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.
