# minerva

[from Gene Ward Smith, Yahoo tuning-math message 11417 (Sat Aug 28, 2004 9:18 pm)]

225/32 gives us a 5-limit version of 7, and 5625/512 a 5-limit version of 11. Tempering out both 225/224 and 5625/5632 gives what I've called "minerva", the 11-limit planar temperament with TM basis {99/98, 176/175}.

It has wedgie <<<1 2 4 -2 -2 4 -5 -9 2 8|| and mapping [ <1 0 0 -5 -9|, <0 1 0 2 2|, <0 0 1 2 4| ] .

Written as a matrix of five rows and three columns,

```[  1  0  0 ]
[  0  1  0 ]
[  0  0  1 ]
[ -5  2  2 ]
[ -9  2  4 ]
```

The bottom two rows are simply 225/32 and 5625/512, which is typical behavior in these planar mappings using Hermite reduction.

As often happens with planar temperaments, the optimized tunings can be rather variable. The minimax tuning is particularly interesting, since it equates the error of 5 and 7, and hence makes 7/5 pure.

In other words, the minimax tuning for minerva, an extension of 7-limit marvel, is close to Bigler 1/2-kleismic marvel. As you may recall, this is marvel sneaking off in the direction of meantone; 81/80 does not vanish but it shrinks to a mere 4.47 cents; nor does 385/384 vanish, but it shrinks also to 1.45 cents.

We don't get the huygens version of 11-limit meantone, but we draw close to it--close enough to make a difference. We end up in a system which can be regarded as either irregular, inconsistent, or just plain lucky, depending on how you want to conceptualize it. Whatever you call it, Bigler marvel/minimax minerva seems like a useful way to temper Fokker blocks and genera.

The rms tuning for minerva is also interesting; the value it gives to 7/6 makes it a good orwell generator; in fact, pretty close to the orwell of 84+53 = 137-equal.

In terms of lattices and euler-fokker-genera, the 225/32 approximate 7 lies next to 375/256, an approximate 16/11. We have 16/11 ~ 35/24 via 385/384, and 35/24 ~ 375/256 via 225/224, so this is an 11-limit marvel approximation. It isn't as good as 5625/512 for producing complete 11-limit chords, since it is a 16/11 and not an 11/8, but it is excellent at adding in 11-limit intervals.

If we decide we want a 225/224 system in which both 5625/512 is an approximate 11 *and* 375/256 is an approximate 16/11, we want all of 225/224, 385/384, and 5625/5632 to vanish; this combined marvel-minerva is 11-limit orwell, 22&31. The rms tuning, and orwell for that matter, makes the 5-limit more in tune; the rms minerva tuning has a fifth 1.29 cents flat and a major third only 0.34 cents sharp. On the other hand, it doesn't shrink 385/384, but actually makes it larger--8.71 cents. Nor does it do a very good job with 81/80, managing only to cut it down to 16 cents.

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