(2nd batch) Qs on Paul's "Tuning, Tonality, 22..."

Here are some more questions that I have in regards to this paper:

[page 1] After communicating with Paul, I realize that by "5 limit",
he really means "5 odd limit" which is specifically the set
of fractions
{ 1/1 6/5 5/4 4/3 3/2 8/5 5/3 2/1 }
I notice that this is missing Bb = 9/5 ... Is this a problem?

[page 5] There is the line
"For example, any meantone system has 4Q = T".
Shouldn't that be Q^4 = T?

[page 6] "A characteristic dissonance may share a note with the
tonic chord if, when played together, they form a
consonant chord of the next
higher limit ( 3 ==> 5, 5==> 7, 7 ==> 9)"
Could someone please give me an example of this?

[page 7] This is just an observation, but I had a lot of trouble
here until I drew the diatonic major scale as a circle
and then drew lines to represent all the intervals.
Ah, I see the characteristic dissonance b - f is 3L!
I wonder if this mental model will serve me well for
the rest of the paper? I seems to be breaking down
when I draw the corresponding one for the pentatonic
scale on page 8, but perhaps that is because I ran
out of coffee ...

[page 8] "The major ( s s L s L ) and minor ( L s s L s)
pentatonic modes..."
How do we identify those as major and minor?

thanks,
Jeff

I will just comment on p. 8's enquiry.

--- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...> wrote:
> Here are some more questions that I have in regards to this paper:
>
> >
> [page 8] "The major ( s s L s L ) and minor ( L s s L s)
> pentatonic modes..."
> How do we identify those as major and minor?
>
>
> thanks,
> Jeff

If we make s stand for short interval (2 semitones) and L for long
interval (3 semitones), we get (taking C as fundamental):

C-D-E-G-A-C
C-Eb-F-G-Bb-C

The first has of course a major flavour, and respectively, the second
one sounds minor. If examined with reference to the tonic, the modal
(eg, minor or major) intervals of each scale are major or minor,
respectively. For instance, the first scale lists

CD major 2nd
CE major 3rd
CG (perfect fifth)
CA major 6th
CC (perfect octave)

The second:
CEb minor 3rd
CF (perfect fourth)
CG (perfect fifth)
CBb minor 7th
CC (perfect octave)

To be more clear, let's take A as root for the second scale. Thus you
get:
Major CDEGAC
Minor ACDEGA

You could see that the second is a translation of the first by 3
semitones downwards. The distance 3ST is equivalent to the distance
between the minor-major relatives in heptatonic scales.

--- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...> wrote:

> Here are some more questions that I have in regards to this paper:
>
> [page 1] After communicating with Paul, I realize that by "5 limit",
> he really means "5 odd limit" which is specifically the set
> of fractions
> { 1/1 6/5 5/4 4/3 3/2 8/5 5/3 2/1 }
> I notice that this is missing Bb = 9/5 ... Is this a
>problem?

These fractions are not to be interpreted as pitches, but only as
intervals. For example the "scale" C-E-G-c-e-g-c'-e'-g', though
having only three pitches (per octave), contains all the intervals in
the set above.

9/5 would be in the "9 odd limit" and above but not in the "5 odd
limit" because the largest odd factor in the ratio is 9.

> [page 5] There is the line
> "For example, any meantone system has 4Q = T".
> Shouldn't that be Q^4 = T?

Not if you measure in scale degrees or generators, etc.

> [page 6] "A characteristic dissonance may share a note with the
> tonic chord if, when played together, they form a
> consonant chord of the next
> higher limit ( 3 ==> 5, 5==> 7, 7 ==> 9)"
> Could someone please give me an example of this?

Using the mixolydian mode on G, the characteristic dissonance b-f
shares a note with the tonic chord g-b-d and together they may
approximate the 4:5:6:7 chord (7-limit consonant chord).

>
> [page 8] "The major ( s s L s L ) and minor ( L s s L s)
> pentatonic modes..."
> How do we identify those as major and minor?

This is common terminology in all references I've seen -- on C, the
major pentatonic is C-D-E-G-A-(C), and the minor pentatonic is C-Eb-F-
G-Bb-(C).

--- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
<giordanobruno76@y...> wrote:
> I will just comment on p. 8's enquiry.
>
> --- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...>
wrote:
> > Here are some more questions that I have in regards to this paper:
> >
> > >
> > [page 8] "The major ( s s L s L ) and minor ( L s s L s)
> > pentatonic modes..."
> > How do we identify those as major and minor?
> >
> >
> > thanks,
> > Jeff
>
> If we make s stand for short interval (2 semitones) and L for long
> interval (3 semitones), we get (taking C as fundamental):
>
> C-D-E-G-A-C
> C-Eb-F-G-Bb-C
>
> The first has of course a major flavour, and respectively, the
second
> one sounds minor. If examined with reference to the tonic, the
modal
> (eg, minor or major) intervals of each scale are major or minor,
> respectively. For instance, the first scale lists
>
> CD major 2nd
> CE major 3rd
> CG (perfect fifth)
> CA major 6th
> CC (perfect octave)
>
> The second:
> CEb minor 3rd
> CF (perfect fourth)
> CG (perfect fifth)
> CBb minor 7th
> CC (perfect octave)
>
> To be more clear, let's take A as root for the second scale. Thus
you
> get:
> Major CDEGAC
> Minor ACDEGA
>
> You could see that the second is a translation of the first by 3
> semitones downwards. The distance 3ST is equivalent to the distance
> between the minor-major relatives in heptatonic scales.

Thanks for your reply, and sorry it has taken me so long to respond.
I just want to say that Margo Schulter has posted some interesting
insight into this paper in a current thread over in

http://groups.google.com/groups?group=rec.music.theory

--Jeff