Asymmetrical Tuning "Scale-Step" Inversions:

While only symmetrical and/or EDO tunings
are capable of doing a genuine inversion, you
could do a pseudo-inversion with asymmetrical
tunings by doing what I call a "scale step inversion".

As an example, take the 7-tone just intonation
tuning with the following pitch classes ascending:

1-2-3-4-5-6-7

The prime 7-note/tone melody will be:

4-3-5-6-2-7-1

The scale step inversion would be:

4-5-3-2-6-1-7

These could also be "rotated" 7 different
ways in order to mimic the operation of
transposition in 12EDO.

Bill.Flavell at GMail.com

--- In tuning@yahoogroups.com, "Bill Flavell" <musictheorybill@...>
wrote:
>
>
> While only symmetrical and/or EDO tunings
> are capable of doing a genuine inversion, you
> could do a pseudo-inversion with asymmetrical
> tunings by doing what I call a "scale step inversion".
>
> As an example, take the 7-tone just intonation
> tuning with the following pitch classes ascending:
>
> 1-2-3-4-5-6-7
>
>
> The prime 7-note/tone melody will be:
>
> 4-3-5-6-2-7-1
>
> The scale step inversion would be:
>
> 4-5-3-2-6-1-7

That's a "tonal inversion", which would alter notes from literal
intervallic movement if necessary, to conform to tonal or modal
considerations, if you are referring to actual scale steps.

A "literal inversion" would be based on intervallic movement, damn
the torpedoes.

The way I look at it, the "most genuine" inversions don't happen in
equal temperaments, but in rational tunings, where harmonic series,
sum and difference tones, and the whole ball of wax can be flipped
around an axis while maintaining literal harmonic relationships, as
in coincidences in the partials. The way I listen to it, I should
say- the audible balance of this kind of (rational) symmetry is like
the big stone head floating in the air in the movie "Zardoz",
something that doesn't happen in 12-EDO.

-Cameron Bobro

> While only symmetrical and/or EDO tunings
> are capable of doing a genuine inversion, you
> could do a pseudo-inversion with asymmetrical
> tunings by doing what I call a "scale step inversion".

In common-practice theroy, this is what is usually
meant by "inversion". Exact inversion is called
mirror inversion.

Some just and other scales allow mirror inversion
in a limited number of keys. ETs are the only scales
to allow it in all keys.

-Carl

Thanks for the response, Cameron, although I don't understand what
you're saying! :)

Bill Flavell

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Bill Flavell" <musictheorybill@>
> wrote:
> >
> >
> > While only symmetrical and/or EDO tunings
> > are capable of doing a genuine inversion, you
> > could do a pseudo-inversion with asymmetrical
> > tunings by doing what I call a "scale step inversion".
> >
> > As an example, take the 7-tone just intonation
> > tuning with the following pitch classes ascending:
> >
> > 1-2-3-4-5-6-7
> >
> >
> > The prime 7-note/tone melody will be:
> >
> > 4-3-5-6-2-7-1
> >
> > The scale step inversion would be:
> >
> > 4-5-3-2-6-1-7
>
> That's a "tonal inversion", which would alter notes from literal
> intervallic movement if necessary, to conform to tonal or modal
> considerations, if you are referring to actual scale steps.
>
> A "literal inversion" would be based on intervallic movement, damn
> the torpedoes.
>
> The way I look at it, the "most genuine" inversions don't happen in
> equal temperaments, but in rational tunings, where harmonic series,
> sum and difference tones, and the whole ball of wax can be flipped
> around an axis while maintaining literal harmonic relationships, as
> in coincidences in the partials. The way I listen to it, I should
> say- the audible balance of this kind of (rational) symmetry is
like
> the big stone head floating in the air in the movie "Zardoz",
> something that doesn't happen in 12-EDO.
>
> -Cameron Bobro
>

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> the "most genuine" inversions don't happen in
> equal temperaments, but in rational tunings, where harmonic series,
> sum and difference tones, and the whole ball of wax can be flipped
> around an axis while maintaining literal harmonic relationships, as
> in coincidences in the partials. The way I listen to it, I should
> say- the audible balance of this kind of (rational) symmetry is like
> the big stone head floating in the air in the movie "Zardoz",
> something that doesn't happen in 12-EDO.
>
> -Cameron Bobro
>

I'm kind of lost here, apart from agreeing that 12-EDO does not
involve big stone heads floating in the air. You mean transforming
'normal' harmony into music composed of sounds that have a subharmonic
series of partials, rather than a harmonic one?

That's certainly a novel idea... but how to realize it?

Wouldn't the low-frequency end get rather crowded??

~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

>
> I'm kind of lost here, apart from agreeing that 12-EDO does not
> involve big stone heads floating in the air. You mean transforming
> 'normal' harmony into music composed of sounds that have a
subharmonic
> series of partials, rather than a harmonic one?
>
> That's certainly a novel idea... but how to realize it?
>
> Wouldn't the low-frequency end get rather crowded??
>
> ~~~T~~~
>

Well, sum and difference tones can create psychoacoustic subharmonics,
and close voicing can crowd the percieved lower end, but I meant
something more simple.

Just look at some mirror movement with simple rational proportions,
for example two voices starting in unison: one goes up a perfect fifth,
one goes down a perfect fifth, continuing in exact contrary motion, say
tenor continues back down to 9/8 and bass up to 8/9, etc. A look
through the sum, difference, and what is the most often percieved
resultant AFAIK, 2F1-F2 (F1 being the higher frequency) tones, and the
harmonic series of both voices, will reveal strong and simple harmonic
relationships, which makes the whole thing "float".

-Cameron Bobro