Pr1me Numbers (english version)

Track List:

1. Ulam’s spiral – Club Dance Beat
2. Ulam’s spiral – Hammer and Noise
3. Ulam’s spiral – 808
4. Prime Numbers – Goa Version
5. Prime Numbers

Pr1me Numbers is available on Spotify at the following link: https://open.spotify.com/album/4BJkkdN3uZLHR7OpruqnB7
Pr1me Numbers is available on iTunes at the following link: https://itunes.apple.com/gb/album/id1510795025?app=itunes

Pr1me Numbers was born in April 2020 as a theoretical research on prime numbers and their convertibility into music published on Mathesis Nazionale (http://www.mathesisnazionale.it/2020/04/18/la-musica-dei-numeri-primi/).

These are the simple steps I have adopted for transforming numbers into notes:

1. consider the 12 notes included within an octave (C, C#, D, D#, etc.);

2. starting from number 1, associate each note with a number in ascending order (C=1; C#=2; D=3; etc.);

3. after the twelfth number we start again with the note do (C=13; C#=14; D=15; etc.) and so on to infinity; 4. since there are twelve notes within an octave, I have chosen to use bars with 12 subdivisions and therefore (for pure convenience) the 12/8 meter.

If we only care about the prime numbers, all the others will correspond to pauses so that there is no confusion in the reading. In this way you will also obtain a visual simplification in the search for numbers: you will know a priori that in each of the infinite bars that will generate the note C will always be in the first subdivision of the bar, D in the third and F in the sixth and that to find the same note it will be sufficient to add or remove 12 (or a multiple thereof) to its number.

At this point I proceeded to the creation of the score of the prime numbers (I gave myself as a limit those between 1 and 1000) with the given rules:

If you exclude the 2 and the 3 you will notice that all the other numbers always correspond to the same 4 notes: C, E, F#, A# even if arranged in an apparently random sequence.These four groups of notes that I associated with their relative prime numbers are striking:

C: 13, 37, 61, 73, 97, 109, 157, 181, 193, 229, 241, 277, 313, 337, 349, 373, 397, 409, 421, 433, 457, 541, 577, 601, 613, 661, 673, 709, 733, 757, 769, 829, 853, 877, 937, 997…

E: 5, 17, 29, 41, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, 269, 281, 293, 317, 353, 389, 401, 449, 461, 509, 521, 557, 569, 593, 617, 641, 653, 677, 701, 761, 773, 797, 809, 821, 857, 881, 929, 941, 953, 977…

F#: 7, 19, 31, 43, 67, 79, 103, 127, 139, 151, 163, 199, 211, 271, 283, 307, 331, 367, 379, 439, 463, 487, 499, 523, 547, 571, 607, 619, 631, 643, 691, 727, 739, 751, 787, 811, 823, 859, 883, 907, 919, 967, 991…

A#: 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263, 311, 347, 359, 383, 419, 431, 443, 467, 479, 491, 503, 563, 587, 599, 647, 659, 683, 719, 743, 827, 839, 863, 887, 911, 947, 971, 983…

If we do not consider 1 as the first and omitting 2 and 3 as I said, if we consider the next four prime numbers (5, 7, 11, 13) as the point of origin of the sequences, we will notice that adding 12 or a multiple thereof at these points you will always get a prime number at some point. Consequently, to know whether a prime number belongs to one of the 4 groups, it is sufficient to subtract 12 as many times as necessary to obtain one of the 4 'basic' prime numbers as a result. It is therefore possible that there are relationships between the numbers belonging to the groups and to better visualize a possible law that regulates the alternation of the numbers I thought of simplifying the scheme of the 4 groups by indicating multiples of 12 with progressive numbers such that 12 = 1 , 24 = 2, 36 = 3, etc .; in this way the following sequences will be generated:

C: 2, 2, 1, 2, 1, 4, 2, 1, 3, 1, 3, 3, 2, 1, 2, 2, 1, 1, 1, 2, 7, 3, 2, 1, 4, 1, 3, 2, 2, 1, 5, 2, 2, 5, 5…

E: 1, 1, 1, 1, 3, 1, 1, 2, 1, 2, 2, 3, 2, 1, 1, 1, 2, 3, 3, 1, 4, 1, 4, 1, 3, 1, 2, 2, 2, 1, 2, 2, 5, 1, 2, 1, 1, 3, 2, 4, 1, 1, 2…

F#: 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 3, 1, 5, 1, 2, 2, 3, 1, 5, 2, 2, 1, 2, 2, 2, 3, 1, 1, 1, 4, 3, 1, 1, 3, 2, 1, 3, 2, 2, 1, 4, 2…

A#: 1, 2, 1, 1, 1, 2, 2, 3, 1, 1, 3, 1, 1, 1, 4, 3, 1, 2, 3, 1, 1, 2, 1, 1, 1, 5, 2, 1, 4, 1, 2, 3, 2, 7, 1, 2, 2, 2, 3, 2, 1…

As pointed out to me by prof. Francesco Fournier "it is no coincidence that if we exclude the 2 and the 3 it can be noticed that all the other numbers always correspond to the same 4 notes: C, E, F#, A# even if arranged in an apparently random sequence. Indeed the process continues identical. Translated into numbers, we are saying that if we want to express a prime number like 12k + r (ie a multiple of 12 and a remainder smaller than 12), then r must be equal to 1 (C), 5 (E), 7 (F#) or 11 (A#), with the exceptions of 2 and 3. Here's why: the other numbers between 1 and 12 are all multiples of 2 or 3. Since 12 is a multiple of both, if r is not among these four numbers, then 12k + r is a multiple of 2 or 3. So it cannot be prime unless it equals 2 or 3. In formal language, 1, 5, 7 and 11 are representatives of the modulo 12 units, and any prime that does not divide twelve is a modulo 12 unit. Furthermore, if we do not consider the 1 as the first and leaving out the 2 and the 3, if we consider the following four prime numbers (5, 7, 11, 13) as the point of origin of the sequences, we will notice that adding 12 or a multiple thereof to these numbers you will always get a prime number at some point. This too is no accident, but the reason is much deeper and cannot be explained with an elementary reasoning like the previous one. We have seen that taking the remainder of the division by 12 of a prime number we get 1, 5, 7 or 11. So the prime numbers are divided into these four categories, which are called modulo 12 classes. Now, we know from Euclid that there is a infinity of prime numbers. A much more difficult question is: are there an infinite number of primes in each of these four classes? It is clear that at least one of these classes must contain an infinity of primes, but who knows, maybe the others contain only a finite number. The answer is yes: the general statement is called Dirichlet's Theorem, or theorem of arithmetic progression. It was conjectured by Gauss and demonstrated by Dirichlet in 1835, and according to many that demonstration is the birth of the analytical theory of numbers, which will then explode in 1859 with the text of Riemann in which he places his famous hypothesis".

NOTES ON THE SONGS:

Ulam’s spiral - Club Dance Beat

The spiral of Ulam, born in 1963 probably as a simple pastime of the Polish mathematician Stanislaw Ulam, has provided some additional ideas to the study of prime numbers even if, to date, it has not yet produced clear results nor has it yet been established whether it can be significantly useful. The effect, especially visual, that is generated is fascinating. In a variant of his 'game' Ulam inserted the number 17 (and not the 1) in the center of the square, thus noting that by proceeding in a spiral with all the other subsequent numbers, a decidedly significant diagonal of prime numbers was formed but which does not continues beyond neither upwards nor downwards but regenerates in other sectors. Just from this variant I started for the conversion to music:

Applying the same conversion principle also to the spiral (I chose the 4/8 meter because the square was composed of lines of 16 numbers and therefore the division into 4 bars per line graphically keeps the idea of ​​the diagonal) generate this melodic line:

The sequence generated by following the diagonal from the bottom to the top is also evident, a sequence that I used as an obstinate bass line in all 5 tracks:

All synth, piano and melodic parts rotate around the same scale, the one generated by prime numbers of course:

Ulam’s spiral - Hammer and Noise

The stubborn sound of rhythm is produced by a hammer blow that hits a fork of my son's bicycle. The bichords of Mellotron and the riffs of Clavinet are also composed only of the notes of the scale of the prime numbers.

Ulam’s spiral - 808

Arrangement for Drum Machine and synth of the sequence generated by the Ulam spiral. The solo was performed only with the notes belonging to the scale of prime numbers.

Prime Numbers - Goa Version

The sequence of prime numbers from 2 to 997 is shown by the synth counterpointed at the bottom by the diagonal of the square of Ulam. Mellotron bichords are also composed only of the notes on the prime number scale.

Prime Numbers

The sequence of prime numbers from 2 to 997 is shown by the synth counterpointed at the bottom by the diagonal of the square of Ulam.