Tuning-Math Digests messages 6500 - 6524

This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).

Contents Hide Contents S 7

Previous Next

6000 6050 6100 6150 6200 6250 6300 6350 6400 6450 6500 6550 6600 6650 6700 6750 6800 6850 6900 6950

6500 - 6525 -



top of page bottom of page down


Message: 6500

Date: Sun, 16 Feb 2003 00:44:17

Subject: Re: A common notation for JI and ETs

From: monz

hi paul,



> From: <wallyesterpaulrus@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Saturday, February 15, 2003 1:02 PM
> Subject: [tuning-math] Re: A common notation for JI and ETs
>
>
> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> 
> wrote:
> 
> > and i think you missed this -- which you'd also probably
> > be interested in at least in passing:
> > Onelist Tuning Digest # 483 message 26, (c)2000 by Joe Monzo *
> > 
> > 
> > 
> > -monz
> 
> hi monz,
> 
> i brought this page up to dave and george very recently (last 
> week) here on this list, and they indeed found it very useful for 
> their discussion. 



oops, my bad.  OK, i haven't really been following
this particular list closely, but have only glanced
more-or-less randomly at posts which looked interesting.
i'm glad my page was found useful.



> unfortunately, several erroneous statements 
> persist on this page, most notably:
> 
> "It can be seen easily from the lattice that all the intervals are 
> made up of various combinations of the ones described by 
> Paul."
>
 > of course, we all know you're very busy right now, and i at least 
> appreciate your brief and all too infrequent visits to this list.



thanks, paul.

OK, if you tell me *exactly* what i should do with that
sentence (remove it, edit it, change it? -- and if the
latter two, then replace it with exactly what?), i'll just
copy and paste what you write into the page to replace
my sentence.


(i really do try to stay on top of the accuracy of my webpages.
sorry about falling behind sometimes. ... i know you still
have a slew of stuff that you want me to fix.  i'll do them
one page at a time.)



-monz


top of page bottom of page up down


Message: 6502

Date: Mon, 17 Feb 2003 15:39:04

Subject: Huron Voice Leading

From: Graham Breed

I've finished reading "A Derivation of the Rules of Voice-leading from 
Perceptual Principles".  It seems to be good sense in so far as it goes. 
  Not really a derivation, but a good place for scientists to start when 
learning counterpoint.

Janata et al isn't so interesting.  All it says is that notes closer to 
the key center are closer to the key center, or something.  And there 
are some brain images I don't understand.

That "Pitch Schemata" link I gave might have the background details. 
The first time I looked at it I recognized a lot of Rothenberg's ideas, 
but they were credited to Balzano.  I'll have another look sometime.


                       Graham


top of page bottom of page up down


Message: 6504

Date: Mon, 17 Feb 2003 09:19:20

Subject: Re: Huron Voice Leading

From: Carl Lumma

>That "Pitch Schemata" link I gave might have the background details. 
>The first time I looked at it I recognized a lot of Rothenberg's ideas, 
>but they were credited to Balzano.  I'll have another look sometime.

Balzano independently came up with a lot of Rothenberg's ideas, years
later, and along with some other mush.

-Carl


top of page bottom of page up down


Message: 6506

Date: Mon, 17 Feb 2003 17:34:48

Subject: Pitch Schemata (Was briefly: Huron Voice Leading

From: Graham Breed

Carl Lumma wrote:
> Balzano independently came up with a lot of Rothenberg's ideas, years
> later, and along with some other mush.

So why doesn't Rothenberg get any credit for them?  And it's not only 
Balzano -- Browne's given some of them as well.

Another weird thing is that a Forte 1973 paper is mentioned as a 
"formulation" of Balzano's 1982.  So what was Forte formulating?  It 
must be peculiar if major and minor triads are considered identical.

Pitch Schemata *

It says that Balzano's coherence is different to Rothenberg's propriety 
because it only considers adjacent pairs of intervals.  So a Pythagorean 
diatonic is still coherent because the conflicting sizes of tritones 
aren't considered.  Is that right?


                       Graham


top of page bottom of page up down


Message: 6507

Date: Mon, 17 Feb 2003 10:05:09

Subject: Re: Huron Voice Leading

From: Carl Lumma

[I wrote...]
>>That "Pitch Schemata" link I gave might have the background details. 
>>The first time I looked at it I recognized a lot of Rothenberg's ideas, 
>>but they were credited to Balzano.  I'll have another look sometime.
>
>Balzano independently came up with a lot of Rothenberg's ideas, years
>later, and along with some other mush.

While many of their criteria were the same, many were not.  So it must
be considered a strange coincidence that R. and B. wind up recommending
the same scale, R in 31-tET, and B in 20-tET.  B missed the 31-tET
version because it didn't have all the other nonsense properties that
he was so fascinated with (such as the product of the sizes of the 3rds
giving the number of notes in the embedding et).  Dan Stearns further
independently suggested this scale in 20-tET for his own reasons.  But
AFAIK I'm the first to notice its excellent approximations to 5:3 and
7:4 nicely interleaved on its 8ths.  The 31-tET version gets closer to
JI, but the 20-tET version has higher Lumma stability and already gets
you closer than 12-tET to these intervals.

The lower stability of Rothenberg's scale kicks it down to position 6
on my gd spreadsheet -- Balzano's version is at position 3, just below
the pentatonic and diatonic scales...

http://lumma.org/stuff/gd.xls *

All the scales are available as scala files...

http://lumma.org/stuff/gd-scl.zip *

-Carl


top of page bottom of page up down


Message: 6508

Date: Mon, 17 Feb 2003 10:19:29

Subject: Re: Pitch Schemata

From: Carl Lumma

>Pitch Schemata *
>
>It says that Balzano's coherence is different to Rothenberg's propriety 
>because it only considers adjacent pairs of intervals.  So a Pythagorean 
>diatonic is still coherent because the conflicting sizes of tritones 
>aren't considered.  Is that right?

I think that's wrong.  ;)

No, I remember there being small differences like that.  Here's a bit of
a message I sent to tuning some years ago...

"""
Balzano wants // the scale to be covered with three-note chords that fall
on every-other degree of the scale.  Which means they'll be made of thirds
and fifths.  It is actually a huge mistake to consider them chords, tho,
since Balzano hasn't given any property that defines "chords" (he's
deliberately thrown out the usual one: harmony).  Which means that his
whole idea amounts to a lot of nothing.

Well, not quite.  He does require that the same interval appears as a
fifth in exactly n-1 modes of the scale, when the scale has n notes per
2:1.  Which is no more and no less than MOS when the generator turns out
to be a fifth and the interval of equivalence a 2:1.  So the final list
of stuff is now...

(1) // "coherence" of scale degrees across modes.
	Search list archives for "Rothenberg".

(2) tuning coverage; rank order matrix and interval matrix are the same.
	I don't think this has much to do with anything.  //

(3) linear connectivity; transposing by generator changes one note.
	But not less than one --- closed chains are not allowed.  Makes
	for symmetry at the generator, and prevents the (Rothenberg)
	efficiency from becoming very low (see TD 262.14).

(4) fifths; the generator is a fifth in exactly n-1 modes of the scale.
	// Choice of fifths is completely arbitrary.
"""

-Carl


top of page bottom of page up down


Message: 6510

Date: Mon, 17 Feb 2003 12:18:38

Subject: Re: Pitch Schemata

From: Carl Lumma

>>>It says that Balzano's coherence is different to Rothenberg's 
>>>propriety because it only considers adjacent pairs of intervals.
//
>>I think that's wrong.  ;)
> 
>it is wrong, because balzano, like clough and too many academic 
>theorists, considers all scales to be subsets of some discrete 
>cyclic "universe": 12-tone, 20-tone, etc. balzano defines coherence 
>in terms of units of the "universe set". which means that it, like 
>maximal evenness, is undefined for a random scale given in cents (or 
>ratios, or whatever), without any assumed "universe" in which it is 
>embedded. this is a serious weakness of academic scale theory, in my 
>opinion.

For sure.

But further, it's wrong unless Balzano has a model that justifies
coherence (which I don't remember him having).  Rothenberg assumes
that listeners can rank intervals by size, and gives the condition
required for them to be able to repeatably map what they hear to a
fixed scale.  It's obvious, it's simple, and it's probably true.
I can't imagine how one could doctor this so that only adjacent
intervals matter...

-Carl


top of page bottom of page up down


Message: 6511

Date: Mon, 17 Feb 2003 12:02:58

Subject: Re: Pitch Schemata

From: Carl Lumma

>So why doesn't Rothenberg get any credit for them?

Presumably because "Mathematical Systems Theory" isn't one
of the journals that the exclusive circle of music theorists
watch.

>And it's not only Balzano -- Browne's given some of them as well.

From the details available in the schemata paper, Browne
rediscovers not only something like propriety ("pattern matchng"),
but also something like efficiency ("position finding").

-Carl


top of page bottom of page up down


Message: 6512

Date: Mon, 17 Feb 2003 21:17:57

Subject: Re: Pitch Schemata

From: Graham Breed

Carl Lumma wrote:

> But further, it's wrong unless Balzano has a model that justifies
> coherence (which I don't remember him having).  Rothenberg assumes
> that listeners can rank intervals by size, and gives the condition
> required for them to be able to repeatably map what they hear to a
> fixed scale.  It's obvious, it's simple, and it's probably true.
> I can't imagine how one could doctor this so that only adjacent
> intervals matter...

Then did Balzano so doctor it?  So that the Pythagorean diatonic 
embedded in 53-equal would still be coherent?  I was only asking if the 
Pitch Schemata paper had Balzano correct.


                          Graham


top of page bottom of page up down


Message: 6513

Date: Mon, 17 Feb 2003 13:23:41

Subject: Re: Pitch Schemata

From: Carl Lumma

>> But further, it's wrong unless Balzano has a model that justifies
>> coherence (which I don't remember him having).  Rothenberg assumes
>> that listeners can rank intervals by size, and gives the condition
>> required for them to be able to repeatably map what they hear to a
>> fixed scale.  It's obvious, it's simple, and it's probably true.
>> I can't imagine how one could doctor this so that only adjacent
>> intervals matter...
>
>Then did Balzano so doctor it?  So that the Pythagorean diatonic 
>embedded in 53-equal would still be coherent?  I was only asking if
>the Pitch Schemata paper had Balzano correct.

I was assuming so.  I don't remember that from the Balzano paper,
and my copy is in Montana, so...

-Carl


top of page bottom of page up down


Message: 6514

Date: Mon, 17 Feb 2003 14:13:50

Subject: Re: A common notation for JI and ETs

From: monz

hi paul, 


> From: <wallyesterpaulrus@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, February 16, 2003 6:25 PM
> Subject: [tuning-math] Re: A common notation for JI and ETs
>
>
> Onelist Tuning Digest # 483 message 26, (c)2000 by Joe Monzo *
>
>
> please remove the sentence, and replace it with this:
> 
> '
> It can be seen easily from the lattice that these intervals, as well 
> as some lesser-known 'commas' like 243:250 and 3072:3125, cannot made 
> up of various combinations of the ones described by Paul.
> 
> Western triadic music prior to Beethoven requires "bridging" solely 
> through the syntonic comma, and hence is often performed in meantone 
> temperament. Since Beethoven, "bridging" through syntonic comma and 
> *any* (and therefore, all) of the other 'commas' paul mentions above 
> (in connection with mathieu) has been a feature of western triadic 
> music, hence the use of 12-tone equal (or well) temperament. The 
> other 'commas' can be used for bridging in other, "invented" musical 
> systems, motivating certain corresponding tuning systems as shown at:
> 
> Definitions of tuning terms: equal temperament, (c) 1998 by Joe Monzo *
> 
> for example, you can see from the first chart and table on that page 
> that "bridging" through 243:250 is characteristic of porcupine 
> temperament, through 3072:3125 of magic temperament, and through both 
> of them (and thus also any combination of the two) of 22-tone equal 
> temperament.
> '


OK, i added that.  when i have more time i'd also like to
include what you wrote after that.




-monz


top of page bottom of page up down


Message: 6515

Date: Tue, 18 Feb 2003 19:00:43

Subject: Re: scala show data

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

That bug is fixed now, along with some other ones.

There's a new feature which may be interesting,
in the Chromatic Clavier you can now arpeggiate or
hold a chord, and play with the mouse at the same
time. The chord is entered using the right mouse button,
like it could be done before.
Also new now is that when you open the chord list,
the selected chord is used (actually the nearest
approximation of it in the current scale), so you can 
quickly change chords without having to click-enter them.
Please click on the Help button in the clavier window first
if there's some trouble.

http://www.xs4all.nl/~huygensf/software/Scala_Setup.exe - Ok *

Manuel


top of page bottom of page up down


Message: 6516

Date: Tue, 18 Feb 2003 11:04:50

Subject: Re: scala show data

From: Carl Lumma

>That bug is fixed now, along with some other ones.

With equal 6, I get .2 for Rothenberg stability.
How are you getting that?  If I delete the 3rd
degree, it goes up to .4!

-Carl


top of page bottom of page up down


Message: 6517

Date: Tue, 18 Feb 2003 13:23:38

Subject: lattice diagram "levels" of complexity

From: monz

is there an accepted method of nomenclature for
describing the "level" of complexity of lattice
diagrams?

i know that "stellation" has something to do
with this, but here i'm talking pretty much about
the algorithm used by Partch to fill out the
Tonality Diamond.

in other words, say we have a 7-limit (3-D) lattice.

(use "Expand Messages" if viewing on Yahoo website
for proper formatting of diagram)


here are the two basic tetrads (otonal and utonal)
which have the 1/1 ratio as their 1-identity:


                  5:4
                  /|\
                 / | \
                /  |  \
               /  7:4  \
              /. '   ' .\
4:3---------1:1---------3:2
  \ '.   .' /
   \  8:7  /
    \  |  /
     \ | /
      \|/
      8:5

this would be "level" 1.


now if we build complete otonal tetrads
on all of the notes in the "basic" utonality
tetrad (i.e., 4/3, 8/5, and 8/7 all become
1-odentities of their respective tetrads), and
complete utonal tetrads on all of the notes in
the "basic" otonality tetrad (i.e, 3/2, 5/4, and
7/4 all become 1-udentities of their respective
tetrads), we get this:

      5:3---------5:4
      /|\ '.   .' /|\
     / | \ 10:7  / | \
    /  |  \ /|\ /  |  \
   /  7:6--/-|-\--7:4  \
  /. '   '/.\|/.\'   ' .\
4:3------/--1:1--\------3:2
  \ '.  /.' /|\ '.\   .'/
   \  8:7--/-|-\--12:7 /
    \  |  /  |  \  |  /
     \ | /  7:5  \ | /
      \|/ .'   '. \|/
      8:5---------6:5

which would be "level" 2.


is there already an accepted term for what i'm
calling "level"?

and can someone give a very clear and lucid explanation
of how stellation differs from this, if it does?

thanks.



-monz


top of page bottom of page up down


Message: 6518

Date: Tue, 18 Feb 2003 14:49:48

Subject: Re: lattice diagram "levels" of complexity

From: Carl Lumma

heya monz,

>is there already an accepted term for what i'm
>calling "level"?

Not to my knowledge, though we often talk about
regions of the lattice within some taxicab
radius.  The diamond is r=1, and it sounds like
your levels correspond to successively higher
r values.  You might want to check that and tell
me if it's the case.

I said "lattice region".  Gene has used the term
"ball", and I think that's a convex hull, plus
everything inside.  zthat right, Gene?

>and can someone give a very clear and lucid explanation
>of how stellation differs from this, if it does?

Stellation is Wilson's term.  He borrowed it from
geometry, where the term often refers to the process
of adding points above the faces of a polyhedron,
turning *them* into polyhedra.  This is indeed what
happens when stellating the hexany -- it's an octahedron
being extended so that each face becomes a tetrahedron.
I'm sure there's a more precise definition on a geometry
website somewhere.  Post it here if you are interested
and find it...  The one I'm remembering is 'the compound
of a polyhedron and its dual'.

Back when, there was some debate over what stellation
should include when extending other CPSs (the eikosany
was the main inquiry).  I said that each existing face
should be completed into a saturated n-limit chord, and
that's it.  Others, apparently including Wilson, wanted
to include other points, basically out to the power
set (the compound of all CPSs of a given limit).  I was
never clear on why they wanted to do this.  Maybe Paul
remembers.

I don't think it's related to your levels, really.

-Carl


top of page bottom of page up down


Message: 6523

Date: Tue, 18 Feb 2003 16:26:00

Subject: Re: lattice diagram "levels" of complexity

From: Carl Lumma

>> Others, apparently including Wilson, wanted
>> to include other points, basically out to the power
>> set (the compound of all CPSs of a given limit).
>
>that isn't the case. look over the old discussions on this list about 
>stellation. the compound of all CPSs of a given limit can be 
>constructed in many ways, but most typically (as d'allessandro) as an 
>euler genus, and this (or any of the other ways) clearly does not 
>have the symmetry of the original, unstellated CPS, which the 
>stellation must have by definition.

Right, right, you're extending the tones out to a EF Genus in all
directions.  A stellated EF Genus is what I called it in that thread.

-Carl


top of page bottom of page up

Previous Next

6000 6050 6100 6150 6200 6250 6300 6350 6400 6450 6500 6550 6600 6650 6700 6750 6800 6850 6900 6950

6500 - 6525 -

top of page