Yahoo Tuning Groups Ultimate Backup tuning Music of JP

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Thanks Carl, I'm listening. :-)
>
> BTW: On your website, I found a list of 7-limit 2D temperaments
> and I'm not sure if I've understood some of these; I'm
> particularly interested in "wizard" and I'm unable to figure out
> the mapping ... Perhaps you could give me a suggestion?
> Thanks again.
>
> Petr

My website? Must be Gene's website. What's the link?

-Carl

🔗Petr Parízek <p.parizek@...>

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Of course, you were right. It's here:
> http://lumma.org/tuning/gws/sevnames.htm
>
> Petr

A 7-limit mapping is a pair of vals.

< a b c |
< d e f |

Given on this page are "wedgies", which are bivals -- the wedge
product of two vals. They specify a temperament uniquely,
whereas many different pairs of vals will wedge to the same
bival. So if you want to transform a wedgie back into a pair
of vals, you need a way to pick which pair. Gene used something
called Hermite normal form to do this -- there's only one pair
of vals in hermite normal form for each wedgie. Graham I think
can do it, but I don't know how.

-Carl

🔗Graham Breed <gbreed@...>

2008/12/14 Petr Parízek <p.parizek@chello.cz>:
> Thanks Carl, I'm listening. :-)
>
> BTW: On your website, I found a list of 7-limit 2D temperaments and I'm not
> sure if I've understood some of these; I'm particularly interested
> in "wizard" and I'm unable to figure out the mapping ... Perhaps you could
> give me a suggestion?
> Thanks again.

I have it in the 11-limit as

| <2, 1, 5, 2, 8| , <0, 6, -1, 10, -3| >

It works with 22, 50, and 72 note equal temperaments.

Graham

🔗Graham Breed <gbreed@...>

2008/12/15 Carl Lumma <carl@lumma.org>:
> --- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>>
>> Of course, you were right. It's here:
>> http://lumma.org/tuning/gws/sevnames.htm
>>
>> Petr
>
> A 7-limit mapping is a pair of vals.
>
> < a b c |
> < d e f |
>
> Given on this page are "wedgies", which are bivals -- the wedge
> product of two vals. They specify a temperament uniquely,
> whereas many different pairs of vals will wedge to the same
> bival. So if you want to transform a wedgie back into a pair
> of vals, you need a way to pick which pair. Gene used something
> called Hermite normal form to do this -- there's only one pair
> of vals in hermite normal form for each wedgie. Graham I think
> can do it, but I don't know how.

Lets talk about an integer version of Hermite normal form. By
definition, d is zero, so you have

< a b c |
< 0 e f |

The numbers e and f are always the first entries in the wedgie, and
this generalizes for a rank 2 temperament with any number of primes.
This is the "generator mapping" multiplied by the number of periods to
the octave. That "multiplied by" tells you that this form can
introduce torsion. But at least having the generator mapping and
period gives you a big clue about the temperament. Usually the normal
form will be

< a 0 c |
< 0 e f |

but there's also a case where e=0 so you have

< a b 0 |
< 0 0 f |

so it isn't so easy to relate such forms to the wedgie. What you can
do, though, is normalize to

<-e 0 c |
< 0 e f |

This does always exist, and is easy to calculate. It has the same
numbers as the wedgie, which would be <<e f c||. For the 7-limit I'll
have to change the letters, so call it

<-a 0 d e |
< 0 a b c |

and the wedgie is <<a b c d e ...||

To get a mapping from the wedgie you can set the numbers like this.
Unless you have a column of zeros in which case I'm not sure what
happens. And the result can have torsion. To remove it you have to
keep adding the generator mapping to the first row until you get a
common factor to divide through by.

Graham

Music of JP

🔗Petr Pařízek <p.parizek@...>

🔗Carl Lumma <carl@...>

🔗Petr Parízek <p.parizek@...>

🔗Carl Lumma <carl@...>

🔗Petr Parízek <p.parizek@...>

🔗Carl Lumma <carl@...>

🔗Graham Breed <gbreed@...>

🔗Graham Breed <gbreed@...>