Friendly guide to what we were talking about here, part 1

I want to say hello and welcome to Marc Jones (who happened to play
in the same "key" in 32 as I did in 22 on the AFMM radio broadcast);
to Gary Morrison, a fine mind whose voice had been regrettably absent
for a while; and to all the silent people out there who are crazy
enough to stay subscribed.

Undoubtedly, many people are lost in the lingo (I'm imagining myself
day-trading) that comes up in the technical tuning discussions, and
so stop reading those threads after a while. So I thought I'd try to
outline the basic thinking for those who don't care to get
mathematical, and sum up what we've learned since that hasn't been
done yet.

There is a long history of theorists whose work anticipates these
ideas, but for convenience I'll stick to just four: Partch and
Wilson, Euler and Fokker.
I invite Kraig Grady to correct any errors in what follows and
provide an alternative viewpoint should he so desire.

Somewhere between Partch's "One-Footed Bride" graph and his numerical
theories, Partch enunciated a theory of consonance. It seems to be
the perfect way of applying psychoacoustics to the problem of
intervals in an octave-repeating tuning system. That is, if trying to
decide what set of pitches you want, given that they will repeat
every 2:1 octave, Partch gives you the measure of consonance to use.

Partch's conception of "tonality" was that one pitch was at the
center of all the consonances, i. e. it should be accompanied by all
the pitches consonant with it. This concept, married to his
consonance model, bears child in the form of the Tonality Diamond
scales.

Erv Wilson showed how Partch's scales could be geometrically
depicted, with the tonic lying in the center of a fantastically
symmetrical shape. Erv's constructions relied on a concept called the
lattice: every pitch in a JI tuning system can be mapped to a point
in space (sometimes higher-dimensional space), such that its nearest
neigbors are the pitches most consonant with it.

Erv Wilson also created new scales which were the perfect theoretical
complement to Partch's Tonality Diamonds. Rather than containing one
central pitch, the new scales put every note in the scale in an equal
position, each sharing an equal number of consonances with its
neighbors. Yet there is no "space" in the lattice into which any
central (or more central) pitch could be inserted. He called these
scales "CPS scales". Very mathematical.

Getting back to Partch, Partch realized that his Tonality Diamond,
though symmetrically centered in the lattice, had some
melodic "holes" in it when ordered as an ascending or descending
scale. Thus he added some extra pitches to his scale, which partook
of about as many strong consonances as possible in the expanded scale.

Wilson, designing keyboards and instruments to play his CPS scales,
found it similarly logical to fill the melodic holes in his scale
with extra, consonant pitches. He thought of this process largely in
terms of a closed cycle of fifths, mainly the consonant 3:2 fifths,
but with a some of the fifths "mutated" to be more complex JI ratios,
in order to accomodate the CPS scale. He used the same process when
designing keyboards for Partch's scale, and doubtlessly others of
which I am unaware.

Now we bring in one more theorist, Fokker. Fokker's main realization
was a simple but elusive one. Fokker studied the JI lattice. He
showed that by defining two or three or four (depending on the number
of dimensions in the lattice) small JI intervals to be "unisons", one
could divide the lattice into congruent "chunks" each of which, as a
scale:
has a lot of consonant intervals (by virtue of it being a chunk);
is melodically pretty smooth;
provides a "unison" to each of the infinity of pitches in the lattice;
tiles with "unison" copies of itself indefinitely to fill the lattice.
He called these scales PBs.

Fokker created just a few PBs of his own. But we've found that most
of Wilson's scales, including his CPS scales, and nearly enough
Partch's scale, actually _are_ PBs! Undoubtedly, Wilson understood
the basic PB insight in many different mathematical forms. For
example, the cycle of fifths in Wilson's keyboard designs can be
visualized as a straight line through the infinite lattice, along the
direction representing the 3:2 interval. Meanwhile, the lattice is
divided into PB chunks (chunky peanut butter) sliced along the
directions of the "unisons". Each time the line of fifths crosses the
boundary from one PB to the next, it must "mutate" by a "unison" in
order to compensate for the "unison" pitch shift between one PB and
its neighbor. Eventually, it passes through a note in a distant PB
that is the "image" of the beginning note in the beginning PB, and
the cycle is closed.

Another insight of Wilson had to do with another way of slicing the
infinite lattice, not into "unisonious" pieces but into pieces
aligned along the directions of the "prime" consonances (those
relating a fundamental and an overtone). This construction was found
by Euler centuries ago, and he called each piece a "genus". Fokker
extended the list of "primes", much like Partch did, and the higher-
dimensional "genus" became known as an EFG (Euler-Fokker genus).
Wilson found that each of his "families" of CPS scales,
suitably "housed", would fit with no gaps into an EFG. Thus the
infinite lattice could be seen as an infinite number of CPS scales,
fitting together with no gaps. Or alternatively, as Carl Lumma (I
think) pointed out, as an infinite number of Tonality Diamonds, with
some CPSs filling the gaps.

So a PB, being a chunk of the lattice, has to contain a Tonality
Diamond, a CPS scale, some of both, or parts of both. Regardless,
there will generally be lots of consonant intervals. Partch (guided
more by judgment than by mathematics) and Wilson carefully
constructed each of their PBs to carefully fit around one Tonality
Diamond or around one CPS or one EFG, with as few extra notes as
possible.

(to be continued . . .)

Paul!
I would add the concept of Moments of Symmetry (with constant structures as resulting in much
of the same material as PB) , the expansion of the Yasser/ Kornerup tree into the scale tree as
applicable to music (now known as the Stern Brocot tree in math). Also taking Kornerup as a model
to take the noble numbers resulting from the scale tree and illustrating their Moments of
Symmetries. Also the development of Keyboard designs that encompass the entire scale tree. Also
the Marwa permutations and the Purvi modulations as ways to develop the tetrachord- (could be seen
as an appendix to Chalmers work). There is also some recent work on the Lambdoma involving series
besides the harmonic subharmonic series.
Just to be clear, Erv is not concerned with the consonance question. His choice of
investigation is more along the idea of archetypes as a basis for further exploration. This view
is attributed to Yasser who first proposed a theory as that tackles to the way scales evolve over
time. With the consonance question he has not been satisfied with any of the explanation as being
universal. In particular he will acknowledge them as pointing toward certain solutions.
In regard to the CPS as being purely mathematical i do take issue. It is a highly perceptible
structure that as easy to learn as the major scale. As someone who worked with it for years, i
have no trouble hearing where i am any more than when i hear a V7 chord. Its logic is based on how
the mind hears. It also has the notoriety of being the first structures of organizing simple
ratios in a way free from a tonal center yet self-containing and defining. Possibly it solves
exactly what the atonalist wanted to do without having to avoid acoustical relation.

paul@stretch-music.com wrote:

> I want to say hello and welcome to Marc Jones (who happened to play
> in the same "key" in 32 as I did in 22 on the AFMM radio broadcast);
> to Gary Morrison, a fine mind whose voice had been regrettably absent
> for a while; and to all the silent people out there who are crazy
> enough to stay subscribed.
>
> Undoubtedly, many people are lost in the lingo (I'm imagining myself
> day-trading) that comes up in the technical tuning discussions, and
> so stop reading those threads after a while. So I thought I'd try to
> outline the basic thinking for those who don't care to get
> mathematical, and sum up what we've learned since that hasn't been
> done yet.
>
> There is a long history of theorists whose work anticipates these
> ideas, but for convenience I'll stick to just four: Partch and
> Wilson, Euler and Fokker.
> I invite Kraig Grady to correct any errors in what follows and
> provide an alternative viewpoint should he so desire.
>
> Somewhere between Partch's "One-Footed Bride" graph and his numerical
> theories, Partch enunciated a theory of consonance. It seems to be
> the perfect way of applying psychoacoustics to the problem of
> intervals in an octave-repeating tuning system. That is, if trying to
> decide what set of pitches you want, given that they will repeat
> every 2:1 octave, Partch gives you the measure of consonance to use.
>
> Partch's conception of "tonality" was that one pitch was at the
> center of all the consonances, i. e. it should be accompanied by all
> the pitches consonant with it. This concept, married to his
> consonance model, bears child in the form of the Tonality Diamond
> scales.
>
> Erv Wilson showed how Partch's scales could be geometrically
> depicted, with the tonic lying in the center of a fantastically
> symmetrical shape. Erv's constructions relied on a concept called the
> lattice: every pitch in a JI tuning system can be mapped to a point
> in space (sometimes higher-dimensional space), such that its nearest
> neigbors are the pitches most consonant with it.
>
> Erv Wilson also created new scales which were the perfect theoretical
> complement to Partch's Tonality Diamonds. Rather than containing one
> central pitch, the new scales put every note in the scale in an equal
> position, each sharing an equal number of consonances with its
> neighbors. Yet there is no "space" in the lattice into which any
> central (or more central) pitch could be inserted. He called these
> scales "CPS scales". Very mathematical.
>
> Getting back to Partch, Partch realized that his Tonality Diamond,
> though symmetrically centered in the lattice, had some
> melodic "holes" in it when ordered as an ascending or descending
> scale. Thus he added some extra pitches to his scale, which partook
> of about as many strong consonances as possible in the expanded scale.
>
> Wilson, designing keyboards and instruments to play his CPS scales,
> found it similarly logical to fill the melodic holes in his scale
> with extra, consonant pitches. He thought of this process largely in
> terms of a closed cycle of fifths, mainly the consonant 3:2 fifths,
> but with a some of the fifths "mutated" to be more complex JI ratios,
> in order to accomodate the CPS scale. He used the same process when
> designing keyboards for Partch's scale, and doubtlessly others of
> which I am unaware.
>
> Now we bring in one more theorist, Fokker. Fokker's main realization
> was a simple but elusive one. Fokker studied the JI lattice. He
> showed that by defining two or three or four (depending on the number
> of dimensions in the lattice) small JI intervals to be "unisons", one
> could divide the lattice into congruent "chunks" each of which, as a
> scale:
> has a lot of consonant intervals (by virtue of it being a chunk);
> is melodically pretty smooth;
> provides a "unison" to each of the infinity of pitches in the lattice;
> tiles with "unison" copies of itself indefinitely to fill the lattice.
> He called these scales PBs.
>
> Fokker created just a few PBs of his own. But we've found that most
> of Wilson's scales, including his CPS scales, and nearly enough
> Partch's scale, actually _are_ PBs! Undoubtedly, Wilson understood
> the basic PB insight in many different mathematical forms. For
> example, the cycle of fifths in Wilson's keyboard designs can be
> visualized as a straight line through the infinite lattice, along the
> direction representing the 3:2 interval. Meanwhile, the lattice is
> divided into PB chunks (chunky peanut butter) sliced along the
> directions of the "unisons". Each time the line of fifths crosses the
> boundary from one PB to the next, it must "mutate" by a "unison" in
> order to compensate for the "unison" pitch shift between one PB and
> its neighbor. Eventually, it passes through a note in a distant PB
> that is the "image" of the beginning note in the beginning PB, and
> the cycle is closed.
>
> Another insight of Wilson had to do with another way of slicing the
> infinite lattice, not into "unisonious" pieces but into pieces
> aligned along the directions of the "prime" consonances (those
> relating a fundamental and an overtone). This construction was found
> by Euler centuries ago, and he called each piece a "genus". Fokker
> extended the list of "primes", much like Partch did, and the higher-
> dimensional "genus" became known as an EFG (Euler-Fokker genus).
> Wilson found that each of his "families" of CPS scales,
> suitably "housed", would fit with no gaps into an EFG. Thus the
> infinite lattice could be seen as an infinite number of CPS scales,
> fitting together with no gaps. Or alternatively, as Carl Lumma (I
> think) pointed out, as an infinite number of Tonality Diamonds, with
> some CPSs filling the gaps.
>
> So a PB, being a chunk of the lattice, has to contain a Tonality
> Diamond, a CPS scale, some of both, or parts of both. Regardless,
> there will generally be lots of consonant intervals. Partch (guided
> more by judgment than by mathematics) and Wilson carefully
> constructed each of their PBs to carefully fit around one Tonality
> Diamond or around one CPS or one EFG, with as few extra notes as
> possible.
>
> (to be continued . . .)
>
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-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> Paul!
> I would add the concept of Moments of Symmetry

whoa . . . that's coming up in part 2, or 3 . . .

>(with constant structures as resulting in much
> of the same material as PB)

Actually I did go over that . . .

> , the expansion of the Yasser/ Kornerup tree into the scale tree as
> applicable to music (now known as the Stern Brocot tree in math).
> Also taking Kornerup as a model
> to take the noble numbers resulting from the scale tree and
illustrating their Moments of
> Symmetries. Also the development of Keyboard designs that encompass
the entire scale tree. Also
> the Marwa permutations and the Purvi modulations as ways to develop
the tetrachord- (could be seen
> as an appendix to Chalmers work). There is also some recent work on
the Lambdoma involving series
> besides the harmonic subharmonic series.

Now I didn't say I was going to cover _all_ of Erv's theories, just
the stuff that's relevant to the big wad of material we just created.

> Just to be clear, Erv is not concerned with the consonance
question. His choice of
> investigation is more along the idea of archetypes as a basis for
further exploration. This view
> is attributed to Yasser who first proposed a theory as that tackles
to the way scales evolve over
> time. With the consonance question he has not been satisfied with
any of the explanation as being
> universal. In particular he will acknowledge them as pointing
toward certain solutions.

I'm glad you clarified that.

> In regard to the CPS as being purely mathematical i do take
issue.

I did not say it way purely mathematical . . . I just said "Very
mathematical." as a way of ending the paragraph by saying, "well, all
you non-math folks, there's a lot of math involved, so we'll leave it
at that".

> It is a highly perceptible
> structure that as easy to learn as the major scale. As someone who
worked with it for years, i
> have no trouble hearing where i am any more than when i hear a V7
chord. Its logic is based on how
> the mind hears.

Feel free to go into it in more depth -- I'm fascinated.

> It also has the notoriety of being the first structures of
organizing simple
> ratios in a way free from a tonal center yet self-containing and
defining. Possibly it solves
> exactly what the atonalist wanted to do without having to avoid
acoustical relation.

Thanks for bringing that up.

Paul! (exclamation point on loan from KG...)

--- In tuning@y..., paul@s... wrote:
> Undoubtedly, many people are lost in the lingo (I'm imagining
> myself day-trading) that comes up in the technical tuning
> discussions, and so stop reading those threads after a while.
> So I thought I'd try to outline the basic thinking for those who
> don't care to get mathematical, and sum up what we've learned since
> that hasn't been done yet.

...and the series to come. Thank You So Much for this, as it is
almost as if you were writing it for me! I look forward to the
upcoming installments, and I plan on collating them and doing editing
with Kraig's (and anyone else's) commentary, and saving it for those
all-too-common moments when I feel like I don't know a f***ing thing
about tuning theory!

And, in case anyone might get the wrong idea, this is a very sincere
thanks, to you and to all those that will help crystalize and clarify
such a basic set of notions. No sarcasm, no tongue-in-cheek.

But, also, can this count as a positive or constructive posting?
Please??

With gratitude,
Jon

--- In tuning@y..., paul@s... wrote:

/tuning/topicId_23492.html#23492

> Partch's conception of "tonality" was that one pitch was at
> the center of all the consonances, i. e. it should be
> accompanied by all the pitches consonant with it. This concept,
> married to his consonance model, bears child in the form of
> the Tonality Diamond scales.

Paul, thanks for a terrific start to the summary of what's
been going on here lately. Looking forward to the rest.

I've recently mentioned several times my unpublished paper
_Similarities Between Partch and Schoenberg as Originators
of New Harmonic Principles_. This is yet another.

In his 1949 book _Structural Functions of Harmony_,
Schoenberg presents his concept of "monotonality", the
basic idea of which is that any piece or movement, regardless
of how often or remotely it modulates, is centered on one
specific pitch, and every other harmony has a classifiable
relationship to it, in various combinations of direct/indirect
and close/remote.

The similarity to Partch's "Monophony" is striking.
Note that Partch's book was published two years *before*
Schoenberg's. But my hunch is that Schoenberg had this
idea in mind for a long time before his book was published.

-monz
http://www.monz.org
"All roads lead to n^0"

Joe,

--- In tuning@y..., "monz" <joemonz@y...> wrote:
> I've recently mentioned several times my unpublished paper
> _Similarities Between Partch and Schoenberg as Originators
> of New Harmonic Principles_. This is yet another.

If there are similarities, and if they are significant, how would you
then account for the fact that it would be difficult to find two
people whose artistic output and aesthetic stances are farther apart?

Like, light years apart.

Just throwing this out for thought, though I'm sure you've thought
about it, Joe! I can't wait to go up to Arnies bust in the foyer of
the Hall this weekend and ... polish his pate.

Cheers,
Jon

P.S. Note the Blackburn announcement, and let the SD boys know (are
there any SD 'girls'?)

--- In tuning@y..., JSZANTO@A... wrote:

/tuning/topicId_23492.html#23529

> Joe,
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > I've recently mentioned several times my unpublished paper
> > _Similarities Between Partch and Schoenberg as Originators
> > of New Harmonic Principles_. This is yet another.
>
> If there are similarities, and if they are significant, how
> would you then account for the fact that it would be difficult
> to find two people whose artistic output and aesthetic stances
> are farther apart?
>
> Like, light years apart.
>
> Just throwing this out for thought, though I'm sure you've
> thought about it, Joe!

Yup... It's one of the most profound ironies in all the
tons of musical research I've done!

> I can't wait to go up to Arnies bust in the foyer of
> the Hall this weekend and ... polish his pate.

Now *that* I found *REALLY* ironic!! The Partch centennial
celebration is being held in *SCHOENBERG* Hall!

> P.S. Note the Blackburn announcement,

Already noted and responded.

> and let the SD boys know

Will do.

> (are there any SD 'girls'?)

Yes, believe it or not, there is a small coterie of women
here who are interested in microtonal and "new music" in
general.

Part of the fun of hanging out at Sonic Arts...

(which, I must admit, I haven't done much of lately...
been way too busy.)

-monz
http://www.monz.org
"All roads lead to n^0"

--- In tuning@y..., JSZANTO@A... wrote:

/tuning/topicId_23492.html#23529

> Joe,
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > I've recently mentioned several times my unpublished paper
> > _Similarities Between Partch and Schoenberg as Originators
> > of New Harmonic Principles_. This is yet another.
>
> If there are similarities, and if they are significant, how would
you then account for the fact that it would be difficult to find two
> people whose artistic output and aesthetic stances are farther
apart?
>
> Like, light years apart.

That's simple, Jon. Monz is taking the "overview" and *you're*
NOT.... Even *I* can answer THAT question!

__________ ______ __________ _____
Joseph Pehrson

--- In tuning@y..., JSZANTO@A... wrote:

>
> ...and the series to come. Thank You So Much for this, as it is
> almost as if you were writing it for me! I look forward to the
> upcoming installments, and I plan on collating them and doing
editing
> with Kraig's (and anyone else's) commentary, and saving it for
those
> all-too-common moments when I feel like I don't know a f***ing
thing
> about tuning theory!
>
> And, in case anyone might get the wrong idea, this is a very
sincere
> thanks, to you and to all those that will help crystalize and
clarify
> such a basic set of notions. No sarcasm, no tongue-in-cheek.
>
> But, also, can this count as a positive or constructive posting?
> Please??
>
> With gratitude,
> Jon

You've made my day, Jon. I'm so glad I decided to spend a couple of
hours working on this "bridge" between our different "islands". More
bridgework to come!

--- In tuning@y..., JSZANTO@A... wrote:
> Joe,
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > I've recently mentioned several times my unpublished paper
> > _Similarities Between Partch and Schoenberg as Originators
> > of New Harmonic Principles_. This is yet another.
>
> If there are similarities, and if they are significant, how would
you
> then account for the fact that it would be difficult to find two
> people whose artistic output and aesthetic stances are farther
apart?
>
> Like, light years apart.

Well, Schoenberg wasn't referring to his _own_ music (except his
early, tonal works) in his analyses of works where the concept of a
central pitch seemed to be operative. His own music sought to break
out of that paradigm.

Joe,

--- In tuning@y..., jpehrson@r... wrote:
> That's simple, Jon. Monz is taking the "overview" and *you're*
> NOT.... Even *I* can answer THAT question!

Well, at least you can answer it incorrectly. Joe has already replied
that, indeed, it is a difficult assertation to hold onto. My problem
is that to take two entirely different perspectives on ... *lots* of
things and put them in bed together just because they have one very
small common context...well, that is very misleading to people who
don't know other background material on the two composers.

Dig out "Genesis of a Music" and read the section on Schoenberg in
the chapter "From Emporer Chun to the Vacant Lot"; Partch is pretty
clear about his disdain, and that is enough for me.

Broad overview? They were both composers, and how much is *that*
statement worth.

Simplifications are dangerous, but I await Joe's paper in full before
the final judgement. And then, fortunately, he lives close enough to
me that I can go lob water balloons at him for tepid scholasticism.

Cheers,
Jon

Paul,

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> Well, Schoenberg wasn't referring to his _own_ music (except his
> early, tonal works) in his analyses of works where the concept of a
> central pitch seemed to be operative. His own music sought to break
> out of that paradigm.

Then I would argue that "...Schoenberg as Originator(s) of New
Harmonic Principles" is a misleading title.

Besides, what's the deal? I asked Monz to explain himself, not you
and Joseph! And I just have a particular problem with finding two
composers that had nose hair that curled to the left and nothing more
in common as a basis for a comparative study. But it's a big world
(with lots of nose hair out there...).

Cheers,
Jon

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

/tuning/topicId_23492.html#23559

> --- In tuning@y..., JSZANTO@A... wrote:
> > Joe,
> >
> > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > > I've recently mentioned several times my unpublished paper
> > > _Similarities Between Partch and Schoenberg as Originators
> > > of New Harmonic Principles_. This is yet another.
> >
> > If there are similarities, and if they are significant,
> > how would you then account for the fact that it would be
> > difficult to find two people whose artistic output and
> > aesthetic stances are farther apart?
> >
> > Like, light years apart.
>
> Well, Schoenberg wasn't referring to his _own_ music (except
> his early, tonal works) in his analyses of works where the
> concept of a central pitch seemed to be operative. His own
> music sought to break out of that paradigm.

Thanks, Paul! How could I have missed clarifying that myself?!
You're absolutely right.

He spoke of all of his post-1908 work as being "pantonal".
His "monotonality" concept, and indeed all of the books he
wrote after coming to America in 1934, were written strictly
for the purpose of instructing his students in "classical"
harmony.

-monz
http://www.monz.org
"All roads lead to n^0"

--- In tuning@y..., JSZANTO@A... wrote:

/tuning/topicId_23492.html#23574

> Paul,
>
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Well, Schoenberg wasn't referring to his _own_ music
> > (except his early, tonal works) in his analyses of works
> > where the concept of a central pitch seemed to be
> > operative. His own music sought to break out of that
> > paradigm.
>
> Then I would argue that "...Schoenberg as Originator(s) of New
> Harmonic Principles" is a misleading title.

Well, Jon... Schoenberg may have been examining old music,
but he certainly did create a new way of looking at it!

I believe the "monotonality" concept originated with him,
and the related "Monophony" with Partch, both at the same
time, around the 1940s.

> ...I just have a particular problem with finding two
> composers that had nose hair that curled to the left and
> nothing more in common as a basis for a comparative study.

OK, I withhold any further comment on this subject until
after I *have* made a webpage out of my old essay.

Just please note that I *am* discussing what the *title* states:

"Similarites between Partch and Schoenberg as Originators
of NEW HARMONIC PRINCIPLES".

Nothing at all in there about corporealism-vs.-abstract,
integrity of the spoken word, etc. etc.

But guess what?... Schoenberg did indeed write stuff about
all this that *also* echos what Partch said to a great extent.

I suppose, Jon, that you'll only be able to get *this* whole
picture whenever I get around to assembling my second book,
which will be devoted to Schoenberg. (I'm still working on
the first one...)

PS - I'm ready for those water balloons! It was a hot one today!

-monz
http://www.monz.org
"All roads lead to n^0"

Joe,

--- In tuning@y..., "monz" <joemonz@y...> wrote:
> I suppose, Jon, that you'll only be able to get *this* whole
> picture whenever I get around to assembling my second book,
> which will be devoted to Schoenberg. (I'm still working on
> the first one...)

Hey, it was only a gentle tweak anyway. You can bark up any tree you
want to!

> PS - I'm ready for those water balloons! It was a hot one today!

Wait til today...

Cheers,
Jon

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

> Well, Jon... Schoenberg may have been examining old music,
> but he certainly did create a new way of looking at it!
>
> I believe the "monotonality" concept originated with him,

Didn't Riemann and even Rameau see musical compositions as revolving
around a central pitch in the lattice?

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

/tuning/topicId_23492.html#23622

> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > Well, Jon... Schoenberg may have been examining old music,
> > but he certainly did create a new way of looking at it!
> >
> > I believe the "monotonality" concept originated with him,
>
> Didn't Riemann and even Rameau see musical compositions
> as revolving around a central pitch in the lattice?

Hi Paul. I think you may be right about both of those.
My Riemann studies are quite rusty, and my Rameau studies
even hazier about 10 years down the line. Wish I had the
photographic memory you possess!

I try to look into this more; I have the books at hand.
(Just not the time...)

-monz
http://www.monz.org
"All roads lead to n^0"