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Tony Deff
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Minor Mystery - solved  
« on: Nov 23rd, 2019, 8:18pm »
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If you read an article about the Just Scale, you learn that the three major chords each have the simple frequency ratio 4:5:6

 1 9/8 5/4 4/3 3/2 5/3 15/8  2  9/4 5/2  
 C  D   E   F   G   A   B   C   D   E  
 
 
C : E : G  =  4/4 : 5/4 : 6/4    
;    F : A : C  =  4/3 : 5/3 : 6/3
 
G : B : D  =  3/2 : 15/8 : 9/4
  (x 8/3) = 4 : 5 : 6
 
but that for the minor chords ...  
 
E : G : B  =  5/4 : 3/2 : 15/8  (x 8 ) = 10 : 12 : 15
 
A : C : E  =  5/3 : 6/3 : 5/2   (x 6 ) = 10 : 12 : 15
 
For Nature's most harmonious sound of a major chord, the simple ratio (1:2):3:4:5 makes joyful sense  
But a minor chord is Nature's next most harmonious sound, so why the more complex ratio 10:12:15  
 
If we take the reciprocal (inverse) ratios and multiply them by 60 …
 
1/10 : 1/12 : 1/15  (x 60) = 6 : 5 : 4   and   1/4 : 1/5 : 1/6  (x 60) = 15 : 12 : 10
 
Major and Minor chords are the inverse of each other
(Maybe that should have been obvious from the fact that the major 3rd and minor 3rds are inverted)  
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Anglophone, HA 64 Win.10, amateur composer.
La musique classique est ce dont vous attendez toujours qu'elle se transforme en mélodie
ANdre_B
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Re: Minor Mystery - solved  
« Reply #1 on: Nov 24th, 2019, 5:44pm »
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Oh Tony, you solve the question before raising it!  
 
In addition, a 7th chord (C E G Bb) présents the ratio 4:5:6:7
 
This "just scale" aspect is very interesting indeed (I found that out a couple of years ago); as a mathematician, I enjoy to see something physical using integer numbers (yes rational (fractions) ones, but at least not real ones).
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André Baeck, de retour en Belgique après 12 ans passés dans le Gard.
Windows 11, HA 997e (et précédents)
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