Tetrads :) Do you promote dualism ?

Paul,

It's funny that harmonic-minded acoustico-physiologists like Helmholtz,
Plomp ... (and Erlich ??) produce harmonic-subharmonic symmetrical results
with harmonic-only hypothesis. Beating of harmonics in Helmholtz don't give
asymmetric "consonance basin" nor really better "computed consonance" for
given ratio compared to its inversion (difference between 4/3 and 3/2 is
due, by example, to choice of reference octave : above unison).

At acoustical level, I don't deny that harmonic asymmetry is a necessity.
Without that, there would be no periodicity and then no perceptible pitch
with which are constructed intervals. At macrotonal level, however, it is
clear that harmonic-subharmonic symmetry is perfect. Between them, at
microtonal level, we have to be very critical vis-à-vis melting aspects of
sonance question. Progress are'nt easy with present degree of confusion.

I don't have time to deepen now. The following is only a remark. Although
starting point seems harmonic series, according to your comments, I
could'nt find better example of dualism than your list of 34 tetrads ranked
by harmonic entropy sum on six dyadic components. (Obviously, reason is in
dyadic approach, but other ways are yet speculation).

Here, (a b c d) or (a b c d)* are chords. Sign _*_ indicate that values are
subharmonics rather than harmonics. Chords (a b c d) and (d c b a)* are
dual chords. Sign _SYM_ indicate that chord is identical with its dual.
This symmetrical type of chord is clearly favoured here. Sign _ROT_
indicate that tied tetrads are not only dual chords, but they have also
symmetry by rotation.

I made correction at number 10 and added just ratios where missing except
at number 19 which is irrelevant.

1) 0-498-886-1384 9:12:15:20 (9 3 15 5) == (5 15 3 9)* SYM

2) 0-492-980-1472 12:16:21:28 (3 1 21 7) == (7 21 1 3)* SYM

3) 0-386-702-1088 8:10:12:15 (1 5 3 15) == (15 3 5 1)* SYM

4) 0-502-1002-1390 9:12:16:20 (9 3 1 5) == (5 15 45 9)*
0-388-888-1390 36:45:60:80 (9 45 15 5) == (5 1 3 9)*

5) 0-306-702-1008 10:12:15:18 (5 3 15 9) == (9 15 3 5)* SYM

6) 0-204-702-1088 8:9:12:15 (1 9 3 15) == (45 5 15 3)*
0-386-884-1088 24:30:40:45 (3 15 5 45) == (15 3 9 1)*

7) 0-388-886-1274 12:15:20:25 (3 15 5 25) == (25 5 15 3)* SYM

8) 0-268-702-970 12:14:18:21 (3 7 9 21) == (21 9 7 3)* SYM

9) 0-318-818-1320 15:18:24:32 (15 9 3 1) == (3 5 15 45)*
0-502-1002-1320 45:60:80:96 (45 15 5 3) == (1 3 9 15)*

10) 0-388-702-886 12:15:18:20 (3 15 9 5) == (15 3 5 9)* ROT
0-184-498-886 9:10:12:15 (9 5 3 15) == (5 9 15 3)* ROT

11) 0-434-820-1320 25:32:40:60 (25 1 15 5) == (1 25 5 15)* ROT
0-500-886-1320 15:20:25:32 (15 5 25 1) == (5 15 1 25)* ROT

12) 0-302-502-1004 27:32:36:48 (27 1 9 3) == (1 27 3 9)* ROT
0-502-702-1004 18:24:27:32 (9 3 27 1) == (3 9 1 27)* ROT

13) 0-318-816-1020 5:6:8:9 (5 3 1 9) == (9 15 45 5)*
0-204-702-1020 40:45:60:72 (5 45 15 9) == (9 1 5 3)*

14) 0-498-888-1282 12:16:20:25 (3 1 5 25) == (25 75 15 3)*
0-394-784-1282 48:60:75:100 (3 15 75 25) == (25 5 1 3)*

15) 0-384-588-1086 56:70:80:105 (7 35 5 105) == (15 3 21 1)*
0-498-702-1086 16:21:24:30 (1 21 3 15) == (105 5 35 7)*

16) 0-500-816-1316 15:20:24:32 (15 5 3 1) == (1 5 3 15)* SYM

17) 0-388-776-1090 16:20:25:30 (1 5 25 15) == (75 15 3 5)*
0-314-702-1090 40:48:60:75 (5 3 15 75) == (15 25 5 1)*

18) 0-498-702-886 6:8:9:10 (3 1 9 5) == (15 45 5 9)*
0-184-388-886 36:40:45:60 (9 5 45 15) == (5 9 1 3)*

19) 0-442-884-1326 (Cf. 442 stack)

20) 0-186-576-888 18:20:25:30 (9 5 25 15) == (25 45 9 15)*
0-312-702-888 30:36:45:50 (15 9 45 25) == (15 25 5 9)*

21) 0-388-702-970 4:5:6:7 (1 5 3 7) == (15 35 21 105)*
0-268-582-970 60:70:84:105 (15 35 21 105) == (7 3 5 1)*

These 34 tetrads are ranked now in accordance with internal structure
rather than possible perceptibility.

111-1 (1 3 5 7) 21a - 21b
121-1 (1 3 5 9) 4a - 4b - 13a - 13b - 18a - 18b
223-1 (1 3 5 15) 3 - 16
133-1 (1 3 9 15) 6a - 6b - 9a - 9b
123-3 (3 5 9 15) 1 - 5 - 10a - 10b
225-3 (1 3 7 21) 2
243-3 (3 7 9 21) 8
315-3 (1 3 15 21) 15a - 15b
145-5 (1 5 15 25) 11a - 11b - 17a - 17b
541-5 (3 5 15 25) 7
325-5 (5 9 15 25) 20a - 20b
441-7 (1 3 5 25) 14a - 14b
263-5 (1 3 9 27) 12a -12b
425-9 (3 5 13 31) 19 (nearest)

I use incremental generator value abc-d in first column for chords
comparaison. It's not a deep theoritical concept, it's only useful. Values
are differentials in most compact form.

abc-d == [a+b+c+d]:[2a+b+c+d]:[2a+2b+c+d]:[2a+2b+2c+d]

If a:b:c:d is the most compact form then

a:b:c:d == [b-a][c-b][d-c]-[2a-d]

Examples :

111-1 == 4:5:6:7 == (1 3 5 7)
121-1 == 5:6:8:9 == (1 3 5 9)

123-3 == 9:10:12:15 == (3 5 9 15) 10:12:15:18 (m7 Chord)
12:15:18:20 (6 Chord)

425-9 == 20:24:26:31 == (3 5 13 31) 24:31:40:52 0-441-884-1326

Pierre Lamothe

--- In tuning@egroups.com, Pierre Lamothe <plamothe@a...> wrote:

http://www.egroups.com/message/tuning/12905

Please note that many of the tetrads that Mr. Lamothe (welcome back,
Pierre) posted will very soon be available for LISTENING in the
Tuning Lab. They have been posted... but will take a few days to
appear. I will notify the list. Thank you!
_________ ____ __ __ _ _
Joseph Pehrson

Paul,

You wrote:

<< The only difference is that I've produced numbers for the dyadic case but
not the chordal case. If I give you some numbers, will you reconsider
dualism? Everyone, listen to the experiment Joseph Pehrson put up. Who
thinks 1/7:1/6:1/5:1/4 and 4:5:6:7 are equally consonant? >>

I didn't want to defend dualism but show that total sum of entropy give
dual results. It's just funny that acousticians can't yet produce
asymmetric results. I'm not in doubt for perception but seek for principle
which will permit asymmetrical results.

Pierre Lamothe

Paul,

You wrote :

<< The principle is, chordal harmonic entropy. We need a multidimensional
analogue of mediants and then we can calculate it. CAN YOU HELP?? >>

Hermite would have say that this problem (analogue of mediants in Z^3)
didn't cease worring and desesperating him during 50 years. I would seek a
less direct approach but I need to read about harmonic entropy. I thank you
for following reference :

<< See Parncutt's book _Harmony -- A Psychoacostic Approach_
(Springer-Verlag) for a taste of where I'm coming from. >>

I can't buy books but I'll try to find it at library. For the moment I take
information only on web. Also, if you know good reference on web about
that, I would appreciate to know it.

Pierre Lamothe

> << See Parncutt's book _Harmony -- A Psychoacostic Approach_
> (Springer-Verlag) for a taste of where I'm coming from. >>
This has gone out of print. None of the on line bookstores, even the out of
print kind, show it as available.

--
M. Edward (Ed) Borasky
znmeb@teleport.com
http://www.borasky-research.com/

Monz wrote,

>Paul, every time I see something like this my brain keeps
>insisting that something resembling our multi-dimensional
>trigonometric prime- or odd-based lattice formulas is the key.

It's part of the answer: distance. But we need more than just distance -- we
need SHAPE.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> http://www.egroups.com/message/tuning/12978
>
> Monz wrote,
>
> > Paul, every time I see something like this my brain keeps
> > insisting that something resembling our multi-dimensional
> > trigonometric prime- or odd-based lattice formulas is the key.
>
> It's part of the answer: distance. But we need more than just
> distance -- we need SHAPE.

Ahh... so I'm not necessarily on the right track, but I am in
the right neighborhood! Give me a good map and I swear I'll
find it :)

Can you elaborate more on what you mean by shape? I thought
I've been defending my lattice formula not only on grounds of
distance, but also on those of shape. ...?

What do you think of the modification of my formula to use the
1/2-circle instead of the whole circle (so that utonal relations
are plotted the same way as otonal) in terms of this 'shape'?

PS - could you also give me your triangular formula in a format
that I can simply paste into an Excel spreadsheet cell?

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

Joe Monzo wrote,

>Can you elaborate more on what you mean by shape?

I mean the shape of the cell surrounding each ratio. In the diadic case, the
answer is simple -- a line segment. In the triadic case, one possible answer
is Voronoi cells -- see http://www.egroups.com/files/tuning/triads.jpg. If I
had the programming skills, I could use this right now to calculate true
triadic harmonic entropy.

>What do you think of the modification of my formula to use the
>1/2-circle instead of the whole circle (so that utonal relations
>are plotted the same way as otonal) in terms of this 'shape'?

I think that would be an improvement in terms of 'distance'. Actually, I got
confused when I replied to you just a moment ago (Herbert came by). The
distance involved in harmonic entropy is different from that in
multidimensional lattice diagrams. The dimensions in the former are pitch
intervals; the dimensions in the latter are prime factors.

>What do you think of the modification of my formula to use the
>1/2-circle instead of the whole circle (so that utonal relations
>are plotted the same way as otonal) in terms of this 'shape'?

That would be an improvement as concerns distance in the lattice sense.
Again, I got a little confused when I said it would apply to the entropy
problem.

>PS - could you also give me your triangular formula in a format
>that I can simply paste into an Excel spreadsheet cell?

Taking wormholes into account, the distance on the triangular lattice is
very simply the log of the odd limit of the ratio between two pitches --
unless you set an overall odd-limit on the ratios used as consonances in the
music. Then it's not so much a formula but an algorithm that determines the
distance -- I refer you to Paul Hahn.

Ultimately, I see it this way -- calculate harmonic entropy first, as well
as other components of sonance, and then design your lattice to be as
reflective of sonance as possible. In my view, on the assumption of
octave-equivalence, one adopts an odd-limit view similar to Partch, but with
a limitation on perceptual accuracy that puts a limit on the limit. Without
octave-equivalence, a Tenney-like view, but again with perceptual accuracy
limiting the limit, emerges.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> http://www.egroups.com/message/tuning/12987
>
> Actually, I got confused when I replied to you just a moment ago
> (Herbert came by).

Yeah, his visits are never indifferent... he always either hurts
or helps.

> Ultimately, I see it this way -- calculate harmonic entropy
> first, as well as other components of sonance, and then design
> your lattice to be as reflective of sonance as possible. In my
> view, on the assumption of octave-equivalence, one adopts an
> odd-limit view similar to Partch, but with a limitation on
> perceptual accuracy that puts a limit on the limit. Without
> octave-equivalence, a Tenney-like view, but again with perceptual
> accuracy limiting the limit, emerges.

This sounds really good, and I'd love to see what they look like.
Someone *please* post the actual math so that I can plug the
formulas into Excel and create some lattices!

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

Monz wrote,

>This sounds really good, and I'd love to see what they look like.
>Someone *please* post the actual math so that I can plug the
>formulas into Excel and create some lattices!

Well, if you're looking to draw the lattices, you'll run against the
limitation that Tenney's lattice for the 5-limit, and the triangular lattice
for the 7-limit, are inherently 3-dimensional and would have to be projected
onto a 2-d plane. In the triangular case, this projection would essentially
make the angles and lengths arbitary (well, one would determine the other)
and so you can use whatever you like, so long as the basic structure of the
lattices is preserved. For example, you could use your half-circle idea as a
beginning for the triangular lattice, adding extra connections where
necessary. For Tenney's idea, if you represented the octave as the interval
at 0 degrees, with length proportional to log(2), and all the other primes
with lengths proportional to log(p), the city-block distance metric Tenney
proposed would come out right no matter what angles you chose in the
"projection" down to 2-d -- so you could even use your half-circle idea.

Joseph Pehrson wrote,

>You know, if David Keenan can't figure this out,

Uh, that was Monz's post you were replying to . . .

>maybe we could post
>to some scientific or mathematics list for a "cross post"(??) Maybe
>somebody would "dig" working on a new problem related to numbers and
>music... Just a thought.

Joseph, did you see Pierre's post:

>Hermite would have say that this problem (analogue of mediants in Z^3)
>didn't cease worring and desesperating him during 50 years.

From what I have found on the Internet (I posted some stuff, I think, early
this year), it doesn't look like anyone has really found the "key" since
this great mathematician's desparate attempts -- in fact it may have been
proved that no solution that captures all the propertied of 1-d mediants
exists, though I didn't really understand all the stuff I found. I've also
tried to contact mathematicians about it; the profs I spoke to about it
while at Yale were stumped, and I've gotten no reply back from John Conway
and others. But yes, I'd welcome any attempts to address this with
cross-posting or whatever . . . I think ultimately, though, a simple Voronoi
cell approach, or some clever modification of it, will be good enough . . .

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

http://www.egroups.com/message/tuning/13030

>
> Joseph, did you see Pierre's post:
>
> >Hermite would have say that this problem (analogue of mediants in
Z^3)>didn't cease worring and desesperating him during 50 years.
>
> From what I have found on the Internet (I posted some stuff, I
think, early this year), it doesn't look like anyone has really found
the "key" since this great mathematician's desparate attempts --

....Paul, if I couldn't laugh about this, I might be depressed...
__________ ____ __ _
Joseph Pehrson

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> http://www.egroups.com/message/tuning/13030
>
> But yes, I'd welcome any attempts to address this with
> cross-posting or whatever . . .

I think cross-posting can be extremely helpful. I never
hesitated in the past to write to scientists or mathematicians
or their internet lists for help about lattices, prime numbers,
spiders, etc., which all went into pre-compositional work for
pieces, webpages, or commentary to this list.

My opinion is that it's not a good thing to break up this list
by creating new ones UNLESS there is really a significantly big
population with strong similar interests. - I leave it to the
reader to decide what scale to use to determine 'significance' ;-)

I've written here in the past celebrating the disparity of
approach and vision exhibited by the subscribers on this list
who post. It may be impossible to resolve any clear superiority
between a variety of perspective and a widespread narrow focus
(which I see as opposite poles); in *both cases* the contributions
posted create an exciting synergy between corresponents which
results in a plethora of discovery, invention and creativity,
and perhaps most importantly of all (as we've just seen with the
Schulter/Keenan paper), cooperation.

I say BRAVO! to the whole list! A terrific example of a cyber-
*community*! (now if only we never had those lapses into
flame-wars....)

But still the feeling nags away at me that variety of perspective
should be encouraged at all costs. The disparity of different
opinions often creates factionism, with people siding with all
different kinds of camps on various topics. As long as the
societal pressure to be civilized remains strong on the list,
this is all good. Woe to the day if the majority ever prefers
to fan the flames - I'll be leaving then for sure.

I've often been inspired to some of my best work, in compositions
and webpages, by ideas catalyzed thru interaction on this list.

(Boy, I'm really rambling tonight...)

On another subject: I totally agree with Jon Szanto and Paul
Erlich that really great compositions and composers can sound
great in various diverse tunings.

(OK, back to the original thread...)

> ... I think ultimately, though, a simple Voronoi cell approach,
> or some clever modification of it, will be good enough . . .

Yes, Paul, my intution tells me that you're definitely on the
right track with this. I've examined your Voronoi diagram and
like it a lot. I'd be happy to help you scheme up some
'clever modification of it'... (my guess is that consultations
with Herbert would be especially beneficial here!)

I have some programming skills in VB if that can help your
harmonic entropy research, and am willing to make MIDI-files
of audio examples that John Starrett or Joe Pehrson would
probably be happy to turn into mp3.

I get more and more excited about the sense of collaborative
research and creativity evolving here!

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