youtube video of 2-d tunings

Jim Plamondon (the guy who is trying to get the "thummer", an
inexpensive 2-d hex controller to market) has put up a youtube video
showing some of the possibilities. The video is here:

http://www.youtube.com/watch?v=Nd4h8vmEsQM#SAFsCUTZ470

Here's the website for the thummmer controller (also with videos)

http://www.thummer.com/

Here's a paper out recently in the Computer Music Journal describing
the methods and system of the thummer (co-authored by Plamondon, Andy
Milne, and myself)

http://eceserv0.ece.wisc.edu/~sethares/paperspdf/
InvariantFingering.pdf

For the more mathematically inclined, here's a more complete version
of the material that will appear shortly in the J. Math and Music:

http://eceserv0.ece.wisc.edu/~sethares/paperspdf/tuningcontinua.pdf

Your comments, critiques, and/or general thoughts are appreciated!

-- Bill Sethares

Wow, I'm out of words ... That's amazing! Do I understand it right that the period is meant to be 2/1 and that what is actually changed there is the size of the fifth? If so, I wonder what could be the smallest or the largest fifth possible to use as the generator.

I don't understand what was meant there by saying that the partials of the played tones are made in tune with the intervals of the tuning. Supposing that 5/4 is mapped to periods and generators as "0 4", what happens if I go to schismatic temperaments where the mapping should be "7 -8"?

Petr

PS: Who was the speaker?

Oops, that was a typo again ... I meant 5/1, of course.

Petr

Bill Sethares wrote:
> Jim Plamondon (the guy who is trying to get the "thummer", an > inexpensive 2-d hex controller to market) has put up a youtube video > showing some of the possibilities. The video is here:
> > http://www.youtube.com/watch?v=Nd4h8vmEsQM#SAFsCUTZ470
> > Here's the website for the thummmer controller (also with videos)
> > http://www.thummer.com/
> > Here's a paper out recently in the Computer Music Journal describing > the methods and system of the thummer (co-authored by Plamondon, Andy > Milne, and myself)
> > http://eceserv0.ece.wisc.edu/~sethares/paperspdf/
> InvariantFingering.pdf
> > For the more mathematically inclined, here's a more complete version > of the material that will appear shortly in the J. Math and Music:
> > http://eceserv0.ece.wisc.edu/~sethares/paperspdf/tuningcontinua.pdf
> > Your comments, critiques, and/or general thoughts are appreciated!
> > -- Bill Sethares

This looks interesting and I'll have to remind myself to come back and look at it when I have some spare time. One comment, glancing over these papers I didn't notice if there's anything on half-octave period temperaments like srutal (diaschismic) or other multiples of periods to the octave (augmented, diminished, blackwood). Is the size of the period settable?

Bill Sethares wrote:
> Jim Plamondon (the guy who is trying to get the "thummer", an > inexpensive 2-d hex controller to market) has put up a youtube video > showing some of the possibilities. The video is here:
> > http://www.youtube.com/watch?v=Nd4h8vmEsQM#SAFsCUTZ470
> > Here's the website for the thummmer controller (also with videos)
> > http://www.thummer.com/
> > Here's a paper out recently in the Computer Music Journal describing > the methods and system of the thummer (co-authored by Plamondon, Andy > Milne, and myself)
> > http://eceserv0.ece.wisc.edu/~sethares/paperspdf/
> InvariantFingering.pdf

I see that's also available as a free download from the CMJ website:

http://www.mitpressjournals.org/toc/comj/31/4

Thank you for making that the case!

That official article (but not the one on Bill's website) mentions magic temperament. So in response a question from a while back about where the term is used in peer reviewed literature, here it is.

Also on terminology, this article generally uses "regular temperament" to mean what I'd call "regular temperament class" but not consistently so (pp.7,9). It also introduces the term "tuning continuum" (p.2) with a similar meaning. I think a "tuning continuum with tuning invariance" is exactly what I'd call a "temperament class". Or maybe it should be a "regular tuning continuum" because a "regular tuning system" or "regular temperament" (p.3) entails a specific mapping from just intonation. And maybe if the promised definition of "regular temperament" were supplied, it'd match my "regular temperament class".

> For the more mathematically inclined, here's a more complete version > of the material that will appear shortly in the J. Math and Music:
> > http://eceserv0.ece.wisc.edu/~sethares/paperspdf/tuningcontinua.pdf

Includes the term "tempered tuning" :-O

I missed the CMJ publication, but I did check this journal's website last night, as it happens. They promise 3 issues a year so I hope this really does appear shortly.

There's a term "valid tuning range" (VTR) here which is similar to imposing an error cutoff. The "size of a harmonic-melodic gamut" is also similar to complexity. Some readers may find that interesting.

Graham

Petr wrote:

> Wow, I'm out of words ... That's amazing!

Thanks!

> Do I understand it right that the period is meant to be
2/1 and that what is actually changed
there is the size of the fifth? If so, I wonder what could be the
smallest or the largest fifth possible to use as the generator.

Almost -- there are two generators -- one near 2/1 (or near
the half octave sqrt(2)/1, etc) and one that defines the remaining
terms. This is most familiarly the fifth, though it could be
some other interval. Fixing the two generators at any
specific values (like 2/1 and 3/2) defines a specific
tuning of the intervals (in this case pythagorean) though
as you can see from the slider in the video, there are many
different tuning systems (some have names, and some don't)
that can occur as you move the slider (i.e, as you change
the values of the generators.

Petr continued:

> I don't understand what was meant there by saying that the
partials of the played tones are made in tune with the intervals of
the tuning. Supposing that 5/4 is mapped to periods and generators
as "0 4", what happens if I go to schismatic temperaments where the
mapping should be "7 -8"?

In TTSS, I talk about moving the partials of a sound in order
to make them lie on the scale steps of a given tuning -- so for
instance, you might move the partials of a sound from their harmonic
position to the steps of the 10-edo scale -- this would tend to
make some of the 10-edo intervals less dissonant (at least in
"sensory dissonance" measure). Here, we're parameterizing
both the tuning (via the two generators) and the timbre
(also via the same two generators). So the partials of the
sound change slightly as you change the generators, along with
the tuning of the notes.

> PS: Who was the speaker?

That's Jim Plamondon. I think he actually sounds like that in
real life, too.

Herman Miller wrote:

> This looks interesting and I'll have to remind myself to come back
and look at it when I have some spare time. One comment, glancing
over these papers I didn't notice if there's anything on half-octave
period temperaments like srutal (diaschismic) or other multiples of
periods to the octave (augmented, diminished, blackwood). Is the size
of the period settable?

Yes... you're not restricted to octave-based generators.
You can move the generator slightly ("tempering the octave" to
something like 2.03, for instance) or you can base it on the half
octave, quarter octave... ) Basically there are just two parameters
and in principle they can take on any values. The software does place
limits on the values so that thigns are more easily diplayable.

Graham wrote:

>I see that's also available as a free download from the CMJ
website:

http://www.mitpressjournals.org/toc/comj/31/4

Thank you for making that the case!

Actually, they (meaning CMJ) do not allow authors to post
the final versions on their own website.
So the version on my website is a slightly older version.

> That official article (but not the one on Bill's website)
mentions magic temperament. So in response a question from
a while back about where the term is used in peer reviewed
literature, here it is.

We tried to use established terminology wherever
possible -- Graham will remember that I contacted him
earlier to try and sort out some of these issues.
I also emailed several times with GWS in order to
try and get it straight.
The final terms we used were also greatly influenced
by the reviewers of the JMM paper - they had some strong
opnions about naming conventions and in the end we had to bow
to their will. Because we were writing the two papers more
or less simultaneously, we also tried to keep the terminology
consistent between the two papers.
But to answer less doulfully, yes, there are a
number of terms coined in this group that are now in
peer-reviewed journals. Of course, we referenced people here
as much as possible... websites and even a few conversations
on this forum.

> Also on terminology, this article generally uses "regular
temperament" to mean what I'd call "regular temperament
class" but not consistently so (pp.7,9). It also introduces
the term "tuning continuum" (p.2) with a similar meaning. I
think a "tuning continuum with tuning invariance" is exactly
what I'd call a "temperament class". Or maybe it should be
a "regular tuning continuum" because a "regular tuning
system" or "regular temperament" (p.3) entails a specific
mapping from just intonation. And maybe if the promised
definition of "regular temperament" were supplied, it'd
match my "regular temperament class".

The "continuum" term is a compromise. What's good about it
is that it emphasizes that there are a continuum of possible
values for the intervals as the generators are changed.
I think what's bad about the term should be self evident
in this forum!

Graham continues:

> I missed the CMJ publication, but I did check this journal's
website last night, as it happens. They promise 3 issues a
year so I hope this really does appear shortly.

JMM has indeed published the three issues from 2007.
Our paper will be in the first issue of 2008, and we have
already been through the page proofs stage. Again, the paper
on my website is a slightly older version.
I would encourage anyone in this group with a strong math
background to consider publishing here...

> There's a term "valid tuning range" (VTR) here which is
similar to imposing an error cutoff. The "size of a
harmonic-melodic gamut" is also similar to complexity. Some
readers may find that interesting.

The VTR serves a similar purpose to imposing a cutoff, but it
does not require a choice of value for the cutoff. Rather, the VTR
is the range over which the parameters can vay without changing
the order of the intervals in the scale For instance, the VTR
defines the range of parameter values over which
the second remains smaller than the minor third, and over which
the minor third remains smaller than the major third, etc.).

Bill Sethares wrote:

> JMM has indeed published the three issues from 2007.
> Our paper will be in the first issue of 2008, and we have
> already been through the page proofs stage. Again, the paper
> on my website is a slightly older version.
> I would encourage anyone in this group with a strong math
> background to consider publishing here... The trouble is, most of us *don't* have a strong math background.

I've signed on and got my free introductory issue, anyway. There's a Douthett and Krantz paper where they go to great lengths to measure the errors of equal temperaments. They mention something called a "Desirability Function" which they've published before. It's essentially the opposite of average error as a function of step size. Except, after all these years, they still don't enforce a consistent mapping.

I don't know if anything in the next two issues advanced from there because I can't read them. So maybe somebody's discovered consistency, log-flat badness, or TOP errors.

Graham

>
> Bill Sethares wrote:
>
> > JMM has indeed published the three issues from 2007.
> > Our paper will be in the first issue of 2008, and we have
> > already been through the page proofs stage. Again, the paper
> > on my website is a slightly older version.
> > I would encourage anyone in this group with a strong math
> > background to consider publishing here...

to which Graham replied:

> The trouble is, most of us *don't* have a strong math
> background.

I think there has been a lot of very significant and
original mathematical work done on these lists over the
past several years. For instance:

(1) Monz's work that underlies his tuning software

(2) GWS, Graham, Carl, (and many others) who developed the
ideas behind two-dimensional tuning systems.

(3) Paul Erlich's work as exemplified in the "Middle Path"
paper.

(4) The various measures of fidelity/error/goodness such as as TOP
and consistency and others...

I am no doubt forgetting some important conversations, but the
point is that any of these topics could be presented to the
math/music community and published, if done right.
Many of the people publishing in JMM are not professional
mathematicians, and many are not professional musicians/composers.
Practically no one is both!

Graham continues:

> I've signed on and got my free introductory issue, anyway.
> There's a Douthett and Krantz paper where they go to great
> lengths to measure the errors of equal temperaments. They
> mention something called a "Desirability Function" which
> they've published before. It's essentially the opposite of
> average error as a function of step size. Except, after all
> these years, they still don't enforce a consistent mapping.
>
> I don't know if anything in the next two issues advanced
> from there because I can't read them. So maybe somebody's
> discovered consistency, log-flat badness, or TOP errors.

I'm looking through the table of contents.
Issue 2 has a paper by Carey called "Coherence and sameness in
well-formed pairwise well-formed scales" by Normal Carey.

Thomas Noll has one on: "Musical intervals and special linear
transformations".

In issue 3 the most relevant paper is: Amoit's "David Lewin and
maximally even sets"

So you can see the focus is not directly on microtonal things,
though this is not precluded. Most of the above are about the
structure of scales, and not at the level of tunings, as we talk
about them.

--Bill Sethares

> I am no doubt forgetting some important conversations, but the
> point is that any of these topics could be presented to the
> math/music community and published, if done right.

For starters, I would like to do a review of the best-selling
music set theory from the past 30 years. Pick the most cited
10 papers and find for each one a central, falsifiable
prediction about some simple chord progression. Then tune
the progression in 31-ET and watch 8 or 9 of them fail, under
the assumption that people can listen to the progression and
observe that it sounds fine.

-Carl

Hello Bill~

Wilson's concept of Moments of Symmetry covers much of this ground of scale as differentiated from any particular tuning
first article on this page
http://anaphoria.com/wilson.html

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Bill Sethares wrote:
>
>
>
>
> So you can see the focus is not directly on microtonal things,
> though this is not precluded. Most of the above are about the
> structure of scales, and not at the level of tunings, as we talk
> about them.
>
> --Bill Sethares
>
>

Carl Lumma wrote:
>> I am no doubt forgetting some important conversations, but the
>> point is that any of these topics could be presented to the
>> math/music community and published, if done right.
> > For starters, I would like to do a review of the best-selling
> music set theory from the past 30 years. Pick the most cited
> 10 papers and find for each one a central, falsifiable
> prediction about some simple chord progression. Then tune
> the progression in 31-ET and watch 8 or 9 of them fail, under
> the assumption that people can listen to the progression and
> observe that it sounds fine.

Musical set theory makes falsifiable predictions?

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Carl Lumma wrote:
> >> I am no doubt forgetting some important conversations, but the
> >> point is that any of these topics could be presented to the
> >> math/music community and published, if done right.
> >
> > For starters, I would like to do a review of the best-selling
> > music set theory from the past 30 years. Pick the most cited
> > 10 papers and find for each one a central, falsifiable
> > prediction about some simple chord progression. Then tune
> > the progression in 31-ET and watch 8 or 9 of them fail, under
> > the assumption that people can listen to the progression and
> > observe that it sounds fine.
>
> Musical set theory makes falsifiable predictions?
>

If it does not, we could hardly file it under "science", could we?
If it does make falsifiable predictions, then a test such as Carl
proposes isn't just a sardonic idea, but mandatory.

Making statements which are not falsifiable is just fine for the
arts. Cooking them up along with paradoxes and so on may even be a
vital function of art. But making falsifiable predictions and not
testing them.... there are entertaining names, not all of which are
suitable for a fine family forum such as this, for that kind of thing.

-Cameron Bobro

Bill Sethares wrote:
>> Bill Sethares wrote:
>>
>>> JMM has indeed published the three issues from 2007.
>>> Our paper will be in the first issue of 2008, and we have
>>> already been through the page proofs stage. Again, the paper
>>> on my website is a slightly older version.
>>> I would encourage anyone in this group with a strong math
>>> background to consider publishing here... > > to which Graham replied:
> >> The trouble is, most of us *don't* have a strong math >> background.
> > I think there has been a lot of very significant and
> original mathematical work done on these lists over the > past several years. For instance:
> > (1) Monz's work that underlies his tuning software

As he wants to sell the software maybe he'll try publishing the theory.

> (2) GWS, Graham, Carl, (and many others) who developed the
> ideas behind two-dimensional tuning systems.

Of course you're getting that published now, which is where we started.

I'm trying to write up what I know this year, and it's taking a lot of effort. The most interesting mathematical result so far is

http://x31eq.com/complete.pdf

Which is about the right length for JMM. I don't have any moral objections to packaging it up for them but I haven't done so yet.

> (3) Paul Erlich's work as exemplified in the "Middle Path"
> paper.

And published in the Middle Path paper, which so far only you've cited.

> (4) The various measures of fidelity/error/goodness such as as TOP > and consistency and others...

Consistency was published in Paul Erlich's 22-equal paper from 10 years ago. So far it's collected one -- count it -- one citation. (By Daniel Wolf, according to Google Scholar.)

> I am no doubt forgetting some important conversations, but the
> point is that any of these topics could be presented to the
> math/music community and published, if done right.
> Many of the people publishing in JMM are not professional
> mathematicians, and many are not professional musicians/composers.
> Practically no one is both!

Most of them are one or the other. Some papers are a collaboration between a musician and a mathematician. Some of them must be getting funding. Some of them meet each other at conferences. And yet they don't have time to discuss it here and don't follow our work.

> I'm looking through the table of contents.
> Issue 2 has a paper by Carey called "Coherence and sameness in > well-formed pairwise well-formed scales" by Normal Carey.
> > Thomas Noll has one on: "Musical intervals and special linear > transformations".
> > In issue 3 the most relevant paper is: Amoit's "David Lewin and > maximally even sets"

Sure, but how can I read them? I looked for preprints yesterday and couldn't find any. In the physical sciences I believe it's normal for preprints to remain available and, despite the journals tolerating this, scientists are getting together to found new journals with open access. And yet here's a new journal with closed access and no preprints for the first three issues. (One -- yours -- for the forthcoming issue.) I'm supposed to pay $28 for the Amiot paper. That's affordable but expensive when you add up all the papers I might want to read. Until I buy them I don't know how interesting they'll be. They're really not making it easy for amateurs to join in.

> So you can see the focus is not directly on microtonal things, > though this is not precluded. Most of the above are about the
> structure of scales, and not at the level of tunings, as we talk
> about them.

Complexity is a property of scales rather than tunings. I haven't seen any mention of scalar complexity in the literature -- either for temperament classes or simultaneous linear Diophantine approximations or linear least squares approximations. I'd still like to find a tame mathematician to discuss this with (Gene should know about it but didn't show any interest). It's essentially the area of your button lattices.

Graham

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> > I am no doubt forgetting some important conversations, but the
> > point is that any of these topics could be presented to the
> > math/music community and published, if done right.
>
> For starters, I would like to do a review of the best-selling
> music set theory from the past 30 years. Pick the most cited
> 10 papers and find for each one a central, falsifiable
> prediction about some simple chord progression. Then tune
> the progression in 31-ET and watch 8 or 9 of them fail, under
> the assumption that people can listen to the progression and
> observe that it sounds fine.
>
> -Carl
>

I'm not a great fan of musical set theory, but you're making a
ridiculous request. Set theory doesn't make claims or predictions
about tonal chord progression; rather it is a serious of observed
properties regarding the partitions of a collection of n tones,
ordered or unordered and subject to a number of operations
(transposition, inversion, m5/7 etc). The terminology used is, in
many cases, mathemtically unfortunate with its mixture of set and
group theory due to some historical baggage and terminological
inertia, but the observations are true. There may well be questions
about whether they are trivially true and, morever, for the utility
of this for real music, but that's a question that can only be
settled by composers and listeners. In any case, they are not
predictions about chord progressions.

A more interesting question might be found in some other area of
contemporary music theory, for example, in neo-Riemannian voice
leadings. What happens when they are mapped to a tuning with a metric
substantially different from 12tet? Again, there are no predictions
about chord progressions made by this, but one might extrapolate some
predictions about the fit between voice leadings and tunings. Could
these in turn become a set of rules or even predictions for, example,
the most effective mapping of a piece of music from one temperament
to another? I suspect so, but the intermediate steps are probably
quite delicate, and the end result probably a matter of taste.

Which, of course, brings us back to an essential problem with the
term "music theory". Generally speaking, most of what is taught as
"music theory" is not theory in the sense that a scientist would
recognize (i.e. falsifiability) but rather the practice of real music
making, and at a somewhat more abstract level, the description of
that practice. Questions about which possible description is more or
less accurate or efficient and work in more conventional scientific
domains (like the psychophysics or neuroscience of music) are indeed
of "real" theoretical interest, but music itself is an aesthetic
domain and perhaps it's more correct to identify composition as the
falsifiable experimental domain for music with speculative (or
precompositional) music theory as its theoretical background.

The question of whether a body of speculative theory -- be it set
theory or neo-Riemannian voice leading or tuning theory -- has
predictive power, to the extent that it predicts that a particular
set of materials will be musically useful, has to be settled
compositionally. But is anything compositional ever settled to
universal satisfaction? I hope not...

Daniel Wolf

> Complexity is a property of scales rather than tunings. I
> haven't seen any mention of scalar complexity in the
> literature -- either for temperament classes or simultaneous
> linear Diophantine approximations or linear least squares
> approximations. I'd still like to find a tame mathematician
> to discuss this with (Gene should know about it but didn't
> show any interest). It's essentially the area of your
> button lattices.
>
> Graham

Incidentally, the Tymoczko paper I pointed to recently shows
an image of "Erlich's unpublished continuous" triad plot (from
the harmonic entropy list) without further comment. However,
there's nothing of Erv's that I can see. How anyone can write
a paper on geometrical in music theory and not cite Wilson is
beyond me.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> > Complexity is a property of scales rather than tunings. I
> > haven't seen any mention of scalar complexity in the
> > literature -- either for temperament classes or simultaneous
> > linear Diophantine approximations or linear least squares
> > approximations. I'd still like to find a tame mathematician
> > to discuss this with (Gene should know about it but didn't
> > show any interest). It's essentially the area of your
> > button lattices.
> >
> > Graham
>
> Incidentally, the Tymoczko paper I pointed to recently shows
> an image of "Erlich's unpublished continuous" triad plot (from
> the harmonic entropy list) without further comment. However,
> there's nothing of Erv's that I can see. How anyone can write
> a paper on geometrical in music theory and not cite Wilson is
> beyond me.
>
> -Carl
>

And there will always be some troublemaker in the back row muttering
"Schillinger..."

Oops, that would be me.

-Cameron Bobro

--- In tuning@yahoogroups.com, "djwolf_frankfurt" <djwolf@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > > I am no doubt forgetting some important conversations, but the
> > > point is that any of these topics could be presented to the
> > > math/music community and published, if done right.
> >
> > For starters, I would like to do a review of the best-selling
> > music set theory from the past 30 years. Pick the most cited
> > 10 papers and find for each one a central, falsifiable
> > prediction about some simple chord progression. Then tune
> > the progression in 31-ET and watch 8 or 9 of them fail, under
> > the assumption that people can listen to the progression and
> > observe that it sounds fine.
> >
> > -Carl
>
> I'm not a great fan of musical set theory, but you're making a
> ridiculous request. Set theory doesn't make claims or predictions
> about tonal chord progression; rather it is a serious of observed
> properties regarding the partitions of a collection of n tones,
> ordered or unordered and subject to a number of operations
> (transposition, inversion, m5/7 etc).

I was using the term "set theory" very broadly there. I just
mean any of the rubbish music theory I've seen, from Agmon to
Tymoczko.

> In any case, they are not
> predictions about chord progressions.

Tymoczko makes predictions about chord progressions, and I've
seen dozens of papers purporting to explain why the diatonic
scale is so popular, or why V-I cadences are so good, based on
some tiny melodic factoid that's usually based on some
inaudible group transformation that usually doesn't work
outside of 12-ET. All of these papers involve strong implicit
(and often explicit) assumptions or predictions. "Set theory"
is somehow the perfect term for the lot of it. You know, it's
nice to mind one's own business, but at some point we just need
a Richard Dawkins to call shenanigans on the whole affair.

-Carl

--- In tuning@yahoogroups.com, "djwolf_frankfurt" <djwolf@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > > I am no doubt forgetting some important conversations, but the
> > > point is that any of these topics could be presented to the
> > > math/music community and published, if done right.
> >
> > For starters, I would like to do a review of the best-selling
> > music set theory from the past 30 years. Pick the most cited
> > 10 papers and find for each one a central, falsifiable
> > prediction about some simple chord progression. Then tune
> > the progression in 31-ET and watch 8 or 9 of them fail, under
> > the assumption that people can listen to the progression and
> > observe that it sounds fine.
> >
> > -Carl
> >
>
> I'm not a great fan of musical set theory, but you're making a
> ridiculous request.

Let's see... "sciencemag.org" is where the article in the spotlight
shows up... hey Carl, I didn't realize that it was you who gave a
musical set theory article the green light at the "science" turnpike!
Are you an editor? Maybe my monograph "Firm Buttocks and Epaulettes:
Transformational Invariance and Incarceration in "our German music"
would be of interest for your next issue!

> Set theory doesn't make claims or predictions
> about tonal chord progression; rather it is a serious of observed
> properties regarding the partitions of a collection of n tones,
> ordered or unordered and subject to a number of operations
> (transposition, inversion, m5/7 etc).
>The terminology used is, in
> many cases, mathemtically unfortunate with its mixture of set and
> group theory due to some historical baggage and terminological
> inertia, but the observations are true. There may well be questions
> about whether they are trivially true and, morever, for the utility
> of this for real music, but that's a question that can only be
> settled by composers and listeners. In any case, they are not
> predictions about chord progressions.
>
> A more interesting question might be found in some other area of
> contemporary music theory, for example, in neo-Riemannian voice
> leadings. What happens when they are mapped to a tuning with a
>metric
> substantially different from 12tet? Again, there are no predictions
> about chord progressions made by this, but one might extrapolate
some
> predictions about the fit between voice leadings and tunings.
>Could
> these in turn become a set of rules or even predictions for,
>example,
> the most effective mapping of a piece of music from one temperament
> to another? I suspect so, but the intermediate steps are probably
> quite delicate, and the end result probably a matter of taste.
>
> Which, of course, brings us back to an essential problem with the
> term "music theory". Generally speaking, most of what is taught as
> "music theory" is not theory in the sense that a scientist would
> recognize (i.e. falsifiability) but rather the practice of real
>music
> making, and at a somewhat more abstract level, the description of
> that practice. Questions about which possible description is more
>or
> less accurate or efficient and work in more conventional scientific
> domains (like the psychophysics or neuroscience of music) are
indeed
> of "real" theoretical interest, but music itself is an aesthetic
> domain and perhaps it's more correct to identify composition as the
> falsifiable experimental domain for music with speculative (or
> precompositional) music theory as its theoretical background.
>
> The question of whether a body of speculative theory -- be it set
> theory or neo-Riemannian voice leading or tuning theory -- has
> predictive power, to the extent that it predicts that a particular
> set of materials will be musically useful, has to be settled
> compositionally. But is anything compositional ever settled to
> universal satisfaction? I hope not...
>
> Daniel Wolf
>

I just realized that I might still have Forte in a box somewhere,
LOL.

Anyway, it seems to me that if a series of observations about a
manufacturing process consists of measurements and inventories which
cannot be run backwards, so to speak, or implemented as a blueprint,
in order to create the same manufacturing process, it is patently
trivial.

But a more accurate description would be "cargo cult".

At any rate, if you peek backstage, behind the sets of "set theory",
you'll find Hanson, which I think would be interesting to many here
because his "sets" are ordered according to consonance and the
harmonic series. From this kind of starting point, and given some
simple "givens", falsifiable, non-trivial, and musically audible
observations and experiments could be conducted in the field of
"microtonality", I believe. In fact Graham Breed, here, gave an
example of this kind of thing a few weeks ago. (Non-trivial things
that are true about a harmony found in the lower harmonics,
transposed to the nearest higher point in the harmonic series where
they are still true).

-Cameron Bobro

Tymoczko has referred to Wilson as an amateur.
where as i pointed out when he was on the list he failed to deal with 4 voice harmony which he wants to ignore so he can impress people which as far as i can tell is the sole purpose of his selfish work.
the CPS work is almost a half century old, it has been used by numerous composers in important concerts. instruments have been built in his tunings. Who in the world has ever applied a single idea of this charlatan to any real music. If one can find something to apply. Mediant progressions by steps, i can barely stay awake.
/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Carl Lumma wrote:
>
> > Complexity is a property of scales rather than tunings. I
> > haven't seen any mention of scalar complexity in the
> > literature -- either for temperament classes or simultaneous
> > linear Diophantine approximations or linear least squares
> > approximations. I'd still like to find a tame mathematician
> > to discuss this with (Gene should know about it but didn't
> > show any interest). It's essentially the area of your
> > button lattices.
> >
> > Graham
>
> Incidentally, the Tymoczko paper I pointed to recently shows
> an image of "Erlich's unpublished continuous" triad plot (from
> the harmonic entropy list) without further comment. However,
> there's nothing of Erv's that I can see. How anyone can write
> a paper on geometrical in music theory and not cite Wilson is
> beyond me.
>
> -Carl
>
>

I would be interested in this :)

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Cameron Bobro wrote:
>
> Maybe my monograph "Firm Buttocks and Epaulettes:
> Transformational Invariance and Incarceration in "our German music"
> would be of interest for your next issue!
>
>
>
>

Hi Kraig -- of course -- our paper references Wilson's letter (and
indeed, your website's hosting of the letter) as one of the initial
pieces of this puzzle. It wasn't in the list of my previous email
because it wasn't developped as part of (or as a result of) the
tuning forum's discussions...

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> Hello Bill~
>
> Wilson's concept of Moments of Symmetry covers much of this ground
of
> scale as differentiated from any particular tuning
> first article on this page
> http://anaphoria.com/wilson.html
>
>
> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> _'''''''_ ^North/Western Hemisphere:
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
>
> _'''''''_ ^South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria <http://
anaphoriasouth.blogspot.com/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
>
>
>
> Bill Sethares wrote:
> >
> >
> >
> >
> > So you can see the focus is not directly on microtonal things,
> > though this is not precluded. Most of the above are about the
> > structure of scales, and not at the level of tunings, as we talk
> > about them.
> >
> > --Bill Sethares
> >
> >
>

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> I'm trying to write up what I know this year, and it's
> taking a lot of effort. The most interesting mathematical
> result so far is
>
> http://x31eq.com/complete.pdf
>
> Which is about the right length for JMM. I don't have any
> moral objections to packaging it up for them but I haven't
> done so yet.

This is great, Graham. I think with a few formatting (and other
superficial) changes, you'll be good to go. I particularly like the
way you relate the various measures and the concrete bounds that you
derive.

> > (3) Paul Erlich's work as exemplified in the "Middle Path"
> > paper.
>
> And published in the Middle Path paper, which so far only
> you've cited.

The publication date is 2006 on this paper, and Xenharmonikon is not
a mainstream peer reviewed journal. I would anticipate Paul's paper
will become better referenced (like Wilson's letter) as it becomes
better known.
>
> > I'm looking through the table of contents.
> > Issue 2 has a paper by Carey called "Coherence and sameness in
> > well-formed pairwise well-formed scales" by Normal Carey.
> >
> > Thomas Noll has one on: "Musical intervals and special linear
> > transformations".
> >
> > In issue 3 the most relevant paper is: Amoit's "David Lewin and
> > maximally even sets"
>
> Sure, but how can I read them? I looked for preprints
> yesterday and couldn't find any. In the physical sciences I
> believe it's normal for preprints to remain available and,
> despite the journals tolerating this, scientists are getting
> together to found new journals with open access. And yet
> here's a new journal with closed access and no preprints for
> the first three issues. (One -- yours -- for the
> forthcoming issue.) I'm supposed to pay $28 for the Amiot
> paper. That's affordable but expensive when you add up all
> the papers I might want to read. Until I buy them I don't
> know how interesting they'll be. They're really not making
> it easy for amateurs to join in.

I would email the authors and ask for a copy. While the journal
forbids authors to put final copies up on the web for public access,
there's no way they can forbid an author sending a paper or email
copy to a colleague.

--Bill

Dear Carl,

[Please let me know if this post if off-topic in this list. This post is not about tunings as such, but about creating chord progressions in non-12 ET tunings. If this is off-topic here, is there perhaps some forum where such subjects are more suitable? Thank you.]

could you be more specific about your mentioned predictions on chord progressions, both too-restricted ones which only work for 12 ET and possibly also more general ones?

Here is a suggestion I can make for guidelines on good chord progressions, which is indeed based on the notion of pitch class sets, but which also works beyond 12 ET (e.g., I did examples using them with 31 ET). This suggestion is based on Schoenberg's guidelines on good chord root progressions where he introduces the notion of ascending (strong), descending ('weak') and superstrong progressions (see the respective chapter in his Theory of Harmony). The summary of Schoenberg's guidelines/rules is based on 12 ET, but his explanation is actually more general. A formalisation of his explanations (instead of his actual rules) does work beyond 12 ET.

The main difference between Schoenberg's actual guidelines and their formalised generalisation is that Schoenberg's guidelines are based on scale degree intervals between chord roots, whereas the generalisation exploits whether the root pitch class of some chord is contained in the pitch class set of another chord. OK, it follows the core of is the formalisation (I hope the notation explains itself: defined are Boolean functions expecting two neighbouring chords).

/* Chord1 and Chord2 have common pitch classes, but the root of Chord2 does not occur in the set of Chord1's pitchclasses. */

isAscendingProgression(Chord1, Chord2) :=
( NOT ( getRoot(Chord2) \in getPitchClasses(Chord1) ) )
AND ( intersection(getPitchClasses(Chord1), getPitchClasses(Chord2)) \= emptySet )

/* A non-root pitchclass of Chord1 is root in Chord2. */

isDescendingProgression(Chord1, Chord2) :=
( getRoot(Chord2) \in getPitchClasses(Chord1) )
AND ( getRoot(Chord1) \= getRoot(Chord2) )

/* Chord1 and Chord2 have no common pitch classes */

isSuperstrongProgression(Chord1, Chord2) :=
intersection(getPitchClasses(Chord1), getPitchClasses(Chord2)) = emptySet

If you are interested I can post further formalisations/generalisations of Schoenberg's rules. For example, Schoenberg recommends that descending progressions should "resolve". For any three successive chords, if the first two chords form a descending progression, then the progression from the first to the third chord should form a strong or superstrong progression (so the middle chord is quasi a 'passing chord'). Also, Schoenberg observes that the more common pitches two neighbouring chords share the more "weak" this progression is. Based on that idea I defined a progression strength measurement which combines the definitions above and the cardinality of pitch class set intersections.

Anyway, for chord progressions of diatonic triads in major, the generalised formalisation and Schoenberg's rules are equivalent (e.g., the function isAscendingProgression above returns true for progressions which Schoenberg calls ascending). Still, the behaviour of the constraints and Schoenberg's rules differ for more complex cases. According to Schoenberg, a progression is superstrong if the root interval is a step up or down. For example, the progression V7 IV is superstrong according to Schoenberg. For the definitions above, however, this progression is descending (!), because the root of IV is contained in V7 (e.g. in G7 F, the F's root pitchclass f is already contained in G7). Indeed, this progression is rare in music. By contrast, the progression I IIIb (e.g., C Eb) is a descending progression in Schoenbergs original definition. For the definitions above, however, this is an ascending progression (the root of Es is not contained in C), and indeed for me the progression feels strong.

As I said, I used these definitions for generating chord progressions beyond 12 ET. As an example, I will sketch a simple rule set on a chord progression in 31 ET. Let us create a sequence of 5 chords and allow only for diatonic triads in C-major. The first and last chord should be the tonic. Following Schoenberg let us specify that ascending progressions are "resolved" (see above), and we don't allow for superstrong progressions. Finally, the union of the last three chords must contain all pitch classes of C-major (i.e. these chords form a cadence).

This problem has only two solutions (I used the computer :). These solutions are as follows. They happen to contain only ascending progressions. Such solutions are particularly convincing. For simplicity I given absolute chord names.

C-maj F-maj D-min G-maj C-maj (only ascending progressions)
C-maj A-min D-min G-maj C-maj (only ascending progressions)

If instead of C-major we use the natural C-minor scale (and don't change anything else), there are only three solutions. Interestingly, these result in plagal cadences. Plagal cadences contain descending ("weak") progressions, but they are all "resolved" by superstrong progressions. There exists no purely ascending progressions which form a cadence (where the union of chord pitch classes contains all scale notes).

C-min G-min Bb-maj F-maj C-min
C-min F-min Bb-maj F-min C-min
C-min Eb-maj Bb-maj F-min C-min

Finally, here are the three only solutions in harmonic C-minor. The subdim chord here is 5:6:7 (root is 5), transposed to the specified root. Note that a progression F-dim G-maj would be descending according to Schoenberg (root progresses stepwise), but according to the definition above F-dim G-maj (and the F-subdim G-maj here) are strong progressions (the two chords share the pitch class B, but the root G is not contained in the former chord).

C-min Ab-maj F-subdim G-maj C-min (only ascending progressions)
C-min Eb-aug F-subdim G-maj C-min
C-min G-maj F-subdim G-maj C-min

Perhaps I should repeat that I did all this in 31 ET. These definitions work in principle with any pitch class sets, including just intonation where pitch class sets are ratios.

Comments are appreciated. Thank you!

Best
Torsten

PS: I use the terms "guidelines" (and sometimes "rules") instead of "predictions", because I feel the latter is too strong (as others pointed out already here). I don't believe that there exist any strict laws concerning harmony or composition in general :)

PPS: Shameless self-promotion: the examples above are part of my composition system Strasheela (in examples/HarmonicProgressions-31ET.oz the first example).

On Apr 22, 2008, at 9:34 AM, Carl Lumma wrote:
--- In tuning@yahoogroups.com, "djwolf_frankfurt" <djwolf@...> wrote:
> > --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> > >
> > > > I am no doubt forgetting some important conversations, but the
> > > > point is that any of these topics could be presented to the
> > > > math/music community and published, if done right.
> > >
> > > For starters, I would like to do a review of the best-selling
> > > music set theory from the past 30 years. Pick the most cited
> > > 10 papers and find for each one a central, falsifiable
> > > prediction about some simple chord progression. Then tune
> > > the progression in 31-ET and watch 8 or 9 of them fail, under
> > > the assumption that people can listen to the progression and
> > > observe that it sounds fine.
> > >
> > > -Carl
> >
> > I'm not a great fan of musical set theory, but you're making a
> > ridiculous request. Set theory doesn't make claims or predictions
> > about tonal chord progression; rather it is a serious of observed
> > properties regarding the partitions of a collection of n tones,
> > ordered or unordered and subject to a number of operations
> > (transposition, inversion, m5/7 etc).
>
> I was using the term "set theory" very broadly there. I just
> mean any of the rubbish music theory I've seen, from Agmon to
> Tymoczko.
>
> > In any case, they are not
> > predictions about chord progressions.
>
> Tymoczko makes predictions about chord progressions, and I've
> seen dozens of papers purporting to explain why the diatonic
> scale is so popular, or why V-I cadences are so good, based on
> some tiny melodic factoid that's usually based on some
> inaudible group transformation that usually doesn't work
> outside of 12-ET. All of these papers involve strong implicit
> (and often explicit) assumptions or predictions. "Set theory"
> is somehow the perfect term for the lot of it. You know, it's
> nice to mind one's own business, but at some point we just need
> a Richard Dawkins to call shenanigans on the whole affair.
>
> -Carl
>

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

Hi Torsten,

> could you be more specific about your mentioned predictions
> on chord progressions, both too-restricted ones which only
> work for 12 ET and possibly also more general ones?

I will have to cop out with an excuse: I don't have time
at the moment. I do not archive the bad music theory papers
I've read over the years.

> Here is a suggestion I can make for guidelines on good chord
> progressions, which is indeed based on the notion of pitch class
> sets, but which also works beyond 12 ET (e.g., I did examples
> using them with 31 ET). This suggestion is based on Schoenberg's
> guidelines on good chord root progressions where he introduces
> the notion of ascending (strong), descending ('weak') and
> superstrong progressions (see the respective chapter in his
> Theory of Harmony). The summary of Schoenberg's guidelines/rules
> is based on 12 ET, but his explanation is actually more general.
> A formalisation of his explanations (instead of his actual rules)
> does work beyond 12 ET.
>
> The main difference between Schoenberg's actual guidelines and
> their formalised generalisation is that Schoenberg's guidelines
> are based on scale degree intervals between chord roots, whereas
> the generalisation exploits whether the root pitch class of some
> chord is contained in the pitch class set of another chord.

I haven't read Schoenberg's book but it sounds like you've
done a bit of work for him. Yes, I could probably generalize
many (though not all) of the "bad music theory papers" if I
worked at it. And some such minimal effort would only be
fair in the meta-analysis I proposed. Where to draw the line
is another question. Did Schoenberg say the important thing
is whether the root is a member of the new chord (which is
a reasonable thing)? Or did he say something else?

> OK, it follows the
> core of is the formalisation (I hope the notation explains itself:
> defined are Boolean functions expecting two neighbouring chords).
>
> /* Chord1 and Chord2 have common pitch classes, but the root of
> Chord2 does not occur in the set of Chord1's pitchclasses. */
>
> isAscendingProgression(Chord1, Chord2) :=
> ( NOT ( getRoot(Chord2) \in getPitchClasses(Chord1) ) )
> AND ( intersection(getPitchClasses(Chord1), getPitchClasses
> (Chord2)) \= emptySet )
>
> /* A non-root pitchclass of Chord1 is root in Chord2. */
>
> isDescendingProgression(Chord1, Chord2) :=
> ( getRoot(Chord2) \in getPitchClasses(Chord1) )
> AND ( getRoot(Chord1) \= getRoot(Chord2) )
>
> /* Chord1 and Chord2 have no common pitch classes */
>
> isSuperstrongProgression(Chord1, Chord2) :=
> intersection(getPitchClasses(Chord1), getPitchClasses(Chord2)) =
> emptySet

Yes, perfectly understandable, though I find the terms
"ascending" and "descending" extremely odd here.

> If you are interested I can post further formalisations/
> generalisations of Schoenberg's rules.

I would be interested to read them, especially if you
show how you generalized them. And if monz is reading,
I bet he'd be interested, too.

> For example, Schoenberg
> recommends that descending progressions should "resolve". For any
> three successive chords, if the first two chords form a descending
> progression, then the progression from the first to the third chord
> should form a strong or superstrong progression (so the middle chord
> is quasi a 'passing chord').

Hm... this seems kindof strange.

> Also, Schoenberg observes that the more
> common pitches two neighbouring chords share the more "weak" this
> progression is. Based on that idea I defined a progression strength
> measurement which combines the definitions above and the cardinality
> of pitch class set intersections.

That makes sense.

> Anyway, for chord progressions of diatonic triads in major, the
> generalised formalisation and Schoenberg's rules are equivalent
> (e.g., the function isAscendingProgression above returns true for
> progressions which Schoenberg calls ascending). Still, the behaviour
> of the constraints and Schoenberg's rules differ for more complex
> cases. According to Schoenberg, a progression is superstrong if the
> root interval is a step up or down.

Bzzz. :)

> For example, the progression V7 IV is superstrong according to
> Schoenberg. For the definitions above, however, this progression
> is descending (!), because the root of IV is contained in V7
> (e.g. in G7 F, the F's root pitchclass f is already contained
> in G7). Indeed, this progression is rare in music.
> By contrast, the progression I IIIb (e.g., C Eb) is a descending
> progression in Schoenbergs original definition. For the
> definitions above, however, this is an ascending progression
> (the root of Es is not contained in C), and indeed for me the
> progression feels strong.

Sounds like we are really dealing with the Anders theory
of chord progressions. Which is probably a blessing.

> Let us create a sequence of 5 chords and
> allow only for diatonic triads in C-major. The first and last
> chord should be the tonic. Following Schoenberg let us specify
> that ascending progressions are "resolved" (see above), and we
> don't allow for superstrong progressions. Finally, the union
> of the last three chords must contain all pitch classes of
> C-major (i.e. these chords form a cadence).
>
> This problem has only two solutions (I used the computer :).
> These solutions are as follows. They happen to contain only
> ascending progressions. Such solutions are particularly
> convincing. For simplicity I given absolute chord names.
>
> C-maj F-maj D-min G-maj C-maj (only ascending progressions)
> C-maj A-min D-min G-maj C-maj (only ascending progressions)

Only allowing for diatonic triads is C-major is a very
helpful constraint. Howabout we allow all 7-limit triads
in 31-ET, for a start?

-Carl

Dear Carl,

On Apr 22, 2008, at 7:19 PM, Carl Lumma wrote:
> > Here is a suggestion I can make for guidelines on good chord
> > progressions, which is indeed based on the notion of pitch class
> > sets, but which also works beyond 12 ET (e.g., I did examples
> > using them with 31 ET). This suggestion is based on Schoenberg's
> > guidelines on good chord root progressions where he introduces
> > the notion of ascending (strong), descending ('weak') and
> > superstrong progressions (see the respective chapter in his
> > Theory of Harmony). The summary of Schoenberg's guidelines/rules
> > is based on 12 ET, but his explanation is actually more general.
> > A formalisation of his explanations (instead of his actual rules)
> > does work beyond 12 ET.
> >
> > The main difference between Schoenberg's actual guidelines and
> > their formalised generalisation is that Schoenberg's guidelines
> > are based on scale degree intervals between chord roots, whereas
> > the generalisation exploits whether the root pitch class of some
> > chord is contained in the pitch class set of another chord.
>
> I haven't read Schoenberg's book but it sounds like you've
> done a bit of work for him. Yes, I could probably generalize
> many (though not all) of the "bad music theory papers" if I
> worked at it. And some such minimal effort would only be
> fair in the meta-analysis I proposed. Where to draw the line
> is another question.
>

> Did Schoenberg say the important thing
> is whether the root is a member of the new chord (which is
> a reasonable thing)? Or did he say something else?
>

Sorry, I don't fully understand your question. In an ascending progression, the root of the preceding chord is a pitch in the following chord. Did you mean that?

> > OK, it follows the
> > core of is the formalisation (I hope the notation explains itself:
> > defined are Boolean functions expecting two neighbouring chords).
> >
> > /* Chord1 and Chord2 have common pitch classes, but the root of
> > Chord2 does not occur in the set of Chord1's pitchclasses. */
> >
> > isAscendingProgression(Chord1, Chord2) :=
> > ( NOT ( getRoot(Chord2) \in getPitchClasses(Chord1) ) )
> > AND ( intersection(getPitchClasses(Chord1), getPitchClasses
> > (Chord2)) \= emptySet )
> >
> > /* A non-root pitchclass of Chord1 is root in Chord2. */
> >
> > isDescendingProgression(Chord1, Chord2) :=
> > ( getRoot(Chord2) \in getPitchClasses(Chord1) )
> > AND ( getRoot(Chord1) \= getRoot(Chord2) )
> >
> > /* Chord1 and Chord2 have no common pitch classes */
> >
> > isSuperstrongProgression(Chord1, Chord2) :=
> > intersection(getPitchClasses(Chord1), getPitchClasses(Chord2)) =
> > emptySet
>
> Yes, perfectly understandable, though I find the terms
> "ascending" and "descending" extremely odd here.
>
These are Schoenberg's terms. He mentions that H. Schenker uses these terms as well ("aufsteigend" and "fallend") though in opposite meaning. He remarks something like that anyone who loves Brahmsian harmony would likely come up with similar concepts.

> > If you are interested I can post further formalisations/
> > generalisations of Schoenberg's rules.
>
> I would be interested to read them, especially if you
> show how you generalized them. And if monz is reading,
> I bet he'd be interested, too.
>
I'll write a new mail soon..

> > For example, Schoenberg
> > recommends that descending progressions should "resolve". For any
> > three successive chords, if the first two chords form a descending
> > progression, then the progression from the first to the third chord
> > should form a strong or superstrong progression (so the middle chord
> > is quasi a 'passing chord').
>
> Hm... this seems kindof strange.
>
Schoenberg is always very exhaustive :) He does not want to ban descending progressions, so he looks for a way how he can allow for them in a way which is musically convincing in most cases. So, he comes up with the idea of a 'passing chord'.

Please note that he makes is always very clear that his rules are guidelines for the pupil, but no laws for masterworks. This is why he prefers the term "descending" instead of "weak" progressions (he uses the term "strong" for "ascending" progressions a lot). Still, he does not discuss where descending progressions are used to good purpose in any masterwork, but such considerations have hardly a place anyway in his systematic approach.

> > Anyway, for chord progressions of diatonic triads in major, the
> > generalised formalisation and Schoenberg's rules are equivalent
> > (e.g., the function isAscendingProgression above returns true for
> > progressions which Schoenberg calls ascending). Still, the behaviour
> > of the constraints and Schoenberg's rules differ for more complex
> > cases. According to Schoenberg, a progression is superstrong if the
> > root interval is a step up or down.
>
> Bzzz. :)
>
> > For example, the progression V7 IV is superstrong according to
> > Schoenberg. For the definitions above, however, this progression
> > is descending (!), because the root of IV is contained in V7
> > (e.g. in G7 F, the F's root pitchclass f is already contained
> > in G7). Indeed, this progression is rare in music.
> > By contrast, the progression I IIIb (e.g., C Eb) is a descending
> > progression in Schoenbergs original definition. For the
> > definitions above, however, this is an ascending progression
> > (the root of Es is not contained in C), and indeed for me the
> > progression feels strong.
>
> Sounds like we are really dealing with the Anders theory
> of chord progressions. Which is probably a blessing.
>
:) As I said, I simply implemented Schoenberg's rule explanation instead of his actual rules. So, I cannot accept the honour you are implying.

> > Let us create a sequence of 5 chords and
> > allow only for diatonic triads in C-major. The first and last
> > chord should be the tonic. Following Schoenberg let us specify
> > that ascending progressions are "resolved" (see above), and we
> > don't allow for superstrong progressions. Finally, the union
> > of the last three chords must contain all pitch classes of
> > C-major (i.e. these chords form a cadence).
> >
> > This problem has only two solutions (I used the computer :).
> > These solutions are as follows. They happen to contain only
> > ascending progressions. Such solutions are particularly
> > convincing. For simplicity I given absolute chord names.
> >
> > C-maj F-maj D-min G-maj C-maj (only ascending progressions)
> > C-maj A-min D-min G-maj C-maj (only ascending progressions)
>
> Only allowing for diatonic triads is C-major is a very
> helpful constraint. Howabout we allow all 7-limit triads
> in 31-ET, for a start?
>

The point of these generalised definitions is exactly that they allow for such cases :) I only simplified for the sake of an argument. So, here is a 7-limit variant. Let the scale be septimal natural minor (Scala term, in 31 ET the pitches C D D# F G G# A#). Let us allow for the following septimal triads (please correct my terminology if necessary): harmonic diminished (5/5 6/5 7/5), subharmonic diminished (7/7 7/6 7/5), subminor (6/1 7/1 9/1), supermajor (1/9 1/7 1/6). Note that I consider all these chords as consonances for simplicity which require neither preparation nor resolution. Naturally, we could define extra rules which care for that.

There are 2 solutions of 5 chords with this scale, these chords and the rule set specified before. For convenience, I attached these solutions as PDF files (please let me know if doing so violates some policies of this list).

The solutions are shown by two staffs. The upper stuff shows the actual chord pitches (I didn't bother to implement voice leading rules here, so things like Bruckner's law of the shortest path are violated). The staff notation is in 31 ET (i.e. the interval C D# is 7/6). The lower staff shows the chord roots in staff notation plus there ratios expressed as a product of the untransposed chord ratios and a root factor for each chord ratio. Some of these factors may look odd at first, e.g., the first chord has the factor 4/3 for the root C, but this is due to that fact that the untransposed chord ratios don't contain 1/1 -- the resulting product ratios do contain 1/1.

If you are interested in other chord sets, scales, or longer solutions then just say so -- its easy to change some numbers in a computer program :)

Best
Torsten

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

What i remember which goes quite far back in time and my copy is still unpacked is strong progressions are where more important note/s become less and vice versa. i have actually played with this in structures such as the eikosany. Anyway it seems if you allow the 7th and 9th harmonics in chords, even triads using them, it seems one would have further progressions to choose from although this would quickly lead outside of any particular scale. One could also use the 11th.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Torsten Anders wrote:
>
> Dear Carl,
>
> On Apr 22, 2008, at 7:19 PM, Carl Lumma wrote:
> > > Here is a suggestion I can make for guidelines on good chord
> > > progressions, which is indeed based on the notion of pitch class
> > > sets, but which also works beyond 12 ET (e.g., I did examples
> > > using them with 31 ET). This suggestion is based on Schoenberg's
> > > guidelines on good chord root progressions where he introduces
> > > the notion of ascending (strong), descending ('weak') and
> > > superstrong progressions (see the respective chapter in his
> > > Theory of Harmony). The summary of Schoenberg's guidelines/rules
> > > is based on 12 ET, but his explanation is actually more general.
> > > A formalisation of his explanations (instead of his actual rules)
> > > does work beyond 12 ET.
> > >
> > > The main difference between Schoenberg's actual guidelines and
> > > their formalised generalisation is that Schoenberg's guidelines
> > > are based on scale degree intervals between chord roots, whereas
> > > the generalisation exploits whether the root pitch class of some
> > > chord is contained in the pitch class set of another chord.
> >
> > I haven't read Schoenberg's book but it sounds like you've
> > done a bit of work for him. Yes, I could probably generalize
> > many (though not all) of the "bad music theory papers" if I
> > worked at it. And some such minimal effort would only be
> > fair in the meta-analysis I proposed. Where to draw the line
> > is another question.
> >
>
> > Did Schoenberg say the important thing
> > is whether the root is a member of the new chord (which is
> > a reasonable thing)? Or did he say something else?
> >
>
> Sorry, I don't fully understand your question. In an ascending
> progression, the root of the preceding chord is a pitch in the
> following chord. Did you mean that?
>
> > > OK, it follows the
> > > core of is the formalisation (I hope the notation explains itself:
> > > defined are Boolean functions expecting two neighbouring chords).
> > >
> > > /* Chord1 and Chord2 have common pitch classes, but the root of
> > > Chord2 does not occur in the set of Chord1's pitchclasses. */
> > >
> > > isAscendingProgression(Chord1, Chord2) :=
> > > ( NOT ( getRoot(Chord2) \in getPitchClasses(Chord1) ) )
> > > AND ( intersection(getPitchClasses(Chord1), getPitchClasses
> > > (Chord2)) \= emptySet )
> > >
> > > /* A non-root pitchclass of Chord1 is root in Chord2. */
> > >
> > > isDescendingProgression(Chord1, Chord2) :=
> > > ( getRoot(Chord2) \in getPitchClasses(Chord1) )
> > > AND ( getRoot(Chord1) \= getRoot(Chord2) )
> > >
> > > /* Chord1 and Chord2 have no common pitch classes */
> > >
> > > isSuperstrongProgression(Chord1, Chord2) :=
> > > intersection(getPitchClasses(Chord1), getPitchClasses(Chord2)) =
> > > emptySet
> >
> > Yes, perfectly understandable, though I find the terms
> > "ascending" and "descending" extremely odd here.
> >
> These are Schoenberg's terms. He mentions that H. Schenker uses these
> terms as well ("aufsteigend" and "fallend") though in opposite
> meaning. He remarks something like that anyone who loves Brahmsian
> harmony would likely come up with similar concepts.
>
> > > If you are interested I can post further formalisations/
> > > generalisations of Schoenberg's rules.
> >
> > I would be interested to read them, especially if you
> > show how you generalized them. And if monz is reading,
> > I bet he'd be interested, too.
> >
> I'll write a new mail soon..
>
> > > For example, Schoenberg
> > > recommends that descending progressions should "resolve". For any
> > > three successive chords, if the first two chords form a descending
> > > progression, then the progression from the first to the third chord
> > > should form a strong or superstrong progression (so the middle chord
> > > is quasi a 'passing chord').
> >
> > Hm... this seems kindof strange.
> >
> Schoenberg is always very exhaustive :) He does not want to ban
> descending progressions, so he looks for a way how he can allow for
> them in a way which is musically convincing in most cases. So, he
> comes up with the idea of a 'passing chord'.
>
> Please note that he makes is always very clear that his rules are
> guidelines for the pupil, but no laws for masterworks. This is why he
> prefers the term "descending" instead of "weak" progressions (he uses
> the term "strong" for "ascending" progressions a lot). Still, he does
> not discuss where descending progressions are used to good purpose in
> any masterwork, but such considerations have hardly a place anyway in
> his systematic approach.
>
> > > Anyway, for chord progressions of diatonic triads in major, the
> > > generalised formalisation and Schoenberg's rules are equivalent
> > > (e.g., the function isAscendingProgression above returns true for
> > > progressions which Schoenberg calls ascending). Still, the behaviour
> > > of the constraints and Schoenberg's rules differ for more complex
> > > cases. According to Schoenberg, a progression is superstrong if the
> > > root interval is a step up or down.
> >
> > Bzzz. :)
> >
> > > For example, the progression V7 IV is superstrong according to
> > > Schoenberg. For the definitions above, however, this progression
> > > is descending (!), because the root of IV is contained in V7
> > > (e.g. in G7 F, the F's root pitchclass f is already contained
> > > in G7). Indeed, this progression is rare in music.
> > > By contrast, the progression I IIIb (e.g., C Eb) is a descending
> > > progression in Schoenbergs original definition. For the
> > > definitions above, however, this is an ascending progression
> > > (the root of Es is not contained in C), and indeed for me the
> > > progression feels strong.
> >
> > Sounds like we are really dealing with the Anders theory
> > of chord progressions. Which is probably a blessing.
> >
> :) As I said, I simply implemented Schoenberg's rule explanation
> instead of his actual rules. So, I cannot accept the honour you are
> implying.
>
> > > Let us create a sequence of 5 chords and
> > > allow only for diatonic triads in C-major. The first and last
> > > chord should be the tonic. Following Schoenberg let us specify
> > > that ascending progressions are "resolved" (see above), and we
> > > don't allow for superstrong progressions. Finally, the union
> > > of the last three chords must contain all pitch classes of
> > > C-major (i.e. these chords form a cadence).
> > >
> > > This problem has only two solutions (I used the computer :).
> > > These solutions are as follows. They happen to contain only
> > > ascending progressions. Such solutions are particularly
> > > convincing. For simplicity I given absolute chord names.
> > >
> > > C-maj F-maj D-min G-maj C-maj (only ascending progressions)
> > > C-maj A-min D-min G-maj C-maj (only ascending progressions)
> >
> > Only allowing for diatonic triads is C-major is a very
> > helpful constraint. Howabout we allow all 7-limit triads
> > in 31-ET, for a start?
> >
>
> The point of these generalised definitions is exactly that they allow
> for such cases :) I only simplified for the sake of an argument. So,
> here is a 7-limit variant. Let the scale be septimal natural minor
> (Scala term, in 31 ET the pitches C D D# F G G# A#). Let us allow for
> the following septimal triads (please correct my terminology if
> necessary): harmonic diminished (5/5 6/5 7/5), subharmonic diminished
> (7/7 7/6 7/5), subminor (6/1 7/1 9/1), supermajor (1/9 1/7 1/6). Note
> that I consider all these chords as consonances for simplicity which
> require neither preparation nor resolution. Naturally, we could
> define extra rules which care for that.
>
> There are 2 solutions of 5 chords with this scale, these chords and
> the rule set specified before. For convenience, I attached these
> solutions as PDF files (please let me know if doing so violates some
> policies of this list).
>
> The solutions are shown by two staffs. The upper stuff shows the
> actual chord pitches (I didn't bother to implement voice leading
> rules here, so things like Bruckner's law of the shortest path are
> violated). The staff notation is in 31 ET (i.e. the interval C D# is
> 7/6). The lower staff shows the chord roots in staff notation plus
> there ratios expressed as a product of the untransposed chord ratios
> and a root factor for each chord ratio. Some of these factors may
> look odd at first, e.g., the first chord has the factor 4/3 for the
> root C, but this is due to that fact that the untransposed chord
> ratios don't contain 1/1 -- the resulting product ratios do contain 1/1.
>
> If you are interested in other chord sets, scales, or longer
> solutions then just say so -- its easy to change some numbers in a
> computer program :)
>
> Best
> Torsten
>
> > -- > Torsten Anders
> Interdisciplinary Centre for Computer Music Research
> University of Plymouth
> Office: +44-1752-233667
> Private: +44-1752-558917
> http://strasheela.sourceforge.net
> http://www.torsten-anders.de
>
>
>
>

Hi Torsten,

> > > For example, Schoenberg
> > > recommends that descending progressions should "resolve".
> > > For any three successive chords, if the first two chords
> > > form a descending progression, then the progression from
> > > the first to the third chord should form a strong or
> > > superstrong progression (so the middle chord
> > > is quasi a 'passing chord').
> >
> > Hm... this seems kindof strange.
> >
> Schoenberg is always very exhaustive :) He does not want to
> ban descending progressions, so he looks for a way how he can
> allow for them in a way which is musically convincing in most
> cases. So, he comes up with the idea of a 'passing chord'.

Let's not give Schoenberg too much credit as a music theorist.
He was on the other hand one of the best composers ever
known (at least until he started using his theory!).

Here is how I would do it:

root -> root
CM -> Cmin (2 common tones)

root -> tone
CM -> Amin (2 common tones)
CM -> FM (1 common tone)
CM -> Fmin (1 common tone)
CM -> AbM (1 common tone)

tone -> root
CM -> Emin (2 common tones)
CM -> EM (1 common tone)
CM -> GM (1 common tone)
CM -> Gmin (1 common tone)

tone -> tone
CM -> AM (1 common tone)
CM -> C#min (1 common tone)
CM -> EbM (1 common tone)

Now: how would we rate these progressions using our ears?
I would be interested in your opinion, but already it is
obvious that this theory cannot distinguish CM -> EM from
CM -> GM, so it is missing something.

> > > Let us create a sequence of 5 chords and
> > > allow only for diatonic triads in C-major. The first and last
> > > chord should be the tonic. Following Schoenberg let us specify
> > > that ascending progressions are "resolved" (see above), and we
> > > don't allow for superstrong progressions. Finally, the union
> > > of the last three chords must contain all pitch classes of
> > > C-major (i.e. these chords form a cadence).
> > >
> > > This problem has only two solutions (I used the computer :).
> > > These solutions are as follows. They happen to contain only
> > > ascending progressions. Such solutions are particularly
> > > convincing. For simplicity I given absolute chord names.
> > >
> > > C-maj F-maj D-min G-maj C-maj (only ascending progressions)
> > > C-maj A-min D-min G-maj C-maj (only ascending progressions)
> >
> > Only allowing for diatonic triads is C-major is a very
> > helpful constraint. Howabout we allow all 7-limit triads
> > in 31-ET, for a start?
>
> The point of these generalised definitions is exactly that
> they allow for such cases :) I only simplified for the sake
> of an argument. So, here is a 7-limit variant. Let the scale
> be septimal natural minor

By assuming a scale you're guaranteeing valid results.
Or at least limiting the poorness of the results. I meant
all 7-limit chords in 31-ET. This is the kind of stuff
we cut our teeth on around here. :)

> There are 2 solutions of 5 chords with this scale, these chords
> and the rule set specified before. For convenience, I attached
> these solutions as PDF files (please let me know if doing so
> violates some policies of this list).

It's not so much a policy as it is a stupid limitation of
the Yahoo system.
Do you have a web or ftp account you can post to? Some people
use services like yousendit.
As a last resort you can use the "Files" area of the group:

/tuning/files

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Let's not give Schoenberg too much credit as a music theorist.
> He was on the other hand one of the best composers ever
> known (at least until he started using his theory!).
>

Well, composing is basically what this thread is about, and the
situation here is sort of comparable to Schoenberg's: looking for
harmony concepts/chord progressions in new contexts. So it is quite
natural that Schonberg's work can be applied, and I think he deserves
in any case the credit for his approach (if not for his theory...).

Maybe I should mention Mazzola's concept of "cadencial set"
(http://home.datacomm.ch/straub/mamuth/mamufaq.html#Q_cadencialset),
which integrates so well with Schoenberg's modulation model
(http://home.datacomm.ch/straub/mamuth/mamufaq.html#Q_modul), and
which I already have used successfully in microtonal contexts.

Unless, of course, you consider this one of the "bad music theory
papers"...
--
Hans Straub

Torsten Anders wrote:

> Here is a suggestion I can make for guidelines on good chord > progressions, which is indeed based on the notion of pitch class > sets, but which also works beyond 12 ET (e.g., I did examples using > them with 31 ET). This suggestion is based on Schoenberg's guidelines > on good chord root progressions where he introduces the notion of > ascending (strong), descending ('weak') and superstrong progressions > (see the respective chapter in his Theory of Harmony). The summary of > Schoenberg's guidelines/rules is based on 12 ET, but his explanation > is actually more general. A formalisation of his explanations > (instead of his actual rules) does work beyond 12 ET.

I read a lot of Theory of Harmony several years ago. (In fact I remember reading it during the boring lectures on matrices that have since turned out to be useful.) I do remember these rules. But at the time I applied them to 31 ET and didn't see any problem doing so. Pitch class sets are a more modern concept that I don't see the need to bring in.

Now, Schoenberg's rules, and traditional theory in general, are tied to meantone temperament or something like it and a particular style of music assuming particular scales and conventions. The don't automatically generalize to other contexts. Schoenberg himself could hardly be accused of following the rules of 19th Century harmony too closely in his music.

I think it's an interesting idea to think about how harmony can be generalized. One thing I've concluded, after all these years, is that the rules given in harmony textbooks are often rules of thumb that happen to work within that context rather than rules of nature that you can expect to be more generally valid. Behind it all I can identify two general rules:

1) Prefer descending fifths

2) Resolve by stepwise contrary motion

So, which is more important?

The issue of fifths is a tricky one. For a wide range of music, from madrigals to blues, there's an expectation that chords are related by fifths. Whether it's natural or not it's something listeners will expect. But is it possible to establish other relationships that work as well in scales that don't have long chains of fifths?

Next, there's the issue of "what is a step"? In miracle temperament (31&41) I tried to follow the decimal scale. I had rules to ensure "steps" were smaller than "leaps" and I avoided smaller intervals that could be heard as "mistunings". This lead to decimal counterpoint, for which I've tracked down the examples now:

http://x31eq.com/music/counterpoint.html

I still haven't explained the theory but it's a generalization of Palestrina counterpoint as explained by Jeppesen and Fux. I think it makes sense but it's still counterpoint rather than harmony. There is a tendency for chords or modulations to be related by secors (the large semitones) because they're the generators of the scales.

Something I've been thinking about lately is harmony in magic temperament (19&22). There, the simple scales naturally have lots of small semitones. They're around the size George Secor identified as optimal for melody (1/22 to 1/19 instead of 1/17). But the simple scales are very uneven, so you might have a few semitones and then a leap of a minor third. I'm wondering how much of a problem this would be in practice. Unfortunately my keyboard's in another city so I'm only speculating.

Chords will naturally be related by thirds rather than fifths. It's difficult to get two consonances to resolve by these small semitones without parallel fifths but it's easy to resolve an augmented triad onto major or minor. That violates the rule that a perfect cadence works because it relates two strong consonances.

All of which takes us a long way from Schoenberg. I eventually gave up on the book because he spent all the time talking about particular chords when I'm much more interested in general rules. So, what rules do people find are helpful in navigating microtonal harmony?

Graham

Dear Carl,

On Apr 23, 2008, at 12:35 AM, Carl Lumma wrote:
>
> By assuming a scale you're guaranteeing valid results.
> Or at least limiting the poorness of the results. I meant
> all 7-limit chords in 31-ET. This is the kind of stuff
> we cut our teeth on around here. :)
>
Sorry for contradicting, but using a scale is by no means "guaranteeing valid results". Here is an extreme counter-example in C-maj, which is only uses diatonic chords but does not follow the mentioned guidelines. This progression has not even a "harmonic band" (common tones) between chords (it still does a "cadence" in the sense that the last three chords express all scale degrees).

CM Dmin Emin Dmin CM

More to the point, there are quite a lot of different diatonic progressions of 5 chords possible in C-major. I was surprised when I realised that only two (!) 5-chord progressions follow the rule set I described. I would not argue that there are no other good sounding progressions, but I find it remarkable that this rule set only finds particularly common progressions.

Anyway, we can certainly also do non-diatonic 7-limit chord progressions. I did some examples which start in C, use only ascending/strong progressions, wander through keys so to speak, but end again in C major with some sort of a cadence. We could use, e.g., some septimal scale instead for a cadence, but leaving any cadence off entirely or ending in another key feels less convincing (less closed) for me. However, I would very much welcome alternative suggestions concerning endings :) Also, there are some inversions now so that the bass sometimes progresses stepwise.

Naturally, there exist many such progressions. I uploaded a few examples to

/tuning/files/TorstenAnders/ChordProgressions/ChordProgression-*

The same limitations are mentioned before apply: no voice-leading rules applied for simplicity and no preparation or resolution of any dissonances.

In an actual composition, I would likely further structure a progression, e.g., have some sub-progressions which are repeated in different transpositions. These may then coincide with motif relations. In a first attempt, I did chord progressions where the transposition interval and the chord "type" (i.e. in intervallic relations within the chord) repeat every four chords as in a sequence.

I uploaded these examples to

/tuning/files/TorstenAnders/ChordProgressions/ChordProg-wSequence-*

Comments and recommendations for better approaches are of course appreciated :)

> > > > For example, Schoenberg
> > > > recommends that descending progressions should "resolve".
> > > > For any three successive chords, if the first two chords
> > > > form a descending progression, then the progression from
> > > > the first to the third chord should form a strong or
> > > > superstrong progression (so the middle chord
> > > > is quasi a 'passing chord').
> > >
> > > Hm... this seems kindof strange.
> > >
> > Schoenberg is always very exhaustive :) He does not want to
> > ban descending progressions, so he looks for a way how he can
> > allow for them in a way which is musically convincing in most
> > cases. So, he comes up with the idea of a 'passing chord'.
>
> Let's not give Schoenberg too much credit as a music theorist.
> He was on the other hand one of the best composers ever
> known (at least until he started using his theory!).
>

My remarks on why/how he introduced 'passing chord' (my term, but he says something like "the effect is as if some chord was inserted for merely melodic reasons") above are purely speculative and perhaps misleading. Nevertheless, I feel you shouldn't make such a statement if you haven't read his theoretical works (as you mentioned before). Many theorists agree that Schoenberg is one of the most important music theorists of the 20th century. BTW, he certainly did not directly "use" any of his published theoretical works for his own compositions, as these books are meant for teaching and are much more conservative than his compositions. Nevertheless, these writings try to understand why common practise works the way it works and he uses this also extrapolate from common practise to new possibilities.

> Here is how I would do it:
>
> root -> root
> CM -> Cmin (2 common tones)
>
> root -> tone
> CM -> Amin (2 common tones)
> CM -> FM (1 common tone)
> CM -> Fmin (1 common tone)
> CM -> AbM (1 common tone)
>
> tone -> root
> CM -> Emin (2 common tones)
> CM -> EM (1 common tone)
> CM -> GM (1 common tone)
> CM -> Gmin (1 common tone)
>
> tone -> tone
> CM -> AM (1 common tone)
> CM -> C#min (1 common tone)
> CM -> EbM (1 common tone)
>

I am not sure I follow. I assume you mean by "root -> tone" that the root of the second chord was no chord tone in the previous chord, and by "tone -> root" that the root of the second chord was some non-root tone of the previous. But what is then "tone -> tone"? Could you please explain?

Just as an illustration, I resort your chords according to the category formalisation I discussed in my first mail (I might have forgotten some). Its not meant to improve your list (as I did not understand yours), only to make things more clear. I also added where my proposed formalisation and Schoenberg's original rules diverge. I should mention that Schoenberg discusses root progressions only in the context of diatonic progressions, and in those cases my formalism and his rules behave the same. I extrapolated Schoenberg's rules for non-diatonic cases (e.g., CM -> C#min), but I don't feel 100% sure whether I am correct about his opinion for such non-diatonic cases as he does not discuss them explicitly in terms of root progressions (as far as I remember).

constant root "progression" (not considered by Schoenberg when discussing root progressions)
CM -> Cmin (2 common tones)

ascending progressions (strong)
CM -> Amin (2 common tones)
CM -> FM (1 common tone)
CM -> Fmin (1 common tone)
CM -> AbM (1 common tone)
CM -> AM (1 common tone)
CM -> C#min (1 common tone) [is superstrong after Schoenberg]
CM -> EbM (1 common tone) [is descending after Schoenberg]

descending progressions
CM -> Emin (2 common tones)
CM -> EM (1 common tone)
CM -> GM (1 common tone)
CM -> Gmin (1 common tone)

superstrong progressions (e.g., used in deceptive cadence)
CM -> DM
CM -> Dmin
CM -> C#M
CM -> BM
CM -> Bmin
CM -> BbM
CM -> Bbmin
CM -> Ebmin [descending after Schoenberg]
CM -> F#M [probably ascending after Schoenberg]
CM -> F#min [probably ascending after Schoenberg]
CM -> Abmin [ascending after Schoenberg]

> Now: how would we rate these progressions using our ears?
> I would be interested in your opinion, but already it is
> obvious that this theory cannot distinguish CM -> EM from
> CM -> GM, so it is missing something.

I see what you mean. Shall we complement the formalisation by some harmonic distance, e.g., distance of the chords on the lattice? (e.g., sum of Tenney harmonic distance of all chord tones divided by the tone number??). EMaj is more remote on the lattice from CMaj than GMaj is.

Comments appreciated.

Best
Torsten

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

Harmonic structures in microtuning particularly interest me, and I have found that many of the Schoenberg concepts prior to his sinking into serialism and atonality can be useful.

I have found that the less "classical" approach to be more practical, as it avoids much of the ambiguous and obscure terminology. Although most studies of harmony assume 12edo, Beside the well-known Piston etc. I have found two writers/books particularly instructive:

Joseph Schillinger

http://josephschillinger.com/books.htm

For (jazz) guitar "Chord Chemistry" by Ted Greene

http://www.amazon.com/Chord-Chemistry-Ted-Greene/dp/0898986966

Greene has a very useful listing of chord progressions consisting of four contiguous chords, which can be grouped/joined together to provide more complex patterns.

By initially considering the harmony in 12edo terms, this can easily be extended to cover an infinite number of steps of fourths (towards the flat keys) and fifths (sharps); the direction of the steps being determined by the "level" of consonance/dissonance that you wish to use.

You may find these pages of chords and scales helpful:

http://www.lucytune.com/new_to_lt/chords.html

http://www.lucytune.com/scales/

On 23 Apr 2008, at 14:22, Torsten Anders wrote:

> Sorry for contradicting, but using a scale is by no means
> "guaranteeing valid results". Here is an extreme counter-example in C-
> maj, which is only uses diatonic chords but does not follow the
> mentioned guidelines. This progression has not even a "harmonic
> band" (common tones) between chords (it still does a "cadence" in the
> sense that the last three chords express all scale degrees).
>
> CM Dmin Emin Dmin CM
>
> M
>

Charles Lucy
lucy@lucytune.com

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

Dear Graham,

thanks for your reply.

On Apr 23, 2008, at 1:30 PM, Graham Breed wrote:
> Torsten Anders wrote:
>
> > Here is a suggestion I can make for guidelines on good chord
> > progressions, which is indeed based on the notion of pitch class
> > sets, but which also works beyond 12 ET (e.g., I did examples using
> > them with 31 ET). This suggestion is based on Schoenberg's > guidelines
> > on good chord root progressions where he introduces the notion of
> > ascending (strong), descending ('weak') and superstrong progressions
> > (see the respective chapter in his Theory of Harmony). The > summary of
> > Schoenberg's guidelines/rules is based on 12 ET, but his explanation
> > is actually more general. A formalisation of his explanations
> > (instead of his actual rules) does work beyond 12 ET.
>
> I read a lot of Theory of Harmony several years ago. (In
> fact I remember reading it during the boring lectures on
> matrices that have since turned out to be useful.) I do
> remember these rules. But at the time I applied them to 31
> ET and didn't see any problem doing so.
>

> Now, Schoenberg's rules, and traditional theory in general,
> are tied to meantone temperament or something like it and a
> particular style of music assuming particular scales and
> conventions. The don't automatically generalize to other
> contexts. Schoenberg himself could hardly be accused of
> following the rules of 19th Century harmony too closely in
> his music.
>
> I think it's an interesting idea to think about how harmony
> can be generalized. One thing I've concluded, after all
> these years, is that the rules given in harmony textbooks
> are often rules of thumb that happen to work within that
> context rather than rules of nature that you can expect to
> be more generally valid. Behind it all I can identify two
> general rules:
>
> 1) Prefer descending fifths
>

Sure, decending fifths are the most important root progression in many musical styles. For example, in Jazz you can have astonishingly complex chord progressions which are all understood as II V I progressions.

We can directly apply the idea to microtonal harmony where we could have root progressions only along the 3-dimension of the lattice and then stagger various chords on top of these roots. Of course, we cannot have only descending fifths in JI (we would never get "back"), but we can write music where most root progressions are of this kind.

It would be interesting to explore how we can create convincing chord progressions where the roots are not all "stringed" on the 3-dimension of the lattice. 19th century harmony did this a lot, and this is also a domain explored in Schoenberg's harmony (his guidelines are not restricted to fifths progressions).

Yet, 19th century harmony is of course restricted to 5-limit -- its root progressions certainly are. Meanwhile, we realised that 7-limit and beyond can be fruitful to for harmony. So, my question is how can we create convincing chord progressions beyond 5-limit? Can we find some guidelines which simplify the composition process, much like the guidelines of 5-limit harmony did for former centuries.

I feel it can be a good idea trying to understand the underlying principles of previous guidelines so that these can be generalised for new music. A comparable example may be Erlich's decatonic scales which generalise principles underlying diatonic scales (and meantone temperament) in order to find scales for 7-limit.

I really like such an approach. So, I tried to generalise some principles underlying 5-limit chord progressions. The most thorough discussion of these which I could find so far are Schoenberg's two harmony books. Some underlying principles I hope I have found in Schoenberg's explanation of his guidelines for root progressions. Therefore, I formalised his explanation, and not his actual rules.

You mention that you could apply his rules to 31 ET. That's of could right, you can do that (in my former mail I mentioned 12 ET, because I reacted to Carl dispraise of music theory approaches restricted to 12 ET). However, you can only create 5-limit root progressions that way (naturally, you may stagger higher-limit chords on top). My generalised guidelines, on the other hand, at least formally also work for 7-limit (or any other limit).

Now, my question is of could do these generalised guidelines also result in chord progressions which convince musically? Or did I just some formal exercise here?

I would greatly appreciate comments in critics in this regard (e.g., by criticising the example progressions I sent in multiple mails before).

Even more so, I would like to hear alternative guidelines for creating chord progression where root relations are beyond 5-limit.

I should perhaps mention some guideline related to my generalised guidelines: prefer chord progressions which share common tones (a harmonic band in Schoenberg's terminology). The guidelines I detailed before are more precise that this harmonic-band-guideline.

> 2) Resolve by stepwise contrary motion
>
I'm sorry, but we may talk at cross-purposes here. Melodic motion is of course also important for chord progressions, but is a different matter (and you simplify it too :). So far, I was solely talking about chord root progressions, only a sub-aspect of harmony in general. Of course, there are other important aspects in harmony, which I left out for simplicity. For example, there is the treatment of dissonances (e.g., what is a dissonance, are they prepared and how, are they resolved and how). This matter is related to root progressions in the sense that dissonances are often resolved by strong harmonic progressions (V7 -> 1). There are also melodic considerations concerning the treatment of dissonances (e.g., dissonances are most smoothly resolved by a step downwards), but these might be somewhat less crucial than the harmonic progression resolving a dissonance (e.g., if V7 is resolved in I but the 7th is not directly leading into the 3th of I, we may still feel it is OK). Moreover, there is the aspect of how chords are related to a scale and how the underlying scale can change (modulation). And there are further aspects (e.g., chords can be inserted or left out of a progression etc.).

Now, besides all this there is of course also the field of counterpoint. I assume you are referring to that.

> So, which is more important?
>

If your where referring to counterpoint with your remark, then there can be no definite answer to this question. In some historical periods, the contrapuntal organisation of melodic motion was more important. At others, harmony dominated. I might say that it is a good thing to have both, but that is only a personal preference.

> The issue of fifths is a tricky one. For a wide range of
> music, from madrigals to blues, there's an expectation that
> chords are related by fifths. Whether it's natural or not
> it's something listeners will expect. But is it possible to
> establish other relationships that work as well in scales
> that don't have long chains of fifths?
>
I agree in general and certainly for the blues (see also discussion above). I am not so sure about madrigals though, many go beyond that. For example, many root progressions in Gesualdo's work are 5-limit.

> Next, there's the issue of "what is a step"? In miracle
> temperament (31&41) I tried to follow the decimal scale. I
> had rules to ensure "steps" were smaller than "leaps" and I
> avoided smaller intervals that could be heard as
> "mistunings". This lead to decimal counterpoint, for which
> I've tracked down the examples now:
>
> http://x31eq.com/music/counterpoint.html
>
> I still haven't explained the theory but it's a
> generalization of Palestrina counterpoint as explained by
> Jeppesen and Fux. I think it makes sense but it's still
> counterpoint rather than harmony. There is a tendency for
> chords or modulations to be related by secors (the large
> semitones) because they're the generators of the scales.
>
I would like to read the theory behind these.

> Something I've been thinking about lately is harmony in
> magic temperament (19&22). There, the simple scales
> naturally have lots of small semitones. They're around the
> size George Secor identified as optimal for melody (1/22 to
> 1/19 instead of 1/17). But the simple scales are very
> uneven, so you might have a few semitones and then a leap of
> a minor third. I'm wondering how much of a problem this
> would be in practice. Unfortunately my keyboard's in
> another city so I'm only speculating.
>
> Chords will naturally be related by thirds rather than
> fifths. It's difficult to get two consonances to resolve by
> these small semitones without parallel fifths but it's easy
> to resolve an augmented triad onto major or minor. That
> violates the rule that a perfect cadence works because it
> relates two strong consonances.
>
> All of which takes us a long way from Schoenberg. I
> eventually gave up on the book because he spent all the time
> talking about particular chords when I'm much more
> interested in general rules.
>
I understand why you felt that way, but you meanwhile may also understand why I don't agree here :)

> Pitch class sets
> are a more modern concept that I don't see the need to bring in.
>
I am using the pitch class notion in a generalised sense: a pitch where the octave component is neglected. I understood that this notion is common in microtonal music as well. For example, it means exactly the same as Doty's "Primer" denotes with the term "identity" (e.g., 5/1 is an identity and a pitch class). Using the notion of sets to refer to multiple pitch classes (identities) allows then to speak precisely about relations of multiple chords etc. and formalise such relations. Would you suggest an alternative concept instead for this purpose?

I assume Schoenberg did not use the term pitch class. But the composer who likely composed the most important dodecaphonic pieces we have certainly knew exactly what is meant by this term.

> So, what rules do people find
> are helpful in navigating microtonal harmony?
>
Exactly the question I was addressing :) I would greatly welcome alternative proposals.

Best
Torsten

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

Dear Hans Straub,

thank you for your links (and for compiling these FAQs in the first place). I will look at them more closely soon.

On Apr 23, 2008, at 8:34 AM, hstraub64 wrote:
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> >
> > Let's not give Schoenberg too much credit as a music theorist.
> > He was on the other hand one of the best composers ever
> > known (at least until he started using his theory!).
> >
>
> Well, composing is basically what this thread is about, and the
> situation here is sort of comparable to Schoenberg's: looking for
> harmony concepts/chord progressions in new contexts. So it is quite
> natural that Schonberg's work can be applied, and I think he deserves
> in any case the credit for his approach (if not for his theory...).
>
> Maybe I should mention Mazzola's concept of "cadencial set"
> (http://home.datacomm.ch/straub/mamuth/mamufaq.html#Q_cadencialset),
> which integrates so well with Schoenberg's modulation model
> (http://home.datacomm.ch/straub/mamuth/mamufaq.html#Q_modul), and
> which I already have used successfully in microtonal contexts.
>
That sounds very interesting. Could you please detail this a bit more?

Thank you!

Best
Torsten

> Unless, of course, you consider this one of the "bad music theory
> papers"...
> --> Hans Straub
>
>
>
--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

Dear Kraig,

On Apr 22, 2008, at 10:33 PM, Kraig Grady wrote:
> What i remember which goes quite far back in time and my copy is still
> unpacked is strong progressions are where more important note/s become
> less and vice versa. i have actually played with this in structures > such
> as the eikosany. Anyway it seems if you allow the 7th and 9th > harmonics
> in chords, even triads using them, it seems one would have further
> progressions to choose from although this would quickly lead > outside of
> any particular scale. One could also use the 11th.
>

thanks for your mail. Unfortunately, I don't quite understand what you mean by "strong progressions are where more important note/s become less and vice versa". If you are referring to the chord root as the only important note, then this guideline is closely related to what I was suggesting (but not the same). I said that in a strong progression there appears new "important note" -- the root of the new chord was not in the previous chord.

But it seems your remark is more general. In particular, it appears to are referring to specific higher harmonics. Could you please detail this further?

Thank you!

Best
Torsten

>
>
> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> _'''''''_ ^North/Western Hemisphere:
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
>
> _'''''''_ ^South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria <http://> anaphoriasouth.blogspot.com/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
> Torsten Anders wrote:
> >
> > Dear Carl,
> >
> > On Apr 22, 2008, at 7:19 PM, Carl Lumma wrote:
> > > > Here is a suggestion I can make for guidelines on good chord
> > > > progressions, which is indeed based on the notion of pitch class
> > > > sets, but which also works beyond 12 ET (e.g., I did examples
> > > > using them with 31 ET). This suggestion is based on Schoenberg's
> > > > guidelines on good chord root progressions where he introduces
> > > > the notion of ascending (strong), descending ('weak') and
> > > > superstrong progressions (see the respective chapter in his
> > > > Theory of Harmony). The summary of Schoenberg's guidelines/rules
> > > > is based on 12 ET, but his explanation is actually more general.
> > > > A formalisation of his explanations (instead of his actual > rules)
> > > > does work beyond 12 ET.
> > > >
> > > > The main difference between Schoenberg's actual guidelines and
> > > > their formalised generalisation is that Schoenberg's guidelines
> > > > are based on scale degree intervals between chord roots, whereas
> > > > the generalisation exploits whether the root pitch class of some
> > > > chord is contained in the pitch class set of another chord.
> > >
> > > I haven't read Schoenberg's book but it sounds like you've
> > > done a bit of work for him. Yes, I could probably generalize
> > > many (though not all) of the "bad music theory papers" if I
> > > worked at it. And some such minimal effort would only be
> > > fair in the meta-analysis I proposed. Where to draw the line
> > > is another question.
> > >
> >
> > > Did Schoenberg say the important thing
> > > is whether the root is a member of the new chord (which is
> > > a reasonable thing)? Or did he say something else?
> > >
> >
> > Sorry, I don't fully understand your question. In an ascending
> > progression, the root of the preceding chord is a pitch in the
> > following chord. Did you mean that?
> >
> > > > OK, it follows the
> > > > core of is the formalisation (I hope the notation explains > itself:
> > > > defined are Boolean functions expecting two neighbouring > chords).
> > > >
> > > > /* Chord1 and Chord2 have common pitch classes, but the root of
> > > > Chord2 does not occur in the set of Chord1's pitchclasses. */
> > > >
> > > > isAscendingProgression(Chord1, Chord2) :=
> > > > ( NOT ( getRoot(Chord2) \in getPitchClasses(Chord1) ) )
> > > > AND ( intersection(getPitchClasses(Chord1), getPitchClasses
> > > > (Chord2)) \= emptySet )
> > > >
> > > > /* A non-root pitchclass of Chord1 is root in Chord2. */
> > > >
> > > > isDescendingProgression(Chord1, Chord2) :=
> > > > ( getRoot(Chord2) \in getPitchClasses(Chord1) )
> > > > AND ( getRoot(Chord1) \= getRoot(Chord2) )
> > > >
> > > > /* Chord1 and Chord2 have no common pitch classes */
> > > >
> > > > isSuperstrongProgression(Chord1, Chord2) :=
> > > > intersection(getPitchClasses(Chord1), getPitchClasses(Chord2)) =
> > > > emptySet
> > >
> > > Yes, perfectly understandable, though I find the terms
> > > "ascending" and "descending" extremely odd here.
> > >
> > These are Schoenberg's terms. He mentions that H. Schenker uses > these
> > terms as well ("aufsteigend" and "fallend") though in opposite
> > meaning. He remarks something like that anyone who loves Brahmsian
> > harmony would likely come up with similar concepts.
> >
> > > > If you are interested I can post further formalisations/
> > > > generalisations of Schoenberg's rules.
> > >
> > > I would be interested to read them, especially if you
> > > show how you generalized them. And if monz is reading,
> > > I bet he'd be interested, too.
> > >
> > I'll write a new mail soon..
> >
> > > > For example, Schoenberg
> > > > recommends that descending progressions should "resolve". For > any
> > > > three successive chords, if the first two chords form a > descending
> > > > progression, then the progression from the first to the third > chord
> > > > should form a strong or superstrong progression (so the > middle chord
> > > > is quasi a 'passing chord').
> > >
> > > Hm... this seems kindof strange.
> > >
> > Schoenberg is always very exhaustive :) He does not want to ban
> > descending progressions, so he looks for a way how he can allow for
> > them in a way which is musically convincing in most cases. So, he
> > comes up with the idea of a 'passing chord'.
> >
> > Please note that he makes is always very clear that his rules are
> > guidelines for the pupil, but no laws for masterworks. This is > why he
> > prefers the term "descending" instead of "weak" progressions (he > uses
> > the term "strong" for "ascending" progressions a lot). Still, he > does
> > not discuss where descending progressions are used to good > purpose in
> > any masterwork, but such considerations have hardly a place > anyway in
> > his systematic approach.
> >
> > > > Anyway, for chord progressions of diatonic triads in major, the
> > > > generalised formalisation and Schoenberg's rules are equivalent
> > > > (e.g., the function isAscendingProgression above returns true > for
> > > > progressions which Schoenberg calls ascending). Still, the > behaviour
> > > > of the constraints and Schoenberg's rules differ for more > complex
> > > > cases. According to Schoenberg, a progression is superstrong > if the
> > > > root interval is a step up or down.
> > >
> > > Bzzz. :)
> > >
> > > > For example, the progression V7 IV is superstrong according to
> > > > Schoenberg. For the definitions above, however, this progression
> > > > is descending (!), because the root of IV is contained in V7
> > > > (e.g. in G7 F, the F's root pitchclass f is already contained
> > > > in G7). Indeed, this progression is rare in music.
> > > > By contrast, the progression I IIIb (e.g., C Eb) is a descending
> > > > progression in Schoenbergs original definition. For the
> > > > definitions above, however, this is an ascending progression
> > > > (the root of Es is not contained in C), and indeed for me the
> > > > progression feels strong.
> > >
> > > Sounds like we are really dealing with the Anders theory
> > > of chord progressions. Which is probably a blessing.
> > >
> > :) As I said, I simply implemented Schoenberg's rule explanation
> > instead of his actual rules. So, I cannot accept the honour you are
> > implying.
> >
> > > > Let us create a sequence of 5 chords and
> > > > allow only for diatonic triads in C-major. The first and last
> > > > chord should be the tonic. Following Schoenberg let us specify
> > > > that ascending progressions are "resolved" (see above), and we
> > > > don't allow for superstrong progressions. Finally, the union
> > > > of the last three chords must contain all pitch classes of
> > > > C-major (i.e. these chords form a cadence).
> > > >
> > > > This problem has only two solutions (I used the computer :).
> > > > These solutions are as follows. They happen to contain only
> > > > ascending progressions. Such solutions are particularly
> > > > convincing. For simplicity I given absolute chord names.
> > > >
> > > > C-maj F-maj D-min G-maj C-maj (only ascending progressions)
> > > > C-maj A-min D-min G-maj C-maj (only ascending progressions)
> > >
> > > Only allowing for diatonic triads is C-major is a very
> > > helpful constraint. Howabout we allow all 7-limit triads
> > > in 31-ET, for a start?
> > >
> >
> > The point of these generalised definitions is exactly that they > allow
> > for such cases :) I only simplified for the sake of an argument. So,
> > here is a 7-limit variant. Let the scale be septimal natural minor
> > (Scala term, in 31 ET the pitches C D D# F G G# A#). Let us allow > for
> > the following septimal triads (please correct my terminology if
> > necessary): harmonic diminished (5/5 6/5 7/5), subharmonic > diminished
> > (7/7 7/6 7/5), subminor (6/1 7/1 9/1), supermajor (1/9 1/7 1/6). > Note
> > that I consider all these chords as consonances for simplicity which
> > require neither preparation nor resolution. Naturally, we could
> > define extra rules which care for that.
> >
> > There are 2 solutions of 5 chords with this scale, these chords and
> > the rule set specified before. For convenience, I attached these
> > solutions as PDF files (please let me know if doing so violates some
> > policies of this list).
> >
> > The solutions are shown by two staffs. The upper stuff shows the
> > actual chord pitches (I didn't bother to implement voice leading
> > rules here, so things like Bruckner's law of the shortest path are
> > violated). The staff notation is in 31 ET (i.e. the interval C D# is
> > 7/6). The lower staff shows the chord roots in staff notation plus
> > there ratios expressed as a product of the untransposed chord ratios
> > and a root factor for each chord ratio. Some of these factors may
> > look odd at first, e.g., the first chord has the factor 4/3 for the
> > root C, but this is due to that fact that the untransposed chord
> > ratios don't contain 1/1 -- the resulting product ratios do > contain 1/1.
> >
> > If you are interested in other chord sets, scales, or longer
> > solutions then just say so -- its easy to change some numbers in a
> > computer program :)
> >
> > Best
> > Torsten
> >
> >
> > --
> > Torsten Anders
> > Interdisciplinary Centre for Computer Music Research
> > University of Plymouth
> > Office: +44-1752-233667
> > Private: +44-1752-558917
> > http://strasheela.sourceforge.net
> > http://www.torsten-anders.de
> >
> >
> >
> >
>
>
--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

Hi Torsten,

> Even more so, I would like to hear alternative guidelines
> for creating chord progression where root relations are
> beyond 5-limit.

Don't want to butt in here between you and Graham, but I
thought I'd point out that the roots of many higher-limit
chords are ill-defined by current theory.

-Carl

Hi Torsten,

> > By assuming a scale you're guaranteeing valid results.
> > Or at least limiting the poorness of the results. I meant
> > all 7-limit chords in 31-ET. This is the kind of stuff
> > we cut our teeth on around here. :)
>
> Sorry for contradicting, but using a scale is by no means
> "guaranteeing valid results". Here is an extreme counter-example
> in C-maj, which is only uses diatonic chords but does not follow
> the mentioned guidelines. This progression has not even a
> "harmonic band" (common tones) between chords (it still does a
> "cadence" in the sense that the last three chords express all
> scale degrees).
>
> CM Dmin Emin Dmin CM
>
> More to the point, there are quite a lot of different diatonic
> progressions of 5 chords possible in C-major. I was surprised
> when I realised that only two (!) 5-chord progressions follow
> the rule set I described. I would not argue that there are no
> other good sounding progressions, but I find it remarkable that
> this rule set only finds particularly common progressions.

If I had a nickel for every paper that starts with the diatonic
scale, applies some rules, and winds up with the central
cadences of common practice music, I'd be a rich man. :)
Even Rothenberg had a way to get them, based on his pattern
recognition model.

> Anyway, we can certainly also do non-diatonic 7-limit chord
> progressions. I did some examples which start in C, use only
> ascending/strong progressions, wander through keys so to speak,
> but end again in C major with some sort of a cadence.//
> Naturally, there exist many such progressions. I uploaded a
> few examples to
>
> /tuning/files/TorstenAnders/
> ChordProgressions/ChordProgression-*
>
> The same limitations are mentioned before apply: no voice-leading
> rules applied for simplicity and no preparation or resolution of
> any dissonances.
>
> In an actual composition, I would likely further structure a
> progression, e.g., have some sub-progressions which are repeated
> in different transpositions. These may then coincide with motif
> relations. In a first attempt, I did chord progressions where
> the transposition interval and the chord "type" (i.e. in
> intervallic relations within the chord) repeat every four
> chords as in a sequence.
>
> I uploaded these examples to
>
> /tuning/files/TorstenAnders/
> ChordProgressions/ChordProg-wSequence-*
>
> Comments and recommendations for better approaches are of
> course appreciated :)

Wonderful work, simply fantastic. One could spend several
lifetimes writing music around such progressions. I would
probably have put the examples at a somewhat lower pitch...
What is your workflow for producing the notation?

"These ratios automatically derived from the 31 ET pitch
classes"... using Strasheela, I take it. Wow, very powerful.

> > > > > For example, Schoenberg
> > > > > recommends that descending progressions should "resolve".
> > > > > For any three successive chords, if the first two chords
> > > > > form a descending progression, then the progression from
> > > > > the first to the third chord should form a strong or
> > > > > superstrong progression (so the middle chord
> > > > > is quasi a 'passing chord').
> > > >
> > > > Hm... this seems kindof strange.
> > > >
> > > Schoenberg is always very exhaustive :) He does not want to
> > > ban descending progressions, so he looks for a way how he
> > > can allow for them in a way which is musically convincing
> > > in most cases. So, he comes up with the idea of a 'passing
> > > chord'.
> >
> > Let's not give Schoenberg too much credit as a music theorist.
> > He was on the other hand one of the best composers ever
> > known (at least until he started using his theory!).
//
> I feel you shouldn't make such a statement if you haven't
> read his theoretical works (as you mentioned before).

You're right. I did actually read a few pages from his
book on one occasion, but it isn't enough to go on.

> Many theorists agree that Schoenberg is one of the most
> important music theorists of the 20th century. BTW, he
> certainly did not directly "use" any of his published
> theoretical works for his own compositions, as these books
> are meant for teaching and are much more conservative than
> his compositions. Nevertheless, these writings try to
> understand why common practise works the way it works and
> he uses this also extrapolate from common practise to new
> possibilities.

I thought his works could (roughly) be divided into
pre- and post- 12-tone technique periods (?).

> > Here is how I would do it:
> >
> > root -> root
> > CM -> Cmin (2 common tones)
> >
> > root -> tone
> > CM -> Amin (2 common tones)
> > CM -> FM (1 common tone)
> > CM -> Fmin (1 common tone)
> > CM -> AbM (1 common tone)
> >
> > tone -> root
> > CM -> Emin (2 common tones)
> > CM -> EM (1 common tone)
> > CM -> GM (1 common tone)
> > CM -> Gmin (1 common tone)
> >
> > tone -> tone
> > CM -> AM (1 common tone)
> > CM -> C#min (1 common tone)
> > CM -> EbM (1 common tone)
>
> I am not sure I follow. I assume you mean by "root -> tone"
> that the root of the second chord was no chord tone in the
> previous chord,

That the second chord contains the root of the first as a
non-root tone. It happens that this also means what you say
here is true.

> and by "tone -> root" that the root of the second chord was
> some non-root tone of the previous.

Yes.

> But what is then "tone -> tone"? Could you please explain?

Some non-root tone of the first chord is a non-root tone
in the second. Hopefully the examples will help.

> Just as an illustration, I resort your chords according to
> the category formalisation I discussed in my first mail
//
> constant root "progression"
> CM -> Cmin (2 common tones)
>
> ascending progressions (strong)
> CM -> Amin (2 common tones)
> CM -> FM (1 common tone)
> CM -> Fmin (1 common tone)
> CM -> AbM (1 common tone)
> CM -> AM (1 common tone)
> CM -> C#min (1 common tone) [is superstrong after Schoenberg]
> CM -> EbM (1 common tone) [is descending after Schoenberg]
>
> descending progressions
> CM -> Emin (2 common tones)
> CM -> EM (1 common tone)
> CM -> GM (1 common tone)
> CM -> Gmin (1 common tone)

Yes. The classification I gave is directional; your
implementation of Schoenberg's is not. This means that
whereas I have distinguished between root->tone and
tone->root, S. considers them together. So it seems the
simplest definitions for S.'s terms are:

ascending progression - two chords having at least one
common tone which is a root in one of the chords

descending progression - two chords having at least
one common tone which are not roots of either chord.

Listening to the examples, I don't hear evidence that the
direction matters, but I thought I should expose it to
test this.

> > Now: how would we rate these progressions using our ears?
> > I would be interested in your opinion, but already it is
> > obvious that this theory cannot distinguish CM -> EM from
> > CM -> GM, so it is missing something.
>
> I see what you mean. Shall we complement the formalisation
> by some harmonic distance, e.g., distance of the chords on
> the lattice? (e.g., sum of Tenney harmonic distance of all
> chord tones divided by the tone number??). EMaj is more
> remote on the lattice from CMaj than GMaj is.

Yes, this would be one sensible approach.

In fact I would suggest that the strength of the perceived
change between two chords is the sum of the Tenney-weighted
harmonic distances between pairs of changing tones in the
ensemble, arranged so that this sum is minimal. Let me
explain...

The "ensemble" is the set of pitches needed to perform
both chords.

"changing pitches": every pitch in the ensemble not
occurring in both chords.

For a pair of changing pitches, the Tenney-weighted
harmonic distance between them is
log2(TenneyHeight(pitchA - pitchB))
where
pitchA >= pitchB
and
TenneyHeight(p/q) = p*q

For temperaments, we mean the approximations to pitchA
and pitchB in just intonation.

So in the example CM -> Cmin, the changing pitches are
E <-> Eb, which are approximately 5/4 <-> 6/5 in just
intonation. 5/4 - 6/5 = 25/24, and log2(600) = 9.23.
Compare this to CM -> Amin, where log2(10*9) = 6.49.

If there is only 1 changing pitch in the enemble (for
example C-E-G-Bb -> C-E-G, where the changing pitch
is Bb), then the "perceived change strength" between
the chords shall be zero.

If there are more than 2 changing pitches, we find the
pairing for which the sum of the harmonic distances is
minimal. This includes the ability to exclude the most
distant pitch in case there are an odd number of pitches.

Back in the original example (CM->GM vs. CM->EM), we
have
C<->D + E<->B = 8.75 for CM->GM
and
G<->B + C<->G# = 12.97 for CM->EM
So perhaps it's working. We should calculate values for
all of the progressions in my table and see if they make
sense.

This method should be completely general for higher limits,
and would even claim to measure the strength of progressions
between chords of different sizes (tetrads, triads, etc.)
on the same scale.

-Carl

Torsten Anders wrote:

>> I think it's an interesting idea to think about how harmony
>> can be generalized. One thing I've concluded, after all
>> these years, is that the rules given in harmony textbooks
>> are often rules of thumb that happen to work within that
>> context rather than rules of nature that you can expect to
>> be more generally valid. Behind it all I can identify two
>> general rules:
>>
>> 1) Prefer descending fifths
> > Sure, decending fifths are the most important root progression in > many musical styles. For example, in Jazz you can have astonishingly > complex chord progressions which are all understood as II V I > progressions.
> > We can directly apply the idea to microtonal harmony where we could > have root progressions only along the 3-dimension of the lattice and > then stagger various chords on top of these roots. Of course, we > cannot have only descending fifths in JI (we would never get "back"), > but we can write music where most root progressions are of this kind.

Conventional harmony doesn't use only descending fifths anyway. But you can work forwards and backwards until you have chords with common notes and make the join that way. With other temperament classes you may not even have enough fifths to do that given a reasonable number of notes.

> It would be interesting to explore how we can create convincing chord > progressions where the roots are not all "stringed" on the 3- > dimension of the lattice. 19th century harmony did this a lot, and > this is also a domain explored in Schoenberg's harmony (his > guidelines are not restricted to fifths progressions).

The simplest thing in magic is to go up in thirds and then down a fifth.

> Yet, 19th century harmony is of course restricted to 5-limit -- its > root progressions certainly are. Meanwhile, we realised that 7-limit > and beyond can be fruitful to for harmony. So, my question is how can > we create convincing chord progressions beyond 5-limit? Can we find > some guidelines which simplify the composition process, much like the > guidelines of 5-limit harmony did for former centuries.

9-limit chords are interesting. You get a 6:7:9 subminor triad with a bluesey feel that seems to work well enough as well as its opposite, the car horn supermajor triad. Because they're still traids they have clear roots and follow voice leading rules. 7 odd-limit triads (other than the 5-limit) don't fit so well.

> I feel it can be a good idea trying to understand the underlying > principles of previous guidelines so that these can be generalised > for new music. A comparable example may be Erlich's decatonic scales > which generalise principles underlying diatonic scales (and meantone > temperament) in order to find scales for 7-limit.

I think the decatonics are the best theory we have so far and the only way to improve on them is with more experience.

> I really like such an approach. So, I tried to generalise some > principles underlying 5-limit chord progressions. The most thorough > discussion of these which I could find so far are Schoenberg's two > harmony books. Some underlying principles I hope I have found in > Schoenberg's explanation of his guidelines for root progressions. > Therefore, I formalised his explanation, and not his actual rules.
>
> You mention that you could apply his rules to 31 ET. That's of could > right, you can do that (in my former mail I mentioned 12 ET, because > I reacted to Carl dispraise of music theory approaches restricted to > 12 ET). However, you can only create 5-limit root progressions that > way (naturally, you may stagger higher-limit chords on top). My > generalised guidelines, on the other hand, at least formally also > work for 7-limit (or any other limit).

Carl specifically mentioned musical set theory. I don't think that objection applies to music theory in general. Although some of S's ideas led to musical set theory I don't think he was guilty of that way of thinking.

> Now, my question is of could do these generalised guidelines also > result in chord progressions which convince musically? Or did I just > some formal exercise here?
> > I would greatly appreciate comments in critics in this regard (e.g., > by criticising the example progressions I sent in multiple mails > before).

I have PDFs but not audio examples. Are they the optimal voicings?

> Even more so, I would like to hear alternative guidelines for > creating chord progression where root relations are beyond 5-limit.

One of the ideas we had was to follow chords with common tones round the lattice, and arrive at a distinct interval in JI that you temper out. These are "comma pumps". I don't know if it's a way to find good progressions but it's a great way to confuse people who are listening for the roots!

> I should perhaps mention some guideline related to my generalised > guidelines: prefer chord progressions which share common tones (a > harmonic band in Schoenberg's terminology). The guidelines I detailed > before are more precise that this harmonic-band-guideline.

But there's another view, from Rothenberg, that cadences should include all notes in the scale. So it's actually best to avoid common tones where possible.

>> 2) Resolve by stepwise contrary motion
>>
> I'm sorry, but we may talk at cross-purposes here. Melodic motion is > of course also important for chord progressions, but is a different > matter (and you simplify it too :). So far, I was solely talking > about chord root progressions, only a sub-aspect of harmony in > general. Of course, there are other important aspects in harmony, > which I left out for simplicity. For example, there is the treatment > of dissonances (e.g., what is a dissonance, are they prepared and > how, are they resolved and how). This matter is related to root > progressions in the sense that dissonances are often resolved by > strong harmonic progressions (V7 -> 1). There are also melodic > considerations concerning the treatment of dissonances (e.g., > dissonances are most smoothly resolved by a step downwards), but > these might be somewhat less crucial than the harmonic progression > resolving a dissonance (e.g., if V7 is resolved in I but the 7th is > not directly leading into the 3th of I, we may still feel it is OK). > Moreover, there is the aspect of how chords are related to a scale > and how the underlying scale can change (modulation). And there are > further aspects (e.g., chords can be inserted or left out of a > progression etc.).

Margo Schulter gave some medieval examples and stepwise contrary motion was the core idea. I've held that as the unifying rule for harmony in different styles. Yes, root progressions are different, but the Rules of Harmony grow out of good practice that takes account of melody. Sometimes the rules that make good root progressions are there because they lead to good melody. Rather than take those rules literally to new systems you should find new rules that lead to good melody.

> Now, besides all this there is of course also the field of > counterpoint. I assume you are referring to that.

Harmony and counterpoint are different ways of looking at the same thing a lot of the time.

>> The issue of fifths is a tricky one. For a wide range of
>> music, from madrigals to blues, there's an expectation that
>> chords are related by fifths. Whether it's natural or not
>> it's something listeners will expect. But is it possible to
>> establish other relationships that work as well in scales
>> that don't have long chains of fifths?
>>
> I agree in general and certainly for the blues (see also discussion > above). I am not so sure about madrigals though, many go beyond that. > For example, many root progressions in Gesualdo's work are 5-limit.

Gesualdo used different cadences in his madrigals but he did prefer V-I. Some people find his harmony aimless, of course, partly because they hold him to rules that hadn't been invented.

An interesting thing about Gesualdo, though, there are some examples where he implies intervals of a diesis, which would have been 1 step of 31-equal (a plausible tuning if he had a 19 note harpsichord). Easley Blackwood gives some examples. You could also interpret them as being progressions from equal temperament that don't work with common tones in meantone. However, you can mis-spell one of the notes and call that chord a 9-limit triad. That works because the "bad" thirds in meantone have a 9-limit interpretation. So, whatever Gesualdo intended, this shows that there's a family of progressions for which 9-limit harmony is the *solution* to a common tone modulation problem.

>> Next, there's the issue of "what is a step"? In miracle
>> temperament (31&41) I tried to follow the decimal scale. I
>> had rules to ensure "steps" were smaller than "leaps" and I
>> avoided smaller intervals that could be heard as
>> "mistunings". This lead to decimal counterpoint, for which
>> I've tracked down the examples now:
>>
>> http://x31eq.com/music/counterpoint.html
>>
>> I still haven't explained the theory but it's a
>> generalization of Palestrina counterpoint as explained by
>> Jeppesen and Fux. I think it makes sense but it's still
>> counterpoint rather than harmony. There is a tendency for
>> chords or modulations to be related by secors (the large
>> semitones) because they're the generators of the scales.
>>
> I would like to read the theory behind these.

You need to be familiar with decimal notation:

http://x31eq.com/decimal_notation.htm

I have a table of intervals as follows:

C1 | 0* 6* 3^+
C2 | 4^ 3^ 3v 7 8^* 2^* 7^^*
C3 | 4^+ 3v+ 7+ 2^ 5* 5^ 2 2v+ 9v 5v+ 9+
D1 | 2+ 5^+ 2v 1^^* 4v* 6^^* 8* 8^^* 9v+ 2vv+ 3 5v 9 7v
D2 | 1^ 2vv 3+ 3^^* 7v+ 7^* 8^^^* 4+ 9^ 5^^*

The meaning is

C/Dn -- level of consonance/dissonance
Digit -- number of secors (16:15 or 15:14)
^ -- raise by a quomma (between 10 secors and an octave)
v -- lower by a secor
* -- or equivalent in any octave
+ -- only equivalents greater than an octave

The idea is that you shouldn't jump directly from C1 to a dissonance. When you're in C1 then C3 intervals have to be treated as dissonances. Generally with two parts C3 counts as mild dissonance. D2 intervals aren't used much.

I also have an alteration moving 2 from C3 do D1 because I changed my mind at some point.

The simplest melodic rule to explain is that you have a mohajira scale in the decimal nominals:

1 1 2 1 2 1 2

Each note has two or three alternative spellings differing by a quomma. The simplest way of thinking about that is that the whole piece has to be playable within one blackjack scale. You also have to be careful that the 1 1 interval is heard as two steps whereas a 2 interval is a single step. So if you ever leap over the 1 1 you have to immediately fill it in.

Given all that you can generalize your favourite rules of counterpoint in the obvious way.

>> Pitch class sets
>> are a more modern concept that I don't see the need to bring in.
>>
> I am using the pitch class notion in a generalised sense: a pitch > where the octave component is neglected. I understood that this > notion is common in microtonal music as well. For example, it means > exactly the same as Doty's "Primer" denotes with the term > "identity" (e.g., 5/1 is an identity and a pitch class). Using the > notion of sets to refer to multiple pitch classes (identities) allows > then to speak precisely about relations of multiple chords etc. and > formalise such relations. Would you suggest an alternative concept > instead for this purpose?

I'd rather pitch classes included a sense of tuning imprecision but I'm fighting a losing battle on that one.

Graham

--- In tuning@yahoogroups.com, Torsten Anders <torstenanders@...>
wrote:

> On Apr 23, 2008, at 8:34 AM, hstraub64 wrote:

> >
> > Maybe I should mention Mazzola's concept of "cadencial set"
> >
(http://home.datacomm.ch/straub/mamuth/mamufaq.html#Q_cadencialset),
> > which integrates so well with Schoenberg's modulation model
> > (http://home.datacomm.ch/straub/mamuth/mamufaq.html#Q_modul), and
> > which I already have used successfully in microtonal contexts.
> >
> That sounds very interesting. Could you please detail this a bit
> more?
>

The basic idea behind it is actually quite simple and primitive. I
would say it is almost pure musical set theory. A step-by step
introduction of how I use it can be found on

http://home.datacomm.ch/straub/mamuth/modul/ontosu_e.html

(There is also a german version of this page). The example there is
in 12EDO, but the generalization to n-EDO should obvious. There is an
example in 19EDO on

http://home.datacomm.ch/straub/mamuth/modul/wt19_e.html
--
Hans Straub

Graham wrote...

> But there's another view, from Rothenberg, that cadences
> should include all notes in the scale. So it's actually
> best to avoid common tones where possible.

Are you sure that's one of his? He discusses a lot of
things in his papers, but his main thing with cadences
is that they should tell you what key you're in. For
many scales, that doesn't require hearing all the scale
tones.

> Margo Schulter gave some medieval examples and stepwise
> contrary motion was the core idea. I've held that as the
> unifying rule for harmony in different styles. Yes, root
> progressions are different, but the Rules of Harmony grow
> out of good practice that takes account of melody.
> Sometimes the rules that make good root progressions are
> there because they lead to good melody. Rather than take
> those rules literally to new systems you should find new
> rules that lead to good melody.

I do like the notion of voice leading distance (though I
would make it directionless). One interesting thing would
be to find chord progressions which have minimal voice
leading distance but maximal strength according to the
definition I just gave.

-Carl

Dear Carl,

On Apr 23, 2008, at 9:07 PM, Carl Lumma wrote:
> > Even more so, I would like to hear alternative guidelines
> > for creating chord progression where root relations are
> > beyond 5-limit.
>
> Don't want to butt in here between you and Graham, but I
> thought I'd point out that the roots of many higher-limit
> chords are ill-defined by current theory.
>
Thanks for pointing that out, this is certainly a valid point. Still, there are a lot of nice chords beyond 5-limit for which the root is pretty clear.

Also, we might consider to base chord progressions on the prime/fundamental (1/1). That would be less ill-defined. That would change things a lot though. We also would have to decide how to deal with utonalities. Shall we consider them as otonalities (e.g. is it 4/4 4/5 4/6 or 10/8 12/8 15/8) and use the 1/1 of this otonal chord, or should we use the prime of the utonal chord? Also, there are chords which are temperament-specific (i.e. not JI) and for which we cannot find a 1/1.

Best
Torsten

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

--- In tuning@yahoogroups.com, Torsten Anders <torstenanders@...> wrote:
>
> Dear Carl,
>
> On Apr 23, 2008, at 9:07 PM, Carl Lumma wrote:
> > > Even more so, I would like to hear alternative guidelines
> > > for creating chord progression where root relations are
> > > beyond 5-limit.
> >
> > Don't want to butt in here between you and Graham, but I
> > thought I'd point out that the roots of many higher-limit
> > chords are ill-defined by current theory.
>
> Thanks for pointing that out, this is certainly a valid point.
> Still, there are a lot of nice chords beyond 5-limit for which
> the root is pretty clear.

I'm not sure I agree. Already in the 7-limit, the primary
utonal chord is 1/1-7/6-7/5-7/4 according to Partch but
1/1-6/5-3/2-12/7 according to Erlich.

> Also, we might consider to base chord progressions on the
> prime/fundamental (1/1).

What's the 1/1 of 7:9:11? Is it 1, 4, 8, 7 or ...?

-Carl

On Apr 23, 2008, at 9:31 PM, Carl Lumma wrote:
>> I was surprised
> > when I realised that only two (!) 5-chord progressions follow
> > the rule set I described. I would not argue that there are no
> > other good sounding progressions, but I find it remarkable that
> > this rule set only finds particularly common progressions.
>
> If I had a nickel for every paper that starts with the diatonic
> scale, applies some rules, and winds up with the central
> cadences of common practice music, I'd be a rich man. :)
>
Point taken :)

> > Anyway, we can certainly also do non-diatonic 7-limit chord
> > progressions. I did some examples which start in C, use only
> > ascending/strong progressions, wander through keys so to speak,
> > but end again in C major with some sort of a cadence.//
> > Naturally, there exist many such progressions. I uploaded a
> > few examples to
> >
> > /tuning/files/TorstenAnders/
> > ChordProgressions/ChordProgression-*
> >
> > The same limitations are mentioned before apply: no voice-leading
> > rules applied for simplicity and no preparation or resolution of
> > any dissonances.
> >
> > In an actual composition, I would likely further structure a
> > progression, e.g., have some sub-progressions which are repeated
> > in different transpositions. These may then coincide with motif
> > relations. In a first attempt, I did chord progressions where
> > the transposition interval and the chord "type" (i.e. in
> > intervallic relations within the chord) repeat every four
> > chords as in a sequence.
> >
> > I uploaded these examples to
> >
> > /tuning/files/TorstenAnders/
> > ChordProgressions/ChordProg-wSequence-*
> >
> > Comments and recommendations for better approaches are of
> > course appreciated :)
>
> Wonderful work, simply fantastic. One could spend several
> lifetimes writing music around such progressions. I would
> probably have put the examples at a somewhat lower pitch...
> What is your workflow for producing the notation?
>
> "These ratios automatically derived from the 31 ET pitch
> classes"... using Strasheela, I take it. Wow, very powerful.

These examples are just meant as a demonstration of ascending chord progressions which are not restricted by a scale and are beyond 7-limit (which is what you asked for :). I like listening to them too, but already pointed out that they do have problems (e.g., no proper voice-leading, the treatment of inversions is perhaps too simplistic, no dissonance treatment -- although I actually don't feel that the latter is a problem here).

The notation and sound synthesis was created automatically once I had a solution (just a mouse click..).

Here is briefly how this works. The Strasheela user defines a music theory model (a musical constraint problem, a combinatorial problem) by specifying a music representation which contains variables (unknowns) and by applying restrictions (constraints) on these variables. Many ready-to-use score object types are already available for the music representation. In this example, I created a plain sequence of chords. The variables in these chords are their type (e.g., the intervallic relations of the untransposed pitch classes and the untransposed root, can be much more), transposition, root, and the transposed pitch classes. I applied harmonic constraints to these variables (e.g., all chord progression are "ascending", it ends in a cadence etc). The problem is then handed to a constraint solver in order to find one or more solutions. Once we have a solution we can output it into various formats. Here, I used Lilypond for the notation and Csound for the synthesis.

For generating the output, all the information contained in the music representation is available. The Strasheela user has much control over this output process. For example, you can define in a clause-like manner what Lilypond code is generated for a note and what code for a chord etc. For 31 ET I predefined clauses for chord output, and these were used in the examples you saw. Actually, these clauses are pretty simple. The clause has access to the untransposed pitch classes of the chord objects and their root. So, it outputs the root in staff notation and additionally creates these ratios from the untransposed pitch classes and the root. There exists a mapping from 31 ET pitch classes to ratios. As I pointed out in the readme file, this approach can fail, as tempered pitches are not mapped unambiguously to JI pitches. For example, there are some 11-limit root ratios in the notation, but all intervals between chord roots are 9-limit (because there are only 9-limit chords and all chords share common pitch classes). The 11-limit ratios are the result of the temperament ambiguity...

> > BTW, Schoenberg
> > certainly did not directly "use" any of his published
> > theoretical works for his own compositions, as these books
> > are meant for teaching and are much more conservative than
> > his compositions.
>
> I thought his works could (roughly) be divided into
> pre- and post- 12-tone technique periods (?).
>
That is the overall tendency. However, there also exist many tonal compositions he wrote late in his life -- besides dodecaphonic. Also, there exist compositions which are neither tonal nor dodecaphonic, or at the borderline between tonal and non-tonal, and these are among the most interesting I feel.

> > > Here is how I would do it:
> > >
> > > root -> root
> > > CM -> Cmin (2 common tones)
> > >
> > > root -> tone
> > > CM -> Amin (2 common tones)
> > > CM -> FM (1 common tone)
> > > CM -> Fmin (1 common tone)
> > > CM -> AbM (1 common tone)
> > >
> > > tone -> root
> > > CM -> Emin (2 common tones)
> > > CM -> EM (1 common tone)
> > > CM -> GM (1 common tone)
> > > CM -> Gmin (1 common tone)
> > >
> > > tone -> tone
> > > CM -> AM (1 common tone)
> > > CM -> C#min (1 common tone)
> > > CM -> EbM (1 common tone)
> >
> > I am not sure I follow. I assume you mean by "root -> tone"
> > that the root of the second chord was no chord tone in the
> > previous chord,
>
> That the second chord contains the root of the first as a
> non-root tone. It happens that this also means what you say
> here is true.
>
> > and by "tone -> root" that the root of the second chord was
> > some non-root tone of the previous.
>
> Yes.
>
> > But what is then "tone -> tone"? Could you please explain?
>
> Some non-root tone of the first chord is a non-root tone
> in the second. Hopefully the examples will help.
>

If two chords have a different root but common pitches, then their are only two cases possible. Either the root of the new chord did exists (as non-root) in the previous chord (descending), or it did not (ascending or strong progression). All your "tone -> tone" chords can be sorted into these two categories, just try it out :)

> > Just as an illustration, I resort your chords according to
> > the category formalisation I discussed in my first mail
> //
> > constant root "progression"
> > CM -> Cmin (2 common tones)
> >
> > ascending progressions (strong)
> > CM -> Amin (2 common tones)
> > CM -> FM (1 common tone)
> > CM -> Fmin (1 common tone)
> > CM -> AbM (1 common tone)
> > CM -> AM (1 common tone)
> > CM -> C#min (1 common tone) [is superstrong after Schoenberg]
> > CM -> EbM (1 common tone) [is descending after Schoenberg]
> >
> > descending progressions
> > CM -> Emin (2 common tones)
> > CM -> EM (1 common tone)
> > CM -> GM (1 common tone)
> > CM -> Gmin (1 common tone)
>
> Yes. The classification I gave is directional; your
> implementation of Schoenberg's is not.
>
Perhaps I misunderstand you here. However, Schoenberg's chord progression categories are directional. If you reverse an ascending progression it becomes descending and vice versa.

> This means that
> whereas I have distinguished between root->tone and
> tone->root, S. considers them together. So it seems the
> simplest definitions for S.'s terms are:
>
> ascending progression - two chords having at least one
> common tone which is a root in one of the chords
>
Two chords have at least one common tone, but the root of the second is new.

> descending progression - two chords having at least
> one common tone which are not roots of either chord.
>

> Listening to the examples, I don't hear evidence that the
> direction matters, but I thought I should expose it to
> test this.
>
Try playing either II V I or I V II ..

> > > Now: how would we rate these progressions using our ears?
> > > I would be interested in your opinion, but already it is
> > > obvious that this theory cannot distinguish CM -> EM from
> > > CM -> GM, so it is missing something.
> >
> > I see what you mean. Shall we complement the formalisation
> > by some harmonic distance, e.g., distance of the chords on
> > the lattice? (e.g., sum of Tenney harmonic distance of all
> > chord tones divided by the tone number??). EMaj is more
> > remote on the lattice from CMaj than GMaj is.
>
> Yes, this would be one sensible approach.
>
> In fact I would suggest that the strength of the perceived
> change between two chords is the sum of the Tenney-weighted
> harmonic distances between pairs of changing tones in the
> ensemble, arranged so that this sum is minimal. Let me
> explain...
>
> The "ensemble" is the set of pitches needed to perform
> both chords.
>
> "changing pitches": every pitch in the ensemble not
> occurring in both chords.
>
> For a pair of changing pitches, the Tenney-weighted
> harmonic distance between them is
> log2(TenneyHeight(pitchA - pitchB))
> where
> pitchA >= pitchB
> and
> TenneyHeight(p/q) = p*q
>
> For temperaments, we mean the approximations to pitchA
> and pitchB in just intonation.
>
> So in the example CM -> Cmin, the changing pitches are
> E <-> Eb, which are approximately 5/4 <-> 6/5 in just
> intonation. 5/4 - 6/5 = 25/24, and log2(600) = 9.23.
> Compare this to CM -> Amin, where log2(10*9) = 6.49.
>
> If there is only 1 changing pitch in the enemble (for
> example C-E-G-Bb -> C-E-G, where the changing pitch
> is Bb), then the "perceived change strength" between
> the chords shall be zero.
>
> If there are more than 2 changing pitches, we find the
> pairing for which the sum of the harmonic distances is
> minimal. This includes the ability to exclude the most
> distant pitch in case there are an odd number of pitches.
>
> Back in the original example (CM->GM vs. CM->EM), we
> have
> C<->D + E<->B = 8.75 for CM->GM
> and
> G<->B + C<->G# = 12.97 for CM->EM
> So perhaps it's working. We should calculate values for
> all of the progressions in my table and see if they make
> sense.
>
> This method should be completely general for higher limits,
> and would even claim to measure the strength of progressions
> between chords of different sizes (tetrads, triads, etc.)
> on the same scale.
>

Thank you very much for this lecture, I really like it :)

I am not quite convinced yet by (or I don't understand) your proposal how you access the pair(s) of pitches used for computing the "change strength". For example, I feel G79 -> G should also result into 0, even if there are 2 changing pitches. Also, using only the pair with the minimal value results in equal values for the following two chord progressions: CM->Amin vs CM -> F#min. If I am not mistaken, the interval with minimal harmonic distance in both cases is G-A (9/10). Perhaps we better consider some mean (arithmetic mean?) of all pairs of changing pitches? The exception would be chord progressions which simply add or remove pitches (i.e. one set of pitches is a subset of the other). Those should be 0, as you suggested already.

Best
Torsten

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

Dear Graham and Carl,

On Apr 24, 2008, at 4:58 PM, Carl Lumma wrote:
> Graham wrote...
>
> > But there's another view, from Rothenberg, that cadences
> > should include all notes in the scale. So it's actually
> > best to avoid common tones where possible.
>
> Are you sure that's one of his? He discusses a lot of
> things in his papers, but his main thing with cadences
> is that they should tell you what key you're in. For
> many scales, that doesn't require hearing all the scale
> tones.
>
Could you please point me to the reference of this text?

> > Margo Schulter gave some medieval examples and stepwise
> > contrary motion was the core idea. I've held that as the
> > unifying rule for harmony in different styles. Yes, root
> > progressions are different, but the Rules of Harmony grow
> > out of good practice that takes account of melody.
> > Sometimes the rules that make good root progressions are
> > there because they lead to good melody. Rather than take
> > those rules literally to new systems you should find new
> > rules that lead to good melody.
>
> I do like the notion of voice leading distance (though I
> would make it directionless). One interesting thing would
> be to find chord progressions which have minimal voice
> leading distance but maximal strength according to the
> definition I just gave.
>
Sorry, I don't follow. What do you mean by voice leading distance?

Thanks!

Torsten

>
>
> -Carl
>
>
>
--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

Dear Graham,

On Apr 24, 2008, at 2:46 PM, Graham Breed wrote:
> Torsten Anders wrote:
> > It would be interesting to explore how we can create convincing > chord
> > progressions where the roots are not all "stringed" on the 3-
> > dimension of the lattice. 19th century harmony did this a lot, and
> > this is also a domain explored in Schoenberg's harmony (his
> > guidelines are not restricted to fifths progressions).
>
> The simplest thing in magic is to go up in thirds and then
> down a fifth.
>
?? I take it you mean temperament here. However, you can do that in other temperaments too like 12 ET or meantone. Or am I missing something here?

> > Yet, 19th century harmony is of course restricted to 5-limit -- its
> > root progressions certainly are. Meanwhile, we realised that 7-limit
> > and beyond can be fruitful to for harmony. So, my question is how > can
> > we create convincing chord progressions beyond 5-limit? Can we find
> > some guidelines which simplify the composition process, much like > the
> > guidelines of 5-limit harmony did for former centuries.
>
> 9-limit chords are interesting. You get a 6:7:9 subminor
> triad with a bluesey feel that seems to work well enough as
> well as its opposite, the car horn supermajor triad.
> Because they're still traids they have clear roots and
> follow voice leading rules. 7 odd-limit triads (other than
> the 5-limit) don't fit so well.
>

I love the 6:7:9 subminor triad :) Supermajor needs getting used to though.. I assume they have a clear root because they have a fifths (Hindemith's explanation). We can have 7-limit tetrads with clear root too.

However, I actually meant chord progressions where the intervals between chord roots (or alternatively their fundamentals, their relative 1/1) are beyond 7-limit. A comparative 5-limit example are mediant chords: in C-major going to Ab-maj, Eb-maj and similar progressions. I am looking for guidelines to creating convincing 7-limit root progressions.

I proposed some and showed some examples, but I feel this can be improved...

> > Now, my question is of could do these generalised guidelines also
> > result in chord progressions which convince musically? Or did I just
> > some formal exercise here?
> >
> > I would greatly appreciate comments in critics in this regard (e.g.,
> > by criticising the example progressions I sent in multiple mails
> > before).
>
> I have PDFs but not audio examples. Are they the optimal
> voicings?
>
The audio examples are available just next to the PDF files. And I agree (as discussed before), they are not optimal voicings. Perhaps for demonstrations like this I better bring all chords in root position, as I did in the first examples I posted.

> > I should perhaps mention some guideline related to my generalised
> > guidelines: prefer chord progressions which share common tones (a
> > harmonic band in Schoenberg's terminology). The guidelines I > detailed
> > before are more precise that this harmonic-band-guideline.
>
> But there's another view, from Rothenberg, that cadences
> should include all notes in the scale. So it's actually
> best to avoid common tones where possible.
>
It turns out, these two guidelines work well together: all my examples I posted in this thread (including the diatonic progressions in the beginning) end in a cadence defined exactly like that: bring all tones of the scale. Still, these chord progressions are ascending (e.g. share common pitches but bring a new root pitch), and these are the most common cadences, e.g., II V I.

BTW: what is the reference for Rothenberg paper/book you are referring to?

> >> 2) Resolve by stepwise contrary motion
> >>
> > I'm sorry, but we may talk at cross-purposes here. Melodic motion is
> > of course also important for chord progressions, but is a different
> > matter (and you simplify it too :). So far, I was solely talking
> > about chord root progressions, only a sub-aspect of harmony in
> > general. Of course, there are other important aspects in harmony,
> > which I left out for simplicity. For example, there is the treatment
> > of dissonances (e.g., what is a dissonance, are they prepared and
> > how, are they resolved and how). This matter is related to root
> > progressions in the sense that dissonances are often resolved by
> > strong harmonic progressions (V7 -> 1). There are also melodic
> > considerations concerning the treatment of dissonances (e.g.,
> > dissonances are most smoothly resolved by a step downwards), but
> > these might be somewhat less crucial than the harmonic progression
> > resolving a dissonance (e.g., if V7 is resolved in I but the 7th is
> > not directly leading into the 3th of I, we may still feel it is OK).
> > Moreover, there is the aspect of how chords are related to a scale
> > and how the underlying scale can change (modulation). And there are
> > further aspects (e.g., chords can be inserted or left out of a
> > progression etc.).
>
> Margo Schulter gave some medieval examples and stepwise
> contrary motion was the core idea. I've held that as the
> unifying rule for harmony in different styles. Yes, root
> progressions are different, but the Rules of Harmony grow
> out of good practice that takes account of melody.
> Sometimes the rules that make good root progressions are
> there because they lead to good melody. Rather than take
> those rules literally to new systems you should find new
> rules that lead to good melody.
>

I feel you over-simplify a bit here. By referring to medieval examples you made it rather specific that you are talking about counterpoint (that is, the family of medieval "counterpoints", as there are multiple medieval contrapuntal styles with different rule sets). As you certainly know, harmony is a much younger idea, not arriving before early Baroque. Certainly, some aspects of pre-Baroque counterpoint already contain harmonic considerations, in particular the cadences. Also, in Josquin music I found that ascending "chord" progressions are more likely than descending progressions.

Nevertheless, I feel it is a good idea to look at each aspect carefully in his own right. Yes, there are some chord progressions which are closely couples with melodic progressions. The Neapolitanian sixth chord is a good example. Still, it is somewhat simplifying to quasi say "focus on writing good melody, that will lead to good harmony" :) Actually, in much Western music there is a tendency the other way round, take Blues, cadentical 18-century music or 19th century music like Wagner.

> Harmony and counterpoint are different ways of looking at
> the same thing a lot of the time.
>
:) Certainly, harmony and counterpoint influence each other. However, they are precisely looking at the same thing from different points of view. Consequently, we can end up with rather different results depending on whether counterpoint or harmony played a more important role -- as can be seen throughout Western music history. But this kind of discussion is perhaps leading nowhere, because we are not specific here :)

> >> The issue of fifths is a tricky one. For a wide range of
> >> music, from madrigals to blues, there's an expectation that
> >> chords are related by fifths. Whether it's natural or not
> >> it's something listeners will expect. But is it possible to
> >> establish other relationships that work as well in scales
> >> that don't have long chains of fifths?
> >>
> > I agree in general and certainly for the blues (see also discussion
> > above). I am not so sure about madrigals though, many go beyond > that.
> > For example, many root progressions in Gesualdo's work are 5-limit.
>
> Gesualdo used different cadences in his madrigals but he did
> prefer V-I. Some people find his harmony aimless, of
> course, partly because they hold him to rules that hadn't
> been invented.
>
> An interesting thing about Gesualdo, though, there are some
> examples where he implies intervals of a diesis, which would
> have been 1 step of 31-equal (a plausible tuning if he had a
> 19 note harpsichord). Easley Blackwood gives some examples.
> You could also interpret them as being progressions from
> equal temperament that don't work with common tones in
> meantone. However, you can mis-spell one of the notes and
> call that chord a 9-limit triad. That works because the
> "bad" thirds in meantone have a 9-limit interpretation. So,
> whatever Gesualdo intended, this shows that there's a family
> of progressions for which 9-limit harmony is the *solution*
> to a common tone modulation problem.
>
Thanks for pointing this out! Could you perhaps give a reference of the Blackwood text, or provide the chord progression so that I can follow you better. What 9-limit chords do you find in Gesualdo, 6:7:9 ??

Also, I don't quite understand what you mean by "a family
of progressions for which 9-limit harmony is the *solution*
to a common tone modulation problem."

> >> Next, there's the issue of "what is a step"? In miracle
> >> temperament (31&41) I tried to follow the decimal scale. I
> >> had rules to ensure "steps" were smaller than "leaps" and I
> >> avoided smaller intervals that could be heard as
> >> "mistunings". This lead to decimal counterpoint, for which
> >> I've tracked down the examples now:
> >>
> >> http://x31eq.com/music/counterpoint.html
> >>
> >> I still haven't explained the theory but it's a
> >> generalization of Palestrina counterpoint as explained by
> >> Jeppesen and Fux. I think it makes sense but it's still
> >> counterpoint rather than harmony. There is a tendency for
> >> chords or modulations to be related by secors (the large
> >> semitones) because they're the generators of the scales.
> >>
> > I would like to read the theory behind these.
>
> You need to be familiar with decimal notation:
>
> http://x31eq.com/decimal_notation.htm
>
> I have a table of intervals as follows:
>
> C1 | 0* 6* 3^+
> C2 | 4^ 3^ 3v 7 8^* 2^* 7^^*
> C3 | 4^+ 3v+ 7+ 2^ 5* 5^ 2 2v+ 9v 5v+ 9+
> D1 | 2+ 5^+ 2v 1^^* 4v* 6^^* 8* 8^^* 9v+ 2vv+ 3 5v 9 7v
> D2 | 1^ 2vv 3+ 3^^* 7v+ 7^* 8^^^* 4+ 9^ 5^^*
>
> The meaning is
>
> C/Dn -- level of consonance/dissonance
> Digit -- number of secors (16:15 or 15:14)
> ^ -- raise by a quomma (between 10 secors and an octave)
> v -- lower by a secor
> * -- or equivalent in any octave
> + -- only equivalents greater than an octave
>
> The idea is that you shouldn't jump directly from C1 to a
> dissonance. When you're in C1 then C3 intervals have to be
> treated as dissonances. Generally with two parts C3 counts
> as mild dissonance. D2 intervals aren't used much.
>
> I also have an alteration moving 2 from C3 do D1 because I
> changed my mind at some point.
>
> The simplest melodic rule to explain is that you have a
> mohajira scale in the decimal nominals:
>
> 1 1 2 1 2 1 2
>
> Each note has two or three alternative spellings differing
> by a quomma. The simplest way of thinking about that is
> that the whole piece has to be playable within one blackjack
> scale. You also have to be careful that the 1 1 interval is
> heard as two steps whereas a 2 interval is a single step.
> So if you ever leap over the 1 1 you have to immediately
> fill it in.
>
> Given all that you can generalize your favourite rules of
> counterpoint in the obvious way.
>
Thanks for these details. Slowly changing the degree of dissonance makes perfectly sense to me. I don't understand your scale example though: if these are nominals, does the scale consist of only two different and repeated pitches? I don't understand why 1 1 is an interval larger than unison and heard as two steps.

> >> Pitch class sets
> >> are a more modern concept that I don't see the need to bring in.
> >>
> > I am using the pitch class notion in a generalised sense: a pitch
> > where the octave component is neglected. I understood that this
> > notion is common in microtonal music as well. For example, it means
> > exactly the same as Doty's "Primer" denotes with the term
> > "identity" (e.g., 5/1 is an identity and a pitch class). Using the
> > notion of sets to refer to multiple pitch classes (identities) > allows
> > then to speak precisely about relations of multiple chords etc. and
> > formalise such relations. Would you suggest an alternative concept
> > instead for this purpose?
>
> I'd rather pitch classes included a sense of tuning
> imprecision but I'm fighting a losing battle on that one.
>
Oh, I have no problem with that idea. I agree that we can tune the identity/pitchclass 5/1 in various ways and still recognise it as third over 1/1.

Best
Torsten

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

Hi Torsten,

> > > > Here is how I would do it:
> > > >
> > > > root -> root
> > > > CM -> Cmin (2 common tones)
> > > >
> > > > root -> tone
> > > > CM -> Amin (2 common tones)
> > > > CM -> FM (1 common tone)
> > > > CM -> Fmin (1 common tone)
> > > > CM -> AbM (1 common tone)
> > > >
> > > > tone -> root
> > > > CM -> Emin (2 common tones)
> > > > CM -> EM (1 common tone)
> > > > CM -> GM (1 common tone)
> > > > CM -> Gmin (1 common tone)
> > > >
> > > > tone -> tone
> > > > CM -> AM (1 common tone)
> > > > CM -> C#min (1 common tone)
> > > > CM -> EbM (1 common tone)
//
> > > Just as an illustration, I resort your chords according to
> > > the category formalisation I discussed in my first mail
> > //
> > > constant root "progression"
> > > CM -> Cmin (2 common tones)
> > >
> > > ascending progressions (strong)
> > > CM -> Amin (2 common tones)
> > > CM -> FM (1 common tone)
> > > CM -> Fmin (1 common tone)
> > > CM -> AbM (1 common tone)
> > > CM -> AM (1 common tone)
> > > CM -> C#min (1 common tone) [is superstrong after Schoenberg]
> > > CM -> EbM (1 common tone) [is descending after Schoenberg]
> > >
> > > descending progressions
> > > CM -> Emin (2 common tones)
> > > CM -> EM (1 common tone)
> > > CM -> GM (1 common tone)
> > > CM -> Gmin (1 common tone)
> >
> > Yes. The classification I gave is directional; your
> > implementation of Schoenberg's is not.
> >
> Perhaps I misunderstand you here. However, Schoenberg's chord
> progression categories are directional. If you reverse an
> ascending progression it becomes descending and vice versa.

I meant, the difference between the two lists above
is only that my root->tone and tone->root groups were
merged into "ascending". My other groups are like->like,
so they can't be directional with respect to the type of
the common tone(s).

> > Listening to the examples, I don't hear evidence that the
> > direction matters, but I thought I should expose it to
> > test this.
>
> Try playing either II V I or I V II ..

That's a 3-chord progression. I V and V I sound the same.

> > In fact I would suggest that the strength of the perceived
> > change between two chords is the sum of the Tenney-weighted
> > harmonic distances between pairs of changing tones in the
> > ensemble, arranged so that this sum is minimal. Let me
> > explain...
> >
> > The "ensemble" is the set of pitches needed to perform
> > both chords.
> >
> > "changing pitches": every pitch in the ensemble not
> > occurring in both chords.
> >
> > For a pair of changing pitches, the Tenney-weighted
> > harmonic distance between them is
> > log2(TenneyHeight(pitchA - pitchB))
> > where
> > pitchA >= pitchB
> > and
> > TenneyHeight(p/q) = p*q
> >
> > For temperaments, we mean the approximations to pitchA
> > and pitchB in just intonation.
> >
> > So in the example CM -> Cmin, the changing pitches are
> > E <-> Eb, which are approximately 5/4 <-> 6/5 in just
> > intonation. 5/4 - 6/5 = 25/24, and log2(600) = 9.23.
> > Compare this to CM -> Amin, where log2(10*9) = 6.49.
> >
> > If there is only 1 changing pitch in the enemble (for
> > example C-E-G-Bb -> C-E-G, where the changing pitch
> > is Bb), then the "perceived change strength" between
> > the chords shall be zero.
> >
> > If there are more than 2 changing pitches, we find the
> > pairing for which the sum of the harmonic distances is
> > minimal. This includes the ability to exclude the most
> > distant pitch in case there are an odd number of pitches.
> >
> > Back in the original example (CM->GM vs. CM->EM), we
> > have
> > C<->D + E<->B = 8.75 for CM->GM
> > and
> > G<->B + C<->G# = 12.97 for CM->EM
> > So perhaps it's working. We should calculate values for
> > all of the progressions in my table and see if they make
> > sense.
> >
> > This method should be completely general for higher limits,
> > and would even claim to measure the strength of progressions
> > between chords of different sizes (tetrads, triads, etc.)
> > on the same scale.
>
> Thank you very much for this lecture, I really like it :)
>
> I am not quite convinced yet by (or I don't understand) your
> proposal how you access the pair(s) of pitches used for
> computing the "change strength". For example, I feel
> G79 -> G should also result into 0, even if there are
> 2 changing pitches.

You're right -- pooling all the changing pitches together
isn't enough. One must keep track of them according to the
chord they came from, making sure to take one from each
pool for every pair. So it your example, Bb and D would be
cast out and the "change strength" would be zero.

> Also, using only the pair with the minimal value results
> in equal values for the following two chord progressions:
> CM->Amin vs CM -> F#min. If I am not mistaken, the
> interval with minimal harmonic distance in both cases is
> G-A (9/10).

So CM->Amin is log2(9*10) = 6.49 (as above). CM->F#min
has no common tones, so the sum of pairs of changing tones
will surely be > 7. You'll have to tell me the ratio of
F# to C and then I can calculate it exactly.

> Perhaps we better consider some mean (arithmetic
> mean?) of all pairs of changing pitches?

I don't think this would add information. Instead, it would
put triads, tetrads, etc. each on their own scale. Doing
it my way, we can compare chords of different sizes directly.
In other words, the strength of triad->triad could be
directly compared to the strength of tetrad->tetrad.

-Carl

Dear Hans Straub,

thanks for pointing me to this explanation.

I may come back later to you with more questions concerning MaMuTh. As my background is not math, I have much difficulties to understand the original papers of Mazzola and his followers and colleagues. Thomas Noll once spend quite some time kindly explaining me some fundamentals, but without much success unfortunately. Your writing, on the other hand, is very accessible to me :)

Best
Torsten

On Apr 24, 2008, at 4:05 PM, hstraub64 wrote:
--- In tuning@yahoogroups.com, Torsten Anders <torstenanders@...>
> wrote:
>
> > On Apr 23, 2008, at 8:34 AM, hstraub64 wrote:
>
> > >
> > > Maybe I should mention Mazzola's concept of "cadencial set"
> > >
> (http://home.datacomm.ch/straub/mamuth/mamufaq.html#Q_cadencialset),
> > > which integrates so well with Schoenberg's modulation model
> > > (http://home.datacomm.ch/straub/mamuth/mamufaq.html#Q_modul), and
> > > which I already have used successfully in microtonal contexts.
> > >
> > That sounds very interesting. Could you please detail this a bit
> > more?
> >
>
> The basic idea behind it is actually quite simple and primitive. I
> would say it is almost pure musical set theory. A step-by step
> introduction of how I use it can be found on
>
> http://home.datacomm.ch/straub/mamuth/modul/ontosu_e.html
>
> (There is also a german version of this page). The example there is
> in 12EDO, but the generalization to n-EDO should obvious. There is an
> example in 19EDO on
>
> http://home.datacomm.ch/straub/mamuth/modul/wt19_e.html
> --> Hans Straub
>
>
>
--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

On Apr 24, 2008, at 9:42 PM, Carl Lumma wrote:
> --- In tuning@yahoogroups.com, Torsten Anders <torstenanders@...> > wrote:
> >
> > Dear Carl,
> >
> > On Apr 23, 2008, at 9:07 PM, Carl Lumma wrote:
> > > > Even more so, I would like to hear alternative guidelines
> > > > for creating chord progression where root relations are
> > > > beyond 5-limit.
> > >
> > > Don't want to butt in here between you and Graham, but I
> > > thought I'd point out that the roots of many higher-limit
> > > chords are ill-defined by current theory.
> >
> > Thanks for pointing that out, this is certainly a valid point.
> > Still, there are a lot of nice chords beyond 5-limit for which
> > the root is pretty clear.
>
> I'm not sure I agree. Already in the 7-limit, the primary
> utonal chord is 1/1-7/6-7/5-7/4 according to Partch but
> 1/1-6/5-3/2-12/7 according to Erlich.
>
> > Also, we might consider to base chord progressions on the
> > prime/fundamental (1/1).
>
> What's the 1/1 of 7:9:11? Is it 1, 4, 8, 7 or ...?
>

Following Doty's Primer (and 19th/20th century dualistic harmony, e.g., Karg-Elert and I assume also Riemann), I suggest we should distinguish between the root and the fundamental. Doty's root definition: "the lowest voice when the chord is arranged in the order which yields the lowest [integer] numbers from the harmonic series". This definition is related to Hindemith's definition based on the lowest most simple interval in the chord. Also, for the root we should assume octave equivalence (i.e. the root is a pitch class, expressed by a ratio). So, all ratios 1, 4, and 8 are equivalent.

So, following Doty's definition, the root of 7:9:11 is 7, but the fundamental is 1 (= 4 = 8). For me, the problem remain utonal chords like 1/1-7/6-7/5-7/4. I would agree with Erlich that the root is the lower tone of the fifth (i.e. 7/6 in the utonal setting). However, there are two candidates for the fundamental: the 1/1 of the utonal setting and 1/1 of the otonal setting where the chord is only expressed in terms of integers. Anyway, for Partch you are referring to the utonal fundamental above.

Best
Torsten

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

On Apr 25, 2008, at 12:23 AM, Carl Lumma wrote:
> > Try playing either II V I or I V II ..
>
> That's a 3-chord progression. I V and V I sound the same.
>
??
> > Also, using only the pair with the minimal value results
> > in equal values for the following two chord progressions:
> > CM->Amin vs CM -> F#min. If I am not mistaken, the
> > interval with minimal harmonic distance in both cases is
> > G-A (9/10).
>
> So CM->Amin is log2(9*10) = 6.49 (as above). CM->F#min
> has no common tones, so the sum of pairs of changing tones
> will surely be > 7. You'll have to tell me the ratio of
> F# to C and then I can calculate it exactly.
>
Sorry if I misunderstood you. But as far as I understood you, the minimal harmonic distance for CM -> F#min is the interval between the two notes G-A (9/10). So, we would have the same "change strength" as for CM->Amin.

What am I missing?

Thank you!

Best
Torsten
>
>
> -Carl
>
>
>
--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

> > > But there's another view, from Rothenberg, that cadences
> > > should include all notes in the scale. So it's actually
> > > best to avoid common tones where possible.
> >
> > Are you sure that's one of his? He discusses a lot of
> > things in his papers, but his main thing with cadences
> > is that they should tell you what key you're in. For
> > many scales, that doesn't require hearing all the scale
> > tones.
>
> Could you please point me to the reference of this text?

I can do better than that!
http://lumma.org/tuning/rothenberg/AModelForPatternPerception.pdf

> > > Margo Schulter gave some medieval examples and stepwise
> > > contrary motion was the core idea. I've held that as the
> > > unifying rule for harmony in different styles. Yes, root
> > > progressions are different, but the Rules of Harmony grow
> > > out of good practice that takes account of melody.
> > > Sometimes the rules that make good root progressions are
> > > there because they lead to good melody. Rather than take
> > > those rules literally to new systems you should find new
> > > rules that lead to good melody.
> >
> > I do like the notion of voice leading distance (though I
> > would make it directionless). One interesting thing would
> > be to find chord progressions which have minimal voice
> > leading distance but maximal strength according to the
> > definition I just gave.
>
> Sorry, I don't follow. What do you mean by voice leading
> distance?

I thought this was a pretty well-established thing, but
maybe not. It's the sum of the cents change between pairs
of changing pitches, arranged so that the sum is minimal.

So CM->GM is C<->B + E<->D = 300 cents
CM->EM is C<->B + G<->G# = 200 cents
(in 12-ET)

This minimal sum also tells the smoothest voice leading.

So here's a workflow:

1. Find all p-limit n-ads.
2. Find all pairs of the chords from step 1 (2-combinations).
3. Calculate the "change strength" and voice leading
distance for each of the pairs from step 2.
4. Sort the chord pairs from step 2 by increasing voice
leading distance, and keep only the top m chords.
5. Divide the chords from step 4 into two groups of
equal size around the median chord "change strength".
6. Construct progressions by connecting pairs together,
e.g. C->G + G->F = C->G->F. You can make it so that the
pairs connected are all from the group with high change
strength, all from the group with low change strength,
or alternating between the two groups, etc.

So what I suggested in my previous method would be to
complete step 6 with the choice of high change strength.
Melodically it would be smooth, but harmonically it
would be jarring. ;)

-Carl

Torsten wrote...

> > Already in the 7-limit, the primary
> > utonal chord is 1/1-7/6-7/5-7/4 according to Partch but
> > 1/1-6/5-3/2-12/7 according to Erlich.
//
> > What's the 1/1 of 7:9:11? Is it 1, 4, 8, 7 or ...?
>
> Following Doty's Primer (and 19th/20th century dualistic
> harmony, e.g., Karg-Elert and I assume also Riemann), I
> suggest we should distinguish between the root and the
> fundamental. Doty's root definition: "the lowest voice
> when the chord is arranged in the order which yields the
> lowest [integer] numbers from the harmonic series".

Nice definition but I think it fails on the utonal 7-limit
tetrad (if you're like me and agree with Erlich).

Partch: 60:72:90:105
Erlich: 70:84:105:120

I hope I did that right. The problem is that, precisely
as you say with Hindemith's fifths, very simple (and
therefore strong) dyads can hang out in high harmonic
series representations.

> For me, the problem remain utonal chords
> like 1/1-7/6-7/5-7/4. I would agree with Erlich that the
> root is the lower tone of the fifth (i.e. 7/6 in the utonal
> setting). However, there are two candidates for the
> fundamental:

That too.

-Carl

Torsten Anders <torstenanders@...> wrote:
> > > Try playing either II V I or I V II ..
> >
> > That's a 3-chord progression. I V and V I sound the same.
>
> ??

I was proposing that 2-chord progressions are not
directional. I->V and V->I both sound the same to me.
Or to put it another way, IV->I and V->I sound the
same (in terms of change strength).

> > > Also, using only the pair with the minimal value results
> > > in equal values for the following two chord progressions:
> > > CM->Amin vs CM -> F#min. If I am not mistaken, the
> > > interval with minimal harmonic distance in both cases is
> > > G-A (9/10).
> >
> > So CM->Amin is log2(9*10) = 6.49 (as above). CM->F#min
> > has no common tones, so the sum of pairs of changing tones
> > will surely be > 7. You'll have to tell me the ratio of
> > F# to C and then I can calculate it exactly.
>
> Sorry if I misunderstood you. But as far as I understood you,
> the minimal harmonic distance for CM -> F#min is the interval
> between the two notes G-A (9/10). So, we would have the same
> "change strength" as for CM->Amin.
>
> What am I missing?

Sorry for explaining poorly! It's not just G<->A,
but the sum
C<->C# + E<->F# + G<->A
or the sum
C<->F# + E<->A + G<->C#
or any other similar sum; whichever is least.

-Carl

Chord progressions are what kept me interested in music as long as I have.
So this conversation got my attention. It's kind of interesting that an
atonal composer such as Shoenberg wrote papers on tonal music theory.

Much of the rules I read are from the books "Elementary Harmony" and
"Advanced Harmony" which gives a kind of conservative approach that starts
you out on the "commmon practice" (baroque) harmonies and progresses to
music verging on jazz.

By the way the 2nd of the two books mentioned the Italian sixth (FAD#),
German sixth (FACD#) and French sixth (FABD#) chords, and how they resolve.
The only chord neither they nor anyone else have named is the mirror image
of the German sixth chord, which would resolve FG#BD# -> EACE

But what I would like to see is a document of "rules of music" that can make
sense of why certain chord progressions affect us the way they do. For
instance, why is the German sixth and its resolution so utterly beautiful?
The power of the authentic cadence could be explained mathematically simply
by observing that the root moves up in a 3:4 ratio. The German sixth
resolves to major by a root movement of 4:5. And resolution of a German
sixth to minor is strikingly simple from a mathmatical standpoint in that
the real fundamental pitch class doesn't even move. Does Shoenberg attempt
to explain mathematically why certain musical progressions make more sense
to the western ear and others don't? Such a work I'd like to see.

<<< For a wide range of music, from madrigals to blues, there's an
expectation that chords are related by fifths. Whether it's natural or not
it's something listeners will expect. >>>

The circle of fifths do convey a strong sense of a musical statement.
Non-cadencing non-tertian harmonies of the 20th century sound diluted to me
because of the lack of that powerful tension and resolution. It's like
hearing a language you've never learned and you can't hear the
subject-predicate relationship.

<<< Sure, decending fifths are the most important root progression in many
musical styles. For example, in Jazz you can have astonishingly complex
chord progressions which are all understood as II V I progressions. >>>

And some of these can be found disguised. Using the "altered" jazz chord,
which merely puts a new root a tritone under a chord to create a new one,
you hear a bass going down by half-steps in what is in essence a
circle-of-fifths progression.

<<< We can directly apply the idea to microtonal harmony where we could have
root progressions only along the 3-dimension of the lattice and then stagger
various chords on top of these roots. Of course, we cannot have only
descending fifths in JI (we would never get "back"), but we can write music
where most root progressions are of this kind. >>>

When applying songs that have a strong circle of 5ths progression to JI
interpretations, such as "Five Foot Two", use of Pythagorean tunings for
chord roots lets you use just fifths to "take you home". If you use 5-limit
tunings for 3 or 6 chords, you will need at least one wolf 5th on the return
trip. Maybe a person with a well-tuned ear will sense if the III chord will
be going the circle-of-5ths route to tonic, and sharp it to a Pythagorean
3rd unconsciously, knowing that this is in fact a V/V/V/V chord.

<<< Yet, 19th century harmony is of course restricted to 5-limit -- its root
progressions certainly are. Meanwhile, we realised that 7-limit and beyond
can be fruitful to for harmony. So, my question is how can we create
convincing chord progressions beyond 5-limit? Can we find some guidelines
which simplify the composition process, much like the guidelines of 5-limit
harmony did for former centuries. >>>

I but wonder if some of the blues-style harmonies may be an example of this.
For instance, Mood Indigo has ended in a dominant 7th chord that doesn't
need to be resolved.

<<< I really like such an approach. So, I tried to generalise some
principles underlying 5-limit chord progressions. The most thorough
discussion of these which I could find so far are Schoenberg's two harmony
books. Some underlying principles I hope I have found in Schoenberg's
explanation of his guidelines for root progressions. Therefore, I formalised
his explanation, and not his actual rules. >>>

What is the names of his two harmony books. Surely those aren't the two I
read are they? It was so long.

<<< Even more so, I would like to hear alternative guidelines for creating
chord progression where root relations are beyond 5-limit. >>>

I see examples of this in vocal music that features a pedaltone in one of
the parts, such as a barbershop tag. For instance, if one part is holding
the tonic note while the chord moves from a tonic to a II7, the root will
need to move up a Septimal-Major second (8/7) to keep the chords tuned
without having to bend the pedaltone.

Billy

Torsten Anders wrote:
> Dear Graham,
> > On Apr 24, 2008, at 2:46 PM, Graham Breed wrote:
>> Torsten Anders wrote:
>>> It would be interesting to explore how we can create convincing >> chord
>>> progressions where the roots are not all "stringed" on the 3-
>>> dimension of the lattice. 19th century harmony did this a lot, and
>>> this is also a domain explored in Schoenberg's harmony (his
>>> guidelines are not restricted to fifths progressions).
>> The simplest thing in magic is to go up in thirds and then
>> down a fifth.
>>
> ?? I take it you mean temperament here. However, you can do that in > other temperaments too like 12 ET or meantone. Or am I missing > something here?

I mean magic temperament, consistent with 19, 22, or 41 ET, where the generator is a 5:4 and five 5:4s give one 3:2 (octave equivalently speaking). So after going up fifth thirds you naturally find yourself a fifth above where you started. In meantone if you follow 5 major thirds you'll fall of the end of the keyboard. In 12ET you'll end up a third below where you started.

> I love the 6:7:9 subminor triad :) Supermajor needs getting used to > though.. I assume they have a clear root because they have a fifths > (Hindemith's explanation). We can have 7-limit tetrads with clear > root too.

The best rule I know is that the fifth points to the root. That's following this argument:

http://www.mmk.e-technik.tu-muenchen.de/persons/ter/top/basse.html

It leaves a lot of chords without a clear root and I prefer to keep them ambiguous than to give them an arbitrary root. Hence augmented triads seem to be important in magic so they're likely to be used in cadences. But how you choose the root progression is arbitrary. The same applies to diminished seventh chords in 12-equal.

> However, I actually meant chord progressions where the intervals > between chord roots (or alternatively their fundamentals, their > relative 1/1) are beyond 7-limit. A comparative 5-limit example are > mediant chords: in C-major going to Ab-maj, Eb-maj and similar > progressions. I am looking for guidelines to creating convincing 7- > limit root progressions.

That's where it's useful to start with scales or tunes that naturally include 7-limit intervals. Make the harmony serve the melody.

> I proposed some and showed some examples, but I feel this can be > improved...
> > >>> Now, my question is of could do these generalised guidelines also
>>> result in chord progressions which convince musically? Or did I just
>>> some formal exercise here?
>>>
>>> I would greatly appreciate comments in critics in this regard (e.g.,
>>> by criticising the example progressions I sent in multiple mails
>>> before).
>> I have PDFs but not audio examples. Are they the optimal
>> voicings?
>>
> The audio examples are available just next to the PDF files. And I > agree (as discussed before), they are not optimal voicings. Perhaps > for demonstrations like this I better bring all chords in root > position, as I did in the first examples I posted.

Oh, in the Files section! They're good enough to make music with, I'm sure. But they're quite long and I don't get a sense of returning home at the end. Maybe there should be shorter excursions from the tonic at the start.

I like the idea of your software. Choose some rules, generate some music, and see if it makes sense. That saves having to write a lot of music without knowing what rules to follow and helps us to bootstrap a microtonal tradition. I'm more interested in optimizing a loss function than obeying constraints, though. Perhaps one day I'll be hubristic enough to write my own software to do that.

>>> I should perhaps mention some guideline related to my generalised
>>> guidelines: prefer chord progressions which share common tones (a
>>> harmonic band in Schoenberg's terminology). The guidelines I >> detailed
>>> before are more precise that this harmonic-band-guideline.
>> But there's another view, from Rothenberg, that cadences
>> should include all notes in the scale. So it's actually
>> best to avoid common tones where possible.
>>
> It turns out, these two guidelines work well together: all my > examples I posted in this thread (including the diatonic progressions > in the beginning) end in a cadence defined exactly like that: bring > all tones of the scale. Still, these chord progressions are ascending > (e.g. share common pitches but bring a new root pitch), and these are > the most common cadences, e.g., II V I.
> > BTW: what is the reference for Rothenberg paper/book you are > referring to?

Carl's given the URL for his web copy. And he's right that I had Rothenberg wrong. He said you need to include the notes that uniquely identify the scale, not all notes.

>> Margo Schulter gave some medieval examples and stepwise
>> contrary motion was the core idea. I've held that as the
>> unifying rule for harmony in different styles. Yes, root
>> progressions are different, but the Rules of Harmony grow
>> out of good practice that takes account of melody.
>> Sometimes the rules that make good root progressions are
>> there because they lead to good melody. Rather than take
>> those rules literally to new systems you should find new
>> rules that lead to good melody.
> > I feel you over-simplify a bit here. By referring to medieval > examples you made it rather specific that you are talking about > counterpoint (that is, the family of medieval "counterpoints", as > there are multiple medieval contrapuntal styles with different rule > sets). As you certainly know, harmony is a much younger idea, not > arriving before early Baroque. Certainly, some aspects of pre-Baroque > counterpoint already contain harmonic considerations, in particular > the cadences. Also, in Josquin music I found that ascending "chord" > progressions are more likely than descending progressions.

The Rules of Harmony were newer, but grew out of counterpoint. And sometimes the Rules were codifying things that the Rules of Counterpoint didn't specify but composers did anyway.

Probably ascending/descending is an arbitrary convention.

> Nevertheless, I feel it is a good idea to look at each aspect > carefully in his own right. Yes, there are some chord progressions > which are closely couples with melodic progressions. The > Neapolitanian sixth chord is a good example. Still, it is somewhat > simplifying to quasi say "focus on writing good melody, that will > lead to good harmony" :) Actually, in much Western music there is a > tendency the other way round, take Blues, cadentical 18-century music > or 19th century music like Wagner.

In blues the chords still serve the melody, but you don't think about counterpoint between notes of the chords. I'd very much like to understand exactly what Wagner *was* doing, at a fundamental level. He was using the Rules of Harmony that grew out of the Rules of Counterpoint that were designed to give independent melodies that moved in harmony.

>> Harmony and counterpoint are different ways of looking at
>> the same thing a lot of the time.
>>
> :) Certainly, harmony and counterpoint influence each other. However, > they are precisely looking at the same thing from different points of > view. Consequently, we can end up with rather different results > depending on whether counterpoint or harmony played a more important > role -- as can be seen throughout Western music history. But this > kind of discussion is perhaps leading nowhere, because we are not > specific here :)

That's why I started with specific rules -- stepwise contrary motion, which seems to be common to different schools of counterpoint, and root motion by fifths, which seems to be common to different rules of harmony. Efficiently stating the key is a more sophisticated rule that might generally hold. I don't know about common tones.

> Thanks for pointing this out! Could you perhaps give a reference of > the Blackwood text, or provide the chord progression so that I can > follow you better. What 9-limit chords do you find in Gesualdo, 6:7:9 ??

Blackwood, E. 1985. The Structure of Recognizable Diatonic Tunings. Princeton, New Jersey: Princeton University Press.

I don't find any 9-limit chords in Gesualdo. What I find are chords that could be re-spelled as 9-limit to avoid intervals a diesis apart within the same bar. One example from Anchor che per amarti:

F# F# C# A#
G E B B
Bb E G D
Bb D G D
A E A C#

The A# and Bb are enharmonically equivalent in 12-equal. Of course you can re-spell the first chord from F# to Gb but in context it has to tie in to what came before. So an alternative is

F# F# C# Bb
G E B B
Bb E G D
Bb D G D
A E A C#

If Gesualdo had a 19 note harpsichord that would sound different to what he wrote, so we can assume he didn't intend it. But still, if you wrote something in 12-equal and wanted to turn it into meantone, this will work. It has a 9-limit interpretation where the car horn triad resolves onto E minor.

> Also, I don't quite understand what you mean by "a family
> of progressions for which 9-limit harmony is the *solution*
> to a common tone modulation problem."

Having two ways to spell a third gives you more freedom in choosing chord progressions with common tones.

> Thanks for these details. Slowly changing the degree of dissonance > makes perfectly sense to me. I don't understand your scale example > though: if these are nominals, does the scale consist of only two > different and repeated pitches? I don't understand why 1 1 is an > interval larger than unison and heard as two steps.

They're intervals between nominals. With the nominals (and a better tonic) it's

0 1 3 4 6 7 9 0
1 2 1 2 1 2 1

That isn't Rothenberg proper because intervals of one and two diatonic steps both have 2 chromatic steps. With this specific scale that doesn't matter because 9->1 is 2^ rather than 2. But 1->3 may sometimes be written as 1->3^ and 9->1 as 9^->1 so the interval classes overlap. That means you need special rules to establish whether 2 chromatic steps are interpreted as a step or a leap.

Another rule I forgot: you shouldn't use quommas (0^) as melodic steps, or expose notes that differ by a quomma or octave equivalents. I think this captures the spirit of Palestrina counterpoint. Maybe you wouldn't follow it in real microtonal music because if you paid for all those small intervals you may as well use them.

Graham

> > What am I missing?
>
> Sorry for explaining poorly! It's not just G<->A,
> but the sum
> C<->C# + E<->F# + G<->A
> or the sum
> C<->F# + E<->A + G<->C#
> or any other similar sum; whichever is least.

I've placed a better explanation of the measure here:

http://lumma.org/music/theory/ChordProgressionStrength.txt

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> I've placed a better explanation of the measure here:
>
> http://lumma.org/music/theory/ChordProgressionStrength.txt

Er, make that
http://lumma.org/music/theory/ModulationStrength.txt

-Carl

On Apr 25, 2008, at 2:46 AM, Billy Gard wrote:
> <<< Even more so, I would like to hear alternative guidelines for > creating
> chord progression where root relations are beyond 5-limit. >>>
>
> I see examples of this in vocal music that features a pedaltone in > one of
> the parts, such as a barbershop tag. For instance, if one part is > holding
> the tonic note while the chord moves from a tonic to a II7, the > root will
> need to move up a Septimal-Major second (8/7) to keep the chords tuned
> without having to bend the pedaltone.
>
Wow, do they do such things? I like that.

> <<< The most thorough
> discussion of these which I could find so far are Schoenberg's two > harmony
> books. Some underlying principles I hope I have found in Schoenberg's
> explanation of his guidelines for root progressions. Therefore, I > formalised
> his explanation, and not his actual rules. >>>
>
> What is the names of his two harmony books. Surely those aren't the > two I
> read are they? It was so long.

Theory of Harmony (orig. Harmonielehre)
http://www.amazon.com/Theory-Harmony-California-Library-Reprint/dp/0520049446/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1209110353&sr=8-1

Structural Functions of Harmony
http://www.amazon.com/Structural-Functions-Harmony-Arnold-Schonberg/dp/0393004783/ref=sr_1_3?ie=UTF8&s=books&qid=1209110353&sr=8-3
> Does Shoenberg attempt
> to explain mathematically why certain musical progressions make > more sense
> to the western ear and others don't? Such a work I'd like to see.
>
He does indeed try to explain a lot, but not mathematically.

Best
Torsten

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

On Apr 25, 2008, at 3:50 AM, Carl Lumma wrote:
> > > What am I missing?
> >
> > Sorry for explaining poorly! It's not just G<->A,
> > but the sum
> > C<->C# + E<->F# + G<->A
> > or the sum
> > C<->F# + E<->A + G<->C#
> > or any other similar sum; whichever is least.
>
> I've placed a better explanation of the measure here:
>
> http://lumma.org/music/theory/ChordProgressionStrength.txt
>
>

Thank you, but I cannot find that file (Error 404 - File not Found). BTW: many links on your site top-level http://lumma.org/ result in the same error.

Best
Torsten

> -Carl
>
>
>
--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

OK, I found this link. Thank you!

Best
Torsten

On Apr 25, 2008, at 7:25 AM, Carl Lumma wrote:

> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> > I've placed a better explanation of the measure here:
> >
> > http://lumma.org/music/theory/ChordProgressionStrength.txt
>
> Er, make that
> http://lumma.org/music/theory/ModulationStrength.txt
>
> -Carl
>
>
>
--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

Dear Hans Straub,

again, thanks for pointing me to your text. After reading it, my main question is this: how are cadencial sets (the pitches that determine a key completely) computed? Or to put it the other way round: which pitches of the key are dispensable for that purpose? You kindly explain this idea in your FAQ by examples. For example, in major the chords IV and V form a cadencial set, in other words the tone on degree III is dispensable. I see that having III or IIIb does not change the root of the scale (not considering different modes). But how is III isolated formally here?

In your composition, you applied this idea to your scale C Db Eb E F G Ab. Do you know whether this idea has been applied to scales outside 12 ET? For example, what would be the set of "dispensable" pitches (for tonal unambiguousness) of Erlich's decatonic scales?

Thank you!

Some other points.

It is interesting to see that you (or Mazzola) are referring to Schoenberg's explanation of modulations (neutral phase, fundamental step, cadence) but then seemingly use minor in a sense which greatly differs from the way Schoenberg introduces minor. You are discussing the cadencial sets of harmonic minor. Schoenberg introduces minor in a way which cannot be reduced to a simple scale like that. It is related to melodic minor, but less simplifying. According to him, minor is natural minor plus the raised VI and VII degree. Additionally, the degrees VI and VII are treated with special care. In the strict form, VI# always leads to VII#, VII# -> I, VII natural - > VI natural and VI natural to V. On no account are chromatic intervals permitted. BTW: can we define cadencial sets for this concept of minor? How?

Several times, you mention some hypothetical listener who grew up with your scale and hears your modulations. I wonder whether the diatonic scales are just education or not something far deeper, but that's a big issue :)

Translation of Schoenberg terminology into English: I assume your term "harmonic braid" is a translation of Schoenberg's term "harmonisches Band" (i.e. common pitches between two chords). Unfortunately, I have no English translation of his book. Any idea what term is used in the "official" English translation?

Best
Torsten

On Apr 24, 2008, at 4:05 PM, hstraub64 wrote:
> --- In tuning@yahoogroups.com, Torsten Anders <torstenanders@...>
> wrote:
>
> > On Apr 23, 2008, at 8:34 AM, hstraub64 wrote:
>
> > >
> > > Maybe I should mention Mazzola's concept of "cadencial set"
> > >
> (http://home.datacomm.ch/straub/mamuth/mamufaq.html#Q_cadencialset),
> > > which integrates so well with Schoenberg's modulation model
> > > (http://home.datacomm.ch/straub/mamuth/mamufaq.html#Q_modul), and
> > > which I already have used successfully in microtonal contexts.
> > >
> > That sounds very interesting. Could you please detail this a bit
> > more?
> >
>
> The basic idea behind it is actually quite simple and primitive. I
> would say it is almost pure musical set theory. A step-by step
> introduction of how I use it can be found on
>
> http://home.datacomm.ch/straub/mamuth/modul/ontosu_e.html
>
> (There is also a german version of this page). The example there is
> in 12EDO, but the generalization to n-EDO should obvious. There is an
> example in 19EDO on
>
> http://home.datacomm.ch/straub/mamuth/modul/wt19_e.html
> --> Hans Straub
>
>
>
--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

Dear Carl,

On Apr 25, 2008, at 7:25 AM, Carl Lumma wrote:
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> > I've placed a better explanation of the measure here:
> >
> > http://lumma.org/music/theory/ChordProgressionStrength.txt
>
> Er, make that
> http://lumma.org/music/theory/ModulationStrength.txt
>

thanks for this explanation. I now better understand you. I will think about some way to implement this "harmonic strength" in Strasheela :)

Open question still: what happens in cases where the pairs of changing pitches can not be distributed evenly without "overlapping". Say, what there are three changing pitches?

Minor point: I am not quite convinced by the term "modulation strength", as modulations don't necessarily occur here (we don't know whether any new key would be installed). Alternative proposal (rather long): harmonic distance of chords (Tenney-weighted).

Thanks also for detailing the voice leading subject. However, I don't quite understand this yet. I figure you could simply have an absolute distance like

| pitchA - pitchB |

Having a logarithm doesn't so much "penalise" larger distances, like in the Tenney formular, right? But why having the factor 1200? Concerning your "chord generation regime", what are n-ads, chords of n pitches? Also, I don't understand your step 5. I see that with your approach you minimise the voice-leading distance but allow for larger "chord harmonic distance" (your modulation strength). I like this idea.

However, using Strasheela I would realise it declaratively instead of procedurally as you do here. For example, I may specify that low voice-leading distances are used for specific or all chord progressions (neighbouring chord pairs), and at the same time that the "chord harmonic distance" should be low or high for all or for specific chord progressions. We might also add various other constraints (e.g., the tonic chord should re-appear from time to time etc.). That's what is nice about constraint programming: you just define a conjunction of conditions (other connectives like disjunction or implication and their nesting can be used as well).

Best
Torsten

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

On Apr 25, 2008, at 1:29 AM, Carl Lumma wrote:
> > Could you please point me to the reference of this text?
>
> I can do better than that!
> http://lumma.org/tuning/rothenberg/AModelForPatternPerception.pdf

Thank you very much!

Could you perhaps even summarise Rothenberg's idea concerning which scale notes are required to recognise a scale. Is this related to Mazzola's cadencial sets brought up by Hans?

Thank you!

Best
Torsten

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

Dear Carl,

On Apr 25, 2008, at 12:46 PM, Torsten Anders wrote:
> > > I've placed a better explanation of the measure here:
> > >
> > > http://lumma.org/music/theory/ChordProgressionStrength.txt
> >
> > Er, make that
> > http://lumma.org/music/theory/ModulationStrength.txt

here is a related idea: instead of measuring the changing intervals between two chords (using the Tenney harmonic distance), we could measure the intervals within a chord in order to measure the harmonic quality of a single chord.

I guess this has been done already. Has it?

Thank you!

Best
Torsten

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

I remember reading this by Schoenberg somewhere. i mean we are talking 30 years but it did strike me. where the root becomes the fifth or the fifth the third for example.
Actually i remember the reverse as being the strong. c to g seems strong while G to C seems week because it it is just resolving, so a release of energy. interesting the example of a II7 chord would be an example of of a root becoming a 7th.
I am surprised we haven't heard from Monz on this one.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Torsten Anders wrote:
>
> Dear Kraig,
>
> On Apr 22, 2008, at 10:33 PM, Kraig Grady wrote:
> > What i remember which goes quite far back in time and my copy is still
> > unpacked is strong progressions are where more important note/s become
> > less and vice versa. i have actually played with this in structures
> > such
> > as the eikosany. Anyway it seems if you allow the 7th and 9th
> > harmonics
> > in chords, even triads using them, it seems one would have further
> > progressions to choose from although this would quickly lead
> > outside of
> > any particular scale. One could also use the 11th.
> >
>
> thanks for your mail. Unfortunately, I don't quite understand what
> you mean by "strong progressions are where more important note/s
> become less and vice versa". If you are referring to the chord root
> as the only important note, then this guideline is closely related to
> what I was suggesting (but not the same). I said that in a strong
> progression there appears new "important note" -- the root of the new
> chord was not in the previous chord.
>
> But it seems your remark is more general. In particular, it appears
> to are referring to specific higher harmonics. Could you please
> detail this further?
>
> Thank you!
>
> Best
> Torsten
>
> >
> >
> > /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> > _'''''''_ ^North/Western Hemisphere:
> > North American Embassy of Anaphoria Island <http://anaphoria.com/ > <http://anaphoria.com/>>
> >
> > _'''''''_ ^South/Eastern Hemisphere:
> > Austronesian Outpost of Anaphoria <http://
> > anaphoriasouth.blogspot.com/>
> >
> > ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
> >
> > Torsten Anders wrote:
> > >
> > > Dear Carl,
> > >
> > > On Apr 22, 2008, at 7:19 PM, Carl Lumma wrote:
> > > > > Here is a suggestion I can make for guidelines on good chord
> > > > > progressions, which is indeed based on the notion of pitch class
> > > > > sets, but which also works beyond 12 ET (e.g., I did examples
> > > > > using them with 31 ET). This suggestion is based on Schoenberg's
> > > > > guidelines on good chord root progressions where he introduces
> > > > > the notion of ascending (strong), descending ('weak') and
> > > > > superstrong progressions (see the respective chapter in his
> > > > > Theory of Harmony). The summary of Schoenberg's guidelines/rules
> > > > > is based on 12 ET, but his explanation is actually more general.
> > > > > A formalisation of his explanations (instead of his actual
> > rules)
> > > > > does work beyond 12 ET.
> > > > >
> > > > > The main difference between Schoenberg's actual guidelines and
> > > > > their formalised generalisation is that Schoenberg's guidelines
> > > > > are based on scale degree intervals between chord roots, whereas
> > > > > the generalisation exploits whether the root pitch class of some
> > > > > chord is contained in the pitch class set of another chord.
> > > >
> > > > I haven't read Schoenberg's book but it sounds like you've
> > > > done a bit of work for him. Yes, I could probably generalize
> > > > many (though not all) of the "bad music theory papers" if I
> > > > worked at it. And some such minimal effort would only be
> > > > fair in the meta-analysis I proposed. Where to draw the line
> > > > is another question.
> > > >
> > >
> > > > Did Schoenberg say the important thing
> > > > is whether the root is a member of the new chord (which is
> > > > a reasonable thing)? Or did he say something else?
> > > >
> > >
> > > Sorry, I don't fully understand your question. In an ascending
> > > progression, the root of the preceding chord is a pitch in the
> > > following chord. Did you mean that?
> > >
> > > > > OK, it follows the
> > > > > core of is the formalisation (I hope the notation explains
> > itself:
> > > > > defined are Boolean functions expecting two neighbouring
> > chords).
> > > > >
> > > > > /* Chord1 and Chord2 have common pitch classes, but the root of
> > > > > Chord2 does not occur in the set of Chord1's pitchclasses. */
> > > > >
> > > > > isAscendingProgression(Chord1, Chord2) :=
> > > > > ( NOT ( getRoot(Chord2) \in getPitchClasses(Chord1) ) )
> > > > > AND ( intersection(getPitchClasses(Chord1), getPitchClasses
> > > > > (Chord2)) \= emptySet )
> > > > >
> > > > > /* A non-root pitchclass of Chord1 is root in Chord2. */
> > > > >
> > > > > isDescendingProgression(Chord1, Chord2) :=
> > > > > ( getRoot(Chord2) \in getPitchClasses(Chord1) )
> > > > > AND ( getRoot(Chord1) \= getRoot(Chord2) )
> > > > >
> > > > > /* Chord1 and Chord2 have no common pitch classes */
> > > > >
> > > > > isSuperstrongProgression(Chord1, Chord2) :=
> > > > > intersection(getPitchClasses(Chord1), getPitchClasses(Chord2)) =
> > > > > emptySet
> > > >
> > > > Yes, perfectly understandable, though I find the terms
> > > > "ascending" and "descending" extremely odd here.
> > > >
> > > These are Schoenberg's terms. He mentions that H. Schenker uses
> > these
> > > terms as well ("aufsteigend" and "fallend") though in opposite
> > > meaning. He remarks something like that anyone who loves Brahmsian
> > > harmony would likely come up with similar concepts.
> > >
> > > > > If you are interested I can post further formalisations/
> > > > > generalisations of Schoenberg's rules.
> > > >
> > > > I would be interested to read them, especially if you
> > > > show how you generalized them. And if monz is reading,
> > > > I bet he'd be interested, too.
> > > >
> > > I'll write a new mail soon..
> > >
> > > > > For example, Schoenberg
> > > > > recommends that descending progressions should "resolve". For
> > any
> > > > > three successive chords, if the first two chords form a
> > descending
> > > > > progression, then the progression from the first to the third
> > chord
> > > > > should form a strong or superstrong progression (so the
> > middle chord
> > > > > is quasi a 'passing chord').
> > > >
> > > > Hm... this seems kindof strange.
> > > >
> > > Schoenberg is always very exhaustive :) He does not want to ban
> > > descending progressions, so he looks for a way how he can allow for
> > > them in a way which is musically convincing in most cases. So, he
> > > comes up with the idea of a 'passing chord'.
> > >
> > > Please note that he makes is always very clear that his rules are
> > > guidelines for the pupil, but no laws for masterworks. This is
> > why he
> > > prefers the term "descending" instead of "weak" progressions (he
> > uses
> > > the term "strong" for "ascending" progressions a lot). Still, he
> > does
> > > not discuss where descending progressions are used to good
> > purpose in
> > > any masterwork, but such considerations have hardly a place
> > anyway in
> > > his systematic approach.
> > >
> > > > > Anyway, for chord progressions of diatonic triads in major, the
> > > > > generalised formalisation and Schoenberg's rules are equivalent
> > > > > (e.g., the function isAscendingProgression above returns true
> > for
> > > > > progressions which Schoenberg calls ascending). Still, the
> > behaviour
> > > > > of the constraints and Schoenberg's rules differ for more
> > complex
> > > > > cases. According to Schoenberg, a progression is superstrong
> > if the
> > > > > root interval is a step up or down.
> > > >
> > > > Bzzz. :)
> > > >
> > > > > For example, the progression V7 IV is superstrong according to
> > > > > Schoenberg. For the definitions above, however, this progression
> > > > > is descending (!), because the root of IV is contained in V7
> > > > > (e.g. in G7 F, the F's root pitchclass f is already contained
> > > > > in G7). Indeed, this progression is rare in music.
> > > > > By contrast, the progression I IIIb (e.g., C Eb) is a descending
> > > > > progression in Schoenbergs original definition. For the
> > > > > definitions above, however, this is an ascending progression
> > > > > (the root of Es is not contained in C), and indeed for me the
> > > > > progression feels strong.
> > > >
> > > > Sounds like we are really dealing with the Anders theory
> > > > of chord progressions. Which is probably a blessing.
> > > >
> > > :) As I said, I simply implemented Schoenberg's rule explanation
> > > instead of his actual rules. So, I cannot accept the honour you are
> > > implying.
> > >
> > > > > Let us create a sequence of 5 chords and
> > > > > allow only for diatonic triads in C-major. The first and last
> > > > > chord should be the tonic. Following Schoenberg let us specify
> > > > > that ascending progressions are "resolved" (see above), and we
> > > > > don't allow for superstrong progressions. Finally, the union
> > > > > of the last three chords must contain all pitch classes of
> > > > > C-major (i.e. these chords form a cadence).
> > > > >
> > > > > This problem has only two solutions (I used the computer :).
> > > > > These solutions are as follows. They happen to contain only
> > > > > ascending progressions. Such solutions are particularly
> > > > > convincing. For simplicity I given absolute chord names.
> > > > >
> > > > > C-maj F-maj D-min G-maj C-maj (only ascending progressions)
> > > > > C-maj A-min D-min G-maj C-maj (only ascending progressions)
> > > >
> > > > Only allowing for diatonic triads is C-major is a very
> > > > helpful constraint. Howabout we allow all 7-limit triads
> > > > in 31-ET, for a start?
> > > >
> > >
> > > The point of these generalised definitions is exactly that they
> > allow
> > > for such cases :) I only simplified for the sake of an argument. So,
> > > here is a 7-limit variant. Let the scale be septimal natural minor
> > > (Scala term, in 31 ET the pitches C D D# F G G# A#). Let us allow
> > for
> > > the following septimal triads (please correct my terminology if
> > > necessary): harmonic diminished (5/5 6/5 7/5), subharmonic
> > diminished
> > > (7/7 7/6 7/5), subminor (6/1 7/1 9/1), supermajor (1/9 1/7 1/6).
> > Note
> > > that I consider all these chords as consonances for simplicity which
> > > require neither preparation nor resolution. Naturally, we could
> > > define extra rules which care for that.
> > >
> > > There are 2 solutions of 5 chords with this scale, these chords and
> > > the rule set specified before. For convenience, I attached these
> > > solutions as PDF files (please let me know if doing so violates some
> > > policies of this list).
> > >
> > > The solutions are shown by two staffs. The upper stuff shows the
> > > actual chord pitches (I didn't bother to implement voice leading
> > > rules here, so things like Bruckner's law of the shortest path are
> > > violated). The staff notation is in 31 ET (i.e. the interval C D# is
> > > 7/6). The lower staff shows the chord roots in staff notation plus
> > > there ratios expressed as a product of the untransposed chord ratios
> > > and a root factor for each chord ratio. Some of these factors may
> > > look odd at first, e.g., the first chord has the factor 4/3 for the
> > > root C, but this is due to that fact that the untransposed chord
> > > ratios don't contain 1/1 -- the resulting product ratios do
> > contain 1/1.
> > >
> > > If you are interested in other chord sets, scales, or longer
> > > solutions then just say so -- its easy to change some numbers in a
> > > computer program :)
> > >
> > > Best
> > > Torsten
> > >
> > >
> > > --
> > > Torsten Anders
> > > Interdisciplinary Centre for Computer Music Research
> > > University of Plymouth
> > > Office: +44-1752-233667
> > > Private: +44-1752-558917
> > > http://strasheela.sourceforge.net <http://strasheela.sourceforge.net>
> > > http://www.torsten-anders.de <http://www.torsten-anders.de>
> > >
> > >
> > >
> > >
> >
> >
>
> --
> Torsten Anders
> Interdisciplinary Centre for Computer Music Research
> University of Plymouth
> Office: +44-1752-233667
> Private: +44-1752-558917
> http://strasheela.sourceforge.net <http://strasheela.sourceforge.net>
> http://www.torsten-anders.de <http://www.torsten-anders.de>
>
>

On Apr 25, 2008, at 1:16 PM, Kraig Grady wrote:
> I remember reading this by Schoenberg somewhere. i mean we are talking
> 30 years but it did strike me. where the root becomes the fifth or the
> fifth the third for example.
> Actually i remember the reverse as being the strong. c to g seems > strong
> while G to C seems week because it it is just resolving, so a > release of
> energy.
>
Schoenbergs terminology is definitely the reverse (although he tries to avoid the term weak, uses descending instead). But he remarks in a footnote that Schenker uses the terminology the other way round. He mentions in a footnote that for a while he though he got this idea from Schenker, but his pupils told him that he did in fact use these terms in classes already, many years before he read Schenker.

Best
Torsten
> interesting the example of a II7 chord would be an example of of
> a root becoming a 7th.
> I am surprised we haven't heard from Monz on this one.
>

>
>
> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> _'''''''_ ^North/Western Hemisphere:
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
>
> _'''''''_ ^South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria <http://> anaphoriasouth.blogspot.com/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
> Torsten Anders wrote:
> >
> > Dear Kraig,
> >
> > On Apr 22, 2008, at 10:33 PM, Kraig Grady wrote:
> > > What i remember which goes quite far back in time and my copy > is still
> > > unpacked is strong progressions are where more important note/s > become
> > > less and vice versa. i have actually played with this in > structures
> > > such
> > > as the eikosany. Anyway it seems if you allow the 7th and 9th
> > > harmonics
> > > in chords, even triads using them, it seems one would have further
> > > progressions to choose from although this would quickly lead
> > > outside of
> > > any particular scale. One could also use the 11th.
> > >
> >
> > thanks for your mail. Unfortunately, I don't quite understand what
> > you mean by "strong progressions are where more important note/s
> > become less and vice versa". If you are referring to the chord root
> > as the only important note, then this guideline is closely > related to
> > what I was suggesting (but not the same). I said that in a strong
> > progression there appears new "important note" -- the root of the > new
> > chord was not in the previous chord.
> >
> > But it seems your remark is more general. In particular, it appears
> > to are referring to specific higher harmonics. Could you please
> > detail this further?
> >
> > Thank you!
> >
> > Best
> > Torsten
> >
> > >
> > >
> > > /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> > > _'''''''_ ^North/Western Hemisphere:
> > > North American Embassy of Anaphoria Island <http://anaphoria.com/
> > <http://anaphoria.com/>>
> > >
> > > _'''''''_ ^South/Eastern Hemisphere:
> > > Austronesian Outpost of Anaphoria <http://
> > > anaphoriasouth.blogspot.com/>
> > >
> > > ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
> > >
> > > Torsten Anders wrote:
> > > >
> > > > Dear Carl,
> > > >
> > > > On Apr 22, 2008, at 7:19 PM, Carl Lumma wrote:
> > > > > > Here is a suggestion I can make for guidelines on good chord
> > > > > > progressions, which is indeed based on the notion of > pitch class
> > > > > > sets, but which also works beyond 12 ET (e.g., I did > examples
> > > > > > using them with 31 ET). This suggestion is based on > Schoenberg's
> > > > > > guidelines on good chord root progressions where he > introduces
> > > > > > the notion of ascending (strong), descending ('weak') and
> > > > > > superstrong progressions (see the respective chapter in his
> > > > > > Theory of Harmony). The summary of Schoenberg's > guidelines/rules
> > > > > > is based on 12 ET, but his explanation is actually more > general.
> > > > > > A formalisation of his explanations (instead of his actual
> > > rules)
> > > > > > does work beyond 12 ET.
> > > > > >
> > > > > > The main difference between Schoenberg's actual > guidelines and
> > > > > > their formalised generalisation is that Schoenberg's > guidelines
> > > > > > are based on scale degree intervals between chord roots, > whereas
> > > > > > the generalisation exploits whether the root pitch class > of some
> > > > > > chord is contained in the pitch class set of another chord.
> > > > >
> > > > > I haven't read Schoenberg's book but it sounds like you've
> > > > > done a bit of work for him. Yes, I could probably generalize
> > > > > many (though not all) of the "bad music theory papers" if I
> > > > > worked at it. And some such minimal effort would only be
> > > > > fair in the meta-analysis I proposed. Where to draw the line
> > > > > is another question.
> > > > >
> > > >
> > > > > Did Schoenberg say the important thing
> > > > > is whether the root is a member of the new chord (which is
> > > > > a reasonable thing)? Or did he say something else?
> > > > >
> > > >
> > > > Sorry, I don't fully understand your question. In an ascending
> > > > progression, the root of the preceding chord is a pitch in the
> > > > following chord. Did you mean that?
> > > >
> > > > > > OK, it follows the
> > > > > > core of is the formalisation (I hope the notation explains
> > > itself:
> > > > > > defined are Boolean functions expecting two neighbouring
> > > chords).
> > > > > >
> > > > > > /* Chord1 and Chord2 have common pitch classes, but the > root of
> > > > > > Chord2 does not occur in the set of Chord1's > pitchclasses. */
> > > > > >
> > > > > > isAscendingProgression(Chord1, Chord2) :=
> > > > > > ( NOT ( getRoot(Chord2) \in getPitchClasses(Chord1) ) )
> > > > > > AND ( intersection(getPitchClasses(Chord1), getPitchClasses
> > > > > > (Chord2)) \= emptySet )
> > > > > >
> > > > > > /* A non-root pitchclass of Chord1 is root in Chord2. */
> > > > > >
> > > > > > isDescendingProgression(Chord1, Chord2) :=
> > > > > > ( getRoot(Chord2) \in getPitchClasses(Chord1) )
> > > > > > AND ( getRoot(Chord1) \= getRoot(Chord2) )
> > > > > >
> > > > > > /* Chord1 and Chord2 have no common pitch classes */
> > > > > >
> > > > > > isSuperstrongProgression(Chord1, Chord2) :=
> > > > > > intersection(getPitchClasses(Chord1), getPitchClasses> (Chord2)) =
> > > > > > emptySet
> > > > >
> > > > > Yes, perfectly understandable, though I find the terms
> > > > > "ascending" and "descending" extremely odd here.
> > > > >
> > > > These are Schoenberg's terms. He mentions that H. Schenker uses
> > > these
> > > > terms as well ("aufsteigend" and "fallend") though in opposite
> > > > meaning. He remarks something like that anyone who loves > Brahmsian
> > > > harmony would likely come up with similar concepts.
> > > >
> > > > > > If you are interested I can post further formalisations/
> > > > > > generalisations of Schoenberg's rules.
> > > > >
> > > > > I would be interested to read them, especially if you
> > > > > show how you generalized them. And if monz is reading,
> > > > > I bet he'd be interested, too.
> > > > >
> > > > I'll write a new mail soon..
> > > >
> > > > > > For example, Schoenberg
> > > > > > recommends that descending progressions should "resolve". > For
> > > any
> > > > > > three successive chords, if the first two chords form a
> > > descending
> > > > > > progression, then the progression from the first to the > third
> > > chord
> > > > > > should form a strong or superstrong progression (so the
> > > middle chord
> > > > > > is quasi a 'passing chord').
> > > > >
> > > > > Hm... this seems kindof strange.
> > > > >
> > > > Schoenberg is always very exhaustive :) He does not want to ban
> > > > descending progressions, so he looks for a way how he can > allow for
> > > > them in a way which is musically convincing in most cases. > So, he
> > > > comes up with the idea of a 'passing chord'.
> > > >
> > > > Please note that he makes is always very clear that his rules > are
> > > > guidelines for the pupil, but no laws for masterworks. This is
> > > why he
> > > > prefers the term "descending" instead of "weak" progressions (he
> > > uses
> > > > the term "strong" for "ascending" progressions a lot). Still, he
> > > does
> > > > not discuss where descending progressions are used to good
> > > purpose in
> > > > any masterwork, but such considerations have hardly a place
> > > anyway in
> > > > his systematic approach.
> > > >
> > > > > > Anyway, for chord progressions of diatonic triads in > major, the
> > > > > > generalised formalisation and Schoenberg's rules are > equivalent
> > > > > > (e.g., the function isAscendingProgression above returns > true
> > > for
> > > > > > progressions which Schoenberg calls ascending). Still, the
> > > behaviour
> > > > > > of the constraints and Schoenberg's rules differ for more
> > > complex
> > > > > > cases. According to Schoenberg, a progression is superstrong
> > > if the
> > > > > > root interval is a step up or down.
> > > > >
> > > > > Bzzz. :)
> > > > >
> > > > > > For example, the progression V7 IV is superstrong > according to
> > > > > > Schoenberg. For the definitions above, however, this > progression
> > > > > > is descending (!), because the root of IV is contained in V7
> > > > > > (e.g. in G7 F, the F's root pitchclass f is already > contained
> > > > > > in G7). Indeed, this progression is rare in music.
> > > > > > By contrast, the progression I IIIb (e.g., C Eb) is a > descending
> > > > > > progression in Schoenbergs original definition. For the
> > > > > > definitions above, however, this is an ascending progression
> > > > > > (the root of Es is not contained in C), and indeed for me > the
> > > > > > progression feels strong.
> > > > >
> > > > > Sounds like we are really dealing with the Anders theory
> > > > > of chord progressions. Which is probably a blessing.
> > > > >
> > > > :) As I said, I simply implemented Schoenberg's rule explanation
> > > > instead of his actual rules. So, I cannot accept the honour > you are
> > > > implying.
> > > >
> > > > > > Let us create a sequence of 5 chords and
> > > > > > allow only for diatonic triads in C-major. The first and > last
> > > > > > chord should be the tonic. Following Schoenberg let us > specify
> > > > > > that ascending progressions are "resolved" (see above), > and we
> > > > > > don't allow for superstrong progressions. Finally, the union
> > > > > > of the last three chords must contain all pitch classes of
> > > > > > C-major (i.e. these chords form a cadence).
> > > > > >
> > > > > > This problem has only two solutions (I used the computer :).
> > > > > > These solutions are as follows. They happen to contain only
> > > > > > ascending progressions. Such solutions are particularly
> > > > > > convincing. For simplicity I given absolute chord names.
> > > > > >
> > > > > > C-maj F-maj D-min G-maj C-maj (only ascending progressions)
> > > > > > C-maj A-min D-min G-maj C-maj (only ascending progressions)
> > > > >
> > > > > Only allowing for diatonic triads is C-major is a very
> > > > > helpful constraint. Howabout we allow all 7-limit triads
> > > > > in 31-ET, for a start?
> > > > >
> > > >
> > > > The point of these generalised definitions is exactly that they
> > > allow
> > > > for such cases :) I only simplified for the sake of an > argument. So,
> > > > here is a 7-limit variant. Let the scale be septimal natural > minor
> > > > (Scala term, in 31 ET the pitches C D D# F G G# A#). Let us > allow
> > > for
> > > > the following septimal triads (please correct my terminology if
> > > > necessary): harmonic diminished (5/5 6/5 7/5), subharmonic
> > > diminished
> > > > (7/7 7/6 7/5), subminor (6/1 7/1 9/1), supermajor (1/9 1/7 1/6).
> > > Note
> > > > that I consider all these chords as consonances for > simplicity which
> > > > require neither preparation nor resolution. Naturally, we could
> > > > define extra rules which care for that.
> > > >
> > > > There are 2 solutions of 5 chords with this scale, these > chords and
> > > > the rule set specified before. For convenience, I attached these
> > > > solutions as PDF files (please let me know if doing so > violates some
> > > > policies of this list).
> > > >
> > > > The solutions are shown by two staffs. The upper stuff shows the
> > > > actual chord pitches (I didn't bother to implement voice leading
> > > > rules here, so things like Bruckner's law of the shortest > path are
> > > > violated). The staff notation is in 31 ET (i.e. the interval > C D# is
> > > > 7/6). The lower staff shows the chord roots in staff notation > plus
> > > > there ratios expressed as a product of the untransposed chord > ratios
> > > > and a root factor for each chord ratio. Some of these factors > may
> > > > look odd at first, e.g., the first chord has the factor 4/3 > for the
> > > > root C, but this is due to that fact that the untransposed chord
> > > > ratios don't contain 1/1 -- the resulting product ratios do
> > > contain 1/1.
> > > >
> > > > If you are interested in other chord sets, scales, or longer
> > > > solutions then just say so -- its easy to change some numbers > in a
> > > > computer program :)
> > > >
> > > > Best
> > > > Torsten
> > > >
> > > >
> > > > --
> > > > Torsten Anders
> > > > Interdisciplinary Centre for Computer Music Research
> > > > University of Plymouth
> > > > Office: +44-1752-233667
> > > > Private: +44-1752-558917
> > > > http://strasheela.sourceforge.net <http://> strasheela.sourceforge.net>
> > > > http://www.torsten-anders.de <http://www.torsten-anders.de>
> > > >
> > > >
> > > >
> > > >
> > >
> > >
> >
> > --
> > Torsten Anders
> > Interdisciplinary Centre for Computer Music Research
> > University of Plymouth
> > Office: +44-1752-233667
> > Private: +44-1752-558917
> > http://strasheela.sourceforge.net <http://> strasheela.sourceforge.net>
> > http://www.torsten-anders.de <http://www.torsten-anders.de>
> >
> >
>
>
--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

i am sure translation is part of the problem. Still one could use higher harmonics in this way. introducing them as lower harmonics in preceding chords.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Torsten Anders wrote:
>
>
> On Apr 25, 2008, at 1:16 PM, Kraig Grady wrote:
> > I remember reading this by Schoenberg somewhere. i mean we are talking
> > 30 years but it did strike me. where the root becomes the fifth or the
> > fifth the third for example.
> > Actually i remember the reverse as being the strong. c to g seems
> > strong
> > while G to C seems week because it it is just resolving, so a
> > release of
> > energy.
> >
> Schoenbergs terminology is definitely the reverse (although he tries
> to avoid the term weak, uses descending instead). But he remarks in a
> footnote that Schenker uses the terminology the other way round. He
> mentions in a footnote that for a while he though he got this idea
> from Schenker, but his pupils told him that he did in fact use these
> terms in classes already, many years before he read Schenker.
>
> Best
> Torsten
> > interesting the example of a II7 chord would be an example of of
> > a root becoming a 7th.
> > I am surprised we haven't heard from Monz on this one.
> >
>
> >
> >
> > /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> > _'''''''_ ^North/Western Hemisphere:
> > North American Embassy of Anaphoria Island <http://anaphoria.com/ > <http://anaphoria.com/>>
> >
> > _'''''''_ ^South/Eastern Hemisphere:
> > Austronesian Outpost of Anaphoria <http://
> > anaphoriasouth.blogspot.com/>
> >
> > ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
> >
> > Torsten Anders wrote:
> > >
> > > Dear Kraig,
> > >
> > > On Apr 22, 2008, at 10:33 PM, Kraig Grady wrote:
> > > > What i remember which goes quite far back in time and my copy
> > is still
> > > > unpacked is strong progressions are where more important note/s
> > become
> > > > less and vice versa. i have actually played with this in
> > structures
> > > > such
> > > > as the eikosany. Anyway it seems if you allow the 7th and 9th
> > > > harmonics
> > > > in chords, even triads using them, it seems one would have further
> > > > progressions to choose from although this would quickly lead
> > > > outside of
> > > > any particular scale. One could also use the 11th.
> > > >
> > >
> > > thanks for your mail. Unfortunately, I don't quite understand what
> > > you mean by "strong progressions are where more important note/s
> > > become less and vice versa". If you are referring to the chord root
> > > as the only important note, then this guideline is closely
> > related to
> > > what I was suggesting (but not the same). I said that in a strong
> > > progression there appears new "important note" -- the root of the
> > new
> > > chord was not in the previous chord.
> > >
> > > But it seems your remark is more general. In particular, it appears
> > > to are referring to specific higher harmonics. Could you please
> > > detail this further?
> > >
> > > Thank you!
> > >
> > > Best
> > > Torsten
> > >
> > > >
> > > >
> > > > /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> > > > _'''''''_ ^North/Western Hemisphere:
> > > > North American Embassy of Anaphoria Island > <http://anaphoria.com/ <http://anaphoria.com/>
> > > <http://anaphoria.com/ <http://anaphoria.com/>>>
> > > >
> > > > _'''''''_ ^South/Eastern Hemisphere:
> > > > Austronesian Outpost of Anaphoria <http://
> > > > anaphoriasouth.blogspot.com/>
> > > >
> > > > ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
> > > >
> > > > Torsten Anders wrote:
> > > > >
> > > > > Dear Carl,
> > > > >
> > > > > On Apr 22, 2008, at 7:19 PM, Carl Lumma wrote:
> > > > > > > Here is a suggestion I can make for guidelines on good chord
> > > > > > > progressions, which is indeed based on the notion of
> > pitch class
> > > > > > > sets, but which also works beyond 12 ET (e.g., I did
> > examples
> > > > > > > using them with 31 ET). This suggestion is based on
> > Schoenberg's
> > > > > > > guidelines on good chord root progressions where he
> > introduces
> > > > > > > the notion of ascending (strong), descending ('weak') and
> > > > > > > superstrong progressions (see the respective chapter in his
> > > > > > > Theory of Harmony). The summary of Schoenberg's
> > guidelines/rules
> > > > > > > is based on 12 ET, but his explanation is actually more
> > general.
> > > > > > > A formalisation of his explanations (instead of his actual
> > > > rules)
> > > > > > > does work beyond 12 ET.
> > > > > > >
> > > > > > > The main difference between Schoenberg's actual
> > guidelines and
> > > > > > > their formalised generalisation is that Schoenberg's
> > guidelines
> > > > > > > are based on scale degree intervals between chord roots,
> > whereas
> > > > > > > the generalisation exploits whether the root pitch class
> > of some
> > > > > > > chord is contained in the pitch class set of another chord.
> > > > > >
> > > > > > I haven't read Schoenberg's book but it sounds like you've
> > > > > > done a bit of work for him. Yes, I could probably generalize
> > > > > > many (though not all) of the "bad music theory papers" if I
> > > > > > worked at it. And some such minimal effort would only be
> > > > > > fair in the meta-analysis I proposed. Where to draw the line
> > > > > > is another question.
> > > > > >
> > > > >
> > > > > > Did Schoenberg say the important thing
> > > > > > is whether the root is a member of the new chord (which is
> > > > > > a reasonable thing)? Or did he say something else?
> > > > > >
> > > > >
> > > > > Sorry, I don't fully understand your question. In an ascending
> > > > > progression, the root of the preceding chord is a pitch in the
> > > > > following chord. Did you mean that?
> > > > >
> > > > > > > OK, it follows the
> > > > > > > core of is the formalisation (I hope the notation explains
> > > > itself:
> > > > > > > defined are Boolean functions expecting two neighbouring
> > > > chords).
> > > > > > >
> > > > > > > /* Chord1 and Chord2 have common pitch classes, but the
> > root of
> > > > > > > Chord2 does not occur in the set of Chord1's
> > pitchclasses. */
> > > > > > >
> > > > > > > isAscendingProgression(Chord1, Chord2) :=
> > > > > > > ( NOT ( getRoot(Chord2) \in getPitchClasses(Chord1) ) )
> > > > > > > AND ( intersection(getPitchClasses(Chord1), getPitchClasses
> > > > > > > (Chord2)) \= emptySet )
> > > > > > >
> > > > > > > /* A non-root pitchclass of Chord1 is root in Chord2. */
> > > > > > >
> > > > > > > isDescendingProgression(Chord1, Chord2) :=
> > > > > > > ( getRoot(Chord2) \in getPitchClasses(Chord1) )
> > > > > > > AND ( getRoot(Chord1) \= getRoot(Chord2) )
> > > > > > >
> > > > > > > /* Chord1 and Chord2 have no common pitch classes */
> > > > > > >
> > > > > > > isSuperstrongProgression(Chord1, Chord2) :=
> > > > > > > intersection(getPitchClasses(Chord1), getPitchClasses
> > (Chord2)) =
> > > > > > > emptySet
> > > > > >
> > > > > > Yes, perfectly understandable, though I find the terms
> > > > > > "ascending" and "descending" extremely odd here.
> > > > > >
> > > > > These are Schoenberg's terms. He mentions that H. Schenker uses
> > > > these
> > > > > terms as well ("aufsteigend" and "fallend") though in opposite
> > > > > meaning. He remarks something like that anyone who loves
> > Brahmsian
> > > > > harmony would likely come up with similar concepts.
> > > > >
> > > > > > > If you are interested I can post further formalisations/
> > > > > > > generalisations of Schoenberg's rules.
> > > > > >
> > > > > > I would be interested to read them, especially if you
> > > > > > show how you generalized them. And if monz is reading,
> > > > > > I bet he'd be interested, too.
> > > > > >
> > > > > I'll write a new mail soon..
> > > > >
> > > > > > > For example, Schoenberg
> > > > > > > recommends that descending progressions should "resolve".
> > For
> > > > any
> > > > > > > three successive chords, if the first two chords form a
> > > > descending
> > > > > > > progression, then the progression from the first to the
> > third
> > > > chord
> > > > > > > should form a strong or superstrong progression (so the
> > > > middle chord
> > > > > > > is quasi a 'passing chord').
> > > > > >
> > > > > > Hm... this seems kindof strange.
> > > > > >
> > > > > Schoenberg is always very exhaustive :) He does not want to ban
> > > > > descending progressions, so he looks for a way how he can
> > allow for
> > > > > them in a way which is musically convincing in most cases.
> > So, he
> > > > > comes up with the idea of a 'passing chord'.
> > > > >
> > > > > Please note that he makes is always very clear that his rules
> > are
> > > > > guidelines for the pupil, but no laws for masterworks. This is
> > > > why he
> > > > > prefers the term "descending" instead of "weak" progressions (he
> > > > uses
> > > > > the term "strong" for "ascending" progressions a lot). Still, he
> > > > does
> > > > > not discuss where descending progressions are used to good
> > > > purpose in
> > > > > any masterwork, but such considerations have hardly a place
> > > > anyway in
> > > > > his systematic approach.
> > > > >
> > > > > > > Anyway, for chord progressions of diatonic triads in
> > major, the
> > > > > > > generalised formalisation and Schoenberg's rules are
> > equivalent
> > > > > > > (e.g., the function isAscendingProgression above returns
> > true
> > > > for
> > > > > > > progressions which Schoenberg calls ascending). Still, the
> > > > behaviour
> > > > > > > of the constraints and Schoenberg's rules differ for more
> > > > complex
> > > > > > > cases. According to Schoenberg, a progression is superstrong
> > > > if the
> > > > > > > root interval is a step up or down.
> > > > > >
> > > > > > Bzzz. :)
> > > > > >
> > > > > > > For example, the progression V7 IV is superstrong
> > according to
> > > > > > > Schoenberg. For the definitions above, however, this
> > progression
> > > > > > > is descending (!), because the root of IV is contained in V7
> > > > > > > (e.g. in G7 F, the F's root pitchclass f is already
> > contained
> > > > > > > in G7). Indeed, this progression is rare in music.
> > > > > > > By contrast, the progression I IIIb (e.g., C Eb) is a
> > descending
> > > > > > > progression in Schoenbergs original definition. For the
> > > > > > > definitions above, however, this is an ascending progression
> > > > > > > (the root of Es is not contained in C), and indeed for me
> > the
> > > > > > > progression feels strong.
> > > > > >
> > > > > > Sounds like we are really dealing with the Anders theory
> > > > > > of chord progressions. Which is probably a blessing.
> > > > > >
> > > > > :) As I said, I simply implemented Schoenberg's rule explanation
> > > > > instead of his actual rules. So, I cannot accept the honour
> > you are
> > > > > implying.
> > > > >
> > > > > > > Let us create a sequence of 5 chords and
> > > > > > > allow only for diatonic triads in C-major. The first and
> > last
> > > > > > > chord should be the tonic. Following Schoenberg let us
> > specify
> > > > > > > that ascending progressions are "resolved" (see above),
> > and we
> > > > > > > don't allow for superstrong progressions. Finally, the union
> > > > > > > of the last three chords must contain all pitch classes of
> > > > > > > C-major (i.e. these chords form a cadence).
> > > > > > >
> > > > > > > This problem has only two solutions (I used the computer :).
> > > > > > > These solutions are as follows. They happen to contain only
> > > > > > > ascending progressions. Such solutions are particularly
> > > > > > > convincing. For simplicity I given absolute chord names.
> > > > > > >
> > > > > > > C-maj F-maj D-min G-maj C-maj (only ascending progressions)
> > > > > > > C-maj A-min D-min G-maj C-maj (only ascending progressions)
> > > > > >
> > > > > > Only allowing for diatonic triads is C-major is a very
> > > > > > helpful constraint. Howabout we allow all 7-limit triads
> > > > > > in 31-ET, for a start?
> > > > > >
> > > > >
> > > > > The point of these generalised definitions is exactly that they
> > > > allow
> > > > > for such cases :) I only simplified for the sake of an
> > argument. So,
> > > > > here is a 7-limit variant. Let the scale be septimal natural
> > minor
> > > > > (Scala term, in 31 ET the pitches C D D# F G G# A#). Let us
> > allow
> > > > for
> > > > > the following septimal triads (please correct my terminology if
> > > > > necessary): harmonic diminished (5/5 6/5 7/5), subharmonic
> > > > diminished
> > > > > (7/7 7/6 7/5), subminor (6/1 7/1 9/1), supermajor (1/9 1/7 1/6).
> > > > Note
> > > > > that I consider all these chords as consonances for
> > simplicity which
> > > > > require neither preparation nor resolution. Naturally, we could
> > > > > define extra rules which care for that.
> > > > >
> > > > > There are 2 solutions of 5 chords with this scale, these
> > chords and
> > > > > the rule set specified before. For convenience, I attached these
> > > > > solutions as PDF files (please let me know if doing so
> > violates some
> > > > > policies of this list).
> > > > >
> > > > > The solutions are shown by two staffs. The upper stuff shows the
> > > > > actual chord pitches (I didn't bother to implement voice leading
> > > > > rules here, so things like Bruckner's law of the shortest
> > path are
> > > > > violated). The staff notation is in 31 ET (i.e. the interval
> > C D# is
> > > > > 7/6). The lower staff shows the chord roots in staff notation
> > plus
> > > > > there ratios expressed as a product of the untransposed chord
> > ratios
> > > > > and a root factor for each chord ratio. Some of these factors
> > may
> > > > > look odd at first, e.g., the first chord has the factor 4/3
> > for the
> > > > > root C, but this is due to that fact that the untransposed chord
> > > > > ratios don't contain 1/1 -- the resulting product ratios do
> > > > contain 1/1.
> > > > >
> > > > > If you are interested in other chord sets, scales, or longer
> > > > > solutions then just say so -- its easy to change some numbers
> > in a
> > > > > computer program :)
> > > > >
> > > > > Best
> > > > > Torsten
> > > > >
> > > > >
> > > > > --
> > > > > Torsten Anders
> > > > > Interdisciplinary Centre for Computer Music Research
> > > > > University of Plymouth
> > > > > Office: +44-1752-233667
> > > > > Private: +44-1752-558917
> > > > > http://strasheela.sourceforge.net > <http://strasheela.sourceforge.net> <http://
> > strasheela.sourceforge.net>
> > > > > http://www.torsten-anders.de <http://www.torsten-anders.de> > <http://www.torsten-anders.de <http://www.torsten-anders.de>>
> > > > >
> > > > >
> > > > >
> > > > >
> > > >
> > > >
> > >
> > > --
> > > Torsten Anders
> > > Interdisciplinary Centre for Computer Music Research
> > > University of Plymouth
> > > Office: +44-1752-233667
> > > Private: +44-1752-558917
> > > http://strasheela.sourceforge.net > <http://strasheela.sourceforge.net> <http://
> > strasheela.sourceforge.net>
> > > http://www.torsten-anders.de <http://www.torsten-anders.de> > <http://www.torsten-anders.de <http://www.torsten-anders.de>>
> > >
> > >
> >
> >
>
> --
> Torsten Anders
> Interdisciplinary Centre for Computer Music Research
> University of Plymouth
> Office: +44-1752-233667
> Private: +44-1752-558917
> http://strasheela.sourceforge.net <http://strasheela.sourceforge.net>
> http://www.torsten-anders.de <http://www.torsten-anders.de>
>
>

Dear Carl

Concerning your explanation at
http://lumma.org/music/theory/ModulationStrength.txt

> |log2(pitchA)-log2(pitchB)| * 1200 Eq.2

Why not something like the following?

log2( | pitchA - pitchB | )

Best
Torsten

On Apr 25, 2008, at 12:46 PM, Torsten Anders wrote:
> Thanks also for detailing the voice leading subject. However, I don't
> quite understand this yet. I figure you could simply have an absolute
> distance like
>
> | pitchA - pitchB |
>
> Having a logarithm doesn't so much "penalise" larger distances, like
> in the Tenney formular, right? But why having the factor 1200?

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

On Apr 25, 2008, at 1:37 PM, Kraig Grady wrote:
> i am sure translation is part of the problem. Still one could use
> higher harmonics in this way. introducing them as lower harmonics in
> preceding chords.
>
You mean, a preparation of higher harmonics -- much like the preparation of dissonances in common practise music? That's a very nice idea. Not the same as the matter of ascending/descending progressions, but these ideas are compatible I take it (as they are in common practise).

Best
Torsten

> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> _'''''''_ ^North/Western Hemisphere:
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
>
> _'''''''_ ^South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria <http://> anaphoriasouth.blogspot.com/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
> Torsten Anders wrote:
> >
> >
> > On Apr 25, 2008, at 1:16 PM, Kraig Grady wrote:
> > > I remember reading this by Schoenberg somewhere. i mean we are > talking
> > > 30 years but it did strike me. where the root becomes the fifth > or the
> > > fifth the third for example.
> > > Actually i remember the reverse as being the strong. c to g seems
> > > strong
> > > while G to C seems week because it it is just resolving, so a
> > > release of
> > > energy.
> > >
> > Schoenbergs terminology is definitely the reverse (although he tries
> > to avoid the term weak, uses descending instead). But he remarks > in a
> > footnote that Schenker uses the terminology the other way round. He
> > mentions in a footnote that for a while he though he got this idea
> > from Schenker, but his pupils told him that he did in fact use these
> > terms in classes already, many years before he read Schenker.
> >
> > Best
> > Torsten
> > > interesting the example of a II7 chord would be an example of of
> > > a root becoming a 7th.
> > > I am surprised we haven't heard from Monz on this one.
> > >
> >
> > >
> > >
> > > /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> > > _'''''''_ ^North/Western Hemisphere:
> > > North American Embassy of Anaphoria Island <http://anaphoria.com/
> > <http://anaphoria.com/>>
> > >
> > > _'''''''_ ^South/Eastern Hemisphere:
> > > Austronesian Outpost of Anaphoria <http://
> > > anaphoriasouth.blogspot.com/>
> > >
> > > ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
> > >
> > > Torsten Anders wrote:
> > > >
> > > > Dear Kraig,
> > > >
> > > > On Apr 22, 2008, at 10:33 PM, Kraig Grady wrote:
> > > > > What i remember which goes quite far back in time and my copy
> > > is still
> > > > > unpacked is strong progressions are where more important > note/s
> > > become
> > > > > less and vice versa. i have actually played with this in
> > > structures
> > > > > such
> > > > > as the eikosany. Anyway it seems if you allow the 7th and 9th
> > > > > harmonics
> > > > > in chords, even triads using them, it seems one would have > further
> > > > > progressions to choose from although this would quickly lead
> > > > > outside of
> > > > > any particular scale. One could also use the 11th.
> > > > >
> > > >
> > > > thanks for your mail. Unfortunately, I don't quite understand > what
> > > > you mean by "strong progressions are where more important note/s
> > > > become less and vice versa". If you are referring to the > chord root
> > > > as the only important note, then this guideline is closely
> > > related to
> > > > what I was suggesting (but not the same). I said that in a > strong
> > > > progression there appears new "important note" -- the root of > the
> > > new
> > > > chord was not in the previous chord.
> > > >
> > > > But it seems your remark is more general. In particular, it > appears
> > > > to are referring to specific higher harmonics. Could you please
> > > > detail this further?
> > > >
> > > > Thank you!
> > > >
> > > > Best
> > > > Torsten
> > > >
> > > > >
> > > > >
> > > > > /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> > > > > _'''''''_ ^North/Western Hemisphere:
> > > > > North American Embassy of Anaphoria Island
> > <http://anaphoria.com/ <http://anaphoria.com/>
> > > > <http://anaphoria.com/ <http://anaphoria.com/>>>
> > > > >
> > > > > _'''''''_ ^South/Eastern Hemisphere:
> > > > > Austronesian Outpost of Anaphoria <http://
> > > > > anaphoriasouth.blogspot.com/>
> > > > >
> > > > > ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
> > > > >
> > > > > Torsten Anders wrote:
> > > > > >
> > > > > > Dear Carl,
> > > > > >
> > > > > > On Apr 22, 2008, at 7:19 PM, Carl Lumma wrote:
> > > > > > > > Here is a suggestion I can make for guidelines on > good chord
> > > > > > > > progressions, which is indeed based on the notion of
> > > pitch class
> > > > > > > > sets, but which also works beyond 12 ET (e.g., I did
> > > examples
> > > > > > > > using them with 31 ET). This suggestion is based on
> > > Schoenberg's
> > > > > > > > guidelines on good chord root progressions where he
> > > introduces
> > > > > > > > the notion of ascending (strong), descending ('weak') > and
> > > > > > > > superstrong progressions (see the respective chapter > in his
> > > > > > > > Theory of Harmony). The summary of Schoenberg's
> > > guidelines/rules
> > > > > > > > is based on 12 ET, but his explanation is actually more
> > > general.
> > > > > > > > A formalisation of his explanations (instead of his > actual
> > > > > rules)
> > > > > > > > does work beyond 12 ET.
> > > > > > > >
> > > > > > > > The main difference between Schoenberg's actual
> > > guidelines and
> > > > > > > > their formalised generalisation is that Schoenberg's
> > > guidelines
> > > > > > > > are based on scale degree intervals between chord roots,
> > > whereas
> > > > > > > > the generalisation exploits whether the root pitch class
> > > of some
> > > > > > > > chord is contained in the pitch class set of another > chord.
> > > > > > >
> > > > > > > I haven't read Schoenberg's book but it sounds like you've
> > > > > > > done a bit of work for him. Yes, I could probably > generalize
> > > > > > > many (though not all) of the "bad music theory papers" > if I
> > > > > > > worked at it. And some such minimal effort would only be
> > > > > > > fair in the meta-analysis I proposed. Where to draw the > line
> > > > > > > is another question.
> > > > > > >
> > > > > >
> > > > > > > Did Schoenberg say the important thing
> > > > > > > is whether the root is a member of the new chord (which is
> > > > > > > a reasonable thing)? Or did he say something else?
> > > > > > >
> > > > > >
> > > > > > Sorry, I don't fully understand your question. In an > ascending
> > > > > > progression, the root of the preceding chord is a pitch > in the
> > > > > > following chord. Did you mean that?
> > > > > >
> > > > > > > > OK, it follows the
> > > > > > > > core of is the formalisation (I hope the notation > explains
> > > > > itself:
> > > > > > > > defined are Boolean functions expecting two neighbouring
> > > > > chords).
> > > > > > > >
> > > > > > > > /* Chord1 and Chord2 have common pitch classes, but the
> > > root of
> > > > > > > > Chord2 does not occur in the set of Chord1's
> > > pitchclasses. */
> > > > > > > >
> > > > > > > > isAscendingProgression(Chord1, Chord2) :=
> > > > > > > > ( NOT ( getRoot(Chord2) \in getPitchClasses(Chord1) ) )
> > > > > > > > AND ( intersection(getPitchClasses(Chord1), > getPitchClasses
> > > > > > > > (Chord2)) \= emptySet )
> > > > > > > >
> > > > > > > > /* A non-root pitchclass of Chord1 is root in Chord2. */
> > > > > > > >
> > > > > > > > isDescendingProgression(Chord1, Chord2) :=
> > > > > > > > ( getRoot(Chord2) \in getPitchClasses(Chord1) )
> > > > > > > > AND ( getRoot(Chord1) \= getRoot(Chord2) )
> > > > > > > >
> > > > > > > > /* Chord1 and Chord2 have no common pitch classes */
> > > > > > > >
> > > > > > > > isSuperstrongProgression(Chord1, Chord2) :=
> > > > > > > > intersection(getPitchClasses(Chord1), getPitchClasses
> > > (Chord2)) =
> > > > > > > > emptySet
> > > > > > >
> > > > > > > Yes, perfectly understandable, though I find the terms
> > > > > > > "ascending" and "descending" extremely odd here.
> > > > > > >
> > > > > > These are Schoenberg's terms. He mentions that H. > Schenker uses
> > > > > these
> > > > > > terms as well ("aufsteigend" and "fallend") though in > opposite
> > > > > > meaning. He remarks something like that anyone who loves
> > > Brahmsian
> > > > > > harmony would likely come up with similar concepts.
> > > > > >
> > > > > > > > If you are interested I can post further formalisations/
> > > > > > > > generalisations of Schoenberg's rules.
> > > > > > >
> > > > > > > I would be interested to read them, especially if you
> > > > > > > show how you generalized them. And if monz is reading,
> > > > > > > I bet he'd be interested, too.
> > > > > > >
> > > > > > I'll write a new mail soon..
> > > > > >
> > > > > > > > For example, Schoenberg
> > > > > > > > recommends that descending progressions should > "resolve".
> > > For
> > > > > any
> > > > > > > > three successive chords, if the first two chords form a
> > > > > descending
> > > > > > > > progression, then the progression from the first to the
> > > third
> > > > > chord
> > > > > > > > should form a strong or superstrong progression (so the
> > > > > middle chord
> > > > > > > > is quasi a 'passing chord').
> > > > > > >
> > > > > > > Hm... this seems kindof strange.
> > > > > > >
> > > > > > Schoenberg is always very exhaustive :) He does not want > to ban
> > > > > > descending progressions, so he looks for a way how he can
> > > allow for
> > > > > > them in a way which is musically convincing in most cases.
> > > So, he
> > > > > > comes up with the idea of a 'passing chord'.
> > > > > >
> > > > > > Please note that he makes is always very clear that his > rules
> > > are
> > > > > > guidelines for the pupil, but no laws for masterworks. > This is
> > > > > why he
> > > > > > prefers the term "descending" instead of "weak" > progressions (he
> > > > > uses
> > > > > > the term "strong" for "ascending" progressions a lot). > Still, he
> > > > > does
> > > > > > not discuss where descending progressions are used to good
> > > > > purpose in
> > > > > > any masterwork, but such considerations have hardly a place
> > > > > anyway in
> > > > > > his systematic approach.
> > > > > >
> > > > > > > > Anyway, for chord progressions of diatonic triads in
> > > major, the
> > > > > > > > generalised formalisation and Schoenberg's rules are
> > > equivalent
> > > > > > > > (e.g., the function isAscendingProgression above returns
> > > true
> > > > > for
> > > > > > > > progressions which Schoenberg calls ascending). > Still, the
> > > > > behaviour
> > > > > > > > of the constraints and Schoenberg's rules differ for > more
> > > > > complex
> > > > > > > > cases. According to Schoenberg, a progression is > superstrong
> > > > > if the
> > > > > > > > root interval is a step up or down.
> > > > > > >
> > > > > > > Bzzz. :)
> > > > > > >
> > > > > > > > For example, the progression V7 IV is superstrong
> > > according to
> > > > > > > > Schoenberg. For the definitions above, however, this
> > > progression
> > > > > > > > is descending (!), because the root of IV is > contained in V7
> > > > > > > > (e.g. in G7 F, the F's root pitchclass f is already
> > > contained
> > > > > > > > in G7). Indeed, this progression is rare in music.
> > > > > > > > By contrast, the progression I IIIb (e.g., C Eb) is a
> > > descending
> > > > > > > > progression in Schoenbergs original definition. For the
> > > > > > > > definitions above, however, this is an ascending > progression
> > > > > > > > (the root of Es is not contained in C), and indeed > for me
> > > the
> > > > > > > > progression feels strong.
> > > > > > >
> > > > > > > Sounds like we are really dealing with the Anders theory
> > > > > > > of chord progressions. Which is probably a blessing.
> > > > > > >
> > > > > > :) As I said, I simply implemented Schoenberg's rule > explanation
> > > > > > instead of his actual rules. So, I cannot accept the honour
> > > you are
> > > > > > implying.
> > > > > >
> > > > > > > > Let us create a sequence of 5 chords and
> > > > > > > > allow only for diatonic triads in C-major. The first and
> > > last
> > > > > > > > chord should be the tonic. Following Schoenberg let us
> > > specify
> > > > > > > > that ascending progressions are "resolved" (see above),
> > > and we
> > > > > > > > don't allow for superstrong progressions. Finally, > the union
> > > > > > > > of the last three chords must contain all pitch > classes of
> > > > > > > > C-major (i.e. these chords form a cadence).
> > > > > > > >
> > > > > > > > This problem has only two solutions (I used the > computer :).
> > > > > > > > These solutions are as follows. They happen to > contain only
> > > > > > > > ascending progressions. Such solutions are particularly
> > > > > > > > convincing. For simplicity I given absolute chord names.
> > > > > > > >
> > > > > > > > C-maj F-maj D-min G-maj C-maj (only ascending > progressions)
> > > > > > > > C-maj A-min D-min G-maj C-maj (only ascending > progressions)
> > > > > > >
> > > > > > > Only allowing for diatonic triads is C-major is a very
> > > > > > > helpful constraint. Howabout we allow all 7-limit triads
> > > > > > > in 31-ET, for a start?
> > > > > > >
> > > > > >
> > > > > > The point of these generalised definitions is exactly > that they
> > > > > allow
> > > > > > for such cases :) I only simplified for the sake of an
> > > argument. So,
> > > > > > here is a 7-limit variant. Let the scale be septimal natural
> > > minor
> > > > > > (Scala term, in 31 ET the pitches C D D# F G G# A#). Let us
> > > allow
> > > > > for
> > > > > > the following septimal triads (please correct my > terminology if
> > > > > > necessary): harmonic diminished (5/5 6/5 7/5), subharmonic
> > > > > diminished
> > > > > > (7/7 7/6 7/5), subminor (6/1 7/1 9/1), supermajor (1/9 > 1/7 1/6).
> > > > > Note
> > > > > > that I consider all these chords as consonances for
> > > simplicity which
> > > > > > require neither preparation nor resolution. Naturally, we > could
> > > > > > define extra rules which care for that.
> > > > > >
> > > > > > There are 2 solutions of 5 chords with this scale, these
> > > chords and
> > > > > > the rule set specified before. For convenience, I > attached these
> > > > > > solutions as PDF files (please let me know if doing so
> > > violates some
> > > > > > policies of this list).
> > > > > >
> > > > > > The solutions are shown by two staffs. The upper stuff > shows the
> > > > > > actual chord pitches (I didn't bother to implement voice > leading
> > > > > > rules here, so things like Bruckner's law of the shortest
> > > path are
> > > > > > violated). The staff notation is in 31 ET (i.e. the interval
> > > C D# is
> > > > > > 7/6). The lower staff shows the chord roots in staff > notation
> > > plus
> > > > > > there ratios expressed as a product of the untransposed > chord
> > > ratios
> > > > > > and a root factor for each chord ratio. Some of these > factors
> > > may
> > > > > > look odd at first, e.g., the first chord has the factor 4/3
> > > for the
> > > > > > root C, but this is due to that fact that the > untransposed chord
> > > > > > ratios don't contain 1/1 -- the resulting product ratios do
> > > > > contain 1/1.
> > > > > >
> > > > > > If you are interested in other chord sets, scales, or longer
> > > > > > solutions then just say so -- its easy to change some > numbers
> > > in a
> > > > > > computer program :)
> > > > > >
> > > > > > Best
> > > > > > Torsten
> > > > > >
> > > > > >
> > > > > > --
> > > > > > Torsten Anders
> > > > > > Interdisciplinary Centre for Computer Music Research
> > > > > > University of Plymouth
> > > > > > Office: +44-1752-233667
> > > > > > Private: +44-1752-558917
> > > > > > http://strasheela.sourceforge.net
> > <http://strasheela.sourceforge.net> <http://
> > > strasheela.sourceforge.net>
> > > > > > http://www.torsten-anders.de <http://www.torsten-anders.de>
> > <http://www.torsten-anders.de <http://www.torsten-anders.de>>
> > > > > >
> > > > > >
> > > > > >
> > > > > >
> > > > >
> > > > >
> > > >
> > > > --
> > > > Torsten Anders
> > > > Interdisciplinary Centre for Computer Music Research
> > > > University of Plymouth
> > > > Office: +44-1752-233667
> > > > Private: +44-1752-558917
> > > > http://strasheela.sourceforge.net
> > <http://strasheela.sourceforge.net> <http://
> > > strasheela.sourceforge.net>
> > > > http://www.torsten-anders.de <http://www.torsten-anders.de>
> > <http://www.torsten-anders.de <http://www.torsten-anders.de>>
> > > >
> > > >
> > >
> > >
> >
> > --
> > Torsten Anders
> > Interdisciplinary Centre for Computer Music Research
> > University of Plymouth
> > Office: +44-1752-233667
> > Private: +44-1752-558917
> > http://strasheela.sourceforge.net <http://> strasheela.sourceforge.net>
> > http://www.torsten-anders.de <http://www.torsten-anders.de>
> >
> >
>
>
--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

I was thinking of the higher harmonics as consonances ( which they are more so than not). So the idea is to expand the idea of first third and fifth harmonic as strong progressions, to include seven , nine and eleven. The eleven is hard for many because of where it lies implies scales outside of 12 and for this reason, it more difficult to use for most

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Torsten Anders wrote:
>
>
> On Apr 25, 2008, at 1:37 PM, Kraig Grady wrote:
> > i am sure translation is part of the problem. Still one could use
> > higher harmonics in this way. introducing them as lower harmonics in
> > preceding chords.
> >
> You mean, a preparation of higher harmonics -- much like the
> preparation of dissonances in common practise music? That's a very
> nice idea. Not the same as the matter of ascending/descending
> progressions, but these ideas are compatible I take it (as they are
> in common practise).
>
> Best
> Torsten
>
> > /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> > _'''''''_ ^North/Western Hemisphere:
> > North American Embassy of Anaphoria Island <http://anaphoria.com/ > <http://anaphoria.com/>>
> >
> > _'''''''_ ^South/Eastern Hemisphere:
> > Austronesian Outpost of Anaphoria <http://
> > anaphoriasouth.blogspot.com/>
> >
> > ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
> >
> > Torsten Anders wrote:
> > >
> > >
> > > On Apr 25, 2008, at 1:16 PM, Kraig Grady wrote:
> > > > I remember reading this by Schoenberg somewhere. i mean we are
> > talking
> > > > 30 years but it did strike me. where the root becomes the fifth
> > or the
> > > > fifth the third for example.
> > > > Actually i remember the reverse as being the strong. c to g seems
> > > > strong
> > > > while G to C seems week because it it is just resolving, so a
> > > > release of
> > > > energy.
> > > >
> > > Schoenbergs terminology is definitely the reverse (although he tries
> > > to avoid the term weak, uses descending instead). But he remarks
> > in a
> > > footnote that Schenker uses the terminology the other way round. He
> > > mentions in a footnote that for a while he though he got this idea
> > > from Schenker, but his pupils told him that he did in fact use these
> > > terms in classes already, many years before he read Schenker.
> > >
> > > Best
> > > Torsten
> > > > interesting the example of a II7 chord would be an example of of
> > > > a root becoming a 7th.
> > > > I am surprised we haven't heard from Monz on this one.
> > > >
> > >
> > > >
> > > >
> > > > /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> > > > _'''''''_ ^North/Western Hemisphere:
> > > > North American Embassy of Anaphoria Island > <http://anaphoria.com/ <http://anaphoria.com/>
> > > <http://anaphoria.com/ <http://anaphoria.com/>>>
> > > >
> > > > _'''''''_ ^South/Eastern Hemisphere:
> > > > Austronesian Outpost of Anaphoria <http://
> > > > anaphoriasouth.blogspot.com/>
> > > >
> > > > ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
> > > >
> > > > Torsten Anders wrote:
> > > > >
> > > > > Dear Kraig,
> > > > >
> > > > > On Apr 22, 2008, at 10:33 PM, Kraig Grady wrote:
> > > > > > What i remember which goes quite far back in time and my copy
> > > > is still
> > > > > > unpacked is strong progressions are where more important
> > note/s
> > > > become
> > > > > > less and vice versa. i have actually played with this in
> > > > structures
> > > > > > such
> > > > > > as the eikosany. Anyway it seems if you allow the 7th and 9th
> > > > > > harmonics
> > > > > > in chords, even triads using them, it seems one would have
> > further
> > > > > > progressions to choose from although this would quickly lead
> > > > > > outside of
> > > > > > any particular scale. One could also use the 11th.
> > > > > >
> > > > >
> > > > > thanks for your mail. Unfortunately, I don't quite understand
> > what
> > > > > you mean by "strong progressions are where more important note/s
> > > > > become less and vice versa". If you are referring to the
> > chord root
> > > > > as the only important note, then this guideline is closely
> > > > related to
> > > > > what I was suggesting (but not the same). I said that in a
> > strong
> > > > > progression there appears new "important note" -- the root of
> > the
> > > > new
> > > > > chord was not in the previous chord.
> > > > >
> > > > > But it seems your remark is more general. In particular, it
> > appears
> > > > > to are referring to specific higher harmonics. Could you please
> > > > > detail this further?
> > > > >
> > > > > Thank you!
> > > > >
> > > > > Best
> > > > > Torsten
> > > > >
> > > > > >
> > > > > >
> > > > > > /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> > > > > > _'''''''_ ^North/Western Hemisphere:
> > > > > > North American Embassy of Anaphoria Island
> > > <http://anaphoria.com/ <http://anaphoria.com/> > <http://anaphoria.com/ <http://anaphoria.com/>>
> > > > > <http://anaphoria.com/ <http://anaphoria.com/> > <http://anaphoria.com/ <http://anaphoria.com/>>>>
> > > > > >
> > > > > > _'''''''_ ^South/Eastern Hemisphere:
> > > > > > Austronesian Outpost of Anaphoria <http://
> > > > > > anaphoriasouth.blogspot.com/>
> > > > > >
> > > > > > ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
> > > > > >
> > > > > > Torsten Anders wrote:
> > > > > > >
> > > > > > > Dear Carl,
> > > > > > >
> > > > > > > On Apr 22, 2008, at 7:19 PM, Carl Lumma wrote:
> > > > > > > > > Here is a suggestion I can make for guidelines on
> > good chord
> > > > > > > > > progressions, which is indeed based on the notion of
> > > > pitch class
> > > > > > > > > sets, but which also works beyond 12 ET (e.g., I did
> > > > examples
> > > > > > > > > using them with 31 ET). This suggestion is based on
> > > > Schoenberg's
> > > > > > > > > guidelines on good chord root progressions where he
> > > > introduces
> > > > > > > > > the notion of ascending (strong), descending ('weak')
> > and
> > > > > > > > > superstrong progressions (see the respective chapter
> > in his
> > > > > > > > > Theory of Harmony). The summary of Schoenberg's
> > > > guidelines/rules
> > > > > > > > > is based on 12 ET, but his explanation is actually more
> > > > general.
> > > > > > > > > A formalisation of his explanations (instead of his
> > actual
> > > > > > rules)
> > > > > > > > > does work beyond 12 ET.
> > > > > > > > >
> > > > > > > > > The main difference between Schoenberg's actual
> > > > guidelines and
> > > > > > > > > their formalised generalisation is that Schoenberg's
> > > > guidelines
> > > > > > > > > are based on scale degree intervals between chord roots,
> > > > whereas
> > > > > > > > > the generalisation exploits whether the root pitch class
> > > > of some
> > > > > > > > > chord is contained in the pitch class set of another
> > chord.
> > > > > > > >
> > > > > > > > I haven't read Schoenberg's book but it sounds like you've
> > > > > > > > done a bit of work for him. Yes, I could probably
> > generalize
> > > > > > > > many (though not all) of the "bad music theory papers"
> > if I
> > > > > > > > worked at it. And some such minimal effort would only be
> > > > > > > > fair in the meta-analysis I proposed. Where to draw the
> > line
> > > > > > > > is another question.
> > > > > > > >
> > > > > > >
> > > > > > > > Did Schoenberg say the important thing
> > > > > > > > is whether the root is a member of the new chord (which is
> > > > > > > > a reasonable thing)? Or did he say something else?
> > > > > > > >
> > > > > > >
> > > > > > > Sorry, I don't fully understand your question. In an
> > ascending
> > > > > > > progression, the root of the preceding chord is a pitch
> > in the
> > > > > > > following chord. Did you mean that?
> > > > > > >
> > > > > > > > > OK, it follows the
> > > > > > > > > core of is the formalisation (I hope the notation
> > explains
> > > > > > itself:
> > > > > > > > > defined are Boolean functions expecting two neighbouring
> > > > > > chords).
> > > > > > > > >
> > > > > > > > > /* Chord1 and Chord2 have common pitch classes, but the
> > > > root of
> > > > > > > > > Chord2 does not occur in the set of Chord1's
> > > > pitchclasses. */
> > > > > > > > >
> > > > > > > > > isAscendingProgression(Chord1, Chord2) :=
> > > > > > > > > ( NOT ( getRoot(Chord2) \in getPitchClasses(Chord1) ) )
> > > > > > > > > AND ( intersection(getPitchClasses(Chord1),
> > getPitchClasses
> > > > > > > > > (Chord2)) \= emptySet )
> > > > > > > > >
> > > > > > > > > /* A non-root pitchclass of Chord1 is root in Chord2. */
> > > > > > > > >
> > > > > > > > > isDescendingProgression(Chord1, Chord2) :=
> > > > > > > > > ( getRoot(Chord2) \in getPitchClasses(Chord1) )
> > > > > > > > > AND ( getRoot(Chord1) \= getRoot(Chord2) )
> > > > > > > > >
> > > > > > > > > /* Chord1 and Chord2 have no common pitch classes */
> > > > > > > > >
> > > > > > > > > isSuperstrongProgression(Chord1, Chord2) :=
> > > > > > > > > intersection(getPitchClasses(Chord1), getPitchClasses
> > > > (Chord2)) =
> > > > > > > > > emptySet
> > > > > > > >
> > > > > > > > Yes, perfectly understandable, though I find the terms
> > > > > > > > "ascending" and "descending" extremely odd here.
> > > > > > > >
> > > > > > > These are Schoenberg's terms. He mentions that H.
> > Schenker uses
> > > > > > these
> > > > > > > terms as well ("aufsteigend" and "fallend") though in
> > opposite
> > > > > > > meaning. He remarks something like that anyone who loves
> > > > Brahmsian
> > > > > > > harmony would likely come up with similar concepts.
> > > > > > >
> > > > > > > > > If you are interested I can post further formalisations/
> > > > > > > > > generalisations of Schoenberg's rules.
> > > > > > > >
> > > > > > > > I would be interested to read them, especially if you
> > > > > > > > show how you generalized them. And if monz is reading,
> > > > > > > > I bet he'd be interested, too.
> > > > > > > >
> > > > > > > I'll write a new mail soon..
> > > > > > >
> > > > > > > > > For example, Schoenberg
> > > > > > > > > recommends that descending progressions should
> > "resolve".
> > > > For
> > > > > > any
> > > > > > > > > three successive chords, if the first two chords form a
> > > > > > descending
> > > > > > > > > progression, then the progression from the first to the
> > > > third
> > > > > > chord
> > > > > > > > > should form a strong or superstrong progression (so the
> > > > > > middle chord
> > > > > > > > > is quasi a 'passing chord').
> > > > > > > >
> > > > > > > > Hm... this seems kindof strange.
> > > > > > > >
> > > > > > > Schoenberg is always very exhaustive :) He does not want
> > to ban
> > > > > > > descending progressions, so he looks for a way how he can
> > > > allow for
> > > > > > > them in a way which is musically convincing in most cases.
> > > > So, he
> > > > > > > comes up with the idea of a 'passing chord'.
> > > > > > >
> > > > > > > Please note that he makes is always very clear that his
> > rules
> > > > are
> > > > > > > guidelines for the pupil, but no laws for masterworks.
> > This is
> > > > > > why he
> > > > > > > prefers the term "descending" instead of "weak"
> > progressions (he
> > > > > > uses
> > > > > > > the term "strong" for "ascending" progressions a lot).
> > Still, he
> > > > > > does
> > > > > > > not discuss where descending progressions are used to good
> > > > > > purpose in
> > > > > > > any masterwork, but such considerations have hardly a place
> > > > > > anyway in
> > > > > > > his systematic approach.
> > > > > > >
> > > > > > > > > Anyway, for chord progressions of diatonic triads in
> > > > major, the
> > > > > > > > > generalised formalisation and Schoenberg's rules are
> > > > equivalent
> > > > > > > > > (e.g., the function isAscendingProgression above returns
> > > > true
> > > > > > for
> > > > > > > > > progressions which Schoenberg calls ascending).
> > Still, the
> > > > > > behaviour
> > > > > > > > > of the constraints and Schoenberg's rules differ for
> > more
> > > > > > complex
> > > > > > > > > cases. According to Schoenberg, a progression is
> > superstrong
> > > > > > if the
> > > > > > > > > root interval is a step up or down.
> > > > > > > >
> > > > > > > > Bzzz. :)
> > > > > > > >
> > > > > > > > > For example, the progression V7 IV is superstrong
> > > > according to
> > > > > > > > > Schoenberg. For the definitions above, however, this
> > > > progression
> > > > > > > > > is descending (!), because the root of IV is
> > contained in V7
> > > > > > > > > (e.g. in G7 F, the F's root pitchclass f is already
> > > > contained
> > > > > > > > > in G7). Indeed, this progression is rare in music.
> > > > > > > > > By contrast, the progression I IIIb (e.g., C Eb) is a
> > > > descending
> > > > > > > > > progression in Schoenbergs original definition. For the
> > > > > > > > > definitions above, however, this is an ascending
> > progression
> > > > > > > > > (the root of Es is not contained in C), and indeed
> > for me
> > > > the
> > > > > > > > > progression feels strong.
> > > > > > > >
> > > > > > > > Sounds like we are really dealing with the Anders theory
> > > > > > > > of chord progressions. Which is probably a blessing.
> > > > > > > >
> > > > > > > :) As I said, I simply implemented Schoenberg's rule
> > explanation
> > > > > > > instead of his actual rules. So, I cannot accept the honour
> > > > you are
> > > > > > > implying.
> > > > > > >
> > > > > > > > > Let us create a sequence of 5 chords and
> > > > > > > > > allow only for diatonic triads in C-major. The first and
> > > > last
> > > > > > > > > chord should be the tonic. Following Schoenberg let us
> > > > specify
> > > > > > > > > that ascending progressions are "resolved" (see above),
> > > > and we
> > > > > > > > > don't allow for superstrong progressions. Finally,
> > the union
> > > > > > > > > of the last three chords must contain all pitch
> > classes of
> > > > > > > > > C-major (i.e. these chords form a cadence).
> > > > > > > > >
> > > > > > > > > This problem has only two solutions (I used the
> > computer :).
> > > > > > > > > These solutions are as follows. They happen to
> > contain only
> > > > > > > > > ascending progressions. Such solutions are particularly
> > > > > > > > > convincing. For simplicity I given absolute chord names.
> > > > > > > > >
> > > > > > > > > C-maj F-maj D-min G-maj C-maj (only ascending
> > progressions)
> > > > > > > > > C-maj A-min D-min G-maj C-maj (only ascending
> > progressions)
> > > > > > > >
> > > > > > > > Only allowing for diatonic triads is C-major is a very
> > > > > > > > helpful constraint. Howabout we allow all 7-limit triads
> > > > > > > > in 31-ET, for a start?
> > > > > > > >
> > > > > > >
> > > > > > > The point of these generalised definitions is exactly
> > that they
> > > > > > allow
> > > > > > > for such cases :) I only simplified for the sake of an
> > > > argument. So,
> > > > > > > here is a 7-limit variant. Let the scale be septimal natural
> > > > minor
> > > > > > > (Scala term, in 31 ET the pitches C D D# F G G# A#). Let us
> > > > allow
> > > > > > for
> > > > > > > the following septimal triads (please correct my
> > terminology if
> > > > > > > necessary): harmonic diminished (5/5 6/5 7/5), subharmonic
> > > > > > diminished
> > > > > > > (7/7 7/6 7/5), subminor (6/1 7/1 9/1), supermajor (1/9
> > 1/7 1/6).
> > > > > > Note
> > > > > > > that I consider all these chords as consonances for
> > > > simplicity which
> > > > > > > require neither preparation nor resolution. Naturally, we
> > could
> > > > > > > define extra rules which care for that.
> > > > > > >
> > > > > > > There are 2 solutions of 5 chords with this scale, these
> > > > chords and
> > > > > > > the rule set specified before. For convenience, I
> > attached these
> > > > > > > solutions as PDF files (please let me know if doing so
> > > > violates some
> > > > > > > policies of this list).
> > > > > > >
> > > > > > > The solutions are shown by two staffs. The upper stuff
> > shows the
> > > > > > > actual chord pitches (I didn't bother to implement voice
> > leading
> > > > > > > rules here, so things like Bruckner's law of the shortest
> > > > path are
> > > > > > > violated). The staff notation is in 31 ET (i.e. the interval
> > > > C D# is
> > > > > > > 7/6). The lower staff shows the chord roots in staff
> > notation
> > > > plus
> > > > > > > there ratios expressed as a product of the untransposed
> > chord
> > > > ratios
> > > > > > > and a root factor for each chord ratio. Some of these
> > factors
> > > > may
> > > > > > > look odd at first, e.g., the first chord has the factor 4/3
> > > > for the
> > > > > > > root C, but this is due to that fact that the
> > untransposed chord
> > > > > > > ratios don't contain 1/1 -- the resulting product ratios do
> > > > > > contain 1/1.
> > > > > > >
> > > > > > > If you are interested in other chord sets, scales, or longer
> > > > > > > solutions then just say so -- its easy to change some
> > numbers
> > > > in a
> > > > > > > computer program :)
> > > > > > >
> > > > > > > Best
> > > > > > > Torsten
> > > > > > >
> > > > > > >
> > > > > > > --
> > > > > > > Torsten Anders
> > > > > > > Interdisciplinary Centre for Computer Music Research
> > > > > > > University of Plymouth
> > > > > > > Office: +44-1752-233667
> > > > > > > Private: +44-1752-558917
> > > > > > > http://strasheela.sourceforge.net > <http://strasheela.sourceforge.net>
> > > <http://strasheela.sourceforge.net > <http://strasheela.sourceforge.net>> <http://
> > > > strasheela.sourceforge.net>
> > > > > > > http://www.torsten-anders.de > <http://www.torsten-anders.de> <http://www.torsten-anders.de > <http://www.torsten-anders.de>>
> > > <http://www.torsten-anders.de <http://www.torsten-anders.de> > <http://www.torsten-anders.de <http://www.torsten-anders.de>>>
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > >
> > > > > >
> > > > >
> > > > > --
> > > > > Torsten Anders
> > > > > Interdisciplinary Centre for Computer Music Research
> > > > > University of Plymouth
> > > > > Office: +44-1752-233667
> > > > > Private: +44-1752-558917
> > > > > http://strasheela.sourceforge.net > <http://strasheela.sourceforge.net>
> > > <http://strasheela.sourceforge.net > <http://strasheela.sourceforge.net>> <http://
> > > > strasheela.sourceforge.net>
> > > > > http://www.torsten-anders.de <http://www.torsten-anders.de> > <http://www.torsten-anders.de <http://www.torsten-anders.de>>
> > > <http://www.torsten-anders.de <http://www.torsten-anders.de> > <http://www.torsten-anders.de <http://www.torsten-anders.de>>>
> > > > >
> > > > >
> > > >
> > > >
> > >
> > > --
> > > Torsten Anders
> > > Interdisciplinary Centre for Computer Music Research
> > > University of Plymouth
> > > Office: +44-1752-233667
> > > Private: +44-1752-558917
> > > http://strasheela.sourceforge.net > <http://strasheela.sourceforge.net> <http://
> > strasheela.sourceforge.net>
> > > http://www.torsten-anders.de <http://www.torsten-anders.de> > <http://www.torsten-anders.de <http://www.torsten-anders.de>>
> > >
> > >
> >
> >
>
> --
> Torsten Anders
> Interdisciplinary Centre for Computer Music Research
> University of Plymouth
> Office: +44-1752-233667
> Private: +44-1752-558917
> http://strasheela.sourceforge.net <http://strasheela.sourceforge.net>
> http://www.torsten-anders.de <http://www.torsten-anders.de>
>
>

Torsten wrote...

> > I see examples of this in vocal music that features a pedaltone
> > in one of the parts, such as a barbershop tag. For instance, if
> > one part is holding the tonic note while the chord moves from a
> > tonic to a II7, the root will need to move up a Septimal-Major
> > second (8/7) to keep the chords tuned without having to bend the
> > pedaltone.
>
> Wow, do they do such things? I like that.

They do (I'm a recovering barbershoper). You should get some
recordings!

-Carl

Hi Torsten,

> > --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > > I've placed a better explanation of the measure here:
> > >
> > > http://lumma.org/music/theory/ChordProgressionStrength.txt
> >
> > Er, make that
> > http://lumma.org/music/theory/ModulationStrength.txt
>
> thanks for this explanation. I now better understand you. I will
> think about some way to implement this "harmonic strength" in
> Strasheela :)
>
> Open question still: what happens in cases where the pairs of
> changing pitches can not be distributed evenly without
> "overlapping".
> Say, what there are three changing pitches?

Can you give an example?

> Minor point: I am not quite convinced by the term "modulation
> strength", as modulations don't necessarily occur here (we don't
> know whether any new key would be installed).

I thought about that. In popular music, though, I've often
seen the term used to mean a chord change, not a key change.
Anybody have an opinion on that?

> Thanks also for detailing the voice leading subject. However,
> I don't quite understand this yet. I figure you could simply
> have an absolute distance like
>
> | pitchA - pitchB |
>
> Having a logarithm doesn't so much "penalise" larger distances,
> like in the Tenney formular, right? But why having the factor
> 1200?

Remember pitchA and pitchB are ratios. So log is needed to
make subtraction work. (In my original exposition on modulation
strength I used subtraction on ratios informally... you can
see in the current revision of the document I use division.)
The factor of 1200 simply makes the units cents.

> Concerning your "chord generation regime", what are n-ads,
> chords of n pitches?

Yes.

> Also, I don't understand your step 5. I see that with your
> approach you minimise the voice-leading distance but allow for
> larger "chord harmonic distance" (your modulation strength).
> I like this idea.

For every chord pair you know the modulation strength. Simply
group all the chord pairs into two groups -- those with mod.
strength higher than the median mod. strength, and those
mod. strength below. The two groups will be of equal size
according to the definition of "median". Now when building
a progression, you can choose only from one group or the
other, or alternate between groups, etc.

> However, using Strasheela I would realise it declaratively
> instead of procedurally as you do here. For example, I may
> specify that low voice-leading distances are used for specific
> or all chord progressions (neighbouring chord pairs), and at
> the same time that the "chord harmonic distance" should be
> low or high for all or for specific chord progressions. We
> might also add various other constraints (e.g., the tonic
> chord should re-appear from time to time etc.). That's what
> is nice about constraint programming: you just define a
> conjunction of conditions (other connectives like disjunction
> or implication and their nesting can be used as well).

Sure, I am familiar with constraint programming. However
my brain is stuck in procedural land!

-Carl

Torsten wrote...

> > I can do better than that!
> > http://lumma.org/tuning/rothenberg/AModelForPatternPerception.pdf
>
> Thank you very much!

I am proud to say that this is the only source of these
documents I know of on the internet. Of course that is
also rather unfortunate from my point of view!

> Could you perhaps even summarise Rothenberg's idea concerning
> which scale notes are required to recognise a scale. Is this
> related to Mazzola's cadencial sets brought up by Hans?

I'm not familiar with the concept of Mazzola/Hans, but it
certainly sounds similar.
Your PDF reader should be able to show you "bookmarks" in
the file. These take you to the start of each of the three
papers (originally published separately) in the file.
Go to "Rothenberg2", then scroll to pg. 368, and start
reading at section 14 ("Sufficient Sets; The Coding of P").
Read through section 15 ("Efficiency") until you reach
the end of the paper. Let me know what you think.

-Carl

Dear Carl,

On Apr 25, 2008, at 7:13 PM, Carl Lumma wrote:
> > > --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> > >
> > > > I've placed a better explanation of the measure here:
> > > >
> > > > http://lumma.org/music/theory/ChordProgressionStrength.txt
> > >
> > > Er, make that
> > > http://lumma.org/music/theory/ModulationStrength.txt
> >
> > thanks for this explanation. I now better understand you. I will
> > think about some way to implement this "harmonic strength" in
> > Strasheela :)
> >
> > Open question still: what happens in cases where the pairs of
> > changing pitches can not be distributed evenly without
> > "overlapping".
> > Say, what there are three changing pitches?
>
> Can you give an example?
>
In V7 -> I there are 5 changing pitches. How are you ordering them in pairs? Or, if you would discard some pitch here, how to decide which without doing some pairs first?

> > Thanks also for detailing the voice leading subject. However,
> > I don't quite understand this yet. I figure you could simply
> > have an absolute distance like
> >
> > | pitchA - pitchB |
> >
> > Having a logarithm doesn't so much "penalise" larger distances,
> > like in the Tenney formular, right? But why having the factor
> > 1200?
>
> Remember pitchA and pitchB are ratios. So log is needed to
> make subtraction work. (In my original exposition on modulation
> strength I used subtraction on ratios informally... you can
> see in the current revision of the document I use division.)
> The factor of 1200 simply makes the units cents.
>
You said at http://lumma.org/music/theory/ModulationStrength.txt

"The units are cents, and pitchA and pitchB do not have to be
expressed in just intonation."

So, I assumed that pitchA and pitchB are in cents here (either specifying a JI pitch or something else). Did I misread this sentence?

Thanks

Torsten

> > Also, I don't understand your step 5. I see that with your
> > approach you minimise the voice-leading distance but allow for
> > larger "chord harmonic distance" (your modulation strength).
> > I like this idea.
>
> For every chord pair you know the modulation strength. Simply
> group all the chord pairs into two groups -- those with mod.
> strength higher than the median mod. strength, and those
> mod. strength below. The two groups will be of equal size
> according to the definition of "median". Now when building
> a progression, you can choose only from one group or the
> other, or alternate between groups, etc.
>
> > However, using Strasheela I would realise it declaratively
> > instead of procedurally as you do here. For example, I may
> > specify that low voice-leading distances are used for specific
> > or all chord progressions (neighbouring chord pairs), and at
> > the same time that the "chord harmonic distance" should be
> > low or high for all or for specific chord progressions. We
> > might also add various other constraints (e.g., the tonic
> > chord should re-appear from time to time etc.). That's what
> > is nice about constraint programming: you just define a
> > conjunction of conditions (other connectives like disjunction
> > or implication and their nesting can be used as well).
>
> Sure, I am familiar with constraint programming. However
> my brain is stuck in procedural land!
>
> -Carl
>

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-233667
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

Torsten wrote...

> here is a related idea: instead of measuring the changing
> intervals between two chords (using the Tenney harmonic
> distance), we could measure the intervals within a chord
> in order to measure the harmonic quality of a single chord.
>
> I guess this has been done already. Has it?

The dissonance of a dyad p/q in just intonation is

sqrt(p*q)

For larger chords, find their lowest harmonic series
representation a:b:c... and then find the geometric
mean of a, b, c....

For tempered chords, use harmonic entropy.

But the way chords are usually classified in practice is
as belonging to a certain odd limit. A chord belongs to
the o-limit when all its dyads belong to the o-limit. If
this chord is tempered, the error of the approximation is
considered to slightly weaken the consonance of the just
tuning. The best error method to use is hotly contested,
but I prefer the RMS error of all the dyads in the chord.

-Carl

Torsten wrote...
> > > Open question still: what happens in cases where the pairs of
> > > changing pitches can not be distributed evenly without
> > > "overlapping".
> > > Say, what there are three changing pitches?
> >
> > Can you give an example?
> >
> In V7 -> I there are 5 changing pitches. How are you ordering
> them in pairs? Or, if you would discard some pitch here, how to
> decide which without doing some pairs first?

One pitch must be discarded. All pairings / choice of which
pitch to discard must be tested.

> > > Thanks also for detailing the voice leading subject. However,
> > > I don't quite understand this yet. I figure you could simply
> > > have an absolute distance like
> > >
> > > | pitchA - pitchB |
> > >
> > > Having a logarithm doesn't so much "penalise" larger distances,
> > > like in the Tenney formular, right? But why having the factor
> > > 1200?
> >
> > Remember pitchA and pitchB are ratios. So log is needed to
> > make subtraction work. (In my original exposition on modulation
> > strength I used subtraction on ratios informally... you can
> > see in the current revision of the document I use division.)
> > The factor of 1200 simply makes the units cents.
>
> You said at http://lumma.org/music/theory/ModulationStrength.txt
>
> "The units are cents, and pitchA and pitchB do not have to be
> expressed in just intonation."

The units are cents because of the 1200!

The pitches don't have to be expressed in just intonation,
but they do have to be expressed as ratios. That is, they
may be expressed as complex ratios (of some temperament).

But I should probably clarify this.

-Carl

--- In tuning@yahoogroups.com, Torsten Anders <torstenanders@...> wrote:
>
> Dear Hans Straub,
>
> again, thanks for pointing me to your text. After reading it, my
> main question is this: how are cadencial sets (the pitches that
> determine a key completely) computed? Or to put it the other way
> round: which pitches of the key are dispensable for that purpose?
> You kindly explain this idea in your FAQ by examples. For example,
> in major the chords IV and V form a cadencial set, in other words
> the tone on degree III is dispensable. I see that having III or IIIb
> does not change the root of the scale (not considering different
> modes). But how is III isolated formally here?
>

The point is that when you perform an arbitrary transposition (pitch
shift) on the chord set {IV, V}, you necessarily leave the tonality -
where as, e.g., {V, I} can be transposed up one forth, resulting in
{I, IV} of the same major key. So if you hear only V-I or I-V in,
e.g., C major, you cannot be 100% sure whether it is not I-IV (or
IV-I, respectively) in G major - a thing that cannot happen with IV-V.

> In your composition, you applied this idea to your scale C Db Eb E F
> G Ab. Do you know whether this idea has been applied to scales
> outside 12 ET? For example, what would be the set of "dispensable"
> pitches (for tonal unambiguousness) of Erlich's decatonic scales?
>

I once calculated the cadence-sets for the pentachordal decatonic
scale. I will look up my notes, if you wish.

I must assume that other people have applied parts of the modulation
theory as well - Jan Beran might be one - but I do not know. I am a
little out of the MaMuTh scene by now... Maybe Thomas Noll knows.

>
> Some other points.
>
> It is interesting to see that you (or Mazzola) are referring to
> Schoenberg's explanation of modulations (neutral phase, fundamental
> step, cadence) but then seemingly use minor in a sense which greatly
> differs from the way Schoenberg introduces minor. You are discussing
> the cadencial sets of harmonic minor. Schoenberg introduces minor in
> a way which cannot be reduced to a simple scale like that. It is
> related to melodic minor, but less simplifying.

Well, yes, Mazzola's modulation theory does not do a perfect
description of traditional harmony as used by composers. It is a
highly simplified model covering only certain aspects.

>
> Several times, you mention some hypothetical listener who grew up
> with your scale and hears your modulations. I wonder whether the
> diatonic scales are just education or not something far deeper, but
> that's a big issue :)
>

That is one question behind all this! And the idea is: to find out why
rules are like they are, just break the rules and look what happens...

> Translation of Schoenberg terminology into English: I assume your
> term "harmonic braid" is a translation of Schoenberg's term
> "harmonisches Band" (i.e. common pitches between two chords).
> Unfortunately, I have no English translation of his book. Any idea
> what term is used in the "official" English translation?
>

Yes, "Harmonic braid" is my translation, which may not be alltogether
correct. AFAIK, in Mazzola's book, the term "harmonic band" is used.
--
Hans Straub

<<< They do (I'm a recovering barbershoper). You should get some recordings!
>>>

According to the psychological studies, you will always be a recovering
barbershopper, as this is considered incurable. It's kind of like at AA you
are encouraged to always consider yourself a recovering alcaholic, unlike
Shick Shadel which claims to actually cure you. When I was diagnosed as
SPEBSQSA positive, I decided to save money and forego all treatment. I am
terminal.

Billy

--- In tuning@yahoogroups.com, "Billy Gard" <billygard@...> wrote:
> According to the psychological studies, you will always be a
> recovering barbershopper, as this is considered incurable. It's
> kind of like at AA you are encouraged to always consider yourself
> a recovering alcaholic, unlike Shick Shadel which claims to
> actually cure you. When I was diagnosed as SPEBSQSA positive,
> I decided to save money and forego all treatment. I am terminal.

:)

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Torsten wrote...
>
> > > I can do better than that!
> > >
http://lumma.org/tuning/rothenberg/AModelForPatternPerception.pdf
> >
> > Thank you very much!
>
> I am proud to say that this is the only source of these
> documents I know of on the internet. Of course that is
> also rather unfortunate from my point of view!
>
> > Could you perhaps even summarise Rothenberg's idea concerning
> > which scale notes are required to recognise a scale. Is this
> > related to Mazzola's cadencial sets brought up by Hans?
>
> I'm not familiar with the concept of Mazzola/Hans, but it
> certainly sounds similar.
> Your PDF reader should be able to show you "bookmarks" in
> the file. These take you to the start of each of the three
> papers (originally published separately) in the file.
> Go to "Rothenberg2", then scroll to pg. 368, and start
> reading at section 14 ("Sufficient Sets; The Coding of P").
> Read through section 15 ("Efficiency") until you reach
> the end of the paper. Let me know what you think.
>

It's definitely related!
From first sight, it looks like the idea is actually the same - just
Rothenbergs "sufficient set" is based on single scale tones while
Mazzola's "cadence-set" is based on chords.
I am digging further into the paper to verify...
--
Hans Straub

Isn't it always considered that the dominant is the defining element of a key?
One could say the disjunction plus one more tone for B F A would work as well as G B F

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

hstraub64 wrote:
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, "Carl > Lumma" <carl@...> wrote:
> >
> > Torsten wrote...
> >
> > > > I can do better than that!
> > > >
> http://lumma.org/tuning/rothenberg/AModelForPatternPerception.pdf > <http://lumma.org/tuning/rothenberg/AModelForPatternPerception.pdf>
> > >
> > > Thank you very much!
> >
> > I am proud to say that this is the only source of these
> > documents I know of on the internet. Of course that is
> > also rather unfortunate from my point of view!
> >
> > > Could you perhaps even summarise Rothenberg's idea concerning
> > > which scale notes are required to recognise a scale. Is this
> > > related to Mazzola's cadencial sets brought up by Hans?
> >
> > I'm not familiar with the concept of Mazzola/Hans, but it
> > certainly sounds similar.
> > Your PDF reader should be able to show you "bookmarks" in
> > the file. These take you to the start of each of the three
> > papers (originally published separately) in the file.
> > Go to "Rothenberg2", then scroll to pg. 368, and start
> > reading at section 14 ("Sufficient Sets; The Coding of P").
> > Read through section 15 ("Efficiency") until you reach
> > the end of the paper. Let me know what you think.
> >
>
> It's definitely related!
> >From first sight, it looks like the idea is actually the same - just
> Rothenbergs "sufficient set" is based on single scale tones while
> Mazzola's "cadence-set" is based on chords.
> I am digging further into the paper to verify...
> -- > Hans Straub
>
>

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> Isn't it always considered that the dominant is the defining
> element of a key?
> One could say the disjunction plus one more tone for B F A
> would work as well as G B F

IIRC G B D F is a sufficient subset of the diatonic scale.

-Carl

wouldn't any 5 notes work that had disjunction in it to define the diatonic (mode would be something else)?
It seems the same with the pentatonic except 3 notes - the disjunction and one other

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Carl Lumma wrote:
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, Kraig > Grady <kraiggrady@...> wrote:
> >
> > Isn't it always considered that the dominant is the defining
> > element of a key?
> > One could say the disjunction plus one more tone for B F A
> > would work as well as G B F
>
> IIRC G B D F is a sufficient subset of the diatonic scale.
>
> -Carl
>
>

I don't know what a disjunction is, but G B D F is 4 notes.

-Carl

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> wouldn't any 5 notes work that had disjunction in it to define
> the diatonic (mode would be something else)?
> It seems the same with the pentatonic except 3 notes - the
> disjunction and one other

> > IIRC G B D F is a sufficient subset of the diatonic scale.
> >
> > -Carl

Sorry i mistook the C as part of the series.
With an MOS there is always a closing interval of unique size. a series of Fifths make the diatonic with b to f being a diminished 5 and the disjunction.
So with 4 notes why not B D F A?

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Carl Lumma wrote:
>
> I don't know what a disjunction is, but G B D F is 4 notes.
>
> -Carl
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, Kraig > Grady <kraiggrady@...> wrote:
> >
> > wouldn't any 5 notes work that had disjunction in it to define
> > the diatonic (mode would be something else)?
> > It seems the same with the pentatonic except 3 notes - the
> > disjunction and one other
>
> > > IIRC G B D F is a sufficient subset of the diatonic scale.
> > >
> > > -Carl
>
>

Carl Lumma wrote:
> --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>> Isn't it always considered that the dominant is the defining
>> element of a key?
>> One could say the disjunction plus one more tone for B F A
>> would work as well as G B F
> > IIRC G B D F is a sufficient subset of the diatonic scale.

B F is a sufficient subset provided you can distinguish B from Cb, etc. In equal temperament the smallest sufficient subset is B and F plus one other note.

Graham

that is what one would think. i am trying to see why the others want more.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Graham Breed wrote:
>
> Carl Lumma wrote:
> > --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, > Kraig Grady <kraiggrady@...> wrote:
> >> Isn't it always considered that the dominant is the defining
> >> element of a key?
> >> One could say the disjunction plus one more tone for B F A
> >> would work as well as G B F
> >
> > IIRC G B D F is a sufficient subset of the diatonic scale.
>
> B F is a sufficient subset provided you can distinguish B
> from Cb, etc. In equal temperament the smallest sufficient
> subset is B and F plus one other note.
>
> Graham
>
>