Euler's j Function
j(n) means the number of positive integers, not greater than n, that is relatively prime to n. For example j(10) = 4, since 10 is relatively prime to 1, 3, 7 and 9. Here are the first 30 j values:
1 1 2 2 4 2 6 4 6 4 10 4 12 6 8 8 16 6 18 8 12 10 22 8 20 12 16 12 28 8
The piece of music has three distinct parts:
1) A treble pattern determined by the digit sum of j(n) mapped to the B harmonic minor scale, where 1 is the low B. Here, n ranges from 1 to 810 and all tones are sixteenths. Equal consecutive tones are tied. No pauses in this part.
2) A bass part determined by the smallest digit sum of n mod j(n), where n ranges from 1 to 512 and all tones are eighths. In this part, 0 is mapped to the low B. All prime n:s are pauses.
3) A melody where the hundredth digit of j(n) determine the number of tied eights and the digit sum of the tenth and unit digits determine the pitch. 0 is mapped to the low B. Here, n ranges from 101 to 432 and the part is initiated when n = 101 in the first part above.
Playing time: 2' 23".
Möbius' m Function
Möbius' µ function is defined as follows:
µ(n) = 1, if n =1
µ(n) = 0, if n is divisible by a square
µ(n) = (-1)r, for all other n where r is the number of distinct prime factors of n
For example, µ(18) = 0, since 18 is divisible by the square 9 = 32. Moreover, µ(26) = 1, since 26 has the two distinct prime factors 2 and 13, and (-1)2 = 1. Here are the first 30 µ values:
1 -1 -1 0 -1 1 -1 0 0 1 -1 0 -1 1 1 0 -1 0 -1 0 1 1 -1 0 0 1 0 0 -1 -1
This piece emanated from a skeleton part where 1 was mapped to the D of a bass staff. If µ(n) = 1, ascend to the next tone of the A harmonic minor scale, if µ(n) =-1, descend to the preceding tone of the scale and if µ(n) = 0, stay on the same pitch. This part was then mercilessly discarded after it had been divided into three new parts:
1) The tones got from µ(n) = 1 were shifted 6 scale steps up (so a D would become C) and sustained to the next such tone of the skeleton part.
2) The tones got from µ(n) = -1 were shifted 8 scale steps up and sustained like the first part.
3) The tones got from µ(n) = 0 stayed the same, but were sustained in the same manner as the above parts.
4) Whenever the sum of all numbers from µ(1) to µ(n) equals 0, the tone in one of the three parts which coincides with this event will be accented by this part.
Playing time: 9' 23".
The t Function
This short piece of music consists of all values of t(n) from n = 1 to n = 720. The tau function, t(n), where n is a natural number, means "the number of positive divisors of n". E.g. t(20) = 6, since 20 has the divisors 1, 2, 4, 5, 10 and 20. Here are the first 30 values of t:
1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 5 2 6 2 6 4 4 2 8 3 4 4 6 2 8
In the piece of music, the pitches are determined by the value of t(n). Its value ranges from 1 to 30 in the interval from n = 1 to n = 720. The C whole tone scale with was used. Equal consecutive tones were tied. In the table below, k = 0, 1, 2, 3, 4. Here k stands for the octave.
|t(n)||1 + 6k||2 + 6k||3 + 6k||4 + 6k||5 + 6k||6 + 6k|
When t(n) is greater than both t(n - 1) and t(n + 1) in the piece, it is accented by an octave delayed by one unit.
Playing time: 1' 30".
The integer values of the trigonometric function f(x) = 7sin(x) + 7, its derivative f'(x) = 7cos(x) and its second derivative f''(x) = -7sin(x) were mapped to eighth triplets, using tones of the A natural minor scale. The value, 1 to 14, of f(x) was mapped to a pedalled first tone of each triplet, ranging within two octaves. 0's were discarded (Why? Oh, I just felt like it.). The value, -7 to 7, of f'(x) is the number of scale steps to the second tone of the triplet, and the value, also -7 to 7, of f''(x) is the number of scale steps from the second tone to the third tone of the triplet. The input value x was determined iteratively by x1 = f(x0) + x0 + 1. With all f(x) = 0 discarded, 151 iterations gave the 139 triplets in the piece.
Playing time: 1' 22".
The values of the trigonometric function f(n) = 7sin(n/2), its derivative f'(n) = (7/2)cos(n/2) and its second derivative f"(n) = -(7/4)sin(n/2) were rounded off to the nearest integer and then mapped to the A melodic minor scale. Here, n are the natural numbers from 0 to 337.
The piece of music is has the structure A1 B A2. In A1, the sign of the values of the trigonometric functions, positive or negative, determine if the pitches should be taken from the ascending or descending version of the scale. In B and A2, the scale is used in the normal manner.
There are three parts:
1) A part where f(n) is mapped directly to a walking bass line.
2) A treble part where f'(n) are eighths on the beats and mapped directly to tones. The values of f"(n) are the number of scale steps relative to f'(n) and are eighths off the beats.
3) This part doubles the second part until n = 142, where B starts. Then, f(n) is mapped to the first tone of a phrase: f'(n) determine the number of tones in the phrase, the sign of f'(n) determine if the phrase is ascending or descending and the absolute value of f"(n) determine the intervals between the tones. Equal consecutive tones are tied. At n = 185, A2 starts, and this part doubles the second part until the end like before.
Playing time: 2' 49".