Extending temperaments using TOP tuning

I havn't made use of the "brute force" approach Dave and Herman have
had success with as a temperament finding algorithm, being an
algebraist by training. However, it occurs to me that if I want to
find higher limit tunings which accord with a lower-limit tuning and
hence deserve the same name, I could as an alternative to my algebraic
approach simply use the TOP tuned generators and search for good
approximations to the higher-limit primes. If I use the 5-limit TOP
values for 2 and 3, and if I search for exponents for each in the
range -100 to 100, I get only one value within 2 cents for 7, namely
3^10/2^13 (leading to 7-limit meantone), and only one value for 11
within 2 cents, 2^24/3^13, leading to 11-limit meantone, or "meanpop".
I find nothing for 13 under 2 cents, and it hardly makes sense to
increase the range of the exponents past a complexity of 100. Pushing
the error limit up to 9 cents gives 3^46/2^69, which isn't going to
break any records for wonderfulness.

The wedgie for this 13-limit meantone is

<<1 4 10 -13 46 4 13 -24 69 12 -44 92 -71 92 207||

The mapping is

[<1 2 4 7 -2 23|, <0 -1 -4 -10 13 -46|]

The TM basis is {81/80, 126/125, 385/384, 1573/1568}

The TOP generators are only slightly different. 112 equal is one
tuning choice, but the temperament really makes no sense, since you
may as well simply use 31.

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

> The TOP generators are only slightly different. 112 equal is one
> tuning choice, but the temperament really makes no sense, since you
> may as well simply use 31.

The lack of logic could be the basis for a naming system; since
nothing makes sense as a 13-limit meantone, don't call any of them
"meantone". In the same way, nothing seems to make sense as 13-limit
miracle, or 11-limit ennealimmal, etc, since the complexities are so
high you may as well be in an equal temperament.

Gene Ward Smith wrote:

> I havn't made use of the "brute force" approach Dave and Herman have
> had success with as a temperament finding algorithm, being an
> algebraist by training. However, it occurs to me that if I want to
> find higher limit tunings which accord with a lower-limit tuning and
> hence deserve the same name, I could as an alternative to my algebraic
> approach simply use the TOP tuned generators and search for good
> approximations to the higher-limit primes. If I use the 5-limit TOP
> values for 2 and 3, and if I search for exponents for each in the
> range -100 to 100, I get only one value within 2 cents for 7, namely
> 3^10/2^13 (leading to 7-limit meantone), and only one value for 11
> within 2 cents, 2^24/3^13, leading to 11-limit meantone, or "meanpop".
> I find nothing for 13 under 2 cents, and it hardly makes sense to
> increase the range of the exponents past a complexity of 100. Pushing
> the error limit up to 9 cents gives 3^46/2^69, which isn't going to
> break any records for wonderfulness.

Hmm... if you apply this to 5-limit TOP mavila, you get a couple of potential mappings for 7-limit mavila variants:

[[1, 2, 1, -5], [0, -1, 3, 18]] <<1, -3, -18, -7, -31, -33]]
[[1, 2, 1, 11], [0, -1, 3, -19]] <<1, -3, 19, -7, 27, 52]]

I can see this isn't going to be anyone's favorite 7-limit temperament. But if you skip 7, you have a pretty good 11 (10.15 cents flat). Of course, by the time you get it down into the range of 11/8, it ends up being exactly the same as a fourth. :-) It's only a good 11 because the octaves are wide.

Actually, because of the wide octaves, probably a better mapping is

[[1, 2, 1, 8], [0, -1, 3, -12]] <<1, -3, 12, -7, 16, 36]]

This ends up with a 7:4 that's 12.2 cents sharp, and you'd have a nice 16-note MOS that could use this mapping.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@...> wrote:
>
> I havn't made use of the "brute force" approach Dave and Herman have
> had success with as a temperament finding algorithm, being an
> algebraist by training. However, it occurs to me that if I want to
> find higher limit tunings which accord with a lower-limit tuning and
> hence deserve the same name, I could as an alternative to my algebraic
> approach simply use the TOP tuned generators and search for good
> approximations to the higher-limit primes. If I use the 5-limit TOP
> values for 2 and 3, and if I search for exponents for each in the
> range -100 to 100, I get only one value within 2 cents for 7, namely
> 3^10/2^13 (leading to 7-limit meantone), and only one value for 11
> within 2 cents, 2^24/3^13, leading to 11-limit meantone, or "meanpop".
> I find nothing for 13 under 2 cents, and it hardly makes sense to
> increase the range of the exponents past a complexity of 100. Pushing
> the error limit up to 9 cents gives 3^46/2^69, which isn't going to
> break any records for wonderfulness.
>
> The wedgie for this 13-limit meantone is
>
> <<1 4 10 -13 46 4 13 -24 69 12 -44 92 -71 92 207||
>
> The mapping is
>
> [<1 2 4 7 -2 23|, <0 -1 -4 -10 13 -46|]
>
> The TM basis is {81/80, 126/125, 385/384, 1573/1568}
>
> The TOP generators are only slightly different. 112 equal is one
> tuning choice, but the temperament really makes no sense, since you
> may as well simply use 31.

I found this old post of Gene.

I think it would be reasonable if the higher-limit primes had the same
max damage as the original tuning and using for every prime the first
approximation in the chain of generators that has Tenney-weighted
error under that damage. This also means that the tuning doesn't
change.

As an example 7-limit Pajara extends like this

period 598.4467109449759
generator 106.5665459263645
damage 3.106578110048495

2 2, 0
3 3, 1
5 5, -2
7 6, -2
11 8, -6
13 6, 8
17 8, 1
19 8, 3
23 8, 6
29 11, -7
31 11, -6

Kalle