Chains and lattices

Dumb question:

have there been any explorations of scales formed from constant ratio pairs:

sequences of 10/9 or 17/16 or 13/12? Octave-less endless chains of intervals
of constant size?

I am wondering if some of the lattices drawn on monz' site could be better
done in vrml? I'm an ex mechanical engineer, and am reasonable with CAD
packages. I could steal the use of one at work sometime to draw one or two
up and export them as VRML. Limits requiring higher dimensions could be
layered, so that moving around the space, one could take 3D sections through
the lattices.

I feel JI-superstring theory coming on (31-limit is a 9-dimensional lattice,
I think)

M

Hi Mark,

> From: Mark Gould <mark.gould@argonet.co.uk>
> To: Tuning List <tuning@yahoogroups.com>
> Sent: Saturday, February 09, 2002 1:15 PM
> Subject: [tuning] Chains and lattices
>
>
> Dumb question:
>
> have there been any explorations of scales formed from constant ratio
pairs:
>
> sequences of 10/9 or 17/16 or 13/12? Octave-less endless chains of
intervals
> of constant size?

yes, but the only thing i can refer to off the top of my
head are Jacky Ligon's wild explorations into "8ve"-less JI.
i think he and his work will mostly be found on the
MakeMicroMusic group.

i've done some lattices of these, for each basic prime interval
up to 31 ... they're at the back of the current version of my book.

> I am wondering if some of the lattices drawn on monz' site could be better
> done in vrml?

i began getting ideas for this in the fall of 1998.
Robert Walker has actually created VRML hexanies and
other lattice constructions of Erv Wilson's, in which
the user can rotate his point of view all around the
lattice. Dave Keenan also created the "tumbling dekany"
movie, along similar lines, and Paul Erlich made some
too.

> I'm an ex mechanical engineer, and am reasonable with CAD
> packages. I could steal the use of one at work sometime to draw one or two
> up and export them as VRML. Limits requiring higher dimensions could be
> layered, so that moving around the space, one could take 3D sections
through
> the lattices.

i'm really glad to know all of this, and happy that it
seems you're interested in my project. i encourage you
to join the JustMusic group, which is looking for software
developers to get the app out of its rut and going again:
/justmusic

send me a request to join and i'll approve it.

> I feel JI-superstring theory coming on (31-limit is a 9-dimensional
lattice,
> I think)

depends on exactly which primes you admit from the prime series.

on my usual lattices, which are "8ve"-equivalent and thus
leave out 2, we get:

1 2 3 4 5 6 7 8 9 dimension
3 5 7 11 13 17 19 23 29 prime

so 29-limit is 9-dimensional, 31-limit is 10-d.

note that it's not necessary to keep all the primes
which appear in succession. La Monte Young uses
3,7,17,31, ... and way on up, skipping many; Margo
Schulter has devised many 3-and-something-higher-limit
tunings which can act as multiple Pythagorean chains;
Dan Stearns likewise makes 2- and 3-d lattices of
scattered prime-factors; ... etc. so, number of
dimensions can vary in interesting ways.

note that my lattice formula plots each basic
prime interval at an angle around a circle which
corresponds to its pitch-height within the "8ve".
every prime axis has a unique angle, step length,
and thickness. on my most artistic lattice i
colored in each plane as though light were passing
thru panes of glass, to give the 4-d effect; if this
link doesn't take you right to it, it's about 2/3 of
the way down my lattice page:
http://www.ixpres.com/interval/monzo/lattices/lattices.htm#shaded

-monz

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

>have there been any explorations of scales formed from constant
>ratio pairs:
>
>sequences of 10/9 or 17/16 or 13/12? Octave-less endless chains
>of intervals of constant size?

Any scale made from a chain of a single interval we call a
"linear series" or "linear scale" or "chain-of-x scale", where
x is the interval being chained (called the generator of the
scale). Usually, the term becomes "linear temperament", because
some members of the chain will get close to consonant intervals,
as when four 3:2's get close to a 5:4 -- even though we haven't
tempered the generator, if we treat the near-5:4 as a consonance
musically, we've "tempered" it. All equal temperaments are
linear temperaments when their step size is the generator, and
also when the generator is made of a number of steps relatively
prime with the et number (so prime ets like 17 and 19 are linear
temperaments of all their intervals). The subject of linear
temperaments has been in the spotlight on the tuning-math list
for some time.

Chains of two different generators are less explored... if they
alternate, they can be viewed as a superposition of two chains
of their sum offset by either of them. Scales with alternating
"2nds" are often called "symmetric" in melody-land (such as
the octatonic scale in 12-equal or Blackwood's symmetric decatonic
in 15-equal).

Am I answering your question?

>I am wondering if some of the lattices drawn on monz' site could
>be better done in vrml?

I would imagine so!

>I'm an ex mechanical engineer, and am reasonable with CAD
> packages. I could steal the use of one at work sometime
>to draw one or two up and export them as VRML. Limits requiring
>higher dimensions could be layered, so that moving around the
>space, one could take 3D sections through the lattices.

That would be cool! Have you seen Dave Keenan's stereo
dekany (done in a spreadsheet!)?

http://www.uq.net.au/~zzdkeena/Music/StereoDekany.htm

>I feel JI-superstring theory coming on (31-limit is a 9-
>dimensional lattice, I think).

It depends... many of us here (including myself) would like a
dimension for each odd factor, not just the primes, which would
make the 31-limit 15-dimensional. Fortunately, the 31-limit is
of doubtful psychoacoustical meaning... most of us tend to think
the limit of limits is somewhere between 13 and 25, depending
on the context.

-Carl

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

/tuning/topicId_33902.html#33906

>
>
> i began getting ideas for this in the fall of 1998.
> Robert Walker has actually created VRML hexanies and
> other lattice constructions of Erv Wilson's, in which
> the user can rotate his point of view all around the
> lattice.

****Robert, would you mind posting the link to these VRML Wilson
objects?

I remember them, but haven't seen them for a long time.

best,

Joseph

>>Fortunately, the 31-limit is of doubtful psychoacoustical
>>meaning... most of us tend to think the limit of limits is
>>somewhere between 13 and 25, depending on the context.
/.../
> May I ask how many "most of us" have:
>
> 1. Ever *heard* 31 Limit JI?

I meant most of us who have expressed an opinion. This was
based on years of reading just about everything on these
lists, and a lot of printed literature.

I assume those who have expressed an opinion have heard it. I
myself have heard it, and played it.

Notice I said "psychoacoustic meaning" not "musical value".

>2. (And most importantly addressed to you) Have ever taken the time
>to actually make music with 31 Limit JI?

No, I've never "taken the time" to make a finished recording
in "31-Limit" JI.

> Well - I'd like to hear the music which backs up "most of us
> tend to think the limit of limits is somewhere between 13 and
> 25, depending on the context."

All music backs up that statement. Or refutes it, if you like.

> If this is based on conjecture and from *not hearing* it, then it
> does not reflect much about its musical worth.

I didn't say anything about musical worth.

> I have heard it, I've played it (for over 10 years) and I can
> produce music for you to hear in it - and up to the 67 Limit
> too. My ears doubt what you say here because I've been doing
> it with success for too long to doubt it.

I have a cd of yours with a piece in 31-limit JI.

>Do you ever look at the complexity of ET intervals in the same
>light as JI?

Not sure what you're asking.

>What does their complexity show relative to "doubtful
>psychoacoustical meaning". For the record - I like it
>all. But then - I always to play the scales I discuss.
>
>*Meaning* is in music. May we all eventually discover its Joy.

It's too late for me... save yourself!

-Carl

--- In tuning@y..., "clumma" <carl@l...> wrote:

> Any scale made from a chain of a single interval we call a
> "linear series" or "linear scale" or "chain-of-x scale", where
> x is the interval being chained (called the generator of the
> scale). Usually, the term becomes "linear temperament", because
> some members of the chain will get close to consonant intervals,
> as when four 3:2's get close to a 5:4 -- even though we haven't
> tempered the generator, if we treat the near-5:4 as a consonance
> musically, we've "tempered" it.

Actually, a "linear" temperament has two generators, something with only one generator would be zero dimensional, I suppose. The typical example is an equal division of the octave, but you can divide something else or not divide anything.

--- In tuning@y..., Mark Gould <mark.gould@a...> wrote:

/tuning/topicId_33902.html#33902

> Dumb question:
>
> have there been any explorations of scales formed from constant
ratio pairs:
>
> sequences of 10/9 or 17/16 or 13/12? Octave-less endless chains of
intervals of constant size?
>

****Hello Mark! Well, this question isn't dumb, but there's still a
chance that the *answer* might be... :)

We had a *lengthy* discussion and development of a whole series of
scales, based upon a "Miracle" generator that, I believe, would fall
within your criterion, since there is a constant "generator..."

Joe Monzo has all the information on these scales posted online:

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

J. Pehrson

--- In tuning@y..., "clumma" <carl@l...> wrote:

/tuning/topicId_33902.html#33909

> Any scale made from a chain of a single interval we call a
> "linear series" or "linear scale" or "chain-of-x scale",

****Thank you so much, Carl, for your synopsis on this topic. I hate
to admit there were a couple things in there I didn't know
previously, but so be it! :)

I did have *one* question though:

(so prime ets like 17 and 19 are linear
> temperaments of all their intervals).

****Could you please *illustrate* this part for me??

thanks!

J. Pehrson

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

/tuning/topicId_33902.html#33917

> --- In tuning@y..., "clumma" <carl@l...> wrote:
>
> > Any scale made from a chain of a single interval we call a
> > "linear series" or "linear scale" or "chain-of-x scale", where
> > x is the interval being chained (called the generator of the
> > scale). Usually, the term becomes "linear temperament", because
> > some members of the chain will get close to consonant intervals,
> > as when four 3:2's get close to a 5:4 -- even though we haven't
> > tempered the generator, if we treat the near-5:4 as a consonance
> > musically, we've "tempered" it.
>
> Actually, a "linear" temperament has two generators, something with
only one generator would be zero dimensional, I suppose. The typical
example is an equal division of the octave, but you can divide
something else or not divide anything.

****Hi Gene!

So, then, the "Miracle" scales and Blackjack are *not* "linear"
temperaments.

Yes, no? Yesnoyesno?

J. Pehrson

>>Any scale made from a chain of a single interval we call a
>>"linear series" or "linear scale" or "chain-of-x scale", where
>>x is the interval being chained (called the generator of the
>>scale). Usually, the term becomes "linear temperament", because
>>some members of the chain will get close to consonant intervals,
>>as when four 3:2's get close to a 5:4 -- even though we haven't
>>tempered the generator, if we treat the near-5:4 as a consonance
>>musically, we've "tempered" it.
>
>Actually, a "linear" temperament has two generators, something
>with only one generator would be zero dimensional, I suppose. The
>typical example is an equal division of the octave, but you can
>divide something else or not divide anything.

Technically true, but since this second interval (called the
"interval of equivalence") is almost always the octave, it is
typically just assumed.

I'm sure we'd all like to see any scales you had in mind, Mark!

-Carl

>I did have *one* question though:
>
>>(so prime ets like 17 and 19 are linear
>> temperaments of all their intervals).
>
> ****Could you please *illustrate* this part for me??

No matter what interval you pick as the generator in
19-tET, you'll need 19 of them before you get back to
the starting pitch, since 19 isn't divisible by anything
except 1 (unlike 12, which is divisible by 3 and 4).

-Carl

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> So, then, the "Miracle" scales and Blackjack are *not* "linear"
> temperaments.
>
> Yes, no? Yesnoyesno?

Miracle is a linear temperament, because it has two generators--the secor and the octave. You subtract one from the number of generators to find the dimension, and so get a so-called "linear" temperament, which is something of a misnomer but we are stuck with it. Blackjack is a MOS based on miracle, and not a temperament per se.

--- In tuning@y..., "clumma" <carl@l...> wrote:

> >Actually, a "linear" temperament has two generators, something
> >with only one generator would be zero dimensional, I suppose. The
> >typical example is an equal division of the octave, but you can
> >divide something else or not divide anything.
>
> Technically true, but since this second interval (called the
> "interval of equivalence") is almost always the octave, it is
> typically just assumed.

Since it very often isn't an octave (eg, pajara), I think the assumption is a bad one. Even when it is an octave it doesn't need to be, the octave is simply a very common choice.

--- In tuning@y..., "clumma" <carl@l...> wrote:

/tuning/topicId_33902.html#33930

> >I did have *one* question though:
> >
> >>(so prime ets like 17 and 19 are linear
> >> temperaments of all their intervals).
> >
> > ****Could you please *illustrate* this part for me??
>
> No matter what interval you pick as the generator in
> 19-tET, you'll need 19 of them before you get back to
> the starting pitch, since 19 isn't divisible by anything
> except 1 (unlike 12, which is divisible by 3 and 4).
>
> -Carl

***Oh! Thanks, Carl. I didn't understand that it was this simple an
observation...

JP

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

/tuning/topicId_33902.html#33939

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> > So, then, the "Miracle" scales and Blackjack are *not* "linear"
> > temperaments.
> >
> > Yes, no? Yesnoyesno?
>
> Miracle is a linear temperament, because it has two generators--the
secor and the octave. You subtract one from the number of generators
to find the dimension, and so get a so-called "linear" temperament,
which is something of a misnomer but we are stuck with it. Blackjack
is a MOS based on miracle, and not a temperament per se.

***Thanks, Gene.

JP

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

> > Technically true, but since this second interval (called the
> > "interval of equivalence") is almost always the octave, it is
> > typically just assumed.
>
> Since it very often isn't an octave (eg, pajara), I think the
>assumption is a bad one. Even when it is an octave it doesn't need
>to be, the octave is simply a very common choice.

moreover, in pajara the interval of equivalence _is_ the octave but
the 'other generator' is really the _period_, a half-octave. so carl
should have said the second interval is called the 'period'. the
two 'generators' are not really on a par with one another because the
real 'generator' of the linear temperament is normally only iterated
a finite number of times, while the 'period' extends conceptually
infinitely, practically to the extremes of human hearing.

>>> Technically true, but since this second interval (called the
>>> "interval of equivalence") is almost always the octave, it is
>>> typically just assumed.
>>
>> Since it very often isn't an octave (eg, pajara), I think the
>>assumption is a bad one. Even when it is an octave it doesn't need
>>to be, the octave is simply a very common choice.
>
>moreover, in pajara the interval of equivalence _is_ the octave but
>the 'other generator' is really the _period_, a half-octave. so carl
>should have said the second interval is called the 'period'.

I wasn't talking about pajara. I was talking about MOS in general,
where there is typically only one generator, and one interval of
equivalence, that I've ever heard of. The term "period" is quite
new, and until there's agreement on and documentation of these
recent efforts it can hardly be considered seriously by the
community at large.

>the two 'generators' are not really on a par with one another
>because the real 'generator' of the linear temperament is normally
>only iterated a finite number of times, while the 'period' extends
>conceptually infinitely, practically to the extremes of human
>hearing.

Which is why I didn't mention it the first time around, as I
was trying to explain things as briefly and simply as possible --
actually, I was trying to understand what Mark was asking by
providing a sample answer of one possible question, before
descending into a mire of details. Still there, Mark?

-Carl

--- In tuning@y..., "clumma" <carl@l...> wrote:
> >>> Technically true, but since this second interval (called the
> >>> "interval of equivalence") is almost always the octave, it is
> >>> typically just assumed.
> >>
> >> Since it very often isn't an octave (eg, pajara), I think the
> >>assumption is a bad one. Even when it is an octave it doesn't
need
> >>to be, the octave is simply a very common choice.
> >
> >moreover, in pajara the interval of equivalence _is_ the octave
but
> >the 'other generator' is really the _period_, a half-octave. so
carl
> >should have said the second interval is called the 'period'.
>
> I wasn't talking about pajara. I was talking about MOS in general,
> where there is typically only one generator, and one interval of
> equivalence, that I've ever heard of.

kraig grady assures me that l s s s s l s s s s within an octave is
indeed an mos.

> >the two 'generators' are not really on a par with one another
> >because the real 'generator' of the linear temperament is normally
> >only iterated a finite number of times, while the 'period' extends
> >conceptually infinitely, practically to the extremes of human
> >hearing.
>
> Which is why I didn't mention it the first time around, as I
> was trying to explain things as briefly and simply as possible

yes that was directed more in gene's direction.

>>I wasn't talking about pajara. I was talking about MOS in general,
>>where there is typically only one generator, and one interval of
>>equivalence, that I've ever heard of.
>
>kraig grady assures me that l s s s s l s s s s within an octave
>is indeed an mos.

Sure, but you don't need the concept of period to get it. The
traditional explanation of MOS does not include peroid. I'd
like very much to add it, but I don't have the necessary
information.

-Carl

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

> moreover, in pajara the interval of equivalence _is_ the octave but
> the 'other generator' is really the _period_, a half-octave.

It seems to me the "interval of equivalence" is not a part of the temperament at all, if by it you simply mean octave equivalence. On the one hand you have a temperament, and on the other a psychoacoustic phenomenon. Where's the nexus?

so carl
> should have said the second interval is called the 'period'. the
> two 'generators' are not really on a par with one another because the
> real 'generator' of the linear temperament is normally only iterated
> a finite number of times, while the 'period' extends conceptually
> infinitely, practically to the extremes of human hearing.

Don't both generators extend to the limits of hearing or your instrument? I see no distinction.

--- In tuning@y..., "clumma" <carl@l...> wrote:

> I wasn't talking about pajara. I was talking about MOS in general,
> where there is typically only one generator, and one interval of
> equivalence, that I've ever heard of.

I thought we were talking about temperaments.

>>moreover, in pajara the interval of equivalence _is_ the octave but
>>the 'other generator' is really the _period_, a half-octave.
>
>It seems to me the "interval of equivalence" is not a part of the
>temperament at all, if by it you simply mean octave equivalence.
>On the one hand you have a temperament, and on the other a
>psychoacoustic phenomenon. Where's the nexus?

They are slightly different, which again is why I didn't mention
the octave at first. But once the chain spans an octave, you'll
get a different ordering if you actually reduce by octaves than
if you leave the octave reductions as virtual, psychoacoustic
things.

Unless I'm misunderstanding. Besides the octave and the generator,
what's the 'second generator'? I assumed you meant the period,
but not all MOSs have secondary periods, or is this not correct?

>>so carl should have said the second interval is called the
>>'period'. the two 'generators' are not really on a par with one
>>another because the real 'generator' of the linear temperament is
>>normally only iterated a finite number of times, while the 'period'
>>extends conceptually infinitely, practically to the extremes of
>>human hearing.
>
>Don't both generators extend to the limits of hearing or your
>instrument? I see no distinction.

I think Paul just meant that while you can choose how far to
carry out the chain of generators, when assuming octave equiv.,
you assume an infinite reduction/expansion of every note you
have.

-Carl

>>I wasn't talking about pajara. I was talking about MOS in general,
>>where there is typically only one generator, and one interval of
>>equivalence, that I've ever heard of.
>
>I thought we were talking about temperaments.

I was trying to address Mark's question about chains of intervals.
Mark gave examples of 9:8, 13:12, etc.

I mentioned that we usually consider these temperaments, even when
the generator is just. At least, those of us in the strong
temperament school.

Are we all on the same page now?

-Carl

--- In tuning@y..., "clumma" <carl@l...> wrote:

> Unless I'm misunderstanding. Besides the octave and the generator,
> what's the 'second generator'? I assumed you meant the period,
> but not all MOSs have secondary periods, or is this not correct?

Again we seem at cross purposes, because I wasn't talking about scales, but about temperaments. A linear temperament has two generators, which can be any two which define the given temperament. One of them may or may not be an octave or fraction of an octave.

--- In tuning@y..., "clumma" <carl@l...> wrote:
> >>I wasn't talking about pajara. I was talking about MOS in
general,
> >>where there is typically only one generator, and one interval of
> >>equivalence, that I've ever heard of.
> >
> >kraig grady assures me that l s s s s l s s s s within an octave
> >is indeed an mos.
>
> Sure, but you don't need the concept of period to get it. The
> traditional explanation of MOS does not include peroid. I'd
> like very much to add it, but I don't have the necessary
> information.

???

>Unless I'm misunderstanding. Besides the octave and the generator,
>what's the 'second generator'? I assumed you meant the period,
>but not all MOSs have secondary periods, or is this not correct?
>
>Again we seem at cross purposes, because I wasn't talking about
>scales, but about temperaments.

What's the difference? We may impose a limit of cardinality
on a melodic scale, or abstract the melodic scale from its
actual tuning, but the generative process in the case of the
linear series is the same.

>A linear temperament has two generators, which can be any two
>which define the given temperament. One of them may or may not
>be an octave or fraction of an octave.

Sure. And in some cases, it has a subperiod. Right?

-Carl

>>Sure, but you don't need the concept of period to get it. The
>>traditional explanation of MOS does not include peroid. I'd
>>like very much to add it, but I don't have the necessary
>>information.
>
>???

You only need to specify two intervals to get an MOS: a
generator and an interval of equivalence. Often on these
lists, people will specify only the generator and assume
an I.E. of an octave. I mentioned this to Mark so he could
read stuff here -- as opposed to forging some sort of
group-theoretical definition of linear temperaments.

Gene points out that the I.E. functions just like the
generator. Fine.

Somebody points out that in certain MOSs, there are sub-periods.
Fine.

Where did I go wrong?

-Carl

--- In tuning@y..., "clumma" <carl@l...> wrote:

> Besides the octave and the generator,
> what's the 'second generator'? I assumed you meant the period,
> but not all MOSs have secondary periods, or is this not correct?

the 'second generator' or period is very often the octave.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > moreover, in pajara the interval of equivalence _is_ the octave
but
> > the 'other generator' is really the _period_, a half-octave.
>
> It seems to me the "interval of equivalence" is not a part of the
>temperament at all, if by it you simply mean octave equivalence. On
>the one hand you have a temperament, and on the other a
>psychoacoustic phenomenon. Where's the nexus?

the interval of equivalence, as regards temperament, is an interval
that you demand that any scales derived from the temperament will
repeat exactly at, with an implied repetition in nomenclature and
musical function. the 'normal' generator is considered to have many
different equivalent forms via modification by the 'interval of
equivalence', while the converse is not the case. the interval of
equivalence is usually the octave but bohlen-pierce temperaments for
example use the 3:1 for this purpose. when you proceed to construct a
particular temperament, you may end up with a period that is equal to
the interval of equivalence, or ratio-wise its square root, cube
root, etc. . . . this period is not assumed a priori but is a result
of how the particular temperament is defined.

> Don't both generators extend to the limits of hearing

not really -- certainly many notes that would be perfectly audible
are typically 'cut off' because they involve too many iterations of
the 'normal' generator.

> or your instrument?

well, yes, i suppose the latter could be considered to have to do
with the limits of your instrument.

p.s. i tried, on tuning-math, to draw out of you the implications of
*not* considering the octave as one of the generators, or a power of
a generator, for a 2-dimensional ('linear') temperament. you appeared
to have derived things in this form in the past, but then 'restated'
them so that 'octave-equivalence of the temperament', as i tried to
define for you above, arose. so it appears you already know what it
means for a _temperament_ to possess octave-equivalence. but what
would be more helpful to people like jeff scott, for example, would
be for you to produce results in which this latter step did not
occur. you appeared to doubt its musical value, if i understood you
correctly?

--- In tuning@y..., "clumma" <carl@l...> wrote:
> >>Sure, but you don't need the concept of period to get it. The
> >>traditional explanation of MOS does not include peroid. I'd
> >>like very much to add it, but I don't have the necessary
> >>information.
> >
> >???
>
> You only need to specify two intervals to get an MOS: a
> generator and an interval of equivalence.

i'd prefer to say, a generator and a period. because it may still be
meaningful to refer to the octave as the interval of equivalence,
even if the period is a fraction (or ratio-wise, a root) of the
octave.

> Somebody points out that in certain MOSs, there are sub-periods.
> Fine.

it's not so fine. you could say 'the interval of equivalence is the
octave, and the generator is 709 cents', but totally miss out on the
10-tone mos, if you don't say that the period is 600 cents. daniel
wolf appears to have made this error when looking at the resources of
22-equal.

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., Mark Gould <mark.gould@a...> wrote:
>
> /tuning/topicId_33902.html#33902
>
> > Dumb question:
> >
> > have there been any explorations of scales formed from constant
> ratio pairs:
> >
> > sequences of 10/9 or 17/16 or 13/12? Octave-less endless chains
of
> intervals of constant size?
> >
>
> ****Hello Mark! Well, this question isn't dumb, but there's still
a
> chance that the *answer* might be... :)
>
> We had a *lengthy* discussion and development of a whole series of
> scales, based upon a "Miracle" generator that, I believe, would
fall
> within your criterion, since there is a constant "generator..."
>
> Joe Monzo has all the information on these scales posted online:
>
> http://www.ixpres.com/interval/dict/miracle.htm
>
> J. Pehrson

this tuning system has created so much traffic on this list for the
past year that we've had, for about 10 days now, a list devoted to it
alone:

/miracle_tuning/

miracle_tuning@yahoogroups.com

the miracle-tuning list has only 6 members now . . . and it doesn't
seem yet to be contributing to a reduction of 'miracle' traffic on
this (tuning) list . . . hopefully it will, as the amount of traffic
here is getting kinda scary . . .

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, February 11, 2002 3:45 PM
> Subject: [tuning] Re: Chains and lattices
>
>
> --- In tuning@y..., "clumma" <carl@l...> wrote:
>
> > You only need to specify two intervals to get an MOS: a
> > generator and an interval of equivalence.
>
> i'd prefer to say, a generator and a period. because it may still be
> meaningful to refer to the octave as the interval of equivalence,
> even if the period is a fraction (or ratio-wise, a root) of the
> octave.
>
> > Somebody points out that in certain MOSs, there are sub-periods.
> > Fine.
>
> it's not so fine. you could say 'the interval of equivalence is the
> octave, and the generator is 709 cents', but totally miss out on the
> 10-tone mos, if you don't say that the period is 600 cents. daniel
> wolf appears to have made this error when looking at the resources of
> 22-equal.

i tried for several weeks to resolve my confusion over the differences
between generator, period, and interval of equivalence on tuning-math,
without success.

now i think i'm beginning to get it. paul, can you please stick with
this example and explain how it works? thanks.

-monz

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

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning@y...>
> > Sent: Monday, February 11, 2002 3:45 PM
> > Subject: [tuning] Re: Chains and lattices
> >
> >
> > --- In tuning@y..., "clumma" <carl@l...> wrote:
> >
> > > You only need to specify two intervals to get an MOS: a
> > > generator and an interval of equivalence.
> >
> > i'd prefer to say, a generator and a period. because it may still
be
> > meaningful to refer to the octave as the interval of equivalence,
> > even if the period is a fraction (or ratio-wise, a root) of the
> > octave.
> >
> > > Somebody points out that in certain MOSs, there are sub-periods.
> > > Fine.
> >
> > it's not so fine. you could say 'the interval of equivalence is
the
> > octave, and the generator is 709 cents', but totally miss out on
the
> > 10-tone mos, if you don't say that the period is 600 cents.
daniel
> > wolf appears to have made this error when looking at the
resources of
> > 22-equal.
>
>
> i tried for several weeks to resolve my confusion over the
differences
> between generator, period, and interval of equivalence on tuning-
math,
> without success.

i thought we straightened you out, or at least i did off-list. why
didn't you stick us with more questions?

> now i think i'm beginning to get it. paul, can you please stick
with
> this example and explain how it works? thanks.

well, by implicitly assuming that the period had to be equal to the
interval of equivalence (the octave), daniel wolf looked at the mos
possibilities of the '4:3' (actually 491 cents, or 9 steps in 22-
equal) generator and only saw the 5-, 7-, 12-, and 17-tone ones.
however, since 22 is not a prime number, it is possible to have the
octave play the role of a *multiple* of the period -- in this case,
*two* periods. so with a period of 600 cents (11 steps in 22-equal),
the same generator produces mos scales with 5 or 6 notes per period,
or 10 or 12 notes per octave (and this 12-tone mos is different from
the one daniel found).

note that you don't need to start with the assumption of an non-prime
equal temperament to 'organically' end up with periods that are a
fraction of an octave. for example, these latter 10 or 12 tone mos
scales (possibly tuned differently, though not more than a few cents
in any pitch) come up immediately once you specify that 50:49 and
64:63 are to be tempered out.

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

> p.s. i tried, on tuning-math, to draw out of you the implications of
> *not* considering the octave as one of the generators, or a power of
> a generator, for a 2-dimensional ('linear') temperament.

I thought I answered that, but there were two separate questions--one the octave or fraction thereof as a generator, and the other *scales* constructed so that they do or do not repeat at octaves. The two questions are really distinct, though it is convenient with a linear temperament to make the interval of equivalence for such scales (normally, an octave) to be g^n for one of the generators g.

you appeared
> to have derived things in this form in the past, but then 'restated'
> them so that 'octave-equivalence of the temperament', as i tried to
> define for you above, arose.

I don't think temperaments have octave equivalence per se, though scales constructed with a given temperament in mind normally will.

so it appears you already know what it
> means for a _temperament_ to possess octave-equivalence.

Not really.

but what
> would be more helpful to people like jeff scott, for example, would
> be for you to produce results in which this latter step did not
> occur. you appeared to doubt its musical value, if i understood you
> correctly?

I think we are at cross-purposes as I was with Carl, over the difference between scales, and MOS in particular, and temperaments. I don't equate them--are you regarding them as the same?

--- In tuning@y..., Mark Gould <mark.gould@a...> wrote:
> Dumb question:
>
> have there been any explorations of scales formed from constant
ratio pairs:
>
> sequences of 10/9 or 17/16 or 13/12? Octave-less endless chains of
intervals
> of constant size?

Not dumb. I'm not sure anyone actually answered this question yet.
Some seem to have missed the "octave-less" part and started on about
Miracle temperament.

If the aim was to generate many just or near-just intervals, it would
be rare that the best generators would themselves be simple ratios.
Typically the generator is given in cents and the shorthand "cET" for
"cents equal temperament" is used.

The one I'm most familiar with is Gary Morrison's 88-cET. But you'll
find many others in the Scala archive, including close approximations
to those you mention. File names: cet182.scl, cet105.scl, cet140.scl.
http://www.xs4all.nl/~huygensf/scala/

These can also be considered as equal divisions of a stretched or
compressed octave or some other interval.

Of course, as has been pointed out before, the approx 117 cent
generator of Miracle temperament could be applied in this manner. It's
between 14:15 and 15:16, a mean-semitone. Has anyone worked out what
the 12-integer-limit RMS and MA optima are for this?

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > p.s. i tried, on tuning-math, to draw out of you the implications
of
> > *not* considering the octave as one of the generators, or a power
of
> > a generator, for a 2-dimensional ('linear') temperament.
>
> I thought I answered that, but there were two separate questions--
>one the octave or fraction thereof as a generator, and the other
>*scales* constructed so that they do or do not repeat at octaves.
>The two questions are really distinct, though it is convenient with
>a linear temperament to make the interval of equivalence for such
>scales (normally, an octave) to be g^n for one of the generators g.

well i guess i'm referring to the second now.

> > you appeared
> > to have derived things in this form in the past, but
then 'restated'
> > them so that 'octave-equivalence of the temperament', as i tried
to
> > define for you above, arose.
>
> I don't think temperaments have octave equivalence per se, though
>scales constructed with a given temperament in mind normally will.

ok

> > so it appears you already know what it
> > means for a _temperament_ to possess octave-equivalence.
>
> Not really.

well let's say then, a temperament *plus* enough notational or
whatever kind baggage to determine what its 'normally defined'
generator is, and thus what its mos scale are.

> > but what
> > would be more helpful to people like jeff scott, for example,
would
> > be for you to produce results in which this latter step did not
> > occur. you appeared to doubt its musical value, if i understood
you
> > correctly?
>
> I think we are at cross-purposes as I was with Carl, over the
>difference between scales, and MOS in particular, and temperaments.
>I don't equate them--are you regarding them as the same?

well let's talk about some concept, similar to temperament, which
however serves as a 'master set' for a particular set of mos scales.
any suggestions for a name? the point being that there seems to be an
implication in your work, that if you don't care to make use of
octave-equivalence in your music, there may very well be a way of
choosing a good 'period' -- often having nothing to do with the
octave -- for a particular temperament, say to maximize the occurence
of consonant intervals. am i making sense? can we flesh this stuff
out a bit?

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

> If the aim was to generate many just or near-just intervals, it
would
> be rare that the best generators would themselves be simple ratios.

very good point.

> Typically the generator is given in cents and the shorthand "cET"
for
> "cents equal temperament" is used.
>
> The one I'm most familiar with is Gary Morrison's 88-cET. But
you'll
> find many others in the Scala archive, including close
approximations
> to those you mention. File names: cet182.scl, cet105.scl,
cet140.scl.
> http://www.xs4all.nl/~huygensf/scala/

other, even more popular, non-octave scales are mentioned in the
third sentence here:

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

>>You only need to specify two intervals to get an MOS: a
>>generator and an interval of equivalence.
>
>i'd prefer to say, a generator and a period. because it may still
>be meaningful to refer to the octave as the interval of equivalence,
>even if the period is a fraction (or ratio-wise, a root) of the
>octave.

ok.

>>>Besides the octave and the generator,
>>>what's the 'second generator'? I assumed you meant the period,
>>>but not all MOSs have secondary periods, or is this not correct?
>>
/.../
>when you proceed to construct a particular temperament, you may
>end up with a period that is equal to the interval of equivalence,
>or ratio-wise its square root, cube root, etc. . . . this period
>is not assumed a priori but is a result of how the particular
>temperament is defined.

I thought that's what I've been saying.

-Carl