One of the most fundamental part which is missing in most music books is precicely the topics provided in this web sitr. Congratulation and thanks. Now I suggest we go further and analize music in terms of numbers. A tempered chromatic scale means there is no flat or sharp notes, but instead, there is a group of musical notes each one of equal importance, it is like a stair with the same steps no matter how high or how low we are in the stair. If we keep constant the number of crhomatic steps between two musical notes then we maintain the type of musical interval. The same applies when dealing with a bigger group of musical notes (a chord). I learned and analyze music in terms of numbers. A "Maj" chord for me es just (4,3) which means you play the tonic ( the lowest note in this particular case), then you move 4 chromatic steps higrher and play the resulting note, and finally go 3 steps further and play the third note, and there you have...a major chord. The most common chords are of course: (4,3), (3,4), (2,5), (5,2), (4,3,2), (3,4,2), (4,3,3), (3,4,3), (4,3,4), (4,4,3), ... do you want to go further...then do this: Make a list of numbers by using only numbers 3 or 4. Ii.e. (3,4,3,4),...... try all possible mathematical combinations priorizing those combiations which do not repeat tonalities....there you have...all chords used in tertial harmony...just play with numbers, just make music by combining numbers, and finally, by algebraical operations among numbers. How many muslcal scales de we have, let say for now, with 7 tonalities per octave ? the number of mathematical solutions is ( 12 above 7 )= 7 ! / ( 12 ! * ( 12 - 7 ) ! ) ( note that n ! = 1 x 2 x 3 x 4 ...x n ). I trully recommend to think in numbers when you think in music. Consonance and dosonance, order and disorder, ..both can represented as number values and sorted by number values ( increasing or decreasing ).
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