From 17/29 to 97/165

On way to investigate the sort of thing Margo is interested in, where we look at tuning systems with a fifth generator in a certain range, is to look at the Farey sequence. Below I look at the Farey sequence with denominator <= 200 over a range of sharp fifths from 17/29 to 97/165 of an octave, giving the tuning, its value in cents, and the mapping to primes up to 17 which it suggests. I would say 78/133,
61/104 and 27/46 all give 13-limit mappings which Margo might want to consider if indeed she hasn't already. 47/80 is another to look at.

17/29 703.448276 [0, 1, -8, -14, 11, 8, 7]

112/191 703.664922 [0, 1, 50, 73, 69, 66, -80]

95/162 703.703704 [0, 1, 50, 73, 40, 37, -80]

78/133 703.759399 [0, 1, 50, 44, 40, 37, -51]

61/104 703.846154 [0, 1, 21, 44, 40, 37, -51]

105/179 703.910615 [0, 1, -54, 44, -64, -67, 53]

44/75 704.000000 [0, 1, 21, -31, 11, 37, -22]

115/196 704.081633 [0, 1, 21, 90, 86, 83, -97]

71/121 704.132231 [0, 1, 21, -31, -35, -38, -22]

98/167 704.191617 [0, 1, 21, -31, -35, -38, -22]

27/46 704.347826 [0, 1, 21, 15, 11, 8, -22]

91/155 704.516129 [0, 1, -25, 15, 11, 54, 24]

64/109 704.587156 [0, 1, -25, 15, 11, 8, 24]

101/172 704.651163 [0, 1, -25, 15, 11, 8, -85]

37/63 704.761905 [0, 1, -25, 15, 11, 8, 24]

84/143 704.895105 [0, 1, 38, -48, 11, 8, -39]

47/80 705.000000 [0, 1, 38, 15, 11, 8, -39]

104/177 705.084746 [0, 1, -42, -65, -69, 8, -39]

57/97 705.154639 [0, 1, -42, 32, 11, 8, -39]

67/114 705.263158 [0, 1, 55, 32, 28, 8, -56]

77/131 705.343512 [0, 1, -59, 32, 28, 8, -56]

97/165 705.454546 [0, 1, -76, 49, 28, 8, -73]

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

> 78/133 703.759399 [0, 1, 50, 44, 40, 37]

Let's pick on this for an example. If we take the 13-limit 46-et system with generator a fifth, and move everything but the fifth
an additional 29 generators out, we get this. The rms value for the fifth is 703.8181 cents, and we can use two keyboards 46 generators apart, or in other words, 24.367 cents apart. The tuning does improve quite a bit on the 46-et values.

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
>
> > 78/133 703.759399 [0, 1, 50, 44, 40, 37]

Another way to approach this business, which makes sense in terms of how Margo seems to be using these tunings, is to look at planar rather than linear temperaments. As a 13-limit system, the above is defined by <196/195, 352/351, 364/363>. A period matrix for this, based on approximate 2, 3 and 5 generators, is

[[1, 0, 0, 10, 17, 22], [0, 1, 0, -6, -10, -13], [0, 0, 1, 1, 1, 1]]

The rms values we obtain for the 3,5,7,11 and 13 are

1903.532647 2788.935642 3367.739759 4153.609170 4443.011226

respectively. 81/80 is mapped to 25.194947 cents, and one way of
working with this system would be two keyboards, each with sharp fifths of 703.532647 cents, a comma of size 25.194947 cents apart.