The chromatic scale is defined, by the least number of repetitions of the perfect Pythagorean interval of 3/2 times the root frequency (5th interval) that will return to a multiple (here 7 octaves) of an octave interval. It turns out that it is required 12 repetitions. (3/2)^12 =(almost)=2^7 There is slight difference of (3/2)^12=129.74 and 2^7=128 when reduced inside single octave. To be more precise: Inside one octave , the comparison of the interval of octave (2/1), to the pythagoream (3/2) as listened by the human ear which is equivalent by taking the logarithm is ln(3/2)/ln(2)=0.584962501.....while if measured in semitones it will be 7/12=0.5833333...This difference pc=(ln(3/2)/ln(2))-7/12 is called the Pythagorean comma.
Therefore in the diatonic scale (and mode) each note of it if shifted by a pythagorean interval higher or lower (interval of 5ths) is still a note of the scale.
The diatonic scales have also the optimality among all other 7-tone scales, that they contain the maximum number of 3-notes major and minor chords.
see also https://www.youtube.com/watch?v=TRz73-lSKZA
About the Pythagorean tuning see
https://en.wikipedia.org/wiki/Pythagorean_tuning
(The post has not been written yet completely)
see also https://www.youtube.com/watch?v=TRz73-lSKZA
About the Pythagorean tuning see
https://en.wikipedia.org/wiki/Pythagorean_tuning
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.
C---1 harmonic
D---9
E---5
F---11
G---3
A---27
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 simple 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 is a relevant video
https://www.youtube.com/watch?v=IT9CPoe5LnM
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 as well (https://en.wikipedia.org/wiki/Enharmonic_scale)
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
Here is a relevant video
https://www.youtube.com/watch?v=IT9CPoe5LnM
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.
(The post has not been written yet completely)
