transposition question thing

>
> That's also what I'd expect they'd do! Assuming you meant A=5/3, I've
> taken the liberty of doing up a version showing relative ratios on the
> left of each chord, and common tones with ----, showing cents difference
> in the adjusted notes. . .
>
> 9/5 C 1/1 5 B 15/8 ------ 15 B 15/8
> 3 A 5/3 1 G 3/2 -------- 3 G 3/2
> 6/5 F 4/3 -(-27)-- 7 F- 21/16 5 E 5/4
> 1 D 10/9 -+22--- 3 D+ 9/8 1 C 1/1
>

...and why not

7/4 C 63/64 5 B 15/8 15 B 15/8
3 A 27/16 1 G 3/2 3 G 3/2
7/6 F 21/16 7 F 21/16 5 E 5/4
1 D 9/8 3 D 9/8 1 C 1/1

...or

16/9 C 1/1 5 B 15/8 15 B 15/8
3 A 27/16 1 G 3/2 3 G 3/2
32/27 F 4/3 16/9 F 4/3 5 E 5/4
1 D 9/8 3 D 9/8 1 C 1/1

...or

9/5 C 1/1 5 B 15/8 15 B 15/8
3 A 5/3 1 G 3/2 3 G 3/2
6/5 F 4/3 16/9 F 4/3 5 E 5/4
1 D 10/9 -+22--- 3 D+ 9/8 1 C 1/1

Each one has different warps which may be more or less
offensive depending on what the goal is. I tend to think
that the second option I pose is the most likely... It
doesn't require any drift, there is good tuning at the 3^n
limits, and I don't think the 7/4 on a dominant chord
is that important. [I would contend that modal useage
of the seventh chord is different, including 'the blues'].

Bob Valentine

>From: Robert C Valentine <bval@iil.intel.com>
>
>>
>> That's also what I'd expect they'd do! Assuming you meant A=5/3, I've
>> taken the liberty of doing up a version showing relative ratios on the
>> left of each chord, and common tones with ----, showing cents difference
>> in the adjusted notes. . .
>>
>> 9/5 C 1/1 5 B 15/8 ------ 15 B 15/8
>> 3 A 5/3 1 G 3/2 -------- 3 G 3/2
>> 6/5 F 4/3 -(-27)-- 7 F- 21/16 5 E 5/4
>> 1 D 10/9 -+22--- 3 D+ 9/8 1 C 1/1
>>
>
>...and why not
>
> 7/4 C 63/64 5 B 15/8 15 B 15/8
> 3 A 27/16 1 G 3/2 3 G 3/2
> 7/6 F 21/16 7 F 21/16 5 E 5/4
> 1 D 9/8 3 D 9/8 1 C 1/1

Because that D-7 would sound out of tune, not having the F triad that is
the basic subdominant chord in the key of C. Even the *note* F is different
from the 4/3 root of the basic subdominant triad. Also, I believe that
C=63/64 would tend to loose the tonality of C=1/1. That is, if you sang D-7
that way, I don't think it would be likely to get to the next chords as you
indicate. Instead you might land in G7 with G=(3/2)*(63/64) and B, D, and F
all lowered by 63/64, and then Cmaj7 would also be 63/64 of what you
indicate.

>...or
>
> 16/9 C 1/1 5 B 15/8 15 B 15/8
> 3 A 27/16 1 G 3/2 3 G 3/2
> 32/27 F 4/3 16/9 F 4/3 5 E 5/4
> 1 D 9/8 3 D 9/8 1 C 1/1

Well, D-7 has fifths in tune but thirds out, so the F triad that is the
basic subdominant chord in C major is out of tune. The G7 is possible but
you'd but I don't think good singers would produce that F.

>...or
>
> 9/5 C 1/1 5 B 15/8 15 B 15/8
> 3 A 5/3 1 G 3/2 3 G 3/2
> 6/5 F 4/3 16/9 F 4/3 5 E 5/4
> 1 D 10/9 -+22--- 3 D+ 9/8 1 C 1/1
>

This is what I expect except for the F in G7, for the same reason as in the
example above.

>Each one has different warps which may be more or less
>offensive depending on what the goal is.

I agree that the different options you listed would be more or less
offensive, but is not necessary to make ANYTHING offensive in this simple
progression!

>I tend to think
>that the second option I pose is the most likely... It
>doesn't require any drift,

You mean it doesn't require any *flexibility*. None of the examples drift
(except that if you sang D-7 as in the first example you might not complete
the sequence because of drift).

>there is good tuning at the 3^n
>limits, and I don't think the 7/4 on a dominant chord
>is that important.

I consider your third option to be the least offensive. I'd rather have my
singers do that than either of the others. But they'd still be out of tune!
If you actually SANG an F in G7 as 16/9 relative to C then I suspect that
you wouldn't make the cut at an audition for Gerald Eskelin's L.A. Jazz
Choir, or any good barbershop quartet.

> [I would contend that modal useage
>of the seventh chord is different, including 'the blues']

I'm not yet prepared to write about the blues, but I will sometime soon.

John Link
ALMOST ACAPPELLA

Robert C Valentine wrote,

>there is good tuning at the 3^n
>limits

What exactly do you mean by that?

>
> >there is good tuning at the 3^n
> >limits
>
> What exactly do you mean by that?
>

I am still undecided whether I believe
the existance of the universe is based on the overtone
series, prime limits, or odd limits. A slight
exageration...

I believe in octave equivalence, which conjures up
a vision of a strong 'pocket' at 2^n ratios.

Whether my thinking is based on overtones, prime or odd
limits, 3/2^n has a very strong pocket.

(Of course these results are right in line with the
harmonic entropy graphs you produced).

A prime limit thinker would say that "a 3 of a 3" will
also have a strong pocket. The first of these is 9/8
which is strong from an overtone series point of
view. [I suppose a test is to place tones between
9/8 and 8/7 and see if there is a stronger attraction
to one or the other].

So, my statement was basically saying, that lots of
fifths were in tune, and lots of "fifths of fifths" were
in tune, so the whole scale sat in a lot of overtone and
prime limit pockets.

The examples discussed in that thread may have been flawed
since we were discussing a scale as if it were being sung
or perceived by the singers as a cluster. Jim tended to
think they would tune all notes to an overtone series from
the root of the chord, I thought they would fall into the
various "fifths and major triads pockets" that were also
in the sound.

Bob Valentine

Robert Valentine wrote,

>Whether my thinking is based on overtones, prime or odd
>limits, 3/2^n has a very strong pocket.

>(Of course these results are right in line with the
>harmonic entropy graphs you produced).

I don't know what you mean. I have seen no evidence in any psychoacoustic
model of consonance, whether based on roughness, harmonic entropy, or
combination tones, to suggest that ratios of higher powers of low primes
could be more consonant that ratios of lower numbers based on higher primes.
Certainly one will find the former in many chords built mainly of consonant
intervals, but if you compare the intervals themselves, and really try to
listen for "objective" dissonance, I think you'll find that I'm right.

Robert Valentine wrote,

>Jim tended to
>think they would tune all notes to an overtone series from
>the root of the chord, I thought they would fall into the
>various "fifths and major triads pockets" that were also
>in the sound.

I would agree with you (Robert) there.