Cursory appraisal of concordance (H.E.)

That's just what we need... another subjective cursory appraisal of
concordance...

HOWEVER, in answer to Paul's question, this is what I'm coming up
with:

PLEASE NOTE -- that the file numberings on the Tuning Lab are a
little "messed up" right now...

http://artists.mp3s.com/artists/140/tuning_lab.html

This has happened before... it takes them a while to update the
pages...

Well, *IN GENERAL* it does seem like the Tenney method DOES present a
gradual panorama of increasingly discordant tetrads.

HOWEVER, after a gradual increase in discordance, I sense a bit of a
"return" with the following tetrads:

Tetrad #9-10: 0__204__702__1088

And the following pair seem also "out of place":

Tetrad #11-12: 0__302__502__1004 and
Tetrad #11-12: 0__502__702__1004

Also:

Tetrad #14-15: 0__186__576__888

Then, I'm finding the JI 5:6:8:9 to be VERY concordant... much more
so than many of the preceding:

Tetrad #20-21: 0__318__816__1020

FINALLY, the one we discussed, which REALLY seems pure, practically
beat-free, and 'way down the list at #22-23 is the just 4:5:6:7:

Tetrad #22-23: 0__388__702__970

Other than that, the gradations seem reasonably appropriate.

This is really, though, just a "first take." I want to spend more
time with this when mp3.come gets their "act together," if they do,
and ORDERS these tetrads like I have asked them to.

Their website must be behind with change requests.

More on this later...
____________ ____ __ __
Joseph Pehrson

Joseph wrote,

Well, *IN GENERAL* it does seem like the Tenney method DOES present a
gradual panorama of increasingly discordant tetrads.

HOWEVER, after a gradual increase in discordance, I sense a bit of a
"return" with the following tetrads:

Tetrad #9-10: 0__204__702__1088

That one contains a major triad, so otonality is working in its favor.

>And the following pair seem also "out of place":

>Tetrad #11-12: 0__302__502__1004 and
>Tetrad #11-12: 0__502__702__1004

>Also:

>Tetrad #14-15: 0__186__576__888

Out of place how? Too high or too low in the rankings?

>Then, I'm finding the JI 5:6:8:9 to be VERY concordant... much more
>so than many of the preceding:

>Tetrad #20-21: 0__318__816__1020

Again, otonality is working in its favor.

>FINALLY, the one we discussed, which REALLY seems pure, practically
>beat-free, and 'way down the list at #22-23 is the just 4:5:6:7:

>Tetrad #22-23: 0__388__702__970

The ultimate in tetradic otonality.

>Other than that, the gradations seem reasonably appropriate.

>This is really, though, just a "first take." I want to spend more
>time with this when mp3.come gets their "act together," if they do,
>and ORDERS these tetrads like I have asked them to.

>Their website must be behind with change requests.

>More on this later...

Can't wait . . . for now, it looks like the exceptions you've found can all
be chalked up to the diadic model's failure to see otonal synergies between
the intervals in otonal triads and tetrads, as we've discussed quite a bit.
Perhaps when you post something a little more detailed this impression will
change . . . ?

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/13485

> Joseph wrote,
>
> Well, *IN GENERAL* it does seem like the Tenney method DOES present
a
> gradual panorama of increasingly discordant tetrads.
>
> HOWEVER, after a gradual increase in discordance, I sense a bit of
a
> "return" with the following tetrads:
>
> Tetrad #9-10: 0__204__702__1088
>
> That one contains a major triad, so otonality is working in its
favor.
>
> >And the following pair seem also "out of place":
>
> >Tetrad #11-12: 0__302__502__1004 and
> >Tetrad #11-12: 0__502__702__1004
>
> >Also:
>
> >Tetrad #14-15: 0__186__576__888
>
> Out of place how? Too high or too low in the rankings?
>

These all seem to be ranked as more discordant (too high) than I
would have immediately presumed...

> >Then, I'm finding the JI 5:6:8:9 to be VERY concordant... much more
> >so than many of the preceding:
>
> >Tetrad #20-21: 0__318__816__1020
>
> Again, otonality is working in its favor.
>
> >FINALLY, the one we discussed, which REALLY seems pure,
practically
> >beat-free, and 'way down the list at #22-23 is the just 4:5:6:7:
>
> >Tetrad #22-23: 0__388__702__970
>
> The ultimate in tetradic otonality.
>
> >Other than that, the gradations seem reasonably appropriate.
>
> >This is really, though, just a "first take." I want to spend more
> >time with this when mp3.come gets their "act together," if they
do,
> >and ORDERS these tetrads like I have asked them to.
>
> >Their website must be behind with change requests.
>
> >More on this later...
>
> Can't wait . . . for now, it looks like the exceptions you've found
can all be chalked up to the diadic model's failure to see otonal
synergies between the intervals in otonal triads and tetrads, as
we've discussed quite a bit.

Well, that would make sense, since it seems always like the OTONAL of
the pair that I'm having trouble with... In fact, it really seems
like the answer....

Could it be that the UTONAL chord of the pair is REALLY the one
determining the order?... or is it kind of a COMBINATION of both? I
guess I'm STILL a little unclear on this...

> Perhaps when you post something a little more detailed this
impression will change . . . ?

Yes... if mp3.com could only get its "act" together and give me the
correcting order... it would make things simpler.

Once the page is together, rather than skattered all over the place,
I will try to lend a little more detail!

__________ ___ __ __ _
Joseph Pehrson

Joseph wrote,

>> >And the following pair seem also "out of place":
>
>>>Tetrad #11-12: 0__302__502__1004 and
>>>Tetrad #11-12: 0__502__702__1004
>
>>>Also:
>
>>>Tetrad #14-15: 0__186__576__888
>
>>Out of place how? Too high or too low in the rankings?
>

>These all seem to be ranked as more discordant (too high) than I
>would have immediately presumed...

The first two have two rather dissonant intervals each -- 200 cents and 1004
cents. So this sure contradicts what you were saying before about the one or
two dissonant intervals in the chord dominating the perception. Perhaps we
should go back to the ordering that came out before I decided to use the
exponential of the entropy values.

The third one is a dominant seventh chord, in 5-limit rather than 7-limit
tuning. If nothing else, the major triad in there gives the chord some added
otonal synergy.

>Well, that would make sense, since it seems always like the OTONAL of
>the pair that I'm having trouble with... In fact, it really seems
>like the answer....

Cool!

>Could it be that the UTONAL chord of the pair is REALLY the one
>determining the order?

That would make a lot of sense. Look at the old post of mine linked to from
Graham's "ass" page. In it, I say, _the only reason utonalities are
consonant is because the individual intervals are consonant_. Now since this
most recent model, using only diadic discordances, only looks at the
individual intervals, it does a great job assessing the overal discordance
of utonal chord. But as soon as three or more of the notes enter into an
otonal relationship, the model is missing out on an important component of
their concordance.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/13510

> >Could it be that the UTONAL chord of the pair is REALLY the one
> >determining the order?
>
> That would make a lot of sense. Look at the old post of mine linked
to from Graham's "ass" page. In it, I say, _the only reason
utonalities are consonant is because the individual intervals are
consonant_. Now since this most recent model, using only diadic
discordances, only looks at the individual intervals, it does a great
job assessing the overal discordance of utonal chord. But as soon as
>three or more of the notes enter into an otonal relationship, the
>model is missing out on an important component of their concordance.

Well, this has to be the reason for the discrepancies, then... since
a "cursory appraisal" of the discordance of the UTONAL chords seems
very much "in order." No problem there. In fact, I'm going to
listen to this aspect of the ordering even more carefully.

All the problems I had were with the OTONAL chords... it seems as
though they weren't quite evaluated properly by this method...as you
mention...

__________ ___ __ __
Joseph Pehrson

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> http://www.egroups.com/message/tuning/13510
>
> > [Joseph Pehrson]
> > Could it be that the UTONAL chord of the pair is REALLY the
> > one determining the order?
>
> That would make a lot of sense. Look at the old post of mine
> linked to from Graham's "ass" page. In it, I say, _the only
> reason utonalities are consonant is because the individual
> intervals are consonant_. Now since this most recent model,
> using only diadic discordances, only looks at the individual
> intervals, it does a great job assessing the overal discordance
> of utonal chord. But as soon as three or more of the notes enter
> into an otonal relationship, the model is missing out on an
> important component of their concordance.

Aha! Paul, that last sentence describes *exactly* what I was
getting at: placing the pitches of a tri-/higher-ad on a
prime-factor lattice *shows you the fundamental*, even when
it's not present in the chord itself!

Looking at the ratios of the chord as they appear on the lattice,
points directly to the fundamental, which represents the numerary
nexus. All prime-axes represented (or perhaps more importantly,
*implied*) in the chord, converge at the point where the
fundamental lies.

My guess is, the closer the lattice-points are (in steps
along the prime-axis, i.e., exponents) to that fundamental,
the more strongly the fundamental is perceived. 2 or 3 steps
away is probably too far to give a clear perception of the
fundamental, and so this is probably where chordal harmonic
entropy takes over and tries to imply a prime with a lower
exponent, i.e., a closer connection to the fundamental.

Does this make sense to anyone?

-monz
http://www.ixpres.com/interval/monzo/homepage.html

Monz wrote,

>Aha! Paul, that last sentence describes *exactly* what I was
>getting at: placing the pitches of a tri-/higher-ad on a
>prime-factor lattice *shows you the fundamental*, even when
>it's not present in the chord itself!

A lot of things can show you the fundamental -- just look at the numbers,
the power of two is the fundamental.

>My guess is, the closer the lattice-points are (in steps
>along the prime-axis, i.e., exponents) to that fundamental,
>the more strongly the fundamental is perceived. 2 or 3 steps
>away is probably too far to give a clear perception of the
>fundamental, and so this is probably where chordal harmonic
>entropy takes over and tries to imply a prime with a lower
>exponent, i.e., a closer connection to the fundamental.

Again, prime limit is irrelevant -- for example, when 24:17 tries to imply
17:12, the prime limit doesn't change, but the process is just as you
describe.

**BUT**

Your description can be made to make sense if you use the Tenney lattice.
The 2-axis has a length of log(2), the 3-axis has a length of log(3), etc.
What I found for diads was a best-case Tenney limit of log(108), and a
typical-case one of log(67). The latter would mean that you could go a
city-block distance of 6 steps on the 2-axis, 3 steps on the 3-axis, 2 steps
on the 5- or 7-axis, or 1 step on the 11-, 13-, ... or 61- axis, before
harmonic entropy "takes over". Note that a single step on the 61-axis can
only result in an interval of 1:61, not 32:61 or anything like that . . . if
you do want to impose octave equivalence then odd-limit comes into play.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> http://www.egroups.com/message/tuning/13535
>
> Your description can be made to make sense if you use the
> Tenney lattice. The 2-axis has a length of log(2), the 3-axis
> has a length of log(3), etc. What I found for diads was a
> best-case Tenney limit of log(108), and a typical-case one of
> log(67). The latter would mean that you could go a city-block
> distance of 6 steps on the 2-axis, 3 steps on the 3-axis,
> 2 steps on the 5- or 7-axis, or 1 step on the 11-, 13-, ...
> or 61- axis, before harmonic entropy "takes over". Note that
> a single step on the 61-axis can only result in an interval
> of 1:61, not 32:61 or anything like that . . . if you do want
> to impose octave equivalence then odd-limit comes into play.

This sounds good, Paul! So your chordal harmonic entropy *has*
given more-or-less the same results that I intuited.

Are the angles of the lattice axes relevant in any way?
I'm thinking that the best way to construct a Tenney lattice
like this is to make use of Dave Canright's method, where
one axis gives an indication of pitch-height. Is that
possible with step-sizes on the axis of log(prime)?

Actually, it's not clear to me if the axes here are prime-
or odd-based; it looks like you're using prime-axes. ...?

Give me some help on how to modify my lattice formula into
this, so I can produce some actual lattices with Excel.
I wanna see what they look like.

-monz
http://www.ixpres.com/interval/monzo/homepage.html

I wrote,

>> Your description can be made to make sense if you use the
>> Tenney lattice. The 2-axis has a length of log(2), the 3-axis
>> has a length of log(3), etc. What I found for diads was a
>> best-case Tenney limit of log(108), and a typical-case one of
>> log(67). The latter would mean that you could go a city-block
>> distance of 6 steps on the 2-axis, 3 steps on the 3-axis,
>> 2 steps on the 5- or 7-axis, or 1 step on the 11-, 13-, ...
>> or 61- axis, before harmonic entropy "takes over". Note that
>> a single step on the 61-axis can only result in an interval
>> of 1:61, not 32:61 or anything like that . . . if you do want
>> to impose octave equivalence then odd-limit comes into play.

>This sounds good, Paul! So your chordal harmonic entropy *has*
>given more-or-less the same results that I intuited.

Right -- the differences being the inclusion of a 2-axis, the use of a
city-block metric, and the use of logs for the distances.

>Are the angles of the lattice axes relevant in any way?

No, because the city-block metric is being used. You can choose your angles
for convenience, for example:

>I'm thinking that the best way to construct a Tenney lattice
>like this is to make use of Dave Canright's method, where
>one axis gives an indication of pitch-height. Is that
>possible with step-sizes on the axis of log(prime)?

Yes -- Tenney himself uses the same approach. See
http://www.music.mcgill.ca/~gems/tenney/theory.html.

>Actually, it's not clear to me if the axes here are prime-
>or odd-based; it looks like you're using prime-axes. ...?

Yes, Monz, it satisfies both of us because log(3)+log(3)=log(9),
log(3)+log(5)=log(15), etc. . . . only prime axes need be used but both
prime and composite numbers end up being represented based on the size of
the number and primes aren't "special" (either longer or shorter) compared
with other numbers their size.

>Give me some help on how to modify my lattice formula into
>this, so I can produce some actual lattices with Excel.
>I wanna see what they look like.

Sure, Monz, why don't you send me your Excel file off-list and I'll see what
I can do.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> http://www.egroups.com/message/tuning/13549
>
> [me, monz]
> >Actually, it's not clear to me if the axes here are prime-
> >or odd-based; it looks like you're using prime-axes. ...?
>
> [Paul]
> Yes, Monz, it satisfies both of us because log(3)+log(3)=log(9),
> log(3)+log(5)=log(15), etc. . . . only prime axes need be used
> but both prime and composite numbers end up being represented
> based on the size of the number and primes aren't "special"
> (either longer or shorter) compared with other numbers their
> size.

Ah... I get it. The primes aren't 'special' in terms of theory,
complexity, or any of that stuff, but they *are* special in that
they allow a unique and at the same time unambiguous representation
of each ratio. So their importance lies in their ability to
simplify the visual representation. Right?

-monz
http://www.ixpres.com/interval/monzo/homepage.html

Monz wrote,

>Ah... I get it. The primes aren't 'special' in terms of theory,
>complexity, or any of that stuff, but they *are* special in that
>they allow a unique and at the same time unambiguous representation
>of each ratio.

Yup!

>So their importance lies in their ability to
>simplify the visual representation. Right?

Visually, and also in terms of understanding the resources of a tuning, the
uniqueness is nice.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> http://www.egroups.com/message/tuning/13587
>
> > So their importance [of prime numbers] lies in their ability
> > to simplify the visual representation. Right?
>
> Visually, and also in terms of understanding the resources of
> a tuning, the uniqueness is nice.

Whew! Seems like you and I are finally reaching pretty good
agreement on the whole prime-vs-odd thing, Paul!

Of course, for me, the aspect of visual representation is of
utmost importance, because it's the basis of my JustMusic
software project. Guess maybe that's why I've laid such
importance on the primes all along.

-monz
http://www.ixpres.com/interval/monzo/homepage.html