Octave Invariance by Nurture?

At the site: http://www.philomel.com/description.html#tritone

One finds the following pair of WAV files:

WAV FILE (1) Described as "a well-kown tune", and, "the tones are distributed haphazardly among three different octaves".

No biggie, I said, "octave invariance" (either "inborn" or "nurtured") will allow me to figure it out!... :) Try it yourself by clicking below:

http://www.philomel.com/demos/wav/mys_scr.wav

Then (thoroughly) *stumped*, I finally listened to the "non-octave-scrambled" tune (a bit wierd sounding itself due to synth source).

http://www.philomel.com/demos/wav/mys_un.wav

Hoewever, once I "recalled" the melody (from "musical memory"), the *first* first WAV files' melody became (and remains, whether I like it or not) clearly recognizable.

By nature or nurture, my *own* (cerebral?) "octave-invariant algorithm" [without the presence of other simultaneously sounded tone(s) from which to gain a sense of reference to other relative pitch (or pitches)], fails me completely in the above case of (octave-scrambled) serial melody...

*until* the "familiar melody" was (re-, in this case) *learned* in the context of octave stretching/shrinking. Now, once "learned", that recognition (the "target metaphor" indentified) will not go away!

The "contextual framework" (of the melodic "identity") formed, it will not recede (by will or otherwise) in my "aural mind".

What does the fact that a (formerly) recognizable melody in pitch can exist in numerous other (octave stretched/shrunk) arrangements of the various notes as they are played in time and never be recognized say about the *relevance* of "octave-invariance* to serial melody???

Curiously, J Gill

--- In tuning@y..., J Gill <JGill99@i...> wrote:

> What does the fact that a (formerly) recognizable melody in pitch
can exist
> in numerous other (octave stretched/shrunk) arrangements of the
various
> notes as they are played in time and never be recognized say about
the
> *relevance* of "octave-invariance* to serial melody???

It's clearly limited, a fact that Schoenberg seemed particularly
interested in. Some of the most salient features of melody are _small
steps_ and _direction of movement_, and these features are totally
perverted by the version given here. However, if you transpose the
_whole_ melody by an octave, won't it appear more like the original
that if transposed by any other interval (save a tiny microtonal one)?

--- In tuning@y..., J Gill <JGill99@i...> wrote:

/tuning/topicId_31804.html#31804

>
> The "contextual framework" (of the melodic "identity") formed, it
will not
> recede (by will or otherwise) in my "aural mind".
>
> What does the fact that a (formerly) recognizable melody in pitch
can exist
> in numerous other (octave stretched/shrunk) arrangements of the
various
> notes as they are played in time and never be recognized say about
the
> *relevance* of "octave-invariance* to serial melody???
>
>
> Curiously, J Gill

This is an interesting discussion, J. Gill, because it happens to
also be the centerpiece of the chapter on Schoenberg by the theorist
Hugo Leichtentritt in has fine work, _Musical Form._

Essentially, Leichtentritt shows that Schoenberg's music really
shouldn't be that difficult to understand if the octave displacements
are taken away.

The melodies, without these displacement, are rather commonplace...
I'm not saying not of quality, but much more "expected" than the
final version with the octave displacements.

So, in essence, Schoenberg uses octave displacements as a kind of
*elaboration* technique, as an integral part of his melodic process.
Essentially, the melody becomes something *different* with the
displacements, so it's not surprising that it wouldn't be readily
identifiable.

Paul Erlich, however, makes a very good point in his response to your
post, which I already read ahead. If you were to transpose the
*entire* melody, with the octave additions and all, that would be a
different and more significant instance of *octave invariance* than
the use of the octaves within a melody as a kind of "variation"
process.

Thanks for bringing this up and I just wanted to point out that it is
a topic that has been previously discussed regarding Schoenberg's
music.

Maybe Monz, who is a Schoenberg "expert" will have more to say on
this subject...

best,

Joe Pehrson

Well, I've been hanging out on this one waiting to see who else would
say it first. Paul Erlich has come the closest. Melodic perception
has everything to do with DIRECTION OF MELODIC MOVEMENT!!! Displacing
some notes and not others to create a vastly different melodic
terrain
spite of the notes being otherwise identical with the original melody
is therefore NOT A VALID TEST of the principle of octave invariance.

Who would say that a major third sounds the same even harmonically as
it's first cousin and inversion, the minor sixth?!? This and other
intervals and their inversions are even less alike melodically than
harmonically. You take the whole thing and transpose ALL of it by an
octave or two or you've changed the whole game completely in terms of
melodic integrity.

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., J Gill <JGill99@i...> wrote:
>
> /tuning/topicId_31804.html#31804
>
> >
> > The "contextual framework" (of the melodic "identity") formed, it
> will not
> > recede (by will or otherwise) in my "aural mind".
> >
> > What does the fact that a (formerly) recognizable melody in pitch
> can exist
> > in numerous other (octave stretched/shrunk) arrangements of the
> various
> > notes as they are played in time and never be recognized say
about
> the
> > *relevance* of "octave-invariance* to serial melody???
> >
> >
> > Curiously, J Gill
>
>
> This is an interesting discussion, J. Gill, because it happens to
> also be the centerpiece of the chapter on Schoenberg by the
theorist
> Hugo Leichtentritt in has fine work, _Musical Form._
>
> Essentially, Leichtentritt shows that Schoenberg's music really
> shouldn't be that difficult to understand if the octave
displacements
> are taken away.
>
> The melodies, without these displacement, are rather commonplace...
> I'm not saying not of quality, but much more "expected" than the
> final version with the octave displacements.
>
> So, in essence, Schoenberg uses octave displacements as a kind of
> *elaboration* technique, as an integral part of his melodic
process.
> Essentially, the melody becomes something *different* with the
> displacements, so it's not surprising that it wouldn't be readily
> identifiable.
>
> Paul Erlich, however, makes a very good point in his response to
your
> post, which I already read ahead. If you were to transpose the
> *entire* melody, with the octave additions and all, that would be a
> different and more significant instance of *octave invariance* than
> the use of the octaves within a melody as a kind of "variation"
> process.
>
> Thanks for bringing this up and I just wanted to point out that it
is
> a topic that has been previously discussed regarding Schoenberg's
> music.
>
> Maybe Monz, who is a Schoenberg "expert" will have more to say on
> this subject...
>
> best,
>
> Joe Pehrson

Joe,

It's a pleasure to hear from you, my friend!

In relation to the comment you mentioned:

> If you were to transpose the
> *entire* melody, with the octave additions and all, that would be a
> different and more significant instance of *octave invariance* than
> the use of the octaves within a melody as a kind of "variation"
> process.

It strikes me that such is a statement to the effect that:
(if one *already* has familiarity with a given melody)
*then* an "octave invariance" applies in perception...

It seems (to me) that to such a concept of "variation"
only has meaning relative to what is already *expected*
(as in the case of the recognition, *after* hearing the
non-transposed melody, of that melodies "prescence"
within the transposed melody).

After all, there exists no "sense" of what would constitute
a "variation" until we aquire a "sense" of normalcy from
the process of our minds "learning" a melody.

There is no specific "rule" that a given melody restrict
itself to sounding notes in "this or that" octave...

While we might well be able to "pick out" notes which
are individually sounded (in a serial manner) which
have an octave relation to the other notes so sounded,
it appears that perhaps such learned abilities fall
short (at least in my case) where it comes to the
ability to (in a like manner)"juggle" numerous scale
degrees (and their octaves/sub-octaves) in a melody.

Therefore, it seems (to me) that - if such a concept of
"invariance" only applies *after* the specific melody
has been "nurtured", then such "invariance" does not
exist by "nature". I would think that to simply hold
that "if we already know what we are looking for, then
we are able to recognize it" only makes the case for
the proposition that an (unlearned) "octave invariance"
(existing "by nature") perhaps exists not...

Best Regards, J Gill :)

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., J Gill <JGill99@i...> wrote:
>
> /tuning/topicId_31804.html#31804
>
> >
> > The "contextual framework" (of the melodic "identity") formed, it
> will not
> > recede (by will or otherwise) in my "aural mind".
> >
> > What does the fact that a (formerly) recognizable melody in pitch
> can exist
> > in numerous other (octave stretched/shrunk) arrangements of the
> various
> > notes as they are played in time and never be recognized say about
> the
> > *relevance* of "octave-invariance* to serial melody???
> >
> >
> > Curiously, J Gill
>
>
> This is an interesting discussion, J. Gill, because it happens to
> also be the centerpiece of the chapter on Schoenberg by the theorist
> Hugo Leichtentritt in has fine work, _Musical Form._
>
> Essentially, Leichtentritt shows that Schoenberg's music really
> shouldn't be that difficult to understand if the octave displacements
> are taken away.
>
> The melodies, without these displacement, are rather commonplace...
> I'm not saying not of quality, but much more "expected" than the
> final version with the octave displacements.
>
> So, in essence, Schoenberg uses octave displacements as a kind of
> *elaboration* technique, as an integral part of his melodic process.
> Essentially, the melody becomes something *different* with the
> displacements, so it's not surprising that it wouldn't be readily
> identifiable.
>
> Paul Erlich, however, makes a very good point in his response to your
> post, which I already read ahead. If you were to transpose the
> *entire* melody, with the octave additions and all, that would be a
> different and more significant instance of *octave invariance* than
> the use of the octaves within a melody as a kind of "variation"
> process.
>
> Thanks for bringing this up and I just wanted to point out that it is
> a topic that has been previously discussed regarding Schoenberg's
> music.
>
> Maybe Monz, who is a Schoenberg "expert" will have more to say on
> this subject...
>
> best,
>
> Joe Pehrson

--- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> Well, I've been hanging out on this one waiting to see who else >would
> say it first. Paul Erlich has come the closest. Melodic perception
> has everything to do with DIRECTION OF MELODIC MOVEMENT!!!

J Gill: This is very interesting, Bob.
Could you explain this "directional effect"?

> Displacing
> some notes and not others to create a vastly different melodic
> terrain
> spite of the notes being otherwise identical with the original >melody
> is therefore NOT A VALID TEST of the principle of octave invariance.

J Gill: On the basis of a "directional effect"
existing (only)?

> Who would say that a major third sounds the same even harmonically >as
> it's first cousin and inversion, the minor sixth?!? This and other
> intervals and their inversions are even less alike melodically

J Gill: Now, this *does* "resonate" with me.
With or without any kind of "directional effect"
at play, and despite what some may say (about
an interval and its algebraic inversion being
"equivalent" because they represent the same
"interval"), *my ear* does not hear a 2/3 pitch
played *below* a 1/1 pitch as "One and Five"
relationship (*unless* the "implied root", or
"virtual fundamental" has flipped downward to
the 2/3 pitch (from the 1/1 pitch). As they say,
"you can't hear it both ways...". :)

> than
> harmonically.

J Gill: Once again, you "resonate". Would anyone claim
that a chord made up of the pitches 1/7, 1/6, 1/5, 1/4
has harmonic characteristics remotely resembling a chord
made up of the pitches 4/1, 5/1, 6/1, 7/1 ???

> You take the whole thing and transpose ALL of it by an
> octave or two or you've changed the whole game completely in terms >of
> melodic integrity.

J Gill: But if you were to transpose *all* of the notes
of a given scale or chord upwards/downwards "by an octave
or two", and you have "changed the whole game completely
in terms of melodic integrity" in doing so - is that not
an argument that "octave equivalence" is non-existent,
period (in any of the possible circumstances heretofore
discussed)???

Regards, J Gill

Well, first of all, by inversion I meant the classical concept of
interval inversion and not a fifth below as opposed to a fifth
above. Those are both fifths. I'm referring to the inversion of the
fifth, which is the fourth, and which has a different melodic span in
terms of pitch difference.

This is much more relevant to the issue of octave displacement of
selected notes in a melody. In that case, both the DIRECTION of
motion is reversed, and the SPAN (PITCH DISTANCE) of the interval is
changed. The further removed from a tritone the interval to be
inverted is, the greater the change in pitch distance.

I'm simply saying that Frere Jacque on C, for example, based on a
motive with a pitch sequence of CDEC in the same octave, is a
sequence of two upward stepwise motions and then a drop to the
orginal pitch a third below. That is how we hear melody: by melodic
contour!

If you transpose the initial C up an octave, you get an initial
DOWNWARD motion of a seventh before proceeding to the E which, if we
transpose it down an octave, entails another DOWNWARD motion of a
seventh before returning to the untransposed final C a minor sixth
ABOVE it. The melodic contour has been RADICALLY ALTERED by the
octave transposition of two notes. This is why such a melody sounds
so entirely different. We've radically changed the melodic CONTOUR in
terms of both directin of melodic movement and the DISTANCE of that
motion.

On the "expectation" issue, if you take an unfamiliar melody and
repeat it in its entirety with a whole octave displacement for every
note without advising anyone that this is what you're going to do,
anyone with the slightest quantum of musical perception will identify
it as the same melody even if they don't know what an octave is.

They may do the same thing for the same melody tranposed a fifth up
or down, or any other interval, but it won't sound as if it were in
the same key. I don't understand how anyone can seriously argue
against the concept of octave equivalence. I've never seen any
evidence of its being a culturally conditioned phenomenon. It seems
to be quite universal that men and women sing the same melody an
octave apart and perceive each other as singing the same melody.

--- In tuning@y..., "unidala" <JGill99@i...> wrote:
> --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> > Well, I've been hanging out on this one waiting to see who else
>would
> > say it first. Paul Erlich has come the closest. Melodic
perception
> > has everything to do with DIRECTION OF MELODIC MOVEMENT!!!
>
> J Gill: This is very interesting, Bob.
> Could you explain this "directional effect"?
>
> > Displacing
> > some notes and not others to create a vastly different melodic
> > terrain
> > spite of the notes being otherwise identical with the original
>melody
> > is therefore NOT A VALID TEST of the principle of octave
invariance.
>
> J Gill: On the basis of a "directional effect"
> existing (only)?
>
> > Who would say that a major third sounds the same even
harmonically >as
> > it's first cousin and inversion, the minor sixth?!? This and
other
> > intervals and their inversions are even less alike melodically
>
> J Gill: Now, this *does* "resonate" with me.
> With or without any kind of "directional effect"
> at play, and despite what some may say (about
> an interval and its algebraic inversion being
> "equivalent" because they represent the same
> "interval"), *my ear* does not hear a 2/3 pitch
> played *below* a 1/1 pitch as "One and Five"
> relationship (*unless* the "implied root", or
> "virtual fundamental" has flipped downward to
> the 2/3 pitch (from the 1/1 pitch). As they say,
> "you can't hear it both ways...". :)
>
>
> > than
> > harmonically.
>
> J Gill: Once again, you "resonate". Would anyone claim
> that a chord made up of the pitches 1/7, 1/6, 1/5, 1/4
> has harmonic characteristics remotely resembling a chord
> made up of the pitches 4/1, 5/1, 6/1, 7/1 ???
>
>
> > You take the whole thing and transpose ALL of it by an
> > octave or two or you've changed the whole game completely in
terms >of
> > melodic integrity.
>
>
> J Gill: But if you were to transpose *all* of the notes
> of a given scale or chord upwards/downwards "by an octave
> or two", and you have "changed the whole game completely
> in terms of melodic integrity" in doing so - is that not
> an argument that "octave equivalence" is non-existent,
> period (in any of the possible circumstances heretofore
> discussed)???
>
>
> Regards, J Gill

--- In tuning@y..., "unidala" <JGill99@i...> wrote:

> While we might well be able to "pick out" notes which
> are individually sounded (in a serial manner) which
> have an octave relation to the other notes so sounded,
> it appears that perhaps such learned abilities fall
> short (at least in my case) where it comes to the
> ability to (in a like manner)"juggle" numerous scale
> degrees (and their octaves/sub-octaves) in a melody.
>
> Therefore, it seems (to me) that - if such a concept of
> "invariance" only applies *after* the specific melody
> has been "nurtured", then such "invariance" does not
> exist by "nature". I would think that to simply hold
> that "if we already know what we are looking for, then
> we are able to recognize it" only makes the case for
> the proposition that an (unlearned) "octave invariance"
> (existing "by nature") perhaps exists not...

No, I think it just means that octave similarity is a limited
phenomenon, that can only stand up to so much 'juggling' . . . but
there seems to be no question in any musician's mind, from
practically whatever culture, that a "C" is a "c" is a "c'" -- though
of course you'd replace "C" with some other symbol in most cases.

--- In tuning@y..., "unidala" <JGill99@i...> wrote:

> J Gill: Now, this *does* "resonate" with me.
> With or without any kind of "directional effect"
> at play, and despite what some may say (about
> an interval and its algebraic inversion being
> "equivalent" because they represent the same
> "interval"), *my ear* does not hear a 2/3 pitch
> played *below* a 1/1 pitch as "One and Five"
> relationship (*unless* the "implied root", or
> "virtual fundamental" has flipped downward to
> the 2/3 pitch (from the 1/1 pitch). As they say,
> "you can't hear it both ways...". :)

I think you're misunderstanding how octave equivalence is supposed to
work. According to octave-equivalence, you would hear the 2/3 pitch
as equivalent to the 4/3 pitch, thus "One and Four", I believe, is
the relationship you'd hear as you'd describe it. The implications of
this, to the question of intervals, is that the interval of the fifth
(downward, if you wish) and the interval of the fourth (upward, if
you wish) can both describe the relationship between the _same
functional_ pair of pitches, and therefore when considering
pitches "functionally" and abstracted away from "octave specificity",
an important classification of intervals would place the fifth and
the fourth in the _same_ class. Let me know if this needs
clarification.

--- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:

> I don't understand how anyone can seriously argue
> against the concept of octave equivalence. I've never seen any
> evidence of its being a culturally conditioned phenomenon. It seems
> to be quite universal that men and women sing the same melody an
> octave apart and perceive each other as singing the same melody.

That's my view too, but I'm all in favor of composers trying to
compose based on 3/1-equivalence, or whatever, and seeing if they can
evoke a similar experience. Some claim that they have. Well, live and
let live.

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> > J Gill: Now, this *does* "resonate" with me.
> > With or without any kind of "directional effect"
> > at play, and despite what some may say (about
> > an interval and its algebraic inversion being
> > "equivalent" because they represent the same
> > "interval"), *my ear* does not hear a 2/3 pitch
> > played *below* a 1/1 pitch as "One and Five"
> > relationship (*unless* the "implied root", or
> > "virtual fundamental" has flipped downward to
> > the 2/3 pitch (from the 1/1 pitch). As they say,
> > "you can't hear it both ways...". :)
>
>PE: I think you're misunderstanding how octave equivalence is >supposed to
> work. According to octave-equivalence, you would hear the 2/3 pitch
> as equivalent to the 4/3 pitch, thus "One and Four", I believe, is
> the relationship you'd hear as you'd describe it.

JG: Understood, and agreed (where the "implied root" remains the reference pitch), and the ear is not persuaded that the 2/3 pitch is not the (new) "implied root" [since it is now the lower frequency tone, and (sounded simultaneously in isolation) with the 1/1 pitch, does not, then, assume the status of the "implied fundamental"]?

I have been using the phrase "octave equivalence" in reference to the phenomenon of the "perceived similarity" of an *individual tone* being raised/lowered in pitch by powers of 2 to the shifted tone.

> PE:The implications of
> this, to the question of intervals, is that the interval of the >fifth
> (downward, if you wish) and the interval of the fourth (upward, if
> you wish) can both describe the relationship between the _same
> functional_ pair of pitches,

JG: In situations where the pitches separated by that "interval" are related in a manner in which any "implied fundamental" resulting from the (serial or simultaneous) sounding of the two individual pitches is free to be interpreted as *either* the *lowest* pitch tone, or the *highest tone*. How does that comport with this "implied fundamental" business typically (at least in such a simple case) being identified as relating to the *lower* frequency tone, only?

> and therefore when considering
> pitches "functionally" and abstracted away from "octave >specificity",

JG: Wouldn't you also have to say, "abstracted away from the concept of the 'implied fundamental' in the case of an interval between two pitches under consideration?

Does that mean "implied fundamental" can *only* be applied to (serially or simultaneously sounded) combinations of *greater than* two tones?

> an important classification of intervals would place the fifth and
> the fourth in the _same_ class. Let me know if this needs
> clarification.

JG: Yes. See questions above.
If possible, could you also relate the meaning as you understand it of the phrase "interval of equivalence", to this situation, or is the phrase "interval of equivalence" *only* applicable where a group of "intervals" exists (as "step-sizes" between various pitches) within a given scale?

Have been searched through the ATL archives, and am still hazy (though ... gradually ... converging) as to the utilization of the phrases in this "tuning vernacular".

While you are at it (though this may seem like the Nth time some moron hasn't "gotten it"), could you give a stab at (briefly, and in your own language) the phrases:

"Moment of Symmetry" (MOS)

"Constant Product Set" (CPS)

"sub-tends" (everyone seems to assume that the reader understands the intended meaning of this; I'm still scratching my head about it).

I've been through the archives a bunch of times, checked Monz's definitions, read a bunch of people's posts, etc. Still a bit hazy...

Curiously, J Gill

--- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:

/tuning/topicId_31804.html#31971

> Well, I've been hanging out on this one waiting to see who else
would
> say it first. Paul Erlich has come the closest. Melodic perception
> has everything to do with DIRECTION OF MELODIC MOVEMENT!!!
Displacing
> some notes and not others to create a vastly different melodic
> terrain
> spite of the notes being otherwise identical with the original
melody
> is therefore NOT A VALID TEST of the principle of octave invariance.
>
> Who would say that a major third sounds the same even harmonically
as
> it's first cousin and inversion, the minor sixth?!? This and other
> intervals and their inversions are even less alike melodically than
> harmonically. You take the whole thing and transpose ALL of it by
an
> octave or two or you've changed the whole game completely in terms
of
> melodic integrity.
>
>

Hello Bob...

Thanks for your contribution to this interesting discussion. I guess
from an *artistic* point of view there really isn't something
explicitly called "octave equivalence." There would be no trick to
*orchestration* otherwise! :)

Scriabin, for instance, creates "new" music with unusual spacing of
his harmonies. Arvo Part as well... and many others. I hear "new"
music all the time that basically uses traditional elements with
different octave displacements....

I believe Kraig Grady has also commented on this list that
*inversions* of chords are very different from one another and, if I
remember correctly, he cited Stravinsky as mentioning the same.

So, if there is such a thing as "octave equivalence" it certainly has
to be modified in an artistic context. Probably nobody is going to
disagree with that...

Best,

J. Pehrson

--- In tuning@y..., "unidala" <JGill99@i...> wrote:

/tuning/topicId_31804.html#31988

>
> Therefore, it seems (to me) that - if such a concept of
> "invariance" only applies *after* the specific melody
> has been "nurtured", then such "invariance" does not
> exist by "nature". I would think that to simply hold
> that "if we already know what we are looking for, then
> we are able to recognize it" only makes the case for
> the proposition that an (unlearned) "octave invariance"
> (existing "by nature") perhaps exists not...
>
>
> Best Regards, J Gill :)
>

Hello J. Gill!

Well, perhaps it can be reduced to an idea that in an *artistic*
sense there is really no such thing as "octave equivalence."

In an *acoustical* sense, though, it seems clear that multiplying
something by 2 will result in something more similar to the original
than anything else...

Short of multiplying by 1... :)

best,

J. Pehrson

--- In tuning@y..., "unidala" <JGill99@i...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning@y..., "unidala" <JGill99@i...> wrote:
> >
> > > J Gill: Now, this *does* "resonate" with me.
> > > With or without any kind of "directional effect"
> > > at play, and despite what some may say (about
> > > an interval and its algebraic inversion being
> > > "equivalent" because they represent the same
> > > "interval"), *my ear* does not hear a 2/3 pitch
> > > played *below* a 1/1 pitch as "One and Five"
> > > relationship (*unless* the "implied root", or
> > > "virtual fundamental" has flipped downward to
> > > the 2/3 pitch (from the 1/1 pitch). As they say,
> > > "you can't hear it both ways...". :)
> >
> >PE: I think you're misunderstanding how octave equivalence is
>supposed to
> > work. According to octave-equivalence, you would hear the 2/3
pitch
> > as equivalent to the 4/3 pitch, thus "One and Four", I believe,
is
> > the relationship you'd hear as you'd describe it.
>
> JG: Understood, and agreed (where the "implied root" remains the
reference pitch), and the ear is not persuaded that the 2/3 pitch is
not the (new) "implied root" [since it is now the lower frequency
tone, and (sounded simultaneously in isolation) with the 1/1 pitch,
does not, then, assume the status of the "implied fundamental"]?

Well, the question of what the implied fundamental is of the dyad One
and Four is a separate one, though I'd tend to say Four in most cases.

>
> I have been using the phrase "octave equivalence" in reference to
>the phenomenon of the "perceived similarity" of an *individual tone*
>being raised/lowered in pitch by powers of 2 to the shifted tone.

There is such similarity, seemingly recognized in all cultures whose
music spans more than an octave.
>
> > PE:The implications of
> > this, to the question of intervals, is that the interval of the
>fifth
> > (downward, if you wish) and the interval of the fourth (upward,
if
> > you wish) can both describe the relationship between the _same
> > functional_ pair of pitches,
>
> JG: In situations where the pitches separated by that "interval"
>are related in a manner in which any "implied fundamental" resulting
>from the (serial or simultaneous) sounding of the two individual
>pitches is free to be interpreted as *either* the *lowest* pitch
>tone, or the *highest tone*. How does that comport with
>this "implied fundamental" business typically (at least in such a
>simple case) being identified as relating to the *lower* frequency
>tone, only?

Again, questions of "implied fundamental" are a separate issue, but
certainly I wouldn't call the lower tone the implied fundamental in
the case of the dyad 1/1 4/3.

> > and therefore when considering
> > pitches "functionally" and abstracted away from "octave
>>specificity",
>
> JG: Wouldn't you also have to say, "abstracted away from the
>concept of the 'implied fundamental' in the case of an interval
>between two pitches under consideration?

No.
>
> Does that mean "implied fundamental" can *only* be applied to
(serially or simultaneously sounded) combinations of *greater than*
two tones?

No.

> If possible, could you also relate the meaning as you understand it
> of the phrase "interval of equivalence", to this situation,

The octave is the interval of equivalence. Seems clear.
>
> While you are at it (though this may seem like the Nth time some
moron hasn't "gotten it"), could you give a stab at (briefly, and in
your own language) the phrases:
>
> "Moment of Symmetry" (MOS)
>
> "Constant Product Set" (CPS)
>
> "sub-tends" (everyone seems to assume that the reader understands
the intended meaning of this; I'm still scratching my head about it).
>
> I've been through the archives a bunch of times, checked Monz's
definitions, read a bunch of people's posts, etc. Still a bit hazy...

Why don't you give me some _contexts_ where you found these terms
used and I'll go over the specific cases with you. It usually doesn't
help to stare at a generalized definition if you haven't absorbed a
few specific cases yet.

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> Thanks for your contribution to this interesting discussion. I
guess
> from an *artistic* point of view there really isn't something
> explicitly called "octave equivalence." There would be no trick to
> *orchestration* otherwise! :)
>
> Scriabin, for instance, creates "new" music with unusual spacing of
> his harmonies. Arvo Part as well... and many others. I hear "new"
> music all the time that basically uses traditional elements with
> different octave displacements....
>
> I believe Kraig Grady has also commented on this list that
> *inversions* of chords are very different from one another and, if
I
> remember correctly, he cited Stravinsky as mentioning the same.
>
> So, if there is such a thing as "octave equivalence" it certainly
has
> to be modified in an artistic context. Probably nobody is going to
> disagree with that...
>
> Best,
>
> J. Pehrson

Joseph, musicians give pitches an octave apart the same name. OK,
that isn't a TOTAL equivalence, but what would you call it? How about
octave similarity?

--- In tuning@y..., "paulerlich" <paul@s...> wrote:

/tuning/topicId_31804.html#32134

>
> Joseph, musicians give pitches an octave apart the same name. OK,
> that isn't a TOTAL equivalence, but what would you call it? How
about
> octave similarity?

Hi Paul...

Well, of course... that's the whole notion of "pitch class"
in "traditional" set theory, etc., etc.

However, J. Gill seemed to want to make a bigger case for "octave
equivalence" than really exists... at least it seemed that way to me.

I think you summed the whole issue up in about five lines back aways
when you said that the concept of "octave equivalence" was somewhat
limited...

JP

> From: jpehrson2 <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, December 30, 2001 12:11 PM
> Subject: [tuning] Re: Schoenberg and octave displacement
>
>
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> /tuning/topicId_31804.html#32134
>
> >
> > Joseph, musicians give pitches an octave apart the same name. OK,
> > that isn't a TOTAL equivalence, but what would you call it? How
> > about octave similarity?
>
> Hi Paul...
>
> Well, of course... that's the whole notion of "pitch class"
> in "traditional" set theory, etc., etc.
>
> However, J. Gill seemed to want to make a bigger case for "octave
> equivalence" than really exists... at least it seemed that way to me.
>
> I think you summed the whole issue up in about five lines back aways
> when you said that the concept of "octave equivalence" was somewhat
> limited...
>
> JP

Just tossing in my two cents... To me, 8ve-equivalence or similarity
is simply a result of the acoustical properties of prime-factor 2.
It's not essentially different in kind, but only in degree, from
those of the higher prime-factors. I find that the ratio 3:2, and
to some extent 9:8, also have this effect of "similarity" but to
a much lesser degree than 2:1. The 5-limit ratios still have
a nice consonant blending effect, but the "similarity" relation
is *far* weaker than with 2:1 or the 3-limit ratios. Etc. ...

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_31804.html#32158

>
> Just tossing in my two cents... To me, 8ve-equivalence or
similarity
> is simply a result of the acoustical properties of prime-factor 2.
> It's not essentially different in kind, but only in degree, from
> those of the higher prime-factors. I find that the ratio 3:2, and
> to some extent 9:8, also have this effect of "similarity" but to
> a much lesser degree than 2:1. The 5-limit ratios still have
> a nice consonant blending effect, but the "similarity" relation
> is *far* weaker than with 2:1 or the 3-limit ratios. Etc. ...
>
>
>
> -monz
>

Hi Monz!

Well, that makes sense, but I think the point in this thread was that
J. Gill was "surprised" that "octave equivalence" didn't work so well
when a melody was subjected to "octave displacements."

So really, as Paul states, the "equivalence" is only a "similarity" a
shorthand for pitch classes and a way to set up scales but, of
course, quite different in compositional and thematic usage.

Otherwise, I think "orchestrators" would be put out of business... :)

JP

Have some of us failed to tune into the oservation pointed out
earlier that the DIRECTION and SPAN of melodic MOTION has everything
to do with melodic perception and that we are able to disguise a
melody beyond recognition by shifting isolated pieces of it up or
down an octave is a simple reflection of this basic fact? Octave
equivalence is not even being tested, let alone challenged by such an
experiment, so why are we still barking up that tree is if it were?

You have to move ALL the pitches in the melody up or down an octave
to get a valid example of "octave equivalence". Otherwise you
radically alter the melodic CONTOUR, and the resulting NEW MELODIC
CONTOUR is resposnble for the unrecognizability! this has ZERO, YES
ZERO, to do with whether or not "octave equivalence" exists.

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> /tuning/topicId_31804.html#32158
>
> >
> > Just tossing in my two cents... To me, 8ve-equivalence or
> similarity
> > is simply a result of the acoustical properties of prime-factor 2.
> > It's not essentially different in kind, but only in degree, from
> > those of the higher prime-factors. I find that the ratio 3:2, and
> > to some extent 9:8, also have this effect of "similarity" but to
> > a much lesser degree than 2:1. The 5-limit ratios still have
> > a nice consonant blending effect, but the "similarity" relation
> > is *far* weaker than with 2:1 or the 3-limit ratios. Etc. ...
> >
> >
> >
> > -monz
> >
>
> Hi Monz!
>
> Well, that makes sense, but I think the point in this thread was
that
> J. Gill was "surprised" that "octave equivalence" didn't work so
well
> when a melody was subjected to "octave displacements."
>
> So really, as Paul states, the "equivalence" is only a "similarity"
a
> shorthand for pitch classes and a way to set up scales but, of
> course, quite different in compositional and thematic usage.
>
> Otherwise, I think "orchestrators" would be put out of
business... :)
>
> JP

Have some of us failed to tune into the oservation pointed out
earlier that the DIRECTION and SPAN of melodic MOTION has everything
to do with melodic perception and that we are able to disguise a
melody beyond recognition by shifting isolated pieces of it up or
down an octave is a simple reflection of this basic fact? Octave
equivalence is not even being tested, let alone challenged by such an
experiment, so why are we still barking up that tree is if it were?

You have to move ALL the pitches in the melody up or down an octave
to get a valid example of "octave equivalence". Otherwise you
radically alter the melodic CONTOUR, and the resulting NEW MELODIC
CONTOUR is resposnble for the unrecognizability! this has ZERO, YES
ZERO, to do with whether or not "octave equivalence" exists.

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> /tuning/topicId_31804.html#32158
>
> >
> > Just tossing in my two cents... To me, 8ve-equivalence or
> similarity
> > is simply a result of the acoustical properties of prime-factor 2.
> > It's not essentially different in kind, but only in degree, from
> > those of the higher prime-factors. I find that the ratio 3:2, and
> > to some extent 9:8, also have this effect of "similarity" but to
> > a much lesser degree than 2:1. The 5-limit ratios still have
> > a nice consonant blending effect, but the "similarity" relation
> > is *far* weaker than with 2:1 or the 3-limit ratios. Etc. ...
> >
> >
> >
> > -monz
> >
>
> Hi Monz!
>
> Well, that makes sense, but I think the point in this thread was
that
> J. Gill was "surprised" that "octave equivalence" didn't work so
well
> when a melody was subjected to "octave displacements."
>
> So really, as Paul states, the "equivalence" is only a "similarity"
a
> shorthand for pitch classes and a way to set up scales but, of
> course, quite different in compositional and thematic usage.
>
> Otherwise, I think "orchestrators" would be put out of
business... :)
>
> JP

--- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> Have some of us failed to tune into the oservation pointed out
> earlier that the DIRECTION and SPAN of melodic MOTION has
everything
> to do with melodic perception and that we are able to disguise
a
> melody beyond recognition by shifting isolated pieces of it up
or
> down an octave is a simple reflection of this basic fact? Octave
> equivalence is not even being tested, let alone challenged by
such an
> experiment, so why are we still barking up that tree is if it were?
>
> You have to move ALL the pitches in the melody up or down an
octave
> to get a valid example of "octave equivalence". Otherwise you
> radically alter the melodic CONTOUR, and the resulting NEW
MELODIC
> CONTOUR is resposnble for the unrecognizability! this has
ZERO, YES
> ZERO, to do with whether or not "octave equivalence" exists.

Good point, Bob. It seems, though, that certain people might find
it more comfortable to set up some kind of "straw man" to
associate with those for whom or for whose ideas, for one
reason or another, they seem to have acquired a distaste, and
then attack the straw man, rather than approaching the relevant
parties directly in the spirit of dialogue and debate . . . (???)

Peace and Love to all and Happy New Year,
Paul

J Gill: Good thoughts, Robert. Thanks for sharing your
outlook on melodic perception.

--- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:

> Well, first of all, by inversion I meant the classical concept of
> interval inversion and not a fifth below as opposed to a fifth
> above. Those are both fifths. I'm referring to the inversion of the
> fifth, which is the fourth, and which has a different melodic span > in
> terms of pitch difference.
>
> This is much more relevant to the issue of octave displacement of
> selected notes in a melody. In that case, both the DIRECTION of
> motion is reversed, and the SPAN (PITCH DISTANCE) of the interval
> is changed. The further removed from a tritone the interval to be
> inverted is, the greater the change in pitch distance.
>
> I'm simply saying that Frere Jacque on C, for example, based on a
> motive with a pitch sequence of CDEC in the same octave, is a
> sequence of two upward stepwise motions and then a drop to the
> orginal pitch a third below. That is how we hear melody: by melodic
> contour!
>
> If you transpose the initial C up an octave, you get an initial
> DOWNWARD motion of a seventh before proceeding to the E which, if
> we
> transpose it down an octave, entails another DOWNWARD motion of a
> seventh before returning to the untransposed final C a minor sixth
> ABOVE it. The melodic contour has been RADICALLY ALTERED by the
> octave transposition of two notes. This is why such a melody sounds
> so entirely different. We've radically changed the melodic CONTOUR
> in
> terms of both directin of melodic movement and the DISTANCE of that
> motion.
>
> On the "expectation" issue, if you take an unfamiliar melody and
> repeat it in its entirety with a whole octave displacement for
> every
> note without advising anyone that this is what you're going to do,
> anyone with the slightest quantum of musical perception will
> identify
> it as the same melody even if they don't know what an octave is.
>
> They may do the same thing for the same melody tranposed a fifth up
> or down, or any other interval, but it won't sound as if it were in
> the same key. I don't understand how anyone can seriously argue
> against the concept of octave equivalence. I've never seen any
> evidence of its being a culturally conditioned phenomenon. It seems
> to be quite universal that men and women sing the same melody an
> octave apart and perceive each other as singing the same melody.
>
>
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
> > --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> > > Well, I've been hanging out on this one waiting to see who else
> >would
> > > say it first. Paul Erlich has come the closest. Melodic
> perception
> > > has everything to do with DIRECTION OF MELODIC MOVEMENT!!!
> >
> > J Gill: This is very interesting, Bob.
> > Could you explain this "directional effect"?
> >
> > > Displacing
> > > some notes and not others to create a vastly different melodic
> > > terrain
> > > spite of the notes being otherwise identical with the original
> >melody
> > > is therefore NOT A VALID TEST of the principle of octave
> invariance.
> >
> > J Gill: On the basis of a "directional effect"
> > existing (only)?
> >
> > > Who would say that a major third sounds the same even
> harmonically >as
> > > it's first cousin and inversion, the minor sixth?!? This and
> other
> > > intervals and their inversions are even less alike melodically
> >
> > J Gill: Now, this *does* "resonate" with me.
> > With or without any kind of "directional effect"
> > at play, and despite what some may say (about
> > an interval and its algebraic inversion being
> > "equivalent" because they represent the same
> > "interval"), *my ear* does not hear a 2/3 pitch
> > played *below* a 1/1 pitch as "One and Five"
> > relationship (*unless* the "implied root", or
> > "virtual fundamental" has flipped downward to
> > the 2/3 pitch (from the 1/1 pitch). As they say,
> > "you can't hear it both ways...". :)
> >
> >
> > > than
> > > harmonically.
> >
> > J Gill: Once again, you "resonate". Would anyone claim
> > that a chord made up of the pitches 1/7, 1/6, 1/5, 1/4
> > has harmonic characteristics remotely resembling a chord
> > made up of the pitches 4/1, 5/1, 6/1, 7/1 ???
> >
> >
> > > You take the whole thing and transpose ALL of it by an
> > > octave or two or you've changed the whole game completely in
> terms >of
> > > melodic integrity.
> >
> >
> > J Gill: But if you were to transpose *all* of the notes
> > of a given scale or chord upwards/downwards "by an octave
> > or two", and you have "changed the whole game completely
> > in terms of melodic integrity" in doing so - is that not
> > an argument that "octave equivalence" is non-existent,
> > period (in any of the possible circumstances heretofore
> > discussed)???
> >
> >
> > Regards, J Gill