Yahoo Tuning Groups Ultimate Backup tuning lattices of meantone JI implications

> From: monz <joemonz@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, December 09, 2001 3:05 PM
> Subject: [tuning] lattices of meantone JI implications
>
>
> I've now made a webpage comparing the "true" meantone (1/4-comma)
> and a variety of different quasi-meantone systems:
>
> http://www.ixpres.com/interval/monzo/meantone/lattices/lattices.htm
>

I added some important points to the text on that page, and
reproduce it below.

[in the introductory part]

In every case, the list of implied JI ratios includes both of the two
nearest, except for instances where the meantone tuning gives one exact
ratio (as, for example, the +3, +6, +9 generators A, F#, D# of 1/3-comma;
the +7 generator C# of 2/7-comma; the +4, +8, +12 generators E, G#, B# of
1/4-comma; the +5, +10 generators B, A# of 1/5-comma; etc.).

The lattice diagrams, however, only give two ratios in cases where the
meantone is exactly at the midpoint between them (as with the +2 generator
"D" in 1/4-comma meantone, and with the +3 generator "A" in 1/6-comma
meantone); otherwise only the nearest JI implied ratio is given.

The meantone chains could be extended beyond my diagrams; I chose a 27-note
chain of +/- 13 generators (Gbb to Fx) as an arbitrary limit in every
example.

...

[in the text describing 1/11-comma]

Note that because the JI lattice implied by 1/11-comma meantone includes the
skhisma, the chain effectively closes at 12 tones; a typical version could
be -3 Eb to +8 G#.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
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🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> To me, a lattice of meantone 5-limit JI implications would have to
be
> a _cylinder_.

At Monz's: http://www.ixpres.com/interval/dict/lattice.htm
one finds the following:

lattice diagram [definition]
a visual representation of the mathematical relationships of musical
ratios in 2-, 3-, or multi-dimensional space, consisting of points
which represent the ratios as positions calculated according to the
Fundamental Theorem of Arithmetic....
[J GILL: THIS IMPLIES TO ME THAT SPECIFIC "POINTS", IN THE ABOVE
CONTEXT, WOULD BE APPLICABLE ONLY TO - RATIONAL NUMBER - POSITIONS
CALCULATED FROM THE MULTIPLICATIONS OF VARIOUS PRIME FACTORS.]

At http://hissa.nist.gov/dads/HTML/lattice.html
one finds a definition of the term "lattice":
A point lattice generated by taking integer linear combinations of a
set of basis vectors.
__________________________________________________

At: http://www.math.sfu.ca/~gfee/Math342/L301.html
one finds a more detailed definition of the term "lattice":

There are at least two definitions of a lattice in mathematics[:]

Definition 1. A lattice is a partially ordered set for which each
pair of elements has both a greatest lower bound and a least upper
bound.
[J GILL: COULD MONZ'S "MEANTONE JI IMPLICATIONS" BE SAID TO FALL
UNDER DEFINTION 1 DIRECTLY ABOVE?]

Definition 2. A lattice L, is the set of integer coefficient linear
combinations, of a set of linearly independent vectors in a finite
dimensional real vector space. We call the set of linearly
independent vectors a basis for the the lattice L .

We will be only consider the second definition.
Example: v[1] = (7,13), v[2]=(8,11) in R^2 . Then {v[1],v[2]} are
linearly independent vectors.
Define L as the lattice of integer combinations of {v[1],v[2]}.

[J GILL: COULD PAUL'S OBJECTIONS BE BASED UPON AN OBSERVED LACK
OF "LINEAR INDEPENDENCE" IN THE IMPLIED "LOCATIONS" OF THE ELEMENTS
(OF AN IRRATIONAL NUMERIC VALUE) WHICH ARE DEPICTED IN
MONZ'S "MEANTONE JI IMPLICATIONS" FOUND AT
http://www.ixpres.com/interval/monzo/meantone/lattices/lattices.htm
?]

________________________________________________________

At: http://www.its.bldrdoc.gov/fs-1037/dir-003/_0364.htm
one finds a definition of "array":
array: 1. An arrangement of elements in one or more dimensions.
At:
http://www.vuse.vanderbilt.edu:8888/es130/lectures/lecture9/matwhat.ht
m
one finds a definition of "matrix":
A matrix is a set of numbers arranged in a rectangular grid of rows
and columns.
[J GILL: THUS "MATRIX" AND "ARRAY" APPEAR TO BE INTERCHANGEABLE]

In Monzo's "JustMusic: A New Harmony" (c) 1999, Page 45, Building the
Matrix chapter, Joe's text utilizes the term "matrix" to describe
(specific) positions located "within" arrays (or "matrices") of
("linearly independent", it seems) rational numbers.

While these "locations" of "JI implications" exist at numerical
values which exist "in between" the rational values which surround
them, the numerical "locations" themselves are of a value irrational.

Nevertheless, such juxtapositions seem natural for our conceptual
visualization, and I like what Joe is doing here! As he states:
<<The meantone chain forms its own linear axis on the lattice, and
makes it easy to visualize which JI ratios are acoustically the most
closely implied by the meantone pitches.>>

Paul, however, appears (PE: correct me if I have misinterpreted you)
to point out that such an array, or matrix, (or, perhaps, "linear
axis of points"?) cannot "wrap-around" upon itself, as with
a "cylinder" wrapping in one dimension, or a "torus" as seen at:
http://www.math.unm.edu/~loring/links/top_f01/torus2.swf
resulting from a wrapping in two dimensions - as a result of the
irrational numerical values obtained from the taking of JI ratios to
fractional (as opposed to integer) values in deriving the path of
a "linear axis" (evidently referring to the linear slope of lines
extending through a logarithmic (or geometric, ratiometric) domain.

Question: While such a "linear projection" (in log domain) represents
an "irrational" [in the mathematical, and not judgemental, sense :)]
space existing in a sort of "no integer's land" which is not directly
mathematically juxtaposable within the (rational) space (where direct
meaning appears to exist only "on", and not "near" the JI interval
vector locations) - is the (mathematical) fact that the existence
such irrational values (which cannot be factored into prime numbers
taken to integer powers) precludes their "graphical utility" in
presenting concepts of numerical values which are
(numerically) "intertwined" (whether via a linear or "step-wise
linear" locus) within a particular (mathematically "rational") JI
lattice which *may*, itself, be able to be "wrapped" into
a "cylinder" (in one dimension) or a "torus" (in two dimensions)?

The potential multiplicity of possible roots (of the rational number
to a non-integer power) which inherently accompanies such a concept
does not (to me) necessarily preclude its conceptual usefulness
(though it may require a possible remaning - from lattice, to matrix
or to array - when considering a description of one or more
dimensional "axes of implication")... It does seem quite the hybrid.

Second question: (with all due respect to personal beliefs, and the
First Amendment's guarantee of "freedom of existential thought"):
WILL THE (mathematical) STATE NOT ...LATTICE..ARRAY (let us..pray)?

SHALL WE ALLOW A "MATHEMATOCRACY" (even our own) TO DENY US OUR
VISUAL INTUITIVE DELIGHTS (which may possess visually and semi-
sonically functional, albeit mathematically "irrational",
integrity)? It is understood that *seeing* is not *hearing*... :)

Curiously (but possibly confused/confusing), J Gill

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "unidala" <JGill99@i...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
> > To me, a lattice of meantone 5-limit JI implications would have
to
> be
> > a _cylinder_.
>
> At Monz's: http://www.ixpres.com/interval/dict/lattice.htm
> one finds the following:
>
> lattice diagram [definition]
> a visual representation of the mathematical relationships of
musical
> ratios in 2-, 3-, or multi-dimensional space, consisting of points
> which represent the ratios as positions calculated according to the
> Fundamental Theorem of Arithmetic....
> [J GILL: THIS IMPLIES TO ME THAT SPECIFIC "POINTS", IN THE ABOVE
> CONTEXT, WOULD BE APPLICABLE ONLY TO - RATIONAL NUMBER - POSITIONS
> CALCULATED FROM THE MULTIPLICATIONS OF VARIOUS PRIME FACTORS.]

Obviously Monz doesn't restrict it to that sense, as he is mapping
irrational numbers to his lattice as well.
>
> At http://hissa.nist.gov/dads/HTML/lattice.html
> one finds a definition of the term "lattice":
> A point lattice generated by taking integer linear combinations of
a
> set of basis vectors.

That's pretty good for our purposes.

> __________________________________________________
>
> At: http://www.math.sfu.ca/~gfee/Math342/L301.html
> one finds a more detailed definition of the term "lattice":
>
> There are at least two definitions of a lattice in mathematics[:]
>
> Definition 1. A lattice is a partially ordered set for which each
> pair of elements has both a greatest lower bound and a least upper
> bound.
> [J GILL: COULD MONZ'S "MEANTONE JI IMPLICATIONS" BE SAID TO FALL
> UNDER DEFINTION 1 DIRECTLY ABOVE?]

No. This definition is far removed from the sense in which most of us
music theorists use it.

> Definition 2. A lattice L, is the set of integer coefficient linear
> combinations, of a set of linearly independent vectors in a finite
> dimensional real vector space. We call the set of linearly
> independent vectors a basis for the the lattice L .
>
> We will be only consider the second definition.
> Example: v[1] = (7,13), v[2]=(8,11) in R^2 . Then {v[1],v[2]} are
> linearly independent vectors.
> Define L as the lattice of integer combinations of {v[1],v[2]}.
>
> [J GILL: COULD PAUL'S OBJECTIONS BE BASED UPON AN OBSERVED LACK
> OF "LINEAR INDEPENDENCE" IN THE IMPLIED "LOCATIONS" OF THE
ELEMENTS
> (OF AN IRRATIONAL NUMERIC VALUE) WHICH ARE DEPICTED IN
> MONZ'S "MEANTONE JI IMPLICATIONS" FOUND AT
> http://www.ixpres.com/interval/monzo/meantone/lattices/lattices.htm
> ?]

Well, I've made that sort of objection before, but really my bigger
objection is that the important relationship between meantone and JI,
as understood by me and, seemingly, all musicians of the past 500
years who have thought about it, is in the _simple consonant
intervals_ and not in the complex intervals nor in the _pitches_.

> Paul, however, appears (PE: correct me if I have misinterpreted
you)
> to point out that such an array, or matrix, (or, perhaps, "linear
> axis of points"?) cannot "wrap-around" upon itself, as with
> a "cylinder" wrapping in one dimension, or a "torus" as seen at:
> http://www.math.unm.edu/~loring/links/top_f01/torus2.swf
> resulting from a wrapping in two dimensions - as a result of the
> irrational numerical values obtained from the taking of JI ratios
to
> fractional (as opposed to integer) values in deriving the path of
> a "linear axis" (evidently referring to the linear slope of lines
> extending through a logarithmic (or geometric, ratiometric) domain.

That's a tough sentence to chew on . . . now Monz is finding that
meantone pitches occur in more than one place on his lattice . . . in
fact, the way he's doing things, each meantone pitch will appear in
an infinite number of places in the lattice . . . this infinitude can
be reduced to uniqueness by rolling the lattice into a cylinder . . .
but the JI implications will still be rather confused. Far simpler is
to start with a standard JI 5-limit lattice, and roll it into a
cylinder by overlaying every pair of 81:80-separated ratios . . .
this will be a far better depiction of the JI implications of
meantone, in my view.

> Question: While such a "linear projection" (in log domain)
represents
> an "irrational" [in the mathematical, and not judgemental,
sense :)]
> space existing in a sort of "no integer's land" which is not
directly
> mathematically juxtaposable within the (rational) space (where
direct
> meaning appears to exist only "on", and not "near" the JI interval
> vector locations)

Why do you say "not directly juxtaposable"? Monz is doing just this
juxtaposition (whatever you may think about its musical utility).

> - is the (mathematical) fact that the existence
> such irrational values (which cannot be factored into prime numbers
> taken to integer powers) precludes their "graphical utility" in
> presenting concepts of numerical values which are
> (numerically) "intertwined" (whether via a linear or "step-wise
> linear" locus) within a particular (mathematically "rational") JI
> lattice which *may*, itself, be able to be "wrapped" into
> a "cylinder" (in one dimension) or a "torus" (in two dimensions)?

I don't understand this sentence.

> The potential multiplicity of possible roots (of the rational
number
> to a non-integer power) which inherently accompanies such a concept
> does not (to me) necessarily preclude its conceptual usefulness
> (though it may require a possible remaning - from lattice, to
matrix
> or to array - when considering a description of one or more
> dimensional "axes of implication")... It does seem quite the hybrid.

Whew! Well, let's make this very concrete. To me, JI is desirable
only because it gives you the consonant intervals exactly. This is
why a JI lattice, where the "rungs" are consonant intervals, is so
useful. Meantone is desirable because it gives you the consonant
intervals almost exactly, and a finite meantone system allows you to
chain them together in ways which a finite JI system doesn't. Either
way, it is the _consonant intervals_ which should be the "rungs"
or "basis vectors" of the lattice. Furthermore, whether the meantone
system _happens_ to have some far-flung intervals _exactly_ as they
are in JI (as in 1/6-comma meantone, or any n/m-comma meantone, where
n and m are integers) or doesn't (as in Golden Meantone, or
LucyTuning, or 55-tET) should have essentially _no_ effect on how the
lattice looks, let alone on the question of whether the lattice can
or cannot be produced at all!

> Second question: (with all due respect to personal beliefs, and the
> First Amendment's guarantee of "freedom of existential thought"):
> WILL THE (mathematical) STATE NOT ...LATTICE..ARRAY (let us..pray)?
>
> SHALL WE ALLOW A "MATHEMATOCRACY" (even our own) TO DENY US OUR
> VISUAL INTUITIVE DELIGHTS (which may possess visually and semi-
> sonically functional, albeit mathematically "irrational",
> integrity)? It is understood that *seeing* is not *hearing*... :)

Everyone is free to indulge in their own mathematical flights of
fancy. My interest is in the _practical_ end of things -- what is
most enlightening for the actual creation of music. Joseph Pehrson is
presently composing using a lattice for a certain tuning system, so
he, for one, can attest to the _practical_ utility of a lattice. Many
JI composers have also employed lattices -- witness, for example, the
nice big 7-limit lattice in David Doty's _Just Intonation Primer_;
Ben Johnston's lattice where the factors of 3 are vertical rather
than horizontal; and so on. What makes these useful is this property,
call it "L":

L: One can locate immediately all the pitches bearing audible,
consonant relationships with other pitches. The pitches consonant
with one another tend to be close to one another, while pitches
dissonant with one another tend to be further from one another.

Now tempered systems, such as meantone, are also based on consonant
intervals, but these intervals are tempered so that combinations of
them can lead back to the starting pitch (allowing for a finite set
of meantone pitches to stand in for an infinite set of JI pitches).
Whether can be shown by an actual rolled-up cylinder or by repetition
in the lattice. But what seems essential to me is that, given the
system's ultimate basis in consonant intervals, property L should
hold!

As an example, take 1/4-comma meantone temperament. If I were a
composer seeking a lattice that would be of practical utility in my
task, I would want a lattice that showed a pair of pitches separated
by an interval of a fifth or a third as close to one another; while a
pair of pitches separated by a major second would be further apart.

There's an article by Donald E. Hall that I'd like to send to you and
Monz -- and the article appears to be missing from the Bibliography!

http://www.xs4all.nl/~huygensf/doc/bib.html

I'll post the reference tomorrow -- meanwhile, perhaps you and Monz
can send me your snail-mail addresses . . .

lattices of meantone JI implications

🔗monz <joemonz@yahoo.com>

🔗paulerlich <paul@stretch-music.com>

🔗monz <joemonz@yahoo.com>

🔗monz <joemonz@yahoo.com>

🔗unidala <JGill99@imajis.com>

🔗paulerlich <paul@stretch-music.com>

🔗paulerlich <paul@stretch-music.com>