Base 12 - the octave as foundation for the study of prime numbers
January 14, 2017 05:10PM
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 !
Re: Base 12 - the octave as foundation for the study of prime numbers
June 18, 2021 11:49PM
Hi, I'm very new to music,
I've been creating for a few years, but I've never been able to wrap my head around music theory because I'm very mathematically minded. Like any mathematical system, there needs to be a formula to music. While trying to figure this out, I realized that music is indeed a duodecimal numerical system (radix of 12 as you pointed out). So I've spent the last few hours trying to visualize music numerically well as we all visualize the decimal system. Looking online, you have been the only person so far that I could find who saw music the same way. I'm hoping to exchange and learn ideas from you to better connect the dots in my head. I can't understand music theory in the conventional way...I feel like I need to do it in this way.
Kind regards
Saman currently in south africa
Re: Base 12 - the octave as foundation for the study of prime numbers
June 26, 2021 05:19PM
Hi Treblebypass,

You can email me at fluo2005@gmail.com. Or we can post on this forum or the one below so all can see.

Using base 12, I discovered a way to produce the list of all the prime numbers, and posted it on mathforums.com recently. Not on this site, because base 12 work had no followers.

The posters on mathforums.com basically state that all I've done was repeat the sieve of Eratosthenes, which uses the addition of primes to remove their composites from the list of whole numbers, conceived 2,300 years ago I am told.

They also claim that base 12 is no way better than base 10, or any other base. I disagree, because base 12 can create two musical staffs which show all the notes of the octave, removing the need for sharps and flats. This provides for immediate musical literacy. These two patterns can also be applied to the piano keyboard, modifying its appearance substantially, and improving both note and chord recognition.

Have a look at my latest post on mathforums.com which is a verbal rendition of the algorithm I used. It uses multiplication rather than addition, because using factors rather than the difference with previous values makes it easier to understand what is going on.

Here is the link: mathforums.com/threads/how-to-remove-prime-composites-and-reveal-the-primes-in-base-12.358553/

It will take me a while to answer the last post because it covers so many topics and will require research on my part.

The one important discovery I made with base 12 concerning prime numbers is that base 12 creates the list of all the prime numbers starting with 5 on up, and of all the composites of prime numbers starting with 5 and up. They are all found when you add 12 in layers to intervals 1 5 7 11 in order to create columns of indefinite size.

I call this list the 'list of possible primes', because you don't know just by looking at large numbers which are prime and which are composite, except for those that end with 5.

Prime numbers 2 and 3, and all the composite numbers they create, are all found in intervals 0 2 4 6 8 9 10.

Base 12 in music uses the semitone as the basic interval, and not the tone, which is a variable containing either two semitones or one only (algebra). So it provides a proper name to the five black keys. It also breaks the scales into two half-units containing four notes, but covering 7 note spaces. And it breaks the 12 semitones into two families, the whole tone scales, coloring one family of six tones white, and the other family black.

I am wondering what other discoveries if any can be made in the world of prime numbers using base 12 as a foundation.



Edited 1 time(s). Last edit at 06/26/2021 05:44PM by fluo2005. (view changes)
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