the sharp knife has a white handle

so after a few years with not a lot of interest in tuning forums etc,
I have for whatever reason been working on a few little things from
long ago lately and, well here I am.... basically i have one thing
which i think is a pretty neat little model for defining the upper
and lower limits of a given interval's "field of attraction". This is
nothing new of course, Helmholtz , Partch and, on these very same
forums long ago,p.Erlich (his HE being a nice acronym if there ever
was one, no?) all had nice models of this very same idea. So what
exactly is the idea...? Well, basically the idea is that the simpler
a given interval is, i.e. the octave, the fifth, the more it is both
sensitive to mistuning and likely to extend a wider field of
attraction that leads one to hear other nearby intervals as mistuning
of itself.

The little model i've worked on tries to give a given frequency
interval a rough (but still numerical) lower and upper limit--how
flat and sharp a given interval can be before it crosses into another
interval's space and is liable to be either an ambiguity or a
misidentified simpler ratio. However, i use the word ratio here with
caution, as i think what i'm really looking for is an interval space
occupied by some interval that is not in the magnetic field of
attraction of its neighbor.

To me the math model i'm using looks good, so long as it's seen in
approximate terms as opposed to fixed boundaries. However, i'd like
to ask people here with an interest in this sort of thing to give me
their own input based solely on their ears...okay, so with that in
mind here are a few of the questions i'd like to ask of anyone who
might be interested enough to answer:

1) if you were to allow that the 5/4 and 9/7 occupy completely
independent interval spaces, is there a space in-between that won't
be confused for either of these? The model says no, that the median,
the 14/11 for instance, will most likely be misidentified as a 9/7
interval space rather than a 5/4 interval space; though the model
predicts that this median's lower limit would share the 5/4's upper
limit.

2) the model predicts that the space between thirds, the neutral
third, perhaps even the 11/9 itself, is a clearly defined space.
Interestingly, the model predicts that the median space between an
8/7 and a 7/6 is no less defined than the far more common median
interval spaces of the 11/8 and the 13/8. According to the model the
15/13 overlaps the lower limit of the 7/6 by approximately the same
amount the 11/8 overlaps the lower limit of the 7/5 and the 13/8's
lower limit overlaps the 8/5's upper limit. So the question here is
how does that measure to people's ears….is the median space between
an 8/7 and a 7/6 as independently valid a space as that the 11/8 and
the 13/8 share with their simpler neighbors?

3)According to the model, the 1/1 manifests a field of attraction so
strong and a shadow so long that there is a state of ambiguity all
the way to the 11/10, 10/9 range. The 2/1 similarly back to the
19/10, 17/9 range, and the 3/2 all the way back to the 10/7 and all
the way up to the 11/7 space. How does this stand up ? remember, the
model is not so much saying here's the line where an interval space
definitively changes as it is saying here's the area where ambiguity
coalesces into a defined area of recognition .and all this is
somewhat outside the usual connotations of the perceived complexity
or simplicity of a specific given ratio---->in other words, an 11/9
might be the simplest ratio to define a neutral third, but whether or
not that's THE interval is somewhat besides the point in comparison
to the question of whether this is an independently articulated
interval space outside the field of attraction imposed by the long
shadows of its neighbors.

4) Simple, rule-of-thumb measures for appraising a given ratios
complexity, like N*D ,et al., might not hold up so well. why?,well,
if the model has anything to say in this regard it clearly says that
an interval space such as the 16/9 are uniquely articulated whereas
interval spaces such as the 11/8 and the 13/8 are not. Are complexity
spikes more the result of where we sit in the octave/interval space,
or are they more the result of a given ratio's complexity?

Okay, there's a lot more and I'm only tapping the surface of a thing
that might or might not be valid to begin with anyway....but that's
the way questions work if they're not rhetorical anyway, right? Well
please feel free to contribute any inputs, I am interested!

TIA- http://zebox.com/danstearns_4/

I would say the models are wrong.
i am not sure if Ehrlich's theory treats the octave as being more sensitive to mistuning.
I thought he treated it equally. i don't know
i have yet to hear examples of these ideas in chords spread out to multiple octaves. Say chorale style.
in more than one timbre.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

daniel_anthony_stearns wrote:
>
> so after a few years with not a lot of interest in tuning forums etc,
> I have for whatever reason been working on a few little things from
> long ago lately and, well here I am.... basically i have one thing
> which i think is a pretty neat little model for defining the upper
> and lower limits of a given interval's "field of attraction". This is
> nothing new of course, Helmholtz , Partch and, on these very same
> forums long ago,p.Erlich (his HE being a nice acronym if there ever
> was one, no?) all had nice models of this very same idea. So what
> exactly is the idea...? Well, basically the idea is that the simpler
> a given interval is, i.e. the octave, the fifth, the more it is both
> sensitive to mistuning and likely to extend a wider field of
> attraction that leads one to hear other nearby intervals as mistuning
> of itself.
>
> The little model i've worked on tries to give a given frequency
> interval a rough (but still numerical) lower and upper limit--how
> flat and sharp a given interval can be before it crosses into another
> interval's space and is liable to be either an ambiguity or a
> misidentified simpler ratio. However, i use the word ratio here with
> caution, as i think what i'm really looking for is an interval space
> occupied by some interval that is not in the magnetic field of
> attraction of its neighbor.
>
> To me the math model i'm using looks good, so long as it's seen in
> approximate terms as opposed to fixed boundaries. However, i'd like
> to ask people here with an interest in this sort of thing to give me
> their own input based solely on their ears...okay, so with that in
> mind here are a few of the questions i'd like to ask of anyone who
> might be interested enough to answer:
>
> 1) if you were to allow that the 5/4 and 9/7 occupy completely
> independent interval spaces, is there a space in-between that won't
> be confused for either of these? The model says no, that the median,
> the 14/11 for instance, will most likely be misidentified as a 9/7
> interval space rather than a 5/4 interval space; though the model
> predicts that this median's lower limit would share the 5/4's upper
> limit.
>
> 2) the model predicts that the space between thirds, the neutral
> third, perhaps even the 11/9 itself, is a clearly defined space.
> Interestingly, the model predicts that the median space between an
> 8/7 and a 7/6 is no less defined than the far more common median
> interval spaces of the 11/8 and the 13/8. According to the model the
> 15/13 overlaps the lower limit of the 7/6 by approximately the same
> amount the 11/8 overlaps the lower limit of the 7/5 and the 13/8's
> lower limit overlaps the 8/5's upper limit. So the question here is
> how does that measure to people's ears�.is the median space between
> an 8/7 and a 7/6 as independently valid a space as that the 11/8 and
> the 13/8 share with their simpler neighbors?
>
> 3)According to the model, the 1/1 manifests a field of attraction so
> strong and a shadow so long that there is a state of ambiguity all
> the way to the 11/10, 10/9 range. The 2/1 similarly back to the
> 19/10, 17/9 range, and the 3/2 all the way back to the 10/7 and all
> the way up to the 11/7 space. How does this stand up ? remember, the
> model is not so much saying here's the line where an interval space
> definitively changes as it is saying here's the area where ambiguity
> coalesces into a defined area of recognition .and all this is
> somewhat outside the usual connotations of the perceived complexity
> or simplicity of a specific given ratio---->in other words, an 11/9
> might be the simplest ratio to define a neutral third, but whether or
> not that's THE interval is somewhat besides the point in comparison
> to the question of whether this is an independently articulated
> interval space outside the field of attraction imposed by the long
> shadows of its neighbors.
>
> 4) Simple, rule-of-thumb measures for appraising a given ratios
> complexity, like N*D ,et al., might not hold up so well. why?,well,
> if the model has anything to say in this regard it clearly says that
> an interval space such as the 16/9 are uniquely articulated whereas
> interval spaces such as the 11/8 and the 13/8 are not. Are complexity
> spikes more the result of where we sit in the octave/interval space,
> or are they more the result of a given ratio's complexity?
>
> Okay, there's a lot more and I'm only tapping the surface of a thing
> that might or might not be valid to begin with anyway....but that's
> the way questions work if they're not rhetorical anyway, right? Well
> please feel free to contribute any inputs, I am interested!
>
> TIA- http://zebox.com/danstearns_4/ <http://zebox.com/danstearns_4/>
>
>

hi Kraig, when i say "sensitive to mistuning" i simply mean that your
ear's going to notice when the octave is slightly mistuned more than
your ear's going to notice when the 19/10 is mistuned, no? Anyway, i
only put my imagination in numbers not my faith, so worry not my
friend....anyways, any thoughts on the specific just-o-ramic interval
questions i asked...?in your case i'd be extra interested as your
ear's been on them longer than most.
daniel
--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> I would say the models are wrong.
> i am not sure if Ehrlich's theory treats the octave as being more
> sensitive to mistuning.
> I thought he treated it equally. i don't know
> i have yet to hear examples of these ideas in chords spread out to
> multiple octaves. Say chorale style.
> in more than one timbre.
>
>
> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> _'''''''_ ^North/Western Hemisphere:
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
>
> _'''''''_ ^South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria
<http://anaphoriasouth.blogspot.com/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
>
>
>
> daniel_anthony_stearns wrote:
> >
> > so after a few years with not a lot of interest in tuning forums
etc,
> > I have for whatever reason been working on a few little things
from
> > long ago lately and, well here I am.... basically i have one thing
> > which i think is a pretty neat little model for defining the upper
> > and lower limits of a given interval's "field of attraction".
This is
> > nothing new of course, Helmholtz , Partch and, on these very same
> > forums long ago,p.Erlich (his HE being a nice acronym if there
ever
> > was one, no?) all had nice models of this very same idea. So what
> > exactly is the idea...? Well, basically the idea is that the
simpler
> > a given interval is, i.e. the octave, the fifth, the more it is
both
> > sensitive to mistuning and likely to extend a wider field of
> > attraction that leads one to hear other nearby intervals as
mistuning
> > of itself.
> >
> > The little model i've worked on tries to give a given frequency
> > interval a rough (but still numerical) lower and upper limit--how
> > flat and sharp a given interval can be before it crosses into
another
> > interval's space and is liable to be either an ambiguity or a
> > misidentified simpler ratio. However, i use the word ratio here
with
> > caution, as i think what i'm really looking for is an interval
space
> > occupied by some interval that is not in the magnetic field of
> > attraction of its neighbor.
> >
> > To me the math model i'm using looks good, so long as it's seen in
> > approximate terms as opposed to fixed boundaries. However, i'd
like
> > to ask people here with an interest in this sort of thing to give
me
> > their own input based solely on their ears...okay, so with that in
> > mind here are a few of the questions i'd like to ask of anyone who
> > might be interested enough to answer:
> >
> > 1) if you were to allow that the 5/4 and 9/7 occupy completely
> > independent interval spaces, is there a space in-between that
won't
> > be confused for either of these? The model says no, that the
median,
> > the 14/11 for instance, will most likely be misidentified as a 9/7
> > interval space rather than a 5/4 interval space; though the model
> > predicts that this median's lower limit would share the 5/4's
upper
> > limit.
> >
> > 2) the model predicts that the space between thirds, the neutral
> > third, perhaps even the 11/9 itself, is a clearly defined space.
> > Interestingly, the model predicts that the median space between an
> > 8/7 and a 7/6 is no less defined than the far more common median
> > interval spaces of the 11/8 and the 13/8. According to the model
the
> > 15/13 overlaps the lower limit of the 7/6 by approximately the
same
> > amount the 11/8 overlaps the lower limit of the 7/5 and the 13/8's
> > lower limit overlaps the 8/5's upper limit. So the question here
is
> > how does that measure to people's ears….is the median space
between
> > an 8/7 and a 7/6 as independently valid a space as that the 11/8
and
> > the 13/8 share with their simpler neighbors?
> >
> > 3)According to the model, the 1/1 manifests a field of attraction
so
> > strong and a shadow so long that there is a state of ambiguity all
> > the way to the 11/10, 10/9 range. The 2/1 similarly back to the
> > 19/10, 17/9 range, and the 3/2 all the way back to the 10/7 and
all
> > the way up to the 11/7 space. How does this stand up ? remember,
the
> > model is not so much saying here's the line where an interval
space
> > definitively changes as it is saying here's the area where
ambiguity
> > coalesces into a defined area of recognition .and all this is
> > somewhat outside the usual connotations of the perceived
complexity
> > or simplicity of a specific given ratio---->in other words, an
11/9
> > might be the simplest ratio to define a neutral third, but
whether or
> > not that's THE interval is somewhat besides the point in
comparison
> > to the question of whether this is an independently articulated
> > interval space outside the field of attraction imposed by the long
> > shadows of its neighbors.
> >
> > 4) Simple, rule-of-thumb measures for appraising a given ratios
> > complexity, like N*D ,et al., might not hold up so well.
why?,well,
> > if the model has anything to say in this regard it clearly says
that
> > an interval space such as the 16/9 are uniquely articulated
whereas
> > interval spaces such as the 11/8 and the 13/8 are not. Are
complexity
> > spikes more the result of where we sit in the octave/interval
space,
> > or are they more the result of a given ratio's complexity?
> >
> > Okay, there's a lot more and I'm only tapping the surface of a
thing
> > that might or might not be valid to begin with anyway....but
that's
> > the way questions work if they're not rhetorical anyway, right?
Well
> > please feel free to contribute any inputs, I am interested!
> >
> > TIA- http://zebox.com/danstearns_4/
<http://zebox.com/danstearns_4/>
> >
> >
>

this does seem to be the case, that 'basically , the simpler ratios we seem to notice the mistuning quicker.
I don't know if this is because we are a somewhat 'simpler ratio' culture.
For me it brings up the question is why we hear things as thirds or fourths which are even broader generalizations before tuning even takes place.
It might all be influenced by the scale we were brought up with. but maybe not.

my own theory.
It seems this has to do with the range in which the difference tones falls in terms of octaves and then basic ranges within. compare basic intervals.
these are appoximate depending on your ratios
semitones 4 octave lower
whole tones 3 octave lower
minor thirds 2 octaves and a fifth to a minor third +/-
major thirds 2 Octaves
fourth 1 octave and a fifth
fifth I octave
etc.
It appears there are thresholds of intervals some clearer than not.
for instance we can see how the 5/4 is probably the rounded lower limit if what we might hear as a major third with the difference tone being 2 octave below. above this we hear as major , below moving toward a neutral maybe more quickly. even though 9/7 is not that much more distant than 11/9 from 5/4, there might be more of a tolerance for one direction over the other. sharp octaves over flat ones. which would play into why we have a name for neutral thirds but none for sharp major ones.

To 'quantify' such things might work well when submitting papers, but i prefer to look at the beast for what it is, 'cloudy' in a very Popper like manner.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

daniel_anthony_stearns wrote:
>
> hi Kraig, when i say "sensitive to mistuning" i simply mean that your
> ear's going to notice when the octave is slightly mistuned more than
> your ear's going to notice when the 19/10 is mistuned, no? Anyway, i
> only put my imagination in numbers not my faith, so worry not my
> friend....anyways, any thoughts on the specific just-o-ramic interval
> questions i asked...?in your case i'd be extra interested as your
> ear's been on them longer than most.
> daniel
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, Kraig > Grady <kraiggrady@...> wrote:
> >
> > I would say the models are wrong.
> > i am not sure if Ehrlich's theory treats the octave as being more
> > sensitive to mistuning.
> > I thought he treated it equally. i don't know
> > i have yet to hear examples of these ideas in chords spread out to
> > multiple octaves. Say chorale style.
> > in more than one timbre.
> >
> >
> > /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> > _'''''''_ ^North/Western Hemisphere:
> > North American Embassy of Anaphoria Island <http://anaphoria.com/ > <http://anaphoria.com/>>
> >
> > _'''''''_ ^South/Eastern Hemisphere:
> > Austronesian Outpost of Anaphoria
> <http://anaphoriasouth.blogspot.com/ > <http://anaphoriasouth.blogspot.com/>>
> >
> > ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
> >
> >
> >
> >
> > daniel_anthony_stearns wrote:
> > >
> > > so after a few years with not a lot of interest in tuning forums
> etc,
> > > I have for whatever reason been working on a few little things
> from
> > > long ago lately and, well here I am.... basically i have one thing
> > > which i think is a pretty neat little model for defining the upper
> > > and lower limits of a given interval's "field of attraction".
> This is
> > > nothing new of course, Helmholtz , Partch and, on these very same
> > > forums long ago,p.Erlich (his HE being a nice acronym if there
> ever
> > > was one, no?) all had nice models of this very same idea. So what
> > > exactly is the idea...? Well, basically the idea is that the
> simpler
> > > a given interval is, i.e. the octave, the fifth, the more it is
> both
> > > sensitive to mistuning and likely to extend a wider field of
> > > attraction that leads one to hear other nearby intervals as
> mistuning
> > > of itself.
> > >
> > > The little model i've worked on tries to give a given frequency
> > > interval a rough (but still numerical) lower and upper limit--how
> > > flat and sharp a given interval can be before it crosses into
> another
> > > interval's space and is liable to be either an ambiguity or a
> > > misidentified simpler ratio. However, i use the word ratio here
> with
> > > caution, as i think what i'm really looking for is an interval
> space
> > > occupied by some interval that is not in the magnetic field of
> > > attraction of its neighbor.
> > >
> > > To me the math model i'm using looks good, so long as it's seen in
> > > approximate terms as opposed to fixed boundaries. However, i'd
> like
> > > to ask people here with an interest in this sort of thing to give
> me
> > > their own input based solely on their ears...okay, so with that in
> > > mind here are a few of the questions i'd like to ask of anyone who
> > > might be interested enough to answer:
> > >
> > > 1) if you were to allow that the 5/4 and 9/7 occupy completely
> > > independent interval spaces, is there a space in-between that
> won't
> > > be confused for either of these? The model says no, that the
> median,
> > > the 14/11 for instance, will most likely be misidentified as a 9/7
> > > interval space rather than a 5/4 interval space; though the model
> > > predicts that this median's lower limit would share the 5/4's
> upper
> > > limit.
> > >
> > > 2) the model predicts that the space between thirds, the neutral
> > > third, perhaps even the 11/9 itself, is a clearly defined space.
> > > Interestingly, the model predicts that the median space between an
> > > 8/7 and a 7/6 is no less defined than the far more common median
> > > interval spaces of the 11/8 and the 13/8. According to the model
> the
> > > 15/13 overlaps the lower limit of the 7/6 by approximately the
> same
> > > amount the 11/8 overlaps the lower limit of the 7/5 and the 13/8's
> > > lower limit overlaps the 8/5's upper limit. So the question here
> is
> > > how does that measure to people's ears�.is the median space
> between
> > > an 8/7 and a 7/6 as independently valid a space as that the 11/8
> and
> > > the 13/8 share with their simpler neighbors?
> > >
> > > 3)According to the model, the 1/1 manifests a field of attraction
> so
> > > strong and a shadow so long that there is a state of ambiguity all
> > > the way to the 11/10, 10/9 range. The 2/1 similarly back to the
> > > 19/10, 17/9 range, and the 3/2 all the way back to the 10/7 and
> all
> > > the way up to the 11/7 space. How does this stand up ? remember,
> the
> > > model is not so much saying here's the line where an interval
> space
> > > definitively changes as it is saying here's the area where
> ambiguity
> > > coalesces into a defined area of recognition .and all this is
> > > somewhat outside the usual connotations of the perceived
> complexity
> > > or simplicity of a specific given ratio---->in other words, an
> 11/9
> > > might be the simplest ratio to define a neutral third, but
> whether or
> > > not that's THE interval is somewhat besides the point in
> comparison
> > > to the question of whether this is an independently articulated
> > > interval space outside the field of attraction imposed by the long
> > > shadows of its neighbors.
> > >
> > > 4) Simple, rule-of-thumb measures for appraising a given ratios
> > > complexity, like N*D ,et al., might not hold up so well.
> why?,well,
> > > if the model has anything to say in this regard it clearly says
> that
> > > an interval space such as the 16/9 are uniquely articulated
> whereas
> > > interval spaces such as the 11/8 and the 13/8 are not. Are
> complexity
> > > spikes more the result of where we sit in the octave/interval
> space,
> > > or are they more the result of a given ratio's complexity?
> > >
> > > Okay, there's a lot more and I'm only tapping the surface of a
> thing
> > > that might or might not be valid to begin with anyway....but
> that's
> > > the way questions work if they're not rhetorical anyway, right?
> Well
> > > please feel free to contribute any inputs, I am interested!
> > >
> > > TIA- http://zebox.com/danstearns_4/ <http://zebox.com/danstearns_4/>
> <http://zebox.com/danstearns_4/ <http://zebox.com/danstearns_4/>>
> > >
> > >
> >
>
>

With the cautionary caveat that the ratio names are not as important
as the approximate position they occupy in the confines of the
octave's pitch-continuum, according to this algorithm, the only
interval-spaces that occupy their own territory and neither abut, nor
overlap another's are the following:

http://tinyurl.com/5okqxp

However, a more inclusive view is one that allows for shared
boundaries as is the case with the 11/10 and the 10/9, the 7/5 and
the 10/7, the 11/7 and the 8/5 and the 17/9 and the 19/10:

http://tinyurl.com/6ans6r

As I mentioned earlier, if we were to include such common JI
intervals as the 11th and 13th harmonic, then the algorithm would
suggest that the median space between the 8/7 and the 7/6, the 15/13-
space, is no less of an independent peak positioned between two
stronger fields of attraction:

http://tinyurl.com/6dccru

I've always been somewhat of a pantheist when it comes to tuning
systems, so the idea of an absolute tuning hierarchy is one at odds
with my nature. Yet, Just Intonation's relation to the human auditory
system is a benefit seldom lost on anyone who navigates their
instrument's tuning by their ear… but whoever told Nebraska policeman
Herbert Schirmer that "Someday, Watchman, you will see the universe";
be it himself, some semi-autonomous exponent of himself, or some
intervening Other, I think it's interesting to acknowledge the idea
that experiences not only give us memories and events, but they also
sometimes give us answers. And unfortunately, it's often an
uncomforting fact that our answers are disproportionately taken as
the basis on which we're judged. When Hermann Helmholtz wrote his
epic, On the Sensations of Tone, he was a real-time contributing
editor to the blossoming, golden age of reason. So it's no surprise
he prefaces his work with a lot of rational, raison d'être directed
at the expense of arbitrary assumption. But tuning theory has always
sat on a fence, uncomfortably straddling the dueling influences of a
new religion and a new science. Oddly, both draw on the tools of the
other in such a fashion that their mutual disregard is both puzzling
and—given the peculiar way these things work—predictable.

For as long as I can remember I've been interested in art and music
and math, but I'm neither an artist nor a musician and certainly not
a mathematician. I've always felt that skill was an end to a mean,
not an end in itself. This is a simple thing to say, and everyone
seems to say it too… but I tend to see it as in exaggerated terms
where your skills only have to be good enough to execute the contents
of your imagination, but your imagination has to good enough to say
something worth saying and animate all the intuition and imaginative
possibilities we innately posses and continually foster. Like
everyone, I tend to exaggerate along the lines of our interests, and
I've always been a dreamer who likes to spend his time trying to make
little models out of the leftover pieces of those dreams. It's a
skill I suppose, but certainly not the formal moving part of a
normative science. There's a short scene in Ingmar Bergman's movie
The Seventh Seal where Death is asked "what he knows", and "he knows
nothing" is his answer. Bergman himself tends to fall in and out of
favor precisely because he asked these kinds of questions and wasn't
afraid to answer.

http://zebox.com/danstearns_4/

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> this does seem to be the case, that 'basically , the simpler ratios
we
> seem to notice the mistuning quicker.
> I don't know if this is because we are a somewhat 'simpler ratio'
culture.
> For me it brings up the question is why we hear things as thirds or
> fourths which are even broader generalizations before tuning even
takes
> place.
> It might all be influenced by the scale we were brought up with.
but
> maybe not.
>
> my own theory.
> It seems this has to do with the range in which the difference
tones
> falls in terms of octaves and then basic ranges within. compare
basic
> intervals.
> these are appoximate depending on your ratios
> semitones 4 octave lower
> whole tones 3 octave lower
> minor thirds 2 octaves and a fifth to a minor third +/-
> major thirds 2 Octaves
> fourth 1 octave and a fifth
> fifth I octave
> etc.
> It appears there are thresholds of intervals some clearer than not.
> for instance we can see how the 5/4 is probably the rounded lower
limit
> if what we might hear as a major third with the difference tone
being 2
> octave below. above this we hear as major , below moving toward a
> neutral maybe more quickly. even though 9/7 is not that much more
> distant than 11/9 from 5/4, there might be more of a tolerance for
one
> direction over the other. sharp octaves over flat ones. which would
play
> into why we have a name for neutral thirds but none for sharp major
ones.
>
> To 'quantify' such things might work well when submitting papers,
but i
> prefer to look at the beast for what it is, 'cloudy' in a very
Popper
> like manner.
>
>
>
>
>
>
> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> _'''''''_ ^North/Western Hemisphere:
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
>
> _'''''''_ ^South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria
<http://anaphoriasouth.blogspot.com/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
>
>
>
> daniel_anthony_stearns wrote:
> >
> > hi Kraig, when i say "sensitive to mistuning" i simply mean that
your
> > ear's going to notice when the octave is slightly mistuned more
than
> > your ear's going to notice when the 19/10 is mistuned, no?
Anyway, i
> > only put my imagination in numbers not my faith, so worry not my
> > friend....anyways, any thoughts on the specific just-o-ramic
interval
> > questions i asked...?in your case i'd be extra interested as your
> > ear's been on them longer than most.
> > daniel
> > --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>,
Kraig
> > Grady <kraiggrady@> wrote:
> > >
> > > I would say the models are wrong.
> > > i am not sure if Ehrlich's theory treats the octave as being
more
> > > sensitive to mistuning.
> > > I thought he treated it equally. i don't know
> > > i have yet to hear examples of these ideas in chords spread out
to
> > > multiple octaves. Say chorale style.
> > > in more than one timbre.
> > >
> > >
> > > /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> > > _'''''''_ ^North/Western Hemisphere:
> > > North American Embassy of Anaphoria Island
<http://anaphoria.com/
> > <http://anaphoria.com/>>
> > >
> > > _'''''''_ ^South/Eastern Hemisphere:
> > > Austronesian Outpost of Anaphoria
> > <http://anaphoriasouth.blogspot.com/
> > <http://anaphoriasouth.blogspot.com/>>
> > >
> > > ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
> > >
> > >
> > >
> > >
> > > daniel_anthony_stearns wrote:
> > > >
> > > > so after a few years with not a lot of interest in tuning
forums
> > etc,
> > > > I have for whatever reason been working on a few little things
> > from
> > > > long ago lately and, well here I am.... basically i have one
thing
> > > > which i think is a pretty neat little model for defining the
upper
> > > > and lower limits of a given interval's "field of attraction".
> > This is
> > > > nothing new of course, Helmholtz , Partch and, on these very
same
> > > > forums long ago,p.Erlich (his HE being a nice acronym if there
> > ever
> > > > was one, no?) all had nice models of this very same idea. So
what
> > > > exactly is the idea...? Well, basically the idea is that the
> > simpler
> > > > a given interval is, i.e. the octave, the fifth, the more it
is
> > both
> > > > sensitive to mistuning and likely to extend a wider field of
> > > > attraction that leads one to hear other nearby intervals as
> > mistuning
> > > > of itself.
> > > >
> > > > The little model i've worked on tries to give a given
frequency
> > > > interval a rough (but still numerical) lower and upper limit--
how
> > > > flat and sharp a given interval can be before it crosses into
> > another
> > > > interval's space and is liable to be either an ambiguity or a
> > > > misidentified simpler ratio. However, i use the word ratio
here
> > with
> > > > caution, as i think what i'm really looking for is an interval
> > space
> > > > occupied by some interval that is not in the magnetic field of
> > > > attraction of its neighbor.
> > > >
> > > > To me the math model i'm using looks good, so long as it's
seen in
> > > > approximate terms as opposed to fixed boundaries. However, i'd
> > like
> > > > to ask people here with an interest in this sort of thing to
give
> > me
> > > > their own input based solely on their ears...okay, so with
that in
> > > > mind here are a few of the questions i'd like to ask of
anyone who
> > > > might be interested enough to answer:
> > > >
> > > > 1) if you were to allow that the 5/4 and 9/7 occupy completely
> > > > independent interval spaces, is there a space in-between that
> > won't
> > > > be confused for either of these? The model says no, that the
> > median,
> > > > the 14/11 for instance, will most likely be misidentified as
a 9/7
> > > > interval space rather than a 5/4 interval space; though the
model
> > > > predicts that this median's lower limit would share the 5/4's
> > upper
> > > > limit.
> > > >
> > > > 2) the model predicts that the space between thirds, the
neutral
> > > > third, perhaps even the 11/9 itself, is a clearly defined
space.
> > > > Interestingly, the model predicts that the median space
between an
> > > > 8/7 and a 7/6 is no less defined than the far more common
median
> > > > interval spaces of the 11/8 and the 13/8. According to the
model
> > the
> > > > 15/13 overlaps the lower limit of the 7/6 by approximately the
> > same
> > > > amount the 11/8 overlaps the lower limit of the 7/5 and the
13/8's
> > > > lower limit overlaps the 8/5's upper limit. So the question
here
> > is
> > > > how does that measure to people's ears….is the median space
> > between
> > > > an 8/7 and a 7/6 as independently valid a space as that the
11/8
> > and
> > > > the 13/8 share with their simpler neighbors?
> > > >
> > > > 3)According to the model, the 1/1 manifests a field of
attraction
> > so
> > > > strong and a shadow so long that there is a state of
ambiguity all
> > > > the way to the 11/10, 10/9 range. The 2/1 similarly back to
the
> > > > 19/10, 17/9 range, and the 3/2 all the way back to the 10/7
and
> > all
> > > > the way up to the 11/7 space. How does this stand up ?
remember,
> > the
> > > > model is not so much saying here's the line where an interval
> > space
> > > > definitively changes as it is saying here's the area where
> > ambiguity
> > > > coalesces into a defined area of recognition .and all this is
> > > > somewhat outside the usual connotations of the perceived
> > complexity
> > > > or simplicity of a specific given ratio---->in other words, an
> > 11/9
> > > > might be the simplest ratio to define a neutral third, but
> > whether or
> > > > not that's THE interval is somewhat besides the point in
> > comparison
> > > > to the question of whether this is an independently
articulated
> > > > interval space outside the field of attraction imposed by the
long
> > > > shadows of its neighbors.
> > > >
> > > > 4) Simple, rule-of-thumb measures for appraising a given
ratios
> > > > complexity, like N*D ,et al., might not hold up so well.
> > why?,well,
> > > > if the model has anything to say in this regard it clearly
says
> > that
> > > > an interval space such as the 16/9 are uniquely articulated
> > whereas
> > > > interval spaces such as the 11/8 and the 13/8 are not. Are
> > complexity
> > > > spikes more the result of where we sit in the octave/interval
> > space,
> > > > or are they more the result of a given ratio's complexity?
> > > >
> > > > Okay, there's a lot more and I'm only tapping the surface of a
> > thing
> > > > that might or might not be valid to begin with anyway....but
> > that's
> > > > the way questions work if they're not rhetorical anyway,
right?
> > Well
> > > > please feel free to contribute any inputs, I am interested!
> > > >
> > > > TIA- http://zebox.com/danstearns_4/
<http://zebox.com/danstearns_4/>
> > <http://zebox.com/danstearns_4/ <http://zebox.com/danstearns_4/>>
> > > >
> > > >
> > >
> >
> >
>

btw, for anyone who's interested Robert Walker made a nice applet
awhile back based on another aspect of this algorithm/concept here:

http://tunesmithy.netfirms.com/japplets/uo_non_oct.htm

--- In tuning@yahoogroups.com, "daniel_anthony_stearns"
<daniel_anthony_stearns@...> wrote:
>
> With the cautionary caveat that the ratio names are not as
important
> as the approximate position they occupy in the confines of the
> octave's pitch-continuum, according to this algorithm, the only
> interval-spaces that occupy their own territory and neither abut,
nor
> overlap another's are the following:
>
> http://tinyurl.com/5okqxp
>
> However, a more inclusive view is one that allows for shared
> boundaries as is the case with the 11/10 and the 10/9, the 7/5 and
> the 10/7, the 11/7 and the 8/5 and the 17/9 and the 19/10:
>
> http://tinyurl.com/6ans6r
>
> As I mentioned earlier, if we were to include such common JI
> intervals as the 11th and 13th harmonic, then the algorithm would
> suggest that the median space between the 8/7 and the 7/6, the
15/13-
> space, is no less of an independent peak positioned between two
> stronger fields of attraction:
>
> http://tinyurl.com/6dccru
>
> I've always been somewhat of a pantheist when it comes to tuning
> systems, so the idea of an absolute tuning hierarchy is one at odds
> with my nature. Yet, Just Intonation's relation to the human
auditory
> system is a benefit seldom lost on anyone who navigates their
> instrument's tuning by their ear… but whoever told Nebraska
policeman
> Herbert Schirmer that "Someday, Watchman, you will see the
universe";
> be it himself, some semi-autonomous exponent of himself, or some
> intervening Other, I think it's interesting to acknowledge the idea
> that experiences not only give us memories and events, but they
also
> sometimes give us answers. And unfortunately, it's often an
> uncomforting fact that our answers are disproportionately taken as
> the basis on which we're judged. When Hermann Helmholtz wrote his
> epic, On the Sensations of Tone, he was a real-time contributing
> editor to the blossoming, golden age of reason. So it's no surprise
> he prefaces his work with a lot of rational, raison d'être directed
> at the expense of arbitrary assumption. But tuning theory has
always
> sat on a fence, uncomfortably straddling the dueling influences of
a
> new religion and a new science. Oddly, both draw on the tools of
the
> other in such a fashion that their mutual disregard is both
puzzling
> and—given the peculiar way these things work—predictable.
>
> For as long as I can remember I've been interested in art and music
> and math, but I'm neither an artist nor a musician and certainly
not
> a mathematician. I've always felt that skill was an end to a mean,
> not an end in itself. This is a simple thing to say, and everyone
> seems to say it too… but I tend to see it as in exaggerated terms
> where your skills only have to be good enough to execute the
contents
> of your imagination, but your imagination has to good enough to say
> something worth saying and animate all the intuition and
imaginative
> possibilities we innately posses and continually foster. Like
> everyone, I tend to exaggerate along the lines of our interests,
and
> I've always been a dreamer who likes to spend his time trying to
make
> little models out of the leftover pieces of those dreams. It's a
> skill I suppose, but certainly not the formal moving part of a
> normative science. There's a short scene in Ingmar Bergman's movie
> The Seventh Seal where Death is asked "what he knows", and "he
knows
> nothing" is his answer. Bergman himself tends to fall in and out of
> favor precisely because he asked these kinds of questions and
wasn't
> afraid to answer.
>
> http://zebox.com/danstearns_4/
>
>
> --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@> wrote:
> >
> > this does seem to be the case, that 'basically , the simpler
ratios
> we
> > seem to notice the mistuning quicker.
> > I don't know if this is because we are a somewhat 'simpler ratio'
> culture.
> > For me it brings up the question is why we hear things as thirds
or
> > fourths which are even broader generalizations before tuning even
> takes
> > place.
> > It might all be influenced by the scale we were brought up with.
> but
> > maybe not.
> >
> > my own theory.
> > It seems this has to do with the range in which the difference
> tones
> > falls in terms of octaves and then basic ranges within. compare
> basic
> > intervals.
> > these are appoximate depending on your ratios
> > semitones 4 octave lower
> > whole tones 3 octave lower
> > minor thirds 2 octaves and a fifth to a minor third +/-
> > major thirds 2 Octaves
> > fourth 1 octave and a fifth
> > fifth I octave
> > etc.
> > It appears there are thresholds of intervals some clearer than
not.
> > for instance we can see how the 5/4 is probably the rounded lower
> limit
> > if what we might hear as a major third with the difference tone
> being 2
> > octave below. above this we hear as major , below moving toward a
> > neutral maybe more quickly. even though 9/7 is not that much more
> > distant than 11/9 from 5/4, there might be more of a tolerance
for
> one
> > direction over the other. sharp octaves over flat ones. which
would
> play
> > into why we have a name for neutral thirds but none for sharp
major
> ones.
> >
> > To 'quantify' such things might work well when submitting papers,
> but i
> > prefer to look at the beast for what it is, 'cloudy' in a very
> Popper
> > like manner.
> >
> >
> >
> >
> >
> >
> > /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> > _'''''''_ ^North/Western Hemisphere:
> > North American Embassy of Anaphoria Island <http://anaphoria.com/>
> >
> > _'''''''_ ^South/Eastern Hemisphere:
> > Austronesian Outpost of Anaphoria
> <http://anaphoriasouth.blogspot.com/>
> >
> > ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
> >
> >
> >
> >
> > daniel_anthony_stearns wrote:
> > >
> > > hi Kraig, when i say "sensitive to mistuning" i simply mean
that
> your
> > > ear's going to notice when the octave is slightly mistuned more
> than
> > > your ear's going to notice when the 19/10 is mistuned, no?
> Anyway, i
> > > only put my imagination in numbers not my faith, so worry not my
> > > friend....anyways, any thoughts on the specific just-o-ramic
> interval
> > > questions i asked...?in your case i'd be extra interested as
your
> > > ear's been on them longer than most.
> > > daniel
> > > --- In tuning@yahoogroups.com <mailto:tuning%
40yahoogroups.com>,
> Kraig
> > > Grady <kraiggrady@> wrote:
> > > >
> > > > I would say the models are wrong.
> > > > i am not sure if Ehrlich's theory treats the octave as being
> more
> > > > sensitive to mistuning.
> > > > I thought he treated it equally. i don't know
> > > > i have yet to hear examples of these ideas in chords spread
out
> to
> > > > multiple octaves. Say chorale style.
> > > > in more than one timbre.
> > > >
> > > >
> > > > /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> > > > _'''''''_ ^North/Western Hemisphere:
> > > > North American Embassy of Anaphoria Island
> <http://anaphoria.com/
> > > <http://anaphoria.com/>>
> > > >
> > > > _'''''''_ ^South/Eastern Hemisphere:
> > > > Austronesian Outpost of Anaphoria
> > > <http://anaphoriasouth.blogspot.com/
> > > <http://anaphoriasouth.blogspot.com/>>
> > > >
> > > > ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
> > > >
> > > >
> > > >
> > > >
> > > > daniel_anthony_stearns wrote:
> > > > >
> > > > > so after a few years with not a lot of interest in tuning
> forums
> > > etc,
> > > > > I have for whatever reason been working on a few little
things
> > > from
> > > > > long ago lately and, well here I am.... basically i have
one
> thing
> > > > > which i think is a pretty neat little model for defining
the
> upper
> > > > > and lower limits of a given interval's "field of
attraction".
> > > This is
> > > > > nothing new of course, Helmholtz , Partch and, on these
very
> same
> > > > > forums long ago,p.Erlich (his HE being a nice acronym if
there
> > > ever
> > > > > was one, no?) all had nice models of this very same idea.
So
> what
> > > > > exactly is the idea...? Well, basically the idea is that the
> > > simpler
> > > > > a given interval is, i.e. the octave, the fifth, the more
it
> is
> > > both
> > > > > sensitive to mistuning and likely to extend a wider field of
> > > > > attraction that leads one to hear other nearby intervals as
> > > mistuning
> > > > > of itself.
> > > > >
> > > > > The little model i've worked on tries to give a given
> frequency
> > > > > interval a rough (but still numerical) lower and upper
limit--
> how
> > > > > flat and sharp a given interval can be before it crosses
into
> > > another
> > > > > interval's space and is liable to be either an ambiguity or
a
> > > > > misidentified simpler ratio. However, i use the word ratio
> here
> > > with
> > > > > caution, as i think what i'm really looking for is an
interval
> > > space
> > > > > occupied by some interval that is not in the magnetic field
of
> > > > > attraction of its neighbor.
> > > > >
> > > > > To me the math model i'm using looks good, so long as it's
> seen in
> > > > > approximate terms as opposed to fixed boundaries. However,
i'd
> > > like
> > > > > to ask people here with an interest in this sort of thing
to
> give
> > > me
> > > > > their own input based solely on their ears...okay, so with
> that in
> > > > > mind here are a few of the questions i'd like to ask of
> anyone who
> > > > > might be interested enough to answer:
> > > > >
> > > > > 1) if you were to allow that the 5/4 and 9/7 occupy
completely
> > > > > independent interval spaces, is there a space in-between
that
> > > won't
> > > > > be confused for either of these? The model says no, that the
> > > median,
> > > > > the 14/11 for instance, will most likely be misidentified
as
> a 9/7
> > > > > interval space rather than a 5/4 interval space; though the
> model
> > > > > predicts that this median's lower limit would share the
5/4's
> > > upper
> > > > > limit.
> > > > >
> > > > > 2) the model predicts that the space between thirds, the
> neutral
> > > > > third, perhaps even the 11/9 itself, is a clearly defined
> space.
> > > > > Interestingly, the model predicts that the median space
> between an
> > > > > 8/7 and a 7/6 is no less defined than the far more common
> median
> > > > > interval spaces of the 11/8 and the 13/8. According to the
> model
> > > the
> > > > > 15/13 overlaps the lower limit of the 7/6 by approximately
the
> > > same
> > > > > amount the 11/8 overlaps the lower limit of the 7/5 and the
> 13/8's
> > > > > lower limit overlaps the 8/5's upper limit. So the question
> here
> > > is
> > > > > how does that measure to people's ears….is the median space
> > > between
> > > > > an 8/7 and a 7/6 as independently valid a space as that the
> 11/8
> > > and
> > > > > the 13/8 share with their simpler neighbors?
> > > > >
> > > > > 3)According to the model, the 1/1 manifests a field of
> attraction
> > > so
> > > > > strong and a shadow so long that there is a state of
> ambiguity all
> > > > > the way to the 11/10, 10/9 range. The 2/1 similarly back to
> the
> > > > > 19/10, 17/9 range, and the 3/2 all the way back to the 10/7
> and
> > > all
> > > > > the way up to the 11/7 space. How does this stand up ?
> remember,
> > > the
> > > > > model is not so much saying here's the line where an
interval
> > > space
> > > > > definitively changes as it is saying here's the area where
> > > ambiguity
> > > > > coalesces into a defined area of recognition .and all this
is
> > > > > somewhat outside the usual connotations of the perceived
> > > complexity
> > > > > or simplicity of a specific given ratio---->in other words,
an
> > > 11/9
> > > > > might be the simplest ratio to define a neutral third, but
> > > whether or
> > > > > not that's THE interval is somewhat besides the point in
> > > comparison
> > > > > to the question of whether this is an independently
> articulated
> > > > > interval space outside the field of attraction imposed by
the
> long
> > > > > shadows of its neighbors.
> > > > >
> > > > > 4) Simple, rule-of-thumb measures for appraising a given
> ratios
> > > > > complexity, like N*D ,et al., might not hold up so well.
> > > why?,well,
> > > > > if the model has anything to say in this regard it clearly
> says
> > > that
> > > > > an interval space such as the 16/9 are uniquely articulated
> > > whereas
> > > > > interval spaces such as the 11/8 and the 13/8 are not. Are
> > > complexity
> > > > > spikes more the result of where we sit in the
octave/interval
> > > space,
> > > > > or are they more the result of a given ratio's complexity?
> > > > >
> > > > > Okay, there's a lot more and I'm only tapping the surface
of a
> > > thing
> > > > > that might or might not be valid to begin with anyway....but
> > > that's
> > > > > the way questions work if they're not rhetorical anyway,
> right?
> > > Well
> > > > > please feel free to contribute any inputs, I am interested!
> > > > >
> > > > > TIA- http://zebox.com/danstearns_4/
> <http://zebox.com/danstearns_4/>
> > > <http://zebox.com/danstearns_4/
<http://zebox.com/danstearns_4/>>
> > > > >
> > > > >
> > > >
> > >
> > >
> >
>

I disagree with the notion that it works in such a way that the
"simplest" intervals have a wider range of attraction - I've heard
11:4 be recognizable from a full quarter tone up - C-F# in 12-tet can
pass for 11:4 in certain cases. On the other hand, 5:4 a quarter tone
down or up doesn't sound like 5:4 at all, usually.

-Mike

hello there Mike, I think you might've misunderstood me. What i meant
was that smaller ratios, like the 5/4 you mention, are more apt to be
heard as distinctly different despite very small mistunings, whereas
more complex ratios such as the 11/8 as far less likely to be heard
as decisively different given the same minute degree of mistuning.
Now of course there are ears so rough when it comes to pitch
discrimination, that a 2/1 can be mistaken for a 3/1--say in the
average family' happy birthday singalong--and ears so acute when it
comes to pitch discrimination that they can confidently navigate
extremely fine fractions of a tone, etc.

So many people, myself included, might say that to ask a math model
questions only an ear can answer seem like the wrong approach given
the question! But the ear is not only an instrument that operates
with blind obedience on the physiological level, it's also one that
is more and less acute, more and less biased, more and less
contextually obligated to respond in one way or another of a thousand
different ways.... so the idea with the math here is that i'm only
trying to imagine a generalization, one which is admittedly based
more on intuition and a personal commitment to experiment with ideas
i like than it is on any factual data.

So, with that troubling confession out of the way, what I'm trying do
is to generalize what ratio spaces are unique and which are not by
asking a specific question, that is: which spaces within the octave
are most unlikely to be mistaken for their neighbors? So for example,
it's easy to see from the graph that while a quartertone is miles
away from a semitone, all minor 2nds are unlikely to be discerned
from their next closest neighbors... and it's exactly that kind of
distinction that is really at the heart of this!

To look at it another way, and i'll take another thing you mentioned
as an example, that the 5/4 seems quite different a 1/4 of a tone or
so in either direction, well the experiment here says that that's
true, and it says it's true because there's a space to the left at
11/9 and a space to the right at 9/7 that are also unique from their
nearest neighbors. In the graph, the height (the vertical axis) is an
indication of a given ratios complexity, so the deeper the valley the
simpler the pitch-space, and the higher the spike the more complex
the pitch-space. But this aspect is dependent on the idea that we're
talking about frequency ratios, as the height is solely dependent on
the numerator and denominator of the ratio. What I'm probably more
interested in here is span and especially uniqueness---asking how
wide an interval space is relative to its proposed uniqueness.

thanks,
http://zebox.com/danstearns_4/

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...>
wrote:
>
> I disagree with the notion that it works in such a way that the
> "simplest" intervals have a wider range of attraction - I've heard
> 11:4 be recognizable from a full quarter tone up - C-F# in 12-tet
can
> pass for 11:4 in certain cases. On the other hand, 5:4 a quarter
tone
> down or up doesn't sound like 5:4 at all, usually.
>
> -Mike
>

Basically you are creating or using the scale tree (stern-brocot) tree to define where the further down you go, the less distinct the intervals would be. I think i mentioned at times that 'means' are possibly more important than 'limits'. Of course all this pertains to western culture where the acoustical phenomenon of these interval occupy more attention than others ( who listen for other things, some known, some not it appears). That we work with a 12 tone scale as our tribal tuning would put certain intervals more in the realm of being identified with one over another two so 'scale' also plays into the picture.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

daniel_anthony_stearns wrote:
>
> hello there Mike, I think you might've misunderstood me. What i meant
> was that smaller ratios, like the 5/4 you mention, are more apt to be
> heard as distinctly different despite very small mistunings, whereas
> more complex ratios such as the 11/8 as far less likely to be heard
> as decisively different given the same minute degree of mistuning.
> Now of course there are ears so rough when it comes to pitch
> discrimination, that a 2/1 can be mistaken for a 3/1--say in the
> average family' happy birthday singalong--and ears so acute when it
> comes to pitch discrimination that they can confidently navigate
> extremely fine fractions of a tone, etc.
>
> So many people, myself included, might say that to ask a math model
> questions only an ear can answer seem like the wrong approach given
> the question! But the ear is not only an instrument that operates
> with blind obedience on the physiological level, it's also one that
> is more and less acute, more and less biased, more and less
> contextually obligated to respond in one way or another of a thousand
> different ways.... so the idea with the math here is that i'm only
> trying to imagine a generalization, one which is admittedly based
> more on intuition and a personal commitment to experiment with ideas
> i like than it is on any factual data.
>
> So, with that troubling confession out of the way, what I'm trying do
> is to generalize what ratio spaces are unique and which are not by
> asking a specific question, that is: which spaces within the octave
> are most unlikely to be mistaken for their neighbors? So for example,
> it's easy to see from the graph that while a quartertone is miles
> away from a semitone, all minor 2nds are unlikely to be discerned
> from their next closest neighbors... and it's exactly that kind of
> distinction that is really at the heart of this!
>
> To look at it another way, and i'll take another thing you mentioned
> as an example, that the 5/4 seems quite different a 1/4 of a tone or
> so in either direction, well the experiment here says that that's
> true, and it says it's true because there's a space to the left at
> 11/9 and a space to the right at 9/7 that are also unique from their
> nearest neighbors. In the graph, the height (the vertical axis) is an
> indication of a given ratios complexity, so the deeper the valley the
> simpler the pitch-space, and the higher the spike the more complex
> the pitch-space. But this aspect is dependent on the idea that we're
> talking about frequency ratios, as the height is solely dependent on
> the numerator and denominator of the ratio. What I'm probably more
> interested in here is span and especially uniqueness---asking how
> wide an interval space is relative to its proposed uniqueness.
>
> thanks,
> http://zebox.com/danstearns_4/ <http://zebox.com/danstearns_4/>
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, "Mike > Battaglia" <battaglia01@...>
> wrote:
> >
> > I disagree with the notion that it works in such a way that the
> > "simplest" intervals have a wider range of attraction - I've heard
> > 11:4 be recognizable from a full quarter tone up - C-F# in 12-tet
> can
> > pass for 11:4 in certain cases. On the other hand, 5:4 a quarter
> tone
> > down or up doesn't sound like 5:4 at all, usually.
> >
> > -Mike
> >
>
>