Bridge sequences

Using reduction to Hermite normal form, we can find a basis for the
kernel (the commas of) a val which consists of the 3-limit comma and
commas of the form 2^a 3^b p^c. Very commonly c will be +-1, and these
will be bridge commas. If we leave off the 3-limit comma, the
remaining commas give a temperament which often has a fifth for a
generator.

Here are a few bridge sequences up to the 11 limit:

10
[256/243, 2048/2025, 64/63, 512/495]
[[8, -5, 0, 0, 0], [11, -4, -2, 0, 0], [6, -2, 0, -1, 0], [9, -2, -1,
0, -1]]

12
[531441/524288, 32805/32768, 64/63, 8192/8019]
[[-19, 12, 0, 0, 0], [-15, 8, 1, 0, 0], [6, -2, 0, -1, 0], [13, -6, 0,
0, -1]]

15
[256/243, 128/125, 64/63, 512/495]
[[8, -5, 0, 0, 0], [7, 0, -3, 0, 0], [6, -2, 0, -1, 0], [9, -2, -1, 0,
-1]]

19
[1162261467/1073741824, 71744535/67108864, 137781/131072,
17537553/16777216]
[[-30, 19, 0, 0, 0], [-26, 15, 1, 0, 0], [-17, 9, 0, 1, 0], [-24, 13,
0, 0, 1]]

22
[34359738368/31381059609, 8388608/7971615, 64/63, 8192/8019]
[[35, -22, 0, 0, 0], [23, -13, -1, 0, 0], [6, -2, 0, -1, 0], [13, -6,
0, 0, -1]]

27
[8796093022208/7625597484987, 2147483648/1937102445, 64/63, 8192/8019]
[[43, -27, 0, 0, 0], [31, -18, -1, 0, 0], [6, -2, 0, -1, 0], [13, -6,
0, 0, -1]]

31
[617673396283947/562949953421312, 38127987424935/35184372088832,
73222472421/68719476736, 17537553/16777216]
[[-49, 31, 0, 0, 0], [-45, 27, 1, 0, 0], [-36, 21, 0, 1, 0], [-24, 13,
0, 0, 1]]

I note that 12 and 22 share 64/63 and 8192/8019, leading to a
corresponding planar temperament with basis 64/63 and 99/98.

Here is the standard 12 val taken out to higher primes, in case that
is inspirational to people considering notation systems:

[531441/524288, 32805/32768, 64/63, 8192/8019, 1053/1024, 4131/4096,
513/512, 16767/16384, 261/256, 67797/65536, 1024/999, 269001/262144,
129/128]

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> Using reduction to Hermite normal form, we can find a basis for the
> kernel (the commas of) a val which consists of the 3-limit comma and
> commas of the form 2^a 3^b p^c.

We can then further reduce each comma with the 3-limit comma. In the
case of standard 12, that would give

531441/524288, 81/80, 64/63, 128/121, 169/162, 289/288, 513/512...

We might also require that the non-three limit comma appears in the
product just once, so that we stick to the bridge idea, giving instead

531441/524288, 81/80, 64/63, 729/704, 1053/1024, 2187/2076, 513/512...

instead.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> I note that 12 and 22 share 64/63 and 8192/8019, leading to a
> corresponding planar temperament with basis 64/63 and 99/98.

And 27.