Temperaments for diamonds

If we ask for an r2 temperament which distinguishes all 13 notes of
the 7-limit diamond and has minimal diamond complexity, we find we can
get it down to 19 with some undistingished temperaments, and down to
21 with better ones.

For 19, we can try <<3 -5 -6 -15 -18 0||, <<1 -8 -2 -15 -6 18|| or
<<3 5 9 1 6 7||. These have TM bases of {64/63,392/375},
{64/63,360/343} and {28/27,126/125} respectively. Considerably more
accurate are certain temperaments with a diamond complexity of 21,
which are meantone, myna, augene/tripletone, and another cheesy one,
<<3 0 9 -7 6 21||. Of these, the greatest accuracy is found with myna,
and the smallest Graham complexity ties between augene/tripletone and
the cheesy one. If we are willing to let the complexity go as high as
23, then we have various further choices, the most interesting being
orwell, and another superpyth.

For the 9-limit diamond and diamond complexity, we can bring the
diamond complexity down to 25 using magic, or we have a cheesy
alternative with <<0 5 10 8 16 9|| tempering out {200/189,225/224}.

Detempered versions of the minimal scales will contain tonality
diamonds, and are of possible interest.

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

> For the 9-limit diamond and diamond complexity, we can bring the
> diamond complexity down to 25 using magic, or we have a cheesy
> alternative with <<0 5 10 8 16 9|| tempering out {200/189,225/224}.

In the 11-limit, the clear winner is miracle, with a diamond
complexity of 45; this gets us back to the Miracle[45] scale which has
been discussed before. Runner up, surprisingly to me, is myna;
Myna[51] also containing a diamond. By 11-limit myna I mean 31&58,
with a TM basis of
{126/125, 176/175, 243/242} and a wedgie <<10 9 7 25 -9 -17 5 -9 27 46||
This has a Graham complexity of 25, as opposed to 22 for miracle, so
miracle simply whips the llama's ass in the 11-limit.

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

> In the 11-limit, the clear winner is miracle...

In the 13-limit, garibaldi, or 41&53, has a diamond complexity of
75 and a Graham complexity of 37, and seems to be the winner. We also
have rodan, or 41&46, with a diamond complexity of 79 and a Graham
complexity of 39, only a touch more complex, but also a touch more
accurate in TOP error. Finally there is mystery, or 58&87, which has a
diamond complexity of 87, but a Graham complexity of 29, which is much
better than garibaldi or rodan, and which has a better accuracy figure
to boot.

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

> In the 13-limit, garibaldi, or 41&53, has a diamond complexity of
> 75 and a Graham complexity of 37, and seems to be the winner. We also
> have rodan, or 41&46, with a diamond complexity of 79 and a Graham
> complexity of 39, only a touch more complex, but also a touch more
> accurate in TOP error. Finally there is mystery, or 58&87, which has a
> diamond complexity of 87, but a Graham complexity of 29, which is much
> better than garibaldi or rodan, and which has a better accuracy figure
> to boot.

In the 15-limit, we get the same three suspects, but now rodan climbs
slightly in complexity, to a diamond complexity of 85 and a Graham
complexity of 42. Both rodan and mystery are supported by the 87-et
standard val, and have a MOS of that size, which could be detempered
to give a scale containing a 15-limit diamond. 87-et in the 13-limit has
a TM basis {196/195, 245/243, 352/351, 364/363, 625/624}; leaving off
the 625/624 gives the TM basis for rodan, and leaving off the 245/243
the TM basis for mystery.