Yahoo Tuning Groups Ultimate Backup tuning-math Kees integral tning example

Here's a worked out example of this integral/t(N) limit tuning
business, using Kees rather than Tenney metrics (so I have one less
dimension to worry about when integrating.) Working out the general
formula for various p-limits for these will allow us to simply plug in
the particular generators and get an answer, so I think I'll do that.

The limit as the bound N goes to infinity for the Kees metric, when
scaled, turns into an integral over the Kees unit ball K. So, if we
have <0 1 4| as our generator val, we want to take the integral of the
vector-valued function |0 x*(x+4y) y*(x+4y)> over the ball K. By
symmetry, we need only look at the y>=0 part, and we break that into two
double integrals:

int_{y = 0...1/p5} int_{x = -1/p3...0) |0 x*(x+4y) y*(x+4y)> dx dy

int_{y = 0...1/p5} int_{x = 0...(1-q5*y)/q3} |0 x*(x+4y) y*(x+4y)> dx dy

This gives an eigenvector

|0 5/12*(p5-2*p3)/(p5^2*p3^3) -5/24*(p5-8*p3)/(p5^3*p3^2)>

which in numerical terms is

|0 -.016459950271770033886 .068618437313876562583>

Solving for that as an eigenvector for the pure-octaves projection map
for 81/80 leads to the meantone tuning with a fifth of
696.2354308642524176 cents. This is the 5-limit Kees integral meantone
tuning, I guess you could call it. Similarly, there would be a Tenney
integral tuning.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> Here's a worked out example of this integral/t(N) limit tuning
> business...

I should add that the integral is the limit where you take unreduced
ratios as distinct--you count 3, and if it is below the limit, also 9/3.

The actual author of the strange treatise from which
this comes is unknown.

There seems to have been an actual, and typically
Pythagorean, attempt to state but conceal the main
mystery.

Certainly the material in it, according to the
Barbera, could have been put together in some form in
the fifth century BC or at the turn of the fourth
century BC,35 and reworked some centuries later.36 But
some of the content, and in particular the sly
reference to the Comma of Pythagoras, appear to come
from very ancient and unidentified Pythagorean sources
which cannot be traced today. No overt statement of
the important number is given, and even its
computation requires two successive arithmetical
operations, the carrying out of which would not even
occur to anyone who didn't know what he was looking
for in the first place. The nine-decimal value of the
universal constant, the Comma of Pythagoras, is
therefore concealed in this ancient text in a kind of
code, but one which is entirely unambiguous once it
has been recognized as such. The ancient text is so
extraordinarily dry, technical and boring, that only
expert musical theorists would ever have read it, and
of those, only a handful of initiates would have
deciphered the purposely concealed reference to one of
the greatest discoveries ever made in ancient science
and mathematics. The text therefore seems to have been
intended, amongst its other, more mundane discussions,
to preserve this secret Pythagorean (and originally
Egyptian) knowledge whilst hiding it so carefully that
its preservation would await discovery by the right
kind of person.

A value of the Comma of Pythagoras computable to an
astonishing nine decimal places appears in the form of
an arithmetical fraction preserved in the ancient
Greek Pythagorean treatise Katatomeï¿½ Kanonos (Division
of the Canon).33 There we are told that the number
531,441 is greater than twice 262,144. Twice 262,144
equals 524,288,though this number is not actually
stated. The ratio is not computed in the text either,
but if we carry out the division we obtain the number
1.013643265, namely, the Comma of Pythagoras expressed
to nine decimal places. The Greek text is coy in the
extreme, giving the information in such an obscure
manner that only someone initiated into its
significance could be expected to have any idea what
was being said. The only explanatory comment earlier
in the passage is: 'Six sesquioctave intervals are
greater than one duple interval.' One has to be fairly
well educated in these matters to have any idea at all
of what the author means! Andr'e Barbera, the
immensely learned modern editor and translator of this
text, has apparently not noticed that this passage,
which he has translated from no less than three
separate versions, in fact presents obliquely the
mystery of the Comma of Pythagoras. He does not
mention the Coma, has evidently never carried out the
necessary multiplication and division to arrive at it,
and gives no indication whatever that he is aware of
the special significance of the passage.34 If Barbera,
who is probably the world's expert on this text, has
no inkling of its true importance, then it is no
wonder that no one else until now has either.

I have done a great deal of work on the Comma of
Pythagoras over many years, and I found it necessary
to give a name to the decimal increment 0.0136 itself;
I have named it the Particle of Pythagoras, which I
hope will be found acceptable by others -- should
anyone but myself ever wish to discuss it, of course.
I believe the numerical coefficient of this Particle,
136, is related to the 136 degrees of freedom of the
electron discussed by the famous physicist, the late
Sir Arthur Eddington,37 and that the number plus one
gives the Fine Structure Constant of nuclear physics,
which is 137.38 (The Fine Structure Constant is a
universal natural constantly greatly beloved by
physicists, although hardly anyone else has ever heard
of it.) I have discovered relationships between this
natural constant and several others such as phi, e,
and pi. However, such discussions are too lengthy and
distracting for inclusion here. I mention this only so
readers will understand how important the Particle of
Pythagoras really is. Essentially, one could say that
it expresses the minute discrepancy between the ideal
and the real. For the pyramid builders to incorporate
it as the identical discrepancy just discussed in the
Sirius and pyramid correlation should be interpreted
as their way of signaling to us: 'This is a symbolic
representation of a real cosmic fact.'

Musical theorists will be well aware that the
discrepancy 0.0136 necessitates the tuning technique
known as 'equal tempering'. I have published an
account of the invention of the Equal Temperament
system elsewhere.39 As if to tease us, the builders of
the pyramids appear to have left a microscopic
discrepancy in the correlation precisely equal to a
universal numerical constant. For the Comma of
Pythagoras is implicit in the structure of the
Universe itself, and is absolute throughout the
cosmos.

However, another point should be made about this
correlation. That is, the ratio of 1.053 is actually
the precise value of the sacred fraction 256/243
mentioned by Macrobius at the turn of the fourth/fifth
centuries AD, who describes its use in harmonic theory
by people who to him were 'the ancients'.40 The
fraction was also mentioned in antiquity by the
mathematical, harmonic, and philosophical writers
Theon of Smyrna (second century AD), Gaudentius,
Chalcidius (fourth century AD), and Proclus (fifth
century AD, for whom see Appendix II of this book, as
he seems to have been aware of the Sirius Mystery).41
One must ask how it is that this precise value of
1.053, which we see is astrophysically the precise
ratio between the masses of Sirius B and our sun, was
mentioned so frequently in the works of writers
dealing in esoteric astronomical lore in ancient
times, one of whom (Macrobius) is prominently
identified with the heliocentric theory, and another
of whom (Proclus) appears to have been initiated into
the Sirius Mystery and specifically mentions the
existence of important but invisible heavenly bodies.
Especially in the case of Proclus, who appears to have
known of the existence of Sirius B, to have him also
mentioning this number, exact to three decimal places,
specifying its mass stretches credulity beyond its
limits. Surely the coincidences are multiplying to an
impossible degree if we are to view this as all being
by chance. (As regards him, I recently discovered the
following passage in an old book about the pyramids:
'The hieroglyph for Sirius is, oddly enough, the
triangular face of a pyramid. Dufeu [a
nineteenth-century French author who wrote about the
pyramids* ] and others suppose that the pyramid may
have been dedicated to this venerated star... Proclus
relates the belief in Alexandria that the pyramid was
used for observations of Sirius.'ï¿½ Unfortunately, this
has come to light just before going to press, so I
have not been able to locate the passage in the works
of Proclus.)

--------------------------------------------------------------------------------
If, as Bauval asserts, the pyramids represent the
"belt" of Orion, and the triangular face of each
pyramid is meant to invoke Sirius [in the
constellation of "Canis Major"], then the pyramids
themselves are the literal embodiment of the gods that
those two constellations represent ...
--------------------------------------------------------------------------------

But there is also this purely cosmological question:
why is it that our sun and the star Sirius B have a
mass ratio of 1.053 in any event? For the fraction
256/243 of which 1.053 is the decimal expression does
appear to have a universal harmonic status. So by
stumbling upon this coincidence we may have uncovered
some hitherto unsuspected astrophysical harmonical
value in operation between two neighbouring stars. I
don't believe anyone before has found a precise
numerical correlation which could extend the notion of
a 'harmony of spheres' beyond our solar system, to
link it with a neighboring one. But this appears to be
the case here. Perhaps it has something to do with the
inherent nature of white dwarf stars and their
dimensions vis 'a vis normal stars like our sun, and
this ratio would thus occur throughout the Universe
frequently. It makes more sense to view the
correlation as one which appeals to underlying
fundamentals of cosmic structures than to view it as a
special case applying only to Sirius B and our sun.
But even so, the correlation is extraordinary and so
precise that it suggests whole avenues of research and
offers hope of absolute numerical expressions
recurring in the cosmos where none had been suspected.
And by discovering this, we can only be pleased, since
it enables us to discern some scaling elements of
concealed structure which may be cosmic in scope. I
hope cosmologists will not neglect this observation. I
believe it demonstrates that the Universe has more
structure than we thought, and that that structure can
be so precisely articulated that it can generate an
exact value of this kid as a ratio between neighboring
stellar bodies. For Sirius Band our sun, in terms of
the cosmos, are certainly neighbors. And it all comes
down to the question: how is it that two stars 8.7
light-years apart can have a mass ratio which is not
random but which expresses a universal harmonic value
which is precise to three decimal points? It can only
be because the astrophysics of stars and their
evolutionary development (such as in the formation of
a white dwarf) follow certain harmonic laws which we
have not yet suspected, much less expressed. And we
should not overlook the fact that the universal
harmonic fraction concerned is not one which today
receives a lot of attention. This in turn indicates
that it is ancient harmonic theory that should be
dusted off and studied for clues as to what is going
on. Many of us have believed this for years, even
without this evidence.42 One of my 'hobbies' is trying
to get to grips with ancient harmonic theory, which is
why I took the fraction seriously enough to work out
its decimal expression and notice its importance;
needless to say, the decimal value of the fraction
does not appear in Macrobius, and only someone
actually doing the division and holding up the result
beside the mass ratio value of Sirius B and our sun
for comparison would ever have noticed anything at
all.

The implication of all this is that different types of
stars express different harmonical values in a
surprisingly precise way. But why should stellar
evolution not have a harmonical nature and structure
to it? This will probably be found to be relevant to
the concept of the 'stellar mass function' which
astrophysicists speculate about. It may be found, for
instance, that the difficulties of star formation in
the first place are regularly overcome by some kind of
binary-star formation; in our own solar system we
could view the planet Jupiter as an incipient brown
dwarf star in the making. And in 1983 I published an
account of the possible existence of the possible
existence of another actual small invisible star in
our own solar system, which was first suggested in
1977 by the radioastronomer E. R. Harrison because of
a perturbation which he discovered that our solar
system was exercising on six particular pulsars in a
small region of the sky.43 Star formation might thus
involve a binary process in far more cases than we
think, possibly in all. Binary stars may only be able
to coexist according to specific harmonic relations,
just as certain musical notes when struck together are
consonant when they are in specific proportions such
as the musical fifth or fourth.

Fundamental to an improved theory will be a
realization that star types are expressions or
articulations of harmonical ratios and frequencies and
that however much variance they may show, even those
variations are always methodical and coherent. Any
lack of method and coherence which appears in these
cosmic occurrences is thus due not to any lack of
structure in the Universe, but is due rather to our
own lack of understanding of it. We have learned this
lesson in any case by discovering that even chaos is
ordered, with the marvelous development of Chaos
Theory.

More important is Complexity Theory, which is still in
the process of establishing itself. It deals with the
sudden onset or loss of long-range order by what
scientists call 'phase transitions' and 'symmetry
breaking'. I should point out that the mass ratio of
Sirius B and our sun demonstrates that long-range
order exists between the two solar systems, extending
over a distance of 8.7 light-years, which can only be
explained by conceiving of the two solar systems
inhabiting the same 'cell' of space. And if that is
the case, then we know from Complexity Theory that a
strange form of what resembles 'instantaneous
communication' subsists in such 'cells' whereby huge
macro-regions of space behave as if their elements
were not separated by spatial or temporal distance,
and the 'cell' engages in what is called
'self-organization'. Such a 'cell' turns into what is
called by scientists a "dissipative structure' which
turns disorder into order.

The 1977 Nobel laureate for Chemistry, Professor Ilya
Prigogine, whom I have visited in Brussels, has
stressed that the onset of complexity in a system can
result in the instantaneous extension of long-range
order by a magnitude of ten million or more, as is
easily demonstrated in the onset so-called B'enard
Cells caused by thermal convection in a fluid.44 This
enormous expansion of order is equivalent to one fifth
of the population of Britain suddenly and
spontaneously adopting the same body posture at the
same instant while having no direct contact with one
another. Imagine ten million people suddenly standing
on their heads for no apparent reason. An outside
observer might call this uncontrollable turbulence,
for a hairdresser doing this would start cutting
toenails, drivers would lose control of their
vehicles, tennis players would invariably hit the net
... It would be chaos. But nevertheless, it would not
be denied that ten million people had stood on their
heads at the same time due to some mysterious
long-range ordering principle which extended across
the whole country. This turbulent chaos is in fact a
spontaneous creation of complexity. For a moment the
ten million people had nothing in common about their
posture, but now there is no denying that there is an
immense complexity in existence -- a connection
suddenly exists which did not previously exist -- a
coherence is established. Ten million simultaneous,
complex, intricate and criss-crossing links exist: the
ten million have all suddenly stood on their heads all
just like each other. This is analogous to what
actually happens in a B'enard Cell, where ten million
molecules instantaneously align.

The discovery of the significance of the 1.053 mass
ratio between Sirius B and our sun suggests that our
solar system and the Sirius system are elements of a
larger entity which is a self-organizing open system
-- what is called in thermodynamics a "dissipative
structure far from thermal equilibrium'. But let us
give it an actual name. I propose to call it the
Anubis Cell. The Anubis Cell clearly has long-range
order extending over at least 8.7 light-years. Since
all such structures increase their order and eliminate
their disorder, a continuous ordering process must
have been in operation inside the Anubis Cell since at
least the formation of either our sun or Sirius B's
condensation as a white dwarf, whichever was later.
Long-range order has thus operated between the systems
presumably for billions of years. Under such
circumstances, both solar systems must have a shared
movement in relation to the Galaxy. The two systems
must also be in continuous harmonic resonance with one
another. It may be presumed that a significant
perturbation of one would affect the other, and that
this could apply to very high frequency events
including the 'mental', 'thought' or 'information'
events. Membership of the same cosmic cell implies the
potential for the modulation of some shared field (of
an unknown type, but possibly not unlike the 'quantum
potential' proposed by my friend the late David Bohm
to solve the Einstein-Podolsky-Rosen Paradox in
physics -- a subject we cannot go into here!) for the
purposes of communication between systems. Let us call
it here the 'cell potential'. In other words,
electromagnetic amplitude modulation such as radio,
for signaling in the traditional manner, may be
unnecessary. The strange aspects of long-range order
may mean that in some way yet to be discovered by us,
instantaneous communication between systems might be
possible, which would seem to overcome the limitations
of the speed of light for communication between
them.45 Psychic communication and even nonmaterial
interactions of souls might be possible. The ancient
Egyptians said that the Sirius system was where people
go when they die. The Dogon say the same thing, and
perhaps the Sirius system is the actual location of
'the Other World' in more senses than one. Inspiration
may even come to humans on Earth from the Sirius
system by harmonic resonance articulated by the (still
undefined) Anubis Field of the Anubis Cell, and this
might be instantaneously 'transmitted' not as a signal
but by harmonic resonance response within the
continuous Anubis Field subsisting within this cosmic
cell.

We have similar phenomena throughout nature: even the
lowly sponge has been found to have a physically
impossible 'conduction velocity' for stimulus
transmission from one end of its body to another. So
bizarre were these findings that the three Canadian
scientists involved in studying it were forced to
suggest that a sponge was like a single giant nerve
cell so that: 'the entire conduction system could act
as a single neuron'.46 If a simple sponge can defy
time and space at the bottom of the sea, surely the
Anubis Cell can do so within the Galaxy. The Anubis
Cell may be analogous to a macroscopic 'neuron' seen
from the point of view of Galactic scale. And this
beings us to another possibility: the Anubis Cell may
be alive. The vast Ordering Principle may be an
Entity. Even if it were not an Entity to start with,
it must long ago have spontaneously generated
considerable consciousness, if only by weighted
connections in parallel distributed processing.47 And
we can be sure that it has had a few billion years to
do its thinking.

Returning now to our observations of the pyramid
measurements, the value of 1.0678 given there may thus
also be a double-tease by the builders. For not only
does it vary from the precise mass ratio of Sirius B
and our sun by a tiny amount equal to one harmonic
natural constant, but it varies from another harmonic
natural constant by that same exact amount. One could
then say that the builders were only intending to
express the latter, ignorant of the astrophysical
ratio, but the following additional correlations
relating to the Sirius system discourage such a
notion.

What about the respective radii of Sirius B and our
sun? Are they indicated by the two pyramids? Turning
to a different form of measure, the slope angles of
the respective pyramids, we find that the sides of the
Great Pyramid originally had slope angles of about 51ï¿½
51' to the ground, according to Edwards,48 which is
equal to 51.866ï¿½, whereas the Pyramid of Khephren had
slightly steeper slopes of 52ï¿½20'according to
Edwards,49 with is equal to 52.333ï¿½. The slope of the
Great Pyramid is thus 0.0089 less than the slope of
the Pyramid of Khephren, which yields a value
equivalent to the relative radius of Sirius B to that
of our sun accurate to 0.0011. The appearance of these
two correspond-ences act as a kind of
cross-correlation on each other, since one is accurate
to0.014 and the other is accurate to0.0011. This
significantly reduces the chance of coincidence being
at work in these correlations, as there is not only
one such correlation but a pair. However, there are
two more to come.

I am not insisting that these correlations are
intended, but suggesting that they may be, considering
the established connections already noted between the
pyramids and the Sirius cult.

From the latest information about Sirius C in their
1995 article, Benest and Duvent state that Sirius C
cannot be much more than about 0.05 of the mass of our
sun (and of Sirius B).50 Using one of the simple
length measures of the kind which seemed to indicate
the relative masses of Sirius B and our sun, the mass
of Sirius C may be indicated by the height of the
missing pyramidion (top point) of the Great Pyramid.
For it was 31 feet and the original total height of
the pyramid was 481.4 feet, according to Edwards,51 so
that the height of the pyramidion was 0.0643 of the
total height of the pyramid, corresponding to within
0.01 of the 0.05 of solar mass suggested for Sirius C
in1995.

This is thus the third Sirius astrophysical
measurement correlation accurate to at least 0.01 to
be found in the Giza pyramid complex.

What about the third pyramid in the Giza complex,
known as the Pyramid of Mycerinus? What significance
could it have in this scheme of things? Edwards says
that the Pyramid of Mycerinus originally had a height
of 218 feet.52 The height of the Pyramid of Khephren
was originally 471 feet, according to Edwards.53 The
ratio of these two heights is 2.160. We note from
Benest and Duvant that the latest estimates of the
ratio of the masses of our sun and Sirius A is 2.14.54
The correspondence is thus accurate to within 0.02.
This is a fourth possible correspondence.

Can it be, therefore, that the pyramid complex at Giza
is representing to us, among many other things such as
the value of pi and the dimensions of our Earth, the
relative masses of the three stars in the Sirius
system? They all seem to be there, accurate to the
second or third decimal. Nor is that likely to exhaust
the possibilities. But any further discussions will
have to be left for another time.

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What's it a worked example of? The pure octaves average tuning, but with Kees weighting?

> The limit as the bound N goes to infinity for the Kees metric, when > scaled, turns into an integral over the Kees unit ball K. So, if we
> have <0 1 4| as our generator val, we want to take the integral of the > vector-valued function |0 x*(x+4y) y*(x+4y)> over the ball K. By > symmetry, we need only look at the y>=0 part, and we break that into two
> double integrals:

Where does this come from? x+4y from <0 1 4|?

> int_{y = 0...1/p5} int_{x = -1/p3...0) |0 x*(x+4y) y*(x+4y)> dx dy
> > +
> > int_{y = 0...1/p5} int_{x = 0...(1-q5*y)/q3} |0 x*(x+4y) y*(x+4y)> dx dy
> > This gives an eigenvector > > |0 5/12*(p5-2*p3)/(p5^2*p3^3) -5/24*(p5-8*p3)/(p5^3*p3^2)>

What's it an eigenvector of?

> which in numerical terms is
> > |0 -.016459950271770033886 .068618437313876562583>
> > Solving for that as an eigenvector for the pure-octaves projection map > for 81/80 leads to the meantone tuning with a fifth of > 696.2354308642524176 cents. This is the 5-limit Kees integral meantone > tuning, I guess you could call it. Similarly, there would be a Tenney > integral tuning.

So is it an eigenmonzo or what?

Anyway, I can't follow the above, but I've worked out what the optimal fifth for the Tenney-weighted STD should be on paper:

(x+y+y*x-2*x**2)/2/(x**2+y**2-x*y)

octaves where x=1/log2(3) and y=4/log2(5) octaves (note the 4). (It's a verified Python expression, and so follows Python rules for precedence and so on.) If Maple can solve your eigenvector problem symbolically (as I assume it can) you can check that these really are identical.

Obviously multiply by 1200 to get cents.

Graham

Gene Ward Smith wrote:
> Here's a worked out example of this integral/t(N) limit tuning > business, using Kees rather than Tenney metrics (so I have one less > dimension to worry about when integrating.) Working out the general > formula for various p-limits for these will allow us to simply plug in > the particular generators and get an answer, so I think I'll do that.
> > The limit as the bound N goes to infinity for the Kees metric, when > scaled, turns into an integral over the Kees unit ball K. So, if we
> have <0 1 4| as our generator val, we want to take the integral of the > vector-valued function |0 x*(x+4y) y*(x+4y)> over the ball K. By > symmetry, we need only look at the y>=0 part, and we break that into two
> double integrals:
<snip>

Incidentally, in this post Gene uses x and y. In my reply I defined the fifth in terms of x and y. But my x and y are not related to Gene's x and y.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> What's it a worked example of? The pure octaves average tuning,
but
> with Kees weighting?

It's a worked example of 5-limit meantone in the Kees integral tuning,
which integrates monz's beloved floating point monzos, |0 x y>.

The number of "generator steps" for |0 x y> in meantone is
<0 1 4|0 x y> = x+4*y; generators steps times the floating monzo is
then the scalar product (x+4*y)*|0 x y> = |0 x*(x+4*y) y&*x+4*y)>.
Integrating this over the Kees unit ball gives an eigenvector for the
tuning with eigenvalue 1; alongside the other such eigenvector, which
is |1 0 0> since octaves are pure, and |-4 4 -1> for the zero
eigenvalue since it is meantone, we can define a 3x3 projection
operator matrix which projects 5-limit JI onto a meantone tuning.

This may seem not to make sense because of the floating
point "monzos", but consider that we could take a ten-digit
approximation, so that everything was a rational number with
denominator 10^10. Now multiply through by 10^10, and take the region
we are averaging over to be Kees distance less than log2(10^10), and
it simply scales (since all we care about is the eigenvector up to a
scalar proportionality constant.)

> > The limit as the bound N goes to infinity for the Kees metric,
when
> > scaled, turns into an integral over the Kees unit ball K. So, if
we
> > have <0 1 4| as our generator val, we want to take the integral
of the
> > vector-valued function |0 x*(x+4y) y*(x+4y)> over the ball K. By
> > symmetry, we need only look at the y>=0 part, and we break that
into two
> > double integrals:
>
> Where does this come from? x+4y from <0 1 4|?

Yes.

> > int_{y = 0...1/p5} int_{x = -1/p3...0) |0 x*(x+4y) y*(x+4y)> dx dy
> >
> > +
> >
> > int_{y = 0...1/p5} int_{x = 0...(1-q5*y)/q3} |0 x*(x+4y) y*(x+4y)
> dx dy
> >
> > This gives an eigenvector
> >
> > |0 5/12*(p5-2*p3)/(p5^2*p3^3) -5/24*(p5-8*p3)/(p5^3*p3^2)>
>
> What's it an eigenvector of?

The 3x3 matrix which projects the 5-limit down to this tuning of
meantone.

> So is it an eigenmonzo or what?

A floating point one.

The interest in all this at this point is not to define a new tuning,
or find a good way to compute SDT, but because this way of arriving
at STD has a clear justification in terms of why the tuning might be
considered optimal. Now all that is needed is a proof it *is* SDT.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> What's it a worked example of? The pure octaves average tuning,
but
> with Kees weighting?

It's a worked example of 5-limit meantone in the Kees integral tuning,
which integrates monz's beloved floating point monzos, |0 x y>.

Yes.

The 3x3 matrix which projects the 5-limit down to this tuning of
meantone.

> So is it an eigenmonzo or what?

A floating point one.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> What's it a worked example of? The pure octaves average tuning,
but
> with Kees weighting?

It's a worked example of 5-limit meantone in the Kees integral tuning,
which integrates monz's beloved floating point monzos, |0 x y>.

Yes.

The 3x3 matrix which projects the 5-limit down to this tuning of
meantone.

> So is it an eigenmonzo or what?

A floating point one.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> What's it a worked example of? The pure octaves average tuning,
but
> with Kees weighting?

It's a worked example of 5-limit meantone in the Kees integral tuning,
which integrates monz's beloved floating point monzos, |0 x y>.

Yes.

The 3x3 matrix which projects the 5-limit down to this tuning of
meantone.

> So is it an eigenmonzo or what?

A floating point one.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> What's it a worked example of? The pure octaves average tuning,
but
> with Kees weighting?

It's a worked example of 5-limit meantone in the Kees integral tuning,
which integrates monz's beloved floating point monzos, |0 x y>.

Yes.

The 3x3 matrix which projects the 5-limit down to this tuning of
meantone.

> So is it an eigenmonzo or what?

A floating point one.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> What's it a worked example of? The pure octaves average tuning,
but
> with Kees weighting?

It's a worked example of 5-limit meantone in the Kees integral tuning,
which integrates monz's beloved floating point monzos, |0 x y>.

Yes.

The 3x3 matrix which projects the 5-limit down to this tuning of
meantone.

> So is it an eigenmonzo or what?

A floating point one.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> What's it a worked example of? The pure octaves average tuning,
but
> with Kees weighting?

It's a worked example of 5-limit meantone in the Kees integral tuning,
which integrates monz's beloved floating point monzos, |0 x y>.

Yes.

The 3x3 matrix which projects the 5-limit down to this tuning of
meantone.

> So is it an eigenmonzo or what?

A floating point one.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> What's it a worked example of? The pure octaves average tuning,
but
> with Kees weighting?

It's a worked example of 5-limit meantone in the Kees integral tuning,
which integrates monz's beloved floating point monzos, |0 x y>.

Yes.

The 3x3 matrix which projects the 5-limit down to this tuning of
meantone.

> So is it an eigenmonzo or what?

A floating point one.

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> >> What's it a worked example of? The pure octaves average tuning, > but >> with Kees weighting?
> > It's a worked example of 5-limit meantone in the Kees integral tuning,
> which integrates monz's beloved floating point monzos, |0 x y>.

Is anyone else seeing multiple copies of this and other messages from the Yahoo lists? Can anything be done about it?

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> What's it a worked example of? The pure octaves average tuning,
but
> with Kees weighting?

It's a worked example of 5-limit meantone in the Kees integral tuning,
which integrates monz's beloved floating point monzos, |0 x y>.

Yes.

The 3x3 matrix which projects the 5-limit down to this tuning of
meantone.

> So is it an eigenmonzo or what?

A floating point one.

>> It's a worked example of 5-limit meantone in the Kees integral tuning,
>> which integrates monz's beloved floating point monzos, |0 x y>.
>
>Is anyone else seeing multiple copies of this and other messages from
>the Yahoo lists? Can anything be done about it?

Heh. I got about 30 copies of this message.

I've been seeing it on tuning, too.

-Carl

Is anyone else seeing multiple copies of this and other messages from the Yahoo lists? Can anything be done about it?

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> >>What's it a worked example of? The pure octaves average tuning, > but >>with Kees weighting?
> > It's a worked example of 5-limit meantone in the Kees integral tuning,
> which integrates monz's beloved floating point monzos, |0 x y>.

Sure, but what's Kees integral tuning?

> The number of "generator steps" for |0 x y> in meantone is
> <0 1 4|0 x y> = x+4*y; generators steps times the floating monzo is
> then the scalar product (x+4*y)*|0 x y> = |0 x*(x+4*y) y&*x+4*y)>.
> Integrating this over the Kees unit ball gives an eigenvector for the > tuning with eigenvalue 1; alongside the other such eigenvector, which > is |1 0 0> since octaves are pure, and |-4 4 -1> for the zero > eigenvalue since it is meantone, we can define a 3x3 projection > operator matrix which projects 5-limit JI onto a meantone tuning.

I think I get that. What's the Kees unit ball? And why define this 3x3 matrix? If you know this floating point interval is unchanged by the tempering, surely you can solve very easily for the generator size.

The equation must be

XH = XN0/N0[0] + XN1g

where

X is the eigenvector

N0 is the unweighted period mapping

N0[0] is the number of periods to the octave

N1 is the unweighted generator mapping

g is the generator size

that gives

g = X(H-N0/N0[0])/XN1

from which I can deduce that your p3 is log2(3) and p5 is log2(5). I checked it on paper and, for the meantone example, it's exactly right!

The next step is to find the eigenvector in terms of the mapping, instead of numbers.

I can't solve for X knowing g. How does knowing the eigenvalue is 1 help?

The Tenney STD optimal generator, BTW, is

g = -cov(M0,M1)/[N0[0]*var(M1)]

where cov is the covariance, M0 is the weighted period mapping, M1 is the weighted generator mapping, N0[0] is the number of periods to the octave again, and var is the variance.

> This may seem not to make sense because of the floating > point "monzos", but consider that we could take a ten-digit > approximation, so that everything was a rational number with > denominator 10^10. Now multiply through by 10^10, and take the region > we are averaging over to be Kees distance less than log2(10^10), and > it simply scales (since all we care about is the eigenvector up to a > scalar proportionality constant.)

Fortunately I don't see any reason whey floating point vectors shouldn't work, so I don't have to be confused by this explanation.

<snip>

>>So is it an eigenmonzo or what?
> > A floating point one.
> > The interest in all this at this point is not to define a new tuning, > or find a good way to compute SDT, but because this way of arriving > at STD has a clear justification in terms of why the tuning might be > considered optimal. Now all that is needed is a proof it *is* SDT.

Yes, I've tried to do this kind of thing for TOP-RMS but couldn't get it to work. Partly I was trying to find the error rather than the tuning, and I suppose the tuning is easier to get to converge.

Graham

Heh. I got about 30 copies of this message.

I've been seeing it on tuning, too.

-Carl

Gene Ward Smith wrote:

> This gives an eigenvector > > |0 5/12*(p5-2*p3)/(p5^2*p3^3) -5/24*(p5-8*p3)/(p5^3*p3^2)>

If the only point of this vector is that it behaves like an eigenmonzo (its just and tempered tunings would be the same if it were an interval) we can happily remove common factors. So it becomes

|0 2*(p5-2*p3)/p3 -(p5-8*p3)/p5>

which is much simpler. It may be significant that the denominators are consistent with Tenney weighting.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> What's it a worked example of? The pure octaves average tuning,
but
> with Kees weighting?

It's a worked example of 5-limit meantone in the Kees integral tuning,
which integrates monz's beloved floating point monzos, |0 x y>.

Yes.

The 3x3 matrix which projects the 5-limit down to this tuning of
meantone.

> So is it an eigenmonzo or what?

A floating point one.

Gene Ward Smith wrote:

> The interest in all this at this point is not to define a new tuning, > or find a good way to compute SDT, but because this way of arriving > at STD has a clear justification in terms of why the tuning might be > considered optimal. Now all that is needed is a proof it *is* SDT.

I've done it! It's all on paper, but I've proved the two methods are identical in the 5-limit for any temperament class. Nothing clever about it, but very fiddly.

Graham

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> Is anyone else seeing multiple copies of this and other messages from
> the Yahoo lists? Can anything be done about it?
>

I have the strangest feelings of...deja vu.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Gene Ward Smith wrote:
>
> > The interest in all this at this point is not to define a new
tuning,
> > or find a good way to compute SDT, but because this way of arriving
> > at STD has a clear justification in terms of why the tuning might
be
> > considered optimal. Now all that is needed is a proof it *is* SDT.
>
> I've done it! It's all on paper, but I've proved the two
> methods are identical in the 5-limit for any temperament
> class. Nothing clever about it, but very fiddly.

Great work, Graham! I've been trying to see my way to a clever,
unfiddly proof but to hell with it.

What did you do?

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> >>Gene Ward Smith wrote:
>>
>>
>>>The interest in all this at this point is not to define a new > > tuning, > >>>or find a good way to compute SDT, but because this way of arriving >>>at STD has a clear justification in terms of why the tuning might > > be > >>>considered optimal. Now all that is needed is a proof it *is* SDT.
>>
>>I've done it! It's all on paper, but I've proved the two >>methods are identical in the 5-limit for any temperament >>class. Nothing clever about it, but very fiddly.
> > > Great work, Graham! I've been trying to see my way to a clever, > unfiddly proof but to hell with it.

We still need something cleverer for arbitary prime limits.

> What did you do?

Plugged in two variables for the arbitrary entries in the mapping, solved the integrals, and finding the implied generator size. Then showing it was the same as the generator for the STD tuning.

The interesting intermediate step is that eigenvector. In general, it's

(5/24p3^2p5^2)|0 (2ap5-bp3)/p3 (2bp3-ap5)/p5>

where the generator mapping is <0 a b|. It's still an invariant of the temperament if you renormalize it

|0 (2ap5-bp3)/p3 (2bp3-ap5)/p5>

If it were an interval, the tuning would be

(2ap5-bp3)+(2bp3-ap5) = ap5 + bp3

which is proportional to the sum of the weighted mapping (and so could be made exact by another renormalization).

Graham

--- In tuning-math@yahoogroups.com, Dan Amateur <xamateur_dan@...> wrote:
>
> The actual author of the strange treatise from which
> this comes is unknown.

I gather by what you wrote later that the "strange treatise"
you're referring to is the one known by its Latin name as
_Sectio Canonis_.

> There seems to have been an actual, and typically
> Pythagorean, attempt to state but conceal the main
> mystery.
>
> Certainly the material in it, according to the
> Barbera, could have been put together in some form in
> the fifth century BC or at the turn of the fourth
> century BC,35 and reworked some centuries later.36 But
> some of the content, and in particular the sly
> reference to the Comma of Pythagoras, appear to come
> from very ancient and unidentified Pythagorean sources
> which cannot be traced today. No overt statement of
> the important number is given, and even its
> computation requires two successive arithmetical
> operations, the carrying out of which would not even
> occur to anyone who didn't know what he was looking
> for in the first place. The nine-decimal value of the
> universal constant, the Comma of Pythagoras, is
> therefore concealed in this ancient text in a kind of
> code, but one which is entirely unambiguous once it
> has been recognized as such. The ancient text is so
> extraordinarily dry, technical and boring, that only
> expert musical theorists would ever have read it, and
> of those, only a handful of initiates would have
> deciphered the purposely concealed reference to one of
> the greatest discoveries ever made in ancient science
> and mathematics. The text therefore seems to have been
> intended, amongst its other, more mundane discussions,
> to preserve this secret Pythagorean (and originally
> Egyptian) knowledge whilst hiding it so carefully that
> its preservation would await discovery by the right
> kind of person.
>
>
>
> A value of the Comma of Pythagoras computable to an
> astonishing nine decimal places appears in the form of
> an arithmetical fraction preserved in the ancient
> Greek Pythagorean treatise Katatome Kanonos (Division
> of the Canon).33 There we are told that the number
> 531,441 is greater than twice 262,144. Twice 262,144
> equals 524,288,though this number is not actually
> stated. The ratio is not computed in the text either,
> but if we carry out the division we obtain the number
> 1.013643265, namely, the Comma of Pythagoras expressed
> to nine decimal places.

I don't really understand your point here. The
pythagorean-comma is a rational number, which means
that in decimal form it will either terminate or
repeat some digits infinitely. What's the point of
stopping at nine decimal places?

I calculated 531441/524288 by hand and got the
following decimal value to 19 places with no remainder:
1.0136432647705078125.

> The Greek text is coy in the
> extreme, giving the information in such an obscure
> manner that only someone initiated into its
> significance could be expected to have any idea what
> was being said. The only explanatory comment earlier
> in the passage is: 'Six sesquioctave intervals are
> greater than one duple interval.' One has to be fairly
> well educated in these matters to have any idea at all
> of what the author means! Andr'e Barbera, the
> immensely learned modern editor and translator of this
> text, has apparently not noticed that this passage,
> which he has translated from no less than three
> separate versions, in fact presents obliquely the
> mystery of the Comma of Pythagoras. He does not
> mention the Coma, has evidently never carried out the
> necessary multiplication and division to arrive at it,
> and gives no indication whatever that he is aware of
> the special significance of the passage.34 If Barbera,
> who is probably the world's expert on this text, has
> no inkling of its true importance, then it is no
> wonder that no one else until now has either.

I gave a summary of the _Sectio Canonis_ explanation
of the pythagorean-comma here:

/tuning/topicId_18146.html#22095

I do have more than one version of the Greek text,
but have not really explored it much beyond some of
Berbera's other articles about it, in which he quotes
some of the Greek text.

However, in that post i cite Barker's excellent
English translation of it, and also Bower's English
translation of Boethius, who summarized the discussion
of the comma from the _Sectio Canonis_ in his own treatise.

I fail to see what so "coy" about it. The entire text
of the _Sectio Canonis_ is an explanation of musical
tuning carried out by division of the monochord, and
the presentation of the pythagorean-comma is an essential
part of that.

My copy of the book is not handy right now, so i cannot
consult it. If i get a chance to take a look at it soon,
i'll have more to post here.

-monz
http://tonalsoft.com
Tonescape microtonal music software

Thanks for the feedback, I look forward to any other
thoughts you'd like to share on Pythagoras' Comma!

--- monz <monz@tonalsoft.com> wrote:

>
> --- In tuning-math@yahoogroups.com, Dan Amateur
> <xamateur_dan@...> wrote:
> >
> > The actual author of the strange treatise from
> which

__________________________________________________
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Tired of spam? Yahoo! Mail has the best spam protection around
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Hi Dan,

--- In tuning-math@yahoogroups.com, Dan Amateur <xamateur_dan@...> wrote:

> Thanks for the feedback, I look forward to any other
> thoughts you'd like to share on Pythagoras' Comma!

I'd appreciate some feedback from you too!
(did you read all of my post?)

Particularly, why such emphasis was placed on the
"correct to 9 decimal places" decimal value of the
pythagorean-comma. The totally accurate 19-place
decimal value is 1.0136432647705078125.

-monz
http://tonalsoft.com
Tonescape microtonal music software

Hi Monz,

The article was written by Robert Temple.

I understand he has a lot of related information on hi
website.

Regarding feedback on the comma, I'm afraid I haven't
much to give. Temple seems to indicate that this
particular number of decimal places are significant.
I don't know why however, some might say its simply
the calculations involved to get this value, but
ancient peoples were capable of such if I understand
correctly.

I suspect he is hinting at something such as an
importance of the nine places being related to some
sort of harmonic significance, but your guess is as
good as mine.

I'd like to get answers to the same questions you've
just asked me - hence my post to the forum....

--- monz <monz@tonalsoft.com> wrote:

> Hi Dan,
>
>
> --- In tuning-math@yahoogroups.com, Dan Amateur
> <xamateur_dan@...> wrote:
>
> > Thanks for the feedback, I look forward to any
> other
> > thoughts you'd like to share on Pythagoras' Comma!
>
>
> I'd appreciate some feedback from you too!
> (did you read all of my post?)
>
> Particularly, why such emphasis was placed on the
> "correct to 9 decimal places" decimal value of the
> pythagorean-comma. The totally accurate 19-place
> decimal value is 1.0136432647705078125.
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>
>
>
>

__________________________________________________
Do You Yahoo!?
Tired of spam? Yahoo! Mail has the best spam protection around
http://mail.yahoo.com

Hi Dan,

--- In tuning-math@yahoogroups.com, Dan Amateur <xamateur_dan@...> wrote:

> Regarding feedback on the comma, I'm afraid I haven't
> much to give. Temple seems to indicate that this
> particular number of decimal places are significant.
> I don't know why however, some might say its simply
> the calculations involved to get this value, but
> ancient peoples were capable of such if I understand
> correctly.
>
> I suspect he is hinting at something such as an
> importance of the nine places being related to some
> sort of harmonic significance, but your guess is as
> good as mine.
>
> I'd like to get answers to the same questions you've
> just asked me - hence my post to the forum....
>
>
> --- monz <monz@...> wrote:
>
> > <snip>
> >
> > Particularly, why such emphasis was placed on the
> > "correct to 9 decimal places" decimal value of the
> > pythagorean-comma. The totally accurate 19-place
> > decimal value is 1.0136432647705078125.

Well, as you can see from what i wrote, if you carry out
the division of the pythagorean-comma's ratio, 531441/524288,
as far as it will go, you will end up with that 19-place
decimal number, with no remainder.

Because this is not an irrational number, there is no
special significance whatsoever to being able to calculate
it to 9 places. My guess is that Temple simply used a
calculator that had that limit of accuracy and made no
mention of the rounding that his calculator was doing;
perhaps he didn't realize it himself.

Ancient people were capable of doing extremely accurate
calculations. The Sumerians, the people who invented
writing and therefore the ones who left us the earliest
written records (from c.3000-2000 BC), used a numbering
system in their calculations that was a combination of
base-10 and base-60. This produces results which are
far more accurate than our own entirely base-10 system,
given the same number of "decimal" places ... in that
system it's proper to call them "sexagesimal" places.

This numbering system survived with the Babylonians,
Persians, and Greeks for their calculations which
needed the most accuracy, such as astronomy, and to
this day we still use it for angles and clocks.

-monz
http://tonalsoft.com
Tonescape microtonal music software

Monz, thanks for the reply, sheds more light on the
whole affair, you could be right about Temple.

Regardinging the numbering system of the Sumerians,
wasn't it sexigesimal? You say its a combination of
base ten and base 60 though.... ?

> > >
> > > Particularly, why such emphasis was placed on
> the
> > > "correct to 9 decimal places" decimal value of
> the
> > > pythagorean-comma. The totally accurate 19-place
> > > decimal value is 1.0136432647705078125.
>
>
> Well, as you can see from what i wrote, if you carry
> out
> the division of the pythagorean-comma's ratio,
> 531441/524288,
> as far as it will go, you will end up with that
> 19-place
> decimal number, with no remainder.
>
> Because this is not an irrational number, there is
> no
> special significance whatsoever to being able to
> calculate
> it to 9 places. My guess is that Temple simply used
> a
> calculator that had that limit of accuracy and made
> no
> mention of the rounding that his calculator was
> doing;
> perhaps he didn't realize it himself.
>
> Ancient people were capable of doing extremely
> accurate
> calculations. The Sumerians, the people who invented
> writing and therefore the ones who left us the
> earliest
> written records (from c.3000-2000 BC), used a
> numbering
> system in their calculations that was a combination
> of
> base-10 and base-60. This produces results which are
> far more accurate than our own entirely base-10
> system,
> given the same number of "decimal" places ... in
> that
> system it's proper to call them "sexagesimal"
> places.
>
> This numbering system survived with the Babylonians,
> Persians, and Greeks for their calculations which
> needed the most accuracy, such as astronomy, and to
> this day we still use it for angles and clocks.
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>
>
>
>

Be smarter than spam. See how smart SpamGuard is at giving junk email the boot with the All-new Yahoo! Mail at http://mrd.mail.yahoo.com/try_beta?.intl=ca

Hi Dan,

--- In tuning-math@yahoogroups.com, Dan Amateur <xamateur_dan@...> wrote:
>
> Monz, thanks for the reply, sheds more light on the
> whole affair, you could be right about Temple.
>
> Regardinging the numbering system of the Sumerians,
> wasn't it sexigesimal? You say its a combination of
> base ten and base 60 though.... ?

The Sumerians used ordinary base-10 for everyday stuff,
the same as we do. For their more intricate math
calculations, they used base-60.

Their system for writing numbers was a combination of both,
with symbols for 1 and 10. To write 59, they used 5 "10"s
and 9 "1"s, then 60 would look like "1" again.

-monz
http://tonalsoft.com
Tonescape microtonal music software

Kees integral tning example

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

🔗Dan Amateur <xamateur_dan@yahoo.ca>

🔗Graham Breed <gbreed@gmail.com>

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🔗Carl Lumma <ekin@lumma.org>

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🔗Herman Miller <hmiller@IO.COM>

🔗Herman Miller <hmiller@IO.COM>

🔗Herman Miller <hmiller@IO.COM>

🔗Graham Breed <gbreed@gmail.com>

🔗Carl Lumma <ekin@lumma.org>

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🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

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