"Bi-Directional" Tunings?

If a particular tuning is defined by a particular
set/group of intervals in a particular
sequence/order in musical space, then that
particular set/order cannot be completely
explored unless that set/ordering of intervals
can be sounded IN BOTH THEIR ASCENDING
AND DESCENDING FORMS.

Has anybody in the history of alternative tunings
made this proposition/assertion before?

In terms of a concrete example, I'm going to start
working on a 12-tone bi-directional just tuning that
could serve as a possible "crossover" tuning that
could be mapped onto conventional 12EDO
synthesizer keybords.

I'll report my results later.

Bill Flavell

On 3/4/06, Bill Flavell <bill_flavell@email.com> wrote:
>
> If a particular tuning is defined by a particular
> set/group of intervals in a particular
> sequence/order in musical space, then that
> particular set/order cannot be completely
> explored unless that set/ordering of intervals
> can be sounded IN BOTH THEIR ASCENDING
> AND DESCENDING FORMS.
[...]

I don't understand. What's an example of a set/ordering of intervals
that cannot be sounded in both ascending and descending forms? That
seems like a logical impossibility. You can play any scale up or down.

Keenan

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> I don't understand. What's an example of a set/ordering of intervals
> that cannot be sounded in both ascending and descending forms?

The old 9/8, 10/9, 16/15 7-tone just intonation scale would be an
example. It's asymmetrical, which by definition means that there is a
different ordering of intervals when played ascending as opposed to
descending.

The uni-directional bias is very deeply imbedded in western music
theory and uni-directionality is really more of a
literary/architectural "aesthetic than a musical one.

> That seems like a logical impossibility. You can play any scale up or
>down.

Sure, Keenan, but the intervals aren't in the same order.

I would NOT want to design a tuning that didn't allow for the
inversions of any melody or chord to be played in comparison to
the "prime" forms. That would reduce the amount of musical material in
half!

Bill Flavell

--- In tuning@yahoogroups.com, "Bill Flavell" <bill_flavell@...> wrote:

> The old 9/8, 10/9, 16/15 7-tone just intonation scale would be an
> example. It's asymmetrical, which by definition means that there is a
> different ordering of intervals when played ascending as opposed to
> descending.

The obvious way of producing a scale inversely symmetrical around a
given note is to take the union of the scale with its inverse with
respect to that note. If we do this with the scale Scala calls
"zarlino.scl" and which you refer to above around the 1/1, we get an
11-note inversely symmetrical scale which I give below. It's not too
terribly irregular, has five major and five minor triads, and one each
of each kind of tetrad if we allow marvel tempering.

As always, Scala takes note of the fact that the resulting scale is
inversely symmetrical, and specifies the nature of the symmetry.
Another way of getting inversely symmetrical scales is via diamonds,
and Scala also notes this scale contains the 1-3-5-15 diamond, a
9-note inversely symmetrical scale.

! zorro.scl
zarlino union inverted zarlino
11
!
16/15
9/8
6/5
5/4
4/3
3/2
8/5
5/3
16/9
15/8
2

--- In tuning@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:

Thank you very much for that info, Mr. Smith, although I don't
understand most of it! :)

I don't have access to Scala or anything else right now, so I'm just
plowing away with my own format for the time being. Hopefully the
general principles of construction are valuable enough that they
overshadow my notational shortcomings! :)

Bill Flavell

>
> The obvious way of producing a scale inversely symmetrical around a
> given note is to take the union of the scale with its inverse with
> respect to that note. If we do this with the scale Scala calls
> "zarlino.scl" and which you refer to above around the 1/1, we get an
> 11-note inversely symmetrical scale which I give below. It's not too
> terribly irregular, has five major and five minor triads, and one each
> of each kind of tetrad if we allow marvel tempering.
>
> As always, Scala takes note of the fact that the resulting scale is
> inversely symmetrical, and specifies the nature of the symmetry.
> Another way of getting inversely symmetrical scales is via diamonds,
> and Scala also notes this scale contains the 1-3-5-15 diamond, a
> 9-note inversely symmetrical scale.
>
> ! zorro.scl
> zarlino union inverted zarlino
> 11
> !
> 16/15
> 9/8
> 6/5
> 5/4
> 4/3
> 3/2
> 8/5
> 5/3
> 16/9
> 15/8
> 2
>