Yahoo Tuning Groups Ultimate Backup tuning Model 2: ?(x) convolved with a Gaussian

Running RDD on various MOS's shows that good generators for MOS's are
those which tend to be farther from simple rational divisions of the
octave. It's not difficult to see the rationale for this: (proper)
MOS's can develop ambiguous intervals in one of 3 ways:

1) c gets too small, and you can't tell major and minor intervals apart anymore
2) s gets too small, and you can't tell major seconds and minor thirds
apart anymore (or intervals differing by s in general)
3) d (= |s-c|) gets too small, and you can't tell apart anymore the
intervals which Rothenberg called "ambiguous"

Note that all of these correlate to the MOS generator being tuned as a
subset of some EDO. For example, as c approaches 0, an N note MOS
starts approaching N-EDO. As s approaches 0, an N note MOS starts
approaching whatever EDO it is when L/s = Inf. As d approaches 0, an N
note MOS starts approaching whatever EDO it is when L/s = 2/1.

We can use this insight to consider the space of rank-2 tunings in
general, without considering any particular MOS as being
representative thereof. We'll generalize the above observations as
follows:

1) The generators for MOS's ought to try and avoid simple rational
divisions of the period.
2) Being close to a very simple rational, i.e. a generator close to
1/2-period, is worse than being close to a more complex rational, i.e.
a generator close to 100/171-period, when considering the ambiguity of
the MOS series at large.
3) Very complex rationals don't matter at all, because sooner or later
the MOS series will produce an MOS with so many notes that the whole
thing is unavoidably self-ambiguous.

The first two criteria basically lead us to Erv Wilson's golden
horograms/recurrent sequences. The last criterion changes things a
bit, and implies the following analogy: golden horograms:metastable
intervals::what we want:maxima of HE.

A good function which fits these criteria, taken from one of Gene's
suggestions from a while ago, is to look at Minkowski's question mark
function ?(x). More on that to come...

-Mike

🔗kraiggrady <kraiggrady@...>

It is important to not confuse the noble numbers of golden horograms with his recurrent series found via or derived from the Mt. Meru triangle. They are completely two different series. The latter are less evenly distributed in the continuum.
Since all the nobles are infinite fractions an MOS could only indeed converge at infinity.

On 20/03/12 11:05 PM, Mike Battaglia wrote:
> The first two criteria basically lead us to Erv Wilson's golden
> horograms/recurrent sequences. The last criterion changes things a
> bit, and implies the following analogy: golden horograms:metastable
> intervals::what we want:maxima of HE.

/^_,',',',_ //^/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/>

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

a momentary antenna as i turn to water
this evaporates - an island once again

🔗Mike Battaglia <battaglia01@...>

🔗kraiggrady <kraiggrady@...>

The Golden Horograms all include phi as apart of each formula. This can be found on each of the Horogram pages. Perhaps those more mathematically astute might already understand the pattern. I admit I have not worked with these formulas directly at all not having a context to need them.

Both Finnamore and I separately reorganized the Horograms along the lines of how they occur on the tree with a slight difference between the two of us .
Basically starting at the top of the tree and working its way down.
Erv listed them from smallest to largest as i don't think he pictured making more than the original 32. The problem with this is if you go down another layer to a set of 64 , one has to renumber them all to preserve their order.

If there are others interested in an alternative arrange along side the one that exist, i would be up for making that available. My copy i have numbered with Roman Numerals to tell the two sets apart.

Golden Meantone he did though got off of Mt. Meru as a secondary pattern and not the Horograms. He although surely understood what area it would have been in. It is too easy to confuse some of them. I would imagine that Hanson helped with the math formula found in Meruthree. One can see with the earlier Harrisonian Resonant Triads it required a more drawn out method. Both found on this page http://anaphoria.com/wilsonmeru.html

On 20/03/12 11:41 PM, Mike Battaglia wrote:
> On Tue, Mar 20, 2012 at 8:34 AM, kraiggrady<kraiggrady@...>
> wrote:
>> It is important to not confuse the noble numbers of golden horograms with
>> his recurrent series found via or derived from the Mt. Meru triangle. They
>> are completely two different series. The latter are less evenly distributed
>> in the continuum.
>> Since all the nobles are infinite fractions an MOS could only indeed
>> converge at infinity.
> Yeah, I'm talking only about the golden horograms ones. Like the
> golden MOS's that generalize golden meantone. Is "recurrent sequence"
> not the right word for that?
>
> I know the Mt. Meru thing is something else and has to do with
> proportional beating.
>
> -Mike
>
>
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/^_,',',',_ //^/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/>

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

a momentary antenna as i turn to water
this evaporates - an island once again

🔗kraiggrady <kraiggrady@...>

Happy Spring or Fall as it is here.

/^_,',',',_ //^/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/>

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

a momentary antenna as i turn to water
this evaporates - an island once again

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

On Mar 20, 2012, at 2:38 PM, genewardsmith <genewardsmith@...>
wrote:

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Note that all of these correlate to the MOS generator being tuned as a
> subset of some EDO. For example, as c approaches 0, an N note MOS
> starts approaching N-EDO. As s approaches 0, an N note MOS starts
> approaching whatever EDO it is when L/s = Inf. As d approaches 0, an N
> note MOS starts approaching whatever EDO it is when L/s = 2/1.

Without talking about EDOs, you seem to be saying that this avoids L/s
close to 1, 2, and infinity. Probably L/s near phi is grand.

Right. The point of the EDO thing, though, was to make the further point
that in these cases, L and s become commensurate, so they'll also be a part
of some rank-1 scale.

We can consider "all MOS's" of a generator by considering that the "s" of a
certain MOS is the "c" of the next-smallest MOS, and that the "d" of this
MOS is the "c" of the next MOS. In fact, all s's and c's and d's can be
thought of as Z-linear combinations of s's and c's and d's of the MOS 1L1s, or
most generally, of the period and generator.

Thus, we can simply look at the 1L1s MOS and note that if L/s for this MOS
is near a rational number, for -SOME- MOS down the line, there will be
intervals that are ambiguous. In fact, most generally, we can simply note
the same thing about the ratio of generator/period itself, with the caveat
that simple rationals are worst and larger rationals are OK. I'll call this
the g/p ratio.

L/s = phi is good for various MOS's, but it would be useful to be able to
assign every g/p a scalar quantifying how good it is. ?(x) will give us a
way to do that, as we'll see, and it clearly won't be much of a surprise
that generators leading to lots of L/s=phi MOS's down the line as maxima of
the resulting function (with a twist in the form of a free parameter
corresponding to something like "coarseness of hearing", which sometimes
makes the results diverge notably from L/s = exactly phi).

-Mike

🔗kraiggrady <kraiggrady@...>

🔗Mike Battaglia <battaglia01@...>

On Tue, Mar 20, 2012 at 3:30 PM, Mike Battaglia <battaglia01@...> wrote:
>
> L/s = phi is good for various MOS's, but it would be useful to be able to
> assign every g/p a scalar quantifying how good it is. ?(x) will give us a
> way to do that, as we'll see, and it clearly won't be much of a surprise
> that generators leading to lots of L/s=phi MOS's down the line as maxima of
> the resulting function (with a twist in the form of a free parameter
> corresponding to something like "coarseness of hearing", which sometimes
> makes the results diverge notably from L/s = exactly phi).

Gene suggested a long time ago that I use the derivative of ?(2^x),
convolved with some suitable holonomic function, to see if I could
come up with a quick and dirty curve that looks like Harmonic Entropy.
While the results did end up yielding minima at rationals and so on,
it wasn't a particularly nice looking graph in the sense that its
fields of attraction were a bit lopsided and jumpy. But ?(x) itself
certainly works wonders for analyzing MOS's, where it might be said to
play a role analogous to what the zeta function does for analyzing
harmony.

?(x) is this function here:

http://en.wikipedia.org/wiki/Minkowski's_question_mark_function

It's defined as a function of the continued fraction approximation of
any real number. Note that it's most "flat" around the rationals, and
that each region of flatness has a width inversely proportional to the
complexity of the rational. Unfortunately, we can't differentiate this
curve, because its derivative vanishes over the rationals. The
solution is what Gene suggested a while ago: convolve it first with a
Gaussian curve, which will make it absolutely continuous, and then
differentiate.

The one free parameter in this setup is the standard deviation of the
curve, which controls something like the "fineness" of our melodic
differentiate ability - it specifies how close an MOS needs to be to
an EDO before the listener starts to perceive the ambiguity. This is
vaguely metaphorically similar to the role to the real coordinate of
the zeta function, which specifies the weighting of the primes (and
integers).

Here's what we get if we convolve with a Gaussian of s=5 cents and
differentiate:

/tuning/files/MikeBattaglia/questconvs=5.png

Here are the -maxima- of this curve, e.g. the points maximally far
from rational numbers:

459.96 - father
501.36 - meantone
440.52 - sensi
330.96 - orgone
353.28 - maqamic
524.76 - mavila
257.76 - semaphore
274.2 - orwell
538.56 - avila/wilsec
213.96 - machine
183.84 - glacial
159.48 - opossum
140.88 - bleu

Here's the same thing, but for s=3 cents.

/tuning/files/MikeBattaglia/questconvs=3.png

Here's the list of all prominent maxima for this curve:

457.32 - father
504.24 - meantone
441.36 - sensi
332.52 - orgone in like 18-edo
495.12 - superpyth in like 17-edo
352.92 - maqamic
525.12 - mavila
321.84 - orgone
431.28 - sentry
259.32 - semaphore
365.04 - sephiroth
272.76 - orwell
539.64 - avila/wilsec
212.04 - machine
223.08 - slendroid
549.24 - ????
180.48 - tutone
187.56 - glacial
158.64 - opossum
140.52 - bleu
125.88 - negri
113.88 - miracle

Here's a really coarse curve at s=8 cents. Note the behavior - the
best meantone-sized generator is now being pushed away from the L/s =
phi point and towards the 12-EDO point. In other words, it's now
become more important to avoid really simple rationals, so it doesn't
matter as much if we develop a few ambiguous intervals of lesser
importance down the road. Keep in mind these statements still apply to
entire rank-2 tunings in general, though, and not any particular MOS.

/tuning/files/MikeBattaglia/questconvs=8.png

Minima are

459.24 - father
500.16 - meantone
328.32 - orgone
355.44 - beatles
524.88 - mavila
261.72 - superpelog
217.32 - machine
184.08 - glacial
162.24 - porcupine

Here's a superfine curve at s=1 cents. Too many minima to label:
/tuning/files/MikeBattaglia/questconvs=1.png

-Mike

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

🔗lobawad <lobawad@...>

🔗Mike Battaglia <battaglia01@...>

On Wed, Mar 21, 2012 at 7:41 PM, lobawad <lobawad@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > While the results did end up yielding minima at rationals and so on,
> > it wasn't a particularly nice looking graph in the sense that its
> > fields of attraction were a bit lopsided and jumpy.
>
> Now there is something I'd be very interested in seeing. I have maintained
> for some years now that, assuming we're thinking of "fields of attraction"
> as basically those regions in which we feel rational relationships between
> tones, these regions are "lopsided and jumpy", not nicely symmetrical, and
> it would be interesting to see how much the lumps coincide with the
> lumpinessess I (and others) perceive.

No, they were symmetrical. The graph was just stupid. You can see some
of them here:

/tuning-math/files/MikeBattaglia/%3F%28x%29%20convolved%20with%20g%28x%29/

Not a fan.

-Mike