fareylen — returns the length of a Farey Sequence.
This opcode can be used in conjunction with GENfarey. It calculates the length of Farey Sequence Fn. Its length is given by: |Fn| = 1 + SUM over n phi(m) where phi(m) is Euler's totient function, which gives the number of integers ≤ m that are coprime to m.
Some values for the length of Fn given n:
| n | Fn |
|---|---|
| 1 | 2 |
| 2 | 3 |
| 3 | 5 |
| 4 | 7 |
| 5 | 11 |
| 6 | 13 |
| 7 | 19 |
| 8 | 23 |
| 9 | 29 |
| 10 | 33 |
| 11 | 43 |
| 12 | 47 |
| 13 | 59 |
| 14 | 65 |
| 15 | 73 |
| 16 | 81 |
| 17 | 97 |
| 18 | 103 |
| 19 | 121 |
| 20 | 129 |
The length of the identified Farey sequence is returned.
kfn -- Integer identifying the sequence.
Here is an example of the fareylen opcode. It uses the file fareylen.csd.
Example 294. Example of the fareylen opcode.
See the sections Real-time Audio and Command Line Flags for more information on using command line flags.
<CsoundSynthesizer> <CsOptions> -n </CsOptions> <CsInstruments> sr=44100 ksmps=32 nchnls=1 ; by Stefano Cucchi and Menno Knevel - 2024 instr 10 kfarnf randomi 1, 10, 4 kfarn ceil kfarnf klenght fareylen kfarn printks "farey length of %d = %d\n",.1, kfarn, klenght endin </CsInstruments> <CsScore> i 10 0 3 e </CsScore> </CsoundSynthesizer>