the ambiguous triad

Has anyone played with the ambiguous triad, made by cutting a perfect
fifth into two equal parts? Its third degree is halfway between 6/5
and 5/4, right in no-man's land. With a backbone of the solid fifth,
the chord doesn't sound bad; it sounds ... ambiguous.

With two equal third intervals, each 350.98 cents, it's close to a pair
of 11/9 intervals (347.41 cents each), but this is not really a JI
chord.

Since I have been off in my own coocoon for most of my musical life, I
half expect there to be a deluge of references to work previously done
with this chord. I think it has possibilities...

JdL

Seeing the definition of Ervin Wilson's MOS I remembered an article by
Norman Carey and David Clampitt dealing with related matters. It is
downloadable from the "Perspectives of New Music" site:
"Self-Similar Pitch Structures, Their Duals, and Rhythmic Analogues"
http://weber.u.washington.edu/~pnm/CLAMPITT.pdf
In their process of constructing dual scales, they derive an ambigous scale
as the dual of the diatonic.
In cents:
0, 200, 350, 500, 700, 850, 1050, 1200
In JI:
1/1 9/8 sqrt(3/2) 4/3 3/2 sqrt(8/3) sqrt(27/8) 2/1
It can also be taken as the geometric mean of comparable notes in the
standard harmonic major and minor.

------------------------------------------
Kees van Prooijen
email: kees@dnai.com
web: http://www.dnai.com/~kees

-----Original Message-----
From: John A. deLaubenfels <jadl@idcomm.com>
To: tuning@onelist.com <tuning@onelist.com>
Date: Friday, April 23, 1999 6:45 AM
Subject: [tuning] the ambiguous triad

>From: "John A. deLaubenfels" <jadl@idcomm.com>
>
>
>Has anyone played with the ambiguous triad, made by cutting a perfect
>fifth into two equal parts? Its third degree is halfway between 6/5
>and 5/4, right in no-man's land. With a backbone of the solid fifth,
>the chord doesn't sound bad; it sounds ... ambiguous.
>
>With two equal third intervals, each 350.98 cents, it's close to a pair
>of 11/9 intervals (347.41 cents each), but this is not really a JI
>chord.
>
>Since I have been off in my own coocoon for most of my musical life, I
>half expect there to be a deluge of references to work previously done
>with this chord. I think it has possibilities...
>
>JdL
>

Kees van Prooijen wrote:

>
> Seeing the definition of Ervin Wilson's MOS I remembered an article by
> Norman Carey and David Clampitt dealing with related matters.

If you are referring to the one I put up a few days ago, this was my
understanding and I take the blame for any misunderstanding. I believe
Wilson's work predates the others and can be seen as part of the Wilson
Archive at site listed below!
-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com

Kraig

I don't think there is a misunderstanding. You just triggered me looking at
that paper. And you are absolutely right about Wilson predating their paper.
That fact is even recognised by the authors.

Kees

-----Original Message-----
From: Kraig Grady <kraiggrady@anaphoria.com>
To: tuning@onelist.com <tuning@onelist.com>
Date: Friday, April 23, 1999 10:29 AM
Subject: [tuning] Re: the ambiguous triad

>From: Kraig Grady <kraiggrady@anaphoria.com>
>
>
>
>Kees van Prooijen wrote:
>
>>
>> Seeing the definition of Ervin Wilson's MOS I remembered an article by
>> Norman Carey and David Clampitt dealing with related matters.
>
>If you are referring to the one I put up a few days ago, this was my
>understanding and I take the blame for any misunderstanding. I believe
>Wilson's work predates the others

John A. deLaubenfels wrote:

> Has anyone played with the ambiguous triad, made by cutting a perfect
> fifth into two equal parts? Its third degree is halfway between 6/5
> and 5/4, right in no-man's land. With a backbone of the solid fifth,
> the chord doesn't sound bad; it sounds ... ambiguous.

> Since I have been off in my own coocoon for most of my musical life, I
> half expect there to be a deluge of references to work previously done
> with this chord. I think it has possibilities...

Hi John,

I'm sure that there are a lot of people here who use this
interval a lot since it's present in (3/2)^(1/8).

I've composed with it a lot in my work in 88cET -
which I have been known to tune as (3/2)^(1/8) = 87.745cET.
(eighth root of fifth, can I call it 8rP5?)

I have called it the ambiguous third, the grey third,
and the neutral third.

It is an interesting interval since it sounds like some
kind of third but not any kind in particular. It doesn't
seem to want to move anywhere but doesn't give any trouble
if it does.

Harmonically, I've used it with the 3:2 and 7:4 found in 8rP5.
In 8rP5 you find three thirds:

87.745 cET 88cET
-----------------------------------
7/6 - 3.6 cents | 7/6 - 2.9 cents subminor third, septimal major third
11/9 + 3.6 cents | 11/9 + 4.6 cents neutral third, gray third, ambiguous third
9/7 + 3.6 cents | 9/7 + 4.9 cents supermajor third, septimal major third

That's one of the neat properties of 88cET/87.745cET -
you get three very useful thirds - including both septimal
thirds.

I recall being mostly interested in the 7:6 and 11:9 more
than the 9:7, which - now that I think of it - I thought
of as dissonant when I was working with those tunings.

You've caught me - for some reason I didn't like the
sharp 9:7 in these tunings as much as I did in the fifth-repeating-
stretched-just tuning I mentioned last week. Maybe it's context,
or maybe I've just learned to like 9:7 more since then. (1997 was
my big year for 88cET/87.745cET - since then I've been rolling all my own
nonoctave repeating tunings instead of relying upon the generosity
of others.)

I'm sure Gary and some of the others will have much more to say.

- Jeff

[Kees van Prooijen wrote:]
> In their process of constructing dual scales, they derive an ambigous
> scale as the dual of the diatonic. In cents:
> 0, 200, 350, 500, 700, 850, 1050, 1200

Perfect for 24-tET; ironically it tunes this slightly odd chord so much
more closely than it does a normal tried!

> In JI:
> 1/1 9/8 sqrt(3/2) 4/3 3/2 sqrt(8/3) sqrt(27/8) 2/1

Right. (Assuming that we allow sqrt(anything) to be called "JI"!).

> It can also be taken as the geometric mean of comparable notes in the
> standard harmonic major and minor.

Yes, about three calculations all lead to the same result.

[Jeff Scott wrote:]
> I've composed with it a lot in my work in 88cET -
> which I have been known to tune as (3/2)^(1/8) = 87.745cET.
> (eighth root of fifth, can I call it 8rP5?)

Interesting tuning...

> I have called it the ambiguous third, the grey third,
> and the neutral third.

I agree with these characterizations.

> It is an interesting interval since it sounds like some
> kind of third but not any kind in particular. It doesn't
> seem to want to move anywhere but doesn't give any trouble
> if it does.

I agree again.

> That's one of the neat properties of 88cET/87.745cET -
> you get three very useful thirds - including both septimal
> thirds.

But, you DON'T get the "normal" major and minor thirds, of course...

> I recall being mostly interested in the 7:6 and 11:9 more
> than the 9:7, which - now that I think of it - I thought
> of as dissonant when I was working with those tunings.

> You've caught me

Yes. You're busted. But I agree: 11:9 is much nicer to the ear than
9:7.

JdL

[Gary Morrison wrote:]
> I'm glad to hear discussions about the neutral third, probably best
> characterized as 11:9 or thereabouts. I think that "ambiguous" is
> indeed good way to describe it.

> In particular, I've found that we are so conditioned to think of
> thirds as either major or minor that when we hear something right
> between the two, our ears have a tendency from habit, to characterize
> it as one or the other. The option to portray that kind of ambiguity
> can be a useful to tool for composers; it makes listeners reconsider
> the basis behind our impressions of majors and minors.

I agree.

> I find stacks of four, five, and more neutral thirds more interesting
> than just neutral triads. That's true of 9:7s as well.

Interesting; I'll have to try that!

> I have toyed with the idea of expanding an 88-cent interval size to
> somewhere around 88 1/4 cents to make it less like 41 per
> triple-octave.

I'm afraid it's not clear to me why it's important to miss an n-octave.

> Jeff mentioned not liking 9:7 as much as 11:9 and 7:6. As I've
> mentioned before on the list, I think the key to understanding 9:7 is
> to avoid thinking of it as a major-like third, but instead as a
> shocking or perhaps "freaky"-sounding third instead.

That's about how I'd describe it!

JdL