x31eq: meantone+

This has a step of -2.x cents?

http://x31eq.com/cgi-bin/rt.cgi?ets=5_12_19&limit=7

-Carl

Carl Lumma <carl@lumma.org> wrote:
> This has a step of -2.x cents?
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=5_12_19&limit=7

Yes. And everything's positive if you change the 5 to a 7:

http://x31eq.com/cgi-bin/rt.cgi?ets=7_12_19&limit=7

Graham

>> This has a step of -2.x cents?
>>
>> http://x31eq.com/cgi-bin/rt.cgi?ets=5_12_19&limit=7
>
>Yes. And everything's positive if you change the 5 to a 7:
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=7_12_19&limit=7

I don't understand what's happening here - do you? -Carl

Carl Lumma <carl@lumma.org> wrote:
> >> This has a step of -2.x cents?
> >>
> >> http://x31eq.com/cgi-bin/rt.cgi?ets=5_12_19&limit=7
> >
> >Yes. And everything's positive if you change the 5 to a
> >7:
> >
> > http://x31eq.com/cgi-bin/rt.cgi?ets=7_12_19&limit=7
>
> I don't understand what's happening here - do you? -Carl

225:224 is tempered out by 12 and 19-equal. In 5-equal, it
approximates to 1 step. (16:15 is tempered out, 8:7
approximates to 1 step. 225:224 is the amount by which to
steps of 16:15 fall short of 8:7.) That all means the step
corresponding to 5-equal gets tuned to the approximation of
225:224. It happens that in the optimal temperament,
225:224 approximates to -2.5 cents.

225:224 is also tempered out by the best mapping for
7-equal. That means 7d&12&19 has a rank of 2 and doesn't
get shown by rt.cgi. With that out of the way, 7&12&19 is
unambiguously 7p&12&19. 225:224 approximates to -1 step of
7p (8:7 and 16:15 both approximate to 1 step).

Negative steps are unusual because a good temperament will
usually approximate simple commas so that the error's less
than the size of the comma. The error has to be larger then
the size of the comma for the comma to come out negative.

Graham

Great explanation, thanks.

Sent from my iPhone

On Jul 22, 2011, at 5:02 PM, Graham Breed <gbreed@gmail.com> wrote:

Carl Lumma <carl@lumma.org> wrote:
> >> This has a step of -2.x cents?
> >>
> >> http://x31eq.com/cgi-bin/rt.cgi?ets=5_12_19&limit=7
> >
> >Yes. And everything's positive if you change the 5 to a
> >7:
> >
> > http://x31eq.com/cgi-bin/rt.cgi?ets=7_12_19&limit=7
>
> I don't understand what's happening here - do you? -Carl

225:224 is tempered out by 12 and 19-equal. In 5-equal, it
approximates to 1 step. (16:15 is tempered out, 8:7
approximates to 1 step. 225:224 is the amount by which to
steps of 16:15 fall short of 8:7.) That all means the step
corresponding to 5-equal gets tuned to the approximation of
225:224. It happens that in the optimal temperament,
225:224 approximates to -2.5 cents.

225:224 is also tempered out by the best mapping for
7-equal. That means 7d&12&19 has a rank of 2 and doesn't
get shown by rt.cgi. With that out of the way, 7&12&19 is
unambiguously 7p&12&19. 225:224 approximates to -1 step of
7p (8:7 and 16:15 both approximate to 1 step).

Negative steps are unusual because a good temperament will
usually approximate simple commas so that the error's less
than the size of the comma. The error has to be larger then
the size of the comma for the comma to come out negative.

Graham

Graham wrote:

>> >> This has a step of -2.x cents?
>> >>
>> >> http://x31eq.com/cgi-bin/rt.cgi?ets=5_12_19&limit=7
>> >
>> >Yes. And everything's positive if you change the 5 to a
>> >7:
>> >
>> > http://x31eq.com/cgi-bin/rt.cgi?ets=7_12_19&limit=7
>>
>> I don't understand what's happening here - do you? -Carl
>
>225:224 is tempered out by 12 and 19-equal. In 5-equal, it
>approximates to 1 step. (16:15 is tempered out, 8:7
>approximates to 1 step. 225:224 is the amount by which to
>steps of 16:15 fall short of 8:7.) That all means the step
>corresponding to 5-equal gets tuned to the approximation of
>225:224. It happens that in the optimal temperament,
>225:224 approximates to -2.5 cents.

These two have the same "reduced mapping", which I feel like
I understand. They have different "equal temperament mappings",
which apparently I don't understand. They ought to be the
same temperament and therefore have the same rank. Any ideas
how I can clear up my recalcitrant thinking?

-Carl

Carl Lumma <carl@lumma.org> wrote:

> >> >> http://x31eq.com/cgi-bin/rt.cgi?ets=5_12_19&limit=7

> >> > http://x31eq.com/cgi-bin/rt.cgi?ets=7_12_19&limit=7

> These two have the same "reduced mapping", which I feel
> like I understand. They have different "equal
> temperament mappings", which apparently I don't
> understand. They ought to be the same temperament and
> therefore have the same rank. Any ideas how I can clear
> up my recalcitrant thinking?

They are the same temperament (by any definition of
"temperament" that might apply) and they have the same rank
(3). The equal temperament mappings are different because
different equal temperaments are being mapped to.

Graham

>> >> >> http://x31eq.com/cgi-bin/rt.cgi?ets=5_12_19&limit=7
>> >> >> http://x31eq.com/cgi-bin/rt.cgi?ets=7_12_19&limit=7
>
>> These two have the same "reduced mapping", which I feel
>> like I understand. They have different "equal
>> temperament mappings", which apparently I don't
>> understand. They ought to be the same temperament and
>> therefore have the same rank. Any ideas how I can clear
>> up my recalcitrant thinking?
>
>They are the same temperament (by any definition of
>"temperament" that might apply) and they have the same rank
>(3). The equal temperament mappings are different because
>different equal temperaments are being mapped to.

Thanks, I needed that!

-Carl