2499 comma

I was wondering how to incorporate the 17/10 dim7 into my framework of
septimal 1/3 comma adaptive tuning (provisionally 171edo) and it turns
out to be trivial... at least on the level of formal progression.

The key to the scheme as explained earlier was that every minor third
was either a 6/5 or (if sharing a 10/7 tritone with a 6/5) a 25/21.
Since the difference between these is two steps of 1/3 comma or
171edo, this forces one voice to go up a step and one down a step when
moving from one to the other.

Now if I want to introduce a 17/10 dim7 in place of 12/7, that implies
a 17/12 dim5 and gives another minor third of 119/100 (I don't plan on
dividing 17/12 into 6/5 and 85/72...). This is short of 6/5 by
119/120, which is rather close to the septimal comma 125/126 that gets
mapped onto one step of 171edo. The difference between 25/21 and
119/100 is actually just 2499/2500 = 3.7.7.17/2.2.5.5.5.5, which is a
comma I have never seen mentioned - although it seems to have cropped
up rather naturally here.

I suppose this means there is a family of temperaments which eliminate
it, of which 171edo provides one member...
~~~T~~~

I hope you don't call this comma a 'dent'
but now it is probably going to stick! :)

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Tom Dent wrote:
>
>
> I was wondering how to incorporate the 17/10 dim7 into my framework of
> septimal 1/3 comma adaptive tuning (provisionally 171edo) and it turns
> out to be trivial... at least on the level of formal progression.
>
> The key to the scheme as explained earlier was that every minor third
> was either a 6/5 or (if sharing a 10/7 tritone with a 6/5) a 25/21.
> Since the difference between these is two steps of 1/3 comma or
> 171edo, this forces one voice to go up a step and one down a step when
> moving from one to the other.
>
> Now if I want to introduce a 17/10 dim7 in place of 12/7, that implies
> a 17/12 dim5 and gives another minor third of 119/100 (I don't plan on
> dividing 17/12 into 6/5 and 85/72...). This is short of 6/5 by
> 119/120, which is rather close to the septimal comma 125/126 that gets
> mapped onto one step of 171edo. The difference between 25/21 and
> 119/100 is actually just 2499/2500 = 3.7.7.17/2.2.5.5.5.5, which is a
> comma I have never seen mentioned - although it seems to have cropped
> up rather naturally here.
>
> I suppose this means there is a family of temperaments which eliminate
> it, of which 171edo provides one member...
> ~~~T~~~
>
>