81. HOW TO DERIVE NEW SCALES FROM THE HARMONICS (OVERTONES) OF A SINGLE TONE. THE HARMONIC PENTATONIC, HARMONIC 7-TONE AND HARMONIC 8-TONE SCALES.

HOW TO DERIVE NEW SCALES FROM THE HARMONICS OF A SINGLE TONE. THE HARMONIC PENTATONIC, HARMONIC 7-TONE AND HARMONIC 8-TONE SCALES.

SUCH SCALES OCCUR IN THE OVERTONE FLUTES

The pentatonic scale is supposed to be obtained by the first 9 harmonics when reduced to the first octave with frequencies based on a fundamental,  given by the harmonic order (numerator) and power of 2 which is the reduction in the first octave

C, D, E, G, A, C'

1, 9/8 , 5/4 , 3/2 , 7/4. 



Similarly the 7-notes diatonic scale

C---1 harmonic

D---9

E---5

F---11

G---3

A---7

B---15

C---2

To derive the 7 notes of the diatonic scale in major mode requires more harmonics (of a single note) that one may imagine.

See e.g. https://en.wikipedia.org/wiki/Harmonic_series_(music)

Thus the 7-notes  of the diatonic scale in major mode, require 27 harmonics. And the correspondence is the next. The notes are of course lowered to be within one octave, while the harmonics in many higher.

C---1 harmonic

D---9

E---5

F---11

G---3

A---27 (or 7) 

B---15

C---2

So a major scale like C-E-G, requires only the first 5 harmonics

While a minor scale e.g. C-Eb-G requires 19 harmonics as Eb is obtained as the 19th harmonic.

In the Pythagorean method, we derive the 7-notes diatonic scale, by repeating 7 times, the 3rd harmonic of the previous harmonic (thus 3^6=729 harmonics of the deepest tone which is the F here)

So the correspondence in harmonics would be, starting from F this time

F---1 harmonic
C---3 harmonic

D---3^3=27

E---3^5=243

G---3^2=9

A---3^4=81

B---3^6=729

F---2

If the base note is C then

C---1 harmonic
G---3 harmonic

A---3^3=27

B---3^5=243

D---3^2=9

E---3^4=81

F---3^6=729

C---2

With this Pythagorean method, therefore all frequencies of the scale are simple ratios with numerator powers of 3 and denominator powers 2. The harmonics by 3, 3^7=2187, are close for the first time to harmonics by 2, for 2^11=2048. So after 11+1=12 octaves by harmonics as powers of 2  (+1 because we started lower than C) and after 7 intervals by 5th (harmonics 3^7) the two harmonics differ by an amount very close to the discrimination threshold by the human ear, which is called the Pythagorean comma. More formally the  Pythagorean comma, denoted by pc can be defined as the difference pc= log(3/2)/log(2)-7/12=0.001629167..... and it is an irrational number.

Here an analytic table from (https://en.wikipedia.org/wiki/Pythagorean_tuning )

Note	C		D		E		F		G		A		B		C
Ratio	​1⁄1		​9⁄8		​81⁄64		​4⁄3		​3⁄2		​27⁄16		​243⁄128		​2⁄1
Step	—	​9⁄8		​9⁄8		​256⁄243		​9⁄8		​9⁄8		​9⁄8		​256⁄243		—

In full the Pythagorean enharmonic scale is (from https://en.wikipedia.org/wiki/Enharmonic_scale )

The following Pythagorean scale is enharmonic:

Note	Ratio	Decimal	Cents	Difference (cents)
C	1:1	1	0
D♭	256:243	1.05350	90.225	23.460
C♯	2187:2048	1.06787	113.685
D	9:8	1.125	203.910
E♭	32:27	1.18519	294.135	23.460
D♯	19683:16384	1.20135	317.595
E	81:64	1.26563	407.820
F	4:3	1.33333	498.045
G♭	1024:729	1.40466	588.270	23.460
F♯	729:512	1.42383	611.730
G	3:2	1.5	701.955
A♭	128:81	1.58025	792.180	23.460
G♯	6561:4096	1.60181	815.640
A	27:16	1.6875	905.865
B♭	16:9	1.77778	996.090	23.460
A♯	59049:32768	1.80203	1019.550
B	243:128	1.89844	1109.775
C′	2:1	2	1200

Notice that all the ratios of the 7-notes of the enharmonic Pythagorean diatonic scale are quotients powers that have base 2 or 3

C = 1

D= (3^2)/(2^3)=9/8

E=(3^4)/(2^3)=81/64

F=(2^2)/3=4/3

G=3/2

A=(3^3)/(2^4)=27/16

B=(3^5)/(2^6)=243/128 

This approach is very relevant t the Chinese musical system where all ratios are quotients of powers with base 2 or 3 (thus derived from the 3rd and 2 harmonic and their harmonics)

The ancient Chinese musical system depends on very ancient mathematics used to determine sound frequencies. The easiest way to explain it is to work through a real example.

Suppose that somebody wanted to make a musical instrument that could play any song in the ancient Chinese system. Here are the instructions:

Make a wooden box 105 cm long and 60 cm wide. Put guides for the strings near each end of the box, and fix it so that these two guides are 99 cm apart. Multiply 99 cm by 2/3, which is 66 cm. Place a fret all the way across the box on the 66 cm line.
Multiply 66 cm by 4/3, which is 88 cm. Place a fret along the 88 cm line.
Multiply 88 cm by 2/3, which is 58.66...6 cm. Place a fret along this line.
Multiply 58.66...6 cm by 4/3, which is 78.22...2 cm. Place a fret along this line.
Multiply 78.22...2 cm by 2/3, which is 54.148148...148 cm. Place a fret along this line.
Multiply 54.148148...148 cm by 4/3, which is 69.531 cm. Place a fret along this line.
Multiply 69.531 cm by 2/3, which is 46.354 -- and which is too short, so double it to get 92.708 cm. Place a fret along this line.
Multiply 92.708 cm by 4/3...
Multiply the previous answer by 2/3...
Keep going until you have put down eleven frets.

Counting the frequency on the open string and the frequencies on the fretted strings, for each string there will be 12 defined frequencies.

Tune the bottom string to some basic frequency. Tune the next string to the frequency of the bottom string at the first fret. Tune the third string to the bottom string's second fret. Keep going until you have tuned all twelve strings.

When you pluck these strings at all the fretted and unfretted positions, you will get 144 frequencies. Some of them will be duplicates, but not as many as you might think because this system is not like the Equal tempered system now used for almost all Western music.

Out of each twelve frequencies on a single string, you can make many selections of either five frequencies (for the pentatonic scales) or seven frequencies (for the heptatonic scales).

Since the sequence of 5ths is say the F->C->G->D->A->E->B , it is interesting to define successively larger and larger scales that contain many intervals of 5th in this sequence.

1) One interval of 5th C-G
2) 3 notes C,F,G (it is not a  major chord!) Interval structure 5-2-5
3) 4 notes C, D, F, G Interval structure 2-3-2-5
4) 5 notes C,D, F, G, A  Interval structure 2-3-2-2-3 which is the inverse of the well known minor pentatonic scale.
5) 6 notes C, D, E, F, G, A Interval structure 2-2-1-2-2-3. This may be called the Pyrhagorean 6-notes harmonic scale 

6)The scale F1-C2-G2-D3-A3-E4-B4-F5  or short wheel of intervals of 5.  This scale as sequence of intervals of 5 is e.g. F1-C2-G2-D3-A3-E4-B4-F5 has a span of 4 octaves! From F1 to F5 while the last interval B4-F5 is a not of 7 semitones but of 6.

Similarly if we would use intervals of 4 in the reverse order it would be

7) The scale B1-E2-A2-D3-G3-C4-F4-B4 or short wheel of intervals of 4.  This scale as sequence of intervals of 4 is e.g. F1-C2-G2-D3-A3-E4-B4-F5 has a span of 3 octaves! From B1 to B4 while the last interval F4-B4 is a not of 5 semitones but of 6.

Going back to the order in which the simplest harmonics derive the 12-tone chromatic scale, we may put, the intervals, chords, and scales with the maximum number of simplest harmonics, in the next order

1) A SINGLE TONE C ( ALL SIMPLE HARMONICS )

2) AN INTERVAL OF OCTAVE  C(N)-C(N+1) 2ND HARMONIC)

3) POWER CHORD C-G-C (2ND 3RD HARMONIC)

4) MAJOR TRIAD CHORD C-E-G-C (WITHIN THE FIRST 5 HARMONICS)

5) MAJOR TRIAD SUSPENDED 2 OR ADDED 9TH Cadd9 or Csus2 or Em7#5=Em7+
(WITHIN THE FIRST 9 HARMONICS. HERE A COINCIDENCE OF THE NUMBER 9)
THEY CAN BE CONSIDERED ALSO AS A 4-NOTES HARMONIC SCALE C-D-E-G-C
interval structure 2-2-3-5, which is a very common overtones scale of the overtones flutes! 

6) THE HARMONIC PENTATONIC (AN UNNOTICED SO FAR PENTATONIC SCALE!)

C-D-E-F-G-C (SEMITONE STRUCTURE 2-2-1-2-5)
(WITHIN THE FIRST 11 HARMONICS) which is again sometimes  overtones scale of the overtones flutes!  This scale also is directly playable in the Ney-flute which  has hole spanning a 5th of scale rather than an 8th C-D-(Eb)-E-F-(F#)-G

Nevertheless by combining acoustics of open-open and open-closed pipe with the first 9 harmonics we may get the classical mode of the major pentatonic scale. See  this  post below about overtone flutes.

7) THE HARMONIC 6-TONES SCALE (not to be confused with the Pythagoream 6-notes harmonic scale above)

C-D-E-F-G-Ab-C  (SEMITONE STRUCTURE 2-2-1-2-1-4)
(WITHIN THE FIRST 13  HARMONICS).


8) THE MELODIC MINOR  7-TONES SCALE (Not to be confused with the harmonic minor or major scale!)

C-D-E-F-G-Ab-Bb-C  (SEMITONE STRUCTURE 2-2-1-2-1-2-2 NOTICE THAT IT IS SYMMETRIC RELATIVE TO THE CENTRAL TONE INTERVAL OF 2 SEMITONES ON F-G. THIS SCALE IS KNOWN ALSO AS HINDU SCALE )

(WITHIN THE FIRST 14 HARMONICS).

NOTICE THAT COMPARED TO THE DIATONIC 7-NOTES SCALE, IT IS DERIVED WITHIN THE FIRST 14 HARMONICS WHILE THE 7-NOTES DIATONIC IS DERIVED WITHIN THE FIRST 27 HARMONICS!

9) THE HARMONIC 8-TONES SCALE 

C-D-E-F-G-Ab-Bb-B-C  (SEMITONE STRUCTURE 2-2-1-2-1-2-1-1 )

(WITHIN THE FIRST 15 HARMONICS).

(NOTICE THAT BY ELIMINATING THE Bb, WE RESULT TO THE

7-NOTES  1ST BYZANTINE SCALE OR HARMONIC  MINOR SCALE

WITH AMAZING SOUND

C-D-E-F-G-Ab-B-C  (SEMITONE STRUCTURE 2-2-1-2-1-3-1 ) AGAIN WITHIN THE 15 HARMONICS!

10) THE HARMONIC 9-TONES SCALE 


C-Db-D-E-F-G-Ab-Bb-B-C  (SEMITONE STRUCTURE 1-1-2-1-2-1-2-1-1 )

(WITHIN THE FIRST 17 HARMONICS).

(NOTICE THAT BY ELIMINATING THE D, WE RESULT TO A

SECOND HARMONIC 8-NOTES HARMONIC SCALE

WITH AMAZING SOUND

C-Db-E-F-G-Ab-Bb-B-C  (SEMITONE STRUCTURE 1-3-1-2-1-2-1-1 ) AGAIN WITHIN THE 17 HARMONICS!

AND BY ELIMINATING THE Bb IN THIS SCALE WE GET THE REMARKABLE

C-Db-E-F-G-Ab-B-C  AGAIN WITHIN THE 17 HARMONICS,  WITH SEMITONE STRUCTURE 1-3-1-2-1-3-1  WHICH IS NOTHING ELSE THAN THE 2ND BYZANTINE SCALE OR HARMONIC DOUBLE MINOR OR HUNGARIAN MINOR OR GYPSY MINOR SCALE!



11) THE HARMONIC 10-TONES SCALE 

C-Db-D-Eb-E-F-G-Ab-Bb-B-C  (SEMITONE STRUCTURE 1-1-1-1-1-2-1-2-1-1 )

NOTICE THE BLUE-NOTE Eb-E, THAT ALLOWS BOTH C MAJOR AND C MINOR CHORD.

(WITHIN THE FIRST 19 HARMONICS).

12) THE DIATONIC 7-TONES SCALE 

C-D-E-F-G-A-B-C  (SEMITONE STRUCTURE 2-2-1-2-2-2-1 )


If on the other hand we have an overtone flute based on C4, by alternating open-close-open-close etc and blowing each time more we get the 9 notes sequence of notes spanned on 3 octaves (4th, 5th, 6th) and up to the first 5 harmonic-overtones  by the open-pen acoustics and first 4 harmonic-overtones of the open-closed acoustics, in total 1st 2nd 3rd, 4th, 5th, 7th, 9th harmonics 

C4(open 1st basic harmonic)-G4 a bit sharped(closed 3rd harmonic)-C5(open 2nd harmonic )-E5(closed 5th harmonic)-G5(open 3rd harmonic)-A5 a bit sharped(closed 7nth harmonic )-C6(open 4th harmonic )-D6 a bit sharped (closed 9th harmonic)-E6(open 5th harmonic)

with intervals structure in semitones 

7-5-4-3-2-3-2-2

Main Chords C major C6
               Aminor    etc

which when reduced to a single octave it is the 6-notes scale by overtones

C-D-E-G-A-C  with interval structure  2-2-3-2-3

which is nothing else than the standard (Mongolian)  mode of the (major C) pentatonic scale



(WITHIN THE FIRST 27 HARMONICS).

ALTHOUGH THE DIATONIC SCALE REQUIRES MANY HARMONICS TO BE DEFINED, IT CAN BE PROVED THAT IT HAS THE LARGEST NUMBER OF MAJOR AND MINOR TRIADS COMPARED TO THE OTHER SCALES.

NEVERTHELESS THE STANDARD PENTATONIC SCALE IS THE MAXIMAL SUB-SCALE OF THE DIATONIC WHICH IS CLOSED TO INTERVALS BY 5TH (7 SEMITONES) iN OTHER WORDS STARTING FROM A NOTE OF THE SCALE BY GOING UP OR DOWN A 5THS (7 SEMITONES) WE ARE AGAIN BACK TO A NOTE OF THE SCALE. THE DIATONIC IS NOT CLOSED. IT IS CLOSED ONLY IF WE TOLERATE EITHER AN INTERVAL OF  5TH OR OF  4TH



Here is how it is made willow flutes without holes, playing on scale of...harmonics

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

80. The pitch translation homomorphism between melodic themes and underlying chords.

The pitch translation homomorphism between melodic themes and underlying chords.

When the melody is composed from little pieces called melodic themes M1, M2, M3 etc and each one of them or a small number of them (e.g. M1, M2)  , have the same underlying chord C1, then we have a particular simple and interesting relation between the chords C1, C2 , C3 and the melodic themes (M1, M2), (M3, M4) ,...etc. This is not the case when the melodic themes start at one chord and end to the next, that we usually call in the book, as "external melodic Bridges" . We are in the case of "internal melodic Bridges". This relation is based on the pitch translations of the melodic themes and of the chords. Actually this is also a scheme of composition of melodies based on small melodic themes (see post 9), when the chord progression is given or pre-determined.

So let us say that the melodic themes (M1, M2), have underlying chord C1. Then as we have said there are only 3 possible chord-transition relations in a chord progression (see post 30 ): C2 will be either in resolution relation with C1 which means that the rood of C2 is a 4th lower or higher relative to the root of C1, or the root of C2 is an interval of  3rd  away from the root of C1 , or and interval  of 2nd away from the root of C1. Let us symbolize by tr4(), tr3(), tr2() , where tr() is from the word translation, of these three pitch shifts. Then we may also translate the melodic themes similarly

tr4(M1), tr4(M2), or tr3(M1), tr3(M2), or tr2(M1), tr2(M2), Then automatically the new translated melodic themes will have as appropriate underlying chord the C2. Actually in the case of intervals of 3rd or 5th, the melodic themes tr3(M1), tr3(M2)  or tr5(M1), tr5(M2) may as well as appropriate underlying chord the C1 again as the 3rd and 5th of the chord is a pitch translation that leads to a note again inside the chord. This is the reason we called this relation homomorphism and not isomorphic. In mathematics , and correspondence H is call homomorphism relative to some relations R, if the the objects H(x1), H(x2) are in relation R , if the objects x1, x2 are in relation R. Here H(M1)=C1 and H(tr(M2))=C2 and C2=trn(C1) that is are in distance of interval of n (=2,3,4 etc) if tr(M1) and M1 are in distance of interval of n. It may happen that H(x1)=H(x2). But if when x1 is different from x2 then always also H(x1) is different from H(x2) we say that H is an isomorphism. Here because it may happen that H(M1)=C1, and H(tr(M1))=C1 again the correspondence of melodic themes and chord is not 1-1, that is H is an homomorphism not an isomorphism in general. 

By continuing in this way translating in pitch the initial melodic themes M1, M2 according to the interval shifts of the roots of the chord progression an remaining inside a diatonic scale , we compose a melody (or simplicial sub-melody too, see post 9). Of course in order for the melody not to be too monotonous we may vary also the melodic themes from ascending to descending etc.

Other translations of the melodic themes can be during the same underlying chord, and are obviously of an interval of 3rd.

Now even when we are at external melodic bridges e.g. M1 which starts at underlying chord C1 and ends in underlying chord C2, even then this homomorphism is of use! The way to make it work is to take the range of the melodic theme (usually starting and ending note as simplicial submelody) equal as interval to the interval of the roots of the underlying chords C1, C2. 

Chord progressions that two successive chords  are always either 1) an interval of  4th , that is successive n the wheel of 4ths 2) Relative chords where major turns to minor and vice-versa, thus roots-distance  an interval of 3rd 3) Chromatic relation , in other words the roots differ by a semitone
are best chord progressions for parallel translations of melodic themes by intervals of octave, 4th-5th, 3rd and semitone. 

Here is a video of jazz improvisation which uses this idea. The chord progression id C, F, G, and the of pitch translations of the initial melodic themes are always intervals of 4th or 5th.  

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

79. THE MELODIC-HARMONIC 7 NOTES, 2 OCTAVES SCALE (WHEEL OF 3RDS). The 7 notes 3 and 4 octaves scales of 4ths and 5ths.

Before this post the reader must study the posts 40 (that classifies intervals), the classification of  2-string triads , and  38 (that classifies 3-string triads) and 35. 

This scale is also the Harmonic tuning of the guitar (see post 1) which is optimal when chord playing is mainly the traget and not soloing so much

OPTIMAL GUITAR TUNING FOR CHORDS PLAYING MAINLY

2.) An more optimal but unkown tuning for the 6-string guitar when chord-playing is the main target and not so much solo playing is and even better by alternating minor and major 3rds. In semitones for the 6 strings   4-3-4-3-4 or 3-4-3-4-3
E.g. Bb2- D3-F3-A3-C4-E4 or F2-A2-C3-E3-G3-B3 or A2-C3-E4-G4-B4-D4
THIS MAY BE CALLED THE HARMONIC TUNING OF THE GUITAR AS IT IS BASED ON THE HARMONIC 2-OCTAVES 7-NOTES SCALE (see also post 79)
The latter is the most natural open tuning. There the same shape for major and minor chords and only 3 of them and in only one or frets compared to the 6 in the standard tuning guitar. If we want also dominant and major 7nth chords we use again only 2 frets. The same with the aug chords Only the dim7 chords require 3 frets. Because  of the symmetry of the tuning among the strings, the relations of relative chords and also chords in the wheel of 4ths is immediate to grasp also geometrically. Of course when we say shape of chords as it is standard in jazz, we do not play all 6-strings but only 3 or 4 strings.
Within 3 frets exist all chords of the  8-notes scale with interval structure 2-2-1-2-2-1-1-1 which is an extension and variation of the melodic double minor 2-2-1-1-1-1-2-2 or (1-1-1)-(2-2)-(2-2-1)
But also all chords of diatonic scale!

All the next chords are superposition of 2-notes relatives chords of types major, minor, diminished, and augmented. 


Discussion of the defnition-creation  and roles of the 8 basic such chords (in root position)
433=majVdim=R7
434=majVmin=Rmaj7
343=minVmaj=Rm7=R'6, where R' the root of  a  relative chord
333=dimVdim=Rdim7
334=dimVmin=Rm6=R'm7b5 , called also half diminsihed where R' the root of  a  relative chord
444=augVaug=Raug
344=minVaug=Rminmaj7
443=augVmaj=Rmaj7#5 

4 of the above chords can be 4-notes chords of the diatonic scale e.g. the R7, Rmaj7, Rm7, Rm7b5. But the Rdim7, Raug, Rminmaj7, Rmaj7#5 cannot be 4-note chords of the diatonic scale. Nevertheless there exist in other scales as we shall see like Harmonic minor, and harmonic double minor Melodic minor (Hindu)and Melodic double minor (Arabic) and other oriental scales as in the scales of post 52.

Some of the above  chords appear as chords of the HARMONIC LONG 12 NOTES 7 OCTAVES SCALE  (see post 42, and 34)  Wheel of 3rds
(successive distance in semitones) (434343434343434343434343) 

Besides the long scale we may define the HARMONIC 2 OCTAVES , 7 NOTES SCALE of 7 notes that spans 2 octaves, and every 3 or 4 successive notes is a chord of the above types 

Here it is (STARTING FROM THE  C3)

etc -C3-E3-G3-B3-D4-F4-A4-C5- etc  (all steps intervals of 3rds)

As a structure of intervals over semitones it is the next sequence

-4-3-4-3-3-4-3-

And the sequence of 3-notes chords , by successive 3 notes in this scale is the chord progression

C->Em->G->Bdim->Dm->F->Am

while as sequence of 4-notes chords is the

Cmaj7->Em7->G7->Bm6->Dm7->Fmaj7->Am7 or 

Cmaj7->G6->G7->Bm7b5=Dm6->F6->Fmaj7->C6

THE MELODIC CORRIDOR 

Constructing therapeutic harps, or similar instruments (e. thump  pianos https://en.wikipedia.org/wiki/Mbira or hang and handpans ) or such keyboards with this scale, has the property that almost what ever we play sounds harmonic as 3 or 4 notes in sequence are well known cords, and overlapping such triads , are relative chords. In fact we may design such an harmonic diatonic  double row keyboard

 as follows 

  -D3-F3-A3-C3- E4-G4-B4-

-C4-E3-G3-B3-D4-F4-A4-C5- 

Similarly this can be a beautiful , practical and harmonic tuning for harmonicas.

The blue is blowing and the red is draw. Then with the key we may have the sharps too thus in total

-D3#-F3#-A3#-C3#- E4#-G4#-B4#-

 -D3-  F3- A3-   C3-  E4-  G4-   B4-

-C4-  E3-  G3-  B3  -D4  -F4-  A4-  C5- 
-C4#-E3#-G3#-B3#-D4#-F4#-A4#-C5#- 

OR 

-D3#-F3#-A3#-C3#- E4#-G4#-B4#-

 -D3-  F3- A3-   C3-  E4-  G4-   B4-

-C4#-E3#-G3#-B3#-D4#-F4#-A4#-C5#- 

-C4-  E3-  G3-  B3  -D4  -F4-  A4-  C5- 

This type of tuning can be applied also as setting to 4 strings  frets of  the fretboard of the artiphon the midi-controller, and give it sound of adouble diatonic harp or flute etc.

(for he artiphon see here https://artiphon.com/  )


THIS KIND OF KEYBOARD OR TUNING OF INSTRUMENTS (FLUTES, HARMONICA, ARTIPHON , KEYBOARD ETC)  ALLOWS FOR MELODIES IMPROVISATION IN A FAST  WAY EASILY WITHOUT COMPLICATED FINGERINGS OBSTRUCTIONS AS LONG AS THE MELODY IS A DIATONIC SCALE. IT IS A BIT BETTER THAN PIANO KEYBOARD AS IT FOLLOWS THE SHAPING OF CHORDS BY INTERVALS OF 3RDS. IT IS EVEN AEASIRT THAN SINGING OR WHISTLING FROM THE HARMONY POINT OF VIEW!


The melodic corridor can be also be spotted  on the fretboard of the standard tuning 6-string guitar. It will also define an unusual shape  of the diatonic scale with 2 only notes per string. More on that on the post 94.

Similar to this harmonic scale is the next , with interval pattern 

-4-3-4-4-3-3-3-

or C3-E3-G3-B3-D#4-F#4-A4-C5

And the sequence of 3-notes chords , by successive 3 notes in this scale is the chord progression

C->Em->Gaug->Baug->D#->F#dim->Adim

Which when projected within one octave it is the 

C3-D3#-E3-F3#-G3-A3-B3-C4

which in interval structure it is the

-3-1-2-1-2-2-1

which is nothing else than the Harmonic minor (here we consider a sequence of notes and circular permutation of them the same scale)     

The last remaining permutation of harmonic 2 octaves scale , is the


-4-4-4-3-3-3-3 - or  3-3-3-3-4-4-4- 

e.g. 

C3-E3-G3#-C4-D#4-F#4-A4-C5  or as interval structure within one octave


C3-D#3-E3-F#3-G#3-A3-C4   

3-1-2-2-1-3- 

which is a 6-notes scale which is very close the 7-notes scale

3-1-2-2-1-2-1 

 which is the  2nd Harmonic minor or  Kurdi or Kassigar (see post 72).

Similar to the above 7 notes 2 octaves scale by 3rds are the 7 notes scales by 4ths and 4-octaves scale by 5ths.

In other words the next 7 notes scales

(We may agree to start from the 2nd octave)

7-notes 3 octaves scale by 4ths:

B2-E3-A3-D4-G4-C5-F5-B5

7-notes 4 octaves scale by 5ths:

F2-C3-G3-D4-A4-E5-B5-F6

See also the symmetric scales post 95 and the hexagonal pattern (Terpstra, Keyboards) of harmonies and melodies of the diatomic scales in post 99. 

(This post has not been written completely yet)

78. HARMONIC POLES AND HARMONIC WAVING IN THE CHORD PROGRESSION AND STRUCTURE OF THEMES IN THE MELODY

HARMONIC POLES AND HARMONIC WAVING IN THE CHORD PROGRESSION AND STRUCTURE OF THEMES IN THE MELODY (RULES SO AS TO PLAY ALMOST ANYTHING AND STILL SOUND HARMONIOUS AND BEAUTIFUL) :

The emotional parallel of the harmony of intervals, suggests that the chord progression and harmonic structure of the themes of the melody, waves (Harmonic waves) between the emotion of stress,intervals of 1 or 2 semitones (CHROMATIC CHANGES) to the pole of harmonic serenity ,intervals of 5 (4th) , 7 (5th) or 12 (octave) semitones (HARMONIC CHANGES).

We remind that in the post 30 we described the three relations of chord transitions corresponding to the intervals of 2nd, 3rd and 4th or 5th. 

1) Resolution relation of chords

MELODIC MEANING : When such a pair of chords accompanies a melody then taking the themes of this melody either the theme is inside the chords is or is a bridge theme relating the two chords. This means that either the interval of 5 (4th) or 7 semitones (5th) appears as a shift or translation of the theme (chord-theme) or exists inside the theme (bridge-theme).

2) Relative chords

MELODIC MEANING: When such a pair of chords accompanies a melody then taking the themes of this melody either the theme is inside the chords is or is a bridge theme relating the two chords. This means that either the interval of 3 (minor 3rd) or 4 semitones (major 3rd) appears as a shift or translation of the theme (chord-theme) or exists inside the theme (bridge-theme).


MELODIC MEANING: When such a pair of chords accompanies a melody then taking the themes of this melody either the theme is inside the chords is or is a bridge theme relating the two chords. This means that either the interval of 3 (minor 3rd) or 4 semitones (major 3rd) appears as a shift or translation of the theme (chord-theme) or exists inside the theme (bridge-theme).


3) Complementary transition of chords

MELODIC MEANING: When such a pair of chords accompanies a melody then taking the themes of this melody either the theme is inside the chords is or is a bridge theme relating the two chords. This means that either the interval of 1 (minor 2nd) or 2 semitones (major 2nd) appears as a shift or translation of the theme (chord-theme) or exists inside the theme (bridge-theme).



This harmonic waving layer is more macroscopic than the emotional and melodic meaning of single melodic theme. A single melodic theme has a simple emotional meaning and this is a simple interplay or move inside the duality of emotions (positive-negative emotions).

We remind the reader that 
1) Ascending with larger steps that those of descending indicates favor of joy

2) Accelerating ascending indicates more joy, while decelerating ascending less joy. The converse with descending.

Based on the idea of the three relations of the chords (see post 30) , we may compose beautiful chord progressions. Two general rules that create chord progressions and therefore overlying melodic themes, in such a way that almost anything we play sounds harmonic and beautiful are the next:

A11 . 1st general rule for harmonic chord progressions:  Progressions by arcs in the 12-chord cycle by intervals of 4th, with substitutions with relative chords

This cycle defines by every connected arc of it a chord progression , where a chord may be substituted  with its same root relative major or minor chord , or its lower or upper minor relative chord. Of course  as they are an arc of the above 12-cycle they are successive chords or in the harmonic relation of resolution.

E.g. B7->Em->Am->D7->G->Bm-> etc 

Or B7->Em->Am->D7->G->C->(Am orA7)->D7 etc

E.g. The well known song of Frank Sinatra "Fly me to the moon" is using this technique in its sequence of chords 

https://www.youtube.com/watch?v=U8zvLvfUzCQ

Another example is the song of Nat King Cole L.O.V.E.

(main arc is the (Em or E7)->A7->D7->G(or Bm or Gm7) ->E7 etc with backwards retraces by one chord)

https://www.youtube.com/watch?v=cZ4TfRt1NK0&list=PLCD6F28C13A0A9AC2&index=12

 A12 . 2nd general rule for harmonic chord progressions:  Two arcs  in the 12-chord cycle by intervals of 4th (substituting any of the chords with its minor if it is major or vice versa) that have distance at the closest ends either 1 , or 2 or 3 or 4 semitones!

E.g. D7->G ,(1 semitone apart)  Db7->Fm 

or D7->G, (2 semitones apart) E7->Am 

or D7->G , (3 semitones apart) B7-> Em 

or Am->D7->G, (1 semitone apart) F#7->B7->Em.

77. LONG-SHORT , TWO-NOTES, IN 3-or-4 BEATS , MELODIC THEME PATTERNS TO COMPOSE MELODIES

In the traditional Irish melodies that utilize 2-3 only major chords, while the melodic moves are 4-5 or 6-7 , but also in the traditional Greek music of the Aegean Islands, the starting and ending point of the melodic move is during the duration of a single chord and are notes of the chord! But still the rule 2/3 -1/3 for notes internal and external to the chord still holds, and the starting and ending notes of the melodic move may define the simplicial submelody. The same of course with waltz songs. 

In the harmonic method of composition (see post 9) we conversely start with the chord progression, and its chord transitions, we select the starting and ending points of the melodic moves, and then  the morphological type of the melodic move , their length , their  rhythm , harmonic speeds etc. 

LONG-SHORT TWO-STEP PATTERN
A very simple trick to create beautiful melodies like those of Irish songs is to use a two-step rhythmic and melodic scheme of one short-duration note and one long duration (like in ancient Greek language that vowels are divided into two categories long and short duration). The long duration is, say two (or three) times longer, and the long duration is a note belonging to the chord while the short duration it does not! As the timing of this elementary melodic theme pattern, is 3-beats, some times it is 3 notes but two of them the same note and we may call it also triad-pattern. Usually, the two-step pattern is of diatonic density that is the interval of the two notes is a tine or semitone. But sometimes it may be 3 or 4 semitones, that is, of middle harmonic density. One of the goals of the melody is e.g. to walk down or up an octave or an interval of pure fifth or fourth that is to go from simplicial sub-melody center to another. In this way, both internal and external bridges among the chords can be created. We compose small ripples of this walking up or down (bridges) by the two-step pattern so that the long note is always a note of the underlying chord. Thus given a chord progression we may easily compose such beautiful melodies! If there are lyrics and the lyrics e.g. are in the Greek language we immediately derive an appropriate rippling of the melody.
The transformations of this short-long 2-notes melodic theme patter may be
1) Horizontal, time-order Inversion, where the long is first then the short note
2) Vertical, pitch-order inversion, where if the short is lower note and the long is higher now the long is lower and the short is higher pitch.
3) Elongation, where if the pitch distance of the short and long is say 1 or 2 semitones it may become 3, 4, 5, or 5 semitones etc
4) Half-repetition, where the long or the short not  is repeated in the next pair of short-long. 
5) Unification, where both the short and long notes become of the same pitch and are now a single note.
6) And of course pitch translation , where we shift the pair short-long inside a scale or chord.

Such melodies can be fast e.g. 4 times the 2-notes pattern per measure, or 2 times per chord strumming.
Since the 2-notes pattern is also found in the rhythm of the dance Waltz they can be danced as waltz. That is why sometimes such 2-notes short-long notes created melodies , we may call them drunken waltzes . 

Moving in the melody from intervals of 1,2 semitones to 7 (5th) , 12(octave) is moving from anxiety and stress to joy and serenity.

When composing melodies through bridges (see post 72), the bridges themselves are not sufficient to justify the choice of the chords, and we have to walk through the notes of each chord like arpeggios and between the bridges so as to have a full melody that justifies the particular choice of the chords. But when composing melodies after a chord progression through the 2-step patterns (internal-long, external-short notes for the chord) we walk the octave at the area of each chord, therefore we use all the notes of the chord and the melody immediately justifies the choice of the chords.

Here is an example of such a melody
(Frost is all over Irish melody)

http://ungaretti.racine.ra.it/ireland/music/frost.mid

(More traditional and beautiful Irish tunes in midi files here
http://ungaretti.racine.ra.it/ireland/music/irshmenu.htm )

In this way of composing chord-transition melodic moves, the starting and ending points are of paramount importance. Generally speaking, they are not identical with the centers of the melody, as they do not last in general longer than the other notes. They can be used though to define the simplicial submelody.    


IN MORE DETAIL

Here we concentrate one only simple organization structure which the closest corresponded in the poetic language and lyrics is the word. So we introduce a concept of micro-melodic theme, called

MUSICAL WORD that we may agree to symbolize say by w. It consists of a very small number of beats higher than 2 e.g.  3 or 4, and we may symbolize it with 0,s and 1,s , which means that at this beat if no sound is heard it is zero, while if a sound is heard it is 1. E.g. (0101) or (011) etc Now we divide the word in its LONG PART , that symbolize by L(w) , and SHORT PART . that we symbolize by S(w) and so that in time duration, or beats it holds that L(w)/S(w)>=2 (e.g. L(w)/S(w)=3 etc).

The musical-words or melodic micro-themes need not be by intervals of 2nds! They can be by intervals of 3rds and 5ths or 4ths! 

Such musical words may be ascending, descending or waving. Ascending as excitation may be small (intervals of 2nd) low middle (intervals of 3rds) or high middle (interval of 5th or 4th) or high (intervals of  8th or higher) Of course, as they are combined, they definitely create the effect of waving. BUT the waving is not the very standard by intervals by 2nds but a richer one, that involves many intervals of 3rds and even 5ths, and 8ths. The simplicial sub-melody of such melodies are movements mainly with intervals by 3rds and 5ths. There is also acceleration and deceleration as the melodic theme starts and ends.

Such techniques come also from the syllabic poetry measure and also dancing measure.
E.g. in Homer's Iliad and Odyssey alternate lines of 9 and 8 syllables where the syllables in a each line are grouped by the dactylic hexameter (one long two shorts syllables or 1011 or -UU where - =10 and U=1) This is also the rhythmic tempo of the ancient but also modern Greek dance of syrtos (and laso Cretan Sousta). Thus poetic measure, musical rhythm and dancing rhythm all coincide.  It is  not a coincidence that in the modern syrtos called kalamatianos, the dancer moves 7 steps right and 2 steps left and so one, and at the same time the full measure of a poetic line of Homer's poems are 9 syllables.

E.g. we may descend with a chord say Am and its relative C (out of chords would be notes of G), and ascend with its chromatic-complementary thee G7 (out of chord notes would be those of Am or C ) etc. In other words, we ascend with even or odd notes and descend conversely. Here although we may utilize only 3 chords (Am, C, G) the alternating-changing may be fast covering practically all waving and melodies of the pentatonic or diatonic scale. The scale-completion of the melody (see post 86)  , may be at the next octave rather than in the same octave!
The rhythmic repetition 3 times then the 4th is different is more common than 2 times repeated then 2 times a different. The total range of waving say of the first 3 repetitions may be of size a 5th, while the 4th measure a range of size an 8th, or vice versa.

Let us also assume that the chord progression that underlines the melody is the X(1), X(2) ,...X(n).

As we wrote in previous posts, the melody consists by a progression of melodic themes, that are transformed, by the 4 main transformations or translation, inversion, rotation including dilation and rhythmic transformation. This is indeed happening in to the melodic micro-themes or melodic or musical words during the part of the melody that sounds during say the chord X(i) i=1,2...n, BUT we impose here a very important structure which is the key to the beautiful folk melodies, and makes them compatible with the chord progression that underlines, the melody. And this rule is a 

RULE1 OF TRANSIENT AND CHORD NOTES. Obligatory part: In simple words, each musical-word w , that has underlined chord X(i) has the notes of its long part L(w) , to be notes of the chord X(i), (which includes extended forms of X(i) like X(i)maj7 or X(i)7 or X(i)add9 or X(i)sus4 ) while , the notes of its short part S(w) to be transient and belonging to the notes of the neighboring chord that is X(i-1) or  X(i+1), (which includes extended forms of X(i+1) like X(i+1)maj7 or X(i+1)7 or X(i+1)add9 or or X(i+1)sus4) or and more rarely to the rest of the chords of the chord progression. And if so if it contains a note from a non-adjacent chord Y(j) of the progression, then usually somewhere in the progression there is a transition X(i)->Y(j) or Y(j)->X(i) . We keep the transient notes sound at most 1/3 of the time only and the notes of the chord at least 2/3 of the time, because of the rule of long and short parts of the musical word or micro-theme. No mentioning of any scale is necessary in this definition (as usually there are more than one) but only of the chord progression, which is compatible with our enhanced concept of modern harmony. Nevertheless the chord progression over which this technique produces fast melodies may contain very fast chord changes, and may not be identical with the actual chord progression that the instruments play as background to the melody. This is the concept of "ghost chords" in the melody as described in the post 87. E.g. The full ghost-chord progression may be D G D G D A D. While the chords really played is only D. 

RULE2 An alternative rule is that a musical-word w , that has underlined chord X(i) has the notes of its long part L(w) , to be notes of the chord X(i), (which includes extended forms of X(i) like X(i)maj7 or X(i)7 or X(i)add9 or X(i)sus4 ) while , the notes of its short part S(w) to be transient and is one only intermediate not between the notes of the  chord X(i) (usually a 2nd away from the notes of X(i) and preferably but not obligatory this additional note to be a note of the other chords of the progression, again preferably and if possible of the previous or next chord, rarely on  of other chords. And if so, if it contains a note from a non-adjacent chord Y(j) of the progression, then usually somewhere in the progression there is a transition X(i)->Y(j) or Y(j)->X(i) .In this way we keep the transient notes sound at most 1/3 of the time only and the notes of the chord at least 2/3 of the time, in addition to the rule of long and short parts of the musical word or micro-theme. Even if we did not have the structure of micro-themes as musical-words with long and short notes , and we are playing in a random way the three notes of the chord plus one transient, in equal time in the average, we are still in the harmony of this chord, because of the proportion 3:1. And this would still hold if we used 2 transient notes in which case we would have the time proportion 3:2.  But in addition to this rule if we want also the intervals of 3rds, 4ths, 5th and 8th to be more than 2/3 of all the intervals the way is to apply harping in a chord say with 6 or 8 steps on notes, where it is added only one intermediate note in the chord (e.g. 7nh, 6th, 4th or 2nd) and so that the created intervals of 2nd are only 2 in the 6 or 8 intervals. Then we shift to a relative chord an interval of  3rd away or to a resolution transition which is a chord in an interval  5th or 4th away , or we even shift to a chord a 2nd away in which case we do not use any additional note, and we continue so.  So finally %3rds+%4ths/5ths/8ths>=2*(% 2nds) . Again the chord progression over which this technique produces fast melodies may contain very fast chord changes, and may not be identical with the actual chord progression that the instruments play as background to the melody. This is the concept of "ghost chords" in the melody as described in the post 87. E.g. The full ghost chord progression may be D G D G D A D. While the chords really played is only D. 

76. The 4 basic transformations of the elementary melodic themes: Translation , Inversion, expansion (contraction), rotation (mutation) , . All of them both in pitch or rhythm . Loops and arpeggiators

(This post has not yet being completely written)

THE KEY-WORD HERE IN THE 4TH GENERATION DIGITAL MUSIC FOR THE MUSICAL-THEORETIC IDEAS OF THIS   POST (AS FAR AS MORDEN SOFTWARE FOR MUSIC MAKING IS ) IS MELODY-SEQUENCERS 

THE TERM  SEQUENCER MEANS HERE A LOOP OR RHYTHMIC CYCLE OF   A  MELODIC THEME THAT IS VARIATED INTERACTIVELY BY THE USER  IN A MELODIC SEQUENCER.

THERE MANY GOOD SOFTWARE PROGRAMS FOR THIS COMPOSITION AND IMPROVISATION LIKE FUGUE MACHINE, YAMAHA MOBILE SEQUENCER, THUMPJAM ETC. ALAO ARPIO  AND ARPEGGIONOME FOR GENERAL ARPEGGIOS ALTERNATED WITH MELODIC IMPROVISATIONS

The 3 elementary melodic themes, as we mentioned earlier (e.g. in posts  66 and 69 ) , are  the ascending melodic interval of two notes, the isokratic melodic interval of two equal notes and the descending melodic interval of two notes.

The 4 basic transformations of them are
(we should remark that such transformations may be interpreted not only in one pitch dimension of the scale, but in modern digital musical instruments with 2-dimesional scales layouts (like Musix (see post 12 and post 312) e.g. horizontally by 2nds andvertically by 3rds) we may have 2-dimensional interpretaion of the transformations or melodic themes variations.  In 2-dimensional interprtation except of these 4 we may also have rotations! )

1) The translation (either with intervals of 2nd , (or diatonic density) or intervals of 3rd (or middle harmonic density) or  of intervals of 4th or 5th (or high harmonic density)). Translations with intervals of 3rd, may applied without changing the underlying chord, or changing it to a relative chord. Translations with intervals of 4th and 5ths, occur when the underlying chords are in resolution-relation that is successive chords on the wheel of 4ths. Translation by one semitone or chromatic translation may happen in the cases where the underlying chords are in resolution relation (successive chords on the wheel of 4ths) and the first is a dominant 7th chord, or when the underlying chords also have roots at  distance of one semitone. 

2) The melodic density change, density contraction or expansion  called also similitude   Often it is neither isocratic expansion neither isocratic contraction but rotation in the sense of stationary cyclic waving like an harping in a chord.(see post 68 and 78) 

3) The inversion where the ascending pitch move becomes descending.

Translations and inversion may both relative to pitch or relative to rhythm.

4) Mutation:  this practically means that we give up with the particular (order topological) shape of the melodic theme and shift to a a new waving or move or melodic theme.



The 5 basic melodic moves (see e.g. post 69)  , being more complicated have more types of transformations, as derived by the writing in a pentagram :

1) Translation
2) Inversion relative to a point
3) Reflection relative to an horizontal line
4) Reflection relative to a vertical line.
5) Rhythm transformation
to the above five we may add the
6) Acceleration (e.g. from the diatonic speed or density to the middle harmonic speed or density) or Deceleration (vice-versa).

Bach has often used the above 6 transformations in his fugue.

More complicated  ways to transform a theme are at least the next 5 and combinations of them (see also post 41)
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 isocratic). We prefer spikes and scaling as the main morphological types, while the waving and isocratic as intermediate bridges. 
6) Increase or decrease the size of steps while ascending or descending (pitch dilation)

Often melodic bridges from a chord to the next, may start with harmonic speed or density covering the first chord A and then decelerate to diatonic speed or density when reaching to the next chord B. 

After the chord progression and simplicial submelody we chose, 

THE DEFINITION OF MELODIC BRIDGES THAN LINK TWO SUCCESSIVE CHORDS BETWEEN THEM AND START AND END AT THE NOTES OF  THE SIMPLICIAL SUBMELODY.

1) WHICH CHORD-TRANSITIONS (PAIRS OF CHORDS) WILL HAVE A MELODIC BRIDGE! (Usually the chord-trasnitions that are in resolutional relation, or resolutional-like relation)

2) THEN WHICH BRIDGES WILL BE ISOMORPHIC IN PITCH AND RHYTHMIC DYNAMIC SHAPE AND WHICH DIFFERENT, DEFINING THEREFORE A PARTITIONING IN THE BRIDGES.

3) THEN IF IN EACH EQUIVALENCE CLASS OF  ISOMORPHIC MELODIC BRIDGES IN THIS PARTITIONING, THE BRIDGES ARE  EVENTUALLY ASCENDING OR DESCENDING (This besides the emotional significance, determines also where to play the chord in one of the 3 neighborhoods of the fretboard)

4) FINALLY  HOW IN EACH EQUIVALENCE CLASS OF  ISOMORPHIC MELODIC BRIDGES IN THE PARTITIONING, THE COMPLICATED PITCH DYNAMIC SHAPE  OR WAVING AND RHYTHM WILL BE AS A REPETITION  OF SUCH PATTERNS OF PREVIOUS ISOMORPHIC MELODIC BRIDGES, OR VARIATION OF  SUCH PATTERNAS S SO NOT TO BE TOO BORING. (This pitch dynamic shape has again a significant emotional meaning)


5) THE JUSTIFICATION OF THE CHORD PROGRESSION USUALLY IS NOT DONE BY THE CHOICE OF THE MELODIC BRIDGES (THAT IS GIVEN THE MELODIC BRIDGES MAYBE A SIMPLER CHORD PROGRESSION MAY COVER THEM HARMONICALLY). BUT AN INTERMEDIATE HARPING OR STRUMMING OF EACH CHORD WILL ENHANCE  THE MELODY OF THE BRIDGES SO THAT ONLY THIS CHORD PROGRESSION IS JUSTIFIED!

MELODIC THEMES TRANSFORMATIONS AND SIMPLICIAL SUBMELODY

We have mentioned in post 72 that the simplicial submelody is usually  the starting or ending notes of simple melodic themes, can be external bridges (see post 72) of the chord transitions (of density diatonic or middle harmonic etc). Therefore here we apply the 3 basic transformations and starting from a single melodic theme ending to the first note of the simplicial submelody we translate or invert or vary rhythmically this theme, and make it end (or start) on the next note of the simplicial submelody.  The transformed melodic themes derived in this way cover most often two chords or a chord transition or chord relation

A melodic  theme-move, can easily have four factors that characterize it

1) If it is sad (-) or joyful (+) (we may call it minor or major  melodic move, although its underground chords sometimes , rarely  may be a  major or a minor chord respectively).

2) Its melodic density (see the 4 melodic speeds or densities, chromatic, diatonic, middle harmonic and high harmonic in post 68)

3) Its range as an interval (this is related somehow by inequality to the density as in 2). melodic theme-moves that their range is more than one octave are special in stressing the nature of being sad or joyful. 

4) Its melodic density acceleration or deceleration 

These three parameters still do not define the melodic move-theme even if we know its first note. As we see melodic theme-moves are much more complicated than 3 or 4 notes chords! When creating a melody through melodic theme-moves, ideas similar to those that structure a good chord progression may apply.

We may device a symbolism for a melodic theme move based on the above three factors as follows An1Bn2(-)(x) or An1Bn2(+)(x) where An1 is the first note and Bn2 the last note of the move (n1 n2 denote the piano octave of it) and a minus - or plus + sign if its is sad (minor)  or joyful (major) and (x)=1,2,3,4 denotes the dominating density of it is chromatic x=1, if it is diatonic x=2, if it is middle harmonic x=3 and high harmonic x=4  (see post 68)  e.g. G5A4(-)(2) . In this way we write the dynamics of he melody as a theme-progression ,much like a chord progression.