Yahoo Tuning Groups Ultimate Backup tuning 12EDO Inversions and "Common Practice" Tonality:

I just wanted to point out that one of the many
ingredients that contributes to the continuation
of the theoretically unsupportable bases of
12EDO common practice tonality is the conventional
definition of a 12EDO inversion, which is NOT the
only possible inversion in 12EDO.

There are at least 3 different kinds that I
can think of:

1. the conventional mono-pitch-class-centric
(axis of symmetry) inversion, which starts the
inverted melody on the same pitch class as
the prime form of the melody, and inverts the
direction of all of the melody's intervals.

2. a scalar or pitch-class-set span inversion, which
reduces the melody to a scale/set, and "flips"
the entire scale/set upside down, the axis of
symmetry being the center of the pitch/registral
span of the scale/set.

3. what I call a melodic pitch/registral span
inversion, which "flips" the entire melody
within its pitch/registral space. This one is the
most wholistic/organic/concrete of the three
to me.

Bill Flavell

🔗yahya_melb <yahya@melbpc.org.au>

Hi Bill,

--- In tuning@yahoogroups.com, Bill Flavell wrote:
>
> I just wanted to point out that one of the many
> ingredients that contributes to the continuation
> of the theoretically unsupportable bases of
> 12EDO common practice tonality is the conventional
> definition of a 12EDO inversion, which is NOT the
> only possible inversion in 12EDO.
>
> There are at least 3 different kinds that I
> can think of:
>
> 1. the conventional mono-pitch-class-centric
> (axis of symmetry) inversion, which starts the
> inverted melody on the same pitch class as
> the prime form of the melody, and inverts the
> direction of all of the melody's intervals.

"Inversion in the unison".

> 2. a scalar or pitch-class-set span inversion, which
> reduces the melody to a scale/set, and "flips"
> the entire scale/set upside down, the axis of
> symmetry being the center of the pitch/registral
> span of the scale/set.

"Inversion in the octave".

> 3. what I call a melodic pitch/registral span
> inversion, which "flips" the entire melody
> within its pitch/registral space. This one is the
> most wholistic/organic/concrete of the three
> to me.

"Inversion in the range interval".

There's quite a bit more to inversion thatn just these
three flavours!

I've added possible short descriptions for each of the
above; each description presupposes that the classical
notion of "inversion in an interval" is clear. For
instance, a composer writing in fugal style may invert
a melody in the fifth or fourth, as well as in the octave.
Inversion in other intervals: third or sixth, second or
seventh, is less common; inversion in non-scale intervals
is rarely encountered (but fun to do).

It's worth noting what should be obvious, but seems not
to be: that all inversions in a scale degree depend on
the scale used. For example, the diatonic heptatonic
major scale inversion of a major third in the octave is
a major sixth (in C major, E->A); the chromatic scale
inversion of a major third in the octave is a minor sixth
(in C chromatic, E->Ab); and the pentatonic major scale
inversion of a major third in the octave is a perfect
fifth (in C pentatonic major, E->G).

I find all these inversions equally concrete, but which
is more organic depends on your notion of the whole being
expressed in the piece. For instance, if the range of a
melody is a ninth, inverting it in the ninth would quite
probably lead to a derived melody with a different modal
feel than the original; whereas inverting it in the fifth
may lead to something more in keeping with the first.

For example, in the pentatonic CDEGA:

(1) Original: CEGA|c:d:|cAGE|G:D:|C:::

(2) Invert at 5th: gdcA|G:E:|GAcd|c:e:|g:::

(3) Invert at 9th: dAGE|D:C:|DEGA|G:c:|d:::

Actually, in all near-just pentatonics, most available
intervals now seem consonant enough to us so that we could
set each of these inversions to the same accompaniment
without much damage, eg:

(Tr.) C C E E |A A G EG|E G E C |E E G AG|E : : :
(Bs.) C : : : |E : : : |A : : : |G : : : |C : : :

However, no accompaniment I can come up with that
harmonises well with the first and second also
harmonises as well with the third.

Finally, although I personally find this topic fascinating,
as all aspects of composition, I think it has next to nothing
to do with Tuning, which is what this group is for. So, if
you'd like to continue this conversation, let's please take
it offline.

Regards,
Yahya

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

Why "inversion in..." and not "inversion at..."?

----- Original Message -----
From: "yahya_melb" <yahya@melbpc.org.au>
To: <tuning@yahoogroups.com>
Sent: 27 Kasï¿½m 2006 Pazartesi 15:08
Subject: [tuning] Re: 12EDO Inversions and "Common Practice" Tonality:

> Hi Bill,
>
> --- In tuning@yahoogroups.com, Bill Flavell wrote:
> >
> > I just wanted to point out that one of the many
> > ingredients that contributes to the continuation
> > of the theoretically unsupportable bases of
> > 12EDO common practice tonality is the conventional
> > definition of a 12EDO inversion, which is NOT the
> > only possible inversion in 12EDO.
> >
> > There are at least 3 different kinds that I
> > can think of:
> >
> > 1. the conventional mono-pitch-class-centric
> > (axis of symmetry) inversion, which starts the
> > inverted melody on the same pitch class as
> > the prime form of the melody, and inverts the
> > direction of all of the melody's intervals.
>
> "Inversion in the unison".
>
>
> > 2. a scalar or pitch-class-set span inversion, which
> > reduces the melody to a scale/set, and "flips"
> > the entire scale/set upside down, the axis of
> > symmetry being the center of the pitch/registral
> > span of the scale/set.
>
> "Inversion in the octave".
>
>
> > 3. what I call a melodic pitch/registral span
> > inversion, which "flips" the entire melody
> > within its pitch/registral space. This one is the
> > most wholistic/organic/concrete of the three
> > to me.
>
> "Inversion in the range interval".
>
> There's quite a bit more to inversion thatn just these
> three flavours!
>
> I've added possible short descriptions for each of the
> above; each description presupposes that the classical
> notion of "inversion in an interval" is clear. For
> instance, a composer writing in fugal style may invert
> a melody in the fifth or fourth, as well as in the octave.
> Inversion in other intervals: third or sixth, second or
> seventh, is less common; inversion in non-scale intervals
> is rarely encountered (but fun to do).
>
> It's worth noting what should be obvious, but seems not
> to be: that all inversions in a scale degree depend on
> the scale used. For example, the diatonic heptatonic
> major scale inversion of a major third in the octave is
> a major sixth (in C major, E->A); the chromatic scale
> inversion of a major third in the octave is a minor sixth
> (in C chromatic, E->Ab); and the pentatonic major scale
> inversion of a major third in the octave is a perfect
> fifth (in C pentatonic major, E->G).
>
> I find all these inversions equally concrete, but which
> is more organic depends on your notion of the whole being
> expressed in the piece. For instance, if the range of a
> melody is a ninth, inverting it in the ninth would quite
> probably lead to a derived melody with a different modal
> feel than the original; whereas inverting it in the fifth
> may lead to something more in keeping with the first.
>
> For example, in the pentatonic CDEGA:
>
> (1) Original: CEGA|c:d:|cAGE|G:D:|C:::
>
> (2) Invert at 5th: gdcA|G:E:|GAcd|c:e:|g:::
>
> (3) Invert at 9th: dAGE|D:C:|DEGA|G:c:|d:::
>
> Actually, in all near-just pentatonics, most available
> intervals now seem consonant enough to us so that we could
> set each of these inversions to the same accompaniment
> without much damage, eg:
>
> (Tr.) C C E E |A A G EG|E G E C |E E G AG|E : : :
> (Bs.) C : : : |E : : : |A : : : |G : : : |C : : :
>
> However, no accompaniment I can come up with that
> harmonises well with the first and second also
> harmonises as well with the third.
>
> Finally, although I personally find this topic fascinating,
> as all aspects of composition, I think it has next to nothing
> to do with Tuning, which is what this group is for. So, if
> you'd like to continue this conversation, let's please take
> it offline.
>
> Regards,
> Yahya
>

🔗Bill Flavell <musictheorybill@gmail.com>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

🔗Bill Flavell <musictheorybill@gmail.com>

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, "Bill Flavell" <musictheorybill@...> wrote:
>
>
> I just wanted to point out that one of the many
> ingredients that contributes to the continuation
> of the theoretically unsupportable bases of
> 12EDO common practice tonality is the conventional
> definition of a 12EDO inversion, which is NOT the
> only possible inversion in 12EDO.
>
> There are at least 3 different kinds that I
> can think of:
>
> 1. the conventional mono-pitch-class-centric
> (axis of symmetry) inversion, which starts the
> inverted melody on the same pitch class as
> the prime form of the melody, and inverts the
> direction of all of the melody's intervals.
>
> 2. a scalar or pitch-class-set span inversion, which
> reduces the melody to a scale/set, and "flips"
> the entire scale/set upside down, the axis of
> symmetry being the center of the pitch/registral
> span of the scale/set.
>
> 3. what I call a melodic pitch/registral span
> inversion, which "flips" the entire melody
> within its pitch/registral space. This one is the
> most wholistic/organic/concrete of the three
> to me.
>
>
> Bill Flavell

You'll have to be more explicit. Why is this a point for or against,
or even relevant to, 12-EDO versus another tuning system?

In *any* equal temperament (n-EDO), if one combines any one of these
notions of inversion with the idea of unrestricted transposition, you
get back the full range of possibilities. Inversion in any interval of
the tuning is just inversion in the unison plus transposition.

Historically, the main point of 12-EDO was unrestricted transposition,
which makes any point about inversion secondary. I think what happens
when you transpose is a more fundamental property, because the
properties of melodies and chords under inversion are not usually
important in practical music-making.

What I mean by that is, an inversion of a melody or chord usually has
a musically different character from the original, so there is no
particular reason to demand a specific property of the tuning under
inversion. Whereas in many applications it is convenient to have a
tuning which retains (most of) its properties under transposition.

One exception is invertible counterpoint where one demands that the
inversion of a consonant interval be consonant... but even then it is
more complicated, because the 4th above the bass was reckoned as a
dissonance whereas the 5th above the bass is consonant.

The diatonic, ascending melodic minor, pentatonic and chromatic scales
do allow for exact melodic inversions; the harmonic minor doesn't;
neither does the harmonic series. But so what?

~~~T~~~

🔗Cameron Bobro <misterbobro@yahoo.com>

Tom, there's a whole school of thought that is very heavy
on "symmetry"- check out George Perle's writings on Scriabin and
Bartok for example. There are related writings by a well-known (in
Middle Europe at least) Hungarian composer/academic (not so well
known that I can remember his name, but I'll find it if anyone
insists), and of course the somewhat embarassing but unshakable
shade of Schillinger, demurely lurking like a bottle-lover.

In addition to your comments, there are other unsown seams, one of
which is the whole issue of perception. The ancient Greeks built
subtle curves into stone to create the illusion of straight lines in
architecture, for example.

Bill, have you read the classic "On Growth and Form" by D'Arcy
Thompson? I saw an annotated and updated edition the other day,
that's probably the one to get (footnotes with recent findings pro
and con the original).

-Cameron Bobro

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

> You'll have to be more explicit. Why is this a point for or
against,
> or even relevant to, 12-EDO versus another tuning system?
>
> In *any* equal temperament (n-EDO), if one combines any one of
these
> notions of inversion with the idea of unrestricted transposition,
you
> get back the full range of possibilities. Inversion in any
interval of
> the tuning is just inversion in the unison plus transposition.
>
> Historically, the main point of 12-EDO was unrestricted
transposition,
> which makes any point about inversion secondary. I think what
happens
> when you transpose is a more fundamental property, because the
> properties of melodies and chords under inversion are not usually
> important in practical music-making.
>
> What I mean by that is, an inversion of a melody or chord usually
has
> a musically different character from the original, so there is no
> particular reason to demand a specific property of the tuning under
> inversion. Whereas in many applications it is convenient to have a
> tuning which retains (most of) its properties under transposition.
>
> One exception is invertible counterpoint where one demands that the
> inversion of a consonant interval be consonant... but even then it
is
> more complicated, because the 4th above the bass was reckoned as a
> dissonance whereas the 5th above the bass is consonant.
>
> The diatonic, ascending melodic minor, pentatonic and chromatic
scales
> do allow for exact melodic inversions; the harmonic minor doesn't;
> neither does the harmonic series. But so what?
>
> ~~~T~~~
>

🔗Bill Flavell <musictheorybill@gmail.com>

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, "Bill Flavell" <musictheorybill@...> wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
>
> > Historically, the main point of 12-EDO was unrestricted transposition,
> > which makes any point about inversion secondary. I think what happens
> > when you transpose is a more fundamental property, because the
> > properties of melodies and chords under inversion are not usually
> > important in practical music-making.
>
>
> I definitely don't agree with you there.
> Inversions sound much more different than
> the original melody than transpositions do.
>
>
> Bill Flavell

Why does this not agree with me ?

What I said, right after the point where you cut me off, was

> an inversion of a melody or chord usually has
> a musically different character from the original, so there is no
> particular reason to demand a specific property of the tuning under
> inversion. Whereas in many applications it is convenient to have a
> tuning which retains (most of) its properties under transposition.

Perhaps I shouldn't have used the word 'fundamental', but the meaning
was clear from the message as a whole.

Basically: if your tuning doesn't allow transposition, that usually
has more consequences for composing and performing, than the question
of whether it allows 'exact' inversion (i.e. preserving all intervals).

At least, that was the case historically. I believe the main selling
point of 12-EDO in the 18th century was just the transpositions. Only
serialism demands exact inversion, so far as I know.

Some unequal temperaments (Vallotti, Kirnberger if you ignore the
schisma) also allow exact inversion, but I don't think that has been
an important point for or against them.

~~~T~~~

🔗Cameron Bobro <misterbobro@yahoo.com>

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

> Basically: if your tuning doesn't allow transposition, that usually
> has more consequences for composing and performing, than the
>question
> of whether it allows 'exact' inversion (i.e. preserving all
>intervals).

I agree- literal inversion is a specific thing, as it often obscures
or destroys traditional tonality. I believe that tonality is more
persistent than is usually taught and so more difficult to
eliminate, but that doesn't matter in this discussion: as far
as "common practice", throw in some literal inversions and you're
heading to Shostakovich territory.

>
> At least, that was the case historically. I believe the main
>selling
> point of 12-EDO in the 18th century was just the transpositions.

I agree with this as well, and find the cliche that 12-EDO evolved
to enable modulations humorous, because it radically reduces
differences between keys. A modulation from C major to G major in 12-
EDO is simply a transposition, the only differences are found in
absolute pitches, tessitura, open/stopped strings, etc., ie.
basically timbral. And even these timbral differences are greatly
reduced by playing technique and instrument design in Western music.

>Only
> serialism demands exact inversion, so far as I know.

Well according to some, as I mentioned before, literal inversion and
symmetries in general are an integral part of some non-serial and
tonal musics. Certainly many "serial" techniques predate both
serialism and universal 12-EDO, coming from fugues and such, but I
suspect that analyzing Scriabin as a hall of mirrors is more of a
way to establish a particular compositional approach than a
true "key" to his own approach. Which is okay- George Pearle's music
is fine stuff.

-Cameron Bobro

🔗Bill Flavell <musictheorybill@gmail.com>

Thanks very much for the response, Tom! :)

I think this subject deserves a new thread, and I'll try and get
one posted, if the moderator doesn't veto it! :)

Bill Flavell

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> --- In tuning@yahoogroups.com, "Bill Flavell" <musictheorybill@>
wrote:
> >
> > --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
> >
> > > Historically, the main point of 12-EDO was unrestricted
transposition,
> > > which makes any point about inversion secondary. I think what
happens
> > > when you transpose is a more fundamental property, because the
> > > properties of melodies and chords under inversion are not
usually
> > > important in practical music-making.
> >
> >
> > I definitely don't agree with you there.
> > Inversions sound much more different than
> > the original melody than transpositions do.
> >
> >
> > Bill Flavell
>
>
> Why does this not agree with me ?
>
> What I said, right after the point where you cut me off, was
>
> > an inversion of a melody or chord usually has
> > a musically different character from the original, so there is no
> > particular reason to demand a specific property of the tuning
under
> > inversion. Whereas in many applications it is convenient to have a
> > tuning which retains (most of) its properties under transposition.
>
> Perhaps I shouldn't have used the word 'fundamental', but the
meaning
> was clear from the message as a whole.
>
> Basically: if your tuning doesn't allow transposition, that usually
> has more consequences for composing and performing, than the
question
> of whether it allows 'exact' inversion (i.e. preserving all
intervals).
>
> At least, that was the case historically. I believe the main selling
> point of 12-EDO in the 18th century was just the transpositions.
Only
> serialism demands exact inversion, so far as I know.
>
> Some unequal temperaments (Vallotti, Kirnberger if you ignore the
> schisma) also allow exact inversion, but I don't think that has been
> an important point for or against them.
>
> ~~~T~~~
>