117. Small 7-notes diatonic scale and big 10-notes diatonic (or chromatic-diatonic) scale. 5-notes, 6-notes , 8-notes and 9-notes scales with a maximal number of major or minor chord-triads.

(This post has not been written completely yet)

We have mentioned in previous posts in this book, that the diatonic scale has the optimal mathematical property of being the only 7-notes scale (within the chromatic Bach 12-notes scale) with he maximum number of major and minor chord-triads.

An interesting next question is what are the corresponding such scales with the same maximal property but with a larger number of notes? E.g. 8,9, or 10 notes?

We will answer it for the case of 10-notes scale. It is known that the 7-notes diatonic scale contains 3 major triad-chords , and 3 minor triads-chords plus a diminished triad chord. E.g. the C major mode diatonic scale has the 3 , C, F, G major and the 3, D, E, A minor chords. It is natural to  think of a scale that you can also have these 3 minor as major chords too , or the 3 majors as minor chords too. In other words all the C, F, G, as minor too or all the  D, E, A as major too , but not both!  This is a 10-notes scale C, D, D#, E, F, G, G#, A, Bb, B, with interval pattern in semitones , 2-1-1-1-2-1-1-1-1-1  or a cyclic permutation (mode) of it  C, C# ,D, E, F, F# ,G, G#, A, B again with interval pattern , 

1-1-2-1-1-1-1-1-2-1 which is a cyclic permutation  of the previous thus according to our definition of a scale in this book, the same scale in different modes. Notice that the pattern of sharps is the known f-d-g but with the original non-sharp notes too!

The 11-notes maximal harmonic scale is obviously the 1-1-1-1-1-1-1-1-1-1-1-2 as there is no other 11-notes scale!

There are instrument players e.g. Quena flute players, that have customized their hole system to a 8-front + 1 thump hole system that corresponds to this 10-notes big diatonic (or chromatic-diatonic scale) E.g. Here  https://www.youtube.com/watch?v=6olStHr43aE
Obviously Xiao flutes or Shakuhachi flutes or Irish whistles , or recorders etc,  can be made with such 8+1 holes 10-notes big diatonic scale.

If we take the power chords 1, 4, 5  of a diatonic scale , we get by their notes a 4-notes scale. E,g, in C major it would be C -G -C ,  F-C-F , G-D-G , thus C-D-F-G-C and in semitones 2-3-2-5.

On the other hand if we take a minor chord e.g. C-Eb-G then by adding a major 7nth Bb we get a
Cm7 chord C-Eb-G-B-C which is by itself a 4-notes maximal harmonic scale with interval structure   3-4-3-2 , which already has two chords in it the C minor and the B major

In the next we classify, 5-notes, 6-notes, 8-notes and 9-notes scales with maximum number of major or minor chord-triads.

5-notes maximal harmonic pentatonic (Greek-Richter pentatonic scale)

intervals patterns 4-3-2-2-1 e.g. c4-e4-g4-a4-b4-c5 or I ,III , V, vi, vii, I

(It is essentially A chord with 7nth and with 6th)

Notice that this scale suggests the chord progression by taking chords of the full 7-notes diatonic scale based on the above notes of the maximal harmonic 5-notes scale which has the highest statistical frequency in popular songs:  1(7)  4 5 6m E.g. in C major scale it would be C7, F , G Am.

Analternative version is the inverse 2-2-3-4-1

This inverse version when tuned at an harmonica asin the table below, will give at the blow row a major chord and at the draw row a major an interval of fith apart
E,g, Fmajor-Cmajor

6-notes maximal harmonic  (Celtic minor) 

intervals patterns 2-2-3-2-2-1 e.g. c4-d4-e4-g4-a4-b4-c5  or I ,II , III, V, vi, vii, I

(It is essentially a chord with 7nth, 6th and 2nd. A mode of it  is known also the (raised) Celtic minor scale of the hang-drums )

Notice that this scale also  suggests a  chord progression by taking chords of the full 7-notes diatonic scale based on the above notes of the maximal harmonic 5-notes scale which has  highest statistical frequency in popular songs:  1(7) , 2m,  4 ,5, 6m E.g. in C major scale it would be C7,Dm, F , G Am. This progression includes both the Blues progression 1 2m 5 , and the most popular 1,4,5,6m.

One very close it is os the Cretan minor scale with interval structure 4-1-2-2-2-1 (e.g. c4-e4-g4-a4-b4-c5) with 4 chords 2 major two minor C , G major and Am Em minor.

8-notes maximal harmonic 

intervals patterns 2-2-1-1-1-2-2-1 e.g. c4-d4-e4-f4-f#4-g4-a4-b4-c5

(mode of it known also as the Bebop dominant scale)

It seems that this maximal 8-notes harmonic scale (also known as bebop dominant) has the property of having  higher  density of minor or major chords per number of notes compared to the 7-notes diatonic . Already the maximal harmonic scales have maximum number of chord among other scales with the same number of notes. It seems also that the maximum density of major/minor chords per number of notes has the 12-notes chromatic scale.

9-notes maximal harmonic  

intervals patterns 1-1-2-1-1-1-2-2-1 e.g. c4-c4#-d4-e4-f4-f#4-g4-a4-b4-c5

I have made flutes , from overtone flutes with maximal harmonic such 5-notes, 6-notes,  8-tone , 9-tone, 10-tone , 11-tone scales!

116. COMPOSING CHORD-PROGRESSIONS BY ALTERNATING FORMS OF POWER-CHORDS.

(This post has not been written completely yet)

As we described in the post 35, power chords essentially are composed by 2 only notes, and are mainly of 3 types, minors, majors, and neutrals. In the same post 35 we described a method of improvisation for relaxation music based on power chords.

We extend this method here as a method of composition of chord progressions.

We choose a base of power chords,B= (Xi, i=1,2.3...n) where the Xi are an arc in the wheel of chords by 4ths.   e.g. i=1,2,3 a minor X1, a major X2 and a neutral X3. Then we compose the chord-progressions as a sequence Xi Y1 Xj Y2 Xk Y3 etc where Xi, Xj, Xk are power chords from the base B. In other words we alternate power-chords from the base with any other type of chords. The result is essentially rather harmonic, and easy to listen. 

Of course this method has an easy extension for relaxing improvisation which is , that we alternate power chords with small pieces of melodies by 2-3 or more notes, pieces of melody without simultaneous chord accompaniment without conscious compatibility rules between the power-chords and the pieces of melodies as they are alternating and not simultaneously sounding.

115. WHAT WE LEARN FROM THE SINGING OF NIGHTINGALES AND BLACKBIRDS, FOR MELODIES COMPOSITION.

(This post has not being written completely  yet)

What can we learn from the singing of Nightingales and Blackbirds for melodies composition?

Listened recordings of many hours with the singing of nightingales , Blackbirds and some other singing birds. E.g.

NIGHTINGALES:

https://www.youtube.com/watch?v=NK2_bcQcoD4

https://www.youtube.com/watch?v=s5gND_sd-Bw

https://www.youtube.com/watch?v=aNUk5eQu9o8

etc

BLACKBIRDS:

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

https://www.youtube.com/watch?v=dhpKHf4Wm08

https://www.youtube.com/watch?v=U4X8bjkx5HI

etc

What I learned is I believe very valuable and interesting. When I was listening to their singing, I tried to become aware of order-topological pitch patterns (as words or phrases in the Dolphin language. See post 101).

Very often, the emotional message of their melodic themes was very relevant or even copying the feelings and shapes of their flying patterns, as flying is their best ability, and they are proud of it. And most probably with their singing they describe it. 

Here is what else I remarked

NIGHTINGALES:

1) The order-topological pitch shapes oft heir melodic themes (or Dolphin language words see post 101, and 107) are mainly expanding rather than contracting. Therefore more joyful than sad. Most of the time they end their melodic phrases at high pitch (as folk Irish and other countries folk music often does) , and not as in classical music in a low final note. 

2) From time to time they start with a very high pitch note, which is hold for long, then they move to a low pitch note and trill and finally the end with an up pitch note again.

3) They trills maybe of very high pitch too, and the pitch distance between them is large. Could be more than 2 octaves!

4) Very often, the emotional message of their melodic themes was very relevant or even copying the feelings and shapes of their flying patterns. E.g. "I fly Up, I go down close to the dangerous ground , I escape death and I fly up again" This is not so common pattern in the singing of Blackbirds.

5) Nightingales have larger intervals of silence between their melodies compared ti Blackbirds.

BLACKBIRDS:

1) Blackbirds have more complicated melodies compared to nightingales. Again as with the nightingales, most of the time they end their melodic phrases at high pitch (as folk Irish and other countries music often does)  , and not as in classical music in a low final note. 

2) Blackbirds utilize many layers of waving and trills  in their melodies (see post 114).

3) Their melodic themes as order-topological patterns of pitch (see post 101) are more often expansive (joy freedom) than contracting (sadness  ,immobility).

4) They have many sliding-up trills 

5) They have very often the next pattern Contracting melodic theme-> Expansive melodic theme that goes up.

6) Very often they start with a low pitch waving melodic theme, and then they shift it fast upwards, where it ends.

7) Quite often the have a scattered expanding channel of separated trills, and then they end upwards with an upwards sliding trill.

As a conclusion, we observe that their melodic themes express mainly the emotions of joy, freedom, expansive flying and most of the times end their singing at the upper registers of pitch ranges rather than at their lower registers . etc Trills are very often melodic patterns.

114. MULTI- ORDER SYNTAX OF DOLPHIN LANGUAGE FOR POETICAL MEASURES MELODIES COMPOSITION. DERIVING THE FULL MELODY FROM A SIMPLISTIC SUB-MELODY. TRINARY HARPING COMBINATIONS . SIMPLICIAL DOLPHIN WORDS

(This post has not being written fully yet)


Here in this  post we refer to the pitch-order shape of a mrlodiv theme as "Dolphin Word" . In the next vide we may see how to creat Dolphin words.



 MELODIC MATHS BY MAX MARTIN AND GERM-PATTERN SYSTEMS OF CREATING MELODIC THEMES AND MUTATIONS OF  MELODIC THEMES AND RHYTHMS

In the next videos one can see how melodic themes of notes  (but also of chords) and mutations of them plus repetitive combinations of them, can be created by keeping invariant an  initial germ-pattern 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

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


A connected dolphin words is also germ-pattern of order of pitch in melodic themes, and could be symbolized as  a sequence of plus , zero and minus signs (+ - + 0 ++ -- 00 ) etc
Further informationcan be given as exponents about how many semitones or scale steps up or down are the + , and - signs. 


AN INTERCATIVE MODE OF VARYING  MANUALLY ONTHE TOUCHSCREEN  THE ORDER SHAPE ("DOLPHON WORD" ) OF MELODIC THEMES IS VERY WELL REALIZED WITH THE APPLICATION OSCILAB

https://www.youtube.com/watch?v=_AiDOCG-Vdk



Comparing the melody with a speaking language suggests the next correspondence

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

E.g. 

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

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

As we remarked in post 9 about the simplicial sub-melody , and also in post 72 and in post 69, the simplicial sub-melody can be multi-layered, in other words there are simplicial sub-melodies of simplicial sub-melodies This is an idea of that exists also in other sciences (e.g. stratified-sampling in statistics, multi-scale organisation of data in digital maps like google-maps , fractals with self-similarity , multi-order syntax of languages in linguistics, higher-order formal languages of logic , etc) and is a simplifying organization idea, where similar organization patterns in different scales are used to create an entity.
Here of course this entity is the full melody. Researcher of Bach have proved that he was utilizing also this method in counterpoint, where a single melodic theme, occurs in very slow time and large range, then in faster time-scale and finally in fastest recognizable time-scale as the main theme of the counterpoint.

See also post 282

Examples of such progressions of simplicial melodic themes (or Dolphin words) are the next  (each vector-arrow is an oriented interval that fits to a single or more  underlying chord(s)).

Or

Or

Or

AS THE COMBINATION SIMPLICIAL MELODIC MOVES (ORIENTED INTERVALS, SEE POST 282) CREATE PATTERNS THAT ARE CALLED "DOLPHIN WORDS" , WE MAYS AS WELL CLASSIFY THE "SIMPLICIAL DOLPHIN WORDS" . The simplest such patterns are of course the 3: 1) THE CYCLE 2) THE ASCENDING SEQUENCE 3) THE DESCENDING SEQUENCE.

Here for reasons of simplicity we describe a 2-levels Dolphin language for melody composition.

We showed in post 101 how the order-topological shapes of melodic themes or Dolphin-words are used to compose a melody. We also discussed in post 72, how the simplicial sub-melody can be used to organize a full melody as starting or ending points of melodic themes of the full melody. Nevertheless what we point-out here in this post is that the simplicial sub-melody could have been composed also my order-topological shapes of the Dolphin language, and in particular that it can be one or 1-3 only Dolphin words that are not only longer in time duration but also larger in pitch-range.  Then the next rule may be applied 

Rules of  interaction of simplicial sub-melody and full melody

Rule of melodic centers 

There   Dolphin words, or order-topological shapes of the full melody, that contain (e.g. end but also may can start too or may have in the middle too) ,  a note of the simplicial sub-melody, which is its center or its goal. So there is at least one such a Dolphin word corresponding to every note and chord of the simplicial sub-melody

Rule of harmony of the centers 

The duration of the center or goal note (a note of the simplicial sub-melody) totally in all its occurrences during it, is the longest among the duration of the notes of the Dolphin word or order-topological pattern, and preferably is larger, than the total duration of all the other notes of the Dolphin word. Notice that that we talk about the total duration of this center in ALL its occurrences that can be MANY.

Rule of subdivision  and of trinary harping combinations (by 3rds)  The single note of a simplicial sub-melody and Dolphin-word as in the Rule of melodic centers may be divided in to many smaller Dolphin-words usually 2, 3 or 4. Possibly of long-short part micro-rhythmic  as in post 92 that the simplest order-topological pattern  is an up , down or horizontal arrow, long a note inside the chord and short a note possible but not necessarily outside the chord. The distance of the long-short note is usually a 3rd, and the long is double duration from the short  We call such system of elementary Dolphin-words trinary harping combinations by 3rds. If such micro-words are an up or own or horizontal arrow, then we prefer to have a balance of all proportions of up or down in any such subdivision. By choosing appropriately the proportions of up-down-horizontal any slope ,melody can be composed, that goes from anywhere to anywhere and at the same time being in harmonic-fitness with he underlying chords!
Such melodies that are created in this way by consistent sub-division e.g. to 4, are usually fast rhythmic melodies as in the Irish reels, or Greek Cretan lyre, or as in  Bach in classical music etc , that is,  faster than a usual human voice melody. Nevertheless if we do not utilize the current rule of subdivision, then the previous two rules may created a middle complexity full melody, that a human voice can sing.
There are 3 reasons why the full melody might be in the mode of trinary harping combinations of Dolphin-words of long-short part.
a) It is an easy way to incorporate notes outside the chord in the melody while the melody is fitting to the chord, and at the same time have a rhythmic sounding
b) It gives fast melodies that for skilled instruments players gives an impressive listening 
c) It gives high harmonic statistical profile with high percentage f intervals of 3rds, if the internal distance of the notes in the elementary Dolphin-word is mainly intervals of 3rds
d) All of the above global properties of the melody are succeed from its very small building blocks. 



An example of such a melody is the Irish melody Blacksmith hornpipe http://ungaretti.racine.ra.it/ireland/music/blakhorn.mid

Another example is the Irish melody "The frost is all over"

http://www.contemplator.com/tunebook/midimusic/frost.mid

Usually of course the center-note together with 2 other notes of the Dolphin word, are the chord-notes of an underlying chord of the Dolphin-word, and these 3 notes in total last longer that the total duration of all other notes of the Dolphin word.

Since in post 104 we described how to derive a simplicial sub-melody from a chord-progression, then we may understand that also a chord-progression may have the structure (e.g. at the roots of the chords) of very few Dolphin-words! 

As the simplicial sub-melody is simpler than the full melody it is natural to start composing from the simplicial sub-melody. And this is the main reason that on post 9, I suggest a composition method of songs that starts with the harmony of the chord-progression (which corresponds to a simplicial sub-melody) as this is a simpler setting (It is also the setting of the jazz improvisation).

So here I propose a composition method of melodies starting from the simpler structure of its simplicial sub-melody. 

The starting note a of a Dolphin word that ends  at a note x of a simplicial sub-melody, can be x itself, or x'  an octave higher than x or an octave  lower than x, or it is a 5th or 4th higher or lower etc. It can be a note in the same underlying chord of x , but it can be also a note of the previous chord of the underlying chord of x. 

Of course when improvising, we do simultaneously  the improvisation of the simplicial sub-melody at first which is an easier task as general directions and "stations" of a  "journey" and then the details of the full melody E.g. choosing at first how many octaves and in which octave to start and where to go end E.g. start at he 2nd higher octave on the 5th go down till the root of the first lower octave  but not directly with free chosen waves, and then move up again in the second octave but end at the root of the second higher octave. 

See also post 102 which essentially a similar idea.

113. 2-OCTAVES HARMONIC SCALES (NON-CHROMATIC) WITH MANY CHORDS

2-OCTAVES HARMONIC SCALES (NON-CHROMATIC) WITH MANY CHORDS

Such scales are based on harmonic intervals like those in a chord and its inversions that is 3 or 4 semitones, 4 or 5 semitones  7 semitones etc.

When going up and down or creating order-topological shapes of the Dolphin Language (see post 101 ) in such scales chords are shaped in a natural and direct way by every 3-successive notes . The types of chords are mostly major, minor, diminished, augmented etc.

The melodic corridor as described in post 94 and the geometry of pan-flutes like the Samponas or modern percussion instruments like hand-pans, hung etc  is based on this idea

When such scales are reduced to a single octave may give familiar scales like diatonic , melodic minor  etc.


1) The -4-3-4-3-3-4-3-  which is the diatonic  scale when reduced to a single octave 
and other are the next


2) 3-3-4-4-3-3-4=24 This scale has obviously successive diminished minor , major and augmented chords. It is the melodic minor scale when reduced to a single octave 


3) 3-3-3-4-4-4-3=24

4) 3-5-3-5-3-5=24

5) 3-3-3-5-5-5=24

6) 3-4-3-4-7-7=24

7) 3-7-3-7-4=24

8) 3-3-7-7-4=24

9) 4-5-4-5-4-2=24

10) 4-5-4-5-4-2=24

11) 4-4-5-5-4-2=24

12) 4-4-4-5-5-5-2=24

13) 4-4-4-5-2-5=24

14) 5-7-5-7=24

15) 3-4-5-3-4-5=24

16) 3-4-3-4-5-5=24

17) 3-3-5-4-4=24

18) 4-5-7-4-4=24

19) 3-5-7-3-4-7=24

etc

Also in more than 2 octaves

3-octaves

5-5-5-5-5-5-6-=36

and 4-octaves

7-7-7-7-7-7-6-=48

The last one is close to how one can derive a diatonic scale by exact 5ths (Here the 5ths of 7-semitones are not exact, so the Pythagorean comma becomes a whole semitone)

etc.



(This post has not been completely yet) 

112. THE HARMONIC STATISTICAL PROFILE OF A CHORD PROGRESSION

(This post has not being written completely yet)

This post describes that the harmonic statistical profile of melodies as in post  , applies also to chord progressions, and in this way it is explained why the suggest chord-cycles or chord-wheels or chord-scales as paragraph 89, are the suggested order.

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.

111. INDEPENDENT AND PARALLEL COMPOSITION OF MELODY AND CHORD-PROGRESSION. RULES OF COMPATIBILITY AND FITTING.

(This post has not being written completely yet)

In this post we describe how to combine two different methods of composition
1) The one that starts from the melody first utilizing the Dolphin Language of order-topological melodic shapes (as in post 101) as historically the  melody composition was easier , it come historically first  , and is  closer to the human voice pitch changes,
  and the one
2)  that starts from the chord-progression , as in Jazz improvisation described in posts like 49, 83 which historically came later, after discovering music with  harmonic structure. Still harmony determination at first in composition very often may be a simpler specification than melody shapes.


One of the first rules of compatibility of course of a melodic phrase M during a time interval that a single chord  C is sounding is as we described in post 27.

RULE 1

If the notes as notes of the melodic theme (a piece of the melody that we have not yet found its underlying chord yet) in total do not sound less time  (preferably >2/3 of the total time) compared to the total duration of the notes of the melodic theme that do not belong to the chord then we accept this chord as underlying chord. We may have also a slightly different and less strict rule:  If we divide all notes of this piece of melody to equal smaller duration  notes, and make a statistical histogram of the re-occurrence of their  pitch , then the triad of notes of maximum duration compared to the duration of any other  triad should correspond to the 3 notes of the underlying  chord. Obviously there is no requirement in the second version of the rule that the notes of this triad in total sound more than the total duration of all other notes that do not belong to the chord. Only that they sound more than any other triad.  A third variation of the previous rules involves also not only the time duration but the loudness of he notes in the obvious way. The previous rules , in particular the first one,  of course may determine more than one chord as underlying chord or no chord at all!. And we may chose with criteria of better quality chord progressions relative to the alternatives. Or if one particular chord progression and chord transition is more common in the particular style of music. We may also put a requirement of lest possible  number of underlying chords, which means that if for the previous melodic theme and previous chord, is so that its notes as notes of the melodic theme both current and previous  in total do not sound less (preferably >2/3 of the total time) compared to the total duration of the notes of the two melodic themes that do not belong to the previous chord then we extend the duration of the previous chord to the current melodic theme.

RULE 2 

There mainly two correlations of a piece of a melody with a chord a) The local condition. This is the next: If we divide all notes of this piece of melody to equal smaller duration  notes, and make a statistical histogram of the re-occurrence of their  pitch , then the top 3 peaks of the histogram should correspond to the 3 notes of the chord. b) The global condition. This is the next: The local condition is overruled according to a quantitative weight of significance, if the previous chord defines with maximum transition probabilities (among a great sample of popular chord progressions)  a different chord (e.g. G7 resolves to C rather than Am or Em)