question on meantone

I do have a question on 1/4 comma meantone...

Why does it work out that when all the fifths (except the "wolf" one,
of course) are tempered by 1/4 syntonic comma it works out that
the "D" is midway between 9/8 and 10/9??

Sorry to be so dense on this...

ALSO, when these TWO different sized whole tones are eliminated to
make ONE tempered 1/4 comma whole tone, is that, essentially defining
a "periodicity block" by eliminating a "unison vector...??"

Thanks!!!!!!

___________ ________ ______
Joseph Pehrson

In-Reply-To: <9l0u00+c6fg@eGroups.com>
Joseph Pehrson wrote:

> I do have a question on 1/4 comma meantone...
>
> Why does it work out that when all the fifths (except the "wolf" one,
> of course) are tempered by 1/4 syntonic comma it works out that
> the "D" is midway between 9/8 and 10/9??

The distance between 9/8 and 10/9 is a syntonic comma. A whole tone is
two steps on the spiral of fifths. So, if you temper the fifth by 1/4
comma, the tone will be tempered by 1/2 comma, leaving it halfway between
the two values.

> Sorry to be so dense on this...
>
> ALSO, when these TWO different sized whole tones are eliminated to
> make ONE tempered 1/4 comma whole tone, is that, essentially defining
> a "periodicity block" by eliminating a "unison vector...??"

It's eliminating a unison vector (the syntonic comma) from a periodicity
block, yes.

Graham

Graham answered:
ALSO, when these TWO different sized whole tones are eliminated to
> make ONE tempered 1/4 comma whole tone, is that, essentially
defining
> a "periodicity block" by eliminating a "unison vector...??"
It's eliminating a unison vector (the syntonic comma) from a
periodicity
block, yes. Graham

Gentlemen, please forgive my ignorance, but could you define the
terms "unison vector" and "periodicity block" for me, or refer me to
some source of information of this kind?

Thanks,

Bob

--- In tuning@y..., graham@m... wrote:

/tuning/topicId_26870.html#26871

> In-Reply-To: <9l0u00+c6fg@e...>
> Joseph Pehrson wrote:
>
> > I do have a question on 1/4 comma meantone...
> >
> > Why does it work out that when all the fifths (except the "wolf"
one, of course) are tempered by 1/4 syntonic comma it works out that
> > the "D" is midway between 9/8 and 10/9??
>
> The distance between 9/8 and 10/9 is a syntonic comma. A whole
tone is two steps on the spiral of fifths. So, if you temper the
fifth by 1/4 comma, the tone will be tempered by 1/2 comma, leaving
it halfway between the two values.
>

DUH. THANKS. FINALLY I got this!

I wish this explanation were up on Kyle Gann's otherwise excellent
webpage on meantone tuning. It was the one thing that wasn't clear!

Also, this really isn't explained in any of Joe Monzo's links on
meantone.

I wish Joe would put it in someplace... Joe??

THANKS GRAHAM!

__________ __________ _______
Joseph Pehrson

--- In tuning@y..., BobWendell@t... wrote:

>
> Gentlemen, please forgive my ignorance, but could you define the
> terms "unison vector" and "periodicity block" for me, or refer me
to
> some source of information of this kind?
>
> Thanks,
>
> Bob

Hi Bob!

Please refer to my _Gentle Introduction to Fokker Periodicity Blocks_:

http://www.ixpres.com/interval/td/erlich/intropblock1.htm

Tempering is not covered here, but tempering of periodicity blocks is
one of the main subjects on

tuning-math@yahoogroups.com

Cheers,
Paul