Musical Set Theory in 12-tET

We hold these truths to be self-evident: (Okay, I watched Obama last night)

A. The only scales that can be produced using seven whites (CDEFGAB) ordinary accidentals, and regular mapping (D->D# means E cannot -> Eb, and D->D# means D cannot ->Db) and so that all skips (FACEGBD) are either major or minor thirds are:

1. Major / Minor
2. Melodic Minor
3. Harmonic Minor / Inverse Harmonic Minor.

E#, Fb, Cb, B# are not needed.

B. Same conditions, somewhat more relaxed, allowing augmented and diminished thirds in the skips (FACEGBD), this picks up the Hungarian scale, for example:

1. All 66 except 0123456, because 01234 pentad cannot be expressed using this mapping.

2. These 66 septachord/pentachords types encircle ALL hexachord types (pentad up or septad down)

except these:

012345
012346 and negative
012347 and negative
012348

Interestingly, you can find the 168 scales by going from black to white as easily as white to black. You get two "regions", the
two-black zone and the three-black zone, for 8 x 21 =
168 combinations which span 65 / 66 scale types (the septachord/pentachords)

168 is an important number also of the Fano plane or PGL(2,7).
More for that on tuning-math.

PGH

--- In tuning@yahoogroups.com, "paulhjelmstad" <paul.hjelmstad@...> wrote:
>
> We hold these truths to be self-evident: (Okay, I watched Obama
> last night)

Hiya Paul. I actually mentioned your work offlist to Caleb.

> A. The only scales that can be produced using seven whites
> (CDEFGAB) ordinary accidentals, and regular mapping (D->D#
> means E cannot -> Eb, and D->D# means D cannot ->Db) and so
> that all skips (FACEGBD) are either major or minor thirds are:
>
> 1. Major / Minor
> 2. Melodic Minor
> 3. Harmonic Minor / Inverse Harmonic Minor.
>
> E#, Fb, Cb, B# are not needed.

Where's the rub? You're assuming pretty much everything you
need to get the output you want: 12-ET and diatonic nominals.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "paulhjelmstad" <paul.hjelmstad@> wrote:
> >
> > We hold these truths to be self-evident: (Okay, I watched Obama
> > last night)
>
> Hiya Paul. I actually mentioned your work offlist to Caleb.
>
> > A. The only scales that can be produced using seven whites
> > (CDEFGAB) ordinary accidentals, and regular mapping (D->D#
> > means E cannot -> Eb, and D->D# means D cannot ->Db) and so
> > that all skips (FACEGBD) are either major or minor thirds are:
> >
> > 1. Major / Minor
> > 2. Melodic Minor
> > 3. Harmonic Minor / Inverse Harmonic Minor.
> >
> > E#, Fb, Cb, B# are not needed.
>
> Where's the rub? You're assuming pretty much everything you
> need to get the output you want: 12-ET and diatonic nominals.
>
> -Carl

Yes, true, and there are levels of restriction. It's amazing to
me though you go from 5 scales to 65 scales so rapidly, though,
also, (the subtle part of my stuff) is pentad/septad complexes,
and of course that dreaded Z-relation. (Not to mention M12, Steiner
S(5,6,12), Polya, necklaces...more for tuning-math).

There's a whole lot more, I will jump over the fence to tuning-math.

PGH

> Where's the rub? You're assuming pretty much everything you
> need to get the output you want: 12-ET and diatonic nominals.
>

Yes exactly.
And how is this even related to tuning?

I can guess where you're probably going with this tuning wise though.
Probably going to use something like the tonic and then optimal tempering of
fifths etc for these scales?
This would be yet another wrong assumption in my opinion to get all fifths
as pure as possible.

Or am I guessing things wrong and is there some other use tuning related for
what you're doing?

Marcel

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> > Where's the rub? You're assuming pretty much everything you
> > need to get the output you want: 12-ET and diatonic nominals.
> >
>
> Yes exactly.
> And how is this even related to tuning?
>
> I can guess where you're probably going with this tuning wise though.
> Probably going to use something like the tonic and then optimal tempering of
> fifths etc for these scales?
> This would be yet another wrong assumption in my opinion to get all fifths
> as pure as possible.
>
> Or am I guessing things wrong and is there some other use tuning related for
> what you're doing?
>
> Marcel

As a matter of fact, I might choose not to tune these scales at all,
except for ordinary 12-tET! You saw this as did Carl. Unfortunately,
there is not a "yahoo-musical set theory" group. In a sense, what
I am trying to do, is find meaning in 12-tET based on illumination
of regular temperaments etc. A little like studying Latin or German
to gain more understanding of English I suppose.

However, I would like to integrate tuning theory with musical
set theory at some point. It's a little like integrating
relavity with quantum physics I guess, they seem to have nothing
to do with each other.

PGH

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> > Where's the rub? You're assuming pretty much everything you
> > need to get the output you want: 12-ET and diatonic nominals.
>
> Yes exactly.
> And how is this even related to tuning?

Music set theory
http://en.wikipedia.org/wiki/Set_theory_%28music%29

and the theory of regular temperament
http://x31eq.com/paradigm.html

are two competing explanations for the foundations of music.
One explicitly addresses intonation and the other doesn't, but
Paul has done some microtonal music set theory, and discussion
of the relationship between the two theories is on-topic in
any case.

-Carl

Carl wrote:

> Music set theory
> http://en.wikipedia.org/wiki/Set_theory_%28music%29
> and the theory of regular temperament
> http://x31eq.com/paradigm.html
> are two competing explanations for the foundations of music.

While on one hand, Pythagorean tuning is 2-dimmensional and 5-limit JI is 3-dimmensional, it seems to me, on the other hand, that all the webpages I've found are describing "set theory" as a 1D systém. Putting these two facts together just doesn't fit to my ideas of thinking about harmony or chords. That's why I would probably find regular mappings to be a better defendable explanation.

Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Carl wrote:
>
> > Music set theory
> > http://en.wikipedia.org/wiki/Set_theory_%28music%29
> > and the theory of regular temperament
> > http://x31eq.com/paradigm.html
> > are two competing explanations for the foundations of music.
>
> While on one hand, Pythagorean tuning is 2-dimmensional and 5-limit JI is 3-dimmensional, it seems to me, on the other hand, that all the webpages I've found are describing "set theory" as a 1D systém. Putting these two facts together just doesn't fit to my ideas of thinking about harmony or chords. That's why I would probably find regular mappings to be a better defendable explanation.
>
> Petr

However, if you consider the relationship of 2^1/4 and 2^1/3 in

0.3.6.9
4.7.10.1
8.11.2.5

Instead of just 0.1.2.3.4.5.6.7.8.9.10.11 (1D)

This is intimately related to

x 3^0,3^1,3^2
5^0
5^1
5^2

(Imagine the grid filled out).

3D, by means of 2,3,5 primes involved....

So issues with "musical set theory" (in terms of group theory)
do relate directly to 3D Just tunings, and more.

Part would be studying C4 X C3 semidirect product expansions using Polya Polynoials and forth

PGH

Disclaimer: However, it's true that musical set theory considerations
become very difficult if you don't have a fixed reference point
for unity (1). Which dictates where the 5's and 3's and their
powers would go.

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Carl wrote:
>
> > Music set theory
> > http://en.wikipedia.org/wiki/Set_theory_%28music%29
> > and the theory of regular temperament
> > http://x31eq.com/paradigm.html
> > are two competing explanations for the foundations of music.
>
> While on one hand, Pythagorean tuning is 2-dimmensional and 5-limit JI is 3-dimmensional, it seems to me, on the other hand, that all the webpages I've found are describing "set theory" as a 1D systém. Putting these two facts together just doesn't fit to my ideas of thinking about harmony or chords. That's why I would probably find regular mappings to be a better defendable explanation.
>
> Petr
>

How does the "regular mappings explanation" classify/analyse the subset {B,C,F,Bb} of 12-tET? And how is it compared to {B,C,F,Bb} in Pythagorean tuning?

I would like to have some model which takes in account both acoustical and combinatorial properties of pitch structures.

--- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@...> wrote:

> How does the "regular mappings explanation" classify/analyse the
> subset {B,C,F,Bb} of 12-tET? And how is it compared to {B,C,F,Bb}
> in Pythagorean tuning?

I'm afraid you'll have to be more specific. I'm not sure what
any theory would tell you about 4 pitch classes... if that is
what these letters represent here.

-Carl

Hi Paul,

> However, if you consider the relationship of 2^1/4 and 2^1/3 in
>
> 0.3.6.9
> 4.7.10.1
> 8.11.2.5
>
> Instead of just 0.1.2.3.4.5.6.7.8.9.10.11 (1D)
>
> This is intimately related to
>
> x 3^0,3^1,3^2
> 5^0
> 5^1
> 5^2
>
> (Imagine the grid filled out).
>
> 3D, by means of 2,3,5 primes involved....

Well...
If you're doing this, then why not just call it musical set theory in Just
Intonation :)
It has nothing to do with 12tet anymore.

Btw it may interest you that I have found a means of getting to the same
result taking a completely different path.
By permutations of the harmonic series.

The set you named:

2^1/4 and 2^1/3 in
0.3.6.9
4.7.10.1
8.11.2.5

alternatively:
x 3^0,3^1,3^2
5^0
5^1
5^2

Can also be derived by taking the harmonic series up till 5
so 1/1 2/1 3/1 4/1 5/1
And then take all permutation of this set:
for instance 1/1 3/2 3/1 15/4 5/1 and 8/5 2/1 3/1 4/1 8/1, etc.
The resulting permutation group is 1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5
5/3 9/5 15/8 2/1 when reduced to one octave.
It's the same as the sets you mentioned, like:
2^1/4 and 2^1/3 in
0.3.6.9
4.7.10.1
8.11.2.5

Only in my permutation group 1/1 is where you say 7.

I use this permutation group together with a GCD derived model of tonica.
The permutation structure gives me certain insights into modulations and a
few other things (for instance which chord progressions are possible in a
consonant manner etc).
You can hear an example on my website of completely random permutations sung
by a choir. www.develde.net at the bottom of the page.
My GCD derived model of tonica gives the major mode on the third harmonic
(8/5 in the permutation group), the minor mode tonic on the fifth harmonic
(4/3 on the permutation group).
The combination of the GCD tonica model and permutation group model give me
the ability to tune common practice music to JI without any comma shifts.

Kind regards,

Marcel
www.develde.net

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@> wrote:
>
> > How does the "regular mappings explanation" classify/analyse the
> > subset {B,C,F,Bb} of 12-tET? And how is it compared to {B,C,F,Bb}
> > in Pythagorean tuning?
>
> I'm afraid you'll have to be more specific. I'm not sure what
> any theory would tell you about 4 pitch classes... if that is
> what these letters represent here.
>
> -Carl
>

I mean {B,C,F,Bb} as a set containing pitch classes representatives. You might imagine it as a chord around the middle octave (that was my mental image). It could choose {C,E,G} as subset of 12-ET or 19-ET, but then the set would lead the discussion into a direction I am not interested in...

Set theory can analyse the subset contents (e.g. interval vector), set properties (e.g. symmetry, common tones under transposition), group pc sets in set classes and propose similarity criteria.

My question is: how the subsets of an universe (a scale) are considered in the regular mapping model? You wrote (on regular mapping and set theory):

> One explicitly addresses intonation and the other doesn't, but
> Paul has done some microtonal music set theory, and discussion
> of the relationship between the two theories is on-topic in
> any case.

Or, putting in another way, can the regular mapping model provide information about the stuff -- *any* subsets -- available in a given scale/tuning? Or is it just a way to represent scales?

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> Hi Paul,
>
>
> > However, if you consider the relationship of 2^1/4 and 2^1/3 in
> >
> > 0.3.6.9
> > 4.7.10.1
> > 8.11.2.5
> >
> > Instead of just 0.1.2.3.4.5.6.7.8.9.10.11 (1D)
> >
> > This is intimately related to
> >
> > x 3^0,3^1,3^2
> > 5^0
> > 5^1
> > 5^2
> >
> > (Imagine the grid filled out).
> >
> > 3D, by means of 2,3,5 primes involved....
>
>
>
> Well...
> If you're doing this, then why not just call it musical set theory in Just
> Intonation :)
> It has nothing to do with 12tet anymore.
>
> Btw it may interest you that I have found a means of getting to the same
> result taking a completely different path.
> By permutations of the harmonic series.
>
> The set you named:
>
> 2^1/4 and 2^1/3 in
> 0.3.6.9
> 4.7.10.1
> 8.11.2.5
>
> alternatively:
> x 3^0,3^1,3^2
> 5^0
> 5^1
> 5^2
>
> Can also be derived by taking the harmonic series up till 5
> so 1/1 2/1 3/1 4/1 5/1
> And then take all permutation of this set:
> for instance 1/1 3/2 3/1 15/4 5/1 and 8/5 2/1 3/1 4/1 8/1, etc.
> The resulting permutation group is 1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5
> 5/3 9/5 15/8 2/1 when reduced to one octave.
> It's the same as the sets you mentioned, like:
> 2^1/4 and 2^1/3 in
> 0.3.6.9
> 4.7.10.1
> 8.11.2.5
>
> Only in my permutation group 1/1 is where you say 7.

Exactly, it really is the same thing, when you consider 5^2=cong=5^-1
etc. Interesting how that works out! With 7="G" being 1/1. (Mixolydian I guess)
>
> I use this permutation group together with a GCD derived model of tonica.
> The permutation structure gives me certain insights into modulations and a
> few other things (for instance which chord progressions are possible in a
> consonant manner etc).
> You can hear an example on my website of completely random permutations sung
> by a choir. www.develde.net at the bottom of the page.
> My GCD derived model of tonica gives the major mode on the third harmonic
> (8/5 in the permutation group), the minor mode tonic on the fifth harmonic
> (4/3 on the permutation group).
> The combination of the GCD tonica model and permutation group model give me
> the ability to tune common practice music to JI without any comma shifts.

Could you show the GCD model? Thanks.

(PGH)

> Kind regards,
>
> Marcel
> www.develde.net
>

--- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@...> wrote:

> I mean {B,C,F,Bb} as a set containing pitch classes
> representatives.

Ok. So what question do you have about them, that you want
theory to answer?

> Set theory can analyse the subset contents (e.g. interval vector),
> set properties (e.g. symmetry, common tones under transposition),
> group pc sets in set classes and propose similarity criteria.

Take the index on the chain of fifths
{B,C,F,Bb} -> {5,0,-1,-2}
Multiply each term by 7, for the number of nominals
available
{35,0,-7,-14}
Add 4 because there are 4 pitch classes in play
{39,4,-3,-10}
Mod 12 for the 12 pitch classes available
{3,4,9,2}
Map to chromatic degrees of 12-ET, with 0 degrees being
the pitch class used as the 0th fifth earlier.
{Eb,E,A,D}
This transformation is definitely producing something
similar! (Actually it was my third try stringing arbitrary
operations together, and they all produced something
'similar').

> Or, putting in another way, can the regular mapping model
> provide information about the stuff -- *any* subsets --
> available in a given scale/tuning? Or is it just a way to
> represent scales?

It lets us find optimal tunings for any scale. It lets us
find the scales, out of all possible scales, that are the
natural competitors of the diatonic scale. It shows us how
these scales are related, and how they can be transformed
into one another, or how music written in one can be cast
into another. It tells us when such recasting is likely to
result in a qualitative change in the music vs. a more subtle
sense that the same piece has simply been retuned. etc.
It doesn't tell us what notes to write once a scales and
tunings have been selected.

-Carl

>
> Exactly, it really is the same thing, when you consider 5^2=cong=5^-1
>

The end result is thesame, but only in this specific case of limiting to the
5-th harmonic.
When limiting to for isntance the 4th or 6th harmonic the result is very
different.
Though music is 5-limit :)
I do also think that thinking about it as harmonic permutations will give
more information in many ways, and give more to work with.
Though I can't be sure as I don't know what's out there in other models.

What does =cong= mean btw? :)
I'm autodidact and never learnt math in schools, never came across it.

> etc. Interesting how that works out! With 7="G" being 1/1. (Mixolydian I
> guess)

Yes it's amazing how it works out indeed. :)

I ment the 7 in your:

2^1/4 and 2^1/3 in
0.3.6.9
4.7.10.1
8.11.2.5
This is intimately related to
x 3^0,3^1,3^2
5^0
5^1
5^2

So I translated this to:
1/1 3/2 9/8 27/16
5/4 15/8 45/32 135/128
25/16 75/64 225/128 675/512

Where your 7 would be 15/8

So your 1/1 is on C then the 1/1 of my harmonic permutation set would be on
B.

If we were to put 3/2 of the harmonic series on C (8/5 of the harmonic
permutation set), giving the C major scale on the white keys, then E will
give Phrygian (not mixolydian).
In other words, 1/1 of the harmonic permutation set will give the tonic of
Phrygian at 1/1. The tonic of Minor at 4/3, and the tonic of Major at 8/5.

As for my tonality model.
There isn't much to it.
Unless one modulates (and the harmonic permutations will tell you what a
modulation is) on is in the 12tone subset we talk about above.
A tonic is simply a tonal center by choice.
The harmonic permutation set gives 1/1 135/128 9/8 75/64 5/4 675/512 45/32
3/2 25/16 27/16 225/128 15/8 2/1 seen from the GCD.
As many chords will point with their GCD to the above 1/1, for instance a
dominant 7th, I've found that what is refered to as major mode in normal
music theory corresponds to a tonic of 3/2 from the GCD. Minor to a tonic
5/4 from the GCD, etc.
It seems normal music theory gets the tonic right a lot of the time allowing
easy transcription to JI.
Sometimes not though, but I'm getting better at finding the real tonics and
modulations of a piece.
I'm bussy making a proper website where I'll show all my experiments and
thoughts on these things soon.

Marcel

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@> wrote:
>
> > I mean {B,C,F,Bb} as a set containing pitch classes
> > representatives.
>
> Ok. So what question do you have about them, that you want
> theory to answer?

Each theory has its own range of specific questions and answers. I could not yet see any relation between set theory and regular mappings, so that they can be comparable models.

My question may look rather vague: let us suppose that I like the chord {B,C,F,Bb} and want to find scales in which the chord has interesting features that could be explored in a composition. Or the reverse: suppose I got a scale/tuning at random (from Scala archive, for example) and found an interesting sonority, that I recognize as (or could represent as) the set {B,C,F,Bb} -- then I want to know if this set has a "family" in that scale/tuning: transpositions and other related sets.

As you see, this is the opposite of generating a scale and/or tuning: it is a matter of analysing a scale/tuning and its subsets.

For instance, in 12-ET, the interval vector of {B,C,F,Bb} (or {11,0,5,10}) is <210021>. Some deductions are: transposition at interval classes 1 (semitone) or 5 (fourth) result in 2 common pitch classes; there are no "M/m thirds" in the set, thus transpositions to those intervals don't maintain invariants. The set can be completely invariant under "inversion": {B,Bb,F,C} (an useful information for counterpoint and serial music). From a more auditive approach, there are contrastant subsets, e.g: {C,F,Bb} ("consonant" 4ths) and {B,C,Bb} ("dissonant" semitones) -- but these yet share the interval 2 (Maj.2nd)...

Such informations could be used as hints when exploring new material (scales, chords...).

>
> > Set theory can analyse the subset contents (e.g. interval vector),
> > set properties (e.g. symmetry, common tones under transposition),
> > group pc sets in set classes and propose similarity criteria.
>
> Take the index on the chain of fifths
> {B,C,F,Bb} -> {5,0,-1,-2}

It would be {11,0,5,10} in a chain of semitones...

> Multiply each term by 7, for the number of nominals
> available
> {35,0,-7,-14}
> Add 4 because there are 4 pitch classes in play
> {39,4,-3,-10}

I could not follow that.

> Mod 12 for the 12 pitch classes available
> {3,4,9,2}
> Map to chromatic degrees of 12-ET, with 0 degrees being
> the pitch class used as the 0th fifth earlier.
> {Eb,E,A,D}
> This transformation is definitely producing something
> similar! (Actually it was my third try stringing arbitrary
> operations together, and they all produced something
> 'similar').
>
> > Or, putting in another way, can the regular mapping model
> > provide information about the stuff -- *any* subsets --
> > available in a given scale/tuning? Or is it just a way to
> > represent scales?
>
> It lets us find optimal tunings for any scale. It lets us
> find the scales, out of all possible scales, that are the
> natural competitors of the diatonic scale. It shows us how
> these scales are related, and how they can be transformed
> into one another, or how music written in one can be cast
> into another. It tells us when such recasting is likely to
> result in a qualitative change in the music vs. a more subtle
> sense that the same piece has simply been retuned. etc.

Could the set {B,C,F,Bb} be considered a "scale" in that sense? How it could be related to other scales, or maybe used to generate other scales?

> It doesn't tell us what notes to write once a scales and
> tunings have been selected.

If it can find scales "that are the natural competitors of the diatonic scale", then I suppose it takes in account certain properties of subsets of those scales (for example: the number of subsets approximating 4:5:6 or other ratios).

I am still seeking some common point between the two theorical approaches.

>
> -Carl
>

What I'd like to add to give what I think is the big picture.

5-limit harmonic permutation structure will give a "basis" structure of
mathematical possibilities in the most "consonant / low ratio" manner
without comma shifts.
Harmonic permutation structure is "U-tonal" in a way (though symetrical in
the harmonic series, not symetrical in the octave as is usually also ment by
u-tonal), meaning it doesn't do GCD etc, all harmonic permutation "chords"
have their "mirror" on any of it's "notes".
Harmonic permutation structure is very limited, it only has very limited
"chords". I tend to not even see them as chords but as structures.
In actual music one can combine any of the 12 notes together. Any of those
notes can belong to many structures.
When combining notes belonging to different harmonic permutation "chords"
this must happen within the harmonic permutation set of all permutations of
the harmonic series (limit 5).
This is where most of the music happens. And this is where I see GCD and
tonica come into play.

So I see music as sort of "u-tonal" up till a certain point, and very tonal
beyond that.

Marcel

As for my tonality model.
> There isn't much to it.
> Unless one modulates (and the harmonic permutations will tell you what a
> modulation is) on is in the 12tone subset we talk about above.
> A tonic is simply a tonal center by choice.
> The harmonic permutation set gives 1/1 135/128 9/8 75/64 5/4 675/512 45/32
> 3/2 25/16 27/16 225/128 15/8 2/1 seen from the GCD.
> As many chords will point with their GCD to the above 1/1, for instance a
> dominant 7th, I've found that what is refered to as major mode in normal
> music theory corresponds to a tonic of 3/2 from the GCD. Minor to a tonic
> 5/4 from the GCD, etc.
> It seems normal music theory gets the tonic right a lot of the time
> allowing easy transcription to JI.
> Sometimes not though, but I'm getting better at finding the real tonics and
> modulations of a piece.
> I'm bussy making a proper website where I'll show all my experiments and
> thoughts on these things soon.
>
>
>

> Each theory has its own range of specific questions and answers.
> I could not yet see any relation between set theory and regular
> mappings, so that they can be comparable models.

First of all, I think I'm speaking with Hudson Lacerda (hi!)
composer of the wonderful piece Neo Old, which I assumed owes
nothing to music set theory (correct me if I'm wrong). So I'm
surprised you have an interest in it, but also happy because
maybe I will learn more about it (since unlike its other
proponents, you speak the microtonal language too).

> My question may look rather vague: let us suppose that I like
> the chord {B,C,F,Bb} and want to find scales in which the
> chord has interesting features that could be explored in a
> composition.

Depending what features you find interesting, regular mapping
can certainly help. We usually start with just intonation,
since JI chords have historically been musical targets and are
identifiable by naive listeners in experiments. But one could
start with any chord. The chord is broken down into its
generators, which in this case could be 100cents and 500cents.
The picture looks like this (please view message with a fixed
width font)

B
|
C---F---Bb

| = 100cents --- = 500cents

Then, following Fokker, we find scales that contain many of
these chords. The basic goal is to maximize their number for
a given scale size (cardinality).

> Or the reverse: suppose I got a scale/tuning at random (from
> Scala archive, for example) and found an interesting sonority,
> that I recognize as (or could represent as) the set
> {B,C,F,Bb} -- then I want to know if this set has a "family"
> in that scale/tuning: transpositions and other related sets.

When I said "maximize their number" I mean by transposition.
"Other related sets" is where we may disagree. If we're doing
music theory, I think we should consider only relations that
are audible. If two sets share a relation they should sound
more similar than two sets not sharing the relation. This is
well established for transposition.

> For instance, in 12-ET, the interval vector of {B,C,F,Bb}
> (or {11,0,5,10}) is <210021>.

Can you explain how is this interval vector is calculated
from {11,0,5,10}?

>Some deductions are: transposition at interval classes 1
>(semitone) or 5 (fourth) result in 2 common pitch classes;

If we generate scales that contain many instances of the
chord per scale tone, the chord instances must have many
common tones between them...

>The set can be completely invariant under "inversion":
>{B,Bb,F,C} (an useful information for counterpoint and serial
>music).

This is apparently "inversion" as in normal music theory,
which is really circular permutation. But we were talking
about sets, which are unordered, so I'm confused. They
also contained pitch classes, so it's impossible to change
their order in a chord (unless we say B' or B'' etc). Are
we playing them as a melody in a different order?

> > Take the index on the chain of fifths
> > {B,C,F,Bb} -> {5,0,-1,-2}
>
> It would be {11,0,5,10} in a chain of semitones...
>
> > Multiply each term by 7, for the number of nominals
> > available
> > {35,0,-7,-14}
> > Add 4 because there are 4 pitch classes in play
> > {39,4,-3,-10}
>
> I could not follow that.

I must confess this was a rhetorical example; a meaningless
transformation that produces a transposition of a M3.
I was trying to poke fun at music set theory.

> > It lets us find optimal tunings for any scale. It lets us
> > find the scales, out of all possible scales, that are the
> > natural competitors of the diatonic scale. It shows us how
> > these scales are related, and how they can be transformed
> > into one another, or how music written in one can be cast
> > into another. It tells us when such recasting is likely to
> > result in a qualitative change in the music vs. a more subtle
> > sense that the same piece has simply been retuned. etc.
>
> Could the set {B,C,F,Bb} be considered a "scale" in that
> sense? How it could be related to other scales, or maybe used
> to generate other scales?

The kind of scales I was referring to here must have a mapping
to the generators of the desired chord(s). So {B,C,F,Bb} can
be the desired chord or the scale. If it's both, the
temperament mapping is trivial and not very interesting.

> > It doesn't tell us what notes to write once a scales and
> > tunings have been selected.
>
> If it can find scales "that are the natural competitors of
> the diatonic scale", then I suppose it takes in account
> certain properties of subsets of those scales (for example:
> the number of subsets approximating 4:5:6 or other ratios).

Exactly.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
[...]
> First of all, I think I'm speaking with Hudson Lacerda (hi!)
> composer of the wonderful piece Neo Old, which I assumed owes
> nothing to music set theory (correct me if I'm wrong). So I'm
> surprised you have an interest in it, but also happy because
> maybe I will learn more about it (since unlike its other
> proponents, you speak the microtonal language too).

Hi Carl,

Yes, I am Hudson Lacerda. Glad you liked (and recalled!) Neo-old(19).

That piece is tonal, thus it does not require a set theory reference.

However I use pitch set theory when (pre-)composing atonal music -- never mind: pitch class set theory is an approach to analyse "post-tonal music", where the traditional harmony description hardly would help; it is not a generalized music theory. Thus, if you try apply it to tonal or modal music, I will probably end up with empty hands, because other analytical approaches are much more suitable for that kind of music. (Well, you can find that there are lots of sets with Forte name "3-11"... :-))

When I was composing Neo-old(19) I also analysed a few sets of 19-EDO using set theory. An interesting finding was that the approximation of harmonics 1:3:5:7:9:11:13 is symmetrical and identical to a transposition of a subharmonics set (1:/3:/5:/7:/9:/11:/13). That property allowed me to build a symmetrical 19-tone row around the pitch class 0:

Harmonics/subharmonics:
15 13 11 9 7 5 3 1 | 1 3 5 7 9 11 13 15
3 7 1 5 13 9 15 11 | 11 15 9 13 5 1 7 3
Pitch classes (= for natural, _ for flat, ^ for sharp) as pitch names, with C as central axis:
_D =E _G _B ^D _A =F =C | =C =G =E __B =D ^F _A =B
Pitch classes in numerical form starting from 0:
0 11 6 15 3 9 13 17 1 4 7 10 14 18 5 12 2 16 8

However, I have not used this series for composition so far.

>
> > My question may look rather vague: let us suppose that I like
> > the chord {B,C,F,Bb} and want to find scales in which the
> > chord has interesting features that could be explored in a
> > composition.
>
> Depending what features you find interesting, regular mapping
> can certainly help. We usually start with just intonation,
> since JI chords have historically been musical targets and are
> identifiable by naive listeners in experiments. But one could
> start with any chord. The chord is broken down into its
> generators, which in this case could be 100cents and 500cents.
> The picture looks like this (please view message with a fixed
> width font)
>
> B
> |
> C---F---Bb
>
> | = 100cents --- = 500cents
>
> Then, following Fokker, we find scales that contain many of
> these chords. The basic goal is to maximize their number for
> a given scale size (cardinality).

Indeed. An additional question is: which cardinality is the most efficient in that aspect (embedding instances of a given subset)?

This question may be relevant in closed systems like 12-ET, since the differences between subsets (e.g. scales) are blurred as the cardinality approaches 12. There may be an optimal cardinality, possibly around 5..7.

>
> > Or the reverse: suppose I got a scale/tuning at random (from
> > Scala archive, for example) and found an interesting sonority,
> > that I recognize as (or could represent as) the set
> > {B,C,F,Bb} -- then I want to know if this set has a "family"
> > in that scale/tuning: transpositions and other related sets.
>
> When I said "maximize their number" I mean by transposition.
> "Other related sets" is where we may disagree. If we're doing
> music theory, I think we should consider only relations that
> are audible. If two sets share a relation they should sound
> more similar than two sets not sharing the relation. This is
> well established for transposition.

Yes, the "canonical" similarity/equivalence relation is transposition. But other relations, even those that may arise controversy, can be useful for composition, or may be useful for analysis. For example, the "inversion equivalence" (which maps a minor triad to a major triad) proposed by Allen Forte is questioned by Larry Solomon. Both have good arguments: inversion was extensively used by composers of atonal music as a rather basic derivation; for the other hand, inversion-related sets can sound different enough to deserve different classification in many contexts.

Can you find that {C,E,G} and {C,Eb,G} share something in common, specially when compared to other 3-element subsets of 12-EDO? Yet: Given the sets A={C,Eb,G,Bb}, B={D,Eb,G#,A}, C={C,E,G,B} and D={B,C,F,Bb}, can you group or sort them according to "similarity"?

Then, I assume that there are different degrees of similarity, starting with transpositions and dispositions of a pitch set (not reduced), then reducing pitch classes in octave equivalence, reducing sets to a normal form, etc.

I also use to evaluate pc set operations as of relative or context-dependent usefulness. For instance, chords with six or more elements are difficult to analyse and compare by ear, and the disposition of the pitches is quite important (compare the chords {C,D,E,F,G,A,B}, {F,C,G,D,A,E,B} and {D,F,A,C,E,G,B}, which are the "same" pitch class set). But if six pitches are presented as a melodic sequence, their repetition (even transposed) may be perceived as equivalent. Such issues are, for me, an incentive to seek more similarity criteria (possibly without reduce the pitches/intervals in classes), since octave equivalence, transposition and inversion show their limits.

It is a matter of sense selecting the sets which may be relevant in a musical excerpt. Therefore, I am not willing to discard such or such pitch relations a priori.

I am composing a serial piece for 17-EDO guitar, and I will use something I thought I would never use, because the result is "too dissimilar" to the origin: transformation by multiplication (just multiply the pitch class number by a constant factor). The interval contents of the series is radically changed: the only common point is the special symmetrical organization of the original series. The original series has a lot of "minor seconds" (interval class 1), which are mapped to "fourths" (ic 7). The goal is to change the harmonic color of the piece in certain sections -- I could use an entirely new material, but using a derivation I can maintain some invariant (structural) properties. That may be an example of a rather arbitrary derivation being useful in composition. Formally, the multiplicated serieses might be considered "equivalent", but the are musically useful exactly for the opposite quality: sound different.

>
> > For instance, in 12-ET, the interval vector of {B,C,F,Bb}
> > (or {11,0,5,10}) is <210021>.
>
> Can you explain how is this interval vector is calculated
> from {11,0,5,10}?

It is an accounting of the occurrences of each interval class. In mod12 we have 6 interval classes, from 1 (semitone, M7, m9...) to 6 (tritone). As a consequence of octave equivalence, complementar (unordered) intervals are equivalent (P4th == P5th). Each entry of that "vector" is the number of instances of the corresponding interval class in the set:

A 4-element set has 6 2-element subsets. In this example the dyads are classified and accounted as <210021>:

2: {Bb,B} and {B,C} for interval class 1;
1: {Bb,C} for ic 2;
0: (none ic 3 (m3));
0: (none ic 4 (M3));
2: {C,F} and {F,Bb} for ic 5;
1: {F,B} for ic 6;

Or, using numbers:

----------.--------------.-------------.
| Interval | Class |
11 0 5 10 | (mod12,mod7) | 1 2 3 4 5 6 |
----------+--------------+-------------|
11 0 | 11-0=11=1 | x |
11 5 | 11-5=6 | x |
11 10 | 11-10=1 | x |
0 5 | 0-5=-5=7=5 | x |
0 10 | 0-10=-10=2 | x |
5 10 | 5-10=-5=7=5 | x |
----------+--------------+-------------|
Interval vector: | 2 1 0 0 2 1 |
-------------------------'-------------'

An interesting extension of the interval vector concept may be a "trichord vector", which analyses the 3-element subsets. In mod12 there are 12 trichord classes according to Forte's classification -- thus, the "trichord vector" would have 12 entries.

>
> >Some deductions are: transposition at interval classes 1
> >(semitone) or 5 (fourth) result in 2 common pitch classes;
>
> If we generate scales that contain many instances of the
> chord per scale tone, the chord instances must have many
> common tones between them...

Yes, "many". How many exactly? Given a scale generated that way are there interesting "byproducts" (subsets, chords, embeded scales)? Which are these "byproducts"? How about other chords of the same scale, which could be used as constrastant material? Are there other generator chords which lead to the same scale?

But there are other interesting sets: sets which show some saturation for the non-redundance feature. For instance:

(0,1,4,6), a.k.a. 4-Z15, e.g. {C,Db,E,F#}, {C,D,F,F#}
(0,1,3,7), a.k.a. 4-Z29, e.g. (C,Db,Eb,G}, {C,E,F#,G}

Their interval vector is <111111>: each interval class occur exactly once, and all them are present. They condense the interval resources of 12-EDO, in some way.

The interesting point here is the equal distribution of interval classes, rather than containing the maximum number of a given subset (to say, an interval) in that cardinality. There are no equivalent subsets: all 3-element subsets are different.

How to find (in a given universe) sets with minimum redundance of interval/subset classes?

In mod13 there are also tetrachords with interval vector <111111>; they could be used in Bohlen-Pierce scale, or in 13-EDO. In mod17, there are not such interval vectors, but there are ones like <10111101> or <1111100>, where two interval classes are absent. In mod10 and mod11, we have pentachords with vector <22222>. Interesting symmetry properties may be found in mod16, since 16 is 2*2*2*2 -- or perhaps it is the opposite: the symmetry may be excessive and boring (like the dim7-chord in 12-ET), an then the interesting sets might be the assymetrical ones -- how diverse are the subsets in mod16? How to effectively explore such properties in those uncommon scales?

These questions may show points of contact or complementarity between different theoretical approaches.

>
> >The set can be completely invariant under "inversion":
> >{B,Bb,F,C} (an useful information for counterpoint and serial
> >music).
>
> This is apparently "inversion" as in normal music theory,
> which is really circular permutation. But we were talking
> about sets, which are unordered, so I'm confused. They
> also contained pitch classes, so it's impossible to change
> their order in a chord (unless we say B' or B'' etc). Are
> we playing them as a melody in a different order?

No, I am talking about "inversion" as "mirror", inverting the direction of the intervals:

B C F Bb => +1 +5 +5
B Bb F C => -1 -5 -5

In this case, as you noticed, the result is equivalent to a simple permutation, and that is an interesting feature for counterpoint.

>
> > > Take the index on the chain of fifths
> > > {B,C,F,Bb} -> {5,0,-1,-2}
> >
> > It would be {11,0,5,10} in a chain of semitones...
> >
> > > Multiply each term by 7, for the number of nominals
> > > available
> > > {35,0,-7,-14}
> > > Add 4 because there are 4 pitch classes in play
> > > {39,4,-3,-10}
> >
> > I could not follow that.
>
> I must confess this was a rhetorical example; a meaningless
> transformation that produces a transposition of a M3.
> I was trying to poke fun at music set theory.

It was ironic that the set resisted firmly to your manipulations. :-)

It is completely invariant under multiplication by 7 (which is that "other" generator besides the semitone):

octave:1> S=[11,0,5,10]
S =

11 0 5 10

octave:2> mod(S*7,12)
ans =

5 0 11 10

Furthermore, simple adding does just a transposition:
4 semitones = Major 3rd.

>
> > > It lets us find optimal tunings for any scale. It lets us
> > > find the scales, out of all possible scales, that are the
> > > natural competitors of the diatonic scale. It shows us how
> > > these scales are related, and how they can be transformed
> > > into one another, or how music written in one can be cast
> > > into another. It tells us when such recasting is likely to
> > > result in a qualitative change in the music vs. a more subtle
> > > sense that the same piece has simply been retuned. etc.
> >
> > Could the set {B,C,F,Bb} be considered a "scale" in that
> > sense? How it could be related to other scales, or maybe used
> > to generate other scales?
>
> The kind of scales I was referring to here must have a mapping
> to the generators of the desired chord(s). So {B,C,F,Bb} can
> be the desired chord or the scale. If it's both, the
> temperament mapping is trivial and not very interesting.

I mean: a scale when comparing it to other scales.

A mapping is a group theory operation. I am wondering if pitch class set theory and regular mappings could be used in a complementary way, each one providing useful insights and tools for each other.

>
> > > It doesn't tell us what notes to write once a scales and
> > > tunings have been selected.
> >
> > If it can find scales "that are the natural competitors of
> > the diatonic scale", then I suppose it takes in account
> > certain properties of subsets of those scales (for example:
> > the number of subsets approximating 4:5:6 or other ratios).
>
> Exactly.

Then there may be space for an integrated approach, if some concepts can be harmonized.

>
> -Carl
>

I hadn't worried to sue David Doty for libel, when he cast doubts on my engineering qualifications, in his biased review of LucyTuning back in the eighties, nor Carl for his rantings.

Although this time it may well be worth it. Unlike Doty and Lumma; Apple have assets.

LucyTuning has been included in the Logic Pro 9 version. Apple calls it harrisonj, yet the cent values to five decimal places are a blatant "lift" from my original papers in my patent applications in 1988.

http://www.youtube.com/watch?v=CW0DUg63lqU&fmt=18

Keep posted for developments.

Charles Lucy
lucy@...

-- Promoting global harmony through LucyTuning --

For more information on LucyTuning go to:

http://www.lucytune.com

LucyTuned Lullabies (from around the world) can found at:

http://www.lullabies.co.uk

Has the patent not yet expired?
-Mike

On Wed, Feb 3, 2010 at 7:30 AM, Charles Lucy <lucy@...> wrote:

>
>
>
>
> I hadn't worried to sue David Doty for libel, when he cast doubts on my
> engineering qualifications, in his biased review of LucyTuning back in the
> eighties, nor Carl for his rantings.
>
> Although this time it may well be worth it. Unlike Doty and Lumma; Apple
> have assets.
>
> LucyTuning has been included in the Logic Pro 9 version. Apple calls it
> harrisonj, yet the cent values to five decimal places are a blatant "lift"
> from my original papers in my patent applications in 1988.
>
> http://www.youtube.com/watch?v=CW0DUg63lqU&fmt=18
>
> Keep posted for developments.
>
> Charles Lucy
> lucy@... <lucy%40lucytune.com>
>
> -- Promoting global harmony through LucyTuning --
>
> For more information on LucyTuning go to:
>
> http://www.lucytune.com
>
> LucyTuned Lullabies (from around the world) can found at:
>
> http://www.lullabies.co.uk
>
>
>

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> >
> > Exactly, it really is the same thing, when you consider 5^2=cong=5^-1
> >
>
> The end result is thesame, but only in this specific case of limiting to the
> 5-th harmonic.
> When limiting to for isntance the 4th or 6th harmonic the result is very
> different.
> Though music is 5-limit :)
> I do also think that thinking about it as harmonic permutations will give
> more information in many ways, and give more to work with.
> Though I can't be sure as I don't know what's out there in other models.
>
> What does =cong= mean btw? :)
> I'm autodidact and never learnt math in schools, never came across it.

I use it for "congruent to" which is three lines in the equal sign
instead of two(=).

>
>
>
> > etc. Interesting how that works out! With 7="G" being 1/1. (Mixolydian I
> > guess)
>
>
> Yes it's amazing how it works out indeed. :)
>
> I ment the 7 in your:
>
> 2^1/4 and 2^1/3 in
> 0.3.6.9
> 4.7.10.1
> 8.11.2.5
> This is intimately related to
> x 3^0,3^1,3^2
> 5^0
> 5^1
> 5^2
>
> So I translated this to:
> 1/1 3/2 9/8 27/16
> 5/4 15/8 45/32 135/128
> 25/16 75/64 225/128 675/512
>
> Where your 7 would be 15/8
>
> So your 1/1 is on C then the 1/1 of my harmonic permutation set would be on
> B.
>
> If we were to put 3/2 of the harmonic series on C (8/5 of the harmonic
> permutation set), giving the C major scale on the white keys, then E will
> give Phrygian (not mixolydian).
> In other words, 1/1 of the harmonic permutation set will give the tonic of
> Phrygian at 1/1. The tonic of Minor at 4/3, and the tonic of Major at 8/5.

Guess I'm confused, if my 7th step = G is your 15/8, then your 1/1
should be Ab, not B....I also noticed that using 1,2,3,4,5 in
all permutations requires 3 to be used twice, that is there
are 9's in your table. perhaps if you could (please) give me
your 4 x 3 grid, exactly, with the field of fractions? Thanks.

> As for my tonality model.
> There isn't much to it.
> Unless one modulates (and the harmonic permutations will tell you what a
> modulation is) on is in the 12tone subset we talk about above.
> A tonic is simply a tonal center by choice.
> The harmonic permutation set gives 1/1 135/128 9/8 75/64 5/4 675/512 45/32
> 3/2 25/16 27/16 225/128 15/8 2/1 seen from the GCD.
> As many chords will point with their GCD to the above 1/1, for instance a
> dominant 7th, I've found that what is refered to as major mode in normal
> music theory corresponds to a tonic of 3/2 from the GCD. Minor to a tonic
> 5/4 from the GCD, etc.
> It seems normal music theory gets the tonic right a lot of the time allowing
> easy transcription to JI.
> Sometimes not though, but I'm getting better at finding the real tonics and
> modulations of a piece.
> I'm bussy making a proper website where I'll show all my experiments and
> thoughts on these things soon.

I'll look at your website once it's ready, thanks - pgh

> Marcel
>

--- In tuning@yahoogroups.com, "paulhjelmstad" <paul.hjelmstad@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Marcel de Velde <m.develde@> wrote:
> >
> > >
> > > Exactly, it really is the same thing, when you consider 5^2=cong=5^-1
> > >
> >
> > The end result is thesame, but only in this specific case of limiting to the
> > 5-th harmonic.
> > When limiting to for isntance the 4th or 6th harmonic the result is very
> > different.
> > Though music is 5-limit :)
> > I do also think that thinking about it as harmonic permutations will give
> > more information in many ways, and give more to work with.
> > Though I can't be sure as I don't know what's out there in other models.
> >
> > What does =cong= mean btw? :)
> > I'm autodidact and never learnt math in schools, never came across it.
>
> I use it for "congruent to" which is three lines in the equal sign
> instead of two(=).
>
> >
> >
> >
> > > etc. Interesting how that works out! With 7="G" being 1/1. (Mixolydian I
> > > guess)
> >
> >
> > Yes it's amazing how it works out indeed. :)
> >
> > I ment the 7 in your:
> >
> > 2^1/4 and 2^1/3 in
> > 0.3.6.9
> > 4.7.10.1
> > 8.11.2.5
> > This is intimately related to
> > x 3^0,3^1,3^2
> > 5^0
> > 5^1
> > 5^2
> >
> > So I translated this to:
> > 1/1 3/2 9/8 27/16
> > 5/4 15/8 45/32 135/128
> > 25/16 75/64 225/128 675/512
> >
> > Where your 7 would be 15/8

* * * *

Hold on, I see the problem. My two grids are not a direct match,
because of 2^1/4=6/5 (not just 3) the columns slide around,
actually the first and fourth column stay the same but the
middle columns are shifted. So actually, my 5^n x 3^n is precisely
the same as yours given here. (15/8 would be B). The shift would
turn

0.3.6.9
4.7.10.1
8.11.2.5

into

0.7.2.9
4.11.6.1
8.3.10.5

to match my

x 3^0,3^1,3^2
5^0
5^1
5^2

which is your

1/1 3/2 9/8 27/16
5/4 15/8 45/32 135/128
25/16 75/64 225/128 675/512

* * * *

> > So your 1/1 is on C then the 1/1 of my harmonic permutation set would be on
> > B.
> >
> > If we were to put 3/2 of the harmonic series on C (8/5 of the harmonic
> > permutation set), giving the C major scale on the white keys, then E will
> > give Phrygian (not mixolydian).
> > In other words, 1/1 of the harmonic permutation set will give the tonic of
> > Phrygian at 1/1. The tonic of Minor at 4/3, and the tonic of Major at 8/5.
>
> Guess I'm confused, if my 7th step = G is your 15/8, then your 1/1
> should be Ab, not B....I also noticed that using 1,2,3,4,5 in
> all permutations requires 3 to be used twice, that is there
> are 9's in your table. perhaps if you could (please) give me
> your 4 x 3 grid, exactly, with the field of fractions? Thanks.
>
> > As for my tonality model.
> > There isn't much to it.
> > Unless one modulates (and the harmonic permutations will tell you what a
> > modulation is) on is in the 12tone subset we talk about above.
> > A tonic is simply a tonal center by choice.
> > The harmonic permutation set gives 1/1 135/128 9/8 75/64 5/4 675/512 45/32
> > 3/2 25/16 27/16 225/128 15/8 2/1 seen from the GCD.
> > As many chords will point with their GCD to the above 1/1, for instance a
> > dominant 7th, I've found that what is refered to as major mode in normal
> > music theory corresponds to a tonic of 3/2 from the GCD. Minor to a tonic
> > 5/4 from the GCD, etc.
> > It seems normal music theory gets the tonic right a lot of the time allowing
> > easy transcription to JI.
> > Sometimes not though, but I'm getting better at finding the real tonics and
> > modulations of a piece.
> > I'm bussy making a proper website where I'll show all my experiments and
> > thoughts on these things soon.
>
> I'll look at your website once it's ready, thanks - pgh
>
> > Marcel
> >
>

Hi Hudson,

>Glad you liked (and recalled!) Neo-old(19).

I have the midi and pdf score carefully archived, along with
your 17-EDO piece. Neo-old is one of my favorite microtonal
compositions.

> > The chord is broken down into its
> > generators, which in this case could be 100cents and 500cents.
> > The picture looks like this (please view message with a fixed
> > width font)
> >
> > B
> > |
> > C---F---Bb
> >
> > | = 100cents --- = 500cents
> >
> > Then, following Fokker, we find scales that contain many of
> > these chords. The basic goal is to maximize their number for
> > a given scale size (cardinality).
>
> Indeed. An additional question is: which cardinality is the
> most efficient in that aspect (embedding instances of a given
> subset)?

We have several precise notions of this efficiency. The most
prominent is called "logflat badness". For a given logflat
badness cutoff there is an infinite series of increasingly-
accurate temperaments, with the same number of temperaments
in each logarithmic bin of scale size (10-100, 100-1000 etc).

> Yes, the "canonical" similarity/equivalence relation is
> transposition. But other relations, even those that may arise
> controversy, can be useful for composition, or may be useful
> for analysis. For example, the "inversion equivalence" (which
> maps a minor triad to a major triad) proposed by Allen Forte
> is questioned by Larry Solomon. Both have good arguments:
> inversion was extensively used by composers of atonal music as
> a rather basic derivation; for the other hand, inversion-
> related sets can sound different enough to deserve different
> classification in many contexts.

Of course it would be useful to analyze music generated with
it -- any relation has this property! But inversion is not
audible to ordinary human beings. If we're talking about
harmony, try comparing 4:5:6:7:9:11:13 to its subharmonic
inverse. If we're talking about melody we can be a little
more permissive, but the similarity is not strong enough to
be generally useful in analyzing music.

> Can you find that {C,E,G} and {C,Eb,G} share something in common,
> specially when compared to other 3-element subsets of 12-EDO?

Yes, they share 2 of 3 pitches! I think C-E-G and F-Ab-C
are proper inverses. They sound less similar but are still
related because they share a strong outer interval (3:2).
Inverses on average (out of all possible structures to start
with) have approximately zero audible similarity.

> I also use to evaluate pc set operations as of relative or
> context-dependent usefulness. For instance, chords with six or
> more elements are difficult to analyse and compare by ear, and
> the disposition of the pitches is quite important (compare the
> chords {C,D,E,F,G,A,B}, {F,C,G,D,A,E,B} and {D,F,A,C,E,G,B},
> which are the "same" pitch class set). But if six pitches are
> presented as a melodic sequence, their repetition (even
> transposed) may be perceived as equivalent. Such issues are,
> for me, an incentive to seek more similarity criteria (possibly
> without reduce the pitches/intervals in classes), since octave
> equivalence, transposition and inversion show their limits.

I think rhythm is more important in melodic similarity than
pitch. There are very few pitch transformations that will
produce any similarity by themselves. In fact, only three are
recognized by common-practice music theory:

1. transposition of entire structures by any interval and
2. transposition of individual pitches by octaves ("voicing")
3. modal transposition (e.g. Ionian to Aeolian).

Regular mapping generally only considers #1 since it is by
far the strongest. Octave-equivalent voicings stop sounding
so similar above the 5-limit. Rothenberg stability (the same
as Balzano coherence IIRC) gives us the likelihood that a
melody will be recognizable in another mode. It is the only
scale property not based on psychoacoustic consonance that's
popular among regular mapping theorists.

Serialists often cite things like the arcane rules of the
Ars Nova, the use of retrograde and other transformations in
canons by people like Bach, and the tradition of change
ringing in England, as historical precedents for their craft.

I'm not familiar enough with Ars Nova practices to comment,
but to the extent such rules were followed I submit they were
just as 'academic' as 20th century serialism, and in both
cases made no contribution to the great music composed.

As far as Bach, I recently wrote this on tuning-math:
"Bach used things like retrograde not because it was an audible
transformation, but because it's challenging to do it and still
harmonize the canon. He tells you he did it (in the canonic
riddle) so you're sure to be impressed. If he were to throw
the retrograde on top of the theme without bothering to
harmonize it, that wouldn't have been impressive at all. With
Bach, the transformation was an arbitrary way to make the canon
harder. With serialists, the transformation is said to be a
meaningful way to make harmony obsolete. That's wrong. It's
not a matter of taste. If you can't hear the transformation
it's not meaningful, period."

As for change ringing, the bells were generally limited in
number and had limited harmonicity. They therefore provided
little ability to evoke stable pitches, and in any case too
few pitches to make detailed melodies. The natural type of
music to create with such a resource is an ambient wash of
partials that takes as long as possible to repeat. One could
write such music for orchestra as a novelty, but it would get
boring quickly since it doesn't fully use the orchestra's
resources (whereas it does fully use the belfry's resources).

> I am composing a serial piece for 17-EDO guitar, and I will
> use something I thought I would never use, because the result
> is "too dissimilar" to the origin: transformation by
> multiplication (just multiply the pitch class number by a
> constant factor).

In discussions with serialists on this list in the past, it
has usually come down to, "But I find these techniques inspire
my compositions!" And of course, that's great! Anything
that inspires music needs no further justification.

> > > For instance, in 12-ET, the interval vector of {B,C,F,Bb}
> > > (or {11,0,5,10}) is <210021>.
> >
> > Can you explain how is this interval vector is calculated
> > from {11,0,5,10}?
>
> It is an accounting of the occurrences of each interval class.
> In mod12 we have 6 interval classes, from 1 (semitone, M7, m9...)
> to 6 (tritone). As a consequence of octave equivalence,
> complementar (unordered) intervals are equivalent (P4th == P5th).
> Each entry of that "vector" is the number of instances of the
> corresponding interval class in the set:

I see. Thanks. As you might guess, I consider such a
representation highly dubious as far as human perception is
concerned.

> > If we generate scales that contain many instances of the
> > chord per scale tone, the chord instances must have many
> > common tones between them...
>
> Yes, "many". How many exactly?

Well, we can count them if we need to know. But generally
if you have the best temperament according to logflat
badness you'll get as many or more than in any other scale
of equivalent size.

>Given a scale generated that way are there interesting
>"byproducts" (subsets, chords, embeded scales)? Which are
>these "byproducts"? How about other chords of the same scale,
>which could be used as constrastant material? Are there other
>generator chords which lead to the same scale?

Generally we just optimize the number of 100- and 500-cent
dyads. Then we're also optimizing the number of consonant
structures based on them, whatever they are. The key is
to find the generators (100cents and 500cents in this case)
that most quickly produce the structure(s) we care about.
In the case of just intonation, these are just the prime
numbers (we induce a weighting on the primes to make the
composite numbers come out right according to psychoacoustic
consonance).

> In mod13 there are also tetrachords with interval vector
> <111111>; they could be used in Bohlen-Pierce scale, or in
> 13-EDO. In mod17, there are not such interval vectors, but
> there are ones like <10111101> or <1111100>, where two interval
> classes are absent. In mod10 and mod11, we have pentachords
> with vector <22222>. Interesting symmetry properties may be
> found in mod16, since 16 is 2*2*2*2 -- or perhaps it is the
> opposite: the symmetry may be excessive and boring (like the
> dim7-chord in 12-ET), an then the interesting sets might be
> the assymetrical ones -- how diverse are the subsets in mod16?
> How to effectively explore such properties in those
> uncommon scales?

It seems to me that most of the techniques are adaptable to
microtonal scales. You should definitely get in touch with
Paul Hjelmistad, as he's done work in this area.

> No, I am talking about "inversion" as "mirror", inverting
> the direction of the intervals:
>
> B C F Bb => +1 +5 +5
> B Bb F C => -1 -5 -5
>
> In this case, as you noticed, the result is equivalent to a
> simple permutation, and that is an interesting feature for
> counterpoint.

Ah, OK.

> It is completely invariant under multiplication by 7 (which is
> that "other" generator besides the semitone):
[snip]
> Furthermore, simple adding does just a transposition:
> 4 semitones = Major 3rd.

Indeed. But there is a point to be made here regarding some
music set theory papers I've seen that claim to derive the
diatonic scale from first principles. As I mentioned, this
was my third attempt at mutilating your set with simple
operations based on famous numbers like 7, but none of them
mutilated too much. That's because 12 is a small, highly-
composite number. It's easy to do things in 12 that produce
meaningful scales because almost all scales in 12 are
meaningful, because 12 happens to be an insanely great
temperament in terms of logflat badness. This is my gripe --
it's regular mapping that has truly derived the diatonic
scale (and 12-ET) from first principles, since it could go
wrong with things like 11-ET and so on. If we assume 12 we
cannot go too wrong.

> > > > It lets us find optimal tunings for any scale. It lets us
> > > > find the scales, out of all possible scales, that are the
> > > > natural competitors of the diatonic scale. It shows us how
> > > > these scales are related, and how they can be transformed
> > > > into one another, or how music written in one can be cast
> > > > into another. It tells us when such recasting is likely to
> > > > result in a qualitative change in the music vs. a more subtle
> > > > sense that the same piece has simply been retuned. etc.
> > >
> > > Could the set {B,C,F,Bb} be considered a "scale" in that
> > > sense? How it could be related to other scales, or maybe used
> > > to generate other scales?
> >
> > The kind of scales I was referring to here must have a mapping
> > to the generators of the desired chord(s). So {B,C,F,Bb} can
> > be the desired chord or the scale. If it's both, the
> > temperament mapping is trivial and not very interesting.
>
> I mean: a scale when comparing it to other scales.

Hm. In regular mapping, we compare scales against a target.
Is {B,C,F,Bb} a scale or the target? If 4:3 were the
target {B,C,F,Bb} is doing OK because it contains two 4:3
intervals in four notes. But it would be inferior to
{G,C,F,Bb} which contains three 4:3 intervals.

> A mapping is a group theory operation. I am wondering if pitch
> class set theory and regular mappings could be used in a
> complementary way, each one providing useful insights and tools
> for each other.

I'm not sure. It bears further thought. We are used to
fighting music set theory bitterly. I much prefer the idea
of cooperation.

-Carl

Yes, although a copyright infringement case my lawyers here advise me could "fly".

On 3 Feb 2010, at 17:27, Mike Battaglia wrote:

> Has the patent not yet expired?
> -Mike
>
>
>
> On Wed, Feb 3, 2010 at 7:30 AM, Charles Lucy <lucy@...> wrote:
>
>
>
> I hadn't worried to sue David Doty for libel, when he cast doubts on my engineering qualifications, in his biased review of LucyTuning back in the eighties, nor Carl for his rantings.
>
> Although this time it may well be worth it. Unlike Doty and Lumma; Apple have assets.
>
> LucyTuning has been included in the Logic Pro 9 version. Apple calls it harrisonj, yet the cent values to five decimal places are a blatant "lift" from my original papers in my patent applications in 1988.
>
> http://www.youtube.com/watch?v=CW0DUg63lqU&fmt=18
>
> Keep posted for developments.
>
> Charles Lucy
> lucy@...
>
> -- Promoting global harmony through LucyTuning --
>
> For more information on LucyTuning go to:
>
> http://www.lucytune.com
>
> LucyTuned Lullabies (from around the world) can found at:
>
> http://www.lullabies.co.uk
>
>
>
>

Charles Lucy
lucy@...

-- Promoting global harmony through LucyTuning --

For more information on LucyTuning go to:

http://www.lucytune.com

LucyTuned Lullabies (from around the world) can found at:

http://www.lullabies.co.uk

--- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > --- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@> wrote:
> >
> > > I mean {B,C,F,Bb} as a set containing pitch classes
> > > representatives.
> >
> > Ok. So what question do you have about them, that you want
> > theory to answer?
>
> Each theory has its own range of specific questions and answers. I could not yet see any relation between set theory and regular mappings, so that they can be comparable models.
>
> My question may look rather vague: let us suppose that I like the chord {B,C,F,Bb} and want to find scales in which the chord has interesting features that could be explored in a composition. Or the reverse: suppose I got a scale/tuning at random (from Scala archive, for example) and found an interesting sonority, that I recognize as (or could represent as) the set {B,C,F,Bb} -- then I want to know if this set has a "family" in that scale/tuning: transpositions and other related sets.
>
> As you see, this is the opposite of generating a scale and/or tuning: it is a matter of analysing a scale/tuning and its subsets.
>
> For instance, in 12-ET, the interval vector of {B,C,F,Bb} (or {11,0,5,10}) is <210021>. Some deductions are: transposition at interval classes 1 (semitone) or 5 (fourth) result in 2 common pitch classes; there are no "M/m thirds" in the set, thus transpositions to those intervals don't maintain invariants. The set can be completely invariant under "inversion": {B,Bb,F,C} (an useful information for counterpoint and serial music). From a more auditive approach, there are contrastant subsets, e.g: {C,F,Bb} ("consonant" 4ths) and {B,C,Bb} ("dissonant" semitones) -- but these yet share the interval 2 (Maj.2nd)...
>
> Such informations could be used as hints when exploring new material (scales, chords...).
>
> >
> > > Set theory can analyse the subset contents (e.g. interval vector),
> > > set properties (e.g. symmetry, common tones under transposition),
> > > group pc sets in set classes and propose similarity criteria.
> >
> > Take the index on the chain of fifths
> > {B,C,F,Bb} -> {5,0,-1,-2}
>
> It would be {11,0,5,10} in a chain of semitones...
>
> > Multiply each term by 7, for the number of nominals
> > available
> > {35,0,-7,-14}
> > Add 4 because there are 4 pitch classes in play
> > {39,4,-3,-10}
>
> I could not follow that.
>
> > Mod 12 for the 12 pitch classes available
> > {3,4,9,2}
> > Map to chromatic degrees of 12-ET, with 0 degrees being
> > the pitch class used as the 0th fifth earlier.
> > {Eb,E,A,D}
> > This transformation is definitely producing something
> > similar! (Actually it was my third try stringing arbitrary
> > operations together, and they all produced something
> > 'similar').
> >
> > > Or, putting in another way, can the regular mapping model
> > > provide information about the stuff -- *any* subsets --
> > > available in a given scale/tuning? Or is it just a way to
> > > represent scales?
> >
> > It lets us find optimal tunings for any scale. It lets us
> > find the scales, out of all possible scales, that are the
> > natural competitors of the diatonic scale. It shows us how
> > these scales are related, and how they can be transformed
> > into one another, or how music written in one can be cast
> > into another. It tells us when such recasting is likely to
> > result in a qualitative change in the music vs. a more subtle
> > sense that the same piece has simply been retuned. etc.
>
> Could the set {B,C,F,Bb} be considered a "scale" in that sense? How it could be related to other scales, or maybe used to generate other scales?
>
> > It doesn't tell us what notes to write once a scales and
> > tunings have been selected.
>
> If it can find scales "that are the natural competitors of the diatonic scale", then I suppose it takes in account certain properties of subsets of those scales (for example: the number of subsets approximating 4:5:6 or other ratios).
>
> I am still seeking some common point between the two theorical approaches.
>
> >
> > -Carl

I am a long ways off with unifying the two approaches myself,
but I am making in-roads. And although this is more along lines
of my conversation with Marcel, since it is the same Subject Thread, I'll respond here. Today I found this: (Don't worry about all the details:)

1.) It's possible to canvas all hexachords with one or two tritones
40 + 30) using just 10 pentachords * 7 (note left out) up
to complementation and inversion. (I treat the cases with 0 tritones
or 3 tritones separately). One hexachord in each row will be
a Steiner Set from S(5,6,12) which is the basis of the M12 group.

2. These pentachords will all have one tritone in the 0,6 slots.

3. Map this onto the grid we have been discussing, (Using {3^-1, 3^0, 3^1, 3^2} X {5^-1, 5^0, 5^1} and map 1/1 to 3^0*5^0 as you would expect). For now, lets map the tritone to 1/1 and 45/1 or 45/32,
assuming octave-equivalence.

4. Examine the remaining pitches and map them to fractions. One
solution is this: (With my I-Ching notation)

AAA, (3.4.5)
A-BA, (2.10.5)
AA-A (2.3.5)
BAA (8.3.4)
B--AA (7.4.5)
A-B-A (1.9.5)
A-AA (1.3.4)
BA--A (7.2.5)
AB-A (1.8.4)
AAA (1.2.3)

Which map to these fractions.*

1) 3/5, 5/1, 1/3
2) 9/1, 9/5, 1/3
3) 9/1, 3/5, 1/3
4) 1/5, 3/5, 5/1
5) 3/1, 5/1, 1/3
6)1/15, 5/3, 1/3
7)1/15, 3/5, 5/1
8) 3/1, 9/1, 1/3
9) 1/15, 1/5, 5/1
10) 1/15, 9/1, 3/5

* If anyone can find a unifying pattern here, you win a virtual beer:)

Where pitches are as follows:

Fractions:

1 1/15
2 9
3 3/5
4 5/1
5 1/3
7 3/1
8 1/5
9 5/3
10 9/5

We see that 11 is not used. This is a mapping up to D12 X S2,(inversion and complementation) I have done others, for example D4 X S3 (inversion and M5 relation)and so forth. There are all sorts of patterns, and I examine Z-relations among the pentachords and hexachords and also weakly related structures.

I will have to upload some of my spreadsheets to the Files section.
They are color coded and fairly self-explanatory (using my hexachord system alphabet, together with my I-Ching notation)

PGH

Dear All

I would appreciate�any perspectives, contemplations, praise songs or formulas�about epimoric ratios and their place in tuning,

Steve

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
[...]
> > Can you find that {C,E,G} and {C,Eb,G} share something in common,
> > specially when compared to other 3-element subsets of 12-EDO?
>
> Yes, they share 2 of 3 pitches!

Inclusion and intersection relations are relevant, then.

> I think C-E-G and F-Ab-C
> are proper inverses. They sound less similar but are still
> related because they share a strong outer interval (3:2).
> Inverses on average (out of all possible structures to start
> with) have approximately zero audible similarity.

How about comparing those other sets (as subsets of 12-EDO)?
A={C,Eb,G,Bb}
B={D,Eb,G#,A}
C={C,E,G,B}
D={B,C,F,Bb}

(Usually in set theory, we do not just compare two chords: we compare two chords against every possible chord available in the universe.)

[...]
> Rothenberg stability (the same
> as Balzano coherence IIRC) gives us the likelihood that a
> melody will be recognizable in another mode. It is the only
> scale property not based on psychoacoustic consonance that's
> popular among regular mapping theorists.

Is it a scale/mode property or a melody property?

> Serialists often cite things like the arcane rules of the
> Ars Nova, the use of retrograde and other transformations in
> canons by people like Bach, and the tradition of change
> ringing in England, as historical precedents for their craft.

I think the goals are different, even when there are related "precedents".

[...]
> As far as Bach, I recently wrote this on tuning-math:
> "Bach used things like retrograde not because it was an audible
> transformation, but because it's challenging to do it and still
> harmonize the canon. He tells you he did it (in the canonic
> riddle) so you're sure to be impressed. If he were to throw
> the retrograde on top of the theme without bothering to
> harmonize it, that wouldn't have been impressive at all. With
> Bach, the transformation was an arbitrary way to make the canon
> harder. With serialists, the transformation is said to be a
> meaningful way to make harmony obsolete. That's wrong. It's
> not a matter of taste. If you can't hear the transformation
> it's not meaningful, period."

(What is the source of that obscure phrase "a meaningful way to make harmony obsolete"?)

In my opinion, the usage of retrograde/inversion/etc. by serialists is not too different of Bach's usage: the "arbitrary" transformation needs additional conditions to work musically, it does not matter if the transformation can be (or should be) perceived as such (or not).

There should be superior compositional principles coordinating the application of the transformations: in certain cases, it is completely irrelevant that listeners can hear a transformation as such -- it matters the final effect, not the means. In such cases, the transformation is not used "because it was an audible transformation, but because it's challenging to do it [i.e. to find a strictly canonical transformation of the original material that can meet a musical goal] and still harmonize the canon [i.e. to put it playing the musical game in a successful and organic way]."

But there is another important point: the poetics of serial music usually intends to create complex textures that defy human perception. It sounds chaotic, yet it has a underlaying structure and an organization principle. Its perception is statistical and holistic. The rational techniques used to derivate material to compose it constrast with the stochastical apparence of the result -- a rather complex pitch organization may paradoxally lead to musical textures which do not seem to be pitch oriented, etc. In this poetics, transparent derivations of elemental components may not be very interesting -- they might lead to a local, analytical (and possibly discursive) hearing, instead of a global, textural, holistic hearing. The paradox of the rigorous underlaying organization and the apparently chaotic musical surface is an attractive feature of serial music.

[...]
> > I am composing a serial piece for 17-EDO guitar, and I will
> > use something I thought I would never use, because the result
> > is "too dissimilar" to the origin: transformation by
> > multiplication (just multiply the pitch class number by a
> > constant factor).
>
> In discussions with serialists on this list in the past, it
> has usually come down to, "But I find these techniques inspire
> my compositions!" And of course, that's great! Anything
> that inspires music needs no further justification.

The point of view of a composer is rather different of that of the theoric. Something "works" in a composition because the musical sensitivity and feeling of the composer approve it or require it -- but the fact that it "works" is only a syntactical requirement, subordinated to the compositional idea: it really "works" because it carries and accomplishes the compositional idea. (For example: a cadence dominant->tonic is used not because it "makes sense in itself", but because it can carry certain kinds of musical ideas in a functional way -- it may be "harmonically effective" in a theoretical sense, but yet it may be misused in nonsense "contexts").

It seems that there is no theory able to explain how atonal music really works; set theory can help us to see how it is organized, which is for sure better than nothing.

What are the achievements of regular mapping theory for analysis of atonal music, by the way?

[...]
> > In mod13 there are also tetrachords with interval vector
> > <111111>; they could be used in Bohlen-Pierce scale, or in
> > 13-EDO. In mod17, there are not such interval vectors, but
> > there are ones like <10111101> or <1111100>, where two interval
> > classes are absent. In mod10 and mod11, we have pentachords
> > with vector <22222>. Interesting symmetry properties may be
> > found in mod16, since 16 is 2*2*2*2 -- or perhaps it is the
> > opposite: the symmetry may be excessive and boring (like the
> > dim7-chord in 12-ET), an then the interesting sets might be
> > the assymetrical ones -- how diverse are the subsets in mod16?
> > How to effectively explore such properties in those
> > uncommon scales?
>
> It seems to me that most of the techniques are adaptable to
> microtonal scales. You should definitely get in touch with
> Paul Hjelmistad, as he's done work in this area.

Let's see...

[...]
> > It is completely invariant under multiplication by 7 (which is
> > that "other" generator besides the semitone):
> [snip]
> > Furthermore, simple adding does just a transposition:
> > 4 semitones = Major 3rd.
>
> Indeed. But there is a point to be made here regarding some
> music set theory papers I've seen that claim to derive the
> diatonic scale from first principles. As I mentioned, this
> was my third attempt at mutilating your set with simple
> operations based on famous numbers like 7, but none of them
> mutilated too much. That's because 12 is a small, highly-
> composite number.

That total invariance under multiplication only was possible because: (1) you have multiplied the set by a number which does not share any factors with 12; (2) for coincidence, that set had that property -- while the highly-composite number 12 _makes possible_ the property, I guess there is a reduced number of sets which map themselves by multiplication. Had you multiplied it by 2, 3, 4, 6, 8, 9, 10... you could easily achieve your goal to mutilate it!

> It's easy to do things in 12 that produce
> meaningful scales because almost all scales in 12 are
> meaningful, because 12 happens to be an insanely great
> temperament in terms of logflat badness. This is my gripe --
> it's regular mapping that has truly derived the diatonic
> scale (and 12-ET) from first principles, since it could go
> wrong with things like 11-ET and so on. If we assume 12 we
> cannot go too wrong.

Indeed, if the number of pitches is a prime number, then all factors would result in different sets.

[...]
> > > > Could the set {B,C,F,Bb} be considered a "scale" in that
> > > > sense? How it could be related to other scales, or maybe used
> > > > to generate other scales?
> > >
> > > The kind of scales I was referring to here must have a mapping
> > > to the generators of the desired chord(s). So {B,C,F,Bb} can
> > > be the desired chord or the scale. If it's both, the
> > > temperament mapping is trivial and not very interesting.
> >
> > I mean: a scale when comparing it to other scales.
>
> Hm. In regular mapping, we compare scales against a target.
> Is {B,C,F,Bb} a scale or the target? If 4:3 were the
> target {B,C,F,Bb} is doing OK because it contains two 4:3
> intervals in four notes. But it would be inferior to
> {G,C,F,Bb} which contains three 4:3 intervals.

But -- from another standpoint -- {G,C,F,Bb} is relatively "poor" as it contains only 3 different intervals and it is redundant concerning to interval class 5 (P4th). {B,C,F,Bb} is more rich in the number of different intervals available (4), and {C,E,F#,G} yet more (6).

Maybe I should ask: what can be said comparing 12-EDO against every one of its subsets as target?

>
> > A mapping is a group theory operation. I am wondering if pitch
> > class set theory and regular mappings could be used in a
> > complementary way, each one providing useful insights and tools
> > for each other.
>
> I'm not sure. It bears further thought. We are used to
> fighting music set theory bitterly. I much prefer the idea
> of cooperation.
>
> -Carl
>

Have you read "Introduction to post-tonal theory" by Richard N. Straus? It is interesting because it is very didactic and contains several musical examples and practical exercises for aural comparations of pitch sets as they occur in atonal masterworks.

Hudson

I think the most obvious relationship is that they represent consecutive partials in the harmonic series.
Putting many super-particular ratios in a row in ascending order IE 6/5 7/6 8/7 AKA (6:7:8) gives a "major" chord feel while putting them in descending order (IE 8/7 7/6 6/5) gives a minor chord feel.

I think it mainly has to do with optimizing creation of one chord, though. Sure, you could make a super-particular chord-scale IE 7:8:9:10:11:12:13:14...but you lose variety of tonal color and have some issues with roughness between the closer partials IE 14/13 (which, at least to me, explains why very small intervals like the minor second are so rarely used in chords). So I think it's a valuable point...but only so far as chords with fairly low denominator ratios...beyond that, IMVHO, it falls apart as the issue of excessive roughness begins to outweigh periodicity at very close ratios.

________________________________
From: Steven Grainger <srgrainger@...>
To: tuning@yahoogroups.com
Sent: Wed, February 3, 2010 5:50:38 PM
Subject: [tuning] Epimoric Ratios

Dear All

I would appreciate any perspectives, contemplations, praise songs or formulas about epimoric ratios and their place in tuning,

Steve

--- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@...> wrote:

> > > Can you find that {C,E,G} and {C,Eb,G} share something in
> > > common, specially when compared to other 3-element subsets
> > > of 12-EDO?
> >
> > Yes, they share 2 of 3 pitches!
>
> Inclusion and intersection relations are relevant, then.

Sounds like intersection. What's inclusion?

> > I think C-E-G and F-Ab-C
> > are proper inverses. They sound less similar but are still
> > related because they share a strong outer interval (3:2).
> > Inverses on average (out of all possible structures to start
> > with) have approximately zero audible similarity.
>
> How about comparing those other sets (as subsets of 12-EDO)?
> A={C,Eb,G,Bb}
> B={D,Eb,G#,A}
> C={C,E,G,B}
> D={B,C,F,Bb}
>
> (Usually in set theory, we do not just compare two chords: we
> compare two chords against every possible chord available in
> the universe.)

No, microtonal theory compares every other chord available
in the universe. :) And we find that inversion is inaudible.
At least, I'm aware of no evidence that it is audible. That's
really the difference. If a relation is to be used as an
*explanation* it should be subjected to experiment. You can
use whatever generative methods you wish, but be honest, many
serialists and set theorists have not stopped there as far as
claims about the explanatory power of the techniques.

> > Rothenberg stability (the same
> > as Balzano coherence IIRC) gives us the likelihood that a
> > melody will be recognizable in another mode. It is the only
> > scale property not based on psychoacoustic consonance that's
> > popular among regular mapping theorists.
>
> Is it a scale/mode property or a melody property?

It's a property of a scale, telling us the likelihood that
melodies in the scale will be recognizable under modal
transposition. Not all regular mapping theorists agree
with it.

> > As far as Bach, I recently wrote this on tuning-math:
> > "Bach used things like retrograde not because it was an audible
> > transformation, but because it's challenging to do it and still
> > harmonize the canon. He tells you he did it (in the canonic
> > riddle) so you're sure to be impressed. If he were to throw
> > the retrograde on top of the theme without bothering to
> > harmonize it, that wouldn't have been impressive at all. With
> > Bach, the transformation was an arbitrary way to make the canon
> > harder. With serialists, the transformation is said to be a
> > meaningful way to make harmony obsolete. That's wrong. It's
> > not a matter of taste. If you can't hear the transformation
> > it's not meaningful, period."
>
> (What is the source of that obscure phrase "a meaningful way to
> make harmony obsolete"?)

I wrote it on Friday. Perhaps I shouldn't have written
"is said to be" but I do think it accurately paraphrases
some of the stuff I've read.

> There should be superior compositional principles coordinating
> the application of the transformations: in certain cases, it is
> completely irrelevant that listeners can hear a transformation
> as such -- it matters the final effect, not the means. In such
> cases, the transformation is not used "because it was an audible
> transformation, but because it's challenging to do it [i.e. to
> find a strictly canonical transformation of the original material
> that can meet a musical goal] and still harmonize the canon
> [i.e. to put it playing the musical game in a successful and
> organic way]."

If the final effect is all that matters, why bother with the
fancy transformations? It's challenging to line everything
up so that it harmonizes. But is it challenging to play the
musical game in an organic way? Probably I am lacking some
listening skills that fans of serial music have, but many
serial compositions I've heard do not sound like they would be
particularly harmed by having been put together differently.

> But there is another important point: the poetics of serial
> music usually intends to create complex textures that defy
> human perception. It sounds chaotic, yet it has a underlaying
> structure and an organization principle. Its perception is
> statistical and holistic. The rational techniques used to
> derivate material to compose it constrast with the stochastical
> apparence of the result -- a rather complex pitch organization
> may paradoxally lead to musical textures which do not seem to
> be pitch oriented, etc.

Right, so the underlying technique is not responsible for the
effect. To be completely honest, I think it was a fad. Back
before the era of computing, the idea of experiencing all
permutations of something probably seemed interesting. Today
we are surrounded by generative possibilities like this. And
the idea of not having control appealed to the philosophy of
the day. The wars created a huge backlash against classicism
and a desire for modernity at any cost. The wikipedia entry
on serialism states, "serialism is not only a technique ... of
composition, but also "a philosophy of life" ..."

> It seems that there is no theory able to explain how atonal
> music really works; set theory can help us to see how it is
> organized, which is for sure better than nothing.

If I generate a piece using technique X, it is hardly fair to
say technique X explains it. Of course it does! Then we have
claims of tone rows appearing in Mozart and so on. Nonsense!

> What are the achievements of regular mapping theory for
> analysis of atonal music, by the way?

I'm not aware it's been tried. We have apparently been able
to derive a few popular indigenous scale species, though even
here a systematic review has yet to be done. Keep in mind
most of us have day jobs!

> > In regular mapping, we compare scales against a target.
> > Is {B,C,F,Bb} a scale or the target? If 4:3 were the
> > target {B,C,F,Bb} is doing OK because it contains two 4:3
> > intervals in four notes. But it would be inferior to
> > {G,C,F,Bb} which contains three 4:3 intervals.
>
> But -- from another standpoint -- {G,C,F,Bb} is relatively
> "poor" as it contains only 3 different intervals and it is
> redundant concerning to interval class 5 (P4th).

Note, in this example I established P4 as the target, so
being redundant in it would be considered good.

> Maybe I should ask: what can be said comparing 12-EDO against
> every one of its subsets as target?

If you happen to choose a subset that approximates a 5-limit
consonance, then it does extremely well!

> Have you read "Introduction to post-tonal theory" by Richard
> N. Straus? It is interesting because it is very didactic and
> contains several musical examples and practical exercises for
> aural comparations of pitch sets as they occur in atonal
> masterworks.

No, but I'll add it to my reading list. I should stress that
I'm sure I lack many of the listening skills of an experienced
fan of atonal/serial music. Nonetheless, I believe that at
least, music set theory overstepped its ability in its claims.
I mean, stuff like this is fairly off the wall,

"Any musician who has not experienced - I do not say
understood, but truly experienced - the necessity of
dodecaphonic music is USELESS."

That's a very famous quote but it's worth thinking about.
What kind of artist says this about all forms other than
his own? And this was not a fringe artist, but one of the
leaders of the movement!

-Carl

I wrote:
> I should stress that I'm sure I lack many of the listening
> skills of an experienced fan of atonal/serial music.

For the record, I have an album of Schoenberg's first four
string quartets, and the first is among the most powerful
pieces of music I've ever heard. I also enjoyed these:
http://en.wikipedia.org/wiki/Sechs_kleine_Klavierst%C3%BCcke

After favoring these, I learned they are both atonal but
predate his formal use of serialism...

-Carl

Thanks Michael
 
 

________________________________
From: Michael <djtrancendance@...>
To: tuning@yahoogroups.com
Sent: Thu, 4 February, 2010 1:02:38 PM
Subject: Re: [tuning] Epimoric Ratios

 
I think the most obvious relationship is that they represent consecutive partials in the harmonic series.
  Putting many super-particular ratios in a row in ascending order IE 6/5 7/6 8/7 AKA (6:7:8) gives a "major" chord feel while putting them in descending order (IE 8/7 7/6 6/5) gives a minor chord feel.

    I think it mainly has to do with optimizing creation of one chord, though.  Sure, you could make a super-particular chord-scale IE 7:8:9:10:11: 12:13:14. ..but you lose variety of tonal color and have some issues with roughness between the closer partials IE 14/13 (which, at least to me, explains why very small intervals like the minor second are so rarely used in chords).   So I think it's a valuable point...but only so far as chords with fairly low denominator ratios...beyond that, IMVHO, it falls apart as the issue of excessive roughness begins to outweigh periodicity at very close ratios.

________________________________
From: Steven Grainger <srgrainger@yahoo. com.au>
To: tuning@yahoogroups. com
Sent: Wed, February 3, 2010 5:50:38 PM
Subject: [tuning] Epimoric Ratios

 
Dear All

I would appreciate any perspectives, contemplations, praise songs or formulas about epimoric ratios and their place in tuning,

Steve

__________________________________________________________________________________
Yahoo!7: Catch-up on your favourite Channel 7 TV shows easily, legally, and for free at PLUS7. www.tv.yahoo.com.au/plus7

I would like to add something to what's been said so far.

Sometimes it's useful to know how close a ratio is to an epimoric one. Back in 2002, I began to use a unit which made this possible to me. I called it the "epimoric denominator" or ED. You can get it by "1/(f-1)", where "f" is the linear factor. So an octave of 2/1 has an ED of 1, a fifth of 3/2 has an ED of 2, and a major sixth of 5/3 has an ED of 1.5 (you may also understand it by saying that 5/3 is between 4/2 and 6/4). It also makes it possible to better imagine an approximate amount of "mistuning" of two tones, because it tells you which frequencies you have to mix to get a 1Hz difference. For example, the syntonic comma of 81/80 has an ED of 80 while the Pyth. comma of (3^12)/(2^19) has an ED of ~73.296, which means that it's larger than the syntonic comma. On the other hand, the 5-limit schisma of 32805/32768 has an ED of ~885.622, which can tell you that this must be something pretty small. Similarly, the ED for the ordinary 100-cent semitone is ~16.817, which means that it's closer to 18/17 than to 17/16.

The important thing to remember here is that negative numbers don't represent the same intervals as positive numbers do. For example, while the ED for a rising 5-limit major third is 4, the ED for a falling major third is -5 because the denominator of the ratio is supposed to be always 1 less than the numerator (4/5 is the same as "(-4)/(-5)").

Petr

You patented John Harrison's tuning?

--- In tuning@yahoogroups.com, Charles Lucy <lucy@...> wrote:
>
> Yes, although a copyright infringement case my lawyers here advise me could "fly".
>
>
> On 3 Feb 2010, at 17:27, Mike Battaglia wrote:
>
> > Has the patent not yet expired?
> > -Mike
> >
> >
> >
> > On Wed, Feb 3, 2010 at 7:30 AM, Charles Lucy <lucy@...> wrote:
> >
> >
> >
> > I hadn't worried to sue David Doty for libel, when he cast doubts on my engineering qualifications, in his biased review of LucyTuning back in the eighties, nor Carl for his rantings.
> >
> > Although this time it may well be worth it. Unlike Doty and Lumma; Apple have assets.
> >
> > LucyTuning has been included in the Logic Pro 9 version. Apple calls it harrisonj, yet the cent values to five decimal places are a blatant "lift" from my original papers in my patent applications in 1988.
> >
> > http://www.youtube.com/watch?v=CW0DUg63lqU&fmt=18
> >
> > Keep posted for developments.
> >
> > Charles Lucy
> > lucy@...
> >
> > -- Promoting global harmony through LucyTuning --
> >
> > For more information on LucyTuning go to:
> >
> > http://www.lucytune.com
> >
> > LucyTuned Lullabies (from around the world) can found at:
> >
> > http://www.lullabies.co.uk
> >
> >
> >
> >
>
> Charles Lucy
> lucy@...
>
> -- Promoting global harmony through LucyTuning --
>
> For more information on LucyTuning go to:
>
> http://www.lucytune.com
>
> LucyTuned Lullabies (from around the world) can found at:
>
> http://www.lullabies.co.uk
>

--- In tuning@yahoogroups.com, "Carl Lumma" <c
> > > Yes, they share 2 of 3 pitches!
> >
> > Inclusion and intersection relations are relevant, then.
>
> Sounds like intersection. What's inclusion?

Inclusion: subset/superset relations. {C,E,G} includes (is a superset of) {C,G}, which is also included in (is a subset of) {C,Eb,G}.

There is an intersection between the two chords, since they share a common subset (not regarding the empty set).

> > > I think C-E-G and F-Ab-C
> > > are proper inverses. They sound less similar but are still
> > > related because they share a strong outer interval (3:2).
> > > Inverses on average (out of all possible structures to start
> > > with) have approximately zero audible similarity.
> >
> > How about comparing those other sets (as subsets of 12-EDO)?
> > A={C,Eb,G,Bb}
> > B={D,Eb,G#,A}
> > C={C,E,G,B}
> > D={B,C,F,Bb}
> >
> > (Usually in set theory, we do not just compare two chords: we
> > compare two chords against every possible chord available in
> > the universe.)
>
> No, microtonal theory compares every other chord available
> in the universe. :) And we find that inversion is inaudible.

It is inaudible as an "inversion". But inversely related sets may (or may not, according to the case) sound similar in some degree.

I am not claming that the inversion is generally valid as an equivalence relation, as several set theorists may defend. As I have said, Larry Solomon does not agree with this reduction, and only accepts transposition as canonical transformation. Solomon additionally tries to find similarity relations between sets which are not related by transposition. I think that set similarity outside the defined "set classes" are important and should be more investigated.

BTW, there are sets in atonal music that I recognize by its set class, although I need to think a bit to know if its is a "prime" ou "inverse" form. That is not always due to some "similarity", but I suppose it is mainly due to learnt cathegorization (something like calling "7th chords" structures as different as {G,B,D#,F} and {G,Bb,Db,F#}). In this sense, atonal composers may be somewhat induced by the association of inversely related sets, just because they are used in atonal music.

Please compare the chords A, B, C and D above. I am sure you can find similarities and differences between them, by ear. Sets are not "just" completely different of each other, with nearly zero audible similarity and period; we can go further.

There are a lot of different sets used in atonal music, thus some reduction and grouping is required to handle them. Set theory is frequently accused of excessive reduction at the very first analytical steps: it is a fair criticism. By the other hand, with all its defects, it showed to be useful in analysis of atonal music and inspirative for composition.

> At least, I'm aware of no evidence that it is audible. That's
> really the difference. If a relation is to be used as an
> *explanation* it should be subjected to experiment. You can
> use whatever generative methods you wish, but be honest, many
> serialists and set theorists have not stopped there as far as
> claims about the explanatory power of the techniques.

It can "explain" how they were originated in a compositional elaboration, but it cannot "explain" how it works in the context.

[...]
> > There should be superior compositional principles coordinating
> > the application of the transformations: in certain cases, it is
> > completely irrelevant that listeners can hear a transformation
> > as such -- it matters the final effect, not the means. In such
> > cases, the transformation is not used "because it was an audible
> > transformation, but because it's challenging to do it [i.e. to
> > find a strictly canonical transformation of the original material
> > that can meet a musical goal] and still harmonize the canon
> > [i.e. to put it playing the musical game in a successful and
> > organic way]."
>
> If the final effect is all that matters, why bother with the
> fancy transformations?

To put it shortly, artisany and control.

> It's challenging to line everything
> up so that it harmonizes. But is it challenging to play the
> musical game in an organic way? Probably I am lacking some
> listening skills that fans of serial music have, but many
> serial compositions I've heard do not sound like they would be
> particularly harmed by having been put together differently.

There are "composers" and composers, "compositions" and compositions.

I don't think Schoenberg's String Trio, Berg's Violin Concert, Webern's Sinfonia, Boulez's Éclat/Multiples or Le marteau sans maître, for example, sound gratuitous in any sense.

(By the other side, Boulez's Structures are boring.)

>
> > But there is another important point: the poetics of serial
> > music usually intends to create complex textures that defy
> > human perception. It sounds chaotic, yet it has a underlaying
> > structure and an organization principle. Its perception is
> > statistical and holistic. The rational techniques used to
> > derivate material to compose it constrast with the stochastical
> > apparence of the result -- a rather complex pitch organization
> > may paradoxally lead to musical textures which do not seem to
> > be pitch oriented, etc.
>
> Right, so the underlying technique is not responsible for the
> effect.

It is indirectly responsible for the effect. Indeed, it is a way (or _the_ chosen way) to achieve the effect, although the control may act in a constrained probabilistic way. The probabilities are controled by the technique employed, thus the technique -- and its relation to the specific material -- is not superfluous.

For instance, if the objective is to creating an aparently random texture (stochastical music, not dodecaphonic music), this human limitation should be taken in account:
http://faculty.rhodes.edu/wetzel/random/mainbody.html#imagine

In dodecaphonic music (think Schoenberg), traditional motivic transformations are in place. I only can say that it is rather complex (elaborated) music, made with rigorous choices. But there are not too many skilled and talented composers like Schoenberg out there to master the technique. In Schoenberg's music, the ear and inteligence acts all the time, intensely.

Still about the interaction between technique and material: replace a tone-row for another in a dodecaphonic music by Berg or Boulez, and then you will get real cacophony, because an acidental replacement will cause pitch relations which are inconsistent with the style and structure.

[...]
> > It seems that there is no theory able to explain how atonal
> > music really works; set theory can help us to see how it is
> > organized, which is for sure better than nothing.
>
> If I generate a piece using technique X, it is hardly fair to
> say technique X explains it. Of course it does! [...]

I disagree. The complexity of the "output" in good music is usually greater than the complexitiy of the conscient path percurred to realize it. Intuition is very very important in art.

(Bad serial music is the reverse: musically inconsistent complication results in banal output which does not justify the complication. Here, complexity is not present.)

[...]
> > Have you read "Introduction to post-tonal theory" by Richard
> > N. Straus?

Oops: his name is Joseph N. Straus.

> Nonetheless, I believe that at
> least, music set theory overstepped its ability in its claims.
> I mean, stuff like this is fairly off the wall,
>

I will check Allen Forte's "The structure of atonal music" again for "its claims". I recall the second part of the book is rather abstract.

However, as I wrote before: there are no successful approaches to analyse atonal music, and set theory proved to be at least a positive contribution anyway. I think it is not sufficient and it tends to be too much reductive, ignoring important aspects of human musical perception. But there are too rare explanatory approaches, then it cannot be safely discarded yet.

As with any approach that uses non-musical concepts, it is subject to misuse in hands of people more inclined to maths than to music.

> "Any musician who has not experienced - I do not say
> understood, but truly experienced - the necessity of
> dodecaphonic music is USELESS."
>
> That's a very famous quote but it's worth thinking about.
> What kind of artist says this about all forms other than
> his own? And this was not a fringe artist, but one of the
> leaders of the movement!

Arrogance, ego/euro/centricism and proselitism are not under discussion here. We are better off listening to Boulez's music rather than taking seriously such (out)dated claims.

BTW, I have found this "definition" in a text about regular mappings, which is not constructive either:

"Atonal Music
Music with lots of wrong notes."

You know that many naive attempts of composing microtonal music based on triads are nothing more than out-of-tune tonal/modal music (because they use harmonic material that is inconsistent with the tuning used). From your description, I undertand that regular mapping can catch such errors, allowing one to find suitable tunings for the chosen harmonic material. (I find that Neo-old(19) sounds mistuned, although I am not sure that other tunings can maintain the intended relations, harmonic ambiguities, etc. I have made a few tries with Secor's 19-WT, but I only tested two keys. An adaptative temperament would be welcome, and I implicitely assume that one should be used if the piece were played in acoustical instruments.)

By the other side, as composer, I often feel that some specific pitch (and only that pitch) is required in certain moment of a "free" atonal music. The choice of the components of a chord affects the next chord, and so on; certain pitches do work, other ones don't. Which are the patterns and rules governing those selections? I assume there are (psycho)acoustical principles (can regular mapping theory here?), "motivic" patterns (pitch sets?), as well as other factors. There may be "wrong notes" or pitches with secondary relevance or that are somewhat indifferent, but there is also a selection by ear, so that the music sounds this way rather than in a different way. This cannot be explained by simply referring to a technical method used in the composition (the "technique" may have been "I just selected the notes intuitively by ear").

I am not interested in void polemics, as opposing pitch set theory and regular mapping as one right/good other wrong/bad, or serial music against low-primes just intonation music etc. I am interested in to learn about how microtonal scales can be used to accomplish tonal, modal, chromatic, atonal, serial _and_ other kinds of music.

For example, I am currently interested in pitch structures that are rich in ambiguities and that can suggest multiple potential meanings. That preference leads me to avoid simple consonanses as uninteresting (because they usually can be easily reduced to a single fundamental, inducing the perception to make that reduction), although the presence of some consonances is required to suggest potential harmonic meanings. Such pitch structures should be suitable for tonal ("extended tonality") and atonal ("pantonal") music.

Hudson

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> I wrote:
> > I should stress that I'm sure I lack many of the listening
> > skills of an experienced fan of atonal/serial music.
>
> For the record, I have an album of Schoenberg's first four
> string quartets, and the first is among the most powerful
> pieces of music I've ever heard.
> I also enjoyed these:
> http://en.wikipedia.org/wiki/Sechs_kleine_Klavierst%C3%BCcke

I have made and played a transcription for two guitars of Op.19.

>
> After favoring these, I learned they are both atonal but
> predate his formal use of serialism...

Op. 19 is "free atonal".

But String Quartet N.1 in D minor is tonal (extended tonality); are you referring to another quartet?

The 4th movement of Quartet N.2, with soprano voice, may be considered atonal:
http://en.wikipedia.org/wiki/File:Schoenberg_Quartet_No._2_4th_movement.OGG

Quartets 3 and 4 are dodecaphonic. There is also another (older) tonal quartet, in D Major.

http://en.wikipedia.org/wiki/String_quartets_(Schoenberg)

--- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@...> wrote:

> Op. 19 is "free atonal".
>
> But String Quartet N.1 in D minor is tonal (extended tonality);
> are you referring to another quartet?

Yes, ok, borderline tonal.

> Quartets 3 and 4 are dodecaphonic.

Yes, I felt these were weaker compositions.

-Carl

--- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@...> wrote:

> Inclusion: subset/superset relations. {C,E,G} includes (is a
> superset of) {C,G}, which is also included in (is a subset of)
> {C,Eb,G}.
>
> There is an intersection between the two chords, since they
> share a common subset (not regarding the empty set).

Ah, yes.

> > No, microtonal theory compares every other chord available
> > in the universe. :) And we find that inversion is inaudible.
>
> It is inaudible as an "inversion". But inversely related sets
> may (or may not, according to the case) sound similar in some
> degree.

Any transformation one can dream up may produce two similar
sounding chords in some cases. That doesn't mean anything.
If you're talking about something real, it'll always hold.

> I am not claiming that the inversion is generally valid as
> an equivalence relation, as several set theorists may defend.
> As I have said, Larry Solomon does not agree with this
> reduction, and only accepts transposition as canonical
> transformation. Solomon additionally tries to find similarity
> relations between sets which are not related by transposition.
> I think that set similarity outside the defined "set classes"
> are important and should be more investigated.

Good. My only advice is to always consider the entire spectrum
of pitch. If you stay in 12-ET you can find things that work
only because 12-ET is an extraordinary tuning.

> I only can say that it is rather complex (elaborated) music,
> made with rigorous choices. But there are not too many skilled
> and talented composers like Schoenberg out there to master the
> technique. In Schoenberg's music, the ear and inteligence acts
> all the time, intensely.

I haven't heard much of Schoenberg's music, but anyone who
could write that 1st string quartet could surely make a purse
out of a sow's ear.

A composer friend once recommended Berio to me. I thought it
was terrible!

> Still about the interaction between technique and material:
> replace a tone-row for another in a dodecaphonic music by Berg
> or Boulez, and then you will get real cacophony, because an
> acidental replacement will cause pitch relations which are
> inconsistent with the style and structure.

Now THIS is something I would like to test. Can you do it
easily? Then we could ask people their opinions. Maybe we
could present them as "A" and "B" and ask people which they
think is the original piece.

> However, as I wrote before: there are no successful approaches
> to analyze atonal music, and set theory proved to be at least
> a positive contribution anyway.

Again, isn't this backwards? You speak as if atonal music was
a flourishing natural genre before serialism / set theory came
along to explain it. Hardly! Try set theory on one of the few
19th-century atonal pieces. No, it only works (if you consider
it working at all) when it was used to create the composition.

> You know that many naive attempts of composing microtonal music
> based on triads are nothing more than out-of-tune tonal/modal
> music (because they use harmonic material that is inconsistent
> with the tuning used).

Sure.

>From your description, I undertand that regular mapping can catch
>such errors, allowing one to find suitable tunings for the chosen
>harmonic material.

We've developed techniques to find optimal tunings given the
commas that 'vanish' in the score. We can also do things like
remap 4:5:6 to any other consonant triad like 5:7:9. The
results may not sound good, but they'll sound better than
simply remapping chromatically to a scale containing 5:7:9
chords. And Erv Wilson and Marcus Hobbs have worked out some
other very interesting transformations...

>(I find that Neo-old(19) sounds mistuned,
>although I am not sure that other tunings can maintain the
>intended relations, harmonic ambiguities, etc.

Neo-old is really a miniature masterpiece, in that it uses
19 organically; it is not just triadic music that uses the
better tuning of triads in 19 (as some of Easley Blackwood's
music in 19 does).

>I have made a few tries with Secor's 19-WT, but I only tested
>two keys. An adaptative temperament would be welcome, and I
>implicitely assume that one should be used if the piece were
>played in acoustical instruments.)

Perhaps I will get a chance to take a closer look at the score.

> By the other side, as composer, I often feel that some specific
> pitch (and only that pitch) is required in certain moment of a
> "free" atonal music. The choice of the components of a chord
> affects the next chord, and so on; certain pitches do work,
> other ones don't. Which are the patterns and rules governing
> those selections? I assume there are (psycho)acoustical
> principles (can regular mapping theory here?), "motivic" patterns
> (pitch sets?), as well as other factors. There may be
> "wrong notes" or pitches with secondary relevance or that are
> somewhat indifferent, but there is also a selection by ear, so
> that the music sounds this way rather than in a different way.
> This cannot be explained by simply referring to a technical
> method used in the composition (the "technique" may have been
> "I just selected the notes intuitively by ear").

Regular mapping doesn't help much here. And I'm skeptical
that any music theory can. Music is very closely related to
language, and I suspect that a music theory able to explain
such preferences and choices may approach the level of
difficulty of creating an AI to understand language.

> For example, I am currently interested in pitch structures that
> are rich in ambiguities and that can suggest multiple potential
> meanings. That preference leads me to avoid simple consonanses
> as uninteresting (because they usually can be easily reduced to
> a single fundamental, inducing the perception to make that
> reduction), although the presence of some consonances is
> required to suggest potential harmonic meanings. Such pitch
> structures should be suitable for tonal ("extended tonality")
> and atonal ("pantonal") music.

Have you seen this:

http://dkeenan.com/Music/NobleMediant.txt

?

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
[...]
> > It is inaudible as an "inversion". But inversely related sets
> > may (or may not, according to the case) sound similar in some
> > degree.
>
> Any transformation one can dream up may produce two similar
> sounding chords in some cases. That doesn't mean anything.
> If you're talking about something real, it'll always hold.

Using extreme registers may do a single pitch class set to sound different in different dispositions, using only octave "equivalence". Thus, "always hold" is also relative :-)

How about your comparison of those 4 chords? ?? ???

[...]
> > Still about the interaction between technique and material:
> > replace a tone-row for another in a dodecaphonic music by Berg
> > or Boulez, and then you will get real cacophony, because an
> > acidental replacement will cause pitch relations which are
> > inconsistent with the style and structure.
>
> Now THIS is something I would like to test. Can you do it
> easily? Then we could ask people their opinions. Maybe we
> could present them as "A" and "B" and ask people which they
> think is the original piece.

That would be really interesting. Boulez' music is very hard to analyse, but there are extensive (maybe I should write "obsessive") papers on some works. Remapping the series would be another complex task. But dodecaphonic music is simpler, and the experiment could be accomplished. Perhaps someone already have tried that.

Anyway, I have several issues concerning to the mapping:
-- How to deal with octave instances so that the overall pitch disposition may be held when possible in the modified version?
-- Regarding to series transpositions, specially when inverted, how to select the transposition level in a consistent way?
-- Should substitute tone row share structural features with the original series?
-- Furthermore, how to handle invariance and imbrication between series? (e.g. symmetry in Webern's Op. 27 and most of his pieces)
-- What kind of dodecaphonic music should be suitable for this experiment?

If we choose compositions that explore series resources in consistent and structured ways, there may be not simple ways to remap series instances.

But there are some simple dodecaphonic study pieces that uses a single repeated tone row, and simple transformations. Maybe Schoenberg's theme from Op.31 (Variations for orchestra) can be suitable (just the melody, with no accompaniment?).

>
> > However, as I wrote before: there are no successful approaches
> > to analyze atonal music, and set theory proved to be at least
> > a positive contribution anyway.
>
> Again, isn't this backwards? You speak as if atonal music was
> a flourishing natural genre before serialism / set theory came
> along to explain it. Hardly! Try set theory on one of the few
> 19th-century atonal pieces. No, it only works (if you consider
> it working at all) when it was used to create the composition.

No. As I understand it, pitch class set theory has been developed since around 1960. That is: about 10 years after Schoenberg's death, 40 years after he developed his 12-tone technique, and around 50 (or more) years after early atonal music.

I am with Allen Forte's "The Structure of Atonal Music", one of the most important books on the matter, published in 1973, opened at the References page. The referenced books/papers are dated: 1960, 61, 64, 65, 69, 62, 68, 70, 65 etc. -- and there one -- only one! -- book dated of 1932 (Edwin von der Nüll's "Moderne Harmonik").

Now, quoting from the Preface of "The Structure of Atonal Music":

<<<--- The inclusion of Stravinsky's name in the list above [a short list of major atonal works] suggests that atonal music was not the exclusive province of Schoenberg and his circle, and that is indeed the case. Many other gifted composers contributed to the repertory: Alexander Scriabin, Charles Ives, Carl Ruggles, Ferrucio Busoni, and Karol Szymanowski--to cite only the more familiar names.
The present study draws upon the music of many of the composers mentioned above. It does not, however, deal with 12-tone music, or with what might be described as paratonal music, or with more recent music which is rooted in the atonal tradition. [...]
--->>>

J.N.Straus' book is more recent (1990) and includes analyses 12-tone music.

[...]
> >From your description, I undertand that regular mapping can catch
> >such errors, allowing one to find suitable tunings for the chosen
> >harmonic material.
>
> We've developed techniques to find optimal tunings given the
> commas that 'vanish' in the score. We can also do things like
> remap 4:5:6 to any other consonant triad like 5:7:9. The
> results may not sound good, but they'll sound better than
> simply remapping chromatically to a scale containing 5:7:9
> chords. And Erv Wilson and Marcus Hobbs have worked out some
> other very interesting transformations...

Interesting. Can you point me introductory references, please?

>
> >(I find that Neo-old(19) sounds mistuned,
> >although I am not sure that other tunings can maintain the
> >intended relations, harmonic ambiguities, etc.
>
> Neo-old is really a miniature masterpiece, in that it uses
> 19 organically; it is not just triadic music that uses the
> better tuning of triads in 19 (as some of Easley Blackwood's
> music in 19 does).

I think 12-EDO triads are better (its thirds may be not very good, but its fifths are well approximated enough as to provide stability to the chords).

I feel 19-EDO tempers 2:3 too much, and many basic relations are tempered downwards. Thus, I hear resulting tones ("fundamentals") as higher than their corresponding tones in the scale. I always have found that the last F (tonic) in the higher voice of Neo-old(19) sounds "too high".

>
> >I have made a few tries with Secor's 19-WT, but I only tested
> >two keys. An adaptative temperament would be welcome, and I
> >implicitely assume that one should be used if the piece were
> >played in acoustical instruments.)
>
> Perhaps I will get a chance to take a closer look at the score.

That would be nice!

[...]

> Have you seen this:
>
> http://dkeenan.com/Music/NobleMediant.txt
>
> ?

I have found that 17-EDO (and 17-WT) can be suitable for atonal music.

Thanks.
Hudson

Two interesting texts that address the issue of the perception of similarity between pitch-class sets:

Ian Cross
Cognitive science and music analaysis - the case of musical pitch
http://www.mus.cam.ac.uk/~ic108/PDF/CogSciMus.pdf

Tuire Kuusi
Set-Class and Chord: Examining Connection between Theoretical Resemblance and Perceived Closeness
http://ethesis.siba.fi/ethesis/files/isbn9525531007.pdf

Petra,
I am real curious how it is useful to you to know the closeness of a ratio to an epimoric one. Have you made some music that can be heard on the web.

Could you say more about why this may be important :
It also makes it possible to better imagine an approximate amount of "mistuning" of two tones, because it tells you which frequencies you have to mix to get a 1Hz difference.

My point of reference is learnign to use a Just 5 limit scale and the relationships between the notes. I am trying to deeply understand consonance (as defined in Indian and Greek music) and how to return there from all manner of 'interesting dissonances'. I find the JI scale is made of 5 limit epimoric ratios: 9/8, 10/9, 16/15, 3/2 etc.

I find it interesting that you say 5/3 is between 4/2 and 6/4 (2 & 3/2) yet 7/4 is also 2 and 3/2. Are there different ways of looking at between?

re:
The important thing to remember here is that negative numbers don't represent the same intervals as positive numbers do. For example, while the ED for a rising 5-limit major third is 4, the ED for a falling major third is -5 because the denominator of the ratio is supposed to be always 1 less than the numerator (4/5 is the same as "(-4)/(-5)") .
 
I don't quite get this yet. Could you say more to someone who has just learnt to multiply and divide fractions?
 
Ta
Steve

________________________________
From: Petr Pařízek <p.parizek@...>
To: tuning@yahoogroups.com
Sent: Thu, 4 February, 2010 8:19:02 PM
Subject: Re: [tuning] Epimoric Ratios

 

I would like to add something to what's been said so far.
Sometimes it's useful to know how close a ratio is to an epimoric one. Back in 2002, I began to use a unit which made this possible to me. I called it the "epimoric denominator" or ED. You can get it by "1/(f-1)", where "f" is the linear factor. So an octave of 2/1 has an ED of 1, a fifth of 3/2 has an ED of 2, and a major sixth of 5/3 has an ED of 1.5 (you may also understand it by saying that 5/3 is between 4/2 and 6/4). It also makes it possible to better imagine an approximate amount of "mistuning" of two tones, because it tells you which frequencies you have to mix to get a 1Hz difference. For example, the syntonic comma of 81/80 has an ED of 80 while the Pyth. comma of (3^12)/(2^19) has an ED of ~73.296, which means that it's larger than the syntonic comma. On the other hand, the 5-limit schisma of 32805/32768 has an ED of ~885.622, which can tell you that this must be something pretty small. Similarly, the ED for the ordinary 100-cent semitone is
~16.817, which means that it's closer to 18/17 than to 17/16.
The important thing to remember here is that negative numbers don't represent the same intervals as positive numbers do. For example, while the ED for a rising 5-limit major third is 4, the ED for a falling major third is -5 because the denominator of the ratio is supposed to be always 1 less than the numerator (4/5 is the same as "(-4)/(-5)") .
Petr
 

__________________________________________________________________________________
Yahoo!7: Catch-up on your favourite Channel 7 TV shows easily, legally, and for free at PLUS7. www.tv.yahoo.com.au/plus7

>"I find it interesting that you say 5/3 is between 4/2 and 6/4 (2
& 3/2) yet 7/4 is also 2 and 3/2. Are there different ways of
looking at between?"

Hmm....if you makes 3/2 5/3 4/2 into a chord you get 9/6 10/6 12/6 AKA the chord 9:10:12.
Meanwhile 3/2 7/4 2/1 becomes 6/4 7/4 8/4 AKA the chord 6:7:8.

So far as periodicity you'd think 6:7:8 would be the more consonant chord as it's closer to the root.
It turns out.......the closest ratio in that chord is 8/7, while the closest in the other is 10/9...so again the 6:7:8 chord wins out.

However, think about the 6:7:8 chord and a 14:17:20. chord.
Of course the 6:7:8 chord has the periodicity advantage, then again 8/7 is a closer (and rougher) ratio than 20:17...so it's debatable which one is more consonant (it could well vary by person).

To me it seems obvious that at some point personal preferences comes in and decides which of the two consonance theories (roughness or periodicity) is more valuable for a given listener. So while having bad roughness and/or periodicity may guarantee dissonance, the difference between ok and good consonance, IMVHO, is very much up to debate and, for some people who value roughness more...can be achieved with non-Epimoric ratios that simply are "around the same tonal area" as the Epimoric ones.

Far as the whole "Epimoric denominator" business...if you could clarify it would be nice b/c on the surface it just sounds to me like swapping Epimoric intervals from major to minor (reversed) order.

I will say this much about adhering to Epimoric intervals though: if you want to get many many chords possible within a scale (as opposed to just a few "perfect" ones) deviating from perfect ratios is crucial.

It follows that there has to be a range at which you can deviate without losing too much periodicity and I would say about an 8th tone AKA 12 cents is where it begins to really turn sour. Coincidentally 12TET has its worst interval "only" about 13 cents off (from) a series of perfect Epimoric "major" intervals in a series...go figure (oddly enough, many people can't hear a noticeable difference between 5-limit JI and 12TET).
A main issue I have with 12TET though, is the high-roughness minor second that becomes too rough to be used in chords (despite being highly close to a perfect Epimoric ratio): it's about 17/16 apart while anything closer than 13/12 apart sounds too rough to be useful in chords to my ears...so musicians have less chords and emotional variations at their disposal.

Interesting stuff but, IMVHO, again I worry too many people care so much about periodicity that they throw the idea of roughness enough out the window they end up making scales that, with just a bit of roughness tweaking, could yield many many more chords while keeping the existing ones in still quite good shape periodicity-wise.

Thanks deeply Michael,
I am pretty new to this. It jsut happens that I got interested in Archytas and making scales by harmonic means. So I will have to go and play some 14:17:20 to hear what your talking about. And study your wonderful email some more.

I am real interested in the 'sweet spots of consonance' and deviating from them and returning to them.

Could  you please say more about the roughness and periodicity theories of consonance as these ideas are new to me, being a bit lean on theory, but it sounds highly relevant to my interests.

Steve
 
 

________________________________
From: Michael <djtrancendance@...>
To: tuning@yahoogroups.com
Sent: Fri, 5 February, 2010 2:19:56 PM
Subject: Re: [tuning] Epimoric Ratios

 
>"I find it interesting that you say 5/3 is between 4/2 and 6/4 (2 & 3/2) yet 7/4 is also 2 and 3/2. Are there different ways of looking at between?"

Hmm....if you makes 3/2 5/3 4/2 into a chord you get 9/6 10/6 12/6  AKA the chord 9:10:12.
Meanwhile 3/2 7/4 2/1 becomes 6/4 7/4 8/4 AKA the chord 6:7:8.

   So far as periodicity you'd think 6:7:8 would be the more consonant chord as it's closer to the root.
It turns out.......the closest ratio in that chord is 8/7, while the closest in the other is 10/9...so again the 6:7:8 chord wins out.

   However, think about the 6:7:8 chord and a 14:17:20. chord.
  Of course the 6:7:8 chord has the periodicity advantage, then again 8/7 is a closer (and rougher) ratio than 20:17...so it's debatable which one is more consonant (it could well vary by person).

    To me it seems obvious that at some point personal preferences comes in and decides which of the two consonance theories (roughness or periodicity) is more valuable for a given listener.  So while having bad roughness and/or periodicity may guarantee dissonance, the difference between ok and good consonance, IMVHO, is very much up to debate and, for some people who value roughness more...can be achieved with non-Epimoric ratios that simply are "around the same tonal area" as the Epimoric ones.

   Far as the whole "Epimoric denominator" business...if you could clarify it would be nice b/c on the surface it just sounds to me like swapping Epimoric intervals from major to minor (reversed) order.

  I will say this much about adhering to Epimoric intervals though: if you want to get many many chords possible within a scale (as opposed to just a few "perfect" ones) deviating from perfect ratios is crucial.

  It follows that there has to be a range at which you can deviate without losing too much periodicity and I would say about an 8th tone AKA 12 cents is where it begins to really turn sour.  Coincidentally 12TET has its worst interval "only" about 13 cents off (from) a series of perfect Epimoric "major" intervals in a series...go figure (oddly enough, many people can't hear a noticeable difference between 5-limit JI and 12TET).
   A main issue I have with 12TET though, is the high-roughness minor second that becomes too rough to be used in chords (despite being highly close to a perfect Epimoric ratio): it's about 17/16 apart while anything closer than 13/12 apart sounds too rough to be useful in chords to my ears...so musicians have less chords and emotional variations at their disposal. 

    Interesting stuff but, IMVHO, again I worry too many people care so much about periodicity that they throw the idea of roughness enough out the window they end up making scales that, with just a bit of roughness tweaking, could yield many many more chords while keeping the existing ones in still quite good shape periodicity- wise.

Thanks

__________________________________________________________________________________
Yahoo!7: Catch-up on your favourite Channel 7 TV shows easily, legally, and for free at PLUS7. www.tv.yahoo.com.au/plus7

--- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@...> wrote:

> > Any transformation one can dream up may produce two similar
> > sounding chords in some cases. That doesn't mean anything.
> > If you're talking about something real, it'll always hold.
>
> Using extreme registers may do a single pitch class set to sound
> different in different dispositions, using only octave
> "equivalence". Thus, "always hold" is also relative :-)

I don't think so. Octave equivalence is hardly absolute,
but it always works however well it does. And it scales like
physical phenomena -- a jump of 2 octavse sounds less similar
than a jump of single octave. The failure of inversion is
of a different kind.

> How about your comparison of those 4 chords? ?? ???

I'll play around with them when I get some audio privacy,
maybe tomorrow. But they appear to be all roughly equally
dissimilar from one another.

> > > Still about the interaction between technique and material:
> > > replace a tone-row for another in a dodecaphonic music by
> > > Berg or Boulez, and then you will get real cacophony, because
> > > an acidental replacement will cause pitch relations which are
> > > inconsistent with the style and structure.
> >
> > Now THIS is something I would like to test. Can you do it
> > easily? Then we could ask people their opinions. Maybe we
> > could present them as "A" and "B" and ask people which they
> > think is the original piece.
>
> That would be really interesting. Boulez' music is very hard
> to analyse, but there are extensive (maybe I should write
> "obsessive") papers on some works. Remapping the series would be
> another complex task. But dodecaphonic music is simpler, and the
> experiment could be accomplished. Perhaps someone already have
> tried that.

Very well then, dodecaphonic it is. Following the tradition
of guerrilla theory here, I think we'll do best by doing it
ourselves. And by we I mean you :) since I don't know enough
about dodecaphonic music to help...

> Anyway, I have several issues concerning to the mapping:
> -- How to deal with octave instances so that the overall pitch
> disposition may be held when possible in the modified version?
> -- Regarding to series transpositions, specially when inverted,
> how to select the transposition level in a consistent way?
> -- Should substitute tone row share structural features with the
> original series?
> -- Furthermore, how to handle invariance and imbrication between
> series? (e.g. symmetry in Webern's Op. 27 and most of his pieces)
> -- What kind of dodecaphonic music should be suitable for this
> experiment?

...see what I mean?

> But there are some simple dodecaphonic study pieces that uses
> a single repeated tone row, and simple transformations. Maybe
> Schoenberg's theme from Op.31 (Variations for orchestra) can be
> suitable (just the melody, with no accompaniment?).

Yes, simple is good. But how would we produce cacophony, as
you claim, without the accompaniment?

> > Again, isn't this backwards? You speak as if atonal music was
> > a flourishing natural genre before serialism / set theory came
> > along to explain it. Hardly! Try set theory on one of the few
> > 19th-century atonal pieces. No, it only works (if you consider
> > it working at all) when it was used to create the composition.
>
> No. As I understand it, pitch class set theory has been developed
> since around 1960. That is: about 10 years after Schoenberg's
> death, 40 years after he developed his 12-tone technique, and
> around 50 (or more) years after early atonal music.

I said "serialism / set theory". And categorizing the likes
of Scriabin, Busoni, Stravinksy and Ives into a genre including
Schoenberg and the 2nd Viennese school is a revisionist fantasy,
at best.

> I am with Allen Forte's "The Structure of Atonal Music", one
> of the most important books on the matter, published in 1973,

I suppose I should read this, since everything seems to depend
on it. I've placed it in my Amazon cart. Actually the cover
looks familiar - I may have a copy in storage somewhere. I've
been told to read it on several occasions. As you can probably
guess, I'm doubtful it explains much of anything, let alone the
structure of atonal music.

> > We've developed techniques to find optimal tunings given the
> > commas that 'vanish' in the score. We can also do things like
> > remap 4:5:6 to any other consonant triad like 5:7:9. The
> > results may not sound good, but they'll sound better than
> > simply remapping chromatically to a scale containing 5:7:9
> > chords. And Erv Wilson and Marcus Hobbs have worked out some
> > other very interesting transformations...
>
> Interesting. Can you point me introductory references, please?

I don't know if these are introductory, but you can try

/tuning-math/message/4146
/tuning-math/message/4147
/tuning-math/message/4151

> I think 12-EDO triads are better (its thirds may be not very
> good, but its fifths are well approximated enough as to provide
> stability to the chords).
> I feel 19-EDO tempers 2:3 too much, and many basic relations
> are tempered downwards.

In isolation, 19-ET triads are slightly more consonant than
12-ET triads. They have lower total error, but also lower
TOP damage, which weights the errors on the 5ths more than
those on the 3rds.

The direction of the errors is an interesting point, because
even weighted measures like TOP damage don't take this into
account. Harmonic entropy does, however, and again 19-ET is
better:

13.1644514 JI
13.2392224 12-ET
13.2333286 19-ET

But the improvement is small and the errors are qualitatively
different than 12-ET, and most listeners are a bit numb to the
sound of 12's errors but not yet to the sound of 19's.
Nonetheless with repeated listening I find 19-ET triads
slightly smoother.

-Carl

--- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@...> wrote:
>
> Two interesting texts that address the issue of the perception
> of similarity between pitch-class sets:
>
> Ian Cross
> Cognitive science and music analaysis - the case of musical pitch
> http://www.mus.cam.ac.uk/~ic108/PDF/CogSciMus.pdf
>
> Tuire Kuusi
> Set-Class and Chord: Examining Connection between Theoretical
> Resemblance and Perceived Closeness
> http://ethesis.siba.fi/ethesis/files/isbn9525531007.pdf

pp 33-38 (in the PDF) are perfectly consistent with what
I've said. The original results of the paper are based on MDS,
which is easy to mess up and I don't have the time to review it
carefully.

Well found though; thanks.

-Carl

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> What I'd like to add to give what I think is the big picture.
>
> 5-limit harmonic permutation structure will give a "basis" structure of
> mathematical possibilities in the most "consonant / low ratio" manner
> without comma shifts.
> Harmonic permutation structure is "U-tonal" in a way (though symetrical in
> the harmonic series, not symetrical in the octave as is usually also ment by
> u-tonal), meaning it doesn't do GCD etc, all harmonic permutation "chords"
> have their "mirror" on any of it's "notes".
> Harmonic permutation structure is very limited, it only has very limited
> "chords". I tend to not even see them as chords but as structures.
> In actual music one can combine any of the 12 notes together. Any of those
> notes can belong to many structures.
> When combining notes belonging to different harmonic permutation "chords"
> this must happen within the harmonic permutation set of all permutations of
> the harmonic series (limit 5).
> This is where most of the music happens. And this is where I see GCD and
> tonica come into play.
>
> So I see music as sort of "u-tonal" up till a certain point, and very tonal
> beyond that.
>
> Marcel

Yes, and I am sure you have seen such sundry things as the
Stellated Hexanies and other structures on this site. My goal
is to tie such structures to group theory, which is done in
pure mathematics with such things as the Leech Lattice, but
unfortunately those structures don't have much direct application
to music (that I can see at this time). I work extensively
with the Mathieu groups M12, (M22) and M24 though.

Lately, I have found that 10 septachord/pentachords (with one tritone in the pentachord) can easily canvas (or cover) all 70 hexachord-partitions with one or two tritones, and just a few more (3 with no tritone, and 3 with two) get the rest - those with zero or three.

Tying this into meaningful sets of fractions is a little harder,
I might explore all possible cases (there are several) and find
the best set of fractions to use. These trios are in the middle zone
as I stated before --

1) 3/5, 5/1, 1/3
2) 9/1, 9/5, 1/3
3) 9/1, 3/5, 1/3
4) 1/5, 3/5, 5/1
5) 3/1, 5/1, 1/3
6)1/15, 5/3, 1/3
7)1/15, 3/5, 5/1
8) 3/1, 9/1, 1/3
9) 1/15, 1/5, 5/1
10) 1/15, 9/1, 3/5

PGH

>
> As for my tonality model.
> > There isn't much to it.
> > Unless one modulates (and the harmonic permutations will tell you what a
> > modulation is) on is in the 12tone subset we talk about above.
> > A tonic is simply a tonal center by choice.
> > The harmonic permutation set gives 1/1 135/128 9/8 75/64 5/4 675/512 45/32
> > 3/2 25/16 27/16 225/128 15/8 2/1 seen from the GCD.
> > As many chords will point with their GCD to the above 1/1, for instance a
> > dominant 7th, I've found that what is refered to as major mode in normal
> > music theory corresponds to a tonic of 3/2 from the GCD. Minor to a tonic
> > 5/4 from the GCD, etc.
> > It seems normal music theory gets the tonic right a lot of the time
> > allowing easy transcription to JI.
> > Sometimes not though, but I'm getting better at finding the real tonics and
> > modulations of a piece.
> > I'm bussy making a proper website where I'll show all my experiments and
> > thoughts on these things soon.
> >
> >
> >
>

Steven wrote:

> I am real curious how it is useful to you to know
> the closeness of a ratio to an epimoric one.

Epimoric ratios of low numbers are easily recognizable by ear for two reasons. A) The GCD frequency (which in this case is the same as the difference frequency) is clearly audible. B) There is no beating in these intervals because they have a steady „guide tone“ (which equals to the LCM frequency). Some people might argue that they find the sound of 5/3 as „clear“ as 5/4 or 4/3 but, at least from my subjective listening tests, I don’t. So if I describe an interval using the unit I call ED, I can quickly get a rough mental image of how „tuned“ or „mistuned“ an interval may sound to me. If the ED is an integer, that means the ratio is epimoric.

> Have you made some music that can be heard on the web.

Well, concerning my great interest in epimoric intervals back in 2002, I made something called „In the epimoric world“. At first, it was just a name for a 12-tone scale. Later, it was actually a name for a piece of music. If you have a copy of Manuel’s „Scale archive“, you can find the 12-tone scale there under the name of „parizek_epiworld.scl“. As to the music, maybe I could upload it and let you hear it if you’re interested.

> My point of reference is learnign to use a Just 5 limit scale and the relationships between the notes.
> I am trying to deeply understand consonance (as defined in Indian and Greek music)
> and how to return there from all manner of 'interesting dissonances'. I find the JI scale is
> made of 5 limit epimoric ratios: 9/8, 10/9, 16/15, 3/2 etc.

There are loads of 5-limit JI scales. Do you mean one scale in particular?

> I find it interesting that you say 5/3 is between 4/2 and 6/4 (2 & 3/2)
> yet 7/4 is also 2 and 3/2. Are there different ways of looking at between?

Again, there are many possible answers to what „between“ means. What you’ve done is the „arithmetic mean“ like (a+b)/2. Another way is the „geometric mean“ like sqrt(a*b). And another is the one which I don’t know by its proper name but I call it the „partial mean“. It goes like (2*a*b-a-b) / (a+b-2). That’s the one that comes out when you make an „arithmetic mean“ of two EDs.

> I don't quite get this yet. Could you say more to someone who has just learnt
> to multiply and divide fractions?

Well, imagine a unit that describes intervals as the lower of the two relative frequencies, leaving a constant difference of 1 between them. In the case of epimoric intervals, this equals to the denominator. So in the case of 2/1, the ED is 1 because you get an octave by mixing 1Hz and 2Hz. For 3/2 the ED is 2 because you get a perfect fifth by mixing 2Hz and 3Hz (the difference of 1 again). Now imagine something like 5/3. This is basically the same as 2.5/1.5 and therefore the ED is 1.5 (the difference of 1 is preserved). And now let’s look at a falling perfect fifth of 2/3. The rule for epimoric ratios says that the denominator is 1 lower than the numerator. But since 3 is higher than 2 and not lower, we can’t use this as our starting factor for the ED conversion. Because „minus times minus is plus“, we can substitue our 2/3 with (-2)/(-3) so that the denominator really is 1 lower than the numerator. But, you see, since the 2 and 3 are now swapped, the ED value can’t be the same. So while an ED for 3/2 is 2, the ED for 2/3 is -3.

Petr

Petr>"B) There is no beating in these intervals because they have a
steady „guide tone“ (which equals to the LCM frequency)."

Correct me if I'm wrong, but it seems obvious to me there IS beating, it just happens to beat along the LCM (thus it beats in the form of "periodic distortion", but still beats).

>"we can substitue our 2/3 with (-2)/(-3) so
that the denominator really is 1 lower than the numerator. But, you see, since
the 2 and 3 are now swapped, the ED value can’t be the same. So while an ED for
3/2 is 2, the ED for 2/3 is -3."

Ah so the same ratio in the reverse direction actually can increase the ED...is it always the case that is does increase the absolute value of the ED (it also appears 4/5 would ED at -5 (-4/-5) )? And would it be fair to say that hearing a note and then a note 4/5 lower would sound more dissonant than a note and then a note 5/4 higher?
If true, it would appear to imply that ascending progressions are more melodic than descending ones with the same intervals covered in reverse order.

_,_._,__

Michael wrote:

> Correct me if I'm wrong, but it seems obvious to me there IS beating, it just happens to beat along
> the LCM (thus it beats in the form of "periodic distortion", but still beats).

If you play two tones (rich in overtones) of 50Hz and 60Hz, the LCM of 300Hz is locked in phase; that's why there’s no beating at that frequency. OTOH, if you play 51Hz and 61Hz, then the similar harmonics of 306Hz and 305Hz cause a new kind of interference which might be approximately simulated, for example, by amplitude-modulating (i.e. synchronous multipliing) a sine of 0.5Hz with a sawtooth of 305.5Hz.

> Ah so the same ratio in the reverse direction actually can increase the ED...is it always
> the case that is does increase the absolute value of the ED (it also appears
> 4/5 would ED at -5 (-4/-5) )?

Yes, it does. Suppose you have two tones which keep falling linearly at a constant speed and are 1Hz apart. At the moment where one of them falls to 0, the higher one is at 1, which means that the ED of 0 is essentially nonsense because it represents a ratio of 1/0. If the tones still go on changing the pitches, then later one is at -0.5Hz and the other is at +0.5Hz, which would be expressed as -0.5 in ED. Even later, the „lower“ frequency would reach -1 and the upper one would get to 0, which means that an ED of -1 means a ratio of „0/(-1)“ or 0/1, which is more or less „silence“ rather than a relative pitch. And even later, the „lower“ frequency gets to -2 and the „upper“ one gets to -1, which means that an ED of -2 means a ratio of „(-1)/(-2)“ or 1/2. And as 2/1 has an ED of 1, this may be one possible demonstrative explanation.

> And would it be fair to say that hearing a note and then a note 4/5 lower
> would sound more dissonant than a note and then a note 5/4 higher?

Whoops .. May I ask what’s lead you to this idea?

Petr

>"If
you play two tones (rich in overtones) of 50Hz and 60Hz, the LCM of 300Hz is
locked in phase; that's why there’s no beating at that frequency."
Hmm...ok. I do still hear beating (tested with a piano sound) but, indeed, it is very quick beating...quick to the point it's not easily audible. You can see the beating if you load up the waveform. Then again even if you do the same test with bare sine waves...there's very little beating simple because the tones are so far apart.

Here's a question, what happens if you do the same thing with the much smaller ratio 16/15 IE 160hz and 150hz...is there still no "aubible" beating and, if so, why not? Judging by your example, I'd guess the LCM would be some very high frequency.

>
And would it be fair to say that hearing a note and then a note 4/5
lower
> would sound more dissonant than a note and then a note 5/4
higher?
> Whoops
.. May I ask what’s lead you to this idea?
I'm trying to relate the concept of melodically switching between two tones to, say, your "falling tone" example IE "Suppose you have two tones which keep falling linearly at a constant
speed and are 1Hz apart."....

Michael wrote:

> Hmm...ok. I do still hear beating (tested with a piano sound) but, indeed, it is
> very quick beating...quick to the point it's not easily audible. You can see the beating if you
> load up the waveform. Then again even if you do the same test with bare sine waves...there's
> very little beating simple because the tones are so far apart.

#1. You’re talking about the general periodicity of the dyad, which is, as far as what I’ve read, not called beating. This has a lot to do with the GCD frequency, while the „beating“ I meant has a lot to do with the LCM. If you mix two sawtooth periods of 40Hz and 70Hz, you get another resulting period whose „general“ frequency is 10Hz (I know that saying „general“ is a bit strange but I want to stress the fact that this frequency of 10Hz would also be there whether you added 100Hz or 110Hz or 90Hz or all of these or none of these to that sound). It is actually the GCD of the sounding frequencies and most papers I’ve read call this the „fundamental frequency“ of the dyad. OTOH, the „guide tone“, which equals to the LCM of the sounding frequencies, is 280Hz in this case. Unless one of the two periods changes its frequency, the guide tone is locked in phase. This means that if you start with both periods at a common phase, then the guide tone comes out as loud as possible, including all other integer multiples. In contrast, if one of the two tones starts at 0 degrees while the other starts at 180, the guide tone and its multiples never appear.

#2. I was wrong when I said that the beating in the harmonics of 51Hz and 61Hz could be simulated with amplitude modulation, as this is only true for sine waves and therefore would only count for the actual dyad of 305Hz and 306Hz but not for its overtones. I should have said „by mixing two frequencies of 305Hz and 306Hz“. Anyway, talking of harmonic spectra like sawtooth and such, the process might really be simulated by amplitude modulation but in a very difficult way -- i.e. by modulating a sine of 305.5Hz with a sine of 0.5Hz, then modulating a sine of 611Hz with a sine of 1Hz, and so on, and mixing all of these individual modulated results.

> Here's a question, what happens if you do the same thing with the much smaller ratio 16/15
> IE 160hz and 150hz...is there still no "aubible" beating and, if so, why not?
> Judging by your example, I'd guess the LCM would be some very high frequency.

Of course, the LCM is 2400Hz in this case. It’s important to know where in the spectrum you want to look for the beats. Surely there’s no beating at 2400Hz in this example if we don’t change one of the pitches. But the GCD of 10Hz is audible as a new kind of periodicity often referred to as the fundamental frequency. However, this has nothing to do with the phenomenon of beating that helps you in tuning tempered intervals, for example.

> I'm trying to relate the concept of melodically switching between two tones to, say,
> your "falling tone" example IE "Suppose you have two tones which keep falling linearly
> at a constant speed and are 1Hz apart."....

I’m not sure if there’s much I could say about this.

Petr

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@> wrote:
>
> > > Any transformation one can dream up may produce two similar
> > > sounding chords in some cases. That doesn't mean anything.
> > > If you're talking about something real, it'll always hold.
> >
> > Using extreme registers may do a single pitch class set to sound
> > different in different dispositions, using only octave
> > "equivalence". Thus, "always hold" is also relative :-)
>
> I don't think so. Octave equivalence is hardly absolute,
> but it always works however well it does. And it scales like
> physical phenomena -- a jump of 2 octavse sounds less similar
> than a jump of single octave. The failure of inversion is
> of a different kind.

Some points:

* If you take your chord 4:5:6:7:9:11:13 and permutate the pitches in the pitch range, you can easily get sonorities that does not resemble too much the original disposition, just moving the pitches across octaves. For instance, let 11 or 13 be the bass note, hide the fundamental (4) inside the chord near 7 or 9, put 6 as a semitone of 11 or 13, etc. If you temper the chord, then the similarity of the several dispositions will vary according to the temperament. I think that one can get a (dis)similarity degree not too distant of some dispositions of the corresponding utonal inverse chord.

* Similarity relations between pitch sets are difficult to define and classify. Several year ago I was tempted to explore such possibilities (considering pitch range, register, disposition, not reducing pitches to pitch classes, etc.), and gave up because the untractable large number of possible combinations. If you reduce the number of objects to analyse (e.g. consonant chords built on 5-limit intervals), you can do a refined job; otherwise, as in atonal music, drastical reduction may be justified, given the large amount, complexity and variety of the pitch combinations which are used. "Set theory" is not generally about similarity, but essentially about structural (intervalar) properties of sets, as they are supposed to be used in atonal composition, or as they are found in analysis.

* I think "similarity" is a bad word to describe the relation between inverse pc sets, as it suggests immediate association at listening. Even "equivalence", which is a proper technical name (once the mapping function is defined), can be interpreted in an exaggerated or misled way. I use to think "equivalence under inversion" and other pc set relations as quantitative features which represent _potential_ capabilities or resources of a set or some sets. I mean: you only can write a rigorous mirror using inversely related sets, and you only can have an exact transposition of a chord using transposition related sets, by definition; you can use the sets (in music) to make those relations explicit or not, and you can show them partially; but if the sets are not equivalent, then you only can obtain (at most) partial correspondences.

* If the universe is 12-EDO, according to Forte, _every_ pc set with n=5 or n=7 elements can hold a subset with n-1 elements invariant under inversion, except some sets which are completely invariant under inversion (mapping themselves). In those cases, at least, the similarity between inversion related sets can be strongly represented (e.g. by presenting a high number of common tones between two chords).

* The interval vector resumes important relations (in special the number of common tones between a set and its transposition to every interval class), which are shared by sets with equal interval vectors (transpositions, inversions ans "Z-related" sets); whether these sets can or cannot be presented so that they sound "similar" is another issue: basically they share a common quantitative feature, from which can be extracted potential usages and other conclusions.
For example, Forte have found that _any_ 5-tone subsets of 12-EDO contain at least one M3 (or m6) -- therefore, a major third as subset is of null relevance as distinctive feature of a given pentachord; you need at least 2 M3 to make a difference. (You can select any 5 different pitches in a single octave in a completely random way, play them as a chord (or cluster) and then hold only two pitches in a M3/m6. Or, conversely, you can hold a fixed M3 and superpose to it a succession of different 3-tone chords, eventually presenting every possible 5-tone chord type with no repetition.)
If the interval vector of a set shows a lot of M2s, or a lot of P4s, one can have some idea of how it may sound, etc. Just don't ask the interval vector more than it can answer, and you can find it useful. (You can surely imagine: there is a lot of delirious "extensions" and "applications" to this "set theory" around) :-)

* To be clear: I don't consider "pitch-class set theory" as a "theory"; of course it cannot be called a "music theory" (there is no the "music theory" thing). Its "name" (which I think is not defined anywhere) is just a way to relate concepts and applications, namely: apply _set theory_ concepts to analyse _pitch_ relations in atonal _music_. It only can classify objects (pitch sets) in a manageable way after using several equivalence reductions (essentially: octave and "enharmony", for pitches, and then transposition, for sets); since atonal music uses a lot of the possible pitch combinations available in 12-EDO, additional reductions are required so that sets can be groupped: then one can evoke inversion equivalence, similarity relations, etc. Procedures are used to analyse combinatorial capabilities of 12-EDO, and set properties, etc. and to represent the results in a compact way. The whole collection of concepts and operative tools is applied in atonal music analysis, mainly for statistical analysis. The _musical_ analysis is to be made upon the raised and tabulated data, checking the real musical relevance of the outputs.

[...]
> > That would be really interesting. Boulez' music is very hard
> > to analyse, but there are extensive (maybe I should write
> > "obsessive") papers on some works. Remapping the series would be
> > another complex task. But dodecaphonic music is simpler, and the
> > experiment could be accomplished. Perhaps someone already have
> > tried that.
>
> Very well then, dodecaphonic it is. Following the tradition
> of guerrilla theory here, I think we'll do best by doing it
> ourselves. And by we I mean you :) since I don't know enough
> about dodecaphonic music to help...

:-)

>
> > Anyway, I have several issues concerning to the mapping:
> > -- How to deal with octave instances so that the overall pitch
> > disposition may be held when possible in the modified version?
> > -- Regarding to series transpositions, specially when inverted,
> > how to select the transposition level in a consistent way?
> > -- Should substitute tone row share structural features with the
> > original series?
> > -- Furthermore, how to handle invariance and imbrication between
> > series? (e.g. symmetry in Webern's Op. 27 and most of his pieces)
> > -- What kind of dodecaphonic music should be suitable for this
> > experiment?
>
> ...see what I mean?

A sophisticated form of experiment design would try contemplate several possibilities.

>
> > But there are some simple dodecaphonic study pieces that uses
> > a single repeated tone row, and simple transformations. Maybe
> > Schoenberg's theme from Op.31 (Variations for orchestra) can be
> > suitable (just the melody, with no accompaniment?).
>
> Yes, simple is good. But how would we produce cacophony, as
> you claim, without the accompaniment?

By presenting inconsistent (ilogical) pitch relations, when replacing the original series with another in an arbitrary way. In special, tonal relations may appear in some moments, with no logical consequences, sounding accidental, wrong, without defined style or language. Usually, in atonal music, tonal allusions are carefully controled, for a harmonically balanced result.

But the theme of Op.31 has a chordal accompaniment that I think it is simple -- however, the orchestration is crucial to suavize dissonances; I am not sure a MIDI file can obtain the delicate result.

AFAICR, the theme uses tone rows that complement the melody tones forming aggregates (12-tone sets) -- that is, 6 theme notes are harmonized by the remaining 6 notes, but from another series instance. This simple procedure was target of a harsh criticism by Pierre Boulez. However, it is impressive the way that Schoenberg explores the 12-tone series to obtain sets as diverse as the dim7 chord (with resolution on a P5!) which opens the work, or the BACH sequence (presented at the 2nd part of the introduction, and explored at the finale), for example. The series does not contain either of these sets: they are obtained by combining different serial forms (O,R,I,RI and transpositions).

http://www.youtube.com/watch?v=u6BzLwHLKis&feature=related
http://www.youtube.com/watch?v=8GymJUFFwlI&feature=related

>
> > > Again, isn't this backwards? You speak as if atonal music was
> > > a flourishing natural genre before serialism / set theory came
> > > along to explain it. Hardly! Try set theory on one of the few
> > > 19th-century atonal pieces. No, it only works (if you consider
> > > it working at all) when it was used to create the composition.
> >
> > No. As I understand it, pitch class set theory has been developed
> > since around 1960. That is: about 10 years after Schoenberg's
> > death, 40 years after he developed his 12-tone technique, and
> > around 50 (or more) years after early atonal music.
>
> I said "serialism / set theory".

Then there is some communication noise here, I could not understand you yet.

Atonal music predates serialism, and atonal music predates pitch-class set theory. Serialism is not a theory nor an analytic method, but a collection of procedures used in composition; it is not intended to explain atonal music and cannot do it. Furthermore, pitch-class set theory (a misname, BTW), which is related to 50s-60s serialists (Milton Babbitt defined several of its core concepts), introduces several concepts that did not exist before, and applies them in atonal music analysis.

> And categorizing the likes
> of Scriabin, Busoni, Stravinksy and Ives into a genre including
> Schoenberg and the 2nd Viennese school is a revisionist fantasy,
> at best.

Forte was not simply classifying every and whatever music by Scriabin or Stravisnky as "atonal". Music with non-functional harmony, using 12-tone scale and non-classified chords share characteristics with atonal music. The number of examples from Stravisnky's and Scriabin's pieces is higher than I could expect (but the large majority is from 2nd Viennese school, as one could expect).

>
> > I am with Allen Forte's "The Structure of Atonal Music", one
> > of the most important books on the matter, published in 1973,
>
> I suppose I should read this, since everything seems to depend
> on it. I've placed it in my Amazon cart. Actually the cover
> looks familiar - I may have a copy in storage somewhere. I've
> been told to read it on several occasions. As you can probably
> guess, I'm doubtful it explains much of anything, let alone the
> structure of atonal music.

"Introduction to Post-Tonal Theory" (Joseth N. Straus, 1990) is perhaps more interesting as an introductory text, because it is didactic and more rich in musical examples. There are newer concepts not addressed by Forte's book, and interesting analysis of dodecaphonic pieces. However several concepts and data provided by Forte are not aborded.

Forte is much more abstract and essentially uses short musical examples to ilustrate the concepts. The second part of Forte's book contains larger examples (including reductions of a few complete pieces), addressing the presentation of the concept of "set complexes", going far from the musical surface. He avoids exaggerated "claims", but there are questionable concepts (for instance, his assumption that his similarity relation "R1" is closer than "R2"; I find it is just more restrictive). The musical examples (from Stravinsky, Ives, Scriabin, Schoenberg, Webern, Berg, Bartók, Busoni and others) indicate that there was extensible previous research and analysis of several works.

A more recent book I have about that is "Teoria analítica da música do século XX", by the portuguese composer João Pedro Paiva de Oliveira, published in 2007. Only the most directly relevant concepts are presented, as it is written by a composer point of view rather that by an analyst (analysts tend to abstraction and to overestimate secundary findings -- that may be just coincidences in most cases!). His analysises of dodecaphonic serieses and its applications in music by Schoenberg, Berg and Webern are very interesting.

>
> > > We've developed techniques to find optimal tunings given the
> > > commas that 'vanish' in the score. We can also do things like
> > > remap 4:5:6 to any other consonant triad like 5:7:9. The
> > > results may not sound good, but they'll sound better than
> > > simply remapping chromatically to a scale containing 5:7:9
> > > chords. And Erv Wilson and Marcus Hobbs have worked out some
> > > other very interesting transformations...
> >
> > Interesting. Can you point me introductory references, please?
>
> I don't know if these are introductory, but you can try
>
> /tuning-math/message/4146
> /tuning-math/message/4147
> /tuning-math/message/4151

Thanks.

[...]
> In isolation, 19-ET triads are slightly more consonant than
> 12-ET triads. They have lower total error, but also lower
> TOP damage, which weights the errors on the 5ths more than
> those on the 3rds. [...]

Indeed. It seems that the problem is the (un)stability of the scale, due to the tempering of the fifths, along with some lack of familiarity with the tuning. Anyway, if we are talking about inconsistent usage of a tuning by using simple and common triadic relations, the fact that 19 is not familiar for most people is relevant. A circular temperament with 12-tones (and with better triads than 12-EDO and 19-EDO) would work better -- at the suitable keys, of course.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@> wrote:
> >
> > Two interesting texts that address the issue of the perception
> > of similarity between pitch-class sets:
> >
> > Ian Cross
> > Cognitive science and music analaysis - the case of musical pitch
> > http://www.mus.cam.ac.uk/~ic108/PDF/CogSciMus.pdf
> >
> > Tuire Kuusi
> > Set-Class and Chord: Examining Connection between Theoretical
> > Resemblance and Perceived Closeness
> > http://ethesis.siba.fi/ethesis/files/isbn9525531007.pdf
>
> pp 33-38 (in the PDF) are perfectly consistent with what
> I've said. The original results of the paper are based on MDS,
> which is easy to mess up and I don't have the time to review it
> carefully.

In a quick overview, I also suspected that the results may be not well explained. I have restrictions also for the practical test design. It is interesting that it presents the definitions of many proposed similarity relations. The other paper also refers to several experiments, including one that concluded that musicians can compare interval content (of a pair of chords) when other comparation elements (such as common tones) are missing.

I have concluded that the main difficulty of any comparisons of pitch-class sets is that they try to compare abstract features -- these features are better undertood as representing _potential_ relations between actual sets. The qualitative correspondences of these features (in hearing level) only are evident when involving special sets -- thus it is important to compare every set-class with all other set-classes available in the universe.

Many sets do not have distinctive features for the perception -- but there are set theory operations which can find these special sets, and to evaluate how special they are. I think these can be useful tools to analyse pitch sets of microtonal scales (ETs).

For example, Forte's relation R0 (totally different interval vectors) could find special pentacords in 12-EDO which are dissimilar to each other, including these transitive triples:

5-2, 5-20, 5-33:
5-2: {0,1,2,3,5} <332110>
5-20: {0,1,3,7,8} <211231>
5-33: {0,2,4,6,8} <040402>

5-13, 5-23, 5-33:
5-13: {0,1,2,4,8} <221311>
5-23: {0,2,3,5,7} <132130>
5-33: {0,2,4,6,8} <040402>

Note that it is easy to associate, in each triple, the first set to "quasi-cluster" chromaticism, the second one to diatonism, and the last set to the whole-tone scale (6-EDO).

I find that sets showing a proportionally large number of some interval class are easier to group as similar or qualitatively related (for perception). (This strategy may fail with tunings which intervals are not familiar, but this assumption deserves to be tested; there may be at least a small number of exceptional sets.)

--- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@...> wrote:

> Some points:
>
> * If you take your chord 4:5:6:7:9:11:13 and permutate the
> pitches in the pitch range, you can easily get sonorities that
> does not resemble too much the original disposition, just
> moving the pitches across octaves.

Yes, absolutely. Which is why the canonical version of
regular mapping does little to privilege octaves. They
have harmonic distance log(2) vs. log(3) for perfect 12ths,
log(6) for a perfect 5th, and so on.

> * Similarity relations between pitch sets are difficult to
> define and classify. Several year ago I was tempted to explore
> such possibilities (considering pitch range, register,
> disposition, not reducing pitches to pitch classes, etc.), and
> gave up because the untractable large number of possible
> combinations. If you reduce the number of objects to analyse
> (e.g. consonant chords built on 5-limit intervals), you can do
> a refined job; otherwise, as in atonal music, drastical
> reduction may be justified, given the large amount, complexity
> and variety of the pitch combinations which are used.
> "Set theory" is not generally about similarity, but essentially
> about structural (intervalar) properties of sets, as they are
> supposed to be used in atonal composition, or as they are found
> in analysis.

It seems to me you reached the right conclusion in your
investigation, and that set theory has no answer for it.
Set theory uses and combines relations anyway. Again, here
is the distinction between set theory as a generative method
(which is fine) and an analytical one (which means it must
meet the standards of a scientific theory).

>you only can have an exact transposition of a chord using
>transposition related sets, by definition;

This definition is true but that does not mean anything else
about sets is relevant to music.

> * If the universe is 12-EDO, according to Forte, _every_
> pc set with n=5 or n=7 elements can hold a subset with n-1
> elements invariant under inversion,

Can you give an example?

> * The interval vector resumes important relations (in special
> the number of common tones between a set and its transposition
> to every interval class), which are shared by sets with equal
> interval vectors (transpositions, inversions ans "Z-related"
> sets); whether these sets can or cannot be presented so that
> they sound "similar" is another issue: basically they share a
> common quantitative feature, from which can be extracted
> potential usages and other conclusions.

This sounds generative.

> For example, Forte have found that _any_ 5-tone subsets of
> 12-EDO contain at least one M3 (or m6) -- therefore, a major
> third as subset is of null relevance as distinctive feature
> of a given pentachord; you need at least 2 M3 to make a
> difference.

If such an observation is to be useful in analysis, we must
assume something like 1. the music to be analyzed was generated
in 12-ET and 2. the composer is presenting us a pentachord, in
such a way that we may identify it from all possible pentachords
using facts about pentachords in 12-ET. In other words, it's
useful if the music was composed using set theory!

>... you can hold a fixed M3 and superpose to it a succession
>of different 3-tone chords, eventually presenting every
>possible 5-tone chord type with no repetition.)

Generative (though interesting!)

>If the interval vector of a set shows a lot of M2s, or a lot
>of P4s, one can have some idea of how it may sound, etc. Just
>don't ask the interval vector more than it can answer, and you
>can find it useful. (You can surely imagine: there is a lot of
>delirious "extensions" and "applications" to this "set theory"
>around) :-)

Yes, very many. We are talking about z-relations on interval
vectors a few paragraphs ago!

> * To be clear: I don't consider "pitch-class set theory" as
> a "theory"; of course it cannot be called a "music theory"

Well ok! But that is NOT how it has been presented to
the world.

> > Very well then, dodecaphonic it is. Following the tradition
> > of guerrilla theory here, I think we'll do best by doing it
> > ourselves. And by we I mean you :) since I don't know enough
> > about dodecaphonic music to help...
[...]
> A sophisticated form of experiment design would try contemplate
> several possibilities.

Or just start with the simplest possible example with the
fewest assumptions.

It did occur to me that using any famous piece will invalidate
the results, since subjects may have heard it before. So maybe
an original composition would be best.

> But the theme of Op.31 has a chordal accompaniment that
> I think it is simple -- however, the orchestration is crucial
> to suavize dissonances; I am not sure a MIDI file can obtain
> the delicate result.
>
> http://www.youtube.com/watch?v=u6BzLwHLKis

I see what you mean about orchestration (and I'll happily
admit that Boulez is one of my favorite conductors). Maybe
a piano reduction would be best for our experiment, if indeed
we decide to use an existing piece.

> http://www.youtube.com/watch?v=8GymJUFFwlI

...This piece, though nice, doesn't speak much to me. It is
like good incidental music.

> > I suppose I should read [Forte's book] [...]
>
> "Introduction to Post-Tonal Theory" (Joseth N. Straus, 1990)
> is perhaps more interesting as an introductory text, because
> it is didactic and more rich in musical examples. There are
> newer concepts not addressed by Forte's book, and interesting
> analysis of dodecaphonic pieces. However several concepts and
> data provided by Forte are not aborded.

Ok, thanks. Unfortunately this book is much more expensive!

This reference...

Hans Keller, "Strict Serial Technique in Classical Music",
Tempo, New Series, no. 37 (Autumn, 1955).

... is apparently a big one on 12-tone analysis of
common-practice music, but I don't find it on the web.

> How about comparing those other sets (as subsets of 12-EDO)?
> A={C,Eb,G,Bb}
> B={D,Eb,G#,A}
> C={C,E,G,B}
> D={B,C,F,Bb}

I played the pitechs in ascending order starting as close
to middle C as possible. I tried notes simultaneously and
arpeggiated. I observed that A and C are more consonant
than B or D, as would be predicted by psychoacoustics and
normal music theory. D was most dissonant. Other than that,
I heard no significant attributes.

-Carl

Carl,

I haven't until now
found enough time to answer your last response (Jan. 26) to one of my notes,
mostly because of the time and effort I knew it would take.  These were my concerns:

--You made a series of
wild statements (zero predictability for pitch class theory, for instance; you
also claimed that "no music receives government support in the US",
but this is wildly wrong; you also claim that "we have the best orchestras
in the world", but this is mere flag-waving propaganda, and is no longer
really true);

--you evaded the
consequences of your own statements (earlier you had lumped atonalists and
serialists together, but when I called you on the unreliability of your
statements, you retreated to "the terminology isn't standardized";
you clearly have a Hegelian-type theory of history [i.e., quasi-biological,
involving a necessitarian death of art forms; you condemned half-a-dozen still
existing genres of music to death] yet when I pointed this out, you ran away
from it, claiming that this was simply a narrative you had constructed [yes,
and it was your narrative, and the condemnations were yours]);

--you accused me of
supporting scientific relativism, which has nothing to do with anything in my
note; and

--your entire response
was dismissive (your response to my explanation of Schoenberg's influence on
Stravinsky and Milhaud was that Schoenberg influenced your cat)  and
revealed an astonishing degree of ignorance, considering the degree of
certainty with which you express your opinions.  For example, you offered
a blanket condemnation of experimental and avant-garde music and claimed to be
able to *demonstrate* that it is "baloney";  this isn't even pseudo-scientism, it is in fact the purest
form of baloney.

--I was also put off by
what I read as a hint that I might get kicked off the tuning list if I
disagreed with you ("We've
spit out Agmon and Tymoczko [among others]").

I've been following some
of these posts, and it's become clear to me that you really don't know all that
much about set theory.  I can't address all of the points above, and
there's probably no point to trying, seeing that you're just going to dance
around the consequences of your own statements.  I'm not going to bother filling you in on the dozens of
articles and books that prove that you don't know what you're talking about
with regard to Schoenberg's influence. You haven't even listened to all of his
works, yet have the temerity to pass harsh--but in fact highly
clichéd--judgments on his later music (which isn't in all cases my favorite,
either, but not because it is "atonal"). I suspect that if you were
to discover that you were wrong concerning Schoenberg's influence, you would
accuse the authors of the articles of being sterile academics, or something of
the sort. And you have continued to speak of Schoenberg's
"atonality," although Schoenberg clearly and distinctly distanced
himself from this term, which he thought was nonsensical (he favored
"pantonal").  I'll just talk briefly about set theory.

If you had responded to
my note by saying that there is a low degree of predictability in set theory, I
probably would have agreed with you, and we might have gotten somewhere. But
claiming that there is a zero degree of predictability is plain wrong. I've
been composing using elements of set theory for the last twenty-odd years, and
I can very clearly hear a large number of sets. I can hear these sets, often
without looking at a score, in the elegantly constructed music of Elliott
Carter (who, by the way, independently catalogued all of the sets in Forte's
book years before Forte's book came out and has used these sets for the last
half century. One of your claims in another post implies that set theory was
concurrent with Schoenberg's atonal compositions, but this is simply wrong; Haimo
has demonstrated on the basis of Schoenberg's manuscripts that it is unlikely
that Schoenberg was thinking in terms of any of the larger sets and
set-relations Forte uses to analyze Schoenberg's music; this is not a
scientific disproof of Forte's theory, but a historical proof of the
unreliability of Forte's assertions.).  My sophomore students learn to
identify the main sets by ear.  Thousands of students have learned these
skills. Whatever you think about the theory, you can't just throw out demonstrably
incorrect statements about it and expect to be taken seriously.

I agree with you that
set theory is not a useful analytical theory, but for very different reasons
than you offer. You claim that an analytic theory "must meet the standards of a scientific
theory," but this is precisely where practically all post-WWII
American-influenced analytic methods have started to go wrong. An analytic
theory cannot meet the standards of a scientific theory, because musical
analysis is not a science.  One can apply scientific methods to a
description of aural data, but aural data is not in itself music; once it is
considered an artwork, the mode of analysis has to change.  Although music
analyses tend to use a common language (or at the least dialects--Riemann vs.
Roman numeral analysis, etc.--that can be translated one into another with
relatively little loss of meaning), their value as analyses is not judged
solely on whether it can be proven that they use this language
"correctly"; the ability to use a common language coherently is
merely the condition for considering the author to be professionally competent.
What distinguishes great from mediocre analyses is always the personal
element--the insight that a great musician offers, the explanatory power of an
original approach, and so forth.  Schumann's analysis of Berlioz's
Symphonie Fantastique will continue to be treated as a classic, even though he
got elements of the form "wrong" (according to modern theories) and
had a primitive analytical terminology, simply because Schumann was a great
musician, a fine writer, and a great composer.

Theories like
Forte's are, I believe, very weak as analytic theories (although, as I
mentioned earlier, the labeling system is useful, as are a few of the tools
described early on in the book) because they claim a scientific validity that
is not justified.  This is the pseudo-scientific element that constantly
creeps into American-influenced academic theory.  This is largely because
American academic institutions tend to judge all disciplines by the standards
of science; because the arts and other disciplines can't meet these standards,
pseudo-scientific and reductionist methods are the ones that tend to prosper.

To answer your
charge, no, I am not pushing any sort of scientific relativism, whatever you
may mean by this (I am not sure, however, that you have kept up on the last
fifty years of theories of science--are you claiming, with Aristotle, that
science is "certain knowledge"?).  I am opposed to pseudo-science,
especially when it invades musical analysis. 

If you want to kick
me off this tuning list for not agreeing with  you, that's fine with me.  At least it will be clear what the purpose of this list is
in your eyes.

Franklin

Tuesday,
January 26, 2010 3:20 AM

From:

"Carl
Lumma" <carl@...>

Add sender
to Contacts

To:

tuning@yahoogroups.com

--- In tuning@yahoogroups. com, Cox Franklin <franklincox@ ...>
wrote:

>

> Carl,

> You're
the one who lumped atonalists together with serialists.

The
terminology isn't standardized.

>
Stravinsky was strongly influenced by Schoenberg, as was

> Milhaud.
In fact, practically every major composer in this

> period
responded to the challenge of Schoenberg's music and

> ideas;
even if they didn't like the music, they took the

> challenge
seriously.

My cat was
also influenced by Schoenberg.

> So it
appears you have a Hegelian theory of music in which

> history
is a one-way train progressing from outmoded

> forms
(Western art music) to the new dominant form, American

> popular
music. Anyone who doesn't jump on the train is

>
obsolescent.

Didn't say
that, and I hardly see any resemblance to Hegel

here, and it's
no any pet theory of mine, just a narrative

I put together
in response to Chris' post.

> It's odd,
I thought the musical populists were complaining

> about the
serialists using the model of progress to

>
marginalize them; apparently now the populists have taken

> over the
train.

Hardly.

> I love
Louis Armstrong and many other popular artists; I am

> not,
however, trying to do the same thing as they are. I

> think we
need to have a wide range of very different musics

> in the
world.

Those who know
me know I agree. You're jumping to conclusions.

> Your
argument appears to be that if government offer support

> to the
musical life of a country, the music that is produced

> is
illegitimate. ..is that right?

No.

> Does that
makes the music of all Medieval and Renaissance

>
composers, almost all Baroque composers, most Classical

>
composers, many Romantic composers, and most modern composers

>
illegitimate?

Actually the
early music scene is quite vibrant on its own

merits. In
fact no music receives gov't support in the U.S.

and as you
probably know, we have the best orchestras in

the world
(though they can no longer be described as vibrant).

The best
chamber ensembles come from Europe, but they make

their living
touring and selling albums here. Speaking of

which, Europe
Galante did a nice show on Saturday I was going

to post
about... nearly perfect 5-limit intonation throughout.

Though Biondi
looks tired, frankly.

> It's
an interesting theory, but, if I may say so, a bit daffy.

You may, but
nobody will take you seriously unless you

back it up.

> For your
information, there are hundreds of composers writing

>
experimental and avant-garde music in the United States as

> well,
without government support.

Yes, I was one
(for about a year when I was 19).

> These is
a pretty strong blanket condemnation of a field you

> don't
like.

You're welcome
to defend it. We've got lots of practice here

demonstrating
that it's baloney. Where by we, I apparently

mean me, since
I'm the last of the older members still reading

the list it
seems.

> There
are serious problems in set theory as a theory of analysis,

You mean like
making zero testable predictions? That certainly

is a problem
for any theory.

> but it
does offer a useful labeling system.

We've
demonstrated that approximately 90% of results from this

field depend
on numerology involving 12-ET, and fall apart

when applied
to other tuning systems that nonetheless sound

very similar.
But go ahead, let's hear about the labels.

> There is
a fair amount of excellent music composed using basic

> elements
of set theory.

No doubt. But
it almost certainly owes nothing to the theory.

> It
would be worthwhile having a fruitful discussion of the

> pros and
cons of set theory, but to judge from the tenor of

> your
paragraph, this probably isn't possible with you.

We've spit out
Agmon and Tymoczko (among others), and given

Jon Wild a
pass because he secretly doesn't believe a word

of it. So by
all means. After your post on scientific

relativism it
should be a blast.

-Carl

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@...

--- On Sun, 2/7/10, Carl Lumma <carl@...> wrote:

From: Carl Lumma <carl@...>
Subject: [tuning] Re: Musical Set Theory in 12-tET
To: tuning@yahoogroups.com
Date: Sunday, February 7, 2010, 7:44 AM

--- In tuning@yahoogroups. com, "hfmlacerda" <hfmlacerda@ ...> wrote:

> Some points:

>

> * If you take your chord 4:5:6:7:9:11: 13 and permutate the

> pitches in the pitch range, you can easily get sonorities that

> does not resemble too much the original disposition, just

> moving the pitches across octaves.

Yes, absolutely. Which is why the canonical version of

regular mapping does little to privilege octaves. They

have harmonic distance log(2) vs. log(3) for perfect 12ths,

log(6) for a perfect 5th, and so on.

> * Similarity relations between pitch sets are difficult to

> define and classify. Several year ago I was tempted to explore

> such possibilities (considering pitch range, register,

> disposition, not reducing pitches to pitch classes, etc.), and

> gave up because the untractable large number of possible

> combinations. If you reduce the number of objects to analyse

> (e.g. consonant chords built on 5-limit intervals), you can do

> a refined job; otherwise, as in atonal music, drastical

> reduction may be justified, given the large amount, complexity

> and variety of the pitch combinations which are used.

> "Set theory" is not generally about similarity, but essentially

> about structural (intervalar) properties of sets, as they are

> supposed to be used in atonal composition, or as they are found

> in analysis.

It seems to me you reached the right conclusion in your

investigation, and that set theory has no answer for it.

Set theory uses and combines relations anyway. Again, here

is the distinction between set theory as a generative method

(which is fine) and an analytical one (which means it must

meet the standards of a scientific theory).

>you only can have an exact transposition of a chord using

>transposition related sets, by definition;

This definition is true but that does not mean anything else

about sets is relevant to music.

> * If the universe is 12-EDO, according to Forte, _every_

> pc set with n=5 or n=7 elements can hold a subset with n-1

> elements invariant under inversion,

Can you give an example?

> * The interval vector resumes important relations (in special

> the number of common tones between a set and its transposition

> to every interval class), which are shared by sets with equal

> interval vectors (transpositions, inversions ans "Z-related"

> sets); whether these sets can or cannot be presented so that

> they sound "similar" is another issue: basically they share a

> common quantitative feature, from which can be extracted

> potential usages and other conclusions.

This sounds generative.

> For example, Forte have found that _any_ 5-tone subsets of

> 12-EDO contain at least one M3 (or m6) -- therefore, a major

> third as subset is of null relevance as distinctive feature

> of a given pentachord; you need at least 2 M3 to make a

> difference.

If such an observation is to be useful in analysis, we must

assume something like 1. the music to be analyzed was generated

in 12-ET and 2. the composer is presenting us a pentachord, in

such a way that we may identify it from all possible pentachords

using facts about pentachords in 12-ET. In other words, it's

useful if the music was composed using set theory!

>... you can hold a fixed M3 and superpose to it a succession

>of different 3-tone chords, eventually presenting every

>possible 5-tone chord type with no repetition.)

Generative (though interesting! )

>If the interval vector of a set shows a lot of M2s, or a lot

>of P4s, one can have some idea of how it may sound, etc. Just

>don't ask the interval vector more than it can answer, and you

>can find it useful. (You can surely imagine: there is a lot of

>delirious "extensions" and "applications" to this "set theory"

>around) :-)

Yes, very many. We are talking about z-relations on interval

vectors a few paragraphs ago!

> * To be clear: I don't consider "pitch-class set theory" as

> a "theory"; of course it cannot be called a "music theory"

Well ok! But that is NOT how it has been presented to

the world.

> > Very well then, dodecaphonic it is. Following the tradition

> > of guerrilla theory here, I think we'll do best by doing it

> > ourselves. And by we I mean you :) since I don't know enough

> > about dodecaphonic music to help...

[...]

> A sophisticated form of experiment design would try contemplate

> several possibilities.

Or just start with the simplest possible example with the

fewest assumptions.

It did occur to me that using any famous piece will invalidate

the results, since subjects may have heard it before. So maybe

an original composition would be best.

> But the theme of Op.31 has a chordal accompaniment that

> I think it is simple -- however, the orchestration is crucial

> to suavize dissonances; I am not sure a MIDI file can obtain

> the delicate result.

>

> http://www.youtube. com/watch? v=u6BzLwHLKis

I see what you mean about orchestration (and I'll happily

admit that Boulez is one of my favorite conductors). Maybe

a piano reduction would be best for our experiment, if indeed

we decide to use an existing piece.

> http://www.youtube. com/watch? v=8GymJUFFwlI

...This piece, though nice, doesn't speak much to me. It is

like good incidental music.

> > I suppose I should read [Forte's book] [...]

>

> "Introduction to Post-Tonal Theory" (Joseth N. Straus, 1990)

> is perhaps more interesting as an introductory text, because

> it is didactic and more rich in musical examples. There are

> newer concepts not addressed by Forte's book, and interesting

> analysis of dodecaphonic pieces. However several concepts and

> data provided by Forte are not aborded.

Ok, thanks. Unfortunately this book is much more expensive!

This reference...

Hans Keller, "Strict Serial Technique in Classical Music",

Tempo, New Series, no. 37 (Autumn, 1955).

... is apparently a big one on 12-tone analysis of

common-practice music, but I don't find it on the web.

> How about comparing those other sets (as subsets of 12-EDO)?

> A={C,Eb,G,Bb}

> B={D,Eb,G#,A}

> C={C,E,G,B}

> D={B,C,F,Bb}

I played the pitechs in ascending order starting as close

to middle C as possible. I tried notes simultaneously and

arpeggiated. I observed that A and C are more consonant

than B or D, as would be predicted by psychoacoustics and

normal music theory. D was most dissonant. Other than that,

I heard no significant attributes.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@> wrote:
[...]
> > * Similarity relations between pitch sets are difficult to
> > define and classify. Several year ago I was tempted to explore
> > such possibilities (considering pitch range, register,
> > disposition, not reducing pitches to pitch classes, etc.), and
> > gave up because the untractable large number of possible
> > combinations. If you reduce the number of objects to analyse
> > (e.g. consonant chords built on 5-limit intervals), you can do
> > a refined job; otherwise, as in atonal music, drastical
> > reduction may be justified, given the large amount, complexity
> > and variety of the pitch combinations which are used.
> > "Set theory" is not generally about similarity, but essentially
> > about structural (intervalar) properties of sets, as they are
> > supposed to be used in atonal composition, or as they are found
> > in analysis.
>
> It seems to me you reached the right conclusion in your
> investigation, and that set theory has no answer for it.

Acoustical similarity is another issue, of course. Categorization of intervals is a premise, just like octave equivalence. The similarity between pc sets works in another level. And a vocabulary of sets can be learned by musicians dealing with atonal music (just like jazz musicians can learn to identify many chord types). They are not inexistent relations.

> Set theory uses and combines relations anyway. Again, here
> is the distinction between set theory as a generative method
> (which is fine) and an analytical one (which means it must
> meet the standards of a scientific theory).

My main interest in "set theory" is for composition. Nonetheless I learned a lot about atonal music reading those references. The concepts presented _are_ useful and the reduction of pitches to integer notation allows automated operations.

Can musical analysis procedure meet the standards of a scientific theory? The inexistence of any "musical theory" makes me doubtful.

>
> >you only can have an exact transposition of a chord using
> >transposition related sets, by definition;
>
> This definition is true but that does not mean anything else
> about sets is relevant to music.

Sets are _representations_ of pitch groups. There are both direct and indirect correspondences, and there are limitations one should be aware, just as it occurs with staff notation as a representation. There are correspondences, therefore they are relevant.

Here I should make a comparison: there is a book which explains tonal harmony using set theory (it is absolutely not related to Babbitt/Forte/etc. works). A reasonable number of tonal relations can be represented that way, whereas many other are not aborded. That approach is not very useful, though, specially because _there are_ other expositive systems which are efficient (Schoenberg's book is very good).

For atonal music analysis, there are traditional analytical tools, usually involving tables of pitch collections and schematic representations. But I have no notice of any standardized analytical procedure (this is not a problem, provided that the analyst is competent). "Set theory" presented a representation of most of the traditional operations using numbers, and explored 12-EDO further, contributing with new concepts. Probably, the most important contribution was simply the classification of pc-sets -- the Forte's list itself -- because as sets can be identified, providing a vocabulary (set names and prime forms). AFAIK, "set theory" was (or is?) like an USA-made fashion, not very welcome by European musicians/analysts for some time. Anyway, if you are interested in atonal music, you should study it as obligatory reference.

>
> > * If the universe is 12-EDO, according to Forte, _every_
> > pc set with n=5 or n=7 elements can hold a subset with n-1
> > elements invariant under inversion,
>
> Can you give an example?

Random pitches: 1 8 11 12 4 9 5
Normal form: 8 9 11 0 1 4 5
Ascending order: 0 1 4 5 8 9 11
Inversions (in ascending order):
0 1 3 4 7 8 11
0 1 2 4 5 8 9 *
1 2 3 5 6 9 10
2 3 4 6 7 10 11
0 3 4 5 7 8 11
0 1 4 5 6 8 9 **
1 2 5 6 7 9 10
2 3 6 7 8 10 11
0 3 4 7 8 9 11
0 1 4 5 8 9 10 ***
1 2 5 6 9 10 11
0 2 3 6 7 10 11
For this (random) heptachord, there are 3 transpositions levels of inverse forms that hold 6 elements invariant (marked with asterisks).

In alphabetical representation:
Original: C C# E F G# A B
Inversion *: C C# D E F G# A
Inversion **: C C# E F F# G# A
Inversion ***: C C# E F G# A Bb

Note: the subset {0,1,4,5,8,9} or {C C# E F G# A} is common to all these forms.

> > * The interval vector resumes important relations (in special
> > the number of common tones between a set and its transposition
> > to every interval class), which are shared by sets with equal
> > interval vectors (transpositions, inversions ans "Z-related"
> > sets); whether these sets can or cannot be presented so that
> > they sound "similar" is another issue: basically they share a
> > common quantitative feature, from which can be extracted
> > potential usages and other conclusions.
>
> This sounds generative.

If it is generative, it is useful.

Yet some set possibilities can be analysed in order to see if an atonal composer explored "obvious" or "special" possibilities of a given set, or omitted some possibilities (for example, when a set type is used many times in a work). Forte's book touches such issues.

>
> > For example, Forte have found that _any_ 5-tone subsets of
> > 12-EDO contain at least one M3 (or m6) -- therefore, a major
> > third as subset is of null relevance as distinctive feature
> > of a given pentachord; you need at least 2 M3 to make a
> > difference.
>
> If such an observation is to be useful in analysis, we must
> assume something like 1. the music to be analyzed was generated
> in 12-ET and 2. the composer is presenting us a pentachord, in
> such a way that we may identify it from all possible pentachords
> using facts about pentachords in 12-ET. In other words, it's
> useful if the music was composed using set theory!

Atonal music typically used all the 12 tones of 12-EDO in a short time span. Thus, the way the tones are combined is the relevant thing. What identifies a set is the fact that the tones are selected and combined as to form that specific set (when it is presented as chord, a for example).

When Forte compares penthachords, he is not just comparing it with every other pentachord in 12-EDO, but also considering its relevance and relative frequency in the vocabulary of atonal music. Virtually every pc set is used in this repertory, thus the comparison with every other pentachord of 12-EDO can be justified by the actual practice. Atonal music is rather different of tonal music, in which a small number of set types prevail.

The example above is just an objective conclusion of the possibilities, showing that an analist must be aware of the combinatorial possibilities in 12-EDO in order to evaluate the relevance of a relation found in a music piece.

[...]
> >If the interval vector of a set shows a lot of M2s, or a lot
> >of P4s, one can have some idea of how it may sound, etc. Just
> >don't ask the interval vector more than it can answer, and you
> >can find it useful. (You can surely imagine: there is a lot of
> >delirious "extensions" and "applications" to this "set theory"
> >around) :-)
>
> Yes, very many. We are talking about z-relations on interval
> vectors a few paragraphs ago!

Z-relation is a fact, and it is relevant in 12-tone music, even if the concept did not exist with this funny name.

And yes, we can also imagine that a scientific music theory can be created simply from psychoacoustical experiments on frequency ratios followed by speculation on uncommon and artificial scales (this is generative). Are there any analytical musical applications of regular mapping on microtonal music? Or is it restricted to analyses of scales?

>
> > * To be clear: I don't consider "pitch-class set theory" as
> > a "theory"; of course it cannot be called a "music theory"
>
> Well ok! But that is NOT how it has been presented to
> the world.

But WHO is present it that way? Can you cite the relevant bibliography on the matter? That resembles the straw man argument, and anyway I think this is completely irrelevant for a fruitful discussion.

>
> > > Very well then, dodecaphonic it is. Following the tradition
> > > of guerrilla theory here, I think we'll do best by doing it
> > > ourselves. And by we I mean you :) since I don't know enough
> > > about dodecaphonic music to help...
> [...]
> > A sophisticated form of experiment design would try contemplate
> > several possibilities.
>
> Or just start with the simplest possible example with the
> fewest assumptions.
>
> It did occur to me that using any famous piece will invalidate
> the results, since subjects may have heard it before. So maybe
> an original composition would be best.

I considered this, but I should assume that a masterpiece by a competent composer of the style is much more suitable than a "counterpoint exercise" by a musician with less experience and not so tied to dodecaphonic technique. Secondary sections, or pieces that are not much known may be better.

>
> > But the theme of Op.31 has a chordal accompaniment that
> > I think it is simple -- however, the orchestration is crucial
> > to suavize dissonances; I am not sure a MIDI file can obtain
> > the delicate result.
> >
> > http://www.youtube.com/watch?v=u6BzLwHLKis
>
> I see what you mean about orchestration (and I'll happily
> admit that Boulez is one of my favorite conductors). Maybe
> a piano reduction would be best for our experiment, if indeed
> we decide to use an existing piece.
>
> > http://www.youtube.com/watch?v=8GymJUFFwlI
>
> ...This piece, though nice, doesn't speak much to me. It is
> like good incidental music.

Incidental? :-0

>
> > > I suppose I should read [Forte's book] [...]
> >
> > "Introduction to Post-Tonal Theory" (Joseth N. Straus, 1990)
> > is perhaps more interesting as an introductory text, because
> > it is didactic and more rich in musical examples. There are
> > newer concepts not addressed by Forte's book, and interesting
> > analysis of dodecaphonic pieces. However several concepts and
> > data provided by Forte are not aborded.
>
> Ok, thanks. Unfortunately this book is much more expensive!
>
> This reference...
>
> Hans Keller, "Strict Serial Technique in Classical Music",
> Tempo, New Series, no. 37 (Autumn, 1955).
>
> ... is apparently a big one on 12-tone analysis of
> common-practice music, but I don't find it on the web.

I don't know it.

>
> > How about comparing those other sets (as subsets of 12-EDO)?
> > A={C,Eb,G,Bb}
> > B={D,Eb,G#,A}
> > C={C,E,G,B}
> > D={B,C,F,Bb}
>
> I played the pitechs in ascending order starting as close
> to middle C as possible. I tried notes simultaneously and
> arpeggiated. I observed that A and C are more consonant
> than B or D, as would be predicted by psychoacoustics and
> normal music theory. D was most dissonant. Other than that,
> I heard no significant attributes.

How can these dissonance predictions be made using "normal music theory"? And/or: is there a simplified way to use psychoacoustic principles to compare those chords?

Hi Franklin,

The onus is on proponents of music set theory / serialism &
related disciplines to show something capable of delivering
on claims made, results supposedly obtained, and doctorates
granted. We're waiting. If you wish to engage in sideline
arguments, such as whether my statements are Hegelian or
petty stuff about who backtracked when... I'm not interested.

We don't practice active moderation here, as you can probably
tell by some of the stuff that gets posted. Tymoczko and
Agmon stopped posting of their own accord and probably still
have active accounts. They were "spit out" only in that
their arguments did not withstand scrutiny in the daylight,
outside the insulated circuits they travel in, where they
probably prefer to remain. The posts are all in the archives
for the interested. Except the 2nd half of the Agmon thread,
which he insisted take place offlist (I wonder why). I have
these messages (in which about a dozen people participated)
archived and I may be able to post excerpts or paraphrase if
there is interest.

We are waiting for something constructive. Citations to
"dozens of books" don't count. This is a place where people
present complete arguments for assessment by others. I
openly admit I know next to nothing about music set theory,
and am very grateful to Hudson for explaining what he has.
Note however, he just admitted that set theory is not a
scientific music theory. And I imagine I know more about
set theory than you know about regular temperaments.

I'm completely confident of my position that the great
neoclassical works of Stravinsky, Prokofiev and others owe
nothing to combinatorial techniques. This means there is no
model explaining why combinatorial techniques produce
enjoyable music to which great neoclassical works conform.
NOT that Stravinsky (or better yet, Mozart) didn't use tone
rows here or there. As far as finding sets in Schoenberg,
you must also disprove the null hypothesis -- that average
serial (but not deliberately set-based) music would contain
them. If you can do this, I'm listening.

No one will deny that with 20 years of training, humans can
learn to recognize almost anything. Again, any method will
hopefully succeed in analyzing works created with it!

> I agree with you that set theory is not a useful analytical
> theory, ...

Then we are in complete agreement! Except apparently over
the widely known fact that the NY MET has the best orchestra
in the world. :) Actually I've been completely out of touch
with orchestral happenings since the late '90s, when Chicago
was all the rage. I do know many U.S. franchises have been
hurting financially since then.

> Theories like Forte's are, I believe, very weak as analytic
> theories (although, as I mentioned earlier, the labeling
> system is useful, as are a few of the tools described early
> on in the book) because they claim a scientific validity that
> is not justified. This is the pseudo-scientific element
> that constantly creeps into American-influenced academic
> theory.

Now it seems as though you are writing my posts for me!

> This is largely because American academic institutions tend to
> judge all disciplines by the standards of science; because the
> arts and other disciplines can't meet these standards, pseudo-
> scientific and reductionist methods are the ones that tend to
> prosper.

Well ok, what can we do about the situation? I would love
to have thriving arts programs in American universities.
They should include all forms of music, dance, etc. and truly
prepare students to be practicing artists. I would even have
room for set theorists to say crafty things about atonal music.

> To answer your charge, no, I am not pushing any sort of
> scientific relativism, whatever you may mean by this (I am
> not sure, however, that you have kept up on the last fifty
> years of theories of science--are you claiming, with
> Aristotle, that science is "certain knowledge"?).

I think it was your msg #85832 that gave me this impression.
My apologies. It's safe to say as a first approximation that
I don't agree with Aristotle about anything. I have indeed
kept up with the philosophy of science, as I study machine
learning. The scientific method is essentially Bayesian
inference, and there is no evidence that any other approach
to thinking exists (recently it's been suggested that the
method of maximum relative entropy generalizes Bayesian
inference...).

-Carl

Carl,

If the proponents of set theory don't meet your demand, are you going to walk into each university and force them to fire the professors of set theory?  As little as I like elements of the theory, there it's established in the academic discourse, and no matter how much you complain about it, it's not leaving any time soon. Do you really think tenured professors worry at all what you think about them? If you want to make your point heard anywhere outside of your closed circle, the onus is on you to prove that all of these professors are wrong.  You are welcome to submit papers to peer-review journals proving that all of the work submitted to these journals is of no value.  But until you master the  discourse, nobody is going to take your assertions seriously.

And until you learn a bit more about the composers you're talking about, I have trouble knowing what to make of much of what you say about modern music.  Whatever arguments you and your circle make must adhere to the historical evidence.  Stravinsky knew Schoenberg's music early on and was directly influenced by it in the period between Petroushka and the Rite; haven't you noticed the dramatic increase in dissonance in the latter work?  Stravinsky also used permutational methods throughout his Neoclassical period, hidden canons, cellular construction, and the like, throughout his Neoclassical period, even though the music was centered in keys.  In the late 1940's his procedures grew progressively closer to serial procedures.  Perhaps you've forgotten that he wrote serial music for the last two decades of his life.  He studied the scores of Berg, Schoenberg, and Webern closely, as well as those of Stockhausen, Boulez, Babbitt et al. The composer he
regarded the most highly of all the post-war group was Elliott Carter (who has never been a serialist, by the way---this is a common misunderstanding).

Milhaud knew and admired Schoenberg greatly and wrote about the impact of Pierrot on him. 

What do you mean by "combinatorial techniques"?  Do you mean the type of combinatoriality used by Schoenberg in his 12-tone works? Or do you mean compositional procedures using set classes?  There's a specific meaning to "combinatorial" in 12-tone theory.

I never claimed to know as much about regular temperaments as you.  The problem I have is that it seems to me that you are claiming to know a lot more about certain subjects than you actually do. I will also concede your superior knowledge in scientific methods, although you will find plenty of disagreement among leading theorists to your assertion concerning Bayesian inference.

I never dismissed the work you've dedicated yourself to; I don't know much about it, but I'm happy that there are lots of different approaches out there.  However, you dismissed the sort of music that I've been doing for the last three decades without knowing anything about it.  Maybe you did composer "atonal" music "when you were 19," as you say, but I doubt that you achieved anything of great subtlety and accomplishment in this field.

I don't know what I'm supposed to prove to you; I'm not looking to you or anyone else in the tuning list for validation. I joined the list because I was hoping that some people in this circle might be interested in the Ben Johnston colloquium I'm organizing.  I got involved in this whole discussion because I read the same sort of pseudo-scientific blanket dismissals of "dissonant," "atonal" music that keep popping up everywhere. I responded to your post because you were condemning one genre after another--including the one in which I and hundreds of other composers are working--to obsolescence.  These sorts of statements are harmful and lead nowhere except into childish disputes.  You keep asserting that you have proof that a lot of "this" or "that" field of work is baloney, but you don't provide any specific examples, and I don't see any evidence that you've proven anything. 

You can prove anything you like to yourself, but the validity in the real world is the acceptance of that proof by the people involved. If you want your proof to mean anything outside of your circle, you're going to have to convince other people outside of your circle.
Until you do that, I and hundreds of other composers will continue practicing the craft to which we've devoted our lives.

Franklin

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@...

--- On Mon, 2/8/10, Carl Lumma <carl@...> wrote:

From: Carl Lumma <carl@lumma.org>
Subject: [tuning] Re: Musical Set Theory in 12-tET
To: tuning@yahoogroups.com
Date: Monday, February 8, 2010, 12:35 AM

Hi Franklin,

The onus is on proponents of music set theory / serialism &

related disciplines to show something capable of delivering

on claims made, results supposedly obtained, and doctorates

granted. We're waiting. If you wish to engage in sideline

arguments, such as whether my statements are Hegelian or

petty stuff about who backtracked when... I'm not interested.

We don't practice active moderation here, as you can probably

tell by some of the stuff that gets posted. Tymoczko and

Agmon stopped posting of their own accord and probably still

have active accounts. They were "spit out" only in that

their arguments did not withstand scrutiny in the daylight,

outside the insulated circuits they travel in, where they

probably prefer to remain. The posts are all in the archives

for the interested. Except the 2nd half of the Agmon thread,

which he insisted take place offlist (I wonder why). I have

these messages (in which about a dozen people participated)

archived and I may be able to post excerpts or paraphrase if

there is interest.

We are waiting for something constructive. Citations to

"dozens of books" don't count. This is a place where people

present complete arguments for assessment by others. I

openly admit I know next to nothing about music set theory,

and am very grateful to Hudson for explaining what he has.

Note however, he just admitted that set theory is not a

scientific music theory. And I imagine I know more about

set theory than you know about regular temperaments.

I'm completely confident of my position that the great

neoclassical works of Stravinsky, Prokofiev and others owe

nothing to combinatorial techniques. This means there is no

model explaining why combinatorial techniques produce

enjoyable music to which great neoclassical works conform.

NOT that Stravinsky (or better yet, Mozart) didn't use tone

rows here or there. As far as finding sets in Schoenberg,

you must also disprove the null hypothesis -- that average

serial (but not deliberately set-based) music would contain

them. If you can do this, I'm listening.

No one will deny that with 20 years of training, humans can

learn to recognize almost anything. Again, any method will

hopefully succeed in analyzing works created with it!

> I agree with you that set theory is not a useful analytical

> theory, ...

Then we are in complete agreement! Except apparently over

the widely known fact that the NY MET has the best orchestra

in the world. :) Actually I've been completely out of touch

with orchestral happenings since the late '90s, when Chicago

was all the rage. I do know many U.S. franchises have been

hurting financially since then.

> Theories like Forte's are, I believe, very weak as analytic

> theories (although, as I mentioned earlier, the labeling

> system is useful, as are a few of the tools described early

> on in the book) because they claim a scientific validity that

> is not justified. This is the pseudo-scientific element

> that constantly creeps into American-influenced academic

> theory.

Now it seems as though you are writing my posts for me!

> This is largely because American academic institutions tend to

> judge all disciplines by the standards of science; because the

> arts and other disciplines can't meet these standards, pseudo-

> scientific and reductionist methods are the ones that tend to

> prosper.

Well ok, what can we do about the situation? I would love

to have thriving arts programs in American universities.

They should include all forms of music, dance, etc. and truly

prepare students to be practicing artists. I would even have

room for set theorists to say crafty things about atonal music.

> To answer your charge, no, I am not pushing any sort of

> scientific relativism, whatever you may mean by this (I am

> not sure, however, that you have kept up on the last fifty

> years of theories of science--are you claiming, with

> Aristotle, that science is "certain knowledge"?) .

I think it was your msg #85832 that gave me this impression.

My apologies. It's safe to say as a first approximation that

I don't agree with Aristotle about anything. I have indeed

kept up with the philosophy of science, as I study machine

learning. The scientific method is essentially Bayesian

inference, and there is no evidence that any other approach

to thinking exists (recently it's been suggested that the

method of maximum relative entropy generalizes Bayesian

inference... ).

-Carl

> But until you master the  discourse, nobody is going to take your assertions seriously.

Taruskin learned all that stuff so that when he claims its all
bullshit no one can say he doesn't know what he's talking about.

Dante

Taruskin is a brilliant musicologist (his book on Stravinsky in very impressive) and  a sharp polemicist.  He is a formidable opponent, but one of the reasons people are terrified to take him on is that he is aggressive at a personal level, sending out nasty little notes to anyone he disagrees with; I've gotten reports that he will  destroy the academic careers of people with whom he disagrees.

I find many of Taruskin's arguments weak; if you go beneath the surface brilliance of his writing, you'll find his arguments are pretty thin. For example, he wrote a hate piece about Donald Martino, trashing the entire output of this fine composer. This article has often been cited by people searching to prove the invalidity of serial music. It turns out that Taruskin didn't know much about Martino's music; in fact over half of Martino's music is high-quality tonal music.  Taruskin's entire argument is taken from Boulez, who made excessive claims about the necessity of writing serial music, claims which have nothing to do with Martino.  Neverthelss, Taruskin applied them to Martino.  The whole presentation was fraudulent, attributing views to a composer that the composer has never held, assuming the composer was a diehard serialist when the composer was not a diehard serialist, and so forth. In other words, a bit of scholarship strips the performance
down to a few shoddy tricks on the part of the ciritic. 

I'm always willing to concede the weakness of pitch-class theory; in fact, I jokingly advise my students to find a book, cut out about twenty pages, and ditch the rest.  However, the "it's *all* bullshit" argument tends to discredit the person upholding it, because the attack collapses  if a single bit of evidence can be found that would demonstrate that the value of the theory. And because many fine composers and theorists use aspects of pitch class theory, the evidence is not hard to find. So the attacker either has to deny evidence in order to uphold his or her claim, or make a more subtle and reasonable argument based on the full range of evidence out there.

Franklin

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@...

--- On Mon, 2/8/10, Dante Rosati <danterosati@...> wrote:

From: Dante Rosati <danterosati@...>
Subject: Re: [tuning] Re: Musical Set Theory in 12-tET
To: tuning@yahoogroups.com
Date: Monday, February 8, 2010, 4:20 AM

> But until you master the  discourse, nobody is going to take your assertions seriously.

Taruskin learned all that stuff so that when he claims its all

bullshit no one can say he doesn't know what he's talking about.

In my last paragraph, I meant "a book on set theory".

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@yahoo.com

--- On Mon, 2/8/10, Dante Rosati <danterosati@...> wrote:

From: Dante Rosati <danterosati@...>
Subject: Re: [tuning] Re: Musical Set Theory in 12-tET
To: tuning@yahoogroups.com
Date: Monday, February 8, 2010, 4:20 AM

> But until you master the  discourse, nobody is going to take your assertions seriously.

Taruskin learned all that stuff so that when he claims its all

bullshit no one can say he doesn't know what he's talking about.

the way I look at it is that there are some ideas in music which are
interesting and worth exploring briefly but which exhaust their
possibilities rather quickly. Schoenberg had an interesting idea which
Webern took to its conclusion, Boulez took it to its extremity, and
everything else has been basically rehashing. Babbitt has shown just how
exhausted an idea can be. The really talented and really smart composers
like Ligeti and Xenakis smelled the coffee early on and explored other
directions which is why they turned out to be the major composers of the
second half of the 20th c. Stravinsky's late serial works are interesting
but lets face it theres nothing there to compete with the Rite, except maybe
the Owl and the Pussycat song.

PS- I got as a present Taruskin's Oxford History of Western Music in 5
volumes now that its in paperback. The guy is completely insane to write
something like that singlehandedly in this day and age but its completely
brilliant and I love perusing it.

On Sun, Feb 7, 2010 at 11:54 PM, Cox Franklin <franklincox@...> wrote:

>
>
> Dante
>
> Taruskin is a brilliant musicologist (his book on Stravinsky in very
> impressive) and a sharp polemicist. He is a formidable opponent, but one
> of the reasons people are terrified to take him on is that he is aggressive
> at a personal level, sending out nasty little notes to anyone he disagrees
> with; I've gotten reports that he will destroy the academic careers of
> people with whom he disagrees.
>
> I find many of Taruskin's arguments weak; if you go beneath the surface
> brilliance of his writing, you'll find his arguments are pretty thin. For
> example, he wrote a hate piece about Donald Martino, trashing the entire
> output of this fine composer. This article has often been cited by people
> searching to prove the invalidity of serial music. It turns out that
> Taruskin didn't know much about Martino's music; in fact over half of
> Martino's music is high-quality tonal music. Taruskin's entire argument is
> taken from Boulez, who made excessive claims about the necessity of writing
> serial music, claims which have nothing to do with Martino. Neverthelss,
> Taruskin applied them to Martino. The whole presentation was fraudulent,
> attributing views to a composer that the composer has never held, assuming
> the composer was a diehard serialist when the composer was not a diehard
> serialist, and so forth. In other words, a bit of scholarship strips the
> performance down to a few shoddy tricks on the part of the ciritic.
>
> I'm always willing to concede the weakness of pitch-class theory; in fact,
> I jokingly advise my students to find a book, cut out about twenty pages,
> and ditch the rest. However, the "it's *all* bullshit" argument tends to
> discredit the person upholding it, because the attack collapses if a single
> bit of evidence can be found that would demonstrate that the value of the
> theory. And because many fine composers and theorists use aspects of pitch
> class theory, the evidence is not hard to find. So the attacker either has
> to deny evidence in order to uphold his or her claim, or make a more subtle
> and reasonable argument based on the full range of evidence out there.
>
>
> Franklin
>
>
>
>
> Dr. Franklin Cox
> 1107 Xenia Ave.
> Yellow Springs, OH 45387
> (937) 767-1165
> franklincox@...
>
> --- On *Mon, 2/8/10, Dante Rosati <danterosati@...>* wrote:
>
>
> From: Dante Rosati <danterosati@...>
> Subject: Re: [tuning] Re: Musical Set Theory in 12-tET
> To: tuning@yahoogroups.com
> Date: Monday, February 8, 2010, 4:20 AM
>
>
>
>
> > But until you master the discourse, nobody is going to take your
> assertions seriously.
>
> Taruskin learned all that stuff so that when he claims its all
> bullshit no one can say he doesn't know what he's talking about.
>
>
>

Dante,

Just be aware that Taruskin's history of modern music is based on a relatively narrow range of knowledge about modern music and is very unreliable. He's been called to task for his misrepresentations in the scholarly literature. 

I'll give you an idea of his approach; about twenty years ago when he was attacking Babbitt and Martino, I wrote to him and defended them--not because I necessarily love their music, but because I found Taruskin's methods shoddy.  Bear in mind that Taruskin had toed the party line for years, using Babbitt as the representative of contemporary American composition in Introduction to Music courses. When I taught under him at Columbia University, I suggested including Steve Reich in the course, but that was too anti-establishment a notion for him. A few years later, the leopard had changed his spots; suddently Reich was the greatest American composer of all and Babbitt a useless old composer whose music was based on demonstrably false premises.  Of course Taruskin is free to worship Reich (who's music is pretty and charming but pretty thin), but the nasty tone and the pseudo-scientism were galling, and the wholesale condemnation of an accomplished
composer's entire career I found unseemly.

Taruskin started peppering me with nasty little notes, as he has done to countless others  who have challenged him. I pointed out to him that numerous younger composers--Roger Redgate, Richard Barrett, Marc Andre, Claus-Steffen Mahnkopf, and so forth--were pushing the boundaries in quite different ways than Babbitt, and were not using total serial methods.  His response: "I don't know any of your friends." In other words, a supposed authority on modern music simply didn't want to know what he didn't want to know.  A few years later, he wrote his mammoth opus, and he continued to wage war against the generation preceding him, with not a sign that he had expanded his knowledge to deal with unusual composers of the younger generation. In other words, the entire history  of modern music ends where his interests end.  This is lousy scholarship. Bear in mind that this is not "Taruskin's History of Western Music," but the Oxford History of Western Music, a
work  whose title lays claims to a degree of reliability and impartiality of which Taruskin is wholly incapable.  Regarding modern music, his entire concern is with  previous generations, and I find it his approach tedious and tendentious. I'm fairly confident that in a few years, once he's not around to threaten people, the gloss will come off his reputation.  He can be pretty shaky as a traditional historian as well. His book on Stravinsky was quite wonderful, though.

There's a lot of wonderful music out there; modern music didn't end with Xenakis and friends. I'm co-editing a series--New Music and Aesthetics in the 21st Century (Wolke Verlag)--that focuses on younger composers, and am co-editing a journal, Search, which has a similar focus (a compilation of the first year's writings will be coming out soon from Mellen Press).  There's been a serious gap in coverage of adventurous composers, and most people are still fixated on the serialists vs. non-serialists issue, which is decades past its prime. 

regards,

Franklin

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@...

--- On Mon, 2/8/10, Dante Rosati <danterosati@...> wrote:

From: Dante Rosati <danterosati@gmail.com>
Subject: Re: [tuning] Re: Musical Set Theory in 12-tET
To: tuning@yahoogroups.com
Date: Monday, February 8, 2010, 5:42 AM

the way I look at it is that there are some ideas in music which are interesting and worth exploring briefly but which exhaust their possibilities rather quickly. Schoenberg had an interesting idea which Webern took to its conclusion, Boulez took it to its extremity, and everything else has been basically rehashing. Babbitt has shown just how exhausted an idea can be. The really talented and really smart composers like Ligeti and Xenakis smelled the coffee early on and explored other directions which is why they turned out to be the major composers of the second half of the 20th c. Stravinsky's late serial works are interesting but lets face it theres nothing there to compete with the Rite, except maybe the Owl and the Pussycat song.

PS- I got as a present Taruskin's Oxford History of Western Music in 5 volumes now that its in paperback. The guy is completely insane to write something like that singlehandedly in this day and age but its completely brilliant and I love perusing it.

On Sun, Feb 7, 2010 at 11:54 PM, Cox Franklin <franklincox@ yahoo.com> wrote:

Dante

Taruskin is a brilliant musicologist (his book on Stravinsky in very impressive) and  a sharp polemicist.  He is a formidable opponent, but one of the reasons people are terrified to take him on is that he is aggressive at a personal level, sending out nasty little notes to anyone he disagrees with; I've gotten reports that he will  destroy the academic careers of people with whom he disagrees.

I find many of Taruskin's arguments weak; if you go beneath the surface brilliance of his writing, you'll find his arguments are pretty thin. For example, he wrote a hate piece about Donald Martino, trashing the entire output of this fine composer. This article has often been cited by people searching to prove the invalidity of serial music. It turns out that Taruskin didn't know much about Martino's music; in fact over half of
Martino's music is high-quality tonal music.  Taruskin's entire argument is taken from Boulez, who made excessive claims about the necessity of writing serial music, claims which have nothing to do with Martino.  Neverthelss, Taruskin applied them to Martino.  The whole presentation was fraudulent, attributing views to a composer that the composer has never held, assuming the composer was a diehard serialist when the composer was not a diehard serialist, and so forth. In other words, a bit of scholarship strips the performance down to a few shoddy tricks on the part of the ciritic. 

I'm always willing to concede the weakness of pitch-class theory; in fact, I jokingly advise my students to find a book, cut out about twenty pages, and ditch the rest.  However, the "it's *all* bullshit" argument tends to discredit the person upholding it, because the attack collapses  if a single bit of evidence can be found that would
demonstrate that the value of the theory. And because many fine composers and theorists use aspects of pitch class theory, the evidence is not hard to find. So the attacker either has to deny evidence in order to uphold his or her claim, or make a more subtle and reasonable argument based on the full range of evidence out there.

Franklin

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@ yahoo.com

--- On Mon, 2/8/10, Dante Rosati <danterosati@ gmail.com> wrote:

From: Dante Rosati <danterosati@ gmail.com>
Subject: Re: [tuning] Re: Musical Set Theory in 12-tET

To: tuning@yahoogroups. com
Date: Monday, February 8, 2010, 4:20 AM

> But until you master the  discourse, nobody is going to take your assertions seriously.

Taruskin learned all that stuff so that when he claims its all

bullshit no one can say he doesn't know what he's talking about.

--- In tuning@yahoogroups.com, Cox Franklin <franklincox@...> wrote:

> If the proponents of set theory don't meet your demand, are
> you going to walk into each university and force them to fire
> the professors of set theory?  As little as I like elements of
> the theory, there it's established in the academic discourse,
> and no matter how much you complain about it, it's not leaving
> any time soon. Do you really think tenured professors worry at
> all what you think about them?

What kind of argument is this? If the emperor is naked we
shouldn't bother to say so?

> If you want to make your point heard anywhere outside of your
> closed circle, the onus is on you to prove that all of these
> professors are wrong.

I hardly ever think about them - you brought it up. I can't
prove a non-item wrong. The onus is on anyone who wants to
discuss set theory on any open forum to state something that
makes sense. They prefer to hide out on post tonality or
whatever it's called, or make personal attacks at me, as you
are doing. Meanwhile, the message that there's been a vacuum
in intonation for far too long is being heard widely indeed.
It's reaching musicians of all ages, in all nations, in genres
from medieval to hip-hop, and this list is only playing a
small part. In fact the microtonal movement is probably
already larger than atonal music, and growing much faster.
Though they're really not in competition, except that set
theorists have been remarkably close-minded, if not violently opposed, to tunings other than 12-ET. Microtonal atonal music
would in fact be a natural direction, as Wendy Carlos pointed
out in her letter to the NY Times.

>Stravinsky knew Schoenberg's music early on and was directly
>influenced by it in the period between Petroushka and the
>Rite; haven't you noticed the dramatic increase in dissonance
>in the latter work?

So what?

> Milhaud knew and admired Schoenberg greatly and wrote about
> the impact of Pierrot on him.

So what? I've studied Milhaud's music a fair amount, as he
is one of my favorite composers, though he wrote so much in
so many different styles it's hard to cover it all. I'll have
to refer you to my previous post for the correct assertion to
disagree with, if that's what you're trying to do.

See Dante's response regarding Stravinsky.

> What do you mean by "combinatorial techniques"? Do you mean
> the type of combinatoriality used by Schoenberg in his 12-tone
> works? Or do you mean compositional procedures using set
> classes?  There's a specific meaning to "combinatorial" in
> 12-tone theory.

I must say I find both you and Hudson obtuse on this point.
You both agree the terminology has problems ("set theory"
being the most obvious). I can say that compared to the
differences between serialism and other genres of music,
there is approximately no difference between serialism and
set theory, or anything else that you and Hudson both know
damn well is in the selfsame tradition of intellectual
snake oil. What do you suggest I call it?

> I never dismissed the work you've dedicated yourself to;

You're welcome to do so, since it is, you know, the topic of
this mailing list. There are plenty of open problems, and
arguments currently in hand-waving status, to pick at.

>However, you dismissed the sort of music that I've been doing
>for the last three decades without knowing anything about it.

I didn't dismiss the music, I dismissed the analysis -- and
in fact we seem to be in complete agreement on this point --
but since you asked, I do happen find a much higher proportion
of serial/set music sucks than is typical for musical genres.

>Maybe you did composer "atonal" music "when you were 19," as
>you say,

When did I say that? I've never composed atonal music.

-Carl

--- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@...> wrote:

> Can musical analysis procedure meet the standards of a
> scientific theory? The inexistence of any "musical theory"
> makes me doubtful.

To the extent it cannot, I don't know if it's worth
thinking about.

> Here I should make a comparison: there is a book which explains
> tonal harmony using set theory (it is absolutely not related to
> Babbitt/Forte/etc. works). A reasonable number of tonal relations
> can be represented that way,

Does it work in 31-ET?

> > > * If the universe is 12-EDO, according to Forte, _every_
> > > pc set with n=5 or n=7 elements can hold a subset with n-1
> > > elements invariant under inversion,
> >
> > Can you give an example?
>
> Random pitches: 1 8 11 12 4 9 5
> Normal form: 8 9 11 0 1 4 5
> Ascending order: 0 1 4 5 8 9 11
> Inversions (in ascending order):
> 0 1 3 4 7 8 11
> 0 1 2 4 5 8 9 *
> 1 2 3 5 6 9 10
> 2 3 4 6 7 10 11
> 0 3 4 5 7 8 11
> 0 1 4 5 6 8 9 **
> 1 2 5 6 7 9 10
> 2 3 6 7 8 10 11
> 0 3 4 7 8 9 11
> 0 1 4 5 8 9 10 ***
> 1 2 5 6 9 10 11
> 0 2 3 6 7 10 11
> For this (random) heptachord, there are 3 transpositions levels
> of inverse forms that hold 6 elements invariant (marked with
> asterisks).
>
> In alphabetical representation:
> Original: C C# E F G# A B
> Inversion *: C C# D E F G# A
> Inversion **: C C# E F F# G# A
> Inversion ***: C C# E F G# A Bb
>
> Note: the subset {0,1,4,5,8,9} or {C C# E F G# A} is common to
> all these forms.

Ok, thanks. All this stuff is so scrambled, I have no idea
how anybody got the idea it would be interesting to listen to.
No ordinary musician would take a melody and reorder its
pitches, etc. Yet you talk about it like it's normal. Try
it on Neo-old and see what you get.

> > In other words, it's
> > useful if the music was composed using set theory!
[snip]
> Atonal music is rather different of tonal music, in which a
> small number of set types prevail.

I rest my case.

> And yes, we can also imagine that a scientific music theory can
> be created simply from psychoacoustical experiments on frequency
> ratios followed by speculation on uncommon and artificial scales
> (this is generative). Are there any analytical musical
> applications of regular mapping on microtonal music? Or is it
> restricted to analyses of scales?

Just to scales. As I said, it's a theory of intonation, not
music composition.

> > > http://www.youtube.com/watch?v=u6BzLwHLKis
> >
> > I see what you mean about orchestration (and I'll happily
> > admit that Boulez is one of my favorite conductors). Maybe
> > a piano reduction would be best for our experiment, if indeed
> > we decide to use an existing piece.
> >
> > > http://www.youtube.com/watch?v=8GymJUFFwlI
> >
> > ...This piece, though nice, doesn't speak much to me. It is
> > like good incidental music.
>
> Incidental? :-0

Maybe a language barrier... this means, program music for a
play or movie. It is not derogatory in itself, though I would
hardly call something like the Hammerklavier incidental music.

> > > "Introduction to Post-Tonal Theory" (Joseth N. Straus, 1990)
> > > is perhaps more interesting as an introductory text, because
> > > it is didactic and more rich in musical examples. There are
> > > newer concepts not addressed by Forte's book, and interesting
> > > analysis of dodecaphonic pieces. However several concepts and
> > > data provided by Forte are not aborded.
> >
> > Ok, thanks. Unfortunately this book is much more expensive!

What do you make of this?

http://www.mta.ca/faculty/arts-letters/music/pc-set_project/pc-set_new/pages/introduction/toc.html

> > > How about comparing those other sets (as subsets of 12-EDO)?
> > > A={C,Eb,G,Bb}
> > > B={D,Eb,G#,A}
> > > C={C,E,G,B}
> > > D={B,C,F,Bb}
> >
> > I played the pitches in ascending order starting as close
> > to middle C as possible. I tried notes simultaneously and
> > arpeggiated. I observed that A and C are more consonant
> > than B or D, as would be predicted by psychoacoustics and
> > normal music theory. D was most dissonant. Other than
> > that, I heard no significant attributes.
>
> How can these dissonance predictions be made using "normal
> music theory"? And/or: is there a simplified way to use
> psychoacoustic principles to compare those chords?

Normal music theory explains consonant chords as being made
of 3rds. Minor 2nds are explicitly dissonant, and Maj7 &
min7 tetrads are explicitly consonant.

As for the psychoacoustics approach, we have two options:
1. Error from just intonation
2. Harmonic entropy

Option 1 goes like this:

1a. List all tetrads, expressed w:x:y:z where each term is an
integer and GCD(w,x,y,z) = 1, such that the product w*x*y*z
is < some cutoff.

1b. Compute each tetrad's consonance by geomean(w,x,y,z).

1c. Target chord {W,X,Y,Z} where each term is a real number
representing a frequency ratio above some 1.0, has error
max(
|log(X/W) - log(x/w)| / log(x*w')
|log(Y/W) - log(y/w)| / log(y*w')
|log(Y/X) - log(y/x)| / log(y*x')
etc.
)
to each JI tetrad, where x*w' is x*w when x/w is in lowest
terms.

1d. Find the JI tetrad to which the target chord has the
smallest error (choose the more consonant tetrad in case of
a tie). Target chord is said to "approximate" this chord and
shares its consonance rating when the error is small.

1e. This approach fails somewhat for so-called "magic" chords
like the dim7 in 12-ET, which, it can be argued, have no true
origin in just intonation.

Option 2 goes like this:

2a. Get Microsoft Excel.
2b. Get this spreadsheet:
http://lumma.org/music/theory/DyadicHarmonicEntropyCalc.xls
2c. Plug in the cents values for the target chord.
2d. Get the result.
2e. This is sum of dyadic entropies, whereas what we really
want is tetradic entropy directly. Unfortunately it is an
open problem to compute tetradic harmonic entropy.

-Carl

I'm gonna go back to lurking, but this is a fascinating thread. I'm
learning a lot. Particularly, this quote:

> Music set theory
> http://en.wikipedia.org/wiki/Set_theory_%28music%29
>
> and the theory of regular temperament
> http://x31eq.com/paradigm.html
>
> are two competing explanations for the foundations of music.
> One explicitly addresses intonation and the other doesn't, but
> Paul has done some microtonal music set theory, and discussion
> of the relationship between the two theories is on-topic in
> any case.
>
> -Carl

is a very interesting take on it. Perhaps I need to read up on set
theory more, as I have never thought it aimed to be a descriptive
theory at how "music" emerges, but rather another charted out
serialist-esque trend that isn't worth paying attention to.

-Mike

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@> wrote:
[...]
> > Here I should make a comparison: there is a book which explains
> > tonal harmony using set theory (it is absolutely not related to
> > Babbitt/Forte/etc. works). A reasonable number of tonal relations
> > can be represented that way,
>
> Does it work in 31-ET?

I don't know, I did not read the entire book (since I assumed it could not bring any significant contribution for tonal music, since tonal harmony is already reasonably well understood with traditional conceptual apparatus), but I might borrow it from a library and seek adaptable concepts.

The book assumes 12-EDO, since that is the main reference for common-pratice tonal music, including in special "enharmonic" relations (which are different in 31-EDO). It may be possible that its ideas can be adapted to 31-EDO, just as Torsten Anders did with Schoenberg's guidelines on chord progressions.

>
> > > > * If the universe is 12-EDO, according to Forte, _every_
> > > > pc set with n=5 or n=7 elements can hold a subset with n-1
> > > > elements invariant under inversion,
> > >
> > > Can you give an example?
> >
> > Random pitches: 1 8 11 12 4 9 5
> > Normal form: 8 9 11 0 1 4 5
> > Ascending order: 0 1 4 5 8 9 11
> > Inversions (in ascending order):
> > 0 1 3 4 7 8 11
> > 0 1 2 4 5 8 9 *
> > 1 2 3 5 6 9 10
> > 2 3 4 6 7 10 11
> > 0 3 4 5 7 8 11
> > 0 1 4 5 6 8 9 **
> > 1 2 5 6 7 9 10
> > 2 3 6 7 8 10 11
> > 0 3 4 7 8 9 11
> > 0 1 4 5 8 9 10 ***
> > 1 2 5 6 9 10 11
> > 0 2 3 6 7 10 11
> > For this (random) heptachord, there are 3 transpositions levels
> > of inverse forms that hold 6 elements invariant (marked with
> > asterisks).
> >
> > In alphabetical representation:
> > Original: C C# E F G# A B
> > Inversion *: C C# D E F G# A
> > Inversion **: C C# E F F# G# A
> > Inversion ***: C C# E F G# A Bb
> >
> > Note: the subset {0,1,4,5,8,9} or {C C# E F G# A} is common to
> > all these forms.
>
> Ok, thanks. All this stuff is so scrambled, I have no idea
> how anybody got the idea it would be interesting to listen to.

I am just showing that the relation between inverse forms of 5- and 7-tone sets in 12-EDO is closer than one could expect.

> No ordinary musician would take a melody and reorder its
> pitches, etc.

Anyway, such procedure (permutation) has been used often by (for example) Bartók, Webern, Messiaen and even Stravinsky (there is an example by Stravinsky in Forte's book, section about ordered sets).

By coincidence, the invariant subset above is a mode of limited transposition. This property (total invariance under transposition) can be identified with set theory tools (for example: just look at the interval vectors in the sets list for a number equal to the cardinality of the set).

> Yet you talk about it like it's normal. Try
> it on Neo-old and see what you get.

Neo-old is tonal harmony, that can be better analysed with other tools, since functional relations should be considered. Therefore, I don't think pitch-class set theory can be efficiently used on tonal music. Its "efficiency" on atonal music is limited, but it still provides useful concepts and tools.

[...]
> > And yes, we can also imagine that a scientific music theory can
> > be created simply from psychoacoustical experiments on frequency
> > ratios followed by speculation on uncommon and artificial scales
> > (this is generative). Are there any analytical musical
> > applications of regular mapping on microtonal music? Or is it
> > restricted to analyses of scales?
>
> Just to scales. As I said, it's a theory of intonation, not
> music composition.

You said before that "Music set theory" and "the theory of regular temperament" are "two competing explanations for the foundations of music", in:
/tuning/topicId_85909.html#85929

That is not correct, since their respective objects are far different. I could try to understand that as a aesthetical contend, where a group defends the usage of harmonic elements that are consonant and easy to recognize, while the other group defends the usage of other harmonic material, each group accusing the other as illegitimate; this is also false, and, as I said before, I am not interested in this kind of "polemics".

My point -- and the motive I have engaged myself in this debate -- is: when the analysing the potential of a chosen scale for microtonal composition, one may take advantage of several tools (which include the referred "theories").

Composers who use procedures derived of atonal and serial music, or that want to compose microtonal atonal music, could benefit of the application of set theory to microtonal scales. You know, Scala includes a number of set theory representations and operations (pitch classes, interval classes, intersect, difference, merge, modes represented as number strings, etc.), and I find that useful. Of course, Scala includes a huge number of other resources, which complement pitch-class sets operations, if one starts from this; and one cannot ignore acoustical and qualitative characteristics of sets, scales, tunings and chords, which only are experienced by listening to them.

My aim was (and still is) to bring useful and constructive output from this debate, by removing some prejudices and misconcepts against pitch-class set theory as whole, so that useful things be not thrown in the trash. (The comparison of those 4 chords provided a point for further exploration which I think can be very interesting.)

[...]
> What do you make of this?
>
> http://www.mta.ca/faculty/arts-letters/music/pc-set_project/pc-set_new/pages/introduction/toc.html

It is a very concise introduction to the basic concepts, in the style of J.N.Straus book, but with only a few musical examples. It seems to be based on Straus' book contents.

(There you can find, BTW, that exaggerated assumption of similarity based on Z-relation. I can provide an analysis of the most famous pair of Z-related sets showing interesting "oppositions" between then.)

See also Solomon's site:
http://solomonsmusic.net/setheory.htm
http://solomonsmusic.net/
He also provides a software for pitch class sets analysis (I guess it is "unfortunately" restrict to 12-EDO).
http://solomonsmusic.net/mas.htm

>
> > > > How about comparing those other sets (as subsets of 12-EDO)?
> > > > A={C,Eb,G,Bb}
> > > > B={D,Eb,G#,A}
> > > > C={C,E,G,B}
> > > > D={B,C,F,Bb}
> > >
> > > I played the pitches in ascending order starting as close
> > > to middle C as possible. I tried notes simultaneously and
> > > arpeggiated. I observed that A and C are more consonant
> > > than B or D, as would be predicted by psychoacoustics and
> > > normal music theory. D was most dissonant. Other than
> > > that, I heard no significant attributes.
> >
> > How can these dissonance predictions be made using "normal
> > music theory"? And/or: is there a simplified way to use
> > psychoacoustic principles to compare those chords?
>
> Normal music theory explains consonant chords as being made
> of 3rds. Minor 2nds are explicitly dissonant, and Maj7 &
> min7 tetrads are explicitly consonant.

That (rather simplified description) seems to be in conflict with the old practice which requires special treatment of 7ths. This also ignores that the dim and dim7 chords -- made of 3rds! -- were traditionally considered as dissonant. And you have omitted the P5, which is important.

Nonetheless, you referred to the intervals that constitute a chord, and its interval structure. Here are the respective interval vectors of the chords (analysed as pitch-class sets).

interval class 1 2 3 4 5 6
m2 M2 m3 M3 P4 A4
A={C,Eb,G,Bb} < 0 1 2 1 2 0 >
B={D,Eb,G#,A} < 2 0 0 0 2 2 >
C={C,E,G,B} < 1 0 1 2 2 0 >
D={B,C,F,Bb} < 2 1 0 0 2 1 >

From it one can deduce that A and C are likely more consonant than B and D, assuming, that intervals of the classes of m2, M2 and A4 are more dissonant than intervals of the classes of m3, M3 and P4. All the 4 chords have 2 intervals of class 5 (P4) -- thus we compare only the remaining consonant classes (3 and 4) against the dissonant ones (1, 2 and 6), and find that B and D are more dissonant than A and C, since they do not contain these consonant intervals, but only the dissonant ones. It is, of course, a rough estimative (since, to cite only one problem, it ignores the difference between m2 and M7), but it works within its large error margin -- it might be useful to find set types near the extreme cases.

Further comparison is not possible for consonance/dissonance from this simplified model (Krenek/Hindemith would propose a slightly finer dissonance classification). One could analyse the chords as pitch sets, rather than pitch-class sets, thus considering the difference between m2 and M7, and pitch register, assuming psychoacoustical properties of the intervals in the tuning (12-EDO in this example).

For analytical ears, the presence of only 3 interval classes is significant (in special the 2 tritones, in the comparison above) for the identification of this remarkable set, very often used in atonal music.

>
> As for the psychoacoustics approach, we have two options:
> 1. Error from just intonation
> 2. Harmonic entropy
>
> Option 1 goes like this:
>
> 1a. List all tetrads, expressed w:x:y:z where each term is an
> integer and GCD(w,x,y,z) = 1, such that the product w*x*y*z
> is < some cutoff.
>
> 1b. Compute each tetrad's consonance by geomean(w,x,y,z).
>
> 1c. Target chord {W,X,Y,Z} where each term is a real number
> representing a frequency ratio above some 1.0, has error
> max(
> |log(X/W) - log(x/w)| / log(x*w')
> |log(Y/W) - log(y/w)| / log(y*w')
> |log(Y/X) - log(y/x)| / log(y*x')
> etc.
> )
> to each JI tetrad, where x*w' is x*w when x/w is in lowest
> terms.
>
> 1d. Find the JI tetrad to which the target chord has the
> smallest error (choose the more consonant tetrad in case of
> a tie). Target chord is said to "approximate" this chord and
> shares its consonance rating when the error is small.
>
> 1e. This approach fails somewhat for so-called "magic" chords
> like the dim7 in 12-ET, which, it can be argued, have no true
> origin in just intonation.

It would be very interesting to compare pitch sets in equal temperaments using this approach, and then classify them in pitch-class sets, and look for pitch-class sets which error is maximum or minimum -- that is: which pitch-class sets are more resistent or weak to pitch/interval disposition along the octave registers.

That analysis could be tested in several equal temperaments.

>
> Option 2 goes like this:
>
> 2a. Get Microsoft Excel.
> 2b. Get this spreadsheet:
> http://lumma.org/music/theory/DyadicHarmonicEntropyCalc.xls
> 2c. Plug in the cents values for the target chord.
> 2d. Get the result.
> 2e. This is sum of dyadic entropies, whereas what we really
> want is tetradic entropy directly. Unfortunately it is an
> open problem to compute tetradic harmonic entropy.
>
> -Carl
>

Is there any other implementation, or a detailed description of the algorithm? I cannot use spreadsheets, and I am not sure they can be used on a large number of 12-EDO subsets.

Otherwise, if you can do it, would you compare all chords inside the range of 2 octaves?

Cheers,
Hudson

Hi Franklin-

I agree that the title of the books should be "Taruskin's History of
Western Music", and he clearly explains in the intro that it does not
pretend to be a 5vol Grout. If I want pure factoid music history I
would go to Grove, but for entertainment and intellectual stimulation,
Taruskin brings the bacon.

I can see you have a fraught personal and professional relationship
with him, whereas to me he's just fun to read. His serious scholarship
resides in his books on Russian music, Shostokovich and, as you
mention, Stravinsky.

> There's a lot of wonderful music out there; modern music didn't end with Xenakis and friends. I'm co-editing a series--New Music >and Aesthetics in the 21st Century (Wolke Verlag)--that focuses on younger composers, and am co-editing a journal, Search, >which has a similar focus (a compilation of the first year's writings will be coming out soon from Mellen Press).  There's been a >serious gap in coverage of adventurous composers, and most people are still fixated on the serialists vs. non-serialists issue, >which is decades past its prime.

I'd love to hear some recent interesting music, I find most of it
seems to be posted on these lists anyway. (despite the opinion of our
current resident retard). If you can point me to any listenables on a
website I'd be grateful.

Dante

Hi Franklin,

Let me jump in here with:

1. Like Dante I'd like to hear some examples. And thank you for the
interesting dialogue.

2. For a little bit of discussion - my impression, now after composing for
30+ years, is that writing true atonal music is rather hard - perhaps this
is because the western audience ear is looking for a tonal center out of
habit, or perhaps because it is truly hard to do given the many ways one can
establish a tonal center. Right now I view tonality as a continuum - on the
far left is ultimate tonal music made of one note - on the far right is
ultimate atonality represented by true random tone rows presented in a
random fashion. Given this view once the composer move away from the
ultimate right you have *some* degree of tonality. And this is the reason
for my question / view. (and yes this example is within the realm of 12 EDO
- I guess one could put "noise" on the far right though I think the same
principal would apply.)

Thanks,

Chris Vaisvil

> I'd love to hear some recent interesting music, I find most of it
> seems to be posted on these lists anyway. (despite the opinion of our
> current resident retard). If you can point me to any listenables on a
> website I'd be grateful.
>
> Dante
>
>
>

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:

> 2. For a little bit of discussion - my impression, now after
> composing for 30+ years, is that writing true atonal music is
> rather hard - perhaps this is because the western audience
> ear is looking for a tonal center out of habit, or perhaps
> because it is truly hard to do given the many ways one can
> establish a tonal center.

It's definitely hard in 12-ET, which is chock full of tonality.
As Wendy Carlos points out, it gets a heck of a lot easier in
something like 11-ET (or maybe using Schulter & Keenan's
metastable scales, based on silver ratios).

I suppose the serialist would retort that achieving atonality
despite this difficulty is the whole point. And that isn't
unreasonable, but it's also just not that interesting. I can
program a computer to generate music all day that avoids
tonality. As I said, before computers, this stuff probably
seemed a lot cooler.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@> wrote:
>
> > 2. For a little bit of discussion - my impression, now after
> > composing for 30+ years, is that writing true atonal music is
> > rather hard - perhaps this is because the western audience
> > ear is looking for a tonal center out of habit, or perhaps
> > because it is truly hard to do given the many ways one can
> > establish a tonal center.
>
> It's definitely hard in 12-ET, which is chock full of tonality.

That is why Schoenberg always said that "atonal" music does not exist, and that his music is "pantonal".

> As Wendy Carlos points out, it gets a heck of a lot easier in
> something like 11-ET (or maybe using Schulter & Keenan's
> metastable scales, based on silver ratios).

Schoenberg (again) when abording floating tonality, extended tonality, and pantonality in his Harmonielehre, writes that the tonality may or may not be stablished. Also, pantonality is a continuity or "development" of traditional tonality. There is no the objective to expurge completely the tonal relations. Schoenberg was opened to microtonalism (he cites Busoni), always as an extension of the traditional material, not as a disruption. A new style may carry "histories".

Another issue is that is difficult to use alternative tunings in traditional instruments, and there are few interprets able to play microtonal music. These are severe practical limitations, and yet today this is a problem, for instrumental music.

>
> I suppose the serialist would retort that achieving atonality
> despite this difficulty is the whole point. And that isn't
> unreasonable, but it's also just not that interesting. [...]

Ahahaha! :-)

> I suppose the serialist would retort that achieving atonality
> despite this difficulty is the whole point.

Microtonalism _has been_ used by serialists and by contemporary composers they have influenced. Examples are Stockhausen, Boulez, Ferneyhough, Arthur Kampela and Flo Menezes.

> And that isn't
> unreasonable, but it's also just not that interesting. I can
> program a computer to generate music all day that avoids
> tonality. As I said, before computers, this stuff probably
> seemed a lot cooler.

Computer music (electroacoutics) explores many new possibilities provided by the computer, therefore microtonalism _as a pitch-oriented approach_ (just like tonalism, atonalism and other scale-based possibilities) is usually considered "no that interesting", compared to other possibilities.

Hudson

--- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@...> wrote:

> > It's definitely hard in 12-ET, which is chock full of tonality.
>
> That is why Schoenberg always said that "atonal" music does
> not exist, and that his music is "pantonal".

Just semantics.

> Another issue is that is difficult to use alternative tunings
> in traditional instruments, and there are few interprets able
> to play microtonal music. These are severe practical
> limitations, and yet today this is a problem, for instrumental
> music.

Yes, it's easy to forgive Schoenberg on account of the greater
limitations back then. However, it didn't stop Partch.

> > I suppose the serialist would retort that achieving atonality
> > despite this difficulty is the whole point.
>
> Microtonalism _has been_ used by serialists and by contemporary
> composers they have influenced. Examples are Stockhausen,
> Boulez, Ferneyhough, Arthur Kampela and Flo Menezes.

Stockhausen didn't obtain microtonality through control of
known pitches that I know of. Correct me if I'm wrong.
I stopped paying attention when I saw this nonsense:
http://en.wikipedia.org/wiki/Helikopter-Streichquartett

I'll refrain from suggesting that any of these names are
aptonyms... I've heard Ferneyhough and Boulez but not the
others.

> Computer music (electroacoutics) explores many new
> possibilities provided by the computer, therefore
> microtonalism _as a pitch-oriented approach_ (just like
> tonalism, atonalism and other scale-based possibilities)
> is usually considered "no that interesting", compared to
> other possibilities.

Actually it's music concrete techniques that are boring.
The human auditory system is designed to extract information
from speech, which pitched music strongly resembles. It is
simply impossible to obtain the same bandwidth to the
listener with noise-based techniques.

-Carl

carl, you seem to state your opinions as something like categorical fact in the last paragraph---"it is simply impossible to obtain the same bandwidth to the listener with noise-based techniques", etc.
after over 30 years in music, most of spent pursuing music most people call noise, i'm of a different opinion----or at least a mindset that still says (or at least tries to say) to my similar dislikes, that I'd be better off to holster too much finalized judgment given that a certain group of people seem to hold an honest opinion based on their experiences that is at odds with my own.
Past a certain point---and short of certain horrors---- i can agree to disagree given the idea that some people really are different than others, and as such are likely to embrace, champion and make work things i could and would not and vis-a-vis .
Daniel

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@> wrote:
>
> > > It's definitely hard in 12-ET, which is chock full of tonality.
> >
> > That is why Schoenberg always said that "atonal" music does
> > not exist, and that his music is "pantonal".
>
> Just semantics.
>
> > Another issue is that is difficult to use alternative tunings
> > in traditional instruments, and there are few interprets able
> > to play microtonal music. These are severe practical
> > limitations, and yet today this is a problem, for instrumental
> > music.
>
> Yes, it's easy to forgive Schoenberg on account of the greater
> limitations back then. However, it didn't stop Partch.
>
> > > I suppose the serialist would retort that achieving atonality
> > > despite this difficulty is the whole point.
> >
> > Microtonalism _has been_ used by serialists and by contemporary
> > composers they have influenced. Examples are Stockhausen,
> > Boulez, Ferneyhough, Arthur Kampela and Flo Menezes.
>
> Stockhausen didn't obtain microtonality through control of
> known pitches that I know of. Correct me if I'm wrong.
> I stopped paying attention when I saw this nonsense:
> http://en.wikipedia.org/wiki/Helikopter-Streichquartett
>
> I'll refrain from suggesting that any of these names are
> aptonyms... I've heard Ferneyhough and Boulez but not the
> others.
>
> > Computer music (electroacoutics) explores many new
> > possibilities provided by the computer, therefore
> > microtonalism _as a pitch-oriented approach_ (just like
> > tonalism, atonalism and other scale-based possibilities)
> > is usually considered "no that interesting", compared to
> > other possibilities.
>
> Actually it's music concrete techniques that are boring.
> The human auditory system is designed to extract information
> from speech, which pitched music strongly resembles. It is
> simply impossible to obtain the same bandwidth to the
> listener with noise-based techniques.
>
> -Carl
>

Carl, 
The distinctions made by Schoenberg are indeed semantic distinctions, and correspond to different content. If you say "just semantics," you are implying that all of these distinctions made by Schoenberg were mere illusions. One could just as well turn that around and claim that every distinction you make is illusory, based on an imprecise and/or incoherent grasp on reality. Now if I were to judge between the two sets of claims--those by Schoenberg, one of the great composers, theorists, orchestrators, innovators, and musical influences of the twentieth century, who conducted pioneering performances, who received the higheset respect of many of the great musical minds of the first half of the century--and those made by you, I would tend to give greater credence to Schoenberg's.
All of these distinctions would be a lot more pleasant if 1) you avoided wholesale condemnations and dismissals of other musicians' work, and 2) you didn't run away from the consequences of your own statements, as you've done about a half dozen times in our few interactions.

Franklin

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@...

--- On Mon, 2/8/10, Carl Lumma <carl@...> wrote:

From: Carl Lumma <carl@lumma.org>
Subject: [tuning] Re: Musical Set Theory in 12-tET
To: tuning@yahoogroups.com
Date: Monday, February 8, 2010, 9:32 PM

--- In tuning@yahoogroups. com, "hfmlacerda" <hfmlacerda@ ...> wrote:

> > It's definitely hard in 12-ET, which is chock full of tonality.

>

> That is why Schoenberg always said that "atonal" music does

> not exist, and that his music is "pantonal".

Just semantics.

> Another issue is that is difficult to use alternative tunings

> in traditional instruments, and there are few interprets able

> to play microtonal music. These are severe practical

> limitations, and yet today this is a problem, for instrumental

> music.

Yes, it's easy to forgive Schoenberg on account of the greater

limitations back then. However, it didn't stop Partch.

> > I suppose the serialist would retort that achieving atonality

> > despite this difficulty is the whole point.

>

> Microtonalism _has been_ used by serialists and by contemporary

> composers they have influenced. Examples are Stockhausen,

> Boulez, Ferneyhough, Arthur Kampela and Flo Menezes.

Stockhausen didn't obtain microtonality through control of

known pitches that I know of. Correct me if I'm wrong.

I stopped paying attention when I saw this nonsense:

http://en.wikipedia .org/wiki/ Helikopter- Streichquartett

I'll refrain from suggesting that any of these names are

aptonyms... I've heard Ferneyhough and Boulez but not the

others.

> Computer music (electroacoutics) explores many new

> possibilities provided by the computer, therefore

> microtonalism _as a pitch-oriented approach_ (just like

> tonalism, atonalism and other scale-based possibilities)

> is usually considered "no that interesting" , compared to

> other possibilities.

Actually it's music concrete techniques that are boring.

The human auditory system is designed to extract information

from speech, which pitched music strongly resembles. It is

simply impossible to obtain the same bandwidth to the

listener with noise-based techniques.

-Carl

--- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@...> wrote:

> > Just to scales. As I said, it's a theory of intonation, not
> > music composition.
>
> You said before that "Music set theory" and "the theory of
> regular temperament" are "two competing explanations for the
> foundations of music", in:
> /tuning/topicId_85909.html#85929
>
> That is not correct, since their respective objects are far
> different. I could try to understand that as a aesthetical
> contend, where a group defends the usage of harmonic elements
> that are consonant and easy to recognize, while the other
> group defends the usage of other harmonic material, each group
> accusing the other as illegitimate; this is also false, and,
> as I said before, I am not interested in this kind of
> "polemics".

Have a look:

http://en.wikipedia.org/wiki/Diatonic_scale#Further_reading

You can see a list of authors attempting to derive the
diatonic scale -- the foundation of common practice music --
with group or "set" theory methods. We can add Agmon and
Tymoczko to the list. Rothenberg is an exception. I haven't
read Mark's paper, but if you're reading Mark, I'd like to.

You yourself seem to shift between stating that set theory
is good only as a generative tool (why it would be preferred
to other generative methods is another question) or for
analyzing atonal music (not other forms). And you've said
it doesn't matter if the relations are audible. But you've
also seemed to say that Forte's relations are simple truths,
and audible to a point, and therefore valid or even
inescapable considerations in music. Which is it?

Then you have folks like Keller, apparently saying that
Mozart and Beethoven were serialists. You have Franklin
saying that set theory explains neoclassicism. Tell me
where to draw the line.

Regular mapping starts with psychoacoustics, which is an
experimental science. We find it supports many rules of
common-practice theory. That's interesting. With regular
mapping we can derive the diatonic scale purely from the
consonances/note consideration, along with other scales we
predict should behave similarly. And it obviously works:

http://lumma.org/tuning/erlich/decatonic-swing.mp3

This is in 22, using the pajara scales I just recommended
to Mike.

We can also find tunings ideal for atonality and probably
even measure the tonality in a piece (I expect, proportional
to the average harmonic entropy of its simultaneities, and
inversely proportional to the standard deviation).

>> What do you make of this?
>
>> http://www.mta.ca/faculty/arts-letters/music/pc-set_project/
>> pc-set_new/pages/introduction/toc.html
>
> It is a very concise introduction to the basic concepts, in
> the style of J.N.Straus book, but with only a few musical
> examples. It seems to be based on Straus' book contents.

Then I'll start here, since it's free.

> (There you can find, BTW, that exaggerated assumption of
> similarity based on Z-relation. I can provide an analysis of
> the most famous pair of Z-related sets showing interesting
> "oppositions" between then.)

Have you published on this?

> See also Solomon's site:
> http://solomonsmusic.net/setheory.htm
> http://solomonsmusic.net/

Thanks!

> He also provides a software for pitch class sets analysis
> (I guess it is "unfortunately" restrict to 12-EDO).
> http://solomonsmusic.net/mas.htm

Looks like a DOS program... a bit outdated.

> Nonetheless, you referred to the intervals that constitute
> a chord, and its interval structure. Here are the respective
> interval vectors of the chords (analysed as pitch-class sets).
>
> interval class 1 2 3 4 5 6
> m2 M2 m3 M3 P4 A4
> A={C,Eb,G,Bb} < 0 1 2 1 2 0 >
> B={D,Eb,G#,A} < 2 0 0 0 2 2 >
> C={C,E,G,B} < 1 0 1 2 2 0 >
> D={B,C,F,Bb} < 2 1 0 0 2 1 >
>
> From it one can deduce that A and C are likely more consonant
> than B and D, assuming, that intervals of the classes of m2,
> M2 and A4 are more dissonant than intervals of the classes of
> m3, M3 and P4.

When you get to extended JI, we find that order matters.
For instance, compare 4:5:6:7:9:11 with its subharmonic
inverse.

> > Option 2 goes like this:
> >
> > 2a. Get Microsoft Excel.
> > 2b. Get this spreadsheet:
> > http://lumma.org/music/theory/DyadicHarmonicEntropyCalc.xls
> > 2c. Plug in the cents values for the target chord.
> > 2d. Get the result.
> > 2e. This is sum of dyadic entropies, whereas what we really
> > want is tetradic entropy directly. Unfortunately it is an
> > open problem to compute tetradic harmonic entropy.
>
> Is there any other implementation, or a detailed description
> of the algorithm? I cannot use spreadsheets, and I am not sure
> they can be used on a large number of 12-EDO subsets.

Excel will handle many millions of rows these days. Anyway
this is just a lookup table that sums the entropies of the
dyads in a chord. The entropy calculation itself has been
described on the harmonic_entropy list.

> Otherwise, if you can do it, would you compare all chords
> inside the range of 2 octaves?

Which chords exactly?

-Carl

--- In tuning@yahoogroups.com, Cox Franklin <franklincox@...> wrote:

> The distinctions made by Schoenberg are indeed semantic
> distinctions, and correspond to different content. If you
> say "just semantics," you are implying that all of these
> distinctions made by Schoenberg were mere illusions.

How so? I said the distinction between atonality and
pantonality is just one of semantics. That can of course
be changed with definitions, which you are welcome to
provide.

> Now if I were to judge between the two sets of claims--those
> by Schoenberg, one of the great composers, theorists,
> orchestrators, innovators, and musical influences of the
> twentieth century, who conducted pioneering performances, who
> received the higheset respect of many of the great musical
> minds of the first half of the century--and those made by you,
> I would tend to give greater credence to Schoenberg's.

Appeal to authority. We can put it with the ad hominem
attacks and other cheap shots you've contributed to this
discussion so far.

> you didn't run away from the consequences of your own
> statements, as you've done about a half dozen times in our
> few interactions.

Pardon me, but what on earth are you talking about?

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@> wrote:
>
> > > It's definitely hard in 12-ET, which is chock full of tonality.
> >
> > That is why Schoenberg always said that "atonal" music does
> > not exist, and that his music is "pantonal".
>
> Just semantics.

I would expect you should agree, since 12-EDO "is chock full of tonality". For me, atonal music is also "is chock full of tonality". That may be one reason I feel atonal music as so expressive. It inherits moods from Romanticism harmony.

>
> > Another issue is that is difficult to use alternative tunings
> > in traditional instruments, and there are few interprets able
> > to play microtonal music. These are severe practical
> > limitations, and yet today this is a problem, for instrumental
> > music.
>
> Yes, it's easy to forgive Schoenberg on account of the greater
> limitations back then. However, it didn't stop Partch.

I did not said it is impossible, of course, but you know that are limitations. One needs to build new instruments and have people willing to learn to play them well. That is no trivial.

>
> > > I suppose the serialist would retort that achieving atonality
> > > despite this difficulty is the whole point.
> >
> > Microtonalism _has been_ used by serialists and by contemporary
> > composers they have influenced. Examples are Stockhausen,
> > Boulez, Ferneyhough, Arthur Kampela and Flo Menezes.
>
> Stockhausen didn't obtain microtonality through control of
> known pitches that I know of. Correct me if I'm wrong.

Yes, you are wrong. See:
http://en.wikipedia.org/wiki/Microtonal_music

I also have in hands a book by Flo Menezes (on musical acoustics), which refers to the scale used in Stockhausen's "Studie II": 25 equal divisions of 5/1, on a reference frequency of 100 Hz.

> I stopped paying attention when I saw this nonsense:
> http://en.wikipedia.org/wiki/Helikopter-Streichquartett

Are you suggesting it is microtonal?

>
> I'll refrain from suggesting that any of these names are
> aptonyms... I've heard Ferneyhough and Boulez but not the
> others.

The other ones are Brazilian composers.
http://www.kampela.com/
http://www.flomenezes.mus.br/

>
> > Computer music (electroacoutics) explores many new
> > possibilities provided by the computer, therefore
> > microtonalism _as a pitch-oriented approach_ (just like
> > tonalism, atonalism and other scale-based possibilities)
> > is usually considered "no that interesting", compared to
> > other possibilities.
>
> Actually it's music concrete techniques that are boring.

I usually agree. That is why I delimited _electroacoustic_ music from the larger realm of computer music. By electroacoustic I mean a thread that was more influenced by "elektronische Musik" than by "musique concrète". (It has some roots in serial music.)

> The human auditory system is designed to extract information
> from speech, which pitched music strongly resembles. It is
> simply impossible to obtain the same bandwidth to the
> listener with noise-based techniques.

Your conclusion is overall consistent with what serialist and electroacoustic composers would say, but they would use different arguments.

(I must also to point out that absence of scale does not implies necessarily noise or complete absence of tones. Speech does not implies a scale either, though it assumes one is able to select harmonic spectra from the environment.)

BTW, I have found this Schoenberg's text which abords a related issue this way:

http://tonalsoft.com/monzo/schoenberg/problems/problems.htm
<<<---
...How, after all, can two tones be joined one with another?

My answer is that such a juxtaposition of tones, if a connection is to be brought about from which a piece of music may be the result, is only possible because a relation already exists between the tones themselves.

Logically, we can only join things that are related, directly or indirectly. In a piece of music I cannot establish a relation between a tone and, let us say, an eraser; simply because no musical relation exists.
--->>>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> > Now if I were to judge between the two sets of claims--those
> > by Schoenberg, one of the great composers, theorists,
> > orchestrators, innovators, and musical influences of the
> > twentieth century, who conducted pioneering performances, who
> > received the higheset respect of many of the great musical
> > minds of the first half of the century--and those made by you,
> > I would tend to give greater credence to Schoenberg's.
>
> Appeal to authority. We can put it with the ad hominem
> attacks and other cheap shots you've contributed to this
> discussion so far.
...
>
> -Carl

Maybe we can play the guessing game called:
"Who is or was an actual musician?"

After that maybe we can squeeze in a round of:
"Who is or was an impudent, obnoxious pretender?"

These are just games, mind you – just games.

Have fun!
AAH
=====

--- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@...> wrote:

> > Stockhausen didn't obtain microtonality through control of
> > known pitches that I know of. Correct me if I'm wrong.
>
> Yes, you are wrong. See:
> http://en.wikipedia.org/wiki/Microtonal_music

I stand corrected.

> > I stopped paying attention when I saw this nonsense:
> > http://en.wikipedia.org/wiki/Helikopter-Streichquartett
>
> Are you suggesting it is microtonal?

No, I'm suggesting it's nonsense. OK, it's been probably
10 years since I've heard it. But let me invoke Partch,

"What? - tickling a big brass gong with a toothpick?
Drinking carrot juice with an amplified gullet?
Prepared piano? ... Zen Buddhism? ... Showmanship? Fine.
Innovation? Not for me.
At a private affair someone asked, "What do you think
about Zen Buddhism? I replied, "I don't give a fuck about
Zen Buddhism," and someone at the back of the room
muttered, "There is a true Zen Buddhist!""

And again, this time writing to Cage,

"I sent you a scrap of manuscript because I did not want
to seem difficult... But when you insist on a statement
from me that is exactly 43 words you are being difficult.
You have done an unbelievably fine job of excerpting and
editing. It is probably better than my original statement.
However, if you dare to mention that number 43 you are
deliberately misrepresenting me. It is the one-half truth
of the one-fourth factor. And I shall *curse* you. You
have been cursed before, but never by me, and if you are
cursed by me there will be a difference."

> > I'll refrain from suggesting that any of these names are
> > aptonyms... I've heard Ferneyhough and Boulez but not the
> > others.
>
> The other ones are Brazilian composers.
> http://www.kampela.com/
> http://www.flomenezes.mus.br/

I'll share my favorite Brazilian composer, in case you
are not familiar. He shares your surname:

http://en.wikipedia.org/wiki/Osvaldo_Lacerda

> BTW, I have found this Schoenberg's text which abords a related
> issue this way:
>
> http://tonalsoft.com/monzo/schoenberg/problems/problems.htm
> <<<---
> ...How, after all, can two tones be joined one with another?
>
> My answer is that such a juxtaposition of tones, if a connection
> is to be brought about from which a piece of music may be the
> result, is only possible because a relation already exists
> between the tones themselves.
>
> Logically, we can only join things that are related, directly
> or indirectly. In a piece of music I cannot establish a relation
> between a tone and, let us say, an eraser; simply because no
> musical relation exists.
> --->>>

I couldn't agree more.

-Carl

--- In tuning@yahoogroups.com, "hpiinstruments" <aaronhunt@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> > > Now if I were to judge between the two sets of claims--those
> > > by Schoenberg, one of the great composers, theorists,
> > > orchestrators, innovators, and musical influences of the
> > > twentieth century, who conducted pioneering performances, who
> > > received the higheset respect of many of the great musical
> > > minds of the first half of the century--and those made by you,
> > > I would tend to give greater credence to Schoenberg's.
> >
> > Appeal to authority. We can put it with the ad hominem
> > attacks and other cheap shots you've contributed to this
> > discussion so far.
> ...
> >
> > -Carl
>
> > Maybe we can play the guessing game called:
> "Who is or was an actual musician?"
>
> After that maybe we can squeeze in a round of:
> "Who is or was an impudent, obnoxious pretender?"
>
> These are just games, mind you – just games.
>
> Have fun!
> AAH

Yes, games! Lot's of fun. So nice to hear from you, Aaron.
You've shown your true colors yet again.

-Carl

To the members of the tuning list,
This might be of interest to some of you. Apparently, my comments about Richard Taruskin were passed directly to Dr. Taruskin by someone on this list. A few hours ago I received a rather rude note from Dr. Taruskin. Small world.
Is this the normal practice on this list, to pass comments made in an informal setting on to other parties, without mentioning this to the poster? 
Franklin

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@...

--- On Mon, 2/8/10, hfmlacerda <hfmlacerda@...> wrote:

From: hfmlacerda <hfmlacerda@...>
Subject: [tuning] Re: Musical Set Theory in 12-tET
To: tuning@yahoogroups.com
Date: Monday, February 8, 2010, 10:57 PM

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

>

> --- In tuning@yahoogroups. com, "hfmlacerda" <hfmlacerda@ > wrote:

>

> > > It's definitely hard in 12-ET, which is chock full of tonality.

> >

> > That is why Schoenberg always said that "atonal" music does

> > not exist, and that his music is "pantonal".

>

> Just semantics.

I would expect you should agree, since 12-EDO "is chock full of tonality". For me, atonal music is also "is chock full of tonality". That may be one reason I feel atonal music as so expressive. It inherits moods from Romanticism harmony.

>

> > Another issue is that is difficult to use alternative tunings

> > in traditional instruments, and there are few interprets able

> > to play microtonal music. These are severe practical

> > limitations, and yet today this is a problem, for instrumental

> > music.

>

> Yes, it's easy to forgive Schoenberg on account of the greater

> limitations back then. However, it didn't stop Partch.

I did not said it is impossible, of course, but you know that are limitations. One needs to build new instruments and have people willing to learn to play them well. That is no trivial.

>

> > > I suppose the serialist would retort that achieving atonality

> > > despite this difficulty is the whole point.

> >

> > Microtonalism _has been_ used by serialists and by contemporary

> > composers they have influenced. Examples are Stockhausen,

> > Boulez, Ferneyhough, Arthur Kampela and Flo Menezes.

>

> Stockhausen didn't obtain microtonality through control of

> known pitches that I know of. Correct me if I'm wrong.

Yes, you are wrong. See:

http://en.wikipedia .org/wiki/ Microtonal_ music

I also have in hands a book by Flo Menezes (on musical acoustics), which refers to the scale used in Stockhausen' s "Studie II": 25 equal divisions of 5/1, on a reference frequency of 100 Hz.

> I stopped paying attention when I saw this nonsense:

> http://en.wikipedia .org/wiki/ Helikopter- Streichquartett

Are you suggesting it is microtonal?

>

> I'll refrain from suggesting that any of these names are

> aptonyms... I've heard Ferneyhough and Boulez but not the

> others.

The other ones are Brazilian composers.

http://www.kampela. com/

http://www.flomenez es.mus.br/

>

> > Computer music (electroacoutics) explores many new

> > possibilities provided by the computer, therefore

> > microtonalism _as a pitch-oriented approach_ (just like

> > tonalism, atonalism and other scale-based possibilities)

> > is usually considered "no that interesting" , compared to

> > other possibilities.

>

> Actually it's music concrete techniques that are boring.

I usually agree. That is why I delimited _electroacoustic_ music from the larger realm of computer music. By electroacoustic I mean a thread that was more influenced by "elektronische Musik" than by "musique concrète". (It has some roots in serial music.)

> The human auditory system is designed to extract information

> from speech, which pitched music strongly resembles. It is

> simply impossible to obtain the same bandwidth to the

> listener with noise-based techniques.

Your conclusion is overall consistent with what serialist and electroacoustic composers would say, but they would use different arguments.

(I must also to point out that absence of scale does not implies necessarily noise or complete absence of tones. Speech does not implies a scale either, though it assumes one is able to select harmonic spectra from the environment. )

BTW, I have found this Schoenberg's text which abords a related issue this way:

http://tonalsoft. com/monzo/ schoenberg/ problems/ problems. htm

<<<---

...How, after all, can two tones be joined one with another?

My answer is that such a juxtaposition of tones, if a connection is to be brought about from which a piece of music may be the result, is only possible because a relation already exists between the tones themselves.

Logically, we can only join things that are related, directly or indirectly. In a piece of music I cannot establish a relation between a tone and, let us say, an eraser; simply because no musical relation exists.

--->>>

Hi Franklin,

I'm sorry this happened. How do you know it was someone who
is a member here?

You may know I'm a moderator of this list, but there's nothing
I can do to stop this kind of thing. Posts here are open to
the public.

For what it's worth, I found your review of Taruskin (whom
I've never read) to be apparently fair... nothing that would
warrant a rude comment from him.

-Carl

--- In tuning@yahoogroups.com, Cox Franklin <franklincox@...> wrote:
>
> To the members of the tuning list,
> This might be of interest to some of you. Apparently, my comments
> about Richard Taruskin were passed directly to Dr. Taruskin by
> someone on this list. A few hours ago I received a rather rude
> note from Dr. Taruskin. Small world.
> Is this the normal practice on this list, to pass comments made
> in an informal setting on to other parties, without mentioning
> this to the poster?
> Franklin
>
> Dr. Franklin Cox
> 1107 Xenia Ave.
> Yellow Springs, OH 45387
> (937) 767-1165
>
> franklincox@...

that sucks. does that mean its the end of our conversation about him?

On Mon, Feb 8, 2010 at 6:45 PM, Cox Franklin <franklincox@...> wrote:

>
>
> To the members of the tuning list,
>
> This might be of interest to some of you. Apparently, my comments about
> Richard Taruskin were passed directly to Dr. Taruskin by someone on this
> list. A few hours ago I received a rather rude note from Dr. Taruskin. Small
> world.
>
> Is this the normal practice on this list, to pass comments made in an
> informal setting on to other parties, without mentioning this to the
> poster?
>
> Franklin
>
>
> Dr. Franklin Cox
> 1107 Xenia Ave.
> Yellow Springs, OH 45387
> (937) 767-1165
> franklincox@...
>
> --- On *Mon, 2/8/10, hfmlacerda <hfmlacerda@...>* wrote:
>
>
> From: hfmlacerda <hfmlacerda@...>
> Subject: [tuning] Re: Musical Set Theory in 12-tET
> To: tuning@yahoogroups.com
> Date: Monday, February 8, 2010, 10:57 PM
>
>
>
> --- In tuning@yahoogroups. com<http://mc/compose?to=tuning%40yahoogroups.com>,
> "Carl Lumma" <carl@...> wrote:
> >
> > --- In tuning@yahoogroups. com<http://mc/compose?to=tuning%40yahoogroups.com>,
> "hfmlacerda" <hfmlacerda@ > wrote:
> >
> > > > It's definitely hard in 12-ET, which is chock full of tonality.
> > >
> > > That is why Schoenberg always said that "atonal" music does
> > > not exist, and that his music is "pantonal".
> >
> > Just semantics.
>
> I would expect you should agree, since 12-EDO "is chock full of tonality".
> For me, atonal music is also "is chock full of tonality". That may be one
> reason I feel atonal music as so expressive. It inherits moods from
> Romanticism harmony.
>
> >
> > > Another issue is that is difficult to use alternative tunings
> > > in traditional instruments, and there are few interprets able
> > > to play microtonal music. These are severe practical
> > > limitations, and yet today this is a problem, for instrumental
> > > music.
> >
> > Yes, it's easy to forgive Schoenberg on account of the greater
> > limitations back then. However, it didn't stop Partch.
>
> I did not said it is impossible, of course, but you know that are
> limitations. One needs to build new instruments and have people willing to
> learn to play them well. That is no trivial.
>
> >
> > > > I suppose the serialist would retort that achieving atonality
> > > > despite this difficulty is the whole point.
> > >
> > > Microtonalism _has been_ used by serialists and by contemporary
> > > composers they have influenced. Examples are Stockhausen,
> > > Boulez, Ferneyhough, Arthur Kampela and Flo Menezes.
> >
> > Stockhausen didn't obtain microtonality through control of
> > known pitches that I know of. Correct me if I'm wrong.
>
> Yes, you are wrong. See:
> http://en.wikipedia .org/wiki/ Microtonal_ music<http://en.wikipedia.org/wiki/Microtonal_music>
>
> I also have in hands a book by Flo Menezes (on musical acoustics), which
> refers to the scale used in Stockhausen' s "Studie II": 25 equal divisions
> of 5/1, on a reference frequency of 100 Hz.
>
> > I stopped paying attention when I saw this nonsense:
> > http://en.wikipedia .org/wiki/ Helikopter- Streichquartett<http://en.wikipedia.org/wiki/Helikopter-Streichquartett>
>
> Are you suggesting it is microtonal?
>
> >
> > I'll refrain from suggesting that any of these names are
> > aptonyms... I've heard Ferneyhough and Boulez but not the
> > others.
>
> The other ones are Brazilian composers.
> http://www.kampela. com/ <http://www.kampela.com/>
> http://www.flomenez es.mus.br/ <http://www.flomenezes.mus.br/>
>
> >
> > > Computer music (electroacoutics) explores many new
> > > possibilities provided by the computer, therefore
> > > microtonalism _as a pitch-oriented approach_ (just like
> > > tonalism, atonalism and other scale-based possibilities)
> > > is usually considered "no that interesting" , compared to
> > > other possibilities.
> >
> > Actually it's music concrete techniques that are boring.
>
> I usually agree. That is why I delimited _electroacoustic_ music from the
> larger realm of computer music. By electroacoustic I mean a thread that was
> more influenced by "elektronische Musik" than by "musique concrète". (It has
> some roots in serial music.)
>
> > The human auditory system is designed to extract information
> > from speech, which pitched music strongly resembles. It is
> > simply impossible to obtain the same bandwidth to the
> > listener with noise-based techniques.
>
> Your conclusion is overall consistent with what serialist and
> electroacoustic composers would say, but they would use different arguments.
>
> (I must also to point out that absence of scale does not implies
> necessarily noise or complete absence of tones. Speech does not implies a
> scale either, though it assumes one is able to select harmonic spectra from
> the environment. )
>
> BTW, I have found this Schoenberg's text which abords a related issue this
> way:
>
> http://tonalsoft. com/monzo/ schoenberg/ problems/ problems. htm<http://tonalsoft.com/monzo/schoenberg/problems/problems.htm>
> <<<---
> ...How, after all, can two tones be joined one with another?
>
> My answer is that such a juxtaposition of tones, if a connection is to be
> brought about from which a piece of music may be the result, is only
> possible because a relation already exists between the tones themselves.
>
> Logically, we can only join things that are related, directly or
> indirectly. In a piece of music I cannot establish a relation between a tone
> and, let us say, an eraser; simply because no musical relation exists.
> --->>>
>
>
>

Carl,
My posting on Taruskin referred to my TAing under him at Columbia University; I have not posted this information anywhere else. He referred specifically to Columbia University in in that time period, so he must have received my posting from someone on this list.
Franklin 

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@yahoo.com

--- On Mon, 2/8/10, Carl Lumma <carl@...> wrote:

From: Carl Lumma <carl@...>
Subject: [tuning] Re: Musical Set Theory in 12-tET
To: tuning@yahoogroups.com
Date: Monday, February 8, 2010, 11:50 PM

Hi Franklin,

I'm sorry this happened. How do you know it was someone who

is a member here?

You may know I'm a moderator of this list, but there's nothing

I can do to stop this kind of thing. Posts here are open to

the public.

For what it's worth, I found your review of Taruskin (whom

I've never read) to be apparently fair... nothing that would

warrant a rude comment from him.

-Carl

--- In tuning@yahoogroups. com, Cox Franklin <franklincox@ ...> wrote:

>

> To the members of the tuning list,

> This might be of interest to some of you. Apparently, my comments

> about Richard Taruskin were passed directly to Dr. Taruskin by

> someone on this list. A few hours ago I received a rather rude

> note from Dr. Taruskin. Small world.

> Is this the normal practice on this list, to pass comments made

> in an informal setting on to other parties, without mentioning

> this to the poster?

> Franklin

>

> Dr. Franklin Cox

> 1107 Xenia Ave.

> Yellow Springs, OH 45387

> (937) 767-1165

>

> franklincox@ ...

Carl,

I'm not sure what you mean by "true colors". I
merely suggested some games to play that might
be fun. If those games I suggested aren't any
fun, then please ignore my suggestion to play
them.

Yours,
AAH
=====

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "hpiinstruments" <aaronhunt@> wrote:
> >
> > --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> > > > Now if I were to judge between the two sets of claims--those
> > > > by Schoenberg, one of the great composers, theorists,
> > > > orchestrators, innovators, and musical influences of the
> > > > twentieth century, who conducted pioneering performances, who
> > > > received the higheset respect of many of the great musical
> > > > minds of the first half of the century--and those made by you,
> > > > I would tend to give greater credence to Schoenberg's.
> > >
> > > Appeal to authority. We can put it with the ad hominem
> > > attacks and other cheap shots you've contributed to this
> > > discussion so far.
> > ...
> > >
> > > -Carl
> >
> > > Maybe we can play the guessing game called:
> > "Who is or was an actual musician?"
> >
> > After that maybe we can squeeze in a round of:
> > "Who is or was an impudent, obnoxious pretender?"
> >
> > These are just games, mind you – just games.
> >
> > Have fun!
> > AAH
>
> Yes, games! Lot's of fun. So nice to hear from you, Aaron.
> You've shown your true colors yet again.
>
> -Carl
>

--- In tuning@yahoogroups.com, Cox Franklin <franklincox@...> wrote:
>
> Carl,
> My posting on Taruskin referred to my TAing under him at
> Columbia University; I have not posted this information anywhere
> else. He referred specifically to Columbia University in in
> that time period, so he must have received my posting from
> someone on this list.
> Franklin

Maybe he just googled his own name and it came up? Remember,
these fora are indexed by search engines.

Or maybe someone here told him his name had come up, without
specifically mentioning you.

Or maybe, hey, he's an avid reader himself?

Just trying to inject some benefit-of-the-doubt here.

-Carl

Is the name of the game you suggest "Calling Carl
an "impudent, obnoxious pretender""?

-Carl

--- In tuning@yahoogroups.com, "hpiinstruments" <aaronhunt@...> wrote:
>
> Carl,
>
> I'm not sure what you mean by "true colors". I
> merely suggested some games to play that might
> be fun. If those games I suggested aren't any
> fun, then please ignore my suggestion to play
> them.
>
> Yours,
> AAH
> =====
>
[snip]
> > > > Maybe we can play the guessing game called:
> > > "Who is or was an actual musician?"
> > >
> > > After that maybe we can squeeze in a round of:
> > > "Who is or was an impudent, obnoxious pretender?"
> > >
> > > These are just games, mind you – just games.
> > >
> > > Have fun!
> > > AAH
> >
> > Yes, games! Lot's of fun. So nice to hear from you, Aaron.
> > You've shown your true colors yet again.
> >
> > -Carl
> >
>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> Is the name of the game you suggest "Calling Carl
> an "impudent, obnoxious pretender""?

No. Read the post again and you'll see the game is called:
"Who is or was an impudent, obnoxious pretender?"
It's a guessing game!

AAH

P.S. Lighten up!
=====

Carl,
He wrote, "I've been reading about myself and my former relationship to you."    I've tried googling his name and mine, but nothing comes up about this posting.
Maybe he is a member of this list.  Who would have thought?
Franklin

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@...

--- On Tue, 2/9/10, Carl Lumma <carl@...> wrote:

From: Carl Lumma <carl@...>
Subject: [tuning] Re: Musical Set Theory in 12-tET
To: tuning@...m
Date: Tuesday, February 9, 2010, 12:01 AM

--- In tuning@yahoogroups. com, Cox Franklin <franklincox@ ...> wrote:

>

> Carl,

> My posting on Taruskin referred to my TAing under him at

> Columbia University; I have not posted this information anywhere

> else. He referred specifically to Columbia University in in

> that time period, so he must have received my posting from

> someone on this list.

> Franklin

Maybe he just googled his own name and it came up? Remember,

these fora are indexed by search engines.

Or maybe someone here told him his name had come up, without

specifically mentioning you.

Or maybe, hey, he's an avid reader himself?

Just trying to inject some benefit-of-the- doubt here.

-Carl

if you google "taruskin" and "cox" together, this thread is the first thing
that comes up. He may have a google alert set to his own name.

On Mon, Feb 8, 2010 at 7:12 PM, Cox Franklin <franklincox@...> wrote:

>
>
> Carl,
>
> He wrote, "I've been reading about myself and my former relationship to
> you." I've tried googling his name and mine, but nothing comes up about
> this posting.
>
> Maybe he is a member of this list. Who would have thought?
>
> Franklin
>
>
>
>
>
>
>
>
>
> Dr. Franklin Cox
> 1107 Xenia Ave.
> Yellow Springs, OH 45387
> (937) 767-1165
> franklincox@...
>
> --- On *Tue, 2/9/10, Carl Lumma <carl@...>* wrote:
>
>
> From: Carl Lumma <carl@...>
> Subject: [tuning] Re: Musical Set Theory in 12-tET
> To: tuning@yahoogroups.com
> Date: Tuesday, February 9, 2010, 12:01 AM
>
>
>
>
> --- In tuning@yahoogroups. com<http://mc/compose?to=tuning%40yahoogroups.com>,
> Cox Franklin <franklincox@ ...> wrote:
> >
> > Carl,
> > My posting on Taruskin referred to my TAing under him at
> > Columbia University; I have not posted this information anywhere
> > else. He referred specifically to Columbia University in in
> > that time period, so he must have received my posting from
> > someone on this list.
> > Franklin
>
> Maybe he just googled his own name and it came up? Remember,
> these fora are indexed by search engines.
>
> Or maybe someone here told him his name had come up, without
> specifically mentioning you.
>
> Or maybe, hey, he's an avid reader himself?
>
> Just trying to inject some benefit-of-the- doubt here.
>
> -Carl
>
>
>

P.P.S.

"Schoenberg was proud of never having studied a
history of music. 'That is too dull for me.'[Schoenberg,
"Harmonielehre", p.350] It apparently never occurred to
Schoenberg that it is impudent to have published a
Theory of Harmony of 470 pages in which one
tantamountly tells the reader that the author himself
would never condescend to read a similar treatise by
anyone else." [Martin Vogel, "On the Relations of Tone",
p.297]

Sigfrid Karg-Elert wrote in the margins of Schoenberg's
Harmony book section on overtones "Heavens, is that
amateurish!" [ibid, p.294]

Like I said, it's a game : )
AAH
=====

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Is the name of the game you suggest "Calling Carl
> an "impudent, obnoxious pretender""?
>
> -Carl
>
> --- In tuning@yahoogroups.com, "hpiinstruments" <aaronhunt@> wrote:
> >
> > Carl,
> >
> > I'm not sure what you mean by "true colors". I
> > merely suggested some games to play that might
> > be fun. If those games I suggested aren't any
> > fun, then please ignore my suggestion to play
> > them.
> >
> > Yours,
> > AAH
> > =====
> >
> [snip]
> > > > > Maybe we can play the guessing game called:
> > > > "Who is or was an actual musician?"
> > > >
> > > > After that maybe we can squeeze in a round of:
> > > > "Who is or was an impudent, obnoxious pretender?"
> > > >
> > > > These are just games, mind you – just games.
> > > >
> > > > Have fun!
> > > > AAH
> > >
> > > Yes, games! Lot's of fun. So nice to hear from you, Aaron.
> > > You've shown your true colors yet again.
> > >
> > > -Carl
> > >
> >
>

--- In tuning@yahoogroups.com, "hpiinstruments" <aaronhunt@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> > Is the name of the game you suggest "Calling Carl
> > an "impudent, obnoxious pretender""?
>
>
> No. Read the post again and you'll see the game is called:
> "Who is or was an impudent, obnoxious pretender?"
> It's a guessing game!
>
> AAH
>
> P.S. Lighten up!
> =====

OK, sorry if I misread. I'll take it down a notch.

-Carl

What's boring on it? Do you mean resulting works? Or working
principles? All depends on the creativity of author. It's just
another esthetics, but results can be very interesting and not at all
boring. It's one of the possibility how to organize sounds (not only
tones) and make an art from it. In this sense there's a connection to
drums & percussions music - there was dramatic increase in number of
drums & percussions (making non-pitched, noise and SFX sounds) used
in orchestra, and composers started to write only for drum groups. As
a pure movement it started with futurists and bruitists in 1910's,
and their influence was great even on main-stream New music - I can
hear them in the music of Antheil, Varese, Partch, Bartók, Ligeti and
more. Concrete music was a logical development, thanks to
technology. And when samplers and later computer audio recording and processing became available, all those techniques were again actual
and living. There are music styles based on noises and sound FX even
in the pop. You can't call all of this boring. All this "futuristic
industrialism" is an important part of music art.

Daniel Forro

On 9 Feb 2010, at 6:32 AM, Carl Lumma wrote:

> Actually it's music concrete techniques that are boring.
> The human auditory system is designed to extract information
> from speech, which pitched music strongly resembles. It is
> simply impossible to obtain the same bandwidth to the
> listener with noise-based techniques.
>
> -Carl
>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
[...]
> You yourself seem to shift between stating that set theory
> is good only as a generative tool [...]

I have not said "only". I have emphasized this aspect, since it may of interest of microtonal composers.

> or for analyzing atonal music (not other forms).

Correct.

> And you've said
> it doesn't matter if the relations are audible.

No, I have not said that. The relations are not always necessarily on the musical surface (easy to identify by ear), but one cannot say they don't exist.

You have read that Franklin can identify set-classes by ear. I don't claim I have so a skilled capability, but I also can do it to a limited extent. The same for 19th-century harmony: not always I can follow remote modulations or chord relations by ear, but I know by experience that my ability for that is greatly improved when I am listening that kind of music with certain frequency. Identifying certain interval relations in Webern's music demand concentration, for me, but most basic relations are on the surface, since Webern explores redundance and uses mainly small sets as basic material. There are also people able to listen to an atonal work once and then speak about it, playing passages on a piano etc.

Most atonal music (Schoenberg in special) are too much _complex_ for most of us -- thus, many of the ideas present may be not perceived for most listeners, even when reading an analysis; but other listeners can perceive those relations.

Furthermore, many constructive relations may be not evident, but can make the music sound organized or at least sound stylistically consistent. They have a, to say, indirect effect on the music. That is my experience with Boulez's harmony, which I find refined and tasteful.

We can test the range of vality and usefulness of set relations, provided they are not discarded sumarily. One can listen to, study and analyse atonal music, and read analysis of pieces (using different approaches, including pc-set-based) and reach conclusions.

I am pretty ready to completely abandon pitch-class set approach in favour of other analytical procedures, if I find they can replace it with advantages.

Unfortunately, I could not yet find that approach, and your position against pitch-class sets -- which is based on pratically null experience with atonal music and approaches to analyse it, as you assumed -- cannot change anything. I am usually skeptical (in several levels, from interpersonal relations to agnostic atheism), and this reflets also on "set theory", which I have questioned since my very first reading of Forte's book a dozen years ago; but there is a practical limit for skepticism or scientific rigour -- in this case, I repeat: an imperfect tool is better than no tool at all. Franklin and I both affirm that Forte's theory has flaws, but also has useful contributions. But you seems to suggest that none is better.

As scientific standards are so important to you (although you repeatedly claimed the historical absurd of Viennese atonal music being based on pitch-class set theory, what is plain wrong, as if Isaac was influenced by Webern or Bach by Schenker): the eventual replacement of a more limited scientific theory for a more efficient is normal in science, just like false concepts can be replaced with correct ones.

Of course, as it is presumable, there is absolutely no barrier against someone (me?) study regular mapping etc. and apply the resulting knowledge to atonal music, bringing out a marvellous theory of the pitch organization in atonal music. Even if someone eventually do that, this does not justify the abandon of pitch-class sets tools until there.

[...]
> You have Franklin
> saying that set theory explains neoclassicism.

Where?

(At least, one can understand neoclassicism as a reference to Schoenberg's dodecaphonic music.) :-)

> Tell me
> where to draw the line.

Well, my references on pitch-class set theory are essentially Forte, Straus and João Pedro Paiva de Oliveira, and all them focus atonal music (and dodecaphonic music, except Forte).

I have read also a few articles on compositional ideas based on pitch-class, but they are not too relevant.

>
> Regular mapping starts with psychoacoustics, which is an
> experimental science. We find it supports many rules of
> common-practice theory. That's interesting.

Indeed it is interesting.

> With regular
> mapping we can derive the diatonic scale purely from the
> consonances/note consideration, along with other scales we
> predict should behave similarly. And it obviously works:
>
> http://lumma.org/tuning/erlich/decatonic-swing.mp3
>
> This is in 22, using the pajara scales I just recommended
> to Mike.

Very very very nice. :-D

>
> We can also find tunings ideal for atonality

That would be interesting, if it was be tried.

> and probably
> even measure the tonality in a piece (I expect, proportional
> to the average harmonic entropy of its simultaneities, and
> inversely proportional to the standard deviation).

That is also interesting.

Just for curiosity: was that tested in pieces based on extended tonality (using wandering and remote chords etc.)? (Neo-old might be an example, or some passages from Schoenberg's Verklaerte Nacht or 1st String Quartet.)

[...]
> > (There you can find, BTW, that exaggerated assumption of
> > similarity based on Z-relation. I can provide an analysis of
> > the most famous pair of Z-related sets showing interesting
> > "oppositions" between then.)
>
> Have you published on this?

That is an observation I have derived reflecting about this debate. I have not written it down.

( BTW, another example of opposition can be found in Forte's book: he defined 4 similarity relations, one based on common subsets (intersection), and others based on interval content; there are set-classes of n elements which are minimally similar with respect to interval content, but are very similar with respect to common subsets (intersection with cardinality n-1). )

>
> > See also Solomon's site:
> > http://solomonsmusic.net/setheory.htm
> > http://solomonsmusic.net/
>
> Thanks!
>
> > He also provides a software for pitch class sets analysis
> > (I guess it is "unfortunately" restrict to 12-EDO).
> > http://solomonsmusic.net/mas.htm
>
> Looks like a DOS program... a bit outdated.

It is a DOS program, with a rather limited user interface. But it seems to be working here on my GNU+Linux dosbox.

>
> > Nonetheless, you referred to the intervals that constitute
> > a chord, and its interval structure. Here are the respective
> > interval vectors of the chords (analysed as pitch-class sets).
> >
> > interval class 1 2 3 4 5 6
> > m2 M2 m3 M3 P4 A4
> > A={C,Eb,G,Bb} < 0 1 2 1 2 0 >
> > B={D,Eb,G#,A} < 2 0 0 0 2 2 >
> > C={C,E,G,B} < 1 0 1 2 2 0 >
> > D={B,C,F,Bb} < 2 1 0 0 2 1 >
> >
> > From it one can deduce that A and C are likely more consonant
> > than B and D, assuming, that intervals of the classes of m2,
> > M2 and A4 are more dissonant than intervals of the classes of
> > m3, M3 and P4.
>
> When you get to extended JI, we find that order matters.
> For instance, compare 4:5:6:7:9:11 with its subharmonic
> inverse.

Agree. That is something one should care when trying to extend pitch-class set theory to other tunings than 12-EDO. Most sets with 6 or more notes work more as scales than as chords.

>
> > > Option 2 goes like this:
> > >
> > > 2a. Get Microsoft Excel.
> > > 2b. Get this spreadsheet:
> > > http://lumma.org/music/theory/DyadicHarmonicEntropyCalc.xls
> > > 2c. Plug in the cents values for the target chord.
> > > 2d. Get the result.
> > > 2e. This is sum of dyadic entropies, whereas what we really
> > > want is tetradic entropy directly. Unfortunately it is an
> > > open problem to compute tetradic harmonic entropy.
> >
> > Is there any other implementation, or a detailed description
> > of the algorithm? I cannot use spreadsheets, and I am not sure
> > they can be used on a large number of 12-EDO subsets.
>
> Excel will handle many millions of rows these days. Anyway
> this is just a lookup table that sums the entropies of the
> dyads in a chord. The entropy calculation itself has been
> described on the harmonic_entropy list.

Have you a URL?

>
> > Otherwise, if you can do it, would you compare all chords
> > inside the range of 2 octaves?
>
> Which chords exactly?

All possible chord types containing no octaves, with 3 to 5 or 6 pitches, for example.

Great, Daniel!

--- In tuning@yahoogroups.com, Daniel Forró <dan.for@...> wrote:
>
> What's boring on it? Do you mean resulting works? Or working
> principles? All depends on the creativity of author. It's just
> another esthetics, but results can be very interesting and not at all
> boring. It's one of the possibility how to organize sounds (not only
> tones) and make an art from it. In this sense there's a connection to
> drums & percussions music - there was dramatic increase in number of
> drums & percussions (making non-pitched, noise and SFX sounds) used
> in orchestra, and composers started to write only for drum groups. As
> a pure movement it started with futurists and bruitists in 1910's,
> and their influence was great even on main-stream New music - I can
> hear them in the music of Antheil, Varese, Partch, Bartók, Ligeti and
> more. Concrete music was a logical development, thanks to
> technology. And when samplers and later computer audio recording and
> processing became available, all those techniques were again actual
> and living. There are music styles based on noises and sound FX even
> in the pop. You can't call all of this boring. All this "futuristic
> industrialism" is an important part of music art.
>
> Daniel Forro
>
>
> On 9 Feb 2010, at 6:32 AM, Carl Lumma wrote:
>
> > Actually it's music concrete techniques that are boring.
> > The human auditory system is designed to extract information
> > from speech, which pitched music strongly resembles. It is
> > simply impossible to obtain the same bandwidth to the
> > listener with noise-based techniques.
> >
> > -Carl
> >
>

I see...I had tried "Taruskin" and "Franklin Cox".

For those on the tuning list, I have had a short exchange with Dr. Taruskin and would like to note that despite my belief that I was "teaching under" (my words) Dr. Taruskin  in a course at Columbia University 24 years ago, apparently I was not officially Dr. Taruskin's TA. I've moved about fifteen times since then and no longer have my papers from that course. I had remembered him as having a supervisory role above the Doctoral TA, I believe, who was on the books for teaching the course.  But I might be mistaken--it is often impossible for graduate TA's to know the precise lines of authority among senior faculty. Dr. Taruskin writes that he did teach the "GS version"  (?)  of Music Humanities at that time; this must have been the source of my error.
Dr. Taruskin does not deny being in the room during the incident mentioned (when I suggested that a piece by Steve Reich be included in the course); he writes concerning this proposal that it "was made not to me, but to a meeting while I was in the room, and I did not oppose the proposal, only failed to support it."  If one understands that a graduate TA does not have the  job security that a tenured professor does, I believe that my proposal  was fairly bold. At any rate, there can be any number of reasons why Dr. Taruskin did not support my proposal at that time, and I  probably should not have characterized his intentions the way I did; this was owing to a lack of time to post my note. For this I apologize to Dr. Taruskin.  
I merely found  it odd when I read in the New York Times a few years later of his belief that Steve Reich was one of the greatest composers of the 20th century. Surely if he had believed this a few years earlier, he as a tenured professor surely could have spoken more forcefully for Reich's inclusion in an introduction to music course.  
Dr. Taruskin writes that he believes that I "deliberately created a false impression." This  is not the case; I had no intention to give the impression that Dr. Taruskin did anything other than what I remember him doing in a meeting 24 years ago. I remember his response clearly and vividly; he did not shoot down my proposal, but he did not support it and was quickly turned down.  My mistake was in not remembering the precise lines of authority for the course and in characterizing Dr. Taruskin's intentions, which I do not know.  
As I am troubled by what I view as an aggressive tone in Dr. Taruskin's notes to me,  I have attempted to break off the correspondence. Because Dr. Taruskin apparently monitors appearances of his name on the web (or is a member of this list?), I do not feel comfortable discussing anything about him or his theories on this tuning list.  
Franklin
Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@...

--- On Tue, 2/9/10, Dante Rosati <danterosati@...> wrote:

From: Dante Rosati <danterosati@...>
Subject: Re: [tuning] Re: Musical Set Theory in 12-tET
To: tuning@yahoogroups.com
Date: Tuesday, February 9, 2010, 12:23 AM

if you google "taruskin" and "cox" together, this thread is the first thing that comes up. He may have a google alert set to his own name.

On Mon, Feb 8, 2010 at 7:12 PM, Cox Franklin <franklincox@ yahoo.com> wrote:

Carl,
He wrote, "I've been reading about myself and my former relationship to you."    I've tried googling his name and mine, but nothing comes up about this posting.

Maybe he is a member of this list.  Who would have thought?
Franklin

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@ yahoo.com

--- On Tue, 2/9/10, Carl Lumma <carl@...> wrote:

From: Carl Lumma <carl@...>
Subject: [tuning] Re: Musical Set Theory in 12-tET

To: tuning@yahoogroups. com
Date: Tuesday, February 9, 2010, 12:01 AM

--- In tuning@yahoogroups. com, Cox Franklin <franklincox@ ...> wrote:

>

> Carl,

> My posting on Taruskin referred to my TAing under him at

> Columbia University; I have not posted this information anywhere

> else. He referred specifically to Columbia University in in

> that time period, so he must have received my posting from

> someone on this list.

> Franklin

Maybe he just googled his own name and it came up? Remember,

these fora are indexed by search engines.

Or maybe someone here told him his name had come up, without

specifically mentioning you.

Or maybe, hey, he's an avid reader himself?

Just trying to inject some benefit-of-the- doubt here.

-Carl

Martin Vogel is scarcely the greatest authority on Schoenberg; he has some interesting theories about just intonation, but these have gained almost no acceptance among music historians. 
One can take any one of hundreds of Schoenberg's statements out of context. He knew the Classical repertory cold, by heart.  That's why gifted musicians had such respect for him.
One must also understand the incredibly high standard musicians held each other to in Vienna at that time.  Musicians and artists with no academic degrees knew more about their art forms than many professors.  

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@...

--- On Tue, 2/9/10, hpiinstruments <aaronhunt@h-pi.com> wrote:

From: hpiinstruments <aaronhunt@...>
Subject: [tuning] Re: Musical Set Theory in 12-tET
To: tuning@yahoogroups.com
Date: Tuesday, February 9, 2010, 12:27 AM

P.P.S.

"Schoenberg was proud of never having studied a

history of music. 'That is too dull for me.'[Schoenberg,

"Harmonielehre" , p.350] It apparently never occurred to

Schoenberg that it is impudent to have published a

Theory of Harmony of 470 pages in which one

tantamountly tells the reader that the author himself

would never condescend to read a similar treatise by

anyone else." [Martin Vogel, "On the Relations of Tone",

p.297]

Sigfrid Karg-Elert wrote in the margins of Schoenberg's

Harmony book section on overtones "Heavens, is that

amateurish!" [ibid, p.294]

Like I said, it's a game : )

AAH

=====

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

>

> Is the name of the game you suggest "Calling Carl

> an "impudent, obnoxious pretender""?

>

> -Carl

>

> --- In tuning@yahoogroups. com, "hpiinstruments" <aaronhunt@> wrote:

> >

> > Carl,

> >

> > I'm not sure what you mean by "true colors". I

> > merely suggested some games to play that might

> > be fun. If those games I suggested aren't any

> > fun, then please ignore my suggestion to play

> > them.

> >

> > Yours,

> > AAH

> > =====

> >

> [snip]

> > > > > Maybe we can play the guessing game called:

> > > > "Who is or was an actual musician?"

> > > >

> > > > After that maybe we can squeeze in a round of:

> > > > "Who is or was an impudent, obnoxious pretender?"

> > > >

> > > > These are just games, mind you – just games.

> > > >

> > > > Have fun!

> > > > AAH

> > >

> > > Yes, games! Lot's of fun. So nice to hear from you, Aaron.

> > > You've shown your true colors yet again.

> > >

> > > -Carl

> > >

> >

>

There's an extra "surely" at the end of the third full paragraph.
Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@...

--- On Tue, 2/9/10, Cox Franklin <franklincox@...> wrote:

From: Cox Franklin <franklincox@...>
Subject: Re: [tuning] Re: Musical Set Theory in 12-tET
To: tuning@yahoogroups.com
Date: Tuesday, February 9, 2010, 1:33 AM

I see...I had tried "Taruskin" and "Franklin Cox".

For those on the tuning list, I have had a short exchange with Dr. Taruskin and would like to note that despite my belief that I was "teaching under" (my words) Dr. Taruskin  in a course at Columbia University 24 years ago, apparently I was not officially Dr. Taruskin's TA. I've moved about fifteen times since then and no longer have my papers from that course. I had remembered him as having a supervisory role above the Doctoral TA, I believe, who was on the books for teaching the course.  But I might be mistaken--it is often impossible for graduate TA's to know the precise lines of authority among senior faculty. Dr. Taruskin writes that he did teach the "GS version"  (?)  of Music Humanities at that time; this must have been the source of my error.
Dr.
Taruskin does not deny being in the room during the incident mentioned (when I suggested that a piece by Steve Reich be included in the course); he writes concerning this proposal that it "was made not to me, but to a meeting while I was in the room, and I did not oppose the proposal, only failed to support it."  If one understands that a graduate TA does not have the  job security that a tenured professor does, I believe that my proposal  was fairly bold. At any rate, there can be any number of reasons why Dr. Taruskin did not support my proposal at that time, and I  probably should not have characterized his intentions the way I did; this was owing to a lack of time to post my note. For this I apologize to Dr. Taruskin.  
I merely found  it odd when I read in the New York Times a few years later of his belief that Steve Reich was one of the greatest composers of the 20th century. Surely if he had believed this a few years earlier, he as a tenured professor surely could have spoken more forcefully for Reich's inclusion in an introduction to music course.  
Dr. Taruskin writes that he believes that I "deliberately created a false impression." This  is not the case; I had no intention to give the impression that Dr. Taruskin did anything other than what I remember him
doing in a meeting 24 years ago. I remember his response clearly and vividly; he did not shoot down my proposal, but he did not support it and was quickly turned down.  My mistake was in not remembering the precise lines of authority for the course and in characterizing Dr. Taruskin's intentions, which I do not know.  
As I am troubled by what I view as an aggressive tone in Dr. Taruskin's notes to me,  I have attempted to break off the correspondence. Because Dr. Taruskin apparently monitors appearances of his name
on the web (or is a member of this list?), I do not feel comfortable discussing anything about him or his theories on this tuning list.  
Franklin
Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@ yahoo.com

--- On Tue, 2/9/10, Dante Rosati <danterosati@ gmail.com> wrote:

From: Dante Rosati <danterosati@ gmail.com>
Subject: Re: [tuning] Re: Musical Set Theory in 12-tET
To: tuning@yahoogroups. com
Date: Tuesday, February 9, 2010, 12:23 AM

if you google "taruskin" and "cox" together, this thread is the first thing that comes up. He may have a google alert set to his own name.

On Mon, Feb 8, 2010 at 7:12 PM, Cox Franklin <franklincox@ yahoo.com> wrote:

Carl,
He wrote, "I've been reading about myself and my former relationship to you."    I've tried googling his name and mine, but nothing comes up about this posting.

Maybe he is a member of this list.  Who would have thought?
Franklin

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@ yahoo.com

--- On Tue, 2/9/10, Carl Lumma <carl@...> wrote:

From: Carl Lumma <carl@...>
Subject: [tuning] Re: Musical Set Theory in 12-tET

To: tuning@yahoogroups. com
Date: Tuesday, February 9, 2010, 12:01 AM

--- In tuning@yahoogroups. com, Cox Franklin <franklincox@ ...> wrote:

>

> Carl,

> My posting on Taruskin referred to my TAing under him at

> Columbia University; I have not posted this information anywhere

> else. He referred specifically to Columbia University in in

> that time period, so he must have received my posting from

> someone on this list.

> Franklin

Maybe he just googled his own name and it came up? Remember,

these fora are indexed by search engines.

Or maybe someone here told him his name had come up, without

specifically mentioning you.

Or maybe, hey, he's an avid reader himself?

Just trying to inject some benefit-of-the- doubt here.

-Carl

--- In tuning@yahoogroups.com, Cox Franklin
<franklincox@...> wrote:
> Martin Vogel is scarcely the greatest authority
> on Schoenberg; he has some interesting theories
> about just intonation, but these have gained almost
> no acceptance among music historians.

To play Devil's Advocate, an opponent would
probably respond to this with something like
"... just as Schoenberg's music has gained almost
no acceptance among audiences." Neither of the
parallel statements is completely true : )

The fact that Vogel's theories haven't gained widespread
acceptance, as you say, I think says absolutely nothing
about their merit. Some people who criticize him do
not actually understand at all what he was talking about.
"Relations" summarizes his life's work, and it really should
not be dismissed with such nonchalance; it is a Magnum
Opus.

Vogel was a musicologist of the highest scholarship,
academically on the level of someone like Heidegger but
in the field of music; fluent in Greek, better studied in
ancient musicology than most of his peers. He was not a
Schoenberg scholar, and as far as I know he never
claimed to be one. I find most of his remarks about
Schoenberg in "Relations" funny and entertaining, and
sometimes insightful, which is why I posted them - for
fun!

> One can take any one of hundreds of Schoenberg's
> statements out of context. He knew the Classical
> repertory cold, by heart. That's why gifted musicians
> had such respect for him.

Sure, but Vogel's statement isn't in fact out of context.
It says what it says, and it's pretty funny. It does I think
mischaracterize Schoenberg's overall attitude, but it
must be admitted that Schoenberg had a huge ego and
sometimes made statements of this kind.

In my opinion, Schoenberg's "Style and Idea", and
"Harmonielehre", should be required reading for all
composers. He was an incredibly insightful autodidact
who knew his place in history well.

> One must also understand the incredibly high standard
> musicians held each other to in Vienna at that time.
> Musicians and artists with no academic degrees knew
> more about their art forms than many professors.

I'm afraid the high standard has since fallen miserably,
but it's still the case that hardworking and experienced
autodidacts can have a lot of insight that eludes
those who concentrate on being purveyors of tradition.
And I do not mean to slight anyone in academia, as that
is my background. There are those that keep the status
quo and those who further their field with innovation.

Both Vogel and Schoenberg were of the latter persuasion.
They were both innovators, and in my opinion they both
deserve the highest respect for their work.

Cheers,
AAH
=====

>
>
>
>
> Dr. Franklin Cox
>
> 1107 Xenia Ave.
>
> Yellow Springs, OH 45387
>
> (937) 767-1165
>
> franklincox@...
>
> --- On Tue, 2/9/10, hpiinstruments <aaronhunt@...> wrote:
>
> From: hpiinstruments <aaronhunt@...>
> Subject: [tuning] Re: Musical Set Theory in 12-tET
> To: tuning@yahoogroups.com
> Date: Tuesday, February 9, 2010, 12:27 AM
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>  
>
>
>
>
>
>
>
>
>
> P.P.S.
>
>
>
> "Schoenberg was proud of never having studied a
>
> history of music. 'That is too dull for me.'[Schoenberg,
>
> "Harmonielehre" , p.350] It apparently never occurred to
>
> Schoenberg that it is impudent to have published a
>
> Theory of Harmony of 470 pages in which one
>
> tantamountly tells the reader that the author himself
>
> would never condescend to read a similar treatise by
>
> anyone else." [Martin Vogel, "On the Relations of Tone",
>
> p.297]
>
>
>
> Sigfrid Karg-Elert wrote in the margins of Schoenberg's
>
> Harmony book section on overtones "Heavens, is that
>
> amateurish!" [ibid, p.294]
>
>
>
> Like I said, it's a game : )
>
> AAH
>
> =====
>
>
>
> --- In tuning@yahoogroups. com, "Carl Lumma" <carl@> wrote:
>
> >
>
> > Is the name of the game you suggest "Calling Carl
>
> > an "impudent, obnoxious pretender""?
>
> >
>
> > -Carl
>
> >
>
> > --- In tuning@yahoogroups. com, "hpiinstruments" <aaronhunt@> wrote:
>
> > >
>
> > > Carl,
>
> > >
>
> > > I'm not sure what you mean by "true colors". I
>
> > > merely suggested some games to play that might
>
> > > be fun. If those games I suggested aren't any
>
> > > fun, then please ignore my suggestion to play
>
> > > them.
>
> > >
>
> > > Yours,
>
> > > AAH
>
> > > =====
>
> > >
>
> > [snip]
>
> > > > > > Maybe we can play the guessing game called:
>
> > > > > "Who is or was an actual musician?"
>
> > > > >
>
> > > > > After that maybe we can squeeze in a round of:
>
> > > > > "Who is or was an impudent, obnoxious pretender?"
>
> > > > >
>
> > > > > These are just games, mind you â€" just games.
>
> > > > >
>
> > > > > Have fun!
>
> > > > > AAH
>
> > > >
>
> > > > Yes, games! Lot's of fun. So nice to hear from you, Aaron.
>
> > > > You've shown your true colors yet again.
>
> > > >
>
> > > > -Carl
>
> > > >
>
> > >
>
> >
>

Hudson wrote:

> You have read that Franklin can identify set-classes by ear.

Although I probably don't understand the terminology, I'd be curious if this is possible to learn. Do you have any suggestions for testing this?

Petr

--- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@...> wrote:

[...]
> > you've said it doesn't matter if the relations are audible.
>
> No, I have not said that. The relations are not always
> necessarily on the musical surface (easy to identify by ear),
> but one cannot say they don't exist.

This is a key point, I wish we could be clear.

Q: Are they valid if inaudible?
Carl's Answer: No.

Q: Are they more than 1% audible to untrained listeners?
Carl's Answer: No.

Q: Are the conceivably audible to highly trained listeners?
Carl's Answer: Yes, but why go to the trouble for these
particular relations?

> You have read that Franklin can identify set-classes by ear.

I don't doubt that he can, but I wonder: how many who claim
this skill can really do it? We know that among wine tasters
for instance, very few if any can back up the claims they
make (audiophiles too). It's an "in" club where the price of
admission is pretending everyone in the club can do it.
I happen to think the majority of atonal music culture operates
on this principle. I've socialized with them in New York and
this is the sense I got. Call me crazy.

Here's one of the world's most renowned wine tasters getting
everything wrong at a blind tasting:
http://www.drvino.com/2009/10/02/blind-tasting-bordeaux-2005-robert-parker/

>The same for 19th-century harmony: not always I can follow
>remote modulations or chord relations by ear, but I know by
>experience that my ability for that is greatly improved when
>I am listening that kind of music with certain frequency.

All worthwhile music challenges listeners. But in Romantic
harmony, one does not have to know the names of the remote
modulations to enjoy them. Naive listeners can tell they're
remote, which is the essential thing. They are usually
contrasted with nearer modulations. And all modulations are
between two tonal keys or two consonant chords, which again
naive listeners can identify from alternatives.

I mentioned before 3 audible relations I think are important
to common practice music: transposition, octave equivalence,
and modal transposition. I'll mention another now: I think
naive listeners know whether a pitch is a 3rd or 5th in a
major chord. They may not know those terms, but in hymnody
for example, when we have F A C A -> D A D F#, listeners can
hear that though the tenor pitch has not changed, its position
among the 'partials' of the root has changed. Such a skill
may be related to adaptions for identifying vowel sounds
in speech.

> As scientific standards are so important to you (although you
> repeatedly claimed the historical absurd of Viennese atonal
> music being based on pitch-class set theory, what is plain
> wrong, as if Isaac was influenced by Webern or Bach by
> Schenker): the eventual replacement of a more limited
> scientific theory for a more efficient is normal in science,
> just like false concepts can be replaced with correct ones.

Exactly. Is Special Relativity based on General Relativity?
Well yes, kind of. I won't be surprised if you tell me that
Lorentz contraction is recoverable from the Einstein field
equations.

> > You have Franklin saying that set theory explains
> > neoclassicism.
>
> Where?

Let's not rehash it.

> (At least, one can understand neoclassicism as a reference
> to Schoenberg's dodecaphonic music.) :-)

Prokofiev started neoclassicism without influence of serialism.

> > With regular
> > mapping we can derive the diatonic scale purely from the
> > consonances/note consideration, along with other scales we
> > predict should behave similarly. And it obviously works:
> >
> > http://lumma.org/tuning/erlich/decatonic-swing.mp3
> >
> > This is in 22, using the pajara scales I just recommended
> > to Mike.
>
> Very very very nice. :-D

This little example is just a demonstration. But have you
ever heard anything sound so familiar, yet so alien?

> > We can also find tunings ideal for atonality
>
> That would be interesting, if it was be tried.

Didn't I provide the URL to Schulter and Keenan's paper on
metastable intervals?

> > and probably
> > even measure the tonality in a piece (I expect, proportional
> > to the average harmonic entropy of its simultaneities, and
> > inversely proportional to the standard deviation).
>
> That is also interesting.
>
> Just for curiosity: was that tested in pieces based on
> extended tonality (using wandering and remote chords etc.)?
> (Neo-old might be an example, or some passages from
> Schoenberg's Verklaerte Nacht or 1st String Quartet.)

I made this idea just now, of average and std deviation of
harmonic entropy. I expect it has a good chance to work,
but we won't know until it is tried.

> > The entropy calculation itself has been
> > described on the harmonic_entropy list.
>
> Have you a URL?

/harmonic_entropy/

Unfortunately, poking through the archives is not fun.
I don't know of a good centralized article on the matter.

> > > Otherwise, if you can do it, would you compare all chords
> > > inside the range of 2 octaves?
> >
> > Which chords exactly?
>
> All possible chord types containing no octaves, with 3 to 5
> or 6 pitches, for example.

They are infinite. You mean, restricted to 12-ET, or...?

-Carl

I suppose you don't mean he was the first neoclassical composer :-) If yes, that's wrong, there were such tendencies years before him.

And Neoclassicism has nothing common with serialism.

Daniel Forro

On 9 Feb 2010, at 6:22 PM, Carl Lumma wrote:

> > (At least, one can understand neoclassicism as a reference
> > to Schoenberg's dodecaphonic music.) :-)
>
> Prokofiev started neoclassicism without influence of serialism.
>
>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@> wrote:
>
> [...]
> > > you've said it doesn't matter if the relations are audible.
> >
> > No, I have not said that. The relations are not always
> > necessarily on the musical surface (easy to identify by ear),
> > but one cannot say they don't exist.
>
> This is a key point, I wish we could be clear.
>
> Q: Are they valid if inaudible?
> Carl's Answer: No.
>
> Q: Are they more than 1% audible to untrained listeners?
> Carl's Answer: No.
>
> Q: Are the conceivably audible to highly trained listeners?
> Carl's Answer: Yes, but why go to the trouble for these
> particular relations?

I must thank you for your questionments on that intriguing kind of music and the use of pitch-class set concept in its analysis. Today I think I may had got an insight on the also intriguing question of why pitch-class sets are a contribution for the understanding of some aspects of atonal music, whereas it is so rough as ignoring important things about harmony (things which should be addressed by the analist on further analysis). BTW, I could understand also why Forte named his book "Structure of atonal music".

It has to do, I think, with what Schoenberg called "emancipation of dissonance" and its practical consequences on the compositional work, for instance, Schoenberg's merging of harmony and theme (Brahms' influence?) under the absence of functional restrictions.

I am going to reflet more on that, and later I would like to explain more clearly what I am thinking on.

[...]
> >The same for 19th-century harmony: not always I can follow
> >remote modulations or chord relations by ear, but I know by
> >experience that my ability for that is greatly improved when
> >I am listening that kind of music with certain frequency.
>
> All worthwhile music challenges listeners. But in Romantic
> harmony, one does not have to know the names of the remote
> modulations to enjoy them.

Atonal music either. Same for serial. I find several pieces from those repertories (and others) lovely even whithout analysing them. (Same for complex tonal music, etc.)

> Naive listeners can tell they're
> remote, which is the essential thing. They are usually
> contrasted with nearer modulations. And all modulations are
> between two tonal keys or two consonant chords, which again
> naive listeners can identify from alternatives.

Yes, harmony lost most of its constructive importance in atonal music (and other modern music). Contrast is achieved with other parameters, including harmony as timbre.

>
> I mentioned before 3 audible relations I think are important
> to common practice music: transposition, octave equivalence,
> and modal transposition. I'll mention another now: I think
> naive listeners know whether a pitch is a 3rd or 5th in a
> major chord. They may not know those terms, but in hymnody
> for example, when we have F A C A -> D A D F#, listeners can
> hear that though the tenor pitch has not changed, its position
> among the 'partials' of the root has changed. Such a skill
> may be related to adaptions for identifying vowel sounds
> in speech.

OK. But art is not under the dictature of Lerhdal (or perhaps Terhardt?). Daniel Forró recalled the relevance of non-pitched percussion music.

[...]
> This little example is just a demonstration. But have you
> ever heard anything sound so familiar, yet so alien?

I liked the piece, as I said before.

But again: that does not exclude other kinds of music. You seems to think only pitch-oriented consonant music is valid. Your questionment is not about pitch-class set theory, but against atonal music as a whole, since you don't like it. I don't like neoclassical music (label which may include Osvaldo Lacerda, as you might expect), because it "doesn't speak much to me" -- it is a personal matter.

>
> > > We can also find tunings ideal for atonality
> >
> > That would be interesting, if it was be tried.
>
> Didn't I provide the URL to Schulter and Keenan's paper on
> metastable intervals?

Yes, though I am not sure of finding the atonal consequences. Anyway 17-EDO includes some intervals aborded in the text, and it seems to be suitable for atonal music. I must study the paper carefuly.

Thanks again for the reference. :-)

[...]
> > > The entropy calculation itself has been
> > > described on the harmonic_entropy list.
> >
> > Have you a URL?
>
> /harmonic_entropy/
>
> Unfortunately, poking through the archives is not fun.
> I don't know of a good centralized article on the matter.

Thanks.

>
> > > > Otherwise, if you can do it, would you compare all chords
> > > > inside the range of 2 octaves?
> > >
> > > Which chords exactly?
> >
> > All possible chord types containing no octaves, with 3 to 5
> > or 6 pitches, for example.
>
> They are infinite. You mean, restricted to 12-ET, or...?

First in 12-EDO, of course. The most interesting ones might be tetrachords. But chords with 3 and 5 tones are sufficient, if 4-tones cannot be compared. They should be generated inside one octave, and then the pitches changed in relative position in every possible disposition -- if necessary, transpose them to fit inside the 2 octaves.

It's simple--go to the piano and play each of the trichord types in various permutations:

012 (symmetrical):

Db
C
B
cluster

Db                        Db                       C                   Db
B                          C
                                                        Db                  C
C                          B                         B                     B

(M2 prominent)    (m 2 prominent)  (M2 prominent) (m2 prominent

Greater registral spread (which makes it harder to hear all three notes as a set):
B                          B

Db                       C

C                         Db  etc.
If you transpose each of these a m3, you'll have all 12 tones--that the next stage in pitch class perception.  Or better, first practice with a hexachord--(012345) -- (013) (245), (012678) -- (012) (678)

(013)
Eb       Eb
D         Db
C         C
 
Eb      Eb         D       Db        Eb    C      Eb    D       Db    Eb   C     Eb
C        C                                 Db                     C       C                     D
                       Eb      Eb                 Eb    D                        Db   Eb
D         Db       C        C          C      Db    C      Eb     Eb    C     D     C

At the next stage, various transposition  or transposition/invervion levels can be used to get all twelve tones (or one could focus on a hexachord--say (012345) --(013) (245) or (023457) -- (023)(457).

And so forth

It's fairly easy to learn the trichords this way.  Tetrachords are trickier, because there are a lot more tetrachords and a lot more ways of parsing them. I tend to feel that accurate perception of sets breaks down after tetrachords, and, ceteris paribus, one is usually perceiving tones and smaller sets. Or perhaps this is more clear: if the larger set is a symmetrical scale (octatonice, whole tone scale, etc), then perception of it is simple, and larger diatonic sets are easy to identify, as are cluster sets.  But perception or more complex hexachords  is what I would call liminal--one can identify a subset and, if a composer is consistently using a small number of hexachords, sense that the other notes belong to one of those hexachords.  I can often recognize the (012478) hexachord, because I know Elliott Carter's music well, and he has been using it for decades. I have used the (012568) hexachord for decades, so I know it cold.  But I can't hear
any six-note combination and instantly name the hexachord.  But this is probably not that important in most contexts. Carter has always insisted that all of his complex work with sets is only a means, not an end, and he is not very much interested in analyses of his music that focus on this aspect.

Franklin

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@...

--- On Tue, 2/9/10, Petr Parízek <p.parizek@...> wrote:

From: Petr Parízek <p.parizek@...>
Subject: Re: [tuning] Re: Musical Set Theory in 12-tET
To: tuning@yahoogroups.com
Date: Tuesday, February 9, 2010, 8:53 AM

Hudson
wrote:
>
You have read that Franklin can identify set-classes by
ear.
Although
I probably don’t understand the terminology, I’d be curious if this is possible
to learn. Do you have any suggestions for testing this?
Petr
 
 

Any modern aural training course that's worth its salt will have
at least part of the term devoted to the identification of pitch
sets. Normal ID requirements stop at trichords.

This book may interest you:

<http://www.amazon.com/Training-Twentieth-Century-Assoc-Michael-Friedmann/dp/0300045379>

It's an ear training text based on pitch set methodology. It's not
really a textbook, but it's interesting. I never used the book
when learning or teaching ear training, but as a teacher I
found it helpful.

If Franklin is reading, I would be interested to hear his take
on that text.

AAH
=====

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Hudson wrote:
>
> > You have read that Franklin can identify set-classes by ear.
>
> Although I probably don't understand the terminology, I'd be curious if this is possible to learn. Do you have any suggestions for testing this?
>
> Petr
>

I don't know that book...thanks for bringing it to my attention.  Most textbooks have a chapter or so on pitch class theory, so I've never needed a separate book.

Franklin

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@...

--- On Tue, 2/9/10, hpiinstruments <aaronhunt@...> wrote:

From: hpiinstruments <aaronhunt@h-pi.com>
Subject: [tuning] Re: Musical Set Theory in 12-tET
To: tuning@yahoogroups.com
Date: Tuesday, February 9, 2010, 6:59 PM

Any modern aural training course that's worth its salt will have

at least part of the term devoted to the identification of pitch

sets. Normal ID requirements stop at trichords.

This book may interest you:

<http://www.amazon. com/Training- Twentieth- Century-Assoc- Michael-Friedman n/dp/0300045379>

It's an ear training text based on pitch set methodology. It's not

really a textbook, but it's interesting. I never used the book

when learning or teaching ear training, but as a teacher I

found it helpful.

If Franklin is reading, I would be interested to hear his take

on that text.

AAH

=====

--- In tuning@yahoogroups. com, Petr Parízek <p.parizek@. ..> wrote:

>

> Hudson wrote:

>

> > You have read that Franklin can identify set-classes by ear.

>

> Although I probably don't understand the terminology, I'd be curious if this is possible to learn. Do you have any suggestions for testing this?

>

> Petr

>

--- In tuning@yahoogroups.com, Daniel Forró <dan.for@...> wrote:

> And Neoclassicism has nothing common with serialism.

Thank you. -Carl

Which Neoclassicism?  Remember that Der Rosenkavalier was a neoclassic opera. Stravinsky's neoclassicism is generally viewed as "Neoclassicism" per se owing to his tremendous influence.  Many view Schoenberg's turn to the 12-tone method as an answer to Stravinsky's neoclassical turn; after years composing free-atonal works that were closely allied to mysticism (he planned a mammoth cycle based on Balzac's Seraphita), suddenly in the 1920's he began writing pieces called "Suite" and "Serenade", composed of movements in classical and baroque forms.  The 12-tone method was instrumental in this change in focus.  My favorite works or Schoenberg's are actually the free atonal works, although I love the Serenade op. 22 (which has one 12-tone movement and other movements based on 10-note series, etc.).
Also look at some of Stravinsky's works from the late 1940's--he is obviously getting very close to serial procedures,  A work like Agon moves effortlessly between the non-serial and serial worlds.  There was a large change in his "sound" when Stravinsky adopted serialism, but the change had been brewing for some time.
Franklin

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@...

--- On Tue, 2/9/10, Carl Lumma <carl@...> wrote:

From: Carl Lumma <carl@lumma.org>
Subject: [tuning] Re: Musical Set Theory in 12-tET
To: tuning@yahoogroups.com
Date: Tuesday, February 9, 2010, 9:09 PM

--- In tuning@yahoogroups. com, Daniel Forró <dan.for@... > wrote:

> And Neoclassicism has nothing common with serialism.

Thank you. -Carl

Your examples show only rows made from 3, 4 or 6 tones subsets, what
about the other possibilities? There's a lot of other 12-tone series
with irregular or assymetric note (and interval) groups, consisting
of more or even all intervals...

In my opinion intervallic structure of the row and directions of
intervals are more important for analysis (and composition) than
sets of consecutive halftones showing only which notes from 12-tone
total were used. This gives basic character to the music based on
rows. But you know this when you talk about "prominent" intervals...

Daniel Forro

On 10 Feb 2010, at 12:55 AM, Cox Franklin wrote:

>
> It's simple--go to the piano and play each of the trichord types in
> various permutations:
>
> 012 (symmetrical):
>
> Db
> C
> B
> cluster
>
>
> Db Db
> C Db
> B C
>
> Db C
> C B
> B B
>
> (M2 prominent) (m 2 prominent) (M2 prominent) (m2 prominent
>
> Greater registral spread (which makes it harder to hear all three
> notes as a set):
> B B
>
> Db C
>
> C Db etc.
> If you transpose each of these a m3, you'll have all 12 tones--that
> the next stage in pitch class perception. Or better, first
> practice with a hexachord--(012345) -- (013) (245), (012678) --
> (012) (678)
>
>
> (013)
> Eb Eb
> D Db
> C C
>
> Eb Eb D Db Eb C Eb D
> Db Eb C Eb
> C C Db
> C C D
> Eb Eb Eb
> D Db Eb
> D Db C C C Db C Eb
> Eb C D C
>
> At the next stage, various transposition or transposition/
> invervion levels can be used to get all twelve tones (or one could
> focus on a hexachord--say (012345) --(013) (245) or (023457) --
> (023)(457).
>
> And so forth
>
> It's fairly easy to learn the trichords this way. Tetrachords are
> trickier, because there are a lot more tetrachords and a lot more
> ways of parsing them. I tend to feel that accurate perception of
> sets breaks down after tetrachords, and, ceteris paribus, one is
> usually perceiving tones and smaller sets. Or perhaps this is more
> clear: if the larger set is a symmetrical scale (octatonice, whole
> tone scale, etc), then perception of it is simple, and larger
> diatonic sets are easy to identify, as are cluster sets. But
> perception or more complex hexachords is what I would call
> liminal--one can identify a subset and, if a composer is
> consistently using a small number of hexachords, sense that the
> other notes belong to one of those hexachords. I can often
> recognize the (012478) hexachord, because I know Elliott Carter's
> music well, and he has been using it for decades. I have used the
> (012568) hexachord for decades, so I know it cold. But I can't
> hear any six-note combination and instantly name the hexachord.
> But this is probably not that important in most contexts. Carter
> has always insisted that all of his complex work with sets is only
> a means, not an end, and he is not very much interested in analyses
> of his music that focus on this aspect.
>
> Franklin
>
>
>
> Dr. Franklin Cox
> 1107 Xenia Ave.
> Yellow Springs, OH 45387
> (937) 767-1165
> franklincox@...
>
> --- On Tue, 2/9/10, Petr Parízek <p.parizek@...> wrote:
>
> From: Petr Parízek <p.parizek@...>
> Subject: Re: [tuning] Re: Musical Set Theory in 12-tET
> To: tuning@yahoogroups.com
> Date: Tuesday, February 9, 2010, 8:53 AM
>
>
> Hudson wrote:
>
> > You have read that Franklin can identify set-classes by ear.
>
> Although I probably don’t understand the terminology, I’d be
> curious if this is possible to learn. Do you have any suggestions
> for testing this?
>
> Petr
>
>

I talked about larger  and asymmetric sets in the end of my note. I understood this to be a question about how to learn to aurally identify different sets, and obviously one should start with smaller sets. It's very easy to get too complicated too quickly.

Franklni

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@...

--- On Wed, 2/10/10, Daniel Forró <dan.for@...> wrote:

From: Daniel Forró <dan.for@...>
Subject: Re: [tuning] Re: Musical Set Theory in 12-tET
To: tuning@yahoogroups.com
Date: Wednesday, February 10, 2010, 1:39 AM

Your examples show only rows made from 3, 4 or 6 tones subsets, what 
about the other possibilities? There's a lot of other 12-tone series 
with irregular or assymetric note (and interval) groups, consisting 
of more or even all intervals...

In my opinion intervallic structure of the row and directions of 
intervals  are more important for analysis (and composition) than 
sets of consecutive halftones showing only which notes from 12-tone 
total were used.  This gives basic character to the music based on 
rows. But you know this when you talk about "prominent" intervals...

Daniel Forro

On 10 Feb 2010, at 12:55 AM, Cox Franklin wrote:

>
> It's simple--go to the piano and play each of the trichord types in 
> various permutations:
>
> 012 (symmetrical):
>
> Db
> C
> B
> cluster
>
>
> Db                        Db                       
> C                   Db
> B                          C
>                                                         
> Db                  C
> C                          B                         
> B                     B
>
> (M2 prominent)    (m 2 prominent)  (M2 prominent) (m2 prominent
>
> Greater registral spread (which makes it harder to hear all three 
> notes as a set):
> B                          B
>
> Db                       C
>
> C                         Db  etc.
> If you transpose each of these a m3, you'll have all 12 tones--that 
> the next stage in pitch class perception.  Or better, first 
> practice with a hexachord--(012345) -- (013) (245), (012678) -- 
> (012) (678)
>
>
> (013)
> Eb       Eb
> D         Db
> C         C
>
> Eb      Eb         D       Db        Eb    C      Eb    D       
> Db    Eb   C     Eb
> C        C                                 Db                     
> C       C                     D
>                        Eb      Eb                 Eb     
> D                        Db   Eb
> D         Db       C        C          C      Db    C      Eb     
> Eb    C     D     C
>
> At the next stage, various transposition  or transposition/
> invervion levels can be used to get all twelve tones (or one could 
> focus on a hexachord--say (012345) --(013) (245) or (023457) -- 
> (023)(457).
>
> And so forth
>
> It's fairly easy to learn the trichords this way.  Tetrachords are 
> trickier, because there are a lot more tetrachords and a lot more 
> ways of parsing them. I tend to feel that accurate perception of 
> sets breaks down after tetrachords, and, ceteris paribus, one is 
> usually perceiving tones and smaller sets. Or perhaps this is more 
> clear: if the larger set is a symmetrical scale (octatonic, whole 
> tone scale, etc), then perception of it is simple, and larger 
> diatonic sets are easy to identify, as are cluster sets.  But 
> perception or more complex hexachords  is what I would call 
> liminal--one can identify a subset and, if a composer is 
> consistently using a small number of hexachords, sense that the 
> other notes belong to one of those hexachords.  I can often 
> recognize the (012478) hexachord, because I know Elliott Carter's 
> music well, and he has been using it for decades. I have used the 
> (012568) hexachord for decades, so I know it cold.  But I can't 
> hear any six-note combination and instantly name the hexachord.   
> But this is probably not that important in most contexts. Carter 
> has always insisted that all of his complex work with sets is only 
> a means, not an end, and he is not very much interested in analyses 
> of his music that focus on this aspect.
>
> Franklin
>
>
>
> Dr. Franklin Cox
> 1107 Xenia Ave.
> Yellow Springs, OH 45387
> (937) 767-1165
> franklincox@...
>
> --- On Tue, 2/9/10, Petr Parízek <p.parizek@...> wrote:
>
> From: Petr Parízek <p.parizek@...>
> Subject: Re: [tuning] Re: Musical Set Theory in 12-tET
> To: tuning@yahoogroups.com
> Date: Tuesday, February 9, 2010, 8:53 AM
>
>
> Hudson wrote:
>
> > You have read that Franklin can identify set-classes by ear.
>
> Although I probably don’t understand the terminology, I’d be 
> curious if this is possible to learn. Do you have any suggestions 
> for testing this?
>
> Petr
>
>

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--- In tuning@yahoogroups.com, Daniel Forró <dan.for@...> wrote:
>
> Your examples show only rows made from 3, 4 or 6 tones subsets, what
> about the other possibilities? There's a lot of other 12-tone series
> with irregular or assymetric note (and interval) groups, consisting
> of more or even all intervals...
>
> In my opinion intervallic structure of the row and directions of
> intervals are more important for analysis (and composition) than
> sets of consecutive halftones showing only which notes from 12-tone
> total were used. This gives basic character to the music based on
> rows. But you know this when you talk about "prominent" intervals...
>
> Daniel Forro
>

"Set theory" isn't really about tone rows. AFAIK it really started with Hanson, which I probably have somewhere come to think of it, and he was dealing first with tonal music. Basically it boils down to naming pitches without regard to modality, in other words, accepting an equal temperament as an "edo", equal division of an octave rather than a temperament. Before Hanson there is that well-known, in America at least, Perle essay critizing the motivic aspect of tone-row composition: this atomizing into pc's seems to have been very much in the air in the middle of the century.

I absolutely agree with you about tone rows by the way, and I don't agree with the atomizing/reduction/reification approach. It is precisely these "basic characters", modalities, that create sense and strength. And it is modality and movement, not the material being moved, that are universal and not tuning-dependent.

By the way, it is not important whether "people" can identify compositional elements consciously and by name, what a pseudo-scientific idea, just laughable. Noone enjoys the shapes of clouds- Brownian motion is far too complex for human perception! Horseshit.

-Cameron Bobro

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
[...]
> I absolutely agree with you about tone rows by the way, and I don't agree with the atomizing/reduction/reification approach. It is precisely these "basic characters", modalities, that create sense and strength. And it is modality and movement, not the material being moved, that are universal and not tuning-dependent.

Yes, analysis of pitch-class sets is essentially about material classification, identification of derivation procedures on the selected material, etc., it is not about functional relations (in a restrict harmonic sense). It is more related to analysis of motifs than to harmonic analysis, but there may occur confusion, since the "motifs" are pitch/interval relations in this case.

Harmonic analysis concerning to movement requires further considerations using other approaches. One can classify the chord types used in a tonal composition (let us suppose it is a music with rich variety of chord types), but the relevant analysis (to obtain useful conclusions) begins when the usage of every chord type is investigated in context. In tonal music, that kind of analysis is not interesting, since the chordal material typically is already widely well known (as well as many of their relations). For atonal music, there is no such a unified approach for the functionality, whereas "set theory" can help in the first stage.

Now, tone rows are essentially a unitary deposit of pre-selected motifs, than can be highlighted or used to form other motifs along a composition; 12-tone serieses were never intended to be identified as such by ear, but that misconception is unfortunately very common. What is important are the actual combinations of the material in the composition, which are governed by "'basic characters', modalities, that create sense and strength', as you wrote above.

> By the way, it is not important whether "people" can identify compositional elements consciously and by name, what a pseudo-scientific idea, just laughable.

I think all people here would agree with that.

> Noone enjoys the shapes of clouds-

I enjoy clouds textures, don't you? Clouds can be classified by their visual aspect, but for sure not by trying to "see" the position of every molecule by naked eyes! Clouds present modality (they present different aspects) and movement (wind transforms them), they are nice to observe.
http://en.wikipedia.org/wiki/Cloud

> Brownian motion is far too complex for human perception! Horseshit.

That holds true if one pretends to be aware of every step, but at a given distance, a qualitative "pattern" can be perceived (its "modality"). If Browninan motion is compared with other motion principles, one should assume that the qualitative difference between them might to be used to achieve modality and movement, the details (identifying every single step or small groups of steps) being unimportant.

For instance, Schoenberg's Piano Piece Op.33a presents contrastant material derived from a single 12-tone row, and that is evident when listening to the piece. This is consistent with Schoenberg's poetics views concerning to "unity" and "variety".

In Five Piano Pieces Op.23 (that are considered a compositional essay leading to his 12-tone technique, used only in the last piece, Waltz), the basic pitch material is presented in several perceptual levels.

In the first piece, there are repetitions of motifs along the music, using similar rhythms, but changing pitches across octaves. The counterpoint of that piece has lots of imitations and other derivations, which degree of similarity varies. There is a wide range of transformations of the material, from closer ones (which are easy to identify) to more remote ones. The qualitative differences between these occurrences are important, but cannot be analysed by just saying that they are derivations the same material. Yet, it is interesting to see how apparently different elements are related and at which extent the similarities between derivated material can be perceived, thus that analysis can be relevant.

By listening to Op.23/I right now (with Maurizio Pollini), I noticed a relation between the main motif and the "quasi-accompaniment" motif of the second section, that I was not aware of so far. Obviously, the derivative motif is intended to bring a constrastant effect with the main one, therefore a very apparent relation would not be suitable. Later, the motifs are merged as an unity.

In conclusion, the strenght and weakness of an analytical approach lies mostly in whether it is used or not in a reasonable way.

P.S.:

Cameron,

I just would like to add that I think my points (below) are essentially in agreement with what you have written, even if apparently it seems to be the reverse. I should remark in special the difference between clouds *shape* vs clouds textural *aspect*. Please take this in consideration when reading. ;-)

Hudson

--- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@...> wrote:
>
> --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> [...]
> > I absolutely agree with you about tone rows by the way, and I don't agree with the atomizing/reduction/reification approach. It is precisely these "basic characters", modalities, that create sense and strength. And it is modality and movement, not the material being moved, that are universal and not tuning-dependent.
>
> Yes, analysis of pitch-class sets is essentially about material classification, identification of derivation procedures on the selected material, etc., it is not about functional relations (in a restrict harmonic sense). It is more related to analysis of motifs than to harmonic analysis, but there may occur confusion, since the "motifs" are pitch/interval relations in this case.
>
> Harmonic analysis concerning to movement requires further considerations using other approaches. One can classify the chord types used in a tonal composition (let us suppose it is a music with rich variety of chord types), but the relevant analysis (to obtain useful conclusions) begins when the usage of every chord type is investigated in context. In tonal music, that kind of analysis is not interesting, since the chordal material typically is already widely well known (as well as many of their relations). For atonal music, there is no such a unified approach for the functionality, whereas "set theory" can help in the first stage.
>
> Now, tone rows are essentially a unitary deposit of pre-selected motifs, than can be highlighted or used to form other motifs along a composition; 12-tone serieses were never intended to be identified as such by ear, but that misconception is unfortunately very common. What is important are the actual combinations of the material in the composition, which are governed by "'basic characters', modalities, that create sense and strength', as you wrote above.
>
> > By the way, it is not important whether "people" can identify compositional elements consciously and by name, what a pseudo-scientific idea, just laughable.
>
> I think all people here would agree with that.
>
> > Noone enjoys the shapes of clouds-
>
> I enjoy clouds textures, don't you? Clouds can be classified by their visual aspect, but for sure not by trying to "see" the position of every molecule by naked eyes! Clouds present modality (they present different aspects) and movement (wind transforms them), they are nice to observe.
> http://en.wikipedia.org/wiki/Cloud
>
> > Brownian motion is far too complex for human perception! Horseshit.
>
> That holds true if one pretends to be aware of every step, but at a given distance, a qualitative "pattern" can be perceived (its "modality"). If Browninan motion is compared with other motion principles, one should assume that the qualitative difference between them might to be used to achieve modality and movement, the details (identifying every single step or small groups of steps) being unimportant.
>
> For instance, Schoenberg's Piano Piece Op.33a presents contrastant material derived from a single 12-tone row, and that is evident when listening to the piece. This is consistent with Schoenberg's poetics views concerning to "unity" and "variety".
>
> In Five Piano Pieces Op.23 (that are considered a compositional essay leading to his 12-tone technique, used only in the last piece, Waltz), the basic pitch material is presented in several perceptual levels.
>
> In the first piece, there are repetitions of motifs along the music, using similar rhythms, but changing pitches across octaves. The counterpoint of that piece has lots of imitations and other derivations, which degree of similarity varies. There is a wide range of transformations of the material, from closer ones (which are easy to identify) to more remote ones. The qualitative differences between these occurrences are important, but cannot be analysed by just saying that they are derivations the same material. Yet, it is interesting to see how apparently different elements are related and at which extent the similarities between derivated material can be perceived, thus that analysis can be relevant.
>
> By listening to Op.23/I right now (with Maurizio Pollini), I noticed a relation between the main motif and the "quasi-accompaniment" motif of the second section, that I was not aware of so far. Obviously, the derivative motif is intended to bring a constrastant effect with the main one, therefore a very apparent relation would not be suitable. Later, the motifs are merged as an unity.
>
> In conclusion, the strenght and weakness of an analytical approach lies mostly in whether it is used or not in a reasonable way.
>

Better quickly explain that I meant the remark about not appreciating clouds sarcastically- of course people enjoy the shapes of cloud without being aware of Brownian motion! That was the point: a structure doesn't have to be something you can put your finger on.

--- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@...> wrote:
>
> --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> [...]
> > I absolutely agree with you about tone rows by the way, and I don't agree with the atomizing/reduction/reification approach. It is precisely these "basic characters", modalities, that create sense and strength. And it is modality and movement, not the material being moved, that are universal and not tuning-dependent.
>
> Yes, analysis of pitch-class sets is essentially about material classification, identification of derivation procedures on the selected material, etc., it is not about functional relations (in a restrict harmonic sense). It is more related to analysis of motifs than to harmonic analysis, but there may occur confusion, since the "motifs" are pitch/interval relations in this case.
>
> Harmonic analysis concerning to movement requires further considerations using other approaches. One can classify the chord types used in a tonal composition (let us suppose it is a music with rich variety of chord types), but the relevant analysis (to obtain useful conclusions) begins when the usage of every chord type is investigated in context. In tonal music, that kind of analysis is not interesting, since the chordal material typically is already widely well known (as well as many of their relations). For atonal music, there is no such a unified approach for the functionality, whereas "set theory" can help in the first stage.
>
> Now, tone rows are essentially a unitary deposit of pre-selected motifs, than can be highlighted or used to form other motifs along a composition; 12-tone serieses were never intended to be identified as such by ear, but that misconception is unfortunately very common. What is important are the actual combinations of the material in the composition, which are governed by "'basic characters', modalities, that create sense and strength', as you wrote above.
>
> > By the way, it is not important whether "people" can identify compositional elements consciously and by name, what a pseudo-scientific idea, just laughable.
>
> I think all people here would agree with that.
>
> > Noone enjoys the shapes of clouds-
>
> I enjoy clouds textures, don't you? Clouds can be classified by their visual aspect, but for sure not by trying to "see" the position of every molecule by naked eyes! Clouds present modality (they present different aspects) and movement (wind transforms them), they are nice to observe.
> http://en.wikipedia.org/wiki/Cloud
>
> > Brownian motion is far too complex for human perception! Horseshit.
>
> That holds true if one pretends to be aware of every step, but at a given distance, a qualitative "pattern" can be perceived (its "modality"). If Browninan motion is compared with other motion principles, one should assume that the qualitative difference between them might to be used to achieve modality and movement, the details (identifying every single step or small groups of steps) being unimportant.
>
> For instance, Schoenberg's Piano Piece Op.33a presents contrastant material derived from a single 12-tone row, and that is evident when listening to the piece. This is consistent with Schoenberg's poetics views concerning to "unity" and "variety".
>
> In Five Piano Pieces Op.23 (that are considered a compositional essay leading to his 12-tone technique, used only in the last piece, Waltz), the basic pitch material is presented in several perceptual levels.
>
> In the first piece, there are repetitions of motifs along the music, using similar rhythms, but changing pitches across octaves. The counterpoint of that piece has lots of imitations and other derivations, which degree of similarity varies. There is a wide range of transformations of the material, from closer ones (which are easy to identify) to more remote ones. The qualitative differences between these occurrences are important, but cannot be analysed by just saying that they are derivations the same material. Yet, it is interesting to see how apparently different elements are related and at which extent the similarities between derivated material can be perceived, thus that analysis can be relevant.
>
> By listening to Op.23/I right now (with Maurizio Pollini), I noticed a relation between the main motif and the "quasi-accompaniment" motif of the second section, that I was not aware of so far. Obviously, the derivative motif is intended to bring a constrastant effect with the main one, therefore a very apparent relation would not be suitable. Later, the motifs are merged as an unity.
>
> In conclusion, the strenght and weakness of an analytical approach lies mostly in whether it is used or not in a reasonable way.
>

the cloud metaphor applies to Xenakis' Stochastic music, not serial
music. Clouds are pleasing to look at and X's stochastic music sounds
good for the same reason, and you dont need to know about his
compositional process to appreciate it.

> Better quickly explain that I meant the remark about not appreciating clouds sarcastically- of course people enjoy the shapes of cloud without being aware of Brownian motion! That was the point: a structure doesn't have to be something you can put your finger on.
>

I've got loads to say on this subject and other going on, just kind of pressed for time- always work and now my child is at home recovering from a mild flu.

So, quickly noting- I also have written plenty of tone-row music, still do sometimes (though not in 12-tET), so I certainly don't categorically dismiss "the whole thing". And as anyone who has worked with such techniques knows, you can't just slap anything anywhere, quite the opposite. But some things that fall under the umbrella of "set theory" I find just plain bogus: the reduction of pcs to 0-6, as Solomon does on his site, is completely unacceptable to me and such an analysis would be wildly off for my own tone-row music, not to mention I'd never be able to write it in the first place if I didn't distinguish between a sixth and a third (more accurately, intervals of these general sizes and feel).

--- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@...> wrote:
>
> P.S.:
>
> Cameron,
>
> I just would like to add that I think my points (below) are essentially in agreement with what you have written, even if apparently it seems to be the reverse. I should remark in special the difference between clouds *shape* vs clouds textural *aspect*. Please take this in consideration when reading. ;-)
>
> Hudson
>
> --- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@> wrote:
> >
> > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > [...]
> > > I absolutely agree with you about tone rows by the way, and I don't agree with the atomizing/reduction/reification approach. It is precisely these "basic characters", modalities, that create sense and strength. And it is modality and movement, not the material being moved, that are universal and not tuning-dependent.
> >
> > Yes, analysis of pitch-class sets is essentially about material classification, identification of derivation procedures on the selected material, etc., it is not about functional relations (in a restrict harmonic sense). It is more related to analysis of motifs than to harmonic analysis, but there may occur confusion, since the "motifs" are pitch/interval relations in this case.
> >
> > Harmonic analysis concerning to movement requires further considerations using other approaches. One can classify the chord types used in a tonal composition (let us suppose it is a music with rich variety of chord types), but the relevant analysis (to obtain useful conclusions) begins when the usage of every chord type is investigated in context. In tonal music, that kind of analysis is not interesting, since the chordal material typically is already widely well known (as well as many of their relations). For atonal music, there is no such a unified approach for the functionality, whereas "set theory" can help in the first stage.
> >
> > Now, tone rows are essentially a unitary deposit of pre-selected motifs, than can be highlighted or used to form other motifs along a composition; 12-tone serieses were never intended to be identified as such by ear, but that misconception is unfortunately very common. What is important are the actual combinations of the material in the composition, which are governed by "'basic characters', modalities, that create sense and strength', as you wrote above.
> >
> > > By the way, it is not important whether "people" can identify compositional elements consciously and by name, what a pseudo-scientific idea, just laughable.
> >
> > I think all people here would agree with that.
> >
> > > Noone enjoys the shapes of clouds-
> >
> > I enjoy clouds textures, don't you? Clouds can be classified by their visual aspect, but for sure not by trying to "see" the position of every molecule by naked eyes! Clouds present modality (they present different aspects) and movement (wind transforms them), they are nice to observe.
> > http://en.wikipedia.org/wiki/Cloud
> >
> > > Brownian motion is far too complex for human perception! Horseshit.
> >
> > That holds true if one pretends to be aware of every step, but at a given distance, a qualitative "pattern" can be perceived (its "modality"). If Browninan motion is compared with other motion principles, one should assume that the qualitative difference between them might to be used to achieve modality and movement, the details (identifying every single step or small groups of steps) being unimportant.
> >
> > For instance, Schoenberg's Piano Piece Op.33a presents contrastant material derived from a single 12-tone row, and that is evident when listening to the piece. This is consistent with Schoenberg's poetics views concerning to "unity" and "variety".
> >
> > In Five Piano Pieces Op.23 (that are considered a compositional essay leading to his 12-tone technique, used only in the last piece, Waltz), the basic pitch material is presented in several perceptual levels.
> >
> > In the first piece, there are repetitions of motifs along the music, using similar rhythms, but changing pitches across octaves. The counterpoint of that piece has lots of imitations and other derivations, which degree of similarity varies. There is a wide range of transformations of the material, from closer ones (which are easy to identify) to more remote ones. The qualitative differences between these occurrences are important, but cannot be analysed by just saying that they are derivations the same material. Yet, it is interesting to see how apparently different elements are related and at which extent the similarities between derivated material can be perceived, thus that analysis can be relevant.
> >
> > By listening to Op.23/I right now (with Maurizio Pollini), I noticed a relation between the main motif and the "quasi-accompaniment" motif of the second section, that I was not aware of so far. Obviously, the derivative motif is intended to bring a constrastant effect with the main one, therefore a very apparent relation would not be suitable. Later, the motifs are merged as an unity.
> >
> > In conclusion, the strenght and weakness of an analytical approach lies mostly in whether it is used or not in a reasonable way.
> >
>

I love clouds and their watching. Have somebody of you ever seen that new type, called Asperatus? Unbelievable shapes and impact. Very microtonal.

http://www.novinky.cz/domaci/170414-novy-druh-mraku-vypada-jako-morska-hladina.html

http://www.cloudappreciationsociety.org/gallery/index.php?x=browse&category=52&pagenum=1

Daniel Forro

On 11 Feb 2010, at 1:40 AM, cameron wrote:

>
> Better quickly explain that I meant the remark about not > appreciating clouds sarcastically- of course people enjoy the > shapes of cloud without being aware of Brownian motion! That was > the point: a structure doesn't have to be something you can put > your finger on.
>
>

Hi Hudson,

> I am going to reflet more on that, and later I would like to
> explain more clearly what I am thinking on.

Great, I look forward to it! Thank you, by the way, for your
very professional and courteous tone through this discussion.

> > This little example is just a demonstration. But have you
> > ever heard anything sound so familiar, yet so alien?
>
> I liked the piece, as I said before.
>
> But again: that does not exclude other kinds of music. You
> seems to think only pitch-oriented consonant music is valid.

No, I don't mean that. However as I said before, I think
there are strong arguments (not "proofs" as my detractors
accuse) that the bandwidth composers can use to communicate
is higher with pitched music. This would not be relevant
for individual pieces. But when we are talking about the
potential of an entire genre of music containing many pieces,
I think it becomes relevant.

I just mentioned the idea of tracking harmonic number in
chords (at least for roots, 3rds, 5ths and I think it might
work with up to six identities at a time drawn from
harmonics < 18).

If we notate a music as a stream of frames (or hierarchy of
frames like Schenker), each frame can have bytes telling
about the relations I mentioned:

1. Address of previous frame which is an exact transposition
of this frame, #FF if none.

2. Address of previous frame which is a modal transposition
of this frame, #FF if none.

These would describe the frame vertically (all voices).
Horizontally (each voice) we have also these two bytes, and:

3. Harmonic number of this pitch above fundamental.

Many of these messages will be available in atonal music.
But in music concrete or noise percussion, we lose some or
all of them. We still have rhythm, and I said rhythm is the
biggest part of music. But to have complex rhythms we need
source separation, and pitch is a very powerful way the mind
assigns sounds to sources. Without pitch we have only
things like interaural timing, interaural loudness, and the
HRTF (Head-Related Transfer Function). These do not work
well in typical stereo recordings, which is the main medium
for music concrete! (live performance is another matter).

Anyway, I don't want to dwell on this too much. I like
Dan Stearns' music and some recent electronic music in the
concrete tradition. But I think it has less dynamic range
as a total artform. Dan may find this offensive, but I don't
think he should. He may disagree of course. And I may be
wrong, of course.

> > They are infinite. You mean, restricted to 12-ET, or...?
>
> First in 12-EDO, of course.

But of course! :P

> The most interesting ones might be tetrachords. But chords
> with 3 and 5 tones are sufficient, if 4-tones cannot be
> compared. They should be generated inside one octave, and
> then the pitches changed in relative position in every
> possible disposition -- if necessary, transpose them to fit
> inside the 2 octaves.

Because I'm lazy I'll give you something slightly different
for now. Paul Erlich looked at all subsets of 600-ET
using a Monte Carlo algorithm. . . .
Turns out it might have been easier to do it myself, than
collecting all those old posts! But here they are:

http://lumma.org/tuning/erlich/2000.08.EntropyMinimizer.txt

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hi Hudson,
>
> > I am going to reflet more on that, and later I would like to
> > explain more clearly what I am thinking on.
>
> Great, I look forward to it! Thank you, by the way, for your
> very professional and courteous tone through this discussion.

Hello Carl,

I think I addressed the essential things in the message linked below. In short, pitch-class set analysis abords interval materials and the way they are combined; it cannot do harmonic analysis in the usual meaning of the term.
/tuning/topicId_85909.html#86220

[...]
> I just mentioned the idea of tracking harmonic number in
> chords (at least for roots, 3rds, 5ths and I think it might
> work with up to six identities at a time drawn from
> harmonics < 18).
>
> If we notate a music as a stream of frames (or hierarchy of
> frames like Schenker), each frame can have bytes telling
> about the relations I mentioned:
>
> 1. Address of previous frame which is an exact transposition
> of this frame, #FF if none.
>
> 2. Address of previous frame which is a modal transposition
> of this frame, #FF if none.
>
> These would describe the frame vertically (all voices).
> Horizontally (each voice) we have also these two bytes, and:
>
> 3. Harmonic number of this pitch above fundamental.
>
> Many of these messages will be available in atonal music.

Interesting representation.

Atonality (pantonality) implies that item 3 is characterized by ambiguity and dynamic (moving) meaning.

> But in music concrete or noise percussion, we lose some or
> all of them. We still have rhythm, and I said rhythm is the
> biggest part of music. But to have complex rhythms we need
> source separation, and pitch is a very powerful way the mind
> assigns sounds to sources. Without pitch we have only
> things like interaural timing, interaural loudness, and the
> HRTF (Head-Related Transfer Function). These do not work
> well in typical stereo recordings, which is the main medium
> for music concrete! (live performance is another matter).

Yes. There are also interesting possibilities relating tones and noises: dense chords may sound like "coloured" or filtered noise, whereas percussion sounds may have a pitch "centroid" (I am not sure the word is proper, but I hope it carries idea) which allows organization in a sort of scale (a set of tom-toms, for example).

[...]
> > The most interesting ones might be tetrachords. But chords
> > with 3 and 5 tones are sufficient, if 4-tones cannot be
> > compared. They should be generated inside one octave, and
> > then the pitches changed in relative position in every
> > possible disposition -- if necessary, transpose them to fit
> > inside the 2 octaves.
>
> Because I'm lazy I'll give you something slightly different
> for now. Paul Erlich looked at all subsets of 600-ET
> using a Monte Carlo algorithm. . . .
> Turns out it might have been easier to do it myself, than
> collecting all those old posts! But here they are:
>
> http://lumma.org/tuning/erlich/2000.08.EntropyMinimizer.txt

Thank you. I am going to have a look.

BTW, is there a way to use Sethares' dissonance measure algorithm to to that comparison? Would you suggest a spectral model for the tests? I have an implementation of Sethares algorithm for GNU Octave, adaptated from Sethares' Matlab code. And I also have implemented some pitch-class sets operations in Octave, thus I could program and run the tests by myself.

Entropy model seems to assume some spectral model for the tones that form an interval, or maybe pure tones -- even in the later case, the specific frequencies should be relevant.

Any suggestions?

P.S.: Why near all links in your main page generate errors 404 and 403?

Cheers,
Hudson

--- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@...> wrote:

> BTW, is there a way to use Sethares' dissonance measure algorithm
> to to that comparison? Would you suggest a spectral model for the
> tests? I have an implementation of Sethares algorithm for GNU
> Octave, adaptated from Sethares' Matlab code. And I also have
> implemented some pitch-class sets operations in Octave, thus I
> could program and run the tests by myself.

Sure, if you can calculate sensory dissonance between two tones,
you can implement a Monte Carlo minimizer like Paul's. Paul
once looked at Sethares' code and found possible pathologies in
it -- or at least had questions about it, which Sethares never
answered. But presumably the results would be similar, since
Sethares' dissonance curves look a lot like those of harmonic
entropy (for normal timbres).

> Entropy model seems to assume some spectral model for the tones
> that form an interval, or maybe pure tones -- even in the later
> case, the specific frequencies should be relevant.

It assumes sine waves, but it's believed the results would be
similar if normal harmonic spectra were used instead.

> Any suggestions?

1 0 dB
2 -6 dB
3 -9.5 dB
4 -12 dB
5 -13.9 dB
6 -15.5 dB
7 -16.8 dB
8 -18 dB

> P.S.: Why near all links in your main page generate errors 404
> and 403?

I really should put those pages up. :)

-Carl

On 8 February 2010 12:27, Carl Lumma <carl@...> wrote:
> --- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@...> wrote:

>> And yes, we can also imagine that a scientific music theory can
>> be created simply from psychoacoustical experiments on frequency
>> ratios followed by speculation on uncommon and artificial scales
>> (this is generative). Are there any analytical musical
>> applications of regular mapping on microtonal music? Or is it
>> restricted to analyses of scales?
>
> Just to scales.  As I said, it's a theory of intonation, not
> music composition.

"Regular mapping" isn't a theory in the sense you accuse m. set theory
of failing to measure up to, so don't over hype it. As a framework
for constructing theories it can handle all kinds of things, including
traditional music theory, as I think you said somewhere else. It's
related to group theory, and so is a natural for some of the
transformations mentioned in this thread, including the dreaded
inversion.

Even theories of 12 equal pitch classes work if you consider one equal
temperament mapping with the octave dimension free. But a theory that
assumes one particular mapping is going to look less valuable than one
that works across mappings, seen through the regular mapping lens. I
don't know if set theory is such and haven't seen the question
answered.

Graham

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> Sure, if you can calculate sensory dissonance between two tones,
> you can implement a Monte Carlo minimizer like Paul's.

I am not interested in optimization. I want to compare the range of dissonance variation of all possible "voicings" (with no doubling) of every chord type inside a pitch range, to know which chord types are more sensible/resistent to inversions.

[...]
> > Any suggestions?
>
> 1 0 dB
> 2 -6 dB
> 3 -9.5 dB
> 4 -12 dB
> 5 -13.9 dB
> 6 -15.5 dB
> 7 -16.8 dB
> 8 -18 dB

Thanks.

Hudson
(Wondering why you have suggested that specific spectral model...)

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> Sure, if you can calculate sensory dissonance between two tones,
> you can implement a Monte Carlo minimizer like Paul's.

I am not interested in optimization. I want to compare the range of dissonance variation of all possible "voicings" (with no doubling) of every chord type inside a pitch range, to know which chord types are more sensible/resistent to inversions.

[...]
> > Any suggestions?
>
> 1 0 dB
> 2 -6 dB
> 3 -9.5 dB
> 4 -12 dB
> 5 -13.9 dB
> 6 -15.5 dB
> 7 -16.8 dB
> 8 -18 dB

Thanks.

Hudson
(Wondering why you have suggested that specific spectral model...)

--- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> > Sure, if you can calculate sensory dissonance between two tones,
> > you can implement a Monte Carlo minimizer like Paul's.
>
> I am not interested in optimization. I want to compare the range
> of dissonance variation of all possible "voicings" (with no
> doubling) of every chord type inside a pitch range, to know which
> chord types are more sensible/resistent to inversions.

Then you'll have to specify your question more precisely (this
isn't what we were talking about before), but it sounds
perfectly amendable to 'sum of diadic discordance' models like
the one Paul was using.

> (Wondering why you have suggested that specific spectral model...)

It's simple, and approximates the human voice.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> --- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@> wrote:
> > I am not interested in optimization. I want to compare the range
> > of dissonance variation of all possible "voicings" (with no
> > doubling) of every chord type inside a pitch range, to know which
> > chord types are more sensible/resistent to inversions.
>
> Then you'll have to specify your question more precisely (this
> isn't what we were talking about before), [...]

It likely occurred a communication problem, due to my limitations in English, specially when trying to use short phrases. (BTW, to be clear, by "inversions" above I mean changing the register of the pitches, exactly as in the ear training described by Franklin.)

Anyway, that comparation can be regarded as optimization problem (with several questions), but this is not necessary when the number of chords to compare is relatively small (e.g. trichords and tetrachords types in 12-EDO inside 2 octaves). When this is feasible, there is the advantage if knowing the complete thing. An optimization approach would be preferred (or required) for larger sets/scales, for computational reasons.

An interesting question would be: how many (and which ones) pc-set classes -- as defined by Solomon in http://solomonsmusic.net/pcsets.htm (that is, only including transpositions) -- show the lesser statistical variance in dissonance of their several pitch dispositions, when compared to other pc-set classes?

The reverse question is also interesting, and I would like yet to know the variance present in the analysed universe.

It is still undefined the way to weight the contribution of the dissonance measure of each disposition of a chord type, in order to obtain a mean value for comparison with other chord types.

Hudson

--- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@...> wrote:

> It is still undefined the way to weight the contribution of the
> dissonance measure of each disposition of a chord type, in order
> to obtain a mean value for comparison with other chord types.
>
> Hudson

I gave you two ways to do it.

-Carl

> > It is still undefined the way to weight the contribution of the
> > dissonance measure of each disposition of a chord type, in order
> > to obtain a mean value for comparison with other chord types.
> >
> > Hudson
>
> I gave you two ways to do it.
>
> -Carl

Entropy model sums up the entropy for every pair of tones of the chord, to get the dissonance (entropy) measure of the chord:
---<<< By "total pairwise harmonic entropy" I mean I added the six harmonic entropy values for the six possible pairs of notes found within the tetrad. >>>--- http://lumma.org/tuning/erlich/2000.08.EntropyMinimizer.txt

That is a measurement for a single chord.

I want to compare all voicings (no doubling) of a chord (taken as a group, or ps-set class) with all voicings of another chord (also taken as a group). In other terms, I am comparing the dissonance of two pc-set classes, not only the dissonance of chords within a class.

Maybe the dissonance within a class could be resumed as a single value, some kind of mean, but which one? That was the indefinition I referred (I don't know statistics techniques). But now I could see: the choice depends of the results I can find, and that is a secondary issue: first I should compute the dissonance measures.

The range of variance within each class could be well dramatized in a simple plot, with the dissonance values of the chords of the same class in the same horizontal position.

Hudson

--- In tuning@yahoogroups.com, "hfmlacerda" <hfmlacerda@...> wrote:

> That is a measurement for a single chord.
>
> I want to compare all voicings (no doubling) of a chord (taken
> as a group, or ps-set class) with all voicings of another chord
> (also taken as a group). In other terms, I am comparing the
> dissonance of two pc-set classes, not only the dissonance of
> chords within a class.

I see. I believe the octave-equivalent harmonic entropy used
by Paul in the thread I linked to will do the job. (It is
different than the harmonic entropy used in my spreadsheet.)

> Maybe the dissonance within a class could be resumed as a
> single value, some kind of mean, but which one?

At first, Paul just averaged the entropy of an interval
(like 5:4) with its inversion (8:5). But then he came up
with a more sophisticated approach:

/tuning/topicId_12722.html#12722?var=0