Yahoo Tuning Groups Ultimate Backup tuning finity (was: A single notation system for any tuning)

hi paul,

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, November 26, 2002 10:34 AM
> Subject: [tuning] Re: A single notation system for any tuning
>
>
> --- In tuning@y..., "monz" <monz@a...> wrote:
>
> > you might like to take a look at my ideas concerning "finity":
> > http://sonic-arts.org/dict/finity.htm
>
> hi folks,
>
> i still maintain that this:
>
> "[Paul Erlich has done important work in this area - see his Tuning
> Digest postings on harmonic entropy]"
>
> is misleading. any important work that i may have done in the field
> of finity has concerned periodicity blocks, the Hypothesis, etc.
> harmonic entropy has nothing, as far as i can see, to do with it.
> harmonic entropy investigates a possible information-theoretic basis
> for choosing the *rungs* of the lattice -- the basic consonant
> intervals from which the lattice is constructed. finity, especially
> in the context of periodicity blocks, concerns the similarity in
> pitch between two vertices in the lattice, normally separated from one
> another by quite a few rungs, and the effect of treating these
> vertices as equivalent pitches, whether by outright tempering of the
> rungs or by notational identification.

hmmm ... sounds like your fencing off your own definition of finity.

whenever i mention finity, i always add the disclaimer that my ideas
on it are fuzzy and not well formulated.

but to my thinking, "a possible information-theoretic basis for
choosing the *rungs* of the lattice" definitely has *something*
to do with the intellectual act of somehow limiting the potentially
infinite palette of pitch resources available within the range of
human hearing.

one of these days when i have time we really need to chat about finity.

-monz

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗monz <monz@attglobal.net>

----- Original Message -----
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: Wednesday, November 27, 2002 11:04 AM
Subject: [tuning] Re: finity (was: A single notation system for any tuning)

> --- In tuning@y..., "monz" <monz@a...> wrote:
>
> > but to my thinking, "a possible information-theoretic basis for
> > choosing the *rungs* of the lattice" definitely has *something*
> > to do with the intellectual act of somehow limiting the potentially
> > infinite palette of pitch resources available within the range of
> > human hearing.
>
> it has something to do with the act of limiting the set of intervals
> which we'd treat as rungs in the lattice. even if there's only one
> type of rung, a perfect fifth, there is still no finity yet, since
> the pythagorean chain can extend infinitely.

mathematically and conceptually, a Pythagorean chain can extend
infinitely, but for practical purposes of instrument-building,
notation, analysis, etc., either the Pythagorean comma (~23.5 cents)
or Mercator's comma (~3.6 cents) would act as a limiter, resulting
in chains of respectively 12 or 53 notes.

my own opinion on this specific example is that the Pythagorean
comma is generally distinguishable but Mercator's comma is not.
of course, whether or not either of these (or possibly other
commas too) are distinguishable depends a great deal on musical
context. but in most cases i think Mercator's comma would not
be noticed by a listener, and in an analysis would be assumed
to vanish.

my concept of finity depends on variables such as this.
so it concerns not only the choices made for the rungs on the
lattice, but also, the choices made for the *boundaries* of
the periodicity-blocks.

-monz

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

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

> > > but to my thinking, "a possible information-theoretic basis for
> > > choosing the *rungs* of the lattice" definitely has *something*
> > > to do with the intellectual act of somehow limiting the
potentially
> > > infinite palette of pitch resources available within the range
of
> > > human hearing.
> >
> > it has something to do with the act of limiting the set of
intervals
> > which we'd treat as rungs in the lattice. even if there's only
one
> > type of rung, a perfect fifth, there is still no finity yet,
since
> > the pythagorean chain can extend infinitely.
>
>
>
> mathematically and conceptually, a Pythagorean chain can extend
> infinitely, but for practical purposes of instrument-building,
> notation, analysis, etc., either the Pythagorean comma (~23.5 cents)
> or Mercator's comma (~3.6 cents) would act as a limiter, resulting
> in chains of respectively 12 or 53 notes.

this is a perfect illustration of how the unison vector / periodicity
block business has everything to do with finity. harmonic entropy
doesn't.

> my own opinion on this specific example is that the Pythagorean
> comma is generally distinguishable but Mercator's comma is not.
> of course, whether or not either of these (or possibly other
> commas too) are distinguishable depends a great deal on musical
> context. but in most cases i think Mercator's comma would not
> be noticed by a listener, and in an analysis would be assumed
> to vanish.

of course, if you temper the fifths, you can make mercator's comma,
the pythagorean comma, or even the pythagorean limma vanish exactly,
making distinguishability irrelevant. and if you make the pythagorean
comma vanish, mercator's comma becomes a full semitone, and would not
be assumed to vanish :)

> my concept of finity depends on variables such as this.
> so it concerns not only the choices made for the rungs on the
> lattice, but also, the choices made for the *boundaries* of
> the periodicity-blocks.

what i was arguing is that finity has everything to do with the
boundaries of the periodicity blocks (aka the unison vectors), so we
appear to be in complete agreement on this. however, the choices made
for the ji intervals comprising the set of rungs is a separate issue,
and whatever they are, they could just as easily form the basis for an
infinite-lattice conception of musical pitch resources, and/or one
that derives scales from symmetrical lattice structures (i.e., the
diamond, the eikosany and other CPS scales, stellated structures,
etc.). thus my contention that the statement on your finity webpage
about harmonic entropy "rung" wrong to my ears, since it suggests
that harmonic entropy relates directly to the choices made for unison
vectors (aka periodicity block boundaries).

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@y..., "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@y..., "monz" <monz@a...> wrote:
> >
> /tuning/topicId_41224.html#41260
>
>
> ***Greetings from an undisclosed offsite Thanksgiving location!
> Best to all!
>
> Well, it sure seems to me that Monzo's concept of "finity" and
Paul
> Erlich's "Harmonic Entropy" cover the same ground...
>
> Could somebody please inform me as to why they are so different??
>
> best,
>
> Joseph Pehrson, gobble, gobble

hi joseph!

i'm in new york, of course . . .

what do the two have to do with one another?

let's say you're looking at a harmonic entropy diagram where you see
local minima (valleys) at consonant intervals within the 7-limit.

then you can choose to use these intervals as your basic "rungs" in
the lattice, and expand the lattice infinitely in 3-dimensional
space, or use a symmetrical structure such as a diamond, a stellated
hexany, or whatever . . . already we're pretty far from the realm of
harmonic entropy.

now, finity seems to usually involve a conscious or unconscious
choice of boundaries in the lattice, and the spacing of these
boundaries generally defines a set of unison vectors (usually real
small intervals) which can be used to formalize the finity as a
particular periodicity block. some or all of these unison vectors can
then be tempered out so that extra instances of the consonant
intervals appear, joining a pitch at one boundary to another pitch on
the opposite boundary. the process of finding suitable small
intervals in the lattice, and the rest of it, doesn't strike me as
having anything to do with harmonic entropy, beyond the indirect
connection traced above.

i'm probably not helping, but if any of the above is unclear to you,
clarifying it can't hurt . . .

let me know,
paul

🔗monz <monz@attglobal.net>

hi paul and Joe,

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, November 28, 2002 1:36 PM
> Subject: [tuning] Re: finity (was: A single notation system for any
tuning)
>
>
> --- In tuning@y..., "Joseph Pehrson" <jpehrson@r...> wrote:
> >
> >
> > Well, it sure seems to me that Monzo's concept of "finity"
> > and Paul Erlich's "Harmonic Entropy" cover the same ground...
> >
> > Could somebody please inform me as to why they are so different??
>
>
>
> what do the two have to do with one another?
>
> let's say you're looking at a harmonic entropy diagram where you see
> local minima (valleys) at consonant intervals within the 7-limit.
>
> then you can choose to use these intervals as your basic "rungs" in
> the lattice, and expand the lattice infinitely in 3-dimensional
> space, or use a symmetrical structure such as a diamond, a stellated
> hexany, or whatever . . . already we're pretty far from the realm of
> harmonic entropy.
>
> now, finity seems to usually involve a conscious or unconscious
> choice of boundaries in the lattice, and the spacing of these
> boundaries generally defines a set of unison vectors (usually real
> small intervals) which can be used to formalize the finity as a
> particular periodicity block. some or all of these unison vectors can
> then be tempered out so that extra instances of the consonant
> intervals appear, joining a pitch at one boundary to another pitch on
> the opposite boundary. the process of finding suitable small
> intervals in the lattice, and the rest of it, doesn't strike me as
> having anything to do with harmonic entropy, beyond the indirect
> connection traced above.
>
> i'm probably not helping, but if any of the above is unclear to you,
> clarifying it can't hurt . . .

this helps to clarify things a bit for me too ... as i say
every time i mention "finity", my own conception of it is
a bit fuzzy. (and i came up with the term!)

let's take an example where a temperament is used as the
actual tuning ... let's say, 1/4-comma meantone.

a 7-limit lattice may be constructed to show a 31-tone
periodicity-block, since two tones separated by 32 generators
in a 1/4-comma meantone chain are separated by a small interval
of ~6.07 cents, which will probably most often act as a
unison-vector. (1/4 of a syntonic comma is ~5.4 cents,
which is why extending the chain past 31 tones results
in an adaptive-JI system).

so this is an example of the process you sketch above,
where that interval of +31 generators is perceived as the
same as the origin pitch (or an "8ve" of it), setting
up one boundary of the periodicity-block. if this
unison-vector is tempered out, we end up with 31edo.

in both 1/4-comma meantone and 31edo, the other boundary
of the periodicity-block is defined by the syntonic comma.

now, in 1/4-comma meantone, the interval of +4 generators
is exactly the 5/4 "major-3rd", but that of +10 generators
is an "augmented 6th" which is a close approximation of
the 7/4 "harmonic minor-7th" (only ~3.04 cents narrower),
and that of +1 generator is ~5.4 cents narrower than the
3/2 "perfect-5th".

so, what is it that allows a listener to perceive 5^(1/4)
as a 3/2 and 5^(10/4) as a 7/4? isn't that harmonic entropy?

-monz

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

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

> this helps to clarify things a bit for me too ... as i say
> every time i mention "finity", my own conception of it is
> a bit fuzzy. (and i came up with the term!)
>
> let's take an example where a temperament is used as the
> actual tuning ... let's say, 1/4-comma meantone.
>
> a 7-limit lattice may be constructed to show a 31-tone
> periodicity-block, since two tones separated by 32 generators
> in a 1/4-comma meantone chain are separated by a small interval
> of ~6.07 cents, which will probably most often act as a
> unison-vector. (1/4 of a syntonic comma is ~5.4 cents,
> which is why extending the chain past 31 tones results
> in an adaptive-JI system).
>
> so this is an example of the process you sketch above,
> where that interval of +31 generators is perceived as the
> same as the origin pitch (or an "8ve" of it), setting
> up one boundary of the periodicity-block. if this
> unison-vector is tempered out, we end up with 31edo.
>
> in both 1/4-comma meantone and 31edo, the other boundary
> of the periodicity-block is defined by the syntonic comma.

but you said 7-limit. the above would be true in the 5-limit. but in
the 7-limit, you need *three* unison vectors (all tempered out) to
define 31-equal.

> now, in 1/4-comma meantone, the interval of +4 generators
> is exactly the 5/4 "major-3rd", but that of +10 generators
> is an "augmented 6th" which is a close approximation of
> the 7/4 "harmonic minor-7th" (only ~3.04 cents narrower),
> and that of +1 generator is ~5.4 cents narrower than the
> 3/2 "perfect-5th".
>
> so, what is it that allows a listener to perceive 5^(1/4)
> as a 3/2 and 5^(10/4) as a 7/4? isn't that harmonic entropy?

sort of (in that we have a finite hearing resolution, and the latter
is one of the inputs into the harmonic entropy model). but this has
nothing to do with finity. even an infinite meantone chain exhibits
these same approximations. meanwhile, you could use the very same set
of three unison vectors to define a 31-tone *just* scale, if you
don't temper out any of the unison vectors. so the finity is the same
regardless of whether you temper or not.

temperament works because of the "s" part of harmonic entropy (among
other things). and finity sometimes, though not always, involves
temperament. so in that sense, they're indirectly connected (in yet
another way). but i still feel the statement on your finity page is a
misrepresentation of what harmonic entropy is about.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗monz <monz@attglobal.net>

to everyone:

should we move this discussion to the harmonic entropy
group, or keep it here?

to paul:

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, November 28, 2002 9:39 PM
> Subject: [tuning] Re: finity (was: A single notation system for any
tuning)
>
>
> --- In tuning@y..., "monz" <monz@a...> wrote:
>
> > this helps to clarify things a bit for me too ... as i say
> > every time i mention "finity", my own conception of it is
> > a bit fuzzy. (and i came up with the term!)
> >
> > let's take an example where a temperament is used as the
> > actual tuning ... let's say, 1/4-comma meantone.
> >
> > a 7-limit lattice may be constructed to show a 31-tone
> > periodicity-block, since two tones separated by 32 generators
> > in a 1/4-comma meantone chain are separated by a small interval
> > of ~6.07 cents, which will probably most often act as a
> > unison-vector. (1/4 of a syntonic comma is ~5.4 cents,
> > which is why extending the chain past 31 tones results
> > in an adaptive-JI system).
> >
> > so this is an example of the process you sketch above,
> > where that interval of +31 generators is perceived as the
> > same as the origin pitch (or an "8ve" of it), setting
> > up one boundary of the periodicity-block. if this
> > unison-vector is tempered out, we end up with 31edo.
> >
> > in both 1/4-comma meantone and 31edo, the other boundary
> > of the periodicity-block is defined by the syntonic comma.
>
> but you said 7-limit. the above would be true in the 5-limit.
> but in the 7-limit, you need *three* unison vectors (all
> tempered out) to define 31-equal.

right you are ... i caught that before i sent that post,
and should have fixed it but didn't. but the point i
make after it (quoted below) still holds -- any fairly
complex set of pitches may be perceived as a less complex
set (i.e., a smaller number of pitch- or interval-gestalts),
and your explanations of harmonic entropy seem to me to
possibly be a determining factor in the choices our brains
make in structuring that set.

> > now, in 1/4-comma meantone, the interval of +4 generators
> > is exactly the 5/4 "major-3rd", but that of +10 generators
> > is an "augmented 6th" which is a close approximation of
> > the 7/4 "harmonic minor-7th" (only ~3.04 cents narrower),
> > and that of +1 generator is ~5.4 cents narrower than the
> > 3/2 "perfect-5th".
> >
> > so, what is it that allows a listener to perceive 5^(1/4)
> > as a 3/2 and 5^(10/4) as a 7/4? isn't that harmonic entropy?
>
> sort of (in that we have a finite hearing resolution, and the
> latter is one of the inputs into the harmonic entropy model).
> but this has nothing to do with finity. even an infinite
> meantone chain exhibits these same approximations. meanwhile,
> you could use the very same set of three unison vectors to
> define a 31-tone *just* scale, if you don't temper out any
> of the unison vectors. so the finity is the same regardless
> of whether you temper or not.

right, i understand that, and it doesn't seem to negate
my claim that harmonic entropy is part of the process.

> temperament works because of the "s" part of harmonic
> entropy (among other things). and finity sometimes, though
> not always, involves temperament. so in that sense, they're
> indirectly connected (in yet another way).

hmmm ... i'd appreciate it if you'd flesh out these
statements with a more detailed explanation of how you
see finity and temperament to be "indirectly connected".

> but i still feel the statement on your finity page is
> a misrepresentation of what harmonic entropy is about.

i'll be happy to change it ... but given the disagreements
in our current discussion concerning it, i'm not sure what
i should put there with your acknowledgment. how about
rewriting the offensive paragraph(s) for me?

-monz

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@y..., "monz" <monz@a...> wrote:
> to everyone:
>
> should we move this discussion to the harmonic entropy
> group, or keep it here?
>
>
> to paul:
>
>
> > From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
> > To: <tuning@y...>
> > Sent: Thursday, November 28, 2002 9:39 PM
> > Subject: [tuning] Re: finity (was: A single notation system for
any
> tuning)
> >
> >
> > --- In tuning@y..., "monz" <monz@a...> wrote:
> >
> > > this helps to clarify things a bit for me too ... as i say
> > > every time i mention "finity", my own conception of it is
> > > a bit fuzzy. (and i came up with the term!)
> > >
> > > let's take an example where a temperament is used as the
> > > actual tuning ... let's say, 1/4-comma meantone.
> > >
> > > a 7-limit lattice may be constructed to show a 31-tone
> > > periodicity-block, since two tones separated by 32 generators
> > > in a 1/4-comma meantone chain are separated by a small interval
> > > of ~6.07 cents, which will probably most often act as a
> > > unison-vector. (1/4 of a syntonic comma is ~5.4 cents,
> > > which is why extending the chain past 31 tones results
> > > in an adaptive-JI system).
> > >
> > > so this is an example of the process you sketch above,
> > > where that interval of +31 generators is perceived as the
> > > same as the origin pitch (or an "8ve" of it), setting
> > > up one boundary of the periodicity-block. if this
> > > unison-vector is tempered out, we end up with 31edo.
> > >
> > > in both 1/4-comma meantone and 31edo, the other boundary
> > > of the periodicity-block is defined by the syntonic comma.
> >
> > but you said 7-limit. the above would be true in the 5-limit.
> > but in the 7-limit, you need *three* unison vectors (all
> > tempered out) to define 31-equal.
>
>
>
> right you are ... i caught that before i sent that post,
> and should have fixed it but didn't. but the point i
> make after it (quoted below) still holds -- any fairly
> complex set of pitches may be perceived as a less complex
> set (i.e., a smaller number of pitch- or interval-gestalts),
> and your explanations of harmonic entropy seem to me to
> possibly be a determining factor in the choices our brains
> make in structuring that set.
>
>
>
> > > now, in 1/4-comma meantone, the interval of +4 generators
> > > is exactly the 5/4 "major-3rd", but that of +10 generators
> > > is an "augmented 6th" which is a close approximation of
> > > the 7/4 "harmonic minor-7th" (only ~3.04 cents narrower),
> > > and that of +1 generator is ~5.4 cents narrower than the
> > > 3/2 "perfect-5th".
> > >
> > > so, what is it that allows a listener to perceive 5^(1/4)
> > > as a 3/2 and 5^(10/4) as a 7/4? isn't that harmonic entropy?
> >
> > sort of (in that we have a finite hearing resolution, and the
> > latter is one of the inputs into the harmonic entropy model).
> > but this has nothing to do with finity. even an infinite
> > meantone chain exhibits these same approximations. meanwhile,
> > you could use the very same set of three unison vectors to
> > define a 31-tone *just* scale, if you don't temper out any
> > of the unison vectors. so the finity is the same regardless
> > of whether you temper or not.
>
>
>
> right, i understand that, and it doesn't seem to negate
> my claim that harmonic entropy is part of the process.
>
when you can have either one without the other, how can you say that
one is part of the process of the other?
>
> > temperament works because of the "s" part of harmonic
> > entropy (among other things). and finity sometimes, though
> > not always, involves temperament. so in that sense, they're
> > indirectly connected (in yet another way).
>
>
> hmmm ... i'd appreciate it if you'd flesh out these
> statements with a more detailed explanation of how you
> see finity and temperament to be "indirectly connected".

this last paragraph was a summary of what i wrote above it. what part
of that isn't detailed enough?

>
> > but i still feel the statement on your finity page is
> > a misrepresentation of what harmonic entropy is about.
>
>
> i'll be happy to change it ... but given the disagreements
> in our current discussion concerning it, i'm not sure what
> i should put there with your acknowledgment. how about
> rewriting the offensive paragraph(s) for me?

just delete the statement:

"[Paul Erlich has done important work in this area - see his Tuning
Digest postings on harmonic entropy]"

it doesn't belong there at all. in fact, the statement above it,

"These boundaries also designate the limits of interval size of
categorical interval perception. They have yet to be conclusively
determined, in part because of the effects of bridging."

smells funny to me. what does "limits of interval size of categorical
interval perception" mean? categorical perception seems to be a
different issue entirely, particularly where the finity is effected
through temperament, since then we're dealing with exact pitch
identity -- bridges that represent interval sizes of exactly zero.
and if "they" (meaning what?) have yet to be determined in part
because of bridging, what's the other part?

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>

wrote:
> --- In tuning@y..., "Joseph Pehrson" <jpehrson@r...> wrote:
>
> > Anyway, I'm getting the idea that "finity" is, essentially,
> > corrupting the "purity" of Harmonic Entropy to a degree.
>
> ????????????? i'm not getting where you're coming from on this.
> >
> > By defining things like that, you're really undermining the
> > flexibility or exactness of the original premise...
>
> once again, ?????????????!
>
> > Wow, interesting stuff, though! More later, hopefully, when you
> get
> > a chance..
>
> i have a chance . . .

***Hi Paul,

A certain part of this discussion seems to revolve around
*semantics* so I think I'll let you an Monz battle out the "finity" -
-"harmonic entropy" discussion, since you're both better versed in
it than I am...

best,

Joseph