tunings that highlight C-G-C# and similar collections

I'm looking to write piece that uses the 3-note pitch-collection that I
dislike the most (I like to make myself compose for instruments or
materials that I hate. . . I like to learn to love things).

the famous "Viennese" trichord in this case: C-G-Db, or B-C-F#, etc.
A tri-tone plus a 4th.

Are there tunings that offer interesting interpretations of this trichord?
or maximize such-and-such? Just, ET, meantone, whatever, I'd be
interested in it.

Any ideas?

C Bailey

http://anaphoria.com/firstpelog.GIF
this is the closest i can thing of off the top of my head although i know other recurrent sequences should work also

Christopher Bailey wrote:

>I'm looking to write piece that uses the 3-note pitch-collection that I >dislike the most (I like to make myself compose for instruments or >materials that I hate. . . I like to learn to love things).
>
>the famous "Viennese" trichord in this case: C-G-Db, or B-C-F#, etc.
>A tri-tone plus a 4th.
>
>Are there tunings that offer interesting interpretations of this trichord? >or maximize such-and-such? Just, ET, meantone, whatever, I'd be >interested in it.
>
>
>Any ideas?
>
>
>C Bailey
>
>
>
>
> >Yahoo! Groups Links
>
>
>
> >
>
>
>
> >

--
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

> http://anaphoria.com/firstpelog.GIF
> this is the closest i can thing of off the top of my head although i
> know other recurrent sequences should work also

Ah, another mysterious Erv Wilson diagram.

So, is the idea that 1.366054etc. is a generator ratio that will make a
scale with lots of 4th/tritone sonotiries?
Obviously, the 5th is 1.5, and the 4th is 1.3333, so this is in
between. Though it seems closer to the 4th . . .

Anyways, then I'm not sure about the bottom part of the diagram, where
that's a'comin' from.

but thanks, this is food for thought.

C Bailey

The recurrent sequence it self, the procession of harmonics as in 8-11- 15 is really where the real meat somehow seems to be. it does converge on a definite size by practice and observation has lead many of us to conclude that those spaces in between before the harmonics actually converge as the point where these scales sounds the best. The lower diagram show the moments of symmetry of the series. I am sure there might be a better answer , Erv has gone through the first 200 diagonals and calculated the recurrent sequence and it convergence.
The chord you mention comes up in Stravinsky's 'Song of the Nightingale' as the generating chord. it is one of my favorites of Igor if not the most so.

Christopher Bailey wrote:

> >
>>http://anaphoria.com/firstpelog.GIF >> this is the closest i can thing of off the top of my head although i
>>know other recurrent sequences should work also >> >>
>
>
>Ah, another mysterious Erv Wilson diagram.
>
>So, is the idea that 1.366054etc. is a generator ratio that will make a
>scale with lots of 4th/tritone sonotiries? >Obviously, the 5th is 1.5, and the 4th is 1.3333, so this is in >between. Though it seems closer to the 4th . . .
>
>
>Anyways, then I'm not sure about the bottom part of the diagram, where >that's a'comin' from.
>
>but thanks, this is food for thought.
>
>C Bailey
>
>
>
>
> >Yahoo! Groups Links
>
>
>
> >
>
>
>
> >

--
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

Christopher Bailey escriba:

> I'm looking to write piece that uses the 3-note pitch-collection that I
> dislike the most (I like to make myself compose for instruments or
> materials that I hate. . . I like to learn to love things).
>
> the famous "Viennese" trichord in this case: C-G-Db, or B-C-F#, etc.
> A tri-tone plus a 4th.
>
> Are there tunings that offer interesting interpretations of this trichord?
> or maximize such-and-such? Just, ET, meantone, whatever, I'd be
> interested in it.
>
>
> Any ideas?

A Viennese trichord would be C-Db-Gb in the key of C, or B-C-F in B. (I like that chord; I've used it more than a few times in fact. Also try adding an additional fourth above the top note, i.e. a diminished octave: C-Db-Gb-Cb or B-C-F-Bb.)

In JI, that would be a tricky one, since we're dealing with Schoebergian 12-tone ideas. C-Db-Gb could also be C-C#-F#. In meantone, the tuning for each would be different.

In 12-tone, the set would be {0, 1, 6}. In 31-tone, the "flat" version would be {0, 3, 16}, while the "sharp" version would be {0, 2, 15}. You'd have to listen to both and see which one you're looking for. In 19-tone, the former is {0, 2, 10}; the latter {0, 1, 9}.

And it gets more complicated in JI and extended Pythagorean (and by implication, 41- and 53-tone), because converted 12-tone music often engages in "comma floating", potentially requiring adaptive JI practive. I only expect the third note to be a perfect fourth higher than the second, but it doesn't have to be if you want some really disturbing harmonization ;). In 53-tone, a minor second tends to be 5 degrees, or "commas" in common terms for that tuning, but 4-comma semitones also occur. Augmented primes can be 3 or 4 commas. A perfect fourth is always 22 commas, or else it's not very perfect....

In 5-limit JI, you can have {1/1, 16/15, 64/45}, {1/1, 135/128, 45/32} or {1/1, 27/25, 36/25}.

Good luck!

hi Christopher and Danny,

--- In MakeMicroMusic@yahoogroups.com,
"Danny Wier" <dawiertx@s...> wrote:

> Christopher Bailey escriba:
>
> > I'm looking to write piece
> > that uses the 3-note
> > pitch-collection that I
> > dislike the most (I like
> > to make myself compose for
> > instruments or materials
> > that I hate. . . I like
> > to learn to love things).
> >
> > the famous "Viennese"
> > trichord in this case:
> > C-G-Db, or B-C-F#, etc.
> > A tri-tone plus a 4th.
> >
> > Are there tunings that
> > offer interesting interpretations
> > of this trichord?
> > or maximize such-and-such?
> > Just, ET, meantone, whatever,
> > I'd be interested in it.
> >
> >
> > Any ideas?
>
> <snip>
>
> In 12-tone, the set would be {0, 1, 6}. In 31-tone,
> the "flat" version would be {0, 3, 16}, while the "sharp"
> version would be {0, 2, 15}. You'd have to listen to
> both and see which one you're looking for. In 19-tone,
> the former is {0, 2, 10}; the latter {0, 1, 9}.
>
> <snip>

here are cents measurements for some of the ET meantones:

ET: ... 12 ... 19 .... 31 .... 43 ..... 55

G:C# _ 600 _ 568.4 _ 580.6 _ 586.05 _ 589.09
G:Db _ 600 _ 631.6 _ 619.4 _ 613.95 _ 610.9
C:G __ 700 _ 694.7 _ 696.8 _ 697.7 __ 698.18

(i measured these with the C# and Db in the higher octave)

-monz

I wrote earlier:

> In 5-limit JI, you can have {1/1, 16/15, 64/45}, {1/1, 135/128, 45/32} or
> {1/1, 27/25, 36/25}.

The last pitch-set is a mistake; I meant to say {1/1 25/24 25/18}. (What I did give you is more like B-Dbb-Gbb, and that is another possibility. In 53-tone, that would be {0 6 28}.)

And a 7-limit alternative for C-C#-F# is {1/1 28/27 112/81}.

~Danny~

of course the other rather obvious option that we all seem to have glossed over is to find a good just version of this trichord and create a scale by making a chain of them. of course you could find the worse intonation for this chord you could find too , and put it in the chain you hate the most
--
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

Christopher,
I see those intervals all over the 15 limit tonality diamond.

Consider the diamond on C, where the overtones are 4:5:6:7:9:11:13:15, and the undertones are 1/(6:7:8:9:10:11:12). See a picture at http://prodgers13.home.comcast.net/Sub_Pages/diamond-15-inc.gif

Take 9:14:20 of the C overtone series, which can be called D 9/8, B 7/4, E 5/4. Not exactly what you were talking about, but a nice "pun" on those intervals. You have an interval of 14:9, close to 3:2, and 10:7, very close to the 64:45 that I think you mean by the C#. It has a nice edge to it. Add a C in the base, and suddenly it sounds like part of an overtone series. Swap the D 9/8 for the E 5/4 and you have a nice jazzy ninth chord, with the C in the bass.

You can do the same with lots of other diamond intervals. Try this one: G 3/2 up by 3:2 to D 9/8, then up by a 13:9 to A 13/8. This is even closer to the C-G-C#.

Then look at the utonality. The best I can do on the C undertone series is F 4/3 up by 3:2 to C1/1 then up by 16:11 to G 16/11. Just a slight amount off from your C-G-C#. There are about five others in that scale that are very close.

When you compose with the tonality diamond, you have the added advantage of different colors for every modulation within the scale. The color of a D 9/8, B 7/4, E 5/4 is far different from the color of G 3/2, D 9/8, A 13/8. Take advantage of the color change to shift the moodes. Then you can modulate to another step of the tonality diamond and have the same color, but a different pitch. There's a whole world between the 2:1!

Prent Rodgers

--

Prent Rodgers
Mercer Island, WA

Music that's "Fake but Accurate"!
Web page: http://prodgers13.home.comcast.net
Podcast: http://podcast1024.blogspot.com
Another Podcast: http://BumperMusic.blogspot.com
Music: http://www.soundclick.com/PrentRodgers

>
>of course the other rather obvious option that we all seem to have
>glossed over is to find a good just version of this trichord and create
>a scale by making a chain of them. of course you could find the worse
>intonation for this chord you could find too , and put it in the chain
>you hate the most
>

True enough.

Actually, though, if I'm gonna go the Just route, might as well create a
scale with varying intervals, so that every "attempt to approximate" this
trichord contains a different interval set. . . that's what I'm after I
think. Imperfection, so to speak.

and I am interested in this as different kinds of sounds.. . some with
"tonal implications", others without. . .C F B, C Cb F#, etc.

CB

--- In MakeMicroMusic@yahoogroups.com, Christopher Bailey <chris@m...>
wrote:

>
> Actually, though, if I'm gonna go the Just route, might as well
create a
> scale with varying intervals, so that every "attempt to
approximate" this
> trichord contains a different interval set. . . that's what I'm
after I
> think. Imperfection, so to speak.
>
> and I am interested in this as different kinds of sounds.. . some
with
> "tonal implications", others without. . .C F B, C Cb F#, etc.

Then I recommend looking again at the Erv Wilson pages that Kraig put
up. These "zig-zags" are convergent series based upon simple formulas
(I believe that Wilson found them initially in paths down Pascal's
triangle), with segments of the series immediately interpretable as
consecutive generating intervals in a scale. Scales generated in this
way have some nice features: some generate difference tones that are
themselves scale tones, some resemble familiar tunings (meantones,
pelogs, slendros etc.), most are expressed as rational numbers, making
them suitable for just intonation. Based on your description of your
project, any series converging on a value between 1.4 (or thereabouts)
and 1.5 should do the trick for you.

Daniel Wolf