilize only open chords, so we want to play the melody together with chord-harping (finger-picking) , only on the first 4-5 frets of the fretboard
e) We want to make melodic butterflying with intervals greater than 4 or 5  semitones (intervals of pure 4th or 5th) (e.g 1 or even two octaves) so we want to shift 1 or 2 strings but close to the frets we are. 

f) Other reasons

4) We play the solo with a rule of n-notes per string. 

Reasons

a) We want a particular type of scale patterns among the strings

b)  We want a particular rule to move the left hand as a whole along the fretboard
c) We want a particular speed to move along the fretboard, so as to change the color and softness-hardness of the sound in a regulated way. 
e) Other reasons

5) We play a solo with  sub-scales 1)  v-vi-vii-i and 2) ii-iii-iv in different strings (the Latin numerals are order of steps in a diatonic scale)

a) because the semitones (vii-i) , (iii-iv) in the diatonic scale  have a distance  of exactly a pure 4th (5 semitones), so they are parallel and in the same fret  in two recessive strings tuned by a pure 4th

b) In general we take as  advantage that strings are tuned by  interval of 5 semitones (pure 4th) to translate melodic themes exactly by 5 semitones to the next string with the same shape on the frets.

6) We take as advantage that 3rd and 2nd strings are tuned by a major 3rd, to play parallel second voices on these two strings.


The panacea rule of successive strings transition in improvisation:

We assume here that improvisation along a single string is easier, and we have to talk a bit about a universal rule of transition from one string to another. We  also assume that the two successive strings are tuned in relative pitch of a perfect 4th (5 semitones). Then a universal rule of  transition to the next string is to do it with an interval of minor or major 3rd (3 or 4 semitones) which is for the two strings  a distance of 2 or 1 frets! The reason is that intervals of 3rd are the distance from a first to a second voice, and so when improvising within a diatonic scale, it is true always that a step of either 3 or 4 semitones will be again inside the diatonic scale! We may even try arbitrary one of the two  and if it sounds wrong we slide it quickly to one semitone left or right which will be the correct, while at the same time we create a nice sound shift to a right interval.

A second even more simple and certain  panacea rule of successive strings transition in improvisation is the next. The idea behind it is to play the diatonic scale along a single string , till the step which is of one semitone (e.g. E-F in the C major diatonic) , and then as the next step is for certain one tone we shift to the next string which will be 3 frets behind. Thus the transition to the next string is only once and always of one tone or 3 frets.

This can be generalized by playing the diatonic scale along a single string, and always we chose to shift to the next string only if he step is of one tone,  which will be 3 frets behind.

Again we do not mention the standard rule, of shifting the left hand one fret to higher notes at the 2nd string. 

THE USUAL 4 WAYS TO WALK INSIDE THE FRETBOARD ARE

1) By knowing patterns of scales

2) By known the shapes of chords, and then walk around the chord shapes

3) By knowing all the names of the notes of all the frets of the fretboard

4) Without any mental image, but simply by the feeling of the desired note, and the feeling-familiarization of the fretboard.

E.g. see https://www.youtube.com/watch?v=d7-ZnzAqt0A

A very useful remark for improvisation of melody within a particular chord is the next.
Suppose we are at a note y1 of the melody which fits the underlying chord with notes x1x2x3 (whatever that may mean), then depending on the particular position of y1 relative to the x1x2x3, a shift by an interval of 3rd, 4th, 5th, and 6th wil lead to a note y2 that will again fit the chord!. This is because the relative positions of the notes x1x2x3 of the chord are intervals of major, minor 3rd and pure 5th, and their complementary intervals relative to the octave are minor or major 6th, and pure 4th 

We may apply the previous to the next

LATIN JAZZ AND HARMONIC BUTTERFLYING 

This butterflying is very often utilizing intervals of 3rds (3 or 4 semitones) and 4ths (5 semitones) and their complementary (6th,  8 and 9 semitones and 5th, 7 semitones when changing octave too),   thus it is ascending or descending chords (chord-scales or chord-arpeggios  , that is why it is called harmonic butterflying) and it is thus chord-harping too, but it involves also intervals of 2nd (1 or 2 semitones) which correspond to chord transitions. A hidden simplicity or invariant in this  butterflying is obviously the underlying chord.  This butterflying maybe of  waving type of melodic move but the amplitudes of the waves may be intervals of 3rds (3 or 4 semitones) and 4ths (5 semitones), instead of intervals of 1 or 2 semitones as in eastern folk music butterflying. And it can be of course of non-waving and monotone scaling type of melodic move . Obviously this butterflying prefers changing strings tuned by 4ths, rather than moving along a single string as in the Greek Bouzouki butterflying.

(The post has not been written yet completely)

42. REGULAR RECURSIVE SEQUENCES OF NOTES (or rule of modulations) versus scales in musical composition and improvisation

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. 

We are accustomed to accept the concept of scale as a panacea in musical theory. But the practice of musical composition and improvisation, proves that the concept of scale has a serious disadvantage:

The scale is defining its identity with notes only inside an octave , and then it is assumed to repeat in higher or lower octaves. In other words it has a single rule of repeating higher or lower than an octave, which may not be the desired rule in proceeding to higher or lower octaves in improvisation or composition. But often in improvisation ,we follow a rule to create a sequence of notes ascending or descending, which span more than one octave  and which locally (that is a few notes back or a few notes forward) have great harmonic regularity, but when reducing all the notes inside one octave, this harmonic regularity is lost. From the point of view of classical musical theory, this would be a sequence of modulations. But the concept of sequence of modulations is redundantly complicated as it involves a  sequence of scales, and thus most often a whole set of notes not at all appearing and relevant while playing the regular sequence in the improvisation. The regular sequence on the other hand is a real sequence of notes locally harmonic, and with symmetric rule of repetition. 
We must notice here that a musical scale by octaves in the classical sense is also a regular recursive sequence of notes. But a regular recursive sequence of notes, in general is not a musical scale in the classical sense as it is a more general concept. 
For example repeating a "tetrachord" C, D,F,G by intervals of 5ths is a regular recursive sequence of notes (and rule of modulations) but it is not a scale in the classical sense repeated by octaves. 
Another example is the alternation of minor and major intervals of 3rd (in semitones it is 3-4-3-4-3-4- etc 24 times in total). It is the basis in defining the 24-cycle of chords as in post 34. This long (generalized ) scale is also a rule of modulations which is by increasing by one sharp the number of sharps of the previous scale (F, C, G, D, A, E, B etc)

An example is when we play a 2-string major triad on a string in the guitar and then repeating it in the next string(an interval of 4th higher) and so one, but going also backwards and utilize in melodic improvisation all the last note played. This sequence locally resemble a diatonic scale, but as it goes one with the same rule it alters some notes and in overall it is not a  diatonic scale. But still it can be defined as rule o produce successive modulations of the initial diatonic scale. On the other hand while playing it in the guitar or on an string instrument with strings tuned by 4ths ,such a regular sequence of notes  has a very symmetric rule of playing it. This regular recursive sequence of notes is in fact easier to play than the rule of a diatonic scale and is  locally diatonic while globally harmonically it is richer than a diatonic scale!  Regular recursive sequences of notes are extremely symmetric to play on  string instruments with uniform tuning (that is always two successive strings with the same interval as distance) like mandolin, violin, cello etc. Such regular recursive sequences of notes are easily created using the arpeggios of consecutive chords in the chord cycles of 5ths or 4ths, or the 24 chord cycle that includes the relative chords etc (see post 32 ) etc. Or they can be created by a part of the diatonic scale which is repeated by fixed intervals like 4ths or 5ths. Such pieces of the diatonic scale that are repeated resemble the tetrachords of the ancient musical systems. And e.g. a tetrachord of the diatonic scale within an interval of 5^th repeated by intervals of 5^th , also creates a rule of modulations of diatonic scales. Again we must not confuse the regular recursive sequence of notes with the improvisational melody. Regular recursive sequence of notes play a role similar to scale, and in them improvisational melodies can be played. 

We define also here the concept of  LONG SCALE.
ASCENDING LONG SCALE=it is an ascending  sequence of notes, so that the first time the starting note repeats as name it is the end note of the long scale, but this end note is more than one octave higher than the first note.
The next sequence (successive distance in semitones) (434343434343434343434343) is a long scale  , and when reduced with name=equivalent notes within an octave it is the 12-notes chromatic scale. We call this long scale the 12-notes HARMONIC LONG SCALE , because every 3 consecutive notes make a major or a minor chord!
The HARMONIC LONG SCALE is used to compose the harmonic parts of the melody in songs, as suggested in the present book, where such melodic themes are called HARMONIC THEMES.
Two different long scales may have the same reduction within an octave. E.g. the long scales (successive distance in semitones) (7,7,7,7,7,7,7,7,7,7,7,7) and (5,5,5,5,5,5,5,5,5,5,5,5) have also reduction within an octave the 12-notes chromatic scale, but they are different long scales!

In the next video for example we see how with a 2-string major triad we can create such a regular recursive sequence of notes. In the 2^nd video we see a jazz player suggesting improvisation on all 6 strings, and one or two successive frets which is a similar concept.

 https://www.youtube.com/watch?v=4zS_-gz37JA&list=RD4zS_-gz37JA#t=3

https://www.youtube.com/watch?v=z4SMuD5hATQ

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.

40. Classification of the intervals in the fretboard, and INTERVALS IMPROVISATION.

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

For the ideas discussed in this post see also post 35 about interval-chords

The place of intervals among other entities like notes, melodic themes and chords is the next table. 
We may compare the harmonic method of musical composition, with the way that we shape sentences of meaning in our minds before we choose the exact words to speak  them. We need at first an analogy between the musical language and speaking language.

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 and Dolphin words as in post 101). Chords	Words that make a simple proposition (subject verb object)
Full melody (propositions) . Chords progressions at a duration that may contain many melodic themes (Phrase).	Propositions . Phrases or sentences from a point to  a next point , that may contain many simple propositions

This classification must be studied before , the classification of triads, chords and scales in the fretboard.

When playing melodies in the guitar what matters more is the relative position of notes rather, than the absolute pitch of them. All our western musical system is based on uniformity of translating (transposing) melodies in lower or higher pitch, which makes the distances of the notes a priority in what matters.

We may also classify the intervals to 

VERY SMALL  in semitones  1, 2 THEY ARE PLAYED IN THE SAME STRING USUALLY
SMALL              in semitones 3, 4 THEY ARE PLAYED IN TWO SECLUSIVE  STRINGS                                                            USUALLY
MEDIUM           in semitones  5,6 7 THEY ARE PLAYED IN TWO SECLUSIVE  STRINGS                                                            USUALLY                                                                         
BIG                  in semitones 8,9,10,11, 12 THEY ARE PLAYED USUALLY IN STRINGS,                                                                         THAT ARE TWO    STRINGS  APART  E.G.6TH-                                                                       4TH, 5TH-3RD ETC  
VERY BIG          in semitones >12            THEY ARE PLAYED USUALLY IN STRINGS,                                                                           THREE    STRINGS  APART  E.G.6TH-3RD,                                                                             5TH-2ND ETC      

After the Helmholtz studies (see post 24  ) how good or not good an intervals of notes sounds depends on how many common harmonics they have. The Helmholtz diagram defines also the harmonic hierarchy  of the intervals, where the best sounding is of course the unison (0 semitones) distance then the 2nd best is that of the octave (12 semitones) and the 3rd  best that of the 5th (7 semitones). These two intervals 0-7-12 define also the R5 chord. The next table gives the harmonic order of the intervals.


Why we have 12 notes in music?


https://www.youtube.com/watch?v=IT9CPoe5LnM&t=409s

Alternative to western system is the chinese musical system

https://simple.wikipedia.org/wiki/Chinese_musical_system

In the Helmholtz just intonation or temperament of intervals (see e.g. https://en.wikipedia.org/wiki/Just_intonation   and http://www.phy.mtu.edu/~suits/scales.html ), the next small number rational ratios of frequencies correspond to the next musical intervals.

The remarkable book by Helmholtz On the sensation of tone is online here


https://archive.org/details/onsensationsofto00helmrich

In page 193 of the book of Helmholtz is found also the experimental diagram of how the intervals of just temperament sound in dissonance or not in the human ear. The higher the score on the vertical axis the greater the dissonance. The lower end of  the horizontal x-axis is the note C, and we see also across the horizontal axis the notes e, f, g, a , b    etc.


The smaller a musical interval within an octave the more it signifies sadness, and the more distant, (with maximum a 4th or 5th, as we consider also their inversions) the closer it is to joy.

But the next harmonic score is not about melodies and sadness-joy feelings but more about harmony and stress-serenity feelings.

Interval   Bach scale semitones        Helmholtz scale rational ratio       Harmonic  score
Unison   (Bach scale 0 semitones)          1                                            1/1
Octave   (12 semitones)                        2                                             1/2
5th perfect  (7 semitones)                       3/2                                         1/(3+2)
4th pure     (5 semitones)                       4/3                                          1/(4+3)
6th major    (9 semitones)                       5/3                                          1/(5+3)
3rd major    (4 semitones)                       5/4                                          1/(5+4)
3rd minor     (3 semitones)                       6/5                                          1/(6+5)
7th minor      (9 semitones)                       9/5                                          1/(9+5)
2nd major      (2 semitones)                       9/8                                          1/(9+8)
7th major      (10 semitones)                     15/18                                       1/(15+18)
6th minor      (8 semitones)                        25/16                                      1/(25+16)
2nd minor       (1 semitone)                        25/24      (or 9/8)                      1/(25+24)
5th diminished  (6 semitones)                       45/32                                    1/(45+32)





The smaller the numerator and denominator of the rational ratio of the pitches (in the Helmholts scale) the larger the number of harmonics that are common, when two notes sound with this ratio and thus the better and more harmonic , the interval sounds. In the Bach scale this is a bit ruined but not too much (see also post 24, and if the reader understands Greek also post 25, in the .pdf  manuscript page 53).

This harmonic  clasification of intervals goes back to ancient Pythagorean ideas, but also to modern discoveries of the nature of  senses. For example the flavors and smells are good or bad according to if the "chord" of ultra high frequency sonic high frequencies of its molecules, is harmonic or not . About this see for example the next video.

http://www.ted.com/talks/luca_turin_on_the_science_of_scent

See also about the Pythagorean tuning

https://en.wikipedia.org/wiki/Pythagorean_tuning

Therefore the order of the intervals in the above table is also the order of how good their harmony is when the interval sounds is isolation of anything  else.

Notice that the intervals of perfect 4th and 6th sound better than the intervals of 3rd, and the interval of 7th minor better than the interval of 2nd major. The worst of all is the interval of 6 semitone (4th augmented or 5th diminsihed).

Utilizing the harmonic scores in the above table we can define also harmonic scores of the various chords, by adding the scores of all of their intrevals in all possible ways they shape, divided by the number of these shaped intervals. Let us calculate for example the harmonic scores of the major, minor triads , the R5 chord and the some more chords.

The major triad has intervals M3 m3, P5 , so its harmonic score H(R)  is 

H(R)=1/3*([1/(5+6)]+[1/(4+5)]+[1/(3+2)])=([1/11]+[1/9]+[1/5])/3=0.134

The harmonic score ofthe minor is the same:

H(Rm)=1/3*([1/(5+6)]+[1/(4+5)]+[1/(3+2)])=([1/11]+[1/9]+[1/5])/3=0.134

Now here one may ask, why then a major triad sounds more harmonious and joyful than a minor triad which sounds more sad? The answer is that the same harmonic score of the minor 3rd interval (3 semitones) sounds less harmonious when it is in lower frequencies, than when it is in higher frequencies. And this explains the different feeling of the major and minor chord, although they have the same harmonic score! 

The harmonic score of the R5 chord is the highest:

H(R5)=([1/(3+2)]+[1/(4+3)]+[1/2])/3=0.28

The harmonic score of the R7 is the average of the harmonic scores of it intervals that is 

M3, m3, P5, m3, m7, m5

H(M3)=[1/(4+5)]=0.111, H(m3)=([1/(5+6)]=0.090, H(P5)=[1/(3+2)]=0.2, H(m7)=[1/(9+5)]=0.071

 H(m5)=[1/(32+45)]=0.012

H(R7)=1/6*([1/(5+6)]+[1/(4+5)]+[1/(3+2)]+[1/(6+5)]+[1/(9+5)]+[1/(45+32)])=0.095

The harmonic score of the Rmaj7 is better  than that of R7:

It is the average of the harmonic scores of its intervals  M3, m3, P5, M3, P5, M7, and H(M7)=1/(15+18)=0.030

H(Rmaj7)=1/6*([1/(5+6)]+[1/(4+5)]+[1/(3+2)]+[1/(15+18)]+[1/(4+5)]+[1/(3+2)])=0.123

The harmonic score of the Rm7 is less than  that of Rmaj7 but greater than tha of  R7, and it is the average of the harmonic scores of its intervals  M3, m3, P5, m3, P5 ,  m7,

H(Rm7)=1/6*([1/(5+6)]+[1/(4+5)]+[1/(3+2)]+[1/(3+2)]+[1/(6+5)]+[1/(18+15)])=0.120

The harmonic score of the  Rm7b5 is the average of  the harmonic scores of its intervals  m3, m3, m3,m5, m5, m7, 

H(Rm7b5)=1/6*([1/(5+6)]+[1/(6+5)]+[1/(6+5)]+[1/(45+32)]+[1/(9+5)]+[1/(45+32)])=0.06

While the harmonic score of the  Rdim7 is slightly better than that of Rm7b5 and it is the average of the harmonic scores of its intervals  m3, m3, M2,m5, P4, m6, 

H(Rdim7)=1/6*([1/(5+6)]+[1/(6+5)]+[1/(24+25)]+[1/(45+32)]+[1/(16+25)]+[1/(4+3)])=0.063

The harmonic score of the Raug is better than that of Rdim7 ,Rm7b5  and  is the average of the harmonic scores of its intervals  M3, M3, m6,

H(Raug)=([1/(4+5)]+[1/(4+5)]+[1/(25+16)])/3=0.082

We notice also here that the more notes, a chord has the less would be its harmonic score, compared to chords with few only notes

Therefore the harmonic order of the above chords are

R5>R, Rm>Rmaj7> Rm7>R7>Raug>Rm7b5>Rdim7

We must remark here that these harmonic scores refer to the root position of the chord. The invesrions will have different harmonic score! 

The harmonic score of the invesrions of the major and minor triads are the next

H(R, 1st invesion 3-5-1)=1/3*(H(m3)+H(P4)+H(P5))=1/3*([1/(5+6)]+[1/(4+3)]+[1/(3+2)])=0.144
H(R, 2nd invesion 5-1-3)=1/3*(H(M3)+H(P4)+H(P5))=1/3*([1/(4+5)]+[1/(4+3)]+[1/(3+2)])=0.151

In othe words for the major triad , the 2nd  invesrion has the highest harmonic score,while the 1st inversion has also higher score than the root position!

H(Rm, 1st invesion 3-5-1)=1/3*(H(M3)+H(P4)+H(P5))=1/3*([1/(5+6)]+[1/(4+3)]+[1/(3+2)])=0.151
H(Rm, 2nd invesion 5-1-3)=1/3*(H(m3)+H(P4)+H(P5))=1/3*([1/(4+5)]+[1/(4+3)]+[1/(3+2)])=0.144

In other words for the minor triad , the 1st  inversion has the highest harmonic score,while the 2nd inversion has also higher score than the root position!

We may as well calculate the harmonic score of the moves of a melody, and produce an oscillating diagram of it (Melody harmonic score oscillator)


Here is a not irrelevant video on microtonal and polychromatic music

https://www.youtube.com/watch?v=yVKIXCH-5gE



All the intervals,  on the guitar fretboard from the root met in the chords R, Rm, R7, Rmaj7, Rm7, Rdim Raug  in other words the m2=1,M2=2,m3=3,M3=4,P4=5,P5=7,M6=9,m7=10,M7=11.

Minor 2nd interval=1 semitone





Major 2nd interval=2 semitones






Minor 3rd intevals= 3 semittones






Major 3rd interval= 4 semitones





Perfect 4th interval =5 semitones




Pute 5th interval=7 semitones





Major 6th interval=9 semitones





Minor 7th interval=10 semitones





Major 7th interval=11 semitones







Summarizing with root on the 6th string the M2, M3, P4, P5 ,M6, M7

See also

https://youtu.be/ooPJNYm299k



https://www.youtube.com/watch?v=nmtYaj59slU

http://www.fretjam.com/guitar-intervals-fretboard.html



FOR INTERVAL IMPROVISATION SEE E.G.

https://www.youtube.com/watch?v=binjvIqAN1s

There are of course more types of interval improvisation , that no root is kept always fixed, but the only rule is that at each time an interval sounds rather than only ne  note.


A very useful remark for improvisation of melody within a particular chord is the next.
Suppose we are at a note y1 of the melody which fits the underlying chord with notes x1x2x3 (whatever that may mean), then depending on the particular position of y1 relative to the x1x2x3, a shift by an interval of 3rd, 4th, 5th, and 6th wil lead to a note y2 that will again fit the chord!. This is because the relative positions of the notes x1x2x3 of the chord are intervals of major, minor 3rd and pure 5th, and their complementary intervals relative to the octave are minor or major 6th, and pure 4th 

SINGING INTERVALS:
https://www.youtube.com/watch?v=Sz-U0X7LBRA