Re: unison vectors

Hi Carl,

Rightio, here is another version:

Unison vector:

Used to reduce an infinite lattice of
pitches to a finite scale, called a periodicity
block. Ex. five limit just intonation twelve tone
scales are normally periodicity blocks obtained
using the syntonic comma 81/80 and diesis
125/128 as unison vectors, because you
only have one each of notes a comma or
diesis apart from each other.
Vis:
16/15...8/5...6/5...9/5...(27/20 ~= 4/3)
4/3...1/1...3/2...9/8...(27/16 ~= 5/3)
5/3...5/4...15/8...45/32...(135/64 ~= 25/24)
25/24..25/16..75/64..225/128..
~= 16/15...8/5....6/5....9/5

What's an example of a twelve tone five
limit scale which doesn't use the syntonic
comma and diesis as unison vectors?

Presumably it needs to have pairs of notes
that are a diesis or comma apart - or
am I missing something here?
I know of course you can make
such scales but was assuming the
reader would take it as meaning
more conventional twelve tone
scales with the notes more or
less evenly spaced.

Of course, pythagorean twelve tone
counts as five limit too in a trivial
sense as any three limit scale is
five limit - perhaps I should
mention that too but it is already
rather long for a dictionary entry.

Anyway this is just pro-tem -
an experiment in trying out a
harder to define mathematical
term to see what happens.

Robert

hi Robert,

--- In tuning@yahoogroups.com, "Robert Walker" <robertwalker@n...>
wrote:

> Hi Carl,
>
> Rightio, here is another version:
>
> Unison vector:
>
> Used to reduce an infinite lattice of
> pitches to a finite scale, called a periodicity
> block.

try this:

Interval used as an identity-interval, used to reduce
an infinite lattice of pitches to a finite scale
called a periodicity-block.

(i like hyphenation ... it's a personal thing ...)

-monz

on 7/25/04 6:15 PM, monz <monz@attglobal.net> wrote:

> --- In tuning@yahoogroups.com, "Robert Walker" <robertwalker@n...>
> wrote:
>
>> Unison vector:
>>
>> Used to reduce an infinite lattice of
>> pitches to a finite scale, called a periodicity
>> block.
>
> try this:
>
> Interval used as an identity-interval, used to reduce
> an infinite lattice of pitches to a finite scale
> called a periodicity-block.

A question to help me learn, and conceivably to refine the definition, if
this proves relevant...

Is it possible/useful to have a reduced lattice that is finite in some
dimensions but still infinite in others? If so, does the term unison vector
still apply? If so the reduced lattice is probably not called a periodicity
block, I would guess.

Also for example the ignored octave is a dimension whose infiniteness has
not really been tampered with, even if it is being ignored, right?

Possibly too abstract for his own good,
Kurt

Hi Kurt,

>Is it possible/useful to have a reduced lattice that is finite in some
>dimensions but still infinite in others? If so, does the term unison
>vector still apply? If so the reduced lattice is probably not called
>a periodicity block, I would guess.

Great question. This is the case for linear temperaments, planar
temperaments, etc. Paul has called the resulting structures
periodicity strips and periodicity sheets, respectively. From
yonder archives...

""
To use the analogy provided by the block, the
meantone (1480-1780) composers were working within a
periodicity "strip" (which can be bent around into a cylinder), while
the 12-tone composers (1780-1980) were working within a
periodicity "sheet" (which can be bent into a torus).
""

-Carl

>
>Also for example the ignored octave is a dimension whose infiniteness has
>not really been tampered with, even if it is being ignored, right?
>
>Possibly too abstract for his own good,
>Kurt
>
>
>
>
>You can configure your subscription by sending an empty email to one
>of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
> tuning-nomail@yahoogroups.com - turn off mail from the group.
> tuning-digest@yahoogroups.com - set group to send daily digests.
> tuning-normal@yahoogroups.com - set group to send individual emails.
> tuning-help@yahoogroups.com - receive general help information.
>
>Yahoo! Groups Links
>
>
>
>

on 7/26/04 12:28 AM, Carl Lumma <ekin@lumma.org> wrote:

> Hi Kurt,
>
>> Is it possible/useful to have a reduced lattice that is finite in some
>> dimensions but still infinite in others? If so, does the term unison
>> vector still apply? If so the reduced lattice is probably not called
>> a periodicity block, I would guess.
>
> Great question. This is the case for linear temperaments, planar
> temperaments, etc. Paul has called the resulting structures
> periodicity strips and periodicity sheets, respectively. From
> yonder archives...
>
> ""
> To use the analogy provided by the block, the
> meantone (1480-1780) composers were working within a
> periodicity "strip" (which can be bent around into a cylinder), while
> the 12-tone composers (1780-1980) were working within a
> periodicity "sheet" (which can be bent into a torus).
> ""

I don't get it. I remember reading this at the time, and didn't catch it.
It seems to me that a strict meantone has only one generator aside from the
octave, the 5th, right?

So it is a one-dimensional structure then, and always an ET in some sense
(if the generator is allowed not to be a semitone), but not necessarily an
EDO, right?. If it maps to an EDO then it is finite, and otherwise it is
infinite, but in any case it is one-dimensional, no?

So I was thinking of something more like (as an arbitrary not very good
example) taking a 2-dimensional infinite 3,5 lattice, and deciding to temper
the major 3rds to an exact 1/3 octave (thus making the 3 dimension finite),
but leaving an infinite expanse of pure fifths (which in practice would
still be used in a finite way, of course). The only reason I can think of
for doing such a thing would be to create a keyboard with certain special
capabilities, and I think the example I picked is not a good one for that
purpose, nor am I sure that there is such a thing as a good one for any
useful purpose, but I was starting to think down this track a couple weeks
ago and never got finished with the idea.

-Kurt

hi Kurt,

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> A question to help me learn, and conceivably to
> refine the definition, if this proves relevant...
>
> Is it possible/useful to have a reduced lattice that
> is finite in some dimensions but still infinite in others?

sure ... the meantone helices/cylinders are a
perfect example. take a look at the lattices with
the blue background near the bottom of this page:

http://tonalsoft.com/enc/meantone.htm

the flat, rectangular 3 x 5 lattice has been
first filtered, so that two parallel slices were
made thru the lattice a syntonic comma apart
(and perpendicular to the vector of that comma),
then the lattice twisted so that the two edges
meet and form the helix.

so the placement of the two parallel edges which
describe the syntonic comma, which are twisted into
ends of the long spiral, is somewhat arbitrary
because theoretically those two ends go on into
infinity.

in practice, real meantones have unison-vectors
in that dimension too, which makes them closed.
for 1/3-comma meantone, this happens after 19 notes,
for 1/4-comma after 31, for 1/5-comma after 43, etc.
thus, the relationships of those ETs with those meantones.

> If so, does the term unison vector still apply?
> If so the reduced lattice is probably not called a
> periodicity block, I would guess.

Paul likes to call them "periodicity sheets" and such.

> Also for example the ignored octave is a dimension
> whose infiniteness has not really been tampered with,
> even if it is being ignored, right?

that's right. on lattices which assume 8ve-equivalence
and which thus ignore 2, that can be assumed to be another
dimension which also theoretically extends both directions
into infinity, and in reality extends up to our limits of
pitch perception. (c. 16 and 16,000 Hz).

> Possibly too abstract for his own good,
> Kurt

probably a good thing after all! ;-)

-monz

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> So I was thinking of something more like (as an arbitrary not very good
> example) taking a 2-dimensional infinite 3,5 lattice, and deciding
to temper
> the major 3rds to an exact 1/3 octave (thus making the 3 dimension
finite),
> but leaving an infinite expanse of pure fifths (which in practice would
> still be used in a finite way, of course).

You've just invented the augmented temperament. If three major thirds
are an octave, it means 2/(5/4)^3 = 128/125 is tempered out. Hence
your period, assuming pure octaves, must be the major third of 400
cents, and the generator can be chosen as a sharp fifth.

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

> in practice, real meantones have unison-vectors
> in that dimension too, which makes them closed.

Depends on what you mean by in practice, I think. A typical meantone
would be 1/4-comma on 12 notes, which certainly doesn't close on 12
and in theory does not close at all. In a practice which no one
actually practices, you could put it down for closed in 205-equal, I
suppose, a fact which would be relevant only for people like me using
computers to make music. Even then after 205 fifths you are shy of 119
octaves by 1.4 cents; it never actually closes since no power of five
can ever equal a power of two.

hi Kurt,

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> It seems to me that a strict meantone has only
> one generator aside from the octave, the 5th, right?

right. in strictly formal math, we use the 4th instead.
but it gives the same results, with the generator =/- signs
switched.

> So it is a one-dimensional structure then, and always
> an ET in some sense (if the generator is allowed not to
> be a semitone), but not necessarily an EDO, right?.
> If it maps to an EDO then it is finite, and otherwise
> it is infinite, but in any case it is one-dimensional, no?

from the perspective of a chain-of-5ths, which is an
extension of Pythgorean thinking, yes, it's one-dimensional.

but the beauty of meantone is that it *is* intended to
imply (at least to some extent) 5-limit harmony. indeed,
that is its whole _raison d'etre_.

from that perspective, meantone is definitely 2-dimensional,
and makes use of the equation 81/64 = 5/4, which defines
its classic 1/4-comma temperament. in 2,3,5-monzos, that's
[-6 4, 0> = [-2 0, 1> , or very simply in 8ve-equivalent
terms, 3^4 = 5^1 .

that equation remains in effect for all other meantones,
but with the stipulation that the two ratios are only
approximately equal to each other.

> So I was thinking of something more like (as an
> arbitrary not very good example) taking a 2-dimensional
> infinite 3,5 lattice, and deciding to temper the
> major 3rds to an exact 1/3 octave (thus making the
> 3 dimension finite), but leaving an infinite expanse
> of pure fifths (which in practice would still be used
> in a finite way, of course). The only reason I can
> think of for doing such a thing would be to create a
> keyboard with certain special capabilities, and I think
> the example I picked is not a good one for that purpose,
> nor am I sure that there is such a thing as a good one
> for any useful purpose, but I was starting to think down
> this track a couple weeks ago and never got finished
> with the idea.

hmm ... i just tried to make a helical lattice of this
temperament for you, but found a bug in our software.
thanks for using that example!

-monz

hi Gene and Kurt,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > in practice, real meantones have unison-vectors
> > in that dimension too, which makes them closed.
>
> Depends on what you mean by in practice, I think.
> A typical meantone would be 1/4-comma on 12 notes,
> which certainly doesn't close on 12 and in theory
> does not close at all.

i knew i should have written more there.

yes, in a meantone of 12 tones, the chain will not
close.

but if 1/4-comma meantone is extended to 32 notes,
the 32nd -- 2,3,5-monzo [18 0, -31/4> -- will be
~6.068717548 cents lower (flatter) than the origin note.

see
http://tonalsoft.com/enc/1-4cmt.htm

it is very possible for an interval that small to qualify
as a unison-vector. if we make it vanish, we get 31-ET.

1/3-comma and 1/5-comma meantone provide even stronger
examples:

in 1/3-comma meantone, the 20th generator is
2,3,5-monzo [-14/3 -19/3, 19/3> = ~0.938515663 cent
below the origin note. thus, 1/3-comma is nearly
identical (within ~1 cent) to 19edo.

in 1/5-comma meantone, the 44th generator is
2,3,5-monzo [168/5 -43/5, -43/5> = ~0.88905332 cent
below the origin note. thus, 1/5-comma is nearly
identical (within ~1 cent) to 43edo.

> In a practice which no one actually practices,

but i bet *you* will!! ;-)

> you could put it down for closed in 205-equal, I
> suppose, a fact which would be relevant only for
> people like me using computers to make music. Even
> then after 205 fifths you are shy of 119 octaves
> by 1.4 cents; it never actually closes since no power
> of five can ever equal a power of two.

205-ET is certainly a more accurate convergent to
1/4-comma meantone, but 31-ET is near the +/- 5 cents
human tuning error normally assumed to be negligible.
IOW, for acoustic instruments, 1/4-comma and 31-ET
are the same.

-monz

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

> 205-ET is certainly a more accurate convergent to
> 1/4-comma meantone, but 31-ET is near the +/- 5 cents
> human tuning error normally assumed to be negligible.
> IOW, for acoustic instruments, 1/4-comma and 31-ET
> are the same.

After 30 fifths of size 5^(1/4), you would get a "wolf" of
size 262144 sqrt(5)/390625, which is 0.69 cents sharp. While of course
that is hardly a really a wolf, a fifth of that size, a little sharper
even than that of 41-et, is by no means impossible to distinguish from
a 1/4 comma meantone fifth. In the case of 205-et, the "wolf" is
another meantone fifth, only 2/11 comma meantone rather than 1/4 comma
meantone.

>> ""
>> To use the analogy provided by the block, the
>> meantone (1480-1780) composers were working within a
>> periodicity "strip" (which can be bent around into a cylinder), while
>> the 12-tone composers (1780-1980) were working within a
>> periodicity "sheet" (which can be bent into a torus).
>> ""
>
>I don't get it. I remember reading this at the time, and didn't
>catch it. It seems to me that a strict meantone has only one
>generator aside from the octave, the 5th, right?

You're thinking of the lattice of generators, but Paul is talking
here about the lattice of just intonation. It would probably be
very instructive for you to draw a 5-limit lattice and the 81:80
unison vector on it, and observe how this warps the lattice into
a tube.

>So it is a one-dimensional structure then, and always an ET in some
>sense (if the generator is allowed not to be a semitone), but not
>necessarily an EDO, right?. If it maps to an EDO then it is finite,
>and otherwise it is infinite, but in any case it is one-dimensional,
>no?

The tuning is 1-dimensional but it maps onto a two-dimensional
JI lattice (5-limit) through a "map" or "prime mapping"...

2 3 5
< 0 1 4 ] fifth
< 1 1 0 ] octave

Note the column and row headings are usually omitted in these
maps.

>So I was thinking of something more like (as an arbitrary not very
>good example) taking a 2-dimensional infinite 3,5 lattice, and
>deciding to temper the major 3rds to an exact 1/3 octave

Giving the "augmented" linear temperament.

>(thus making the 3 dimension finite),

I think you mean 5-dimension.

-Carl

>> It seems to me that a strict meantone has only
>> one generator aside from the octave, the 5th, right?
>
>right. in strictly formal math, we use the 4th instead.
>but it gives the same results, with the generator =/- signs
>switched.

Actually it doesn't matter which you use. There's nothing
more strict about either one.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> ""
> >> To use the analogy provided by the block, the
> >> meantone (1480-1780) composers were working within
> >> a periodicity "strip" (which can be bent around into
> >> a cylinder), while the 12-tone composers (1780-1980)
> >> were working within a periodicity "sheet" (which can
> >> be bent into a torus).
> >> ""
> >
> > I don't get it. I remember reading this at the time,
> > and didn't catch it. It seems to me that a strict
> > meantone has only one generator aside from the octave,
> > the 5th, right?
>
> You're thinking of the lattice of generators, but Paul is talking
> here about the lattice of just intonation. It would probably be
> very instructive for you to draw a 5-limit lattice and the 81:80
> unison vector on it, and observe how this warps the lattice into
> a tube.

you can actually see four 2-D views of this here:

http://tonalsoft.com/enc/meantone.htm#helix

-monz

hi Carl,

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >> It seems to me that a strict meantone has only
> >> one generator aside from the octave, the 5th, right?
> >
> >right. in strictly formal math, we use the 4th instead.
> >but it gives the same results, with the generator =/- signs
> >switched.
>
> Actually it doesn't matter which you use. There's nothing
> more strict about either one.
>
> -Carl

just poor wording on my part. it's just a convention
to use the smallest size of generator ... so for meantone,
it's the 4th rather than the 5th, but adjusting for sign,
they're equivalent.

-monz

on 7/26/04 2:14 AM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
>> So I was thinking of something more like (as an arbitrary not very good
>> example) taking a 2-dimensional infinite 3,5 lattice, and deciding
> to temper
>> the major 3rds to an exact 1/3 octave (thus making the 3 dimension
> finite),
>> but leaving an infinite expanse of pure fifths (which in practice would
>> still be used in a finite way, of course).
>
> You've just invented the augmented temperament. If three major thirds
> are an octave, it means 2/(5/4)^3 = 128/125 is tempered out. Hence
> your period, assuming pure octaves, must be the major third of 400
> cents, and the generator can be chosen as a sharp fifth.

Sharp fifth to work better with the sharp third?

I take it in any case this is common practice for an augmented temerament?

-Kurt

on 7/26/04 2:25 AM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
>> in practice, real meantones have unison-vectors
>> in that dimension too, which makes them closed.
>
> Depends on what you mean by in practice, I think. A typical meantone
> would be 1/4-comma on 12 notes, which certainly doesn't close on 12
> and in theory does not close at all. In a practice which no one
> actually practices, you could put it down for closed in 205-equal, I
> suppose, a fact which would be relevant only for people like me using
> computers to make music. Even then after 205 fifths you are shy of 119
> octaves by 1.4 cents; it never actually closes since no power of five
> can ever equal a power of two.

This is kind of a good point because the "finite"-ness (ET-equivalence) of
1/4 comma etc. is mentioned fairly often here with only rare mention that it
is an approximation, IIRC. In fact I believed it without checking for the
better part of the year, and actually repeated the misinformation to others.

I should have known intuitively that it could not be (because a
rational-base log of a rational will never be a rational except in the case
of log(1)=0), but I just didn't think to question it. This was because I
*liked* the result so much! ;)

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> > You've just invented the augmented temperament. If three major thirds
> > are an octave, it means 2/(5/4)^3 = 128/125 is tempered out. Hence
> > your period, assuming pure octaves, must be the major third of 400
> > cents, and the generator can be chosen as a sharp fifth.
>
> Sharp fifth to work better with the sharp third?

Correct, although in fact a pure fifth would be an interesting choice.
The poptimal "range" consists of a single fifth, (1458/125)^(1/6),
but 39-equal will do the job nicely. It extends nicely to the 12&15
"tripletone" 7-limit version I wanted to call simply "augmented" and
Paul seems to think should be "augene".

> I take it in any case this is common practice for an augmented
temerament?

I don't think augmented is much practiced outside of a 12-equal
context, which of course would not involve a sharp fifth. It's what
theory says to do if octaves are pure; the top tuning would instead
flatten the octave by three cents and make fifths pure, which might we
worth exploring if someone wants to practice this tuning. The TOP
tuning for the 7-limit is not much different; the octave is still 3
cents flat and the fifth now 2/3 cent sharp.

on 7/26/04 8:29 AM, Carl Lumma <ekin@lumma.org> wrote:

>>> ""
>>> To use the analogy provided by the block, the
>>> meantone (1480-1780) composers were working within a
>>> periodicity "strip" (which can be bent around into a cylinder), while
>>> the 12-tone composers (1780-1980) were working within a
>>> periodicity "sheet" (which can be bent into a torus).
>>> ""
>>
>> I don't get it. I remember reading this at the time, and didn't
>> catch it. It seems to me that a strict meantone has only one
>> generator aside from the octave, the 5th, right?
>
> You're thinking of the lattice of generators,

Well, no actually. What happenned is I got rusty on the details and was
left simply with my unconscious intuition based on experience. Experience
with 3,5 lattices (correct term?) keeps telling me 2 is ignored and
therefore I intuitively think 2 doesn't count as a dimension. That is it
keep slooking to me like 5-limit lattices are 2-dimensional because the
octave is always ignored in a just lattice since it adds nothing of
interest.

Ok, so now I'm reminded and maybe I won't make the mistake again until I get
*really* old and forgetful.

>> So I was thinking of something more like (as an arbitrary not very
>> good example) taking a 2-dimensional infinite 3,5 lattice, and
>> deciding to temper the major 3rds to an exact 1/3 octave
>
> Giving the "augmented" linear temperament.
>
>> (thus making the 3 dimension finite),
>
> I think you mean 5-dimension.

Yea, the brain keeps getting confused by 5ths being 3 and 3rds being 5. ;)
Made all the worse by the fact that minor 7ths are 7!

-Kurt

>>>> ""
>>>> To use the analogy provided by the block, the
>>>> meantone (1480-1780) composers were working within a
>>>> periodicity "strip" (which can be bent around into a cylinder),
>>>> while the 12-tone composers (1780-1980) were working within a
>>>> periodicity "sheet" (which can be bent into a torus).
>>>> ""
>>>
>>> I don't get it. I remember reading this at the time, and didn't
>>> catch it. It seems to me that a strict meantone has only one
>>> generator aside from the octave, the 5th, right?
>>
>> You're thinking of the lattice of generators,
>
>Well, no actually. What happenned is I got rusty on the details
>and was left simply with my unconscious intuition based on
>experience. Experience with 3,5 lattices (correct term?)

Sure. Also "5-limit lattice" or just "lattice" when context
makes it clear.

>keeps telling me 2 is ignored and therefore I intuitively think
>2 doesn't count as a dimension.

It can be done either way. Traditionally 2 is left out; not
so in the Tenney lattice. And, as an aside, the maxim is: 2 in,
use triangular. 2 out, rectangular. Tape it to your bathroom
mirror. :)

>Ok, so now I'm reminded and maybe I won't make the mistake again
>until I get *really* old and forgetful.

But this has nothing to do with the confusion you seemed to
be having above -- the lattice of generators (which happen
to be near-just in the meantone case) with the JI lattice.

-Carl

I wrote...

>Traditionally 2 is left out; not
>so in the Tenney lattice. And, as an aside, the maxim is: 2 in,
>use triangular. 2 out, rectangular. Tape it to your bathroom
>mirror. :)

Damn it! I got that backward.

Use rectangular if you keep the 2 axis, triangular if you
throw it out.

-Carl