(The post has not been written yet)
In the symbolism of post 23 ,the two ways are the next
1) With A-shape as root, where the I, and V are on the same fret. In the symbolism of post 23 the (nA)X means at n-th fret play the shape A and it sounds as chord X. Here instead of X we will utilize the Latin symbols of the steps in a major scale, as it is standard in Jazz with small if the chord is minor and capital if the chord is major
So the chords I, ii, iii, IV, V, vi, vii, are played on the fretboard only as shapes A and E as follows
I=(nA)I, ii=((n+2)Am)ii , iii=((n-3)Em)iii, or ((n-1)D)iii, IV=((n-2)E)IV, or (nD)IV, V=(nE)V, vi=((n-3)Am)vi or vi=((n+2)Em)vi vii=((n-1)dim7)vii.
In short the three main major chords I, IV, V are the
I=(nA)I, IV=((n)D)IV, V=(nE)V.
It is easy to remember this 3-chord progression position, as the IV, and V are one tone apart and so they should be in the fretboard, and here have to be the E shape. While the I,is related to V as the successive resolution chord in the cycle of 12 chords (see also post 30) , thus it has to be on the same fret with V, but as an A shape.
2) With E-shape as root, where the I, and IV are on the same fret. In the symbolism of post 23 the (nE)X means at n-th fret play the shape E and it sounds as chord X. Here instead of X we will utilize the Latin symbols of the steps in a major scale, as it is standard in Jazz with small if the chord is minor and capital if the chord is major
So the chords I, ii, iii, IV, V, vi, vii, are played on the fretboard only as shapes A and E as follows
I=(nE)I, ii=((n+3)Am)ii , or ii=((n+2)Em)ii , iii=((n-1)Am)iii, IV=((n)A)IV,
V=((n+2)A)V, vi=((n-3)Em)vi , vii=((n-1)dim7)vii.
In short the three main major chords I, IV, V are the
I=(nE)I, IV=((n)A)IV, V=((n+2)A)V.
It is easy to remember this 3-chord progression position, as the IV, and V are one tone apart and so they should be in the fretboard, and here have to be the A shape. While the IV,is related to I as the successive resolution chord in the cycle of 12 chords (see also post 30) , thus it has to be on the same fret with I , and here the I is an E shape.
It easy to convert these fretboard positioned chords in to figures on the fretboard.
As for the equivalence of chords in different positions and shapes on the fretboard for the barre shapes A and E holds the rules
1) The A shape sounds as the same chord withE shape 5 frets higher , In symbols e.g. (1A)A=(5E)A and in general (nA)X=((n+5)E)X
2) The E shape sounds as the same chord with A shape 7 frets higher , In symbols e.g. (1E)E=(7A)E and in general (nE)X=((n+7)A)X
This book is for learning music composition and improvisation , based on more abstract mathematics of the music, the rhythm and the musical instrument and with new musical practice. It is a new awareness and method to link mental perception-images, the creation of feelings and finger actions. The true goal of composition and improvisation is the existential process of creating and listening it. It is both individually healing and socially celebrating.
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Monday, January 18, 2016
24. The Helmholtz acoustic just, unequal temperament 7-tone scale, and real time software that converts music in it.
see e.g. https://en.wikipedia.org/wiki/Just_intonation and http://www.phy.mtu.edu/~suits/scales.html
The next is a software justonic that gets rid of the "sweet poison" of the equal tempered tuning automatically
https://www.youtube.com/watch?v=6NlI4No3s0M
See also
http://whatmusicreallyis.com/papers/
http://whatmusicreallyis.com/papers/sweet_poison.html
The next is a software justonic that gets rid of the "sweet poison" of the equal tempered tuning automatically
https://www.youtube.com/watch?v=6NlI4No3s0M
See also
http://whatmusicreallyis.com/papers/
http://whatmusicreallyis.com/papers/sweet_poison.html
This harmonic clasification of intervals goes back to ancient Pythagorean ideas, but also to modern discoveries of the nature of senses. For example the flavors and smells are good or bad according to if the "chord" of ultra high frequency sonic high frequencies of its molecules, is harmonic or not . About this see for example the next video.
http://www.ted.com/talks/luca_turin_on_the_science_of_scent
The remarkable book by Helmholtz On the sensation of tone is online here
https://archive.org/details/onsensationsofto00helmrich
In page 193 of the book of Helmholtz is found also the experimental diagram of how the intervals of just temperament sound in dissonance or not in the human ear. The higher the score on the vertical axis the greater the dissonance. The lower end of the horizontal x-axis is the note C, and we see also across the horizontal axis the notes e, f, g, a , b etc.
For a "fluid" piano with user defined variable tuning to allow for alternative temperaments see e.g.
https://www.youtube.com/watch?v=t7Cq3pbcMkI
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
The remarkable book by Helmholtz On the sensation of tone is online here
https://archive.org/details/onsensationsofto00helmrich
In page 193 of the book of Helmholtz is found also the experimental diagram of how the intervals of just temperament sound in dissonance or not in the human ear. The higher the score on the vertical axis the greater the dissonance. The lower end of the horizontal x-axis is the note C, and we see also across the horizontal axis the notes e, f, g, a , b etc.
For a "fluid" piano with user defined variable tuning to allow for alternative temperaments see e.g.
https://www.youtube.com/watch?v=t7Cq3pbcMkI
HOW TO DERIVE NEW SCALES FROM THE HARMONICS OF A SINGLE TONE. THE HARMONIC PENTATONIC, HARMONIC 7-TONE AND HARMONIC 8-TONE SCALES.
To derive the 7 notes of the diatonic scale in major mode requires more harmonics (of a single note) that one may imagine.
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.
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
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
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
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.
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)
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) ND 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 COINCIDENCEe OFTHE NUMBER 9)
6) THE HARMONIC PENTATONIC (AN UNNOTICED SO FAR PENTATONIC SCALE!)
C-D-E-F-G-C (SEMITONE STRUCTURE 2-2-1-5)
(WITHIN THE FIRST 11 HARMONICS).
7) THE HARMONIC 6-TONES SCALE
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 )
(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.
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.
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).
OVERTONES-UNDERTONES AND HARMONICS-SUBHARMONICS
When we utilize the undertones or subharmonics the effect of minor sad chord apprears . In other words if a is a fundamental frequency the undertones are the 1/2a ,1/3 a, 1/4a , 1/5a etc
In a string of length l giving frequency a the undertones will be produced by multiplying the length of the string from l, to 2l , 3l 4l 5l etc.
Similarly a fretboard of n equal length l of frets will produce the n undertones of mini-string of length l (but not oft he whole string of n frets)
WHAT IS VERY INTERESTING IS THAT THE INITIAL MAJOR CHORD IN OVERTONES HAS A CORRESPONDING MINOR CHORD OF UNDERTONES!
This is also significant in understanding the sad emotion correlated with the minor chord as it is by contraction and lowering of a fundamental frequency compared to expansion and raising of fundamental frequency by overtones which gives the major chord.
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) ND 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 COINCIDENCEe OFTHE NUMBER 9)
6) THE HARMONIC PENTATONIC (AN UNNOTICED SO FAR PENTATONIC SCALE!)
C-D-E-F-G-C (SEMITONE STRUCTURE 2-2-1-5)
(WITHIN THE FIRST 11 HARMONICS).
7) THE HARMONIC 6-TONES SCALE
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 )
(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.
23. A system of symbolism of guitar chords that determines the exact fingering , shape and position in the guitar fretboard in the DAE system.
This system of symbolism, utilizes the DAE
system of chords on the Guitar fretboard. This also means that any classical
major or minor guitar chord on the fretboard, is one the shapes D, A, or E.
The Open chord shape C is essentially a D-shape appropriately extended ,
and the open chord G is essentially an A-shape appropriately extended.
The standard symbolism of a musical major
or minor chord is of course R for R-major and Rm for R-minor. Now we extend this
musical symbolism with two more characters that give the information of where in
the guitar fretboard is played this chord (as if shifting the 0th open fretboard) and with which of the three shapes D,
A, or E.
So the full symbol is instead of R, or Rm,
it is (XY)R or (XY)Rm where X is one of the order of fret in the guitar
fretboard , (0 for open), or the corresponding note on the E-string, and Y is
one of the symbols D, A, E denoting the shape with which the chord is played.
We may still retain the symbol C(o) and G(o) for the open chords of C and
G.
For example , the C major chord as
barre, with A-shape on the 3rd fret is (3A)C (or (GA)C) , while the G
chord as barre on the 3rd fret and of E-shape will be (3E)G.
The next is a table showing the symbols of
the open chords in the three equivalent neighbourhoods of the C-major scale
(see post 13).
Open chord (D- neighbourhood)
|
A-neighbourhood (frets 3-7)
|
E-neighbourhood (frets 7-12)
|
C(o)
|
(3A)C
|
(8E)C
|
G(o),
|
(5D)G
|
(10A)G
|
F (=(1E)F)
|
(3D)F
|
(8A)F
|
E(=(0E)E)
|
(2D)E
|
(7A)E
|
A
|
(5E)A
|
(8D)A
|
D
|
(5A)D
|
(10E)D
|
B
|
(7E)B
|
(9D)B
|
Fm (=(1E)Fm)
|
(3D)Fm
|
(8A)Fm
|
Em
|
(2D)Em
|
(7A)Em
|
Am
|
(5E)Am
|
(8D)Am
|
Dm
|
(5A)Dm
|
(10E)Dm
|
Bm
|
(7E)Bm
|
(9D)Bm
|
For triad-chords in the guitar fretboard tuned in all the strings by pure 4ths (like a 6-string bass), the symbolism is even simpler as we only need to write a) the number of the fret where the root is b) the 3 shapes a,b,c depending if its normal position 1st inversion (or cyclic permutation) and 2nd inversion (or cyclic permutation), and finally c) the type of the chord, if it is major, minor with 7th , 6th etc E.g. (5c)G etc
On the other hand, we may think of system of symbolism of triad-chords in music in general independent from the guitar fretboard. We need three elements to define any such triad-chord
1) On which octave is the root.
2) Which of the 3 cyclic permutations it is (normal position, 1st inversion, 2nd inversion)
3) What type of chord it is (major, minor, diminished augmented etc). E.g. (3c)G means it is a Gmajor triad chord with root at the 3rd octave the g3, and it is in the 2nd inversion, that is
d3-g3-b3
Here is an application of the 3 neighbourhoods of the guitar on the chords A, E, F#m, D,
(see also post 13 )
https://www.youtube.com/watch?v=aO1XZJvFXu8
The 5 -triads in successive resolution harmonic relation on the fretboard.
The best way to learn the fretboard is without mental images but only the feeling of the notes at each fret.But this takes too much practice and familiarization with the fretboard.
On the other hand the best way to learn all the fretboard through mental images,rather than feeling,is not by patterns of scales, neither by the names of all the notes of the frets, but rather with sufficient many chord-shapes that almost cover all the fretboard. And even better if these chords are organized in to easy repeating patterns. Here we describe a method, based on the triads of chords in shapes of E, A, D, so that each is relative to its previous, at the harmonic relation of successive resolution in the cycle of 4ths (see also post 30).
Here we list the chords of shapes E, A, D,on the notes of the e4-string
e4, g4, a4, b4, d5,
THE BEST WAY TO LEARN THE FRETBOARD IN ANY OPEN TUNING (E.G, OVERTONES TUNINGS OR THE CURRENT TUNING IN THIS POST) IS BY CONCEIVING THE FRETBOARD AS OF A DIATONIC INSTRUMENT, MARK THE DEFAULT PREFERED DIATONIC SCALE ON THE FRETBOARD, AND LEARN THE 3-NOTE CHORDS NORMAL FORMS (ON 3 CONSECUTIVE STRINGS) IN THIS SCALE AND TUNING.
THEN FIGURE OUT THE BASIC 3 INVERSIONS OF A TRIAD CHORD (EQUIVALENT TO THE DEA-SYSTEM) AND CORRESPOND TO EACH INVERSION D, OR E OR A, THE MODE OF THE DIATONIC SCALE THAT IT GIVES.
THEN LEARN THE MINOR CHORDS HARMONIC TRIPLET OF CHORDS AND MAJOR CHORDS HARMONIC TRIPLET OF CHORDS OF THE DIATONIC SCALE WITH ANY CONVENIENT INVERSION ON THE FREBOARD.
The 5 -triads in successive resolution harmonic relation on the fretboard.
The best way to learn the fretboard is without mental images but only the feeling of the notes at each fret.But this takes too much practice and familiarization with the fretboard.
On the other hand the best way to learn all the fretboard through mental images,rather than feeling,is not by patterns of scales, neither by the names of all the notes of the frets, but rather with sufficient many chord-shapes that almost cover all the fretboard. And even better if these chords are organized in to easy repeating patterns. Here we describe a method, based on the triads of chords in shapes of E, A, D, so that each is relative to its previous, at the harmonic relation of successive resolution in the cycle of 4ths (see also post 30).
Here we list the chords of shapes E, A, D,on the notes of the e4-string
e4, g4, a4, b4, d5,
chords (E- shape)
|
|
| ||
e4
|
(0A)A
|
(0D)D
| ||
g4 (3E)A
| (3A)C | (3D)F | ||
a4 (5E)A
|
(5A)D
|
(5D)G
| ||
b4 (7E)B
|
(7A)E
|
(7D)A
| ||
d5 (10E)D
|
(10A)G
|
(10D)C
| ||
THE BEST WAY TO LEARN THE FRETBOARD IN ANY OPEN TUNING (E.G, OVERTONES TUNINGS OR THE CURRENT TUNING IN THIS POST) IS BY CONCEIVING THE FRETBOARD AS OF A DIATONIC INSTRUMENT, MARK THE DEFAULT PREFERED DIATONIC SCALE ON THE FRETBOARD, AND LEARN THE 3-NOTE CHORDS NORMAL FORMS (ON 3 CONSECUTIVE STRINGS) IN THIS SCALE AND TUNING.
THEN FIGURE OUT THE BASIC 3 INVERSIONS OF A TRIAD CHORD (EQUIVALENT TO THE DEA-SYSTEM) AND CORRESPOND TO EACH INVERSION D, OR E OR A, THE MODE OF THE DIATONIC SCALE THAT IT GIVES.
THEN LEARN THE MINOR CHORDS HARMONIC TRIPLET OF CHORDS AND MAJOR CHORDS HARMONIC TRIPLET OF CHORDS OF THE DIATONIC SCALE WITH ANY CONVENIENT INVERSION ON THE FREBOARD.
Sunday, January 17, 2016
22. Non-equal temperament and adjustable guitars and pianos : Ancient Greek, Byzantine and other ethnic tetrachords. The Aristoxenos segments.
See also
http://ww1.byzantine-musics.com/
http://www.musipedia.org/
Non-equal temperament and adjustable guitar fretboard
https://www.youtube.com/watch?v=MYK_PF9WTRE
Non-equal temperament and adjustable piano tuning
https://www.youtube.com/watch?v=X7ti6HUX5xQ
(The post has not been written yet completely)
21. Composing alternative ethnic scales in western 12-tone equal temperament scale . Flamenco, Gypsy Jazz, Greek Bouzouki, Arabic, Chinese etc scales, The algebra of tetrachords
All three 7-notes scales made only from tones and semitones
At first a nice video about scale emerge from the harmonics of a single note!
https://www.youtube.com/watch?v=JDFa8TSn6vY
1) Western Diatonic=2-2-1-2-2-2-1 (e.g. C D E F G A B /C). This scale is optimal at the next aspect: It has the maximum number of major and minor triad-chords.About playing it it see post 4.
Notice that the diatonic scale is not directly identical with its reverse order , but it is identical with one of its cyclic permutations (e.g. starting from F ), therefore it is considered inverse-symmetric
But are there more scales (not counting their cyclic permutations that would produce their different modes) made also from successive steps of one tone or one semitone? From the mathematical point of view it is easy to find them all.
We are restricting to 7-notes scales
At first we notice that as it is a 7-notes scale it cannot have more than 5 whole notes but neither less than 5 whole notes, as this wold not give in total 12 semitones with 7 addition terms.
Therefore all such scales would be permutations of the pattern 2-2-1-2-2-2-1.
In particular all of them would have 2 semitones!.Now these semitones may be separated by 2 or 3 tones as in the diatonic scale, or by one only semitone or by zero semitones. This gives two other scales.
2) Minor melodic=(1-2-1-2-2-2-2) and all cyclic permutations of it that give its modes (known also as Hindu scale see http://www.scales-chords.com/) , This scale is identical with its inverse order
and the
3) Altered natural or second minor melodic or leading whole tone or Arabian scale or Neopolitan major =(1-1-2-2-2-2-2) and all cyclic permutations of it that give its modes. Again this scale is identical with its inverse order.
I found the term altered natural in the next classification list https://psg.com/~dlamkins/lamkins-guitar/Tools/scales.html and http://www.scales-chords.com/
Nevertheless if we do not restrict to 7-notes scales then of course there are more.
For example we may also classify all 8-notes scales made only from semitones and tones:
There are at least 10 such!
1) The 8-notes scale 1-1-1-1-2-2-2-2 and non-cyclic permutations of them like
2) 2-1-2-1-2-1-2-1 (diminished scale see https://en.wikipedia.org/wiki/Octatonic_scale ) .
3) Also the 8-tones Spanish or Jewish 1-2-1-1-1-2-2-2
4) and the 8-notes Jewish (Abot) (inverse of 8-notes Spanish-Jewish)1-2-1-2-2-2-1-1
(For Spanish-Jewish music see https://www.youtube.com/watch?v=f5cdoq8on1w&list=PLFEBD04189100E6EC&index=29 )
5) and the Japanese 8-tones Ichikosucho 2-2-1-1-1-2-2-1 (it is self inverse) e.g. (C D E F G A A# B C)
6) 1st Alternative of Spanish-Jewish 8-notes 1-1-2-1-1-2-2-2 (again it is self inverse)
7) 2nd Alternative of Spanish-Jewish 8-notes 1-1-2-2-1-1-2-2 (again self inverse)
8) 3rd Alternative of Spanish-Jewish 8-notes An extrapolation of the diatonic scale 2-2-1-2-2-2-1 and of the Harmonic minor scale (2-2-1-3-1-2-1) to an 8 notes scale made by
2-2-1-2-1-1-2-1 (C D E F G G# A B C) (it is self-inverse) In this scale the chord progression of the Andaluzian Cadenza fits exactly
9) 4th Alternative of Spanish-Jewish 8-notes (1-2-1-1)-2-(2-1-2)
10) 5th Alternative of Spanish-Jewish 8-notes 1-2-2-1-1-2-1-2 (inverse of the previous)
11) The 8-notes scale 2-1-1-3-1-2-1-1 (might be called blues arpeggio of triple maj-min-maj chord) which is derived from blues arpeggios of the triple chord maj-min-maj e.g.
from c e g b d (which includes the c-major c e g, the e-minor e g b and the g major g b d)
by extrapolation the middle note of each chord by 1-semitone making it from major minor and vice versa (in other words 3 blues arpeggios of the 3 chords) we get the 8-notes scale
c, d, eb, e, g, g# ,bb,b, c
12) Similarly the 8-notes scale 2-1-1-2-1-3-1-1 (might be called blues arpeggio of triple min-maj-min chord) which is derived from blues arpeggios of the triple chord min-maj-min e.g.
from a, c , e, g (which includes the a-minor a c e, the c-major c-e-g and the e minor e g b)
by extrapolation the middle note of each chord by 1-semitone making it from major minor and vice versa (in other words 3 blues arpeggios of the 3 chords) we get the 8-notes scale
a ,b,c.c#,eb, e, g, g# a
We may compare them with the
1-3-1-1-1-1-3-1
and the 10-notes scale 1-1-1-1-1-1-2-1-2-1
We may also notice the interesting 8-notes scale
2-2-1-1-2-2-1-1 which is very symmetric as it re[etas the pattern 2-2-1-1
A variation of this scale so as to include the chromatic pattern 1-3-1, is the next 8-note scale
2-2-1-1-1-1-3-1.
SEE ALSO THE HARMONIC SCALES DERIVED FROM THE HARMONICS OF A SINGLE TONE IN POST 81.
All the above scales may be considered extrapolations to intervals of only 1,2 semitones, of arpeggios of 3-notes chords of the type major,minor, diminished and augmented.
Finally we mention the COMPREHENSIVE 9-NOTES SCALE, (or CHINESE CHROMATIC DIATONIC 9-NOTES SCALE) which is the minimal extension of the diatonic scale, that contains also the harmonic minor scale (1st Byzantine scale) but also the Harmonic double minor (or 2nd Byzantine scale, or Hungarian or Gypsy minor scale)
We only need to add the D# and G# (2nd and 5th) notes at 7-notes diatonic scale
Thus with root C it is the
C D, D# , E, F, G, G#, A, B C
with interval structure (at a different mode from C)
2-1-1-1-2-1-1-2-1.
(Notice the successive symmetry of increasing the number of successive semitones after a tone 2).
The 7-notes sub-scale C D, E, F, G#, A, B C is the E harmonic minor while the
7-notes sub-scale C D#, E, F, G#, A, B C# with intervals structure 3-1 -1-3-1-2-1-
which is the E double harmonic minor scale.
I have created also a caval Romanian flue with 6 front and 2 thump holes with a permutation of the previous 9-note comprehensive diatonic scale, that is
A3-B3-C4-D4-D4#-E4-F4-G4-G4#-A4
and interval structure 2-1-2-1-1-1-2-1-1
which can play the natural diatonic A minor (A B D D E F G A) , but also the E4 harmonic minor (A B C D E F G# a) and E4 double harmonic minor (A B C D# E F G# A).
A similar scale is used in some 7+1 holes XIAO flutes. E.g.
D D#, E, F#, G, A, A#, B C# with intervals structure 2-1-1-1-2-1-1-2-1
which is cyclic permutation from the previous thus the same scale.
With root C it would be
C-D-D#-E-F-G-G#-A-B-C
A slightly alternative such 9-tones scale with pattern of intervals
1-1-2-1-2-2-1-1-1 or in notes e.g. starting from Bb is the
Bb-B-C-D-D#-F-G-G#-A-Bb which is met in traditional Thailand flutes and the inverse of it by chromatic Quenas.
In more detail chromatic Quenas use t 8+1 holes giving the 9-notes scale
2-1-1-1-1-1-2-2-1 e,g.
G-A-Bb-B-C-C#-D-E-F#-G
See also the maximal harmonic 9-notes scale in post 117. Still there is an alternative 9-tones scale with pattern of intervals is
2-1-1-1--2-1-1-1-2 (which is an expansion by semitones at 3-semitones intervals of the mode of the pentatonic 2-3-2-3-2 known as Egyptian 5-tonic scale ) and is also the maximal harmonic 9-notes scale (which is the 9-notes scale with the maximum number of major or minor triads see post 117)
This scale is sometimes is used in folk bass Armenian duduk-like winds
e.g. Bb2-C3-C#3-D3-D#3-F3-F#3-G3-G#3-Bb3
and it happens that I have one such wind.
All 7-notes scales made from semitones, tones and at least one 3-semitone.
Many such scales are used in the Greek folk music with Buzuki, and have their origin in ancient Greece, Byzantine empire, and Arabic music. Many of them have Arabic names although they are played on the 12-semitone Bach equal temperament scale.
For the names of some of these scales see http://www.scales-chords.com/
Therefore its is worth finding them all. We have already found all of them that they do not contain a 3-semitone (see post 51, the diatonic, the melodic minor and the second melodic minor) . So let us find all that contain at least one 3-semitone We already know so far
1) the Harmonic minor= (1-3-1)-2-(-1-2-2), and
2)the Romani (or Hungarian /Ukrainian/Flamenco) double minor (modes also of Niavent and Hijaskar) also called Byzantine or Harmonic double minor scale (and by some also called abydos Egyptin scale)=(1-3-1)-2-(-1-3-1) (notice that the inverse order of it is identical with it).
The Romanian kaval flutes with 5 holes play the 6-notes scale 2-1-3-1-1-4 E.G. A4-B4-C5-D#5-E5-F5-A5 and by adding a 6th thump hole to play the G5# it becomes a Harmonic double minor scale A4-B4-C5-D#5-E5-F5-G5#-A5 or 2-1-3-1-1-3-1 which is a mode of the E5 harmonic double minor scale.
3) A slight alternation of it is the Persian scale or todi theta scale=(1-3-1-1-2-3-1)
4) Inverse Persian scale or Purvi Theta scale (3-1-1-3-2-1-1) or Byzantine parachromatic scale
The last 3 contain two 3-semitones, but only the Harmonic double minor two tetrachords 1-3-1 ! And the other two scales of post 50 are
This scale (as the harmonic double minor or Byzantine double minor too) too is directly derivable from the pentatonic scale e.g., the Egyptian mode of it 2-3-2-3-2 gives the
2-1-1-3-1-1-3 This scale can be easily played in the Shakuhachi minor pentatonic flutes.
5) the 2nd Harmonic minor or Kurdi or Kassigar=(1-3-1)-2-(-2-1-2) (which is the inverse order of the Harmonic minor),
6) and the 3rd Harmonic minor or Shamba (which is the inverse of the Neopolitan scale below) =(1-3-1)-2-(-2-2-1)
all of them containing the oriental tetra-chord 1-3-1.
We may take the inverse order of the Shamba which is called the Neopolitan scale
7) The 4th Harmonic minor or Neopolitan scale (different from the major and the minor Neopolitan)= (1-3-1)-1-(-2-2-2)
Notice that the Neopolitan scale is made by the inverses of 2 ancient Greek tetrachords the 1-1-3 and the 1-2-2 withe and in between tone 2 (disjunction of tetrachords or divorced tetrachords) . The 1-1-3 was called in ancient Greece the tonal tetrachor of the Chromatic Generation and the 1-2-2 the syntono tetrachord of the Diatonic generation.
Now are there more? Certainly there are! Many of them are modern versions in the 12-semitones scale realizations of ancient Byzantine 7-notes scales or "sounds" (ηχοι). If there are two 3-semitones as in the Romani double minor, we may have a permutation of it , which is not a cyclic permutation (mode of the Romani double minor) which are the next
8) (3-1-3-2-1-1-1)
9) Second Harmonic or Romani double minor=(3-1-3-1-2-1-1) or gypsy hexatonic or Mela Gayakapriya, Raga Kalakanti (see post 227 )
10) Third Harmonic or Romani double minor=(3-1-3-1-1-2-1) inverse of Mela Ganamurti, Raga Ganasamavarali (see post 227)
11) (3-1-3-1-1-1-2)
12) (3-1-2-3-1-1-1)
The next scales do not sound too much as minor scales as the 3-3 , or 3-2-3 is not sad and are mainly extrapolations of the western or Chinese/Mongolian pentatonic !
13) (3-2-3-1-1-1-1) (with a very peculiar sound)
14) (3-2-1-3-1-1-1)
15) (3-3-2-1-1-1-1) (with a very peculiar sound)
16) (3-3-1-2-1-1-1)
17) (3-3-1-1-2-1-1)
18) (3-3-1-1-1-2-1)
19) (3-3-1-1-1-1-2) (this is the inverse order of scale 15)
While if it has only one 3-semitone we may take non-cyclic permutations of the 3) and 4) like the next fr which ma nor aware of names, but maybe there are in modern Arabic music (At first we make all possible combinations of different intervals around 3, and then all possible non-cyclic permutations of the rest of the intervals)
20) (1-3-2)-1-(-2-1-2)
21) (1-3-2)-1-(-1-2-2)
22) (1-3-2)-2-(-1-1-2)
23) Called Enigmatic (1-3-2)-2-(-2-1-1)
24) (1-3-2)-2-(-1-2-1)
25) (1-3-2)-1-(-2-2-1). This scale is by combining the ancient Greek tonal tetrachord of the Chromatic generation 1-1-3 in a disjunctive way -2- ith the syntono tetrachord 1-2-2 of the Diatonic generation.
26) (2-3-1)-1-(-2-1-2)
27) (2-3-1)-1-(-1-2-2)
28) (2-3-1)-2-(-1-1-2)
29) (2-3-1)-2-(-2-1-1)
30) Called Hungarian major scale (2-3-1)-2-(-1-2-1)
31) (2-3-1)-1-(-2-2-1) This scale is
refered in this video https://www.youtube.com/watch?v=mjttaiOq-8Q
as the 7-notes soul scale and is refred as the major pentatonic scale with added flat 3rd and flat 7nth!
32) (2-3-2)-1-(-2-1-1)
33) (2-3-2)-1-(-1-1-2)
34) (2-3-2)-1-(-1-2-1)
35) (2-3-2)-2-(-1-1-1)
Some of the 4 and 5-notes sub-scales (tetra-chords and penta-chords) of the above 7-notes scales have known Arabic names
Tetra-chords (all the next are diatonic tetra-chords)
Rast 2-2-1
Ussak 1-2-2
Kurdi 2-1-2
The next is from the second melodic minor or leading whole tone scale
Shamba 2-1-1
The next contain a 3-semitone
Niavent 2-1-3
Hijazz 1-3-1
Huzam 3-1-1
Piraeus 1-3-2
And the next are 5-notes sub-scales (penta-chords) that are essentially diatonic
Rast 2-2-1-2
Ussak 1-2-2-2
Kurdi 2-1-2-1 (this is from the melodic minor)
Minor 2-1-2-2
The next contain a 3-semitone
Shamba 2-1-1-3
Nikriz 2-1-3-1
Hijazz 1-3-1-2
Huzam 3-1-1-2
See also https://www.youtube.com/watch?v=B6xddWJFmt8
For Arabic names of many of the previous scales see
http://www.maqamworld.com/
Ir is obvious also that by extrapolating the steps 3-semitones to 2+1 we get 8-notes and 9-notes scales that are made only from steps of 1 and 2 semitones.
For a list of scales with their chords see http://www.scales-chords.com/
We may compare these scales with the 6-notes minor blue scale derived from the minor pentatonic scale , with interval structure
3-2-1-1-3-2 (see e.g. http://www.jazzguitar.be/minor-blues-scale.html ) which is analysis of the western pentatonic below.
(for the 6-notes major blues scale derived from the major pentatonic scale see post 54)
The 6-tone scale 1-3-3-1-3-1, the inverse which is 1-3-1-3-3-1 and the
1-3-1-3-1-3
Or the 5-tone Egyptian / Mongolian mode of the western pentatonic 2-3-2-3-2
The 6-tone Prometheus 2-2-2-3-1-2
The 6-tones inverse Prometheus 2-1-3-2-2-2
or the 6-tone 2-3-2-2-1-2 which is extrapolation of the 5-tonic
We may compare these scales with the Western pentatonic scale that has also two 3-semitones, that is of interval step structure 2-2-3-2-3 (also known as Egyptian/Mongolian).
Or compare them with the oriental 6-note scale 1-3-1-3-1-3
Or compare them with other 4-notes scales like 3-3-3-3
the 3-5-3-1 and 2-1-4-5 or 3-2-2-5 or 4-2-1-5, and 2-2-3-5 that extrapolations to 4-notes scales of 3-notes major or minor chords arpeggios-scales
Or the 8-notes Algerian=2-1-2-1-1-1-3-1
Or the 8-notes extrapolation of the Romani double minor
1-3-1-1-1-1-3-1
Or the Chinese 5-notes scale that have 2-tones steps , in other words interval structure
Interval fro the root : 1, 3, #4, 5, 7
Intervals in steps: 4 - 2 - 1 - 4 - 1 (the tetra-chord 1-4-1 is used instead of the oriental
1-3-1)
Formula: Quadra-step, Whole, Half, Quadra-step, Half 4 - 2 - 1 - 4 - 1
See e.g.
http://www.pianoscales.org/chinese.html
C: C, E, F#, G, B, C (we notice that it is a sub-scale of the F-major 7-notes scale)
For example if we take the mode of the diatonic scale starting from F
F G A B C D E F , a Chinese scale will be shaped by keeping the semitones B C,, E, F, but eliminating the notes G, D , Thus it will be
F, A, B, C, E, / F
Somehow all the above scales may be considered extrapolations of arpeggios of 3-notes chords of the type major,minor, diminished and augmented.
The oriental (Hijazz) tetra-chord and the creation of all possible 7-notes scales by combinations with diatonic tetra-chords .
The oriental tetra-chord which is called also Hijazz, is 1 semitone-3 semitones-1 semitone
(1-3-1) in total 5 semitones. Its origin is in ancient Greece and Byzantine empire. Thus if it is to combine it with other tetra-chords of total sum 5 it should be combined with one tone distance to make in total 12 semitones (5+2+5=12).
Now combining it with all possible diatonic tetra-chords that is (1-2-2), (2-2-1), (2-1-2), (2-2-2) it gives the scales
1) Harmonic minor (called also mode of Hijazz or Byzantine minor) = (1-3-1)-2-(-1-2-2),
A mode (cyclic permutation) of the harmonic minor is called also Blue-scale in American folk music, and another cyclic permutation of it the Romani minor scale
Chords of the harmonic minor (intervals 2-1-2-2-1-3-1):
Triads: min dim aug min maj maj dim
Extended
4-notes chord: min/maj7 m7b5 maj7#5 min7 dom7 maj7 dim7
For example for the A harmonic minor(A,B,C,D,E,F,G#) (intervals 2-1-2-2-1-3-1),the chords are
Typical progression
More general the chords that fit to the harmonic minor (not taking necessarily the notes in alternating order , that is take one leave one or 1-3-5 etc) are the next
2) One that may be called 2nd Harmonic minor , called also mode of Kurdi or Kasigar =(1-3-1)-2-(2-1-2) and
3) One that may be called 3rd Harmonic minor, called also mode of Shamba=(1-3-1)-2-(-2-2-1).
4) And one than maybe called 4th Harmonic minor. If we take the Shamba in the reverse order we get the Neopolitan scale= (1-3-1)-1-(-2-2-2)
The Neopolitan scale , from D has the next chords (intervals 1-2-2-2-1-3-1)
5) While if combined by itself it gives what is called Romani (Hungarian or Gypsy or Byzantine double minor or Harmonic double minor or mode of the Niavent)=(1-3-1)-2-(-1-3-1).
These names are of Arabic origin but are used in Greek folk music with Buzuki , as they are played on the western 12-semitones Bach equal temperament scale.
The chords that fit to the Romani or Harmonic double minor scale , we take here as an example the D Romani or Hungarian double minor are the next:
Two variations of the Byzantine double minor scale are the Persian and inverse Persian scales
But only the Byzantine (harmonic) double minor containes two tetra-chords 1-3-1 !
Persian scale or todi theta scale=(1-3-1-1-2-3-1)
E.g. starting from C
We may compare these scales with the melodic minor (e.g. A melodic minor) and its chords
All the above scales may be considered extrapolations to intervals of only 1,2 ,3 semitones, of arpeggios of 3-notes chords of the type major,minor, diminished and augmented.
If in the above 5 scales of 1st 2nd 3rd and 4th Harmonic minor and Harmonic double minor scales we extrapolate the intervals of 3-semitones as ascending to 2+1 semitones, we get the next 8-tones scales made only from steps of 2 and 1 semitones
1) From the 1st Harmonic minor 2-1-2-2-1-3-1 the 8-notes scale 2-1-2-2-1-2-1-1 which is a cyclic permutation or mode of what is called in post 51, 3rd alternative of Spanish-Jewish 8-notes scale that contains in its chords the Andaluzian Cadenza.
2) From the 2nd Harmonic minor (1-3-1)-2-(2-1-2) the 8-notes scale (1-2-1-1)-2-(2-1-2)
which is a cyclic permutation or mode of what is called in post 51, 4th alternative of Spanish-Jewish 8-notes scale
3) From the 3rd Harmonic minor (1-3-1)-2-(-2-2-1). the 8-notes scale (1-2-1-1)-2-(2-2-1)
which is a cyclic permutation or mode of what is called in post 51, 1st alternative of Spanish-Jewish 8-notes scale
4) From the 4th Harmonic minor 1-2-2-2-1-3-1 the 8-notes scale 1-2-2-2-1-2-1-1
which is a cyclic permutation or mode of what is called in post 51, Spanish-Jewish 8-notes scale
5) From the Harmonic double minor 2-1-3-1-1-3-1 the 9-notes scale 2-1-2-1-1-1-2-1-1
Nevertheless if cyclic permutations of them are not considered different then 2-2-1=2-1-2. 1-2-1=2-1-1, 1-3-1=-1-1-3 and so they are only 5 different
Diatonic
At first a nice video about scale emerge from the harmonics of a single note!
https://www.youtube.com/watch?v=JDFa8TSn6vY
As it is known the diatonic scale (major or minor mode) is made from successive steps of one tone or one semitone. In particular of 5 tones and 2 semitones in the next order (not counting their cyclic permutations that produce the different diatonic modes)
1) Western Diatonic=2-2-1-2-2-2-1 (e.g. C D E F G A B /C). This scale is optimal at the next aspect: It has the maximum number of major and minor triad-chords.About playing it it see post 4.
Notice that the diatonic scale is not directly identical with its reverse order , but it is identical with one of its cyclic permutations (e.g. starting from F ), therefore it is considered inverse-symmetric
But are there more scales (not counting their cyclic permutations that would produce their different modes) made also from successive steps of one tone or one semitone? From the mathematical point of view it is easy to find them all.
We are restricting to 7-notes scales
At first we notice that as it is a 7-notes scale it cannot have more than 5 whole notes but neither less than 5 whole notes, as this wold not give in total 12 semitones with 7 addition terms.
Therefore all such scales would be permutations of the pattern 2-2-1-2-2-2-1.
In particular all of them would have 2 semitones!.Now these semitones may be separated by 2 or 3 tones as in the diatonic scale, or by one only semitone or by zero semitones. This gives two other scales.
2) Minor melodic=(1-2-1-2-2-2-2) and all cyclic permutations of it that give its modes (known also as Hindu scale see http://www.scales-chords.com/) , This scale is identical with its inverse order
and the
3) Altered natural or second minor melodic or leading whole tone or Arabian scale or Neopolitan major =(1-1-2-2-2-2-2) and all cyclic permutations of it that give its modes. Again this scale is identical with its inverse order.
I found the term altered natural in the next classification list https://psg.com/~dlamkins/lamkins-guitar/Tools/scales.html and http://www.scales-chords.com/
It seems to me that besides the harmonic minor and Romani double minor, Stephan Grappelli and Django Reinhardt were using these two scales together with 8-notes scales made only from tones and semitones (see 10 of them below) in their improvisational embellishments.
Nevertheless if we do not restrict to 7-notes scales then of course there are more.
For example we may also classify all 8-notes scales made only from semitones and tones:
There are at least 10 such!
1) The 8-notes scale 1-1-1-1-2-2-2-2 and non-cyclic permutations of them like
2) 2-1-2-1-2-1-2-1 (diminished scale see https://en.wikipedia.org/wiki/Octatonic_scale ) .
3) Also the 8-tones Spanish or Jewish 1-2-1-1-1-2-2-2
4) and the 8-notes Jewish (Abot) (inverse of 8-notes Spanish-Jewish)1-2-1-2-2-2-1-1
(For Spanish-Jewish music see https://www.youtube.com/watch?v=f5cdoq8on1w&list=PLFEBD04189100E6EC&index=29 )
5) and the Japanese 8-tones Ichikosucho 2-2-1-1-1-2-2-1 (it is self inverse) e.g. (C D E F G A A# B C)
6) 1st Alternative of Spanish-Jewish 8-notes 1-1-2-1-1-2-2-2 (again it is self inverse)
7) 2nd Alternative of Spanish-Jewish 8-notes 1-1-2-2-1-1-2-2 (again self inverse)
8) 3rd Alternative of Spanish-Jewish 8-notes An extrapolation of the diatonic scale 2-2-1-2-2-2-1 and of the Harmonic minor scale (2-2-1-3-1-2-1) to an 8 notes scale made by
2-2-1-2-1-1-2-1 (C D E F G G# A B C) (it is self-inverse) In this scale the chord progression of the Andaluzian Cadenza fits exactly
9) 4th Alternative of Spanish-Jewish 8-notes (1-2-1-1)-2-(2-1-2)
10) 5th Alternative of Spanish-Jewish 8-notes 1-2-2-1-1-2-1-2 (inverse of the previous)
11) The 8-notes scale 2-1-1-3-1-2-1-1 (might be called blues arpeggio of triple maj-min-maj chord) which is derived from blues arpeggios of the triple chord maj-min-maj e.g.
from c e g b d (which includes the c-major c e g, the e-minor e g b and the g major g b d)
by extrapolation the middle note of each chord by 1-semitone making it from major minor and vice versa (in other words 3 blues arpeggios of the 3 chords) we get the 8-notes scale
c, d, eb, e, g, g# ,bb,b, c
12) Similarly the 8-notes scale 2-1-1-2-1-3-1-1 (might be called blues arpeggio of triple min-maj-min chord) which is derived from blues arpeggios of the triple chord min-maj-min e.g.
from a, c , e, g (which includes the a-minor a c e, the c-major c-e-g and the e minor e g b)
by extrapolation the middle note of each chord by 1-semitone making it from major minor and vice versa (in other words 3 blues arpeggios of the 3 chords) we get the 8-notes scale
a ,b,c.c#,eb, e, g, g# a
We may compare them with the
1-3-1-1-1-1-3-1
and the 10-notes scale 1-1-1-1-1-1-2-1-2-1
We may also notice the interesting 8-notes scale
2-2-1-1-2-2-1-1 which is very symmetric as it re[etas the pattern 2-2-1-1
A variation of this scale so as to include the chromatic pattern 1-3-1, is the next 8-note scale
2-2-1-1-1-1-3-1.
SEE ALSO THE HARMONIC SCALES DERIVED FROM THE HARMONICS OF A SINGLE TONE IN POST 81.
All the above scales may be considered extrapolations to intervals of only 1,2 semitones, of arpeggios of 3-notes chords of the type major,minor, diminished and augmented.
Finally we mention the COMPREHENSIVE 9-NOTES SCALE, (or CHINESE CHROMATIC DIATONIC 9-NOTES SCALE) which is the minimal extension of the diatonic scale, that contains also the harmonic minor scale (1st Byzantine scale) but also the Harmonic double minor (or 2nd Byzantine scale, or Hungarian or Gypsy minor scale)
We only need to add the D# and G# (2nd and 5th) notes at 7-notes diatonic scale
Thus with root C it is the
C D, D# , E, F, G, G#, A, B C
with interval structure (at a different mode from C)
2-1-1-1-2-1-1-2-1.
(Notice the successive symmetry of increasing the number of successive semitones after a tone 2).
The 7-notes sub-scale C D, E, F, G#, A, B C is the E harmonic minor while the
7-notes sub-scale C D#, E, F, G#, A, B C# with intervals structure 3-1 -1-3-1-2-1-
which is the E double harmonic minor scale.
I have created also a caval Romanian flue with 6 front and 2 thump holes with a permutation of the previous 9-note comprehensive diatonic scale, that is
A3-B3-C4-D4-D4#-E4-F4-G4-G4#-A4
and interval structure 2-1-2-1-1-1-2-1-1
which can play the natural diatonic A minor (A B D D E F G A) , but also the E4 harmonic minor (A B C D E F G# a) and E4 double harmonic minor (A B C D# E F G# A).
A similar scale is used in some 7+1 holes XIAO flutes. E.g.
D D#, E, F#, G, A, A#, B C# with intervals structure 2-1-1-1-2-1-1-2-1
which is cyclic permutation from the previous thus the same scale.
With root C it would be
C-D-D#-E-F-G-G#-A-B-C
A slightly alternative such 9-tones scale with pattern of intervals
1-1-2-1-2-2-1-1-1 or in notes e.g. starting from Bb is the
Bb-B-C-D-D#-F-G-G#-A-Bb which is met in traditional Thailand flutes and the inverse of it by chromatic Quenas.
In more detail chromatic Quenas use t 8+1 holes giving the 9-notes scale
2-1-1-1-1-1-2-2-1 e,g.
G-A-Bb-B-C-C#-D-E-F#-G
See also the maximal harmonic 9-notes scale in post 117. Still there is an alternative 9-tones scale with pattern of intervals is
2-1-1-1--2-1-1-1-2 (which is an expansion by semitones at 3-semitones intervals of the mode of the pentatonic 2-3-2-3-2 known as Egyptian 5-tonic scale ) and is also the maximal harmonic 9-notes scale (which is the 9-notes scale with the maximum number of major or minor triads see post 117)
This scale is sometimes is used in folk bass Armenian duduk-like winds
e.g. Bb2-C3-C#3-D3-D#3-F3-F#3-G3-G#3-Bb3
and it happens that I have one such wind.
All 7-notes scales made from semitones, tones and at least one 3-semitone.
Many such scales are used in the Greek folk music with Buzuki, and have their origin in ancient Greece, Byzantine empire, and Arabic music. Many of them have Arabic names although they are played on the 12-semitone Bach equal temperament scale.
For the names of some of these scales see http://www.scales-chords.com/
Therefore its is worth finding them all. We have already found all of them that they do not contain a 3-semitone (see post 51, the diatonic, the melodic minor and the second melodic minor) . So let us find all that contain at least one 3-semitone We already know so far
1) the Harmonic minor= (1-3-1)-2-(-1-2-2), and
2)the Romani (or Hungarian /Ukrainian/Flamenco) double minor (modes also of Niavent and Hijaskar) also called Byzantine or Harmonic double minor scale (and by some also called abydos Egyptin scale)=(1-3-1)-2-(-1-3-1) (notice that the inverse order of it is identical with it).
The Romanian kaval flutes with 5 holes play the 6-notes scale 2-1-3-1-1-4 E.G. A4-B4-C5-D#5-E5-F5-A5 and by adding a 6th thump hole to play the G5# it becomes a Harmonic double minor scale A4-B4-C5-D#5-E5-F5-G5#-A5 or 2-1-3-1-1-3-1 which is a mode of the E5 harmonic double minor scale.
3) A slight alternation of it is the Persian scale or todi theta scale=(1-3-1-1-2-3-1)
4) Inverse Persian scale or Purvi Theta scale (3-1-1-3-2-1-1) or Byzantine parachromatic scale
The last 3 contain two 3-semitones, but only the Harmonic double minor two tetrachords 1-3-1 ! And the other two scales of post 50 are
This scale (as the harmonic double minor or Byzantine double minor too) too is directly derivable from the pentatonic scale e.g., the Egyptian mode of it 2-3-2-3-2 gives the
2-1-1-3-1-1-3 This scale can be easily played in the Shakuhachi minor pentatonic flutes.
5) the 2nd Harmonic minor or Kurdi or Kassigar=(1-3-1)-2-(-2-1-2) (which is the inverse order of the Harmonic minor),
6) and the 3rd Harmonic minor or Shamba (which is the inverse of the Neopolitan scale below) =(1-3-1)-2-(-2-2-1)
all of them containing the oriental tetra-chord 1-3-1.
We may take the inverse order of the Shamba which is called the Neopolitan scale
7) The 4th Harmonic minor or Neopolitan scale (different from the major and the minor Neopolitan)= (1-3-1)-1-(-2-2-2)
Notice that the Neopolitan scale is made by the inverses of 2 ancient Greek tetrachords the 1-1-3 and the 1-2-2 withe and in between tone 2 (disjunction of tetrachords or divorced tetrachords) . The 1-1-3 was called in ancient Greece the tonal tetrachor of the Chromatic Generation and the 1-2-2 the syntono tetrachord of the Diatonic generation.
Now are there more? Certainly there are! Many of them are modern versions in the 12-semitones scale realizations of ancient Byzantine 7-notes scales or "sounds" (ηχοι). If there are two 3-semitones as in the Romani double minor, we may have a permutation of it , which is not a cyclic permutation (mode of the Romani double minor) which are the next
8) (3-1-3-2-1-1-1)
9) Second Harmonic or Romani double minor=(3-1-3-1-2-1-1) or gypsy hexatonic or Mela Gayakapriya, Raga Kalakanti (see post 227 )
10) Third Harmonic or Romani double minor=(3-1-3-1-1-2-1) inverse of Mela Ganamurti, Raga Ganasamavarali (see post 227)
11) (3-1-3-1-1-1-2)
12) (3-1-2-3-1-1-1)
The next scales do not sound too much as minor scales as the 3-3 , or 3-2-3 is not sad and are mainly extrapolations of the western or Chinese/Mongolian pentatonic !
13) (3-2-3-1-1-1-1) (with a very peculiar sound)
14) (3-2-1-3-1-1-1)
15) (3-3-2-1-1-1-1) (with a very peculiar sound)
16) (3-3-1-2-1-1-1)
17) (3-3-1-1-2-1-1)
18) (3-3-1-1-1-2-1)
19) (3-3-1-1-1-1-2) (this is the inverse order of scale 15)
While if it has only one 3-semitone we may take non-cyclic permutations of the 3) and 4) like the next fr which ma nor aware of names, but maybe there are in modern Arabic music (At first we make all possible combinations of different intervals around 3, and then all possible non-cyclic permutations of the rest of the intervals)
20) (1-3-2)-1-(-2-1-2)
21) (1-3-2)-1-(-1-2-2)
22) (1-3-2)-2-(-1-1-2)
23) Called Enigmatic (1-3-2)-2-(-2-1-1)
24) (1-3-2)-2-(-1-2-1)
25) (1-3-2)-1-(-2-2-1). This scale is by combining the ancient Greek tonal tetrachord of the Chromatic generation 1-1-3 in a disjunctive way -2- ith the syntono tetrachord 1-2-2 of the Diatonic generation.
26) (2-3-1)-1-(-2-1-2)
27) (2-3-1)-1-(-1-2-2)
28) (2-3-1)-2-(-1-1-2)
29) (2-3-1)-2-(-2-1-1)
30) Called Hungarian major scale (2-3-1)-2-(-1-2-1)
31) (2-3-1)-1-(-2-2-1) This scale is
refered in this video https://www.youtube.com/watch?v=mjttaiOq-8Q
as the 7-notes soul scale and is refred as the major pentatonic scale with added flat 3rd and flat 7nth!
32) (2-3-2)-1-(-2-1-1)
33) (2-3-2)-1-(-1-1-2)
34) (2-3-2)-1-(-1-2-1)
35) (2-3-2)-2-(-1-1-1)
It is clear that these scales can be ordered according to how many usual chords they define (like major, minor diminished, augmented). The more non-weird chords the higher in the list. In other words they are not all of them the same good from the point of view of harmony of the chords they define, although they may seem very similar from the melodic point of view.
There are also the 6 notes Hirajoshi scale
http://www.flutopedia.com/scale_Hirajoshi_Extended.htm
and the Miyako Bushi extended scale
There are also the 6 notes Hirajoshi scale
http://www.flutopedia.com/scale_Hirajoshi_Extended.htm
and the Miyako Bushi extended scale
http://www.flutopedia.com/scale_MiyakoBushi_Extended.htm
ALL 4-NOTES SCALES THAT CONTAIN THE PATTERN IN SEMITONES 1-3-1 , 1-1-3, or two 3's
1) 1-3-1-7
2) 1-3-7-1
3) 1-3-3-5
4) 1-3-5-3
ALL 5 CHROMATIC 5-NOTES SCALES THAT CONTAIN THE PATTERN IN SEMITONES 1-3-1 , 1-1-3, or two 3's
1) 1-3-1-3-4 (a 6-notes sub-scale of the double harmonic minor)
2) 1-3-1-4-3
3) 1-3-3-1-4
4) 1-1-3-3-4
5) 1-1-3-4-3
ALL 6-NOTES SCALES THAT CONTAIN THE PATTERN IN SEMITONES 1-3-1 .
1) 2-1-3-1-2-3 (the Erik Satie 6-note scale)
2) 1-3-1-3-2-2
3) 1-3-1-3-1-3 (a Messiaen scale)
4) 1-3-1-3-3-1
5) 1-3-1-4-2-1
6) 1-3-1-1-4-2
7) 1-3-1-2-4-1
8) 1-3-1-5-1-1
9) 1-3-1-1-5-1
NOTICE THAT THE FAMOUS BLUES 6-NOTES SCALE CONTAIN IS NOT THE 1-3-1 BUT THE 1-1-3:
3-2-1-1-3-2
ALL 4-NOTES SCALES THAT CONTAIN THE PATTERN IN SEMITONES 1-3-1 , 1-1-3, or two 3's
1) 1-3-1-7
2) 1-3-7-1
3) 1-3-3-5
4) 1-3-5-3
ALL 5 CHROMATIC 5-NOTES SCALES THAT CONTAIN THE PATTERN IN SEMITONES 1-3-1 , 1-1-3, or two 3's
1) 1-3-1-3-4 (a 6-notes sub-scale of the double harmonic minor)
2) 1-3-1-4-3
3) 1-3-3-1-4
4) 1-1-3-3-4
5) 1-1-3-4-3
ALL 6-NOTES SCALES THAT CONTAIN THE PATTERN IN SEMITONES 1-3-1 .
1) 2-1-3-1-2-3 (the Erik Satie 6-note scale)
2) 1-3-1-3-2-2
3) 1-3-1-3-1-3 (a Messiaen scale)
4) 1-3-1-3-3-1
5) 1-3-1-4-2-1
6) 1-3-1-1-4-2
7) 1-3-1-2-4-1
8) 1-3-1-5-1-1
9) 1-3-1-1-5-1
NOTICE THAT THE FAMOUS BLUES 6-NOTES SCALE CONTAIN IS NOT THE 1-3-1 BUT THE 1-1-3:
3-2-1-1-3-2
Tetra-chords (all the next are diatonic tetra-chords)
Rast 2-2-1
Ussak 1-2-2
Kurdi 2-1-2
The next is from the second melodic minor or leading whole tone scale
Shamba 2-1-1
The next contain a 3-semitone
Niavent 2-1-3
Hijazz 1-3-1
Huzam 3-1-1
Piraeus 1-3-2
And the next are 5-notes sub-scales (penta-chords) that are essentially diatonic
Rast 2-2-1-2
Ussak 1-2-2-2
Kurdi 2-1-2-1 (this is from the melodic minor)
Minor 2-1-2-2
The next contain a 3-semitone
Shamba 2-1-1-3
Nikriz 2-1-3-1
Hijazz 1-3-1-2
Huzam 3-1-1-2
See also https://www.youtube.com/watch?v=B6xddWJFmt8
For Arabic names of many of the previous scales see
http://www.maqamworld.com/
And if we restrict to only 4-notes sub-scales (tetra-chords) , having inverse such scales not different, then we are left with a small number of 10 of such characteristic tetra-chords
They are also all such tetra-chords containing intervals of 1,2,3, and where inverses and cyclic permutations of them do not count as different . Obviously all of the above scales are compositions of two of them, with possibly an extra interval between them
Diatonic
They are also all such tetra-chords containing intervals of 1,2,3, and where inverses and cyclic permutations of them do not count as different . Obviously all of the above scales are compositions of two of them, with possibly an extra interval between them
Diatonic
2-2-1, (major, natural minor rast, ussak)
2-2-2, (major, augmented)
Melodic minor, double minor (shabach)
1-2-1
Harmonic minor (Hijazz,Huzam)
1-3-1
Harmonic double minor
1-2-3,
Diminished
Diminished
3-3-3 , (diminished 7nth)
3-3-1,
Pentatonic
3-3-2,
2-2-3
2-2-3
Chromatic
1-1-1
Ir is obvious also that by extrapolating the steps 3-semitones to 2+1 we get 8-notes and 9-notes scales that are made only from steps of 1 and 2 semitones.
For a list of scales with their chords see http://www.scales-chords.com/
We may compare these scales with the 6-notes minor blue scale derived from the minor pentatonic scale , with interval structure
3-2-1-1-3-2 (see e.g. http://www.jazzguitar.be/minor-blues-scale.html ) which is analysis of the western pentatonic below.
(for the 6-notes major blues scale derived from the major pentatonic scale see post 54)
The 6-tone scale 1-3-3-1-3-1, the inverse which is 1-3-1-3-3-1 and the
1-3-1-3-1-3
Or the 5-tone Egyptian / Mongolian mode of the western pentatonic 2-3-2-3-2
The 6-tone Prometheus 2-2-2-3-1-2
The 6-tones inverse Prometheus 2-1-3-2-2-2
or the 6-tone 2-3-2-2-1-2 which is extrapolation of the 5-tonic
We may compare these scales with the Western pentatonic scale that has also two 3-semitones, that is of interval step structure 2-2-3-2-3 (also known as Egyptian/Mongolian).
Or compare them with the oriental 6-note scale 1-3-1-3-1-3
Or compare them with other 4-notes scales like 3-3-3-3
the 3-5-3-1 and 2-1-4-5 or 3-2-2-5 or 4-2-1-5, and 2-2-3-5 that extrapolations to 4-notes scales of 3-notes major or minor chords arpeggios-scales
Or the 8-notes Algerian=2-1-2-1-1-1-3-1
Or the 8-notes extrapolation of the Romani double minor
1-3-1-1-1-1-3-1
Or the Chinese 5-notes scale that have 2-tones steps , in other words interval structure
Interval fro the root : 1, 3, #4, 5, 7
Intervals in steps: 4 - 2 - 1 - 4 - 1 (the tetra-chord 1-4-1 is used instead of the oriental
1-3-1)
Formula: Quadra-step, Whole, Half, Quadra-step, Half 4 - 2 - 1 - 4 - 1
See e.g.
http://www.pianoscales.org/chinese.html
C: C, E, F#, G, B, C (we notice that it is a sub-scale of the F-major 7-notes scale)
For example if we take the mode of the diatonic scale starting from F
F G A B C D E F , a Chinese scale will be shaped by keeping the semitones B C,, E, F, but eliminating the notes G, D , Thus it will be
F, A, B, C, E, / F
Somehow all the above scales may be considered extrapolations of arpeggios of 3-notes chords of the type major,minor, diminished and augmented.
The oriental (Hijazz) tetra-chord and the creation of all possible 7-notes scales by combinations with diatonic tetra-chords .
The oriental tetra-chord which is called also Hijazz, is 1 semitone-3 semitones-1 semitone
(1-3-1) in total 5 semitones. Its origin is in ancient Greece and Byzantine empire. Thus if it is to combine it with other tetra-chords of total sum 5 it should be combined with one tone distance to make in total 12 semitones (5+2+5=12).
Now combining it with all possible diatonic tetra-chords that is (1-2-2), (2-2-1), (2-1-2), (2-2-2) it gives the scales
1) Harmonic minor (called also mode of Hijazz or Byzantine minor) = (1-3-1)-2-(-1-2-2),
A mode (cyclic permutation) of the harmonic minor is called also Blue-scale in American folk music, and another cyclic permutation of it the Romani minor scale
Chords of the harmonic minor (intervals 2-1-2-2-1-3-1):
Triads: min dim aug min maj maj dim
Extended
4-notes chord: min/maj7 m7b5 maj7#5 min7 dom7 maj7 dim7
For example for the A harmonic minor(A,B,C,D,E,F,G#) (intervals 2-1-2-2-1-3-1),the chords are
i | iidim | III | iv | V | VI | VII |
Amin | Bdim | Caug | Dmin | Emaj | Fmaj | G#dim |
Aminmaj7 | Bm7b5 | Cmaj7#5 | Dmin7 | E7 | Fmaj7 | G#dim7 |
Typical progression
Typical chord progressions in A harmonic minor | |
i - iv - V7 | Am - Dm - E7 |
ii - V7 - i | Bm7b5 - E7 - Am |
More general the chords that fit to the harmonic minor (not taking necessarily the notes in alternating order , that is take one leave one or 1-3-5 etc) are the next
Notes of this scale: |
A; B; C; D; E; F; G#/Ab; A; |
Interval structure of this scale: |
2-1-2-2-1-3-1 |
Chords that fit in this scale: |
Normal Triads: Caug Dm Ddim E Eaug F Fm Fdim G#aug G#dim Am Bdim Other Triads: Dsus2 Esus4 Asus4 Asus2 4 Notes Chords: Dm6 Dm7 Dm7b5 Dº7 D7sus2 E7 E7#5 E7sus4 F6 Fm6 Fmaj7 Fm(maj7) Fº7 G#º7 Am(maj7) Bm7b5 Bº7 |
2) One that may be called 2nd Harmonic minor , called also mode of Kurdi or Kasigar =(1-3-1)-2-(2-1-2) and
3) One that may be called 3rd Harmonic minor, called also mode of Shamba=(1-3-1)-2-(-2-2-1).
4) And one than maybe called 4th Harmonic minor. If we take the Shamba in the reverse order we get the Neopolitan scale= (1-3-1)-1-(-2-2-2)
The Neopolitan scale , from D has the next chords (intervals 1-2-2-2-1-3-1)
Notes of this scale: |
D; D#/Eb; F; G; A; A#/Bb; C#/Db; D; |
Interval structure of this scale: |
h W W W h (W+h) h |
Chords that fit in this scale: |
Normal Triads: C#aug Dm D# Faug Gm Gdim Aaug A# A#m Other Triads: Dsus4 D#sus2 Gsus2 A#sus4 4 Notes Chords: Dm(maj7) D#maj7 D#7 D#7b5 D#7sus2 F7#5 Gm7 Gm7b5 G7sus2 A7b5 A7#5 A#6 A#m6 A#maj7 A#m(maj7) |
5) While if combined by itself it gives what is called Romani (Hungarian or Gypsy or Byzantine double minor or Harmonic double minor or mode of the Niavent)=(1-3-1)-2-(-1-3-1).
These names are of Arabic origin but are used in Greek folk music with Buzuki , as they are played on the western 12-semitones Bach equal temperament scale.
The chords that fit to the Romani or Harmonic double minor scale , we take here as an example the D Romani or Hungarian double minor are the next:
Notes of this scale: |
D; E; F; G#/Ab; A; A#/Bb; C#/Db; D; |
Interval structure of this scale: |
2-1-3-1-1-3-1 |
Chords that fit in this scale: |
Normal Triads: C# C#m C#aug Dm Ddim Faug A Aaug A# A#m A#dim Other Triads: Dsus2 Asus4 4 Notes Chords: C#6 C#m6 Dm(maj7) E7b5 Amaj7 A#maj7 A#7 A#7b5 A#m7 A#m(maj7) A#m7b5 |
Two variations of the Byzantine double minor scale are the Persian and inverse Persian scales
But only the Byzantine (harmonic) double minor containes two tetra-chords 1-3-1 !
Persian scale or todi theta scale=(1-3-1-1-2-3-1)
E.g. starting from C
Notes of this scale: |
C; C#/Db; E; F; F#/Gb; G#/Ab; B; C; |
Interval structure of this scale: |
h (W+h) h h W (W+h) h |
Chords that fit in this scale: |
Normal Triads: C# C#m Caug E Eaug Fm Fdim G#aug Other Triads: C#sus4 Esus2 F#sus4 F#sus2 Bsus4 Bsus2 4 Notes Chords: C#maj7 C#7 C#m7 C#m(maj7) C#7sus4 E6 Fm(maj7) F#7sus4 F#7sus2 G#7#5 |
Inverse Persian scale or Purvi Theta scale= (3-1-1-3-2-1-1)
E.g. starting from C
Notes of this scale: |
C; C#/Db; E; F#/Gb; G; G#/Ab; B; C; |
Interval structure of this scale: |
h (W+h) W h h (W+h) h |
Chords that fit in this scale: |
Normal Triads: C C#m Caug C#dim E Em Eaug G#aug Other Triads: C#sus4 Esus2 F#sus4 F#sus2 Bsus4 Bsus2 4 Notes Chords: Cmaj7 C#m7 C#m(maj7) C#m7b5 C#7sus4 E6 Em6 F#7sus4 F#7sus2 G#7#5 C\E C\G |
We may compare these scales with the melodic minor (e.g. A melodic minor) and its chords
The chord chart shows the traid and 4 note extended chords belonging to the key of A Melodic Minor.
- A Melodic Minor scale notes: A B C D E F# G#
Triads: | min | min | aug | maj | maj | dim | dim |
Extended: | min/maj7 | min7 | maj7#5 | dom7 | dom7 | m7b5 | m7b5 |
i | ii | III | IV | V | vi | vii |
---|---|---|---|---|---|---|
Amin | Bmin | Caug | Dmaj | Emaj | F#dim | G#dim |
Aminmaj7 | Bm7 | Cmaj7#5 | D7 | E7 | F#m7b5 | G#m7b5 |
All the above scales may be considered extrapolations to intervals of only 1,2 ,3 semitones, of arpeggios of 3-notes chords of the type major,minor, diminished and augmented.
If in the above 5 scales of 1st 2nd 3rd and 4th Harmonic minor and Harmonic double minor scales we extrapolate the intervals of 3-semitones as ascending to 2+1 semitones, we get the next 8-tones scales made only from steps of 2 and 1 semitones
1) From the 1st Harmonic minor 2-1-2-2-1-3-1 the 8-notes scale 2-1-2-2-1-2-1-1 which is a cyclic permutation or mode of what is called in post 51, 3rd alternative of Spanish-Jewish 8-notes scale that contains in its chords the Andaluzian Cadenza.
2) From the 2nd Harmonic minor (1-3-1)-2-(2-1-2) the 8-notes scale (1-2-1-1)-2-(2-1-2)
which is a cyclic permutation or mode of what is called in post 51, 4th alternative of Spanish-Jewish 8-notes scale
3) From the 3rd Harmonic minor (1-3-1)-2-(-2-2-1). the 8-notes scale (1-2-1-1)-2-(2-2-1)
which is a cyclic permutation or mode of what is called in post 51, 1st alternative of Spanish-Jewish 8-notes scale
4) From the 4th Harmonic minor 1-2-2-2-1-3-1 the 8-notes scale 1-2-2-2-1-2-1-1
which is a cyclic permutation or mode of what is called in post 51, Spanish-Jewish 8-notes scale
5) From the Harmonic double minor 2-1-3-1-1-3-1 the 9-notes scale 2-1-2-1-1-1-2-1-1
If we restrict to only 4-notes sub-scales (tetra-chords) , having inverse such scales not different, then we are left with a small number of 8 of such characteristic tetra-chords met in corresponding 7-notes scales.
Diatonic
2-2-1, (major)
2-2-2, (major)
2-1-2 (natural minor)
Melodic minor
1-2-1
Melodic double minor
2-1-1,
Harmonic minor
1-3-1
Harmonic double minor
1-1-3
1-2-3,Nevertheless if cyclic permutations of them are not considered different then 2-2-1=2-1-2. 1-2-1=2-1-1, 1-3-1=-1-1-3 and so they are only 5 different
There is small number of exactly 10 characteristic tetra-chords (=4-notes sub-scales) containing intervals of 1,2,3, semitones and where inverses and cyclic permutations of them do not count as different
Diatonic
2-2-1, (major, natural minor Rast, Ussak)
2-2-2, (major, augmented)
Melodic minor, double minor (Hindu, Arabic, Shabach)
1-2-1
Harmonic minor (Hijazz,Huzam)
1-3-1
Harmonic double minor
1-2-3,
Diminished
Diminished
3-3-3 , (diminished 7nth)
3-3-1,
Pentatonic
3-3-2,
2-2-3
2-2-3
Chromatic
1-1-1
We should be also familiar with the ways we can play them in 1 , 2 or 3 strings.
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