110. The 7-notes 2-octaves and 10-notes 3-octaves arpeggio-scale of a chord, as sufficient space for rich melodic order-topological shapes composition.

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When spending time with an improvised melody that harmonically fits a chord the best idea is to have the chord in 4-notes form e.g. like a with 7nth or with 6th, and in the current octave or in the next. Then start the melody at a note of the chord and end it again at a note of a chord in this or the next octave. Since the chord has 4-notes and the scale  7 notes the passing or transient notes are only 3, less than the 4 of the chord, therefore, any such melodic theme fits harmonically to this chord.

Here is an example :

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

109. 2ND HARMONIC ORDER MELODIC THEMES BRIDGING TWO SUCCESSIVE CHORDS AND THE ROLE OF THE HARMONIC AND CHROMATIC SIMPLISTIC SUB-MELODIES

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Melodic themes that span from one chord to the next have more complicated harmony as the underlying harmony is two successive chords compared to a melodic theme that sounds during a single chord. That is why they are called 2nd harmonic order

Methods of creating melodic themes during a single chord sounding have been already described at least in post 141. We had described there that one of the simplest methods is the next:

When spending time with the melody with an underlying chord the best idea is to have the chord in 4-notes form e.g. like a with 7nth or with 6th, and in the current octave or in the next. Then start the melody at a note of the chord and end it again at a note of a chord in this or the next octave. For example, we may compose the melody from 3-notes micro-themes, the first and last inside the chords and the middle possible outside the chords. Since the chord has 4-notes and the scale  7 notes the passing or transient notes are only 3, less than the 4 of the chord, therefore, any such melodic theme fits harmonically to this chord.

Here is an example :

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

Now, this technique can be extended when passing from one chord to a next. Instead of having the one chord on two octaves and moving from the one octave to the next, we have two chords and we start from a note of the first chord so as to end with a note of the 2nd chord, and controlling of course that the passing or transient notes that do not belong to either chord, are less or sound  less time than the notes of the chords..

As we analyzed in post 104, the Harmonic simplicial sub-melody is a kind of extreme maximum distances among successive chords, while the chromatic simplicial sub-melody is a kind of minimum distance among two successive chords. As we mentioned the harmonic simplicial submelody has at most one note per chord, while the chromatic simplicial submelody has at most 2 notes per chord. 

Now methods of creating melodic themes even inside single chord as in post 103, can be based on the harmonic and chromatic sub-melodies. 

For example we may create from the melodic seeds  order-topological pattern or shape of a melodic themes, a realization of them . For a chord this creates two themes one that starts from the left (first) note of the chromatic simplicial submelody and ends to the harmonic simplical submelody note, and a second which starts from the harmonic simplicial submelody note and ends to the right (second) note of the chromatic simplicial submelody. These themes concatenated with a chromatic link of the right and left notes of the chromatic simplicial submelody of two successive chords and  may create a full and dense melody for the given chord progression.

If the duration of the chord is rather limited, then obviously we create one only of the two such melodic themes.

Because of the property of maxima of the harmonic submelody, the melodic theme is somehow long enough and between harmonic intervals. While because of the property of minimum distances of the notes of the chromatic simlpicial submelody, such a melody also links in the shortest and most chromatic way two succesive chords. This creates an oscillation or wave between harmonicity and chromaticity in the melody which is a beautiful form of balance.

Since the interval distance of the notes of the harmonic Simplicial submelody for two successive chords is in general quite variable, the initial melodic seed order-topological shape of the initial melodic theme may or may not be preserved. But even if its preserved we have an homeomorphism variation of the melodic theme from chord o chord instead of a standard mode-translation. In this way the contraction-expansions (dilations or hoemorphisms) of the seed melodic themes is created naturally as in conformance with the existing harmony of the initial chord progression.

Obviously this method creates also a constraint of how long or how short the chords should sound, therefore it suggests also a rhythm standard to the chord duration neither too long neither too short so that the melody is neither too slow neither too fast. In other words the rhythmic duration of the chords (poetical measure as it has been called earlier) should be determined only after the creation of the melody.
 Therefore in the suggested above method the order of determinations is the next
1) The chord progression
2) The harmonic simplicial submelody
3) The chromatic simplicial submelody
4) The full melody after the melodic seeds and the 1),2),3)
5) The duration  of each one beat and how many beats per chord. 

MELODY-HARMONY INTERACTIVE COMPOSITION (BY INTERVALS OF 5THS AND 8THS).

The technique of melody composition which is described in this post 109, and which is supposed to require a chord progression in advance, can be applied also for melody composition without a chord progression given in advance, but in recursive way starting from the melody . This means that we start with the first realization of the order-topological theme, and so as to compose the next we compose simultaneously an underlying harmony , in other words a  next chord, and also a next melodic theme and so on. This interactive method for reasons of simplicity may compose as correlated harmony a power chord always in various positions, but the harmonic and chromatic simplicial sub-melody need again calculation. The power-chord play only the role of placing the melodic theme, inside the scale, and requiring that the melody passes from harmonic intervals of 8th or 5th. The actual chords that finally would accompany the melody may be different!.
We may of course predetermine a scale but this is nit always necessary.

Since determining a scale determines also a set of chords but not an ordered sequence of the (chord-progression), we may also conceive such a more lose condition in the composition of the melody : Instead of a predetermined chord progression a predetermined set of chords with no pre-decided order. Then as we want to go to the next melodic theme, w just choose a next chord from the predefined set of chords, and apply the method of the post 109.

The boundaries of the range of the available instruments upper and lower (usually 2 or 3 octaves) serve as reflectors, where the melodic themes may have inversion variations either  in pitch or time.

108. The 4 elementary variation of melodic theme. TRANSLATION, INVERSION, EXPANSION , MUTATION . Total and partial variations. Concatenation of melodic themes.

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The 3 elementary types of variations play to the melodic themes similar role that 3 basic chord-transition relations play for the chords.

107. The 3 basic shapes of melodic themes: EXPANSION, CONTRACTION and CYCLES. The 9 basic 2-sub-moves melodic shapes, and the 21 basic 3-sub-moves melodic shapes. Beatty of melodies based on statistical profile of them. The basic Dolphin-language words

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This post should be read together with post 101.

In the statistics, we may study the shapes of melodic themes bu the polarity of them (similar to the polarity of chords as Power-chords=neutral major chords=positive minor or more exotic chords=negative).


The 3 polarities +, - 0 of a melodic theme, and the 3 basic shapes of them: Expansion, Contraction and Cycles. Ascending, Descending Stationary. 

The 3 polarities + , -, 0 , are the correspondent to the melodic themes that the chord types major, minor and power chord are for harmonic triads.

The 3  basic shapes of them: Expansion, Contraction and Cycles, are the correspondent to the melodic themes that the chord extension  types like with 4th, with 6th with 7nth  are for harmonic triads.

The 9 , 2-moves shapes are the 

a) With + or - polarity

1) Straight 1 move

2) Overacting expansion

3) Counter reaction contracting

b) With 0 polarity

4) Isocratic (flat vector) 

5) Upper cycle

6) Lower cycle

The  12 , 3-moves shapes are 

Expansion

1) Strong expansion

2) Mid expansion

Contraction

3)Strong contraction

4) Mid contraction

5) Counter strong expansion up or down

Balance

6) Upper or down wave.

In total 21 shapes of 2 or 3 moves.

If we do not count as different the + or - polarities then we have only 11 shapes

106. THE VARIATION INDEPENDENT MELODIC SEED AND HARMONIC SEED IN COMPOSING A SONG

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In composing  melodic themes based on melodic triads (see post 208) we should use also the concept and technique  of  variational  independent base of melodic shapes or melodic seed. I other words the melodic themes shapes that are mutually variational independent, in other words neither translation neither , inversion, neither rotation can derive any one of them from the others, and in addition all other melodic themes of the song can be derived with variations from them.

As the melodic seed is usually a melodic theme of the chord-local scale (of the chord-yard melody) and the 3-notes chord in general is denote by 1-3-5, it can be described as number sequence from 1 to 7. E.g. 1-7-1 or 1-5-3-6) etc with the appropriate time duration of course.

See also post 311 about the Melidic maths  of Max Martin

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=1 step

E= 2 steps

I= 3 steps

O=4 steps

OU=5 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)

See also post 106 about melodic seeds

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



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

105. HOW TO CREATE NICE OCTAVE SPLITS AND OSCILLATIONS (or of 7nth,6th, 4th intervals) IN A MELODY

A simple and common way to create such an oscillations is to take for example a simple chord harping-waving that contains also with the previous rules less than 50% of the time also notes outside the chord , and then half of this simple theme translate it one octave higher, and so oscillate between the two octaves. Normally the initial non-translated melody would have intervals of 2nd, 3rd, and 5th. The interval of 3rd will become 6th , the interval of 5th, a 4th and an interval of 2nd , will become 7nth. In this way also the sttaitical profile of such melodies will have more frequently high intervals  of 5th, 6th, 7nth and 8th compared to intervals of 2nds (see post 93 ) .g. the folk Irish melody Kerry Polka below

We remind the reader that this should be read in the context of general method of creating melodies like in posts 92 and 103

•  HOW TO CREATE MELODIES FROM A CHORD PROGRESSION 1/2:

•  HOW TO CREATE MELODIES FROM A CHORD PROGRESSION 2/2:

104. HOW TO CALCULATE THE HARMONIC THE CHROMATIC AND CHORD PROGRESSION SIMPLISTIC SUB-MELODIES OF A CHORD PROGRESSION. THE MELODIC SIMPLICITY SUB-MELODY OF THE CENTERS OF A MELODY

Here we describe a basic technique of the composition method that starts first from the chord progression and then the melody introduced in post 9  (as in jazz improvisation).

A simplicial (or simplistic) sub-melody is a bit more varying than a drone (isocratic)  melody, and is also the  source  of the bass lines. 

1) Chromatic simplicial sub-melody (CSS , minimum distance notes) .  The simplicial submelody is defined by the next rules. 

 When two successive chords of the chord progression have notes that are one semitone distance only, we chose these two notes as notes of the simplicial sub-melody. For reasons of flexibility we allow two notes per chord if necessary. This happens for all cases that the two consecutive chords in a diatonic scale that are at roots distance of an interval of pure 4th (5 semitones) or pure 5th (7 semitones) or if they are mutually complementary chords (with roots of one step of  the scale apart). In general it is a good idea to chose as notes of the simplicial submelody for two successive chords in the chord progression, two notes, one from each chord with the minimum distance in semitones from the notes of the two chords. E.g. if the chords are , the first chord is the C major=(c4,e4,g4) and the 2nd chord is the F major=(f4,a4,c5), then the notes are e4-f4 that is 3rd-1st. If the chords are the first chord is the C major=(c4,e4,g4) and the 2nd chord is the D minor=(d4,f4,a4), then the notes are e4-f3 that is 3rd-3rd. 

 If the two consecutive chords are mutually relative with two common notes, the notes of the simplicial submelody for each chord are either a common note or the note that the other chord does not contain! That is the 1st-5th order notes. E.g. If the first chord is the C major=(c4,e4,g4) and the 2nd chord is the E minor=(e4,g4,b4) then the notes are b4-c5, that is 5th, and higher 1st. But if the chords are  C major=(c4,e4,g4) and the 2nd chord is the A minor=(a3,c4,e4), then the notes are a4-g4  , that is the higher 1st. and the 5th. If the chords are major-minor relatives : C major=(c4,e4,g4) and the 2nd chord is the C minor=(c4,eb4,g4), then the notes are eb4-e4 , that is 3rd-3rd. 

1.3) Chromatic links simplicial submelody (also bass lines) In general we may have the next rule. If X1, X2 are two succesive chords of the chord progression, and we are at X1, a chromatic link or chromatic bridge  is defined by finding two notes a1 in X1, a2 in X2, so that a1-a2 is at the minimum interval distance among all other chord notes of X1 , X2. Then the chromatic link starts with a1, b1,b2....,bn,a2 , and ends with a2 and all the intermediate steps are one semitone distance. 

1.3) Minimal chromatic drone sub-melody (MCD sub-melody).
This simplicial sub-melody is like the chromatic sub-melody, except that we utilize preferably the common notes of the chords, and we require it  
1.3.1) of as few notes as possible and
1..3.2)  of as little distance as possible. 
The rules are the next

Rule 1: We start from the chord and we find a common note with its next chord. If there are two common notes, we look at the next 3rd chord and chose this that is also either a note of the 3rd--next chord or minimal distance of a note of it. We proceed in this way till the last chord of the underlying chord progression. 
It can be proved that if the chord progression are chords of a diatonic scale, then the minimal  chromatic drone melody, can have only some or all of the first 3 notes of the scale (e.g. in a C major mode diatonic scale the c, d, e)  
This is very useful in double flutes or whistles or double reed-winds playing where in the first it is played a minimal chromatic drone sub-melody, and in the 2nd a full melody.

A minimal chromatic drone sub-melody need not be a kind of bass-line! It very well be a kind of very high register or octave simple melodic line. Personally I prefer the latter.

2) Harmonic simplicial sub-melody (HSS, maximum distance notes) . Probably the best method of creating  the simplicial sub-melody which is based on preferring intervals distances of the notes of the simplicial sub-melody (opposite to the previous method) that are large intervals ,namely intervals of 5ths , 4th 6th or 8th.  . The simplicial sub-melody is somehow the centers or oscilaltion boundaries of the final melody and most often it is one note per chord of the chord progression . They may be also the start and end of the melodic themes. Or they can be just centers that the melodic theme must pass from them. It can also be considered as a very simple bass line parallel to the melody. So the rule to choose the simplicial sub-melody is the next

2.1) If we have two successive chords X(1) -> X(2) in the chord progression, and a is the note of the simplicial sub-melody belonging to chord X(1) , and b is the note of the simplicial sub-melody belonging to the chord X(2), then a->b is an interval of maximum distance and the preference in intervals is in the following order of preference 5th, 4th, 8th, 6th. 
For the notes of maximum distance between successive chords we have the next choices : 

If the X(1) -> X(2) are in the relation of resolution (succesive in the wheel by 4ths) e.g. G->C then we have 3 choices for a->b, the g->c, or b->e, or d->g. If the X(1) -> X(2) are in the relation of relative chords (two common notes) e.g. C->Em then we have 2 choices for a->b,

c->g, or e->b. And if the X(1) -> X(2) are in the chromatic or complementary relation of  chords (roots that differ by one step of the scale) e.g. C->Dm, then we have one only best choice of a->b, here the c-> f, and a 2nd best choice the c-> which is an interval of 6th.
The notes of maximum distance would be two notes per chord. The 1st would be the maximum distance from the previous chord and the 2nd the maximum distance from the next chord. We prefer usually to simplify it it in to one only note but either two or one only note  if necessary we shift to the next octave so as to have the rule that two successive notes of two successive chords of the harmonic simplicial sub-melody have always distance large intervals of  5th, 4th, 8th, 6th. 
2.2) After we have defined the simplicial harmonic and the chromatic sub-melody then we may create bridges between its notes by smaller intervals e.g. 3rds or 2nds for a  full melody. The best ways is to start from the first note of the Chromatic Simplicial submelody (CSS) of the chord relevant to the previous chord, pass from the unique note of the Harmonic simplicial submelody (HSS)  of the chord and end at the 2nd note of the chromatic simplicial submelody (CSS) of the chord relevant to the next chord. (See post 109).
The notes of the harmonic submelody of a chord progression may be used to be  somehow the centers or oscilaltion boundaries of a final melody and most often.  They may be also be the start and end of the melodic themes. It depends if we create melodic themes inside the chord and around of a note of it which serves as it center or melodic themes linking two of them  and their successive chords. For the first way , the melodic themes inside the chord and around the note of the harmonic simplicial submelody can be created as in the post 103 using the chord-local 7-notes scale for each one note of the harmonic simplicial submelody.


There are also the 


3) CHORD-PROGRESSION SIMPLICIAL SUB-MELODIES (CPSS) 
 This is defined in the most easy way as consisting from one note per chord of the chord progression and always at the same degree (1st or 3rd or 5th, or 6th, or 7nth or 9nth or 2nd etc) 
Here is relevant video that by extrapolating  this simplicial sub-melody , we get an improvisational melody, an idea of Jerry Bergonzy

https://www.youtube.com/watch?v=2X-WsnWCAaA&t=21s


4) The chord-middle note simplicial sub-melody (CMNSS) This is one of the most simple tupes and most characteristic sub-melodies for the chord progression. The reason is that the middle note characterises a chord of it is major or minor, and thus this sub-melody involves notes that sometimes are the critical notes of modulations e.g. from the natural minor o the harmonic minor or double harmonic minor
IN THE NEXT WE DESCRIBE HOW TO CALCULATE THE SIMPLICIAL SUB-MELODY OF THE MELODIC CENTERS OF A MELODY

The simplicial sub-melody of the melodic centers is defined by the melodic centers of the melody (see also post 65 about the centers ) .

HOW TO FIND THE MELODIC CENTERS OF A MELODY:

The way to do it is the next

1) We partition the melody , to time intervals or connected pieces of it defined by the property that each one of then  has a single underlying chord, and the piece of the melody is maximal with this property

2) Then for each such time interval or piece of the melody, we define as its center, the note of the melody with the maximal time duration. There is one such note for each instance of a chord in the chord progression. The sequence of these notes is the simplicial sub-melody if the melodic centers of the initial melody. 

(This post has not been written completely yet)