Yahoo Tuning Groups Ultimate Backup tuning Dictionary updates: 2/7-comma and 1/3-comma meantones

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> a greatly expanded presentation of
> 50edo and 2/7-comma meantone:
>
> http://sonic-arts.org/dict/2-7cmt.htm
>
>
>
> and a brand new Dictionary entry for 19edo
> and 1/3-comma meantone, done in similar fashion:
>
> http://sonic-arts.org/dict/19edo.htm
>
>
>
> in both of these pages, i try to show how gradual
> expansion of the fraction-of-a-comma meantone chains
> results in candidates for further unison-vectors,
> which have the effect of reducing the linear meantones
> by one more dimension, making them closed circular
systems.
>
>
>
> -monz

i just scrolled down to the end to see what i would find:

>It was noted by Salinas in 1577 that 19-EDO was audibly
>indistinguishable from 1/3-comma meantone.2(11/19) =
>696.7741935 cents.

i though it *wasn't* noted by salinas. where did he note it?

>Using vector addition again to compare the 1/3-comma
>meantone "5th" with the 19-EDO "5th", we get a difference
>between the two of:

> 2^ 3^ 5^

> [ 1/3 -1/3 1/3 ] = ~694.7862377 cents = 1/3-comma
>meantone "5th"
> - [ 11/19 0 0 ] = ~694.7368421 cents = 19-EDO "5th"
> -----------------------
> [ 18/31 0 -1/4 ] = 19edo "5th" "-" 1/3-comma meantone
>"5th"

> = ~0.049395561 cent = ~1/20 cent

>Thus, 2-(14/3) * 3-(19/3) * 5(19/3) acts as a unison-vector

no it doesn't. its third power can, though.

>which is not tempered out in 1/3-comma meantone, and it acts
>as a unison-vector which is tempered out in 31edo.

31? don't you mean 19?

🔗monz <monz@attglobal.net>

hi paul,

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, March 30, 2003 2:41 PM
> Subject: [tuning] Re: Dictionary updates: 2/7-comma and 1/3-comma
meantones
>
>
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > a greatly expanded presentation of
> > 50edo and 2/7-comma meantone:
> >
> > http://sonic-arts.org/dict/2-7cmt.htm
> >
> >
> >
> > and a brand new Dictionary entry for 19edo
> > and 1/3-comma meantone, done in similar fashion:
> >
> > http://sonic-arts.org/dict/19edo.htm
> >
> >
> >
> > in both of these pages, i try to show how gradual
> > expansion of the fraction-of-a-comma meantone chains
> > results in candidates for further unison-vectors,
> > which have the effect of reducing the linear meantones
> > by one more dimension, making them closed circular
> systems.
> >
> >
> >
> > -monz
>
> i just scrolled down to the end to see what i would find:
>
> > It was noted by Salinas in 1577 that 19-EDO was audibly
> > indistinguishable from 1/3-comma meantone.2(11/19) =
> > 696.7741935 cents.
>
> i though[t] it *wasn't* noted by salinas. where did he note it?

i've been away from Salinas for several months and will
have to look into it again, but my memory tells me that
he did note that 19edo was nearly the same as 1/3-comma MT.

> > Thus, 2-(14/3) * 3-(19/3) * 5(19/3) acts as a unison-vector
>
> no it doesn't. its third power can, though.

can you explain that in some detail? thanks.

> > which is not tempered out in 1/3-comma meantone, and it acts
> > as a unison-vector which is tempered out in 31edo.
>
> 31? don't you mean 19?

yup, sure do. thanks for catching that. it's been fixed.

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@attglobal.net>

hi Gene,

> From: "Gene Ward Smith" <gwsmith@svpal.org>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, March 30, 2003 11:22 PM
> Subject: [tuning] Re: Dictionary updates: 2/7-comma and 1/3-comma
meantones
>
>
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > > > Thus, 2^-(14/3) * 3^-(19/3) * 5^(19/3) acts as a unison-vector

[i have supplied the "^" which was missing when
paul simply copied the text from my webpage, which
uses HTML superscript tags instead, but which don't
copy correctly. anyone who does "copy and paste"
from my webpages, or others which use superscripts,
should be on the lookout for this when they post here.]

> > >
> > > no it doesn't. its third power can, though.
> >
> >
> > can you explain that in some detail? thanks.
>
> This is an example of why I object to the term "unison vector". The
> above "vector" actually can be regarded as an element of a vector
> space, but isn't a so-called "unison vector" because it does not
> represent a rational number. To do that, scalar multiply the
> vector [-14/3, -19/3, 19/3] by 3 and get [-14, 19, 19]; this is
> not only a vector but (since it has only integer coordinates) also
> an element of an abelian group (the term now being "Z module" rather
> than "vector space".)

thanks for commenting on this.

unfortunately, my limited grasp of mathematics doesn't
permit me to understand such terms as "Z module", or
the logic or reasoning behind your explanation.

here's how i see it:

i can define some prime-factors for my "prime-space",
where each prime-factor creates an axis running at
a unique angle in the space.

i can imagine a lattice which has points at each integer
coordinate, at regular intervals, to represent the
exponents of those prime-factors, for any rational number.

let's say for the sake of example that i'm using
3,5 prime-space, and that i want to visually explore
the properties of a meantone.

i can plot the points of that meantone also at regular
*but non-integer* intervals, along a linear axis whose
angle may be the same as one of the prime axes (as in
the case of 1/4-comma meantone, which plots entirely
along the 5-axis) or may be different from any of them
(as in all other meantones).

at some point, if i extend the meantone chain out
far enough, i'm going to find two pitches whose frequencies
(i.e., pitch-height) are very close to each other. if
i choose to assume that those two pitches are identical,
and thus that that interval vanishes, then i have in
effect transformed the lattice of my linear meantone chain
into a closed circular lattice.

i may temper out that small interval, which will most
often result in a simple EDO (equal-temperament), or
i may choose not to temper it out. but in either case,
assuming that those two pitches are identical means that
in practice the interval vanishes, and that the meantone
lattice has become circular.

(to be more precise, the meantone chain could already
have been viewed as a cylinder, and constraining it
further in this other dimension is to turn the cylinder
into a torus. but anyway...)

so, 2^-(14/3) * 3^-(19/3) * 5^(19/3) most certainly *can*
be employed to close the 1/3-comma meantone chain and
make it a tuning which bears no sensible difference
from 19edo ... but if it *cannot* "act as a unison-vector",
then either:

1) i'm not understanding something very important, or

2) the definition of "unison-vector" has to be altered, or

3) i need another term to describe the function if this
interval in 1/3-comma meantone, or

4) some combination of all three.

please help! ... without *too* much mathematicalese! ...

-monz

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi paul,
>
>
> > From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
> > To: <tuning@yahoogroups.com>
> > Sent: Sunday, March 30, 2003 2:41 PM
> > Subject: [tuning] Re: Dictionary updates: 2/7-comma and 1/3-comma
> meantones
> >
> >
> > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > >
> > > a greatly expanded presentation of
> > > 50edo and 2/7-comma meantone:
> > >
> > > http://sonic-arts.org/dict/2-7cmt.htm
> > >
> > >
> > >
> > > and a brand new Dictionary entry for 19edo
> > > and 1/3-comma meantone, done in similar fashion:
> > >
> > > http://sonic-arts.org/dict/19edo.htm
> > >
> > >
> > >
> > > in both of these pages, i try to show how gradual
> > > expansion of the fraction-of-a-comma meantone chains
> > > results in candidates for further unison-vectors,
> > > which have the effect of reducing the linear meantones
> > > by one more dimension, making them closed circular
> > systems.
> > >
> > >
> > >
> > > -monz
> >
> > i just scrolled down to the end to see what i would find:
> >
> > > It was noted by Salinas in 1577 that 19-EDO was audibly
> > > indistinguishable from 1/3-comma meantone.2(11/19) =
> > > 696.7741935 cents.
> >
> > i though[t] it *wasn't* noted by salinas. where did he note it?
>
>
> i've been away from Salinas for several months and will
> have to look into it again, but my memory tells me that
> he did note that 19edo was nearly the same as 1/3-comma MT.
>
>
> > > Thus, 2-(14/3) * 3-(19/3) * 5(19/3) acts as a unison-vector
> >
> > no it doesn't. its third power can, though.
>
>
> can you explain that in some detail? thanks.

the kinds of vectors of which unison vectors form an example have
integer exponents. they represent intervals in the lattice of
consonant intervals, or primes, without regard to the specific tuning
system (though knowing the JI value is often useful). if the above
expression to the third power, you get

2-(14) * 3-(19) * 5(19)

this is trivially a unison vector of 19-tone equal temperament, since
moving up or down 19 fifths or 19 major thirds, or 19 minor thirds,
will bring you back to the same 19-equal pitch you started at.

we had a similar discussion once when you were comparing two
meantones, and claiming a certain irrational interval represented
a "xenharmonic bridge" between one and the other. functionally,
though, all meantone tunings have the same unison vector tempered
out -- the syntonic comma -- and thus the same algebraic form. the
unison vector formalism is useful for specifying the algebraic
properties of tuning systems and for transforming between different
tuning systems with different algebraic characterizations. it is not
useful for specifying the specifics of different temperaments, and
quickly falls apart if you attempt to use it that way.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> i can define some prime-factors for my "prime-space",
> where each prime-factor creates an axis running at
> a unique angle in the space.

yup.

> i can imagine a lattice which has points at each integer
> coordinate, at regular intervals, to represent the
> exponents of those prime-factors, for any rational number.

yup.

> let's say for the sake of example that i'm using
> 3,5 prime-space, and that i want to visually explore
> the properties of a meantone.
>
> i can plot the points of that meantone also at regular
> *but non-integer* intervals, along a linear axis whose
> angle may be the same as one of the prime axes (as in
> the case of 1/4-comma meantone, which plots entirely
> along the 5-axis) or may be different from any of them
> (as in all other meantones).

here you're letting the numbers carry you away from their musical
meaning. how would you deal with metameantone, or lucytuning, or
golden meantone? and why do the fifth and/or third acquire totally
new lattice directions and lengths, when their musical functions are
so similar, indeed identical, to those of the just intervals -- and
certainly to those of the intervals in other meantones!! the lattice,
and the vectors derived from it, is meant to represent the
relationships of the notes of a tuning system in terms of musical
function. unison vectors help define temperament/approximation
systems because they can show the equivalence of the result of
different sets of harmonic changes, or moves through the lattice.
these moves occur in discrete steps, made up of the basic intervals
of the tuning (usually fifths and thirds, etc.). trying to invoke a
fraction of a musical interval in this context makes no sense.

> at some point, if i extend the meantone chain out
> far enough, i'm going to find two pitches whose frequencies
> (i.e., pitch-height) are very close to each other. if
> i choose to assume that those two pitches are identical,
> and thus that that interval vanishes, then i have in
> effect transformed the lattice of my linear meantone chain
> into a closed circular lattice.

this operation does eliminate an additional unison vector -- and
there are an infinite number of equivalent candidates, differing by
81:80 from one another, for this interval. to express it as a unison
vector, you must derive it from an integer number of steps along each
prime axis -- i don't see how fractional values have any musical
meaning. in the case of 19-equal, closing the meantone chain implies
that you're tempering out any, and therefore all, of the following
unison vectors:

[-10 -1 5] = 3125:3072
[-6 -5 6] = 15625:15552
[-14 3 4] = 16875:16384
[ 2 9 -7] = 78732:78125
[ 8 14 -13] = 1224440064:1220703125
[-14 -19 19] = 19073486328125:19042491875328

this last one is of course the third power of the interval you
mentioned.

> i may temper out that small interval, which will most
> often result in a simple EDO (equal-temperament), or
> i may choose not to temper it out. but in either case,
> assuming that those two pitches are identical means that
> in practice the interval vanishes, and that the meantone
> lattice has become circular.

yes. and in either case, the pitches are derived by moving an integer
number of steps in the lattice of prime intervals, thus representing
a clear *algebraic* musical function.

> so, 2^-(14/3) * 3^-(19/3) * 5^(19/3) most certainly *can*
> be employed to close the 1/3-comma meantone chain

i think you're misunderstanding the nature of unison vectors. they're
not precise intervals at all, but merely relative positions in the
lattice of consonant intervals. the syntonic comma is what happens
when a piece of music goes up four fifths and down a major third. no
matter how these intervals are tuned! thus it's a 'vector'. now how
do i interpret your proposed vector above, in similar terms??

hopefully, gene can continue to help me explain this to you, without
too much math-ese. the idea is that the unison vector formalism is
extremely useful when it's defined correctly, and loses this utility
when perverted . . .