Re: propriety graph

>>I'd always imagined two other measures, that could work together, 1. What
>>percent of pitch-class space is filled? 2. What percent of pitch class
>>space is more than singly-filled? It may be best to formulate this measure
>>graphically, rather than worry about which numbers to show. If you take a
>>tonality diamond, and look down the diagonals, you have the pitch classes.
>>One could imagine a graph where each of the pitch classes would be
>>color-coded strips, with overlapping areas florescent orange, or so. A
>>little practice looking at these and the user would probably be able to
>>tell more about the melodic properties of a scale, faster, than anybody
>>else.
>
>I don't get it.

I wasn't too clear. I'm after a graphic representation of the Rothenberg interval matrix (tonality diamond). Imagine a line segment representing an octave, pitch ascending from left to right, on a log scale. For each scale degree, plot a point for each occurance in each mode and color a segment connecting all the points. The measures are then: 1. What percent of the line is colored? 2. What percent of the line is more than singly covered? It might be better to ask the second question in terms of the first, 2b. What percent of the colored space contains overlaps? But as I said, just looking at the thing will probably tell the most.

One could even keep the modes distinct with a line graph version. Pitch is on the y axis, and on the x axis, at equally-spaced intervals, are the scale degrees. Then the modes can be shown as lines on the graph by connecting the right dots at each scale degree. If a pair of line cross, there's impropriety. The overlap could be colored in, and the area could be used as length was in the first version.

-C.

Carl Lumma wrote,

>I'm after a graphic representation of the Rothenberg interval matrix
(tonality diamond).

Those are not really the same thing. There may be similarities but don't
take them too far. The tonality diamond according to Partch is a set of
pitches, not intervals. However, the earliest depiction of a "tonality
diamond" may have been Max Meyer's in 1929 (rotated 90 degrees with respect
to Partch's construct) which was indeed an interval matrix.

>Imagine a line segment representing an octave, pitch ascending from left to
right, on a log scale. For each scale >degree, plot a point for each
occurance in each mode and color a segment connecting all the points. The
>measures are then: 1. What percent of the line is colored? 2. What percent
of the line is more than singly >covered? It might be better to ask the
second question in terms of the first, 2b. What percent of the colored
>space contains overlaps? But as I said, just looking at the thing will
probably tell the most.

I think I see this, but it would be nice to see some pictures.

>One could even keep the modes distinct with a line graph version. Pitch is
on the y axis, and on the x axis, at >equally-spaced intervals, are the
scale degrees. Then the modes can be shown as lines on the graph by
>connecting the right dots at each scale degree. If a pair of line cross,
there's impropriety.

The converse is not necessarily true though, is it?

>The overlap could be >colored in, and the area could be used as length was
in the first version.

Wouldn't that depend too strongly on what order you plotted the modes in?

Carl,

I don't think I'll get it until you draw one.

-- Dave Keenan
http://dkeenan.com

>>I'm after a graphic representation of the Rothenberg interval matrix
>>(tonality diamond).
>
>Those are not really the same thing. There may be similarities but don't
>take them too far. The tonality diamond according to Partch is a set of
>pitches, not intervals.

That's just a question of use. The two are the same, with the exception
that the tonality diamond is usually applied only to odd harmonics.

>However, the earliest depiction of a "tonality diamond" may have been Max
>Meyer's in 1929 (rotated 90 degrees with respect to Partch's construct)
>which was indeed an interval matrix.

I think the thing's origin is lost in antiquity.

>>Imagine a line segment representing an octave, pitch ascending from left
>>to right, on a log scale. For each scale degree, plot a point for each
>>occurance in each mode and color a segment connecting all the points. The
>>measures are then: 1. What percent of the line is colored? 2. What
>>percent of the line is more than singly covered? It might be better to
>>ask the second question in terms of the first, 2b. What percent of the
>>colored space contains overlaps? But as I said, just looking at the thing
>>will probably tell the most.
>
>I think I see this, but it would be nice to see some pictures.

I'll get some.

>>One could even keep the modes distinct with a line graph version. Pitch
>>is on the y axis, and on the x axis, at equally-spaced intervals, are the
>>scale degrees. Then the modes can be shown as lines on the graph by
>>connecting the right dots at each scale degree. If a pair of line cross,
>>there's impropriety.
>
>The converse is not necessarily true though, is it?

That's a good question. It seems not, but it may be. I'll get back to you.

>>The overlap could be colored in, and the area could be used as length was
>>in the first version.
>
>Wouldn't that depend too strongly on what order you plotted the modes in?

I don't think you have a choice as far as the order of the modes. The
modes are the lines on the graph...

|
P |
| ,-----.
I | ___./ __.
| / _.----'
T | .--/--/
| /_./
C | /_./
| _/__.
H | _/
| ./
+-------------------------------------
1sts 2nds 3rds 4ths

>>>One could even keep the modes distinct with a line graph version. Pitch
>>>is on the y axis, and on the x axis, at equally-spaced intervals, are the
>>>scale degrees. Then the modes can be shown as lines on the graph by
>>>connecting the right dots at each scale degree. If a pair of line cross,
>>>there's impropriety.
>>
>>The converse is not necessarily true though, is it?
>
>That's a good question. It seems not, but it may be. I'll get back to you.

Indeed lines may cross on the graph of a proper scale. I believe that all
improper scales will have crossing lines, but I'm not sure. In any case,
it isn't crossing that indicates propriety.

What does indicate propriety is the slope of the lines. They've got to go
up to be strictly proper. If two intervals are connected by a line
parallel to the x-axis, then they are "ambiguous". If any go down (from
left to right), there is impropriety.

Here's an Excel 97 spreadsheet of the pentachordal major decatonic scale in
22tET...

http://lumma.org/pentmaj.xls

There are two charts. The larger one is as I described above. The smaller
chart does show impropriety by line crossing. I don't think either of them
are as useful as my orginal, 1D idea. Examples of this type are on the way...

-C.

Carl Lumma wrote,

>>>I'm after a graphic representation of the Rothenberg interval matrix
>>>(tonality diamond).
>
>>Those are not really the same thing. There may be similarities but don't
>>take them too far. The tonality diamond according to Partch is a set of
>>pitches, not intervals.

>That's just a question of use. The two are the same, with the exception
>that the tonality diamond is usually applied only to odd harmonics.

Meyer may have used the tonality diamond as a "Rothenberg interval matrix"
of the odd harmonics, but Partch, who named it "tonality diamond", certainly
did not.