Hello everyone,
Over the past few days, I extended my study of the interval method on the piano keyboard to the whole number system, up to interval 48 semitones or four complete octaves, and discovered that the octave is the natural foundation for the study of prime numbers.
The interval method on the piano consists of a starting note, any note, and a jump by a whole number of semitones, any number from 1 semitones to 12 semitones let's say, to a second note, from which you jump again by the same interval to a third note, and so on, until you once again fall upon the starting note, now one or more octaves above its initial position. (Striking the same note on another octave I call for now 'resonance'.) This must be done for all twelve notes in an octave, and results in a pattern or a series of patterns, that are different for all twelve notes. Here is the interval method applied to the octave.
symbols:
octave 1:
Notice the '*' sign which represents 'prime numbers', numbers which cannot be divided by any smaller number above 1. There are 5 prime numbers in octave 1: 2, 3, 5, 7, 11.
Now we look at the intervals from 13 to 24, which are in the second octave above our starting note:
octave 2:
Here we notice that there are 4 prime numbers, at positions 1, 5, 7, 11 of octave 2. The prime positions contain a single series of 12 notes. Series 1 and 11 jump by 1 and 11 semitones respectively, and are the inverse of each other. Series 5 and 7 jump by 5 and 7 semitones respectively, and are the inverse of each other.
Positions 2, 3, 4, 6, 8, 9, 10 and 12 of octave 2 are composite numbers, as they contain 2, 3, 4, 6, 4, 3, 2 and 12 series of numbers respectively, which contain all twelve distinct notes of the octave.
Octave 2 contains the basic patterns for all octaves. The identical patterns repeat as the octaves increase. The only difference being the addition of the '¦', representing 12 semitones or numbers, within the intervals, as the number of octaves increases by one.
We will see in octave 3 that not all numbers in prime positions turn out to be prime.
octave 3:
Notice the '+' sign instead of the '*' before numbers 25 and 35 which indicates that 25 and 35 are non-primes. Both can be divided by 5. That all numbers ending in 5 are composite is a well known rule for division by whole numbers.
Octave 4 is again a 'prime octave', in that all four prime positions are occupied by prime numbers.
octave 4:
My next task will be to identify the 'non-prime' numbers left out of the prime number list, such as the one below, and then study them to determine the mechanisms for their exclusion. This in the hope that patterns can be found, which can improve upon the search for prime numbers.
verification:
My work so far indicates that the octave should be the foundation for the study of music theory. The fact that it can also be used for the study of whole numbers only makes the case stronger. If music theory can be enriched by using the base 12 whole number system, it seems that number theory in turn can be enriched by the use of the musical octave. What a surprise !