fareyleni
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## fareyleni

fareyleni — returns the length of a Farey Sequence.

## Description

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

## Syntax

    ifl fareyleni ifn

## Initialisation

The length of the identified Farey sequence is returned.

ifn -- Integer identifying the sequence.

## Examples

Here is an example of the fareyleni opcode. It uses the file fareyleni.csd.

Example 284. Example of the fareyleni opcode.

See the sections Real-time Audio and Command Line Flags for more information on using command line flags.

    <CsoundSynthesizer>
    <CsOptions>
    ; Select audio/midi flags here according to platform
    -odac     ;;;realtime audio out
    ;-iadc    ;;;uncomment -iadc for RT audio input as well
    ; For Non-realtime ouput leave only the line below:
    ; -o fareyleni.wav -W ;;; for file output any platform
    </CsOptions>
    <CsInstruments>

    sr = 44100
    ksmps = 32
    nchnls = 2
    0dbfs  = 1

          ; initialise integer n for Farey Sequence F_8
          gifarn init 8

          ; calculate length of F_8, it should return 23 for |F_8|
          gires fareyleni gifarn

          ; convert to negative number for the GEN routine because
          ; the table length won't be a power of 2
          ; (The length of a Farey Sequence of n > 1 is always an odd number)
          gilen init gires * -1

          ; create F_8 with GENfarey, the negative table number prevents
          ; unnecessary normalisation (F_8 IS already normalised)
          ; 	         	    n  	   mode:
          gifarey ftgen 100, 0, gilen, "farey", gifarn, 4
          ; if mode=4 then 1 is added to each element of F_n.
          ; Useful for creating just tuning tables that can be read by the cps2pch opcode.

    instr 1
          ; the very last element of F_n is not needed in the case of tuning tables
          ires = gires - 1
          ; read out and print to file
          kndx init 0
          if (kndx < ires) then
          	 kelem tab kndx, gifarey
          	 fprintks "farey8_tuning.txt", "%2.6f\\n", kelem
          	 kndx = kndx+1
          endif

    endin

    instr 2

    ip   cps2pch p4, -100
    asig poscil .5, ip, 1
    aenv linseg 0, 0.1, 1, p3-0.2, 1, 0.1, 0
         outs asig * aenv, asig * aenv

    endin
    </CsInstruments>
    <CsScore>
    f1 0 8192 10 1	;sine wave

    i1	0     .1

    i2	1	.5	8.00
    i2	+	.	>
    i2	+	.	>
    i2	+	.	>
    i2	+	.	>
    i2	+	.	>
    i2	+	.	>
    i2	+	.	>
    i2	+	.	>
    i2	+	.	>
    i2	+	.	>
    i2	+	.	>
    i2	+	.	>
    i2	+	.	>
    i2	+	.	>
    i2	+	.	>
    i2	+	.	>
    i2	+	.	>
    i2	+	.	>
    i2	+	.	>
    i2	+	.	>
    i2	+	.	>
    i2	+	1	8.22

    e
    </CsScore>
    </CsoundSynthesizer>

## Credits

Author: Georg Boenn
University of Glamorgan, UK

New in Csound version 5.13

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