a question

Here's a math-music creative problem solving type question that's
interested me for a while.

Given any scale, and be that scale rational, equidistant, or what have
you, what methods have folks worked with (if any) that could
generalize and stamp one single rotation as something that could in
one way or another be called the standard rotation -- "standard
rotation" not meaning to imply anything but a possible answer to a
particular question.

Perhaps the most "neutral"?

Perhaps the most "stable"?

Perhaps the most "centered"?

etc.,

--Dan Stearns

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Here's a math-music creative problem solving type question that's
> interested me for a while.
>
> Given any scale, and be that scale rational, equidistant, or what
have
> you, what methods have folks worked with (if any) that could
> generalize and stamp one single rotation as something that could in
> one way or another be called the standard rotation -- "standard
> rotation" not meaning to imply anything but a possible answer to a
> particular question.
>
> Perhaps the most "neutral"?
>
> Perhaps the most "stable"?
>
> Perhaps the most "centered"?
>

In my paper I define "static" and "dynamic" modes of scales. This was
by far the part of my paper that I banged my head against the most,
mainly in my senior year and then again a couple of years later,
before it "crystallized".

The key concept for understanding the "dynamic" modes is
the "characteristic dissonance". This is a rare dissonant interval
which is the same number of scale steps as a common consonant
interval. By common I mean "almost everywhere". So in the diatonic
scale, the tritone is the characteristic interval. In the "dynamic"
modes, the characteristic dissonance is disjoint from (has no notes
in common with) the tonic N-ad, a complete chord of the N*2 - 1
limit. For example, in the diatonic scale, the only modes in which
the tritone is disjoint from the tonic triad are the ionian
(="major") and aeolian (="natural minor") modes. You get "strong
tonality" when the characteristic dissonance resolves by contrary
motion through steps of the rarest step size to two notes of the
tonic N-ad. The ionian/major and aeolian/minor modes are
thus "strongly tonal" modes . . . and they pretty much
define "tonality" as we know it. When neither of the two notes of the
tonic N-ad being resolved to is the tonic note itself, one may need
special means to point to that note in order that the other two notes
don't imply some other tonic N-ad. In the diatonic case, this is why
we have the harmonic minor . . . otherwise minor tonality would
dissolve to easily into the relative major.

The "static" modes are modes in which the tonic N-ad, a complete
chord of the N*2 - 1 limit, can be extended to an N+1-ad, a complete
chord of the N*2 + 1 limit. For example, if you accept 12-tone equal
temperament's 7-limit approximations, the mixolydian mode and the
dorian mode are the examples . . . since the tonic major chord in the
mixolydian mode can be completed into an approximate 4:5:6:7 . . .
and utonally for the dorian mode. Note that a huge percentage
of "static", "modal" improvised rock and jazz-rock music is in these
two modes.

I've found lots of loose interpretations of these rules useful when
working with various scales. If there aren't a lot of consonant
intervals in the scale, even a single 3:2 will define a point of
relative "stability" -- the tonic being the note representing the
number 2 in the ratio 3:2. And even this can be subverted in many
styles of composition . . .

Paul and Jon W.,

Thanks for the replies, I'm going to take a better look at both of
them over the weekend.

One thing I had looked at before was using something like the average,
or arithmetic mean of the rotations in cents and looking for the
closest match to P/2 (or perhaps the "skewness", the degree of
asymmetry around the mean, where the best match is closest to zero).

The idea being that this would give the most palindromic rotation (or
rotations) thereby theoretically giving something like a generalized
dorian mode. But I hadn't really followed this very far.

In any event I was hoping for something that would work (give similar
results) across any given random scale.

--Dan Stearns

I wrote,

<<The idea being that this would give the most palindromic rotation
(or rotations) thereby theoretically giving something like a
generalized dorian mode. But I hadn't really followed this very far.>>

If you allow that any given scale can be given a standardized
"spelling" in cents, taking the rotation with the least degree of
asymmetry around the mean, in other words skewness where the best
match is closest to zero, would work in the generalized sense that I
was trying to get at here.

However "dorian" anything is a bad idea, real bad. For example, the
least skewed rotation of something like the syntonic diatonic would be
the I or ionian mode!

But I still think it's a melodic, or vertical condition, that would
work across any arbitrary scale. My "most centered" suggestion from
the first post seems to be barking up the right tree -- any ideas?

--Dan Stearns

I wrote,

<<But I still think it's a melodic, or vertical condition, that would
work across any arbitrary scale.>>

That should've read "a melodic, or horizontal condition" etc.

--Dan Stearns

> From: "D.Stearns" <STEARNS@CAPECOD.NET>
> Subject: Re: a question
>

I'm not exactly sure what you are looking for in this quest
but in the 'scale generation' program I've been playing
with I end up with lists of modes and need some way to
sort them.

In terms of finding a normalized rotation, my approaches
are biased (for better or worse) by proposed useage models.

One approach is to rotate the scale such that the
'best' consonances grow from the 'root'.

For a scale with a characteristic
interval (I'm playing with L and s MOS structures mostly)
I may try to position it so that the CI is most able to
resolve to important tones within the tonic consonance by
the preferred motion (opposite motion by near equal step
sizes preferred over oblique preferred over parallel by
step preferred over etc...)

See Pauls paper for where I'm stealing and corrupting
these ideas from.

Note that my sorting is manual and motivated by a music
that I don't even know if I like yet, so its pretty
much in the formative stages!

Bob Valentine

Hi Robert,

The kind of a solution I was looking for is much too simple for the
problem perhaps, but I also think it's the only way to tackle such a
broad question -- by the numbers.

The approach I was taking followed from the idea that maximally even
subsets tend towards palindromic symmetry, and that these rotations
are special. Though it's quite a leap from two-stepsize scales where
this is the case to arbitrary scales where this could have some
meaning or could irrelevant.

But note that all two-stepsize hepatonic scales operate in exactly the
same manner as the Pythagorean or twelve equal [2,5] rotations where
the maximally even subset is a mirror inversion of itself while the I
= III, the IV = VII and the V = VI.

This was the kind of a model I was hoping to generalize to the point
where it would accommodate any arbitrary scale.

--Dan Stearns

A picture: <http://www.filmarchiv.at/events/balkan/hasret.htm>

Clark

no not clark, jrtroy65@aol.com take me off mail list

No not dude, or clark or John, just a very confused person as to why I am
receiving all of this mail not addressed to me and the subject is all tuning,
and what are you talking about My group. just don't want all of these emails.

I have been trying to get off (one copy) of the tuning list for some
weeks now -- and there does not seem to be any way of doing so. My
browser does not have a "My Groups" thing, and anyway i do not read
e-mail in a Web browser -- I browse web pages with that software.

The question is HOW DOES ONE GET OFF?

I am getting two copies of all the mail and it is tedious.

How does one "sign in"? I have tried a few times and it just repeats
the page. These browser things are new to me -- and never seem to do
what people say they do.
==John ff

John,

We might be able to help if you tell us which browser you are using
to view the web pages of the Yahoo tuning group. If the browser on
your computer is an old version and doesn't show some of the
features, then maybe you could go to a library or internet cafe, get
to the tuning group site, sign in (there is a link to click on if you
want to sign in on a machine that hasn't stored your user name, etc.)
and then edit your membership.

--- In tuning@y..., jpff@c... wrote:
> How does one "sign in"? I have tried a few times and it just
> repeats the page.

It has obviously stored your username and password.

> These browser things are new to me -- and never seem to do
> what people say they do.

These "browser things" have been around almost a decade; the problem
is, the various versions support different levels of sophistication
in terms of web pages, so if you happen to be on Netscape 1.0 you may
not see some of the items the Yahoo page is serving up.

Like pop-up ads, you lucky guy!

Cheers,
Jon

> I have been trying to get off (one copy) of the tuning list for some
> weeks now -- and there does not seem to be any way of doing so. My
> browser does not have a "My Groups" thing, and anyway i do not read
> e-mail in a Web browser -- I browse web pages with that software.

I notice this is in the footer now:

> tuning-nomail@yahoogroups.com - put your email message delivery on
> hold for the tuning group.

So if you send a message setting nomail, the copy you can't do anything
about will still arrive, but you'll still be subscribed with the other
address.

Make sense?

Graham

>>>>> "Jon" == Jon Szanto <JSZANTO@ADNC.COM> writes:

Jon> John,
Jon> We might be able to help if you tell us which browser you are using
Jon> to view the web pages of the Yahoo tuning group. If the browser on
Jon> your computer is an old version and doesn't show some of the
Jon> features, then maybe you could go to a library or internet cafe, get
Jon> to the tuning group site, sign in (there is a link to click on if you
Jon> want to sign in on a machine that hasn't stored your user name, etc.)
Jon> and then edit your membership.

1) i do not use a browser to view the web pages of the tuning group.
I use a mail program to read the mail that is sent to a mailing list

2) When I use a browser to try to get off the list I use Opera.

Jon> --- In tuning@y..., jpff@c... wrote:
>> How does one "sign in"? I have tried a few times and it just
>> repeats the page.

Jon> It has obviously stored your username and password.

It may have stored it, but it does not let me log in!

>> These browser things are new to me -- and never seem to do
>> what people say they do.

Jon> These "browser things" have been around almost a decade; the problem
Jon> is, the various versions support different levels of sophistication
Jon> in terms of web pages, so if you happen to be on Netscape 1.0 you may
Jon> not see some of the items the Yahoo page is serving up.

E-mail has been part of my life for 33 years, while browsers have been
available for less than a year. I still have not seen the advantage
of them, except to waste time. I still cannot see how to stop
getting two copies of every message; I am spending way too much time
trying to do this simple task. On the mailing lists I run there is an
advertised email address to resolve these problems, and I have a
steady stream of requests for manual intervention, which I do even if
the person could do it with other incantations.

==John ffitch

Hi John,

I'm sure this is no end of frustration to you! I fear we should take
this off-list, so feel free to write me directly. That said, just a
couple of pointed questions:

1. Do both copies come to the same address, i.e, is it addressed the
same in the header?

2. I assumed you have unsubscribed using the format as detailed by
other correspondents, i.e.
"Send an empty email to tuning-unsubscribe@yahoogroups.com"?

3. Written the list owner (tuning-owner@yahoogroups.com) for help?

About what you wrote:
> 2) When I use a browser to try to get off the list I use Opera.

Last time I checked Opera did *not* have support for Javascript,
which may be the problem. Just try unsubbing with another browser (at
the university, library, or internet cafe) that *should* allow you to
log-in.

> E-mail has been part of my life for 33 years, while browsers have
> been available for less than a year.

Hmmm. That isn't even remotely correct, as I'm willing to bet that
the first version of MOSAIC, written for CERN, was over a decade ago.
Certainly they have been ubiquitous for at least 5 years.

> I still have not seen the advantage of them, except to waste
> time.

Then, as much as I treasure (and use!) my email program for email
only (I *HATE* these mailing lists with a web interface), you are
missing an important resource, with the potential to transmit
information in ways and forms unavailable formerly. Naturally, this
includes a fair amount of crap as well, but a small bother to pay for
the wealth of information that is now out there.

End of sermon! <g>

Cheers,
Jon (who invites an off-list mail if you wish...)