Yahoo Tuning Groups Ultimate Backup tuning new equal temperament 5-limit error lattices

i've made another significant update to the
"equal temperament" Dictionary entry:

http://www.ixpres.com/interval/dict/eqtemp.htm

from the webpage:

>> Below are lattice diagrams showing the amount of error
>> of various EDOs from 5-limit JI, in which I've used
>> greyscale to show the amount of error from the just ratio.
>>
>> Prime-factor 3 is along the horizontal axis, and
>> prime-factor 5 is along the vertical. The numbers in
>> the central cells show the degree of the EDO which most
>> closely approximates the ratio, and is given to one
>> decimal place. The actual EDO degree is the integer
>> part of that value, and the amount of error is given
>> to one decimal point as 1/10 of a degree in that EDO.
>>
>> If the decimal part is zero, the cell is white. If the
>> decimal part is .1 or .9, the cell is the lightest shade
>> of grey. If the decimal part is .2 or .8, the cell is
>> the next darker shade of grey; .3 or .7, the next darker
>> shade of grey; .4 or .6, the darkest shade of grey,
>> and if it is .5, the cell is black since that ratio
>> can be equally well approximated by the EDO degree on
>> either side of that value.
>>
>> The patterns of shade shown on these lattices show
>> interesting correspondences with other lattices I've
>> made. See, for instance, my webpage
>> "Lattice diagrams comparing rational implications
>> of various meantone chains"
http://www.ixpres.com/interval/monzo/meantone/lattices/lattices.htm
>> and compare the 19edo lattice here with the
>> 1/3-comma meantone lattice on that page; compare 43edo
>> here with 1/5-comma there, and 12edo here with
>> 1/11-comma there.

the new lattices begin right at halfway down the page.

-monz

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🔗monz <joemonz@yahoo.com>

> From: monz <joemonz@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, February 16, 2002 2:50 AM
> Subject: [tuning] new equal temperament 5-limit error lattices
>
>
> i've made another significant update to the
> "equal temperament" Dictionary entry:
>
> http://www.ixpres.com/interval/dict/eqtemp.htm

well, i've been working all night on yet another
major update to this page.

now, for some EDOs, it now also includes some color lattices
which show whether the deviation of the EDO from JI is positive
or negative for any given interval; and also some EDOs
include lattices for 7^1 and 11^1 to show approximations
to those higher primes.

comments appreciated.

(PS - the page takes a long time to load now, because of
the many graphics. eventually i may break it up into
smaller separate pages.)

-monz

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🔗jpehrson2 <jpehrson@rcn.com>

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: monz <joemonz@y...>
> > To: <tuning@y...>
> > Sent: Saturday, February 16, 2002 2:50 AM
> > Subject: [tuning] new equal temperament 5-limit error lattices
> >
> >
> > i've made another significant update to the
> > "equal temperament" Dictionary entry:
> >
> > http://www.ixpres.com/interval/dict/eqtemp.htm
>
>
>
> well, i've been working all night on yet another
> major update to this page.
>
> now, for some EDOs, it now also includes some color lattices
> which show whether the deviation of the EDO from JI is
positive
> or negative for any given interval; and also some EDOs
> include lattices for 7^1 and 11^1 to show approximations
> to those higher primes.
>
> comments appreciated.

i'm very confused by your grayscale and color tables (not really
lattices), though artistically they're quite attractive. for 12-equal,
for example. you have two tables for '7^1', and two tables for
'11^1'. what are the differences between the members of each
pair? what are the axes on these tables? i can't figure that out
from any of the information on this page. it's good practice in
general to always label your axes.

p.s. i think i understand the 5-limit tables, and i don't subscribe
to this method of evaluating equal temperaments, for the same
reasons i had problems with your 'rational implications of
meantone temperaments' bit. also, you're evaluating errors
relative to the step size of the et, rather than absolutely, but you're
a bit too cavalier about making comparisons without noting this
caveat. in other words, i like this whole section (apart from my
confusion mentioned above), but i believe it represents too
'specialized' a viewpoint to reside in a dictionary definition.

🔗monz <joemonz@yahoo.com>

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, February 18, 2002 3:14 PM
> Subject: [tuning] Re: new equal temperament 5-limit error lattices
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> >
> > i've made another significant update to the
> > "equal temperament" Dictionary entry:
> >
> > http://www.ixpres.com/interval/dict/eqtemp.htm
>
>
> i'm very confused by your grayscale and color tables
> (not really lattices),

huh? why aren't they lattices? Fokker drew cellular
lattices just like this, in his papers.

> though artistically they're quite attractive.

thanks, paul! :) took an awful lot of work.

> for 12-equal, for example. you have two tables for
> '7^1', and two tables for '11^1'. what are the differences
> between the members of each pair?

look again, more carefully ... one is 7^1, the other 7^-1,
likewise 11^1 and 11^-1.

> what are the axes on these tables? i can't figure that out
> from any of the information on this page.

it's in the text at the beginning of this section, where
i explain the lattices:

>> Prime-factor 3 is along the horizontal axis, and
>> prime-factor 5 is along the vertical. In those cases where
>> I illustrate lattices for prime-factors higher than 5,
>> the format resembles the old method I used to use (as in
>> several lattices in my book) wherein prime-factors 3 and 5
>> form a grid, which is then replicated for various exponents
>> of the other primes, somewhat like a ladder.

> it's good practice in general to always label your axes.

that's true ... i suppose someday i'll redo these so that
the axes are labeled clearly. but this new gallery of lattices
has been a tremendous amount of work, and now i need to get
away from it for a while.

> p.s. i think i understand the 5-limit tables, and i don't subscribe
> to this method of evaluating equal temperaments, for the same
> reasons i had problems with your 'rational implications of
> meantone temperaments' bit.

yes, i know ... these EDOs are all closed systems, so ideally
each of these lattices would wrap into a torus so that points
which appear separate on my flat lattices but which are represented
by the same EDO note, would actually occupy the same point on
the torus.

> also, you're evaluating errors relative to the step size
> of the et, rather than absolutely,

which i did deliberately and even pointed out on the webpage,
because i believe that it offers some kind of visual representation
of your "consistency" concept. please correct and clarify
if i'm wrong. (see the comments under 24edo and 36edo)

> but you're a bit too cavalier about making comparisons
> without noting this caveat.

that's probably true ... please refer to the case of burnout
cited above, from which i'm suffering now. keep this tucked away
as a "friendly reminder" and i'll eventually do something about it.

> in other words, i like this whole section (apart from my
> confusion mentioned above), but i believe it represents too
> 'specialized' a viewpoint to reside in a dictionary definition.

ok, i guess that's an opinion worth considering ...

now i'm going out for some fresh air ...

-monz

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🔗monz <joemonz@yahoo.com>

> From: monz <joemonz@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, February 18, 2002 3:59 PM
> Subject: Re: [tuning] Re: new equal temperament 5-limit error lattices
>
>
> > also, you're evaluating errors relative to the step size
> > of the et, rather than absolutely,
>
>
> which i did deliberately and even pointed out on the webpage,
> because i believe that it offers some kind of visual representation
> of your "consistency" concept. please correct and clarify
> if i'm wrong. (see the comments under 24edo and 36edo)

i thought i'd add a little more about this:

referring to the greyscale lattices, it seems to me that the
"dark regions" -- the cells which contain ratios which which
are basically midway two EDO pitches, and which i colored black
-- are consistency boundaries. crossing

for example, let's call our 1/1 "C". the note a 32/27 above
that is Eb, represented on the lattice at coordinates [3 5]^[-3 0],
and that cell's contents is the value 2.9, which means that
the pitch of the 3rd degree of 12edo is just a little sharper
than 32/27.

now look up the column from that cell. it looks like:

6.4
2.5
10.7
6.8
2.9

except that you can't read 2.5 because i colored it black.

the corresponding 5-limit note-names would be:

Fx 6.4
D# 2.5
B 10.7
G 6.8
Eb 2.9

but the 12edo representation of Fx would be G, which
is more than half a degree too sharp. thus, an inconsistency.

right, paul? (i hope so...)

ok, now off to enjoy the sunshine ... i'll be back here later.

-monz

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🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning@y...>
> > Sent: Monday, February 18, 2002 3:14 PM
> > Subject: [tuning] Re: new equal temperament 5-limit error
lattices
> >
> >
> > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > >
> > >
> > > i've made another significant update to the
> > > "equal temperament" Dictionary entry:
> > >
> > > http://www.ixpres.com/interval/dict/eqtemp.htm
> >
> >
> > i'm very confused by your grayscale and color tables
> > (not really lattices),
>
>
> huh? why aren't they lattices? Fokker drew cellular
> lattices just like this, in his papers.

> > though artistically they're quite attractive.
>
>
> thanks, paul! :) took an awful lot of work.
>
>
> > for 12-equal, for example. you have two tables for
> > '7^1', and two tables for '11^1'. what are the differences
> > between the members of each pair?
>
>
> look again, more carefully ... one is 7^1, the other 7^-1,
> likewise 11^1 and 11^-1.
>
>
> > what are the axes on these tables? i can't figure that out
> > from any of the information on this page.
>
> it's in the text at the beginning of this section, where
> i explain the lattices

oh . . . so you're only going out to 7 ^ +/- 1, and 11 ^ +/- 1, but to
much higher powers of 3 and 5 . . . no wonder i was confused!

>
> > p.s. i think i understand the 5-limit tables, and i don't
subscribe
> > to this method of evaluating equal temperaments, for the
same
> > reasons i had problems with your 'rational implications of
> > meantone temperaments' bit.
>
>
> yes, i know ... these EDOs are all closed systems, so ideally
> each of these lattices would wrap into a torus so that points
> which appear separate on my flat lattices but which are
represented
> by the same EDO note, would actually occupy the same point
on
> the torus.

not what i had in mind at all . . . really i was just getting at the fact
that the relevant comparison is with the *consonant* intervals of
ji, and since equal temperaments have every interval available
from every pitch, the rest follows from that. yes, your approach is
"equivalent" to this, but doesn't get this point across so well . . . it
may just be a matter of our different philosophies of interval
perception.

> > also, you're evaluating errors relative to the step size
> > of the et, rather than absolutely,
>
> which i did deliberately and even pointed out on the webpage,
> because i believe that it offers some kind of visual
representation
> of your "consistency" concept. please correct and clarify
> if i'm wrong. (see the comments under 24edo and 36edo)

i suspect my main problem in this regard is that with your
rectangular lattices, you're showing 15:8 as being as 'complex'
an interval as 5:3, while i feel them to be in different classes
altogether (15-limit and 5-limit, respectively).

🔗genewardsmith <genewardsmith@juno.com>

🔗joemonz <joemonz@yahoo.com>

> Message 34443
> From: "genewardsmith" <genewardsmith@j...>
> Date: Mon Feb 18, 2002 4:30 pm
> Subject: Re: new equal temperament 5-limit error lattices
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > huh? why aren't they lattices? Fokker drew cellular
> > lattices just like this, in his papers.
>
> They look like tables to me. Not only would mathematicians
> not call these lattices,

ok, fair enough. but to me, if it acts like a lattice,
it's a lattice.

> crystalographers wouldn't either.

really? i thought i'd seen crystals shaped like that.

>
> On the other hand, they're pretty neat.

thanks. nice to know my effort is appreciated. :)

> It would be nice to see a little more commentary on
> some of the systems

there will be lots more forthcoming. most likely,
i'll eventually split this up giving each EDO its own
page, and then collect all the pictures into a gallery
without text.

> --I think you tend to favor meantone or schismic systems
> over the likes of 34, 46 or 58.

not that i favor them -- only that i know more about them.

i've never played with "the likes of 34, 46 or 58", and
still don't understand a lot of what's been written about
them at tuning-math. hopefully you, paul, dave, and graham
will be able to contribute comments to this page.

-monz

🔗paulerlich <paul@stretch-music.com>

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

> i thought i'd add a little more about this:
>
> referring to the greyscale lattices, it seems to me that the
> "dark regions" -- the cells which contain ratios which which
> are basically midway two EDO pitches, and which i colored black
> -- are consistency boundaries.

sort of.

> crossing
>
>
> for example, let's call our 1/1 "C". the note a 32/27 above
> that is Eb, represented on the lattice at coordinates [3 5]^[-3 0],
> and that cell's contents is the value 2.9, which means that
> the pitch of the 3rd degree of 12edo is just a little sharper
> than 32/27.
>
> now look up the column from that cell. it looks like:
>
> 6.4
> 2.5
> 10.7
> 6.8
> 2.9
>
> except that you can't read 2.5 because i colored it black.
>
>
> the corresponding 5-limit note-names would be:
>
> Fx 6.4
> D# 2.5
> B 10.7
> G 6.8
> Eb 2.9
>
>
> but the 12edo representation of Fx would be G, which
> is more than half a degree too sharp. thus, an inconsistency.
>
> right, paul? (i hope so...)

well, this sure isn't "my" consistency . . . but i suppose there
would be a different kind of inconsistency if you actually tried to
compose a piece of music that was supposed to work exactly the same
way in the et as in ji. because in the et, moving though three of the
4/3s would get you to a _different note_ than the one you are
assigning to 32/27. this is somewhat related to paul hahn's
consistency *levels* -- which i avoid by saying that, in 3-limit or 9-
limit or even 23-limit situations, 32:27 is acoustically indistinct
so it's always used to mean three 4:3s (minus an octave) and never
something different.

ok, i guess it would be "my" consistency if we were operating in the
27-limit or a higher odd limit. rather "conceptual", it seems.

🔗paulerlich <paul@stretch-music.com>

🔗monz <joemonz@yahoo.com>

re: http://www.ixpres.com/interval/dict/eqtemp.htm

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, February 18, 2002 4:12 PM
> Subject: [tuning] Re: new equal temperament 5-limit error lattices
>
> ...
> > > p.s. i think i understand the 5-limit tables,
> > > and i don't subscribe to this method of evaluating
> > > equal temperaments, for the same reasons i had
> > > problems with your 'rational implications of
> > > meantone temperaments' bit.
> >
> >
> > yes, i know ... these EDOs are all closed systems,
> > so ideally each of these lattices would wrap into a
> > torus so that points which appear separate on my flat
> > lattices but which are represented by the same EDO
> > note, would actually occupy the same point on the torus.
>
> not what i had in mind at all . . . really i was just
> getting at the fact that the relevant comparison is
> with the *consonant* intervals of ji, and since equal
> temperaments have every interval available from every
> pitch, the rest follows from that. yes, your approach
> is "equivalent" to this, but doesn't get this point across
> so well . . . it may just be a matter of our different
> philosophies of interval perception.

hmm... no, paul, i think now i know where this leads...

the lattice for each EDO should have t h e s a m e
c a r d i n a l i t y as the EDO has.

so the 12edo lattice should only have 12 cells,
the 31edo should have 31, etc.

it would take a lot of work to redo all of these to
be like that, but that's how they should be done.
since composers sometimes want to use an EDO pitch
to mean a "nonstandard" interval, as in punning,
i'd prefer to use border lines to bound the cardinality
of the EDO, and leave a row or two of extra notes outside
those boundaries for the puns.

> > > also, you're evaluating errors relative to the step
> > > size of the et, rather than absolutely,
> >
> > which i did deliberately and even pointed out on the
> > webpage, because i believe that it offers some kind of
> > visual representation of your "consistency" concept.
> > please correct and clarify if i'm wrong. (see the
> > comments under 24edo and 36edo)
>
> i suspect my main problem in this regard is that with your
> rectangular lattices, you're showing 15:8 as being as 'complex'
> an interval as 5:3, while i feel them to be in different classes
> altogether (15-limit and 5-limit, respectively).
>

hmmm ... actually, the consistency bit arose later,
after i made the diagrams and looked at them a while.

as i state on the webpage, the intention of these
lattices is primarily to attempt to quantify Darreg's
idea of EDO "moods" in some way. i think they succeed
in that respect, altho i'm not entirely sure how. but
the visual appearance relates to the sound of the EDO
somehow, i think.

-monz

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🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "monz" <joemonz@y...> wrote:
> re: http://www.ixpres.com/interval/dict/eqtemp.htm
>
>
> > From: paulerlich <paul@s...>
> > To: <tuning@y...>
> > Sent: Monday, February 18, 2002 4:12 PM
> > Subject: [tuning] Re: new equal temperament 5-limit error lattices
> >
> > ...
> > > > p.s. i think i understand the 5-limit tables,
> > > > and i don't subscribe to this method of evaluating
> > > > equal temperaments, for the same reasons i had
> > > > problems with your 'rational implications of
> > > > meantone temperaments' bit.
> > >
> > >
> > > yes, i know ... these EDOs are all closed systems,
> > > so ideally each of these lattices would wrap into a
> > > torus so that points which appear separate on my flat
> > > lattices but which are represented by the same EDO
> > > note, would actually occupy the same point on the torus.
> >
> > not what i had in mind at all . . . really i was just
> > getting at the fact that the relevant comparison is
> > with the *consonant* intervals of ji, and since equal
> > temperaments have every interval available from every
> > pitch, the rest follows from that. yes, your approach
> > is "equivalent" to this, but doesn't get this point across
> > so well . . . it may just be a matter of our different
> > philosophies of interval perception.
>
>
> hmm... no, paul, i think now i know where this leads...
>
> the lattice for each EDO should have t h e s a m e
> c a r d i n a l i t y as the EDO has.
>
> so the 12edo lattice should only have 12 cells,
> the 31edo should have 31, etc.

again, this is not what i was suggesting at all. read the above.

> it would take a lot of work to redo all of these to
> be like that, but that's how they should be done.
> since composers sometimes want to use an EDO pitch
> to mean a "nonstandard" interval, as in punning,
> i'd prefer to use border lines to bound the cardinality
> of the EDO, and leave a row or two of extra notes outside
> those boundaries for the puns.

that would imply the torus, and is fine if you want to do it, but
doesn't seem important to me at all. but if you care about this,
perhaps putting the appropriate et degree number in each cell would
get this information across even better. kind of like james
mccartney's 'bingo card' on rick tagawa's now-defunct 72-equal home
page, if you remember that at all.

> as i state on the webpage, the intention of these
> lattices is primarily to attempt to quantify Darreg's
> idea of EDO "moods" in some way. i think they succeed
> in that respect, altho i'm not entirely sure how. but
> the visual appearance relates to the sound of the EDO
> somehow, i think.

surely you don't feel that 1000-equal and 1200-equal have
similar 'moods' to the ets with much lower cardinalities but similar
appearances, do you?

🔗monz <joemonz@yahoo.com>

re:
http://www.ixpres.com/interval/dict/eqtemp.htm

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, February 20, 2002 11:25 AM
> Subject: [tuning] Re: new equal temperament 5-limit error lattices
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > hmm... no, paul, i think now i know where this leads...
> >
> > the lattice for each EDO should have t h e s a m e
> > c a r d i n a l i t y as the EDO has.
> >
> > so the 12edo lattice should only have 12 cells,
> > the 31edo should have 31, etc.
>
> again, this is not what i was suggesting at all. read the above.

ok ... i'm still not understanding entirely what you
find objectionable about these.

> > it would take a lot of work to redo all of these to
> > be like that, but that's how they should be done.
> > since composers sometimes want to use an EDO pitch
> > to mean a "nonstandard" interval, as in punning,
> > i'd prefer to use border lines to bound the cardinality
> > of the EDO, and leave a row or two of extra notes outside
> > those boundaries for the puns.
>
> that would imply the torus, and is fine if you want to do it, but
> doesn't seem important to me at all. but if you care about this,
> perhaps putting the appropriate et degree number in each cell would
> get this information across even better. kind of like james
> mccartney's 'bingo card' on rick tagawa's now-defunct 72-equal home
> page, if you remember that at all.

huh? my lattices d o have the degree number in each cell!

> > as i state on the webpage, the intention of these
> > lattices is primarily to attempt to quantify Darreg's
> > idea of EDO "moods" in some way. i think they succeed
> > in that respect, altho i'm not entirely sure how. but
> > the visual appearance relates to the sound of the EDO
> > somehow, i think.
>
> surely you don't feel that 1000-equal and 1200-equal have
> similar 'moods' to the ets with much lower cardinalities but similar
> appearances, do you?

well, i'm going to withhold comment on that, because i'm
not sure if i can give a definite yes or no; i haven't
exactly p l a y e d with either of these a s tunings,
only used them as theoretical measurements. perhaps Marc
can give some insight.

but to pick a tuning which i h a v e played with,
i d o think it's interesting that, for example,
1024edo, which is what i used to get as a bottom-line
tuning on my Yamaha TG77, has an error pattern which
indicates that it gives certain 5-limit ratios much
better than others: it basically gives a pure Pythagorean
tuning, and then replicates that exactly at the interval
of a diesis, but the intervening two horizontal rows of
5-limit ratios are less exact.

i'm not saying that anyone is consciously aware of these
tiny differences in tuning, but i a m saying that i'd
bet our ear-brain systems are sensitive enough to notice
s o m e t h i n g about the sound of 1024edo that's
different from, say, 768edo, which is another popular
tuning resolution for electronic instruments. and i think
these lattices show one aspect of that "something".

-monz

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🔗jpehrson2 <jpehrson@rcn.com>

🔗paulerlich <paul@stretch-music.com>

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

> > that would imply the torus, and is fine if you want to do it, but
> > doesn't seem important to me at all. but if you care about this,
> > perhaps putting the appropriate et degree number in each cell
would
> > get this information across even better. kind of like james
> > mccartney's 'bingo card' on rick tagawa's now-defunct 72-equal
home
> > page, if you remember that at all.
>
>
> huh? my lattices d o have the degree number in each cell!

somehow i didn't see that before. but this is completely different
from what james mccartney did, which made much more sense to me, as
far as punning or everything else. james *consistently* used an
*integer* number of steps in the et for every consonant interval. so
the 'fifths' axis would go

0 42 12 54 24 66 36 6 48 18 60 30 0 42 12 54 . . .

this shows that 12 fifths, i.e. the pythagorean comma, is a "pun" for
the unison in 72-equal. your graph, on the other hand, would have a
1.29 as the value after 12 fifths, obscuring the 'punning' that goes
on in 72-equal.

as to the 'mood' of 1024-equal vs. 768-equal, i think it's pretty far-
fetched that their tiny errors relative to ji are going to make
nearly as much as a difference in mood as, say, 19-equal vs. 22-
equal. seriously. i'm really hoping that monz comes back down to
earth, so that his web pages might be viewed with respect and
interest by the whole world.

🔗paulerlich <paul@stretch-music.com>

🔗jonszanto <JSZANTO@ADNC.COM>

🔗paulerlich <paul@stretch-music.com>

🔗jpehrson2 <jpehrson@rcn.com>

🔗monz <joemonz@yahoo.com>

re:
http://www.ixpres.com/interval/dict/eqtemp.htm

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, February 20, 2002 1:25 PM
> Subject: [tuning] Re: new equal temperament 5-limit error lattices
>
> ...
> as to the 'mood' of 1024-equal vs. 768-equal, i think it's pretty far-
> fetched that their tiny errors relative to ji are going to make
> nearly as much as a difference in mood as, say, 19-equal vs. 22-
> equal. seriously. i'm really hoping that monz comes back down to
> earth, so that his web pages might be viewed with respect and
> interest by the whole world.

paul, your colorful metaphor of monz-as-astronaut fails
to make the impact upon me that i think you hoped for.

much as i respect your criticisms of my work, i daresay
that i needn't worry that the "whole world" won't view
my webpages "with respect and interest" simply because
of your criticisms.

(but you're still cool with me, 'cuz i know your heart's
in the right place)

besides, if the you a n d the "whole world" d o
judge me to be so far "out there" that i've left the
planet, i don't give a damn ...

the view from m y perspective is breathtaking!

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

🔗paulerlich <paul@stretch-music.com>

🔗monz <joemonz@yahoo.com>

🔗jpehrson2 <jpehrson@rcn.com>

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning@y...>
> > Sent: Thursday, February 21, 2002 4:48 AM
> > Subject: [tuning] Re: new equal temperament 5-limit error lattices
> >
> >
> > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > > re:
> > > http://www.ixpres.com/interval/dict/eqtemp.htm
> >
> > this is now my favorite page on the internet!
>
>
> cool! thanks, paul!
>
> ... and i promise you, it's going to keep growing!

oof . . . it took 6 minutes to download and even then it was still
only getting the 29-equal chart up. are you sure you can't split
those off onto a separate page/pages?

and now . . . my comments were about your text for 12-equal. what do
you mean it's 'barely' diesic?? and more important than diesic is the
linear temperament based on 648:625, which should be represented by a
line passing through 12 and 28 in your first graph, and takes into
account practices of many romantic composers but bears even more
directly on the music of liszt, stravinsky, bloch, bartok, late
scriabin, and a great deal of jazz since dizzy gillespie (since the
diminished seventh chord and the octatonic scale both result from
tempering out 648:625).

what to call these linear temperaments? 'diesis' unfortunately refers
to a lot of things, including both 128:125 and 648:625.
perhaps 'trefoil' would be better for the former and 'quatrefoil' for
the latter?

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "monz" <joemonz@y...> wrote:
> re:
> http://www.ixpres.com/interval/dict/eqtemp.htm

thanks for making the changes i suggested. this is now officially my favorite page on the internet!

after 6 minutes, the graph for 29 was just coming up . . . but i was still able to read the page while these were loading, which was nice.

about 12-equal -- what do you mean it's 'just barely' diesic? i can't understand why you would say 'just barely'.

also, even more important than diesic, is the linear temperament based on 648:625, whose mos scales include diminished and octatonic, and which falls on a straight line (in the first graph) connecting 12 with 28. this usage of 12 is extremely important in the music of the romantic period, where diminished sevenths can resolve four ways, and even more important in the music of liszt, stravinsky, bartok, bloch, late scriabin, and most jazz musicians since dizzy gillespie, including john coltrane. meanwhile, the 'diesic' usage of 12 shows up only occasionally, i can only think of two examples -- schubert and coltrane.

🔗jonszanto <JSZANTO@ADNC.COM>

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> seriously. i'm really hoping that monz comes back down to
> earth, so that his web pages might be viewed with respect and
> interest by the whole world.

Considering the time and effort Monz has put into his site,
considering that he constantly corrects things according to other's
wishes, considering so many things (like all the people that *don't*
have centralized information up, where their public presence would be
on display for anyone else to complain about)...

Considering all that, you could certainly have used a little more
generous language in dealing with how *you* would make those 5-limit
error lattices. And your judgement as to how the world might view his
work.

Ah, yes: the difference between a helpful correction and an
insensitive putdown. Since you've made the call, and I've brought the
subject up, I'll do any other kibbitzing about this on metatuning.

Thanks, *Monz*, for your efforts, free of charge and so accepting of
the many, many differing opinions offered to you. I know you do it in
the best of spirits...

Cheers,
Jon

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "jonszanto" <JSZANTO@A...> wrote:

> on display for anyone else to complain about)...

you're the one complaining here. monz asked for comments, so i'm
giving them.

> you could certainly have used a little more
> generous language in dealing with how *you* would make those 5-
limit
> error lattices.

i'd probably have 'bingo cards' james mccartney style (hex grid,
though) for the structural view of the et, and then a look at the
position with respect to the blue axes in the first graph (or a
similar one) for the 'harmonic flavor' view. if you choose an et for
the commas it eliminates, and thus its compositional possibilities,
the former is more important; if you choose an et for the way the
consonant ji harmonies are 'inflected', the latter is more important.
monz is trying to present both sets of information at the same time,
which is admirable, but falls a little short in both areas.

> And your judgement as to how the world might view his
> work.

well, as two grown men who have hung out in person, it might very
well be that monz and i are ok speaking in the terms in which we
speak, 'butting heads' so to speak, until the naked truth remains. or
maybe i'm wrong. but what do *you* think about the idea (you could
probably say from personal experience) that if you play something
written in ji on a 768-equal synth and then on a 1024-equal synth,
the difference in mood will be as great as if you played in 12-eq and
then in [pick your favorite low-number et here]? maybe we should set
up an experiment . . .

> Ah, yes: the difference between a helpful correction and an
> insensitive putdown.

i did say it was my favorite page on the internet . . .

🔗paulerlich <paul@stretch-music.com>

🔗jpehrson2 <jpehrson@rcn.com>

--- In tuning@y..., "paulerlich" <paul@s...> wrote:

/tuning/topicId_34327.html#34639

> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > re:
> > http://www.ixpres.com/interval/dict/eqtemp.htm
>
>
> about 12-equal -- what do you mean it's 'just barely' diesic? i
can't understand why you would say 'just barely'.
>
> also, even more important than diesic, is the linear temperament
based on 648:625, whose mos scales include diminished and octatonic,
and which falls on a straight line (in the first graph) connecting 12
with 28. this usage of 12 is extremely important in the music of the
romantic period, where diminished sevenths can resolve four ways, and
even more important in the music of liszt, stravinsky, bartok, bloch,
late scriabin, and most jazz musicians since dizzy gillespie,
including john coltrane. meanwhile, the 'diesic' usage of 12 shows up
only occasionally, i can only think of two examples -- schubert and
coltrane.

****This is awfully interesting. Or, rather, I would say it *would*
be awfully interesting if I could understand it. Now it's somewhat
*less* interesting.

The problem is that, basically, I'm not "getting" the colorful new
Monz pages. I'm getting the idea that the *darker* the cells the
*badder* be things, but I can't figure out what the axes are...

(I hope not "axes" of evil...)

J. Pehrson

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> /tuning/topicId_34327.html#34639
>
> > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > > re:
> > > http://www.ixpres.com/interval/dict/eqtemp.htm
> >
> >
> > about 12-equal -- what do you mean it's 'just barely' diesic? i
> can't understand why you would say 'just barely'.
> >
> > also, even more important than diesic, is the linear temperament
> based on 648:625, whose mos scales include diminished and
octatonic,
> and which falls on a straight line (in the first graph) connecting
12
> with 28. this usage of 12 is extremely important in the music of
the
> romantic period, where diminished sevenths can resolve four ways,
and
> even more important in the music of liszt, stravinsky, bartok,
bloch,
> late scriabin, and most jazz musicians since dizzy gillespie,
> including john coltrane. meanwhile, the 'diesic' usage of 12 shows
up
> only occasionally, i can only think of two examples -- schubert and
> coltrane.
>
> ****This is awfully interesting. Or, rather, I would say it
*would*
> be awfully interesting if I could understand it. Now it's somewhat
> *less* interesting.

you know how the 80:81 vanishing *defines* pitch usage in common-
practice music? well, in a similar fashion, 648:625 often *defines*
pitch usage in 20th century music. anytime four (non-xenharmonic)
minor thirds add up to an octave, it's at work.

> The problem is that, basically, I'm not "getting" the colorful new
> Monz pages. I'm getting the idea that the *darker* the cells the
> *badder* be things, but I can't figure out what the axes are...
>
> (I hope not "axes" of evil...)

i think you just need to catch up on the exchange of messages monz
and i had on this . . . yes? or do we have a 'time warp' situation
again?

🔗jpehrson2 <jpehrson@rcn.com>

🔗paulerlich <paul@stretch-music.com>

🔗Orphon Soul, Inc. <tuning@orphonsoul.com>

On 2/21/02 8:57 PM, "jpehrson2" <jpehrson@rcn.com> wrote:

Hey Joe...

I can explain the colors quickly enough. I'm guessing you might know that
Monz is mapping the accuracies of harmonic intervals in different equal
temperaments? With the cycle perfect fifths going to the right and the
cycle of just major thirds running up?

Well.

In the black and white models, the white cells are the more accurate
intervals, the black ones are the less accurate.

He extends this with an interesting palette, the "blue" scheme is for notes
that fall flat, and the "red" scheme is for notes that fall sharp.

Marc

🔗jpehrson2 <jpehrson@rcn.com>

--- In tuning@y..., "paulerlich" <paul@s...> wrote:

/tuning/topicId_34327.html#34657

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> > I'm "glazing" on that page. Maybe I'll get it more if I spend
some
> > more time with it, but the significance isn't
immediately "popping
> > out" at this stage...
>
> well, maybe then you're just confirming the reactions *i* had to
> it . . .

***Yes, I think so, but *mine* are considerably *worse* even... :(

Pretty colors, though... if I could figure out the significance.

Regrettably, I think Monz either needs to rethink his material or I
need to get considerably smarter, fast...

🔗jpehrson2 <jpehrson@rcn.com>

--- In tuning@y..., "Orphon Soul, Inc." <tuning@o...> wrote:

/tuning/topicId_34327.html#34658

>
> Hey Joe...
>
> I can explain the colors quickly enough. I'm guessing you might
know that Monz is mapping the accuracies of harmonic intervals in
different equal temperaments?

***Umm. *That* much I knew...

With the cycle perfect fifths going to the right and the
> cycle of just major thirds running up?

***Oh duh.

Somehow the idea of *cycle* didn't click in. That was the "key" I
needed.

Thanks, Marc! That's what I needed.

It pays to ask questions around here. Well, it doesn't pay very
*much* but it *pays...* :)

🔗paulerlich <paul@stretch-music.com>

🔗jpehrson2 <jpehrson@rcn.com>

--- In tuning@y..., "paulerlich" <paul@s...> wrote:

/tuning/topicId_34327.html#34662

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > --- In tuning@y..., "Orphon Soul, Inc." <tuning@o...> wrote:
> >
> > /tuning/topicId_34327.html#34658
> >
> >
> > >
> > > Hey Joe...
> > >
> > > I can explain the colors quickly enough. I'm guessing you
might
> > know that Monz is mapping the accuracies of harmonic intervals in
> > different equal temperaments?
> >
> > ***Umm. *That* much I knew...
> >
> >
> > With the cycle perfect fifths going to the right and the
> > > cycle of just major thirds running up?
> >
> >
> > ***Oh duh.
> >
> > Somehow the idea of *cycle* didn't click in. That was the "key"
I
> > needed.
>
> well it's more a *chain* than a *cycle* . . . if you want to get
into that discussion again . . .

****Hi Paul,

Well, that makes sense, but not a *bicycle* chain! :)

Actually, I'm getting a *little* bit more of the Monz page, but it's
still mostly a "stumper." I think, though, barring a more immediate
presentation of this, I will get on to more "productive" things, like
listening to the tetrads over on the "Miracle..." forum! :)

🔗paulerlich <paul@stretch-music.com>

I wrote,

> and then a look at the
> position with respect to the blue axes in the first graph (or a
> similar one) for the 'harmonic flavor' view.

well, if you were into 11-limit rather than 5-limit, you could plot
the interval matrix of the et on top of the diagonal-band-secor-color
graph, like i did for blackjack . . . an et would of course show up
as a regular square grid . . . this is essentially what monz is
already doing in all but the first graph on this page:

http://www.ixpres.com/interval/dict/ETgraphs.htm

where he uses the simpification from a two-dimensional interval
matrix to the one-dimensional pitch graph made possible by the fact
that an et (and only and et) has the same number and kind of pitches
as intervals. if monz would like, i could make some such square
plots . . . of course monz only went up to 72 with good reason and i
wouldn't want to go much higher, say, past 270 . . .

🔗monz <joemonz@yahoo.com>

hi Joe,

> From: jpehrson2 <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, February 21, 2002 6:26 PM
> Subject: [tuning] duh [Monz page]
>
>
> --- In tuning@y..., "Orphon Soul, Inc." <tuning@o...> wrote:
>
> /tuning/topicId_34327.html#34658
>
>
>
> With the cycle perfect fifths going to the right and the
> > cycle of just major thirds running up?
>
>
> ***Oh duh.
>
> Somehow the idea of *cycle* didn't click in. That was the "key" I
> needed.
>
>
> Thanks, Marc! That's what I needed.

2 things:

1)
when Paul refers to an "axis" on my "equal temperament" page,
he's talking about the first diagram, near the top. this is
based on his plot of how different EDOs approximate the basic
5-limit concords. they group in various ways along linear
axes, and so i drew red lines to "connect the dots" and show
those axes. those linear axes are the result of tempering
out a particular 5-limit "comma" (a generic term covering
several different sizes of small intervals). so, for instance,
the line marked "meantone" tempers out the syntonic comma 81:80.
etc. paul doesn't have a problem with this diagram.

2)
the greyscale and color lattices (in the form of Excel spreadsheet
tables) are the ones to which paul holds an objection. i'm
glad to have his criticism, but i fear that it probably boils
down to some fundamental differences in our conceptions of tuning
and tuning-theory.

but anyway, to try and understand a bit better what i'm getting at,
i suggest studying the 12edo lattices. this is a tuning with which
we're all very familiar, and if you understand how the patterns
of colors and shades on the lattices relate to the relationships
between 12edo and 5-, 7-, and 11-limit JI, then you can apply
that understanding to the other EDOs.

there's no absolute accuracy reference here, as there would
have been if i had always used cents. instead, i'm calculating
the degree of each EDO which most closely approximates the JI
ratio, to one decimal place, and then using the value in the
decimal place to determine the shade (and color in the case of
the color lattices). thus, the reference of accuracy to JI
is always given in terms of the EDO itself.

-monz

i suggest
that

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

🔗monz <joemonz@yahoo.com>

re:
http://www.ixpres.com/interval/dict/eqtemp.htm

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, February 21, 2002 5:38 PM
> Subject: [tuning] Re: new equal temperament 5-limit error lattices
>
>
>
> i'd probably have 'bingo cards' james mccartney style (hex grid,
> though) for the structural view of the et,

paul, i'll do the grunt work to produce an example of this
if you give me all the data i need.

> ... and then a look at the
> position with respect to the blue axes in the first graph (or a
> similar one) for the 'harmonic flavor' view. if you choose an et for
> the commas it eliminates, and thus its compositional possibilities,
> the former is more important; if you choose an et for the way the
> consonant ji harmonies are 'inflected', the latter is more important.
> monz is trying to present both sets of information at the same time,
> which is admirable, but falls a little short in both areas.

it's really nice to have your analysis of what i'm trying to do. :)

(i'm not being sarcastic, i mean this.)

> ... what do *you* think about the idea (you could
> probably say from personal experience) that if you play something
> written in ji on a 768-equal synth and then on a 1024-equal synth,
> the difference in mood will be as great as if you played in 12-eq and
> then in [pick your favorite low-number et here]? maybe we should set
> up an experiment . . .

ah, now here's where i could be a lot more sophisticated!

on the color lattices, i'd add an a b s o l u t e accuracy
reference, such as cents, and let that value control the h u e
of the colors, so that as the EDOs have higher and higher
cardinality, the richness of the color gets less and less.
so the low-cardinality EDOs like 12, 31 etc. would have rather
rich colors, and the high ones like 49152 would be very pale.
eventually (at infinity-tET?) the whole lattice would be white.

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

🔗monz <joemonz@yahoo.com>

re:
http://www.ixpres.com/interval/dict/eqtemp.htm

> From: monz <joemonz@yahoo.com>
> To: <tuning@yahoogroups.com>; <justmusic@yahoogroups.com>;
<Ken.Fasano@shawinc.com>
> Sent: Thursday, February 21, 2002 8:15 PM
> Subject: Re: [tuning] Re: new equal temperament 5-limit error lattices
>
>
> > ... what do *you* think about the idea (you could
> > probably say from personal experience) that if you play something
> > written in ji on a 768-equal synth and then on a 1024-equal synth,
> > the difference in mood will be as great as if you played in 12-eq and
> > then in [pick your favorite low-number et here]? maybe we should set
> > up an experiment . . .
>
>
> ah, now here's where i could be a lot more sophisticated!
>
> on the color lattices, i'd add an a b s o l u t e accuracy
> reference, such as cents, and let that value control the h u e
> of the colors, so that as the EDOs have higher and higher
> cardinality, the richness of the color gets less and less.
> so the low-cardinality EDOs like 12, 31 etc. would have rather
> rich colors, and the high ones like 49152 would be very pale.
> eventually (at infinity-tET?) the whole lattice would be white.

also, of course, i could use accuracy to 2 decimal points
instead of 1, which would require 100 colors. but this
is something that's much better left to a programming
language rather than doing it by hand. lattices like this
will also be a part of JustMusic software, and i'll leave
it up to the user to define the levels of accuracy, colors,
etc.

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

🔗Orphon Soul, Inc. <tuning@orphonsoul.com>

🔗paulerlich <paul@stretch-music.com>

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

> 1)
> when Paul refers to an "axis" on my "equal temperament" page,
> he's talking about the first diagram, near the top. this is
> based on his plot of how different EDOs approximate the basic
> 5-limit concords. they group in various ways along linear
> axes, and so i drew red lines to "connect the dots" and show
> those axes. those linear axes

speak for yourself, monz. :) i would never call these axes.

> paul doesn't have a problem with this diagram.

except that i *really* *really* *really* want the line connecting 12
with 28, the 648:625 line, perhaps the most important one for 20th
century music.

i love you monz!

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "monz" <joemonz@y...> wrote:
> re:
> http://www.ixpres.com/interval/dict/eqtemp.htm
>
>
> > From: paulerlich <paul@s...>
> > To: <tuning@y...>
> > Sent: Thursday, February 21, 2002 5:38 PM
> > Subject: [tuning] Re: new equal temperament 5-limit error lattices
> >
> >
> >
> > i'd probably have 'bingo cards' james mccartney style (hex grid,
> > though) for the structural view of the et,
>
>
> paul, i'll do the grunt work to produce an example of this
> if you give me all the data i need.

ok -- let me know what method you'll be using, so i can fit your
needs best. perhaps we should collaborate off-list.
>
> ah, now here's where i could be a lot more sophisticated!
>
> on the color lattices, i'd add an a b s o l u t e accuracy
> reference, such as cents, and let that value control the h u e

you means saturation? luminosity? hue is position along the
rainbow . . .

> of the colors, so that as the EDOs have higher and higher
> cardinality, the richness of the color gets less and less.
> so the low-cardinality EDOs like 12, 31 etc. would have rather
> rich colors, and the high ones like 49152 would be very pale.
> eventually (at infinity-tET?) the whole lattice would be white.

that would address *one* of my objections, yes.

🔗jpehrson2 <jpehrson@rcn.com>

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

/tuning/topicId_34327.html#34673

> >
> 1)
> when Paul refers to an "axis" on my "equal temperament" page,
> he's talking about the first diagram, near the top. this is
> based on his plot of how different EDOs approximate the basic
> 5-limit concords. they group in various ways along linear
> axes, and so i drew red lines to "connect the dots" and show
> those axes. those linear axes are the result of tempering
> out a particular 5-limit "comma" (a generic term covering
> several different sizes of small intervals). so, for instance,
> the line marked "meantone" tempers out the syntonic comma 81:80.
> etc. paul doesn't have a problem with this diagram.
>

***OH! Well, *that* diagram I don't have as much trouble with. I
remember that Paul's *initial* diagram did have exes on it.

>
> 2)
> the greyscale and color lattices (in the form of Excel spreadsheet
> tables) are the ones to which paul holds an objection. i'm
> glad to have his criticism, but i fear that it probably boils
> down to some fundamental differences in our conceptions of tuning
> and tuning-theory.
>
> but anyway, to try and understand a bit better what i'm getting at,
> i suggest studying the 12edo lattices. this is a tuning with which
> we're all very familiar, and if you understand how the patterns
> of colors and shades on the lattices relate to the relationships
> between 12edo and 5-, 7-, and 11-limit JI, then you can apply
> that understanding to the other EDOs.

***Generally speaking, your web pages have been pretty clear to me,
so I'm surprised I'm having so much trouble with *this* one. I see
you have chains of fifths running along the top and chains of thirds
running vertically, but I'm still not totally piecing it together or
understanding the significance. Maybe if I stare it at some more...

🔗monz <joemonz@yahoo.com>

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Friday, February 22, 2002 11:26 AM
> Subject: [tuning] Re: duh [Monz page]
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > 1)
> > when Paul refers to an "axis" on my "equal temperament" page,
> > he's talking about the first diagram, near the top. this is
> > based on his plot of how different EDOs approximate the basic
> > 5-limit concords. they group in various ways along linear
> > axes, and so i drew red lines to "connect the dots" and show
> > those axes. those linear axes
>
> speak for yourself, monz. :) i would never call these axes.

sorry, my bad. thought you did.

> > paul doesn't have a problem with this diagram.
>
> except that i *really* *really* *really* want the line connecting 12
> with 28, the 648:625 line, perhaps the most important one for 20th
> century music.

done. i hope i remembered correctly that you call it "octatonic".

also, i've labeled "porcupine".

http://www.ixpres.com/interval/dict/eqtemp.htm

-monz

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🔗paulerlich <paul@stretch-music.com>

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

> done.

thank you, thank you, thank you!! you're the best, monz!!!

> i hope i remembered correctly that you call it "octatonic".

i think carl and i just came to the consensus here that this should
be called "diminished", and the one you're calling 'diesic' should be
called "augmented". besides the reasons already given, the period of
the "diminished" linear temperament is 1/4 octave (thus outlining a
diminished seventh chord) and the period of the "augmented" linear
temperament is 1/3 octave (thus outlining an augmented triad).

> also, i've labeled "porcupine".

nice going, monz!