Yahoo Tuning Groups Ultimate Backup tuning Paul Erlich Harmonic Entropy Experiment back Online!

Paul Erlich's harmonic entropy experiment where he evaluates the
concordance of *tetrads* is back online at my Tuning Lab:

http://www.soundclick.com/bands/5/tuninglabmusic.htm

This has been moved from mp3.com which is meeting its demise...

Fortunately, I was also able to find the *text* to this great
experiment. I had a hard copy, so had to retype it... (Time for Paul
to check for typos... :)

Actually, I realize that text appeared *somewhere* on the Tuning
List, but I have no idea where, or how many years ago (I would say
maybe three or so...)

Now, the question I have regards the *ordering* of the tetrads.

I kept the same order as was on the original Tuning Lab page. As I
recall, I had *subjectively* reordered these according to what I had
perceived as concordance.

So, essentially, what do the ranking numbers stand for regarding the
chords. Are they ranked according to concordance according to the
Harmonic Entropy program??

So, tetrad #1 would, obviously, get the ranking for the highest
concordance??

Some of the tetrads received the *same* ranking, meaning that the
concordance level was, obviously, considered similar for them.

I'll move the selections around on the page if something else would
make more sense... it's been a while now...

Thanks!

Joseph

🔗Paul Erlich <paul@stretch-music.com>

I think anyone actually reading this post should be fascinated to
learn that many of the tetrads below which got among the highest
consonant ratings were *necessarily* tempered tetrads

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:

> Paul Erlich's harmonic entropy experiment where he evaluates the
> concordance of *tetrads* is back online at my Tuning Lab:
>
> http://www.soundclick.com/bands/5/tuninglabmusic.htm
>
> This has been moved from mp3.com which is meeting its demise...
>
> Fortunately, I was also able to find the *text* to this great
> experiment. I had a hard copy, so had to retype it... (Time for
Paul
> to check for typos... :)
>
> Actually, I realize that text appeared *somewhere* on the Tuning
> List, but I have no idea where, or how many years ago (I would say
> maybe three or so...)

How about the harmonic entropy list?

> Now, the question I have regards the *ordering* of the tetrads.
>
> I kept the same order as was on the original Tuning Lab page. As I
> recall, I had *subjectively* reordered these according to what I
had
> perceived as concordance.
>
> So, essentially, what do the ranking numbers stand for regarding
the
> chords. Are they ranked according to concordance according to the
> Harmonic Entropy program??

Not really -- the rankings are very out-of-date.

First off, they were created as part of a larger list of *local
minima* (in 3D, 'tetradic' space) of *total diadic harmonic entropy*
where we were using a Farey series to 'seed' the diadic harmonic
entropy function. Already, it was interesting that a number of these
tetrads were *necessarily tempered* ones.

But I later changed harmonic entropy from a 'Farey' to a 'Tenney'
seeding. The only change this has is to remove the overall 'slope' of
the harmonic entropy curve, which formerly was a level *downward*
slope as one moved from the unison to ever-larger intervals. As a
result of this original *downward* slope, a lot of the *closer-
voiced* tetrads got rated more dissonant than the more *open-voiced*
tetrads for no other reason than their "closed-voiced-ness" -- and
this shows up as a bias towards "open-voiced-ness" in the original
rankings we're stuck with on your page.

Now 'total diadic harmonic entropy', like Plomp-Levelt-Sethares
dissonance, but unlike 'tetradic harmonic entropy', rates each chord
as *equally consonant* as its mirror-inverse. This explains the pairs
of *same-ranked* tetrads you refer to below.

However, people's rankings of the 36 tetrads conformed a lot more
closely to what we would predict from 'tetradic harmonic entropy'
than from 'total diadic harmonic entropy'. 'Tetradic harmonic
entropy' hasn't actually been calculated yet, but one feature we know
with 99.9999% certainty is that for any sampling of just tetrads of
the form a:b:c:d, where a*b*c*d isn't too large, a*b*c*d will give
the same rankings as 'tetradic harmonic entropy'. Even with 'holes'
in it for the necessarily tempered tetrads, excessively high numbers
put in for the 'mirror-inverse-of-harmonic' or 'utonal'
or 'subharmonic' tetrads, and other deficiencies, this simple
*product* (we actually looked as the geometric mean, which produces
the exact same ranking as the product) conforms far better to
people's rankings than any formula that rates each chord as equally
consonant as its mirror-inverse ever could.

> So, tetrad #1 would, obviously, get the ranking for the highest
> concordance??
>
> Some of the tetrads received the *same* ranking, meaning that the
> concordance level was, obviously, considered similar for them.
>
> I'll move the selections around on the page if something else would
> make more sense... it's been a while now...
>
> Thanks!
>
> Joseph

Let me know!
-Paul

🔗Carl Lumma <ekin@lumma.org>

Awesome review, guys!

-Carl

>> So, essentially, what do the ranking numbers stand for regarding
>> the chords. Are they ranked according to concordance according to
>> the Harmonic Entropy program??
>
>Not really -- the rankings are very out-of-date.
>
>First off, they were created as part of a larger list of *local
>minima* (in 3D, 'tetradic' space) of *total diadic harmonic entropy*
>where we were using a Farey series to 'seed' the diadic harmonic
>entropy function. Already, it was interesting that a number of these
>tetrads were *necessarily tempered* ones.
>
>But I later changed harmonic entropy from a 'Farey' to a 'Tenney'
>seeding. The only change this has is to remove the overall 'slope' of
>the harmonic entropy curve, which formerly was a level *downward*
>slope as one moved from the unison to ever-larger intervals. As a
>result of this original *downward* slope, a lot of the *closer-
>voiced* tetrads got rated more dissonant than the more *open-voiced*
>tetrads for no other reason than their "closed-voiced-ness" -- and
>this shows up as a bias towards "open-voiced-ness" in the original
>rankings we're stuck with on your page.
>
>Now 'total diadic harmonic entropy', like Plomp-Levelt-Sethares
>dissonance, but unlike 'tetradic harmonic entropy', rates each chord
>as *equally consonant* as its mirror-inverse. This explains the pairs
>of *same-ranked* tetrads you refer to below.
>
>However, people's rankings of the 36 tetrads conformed a lot more
>closely to what we would predict from 'tetradic harmonic entropy'
>than from 'total diadic harmonic entropy'. 'Tetradic harmonic
>entropy' hasn't actually been calculated yet, but one feature we know
>with 99.9999% certainty is that for any sampling of just tetrads of
>the form a:b:c:d, where a*b*c*d isn't too large, a*b*c*d will give
>the same rankings as 'tetradic harmonic entropy'. Even with 'holes'
>in it for the necessarily tempered tetrads, excessively high numbers
>put in for the 'mirror-inverse-of-harmonic' or 'utonal'
>or 'subharmonic' tetrads, and other deficiencies, this simple
>*product* (we actually looked as the geometric mean, which produces
>the exact same ranking as the product) conforms far better to
>people's rankings than any formula that rates each chord as equally
>consonant as its mirror-inverse ever could.

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> I think anyone actually reading this post should be fascinated to
> learn that many of the tetrads below which got among the highest
> consonant ratings were *necessarily* tempered tetrads
>

***Umm, you know, that actually gave me a bit of a pause when I was
reposting these... something our JI enthusiasts might want to take
into consideration...

> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
>
> > Paul Erlich's harmonic entropy experiment where he evaluates the
> > concordance of *tetrads* is back online at my Tuning Lab:
> >
> > http://www.soundclick.com/bands/5/tuninglabmusic.htm
> >
> > This has been moved from mp3.com which is meeting its demise...
> >
> > Fortunately, I was also able to find the *text* to this great
> > experiment. I had a hard copy, so had to retype it... (Time for
> Paul
> > to check for typos... :)
> >
> > Actually, I realize that text appeared *somewhere* on the Tuning
> > List, but I have no idea where, or how many years ago (I would
say
> > maybe three or so...)
>
> How about the harmonic entropy list?
>

***Well, I'm sure that text was on the *main* Tuning List, since the
Harmonic Entropy list hadn't "spintered off" yet...

Speaking of which: Paul, could you please notify us on the *main*
list when there is significant activity on the Harmonic Entropy list??

Noticing the posts over there, sometimes there has been a gap of
maybe three months, sometimes even *SIX* between posts! Is that
still a viable list?? Maybe it should be used only when Harmonic
Entropy discussions get too detailed for this list. (Of course,
that's how I thought the Tuning Math list should also be used, but I
believe I was "outvoted" on both sides by the enthusiasts *and*
detractors... :) :)

> > Now, the question I have regards the *ordering* of the tetrads.
> >
> > I kept the same order as was on the original Tuning Lab page. As
I
> > recall, I had *subjectively* reordered these according to what I
> had
> > perceived as concordance.
> >
> > So, essentially, what do the ranking numbers stand for regarding
> the
> > chords. Are they ranked according to concordance according to
the
> > Harmonic Entropy program??
>
> Not really -- the rankings are very out-of-date.
>

***Yes, I remember now this problem with *diadic* Harmonic Entropy.
I'm assuming from reading your post that *triadic* Harmonic Entropy
has never been properly calculated. It's an immense job, yes? Also
limited by computer power??

In any rate, I've preserved the ordering of the tetrads on the Tuning
Lab page, top to bottom, but I have no idea why I had them in that
particular order. I guess it was just personal preference. In any
case, since the orderings for *diadic* Harmonic Entropy seem to make
as little sense as my *own* orderings, I think I'll keep them "as
is..." :) However, at least the tetrads (and their descriptions)
are there for reference [and I'm glad I managed to salvage this
experiment from soon-to-be-defunct mp3.com...}

> First off, they were created as part of a larger list of *local
> minima* (in 3D, 'tetradic' space) of *total diadic harmonic
entropy*
> where we were using a Farey series to 'seed' the diadic harmonic
> entropy function. Already, it was interesting that a number of
these
> tetrads were *necessarily tempered* ones.
>
> But I later changed harmonic entropy from a 'Farey' to a 'Tenney'
> seeding. The only change this has is to remove the overall 'slope'
of
> the harmonic entropy curve, which formerly was a level *downward*
> slope as one moved from the unison to ever-larger intervals. As a
> result of this original *downward* slope, a lot of the *closer-
> voiced* tetrads got rated more dissonant than the more *open-
voiced*
> tetrads for no other reason than their "closed-voiced-ness" -- and
> this shows up as a bias towards "open-voiced-ness" in the original
> rankings we're stuck with on your page.
>

***Got it. I see, looking back on my notes, that you actually
completely understood that even at the time...

> Now 'total diadic harmonic entropy', like Plomp-Levelt-Sethares
> dissonance, but unlike 'tetradic harmonic entropy', rates each
chord
> as *equally consonant* as its mirror-inverse. This explains the
pairs
> of *same-ranked* tetrads you refer to below.
>
> However, people's rankings of the 36 tetrads conformed a lot more
> closely to what we would predict from 'tetradic harmonic entropy'
> than from 'total diadic harmonic entropy'.

***Oh... do you recall if the rankings that I finally put on the page
were the product of "consensus" or just my *own* ranking. I can't
recall. I would prefer the former, of course...

'Tetradic harmonic
> entropy' hasn't actually been calculated yet, but one feature we
know
> with 99.9999% certainty is that for any sampling of just tetrads of
> the form a:b:c:d, where a*b*c*d isn't too large, a*b*c*d will give
> the same rankings as 'tetradic harmonic entropy'. Even with 'holes'
> in it for the necessarily tempered tetrads, excessively high
numbers
> put in for the 'mirror-inverse-of-harmonic' or 'utonal'
> or 'subharmonic' tetrads, and other deficiencies, this simple
> *product* (we actually looked as the geometric mean, which produces
> the exact same ranking as the product) conforms far better to
> people's rankings than any formula that rates each chord as equally
> consonant as its mirror-inverse ever could.
>

***Right, that seemed a bit peculiar right from the beginning...
rather like ranking chords and their inversions as equally concordant
when we know they don't always *sound* that way...

Joseph

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:

> > > chords. Are they ranked according to concordance according to
> the
> > > Harmonic Entropy program??
> >
> > Not really -- the rankings are very out-of-date.
> >
>
> ***Yes, I remember now this problem with *diadic* Harmonic
Entropy.
> I'm assuming from reading your post that *triadic* Harmonic Entropy
> has never been properly calculated. It's an immense job, yes?
Also
> limited by computer power??

My latest machine is definitely up to the task. I just need some
motivation. I've already posted some preliminaries on triadic
harmonic entropy to the harmonic entropy list. Tetradic harmonic
entropy will be a correspondingly huger task, but I think I can
handle that as well . . . someday.

> > First off, they were created as part of a larger list of *local
> > minima* (in 3D, 'tetradic' space) of *total diadic harmonic
> entropy*
> > where we were using a Farey series to 'seed' the diadic harmonic
> > entropy function. Already, it was interesting that a number of
> these
> > tetrads were *necessarily tempered* ones.
> >
> > But I later changed harmonic entropy from a 'Farey' to a 'Tenney'
> > seeding. The only change this has is to remove the
overall 'slope'
> of
> > the harmonic entropy curve, which formerly was a level *downward*
> > slope as one moved from the unison to ever-larger intervals. As a
> > result of this original *downward* slope, a lot of the *closer-
> > voiced* tetrads got rated more dissonant than the more *open-
> voiced*
> > tetrads for no other reason than their "closed-voiced-ness" --
and
> > this shows up as a bias towards "open-voiced-ness" in the
original
> > rankings we're stuck with on your page.
> >
>
> ***Got it. I see, looking back on my notes, that you actually
> completely understood that even at the time...

Yes, and I remember it confusing you quite a bit, because I was
actually using one form of harmonic entropy to represent the theories
that tetradic harmonic entropy would supposedly be an improvement
over . . .

> > Now 'total diadic harmonic entropy', like Plomp-Levelt-Sethares
> > dissonance, but unlike 'tetradic harmonic entropy', rates each
> chord
> > as *equally consonant* as its mirror-inverse. This explains the
> pairs
> > of *same-ranked* tetrads you refer to below.
> >
> > However, people's rankings of the 36 tetrads conformed a lot more
> > closely to what we would predict from 'tetradic harmonic entropy'
> > than from 'total diadic harmonic entropy'.
>
> ***Oh... do you recall if the rankings that I finally put on the
page
> were the product of "consensus" or just my *own* ranking. I can't
> recall. I would prefer the former, of course...

I have a vague recollection that it was neither.

> 'Tetradic harmonic
> > entropy' hasn't actually been calculated yet, but one feature we
> know
> > with 99.9999% certainty is that for any sampling of just tetrads
of
> > the form a:b:c:d, where a*b*c*d isn't too large, a*b*c*d will
give
> > the same rankings as 'tetradic harmonic entropy'. Even
with 'holes'
> > in it for the necessarily tempered tetrads, excessively high
> numbers
> > put in for the 'mirror-inverse-of-harmonic' or 'utonal'
> > or 'subharmonic' tetrads, and other deficiencies, this simple
> > *product* (we actually looked as the geometric mean, which
produces
> > the exact same ranking as the product) conforms far better to
> > people's rankings than any formula that rates each chord as
equally
> > consonant as its mirror-inverse ever could.
> >
>
> ***Right, that seemed a bit peculiar right from the beginning...
> rather like ranking chords and their inversions as equally
concordant
> when we know they don't always *sound* that way...

Be careful with terminology -- for musicians, the *inversions* of a
major triad are still major triads -- I used the term "mirror-
inverse" to more clearly suggest the "flipping" that would turn a
major triad into a minor triad, inspired perhaps by the use of the
term "mirror harmony" in Persichetti's book . . .

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <paul@stretch-music.com>

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_48802.html#48868

> > ***Oh... do you recall if the rankings that I finally put on the
> page
> > were the product of "consensus" or just my *own* ranking. I
can't
> > recall. I would prefer the former, of course...
>
> I have a vague recollection that it was neither.
>

###They were pretty much *random...?* Well, in any case, I
*preserved* this ranking on the new SoundClick page... :)

> >
> > ***Right, that seemed a bit peculiar right from the beginning...
> > rather like ranking chords and their inversions as equally
> concordant
> > when we know they don't always *sound* that way...
>
> Be careful with terminology -- for musicians, the *inversions* of a
> major triad are still major triads -- I used the term "mirror-
> inverse" to more clearly suggest the "flipping" that would turn a
> major triad into a minor triad, inspired perhaps by the use of the
> term "mirror harmony" in Persichetti's book . . .

****Got it! Yes, thanks... this is a much "broader" world...

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_48802.html#48873

> > > I think anyone actually reading this post should be
> > > fascinated to learn that many of the tetrads below
> > > which got among the highest consonant ratings were
> > > *necessarily* tempered tetrads
>
> Here are the rankings I did...
>
> me intervals g.m. d.e.
> -----------------------------
> 1 388 702 970 1 22-23
> 2 318 816 1020 2 20-21
> 3 498 702 886 3 7-8
> 4 386 702 1088 6 1
> 5 268 702 970 11 6
> 6 316 702 1018 8 2
> 7 184 498 886 7 3-4
> 8 498 886 1384 9 5
> 9 204 702 1088 5 9-10
> 10 202 702 974 4 25-26
> 11 268 582 970 30 22-23
> 12 272 772 974 24 25-26
> 13 388 886 1274 13 24
> 14 318 818 1320 15 32-33
> 15 384 588 1086 23 18-19
> -----------------------------
>
> The "me" column was my blind ranking, while g.m. and d.e.
> were geometric mean and dyadic entropy, IIRC.
>
> Which of these (if any) are "neccessarily tempered"?
>
> -Carl

***I wonder if this would be a better ordering on the Tuning Lab
site... It certainly has to beat "computer alphabetical..." :)

J. Pehrson