Yahoo Tuning Groups Ultimate Backup tuning-math exactly what is a xenharmonic bridge?

please pardon the length of this ... i feel that this
is very important to my own work, and i'm confused ...

an Instant Message discussion i had with Paul ...

> joemonz: i want to discuss this with you via IM:
> the objections you were posting the other day about
> my apparent reinterpretation of "xenharmonic bridges".
>
> paulerlich: oh, the LucyTuning-vs.-3/10-comma meantone?
> that clearly deprives them of all value, at least any
> value that could be relevant for finity. in order for
> finity to begin to occur, two ways of constructing an
> interval from the _same_ set of basis intervals (usually
> primes for you, but can be anything) have to lead to the
> same, or aurally indistinguishable results. but this
> doesn't happen in your latest case.
>
> joemonz: i'm a little confused about this. i think that
> my conception of "xenharmonic bridge" is probably a
> more general umbrella type of thing, which includes
> "unison-vector" under it as a more specific aspect.
> my original idea about "xenharmonic bridges" was that,
> for example, we'd hear something in some ET (which
> approximates a certain type of JI) and interpret it
> as being *in* that JI.
>
> paulerlich: ok . . . that's closer to what harmonic
> entropy is good for. but neither definition would
> seem to justify calling your LucyTuning-vs.-3/10-comma
> meantone thing a xenharmonic bridge.
>
> joemonz: well, that's what you've been maintaining all
> along, but i'm having a hard time understanding why.
> the xenharmonic bridge idea is that finity is an
> essential part of music, because no single person
> can comprehend all possible tonal relationships --
> different tunings quantize the pitch-continuum in
> different ways, but *those differences are often not
> audible*, and that's because of the bridging that's
> going on all the time. perhaps there are exceptions
> -- for example, La Monte Young-style strict JI or
> barbershop, where essentially the listender *does*
> hear the actual tuning. does that help you see where
> i'm coming from?
>
> paulerlich: again, finity won't happen until two pitches
> constructed from _the same set of basis intervals_
> are considered the same. all your examples of xenharmonic
> bridging up to this point have been fine illustrations
> of this -- you're about to send that all down the tubes
> with this error in reasoning.
>
> joemonz: hmm ... but what i just wrote to you was the
> *original* conception i had of xenharmonic bridging,
> from 1998. i'm confused now.
>
> paulerlich: do you want it to aid toward finity, that is,
> to reduce the number of dimensions of infinite extent
> in the lattice?
>
> joemonz: well, yes, by definition xenharmonic bridges
> limit infinity in at least two ways (i'll use a JI example):
> 1) by reducing the total number of dimensions,
> 2) by creating a periodicity *within* the remaining dimensions.
>
> paulerlich: ok . . . now how can either of those things
> happen if the two intervals being 'bridged' between
> have no basis intervals in common?
>
> joemonz: hmm ... i have the mathematical definitions of
> "basis" (which i can't even read), and i'm pretty sure
> that i have an intuitive grasp of the concept, but how's
> about you try explaining it to me in plain english a little?
>
> paulerlich: i just mean the smallest units in the lattice
>
> joemonz: i suppose we'd better create an example ...
> i'm always better with concrete examples and diagrams,
> and not so good with abstract stuff. so create a
> hypothetical situation -- let's say someone who's
> never consciously been exposed to microtonal music
> before gets a chance to hear a 13-limit Ben Johnston
> quartet. my bet is that that person will probably
> hear it mostly in terms of 12-edo, and *maybe* will
> pick up some 5- and possibly 7-limit harmonic
> structures, but will probably fail to comprehend
> the 11- and 13-limit ones, because of lack of exposure
> to them. feel free to argue about or change any of this
> ... the whole purpose is to get me to understand a little
> better where you're coming from.
>
> paulerlich: i don't see this as relevant at all so please
> hold on to my previous explanation. anyway, they will hear
> it in terms of 12-equal because of *categorical perception*,
> which is a powerful phenomenon in all of psychology in its
> own right. this has nothing to do, in my opinion, with
> unison vectors or the like. and, if they simply 'fail to
> comprehend' 11-limit and 13-limit harmonic structures,
> rather than hearing them as something else, then there's
> no problem to discuss, because they are not invoking any
> bridges at all.
>
> joemonz: i don't see it that way. i'd say that there's a
> whole slew of bridges in effect: 13==5, 11==5, 13==7,
> 11==7, as well as the ones that bridge to 12-edo that i
> don't have names for. they'd probably just hear the 11-
> and 13-limit intervals as "out of tune", but they'd still
> be comprehending them as 5/7-limit or 12-edo, which is
> n o t what the music actually is.
>
> joemonz: please realize that i'm not approaching this as
> a debate with you ... i believe that you probably have
> a clearer grasp of this than i do, and i'm just trying
> to understand your criticisms.
>
> paulerlich: 11=5?
>
> joemonz: an 11-limit interval that is close to, and is
> being taken for, a 5-limit one.
>
> paulerlich: such as?
>
> paulerlich: and wouldn't that be hearing them as something
> else, rather than failing to comprehend them?
>
> joemonz: exactly yes ... "hearing them as something else"
> is exactly what i had in mind originally with "xenharmonic
> bridges". ok ... here's an example of 11==5: suppose
> there's a prominent 11:8 in a Johnston chord. especially
> if there are a lot of 5-limit harmonies going on, i think
> there's a very good chance that a lot of listeners will
> perceive that 11:8 as a very out-of-tune 45:32. i'm even
> willing to stick my neck out and claim that they might
> perceive it as 5625:4096 = [2,3,5] [-12,2,4] = ~549.1648572
> cents -- i already know that you're going to argue that
> that's highly unlikely, so we can stick with the 45:32 case.
>
> paulerlich: well, we're talking about three very
> different things:
>
> paulerlich: (1) categorical perception
>
> paulerlich: (2) the hearing of a harmonic interval
> as some other harmonic interval, isolated from context,
> where harmonic entropy applies
>
> paulerlich: (3) the hearing of a tuning-system pitch as
> some other tuning-system pitch, where all the pitches
> are labeled with ratios in order to make their derivation
> from consonant/basis intervals clear. this is probably
> what you are talking about in the current example, and
> is where finity applies.
>
> (paulerlich: it's kinda funny . . . in your book you argue
> that the sharp 11 chord implies the 11th harmonic . . .
> but here you are arguing the exact opposite!)
>
> joemonz: but can you see xenharmonic bridge as a very
> general term that could encompass all three? in my mind,
> they're all related to creating finity.
>
> paulerlich: these three are all related to creating finity,
> yes, but they must be clearly distinguished from one
> another, and mixed with much more care.
>
> (joemonz: re: 11 ... yeah, well, you know, 11 is the
> oddball case, because it's nearly exactly between the
> 12-edo F and F#. besides, i've grown a lot since i wrote
> my book ... thanks largely to you! remember back when i
> didn't say anything about meantone? )
>
> joemonz: ok, well that's good for me to know. but i'm
> still not clear on your answer to my question: is it
> possible that "xenharmonic bridge" can serve as a
> general term that covers all 3 aspects? if that's not
> a good idea, then please, tell me why, and more importantly,
> tell me exactly *what* "xenharmonic bridge" d o e s cover
> ... would that simply be the case where it's like a
> unison-vector but works *between* prime-factors rather
> than *within* them (as a UV does)?
>
> paulerlich: can we use different symbol for ratios that
> are built up from simpler basis intervals -- how about
> a semicolon in the case of intervals and a backslash in
> the case of pitches . . . margo suggested something
> similar involving altering the order of the higher number
> and the lower number, but i think this would be more clear.
>
> joemonz: oh yes, i've always been in full agreement with
> adopting the convention of colon-for-interval and
> slash-for-pitch, so this idea works for me. lay it on me.
>
> paulerlich: so we'd say 11:8 as opposed to 45;32 and
> 5625;4096 in your example, or 11/8 as opposed to 45\32
> and 5625\4096
>
> paulerlich: so as to your question . . .
>
> paulerlich: sense (3) is the sense that means unison vector.
> if you like 'xenharmonic bridge' being a _subcategory_
> of unison vector, then it's awfully strange to apply it to
> sense (2) or sense (1), let alone the lucy-tuning vs.
> 3/10-comma meantone comparison, which is none of the above!

and from an old post ...

> From: paulerlich <paul@s...>
> To: <tuning-math@yahoogroups.com>
> Sent: Thursday, January 03, 2002 9:04 PM
> Subject: Re: the unison-vector<-->determinant relationship
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > > > > > There are other fraction-of-a-comma meantones
> > > > > > which come closer to the center, and it seems
> > > > > > to me that the one which *does* run exactly down
> > > > > > the middle is 8/49-comma.
> > > > > >
> > > > > > Is this derivable from the [19 9],[4 -1] matrix?
> > > > >
> > > > > You should find that the interval corresponding
> > > > > to (19 9), AS IT APPEARS in 8/49-comma meantone,
> > > > > is a very tiny interval.
> > > >
> > > > Ah... so then 8/49-comma meantone does *not* run
> > > > *exactly* down the middle. How could one calculate
> > > > the meantone which *does* run exactly down the middle?
> > >
> > > It's 55-tET.
> >
> >
> > Not if the periodicity-block is a parallelogram.
> > 10/57-comma meantone is much closer to 55-EDO than
> > 1/6-comma meantone, yet it is further away from the
> > center of this periodicity-block.
>
> Hmm . . . the line you want is the vector (19 9). So any
> generator 3^a/b * 5^c/d that is a solution to the equation
>
> a/b * 19 + c/d * 9 = 0
>
> would work. This gives
>
> a/b*19 = -c/d*9
>
> Does this help?

it seems to me that what i'm getting at is that i think
kernels should be definable with matrices of fractions as
well as integers.

i've had a hard time all along understanding why my plots
of fraction-of-a-comma meantones are not unique for each
meantone, because i can see that if the exponents of the
prime-factors are rational, the whole matrix can be multiplied
by the gcd and the matrix will once again be composed of
all integers.

can we please start with an example which compares 19-edo
to 1/3-comma meantone? here are both generators and their
difference:

~cents

[2 3 5] [ 1/3 -1/3 1/3] = 694.7862377 = 1/3cmt "5th"

- [2 3 5] [ 11/19 0 0 ] = 694.7368421 = 19edo "5th"
----------------------------

[2 3 5] [-14/57 -1/3 1/3] = 0.0493956 = 1/3cmt==19edo bridge

now if my goal is to plot b o t h of these temperaments
along with the JI pitches all on the same lattice, with the
same 0,0 origin = 1/1 for all three tunings, then why is this
1/3cmt==19edo bridge (that is, "1/3-comma meantone is equivalent
to 19-EDO") not valid as a basis? n o t necessarily as a
l a t t i c e m e t r i c , but as a mathematical basis
for comparing the set of tunings nonetheless.

as i said, the whole matrix could be multiplied by the gcd:

[2 3 5] ( [ 19 -19 19] * 1/57) = 1/3cmt "5th"

- [2 3 5] ( [ 33 0 0] * 1/57) = 19edo "5th"
----------------------------

[2 3 5] ( [-14 -19 19] * 1/57) = 1/3cmt==19edo bridge

is that right? -- assuming that it is ...

now to get the solution i'm seeking, don't we have to keep
the same exponent of 2 for both the 19edo and the 1/3cmt?

so, for this case, what's the solution for the a,b,c,d of
paul's post? that would be: the rational exponent pair for
prime-factors 3 and 5 which most closely approximates each
generator step of 19-edo, while the exponent of 2 is kept
the same for both that tuning and the equivalent 1/3cmt step.

so what i want to do now is change the exponent numerator
of 2 in the last matrix above to 19 for the 19edo "5th",
and adjust the values of 3 and 5 accordingly.

this would therefore plot that tiny "difference" between
19edo and 1/3cmt, and the vector between those two points
would be a xenharmonic bridge. or ... if i'm restricting
the meaning of xenharmonic bridge as Paul suggests i should,
then it's something else that deserves a unique name in
my theory, because it's expressing an important component
of my theory.

-monz

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🔗paulerlich <paul@stretch-music.com>

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> it seems to me that what i'm getting at is that i think
> kernels should be definable with matrices of fractions as
> well as integers.

my opinion: this is sheer nonsense, the intellectual equivalent of
driving off a cliff in terms of understanding temperaments. gene?

> can we please start with an example which compares 19-edo
> to 1/3-comma meantone? here are both generators and their
> difference:
>
> ~cents
>
> [2 3 5] [ 1/3 -1/3 1/3] = 694.7862377 = 1/3cmt "5th"
>
> - [2 3 5] [ 11/19 0 0 ] = 694.7368421 = 19edo "5th"
> ----------------------------
>
> [2 3 5] [-14/57 -1/3 1/3] = 0.0493956 = 1/3cmt==19edo bridge
>
>
> now if my goal is to plot b o t h of these temperaments
> along with the JI pitches all on the same lattice, with the
> same 0,0 origin = 1/1 for all three tunings, then why is this
> 1/3cmt==19edo bridge (that is, "1/3-comma meantone is equivalent
> to 19-EDO") not valid as a basis?

i don't know what you mean by basis here. your basis is [2 3 5], as i
see it.

> n o t necessarily as a
> l a t t i c e m e t r i c , but as a mathematical basis
> for comparing the set of tunings nonetheless.
>
>
> as i said, the whole matrix could be multiplied by the gcd:
>
>
> [2 3 5] ( [ 19 -19 19] * 1/57) = 1/3cmt "5th"
>
> - [2 3 5] ( [ 33 0 0] * 1/57) = 19edo "5th"
> ----------------------------
>
> [2 3 5] ( [-14 -19 19] * 1/57) = 1/3cmt==19edo bridge
>
>
>
> is that right? -- assuming that it is ...
>
> now to get the solution i'm seeking, don't we have to keep
> the same exponent of 2 for both the 19edo and the 1/3cmt?
>
> so, for this case, what's the solution for the a,b,c,d of
> paul's post? that would be: the rational exponent pair for
> prime-factors 3 and 5 which most closely approximates each
> generator step of 19-edo, while the exponent of 2 is kept
> the same for both that tuning and the equivalent 1/3cmt step.
>
> so what i want to do now is change the exponent numerator
> of 2 in the last matrix above to 19 for the 19edo "5th",
> and adjust the values of 3 and 5 accordingly.
>
> this would therefore plot that tiny "difference" between
> 19edo and 1/3cmt, and the vector between those two points
> would be a xenharmonic bridge. or ... if i'm restricting
> the meaning of xenharmonic bridge as Paul suggests i should,
> then it's something else that deserves a unique name in
> my theory, because it's expressing an important component
> of my theory.

i hope gene will come in with his opinion. this could work out
wonderfully, i don't know.

🔗genewardsmith <genewardsmith@juno.com>

🔗monz <joemonz@yahoo.com>

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Wednesday, February 06, 2002 5:12 PM
> Subject: [tuning-math] Re: exactly what is a xenharmonic bridge?
>
>
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > > it seems to me that what i'm getting at is that i think
> > > kernels should be definable with matrices of fractions as
> > > well as integers.
>
> > my opinion: this is sheer nonsense, the intellectual equivalent of
> > driving off a cliff in terms of understanding temperaments. gene?
>
> It isn't nonsense, but I don't see what value it has. One can
> define kernels for mappings of finite-dimentional vector spaces
> over the rational numbers Q. This produces non-finitely-generated
> abelian group structures, whose musical meaning I don't see.
> If Monz can explain why it makes sense, the math would not be
> a problem.

thanks, Gene. i'm trying hard to explain this to the rest of
you, but Paul and i have been arguing about it for 3 years, so
i won't be surprised if it takes a little longer for me to make
my ideas clear.

i'm sorry that i can't do this abstractly, because i don't have
anywhere near enough understanding of the algebra. the only way i
can even attempt to make my points is by using an example, and
i'd like to stick with the one i already proposed, which is a
comparison of 1/3-comma meantone and 19-edo. so ...

> here are both generators and their difference:
>
> ~cents
>
> [2 3 5] [ 1/3 -1/3 1/3] = 694.7862377 = 1/3cmt "5th"
>
> - [2 3 5] [ 11/19 0 0 ] = 694.7368421 = 19edo "5th"
> ----------------------------
>
> [2 3 5] [-14/57 -1/3 1/3] = 0.0493956 = 1/3cmt==19edo bridge
>
> ...
>
> as i said, the whole matrix could be multiplied by the gcd:
>
>
> [2 3 5] ( [ 19 -19 19] * 1/57) = 1/3cmt "5th"
>
> - [2 3 5] ( [ 33 0 0] * 1/57) = 19edo "5th"
> ----------------------------
>
> [2 3 5] ( [-14 -19 19] * 1/57) = 1/3cmt==19edo bridge
>
>
>
> is that right?

so, i s it correct? i can't proceed until i know that
what i've already done is working right. all i want to
know here is if i'm using the correct notation. can the
numbers in the first matrix be legitimately changed to
the numbers in the second matrix, without altering their
values? if this is incorrect, is there another way to
express what i'm doing here? or is my whole procedure wrong?

> > [me, monz]
> > now if my goal is to plot b o t h of these temperaments
> > along with the JI pitches all on the same lattice, with
> > the same 0,0 origin = 1/1 for all three tunings, then why
> > is this 1/3cmt==19edo bridge (that is, "1/3-comma meantone
> > is equivalent to 19-EDO") not valid as a basis?
>
> [paul]
> i don't know what you mean by basis here. your basis is
> [2 3 5], as i see it.

ok, i t h i n k that what i'm trying to do here is use
a simple transformation to change my basis from [2 3 5] to
[2/57 3/57 5/57] -- is that right?

-monz

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