Yahoo Tuning Groups Ultimate Backup tuning-math graham's linear temperament page

hi Paul,

> From: "emotionaljourney22" <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, May 10, 2002 3:25 PM
> Subject: [tuning-math] graham's linear temperament page
>
>
> this page is great and very important (by contrast, check out joe
> monzo's definition of linear temperament if you want to turn red).

how about a more constructive criticism? post something that
i can use to replace or supplement my definition to make it better.
(and also please specify whether i should be replacing or supplementing!)
thanks.

(BTW, i'm only checking in on the lists sporadically these days.
too much other "life" stuff happening...)

-monz

🔗jonszanto <jonszanto@yahoo.com>

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> this page is great and very important

So great and so important you couldn't take the time to include a link? Sure, we can find it from our collection of links, or do a search, etc. - I thought you were the consummate cross-referenced poster.

> (by contrast, check out joe
> monzo's definition of linear temperament if you want to turn red).

I didn't want to comment on this unless Joe had seen it, and he has. Paul, if you ever want a good example of the kind of thing that makes people really sad and crazy and distrustful, just save that above fragment. And maybe don't write about other people so uncharitably in public when you could just as easily be helpful about it.

Some people never learn...

Jon

🔗graham@microtonal.co.uk

In-Reply-To: <abmetp+mul3@eGroups.com>
jonszanto wrote:

> So great and so important you couldn't take the time to include a link?
> Sure, we can find it from our collection of links, or do a search, etc.
> - I thought you were the consummate cross-referenced poster.

Everybody does or doesn't include links according to their mood. This one
is <http://x31eq.com/catalog.htm>.

I personally would like somebody to include the definitions of the
temperaments I'm supposed to be adding. It's been quite difficult to keep
up over the past year.

> > (by contrast, check out joe
> > monzo's definition of linear temperament if you want to turn red).

After following the links, I thought the "temperament" definition was
lacking, and decided to clarify it. Then scrolled down, and saw I already
had. Although I now disagree with my own definition -- temperaments can
approximate "ideal" tunings other than just intonation.

It looks like the problem with the "linear temperament" definition is that
there isn't one! You simply say it's "another term for unequal
temperaments" which isn't the case. Not all unequal temperaments are
linear.

A linear temperament is fully described by a generator and equivalence
interval. The mapping from generators to consonances is fixed. For
example, meantones are defined by the generator approximating 3:2 and four
generators approximating 5:4. The generator can have different sizes,
giving 1/3-, 1/4-, 1/6- comma meantone, but they're all the same
temperament class. 53-equal, however, isn't a meantone because the best
5:4 is 8 fourths rather than 4 fifths.

Planar temperaments are like linear temperaments, but with two generators
and an equivalence interval. The difference between a generator and an
equivalence interval is only a matter of formalism.

Graham

🔗jonszanto <jonszanto@yahoo.com>

🔗monz <monz@attglobal.net>

many thanks, Graham! so the Tuning Dictionary now has
a couple of new definitions:

linear temeperament
http://www.ixpres.com/interval/dict/lineartemp.htm

planar temperament
http://www.ixpres.com/interval/dict/planartemp.htm

as well as an important addition to an old definition:
http://www.ixpres.com/interval/dict/consonance.htm

see Paul, that's the *right* way to do it!

(and if anyone would care to explain that "matter of formalism"
which differentiates a "generator" from an "equivalence interval",
i'd sure like to read it! after much discussion of it a few months
ago, i'm still not clear about it.)

-monz

----- Original Message -----
From: <graham@microtonal.co.uk>
To: <tuning-math@yahoogroups.com>
Sent: Monday, May 13, 2002 2:57 AM
Subject: [tuning-math] Re: graham's linear temperament page

> In-Reply-To: <abmetp+mul3@eGroups.com>
> jonszanto wrote:
>
> > So great and so important you couldn't take the time to include a link?
> > Sure, we can find it from our collection of links, or do a search, etc.
> > - I thought you were the consummate cross-referenced poster.
>
> Everybody does or doesn't include links according to their mood. This one
> is <http://x31eq.com/catalog.htm>.
>
> I personally would like somebody to include the definitions of the
> temperaments I'm supposed to be adding. It's been quite difficult to keep
> up over the past year.
>
> > > (by contrast, check out joe
> > > monzo's definition of linear temperament if you want to turn red).
>
> After following the links, I thought the "temperament" definition was
> lacking, and decided to clarify it. Then scrolled down, and saw I already
> had. Although I now disagree with my own definition -- temperaments can
> approximate "ideal" tunings other than just intonation.
>
> It looks like the problem with the "linear temperament" definition is that
> there isn't one! You simply say it's "another term for unequal
> temperaments" which isn't the case. Not all unequal temperaments are
> linear.
>
> A linear temperament is fully described by a generator and equivalence
> interval. The mapping from generators to consonances is fixed. For
> example, meantones are defined by the generator approximating 3:2 and four
> generators approximating 5:4. The generator can have different sizes,
> giving 1/3-, 1/4-, 1/6- comma meantone, but they're all the same
> temperament class. 53-equal, however, isn't a meantone because the best
> 5:4 is 8 fourths rather than 4 fifths.
>
> Planar temperaments are like linear temperaments, but with two generators
> and an equivalence interval. The difference between a generator and an
> equivalence interval is only a matter of formalism.
>
>
> Graham

🔗dkeenanuqnetau <d.keenan@uq.net.au>

--- In tuning-math@y..., "monz" <monz@a...> wrote:
> (and if anyone would care to explain that "matter of formalism"
> which differentiates a "generator" from an "equivalence interval",
> i'd sure like to read it! after much discussion of it a few months
> ago, i'm still not clear about it.)

Unfortunately I have a slightly different take on this to Graham's
recent statement. I'm wondering if Graham really meant to say the
following.

A linear temperament is generated by two "generating intervals" (or
generators for short), one of which is distinguished as the "interval
of periodicity" (or "period" for short) and the other is simply
called "the generator". The "period" is distinguished from the other
generator by the fact that the "interval of equivalence" is a whole
number multiple of the period. The interval of equivalence is usually
the octave. A particular scale within a linear temperament (e.g. a
diatonic scale within meantone) will have a fixed number of
(non period) generators but an unspecified number of intervals of
equivalence.

🔗D.Stearns <STEARNS@CAPECOD.NET>

Hi Graham,

Interesting. This definition--"A linear temperament is fully described
by a generator and equivalence interval. The mapping from generators
to consonances is fixed."--is basically the same one I tried to use to
generalize meantone. (The way I looked at the Kleismic in this sense
was as a 3:5 in 1D and a 3:4:5 in 2D.)

So true Meantones according to your definition would be found in ETs
5, 7, 12, 19, 26, 31, 43, 45, 50, 55, etc. and true Kleismics in ETs
4, 15, 19, 23, 34, 49, 53, etc., right?

Hmm, I like it, but there must be a zillion tunings like this!

The 6-tone scale in 11-tet that Margo brought up recently would be one
example. I'd see it as a 4:7 in 1D and a 8:11:14 in 2D with true
whatever-you-want-to-call-thems found in ETs 5, 6, 11, 12, 16, 17, 23,
27, etc.

Then again maybe that's too coarse (11 and then 12)? It seems like it
might be tough to say (exactly) when a simple JI interpretation would
be the 'correct' one?

take care,

--Dan Stearns

----- Original Message -----
From: <graham@microtonal.co.uk>
To: <tuning-math@yahoogroups.com>
Sent: Monday, May 13, 2002 2:57 AM
Subject: [tuning-math] Re: graham's linear temperament page

> In-Reply-To: <abmetp+mul3@eGroups.com>
> jonszanto wrote:
>
> > So great and so important you couldn't take the time to include a
link?
> > Sure, we can find it from our collection of links, or do a search,
etc.
> > - I thought you were the consummate cross-referenced poster.
>
> Everybody does or doesn't include links according to their mood.
This one
> is <http://x31eq.com/catalog.htm>.
>
> I personally would like somebody to include the definitions of the
> temperaments I'm supposed to be adding. It's been quite difficult
to keep
> up over the past year.
>
> > > (by contrast, check out joe
> > > monzo's definition of linear temperament if you want to turn
red).
>
> After following the links, I thought the "temperament" definition
was
> lacking, and decided to clarify it. Then scrolled down, and saw I
already
> had. Although I now disagree with my own definition -- temperaments
can
> approximate "ideal" tunings other than just intonation.
>
> It looks like the problem with the "linear temperament" definition
is that
> there isn't one! You simply say it's "another term for unequal
> temperaments" which isn't the case. Not all unequal temperaments
are
> linear.
>
> A linear temperament is fully described by a generator and
equivalence
> interval. The mapping from generators to consonances is fixed. For
> example, meantones are defined by the generator approximating 3:2
and four
> generators approximating 5:4. The generator can have different
sizes,
> giving 1/3-, 1/4-, 1/6- comma meantone, but they're all the same
> temperament class. 53-equal, however, isn't a meantone because the
best
> 5:4 is 8 fourths rather than 4 fifths.
>
> Planar temperaments are like linear temperaments, but with two
generators
> and an equivalence interval. The difference between a generator and
an
> equivalence interval is only a matter of formalism.
>
>
> Graham
>
>
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>

🔗dkeenanuqnetau <d.keenan@uq.net.au>

--- In tuning-math@y..., graham@m... wrote:
> A linear temperament is fully described by a generator and
equivalence
> interval.

I disagree. I'd be a lot closer to agreeing if you had said "a
generator and a periodicity interval" (which can be a fraction of the
equivalence interval), but I think you want to know the equivalence
interval as well as the period.

But what if there isn't an equivalence interval but there are still
two generators; does it still make sense to call it a "linear"
temperament. That's a tough one.

>The mapping from generators to consonances is fixed. For
> example, meantones are defined by the generator approximating 3:2
and four
> generators approximating 5:4. The generator can have different
sizes,
> giving 1/3-, 1/4-, 1/6- comma meantone, but they're all the same
> temperament class. 53-equal, however, isn't a meantone because the
best
> 5:4 is 8 fourths rather than 4 fifths.

So the mapping defines a "temperament class" while specific generators
define a "temperament (instance)". I like that.

But what do we call it when we have different numbers of notes per
equivalence interval (typically MOS) for the same temperament. I guess
we just call them things like "12 note meantone" or "miracle-21".

> Planar temperaments are like linear temperaments, but with two
generators
> and an equivalence interval. The difference between a generator and
an
> equivalence interval is only a matter of formalism.

Again, substitute "period" for "equivalence interval" and "only a
matter of formalism" for the definitions in my preceeding post.

🔗genewardsmith <genewardsmith@juno.com>

🔗graham@microtonal.co.uk

In-Reply-To: <abq076+gjf0@eGroups.com>
genewardsmith wrote:

> I think it would be nice to include links to music in the various
> temperaments discussed. I have examples for Orwell and Wonder, and am
> working on something in Magic. I also have 126/125 and 64/63 if you
> want to add planar temperaments. Porcupine, Miracle and of course
> Meantone are out there--what else?

I'd prefer to link to a "home page" for each temperament. I already have
these in place for meantone, miracle, schismic and diaschismic. If you
can supply your own pages for these other temperaments, I can link to them
and you can handle the links to examples. I should do something for magic
and I plan to tune up the multiple-29 soon as well. The 29&19 temperament
that we're apparently calling Negri does look promising on its own terms
and as a way of slicing up multiple-29, maybe even giving a planar
temperament.

Graham

🔗graham@microtonal.co.uk

In-Reply-To: <abpt98+61vd@eGroups.com>
dkeenanuqnetau wrote:

> I disagree. I'd be a lot closer to agreeing if you had said "a
> generator and a periodicity interval" (which can be a fraction of the
> equivalence interval), but I think you want to know the equivalence
> interval as well as the period.

I should have said period. It can be fully described by the period and
generator, although it can also be convenient to think in terms of an
equivalence interval. If you put aside contorsion, the number of periods
to an equivalence is the GCD of the generator mapping, but that's not
something the definition should concern itself with. Unless we want to
pronounce on whether temperaments with contorsion really count as
temperaments. But to cover them you need to state the full mapping, in
terms of both period and generator, anyway. In which case we still have
to specify that the mapping should only involve integers.

> But what if there isn't an equivalence interval but there are still
> two generators; does it still make sense to call it a "linear"
> temperament. That's a tough one.

You can always call one of the generators an equivalence interval.
There's really little difference as far as the mathematics is concerned.
But I expect musicians will find it easier to think if the period as a
generalisation of an octave and the generator as the generalisation of a
fifth. They'll find it obvious that a fifth and an octave function
differently in meantone. So that's how the simple definition should be
framed, and mathematical caveats added later.

The word "linear" does strongly suggest a single generator, and music
theory usually assumes octave equivalence, sometimes implicitly. If we
define linear temperaments as having 2 generators, people will get
confused by the off-by-one error.

> But what do we call it when we have different numbers of notes per
> equivalence interval (typically MOS) for the same temperament. I guess
> we just call them things like "12 note meantone" or "miracle-21".

I think those are called "scales". These ones in particular being MOS or
Well Formed.

> Again, substitute "period" for "equivalence interval" and "only a
> matter of formalism" for the definitions in my preceeding post.

Those definitions look fine to me.

Oh, and Joe, when I said "consonances" I meant something much closer to
your definition of "consonance" than "concordance". Musical context is
important for this.

Graham

🔗monz <monz@attglobal.net>

🔗graham@microtonal.co.uk

🔗dkeenanuqnetau <d.keenan@uq.net.au>

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > A linear temperament is generated by two "generating intervals"
(or
> > generators for short), one of which is distinguished as the
"interval
> > of periodicity" (or "period" for short) and the other is simply
> > called "the generator".
>
> One of them may or may not be an interval of peridoicity--there's no
requirement to have one, it's simply always possible to express them
in that way. The key fact is that a linear temperament is rank
two--ie, it has two generators.

I agree, but point out that this is a fairly recent way of looking at
them, or at least a particularly mathematical way. To a musician it is
rank 1 where the arithmetic is modulo the interval of equivalence. If
it is "always possible to express them in that way" then they will
always want you to.

🔗dkeenanuqnetau <d.keenan@uq.net.au>

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > But what if there isn't an equivalence interval but there are
still
> > two generators; does it still make sense to call it a "linear"
> > temperament. That's a tough one.
>
> That's more or less my definition of a linear temperament, except I
would add that the generators are mapped to from some subgroup of the
positive rationals. Making an interval of equivalence part of the
definition gives us one meantone for 2, one for 3/2, one for 3, one
for 5/2, another for 5/3 and so forth. I don't like it, and don't plan
to use it.

OK. I agree with Gene and Graham. Although it doesn't make any _sense_
to call it linear when there is no IoE, we will anyway. If only
because it _can_ be treated the same mathematically (rank two) whether
there is an IoE or not. But I wouldn't want a definition to be based
on that since having no IoE is so rare.

I'd prefer to ignore the question of whether the IoE or the other
consonances _must_ be rational ratios, and thereby avoid a whole can
of worms associated with temperaments for inharmonic timbres.

🔗genewardsmith <genewardsmith@juno.com>

🔗emotionaljourney22 <paul@stretch-music.com>

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., graham@m... wrote:
>
> > I personally would like somebody to include the definitions of
the
> > temperaments I'm supposed to be adding. It's been quite
difficult to keep
> > up over the past year.
>
> I think it would be nice to include links to music in the various
>temperaments discussed. I have examples for Orwell and Wonder, and
>am working on something in Magic. I also have 126/125 and 64/63 if
>you want to add planar temperaments. Porcupine, Miracle and of
>course Meantone are out there--what else?

pajara -- 'decatonic swing'

the piece 'glassic' is <2,3,5[,11]> porcupine in the first section,
then goes to the 64:63 temperament in <2,3,7>.

then again, you may decide that 15-equal or 22-equal or 72-equal
should count as examples of equal temperament (even if linear-based
modes are used), not linear temperament . . .

anyway, ara is working out his system bugs over on makemicromusic --
once he's ready, we'll begin making pajara and other-tempered music
in earnest.

by the way, i'd like to send a cd to anyone who'd like to listen to
some enjoyable (i hope) and professional (finally) music. please send
me your snail-mail address if interested. there may be a slight delay
as i'm awaiting receipt of the cd from mastering in NYC.

🔗emotionaljourney22 <paul@stretch-music.com>

--- In tuning-math@y..., "monz" <monz@a...> wrote:
>
> > From: <graham@m...>
> > To: <tuning-math@y...>
> > Sent: Tuesday, May 14, 2002 3:57 AM
> > Subject: [tuning-math] Re: graham's linear temperament page
> >
> >
> > Oh, and Joe, when I said "consonances" I meant something much
closer to
> > your definition of "consonance" than "concordance". Musical
context is
> > important for this.
>
>
> oops ... OK, it's been fixed.
> http://www.ixpres.com/interval/dict/lineartemp.htm

hi joe, sorry if i seemed mean before . . . hope you can forgive me!
it's just that i've done a lot of work trying to help you correct
your et definition and the pages it links to, and you, quite
understandably, have had little time to consider my many e-mails,
posts, and IM comments to you on these issues. hope you've at least
saved my e-mails, and made note of the relevant posts, for later
consideration -- that's all i can ask. maybe in 10 years we can get
back to this -- i'd be very happy with that.

anyway, on the current topic, it seems you missed an important fix
that everyone here agrees on -- please replace "interval of
equivalence" with "period" in your definitions.

🔗emotionaljourney22 <paul@stretch-music.com>

--- In tuning-math@y..., graham@m... wrote:
> monz wrote:
>
> > and what's "contorsion"? that's a new one for me!
>
> If you define 5-limit meantone using neutral thirds instead of
fifths, you
> find a fifth is two generators and a major third is two
generators.
> Because all consonances involve an even number of steps it has a
> contorsion of 2. Whether it counts as a new temperament or an odd
way of
> mapping meantone isn't currently addressed by the definition.
>
>
> Graham

i thought that, a few months ago, gene and i argued, and dave finally
concurred, that contorsion is disallowed in the definition of linear
temperament -- we seemed able to convince dave that "temperament"
means "temperament of JI", and that contorsion involves a far more
radical departure from JI (thought of as a single, infinitely-
extending lattice of notes, connected by justly-intoned concordant
intervals).

🔗emotionaljourney22 <paul@stretch-music.com>

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> I agree, but point out that this is a fairly recent way of looking
at
> them, or at least a particularly mathematical way. To a musician it
is
> rank 1 where the arithmetic is modulo the interval of equivalence.
If
> it is "always possible to express them in that way" then they will
> always want you to.

i suspect you can go through the tuning list archives and find two or
three or more musicians who thought otherwise; i.e., in register-
specific terms. i see no reason to discriminate against such
musicians/thinkers -- especially if gene's current mathematical
system handles such a view in a nice, natural way . . .

🔗emotionaljourney22 <paul@stretch-music.com>

🔗monz <monz@attglobal.net>

> From: "emotionaljourney22" <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Wednesday, May 15, 2002 2:36 PM
> Subject: [tuning-math] Re: graham's linear temperament page
>
>
> > http://www.ixpres.com/interval/dict/lineartemp.htm
>
> hi joe, sorry if i seemed mean before . . . hope you can forgive me!

no sweat, paul -- i'm used to your style by now. :)
consider it water under the bridge.

the main thing is that i want the Dictionary to be as correct
as possible.

> it's just that i've done a lot of work trying to help you correct
> your et definition and the pages it links to, and you, quite
> understandably, have had little time to consider my many e-mails,
> posts, and IM comments to you on these issues. hope you've at least
> saved my e-mails, and made note of the relevant posts, for later
> consideration -- that's all i can ask. maybe in 10 years we can get
> back to this -- i'd be very happy with that.

well, i hope it doesn't take 10 years! but right now i am
kind of tied up with other projects, and have pretty successfully
weaned myself off the tuning list(s) addiction(s). i'm just
checking in these days and fighting the urge to reply to everything.

>
> anyway, on the current topic, it seems you missed an important fix
> that everyone here agrees on -- please replace "interval of
> equivalence" with "period" in your definitions.

ok, look at the linear temperament definition now.

NOW -- regarding "equivalence interval" and "periodicity interval",
you'll all see that i provided links to definitions for those two
terms, but still don't have any content in them. that's because
i'm *still* not clear on how to define them. can someone(s)
*please* post simply a definition for each of them which i can
put into those Dictionary entries. if others disagree, then
i'll try to keep updating the webpages the way i've done with
"linear temperament" -- but please give me something to start with.

-monz

🔗graham@microtonal.co.uk

In-Reply-To: <000f01c1fca0$7d449ca0$af48620c@dsl.att.net>
monz wrote:

> ok, look at the linear temperament definition now.

Fancy that!

> NOW -- regarding "equivalence interval" and "periodicity interval",
> you'll all see that i provided links to definitions for those two
> terms, but still don't have any content in them. that's because
> i'm *still* not clear on how to define them. can someone(s)
> *please* post simply a definition for each of them which i can
> put into those Dictionary entries. if others disagree, then
> i'll try to keep updating the webpages the way i've done with
> "linear temperament" -- but please give me something to start with.

Dave gave some, and I didn't notice anybody disagreeing. The only quibble
I have is that the period definition uses the terminology differently to
the linear temperament one. And that it's circular, but that's probably
unavoidable.

"""
"interval of equivalence" = "equivalence interval" = "formal octave"
is that interval (much larger than a unison) which, when it occurs
between two pitches, we consider them to be, in some sense, (formally
if not perceptibly) the same note. For most scales this is the octave
1:2, and when it is not the octave it is usually some other highly
consonant interval such as the "tritave" 1:3. But the essential
feature of the interval of equivalence in relation to definitions of
scales and types of scales is that when we describe a scale we
describe only the pitches that fall within a single interval of
equivalence, and we leave it up to the instrument builder to decide
the range of the instrument and therefore how many times (including
fractions) the interval of equivalence should be repeated.

"interval of periodicity" = "periodic interval" = "period"
is that generator of a regular temperament (whether linear, planar, or
n-dimensional) which generates the interval of equivalence all by
itself. This means that the period is either equal to the interval of
equivalence or fits into the interval of equivalence a whole number of
times.
"""

Graham

🔗emotionaljourney22 <paul@stretch-music.com>

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

> > anyway, on the current topic, it seems you missed an important
fix
> > that everyone here agrees on -- please replace "interval of
> > equivalence" with "period" in your definitions.
>
>
> ok, look at the linear temperament definition now.

the line

"The difference between a generator and an equivalence interval is
only a matter of formalism."

should read

"The difference between a generator and a period is only a matter of
formalism."

if you're going to say "interval of periodicity" for "period", i'd
suggest that "interval of repetition" is even better -- it's more to
the point, and doesn't cause confusion with periodicity blocks.