Text: Major Third = 2(2mj) 5:6::25:30::C:E Minor Third = (2mn + 2mj) 4:5::16:20::C:D# A third is composed of 1/12s. See below. Ninth 2(3mj + 3mn) 1.1666666666667 Octave 3(3mj) 1 Octave 4(3mn) 1 Major Seventh 2(3mj) + 3mn 0.91666666666667 Minor Seventh 2(3mn) + 3mj 0.83333333333333 Major Sixth 3(3mn) 0.75 Minor Sixth 2(3mj) 0.66666666666667 Fifth 3mj + 3mn 0.58333333333333 Fourth 2(3mn) 0.5 Major Third 2(2mj) 0.33333333333333 Minor Third 2mn + 2mj 0.25 Major Second 2(2mn) 0.16666666666667 Minor Second 1/12 0.083333333333333 1) G can only be a double tone in the diatonic scale while F# is double in the chromatic. G = 4/7 going up and 3/7 coming down, diatonic. F = 3/7 going up and 4/7 going down, diatonic. F# = 6/12 up and down, chromatic. 2) The diatonic is a scale of whole tones each being a multiple of 7. 3) The chromatic is a scale of half tones each being a multiple of 12. 4) The chromatic scale can be builded with thirds beginning with 3rd mn or 3/12 and 3rd mj as 4/12. This got me to thinking really far afield with: 5) Pythagorean Triangle where Base = 3 Height = 4 Hypotenuse = 5 Area = 6 6) And this reminded me of Quantum Arithmetic where these integers are Square Roots of Squares: sqrt|9 = 3 sqrt|16 = 4 sqrt|25 = 5 sqrt|36 = 6 also (3*4)/2 sqrt|49 = 7 (49 is the denomiator when adding 4:5 (3mn) and 5:6 (3mj)! All music intervals can be builded with these numbers meaning its all geometry of triangles and squares. 7) The Fifth is half of a Ninth: Fifth chromatic is the same as ninth in that the ninth is the beginning of the next octave while the fifth is just beyond the midway point going up. They preform the same service which reminds of the number 13. 13 is prime and cannot be evenly divided but if a ninth is 13 it can! 8) Thirds and Sixths Inversion Major 6 = 3/12 down Minor 6 = 4/12 down Major 3 = 4/12 up Minor 3 = 3/12 up 9) Also cannot progress by 1/2 step with 3rds diatonic because there aren't any diatonic half steps. 10) All 7 and 12 fractions divide unevenly leaving thirds as remainders except: Minor Third Major Fourth and Major Sixth These three divide evenly by 5! Dale Pond, 6/16/02

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