Yahoo Tuning Groups Ultimate Backup tuning re: diatonicity

Hello all, Jacky, Paul,

>Could you explain the "action between the latter two" in laymens
>terms? When you say "results in diatonicity", am I understanding
>correctly that it is the interaction between the Rothenberg property
>and "partial positions" (perception of overtones in the timbre being
>used), is what results in the listener's perception of a scale's
>"diatonicity"?

Correct, except for the definition of "partial positions", and that's
my fault, for writing gibberish. I meant one level higher than timbre
here, a sort of meta-timbre. The perception of the position of the
fundamental of a melody note in the harmonic series of the fundamentals
of the other notes. Kapeesh? Idea is, you can simply play a harmonic
series over a drone and get a good, though primitive, sort of tune.

>Very interesting! Do explain more.

It isn't all that impressive, since I basically _define_ diatonicity
with it. Nevertheless, very few scales satisfy the definition, and I
like the ones that do. Here are all the ones I'm aware of:

o Standard meantone pentatonic
o Standard meantone diatonic
o Standard meantone harmonic minor
o Rimski-Korsakov octatonic, 12-tET (0 1 3 4 6 7 9 10)
o Blackwood decatonic, 15-tET (0 1 3 4 6 7 9 10 12 13)
o Erlich pentachordal decatonic, 22-tET (0 2 4 7 9 11 13 16 18 20)
o Erlich symmetrical decatonic, 22-tET (0 2 4 7 9 11 13 15 18 20)
o Keenan decatonic MOS, 29-tET (0 3 6 9 11 14 17 20 23 26)
o Lumma decatonic, 31-tET (0 4 6 10 12 16 18 22 25 28)
o Erlich double-diatonic, 26-tET (0 2 4 6 8 10 12 13 15 17 19 21 23 25)

The basic theme is that any time your brain/ear can follow something,
there's potential to exploit it for melody. Let's re-name and define
the three things now:

# NAME METRIC
---------------------------------------------------
01......Pitch tracking................Miller limit
02......Scalar interval tracking......Rothenberg propriety
03......Partial tracking..............Odd-limit

These conflict with each other in spots, and diatonicity is the
compromise that results -- that's what I meant by "action between
them". I can get diatonicity from 02 and 03, or from 01 and 03. Let's
do 02 and 03 first, since that's what you asked about.

___________________________________________________ __ __ ___ ___

I. Diatonicity = Scalar interval tracking + Partial tracking
_______________________________________________________________________

We already have "partial position" tracking down, and that gives us
harmonic series segments.

Rothenberg propriety is a metric for an _interval ranking_ process.
R. says we can measure intervals between pitches, and recognize
interval patterns by the order of the sizes of the intervals, ignoring
the absolute sizes of the intervals involved. This allows the modal
transposition of themes. It's how, says R., we can recognize the tune
Happy Birthday, when played in the natural minor.

6-12 is the highest complete octave in the harmonic series that's
proper. After that, a compromise must be made (some harmonics must be
left out). This is the beginning of the "action between the two" I was
talking about.

Further, we want to transpose our theme in harmonic numbers as well
as scale degree distances. Aside from the small amount of natural
repetitiveness in the harmonic series (i.e. 6:9 = 3:2), we'll need to
tweak our harmonic series scales so that different instances of a given
scalar interval can function as a single harmonic interval. Poof,
diatonicity!

Now, let's look at 01 and 03, and get the same result.

___________________________________________________ __ __ ___ ___

II. Diatonicity = Pitch tracking + Partial tracking
_______________________________________________________________________

Again, let's start by assuming partial tracking (I like starting with
partial tracking because I think we get the ability from listening to
speech). This time, though, we'll add pitch tracking instead of
scalar interval tracking...

>A little help with the Miller Limit would be of great help. Got a
>paper, post or web page on that one?

George Miller was a cognitive psychologist who wrote a famous paper on
short-term memory and perception. Basically, it says that people can,
on average, hold up to about 7 things at a time in their short-term
memory. What caused the stir was that this number should be the same
for many different types of things. We'll apply it to pitches. The
URL for the paper has been posted here a few times...

http://www.well.com/user/smalin/miller.html

...my original post on the subject was sent to the Mills server. I can
re-post it if anybody would like.

So Miller basically says we can remember and work with up to 7, give or
take, pitches at a time. So now, rather than just following a note in
a harmonic series, we can take a set of pitches, and assign each of
them a position in our imaginary harmonic series. This gives us common
tone modulation -- holding a pitch constant while changing its position
in a harmonic series. For example, with V-I in C-major, we have G
going from 1 to 3. You can even do more than one pitch at a time if
you go from major to minor, or if you use certain forms of temperament.

So, aside from the natural repetitiveness in the harmonic series (i.e.
6:9 = 3:2, etc.), we'll need to adjust our harmonic series segments to
allow pitches to change their harmonic functions. Poof, diatonicity.

(I'll try to work on filling in the poofs later.)

One thing about saying folks can only keep track of 7 things at a time
is that you have to define "things". Miller allows a "thing" to be
a group of related things. He calls grouping to squeeze more out of
memory "chunking". Interestingly, both octave-equivalence (symmetry
at the 2:1) and tetrachordality (symmetry at the 3:2) can be seen as
chunking the pitch set to reduce the burden on the listener's short
term memory.

Sorry this post is so kludgy, but I'm pressed for time. Please post
questions and comments.

-Carl

🔗Jacky Ligon <jacky_ekstasis@yahoo.com>

--- In tuning@egroups.com, Carl Lumma <CLUMMA@N...> wrote:
>
> I meant one level higher than timbre
> here, a sort of meta-timbre. The perception of the position of the
> fundamental of a melody note in the harmonic series of the
fundamentals
> of the other notes. Kapeesh?

Yes, so in other word's, the fundamental pitch relationships of a
scale (or chord?) are being looked at as members of a small, closed
Harmonic Series. Makes good sense.

>
> The basic theme is that any time your brain/ear can follow
something,
> there's potential to exploit it for melody. Let's re-name and
define
> the three things now:
>
> # NAME METRIC
> ---------------------------------------------------
> 01......Pitch tracking................Miller limit
> 02......Scalar interval tracking......Rothenberg propriety
> 03......Partial tracking..............Odd-limit
>
> These conflict with each other in spots, and diatonicity is the
> compromise that results -- that's what I meant by "action between
> them". I can get diatonicity from 02 and 03, or from 01 and 03.
Let's
> do 02 and 03 first, since that's what you asked about.

This must have allot to do with how well a melodic theme will appeal
to memory.

>
> Rothenberg propriety is a metric for an _interval ranking_ process.
> R. says we can measure intervals between pitches, and recognize
> interval patterns by the order of the sizes of the intervals,
ignoring
> the absolute sizes of the intervals involved. This allows the modal
> transposition of themes. It's how, says R., we can recognize the
tune
> Happy Birthday, when played in the natural minor.

So, in Rothenberg's theory, there is a sort of perceptual averaging
of scale steps, and the recoginzability of a given scale's
intervallic pattern is what is what is referred to as "proper"?
Please explain the difference between "strictly proper" and "proper".

>
> 6-12 is the highest complete octave in the harmonic series that's
> proper. After that, a compromise must be made (some harmonics must
be
> left out). This is the beginning of the "action between the two" I
was
> talking about.

I think I still require more of your kind teaching to get the meaning
of this.

>
> Further, we want to transpose our theme in harmonic numbers as well
> as scale degree distances. Aside from the small amount of natural
> repetitiveness in the harmonic series (i.e. 6:9 = 3:2), we'll need
to
> tweak our harmonic series scales so that different instances of a
given
> scalar interval can function as a single harmonic interval. Poof,
> diatonicity!

Then, the scale degrees may not be too narrow - so that they may
function "as a single harmonic interval"?

> ___________________________________________________ __ __ ___
___
>
> II. Diatonicity = Pitch tracking + Partial tracking
>
______________________________________________________________________
_
>
> George Miller was a cognitive psychologist who wrote a famous paper
on
> short-term memory and perception. Basically, it says that people
can,
> on average, hold up to about 7 things at a time in their short-term
> memory. What caused the stir was that this number should be the
same
> for many different types of things. We'll apply it to pitches. The
> URL for the paper has been posted here a few times...

Oh yes, I did read this when the link was posted recently - and it is
a very thought provoking paper indeed. I didn't realize that's what
you were referring to.

> ...my original post on the subject was sent to the Mills server. I
can
> re-post it if anybody would like.

I would be interested in this - please do repost.

>
> So Miller basically says we can remember and work with up to 7,
give or
> take, pitches at a time. So now, rather than just following a note
in
> a harmonic series, we can take a set of pitches, and assign each of
> them a position in our imaginary harmonic series. This gives us
common
> tone modulation -- holding a pitch constant while changing its
position
> in a harmonic series. For example, with V-I in C-major, we have G
> going from 1 to 3. You can even do more than one pitch at a time if
> you go from major to minor, or if you use certain forms of
temperament.
> So, aside from the natural repetitiveness in the harmonic series
(i.e.
> 6:9 = 3:2, etc.), we'll need to adjust our harmonic series segments
to
> allow pitches to change their harmonic functions. Poof,
diatonicity.
> (I'll try to work on filling in the poofs later.)

This is starting to click. Please do provide proofs of your poofs.

>
> One thing about saying folks can only keep track of 7 things at a
time
> is that you have to define "things". Miller allows a "thing" to be
> a group of related things. He calls grouping to squeeze more out of
> memory "chunking". Interestingly, both octave-equivalence (symmetry
> at the 2:1) and tetrachordality (symmetry at the 3:2) can be seen as
> chunking the pitch set to reduce the burden on the listener's short
> term memory.

So a good measure of Diatonicity is when the listener can
use "chunking" - or perceiving a pitch set (scale/chord) as an
overall Gestalt, and be able to easily recognize the set's members
through their harmonic relationships (?).

Thanks,

Jacky Ligon

🔗Monz <MONZ@JUNO.COM>

🔗Carl Lumma <CLUMMA@NNI.COM>

>>Rothenberg propriety is a metric for an _interval ranking_ process.
>>R. says we can measure intervals between pitches, and recognize
>>interval patterns by the order of the sizes of the intervals, ignoring
>>the absolute sizes of the intervals involved. This allows the modal
>>transposition of themes. It's how, says R., we can recognize the
>>tune Happy Birthday, when played in the natural minor.
>
>So, in Rothenberg's theory, there is a sort of perceptual averaging
>of scale steps, and the recoginzability of a given scale's
>intervallic pattern is what is what is referred to as "proper"?

Yup.

>Please explain the difference between "strictly proper" and "proper".

See Dan's post on that. (Thanks Dan, I've bitten off way more than
I can chew here, time-wise.)

>>6-12 is the highest complete octave in the harmonic series that's
>>proper. After that, a compromise must be made (some harmonics must
>>be left out). This is the beginning of the "action between the two" I
>>was talking about.
>
>I think I still require more of your kind teaching to get the meaning
>of this.

It's more speculatory spouting than kind teaching, I think. I'm not
aware of a proof, but the first five octaves of the harmonic series
are proper, and the rest aren't. So if we want to use harmonics, and
we want our scales to be proper, something's going to have to give.
One solution is to omit some harmonics. Like 8:9:10:11:12:14:16.

>>Further, we want to transpose our theme in harmonic numbers as well
>>as scale degree distances. Aside from the small amount of natural
>>repetitiveness in the harmonic series (i.e. 6:9 = 3:2), we'll need
>>to tweak our harmonic series scales so that different instances of a
>>given scalar interval can function as a single harmonic interval.
>>Poof, diatonicity!
>
>Then, the scale degrees may not be too narrow - so that they may
>function "as a single harmonic interval"?

Not sure what you're asking here. To clarify the quoted paragraph:
Since we can transpose a melody in scale intervals, we also want
to be able to transpose it in harmonic intervals. IOW, we want the
ability to express the same set of harmonic intervals in each mode
of the scale. We don't want the listener hearing the modally-transposed
tune with new scalar intervals, but still hearing the same harmonics
as before the transposition.

>So a good measure of Diatonicity is when the listener can use
>"chunking" - or perceiving a pitch set (scale/chord) as an overall
>Gestalt, and be able to easily recognize the set's members through
>their harmonic relationships (?).

Yes, if we believe that these two lead to the phenomenon of common-tone
modulation.

-Carl

🔗Carl Lumma <CLUMMA@NNI.COM>

>>o Lumma decatonic, 31-tET (0 4 6 10 12 16 18 22 25 28)
>
>Is this new? Can you demonstrate its "diatonicity"?

I think I first posted it in February. Here's the interval matrix...

1 (4 6 10 12 16 18 22 25 28 31)
2 (2 6 8 12 14 18 21 24 27 31)
3 (4 6 10 12 16 19 22 25 29 31)
4 (2 6 8 12 15 18 21 25 27 31)
5 (4 6 10 13 16 19 23 25 29 31)
6 (2 6 9 12 15 19 21 25 27 31)
7 (4 7 10 13 17 19 23 25 29 31)
8 (3 6 9 13 15 19 21 25 27 31)
9 (3 6 10 12 16 18 22 24 28 31)
10 (3 7 9 13 15 19 21 25 28 31)

...showing 5-limit triads on degrees 1, 4, and 7 in four of ten modes,
and strict propriety. 4/10 isn't good, I admit, but I like this
scale (it's a pair of pentatonic 7:4 chains interlaced a 5:4 apart).

>>o Keenan decatonic MOS, 29-tET (0 3 6 9 11 14 17 20 23 26)
>
>Can you remind where that was discussed? How about his 11-note
>chain-of-minor-thirds scale?

Dave found it when he did a computer search for "good" MOS generators,
and I found it on his webpage sometime later. I posted it to the list
twice (expressing the scale in degrees of 29-tET was my contribution,
as was discovering that...

1 (3 6 9 11 14 17 20 23 26 29)
2 (3 6 8 11 14 17 20 23 26 29)
3 (3 5 8 11 14 17 20 23 26 29)
4 (2 5 8 11 14 17 20 23 26 29)
5 (3 6 9 12 15 18 21 24 27 29)
6 (3 6 9 12 15 18 21 24 26 29)
7 (3 6 9 12 15 18 21 23 26 29)
8 (3 6 9 12 15 18 20 23 26 29)
9 (3 6 9 12 15 17 20 23 26 29)
10 (3 6 9 12 14 17 20 23 26 29)

...it contains 5-limit triads on degrees 1, 4, and 7 in six of ten modes.
Six of ten isn't good, but I like this scale.

Dave's 11-tone MOS is cool, but 11 is getting notey for a generalized-
diatonic, and there is no pattern of scale degrees that gives different,
consonant chords in different modes, that I know of.

Though I admit I hadn't counted on the way acoustic inversions of chords
don't necessarily match up with scalar inversions when I did my first
search for gd's. In the "diminished" octatonic scale, for example, I
missed the 3:4:5's and 1/(3:4:5)'s because they don't occupy the same
scale degrees in 1st inversion. My next gd search will rememdy this
problem.

>>o Erlich double-diatonic, 26-tET (0 2 4 6 8 10 12 13 15 17 19 21 23 25)
>
>How about the hexachordal version?

Of the double-diatonic? Do tell. I know I left out your dodecatonic
scales in 22-tET...

-Carl