Equal temperaments of arbitrary prime limits

I've got some updated code now at

http://microtonal.co.uk/temper/mixlimit.zip

One thing I've added since my MMM post is that it weights an equal
temperament according to the complexity of the basis. So, 2.3.5.7 is
simpler than 2.13.17.19. As it was, high primes were effectively
favored because they're allowed to be more out of tune for a given
weighed error.

If you're interested the best thing is to download the code and play
around with it. But here are some results for primes up to 19 with no
more than 28 notes to the 2:1 octave and at least 4 primes in the
basis.

2.3.17.19-limit <12, 19, 49, 51]
2.3.11.17-limit <24, 38, 83, 98]
2.3.11.19-limit <24, 38, 83, 102]
2.3.11.13-limit <17, 27, 59, 63]
2.3.11.17.19-limit <24, 38, 83, 98, 102]
2.11.17.19-limit <24, 83, 98, 102]
2.5.11.19-limit <28, 65, 97, 119]
2.3.17.19-limit <24, 38, 98, 102]
2.5.7.11-limit <6, 14, 17, 21]
2.3.5.13-limit <19, 30, 44, 70]
2.5.7.17-limit <25, 58, 70, 102]
2.3.5.7-limit <19, 30, 44, 53]
2.3.7.11-limit <17, 27, 48, 59]
2.3.5.17-limit <12, 19, 28, 49]
2.11.13.17-limit <13, 45, 48, 53]
2.5.11.17-limit <22, 51, 76, 90]
2.7.13.17-limit <10, 28, 37, 41]
2.3.5.19-limit <12, 19, 28, 51]
2.7.11.17-limit <11, 31, 38, 45]
2.5.17.19-limit <25, 58, 102, 106]
2.3.5.11-limit <7, 11, 16, 24]
2.3.5.7-limit <12, 19, 28, 34]
2.3.11.13-limit <24, 38, 83, 89]
2.5.7.19-limit <25, 58, 70, 106]
2.3.5.7-limit <22, 35, 51, 62]
2.7.11.13-limit <20, 56, 69, 74]
2.3.5.7-limit <27, 43, 63, 76]
2.3.7.13-limit <17, 27, 48, 63]
2.7.11.13-limit <26, 73, 90, 96]
2.13.17.19-limit <13, 48, 53, 55]
2.5.7.17.19-limit <25, 58, 70, 102, 106]
2.3.11.13.19-limit <24, 38, 83, 89, 102]
2.3.7.11.13-limit <17, 27, 48, 59, 63]
2.5.7.13-limit <10, 23, 28, 37]
2.5.7.13-limit <16, 37, 45, 59]
2.3.5.11-limit <22, 35, 51, 76]
2.3.5.17.19-limit <12, 19, 28, 49, 51]

Graham

> http://microtonal.co.uk/temper/mixlimit.zip
>
>One thing I've added since my MMM post is that it weights an equal
>temperament according to the complexity of the basis. So, 2.3.5.7 is
>simpler than 2.13.17.19. As it was, high primes were effectively
>favored because they're allowed to be more out of tune for a given
>weighed error.

Great, I tried to ask for that.

>If you're interested the best thing is to download the code and play
>around with it. But here are some results for primes up to 19 with no
>more than 28 notes to the 2:1 octave and at least 4 primes in the
>basis.
>
> 2.3.17.19-limit <12, 19, 49, 51]
> 2.3.11.17-limit <24, 38, 83, 98]
> 2.3.11.19-limit <24, 38, 83, 102]
> 2.3.11.13-limit <17, 27, 59, 63]
> 2.3.11.17.19-limit <24, 38, 83, 98, 102]
> 2.11.17.19-limit <24, 83, 98, 102]
> 2.5.11.19-limit <28, 65, 97, 119]
> 2.3.17.19-limit <24, 38, 98, 102]
> 2.5.7.11-limit <6, 14, 17, 21]
> 2.3.5.13-limit <19, 30, 44, 70]
> 2.5.7.17-limit <25, 58, 70, 102]

Yessir, there's Paul Rappaport's no-3s 25.

> 2.3.5.7-limit <19, 30, 44, 53]

Interesting that 19 comes out best for this basis.

> 2.3.7.11-limit <17, 27, 48, 59]

Yessir, there's the no-5s 17 I (and no doubt others) have
suggested.

> 2.3.5.17-limit <12, 19, 28, 49]
> 2.11.13.17-limit <13, 45, 48, 53]
> 2.5.11.17-limit <22, 51, 76, 90]
> 2.7.13.17-limit <10, 28, 37, 41]
> 2.3.5.19-limit <12, 19, 28, 51]
> 2.7.11.17-limit <11, 31, 38, 45]
> 2.5.17.19-limit <25, 58, 102, 106]
> 2.3.5.11-limit <7, 11, 16, 24]
> 2.3.5.7-limit <12, 19, 28, 34]
> 2.3.11.13-limit <24, 38, 83, 89]
> 2.5.7.19-limit <25, 58, 70, 106]
> 2.3.5.7-limit <22, 35, 51, 62]
> 2.7.11.13-limit <20, 56, 69, 74]
> 2.3.5.7-limit <27, 43, 63, 76]
> 2.3.7.13-limit <17, 27, 48, 63]
> 2.7.11.13-limit <26, 73, 90, 96]
> 2.13.17.19-limit <13, 48, 53, 55]

I'm not seeing any temperaments > 30. My original hope was
to sort not by limit, but by ET < 100... to give the characteristic
n-ad for each ET.

> 2.5.7.17.19-limit <25, 58, 70, 102, 106]
> 2.3.11.13.19-limit <24, 38, 83, 89, 102]
> 2.3.7.11.13-limit <17, 27, 48, 59, 63]
> 2.5.7.13-limit <10, 23, 28, 37]
> 2.5.7.13-limit <16, 37, 45, 59]
> 2.3.5.11-limit <22, 35, 51, 76]
> 2.3.5.17.19-limit <12, 19, 28, 49, 51]

Looks like there are pentads mixed in this bunch.

-Carl

On 29/10/06, Carl Lumma <ekin@lumma.org> wrote:
> > http://microtonal.co.uk/temper/mixlimit.zip
> >
> >One thing I've added since my MMM post is that it weights an equal
> >temperament according to the complexity of the basis. So, 2.3.5.7 is
> >simpler than 2.13.17.19. As it was, high primes were effectively
> >favored because they're allowed to be more out of tune for a given
> >weighed error.
>
> Great, I tried to ask for that.

Oh, well, now you've got it. You can check the zip file for nonoctave results.

> > 2.3.5.7-limit <19, 30, 44, 53]
>
> Interesting that 19 comes out best for this basis.

I think 22 is best in this search for 2.3.5.7.11

> I'm not seeing any temperaments > 30. My original hope was
> to sort not by limit, but by ET < 100... to give the characteristic
> n-ad for each ET.

Nothing higher than 28, as I said. The list gets cluttered with the
bigger temperaments. If you want them you can change a number in the
source code.

If you don't want to sort then stop it sorting. I did say before I
didn't know which order you wanted the sorting done in.

>
> > 2.5.7.17.19-limit <25, 58, 70, 102, 106]
> > 2.3.11.13.19-limit <24, 38, 83, 89, 102]
> > 2.3.7.11.13-limit <17, 27, 48, 59, 63]
> > 2.5.7.13-limit <10, 23, 28, 37]
> > 2.5.7.13-limit <16, 37, 45, 59]
> > 2.3.5.11-limit <22, 35, 51, 76]
> > 2.3.5.17.19-limit <12, 19, 28, 49, 51]
>
> Looks like there are pentads mixed in this bunch.

Yes, at least 4 primes. 17-equal is one of the best with 5 primes,
using the basis George suggested.

Graham

>Nothing higher than 28, as I said. The list gets cluttered with the
>bigger temperaments. If you want them you can change a number in the
>source code.
>
>If you don't want to sort then stop it sorting. I did say before I
>didn't know which order you wanted the sorting done in.

It's not an order question, it's a key-value reversal. I'll have a
look at the code.

-Carl