JI symmetric scales?

Someone once told me that many of the altered sounds
in jazz harmony are approximations of 7+-limit just
intervals, i.e. a #9 over a dominant chord would be
better expressed (and tuned?) as 7/6 than 6/5. What
would other alterations (b9, #11/b5, b13) be in JI?
Can extensions be extended even higher by treating the
dominant chord like a 13 or 17 limit otonality? What
about doing the same things with other types of
traditional jazz tetrachords?
A lot of players use Diminished scales over dominant
chords to hit the altered tones - What would be the
best way(s) to express a diminished scale in JI?
Would the symmetrical sound/usage of the scale be
destroyed if the intervals were just? What about
other symmetric scales? If retuned, the steps
wouldn't be equal any more, so would the scales even
be 'symmetrical' any more? Are symmetric scales too
reliant on enharmonicity and patterns to be retuned
and fitted into JI environments?
How would you treat the tuning of the 4-tonic system?
(when you add all of the pitches of, say the four
tonic substitutions for G7 - Bb7, Db7, and E7, you get
a half-whole diminished scale. But if we wanted to
retune the diminished scale to get just intervals, how
would it affect the 4-tonic substitution chords?
What about the 3-tonic system and Coletrane changes?
Is there a JI 'reasoning' for why Coletrane changes
'work' - or is it only because of the use of symmetry
and patterns?
One last question (for now) that pretty much
summarizes everything above: Are symmetric
scales/chord progressions exclusive to edo tuning
systems or is there such thing as Just Symmetry?
Thanks to anyone who can help me out with this!
-K

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> Someone once told me that many of the altered sounds
> in jazz harmony are approximations of 7+-limit just
> intervals, i.e. a #9 over a dominant chord would be
> better expressed (and tuned?) as 7/6 than 6/5.

That depends on your perference. On the piano it is
both simultaneously, and I don't know of any rigorous
analysis of ensemble performances. Barbershop quartets
do sing 4:5:6:7 dominant chords. Major 7th chords
are clearly 4:5:6:15...

> What would other alterations (b9, #11/b5, b13) be in JI?

Lots of possibilities...

> Can extensions be extended even higher by treating the
> dominant chord like a 13 or 17 limit otonality?

If it sounds right to you, yes.

> A lot of players use Diminished scales over dominant
> chords to hit the altered tones - What would be the
> best way(s) to express a diminished scale in JI?

Like this (5-limit)

1/1
25/24
6/5
5/4
25/18
3/2
5/3
9/5

or this (7-limit)

1/1
15/14
6/5
5/4
7/5
3/2
12/7
7/4

but the full harmonies of this scale require either
temperament, or more than 8 notes.

> Would the symmetrical sound/usage of the scale be
> destroyed if the intervals were just?

Perhaps.

> What about other symmetric scales?

Ditto.

> Are symmetric
> scales/chord progressions exclusive to edo tuning
> systems or is there such thing as Just Symmetry?
> Thanks to anyone who can help me out with this!

Depends what you mean by symetrical. If you mean,
'has a pattern of steps like 1 2 1 2 1 2', this can
be done in JI. If you mean, 'has limited transpositions
within a larger scale' it can't.

-Carl

> Depends what you mean by symetrical. If you mean,
> 'has a pattern of steps like 1 2 1 2 1 2', this can
> be done in JI. If you mean, 'has limited transpositions
> within a larger scale' it can't.

I should have said "A B A B A B". You can't in JI
have an interval that evenly divides the octave 9 ways.

-Carl

> Someone once told me that many of the altered sounds
> in jazz harmony are approximations of 7+-limit just
> intervals, i.e. a #9 over a dominant chord would be
> better expressed (and tuned?) as 7/6 than 6/5. What
> would other alterations (b9, #11/b5, b13) be in JI?
> Can extensions be extended even higher by treating the
> dominant chord like a 13 or 17 limit otonality?

Well, some would say 13 is approximated by the major sixth in some
complex chords, the minor sixth in others. The 11th harmonic by the
augmented forth, 17 by the minor second, and 19 by the minor third.

> A lot of players use Diminished scales over dominant
> chords to hit the altered tones - What would be the
> best way(s) to express a diminished scale in JI?

The problem is that most of these scales are impossible to render
consistently in JI-as scales. Individual chords and melodies can be,
but not the entire scale itself, because each note has more than one
unique intervallic relation to the others.

> Would the symmetrical sound/usage of the scale be
> destroyed if the intervals were just?

Practically, yes.

> Are symmetric scales too
> reliant on enharmonicity and patterns to be retuned
> and fitted into JI environments?

It all depends on your views. A lot of the music that "uses" symmetric
scales-diminshed, augmented and whole tone-doesn't really rely on them
being symmetrical, so the musical results might not be artistically
inferior to the "original", but it would not be "symmetrical" anymore
when transferred to JI, or meantone for that matter.

> How would you treat the tuning of the 4-tonic system?

You don't. :D

> But if we wanted to
> retune the diminished scale to get just intervals, how
> would it affect the 4-tonic substitution chords?

Some would work and some would be completely destroyed. The only way
to achieve the proper symmetry is to go equal.

> One last question (for now) that pretty much
> summarizes everything above: Are symmetric
> scales/chord progressions exclusive to edo tuning
> systems or is there such thing as Just Symmetry?

It all depends on what you mean by it. As in scales that repeat 2, 3
or 4 times within an octave, that is itself a equal division, so it is
not possible using only rational intervals. Though many JI scales do
have a certain "partial" symmetry, that is most of the notes, or at
least a subset, can be played in several different keys and form the
whole scale when combined. If you mean inversional symmetry however,
many of the (relatively) commonly used JI scales have that property,
as inversion and transposition itself is not dependent on equal
divisions, only the concept of equal divisions themselves are.

/Ö

> > Depends what you mean by symetrical. If you mean,
> > 'has a pattern of steps like 1 2 1 2 1 2', this can
> > be done in JI. If you mean, 'has limited transpositions
> > within a larger scale' it can't.
>
> I should have said "A B A B A B". You can't in JI
> have an interval that evenly divides the octave 9 ways.

Nor any octave-repeating scale with the step sizes you mentioned, for
that matter, as it requires the combined step sizes to equal one third
of an octave. MoS like LLsLs are possible, but none where the number
of instances of both step sizes have a common divisor.

/Ö

The question of symmetry and its value is interesting in itself.
In visual art,
symmetrical work is quite common among schizophrenics.
more often in the work of note, balance is achieved by placing certain tension against each other.
In the 'wild' and in the 'field', scales with uneven size intervals are more common than 'equal'.

In one Bunuel film a character sees a spider on the wall
crushes it and says " I hate symmetry!"
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

Using symmetry can be a way of ensuring imbalance or asymmetry, if
the symmetry exists in a source but not in its manifestation.
The "octatonic" or "whole-tone" scales in a rationally tuned system,
for example, or mirror-movements on the keys of an instrument.

And an interesting sounding, but hardly jazz :-D , kind of symmetry
can be achieved with "just intonation"- expand upward and downward
symmetrically from a central point. Been working on ways to use
this as an approach to modulation between different tuning, back in
about 10 years years on this one, hahaha!

I wonder if there are parallels in visual art, hmmm. Maybe there's
something along the lines of mapping out symmetries on a color wheel
in some way that ensures both asymmetry and a curious cohesive logic.

-Cameron Bobro

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> The question of symmetry and its value is interesting in itself.
> In visual art,
> symmetrical work is quite common among schizophrenics.
> more often in the work of note, balance is achieved by placing
certain
> tension against each other.
>
> In the 'wild' and in the 'field', scales with uneven size
intervals are
> more common than 'equal'.
>
> In one Bunuel film a character sees a spider on the wall
> crushes it and says " I hate symmetry!"
> --
> Kraig Grady
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
> The Wandering Medicine Show
> KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles
>

How would you tune a rational octatonic scale?

1/1, 9/8 (or 10/9?), 5/4, (what tritone could work?),
8/5, 16/9, 2/1 .....?

Kevin

--- misterbobro <misterbobro@yahoo.com> wrote:

> Using symmetry can be a way of ensuring imbalance or
> asymmetry, if
> the symmetry exists in a source but not in its
> manifestation.
> The "octatonic" or "whole-tone" scales in a
> rationally tuned system,
> for example, or mirror-movements on the keys of an
> instrument.
>
> And an interesting sounding, but hardly jazz :-D ,
> kind of symmetry
> can be achieved with "just intonation"- expand
> upward and downward
> symmetrically from a central point. Been working on
> ways to use
> this as an approach to modulation between different
> tuning, back in
> about 10 years years on this one, hahaha!
>
> I wonder if there are parallels in visual art, hmmm.
> Maybe there's
> something along the lines of mapping out symmetries
> on a color wheel
> in some way that ensures both asymmetry and a
> curious cohesive logic.
>
> -Cameron Bobro
>
> --- In tuning@yahoogroups.com, Kraig Grady
> <kraiggrady@...> wrote:
> >
> > The question of symmetry and its value is
> interesting in itself.
> > In visual art,
> > symmetrical work is quite common among
> schizophrenics.
> > more often in the work of note, balance is
> achieved by placing
> certain
> > tension against each other.
> >
> > In the 'wild' and in the 'field', scales with
> uneven size
> intervals are
> > more common than 'equal'.
> >
> > In one Bunuel film a character sees a spider on
> the wall
> > crushes it and says " I hate symmetry!"
> > --
> > Kraig Grady
> > North American Embassy of Anaphoria Island
> <http://anaphoria.com/>
> > The Wandering Medicine Show
> > KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed
> 8-9 pm Los Angeles
> >
>
>
>
>
>
>

__________________________________________________
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hello there, for some interesting ideas about this, you could try
going here:

http://tunesmithy.netfirms.com/japplets/uo_non_oct.htm

and setting the scale for 8 notes at a 2/1 periodicity. then click the
"Set Optimal for et" tab, as this will maximizes palindromic symmetry
at the half octave .

http://www.myspace.com/danstearns

--- In tuning@yahoogroups.com, kevin ryan <bentivi_cdo@...> wrote:
>
> How would you tune a rational octatonic scale?
>
> 1/1, 9/8 (or 10/9?), 5/4, (what tritone could work?),
> 8/5, 16/9, 2/1 .....?
>
> Kevin
>
> --- misterbobro <misterbobro@...> wrote:
>
> > Using symmetry can be a way of ensuring imbalance or
> > asymmetry, if
> > the symmetry exists in a source but not in its
> > manifestation.
> > The "octatonic" or "whole-tone" scales in a
> > rationally tuned system,
> > for example, or mirror-movements on the keys of an
> > instrument.
> >
> > And an interesting sounding, but hardly jazz :-D ,
> > kind of symmetry
> > can be achieved with "just intonation"- expand
> > upward and downward
> > symmetrically from a central point. Been working on
> > ways to use
> > this as an approach to modulation between different
> > tuning, back in
> > about 10 years years on this one, hahaha!
> >
> > I wonder if there are parallels in visual art, hmmm.
> > Maybe there's
> > something along the lines of mapping out symmetries
> > on a color wheel
> > in some way that ensures both asymmetry and a
> > curious cohesive logic.
> >
> > -Cameron Bobro
> >
> > --- In tuning@yahoogroups.com, Kraig Grady
> > <kraiggrady@> wrote:
> > >
> > > The question of symmetry and its value is
> > interesting in itself.
> > > In visual art,
> > > symmetrical work is quite common among
> > schizophrenics.
> > > more often in the work of note, balance is
> > achieved by placing
> > certain
> > > tension against each other.
> > >
> > > In the 'wild' and in the 'field', scales with
> > uneven size
> > intervals are
> > > more common than 'equal'.
> > >
> > > In one Bunuel film a character sees a spider on
> > the wall
> > > crushes it and says " I hate symmetry!"
> > > --
> > > Kraig Grady
> > > North American Embassy of Anaphoria Island
> > <http://anaphoria.com/>
> > > The Wandering Medicine Show
> > > KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed
> > 8-9 pm Los Angeles
> > >
> >
> >
> >
> >
> >
> >
>
>
> __________________________________________________
> Do You Yahoo!?
> Tired of spam? Yahoo! Mail has the best spam protection around
> http://mail.yahoo.com
>

Thank you to everyone who has responded so far. Now I
have even more questions though!
I've read/been taught that symmetry / patterning or
even any type of chord substitution 'works' in jazz
(or other music too) because our ears accept the
'dissonance' in favor of the logic of the patterns
created - we hear the structures created (it's the
'wrong' dominant chord but dominant chords have such a
'strong structure' that it doesn't matter), or we hear
that the notes form a pattern and excuse the fact that
some of them don't 'fit' the chords.
My big question is if in some (or even all) cases,
it's not just the patterns that make sense to our
ears, but also that these 'outside' notes are
opproximations of higher prime limit rational
intervals.

Mats �ljare wrote:

>Well, some would say 13 is approximated by the major
> sixth in some
> complex chords, the minor sixth in others. The 11th
> harmonic by the
> augmented forth, 17 by the minor second, and 19 by
> the minor third.

So If you were representing an altered dominant chord
as rational intervals would it be: 'G' 1/1, 'B' 5/4,
'D' 3/2, 'F' 7/4, 'Ab' 17/8, 'A#' 19/8, 'C#' 11/4,
'Eb/E' 13/4 ?
What are some ways of representing it within 13-limit?
(it would no longer be a strict otonality)

Over a D-/G7/C/C chord progression, Coletrane would
substitute
D- Eb7 / Abmaj7 B7 / Emaj7 G7 / Cmaj7 (or many, many
variations) which would give (Eb, G, Bb, Db) over a D-
chord, (Ab, C, Eb, G) then (B, D#, F#, A) over a G7
chord, and (E, G#, B, D#) over the Cmaj7 chord. Can
these chords make 'rational' sense?

On another level I'm trying to figure out which
aspects of Bebop and modern jazz harmony are tethered
to the history of the piano and which can be freed by
JI.

--- Mats �ljare <oljare@hotmail.com> wrote:

> > Someone once told me that many of the altered
> sounds
> > in jazz harmony are approximations of 7+-limit
> just
> > intervals, i.e. a #9 over a dominant chord would
> be
> > better expressed (and tuned?) as 7/6 than 6/5.
> What
> > would other alterations (b9, #11/b5, b13) be in
> JI?
> > Can extensions be extended even higher by treating
> the
> > dominant chord like a 13 or 17 limit otonality?
>
> Well, some would say 13 is approximated by the major
> sixth in some
> complex chords, the minor sixth in others. The 11th
> harmonic by the
> augmented forth, 17 by the minor second, and 19 by
> the minor third.
>
> > A lot of players use Diminished scales over
> dominant
> > chords to hit the altered tones - What would be
> the
> > best way(s) to express a diminished scale in JI?
>
> The problem is that most of these scales are
> impossible to render
> consistently in JI-as scales. Individual chords and
> melodies can be,
> but not the entire scale itself, because each note
> has more than one
> unique intervallic relation to the others.
>
> > Would the symmetrical sound/usage of the scale be
> > destroyed if the intervals were just?
>
> Practically, yes.
>
>
> > Are symmetric scales too
> > reliant on enharmonicity and patterns to be
> retuned
> > and fitted into JI environments?
>
> It all depends on your views. A lot of the music
> that "uses" symmetric
> scales-diminshed, augmented and whole tone-doesn't
> really rely on them
> being symmetrical, so the musical results might not
> be artistically
> inferior to the "original", but it would not be
> "symmetrical" anymore
> when transferred to JI, or meantone for that matter.
>
> > How would you treat the tuning of the 4-tonic
> system?
>
> You don't. :D
>
> > But if we wanted to
> > retune the diminished scale to get just intervals,
> how
> > would it affect the 4-tonic substitution chords?
>
> Some would work and some would be completely
> destroyed. The only way
> to achieve the proper symmetry is to go equal.
>
> > One last question (for now) that pretty much
> > summarizes everything above: Are symmetric
> > scales/chord progressions exclusive to edo tuning
> > systems or is there such thing as Just Symmetry?
>
> It all depends on what you mean by it. As in scales
> that repeat 2, 3
> or 4 times within an octave, that is itself a equal
> division, so it is
> not possible using only rational intervals. Though
> many JI scales do
> have a certain "partial" symmetry, that is most of
> the notes, or at
> least a subset, can be played in several different
> keys and form the
> whole scale when combined. If you mean inversional
> symmetry however,
> many of the (relatively) commonly used JI scales
> have that property,
> as inversion and transposition itself is not
> dependent on equal
> divisions, only the concept of equal divisions
> themselves are.
>
> /Ö
>
>
>
>

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--- In tuning@yahoogroups.com, kevin ryan <bentivi_cdo@...> wrote:
>
> How would you tune a rational octatonic scale?
>
> 1/1, 9/8 (or 10/9?), 5/4, (what tritone could work?),
> 8/5, 16/9, 2/1 .....?
>
> Kevin

In real life (ie not on a cheesey percussion instrument like a piano
:-)) it all flexes by ear, of course, but at any rate
the "octatonic" is melodic first and foremost.

Alternating between 16/15 and 29/26 gives you an octave stretched by
3 cents and a kind of interplay between calm and bitter. That would
be a maniac's tuning (but sounds good to me).

You can juggle 16/15, 25/24, 10/9 and 9/8 all together to get all
simple "classic" intervals, a straight octave, and an
audibly "octatonic" scale, in which case it's symmetrical as far as
sLsLsLsL... steps. Try to find the oldest recordings of Rimsky
Korsakov you can, preferably on Melodiya- I'm pretty sure this is
how they did it.

-Bobro

> I wonder if there are parallels in visual art, hmmm. Maybe there's
> something along the lines of mapping out symmetries on a color wheel
> in some way that ensures both asymmetry and a curious cohesive logic.
>
> -Cameron Bobro

That idea intrigues me - I'm not entirely sure what you mean, but it
sounds like an interesting challenge. Would you like to explain a bit
more about what you mean?

Is http://www.pheddles.com/display.php?id=200 anything like what
you're talking about?

Cheers,
Patty

> On another level I'm trying to figure out which
> aspects of Bebop and modern jazz harmony are tethered
> to the history of the piano and which can be freed by
> JI.

You have to understand that it is never, or at least in all but a few
cases, not possible to retain all aspects of a scale, or its use in a
composition, when "transferred" to JI. The music you are talking about
is made in 12-TET, not JI. It seems that many here and in other places
are obsessed with "rationalizing" existing music and existing scales
ignoring the fact that their full character comes from properties that
are unseparably linked with 12-TET itself.

Turning 12-TET scales and compositions into ratios always assumes that
each note has only ONE rational relationship to the others, while the
depth and complexity of the compositions and the versatility of the
scales more often than not comes from the fact that a pitch can be
thought of as representing more than one ratio, at the same time. This
is especially noticable in symmetrical scales and other "strange"
modes and harmonies.

For example, the most common "altered chord", C E Ab Bb Eb, can be
thought of as approximating high-limit intervals, the Ab 13/8 and the
Eb 19/8, or 7/3, and the Bb 7/4 with the Ab possibly interpreted as
14/9, forming a sequence of forths from the 7/4. But there is also a
Pythagorean (3-limit) interpretation, where C-Bb-Eb-Ab can be seen as
a chain of fourths with only the F missing.

And if the E is assumed to be 5/4, it may be considered to have a
5-limit relation with the three other tones, as 14/9 is off from 25/16
by just a few cents, enough to completely take its place in practice,
and E-Ab-Eb, with or without the Bb being a strong 5-limit chord with
the C being 5-limit in the "other direction" from the E.

When you go into adding melodic lines that contain notes not found in
the proper "altered" scale, the complexity increases further to what
is simply "unrationalizable".

/Ö

--- In tuning@yahoogroups.com, "p_heddles" <p_heddles@...> wrote:

> That idea intrigues me - I'm not entirely sure what you mean, but it
> sounds like an interesting challenge. Would you like to explain a bit
> more about what you mean?
>
> Is http://www.pheddles.com/display.php?id=200 anything like what
> you're talking about?
>
> Cheers,
> Patty
>

I don't know how it would be done either, how did you plan the color
schema of those pictures?

There was a design theory in the late '60s that basically said all
versions of one color go together- cold blue, warm blue, plastic
blue, whatever blue. Everyone is familiar with the commercial
offshoots of the idea (1971 airport lounge that looks like a snail
trying to digest a cactus). Not sure if that's a "symmetry", but
it's a chohesive, if hideous, example of mapping something out from
a color wheel. Straight symmetry from a color wheel would result
in shocking contrasts, wouldn't it?

-Bobro