Friday, February 2, 2018

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.

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 )

NoteCDEFGABC
Ratio119881644332271624312821
Step9898256243989898256243

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

The following Pythagorean scale is enharmonic:
NoteRatioDecimalCentsDifference
(cents)
C1:110
D256:2431.0535090.22523.460
C2187:20481.06787113.685
D9:81.125203.910
E32:271.18519294.13523.460
D19683:163841.20135317.595
E81:641.26563407.820
F4:31.33333498.045
G1024:7291.40466588.27023.460
F729:5121.42383611.730
G3:21.5701.955
A128:811.58025792.18023.460
G6561:40961.60181815.640
A27:161.6875905.865
B16:91.77778996.09023.460
A59049:327681.802031019.550
B243:1281.898441109.775
C′2:121200

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)

(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