Re: An augmented baker's dozen

Carl Lumma gave these 'root position' augmented triads:

7:9:11
8:10:13
12:15:19
16:20:25

in a post to Paul Erlich ("As for the augmented triad, to which just
version do you "prefer" 3tET...").

I'm wondering if the general consensus (though I somehow doubt there is
one...) finds these JI versions best expressed in their simplest sequential
terms (i.e., as Carl gave them...), or if any of their 'inversions' seem to
better express the augmented triad... (I personally find it very appealing
that one triad can have so many varied interpretations vying for its
'legitimacy...')

Dan

Dan!
I have always been amazed at how good the 7-9-11 sounds (and its inversions)
it is one of the reasons that I tend to observe that consonance is not based on
either odd or prime limits but on "coincidence" within difference tones. In this
case 2,2 and 4! to my ear the 10-13-16 sounds better than 8-10-13. In general
though the terms maj. min. aug. and dim are too removed from my musical language
to preserve them.

"D. Stearns" wrote:

> From: "D. Stearns" <stearns@capecod.net>
>
> Carl Lumma gave these 'root position' augmented triads:
>
> 7:9:11
> 8:10:13
> 12:15:19
> 16:20:25
>
> in a post to Paul Erlich ("As for the augmented triad, to which just
> version do you "prefer" 3tET...").
>
> I'm wondering if the general consensus (though I somehow doubt there is
> one...) finds these JI versions best expressed in their simplest sequential
> terms (i.e., as Carl gave them...), or if any of their 'inversions' seem to
> better express the augmented triad... (I personally find it very appealing
> that one triad can have so many varied interpretations vying for its
> 'legitimacy...')
>
> Dan
>
>

-- Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com

Dan Stearns wrote,

>>7:9:11
>>8:10:13
>>12:15:19
>>16:20:25

>I'm wondering if the general consensus (though I somehow doubt there is
>one...) finds these JI versions best expressed in their simplest sequential

>terms (i.e., as Carl gave them...), or if any of their 'inversions' seem to

>better express the augmented triad... (I personally find it very appealing
>that one triad can have so many varied interpretations vying for its
>'legitimacy...')

In the meantone era, when augmented triads first arose, 16:20:25 and its
inversions would have been the only possibilities (assuming 1/4-comma
meantone; otherwise we're talking about very close approximations). I
believe Margo Schulter found or perhaps guessed that 20:25:32 was the
preferred position in those days; I think I've seen 16:20:25 in a Gesualdo
score . . . Any Renaissance experts out there know which was more common?

Kraig Grady wrote,

>I have always been amazed at how good the 7-9-11 sounds (and its
inversions)
>it is one of the reasons that I tend to observe that consonance is not
based on
>either odd or prime limits but on "coincidence" within difference tones.

The two concepts are not mutually exclusive, of course, but certainly
difference tones point to a huge preference for otonal over utonal
sonorities while "limits" treat them equally (I've discussed this quite
extensively). With metallophone timbres such as the ones Kraig uses, utonal
chords don't have any consonance-maximizing properties, while otonal chords
have plenty (due to difference tones, other combination tones, virtual
pitches, and the interactions between all these). Wilson's CPS scales tend
to favor otonal and utonal chords equally; of course, one can use these
scales and treat the utonal chords as dissonances; but then why not use more
otonally-based scales, which are bound to contain plenty of resources for
dissonance anyway? (I think Kraig is about to describe such scales for us,
as well as their utonal versions.)

> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>

> In the meantone era, when augmented triads first arose, 16:20:25 and its
> inversions would have been the only possibilities (assuming 1/4-comma
> meantone; otherwise we're talking about very close approximations). I
> believe Margo Schulter found or perhaps guessed that 20:25:32 was the
> preferred position in those days; I think I've seen 16:20:25 in a Gesualdo
> score . . . Any Renaissance experts out there know which was more common?

No Ren expertise here, but what about 25:32:40? or would that position be even
too weird for Gesualdo?

D.S.

"Paul H. Erlich" wrote:

> i wrote;
> >it is one of the reasons that I tend to observe that consonance is not
> based on
> >either odd or prime limits but on "coincidence" within difference tones.
>
> The two concepts are not mutually exclusive, of course, but certainly
> difference tones point to a huge preference for otonal over utonal
> sonorities while "limits" treat them equally (I've discussed this quite
> extensively).

The idea of difference tones being a major player goes back to Helmholtz and his
examination of the inversions of the major and minor chord. It seemed a good
starting point.
I was hoping to put up a composition Lullaby that goes from the most
consonance inversions to the most dissonant over a 30min. time. (This piece will
be up as soon as break the code of a few problems, a few days). I wrote for my
daughter to play to her when she was first born to expose her these sounds before
the 12et matrix got a hold of her.More on this later. Having tried a few formulas
over the preceeding 3 years, the best one seem to be adding up the harmonic
analysis of each chord, plus the first order differance tones, omitting any
duplications of unisons. Each chord is then given a number. At the time, I could
guess exactly what number would make my girl friend really tense. It became a
game and was pretty consistent no matter how i snuck up on it. We would laugh.
Anyway when this gets put up people could try as see if they agree or not or if
those around them have the same threshold!

Here is what I don't like or question about this system though.
1) Are the numbers abitrary? for instance is 7 , 2 and a 1/3 more dissonant than
3, or is 4 four times more dissonant than 1? I thought of the square root of
each, but in the end, the order would remain the same.Try it
2) As you would say, the number of the subharmonic versions seems to be higher
than the percieved dissonance. Possibly including the first 12 partials of each
tone of a chord (or determined by actual timbre) might be added which might
balance this out with the subharmonic chords having a high coincidence of higher
harmonics (coinciding with combination tones). Ideally the the difference tones
could be determined by their presence in volume. Maybe with louder sounds the
secound order differance tones would have to be included.etc.
3. Range and timbre are not included.
4. as Ernst Toch pointed out, disonance is often determined by context. with an
f# Major chord sounding pretty dissonant in a C pan-diatonic piece.

Put side by side, otonalities seem more consonant than their subharmonic
counterparts. Regardless, much is lost in treating all inversions as equally
con/dis. Helmholz steered us away from this i thought.
Musically though the U's, have a depth and completeness,while the O's maybe more
simple and perfect. With 12et I always preferred the minor chord so it is no
suprise my preference for subharmonics.

> With metallophone timbres such as the ones Kraig uses, utonal
> chords don't have any consonance-maximizing properties, while otonal chords
> have plenty (due to difference tones, other combination tones, virtual
> pitches, and the interactions between all these). Wilson's CPS scales tend
> to favor otonal and utonal chords equally; of course, one can use these
> scales and treat the utonal chords as dissonances; but then why not use more
> otonally-based scales, which are bound to contain plenty of resources for
> dissonance anyway? (I think Kraig is about to describe such scales for us,
> as well as their utonal versions.)

I hope to put i up my Xenharmonikon article on the CPS's. A small point though,
Wilson does not concider the CPS's as scales but as harmonic constructs. He uses
the term scale only in reference to a melodic construct as outlined in his Moment
of Symetries.

As the CPS's shine best when used in a non tonic context, the arrangment of
harmonies based on tension and release is less advantagious. It seems that a
certain contrast of con/dis is necessary for it to be percieved in a meaningful
way. It is no wonder that the seventh was added to the dominant to heighten this
contrast.

-- Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com

>Wilson's CPS scales tend to favor otonal and utonal chords equally;

Only for CPS's where k:n is 1:2 are otonal and utonal resources equal. Ah- actually, for all k of any n (power set of n), they are equal, that's true...

>of course, one can use these scales and treat the utonal chords as
>dissonances; but then why not use more otonally-based scales, which
>are bound to contain plenty of resources for dissonance anyway?

For bells, sure. But for many harmonic timbres, I find the utonal chords beautifully dark consonances thru the 9-limit. And unsaturated utonal chords, which come naturally from the CPS's, can be consonances to the 17-limit, I think.

>(I think Kraig is about to describe such scales for us, as well as their >utonal versions.)

Please do, Kraig!

>With 12et I always preferred the minor chord so it is no suprise my >preference for subharmonics.

Actually, Kraig, I am beginning to believe that the 12tET minor triad approximates 16:19:24 more often than 10:12:15. You and Wilson both told me about the 19-limit approximations in 12-tone... I think you guys were right.

-C.

Carl Lumma wrote:

>
>
> Actually, Kraig, I am beginning to believe that the 12tET minor triad approximates 16:19:24 more often than 10:12:15. You and Wilson both told me about the 19-limit approximations in 12-tone... I think you guys were right.
>
> -C.

I think Tom stone was one of the first to hear this one, if not the source.

>(I think Kraig is about to describe such scales for us, as well as their >utonal versions.)

Please do, Kraig!

I'm not really sure what you both want described.?

-- Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com

Kraig Grady wrote,

>As you would say, the number of the subharmonic versions seems to be higher
>than the percieved dissonance. Possibly including the first 12 partials of
each
>tone of a chord (or determined by actual timbre) might be added which might
>balance this out with the subharmonic chords having a high coincidence of
higher
>harmonics (coinciding with combination tones).

In subharmonic chords, the higher harmonics do coincide, but they don't
coincide with combination tones.

>Maybe with louder sounds the
>secound order differance tones would have to be included.etc.

There is one second-order difference tone (aka cubic difference tone) that
is the loudest combination tone for _quieter_ sounds.

>With 12et I always preferred the minor chord so it is no
>suprise my preference for subharmonics.

I would think that for a keen-eared high-limit just intonation enthusiast,
the 12-tET minor triad would sound like a nearly exact 16:19:24, which has a
clear root, while a 12-tET major triad would sound like a very out-of-tune
4:5:6.

Carl Lumma wrote,

>>Wilson's CPS scales tend to favor otonal and utonal chords equally;

>Only for CPS's where k:n is 1:2 are otonal and utonal resources equal. Ah-
actually, for all k of any n (power set >of n), they are equal, that's
true...

I meant generally. Clearly some CPSs favor otonality, while others favor
utonality. What does your last sentence mean?

>Actually, Kraig, I am beginning to believe that the 12tET minor triad
approximates 16:19:24 more often than
>10:12:15. You and Wilson both told me about the 19-limit approximations in
12-tone... I think you guys were >right.

I guess my comment to that effect came too late.

Kraig Grady wrote,

>I'm not really sure what you both want described.?

Look at your post in the digest before mine. Something with "helix" in it, I
believe?