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I've been studying the "taxicab" diagrams with the comma/temperment
names. I can convince myself as to why Augmented, Diminished and
Aristoxean fall along straight lines. Could someone explain why the
meantone temperments, for example, fall along a straight line.
(I see why knowing 2 out of 3 of perfect fifth, minor third and major
third determine the third one, pretty obvious,)Could someone give me
a general mathematical formula/description that explains why all the
comma/temperments fall along straigt lines? Thanks

--- In tuning-math@yahoogroups.com, "paulhjelmstad"
<paul.hjelmstad@u...> wrote:

> I've been studying the "taxicab" diagrams

huh??

> with the comma/temperment
> names. I can convince myself as to why Augmented, Diminished and
> Aristoxean fall along straight lines.

oh, you must mean the big equal temperament graph with the linear
temperament lines . . . is the meaning of the axes on that graph
clear to you?

> Could someone explain why the
> meantone temperments, for example, fall along a straight line.
> (I see why knowing 2 out of 3 of perfect fifth, minor third and
major
> third determine the third one, pretty obvious,)Could someone give
me
> a general mathematical formula/description that explains why all
the
> comma/temperments fall along straigt lines? Thanks

the comma vanishing implies a certain linear relation between any two
of the consonances. let's say perfect fifth and major third, why
don't we? then for any meantone, and only for meantones,

4*(perfect fifth) - 2400 cents = major third.

is it clear to you that this is the equation of a straight line in
perfect fifth - major third space?

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning-math@yahoogroups.com, "paulhjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> > I've been studying the "taxicab" diagrams
>
> huh??
>
> > with the comma/temperment
> > names. I can convince myself as to why Augmented, Diminished and
> > Aristoxean fall along straight lines.
>
> oh, you must mean the big equal temperament graph with the linear
> temperament lines . . . is the meaning of the axes on that graph
> clear to you?
>
> > Could someone explain why the
> > meantone temperments, for example, fall along a straight line.
> > (I see why knowing 2 out of 3 of perfect fifth, minor third and
> major
> > third determine the third one, pretty obvious,)Could someone give
> me
> > a general mathematical formula/description that explains why all
> the
> > comma/temperments fall along straigt lines? Thanks
>
> the comma vanishing implies a certain linear relation between any
two
> of the consonances. let's say perfect fifth and major third, why
> don't we? then for any meantone, and only for meantones,
>
> 4*(perfect fifth) - 2400 cents = major third.
>
> is it clear to you that this is the equation of a straight line in
> perfect fifth - major third space? Yes
Thanks, that's kind of what I thought but I am still amazed by it.

--- In tuning-math@yahoogroups.com, "paulhjelmstad"
<paul.hjelmstad@u...> wrote:

> Thanks, that's kind of what I thought but I am still amazed by it.

are you also amazed that the slope of each temperament-line is
identical to the slope of the corresponding comma-vector on the
hexagonal "small 5-limit intervals" graph below?

(i presume we're both looking at http://www.sonic-
arts.org/dict/eqtemp.htm)

--- In tuning-math@yahoogroups.com, "paulhjelmstad"
<paul.hjelmstad@u...> wrote:
> > 4*(perfect fifth) - 2400 cents = major third.
> >
> > is it clear to you that this is the equation of a straight line
in
> > perfect fifth - major third space? Yes

> Thanks, that's kind of what I thought but I am still amazed by it.

/tuning-math/files/dualzoomk.gif
/tuning-math/files/dualzoomn.gif

If you want to get really boggled check out the dual zoomers above,
and the rest of them in the files.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "paulhjelmstad"
> <paul.hjelmstad@u...> wrote:
> > > 4*(perfect fifth) - 2400 cents = major third.
> > >
> > > is it clear to you that this is the equation of a straight line
> in
> > > perfect fifth - major third space? Yes
>
> > Thanks, that's kind of what I thought but I am still amazed by it.
>
> /tuning-math/files/dualzoomk.gif
> /tuning-math/files/dualzoomn.gif
>
> If you want to get really boggled check out the dual zoomers above,
> and the rest of them in the files.
All very interesting. The ones I like best are zoomr.gif, zooms.gif
etc which show commas as lines and temperment numbers as
intersections along lines. Dualzoom graphs have temperment numbers as
lines and commas as intersections. Looks like it works both ways?
Any comments?
Thanks!