Outonal chords

Any otonal chord can be considered a utonal chord, and vice-versa, but
normally one way of looking at it leads to smaller numbers than the
other. Also, inverting a chord typically switches otonality with
utonality. This is not always the case, however.

Consider the 18:22:27:33 chord, which on dividing out the 18 gives
1-11/9-3/2-11/6. The inverted chord is 1-9/11-2/3-6/11, and upon
multiplication by 11/6 that becomes 1-11/9-3/2-11/6 again. Have these
been considered and given a name? If not, "outonal chord" is my proposal.

> Consider the 18:22:27:33 chord, which on dividing out the 18 gives
> 1-11/9-3/2-11/6. The inverted chord is 1-9/11-2/3-6/11, and upon
> multiplication by 11/6 that becomes 1-11/9-3/2-11/6 again. Have these
> been considered and given a name? If not, "outonal chord" is my
> proposal.

Weird. -C.

How about ubi-tonal?

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
To: <tuning@yahoogroups.com>
Sent: 10 Nisan 2006 Pazartesi 8:52
Subject: [tuning] Outonal chords

> Any otonal chord can be considered a utonal chord, and vice-versa, but
> normally one way of looking at it leads to smaller numbers than the
> other. Also, inverting a chord typically switches otonality with
> utonality. This is not always the case, however.
>
> Consider the 18:22:27:33 chord, which on dividing out the 18 gives
> 1-11/9-3/2-11/6. The inverted chord is 1-9/11-2/3-6/11, and upon
> multiplication by 11/6 that becomes 1-11/9-3/2-11/6 again. Have these
> been considered and given a name? If not, "outonal chord" is my proposal.
>
>

On 4/10/06, Gene Ward Smith <genewardsmith@coolgoose.com> wrote:
> Any otonal chord can be considered a utonal chord, and vice-versa, but
> normally one way of looking at it leads to smaller numbers than the
> other. Also, inverting a chord typically switches otonality with
> utonality. This is not always the case, however.
>
> Consider the 18:22:27:33 chord, which on dividing out the 18 gives
> 1-11/9-3/2-11/6. The inverted chord is 1-9/11-2/3-6/11, and upon
> multiplication by 11/6 that becomes 1-11/9-3/2-11/6 again. Have these
> been considered and given a name? If not, "outonal chord" is my proposal.

The simplest example would be 3:5:9:15, right? I thought I remember
someone calling these "anomalous saturated suspensions". They come
from odd composite numbers.

Keenan

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:

> The simplest example would be 3:5:9:15, right? I thought I remember
> someone calling these "anomalous saturated suspensions". They come
> from odd composite numbers.

That has indeed been given the unfortunate name of "ass". I had
concluded this was different, which is why I posted about it, but it
seems it isn't; however I think both outonal and ubitonal are better
names, but especially outonal which brings out the essential property
that it is *equally* otonal and utonal.

If you have a:b:au:bu and invert it, you get 1/a:1/b:1/(au):1/(bu).
Now multiply by abu, and you get a:b:au:bu again. The pattern of steps
will contain two intervals of the same size, b/a and ub/ua = b/a,
which are separated by an interval of different size. Hence, reversing
the order just gets you back to the same thing, and so "outonal". The
idea is that abac, reversed, is caba, and up to a circular permutation
that's abac again. That's also possible with numbers of notes other
than four: for example, ababac reverses to cababa, which is a circular
permuation of ababac.

Anyway, it seems to be that saying something is equally otonal and
utonal is much clearer than saying it is neither, which really doesn't
make sense, so I think outonal is a good term anyway, especially since
"ass" is so unfortunate.

Hey, ubi is much better than eau if you wanna kick ass.

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
To: <tuning@yahoogroups.com>
Sent: 10 Nisan 2006 Pazartesi 21:47
Subject: [tuning] Re: Outonal chords

> --- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> > The simplest example would be 3:5:9:15, right? I thought I remember
> > someone calling these "anomalous saturated suspensions". They come
> > from odd composite numbers.
>
> That has indeed been given the unfortunate name of "ass". I had
> concluded this was different, which is why I posted about it, but it
> seems it isn't; however I think both outonal and ubitonal are better
> names, but especially outonal which brings out the essential property
> that it is *equally* otonal and utonal.
>
> If you have a:b:au:bu and invert it, you get 1/a:1/b:1/(au):1/(bu).
> Now multiply by abu, and you get a:b:au:bu again. The pattern of steps
> will contain two intervals of the same size, b/a and ub/ua = b/a,
> which are separated by an interval of different size. Hence, reversing
> the order just gets you back to the same thing, and so "outonal". The
> idea is that abac, reversed, is caba, and up to a circular permutation
> that's abac again. That's also possible with numbers of notes other
> than four: for example, ababac reverses to cababa, which is a circular
> permuation of ababac.
>
> Anyway, it seems to be that saying something is equally otonal and
> utonal is much clearer than saying it is neither, which really doesn't
> make sense, so I think outonal is a good term anyway, especially since
> "ass" is so unfortunate.
>
>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:

> That's also possible with numbers of notes other
> than four: for example, ababac reverses to cababa, which is a circular
> permuation of ababac.

Hexachords like this are 1:15:25:27:45:225 and 5:9:25:45:125:225.

> > The simplest example would be 3:5:9:15, right? I thought I
> > remember someone calling these "anomalous saturated
> > suspensions". They come from odd composite numbers.
>
> That has indeed been given the unfortunate name of "ass". I had
> concluded this was different, which is why I posted about it, but it
> seems it isn't;

I should have recognized that, since I once posted about it.

ASS isn't unfortunate.

> however I think both outonal and ubitonal are better
> names, but especially outonal which brings out the essential
> property that it is *equally* otonal and utonal.

These names don't seem very good to me.

-Carl

> Hexachords like this are 1:15:25:27:45:225 and 5:9:25:45:125:225.

Graham has a complete list of ASSs through the 17-limit or
something, somewhere.

-C.

That's spelled Ass-es.

----- Original Message -----
From: "Carl Lumma" <clumma@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 11 Nisan 2006 Sal� 0:07
Subject: [tuning] Re: Outonal chords

> > Hexachords like this are 1:15:25:27:45:225 and 5:9:25:45:125:225.
>
> Graham has a complete list of ASSs through the 17-limit or
> something, somewhere.
>
> -C.
>
>

i thought that Paul had mentioned these type of parallelograms before.
a kind of specialized Euler genus of the 11/9 and 3/2

Subject: Re: Outonal chords

> > Consider the 18:22:27:33 chord, which on dividing out the 18 gives
> > 1-11/9-3/2-11/6. The inverted chord is 1-9/11-2/3-6/11, and upon
> > multiplication by 11/6 that becomes 1-11/9-3/2-11/6 again. Have these
> > been considered and given a name? If not, "outonal chord" is my
> > proposal.
> Weird. -C.

> -- 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

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > Hexachords like this are 1:15:25:27:45:225 and 5:9:25:45:125:225.
>
> Graham has a complete list of ASSs through the 17-limit or
> something, somewhere.

Defined how? In terms of prime limits, these aren't finite sets. I've
posted something on tuning-math up to the 13 limit, and it's a lot,
even though I filtered things in several ways.

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:

> ASS isn't unfortunate.

It's assinine. :)

If "ass" makes sense, then people should be talking about saturated
suspensions, and they don't seem to. What does "saturated" mean? Why
should we even consider them to be suspensions? What makes them
anomalous, and where are all the non-anomalous examples?

> > however I think both outonal and ubitonal are better
> > names, but especially outonal which brings out the essential
> > property that it is *equally* otonal and utonal.
>
> These names don't seem very good to me.

"Outonal" defines the property in question; that's a good start.

Carl Lumma wrote:
>>Hexachords like this are 1:15:25:27:45:225 and 5:9:25:45:125:225.
> > > Graham has a complete list of ASSs through the 17-limit or
> something, somewhere.

http://x31eq.com/ass.htm

Neither example's on the list, and an ASS can't start with a 1. (225-limit???)

If you think about them as either an otonality or utonality, instead of neither, then they aren't anomalous and they aren't saturated. But they are symmetric, and one "S" originally stood for "symmetric". It turns out that some ASSs aren't symmemtric.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Carl Lumma wrote:
> >>Hexachords like this are 1:15:25:27:45:225 and 5:9:25:45:125:225.
> >
> >
> > Graham has a complete list of ASSs through the 17-limit or
> > something, somewhere.
>
> http://x31eq.com/ass.htm
>
> Neither example's on the list, and an ASS can't start with a 1.
> (225-limit???)

Then an outonal chord need not be an ass, because they can.

This should have been 1:5:9:25:45:225. This has the outonal property,
in that its inversion gives us the same chord back again. The same
property is also shown by 5:9:25:45:125:225. Reducing to the octave in
Tenney-minimal form, the first is 10/9-5/4-25/18-25/16-16/9-2, and the
second is 10/9-5/4-25/18-8/5-16/9-2.

> > > Hexachords like this are 1:15:25:27:45:225 and 5:9:25:45:125:225.
> >
> > Graham has a complete list of ASSs through the 17-limit or
> > something, somewhere.
>
> Defined how? In terms of prime limits, these aren't finite sets.
> I've posted something on tuning-math up to the 13 limit, and it's
> a lot, even though I filtered things in several ways.

You haven't read the page? I believe he used odd limits.

http://microtonal.co.uk/ass.htm

By the way, I feel limit by itself should mean odd limit, since
that's how Partch defined it.

-Carl

> > > however I think both outonal and ubitonal are better
> > > names, but especially outonal which brings out the essential
> > > property that it is *equally* otonal and utonal.
> >
> > These names don't seem very good to me.
>
> "Outonal" defines the property in question; that's a good start.

I suppose a better point would be this: these things have names,
and hundreds of messages using them. That's 'well established
in the literature', such as it is. What makes you think you
should change it?

-Carl

On 4/10/06, Carl Lumma <clumma@yahoo.com> wrote:
[...]
> I suppose a better point would be this: these things have names,
> and hundreds of messages using them. That's 'well established
> in the literature', such as it is. What makes you think you
> should change it?

It seems like we're talking about two different things here.
"Anomalous saturated suspensions" have to be pairwise consonant in
some reasonable odd limit; Gene's things don't. Gene's things have to
be symmetric; ASSs don't. If they're different, why shouldn't they
have different names?

Personally, I think only two or three of them are musically useful to
me: 3:5:9:15, 3:7:9:21, and possibly 3:9:11:33.

Keenan

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> I suppose a better point would be this: these things have names,
> and hundreds of messages using them. That's 'well established
> in the literature', such as it is. What makes you think you
> should change it?

For one thing, it apparently is *not* the same idea, and I was right
the first time.

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:

> Personally, I think only two or three of them are musically useful to
> me: 3:5:9:15, 3:7:9:21, and possibly 3:9:11:33.

"Then" being asses, outonalies, or both? I started off discussing
9:11:27:33 precisely because I thought it was useful, incidentally.

> > I suppose a better point would be this: these things have names,
> > and hundreds of messages using them. That's 'well established
> > in the literature', such as it is. What makes you think you
> > should change it?
>
> It seems like we're talking about two different things here.
> "Anomalous saturated suspensions" have to be pairwise consonant in
> some reasonable odd limit; Gene's things don't.

Where did he say that?

-Carl

Carl Lumma wrote:
>>>I suppose a better point would be this: these things have names,
>>>and hundreds of messages using them. That's 'well established
>>>in the literature', such as it is. What makes you think you
>>>should change it?
>>
>>It seems like we're talking about two different things here.
>>"Anomalous saturated suspensions" have to be pairwise consonant in
>>some reasonable odd limit; Gene's things don't.
> > > Where did he say that?

Which side are you questioning? An ASS needs all its intervals within a given odd limit. That's part of its definition: "suspensions of notes to which none can be added without raising the odd limit". Gene says he doesn't need this property. So the two concepts are different. Gene's chords are, however, like those previously described as "symmetric".

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> Which side are you questioning? An ASS needs all its intervals
within a
> given odd limit. That's part of its definition: "suspensions of notes
> to which none can be added without raising the odd limit". Gene
says he
> doesn't need this property. So the two concepts are different.

A lot of times I'm considering JI chords as a guide to tempered chords
anyway. For instance, consider the 1-11/9-3/2-11/6 chord tbis thread
started from. It has an evil 27-limit interval, 27/22, between 11/9
and 3/2. However, if 243/242 is tempered out this is the same as 11/9
anyway, and we have an 11/9-11/9-11/9-12/11 chord, an 11-limit "magic"
chord. I suppose going systematically through the chords of whatever
temperament you are considering makes the most sense, but there is
never any shortage of microtemperament chords, and you could spend a
lot of time just listening to examples.

Other examples like this are 1-7/6-10/7-5/3, where the 60/49 from 7/6
to 10/7 is a (smaller) neutral third and can be assimilated down to
11/9, which tempers out 540/539, and 1-7/6-7/5-5/3, where the 25/21 is
302 cents, and so is a minor third of some kind. Equating it to 6/5
tempers out 126/125.

Hi all,

On Tue, 11 Apr 2006, Gene Ward Smith wrote:
[snip]
> If "ass" makes sense, then people should be talking about saturated
> suspensions, and they don't seem to. What does "saturated" mean? Why
> should we even consider them to be suspensions? What makes them
> anomalous, and where are all the non-anomalous examples?
>
> > > however I think both outonal and ubitonal are better
> > > names, but especially outonal which brings out the essential
> > > property that it is *equally* otonal and utonal.
> >
> > These names don't seem very good to me.
>
> "Outonal" defines the property in question; that's a good start.

I agree that -
a) The property is something different from anything
else I've seen discussed, the nearest being chords
which are partially utonal and partially otonal; and

b) the name (visually, if not aurally) combines elements
of the names of its two components, the otonal and
utonal chords. Pronouncing it would necessarily be an
exercise in careful enunciation: "o-u-tonal"; therefore
a more easily pronounceable term might be better were
we to discuss it frequently in speech.

A possible alternative term would be: "ambitonal".
This however, unfortunately suggests that the tonal
universe consists of two poles (utonal and otonal)
and ambivalent mixtures between them. It's trivially
easy to demonstrate a tonality which is none of these
- just play an E major triad on your 12-EDO piano or
guitar.

If Gene is the first to describe this phenomenon -
which I for one did not expect to exist (so never
went looking for it!) - then surely he can name it
whatever he likes.

Regards,
Yahya

--
No virus found in this outgoing message.
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Version: 7.1.385 / Virus Database: 268.4.1/307 - Release Date: 10/4/06

> --- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> > The simplest example would be 3:5:9:15, right? I thought I remember
> > someone calling these "anomalous saturated suspensions". They come
> > from odd composite numbers.

The simplest of these is 2:3:4:6.

Petr

> > --- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
> >
> > > The simplest example would be 3:5:9:15, right? I thought I remember
> > > someone calling these "anomalous saturated suspensions". They come
> > > from odd composite numbers.

1. But 2 is also a prime even though it's an even number.

2. If I stack the same interval several times, does this also count? I mean
those things like 1:2:4, 4:6:9, 9:12:16 or 16:20:25.

Petr

On 4/11/06, Petr Parízek <p.parizek@chello.cz> wrote:
> > --- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
> >
> > > The simplest example would be 3:5:9:15, right? I thought I remember
> > > someone calling these "anomalous saturated suspensions". They come
> > > from odd composite numbers.
>
> The simplest of these is 2:3:4:6.

I was assuming octave equivalence...

Keenan

On 4/11/06, Petr Parízek <p.parizek@chello.cz> wrote:
> 1. But 2 is also a prime even though it's an even number.

So?

> 2. If I stack the same interval several times, does this also count? I mean
> those things like 1:2:4, 4:6:9, 9:12:16 or 16:20:25.

Have you read http://microtonal.co.uk/ass.htm ? ASSs are pairwise
consonant and you can't add any more pitches without making it more
dissonant. 16:25, for example, is pretty dissonant (unless 224:225 is
tempered out).

Keenan

Hi Keenan.

I wrote:

> > 1. But 2 is also a prime even though it's an even number.
>
> So?

Well, you've just explained the matter to me. I wasn't assuming octave
equivalence, that was all.

> > 2. If I stack the same interval several times, does this also count? I
mean
> > those things like 1:2:4, 4:6:9, 9:12:16 or 16:20:25.
>
> Have you read http://microtonal.co.uk/ass.htm ? ASSs are pairwise
> consonant and you can't add any more pitches without making it more
> dissonant. 16:25, for example, is pretty dissonant (unless 224:225 is
> tempered out).

I see. This question was meant perhaps rather for Gene's "outonality" as you
could, in some situations, for example, take a C-E-G# and consider it to be
C-E-E-G# -- i.e. the E is the point of symmetry.

Petr

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@>
wrote:
>
> > The simplest example would be 3:5:9:15, right? I thought I
remember
> > someone calling these "anomalous saturated suspensions". They
come
> > from odd composite numbers.

Yes. And does anyone remember "theoretically indispensably tempered
suspensions", better known as "magic" chords. I hope not. ;-)

> That has indeed been given the unfortunate name of "ass". I had
> concluded this was different, which is why I posted about it, but
it
> seems it isn't; however I think both outonal and ubitonal are
better
> names, but especially outonal which brings out the essential
property
> that it is *equally* otonal and utonal.

What's wrong with "symmetrical". A scale that is its own inversion
is called symmetrical. Seems obvious to apply the same to chords.

The 6th chord being the same as the m7th chord of the relative minor
would have to be the best known example.

ASS seems to carry the additional requirement of saturation (i.e.
containing all intervals of a given odd limit) or does that just
happen automatically?

-- Dave

Oops!

I wrote:

> saturation (i.e. containing all intervals of a given odd limit)

That's not saturation. That's "completeness".

Graham wrote:

"I've decided to call these "anomalous saturated suspensions" as they
are suspensions of notes to which none can be added without raising
the odd limit, and which are outside Partch's original theory."

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:

> Yes. And does anyone remember "theoretically indispensably tempered
> suspensions", better known as "magic" chords. I hope not. ;-)

I've been using "magic" chord a lot, but maybe if you can tell me if I
do so correctly. My usage has been to say that a chord in a tempered
system such that every interval in it is a tempered consonance in some
limit, but such that there is no JI chord with this property it maps
from, is "magic". An example would be 1-6/5-7/5-5/3, which is "magic"
in any system (eg meantone) which tempers out 126/125.

If this isn't a correct use of "magic", or even if it is, I am open to
suggestions for better names.

> What's wrong with "symmetrical". A scale that is its own inversion
> is called symmetrical. Seems obvious to apply the same to chords.

It could certainly be called "inversely symmetrical".

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@> wrote:
>
> > Yes. And does anyone remember "theoretically indispensably
tempered
> > suspensions", better known as "magic" chords. I hope not. ;-)

Actually that was "indespensibly tempered _saturations_".

> I've been using "magic" chord a lot, but maybe if you can tell me
if I
> do so correctly. My usage has been to say that a chord in a
tempered
> system such that every interval in it is a tempered consonance in
some
> limit, but such that there is no JI chord with this property it
maps
> from, is "magic". An example would be 1-6/5-7/5-5/3, which
is "magic"
> in any system (eg meantone) which tempers out 126/125.

Pretty much. Except not _every_ interval needs to be tempered.

It is a chord that is saturated ("pairwise consonant" in Keenan
Pepper's excellent terminology), but only because of some tempering.

By the way, I don't recall anyone pointing out before that Keenan
Pepper's "crunchy" chords are mono-unsaturated. Sounds like a
breakfast cereal or something.

> It could certainly be called "inversely symmetrical".

That's fine by me, but what other kinds of symmetry might people
want to talk about in regard to chords?

-- Dave Keenan