Yahoo Tuning Groups Ultimate Backup harmonic

Ed Borasky wrote,

>What
>troubled me is that when I tried descending amplitudes (1, 1/2, 1/3, 1/4,
>1/5, 1/6) I got a dissonance curve with essentially only one minimum at the
>fifth. If I can hard-code all the amplitudes to 1 it will make my code a
>*lot* simpler.

Hi Ed.

Bill e-mailed me off-list to tell me that the "amplitudes" are in dB and not
really amplitudes at all. The conversion from amplitude to dB is dB =
10*log(amp)/log(10) + c, where c is a level-setting constant.

Now there's something about the results that worries me . . . for example,
if I run the program for a triangle wave (where the "amplitude" of the
harmonics decreases at 6 dB/octave) with a certain fundamental dB and then
for a square wave (where the "amplitude" of the harmonics decreases at 3
dB/octave) with half the fundamental dB, I get identical curves, other than
scaling. However, if I use a fixed waveform and simply adjust the volume up
or down (by adding a constant to all the dBs), I get very different curves.
Is that the way it's supposed to work, Bill?

🔗Paul H. Erlich <PERLICH@...>

Hi Bill.

>I think the normal thing to do when comparing different waveforms is
>to equalize the energy in the signal (add up all the dB numbers, then
>divide by the total to normalize to "1", for instance).
>Then youve got a "fair" comparison between the two sounds.

hmm . . . but the results will be very different depending on whether you
normalize to "1" or to some other constant . . . and what about negative dB
values? They're present in any waveform, their extent depending on your
arbitrary choice of a zero-dB point, but they contribute perversely to the
normalization above (the dB numbers could easily sum to zero, or to a
negative number) . . . and you don't really mean "energy" above, since
energy goes as the square of the amplitude, while dBs measure log-amplitude
-- correct?

>Im a little surprised at this, because a square wave only has odd partials,
>while a triangle has both odd and even.

Nope, a triangle wave(/\/\/\/\/\) has only odd partials. You might be
thinking of a sawtooth wave (|\|\|\|\|\).

>All the comments above, though, only make sense within
>a limited range... certainly using dB to approximate amplitude
>is good to first approximation

You say "amplitude" but of course dB is logarithmic in amplitude . . . the
two are in no way interchangeable, as Ed Borasky's comments show . . . do
you really mean "loudness"?

Thanks for taking the time . . .

-Paul

🔗Paul H. Erlich <PERLICH@...>

>ok - sorry if I seem obtuse here, but what exactly is the
>utonal/otonal question?

Well, among the 36 tetrads I made for Joseph Pehrson, which used timbres
with harmonic partials, Joseph found chords like 4:5:6:7 to be among the
most consonant, and chords like 1/7:1/6:1/5:1/4 to be among the most
dissonant, despite the fact that they contain exactly the same intervals.
While your model does indicate a slight preference for 4:5:6:7 over
1/7:1/6:1/5:1/4, it is small and due to the preference for having larger
intervals in the low register. Meanwhile, your model ranks 1/7:1/6:1/5:1/4
as much more concordant than either 8:10:12:15 or 12:15:18:20, both of which
Joseph thought should be near the top of the list (feel free to jump in
here, Joseph)!

🔗Joseph Pehrson <pehrson@...>

--- In harmonic_entropy@egroups.com, "Paul H. Erlich" <PERLICH@A...>

http://www.egroups.com/message/harmonic_entropy/32

As Paul mentioned, I had a problem with the sequential ordering of
the supposedly increasing discordance for the tetrads on the Tuning
Lab page when the distinctions between otonal and utonal chords were
disregarded:

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

Some of the tetrads that seem to "leap out" as being foreign to the
gradual progression toward discordance were:

Tetrad #9-10: 204__702__1088

Tetrad #11-12: 302__502__1004
Tetrad #11-12: 502__702__1004

Tetrad #14-15: 186__576__888

Tetrad #20-21: 318__816__1020 (the 5:6:8:9)

And Tetrad #22-23: 388__702__970

...

Those seemed "out of place" to me...

Paul, subsequently, made a list for me of increasing discordance
using a simple model of geometric means for JI chords. This method
seemed more suitable to describing the gradual progression and, in
fact, NONE of the tetrads (using the same ones on the page mentioned
above)seemed particularly out of place using this new ordering.

Paul's message and new ranking was the following:

"Here's a ranking of the tetrads according to what I anticipate the
solution to tetradic harmonic entropy for simple JI chords would be
like, the geometric mean of the numbers used in the otonal
representation of the chord, and ignoring roughness (the Sethares
factor) as well as diadic and triadic harmonic entropy:

the approximation only works for simple JI chords."

bass tenor alto soprano otonal_rep g.m.

0 388 702 970 4:5:6:7 5.3836
0 318 816 1020 5:6:8:9 6.8173
0 498 702 886 6:8:9:10 8.1072
0 202 702 974 8:9:12:14 10.4872
0 204 702 1088 8:9:12:15 10.6697
0 386 702 1088 8:10:12:15 10.9545
0 184 498 886 9:10:12:15 11.2818
0 316 702 1018 10:12:15:18 13.4164
0 498 886 1384 9:12:15:20 13.4164
0 502 1002 1390 9:12:16:20 13.6346
0 268 702 970 12:14:18:21 15.8745
0 388 702 886 12:15:18:20 15.9549
0 388 886 1274 12:15:20:25 17.3205
0 498 888 1282 12:16:20:25 17.6022
0 318 818 1320 15:18:24:32 21.3394
0 500 816 1316 15:20:24:32 21.9089
0 388 776 1090 16:20:25:30 22.1336
0 186 576 888 18:20:25:30 22.7951
0 498 702 1086 24:32:36:45 33.3979
0 386 884 1088 24:30:40:45 33.7405
0 312 702 888 30:36:45:50 39.4822
0 184 388 886 36:40:45:50 42.4264
0 384 588 1086 32:40:45:60 43.1165
0 272 772 974 36:42:56:63 48.0585
0 388 888 1390 36:45:60:80 52.8067
0 204 702 1020 40:45:60:72 52.8067
0 314 702 1090 40:48:60:75 54.2161
0 502 1002 1320 45:60:80:96 67.4810
0 394 784 1282 48:60:75:100 68.1732
0 268 582 970 60:70:84:105 78.0152
0 302 502 1004 not JI
0 502 702 1004 not JI
0 492 980 1472 not JI
0 500 886 1320 not JI
0 434 820 1320 not JI
0 442 884 1326 not JI

____________ ___ __ _ _
Joseph Pehrson

🔗Paul H. Erlich <PERLICH@...>

Joseph Pehrson wrote,

>Paul, subsequently, made a list for me of increasing discordance
>using a simple model of geometric means for JI chords. This method
>seemed more suitable to describing the gradual progression and, in
>fact, NONE of the tetrads (using the same ones on the page mentioned
>above)seemed particularly out of place using this new ordering.

But Joseph, you yourself said that the chords

0 302 502 1004
0 502 702 1004

were more consonant than the old ranking suggested . . . but the new ranking
puts them way at the bottom! Evidently, you didn't go all the way down the
list (?)

🔗Joseph Pehrson <pehrson@...>

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

http://www.egroups.com/message/harmonic_entropy/34

> Joseph Pehrson wrote,
>
> >Paul, subsequently, made a list for me of increasing discordance
> >using a simple model of geometric means for JI chords. This
method seemed more suitable to describing the gradual progression
and, in fact, NONE of the tetrads (using the same ones on the page
mentioned above)seemed particularly out of place using this new
ordering.
>
> But Joseph, you yourself said that the chords
>
> 0 302 502 1004
> 0 502 702 1004
>
> were more consonant than the old ranking suggested . . . but the
new ranking puts them way at the bottom! Evidently, you didn't go all
the way down the list (?)

Paul... that's great! You don't miss a trick!

Well... that's funny, because it is true that I didn't pay much
attention to the ones at the bottom, since they didn't seem to
"pertain" to the ranking method you set up.

HOWEVER, I *DID* listen to them... and I remember thinking that there
was something wrong... and that those "non JI" chords really should
be somewhere else in the rank!

But how can you do it?? Not by the same methods as the JI ones,
surely (??!!)
___________ ___ __ _ _
Joseph Pehrson

🔗Paul H. Erlich <PERLICH@...>

>NONE of the tetrads (using the same ones on the page
>mentioned above)seemed particularly out of place using this new
>ordering.

But you agreed that 9:11:13:15, which Monz made a version of, _would_ seem
out of place in it . . .

>Paul... that's great! You don't miss a trick!

When you've got as great a desire to find out the truth as I do, no detail
slips by. However, if you asked me to name any friend of mine and the color
of their car, I'd be stumped!

>But how can you do it?? Not by the same methods as the JI ones,
>surely (??!!)

Well, Joseph, let me throw a few things off you. First, none of the chords
were really in JI anyway, so if I were to treat these chords as rough
approximations of 27:32:36:48 and 18:24:27:32, with geometric means 34.9554
and 24.7172 respectively, what would you think of their new places in my
ordering by geometric mean?

🔗Joseph Pehrson <pehrson@...>

--- In harmonic_entropy@egroups.com, "Paul H. Erlich" <PERLICH@A...>
wrote:
http://www.egroups.com/message/harmonic_entropy/36

> Well, Joseph, let me throw a few things off you. First, none of the
chords were really in JI anyway, so if I were to treat these chords
as rough approximations of 27:32:36:48 and 18:24:27:32, with
geometric
means 34.9554 and 24.7172 respectively, what would you think of their
new places in my ordering by geometric mean?

Hmmm. Well, I know they are not "exact," but I listened to them in
the sequence determined by your method above, and I was quite happy
with them. They seemed to fit into the increasingly discordant array
with no problem... Certainly a better positioning than where they
were...
_________ ____ __ _ _ _
Joseph Pehrson

🔗Paul H. Erlich <PERLICH@...>

🔗Joseph Pehrson <pehrson@...>

--- In harmonic_entropy@egroups.com, "Paul H. Erlich" <PERLICH@A...>
wrote:
> >Hmmm. Well, I know they are not "exact," but I listened to them
in the sequence determined by your method above, and I was quite
happy with them. They seemed to fit into the increasingly discordant
array with no problem... Certainly a better positioning than where
they were...
>
> Wouldn't you say, though, that they should be somewhat _higher_ in
the ranking? Based solely on your previous comments on these chords,
it would seem so . . . and even more so if 9:11:13:15 is included . .
. ?

Hmmmm (I'm not humming the chord, just thinking...) you may be right
here... but how on earth would you place it?? You couldn't use the
"geometric mean" ranking (??!)

___________ ___ __ __
Joseph Pehrson

🔗Paul H. Erlich <PERLICH@...>

Joseph wrote,

>Hmmmm (I'm not humming the chord, just thinking...) you may be right
>here...

So you're agreeing that those two near-12-tET tetrads would be ranked higher
than the Pythagorean just interpretations would indicate?

>but how on earth would you place it?? You couldn't use the
>"geometric mean" ranking (??!)

Well, I was trying to impress upon you, with this and with the 9:11:13:15
example, that _both_ the tetradic harmonic entropy (as proxied for by the
geometric mean) and a diadic measure like either the original ranking or a
Sethares-like ranking, are factors in determining the discordance of a
chord. This is the so-called two-component theory of consonance.

(Eventually I will be able to calculate the tetradic harmonic entropy for
_any_ tetrad, not just simple otonal JI ones, and we will be able to
dispense with the geometric mean proxy.)

🔗Joseph Pehrson <pehrson@...>

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

http://www.egroups.com/message/harmonic_entropy/50

>
> So you're agreeing that those two near-12-tET tetrads would be
ranked higher than the Pythagorean just interpretations would
indicate?
>

Well...all the chords around the geometric mean of approximately 33
(including the new ones when we put them there) seem to have a fairly
similar concordance... so it's a bit difficult to judge specifically.

My guess is, though, that I might be "happy" with the non JI chords
even further "up" the scale toward consonance. I wouldn't be
certain, though, until you could give me a real "tentative" (that's
kind of an oxymoron... isn't it!) ordering to test aurally.

> >but how on earth would you place it?? You couldn't use the
> >"geometric mean" ranking (??!)
>
> Well, I was trying to impress upon you, with this and with the
9:11:13:15 example, that _both_ the tetradic harmonic entropy (as
proxied for by the geometric mean) and a diadic measure like either
the original ranking or a Sethares-like ranking, are factors in
determining the discordance of a chord. This is the so-called
two-component theory of consonance.
>

I guess in a way this would make sense... since one method is
evaluating the otonal equivalents (using the geometric mean) and the
other is more akin to evaluating the concordance of the UTONAL chord
of the pair... So the "truth" is somewhat "inbetween" the two
methods of evaluation (??) Am I "getting" you there?

> (Eventually I will be able to calculate the tetradic harmonic
entropy for _any_ tetrad, not just simple otonal JI ones, and we
will be able to dispense with the geometric mean proxy.)

This will be interesting and I will eventually want to re-order the
sound sample page!
___________ ___ __ __
Joseph Pehrson

🔗Paul H. Erlich <PERLICH@...>

Joseph Pehrson wrote,

>My guess is, though, that I might be "happy" with the non JI chords
>even further "up" the scale toward consonance. I wouldn't be
>certain, though, until you could give me a real "tentative" (that's
>kind of an oxymoron... isn't it!) ordering to test aurally.

I don't get it . . . what is it you need me to give you

> (Eventually I will be able to calculate the tetradic harmonic
>entropy for _any_ tetrad, not just simple otonal JI ones, and we
>will be able to dispense with the geometric mean proxy.)

>This will be interesting and I will eventually want to re-order the
>sound sample page!

Well this will still be only one component of the total discordance . . .
deciding how to combine the two components (or three: dyadic, triadic, and
tetradic) will be an additional challenge.

>This is the so-called
>two-component theory of consonance.

>I guess in a way this would make sense... since one method is
>evaluating the otonal equivalents (using the geometric mean) and the
>other is more akin to evaluating the concordance of the UTONAL chord
>of the pair... So the "truth" is somewhat "inbetween" the two
>methods of evaluation (??) Am I "getting" you there?

Basically, though as you're language indicates, it's not at all a
symmetrical situation between otonal and utonal . . .

🔗Joseph Pehrson <pehrson@...>

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

http://www.egroups.com/message/harmonic_entropy/53

> Joseph Pehrson wrote,
>
> >My guess is, though, that I might be "happy" with the non JI
chords
> >even further "up" the scale toward consonance. I wouldn't be
> >certain, though, until you could give me a real "tentative"
(that's
> >kind of an oxymoron... isn't it!) ordering to test aurally.
>
> I don't get it . . . what is it you need me to give you
>

Well, I guess there isn't anything yet... as far as I know you only
put these two tetrads: 302__502__1004 and 502__702__1004 in ONE place
so far, according to their approximations to 27:32:36:48 for the
former and 18:24:27:32 for the latter, correct??

So far, you haven't suggest any other possible place for these
tetrads "higher up" toward concordance in the listing... right??
_________ ___ __ _
Joseph Pehrson

🔗Paul H. Erlich <PERLICH@...>

Joseph wrote,

>Well, I guess there isn't anything yet... as far as I know you only
>put these two tetrads: 302__502__1004 and 502__702__1004 in ONE place
>so far, according to their approximations to 27:32:36:48 for the
>former and 18:24:27:32 for the latter, correct??

That was meant to show the inadequacy of this kind of single-component,
geometric mean or tetradic harmonic entropy discordance model. 9:11:13:15
also demonstrates this inadequacy. Though a fully worked out tetradic
harmonic entropy model will be able to handle non-JI chords naturally, I
expect the problem to remain essentially the same. Actually, it may place
the 0_302__502__1004 and 0_502__702__1004 tetrads even _lower_ in the
ranking than the just 27:32:36:48 and 18:24:27:32, because there will be
more possibility for confusion with 15:18:20:27 and 20:27:30:36,
respectively. So don't expect future developments to improve this model's
ability to rank according to _total_ discordance -- it will still be only
one component.

🔗Joseph Pehrson <pehrson@...>

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

http://www.egroups.com/message/harmonic_entropy/55

>
> That was meant to show the inadequacy of this kind of
single-component, geometric mean or tetradic harmonic entropy
discordance model. 9:11:13:15 also demonstrates this inadequacy.
Though a fully worked out tetradic harmonic entropy model will be
able
to handle non-JI chords naturally, I expect the problem to remain
essentially the same. Actually, it may place the 0_302__502__1004 and
0_502__702__1004 tetrads even _lower_ in the ranking than the just
27:32:36:48 and 18:24:27:32, because there will be more possibility
for confusion with 15:18:20:27 and 20:27:30:36, respectively. So
don't expect future developments to improve this model's ability to
rank according to _total_ discordance -- it will still be only one
component.

Oh... so that entropic "confusion" with the "lower geometrical mean"
otonal chords is why you suspect that the ranking of those tetrads
might go toward the "lower" or "more concordant" side than where we
already placed them, yes??
____________ ____ __ _ _
Joseph Pehrson

🔗Paul H. Erlich <PERLICH@...>

Quite the opposite, Joseph. entropic confusion -> more discordant.
Meanwhile, you indicated that you might want to move them in the other, more
concordant direction.

-----Original Message-----
From: Joseph Pehrson [mailto:pehrson@...]
Sent: Wednesday, October 04, 2000 5:09 PM
To: harmonic_entropy@egroups.com
Subject: [harmonic_entropy] Re: Sethares algorithm

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

http://www.egroups.com/message/harmonic_entropy/55

To unsubscribe from this group, send an email to:
harmonic_entropy-unsubscribe@egroups.com

🔗Joseph Pehrson <pehrson@...>

🔗Paul H. Erlich <PERLICH@...>

🔗Joseph Pehrson <pehrson@...>

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

http://www.egroups.com/message/harmonic_entropy/59

> Joseph wrote,
>
> >Oh! Well, it does seem like that... but wouldn't that correspond
> >with the new "lower" geometric means that you would be giving to
the chords if you were to "confuse" them??
>
> Joseph, the geometric mean only represents the harmonic entropy for
very simple JI otonal chords, where there is one interpretation of
the
chord as a segment of the harmonic series that is far more likely
than
any other. Otherwise, as in the case of confusion, this approximation
breaks down -- and remember, harmonic entropy _is_ exactly a measure
of the degree of confusion.

Hmmm. I see... Well, the "plot thickens" so to speak... I will await
further developments!

_____ __
JP

🔗Paul H. Erlich <PERLICH@...>

Joseph, I may have misunderstood you, but you may have latched on to the
fact that 15:18:20:27 uses lower numbers than 27:32:36:48 -- but ignored the
fact that 20:27:30:36 uses higher numbers than 18:24:27:32 . . . were you
just "reading halfway" again? In any case, these comparisons are irrelevant
-- what is relevant is that

a) a full tetradic harmonic entropy model would find the two near-12-tET
chords in question to be confused between the pairs of alternatives above,
as well as others,

and

b) these chords are more concordant this model would indicate, since a pure
dyadic model (like my earlier one or Sethares') does have something
important to say about total discordance.

🔗Joseph Pehrson <josephpehrson@...>

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

http://www.egroups.com/message/harmonic_entropy/61

> Joseph, I may have misunderstood you, but you may have latched on
to
the fact that 15:18:20:27 uses lower numbers than 27:32:36:48 -- but
ignored the fact that 20:27:30:36 uses higher numbers than
18:24:27:32
. . . were you just "reading halfway" again?

That's more than possible... I can't even figure out where the
20:27:30:36 and 15:18:20:27 came from!?? They weren't from the
tetrads we were discussing, were they?? Or was that the Monzo tetrad
that you found problematic?

I thought the ones we were placing were 0__302__502__1004 and
0__502__702__1004 (??)

_____ __ __
JP

🔗Carl Lumma <CLUMMA@...>

>That was meant to show the inadequacy of this kind of single-component,
>geometric mean or tetradic harmonic entropy discordance model. 9:11:13:15
>also demonstrates this inadequacy.

Sorry to just join in here, fellas, but how does 9:11:13:15 require the
roughness component? It would seem the "composite" and "crunchy" chords
would be good examples here...

>Though a fully worked out tetradic harmonic entropy model will be able to
>handle non-JI chords naturally, I expect the problem to remain essentially
>the same. Actually, it may place the 0_302__502__1004 and 0_502__702__1004
>tetrads even _lower_ in the ranking than the just 27:32:36:48 and
>18:24:27:32, because there will be more possibility for confusion with
>15:18:20:27 and 20:27:30:36, respectively. So don't expect future
>developments to improve this model's ability to rank according to _total_
>discordance -- it will still be only one component.

Sorry to just step in here guys, but how many chord progressions have there
been? From browsing the thread, I'm guessing two (dyadic harmonic entropy
sum, and geometric mean)? Is there any way I can download these as entire
runs, in a zip file, as I requested long ago? I'm sure I'm not the only one
fed up with mp3.com here. In fact, has _anybody_ listened to this besides
Paul and Joseph?

-Carl

🔗David Finnamore <daeron@...>

--- In harmonic_entropy@egroups.com, Carl Lumma <CLUMMA@N...> wrote:
> I'm sure I'm not the only one
> fed up with mp3.com here. In fact, has _anybody_ listened to this
besides
> Paul and Joseph?

Funny. I love mp3.com. I've had no trouble with the tuning lab page
at all, except that it's kind of awkward scrolling up and down,
comparing various tetrads. I found myself wishing that the links
were in a simple list, one line per tetrad. Other than that, it
works great. I'm in the slow process of comparing all the tetrads
with each other, and will post my perceived most-concordant-to-most-
discordant ordering soon.

Joseph, I thought that the mp3.com pages automatically re-
ordered "songs" according to how often they get played. Isn't that
going to screw up the listing from time to time? (Assuming us 3
aren't the only listeners!)

David Finnamore

🔗Joseph Pehrson <pehrson@...>

--- In harmonic_entropy@egroups.com, Carl Lumma <CLUMMA@N...> wrote:

http://www.egroups.com/message/harmonic_entropy/72

I'm sure I'm not the
only one fed up with mp3.com here. In fact, has _anybody_ listened
to this besides Paul and Joseph?
>
> -Carl

Hi Carl...

I'm sorry you're not happy with the mp3.com site. It was a
recommendation from John Starrett, after I was complaining that I
could not get files to "stream" from another site.

If you can find a site where you can "stream" files... please be my
guest to continue the experiments there....

However, if they don't "stream," there's no point to it... since it
takes too long to listen to them if they have to download....

________ ___ __ _
Joseph

🔗Joseph Pehrson <pehrson@...>

--- In harmonic_entropy@egroups.com, "David Finnamore" <daeron@b...>
wrote:
http://www.egroups.com/message/harmonic_entropy/73

> --- In harmonic_entropy@egroups.com, Carl Lumma <CLUMMA@N...> wrote:
> > I'm sure I'm not the only one
> > fed up with mp3.com here. In fact, has _anybody_ listened to
this
> besides
> > Paul and Joseph?
>
> Funny. I love mp3.com. I've had no trouble with the tuning lab
page at all, except that it's kind of awkward scrolling up and down,
> comparing various tetrads. I found myself wishing that the links
> were in a simple list, one line per tetrad. Other than that, it
> works great. I'm in the slow process of comparing all the tetrads
> with each other, and will post my perceived most-concordant-to-most-
> discordant ordering soon.
>
> Joseph, I thought that the mp3.com pages automatically re-
> ordered "songs" according to how often they get played. Isn't that
> going to screw up the listing from time to time? (Assuming us 3
> aren't the only listeners!)
>
> David Finnamore

Hi David...

Well, I'm happy that at least *somebody* is happy with the mp3.com
site. No, they do *NOT* reorder files, thank god.... that is a
setting that you provide MANUALLY.

The reason the files are not close together is due to Paul's
extensive commentary, which I included. I thought, though, that his
comments were valuable .... (??)
__ ___ __ __ _
Joseph

🔗Joseph Pehrson <pehrson@...>

--- In harmonic_entropy@egroups.com, Carl Lumma <CLUMMA@N...> wrote:

http://www.egroups.com/message/harmonic_entropy/72

> Sorry to just step in here guys, but how many chord progressions
have there been? From browsing the thread, I'm guessing two (dyadic
harmonic entropy sum, and geometric mean)? Is there any way I can
download these as entire runs, in a zip file, as I requested long
ago?
>
> -Carl

This is correct, Carl. These are the two basic orderings. You
should know that the "entire runs" of these tetrads are HUGE. When
Paul sent them separately zipped as .wav files it took 6 HOURS to
download them! My computer worked on it all night! However, as mp3
files they would not take as long.

The ordering on the "Tuning Lab" page uses the dyadid harmonic
entropy sum. However, Paul also developed the following ordering of
JI chords based upon the geometric mean:

bass tenor alto soprano otonal_rep g.m.

___ __ __ _
Joseph

🔗Paul H. Erlich <PERLICH@...>

Carl Lumma wrote,

>Sorry to just join in here, fellas, but how does 9:11:13:15 require the
>roughness component?

Relative to the other chords in the example with similar geometric means, it
is way more dissonant.

>It would seem the "composite" and "crunchy" chords
>would be good examples here...

Well there are plenty of those already in there, making for the contrast
against chords like 9:11:13:15.

Carl -- if you like I'll send you the zipped .wav files, as I did for
Joseph, which could bog down your server for a while -- or why don't you
have Joseph send you the .mp3 files?

🔗Joseph Pehrson <pehrson@...>

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

http://www.egroups.com/message/harmonic_entropy/81

> Carl -- if you like I'll send you the zipped .wav files, as I did
for Joseph, which could bog down your server for a while -- or why
don't you have Joseph send you the .mp3 files?

The easiest thing, really, would be for Carl to go to the "dreaded"
mp3.com site and download the mp3 files from there!

Then he would never have to visit the place again, and could do his
*own* manipulation of the files...

That's what I'd suggest. It also is much easier, since he could
download them "bit by bit" (literally!). Otherwise it's about a 40
minute send, even with mp3s!!!
___ __ __ _
Joseph

🔗Paul H. Erlich <PERLICH@...>

Joseph wrote,

>I guess I'm really not understanding how you arrive at this. How do
>you do it?? How did you figure out the JI chords that would be
>confused with those tetrads? I don't have the vaguest idea...

Well, once I have the model up and running, the computer will do this
automatically, but for now, it's by inspection.

Let's start with 0__302__502__1004. If you tune this, starting with 0, as a
just fourth up, another just fourth up from there, and a just perfect fifth
down _from there_, you get 27:32:36:48. Is that clear, or should I slow
down?

Now if you tune the same chord like this: starting with 0, tune a just
fourth up, then _go back to 0_, then tune a just minor third up, then tune a
perfect fifth up from there, you get 15:18:20:27. Right?

(to be continued)

🔗Paul H. Erlich <PERLICH@...>

🔗Joseph Pehrson <pehrson@...>

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

http://www.egroups.com/message/harmonic_entropy/87

> Let's start with 0__302__502__1004. If you tune this, starting with
0, as a just fourth up, another just fourth up from there, and a just
perfect fifth down _from there_, you get 27:32:36:48. Is that clear,
or should I slow down?
>
Well... I can see you are taking the "cents" value of the just
intervals and tuning from there, by a couple of different methods.

But then, how do you get the ratios?? Sorry to be so slow about
this! I could just lie and say I understand it... but that would be
self-defeating!

Thanks!

Joseph

🔗Paul H. Erlich <PERLICH@...>

Joseph wrote,

>But then, how do you get the ratios??

One way is to write the chords as fractions first. So the first chord, tuned
by two fourths up and one fifth down, is 1/1 : 32/27 : 4/3 : 16/9. Do you
see how I get that, or should I slow down some more? Anyway, once you have
that expression, in order to get the otonal representations, you have to
make all the denominators the same. In this case, 27 is the least common
multiple of the denominators, so you can write the chord as 27/27 : 32/27 :
36/27 : 48/27 (do you see how I got that)? Now, you're free to transpose all
the notes of the chord by the same amount, so of course you multiply them
all by 27, to get 27:32:36:48. . . . how confused are you now?

🔗Joseph Pehrson <pehrson@...>

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

http://www.egroups.com/message/harmonic_entropy/91

> One way is to write the chords as fractions first. So the first
chord, tuned by two fourths up and one fifth down, is 1/1 : 32/27 :
4/3 : 16/9. Do you see how I get that, or should I slow down some
more? Anyway, once you have that expression, in order to get the
otonal representations, you have to make all the denominators the
same. In this case, 27 is the least common multiple of the
denominators, so you can write the chord as 27/27 : 32/27 : 36/27 :
48/27 (do you see how I got that)? Now, you're free to transpose
all the notes of the chord by the same amount, so of course you
multiply them all by 27, to get 27:32:36:48. . . . how confused are
you now?

Uhh. I don't get it...

Just joking. Thanks, so much Paul for taking the time to explain
this. This is pretty rudimentary otonal theory, it seems... so I'm
glad I asked the question!

Of course, then the second "possible" just tuning of the tetrad would
be:

1/1 : 6/5 : 4/3 : 9/5

which, with a denominator of 15 would be:

15:18:20:27

...

I think I've got it...

Thanks again!

______ ___ __ _ _
Joseph Pehrson

🔗Monz <MONZ@...>

--- In harmonic_entropy@egroups.com, "Joseph Pehrson" wrote:
> http://www.egroups.com/message/harmonic_entropy/76
>
>> --- In harmonic_entropy@egroups.com, Carl Lumma wrote:
>>
>> http://www.egroups.com/message/harmonic_entropy/72
>>
>>
>> I'm sure I'm not the
>> only one fed up with mp3.com here. In fact, has _anybody_ listened
>> to this besides Paul and Joseph?
>
>
> I'm sorry you're not happy with the mp3.com site. It was a
> recommendation from John Starrett, after I was complaining that I
> could not get files to "stream" from another site.
>
> If you can find a site where you can "stream" files... please be my
> guest to continue the experiments there....
>
> However, if they don't "stream," there's no point to it... since it
> takes too long to listen to them if they have to download....

On my system, streaming is messed up: quality is lower, and
the sound gets interrupted as my phone line gets clogged with
data. My solution to this problem was to download the whole
set overnight while I slept (using an FTP robot). Now I can
peruse the whole list of chords offline at my leisure.

The problem with that?... I had to rewrite the HTML code from
the Tuning Labs page to have the links pointing to the sound
files correctly, and there was SO MUCH advertising and other
crap on the mp3 page that I still haven't finished it. So when
all that's done, I'll listen and give my observations...

But Carl's right: if all the mp3 files and the Tuning Lab page
were put into one zip file, it would be a lot easier for many
of us to get on with the experiment.

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

🔗Carl Lumma <CLUMMA@...>

>If you can find a site where you can "stream" files... please be my
>guest to continue the experiments there....

Two things:

1. I do not accept streaming as a technology. I have never once used
it, and I never will.

2. Even if I did, I do not have a fast enough connection to stream
content of the fidelity needed for these examples.

>However, if they don't "stream," there's no point to it... since it
>takes too long to listen to them if they have to download....

It is easy to see that the only time streaming saves you is the time
it takes to listen to the thing. And since downloading can be done
when you're not at your computer, streaming doesn't really save any
time at all.

>You should know that the "entire runs" of these tetrads are HUGE. When
>Paul sent them separately zipped as .wav files it took 6 HOURS to
>download them!

Both orderings are 6 hours, or are they 6 hours each?

>My computer worked on it all night!

And did that bother you?

>However, as mp3 files they would not take as long.

Right. They'd take about 36 minutes.

>Carl -- if you like I'll send you the zipped .wav files, as I did for
>Joseph, which could bog down your server for a while -- or why don't you
>have Joseph send you the .mp3 files?

Why doesn't somebody post the zipped mp3's to the egroups files area?
Zipping won't save much over raw mp3, of course, but it will save a
ton of bandwidth by reducing the number of connections from 40+ to 2.

Needless to say, please _don't_ attach them to an e-mail. My ISP will
send hired goons.

>The easiest thing, really, would be for Carl to go to the "dreaded"
>mp3.com site and download the mp3 files from there!

Not going to happen. I gave my e-mail address to mp3.com a year ago,
and they spam me every week. I'd fake the e-mail address, but I'd rather
just never patronize their site again.

-Carl

🔗Paul H. Erlich <PERLICH@...>

🔗Monz <MONZ@...>

I've made a zip file of all the tetrads and the Tuning Lab
webpage (as modified by me, not yet finished), and put it at:

http://www.egroups.com/files/harmonic_entropy/tuninglab.zip

It's almost 4 megs, and so will take a little while to download
(it took about 30 minutes for me to send it). But that will put
all the examples, at their best fidelity, on your own hard-drive.

If anyone else wants to finish the editing of the HTML, feel
free to do so...

Also, I only have 26 tetrads, not 36 as Paul described... have
some been added, and I missed it?...

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

🔗Paul H. Erlich <PERLICH@...>

🔗Joseph Pehrson <josephpehrson@...>

--- In harmonic_entropy@egroups.com, " Monz" <MONZ@J...> wrote:

http://www.egroups.com/message/harmonic_entropy/100

>
> I've made a zip file of all the tetrads and the Tuning Lab
> webpage (as modified by me, not yet finished), and put it at:
>
> http://www.egroups.com/files/harmonic_entropy/tuninglab.zip
>
> It's almost 4 megs, and so will take a little while to download
> (it took about 30 minutes for me to send it). But that will put
> all the examples, at their best fidelity, on your own hard-drive.
>
> If anyone else wants to finish the editing of the HTML, feel
> free to do so...
>
> Also, I only have 26 tetrads, not 36 as Paul described... have
> some been added, and I missed it?...
>

Thanks so much, Joe, for doing this and making them available.
However, you should have 36 files, as Paul mentioned. Are you sure
that isn't just a typo??

Thanks again!!

Joe

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

🔗Monz <MONZ@...>

--- In harmonic_entropy@egroups.com, "Joseph Pehrson" wrote:
>
> http://www.egroups.com/message/harmonic_entropy/103
>
> Thanks so much, Joe, for doing this and making them available.
> However, you should have 36 files, as Paul mentioned. Are you
> sure that isn't just a typo??

Positive. I suppose my FTP broke the connection and so I only
got 26 of them - I thought they were all there, but as I've
said, I haven't finished editing the webpage to correlate it
with the mp3 files, so I haven't listened to all of them yet.

When I get a chance, I'll examine what I have and add the
missing ones.

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

🔗Monz <MONZ@...>

🔗Joseph Pehrson <josephpehrson@...>

--- In harmonic_entropy@egroups.com, Carl Lumma <CLUMMA@N...> wrote:

http://www.egroups.com/message/harmonic_entropy/98

> >If you can find a site where you can "stream" files... please be
my guest to continue the experiments there....
>
> Two things:
>
> 1. I do not accept streaming as a technology. I have never once
used it, and I never will.

Wow, Carl. That certainly do mean it don't exist!

>
> 2. Even if I did, I do not have a fast enough connection to stream
> content of the fidelity needed for these examples.
>

OK, point taken.

>
> Both orderings are 6 hours, or are they 6 hours each?
>

They are 6 hours each, Carl (especially since there is only one set).
Have you ever read WAR AND PEACE...? [just joking]

> >My computer worked on it all night!
>
> And did that bother you?
>

It "clogged" my server, and the server "flipped out" even making
multiple copies of mail, and I had some other important mail to get,
so "yes..." Using CompuServe, I cannot delete mail from the
server... I have to DOWNLOAD everything before I can delete files.
Not ideal.

> >However, as mp3 files they would not take as long.
>
> Right. They'd take about 36 minutes.
>

That's right...

>
> Why doesn't somebody post the zipped mp3's to the egroups files
area?

Well, you can "settle down" Carl... since this has been done, as I'm
sure you've already seen.

It's in the "files" area...

http://www.egroups.com/files/harmonic_entropy/Pehrson/

> >The easiest thing, really, would be for Carl to go to the
"dreaded" mp3.com site and download the mp3 files from there!
>
> Not going to happen. I gave my e-mail address to mp3.com a year
ago,and they spam me every week. I'd fake the e-mail address, but
I'd rather just never patronize their site again.
>
> -Carl

Hmmm. Carl, did I ever ask you if you had a strong opinion on this
subject?? <Dan smiley thing>

________ ___ __ _
Joseph Pehrson

🔗Carl Lumma <CLUMMA@...>

>>1. I do not accept streaming as a technology. I have never once
>>used it, and I never will.
>
>Wow, Carl. That certainly do mean it don't exist!

You betcha.

>It "clogged" my server, and the server "flipped out" even making
>multiple copies of mail, and I had some other important mail to get,
>so "yes..." Using CompuServe, I cannot delete mail from the
>server... I have to DOWNLOAD everything before I can delete files.
>Not ideal.

Ouch. Thought about getting rid of Compuserve?

>>Why doesn't somebody post the zipped mp3's to the egroups files
>>area?
>
>Well, you can "settle down" Carl... since this has been done, as I'm
>sure you've already seen.

That depends on when already is. I feel rather left out as a digest
subscriber... but I only get about 30min a day for e-mail. Thanks to
Joe and Joe, and of course Paul, for making this demo possible.

>Hi Paul...
>Could you please "repost" the ordering for the tetrads according to
>the "utonal" harmonic entropy diadic sum??

Actually, in the interest of this discussion, I might ask Paul to repost
the two orderings for diadic harmonic sum and geometric mean. I haven't
seen the file names yet (downloading now), but I assume these orderings
could be made rather concise by file name. What's the "'utonal' harmonic
entropy diadic sum" ordering??

>I can't seem to find the original e-mail.

S'thinks it was on the tuning list? I'm having trouble finding anything
on this thread, including how the orderings were arrived at. Paul, since
the word document describes non-JI chords, how could the geometric mean
order the same files? And how were the 36 chords found? Were you able
to minimize six variables at once, or is this still the Monte Carlo
method? As for the html file everyone's talking about... I haven't the
foggiest. Assuming it is redundant with the word document, and/or in
the zip archive. Don't you hate the newbie effect?

Our medium seems well-suited for consensus-building, but very messy for
'storing' knowledge -- picking up a thread mid-stream is near impossible,
and having to go through a thread from scratch later on seems very
inefficient. Maybe for the medium to be truly effective it should be
the custom, at the end of a thread, for the participants to pick one
among them to add a summary of the thread to an on-going historical
document.

-Carl

🔗Paul Erlich <PERLICH@...>

--- In harmonic_entropy@egroups.com, Carl Lumma <CLUMMA@N...> wrote:

> Actually, in the interest of this discussion, I might ask Paul to
repost
> the two orderings for diadic harmonic sum and geometric mean. I
haven't
> seen the file names yet (downloading now), but I assume these
orderings
> could be made rather concise by file name.

I'll post the orderings when I get back to the office.

> What's the "'utonal' harmonic
> entropy diadic sum" ordering??

Joseph calls it 'utonal' because it seems correct for all chords that
don't have any otonal synergy.

> Paul, since
> the word document describes non-JI chords, how could the geometric
mean
> order the same files?

I put the non-JI chords together at the bottom of that ordering.

> And how were the 36 chords found? Were you able
> to minimize six variables at once, or is this still the Monte Carlo
> method?

Neither. Monte Carlo was used to find global minima, if you recall.
In this case, I only had to locally minimize a function of three
variables, and I did it by brute force, calculating all possible
tetrads within a certain range with 2¢ increments (8,000,000
tetrads). I found 140 -- 15 with octaves, the 36 in our experiment,
and 89 more. That of course was using a certain, Farey-based model,
but no matter -- there's no reason, when comparing different
orderings, why one would want to use chords that are local minima
according to any of the orderings being compared (in fact, it might
have been better if we didn't start with just local minima -- but I
didn't know that we were going to end up comparing different
dissonance orderings using these tetrads . . .)

🔗Joseph Pehrson <josephpehrson@...>

--- In harmonic_entropy@egroups.com, Carl Lumma <CLUMMA@N...> wrote:

http://www.egroups.com/message/harmonic_entropy/117

> Actually, in the interest of this discussion, I might ask Paul to
repost the two orderings for diadic harmonic sum and geometric mean.
I haven't seen the file names yet (downloading now), but I assume
these orderings could be made rather concise by file name.

I believe some of the orderings were sent to me in private e-mail by
Paul. However, I note below (next messages) that he is going to post
all the orderings on the site... which will be good.

>I'm having trouble finding
anything on this thread, including how the orderings were arrived at.

I'm sure Paul will post a little description about the orderings as
well...

>As for the html file everyone's talking about... I haven't
the foggiest.

This you can "forget" about. It's simply the mp3.com page that you
hate. Monzo saved it in html format, but you don't need it.

All you need are the mp3 files in the "files" section of the egroup,
the "Word" document with Paul's comments on the tetrads, and the
orderings which he is going to repost.

And, of course, I am certain your comments will be most welcome to
all concerned!

>Assuming it is redundant with the word document, and/or
in the zip archive.

Yes, as per above.

Don't you hate the newbie effect?
>

I might, if I weren't one of the newest "biees..."

> Our medium seems well-suited for consensus-building, but very messy
for 'storing' knowledge -- picking up a thread mid-stream is near
impossible, and having to go through a thread from scratch later on
seems very inefficient. Maybe for the medium to be truly effective
it should be
> the custom, at the end of a thread, for the participants to pick one
> among them to add a summary of the thread to an on-going historical
> document.
>
> -Carl

Well, this might be a good idea. However, keeping most of the
messages concerning harmonic entropy on the harmonic entropy site
will probably also be a good was to "consolidate" the info.

Best,
__________ ___ _ _ _ _
Joseph Pehrson

🔗Carl Lumma <CLUMMA@...>

>I'll post the orderings when I get back to the office.

Thanks!

>> What's the "'utonal' harmonic
>> entropy diadic sum" ordering??
>
>Joseph calls it 'utonal' because it seems correct for all chords that
>don't have any otonal synergy.

Is that the geometric mean one, then?

>>Paul, since the word document describes non-JI chords, how could the
>>geometric mean order the same files?
>
>I put the non-JI chords together at the bottom of that ordering.

Gotcha.

>> And how were the 36 chords found? Were you able
>> to minimize six variables at once, or is this still the Monte Carlo
>> method?
>
>Neither. Monte Carlo was used to find global minima, if you recall.
>In this case, I only had to locally minimize a function of three
>variables, and I did it by brute force, calculating all possible
>tetrads within a certain range with 2¢ increments (8,000,000
>tetrads). I found 140 -- 15 with octaves, the 36 in our experiment,
>and 89 more. That of course was using a certain, Farey-based model,
>but no matter -- there's no reason, when comparing different
>orderings, why one would want to use chords that are local minima
>according to any of the orderings being compared (in fact, it might
>have been better if we didn't start with just local minima -- but I
>didn't know that we were going to end up comparing different
>dissonance orderings using these tetrads . . .)

Thanks, that makes perfect sense.

-Carl

🔗Carl Lumma <CLUMMA@...>

>>Assuming it is redundant with the word document, and/or
>>in the zip archive.
>
>Yes, as per above.

Thanks for filling me in (on the html page too).

>>Don't you hate the newbie effect?
>
>I might, if I weren't one of the newest "biees..."

Oh, I actually meant newbie to a thread, not a list.

>>Our medium seems well-suited for consensus-building, but very messy
>>for 'storing' knowledge -- picking up a thread mid-stream is near
>>impossible, and having to go through a thread from scratch later on
>>seems very inefficient. Maybe for the medium to be truly effective
>>it should be the custom, at the end of a thread, for the participants
>>to pick one among them to add a summary of the thread to an on-going
>>historical document.
>
>Well, this might be a good idea. However, keeping most of the
>messages concerning harmonic entropy on the harmonic entropy site
>will probably also be a good was to "consolidate" the info.

Perhaps... in that sense I could also see it making things worse, since
now there are two list archives to search!

In general, I think searching is far more powerful in usenet (mailing
lists, etc.) than in traditional media, but because of inside language
that tends to develope in messaging communities, the media is much
more frail, in terms of what happens if you do miss one thing in your
search.

-Carl

🔗Joseph Pehrson <josephpehrson@...>

--- In harmonic_entropy@egroups.com, Carl Lumma <CLUMMA@N...> wrote:

http://www.egroups.com/message/harmonic_entropy/122

> Perhaps... in that sense I could also see it making things worse,
since now there are two list archives to search!
>

Frankly, I'm a little "pissed" about it too! I now have to follow
THREE Tuning Lists (including Grady's!).

It seems people could develop a little toleration of what *other*
people are interested in, rather than "splintering" off or forcing
*other* people to spliter off...

But, my opinion wasn't asked, and there's nothing I, or anybody else,
can do about it... so that's that!

_________ ____ __ __ _ _
Joseph Pehrson

🔗Paul H. Erlich <PERLICH@...>

🔗Monz <MONZ@...>

🔗Carl Lumma <CLUMMA@...>

🔗Paul Erlich <PERLICH@...>

--- In harmonic_entropy@egroups.com, Carl
Lumma <CLUMMA@N...> wrote:
> Thanks everybody for putting together the tetrad listening lab. I
> haven't listened yet -- just got back from a backpacking trip -- but
> I have the two orderings, and David Finnamore's results, and for
> the record, I'm keeping them all until after I rank the chords
myself.
> That should be this week yet... stay tuned!
>
> -Carl

Not to dampen the excitement here, but if I
knew we were going to do listening
experiments, I would have started with
dyads, then moved on to triads, before
tackling tetrads. There are just too many
variables when it comes to tetrads to hope
that a few people's rankings of some tetrads
not chosen for the purpose will lead to a fully
developed model of concordance.

Perhaps we should create a bunch of dyad
files and see if we can ascertain whether
harmonic entropy can effectively proxy for
all the components of dyadic dissonance. If
so, we could determine what value of s is
appropriate for typical timbres and ranges.
John deLaubenfels' judgment of 9:7 as more
dissonant than 11:9 seems to indicate that s
somewhere around 1.5% is probably most
appropriate for him, while Partch, if he is to
be taken at his word, indicates preferences
that imply an s value of about 0.5%. How we
resolve this discrepancy and whether
harmonic entropy can effectively model
dyadic discordance at all is a question that
needs to be resolved before we can attempt
to address the problem of triadic, let alone
tetradic, discordance in better than the most
general of terms.

However, I think the files that exist now and
the responses that are beginning to
accumulate will at least help show that
neither a purely dyadic approach, nor a
purely otonal approach, is sufficient to take
all the variations into account. If we find a
pattern that runs counter to the predictions
of _both_ models, it would certainly be good
to know that at this early stage. I can't check
right now, but I wonder if there is any pair
of tetrads where the one that David
Finnamore found to be the more discordant
(by a significant margin) is predicted by
_both_ of the rankings I just reproduced to
be the more concordant (also preferably by a
significant margin). If there is _any_ such pair
of tetrads (I know, there are a lot of pairs to
consider), and David verifies his ordering,
then we will have cause to step back and
reconsider our assumptions.

Anyone wanna take a stab at performing this
comparison?

🔗Joseph Pehrson <pehrson@...>

--- In harmonic_entropy@egroups.com, "Paul Erlich" <PERLICH@A...>

http://www.egroups.com/message/harmonic_entropy/141

>
> Not to dampen the excitement here, but if I
> knew we were going to do listening
> experiments, I would have started with
> dyads, then moved on to triads, before
> tackling tetrads. There are just too many
> variables when it comes to tetrads to hope
> that a few people's rankings of some tetrads
> not chosen for the purpose will lead to a fully
> developed model of concordance.
>
Gee, Paul. Maybe I'm missing something, but I still don't understand
the point of the exercise if we are never going to do any listening...

> Perhaps we should create a bunch of dyad
> files and see if we can ascertain whether
> harmonic entropy can effectively proxy for
> all the components of dyadic dissonance. If
> so, we could determine what value of s is
> appropriate for typical timbres and ranges.
> John deLaubenfels' judgment of 9:7 as more
> dissonant than 11:9 seems to indicate that s
> somewhere around 1.5% is probably most
> appropriate for him, while Partch, if he is to
> be taken at his word, indicates preferences
> that imply an s value of about 0.5%. How we
> resolve this discrepancy and whether
> harmonic entropy can effectively model
> dyadic discordance at all is a question that
> needs to be resolved before we can attempt
> to address the problem of triadic, let alone
> tetradic, discordance in better than the most
> general of terms.
>
Well, this seems like a good idea. However, this time perhaps you
can convert your .wav files to .mp3s using your "MusicMatch'
software... Then, they can just be uploaded to the Harmonic Entropy
File site with some accompanying text posted on the list. Then we
can *forget* about the html mess...

> However, I think the files that exist now and
> the responses that are beginning to
> accumulate will at least help show that
> neither a purely dyadic approach, nor a
> purely otonal approach, is sufficient to take
> all the variations into account. If we find a
> pattern that runs counter to the predictions
> of _both_ models, it would certainly be good
> to know that at this early stage. I can't check
> right now, but I wonder if there is any pair
> of tetrads where the one that David
> Finnamore found to be the more discordant
> (by a significant margin) is predicted by
> _both_ of the rankings I just reproduced to
> be the more concordant (also preferably by a
> significant margin). If there is _any_ such pair
> of tetrads (I know, there are a lot of pairs to
> consider), and David verifies his ordering,
> then we will have cause to step back and
> reconsider our assumptions.
>
> Anyone wanna take a stab at performing this
> comparison?

Well, I just posted all three listings together, in case somebody
wants to print them out...

Upon examining the tetrads that David Finnamore found the most
discordant... in virtually all the cases there is at least *ONE*
listing, either the JI geometric mean or the diadic sum ranking that
*ALSO* considers the tetrad significantly discordant.

I only found FOUR of the tetrads on the Finnamore list that seemed at
all anomalous. In all four of these cases, it seems that *BOTH* the
JI geometric mean listing and the diadic sum ranking list these
tetrads as SIGNIFICANTLY more concordant than David Finnamore did.

The four in question are (David's numbering):

#24: 498__702__1086

#26: 312__702___888

#32: 388__776__1090

#33: 184__388___886

____________ ____ __
Joseph Pehrson

🔗Paul H. Erlich <PERLICH@...>

🔗Carl Lumma <CLUMMA@...>

🔗David Finnamore <daeron@...>

--- In harmonic_entropy@egroups.com, "Paul H. Erlich" <PERLICH@A...>
wrote:
> I wonder if David cares to
> comment on his feeling on these chords -- they're all pretty
traditional,
> tonal chords and the second one in the list below certainly sounds
like one
> a tonal piece of music might end on (David's own criterion)

I'd love to. In a word, beating is what shoves most of them down the
list for me. I'd agree that these chords, and practically every
chord in the Lab right now, could be used in tonal music. I'd have a
tough time feeling closure with any of the four below, though.

In all fairness to you, Paul, you've repeatedly stated that the
mathematical rankings offered so far don't take roughness into
account. At least I think that's what you meant. If roughness were
factored in, it would skew things closer to my perception. OK, I'm
going to try to "defend my positions," here, but that doesn't mean
I'm *feeling* defensive. Lord knows, this is about as subjective as
it gets. :-)

> >#24: 498__702__1086

This one has a strong poignancy combined with an air of imminent
danger. It would be great for a psycho-thriller movie scene in which
the victim is just on the verge of recognizing the horror of his
situation. I think it's a wonderful chord. It's just tense as a cat
in a strange house.

> >#26: 312__702___888

Despite the plain minor basis, the 6th gives this one a diminished
character. Basically, it's a third inversion half diminished 7, in
traditional Western parlance. That makes it want to "go somewhere";
it's not stable. And there's a 576 cent interval between the second
and forth notes - that's probably a pretty strong color in anybody's
book. It's not as interesting sounding as most of the chords in its
neighborhood. But it feels like one foot is in mid-air between two
stair steps, on the way down a flight of stairs.

> >#32: 388__776__1090

I can't defend my placement of this one logically. It doesn't really
beat very hard, and it doesn't seem to contain any "zingers." Maybe
it would rank higher for me if it were true JI. I wonder why the
program chose to sharpen the second and fourth tones, while leaving
the third one true? Or with a different timbre, it might jump up the
list a good ways. With some more thought, I can see a piece ending
on this chord but it would probably have to be an extremely tense
piece throughout to land here and feel like the circle had been
closed. Still, after comparing to the ones I listed immediately
ahead of it, I'd keep it at #32 for now.

> >#33: 184__388___886

Whew! It warbles like a drunken nightingale. I don't like the sound
of this one at all. I would use it only in passing, to make the
melodic lines work out better or something. Those that are lower
than this in my ordering have greater emotional tension for me, but I
like them all better than this one. It's always been interesting to
me how a set of consonant intervals can be combined in dissonant
ways. This tetrad is a classic example.

And if anybody disagrees with me, you're wrong, wrong, wrong!!!

Hee, hee.

David Finnamore

🔗Paul Erlich <PERLICH@...>

--- In harmonic_entropy@egroups.com,
"David Finnamore" <daeron@b...> wrote:
> --- In harmonic_entropy@egroups.com, "Paul H. Erlich"
<PERLICH@A...>
> wrote:
>
> I'd love to. In a word, beating is what shoves most of them down
the
> list for me.
>
> In all fairness to you, Paul, you've repeatedly stated that the
> mathematical rankings offered so far don't take roughness into
> account. At least I think that's what you meant. If roughness
were
> factored in, it would skew things closer to my perception.

I'm still waiting for Bill Sethares to clear up
some ambiguities in his roughness formula,
but I expect the _dyadic_ ordering, the one
where otonal and utonal are equal, to proxy
quite well for a Setharian ranking. No
roughness measure will ever give a
significantly different score to two chords
with the same intervals (such as an otonal/
utonal mirror image pair of chords).

> > >#24: [0_]498__702__1086
>
> This one has a strong poignancy combined with an air of imminent
> danger. It would be great for a psycho-thriller movie scene in
which
> the victim is just on the verge of recognizing the horror of his
> situation. I think it's a wonderful chord. It's just tense as a
cat
> in a strange house.

It's a common chord in Mozart, where it
resolves to a chord such as

-1200_386_702_1200.

> > >#26: [0_]312__702___888
>
> Despite the plain minor basis, the 6th gives this one a diminished
> character. Basically, it's a third inversion half diminished 7, in
> traditional Western parlance. That makes it want to "go
somewhere";
> it's not stable.

Also known as a minor sixth chord, it is a
very common ending chord in Salsa and
other latin styles (well, at least the 12-tET
version is).

>And there's a 576 cent interval between the second
> and forth notes - that's probably a pretty strong color in
anybody's
> book.

576 cents sounds a lot like 7:5, only 7 cents
flat. Perhaps I need to use a larger s value so
that 7:5 and its vicinity does not get such a
concordant rating. I'd like to try a dyadic
ranking based on s=1.5%, rather than s=1%,
tomorrow.

> > >#32: [0_]388__776__1090
>
> I can't defend my placement of this one logically. It doesn't
really
> beat very hard, and it doesn't seem to contain any "zingers."
Maybe
> it would rank higher for me if it were true JI. I wonder why the
> program chose to sharpen the second and fourth tones, while leaving
> the third one true?

How is the third one "true"? I see two equal
388-cent major thirds, one 314-cent minor
third, a 702-cent fifth, and two dissonant
intervals. The major thirds were probably
expanded a touch (remember, only 2-cent
increments were allowed) in order to slightly
alleviate the sharp dissonance of the 772-cent
interval that they would produce. Actually,
with a Setharian discordance formula, this
wouldn't have happened, since he gives
sharp minima, rather than rounded ones, at
the simple ratios.

Certainly this chord seems like a poor choice
for ending a piece of music. By the way, I
have a feeling its geometric mean, calculated
from the otonal proportion 16:20:25:30, might
already be above the limit of geometric-
mean-validity that a true tetradic harmonic
entropy model will show to follow from an
assumption of s=1%, let alone s=1.5%.

> > >#33: [0_184__388___886]

Warbling? I'll get back to you tomorrow. I
hope you aren't using "Lo-Fi Play"!

🔗Joseph Pehrson <pehrson@...>

--- In harmonic_entropy@egroups.com, "David Finnamore" <daeron@b...>
wrote:

http://www.egroups.com/message/harmonic_entropy/150

I found the discussion between David Finnamore and Paul regarding the
four "anomalous" tetrachords quite interesting... I found David's
interpretations quite "entertaining..." (That's not deprecatory.)

For me, personally, I am quite satisfied with the "JI geometric mean"
ranking of these tetrads... or, at least, I wouldn't be able to
figure out a ranking that would be much better than that...

I have a little trouble with the concept of defining concordance in
terms of the suitability of "ending a piece of music" on the chord.

Maybe I've been too far removed from traditional tonal music all
these years... but in the 21st century so many pieces end in so many
ways (even Stravinsky, as was pointed out on the Tuning List not so
long ago!) that I could not possibly judge by that basis.

I think I would have to opt for a much more "localized"
interpretation of concordance, based more upon some kind of
"Helmholzian" perceptual interpretation.... (???)
__________ ____ __ _ _
Joseph Pehrson

🔗Paul Erlich <PERLICH@...>

--- In harmonic_entropy@egroups.com, "Joseph Pehrson" <pehrson@p...>
wrote:

> For me, personally, I am quite satisfied with the "JI geometric
mean"
> ranking of these tetrads... or, at least, I wouldn't be able to
> figure out a ranking that would be much better than that...

I pointed out a case where the geometric mean didn't work for you (or
at all) -- the 7sus4 and its mirror reflection. Maybe there are more?

> I think I would have to opt for a much more "localized"
> interpretation of concordance, based more upon some kind of
> "Helmholzian" perceptual interpretation.... (???)

Strictly speaking, Helmholtz-Sethares-dyadic is one approach, while
Rameau-tetradicharmonicentropy-geometricmean is another approach.
Above, you seem to endorse the latter as much more important that the
former, yes?

🔗Carl Lumma <CLUMMA@...>

[I wrote...]
>That should be this week yet... stay tuned!

Bingo! Actually, I only got through half of the chords before I
gave up. I'm not sure how meaningful it is to rank tetrads by one
property alone... it certainly was difficult. Follwoing David
Finnamore, I tried to imagine each one at the end of a piece. But
what does that mean? Today I posted to the tuning list about a
Barbershop tune that ends on a otonal 9th chord. Another B-shop
tune I know ends on a dimished tetrad (involving a 17-identity, I
think -- should check with spectrogram). Anywho, here are my results...

me intervals g.m. d.e.
1 388 702 970 1
2 318 816 1020 2
3 498 702 886 3
4 386 702 1088 6
5 268 702 970 11 6
6 316 702 1018 8
7 184 498 886 7
8 498 886 1394 9
9 204 702 1088 5
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
14 318 818 1320 15
15 384 588 1086 23 18-19

-Carl

🔗David J. Finnamore <daeron@...>

Paul Erlich wrote:

> No roughness measure will ever give a
> significantly different score to two chords
> with the same intervals (such as an otonal/
> utonal mirror image pair of chords).

Uh-oh. Maybe I'm using the term "roughness" too broadly. I mean mainly how much
beating there is between the tones, and how "hard" it beats. For example, empty
fifths in quarter comma meantone are "rough" to me. Take the 7-8 pair from the T
Lab: 498 702 886, and 184 388 886. The first one seems to cause the partials to
mostly line up with each other, while the second one causes them to beat against each
other. Or it may be that they both beat, but the second one beats at a rate that
makes it more obvious. I've noticed that this phenomenon is fairly frequent among
otonal/utonal pairs in general.

> > > >#24: [0_]498__702__1086
>
> It's a common chord in Mozart, where it
> resolves to a chord such as
>
> -1200_386_702_1200.

Where it does what? ;-)

> > > >#26: [0_]312__702___888
>
> Also known as a minor sixth chord, it is a
> very common ending chord in Salsa and
> other latin styles (well, at least the 12-tET
> version is).

Ah, context. Having been reminded, I've heard latin jazz pieces end that way. I
thought it peculiar - guess that reveals my bias. As to the 12-equal version, it
seems to distribute the beating in such a way as to kind of mask it, or to beat at a
rate that draws less attention to it. This tuning of it still sounds more tense to
me than any of those I placed above it, even thinking of it in a latin jazz cadential
context. Perhaps it's a mistake to equate tenseness with discordance? I'm open to
that.

> >And there's a 576 cent interval between the second
> > and forth notes - that's probably a pretty strong color in anybody's
> > book.
>
> 576 cents sounds a lot like 7:5, only 7 cents
> flat. Perhaps I need to use a larger s value so
> that 7:5 and its vicinity does not get such a
> concordant rating.

On your HE charts, 7:5 sits in a shallow dip between two pretty tall peaks. In fact,
it's valley dips only about as low as the top of the peak between 6:5 and 5:4.

> > > >#32: [0_]388__776__1090
> [snip] I wonder why the
> > program chose to sharpen the second and fourth tones, while leaving
> > the third one true?
>
> How is the third one "true"?

Whoops. Miscalculated. It's about 3 cents sharp to 25:16. You know, this one ties
for 27-28 on the T Lab page. My placement of 32 is not that far away.

> > > >#33: [0_184__388___886]
>
> Warbling? I'll get back to you tomorrow. I
> hope you aren't using "Lo-Fi Play"!

Nooooo. D-loaded Joseph's zip file. I've listened on two different systems, a pair
of pretty nice, if soft sounding, speakers at work, and Altec cubes with sub-woofer
at home. When I say "warbling" I mean beating. This tetrad beats harder than any
other in the list, to my ears. It's the only one to which I had a negative
reaction. When I listen to it and it's "partner" back to back, they don't even seem
to be in the same ballpark in terms of the concordance/discordance continuum. I may
be confusing pure concordance somewhat with my acculturated sense of tonality, do you
think?

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

🔗Paul H. Erlich <PERLICH@...>

> > > >#26: [0_]312__702___888

David Finnamore wrote,

>Ah, context. Having been reminded, I've heard latin jazz pieces end that
way. >I
>thought it peculiar - guess that reveals my bias. As to the 12-equal
version, >it
>seems to distribute the beating in such a way as to kind of mask it, or to
>beat at a
>rate that draws less attention to it.

Rather than 30:36:45:50 (this chord), it may be a 10:12:15:17 that would
best reflect the favorable properties of the chord in 12-tET. One might also
suggest 1/6:1/5:1/4:2/7, but utonality seems out of favor these days. :)

>Perhaps it's a mistake to equate tenseness with discordance? I'm open to
>that.

Perhaps tenseness correlates well with harmonic entropy. Joseph did say your
results correlated well with the geometric mean rankings.

> > > >#24: [0_]498__702__1086

>> It's a common chord in Mozart, where it
>> resolves to a chord such as
>
>> -1200_386_702_1200.

>Where it does what? ;-)

You're saying that a need to resolve would contradict your criteria for a
favorable position in your ranking. Perhaps you can listen to the chords
again with different criteria, divorcing yourself, if possible, from
imagined musical context? For a tetrad to have appeared _at all_ in most of
Mozart's works, as I would say this one did, its level of discordance would
have to be relatively low (compared with some of those augmented octave
chords, for example), since Mozart's music operates within the range from
moderate discordance to full concordance, and any violent discord would
sound like a mistake rather than a sonority in need of resolution. Of
course, this tetrad carries special tonal meaning that may have made Mozart
more likely to use it than some other chords of similar discordance . . .
and may have made you less inclined to accept this as a "closing sonority"
than other chords of similar concordance.

🔗Paul H. Erlich <PERLICH@...>

Carl Lumma wrote,

> me intervals g.m. d.e.
> 1 388 702 970 1
> 2 318 816 1020 2
> 3 498 702 886 3
> 4 386 702 1088 6
> 5 268 702 970 11 6
> 6 316 702 1018 8
> 7 184 498 886 7
> 8 498 886 13[8]4 9
> 9 204 702 1088 5
> 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
> 14 318 818 1320 15
> 15 384 588 1086 23 18-19

Is there any special reason you left blanks in the last column? Assuming
there isn't, I'll fill them in:

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 13[8]4 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

Clearly both models would have you place 0_268_702_970 three slots lower
than you did, and so would my ears. But otherwise, it seems that a
combination of both models would account for your rankings quite well.

Too bad you didn't include David's "warbling" chord, 0_184_388_886, in your
listening.

🔗David Finnamore <daeron@...>

--- In harmonic_entropy@egroups.com, "Paul H. Erlich" <PERLICH@A...>
wrote:
> Perhaps you can listen to the chords
> again with different criteria, divorcing yourself, if possible, from
> imagined musical context?

I'll try that. But like Carl said, what does it mean?

> For a tetrad to have appeared _at all_ in most of
> Mozart's works, as I would say this one did, its level of
discordance would
> have to be relatively low (compared with some of those augmented
octave
> chords, for example), since Mozart's music operates within the
range from
> moderate discordance to full concordance, and any violent discord
would
> sound like a mistake rather than a sonority in need of resolution.

Your logic cuts like a knife. Does anyone have any suggestions for a
criterion that is likely to give unbiased results?

🔗Paul Erlich <PERLICH@...>

🔗Carl Lumma <CLUMMA@...>

>Is there any special reason you left blanks in the last column?
>Assuming there isn't, I'll fill them in:

Thanks-- I only filled the d.e. in where the g.m. column seemed
particularly far off.

>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

>Clearly both models would have you place 0_268_702_970 three slots lower
>than you did, and so would my ears.

I won't defend my ranking, since I didn't feel very sure of it. In fact,
I moved several chords around as I was making the ranking -- which ones, I
don't know, since I did the test blind.

>But otherwise, it seems that a combination of both models would account
>for your rankings quite well.

Yes, and I was pleased. Although it's important to note that I didn't
try to rank about half of the chords. Also, it seems the geometric
mean works slightly better overall -- for example, I was sure of the first
three chords, and d.e. has got them all wrong.

Can we imagine a complexity measure based entirely on g.m., that subsets
the given chord until the g.m. is below a certain limit, each time adding
a penalty for dismissing the "noise"? (Since g.m. agrees fairly well
with roughness for dyads.) Perhaps there would even be a way to look at
all the subsets from the start, and determine a sort of entropy of entropy;
how well one subset dominates the others!

>Too bad you didn't include David's "warbling" chord, 0_184_388_886, in
>your listening.

Wow. Here's an example of where d.e. has got it all wrong. Chord #3
above sounds way more consonant. I hear the guide tones beat exactly
as you say in both, but only 184 has the warbling, at about 6 times/sec.
You're right though... it isn't beating.

-Carl

🔗Paul Erlich <PERLICH@...>

--- In harmonic_entropy@egroups.com, Carl Lumma <CLUMMA@N...> wrote:
>
> Can we imagine a complexity measure based entirely on g.m., that
subsets
> the given chord until the g.m. is below a certain limit, each time
adding
> a penalty for dismissing the "noise"? (Since g.m. agrees fairly
well
> with roughness for dyads.) Perhaps there would even be a way to
look at
> all the subsets from the start, and determine a sort of entropy of
entropy;
> how well one subset dominates the others!

All good ideas.

>I hear the guide tones beat exactly
> as you say in both,

In the otonal chord, it's not really a guide tone.

> but only 184 has the warbling, at about 6 times/sec.
> You're right though... it isn't beating.

6 times per second? With rhythmic regularity? That might be beating
of some kind . . . though I didn't hear that when I listened to
it . . . can you elaborate?

🔗David Finnamore <daeron@...>

🔗Paul Erlich <PERLICH@...>

🔗Carl Lumma <CLUMMA@...>

>> but only 184 has the warbling, at about 6 times/sec.
>> You're right though... it isn't beating.
>
>6 times per second? With rhythmic regularity? That might be beating
>of some kind . . . though I didn't hear that when I listened to
>it . . . can you elaborate?

I've seldom heard somebody take the words out of my mouth like
David, here:

>When I try to identify exactly what it's pitch is, it becomes elusive.
>It's like I'm hearing it with my "peripheral hearing" and when I try to
>listen "right at it," it nearly disappears.

The "beating" was clearly at 6 times per second, but every time I tried
to determine the pitch of the "beating" tone, it would blend back in to
the mix, and I would hear other tones "beating" (none of which I could
pinpoint the pitch of). When I would clear my head to start over, the
6/sec. "tone" was clearly the dominant one again, but always it would
resist capture.

>So, if harmonic entropy is the result of trying to fit the notes you
>hear into a harmonic series, maybe it's even creating the sensation of
>the best-fit harmonic series in your "peripheral" hearing for some
>reason. This is a fascinating phenomenon -- and I can't see how the
>11 would be produced by any of the normally occuring difference tones.

Wowzers. Maybe we should test this on another chord...

-Carl

🔗Paul Erlich <PERLICH@...>

🔗Monz <MONZ@...>

--- In harmonic_entropy@egroups.com, "David Finnamore" wrote:
> http://www.egroups.com/message/harmonic_entropy/164
>
> --- In harmonic_entropy@egroups.com, "Paul H. Erlich" wrote:
> > Perhaps you can listen to the chords again with different
> > criteria, divorcing yourself, if possible, from imagined
> > musical context?
>
> I'll try that. But like Carl said, what does it mean?

This is exactly what I was going to suggest before Paul did.
Here's my idea:

Treat each separate chord as a sort of mini La Monte Young piece.
Let it swirl around in your ears for 5 minutes, an hour... you
get the idea. Write down as much as you can think of about
just this one chord.

This is the approach I'm taking... it'll be quite a long while
before I'm ready to post anything on it. Perhaps we should
encourage elaborate descriptions of our peceptions and reactions
to each chord rather than a comparison-ranking, then derive
the ranking later from a collation of all the descriptions.

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

🔗Joseph Pehrson <josephpehrson@...>

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

http://www.egroups.com/message/harmonic_entropy/156

>
> > I think I would have to opt for a much more "localized"
> > interpretation of concordance, based more upon some kind of
> > "Helmholzian" perceptual interpretation.... (???)
>
> Strictly speaking, Helmholtz-Sethares-dyadic is one approach, while
> Rameau-tetradicharmonicentropy-geometricmean is another approach.
> Above, you seem to endorse the latter as much more important that
the
> former, yes?

Umm. Well, somehow I thought the "geometric mean" approach, where
we are ranking the tetrads according to simple otonal just intonation
sonorities would be related to Helmholz... who was so big on ranking
things according to just intonation...

Am I misunderstanding something??

🔗Joseph Pehrson <josephpehrson@...>

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

http://www.egroups.com/message/harmonic_entropy/161

>
> >> It's a common chord in Mozart, where it
> >> resolves to a chord such as
> >
> >> -1200_386_702_1200.
>
> >Where it does what? ;-)
>
> You're saying that a need to resolve would contradict your criteria
for a favorable position in your ranking. Perhaps you can listen to
the chords again with different criteria, divorcing yourself, if
possible, from imagined musical context?

Absolutely. I really suggest we *forget* about trying to determine
concordance through musical context. In that case, according to our
previous definitions we are talking about *consonance* rather than
*concordance* anyway, right??

That's something different and that's ANOTHER really BIG topic! and
I don't believe it has ANYTHING to do with this process!

I think this has to be an evaluation of ISOLATED sonorities, yes??

🔗Joseph Pehrson <josephpehrson@...>

🔗Paul H. Erlich <PERLICH@...>

Joseph Pehrson wrote,

>> > I think I would have to opt for a much more "localized"
>> > interpretation of concordance, based more upon some kind of
>> > "Helmholzian" perceptual interpretation.... (???)
>
>> Strictly speaking, Helmholtz-Sethares-dyadic is one approach, while
>> Rameau-tetradicharmonicentropy-geometricmean is another approach.
>> Above, you seem to endorse the latter as much more important that
the
>> former, yes?

>Umm. Well, somehow I thought the "geometric mean" approach, where
>we are ranking the tetrads according to simple otonal just intonation
>sonorities would be related to Helmholz... who was so big on ranking
>things according to just intonation...

>Am I misunderstanding something??

Both methods are equally big on ranking things according to just intonation.