  # eikosany

[John Chalmers, Divisions of the Tetrachord]

A 20 tone CPS generated by taking the elements or factors of a hexad three at a time.

. . . . . . . . .
[Joe Monzo]

The calculation is made using factorials. The general formula for calculating the number of pitches p in a CPS composed of n odd-factors by finding all combinations of choosing m factors at a time is: p = n!/m!(n-m)!, therefore the formula for the eikosany is: p = 6!/3!(6-3)! = 720/6(6) = 720/36 = 20.

Below is a screenshot of a Tonescape® .tuning file of a 1,3,5,7,9,11 eikosany. The notes are produced by multiplying together each triad of odd-factors.

"Octave"-equivalence (2:1 ratio) is assumed, thus, the Tonescape® .tuning file uses prime-factors 2, 3, 5, 7, and 11, as shown in the Tuning Editor window on the left, but the Lattice window on the right does not show the inclusion of prime-factor 2.

(Download the Tonescape® .tuning file here. Click on the image for a high-resolution image.) The table below is ordered according to the column labeled "13579e" which are the odd-factors, and makes it easy to see how the pitches of this tuning are chosen, using all possible combinations of 3 notes. A one ("1") signifies that that odd-factor is used, and a zero ("0") signifies that that odd-factor is not used.

```  2 3, 5 7 11-monzo 13579e   ratio numerator    ~cents

<-3 1, 1 0 0>   =   111000 = 1 * 3 * 5  =  15 = 1088.3
<-4 1, 0 1 0>   =   110100 = 1 * 3 * 7  =  21 =  470.8
<-4 3, 0 0 0>   =   110010 = 1 * 3 * 9  =  27 =  905.9
<-5 1, 0 0 1>   =   110001 = 1 * 3 * 11 =  33 =   53.3
<-5 0, 1 1 0>   =   101100 = 1 * 5 * 7  =  35 =  155.1
<-5 2, 1 0 0>   =   101010 = 1 * 5 * 9  =  45 =  590.2
<-5 0, 1 0 1>   =   101001 = 1 * 5 * 11 =  55 =  937.6
<-5 2, 0 1 0>   =   100110 = 1 * 7 * 9  =  63 = 1172.7
<-6 0, 0 1 1>   =   100101 = 1 * 7 * 11 =  77 =  320.1
<-6 2, 0 0 1>   =   100011 = 1 * 9 * 11 =  99 =  755.2
<-6 1, 1 1 0>   =   011100 = 3 * 5 * 7  = 105 =  857.1
<-7 3, 1 0 0>   =   011010 = 3 * 5 * 9  = 135 =   92.2
<-7 1, 1 0 1>   =   011001 = 3 * 5 * 11 = 165 =  439.6
<-7 3, 0 1 0>   =   010110 = 3 * 7 * 9  = 189 =  674.7
<-7 1, 0 1 1>   =   010101 = 3 * 7 * 11 = 231 = 1022.1
<-8 3, 0 0 1>   =   010011 = 3 * 9 * 11 = 297 =  257.2
<-8 2, 1 1 0>   =   001110 = 5 * 7 * 9  = 315 =  359.1
<-8 0, 1 1 1>   =   001101 = 5 * 7 * 11 = 385 =  706.5
<-8 2, 1 0 1>   =   001011 = 5 * 9 * 11 = 495 = 1141.5
<-9 2, 0 1 1>   =   000111 = 7 * 9 * 11 = 693 =  524.1
```

The table below puts the same notes in descending order of pitch:

```  2 3, 5 7 11-monzo 13579e   ratio numerator    ~cents

<-5 2, 0 1 0>   =   100110 = 1 * 7 * 9  =  63 = 1172.7
<-8 2, 1 0 1>   =   001011 = 5 * 9 * 11 = 495 = 1141.5
<-3 1, 1 0 0>   =   111000 = 1 * 3 * 5  =  15 = 1088.3
<-7 1, 0 1 1>   =   010101 = 3 * 7 * 11 = 231 = 1022.1
<-5 0, 1 0 1>   =   101001 = 1 * 5 * 11 =  55 =  937.6
<-4 3, 0 0 0>   =   110010 = 1 * 3 * 9  =  27 =  905.9
<-6 1, 1 1 0>   =   011100 = 3 * 5 * 7  = 105 =  857.1
<-6 2, 0 0 1>   =   100011 = 1 * 9 * 11 =  99 =  755.2
<-8 0, 1 1 1>   =   001101 = 5 * 7 * 11 = 385 =  706.5
<-7 3, 0 1 0>   =   010110 = 3 * 7 * 9  = 189 =  674.7
<-5 2, 1 0 0>   =   101010 = 1 * 5 * 9  =  45 =  590.2
<-9 2, 0 1 1>   =   000111 = 7 * 9 * 11 = 693 =  524.1
<-4 1, 0 1 0>   =   110100 = 1 * 3 * 7  =  21 =  470.8
<-7 1, 1 0 1>   =   011001 = 3 * 5 * 11 = 165 =  439.6
<-8 2, 1 1 0>   =   001110 = 5 * 7 * 9  = 315 =  359.1
<-6 0, 0 1 1>   =   100101 = 1 * 7 * 11 =  77 =  320.1
<-8 3, 0 0 1>   =   010011 = 3 * 9 * 11 = 297 =  257.2
<-5 0, 1 1 0>   =   101100 = 1 * 5 * 7  =  35 =  155.1
<-7 3, 1 0 0>   =   011010 = 3 * 5 * 9  = 135 =   92.2
<-5 1, 0 0 1>   =   110001 = 1 * 3 * 11 =  33 =   53.3
```
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