Margo's Miracle note comparisons

Hi Margo,

Enjoyed your post.

I am quite thrilled that you brought up the issues of
equal divisions of the fifth, a topic I rather giddily
enjoy. There are lots of fantastic tunings in here. I've
discussed with Gary before that 88cET is really basically
the 8th root of 3:2, as you noted. He does not exactly
agree and likes to keep in at a precise 88.000 cents,
which keeps some other intervals in the scale tuned more
the way he likes them.

No one has yet brought up, to my awareness, that Alpha,
Beta, Gamma _and_ 88cET -- surely the four most well known
nonoctave tunings -- are all essentially equal divisions
of the fifth.

This is why I have been dividing other intervals up like
septimal intervals. I started with 88cET, moved through
the Carlos tunings, toyed with various other scales hoping
to get a good fifth and then finally realized that I could
abandon both the fifth and the octave and still write
music that expressed whatever it was I was trying to get
at. That's when I moved into the equal divisions of
septimal and higher order intervals for the last three or
four years, as I gave some examples of a couple months
back. That's where it's really _happening_ for me big time
right now in nonoctave equal temperaments.

I amused myself earlier this year by making a catalog of
this and I have uploaded it here:

/tuning/files/Jeff/FIFTHS.HTM

Now, please don't be upset that I named all the unnamed
ones after myself. I do expect that someone will find out
who really first discovered all these, hopfully with dates
of discovery. Anyone who knows, email me and I will update
it.

This is a real good 'reference sheet' to get started in
this area.

I have often thought of making it the basis of an article
for xenharmonikon and actually do encyclopedia entries for
each of these and so forth but you know I am probably not
going to get around to that anytime soon so hey I'll just
put in out here and hope it is of use to some people.

I don't think I will be stating anything unobvious in
saying that an album with one track from each of these
tunings makes a lot of sense. I plan to do it, but I have
about three other concept albums I want to do queued up
ahead of this project.

All the best,

- Jeff

> No one has yet brought up, to my awareness, that Alpha,
> Beta, Gamma _and_ 88cET -- surely the four most well known
> nonoctave tunings -- are all essentially equal divisions
> of the fifth.

Not to nitpick, and with thanks for the vote of confidence about 88CET, but I suspect that
Bohlen/Pierce tuning is probably better known among nonoctave tunings than 88CET. That one
doesn't have a clear representation of a fifth, although it does have an exact twelfth.

>
>
> This is why I have been dividing other intervals up like
> septimal intervals. I started with 88cET

Speaking of which, I often think of it as more basically 11 steps per 7:4 than 8 per 3:2, but
either is a good characterization, I'd say.

--- In tuning@y..., Gary Morrison <MR88CET@T...> wrote:

> >
> > This is why I have been dividing other intervals up like
> > septimal intervals. I started with 88cET
>
> Speaking of which, I often think of it as more basically 11 steps
per 7:4 than 8 per 3:2, but
> either is a good characterization, I'd say.

The Miracle scales, in their simplest non-octave form, can be thought
of as 2 steps per ~8:7, or 3 steps per ~11:9. A non-octave form of
Blackjack would superimpose this scale with its transposition at a
~5:4.