Yahoo Tuning Groups Ultimate Backup tuning new monz page: Eivind Groven's Schismatic Temperament

--- In tuning@y..., "monz" <monz@a...> wrote:
> hello all,
>
>
> here's a new essay i've written on the details of
> Groven's 36-tone approximation of JI:
>
> "Eivind Groven's Schismatic Temperament"
> http://www.ixpres.com/interval/monzo/groven/groven.htm
>
>
> i'm particularly interested in getting feedback from
> the mathematically-oriented list members, on how
> Groven's choice of 36 tones might be characterized
> by a periodicity-block. i'll include that info on
> the webpage. thanks.

it can be characterized as a periodicity block with the two unison
vectors

32805:32768 (schisma);
648:625 (major diesis, etc.).

this can be easily seen by tiling the plane with that 36-tone "block"
from your page.

HOWEVER, this block suffers from the dreaded affliction of TORSION
(which you have a page on at
www.ixpres.com/interval/dict/torsion.htm -- though your site is down
right now).

this can be seen as follows (or by using gene's gcd-of-minors test):
multiply the two unison vectors together, and you get an interval
which is another unison vector of the block -- here it happens to be

531441:512000

BUT, as can easily be seen by factoring, this interval is simply

(81:80)^3 (or three syntonic commas).

as 81:80 (the syntonic comma) is distinguished (though conflated with
the pythagorean comma, into an interval of 20.5 cents), not ignored,
in this tuning, we have a bit of a conceptual contradition -- the
syntonic comma is to be considered a full step in the 36-tone tuning,
while three syntonic commas somehow are to be considered to add up to
zero steps.

torsion has been discussed at great length on tuning-math -- another
classic example is helmholtz's 24-tone tuning system, which is just
like this one except that (81:80)^2, not (81:80)^3, is a unison
vector. either way, you're really just dealing with a 12-tone
periodicity block, but it appears in duplicate or triplicate since
the given unison vectors are members of, but not a basis for, the
kernel of the relevant epimorphism.

one way in which torsional blocks differ from "true" periodicity
blocks is in the CS property
(www.ixpres.com/interval/dict/constant.htm). true PBs are CS,
while "TBs" (torsional blocks) aren't.

a follower of the wilson/grady school (such as justin white) would
prefer a 29-tone schismic scale to either the 24- or 36-tone
varieties that have been proposed, since with 29 tones, the CS
property would hold. also with 17 tones, and indeed the 17-tone
schismic scale is one of erv wilson's most famous (and it seems to
date back to medieval arabic theorists) . . .

both helmholtz and groven, however, were thinking in terms of a 12-
tone keyboard, where the range of key signatures would be the same as
that in 12-equal (with a few "kinks" in the helmholtz case), but
comma alterations would be allowed at will to provide just (to within
1/8-schisma, or 1/4 cent) tuning for every triad. this underscores
the fact that the comma is indeed considered a unison vector in, and
thus a 12-tone periodicity block is actually at the heart of, these
systems.

as adaptive just intonation goes, these systems have their plusses
and minuses. let's start with the minuses. clearly there are the 1/4-
cent deviations from justness in the verticalities, but probably more
serious for most listeners, there are full-comma (that is, 20.5 cent)
shifts required in some of the simplest of diatonic chord
progressions (known as "comma pumps"). on these grounds, one might be
tempted to favor vicentino's second tuning of 1555 (two meantone
chains 1/4-comma apart -- vicentino, coincidentally, also used 36
tones, with both this scheme and his other of that year), in which
the triads can be made absolutely pure, with no shifts greater than
1/4-comma (5.4 cents -- just beneath the limen of melodic
discrimination) for the diatonic "comma pump" chord progressions in
question. for these reasons, i've advocated (with some support from
margo) this latter tuning as an ideal for the performance of
renaissance and baroque polyphony.

now the plusses. beginning with beethoven, it became very common for
composers to treat the pythagorean comma (or, equivalently [since the
syntonic comma was already ignored], the schisma or diaschisma) as a
unison vector -- that is, as an enharmonic equivalence, in the course
of extensively modulating music (a key signature of 5 sharps would,
for example, be suddenly reinterpreted as a key signature of 7
flats). in vicentino's scheme, such a modulation would necessitate a
sudden shift of a full diesis, or 41 cents -- almost a quartertone!
meanwhile, in the helmholtz and groven schemes, the shift involved is
that same comma or 20.5 cents -- exactly half as much. if the 20.5
cent shifts were deemed acceptable for the "comma pump" case, they
would be equally acceptable for this "enharmonic modulation" case,
and the entire repertoire of 12-tone-notated triadic harmonic
practice could be acceptably rendered in virtually just vertical
intonation.

another adaptive tuning system of interest along these lines consists
of two or more 12-equal chains, spaced 15 cents apart from one
another. here, the vertical triads are a full 2 cents off from just
intonation (still not enough to bother me), but the "comma pump"
involves a shift of 15 cents, while an "enharmonic modulation"
involves no shift at all! this could be useful for adaptively
rendering music in which the pythagorean comma plays a more
significant role than the syntonic comma -- the song "awaken" by yes
comes to mind . . .

🔗monz <monz@attglobal.net>

thanks for all your great comments here, paul.

i've uploaded the page to here:
/tuning/files/monzo/groven/groven.htm

-monz
"all roads lead to n^0"

----- Original Message -----
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: Thursday, September 26, 2002 7:37 AM
Subject: [tuning] Re: new monz page: Eivind Groven's Schismatic Temperament

> --- In tuning@y..., "monz" <monz@a...> wrote:
> > hello all,
> >
> >
> > here's a new essay i've written on the details of
> > Groven's 36-tone approximation of JI:
> >
> > "Eivind Groven's Schismatic Temperament"
> > http://www.ixpres.com/interval/monzo/groven/groven.htm
> >
> >
> > i'm particularly interested in getting feedback from
> > the mathematically-oriented list members, on how
> > Groven's choice of 36 tones might be characterized
> > by a periodicity-block. i'll include that info on
> > the webpage. thanks.
>
> it can be characterized as a periodicity block with the two unison
> vectors
>
> 32805:32768 (schisma);
> 648:625 (major diesis, etc.).
>
> this can be easily seen by tiling the plane with that 36-tone "block"
> from your page.
>
> HOWEVER, this block suffers from the dreaded affliction of TORSION
> (which you have a page on at
> www.ixpres.com/interval/dict/torsion.htm -- though your site is down
> right now).
>
> this can be seen as follows (or by using gene's gcd-of-minors test):
> multiply the two unison vectors together, and you get an interval
> which is another unison vector of the block -- here it happens to be
>
> 531441:512000
>
> BUT, as can easily be seen by factoring, this interval is simply
>
> (81:80)^3 (or three syntonic commas).
>
> as 81:80 (the syntonic comma) is distinguished (though conflated with
> the pythagorean comma, into an interval of 20.5 cents), not ignored,
> in this tuning, we have a bit of a conceptual contradition -- the
> syntonic comma is to be considered a full step in the 36-tone tuning,
> while three syntonic commas somehow are to be considered to add up to
> zero steps.
>
> torsion has been discussed at great length on tuning-math -- another
> classic example is helmholtz's 24-tone tuning system, which is just
> like this one except that (81:80)^2, not (81:80)^3, is a unison
> vector. either way, you're really just dealing with a 12-tone
> periodicity block, but it appears in duplicate or triplicate since
> the given unison vectors are members of, but not a basis for, the
> kernel of the relevant epimorphism.
>
> one way in which torsional blocks differ from "true" periodicity
> blocks is in the CS property
> (www.ixpres.com/interval/dict/constant.htm). true PBs are CS,
> while "TBs" (torsional blocks) aren't.
>
> a follower of the wilson/grady school (such as justin white) would
> prefer a 29-tone schismic scale to either the 24- or 36-tone
> varieties that have been proposed, since with 29 tones, the CS
> property would hold. also with 17 tones, and indeed the 17-tone
> schismic scale is one of erv wilson's most famous (and it seems to
> date back to medieval arabic theorists) . . .
>
> both helmholtz and groven, however, were thinking in terms of a 12-
> tone keyboard, where the range of key signatures would be the same as
> that in 12-equal (with a few "kinks" in the helmholtz case), but
> comma alterations would be allowed at will to provide just (to within
> 1/8-schisma, or 1/4 cent) tuning for every triad. this underscores
> the fact that the comma is indeed considered a unison vector in, and
> thus a 12-tone periodicity block is actually at the heart of, these
> systems.
>
> as adaptive just intonation goes, these systems have their plusses
> and minuses. let's start with the minuses. clearly there are the 1/4-
> cent deviations from justness in the verticalities, but probably more
> serious for most listeners, there are full-comma (that is, 20.5 cent)
> shifts required in some of the simplest of diatonic chord
> progressions (known as "comma pumps"). on these grounds, one might be
> tempted to favor vicentino's second tuning of 1555 (two meantone
> chains 1/4-comma apart -- vicentino, coincidentally, also used 36
> tones, with both this scheme and his other of that year), in which
> the triads can be made absolutely pure, with no shifts greater than
> 1/4-comma (5.4 cents -- just beneath the limen of melodic
> discrimination) for the diatonic "comma pump" chord progressions in
> question. for these reasons, i've advocated (with some support from
> margo) this latter tuning as an ideal for the performance of
> renaissance and baroque polyphony.
>
> now the plusses. beginning with beethoven, it became very common for
> composers to treat the pythagorean comma (or, equivalently [since the
> syntonic comma was already ignored], the schisma or diaschisma) as a
> unison vector -- that is, as an enharmonic equivalence, in the course
> of extensively modulating music (a key signature of 5 sharps would,
> for example, be suddenly reinterpreted as a key signature of 7
> flats). in vicentino's scheme, such a modulation would necessitate a
> sudden shift of a full diesis, or 41 cents -- almost a quartertone!
> meanwhile, in the helmholtz and groven schemes, the shift involved is
> that same comma or 20.5 cents -- exactly half as much. if the 20.5
> cent shifts were deemed acceptable for the "comma pump" case, they
> would be equally acceptable for this "enharmonic modulation" case,
> and the entire repertoire of 12-tone-notated triadic harmonic
> practice could be acceptably rendered in virtually just vertical
> intonation.
>
> another adaptive tuning system of interest along these lines consists
> of two or more 12-equal chains, spaced 15 cents apart from one
> another. here, the vertical triads are a full 2 cents off from just
> intonation (still not enough to bother me), but the "comma pump"
> involves a shift of 15 cents, while an "enharmonic modulation"
> involves no shift at all! this could be useful for adaptively
> rendering music in which the pythagorean comma plays a more
> significant role than the syntonic comma -- the song "awaken" by yes
> comes to mind . . .

🔗Mark Rankin <markrankin95511@yahoo.com>

Monz,

Twice I tried to go to your Eivind Groven link, and
both times I got an advertising screen for some
business called Internet Express. As is usual with
your site, once I go (or try to go) to one of your
links, my 'forward' and 'back' buttons go dead. After
viewing your site, all I can do is hit 'Home' and go
back to Yahoo's opening screen, if I'm lucky. I think
there is some internal data missing or incorrect at
your site, at least in relation to the 'forward' and
'back' buttons. It has always been like this, in my
experience.

I was trying to see your entry on Groven. I got his
book from Norway a few years back, and was hoping I
could throw some light on the subject.

--Mark

--- monz <monz@attglobal.net> wrote:
> hello all,
>
>
> here's a new essay i've written on the details of
> Groven's 36-tone approximation of JI:
>
> "Eivind Groven's Schismatic Temperament"
>
http://www.ixpres.com/interval/monzo/groven/groven.htm
>
>
> i'm particularly interested in getting feedback from
> the mathematically-oriented list members, on how
> Groven's choice of 36 tones might be characterized
> by a periodicity-block. i'll include that info on
> the webpage. thanks.
>
>
>
> -monz
> "all roads lead to n^0"
>
>
>
>
>
>
>
>
>
>

__________________________________________________
Do you Yahoo!?
New DSL Internet Access from SBC & Yahoo!
http://sbc.yahoo.com

🔗monz <monz@attglobal.net>

hi Mark,

the Internet Express account is now dead. try this:

/tuning/files/monzo/groven/groven.htm

let me know if there's a problem accessing that page.

in private emails, paul erlich has shed some light on the
"periodicity-block" bit, and mostly it conformed to my
original hunch: Groven's 36-tone tuning can most likely
be considered as either an expansion of a 29-tone MOS
periodicity-block, or (the one i'd put my money on) as
a subset of a 41-tone MOS periodicity-block.

viewed as an "8ve"-invariant 5-prime-limit system, which
is as Groven conceived it, this is a 2-dimensional lattice.
therefore, 2 unison-vectors are required to define the
periodicity-block.

one of them is obviously the skhisma, which is tempered
out in Groven's actual tuning. viewed as a 29- or 41-tone
periodicity-block, the other unison-vector is not tempered
out. anyone care to put forth some eligible candidates?
the kleisma or diaschisma, perhaps?

but another part of the question in which i was interested
is this: is there another interval from the 5-limit lattice
which would define a larger periodicity-block than 41, and
which *is* also tempered out? likely candidates would be
found among the named intervals on the lattice on this page:

/tuning/files/td/monzo/o483-26new5limitnames.htm

if that link is broken, use this one:
http://makeashorterlink.com/?F21C151F1

feedback appreciated.

now, re: my webpages:

the most likely reason for the "forward" and "back"
buttons not working on my webpages is because generally
i commit the sin of having new browser windows open
to look at linked pages. others have complained, but
this is how i prefer to do it.

if the webpage opens in a brand-new window, obviously
the "back" button won't work because you haven't viewed
anything previously in that window. but if you close
that window when you're finished reading the new page,
you'll find that the old window with the old page is
still there, behind it, on your desktop.

-monz
"all roads lead to n^0"

----- Original Message -----
From: "Mark Rankin" <markrankin95511@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: Saturday, September 28, 2002 2:29 PM
Subject: Re: [tuning] new monz page: Eivind Groven's Schismatic Temperament

> Monz,
>
> Twice I tried to go to your Eivind Groven link, and
> both times I got an advertising screen for some
> business called Internet Express. As is usual with
> your site, once I go (or try to go) to one of your
> links, my 'forward' and 'back' buttons go dead. After
> viewing your site, all I can do is hit 'Home' and go
> back to Yahoo's opening screen, if I'm lucky. I think
> there is some internal data missing or incorrect at
> your site, at least in relation to the 'forward' and
> 'back' buttons. It has always been like this, in my
> experience.
>
> I was trying to see your entry on Groven. I got his
> book from Norway a few years back, and was hoping I
> could throw some light on the subject.
>
> --Mark

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@y..., "monz" <monz@a...> wrote:
> hi Mark,
>
> the Internet Express account is now dead. try this:
>
> /tuning/files/monzo/groven/groven.htm
>
> let me know if there's a problem accessing that page.
>
>
> in private emails, paul erlich has shed some light on the
> "periodicity-block" bit, and mostly it conformed to my
> original hunch: Groven's 36-tone tuning can most likely
> be considered as either an expansion of a 29-tone MOS
> periodicity-block, or (the one i'd put my money on) as
> a subset of a 41-tone MOS periodicity-block.

put your money on? i'm not sure how you'd go about "verifying" such a
thing. in any case, i believe my answer posted here on the tuning
list, which you've already seen, supercedes anything in those private
e-mails.

> viewed as an "8ve"-invariant 5-prime-limit system, which
> is as Groven conceived it, this is a 2-dimensional lattice.
> therefore, 2 unison-vectors are required to define the
> periodicity-block.
>
> one of them is obviously the skhisma, which is tempered
> out in Groven's actual tuning. viewed as a 29- or 41-tone
> periodicity-block, the other unison-vector is not tempered
> out. anyone care to put forth some eligible candidates?
> the kleisma or diaschisma, perhaps?

well, i think you already know how to work this out (though perhaps
not as efficiently as someone like gene). recall that the schisma is,
in vector form (we'll ignore factors of 2 for brevity),

[8 1].

now let's try completing the periodicity block with various other
unison vectors to see what cardinalities arise. for example, let's
try your suggestions. the kleisma is [5 -6], and taking the
determinant

|8 1|
| |
|5 -6|

yields -53: a 53-tone periodicity block. the diaschisma you should
already know leads good old 12-equal when combined with the schisma.

if you're clever like gene, you'll know how to start with 29 (or 41)
and the schisma and spit out the other unison vector. this isn't
hard. but you can also use brute force and try lots of unison vectors
(i always go to kees' pages such as his 5-limit page
http://www.kees.cc/tuning/s235.html for such a list), taking
determinants of the matrix each form with the schisma, and end up
with the "porcupine comma" (250:243) for 29, and the "5-limit magic
comma" (3072:3125) for 41.

there is an even easier way for you to figure this out! just look at
the first graph on your et page, which temporarily resides at
/tuning/files/dict/eqtemp.htm

focus in on the red line defined by the schisma being tempered out
(that is, the red line corresponding to schismic temperament. locate
29 and 41 on this line. make note of the other lines intersecting the
schismic line at these points. now you can immediately read off the
other unison vector for these systems! if that other unison vector is
tempered out, you get 29-equal and 41-equal -- if it isn't tempered
out, you get 29- and 41-tone MOSs of schismic temperament.

> but another part of the question in which i was interested
> is this: is there another interval from the 5-limit lattice
> which would define a larger periodicity-block than 41, and
> which *is* also tempered out?

no! a 5-limit tuning system with two unison vectors tempered out
becomes a *closed* system, that is, an equal temperament (if the
tempering-out is done uniformly, or a well-temperament otherwise).
1/8-schisma temperament, however, is clearly an infinite, open
system, not an equal- or well-temperament.

🔗monz <monz@attglobal.net>

hi paul,

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
>To: <tuning@yahoogroups.com>
> Sent: Monday, September 30, 2002 1:08 PM
> Subject: [tuning] Re: new monz page: Eivind Groven's Schismatic
Temperament
>
>
> --- In tuning@y..., "monz" <monz@a...> wrote:
> >
> > /tuning/files/monzo/groven/groven.htm
> >
>
> ... <snip> ...
>
> > viewed as an "8ve"-invariant 5-prime-limit system, which
> > is as Groven conceived it, this is a 2-dimensional lattice.
> > therefore, 2 unison-vectors are required to define the
> > periodicity-block.
> >
> > one of them is obviously the skhisma, which is tempered
> > out in Groven's actual tuning. viewed as a 29- or 41-tone
> > periodicity-block, the other unison-vector is not tempered
> > out. anyone care to put forth some eligible candidates?
> > the kleisma or diaschisma, perhaps?
>
> well, i think you already know how to work this out (though perhaps
> not as efficiently as someone like gene). recall that the schisma is,
> in vector form (we'll ignore factors of 2 for brevity),
>
> [8 1].
>
> now let's try completing the periodicity block with various other
> unison vectors to see what cardinalities arise. for example, let's
> try your suggestions. the kleisma is [5 -6], and taking the
> determinant
>
> |8 1|
> | |
> |5 -6|
>
> yields -53: a 53-tone periodicity block. the diaschisma you should
> already know leads good old 12-equal when combined with the schisma.
>
> ... <snip> ...
>
> > but another part of the question in which i was interested
> > is this: is there another interval from the 5-limit lattice
> > which would define a larger periodicity-block than 41, and
> > which *is* also tempered out?
>
> no! a 5-limit tuning system with two unison vectors tempered out
> becomes a *closed* system, that is, an equal temperament (if the
> tempering-out is done uniformly, or a well-temperament otherwise).
> 1/8-schisma temperament, however, is clearly an infinite, open
> system, not an equal- or well-temperament.

well, i think you really answered my question a few paragraphs above
this. i don't argue with your statement that 1/8-skhisma temperament
is open and infinte, but if Groven's system is viewed as a 36-tone
subset of 53edo, which approximates it very closely, then the two
unison-vectors which are tempered out are the skhisma and the kleisma.

so, Groven's tuning can easily be viewed as a subset of a 41-tone
periodicity-block, either JI (with neither unison-vector tempered
out, which can be argued in the case of the skhisma because its
2 cents is within most "margins of error") or skhismatic temperament.
or, if both unison-vectors are tempered out, it becomes a subset
of 53edo.

any debate over that?

-monz
"all roads lead to n^0"

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

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

> well, i think you really answered my question a few paragraphs above
> this. i don't argue with your statement that 1/8-skhisma
temperament
> is open and infinte, but if Groven's system is viewed as a 36-tone
> subset of 53edo, which approximates it very closely, then the two
> unison-vectors which are tempered out are the skhisma and the
kleisma.

right, but the periodicity block defined by these two would have all
53 notes. i think it's better to think of the kleisma as lying
outside groven's intended field of intervals, so it doesn't really
come into play. really the torsionally-36-but-really-12-element
periodicity block interpretation that i described previously best
reflects groven's musical desiderata.

> so, Groven's tuning can easily be viewed as a subset of a 41-tone
> periodicity-block, either JI (with neither unison-vector tempered
> out, which can be argued in the case of the skhisma because its
> 2 cents is within most "margins of error") or skhismatic
temperament.
> or, if both unison-vectors are tempered out, it becomes a subset
> of 53edo.

yes, all three of these tuning variations would probably be
satisfactory for most of groven's intended purposes. but the 41-tone
periodicity block is chimerical in this case, particularly in the 53-
equal tuning, because to justify 53-equal you need to start with a 53-
tone periodicity block!

🔗Mark Rankin <markrankin95511@yahoo.com>

🔗monz <monz@attglobal.net>

hi paul,

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, October 02, 2002 1:36 PM
> Subject: [tuning] Re: new monz page: Eivind Groven's Schismatic
Temperament

re:
/tuning/files/monzo/groven/groven.htm

>
> --- In tuning@y..., "monz" <monz@a...> wrote:
>
> > well, i think you really answered my question a few paragraphs
> > above this. i don't argue with your statement that 1/8-skhisma
> > temperament is open and infinte, but if Groven's system is viewed
> > as a 36-tone subset of 53edo, which approximates it very closely,
> > then the two unison-vectors which are tempered out are the skhisma
> > and the kleisma.
>
> right, but the periodicity block defined by these two would have all
> 53 notes. i think it's better to think of the kleisma as lying
> outside groven's intended field of intervals, so it doesn't really
> come into play. really the torsionally-36-but-really-12-element
> periodicity block interpretation that i described previously best
> reflects groven's musical desiderata.

ok, i can buy that.

> > so, Groven's tuning can easily be viewed as a subset of a 41-tone
> > periodicity-block, either JI (with neither unison-vector tempered
> > out, which can be argued in the case of the skhisma because its
> > 2 cents is within most "margins of error") or skhismatic
> temperament.
> > or, if both unison-vectors are tempered out, it becomes a subset
> > of 53edo.
>
> yes, all three of these tuning variations would probably be
> satisfactory for most of groven's intended purposes. but the 41-tone
> periodicity block is chimerical in this case, particularly in the 53-
> equal tuning, because to justify 53-equal you need to start with a 53-
> tone periodicity block!

ok, i can buy that too. originally, i thought there might be a pair
of unison-vectors coming into play that made 41 kind of jump out as
a magic number which set the finity of Groven's conception of tuning,
and that he simply lopped off a few notes on either end of the
Pythagorean-chain lattice, so that it would more easily jive with
the already-established 12-cardinality basis of the keyboard.

it makes more sense to view it as a complete 36-tone torsional-block
based on a 12-tone periodicity-block, as there are triplets of note-names
in his nomenclature for example:

generator --- prime-factor vector ---
2 5 2 5 ~cents ratio
12 B#3=C3 [ 28/8 -12/8] 20.5294292
0 C2 [ 0 0 ] 0 1/1
-12 Dbb1=C1=B#1 [-20/8 12/8] 1179.470571

-monz
"all roads lead to n^0"

new monz page: Eivind Groven's Schismatic Temperament

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🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

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