Fractional Comma Manifesto, Part 2

As Jonathan Walker said, my equations were not simplified. (He also
pointed that I forgot the frational part of an exponent.)

The reason I did not simplify the expressions was that I wanted the
isolate the 3^m/2^n factor. But the equation of a chain of meantone or
ET fifths can be expressed in a simplified way. I find that the long
expressions helps us understand where the simplifications come from.

The mth meantone fifth is given by

2 ^ (4m/k - n) * 3 ^ (m - 4m/k) * 5 ^ (-m/k)

where m is an integer, k is the "inverse of the comma fration" (for 1/4
comma meantone, k=4). n is the integer part of log2(3)m. The 4's are the
exponents of 2 and 3 in the syntonic comma, 81/80.

For example, 1/4 comma meantone is

2 ^ (4m/4 - n) * 3 ^ (m - 4m/4) * 5 ^ (m/4)

or

2^(m-n) * 5^(m/4).

Why is a power of 3 included in the general equation? Because the 3's do
not always simplify. Take a look at 1/3 comma meantone :

2 ^ (4m/3 - n) * 3 ^ (m - 4m/3) * 5 ^ (m/3)

When m= 3, we get

2^(4*3/3 -4) * 3^(3- 4*3/3) * 5^(3/3) = 5/3.

*******
Now let's look at ET's. Let a=log2(3) and [x] mean integer part of x.
The equation for the mth ET fifth (from my first post) simplifies to :

2 ^ ( [ak]m/k - [am] )

For the 12TET fifth, we have

2 ^ ( [a*12]*1/12 - [a*1] ) = 2^ ( 19/12 - 1 ) = 2^ 7/12.

Surprised? The ET fifth is not only a random division of the octave, it
is derived from a circle of fifths. For nTET, are there n fifths in the
circle? Not always. When I began my little investigation, I
believed that you could get all the notes from the circle of fifths, but
sometimes this circle closes before you get all the scale degrees. Here
is a list of fifths, and the ET's they are used in, for 0
2^x ET
0/1 1
1/2 2,4,8,10
1/3 3
2/5 5
4/7 7,14,21,28,35,42,49,56,63,70
5/9 9,18,27
6/11 11,22
7/12 12,24,36,48,60,72,84,96,108,120,132,144
7/13 13
8/15 15
9/16 16,32
9/17 17
11/19 19,38,57,76,95,114,133
11/20 20
13/23 23,46
14/25 25
15/26 26,52,78,104
16/29 29
17/30 30
18/31 31,62,93,124
19/33 33,66,99
19/34 34
21/37 37
22/39 39
23/40 40,80
23/41 41
25/43 43,86,129
25/44 44
26/45 45,90,135
27/47 47,94
29/50 50,100
29/51 51
31/53 53,106
31/54 54
32/55 55,110
33/58 58
34/59 59
35/61 61
37/64 64,128
38/65 65,130
39/67 67,134
39/68 68
40/69 69,138
41/71 71
42/73 73
43/74 74
43/75 75
45/77 77
46/79 79
47/81 81
47/82 82
48/83 83
49/85 85
50/87 87
51/88 88
52/89 89
53/91 91
53/92 92
56/97 97
57/98 98
59/101 101
59/102 102
60/103 103
61/105 105
62/107 107
63/109 109
64/111 111
65/112 112
66/113 113
67/115 115
67/116 116
68/117 117
69/118 118
69/119 119
70/121 121
71/122 122
71/123 123
73/125 125
73/126 126
74/127 127
76/131 131
79/136 136
80/137 137
81/139 139
81/140 140
82/141 141
83/142 142
83/143 143

Can anyone on the list explain why this happens? And also, please
correct any errors, typos, and add your comments, your "grain of salt".

-Kami

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