Re: interval of equivalence, unison-vector, period

In-Reply-To: <a3am2v+eddv@eGroups.com>
paulerlich wrote:

> The period is often 1/2-octave, 1/3-octave, 1/4-octave, 1/9-
> octave, . . . so that's clearly not a "unison-vector".
>
> The "interval of equivalence" is a unison vector in Graham's system,
> but Graham's system seems more limited than Gene's. Gene treats it as
> only one of the "constructing" consonant intervals, and then
> somehow "sticks it back in at the end" with some LLL reduction of
> something.

What distinction are you seeing between us? The octave is mathematically
like a chromatic unison vector. Like any other chromatic unison vector,
you "stick it back in at the end" to get your linear temperament. LLL
reduction has nothing to do with this.

Divisions of the period are torsion relating the the equivalence interval
as a unison vector.

For example, if you define 25:24 as a unison vector in Miracle
temperament, you get a decimal scale with torsion. That means, when you
stop tempering out the 25:24 you find the approximation to it is always
divided into two equal steps (the quommas).

If the octave is a chromatic unison vector in twintone, it also gets
divided into two equal parts because of torsion. The only thing stopping
an octave being a real unison vector is that it isn't like a unison.

Graham

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <a3am2v+eddv@e...>
> paulerlich wrote:
>
> > The period is often 1/2-octave, 1/3-octave, 1/4-octave, 1/9-
> > octave, . . . so that's clearly not a "unison-vector".
> >
> > The "interval of equivalence" is a unison vector in Graham's
system,
> > but Graham's system seems more limited than Gene's. Gene treats
it as
> > only one of the "constructing" consonant intervals, and then
> > somehow "sticks it back in at the end" with some LLL reduction of
> > something.
>
> What distinction are you seeing between us? The octave is
mathematically
> like a chromatic unison vector.

Only in your system.

> Like any other chromatic unison vector,
> you "stick it back in at the end" to get your linear temperament.

You're meaning something very different from what I meant
by "sticking it back in at the end".

> LLL
> reduction has nothing to do with this.

Take a look again at the way Gene does it. He initially comes out
with a scale with two generators, neither of which is the octave.
This bears no resemblance to the way you do it.

> If the octave is a chromatic unison vector in twintone,

How odd to call an octave a chromatic unison vector. When a note is
altered by a chromatic unison vector, it is supposed to undergo a
small but nonzero change in pitch. In neither the octave-invariant
nor the octave-specific case is this true for the octave!

> it also gets
> divided into two equal parts because of torsion.

Warning -- this does not agree with the definition of torsion that
Gene was talking about.

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

> > If the octave is a chromatic unison vector in twintone,
>
> How odd to call an octave a chromatic unison vector. When a note is
> altered by a chromatic unison vector, it is supposed to undergo a
> small but nonzero change in pitch. In neither the octave-invariant
> nor the octave-specific case is this true for the octave!
>
> > it also gets
> > divided into two equal parts because of torsion.
>
> Warning -- this does not agree with the definition of torsion that
> Gene was talking about.

If you take a set of unison vectors defining an equal temperament, as for instance {50/49, 64/63, 245/243} and now add 2 to the set, then {2, 50/49, 64/63, 245/243} generates a kernel K such that N7/K = C22--we have a map of the 7-limit to a cyclic group of order 22--which is a torsion group, since everything has finite order. All elements are torsion elements, and we have a finite group, so this is rather different than a block with torsion elements.

If we take 50/49^64/63, the wedgie for twintone, and wedge it with
245/245 we get the 7-limit val h22 of the 22-et, which of course defines a temperament. If we wedge the twintone wedgie with 2 instead, we also get a val--the mapping to generators of the non-octave generator of twintone. This is *not* a temperament, or at least not one I'm interested in hearing, so 2 is not acting as a unison, which is hardly a surprise.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > > If the octave is a chromatic unison vector in twintone,
> >
> > How odd to call an octave a chromatic unison vector. When a note
is
> > altered by a chromatic unison vector, it is supposed to undergo a
> > small but nonzero change in pitch. In neither the octave-
invariant
> > nor the octave-specific case is this true for the octave!
> >
> > > it also gets
> > > divided into two equal parts because of torsion.
> >
> > Warning -- this does not agree with the definition of torsion
that
> > Gene was talking about.
>
> If you take a set of unison vectors defining an equal temperament,
>as for instance {50/49, 64/63, 245/243} and now add 2 to the set,
>then {2, 50/49, 64/63, 245/243} generates a kernel K such that N7/K
>= C22--we have a map of the 7-limit to a cyclic group of order 22--

This, I think, corresponds to how Graham thinks of things, and how I
_used_ to think of things, before I understood torsion in the period-
is-1/2-or-1/9-or-1/N-octave sense.

>which is a torsion group, since everything has finite order. All
>elements are torsion elements, and we have a finite group,

And this could happen just as well for a group with a prime number of
elements, such as {2, 25/24, 81/80} -> C7.

>so this is rather different than a block with torsion elements.

Yes it is. Now we really need to revise the definition of torsion :(,
and think of different names for these two things.

> If we take 50/49^64/63, the wedgie for twintone, and wedge it with
> 245/245 we get the 7-limit val h22 of the 22-et, which of course
>defines a temperament. If we wedge the twintone wedgie with 2
>instead, we also get a val--the mapping to generators of the non-
>octave generator of twintone.

Can you go into this in more detail, pretty please with sugar on top?

>This is *not* a temperament, or at least not one I'm interested in
>hearing, so 2 is not acting as a unison, which is hardly a surprise.

??

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

> And this could happen just as well for a group with a prime number of
> elements, such as {2, 25/24, 81/80} -> C7.

Yes, indeedy.

> >so this is rather different than a block with torsion elements.
>
> Yes it is. Now we really need to revise the definition of torsion :(,
> and think of different names for these two things.

Why do we need to worry about it?

> Can you go into this in more detail, pretty please with sugar on top?

I'm not sure what you are asking for, so let's see if this does it:

The MT reduced basis for 22 et in the 7-limit is
{50/49, 64/63, 245/243}. If I take these in pairs and wedge them, I get three temperaments instead, which can also be thought of as a defining basis for 22-et:

50/49^64/63 = [-2,4,4,-2,-12,11] -- twintone

50/49^245/243 = [6,10,10,-5,1,2] with generators
a = 3.0143/22 = 164.4176 cents and b = 1/2

64/63^245/243 = [1,9,-2,-30,6,12] with generators a = 8.9763/22 =
489.6152 cents and b = 1

I can now wedge these with 2, and get triple wedge products. A triple wedge product of three intervals will be a val, but it doesn't have to be an equal temperament val.

50/49^64/63^2 = [0,2,-4,-4]

50/49^245/243^2 = [0,-6,-10,-10]

64/63^245/243^2 = [0,-1,-9,2]

This is giving us the non-octave part of the generator map. We could also wedge with other intervals of equivalence besides 2, and get what the corresponding temperament would be then; for instance

50/49^64/63^3/2 = [-2,-2,-7,-8]

We can then use this mapping to primes (or [2,2,7,8], which seems nicer and which a different order of the triple product would have given us) to define a version of this temperament based on the fifth as an interval of equivalence. Note that both 2 and 3 are mapped to the same number, in this case 2, which is a consequence of
3 = 2 (3/2), and corresponds to the way that 2 maps to 0 in the previous triple wedge products.

> >This is *not* a temperament, or at least not one I'm interested in
> >hearing, so 2 is not acting as a unison, which is hardly a surprise.

Mapping 2 to 1, and both 5 and 7 to 1/9 does not strike me as much of a temperament.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
>
> > And this could happen just as well for a group with a prime
number of
> > elements, such as {2, 25/24, 81/80} -> C7.
>
> Yes, indeedy.
>
> > >so this is rather different than a block with torsion elements.
> >
> > Yes it is. Now we really need to revise the definition of
torsion :(,
> > and think of different names for these two things.
>
> Why do we need to worry about it?

For the sake of Monz' dictionary, perhaps?

> > Can you go into this in more detail, pretty please with sugar on
top?
>
> I'm not sure what you are asking for, so let's see if this does it:
>
> The MT reduced basis for 22 et in the 7-limit is
> {50/49, 64/63, 245/243}. If I take these in pairs and wedge them, I
get three temperaments instead, which can also be thought of as a
defining basis for 22-et:
>
> 50/49^64/63 = [-2,4,4,-2,-12,11] -- twintone
>
> 50/49^245/243 = [6,10,10,-5,1,2] with generators
> a = 3.0143/22 = 164.4176 cents and b = 1/2

Glassic
>
> 64/63^245/243 = [1,9,-2,-30,6,12] with generators a = 8.9763/22 =
> 489.6152 cents and b = 1

"Big fifth" -- a unique facet of 22

>I can now wedge these with 2, and get triple wedge products. A
>triple wedge product of three intervals will be a val, but it
>doesn't have to be an equal temperament val.

What other kinds are there?
>
> 50/49^64/63^2 = [0,2,-4,-4]
>
> 50/49^245/243^2 = [0,-6,-10,-10]
>
> 64/63^245/243^2 = [0,-1,-9,2]
>
> This is giving us the non-octave part of the generator map. We
could also wedge with other intervals of equivalence besides 2, and
get what the corresponding temperament would be then; for instance
>
> 50/49^64/63^3/2 = [-2,-2,-7,-8]
>
> We can then use this mapping to primes (or [2,2,7,8], which seems
>nicer and which a different order of the triple product would have
>given us) to define a version of this temperament based on the fifth
>as an interval of equivalence.

OK . . .

> > >This is *not* a temperament, or at least not one I'm interested
in
> > >hearing, so 2 is not acting as a unison, which is hardly a
surprise.
>
> Mapping 2 to 1, and both 5 and 7 to 1/9 does not strike me as much
of a temperament.

Well . . . I'm lost . . . does this have anything to do with what you
were once showing about your process, where for a "linear" or 2D
temperament, you started off with two generators, but then found a
different generator basis pair where you forced one member to be an
octave?

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Thursday, January 31, 2002 9:48 PM
> Subject: [tuning-math] Re: interval of equivalence, unison-vector, period
>
>
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> >
> > > > If the octave is a chromatic unison vector in twintone,
> > > > it also gets divided into two equal parts because of torsion.
> > >
> > > Warning -- this does not agree with the definition of
> > > torsion that Gene was talking about.
> >
> > If you take a set of unison vectors defining an equal temperament,
> > as for instance {50/49, 64/63, 245/243} and now add 2 to the set,
> > then {2, 50/49, 64/63, 245/243} generates a kernel K such that N7/K
> > = C22--we have a map of the 7-limit to a cyclic group of order 22--
>
> This, I think, corresponds to how Graham thinks of things, and how I
> _used_ to think of things, before I understood torsion in the period-
> is-1/2-or-1/9-or-1/N-octave sense.

Paul, you're really good at explaining things.
Please elaborate on this until I understand it. :)

> > which is a torsion group, since everything has finite order. All
> > elements are torsion elements, and we have a finite group,
>
> And this could happen just as well for a group with a prime number of
> elements, such as {2, 25/24, 81/80} -> C7.
>
> >so this is rather different than a block with torsion elements.
>
> Yes it is. Now we really need to revise the definition of torsion :(,
> and think of different names for these two things.

I'm really interested in this difference between "a block with
torsion elements" and the finite group where "all elements are
torsion elements".

I don't recall anyone ever responding to the lattice diagram
I made for the torsion definition:
http://www.ixpres.com/interval/dict/torsion.htm

I thought that showing the pairs of pitches that are separated
by two unison-vector candidates that are smaller than the
actual unison-vectors defining the torsional-block might have
been saying something significant about what a torsional-block
is, or maybe at least something about this particular example.

Any thoughts?

> > If we take 50/49^64/63, the wedgie for twintone, and wedge it with
> > 245/245 we get the 7-limit val h22 of the 22-et, which of course
> >defines a temperament. If we wedge the twintone wedgie with 2
> >instead, we also get a val--the mapping to generators of the non-
> >octave generator of twintone.
>
> Can you go into this in more detail, pretty please with sugar on top?

And I second that request.

The fog has still not cleared about the three items in the subject line.

-monz

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

> > This, I think, corresponds to how Graham thinks of things, and
how I
> > _used_ to think of things, before I understood torsion in the
period-
> > is-1/2-or-1/9-or-1/N-octave sense.
>
>
> Paul, you're really good at explaining things.
> Please elaborate on this until I understand it. :)

Oops -- I didn't mean that at all. I meant, before I understood
torsion as it's defined in your dictionary. Thanks for pointing out
my brain fart!
>
> I don't recall anyone ever responding to the lattice diagram
> I made for the torsion definition:
> http://www.ixpres.com/interval/dict/torsion.htm
>
> I thought that showing the pairs of pitches that are separated
> by two unison-vector candidates that are smaller than the
> actual unison-vectors defining the torsional-block might have
> been saying something significant about what a torsional-block
> is, or maybe at least something about this particular example.
>
> Any thoughts?

Well, you're definitely doing something right in this case, since
81:80 and 128:125 are definitely intervals that should represent
equivalences here . . . but it won't necessarily be that case that
smaller intervals in the parallelogram than the defining unison
vectors fall into the "equivalent" category for every torsional block.

> The fog has still not cleared about the three items in the subject
> line.

Really? OK, first of all, period is specific to MOS scales and the
linear temperaments they come from.

Examples:

meantone temperament
unison vector: 81:80
interval of equivalence: octave
period: octave

MIRACLE temperament
unison vectors: 224:225, 385:384, 441:440
interval of equivalence: octave
period: octave

diminished/octatonic in 12-tET or 28-tET
unison vector: 648:625
interval of equivalence: octave
period: 1/4 octave

'paultone'
unison vectors: 50:49, 64:63
interval of equivalence: octave
period: 1/2 octave

Bohlen-Pierce
unison vectors: 245:243, 3087:3125
interval of equivalence: tritave (3:1)
period: tritave (3:1)

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

> Bohlen-Pierce
> unison vectors: 245:243, 3087:3125
> interval of equivalence: tritave (3:1)
> period: tritave (3:1)

Well, what I meant was the BP linear temperament (generated by 7:3,
with interval of equivalence 3:1), so 3087:3125 doesn't belong there.
Should be:

Bohlen-Pierce linear temperament (Stearns/Benson/Keenan)
unison vectors: 245:243
interval of equivalence: tritave (3:1)
period: tritave (3:1)

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, February 01, 2002 12:07 AM
> Subject: [tuning-math] Re: interval of equivalence, unison-vector, period
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > > This, I think, corresponds to how Graham thinks of things,
> > > and how I _used_ to think of things, before I understood
> > > torsion in the period-is-1/2-or-1/9-or-1/N-octave sense.
> >
> >
> > Paul, you're really good at explaining things.
> > Please elaborate on this until I understand it. :)
>
> Oops -- I didn't mean that at all. I meant, before I understood
> torsion as it's defined in your dictionary. Thanks for pointing out
> my brain fart!

Well, OK, you're welcome. But I really thought you were saying
you understood something here about which I'm totally clueless.

> >
> > I don't recall anyone ever responding to the lattice diagram
> > I made for the torsion definition:
> > http://www.ixpres.com/interval/dict/torsion.htm
> >
> > I thought that showing the pairs of pitches that are separated
> > by two unison-vector candidates that are smaller than the
> > actual unison-vectors defining the torsional-block might have
> > been saying something significant about what a torsional-block
> > is, or maybe at least something about this particular example.
> >
> > Any thoughts?
>
> Well, you're definitely doing something right in this case, since
> 81:80 and 128:125 are definitely intervals that should represent
> equivalences here

YES! Good, I'm slowly getting it.

> . . . but it won't necessarily be that case that smaller
> intervals in the parallelogram than the defining unison vectors
> fall into the "equivalent" category for every torsional block.

Well, OK, I'll take your word for it. Examples of this would
be good.

>
> > The fog has still not cleared about the three items in the subject
> > line.
>
> Really? OK, first of all, period is specific to MOS scales and the
> linear temperaments they come from.

Ah! OK, that helps a bit.

> Examples:
>
> meantone temperament
> unison vector: 81:80
> interval of equivalence: octave
> period: octave
>
> MIRACLE temperament
> unison vectors: 224:225, 385:384, 441:440
> interval of equivalence: octave
> period: octave
>
> diminished/octatonic in 12-tET or 28-tET
> unison vector: 648:625
> interval of equivalence: octave
> period: 1/4 octave
>
> 'paultone'
> unison vectors: 50:49, 64:63
> interval of equivalence: octave
> period: 1/2 octave
>
> Bohlen-Pierce
> unison vectors: 245:243, 3087:3125
> interval of equivalence: tritave (3:1)
> period: tritave (3:1)

Examples are good, and thanks much for them all.

You know what I realize now, upon really seriously studying
the tuning-math archives since August? Latticing has
almost completely disappeared. That's a big part of the
reason why I'm having such a hard time following.

Bring back the lattice diagrams! P L E A S E !!!!!

(with truckloads of sugar)

-monz

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

> > Well, you're definitely doing something right in this case, since
> > 81:80 and 128:125 are definitely intervals that should represent
> > equivalences here
>
>
> YES! Good, I'm slowly getting it.
>
>
> > . . . but it won't necessarily be that case that smaller
> > intervals in the parallelogram than the defining unison vectors
> > fall into the "equivalent" category for every torsional block.
>
>
> Well, OK, I'll take your word for it. Examples of this would
> be good.

For example, in the Blackjack block (see for example the JI blackjack
block I just made for you), one of the three defining unison vectors
of the 3-dimensional parallelepiped is 36:35, while 64:63 appears a
lot _within_ the block -- yet there is no torsion.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, February 01, 2002 12:23 AM
> Subject: [tuning-math] Re: interval of equivalence, unison-vector, period
>
>
> > > . . . but it won't necessarily be that case that smaller
> > > intervals in the parallelogram than the defining unison vectors
> > > fall into the "equivalent" category for every torsional block.
> >
> >
> > Well, OK, I'll take your word for it. Examples of this would
> > be good.
>
> For example, in the Blackjack block (see for example the JI blackjack
> block I just made for you), one of the three defining unison vectors
> of the 3-dimensional parallelepiped is 36:35, while 64:63 appears a
> lot _within_ the block -- yet there is no torsion.

Hmmm ... so by "smaller", you mean in interval size.

But what about in taxicab-metric size? It seems to me that a
unison-vector within a torsional block must *always* be smaller
in taxicab-size than the unison-vectors which define the
torsional-block. Yes?

Might I be onto something here that's obvious, but that
might have some special meaning that hasn't been noticed?

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Friday, February 01, 2002 12:23 AM
> > Subject: [tuning-math] Re: interval of equivalence, unison-
vector, period
> >
> >
> > > > . . . but it won't necessarily be that case that smaller
> > > > intervals in the parallelogram than the defining unison
vectors
> > > > fall into the "equivalent" category for every torsional block.
> > >
> > >
> > > Well, OK, I'll take your word for it. Examples of this would
> > > be good.
> >
> > For example, in the Blackjack block (see for example the JI
blackjack
> > block I just made for you), one of the three defining unison
vectors
> > of the 3-dimensional parallelepiped is 36:35, while 64:63 appears
a
> > lot _within_ the block -- yet there is no torsion.
>
>
> Hmmm ... so by "smaller", you mean in interval size.
>
> But what about in taxicab-metric size?

> It seems to me that a
> unison-vector within a torsional block must *always* be smaller
> in taxicab-size than the unison-vectors which define the
> torsional-block. Yes?

Usually, but not necessarily -- for example, pitch two pitches near
the pair of corners of the parallelogram that are furthest from one
another -- this will tend to be longer than at least the shortest
unison vector. However, with the hexagonal periodicity blocks we have
a better attempt to do just what you describe -- especially if the
taxicab distance is evaluated with respect to a triangular lattice
like the ones I tend to use.

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

> > 50/49^245/243 = [6,10,10,-5,1,2] with generators
> > a = 3.0143/22 = 164.4176 cents and b = 1/2
>
> Glassic

Good name--where does it come from?

> > 64/63^245/243 = [1,9,-2,-30,6,12] with generators a = 8.9763/22 =
> > 489.6152 cents and b = 1
>
> "Big fifth" -- a unique facet of 22

Not really; I would say it is even more characteristic of 27 or 49.

> >A triple wedge product of three intervals will be a val, but it
> >doesn't have to be an equal temperament val.

> What other kinds are there?

Vals are the dual concept to intervals. We have prime number intervals, such as octave or twelvth, and the dual to those are the
p-adic valuations vp, which count the number of powers of p (positive or negative) in the prime factorization of a rational number. An interval is a finite Z-linear combination of primes; that is, it is p1^e1 * p2^e2 ... pk^ek for certain primes pn and certain integers en. A val is a finite Z-linear combination of p-adic valuations:
e1 v1 + e2 v2 + ... + ek vk. Dual to the comma, or small interval is, more or less, an et val. Another type of val of interest are the maps of generators to primes.

> > Mapping 2 to 1, and both 5 and 7 to 1/9 does not strike me as much
> of a temperament.
>
> Well . . . I'm lost . . . does this have anything to do with what you
> were once showing about your process, where for a "linear" or 2D
> temperament, you started off with two generators, but then found a
> different generator basis pair where you forced one member to be an
> octave?

Right--this val would be the starting point for that process, not a temperament. I can do the same sort of thing starting from
[-2,-2,-7,-8], where I end up with

[-2 2]
[-2 3]
[-7 5]
[-8 6]

as a mapping from generators to primes; here "b" is a wide fifth and "a" is a tritone below that.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > > 50/49^245/243 = [6,10,10,-5,1,2] with generators
> > > a = 3.0143/22 = 164.4176 cents and b = 1/2
> >
> > Glassic
>
> Good name--where does it come from?

Sorry -- that's the wrong name. Glassic has b = 1 . . . my
piece "Glassic" uses it.

> > > 64/63^245/243 = [1,9,-2,-30,6,12] with generators a = 8.9763/22
=
> > > 489.6152 cents and b = 1
> >
> > "Big fifth" -- a unique facet of 22
>
> Not really; I would say it is even more characteristic of 27 or 49.

OK, you're right.

> > >A triple wedge product of three intervals will be a val, but it
> > >doesn't have to be an equal temperament val.
>
> > What other kinds are there?
>
> Vals are the dual concept to intervals. We have prime number
intervals, such as octave or twelvth, and the dual to those are the
> p-adic valuations vp, which count the number of powers of p
(positive or negative) in the prime factorization of a rational
number. An interval is a finite Z-linear combination of primes; that
is, it is p1^e1 * p2^e2 ... pk^ek for certain primes pn and certain
integers en. A val is a finite Z-linear combination of p-adic
valuations:
> e1 v1 + e2 v2 + ... + ek vk.

Ah!

>Dual to the comma, or small interval is, more or less, an et val.
>Another type of val of interest are the maps of generators to primes.

What's the dual to that kind of val?

> > > Mapping 2 to 1, and both 5 and 7 to 1/9 does not strike me as
much
> > of a temperament.
> >
> > Well . . . I'm lost . . . does this have anything to do with what
you
> > were once showing about your process, where for a "linear" or 2D
> > temperament, you started off with two generators, but then found
a
> > different generator basis pair where you forced one member to be
an
> > octave?
>
> Right

GRAHAM, TAKE NOTE!

>--this val would be the starting point for that process, not a
>temperament.

But it's a kind of non-octave-repeating scale, isn't it?? It's
something that would be of interest if we didn't assume that "octaves
sound the same", right?

> I can do the same sort of thing starting from
> [-2,-2,-7,-8], where I end up with
>
> [-2 2]
> [-2 3]
> [-7 5]
> [-8 6]
>
> as a mapping from generators to primes; here "b" is a wide fifth
>and "a" is a tritone below that.

Wouldn't that just be a non-octave ET, approximately 11 tones per
octave?

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

> > I can do the same sort of thing starting from
> > [-2,-2,-7,-8], where I end up with
> >
> > [-2 2]
> > [-2 3]
> > [-7 5]
> > [-8 6]
> >
> > as a mapping from generators to primes; here "b" is a wide fifth
> >and "a" is a tritone below that.
>
> Wouldn't that just be a non-octave ET, approximately 11 tones per
> octave?

Oops, I meant a non-octave linear temperament, approximately 11
periods per octave?

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

> > > Glassic
> >
> > Good name--where does it come from?
>
> Sorry -- that's the wrong name. Glassic has b = 1 . . . my
> piece "Glassic" uses it.

I recall that piece--I thought perhaps it was named for a temperament.

> >Another type of val of interest are the maps of generators to primes.

> What's the dual to that kind of val?

Generators. For the period matrix (pair of vals) for twintone, it would be two intervals, the first mapped to one and the other to zero, and the second to zero and then one--the simplest example being
4/3 and 7/5.

> > I can do the same sort of thing starting from
> > [-2,-2,-7,-8], where I end up with
> >
> > [-2 2]
> > [-2 3]
> > [-7 5]
> > [-8 6]
> >
> > as a mapping from generators to primes; here "b" is a wide fifth
> >and "a" is a tritone below that.
>
> Wouldn't that just be a non-octave ET, approximately 11 tones per
> octave?

No, it's two generators for twintone, only now instead of making one of them an octave or a fraction of an octave, I've made it a fifth or a fraction of a fifth--in this case, the full fifth. If you wanted an et for it, 22 springs to mind. 11 can't work, because twintone needs a tritone (in this case, the difference between the two generators.)

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

> > >Another type of val of interest are the maps of generators to
primes.
>
> > What's the dual to that kind of val?
>
> Generators.

That's what I was going to guess . . .

>For the period matrix (pair of vals) for twintone, it would be two
>intervals, the first mapped to one and the other to zero, and the
>second to zero and then one

Whoa -- this is a very confusing sentence. Can you clarify?

>--the simplest example being
> 4/3 and 7/5.

Not surprising, as these are normally taken as the period and the
generator of twintone. But there are other possibilities, if you
don't assume octave-equivalence?

> > Wouldn't that just be a non-octave ET, approximately 11 tones per
> > octave?
>
> No,

Well . . . I revised the question in the next message, which I hope
you get to see.

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

> Well . . . I revised the question in the next message, which I hope
> you get to see.

Meaning, the next message after the one where the question appeared,
which is back 11 messages:

/tuning-math/message/3109

In-Reply-To: <a3dlmu+4ceg@eGroups.com>
genewardsmith wrote:

> Vals are the dual concept to intervals. We have prime number intervals,
> such as octave or twelvth, and the dual to those are the p-adic
> valuations vp, which count the number of powers of p (positive or
> negative) in the prime factorization of a rational number. An interval
> is a finite Z-linear combination of primes; that is, it is p1^e1 *
> p2^e2 ... pk^ek for certain primes pn and certain integers en. A val is
> a finite Z-linear combination of p-adic valuations: e1 v1 + e2 v2 + ...
> + ek vk. Dual to the comma, or small interval is, more or less, an et
> val. Another type of val of interest are the maps of generators to
> primes.

Does this mean "dual" is the proper word for what I'm calling the
"complement"?

Graham

In-Reply-To: <a3d5gb+ldpj@eGroups.com>
genewardsmith wrote:

> If you take a set of unison vectors defining an equal temperament, as
> for instance {50/49, 64/63, 245/243} and now add 2 to the set, then {2,
> 50/49, 64/63, 245/243} generates a kernel K such that N7/K = C22--we
> have a map of the 7-limit to a cyclic group of order 22--which is a
> torsion group, since everything has finite order. All elements are
> torsion elements, and we have a finite group, so this is rather
> different than a block with torsion elements.

But it still involves torsion?

> If we take 50/49^64/63, the wedgie for twintone, and wedge it with
> 245/245 we get the 7-limit val h22 of the 22-et, which of course
> defines a temperament.

Okay, I assume you mean 245/243.

>>> h22 = (wedgie^temper.WedgableRatio(245,243)).equalTemperament()
>>> h22.basis
(22, 35, 51, 62)

> If we wedge the twintone wedgie with 2 instead,
> we also get a val--the mapping to generators of the non-octave
> generator of twintone.

Right.

>>> g0 = (wedgie^{(0,):1}).equalTemperament()
>>> g0.basis
(0, 1, -2, -2)

> This is *not* a temperament, or at least not one
> I'm interested in hearing, so 2 is not acting as a unison, which is
> hardly a surprise.

Of course it's a temperament. It's twintone/paultone/pajara.

The octave is acting as a unison, but it's more complicated than that. As
it has torsion, it's actually half an octave that's acting as a commatic
unison vector. This isn't a problem for normal unison vectors because
half a unison is still a unison. All you find is that you get twice as
many notes as you expected. But half an octave is very different from an
octave. So our commatic unison vector is, in a sense, the tritone.

(BTW, in an octave-equivalent system, half a unison is a half-octave as
well as a unison. This is obvious if you think of octave-equivalent
frequency space as a Hilbert space, and remember that half the pitch is
the same as the square root of the frequency.)

The first entry being zero tells us that the octave is clearly acting as a
unison. In fact, two tritone-unisons.

The second entry tells us that the 3:1 is a generator. That's a semitone
larger than an octave plus a tritone. As octaves and tritones are
unisons, that means it's a semitone larger than a unison. And two notes
differing by a unison must be the same, so the generator is a semitone.

The third entry tells us that 5:1 is -2 generators. That means it's a
unison less two semitones. In this case, octaves and tritones are
unisons. 5:4 and 5:1 only differ by octaves, which means they only differ
by unisons, so 5:4 is also a unison less two semitones. A tritone is a
unison. So 5:4 is a tritone less two semitones.

The fourth entry tells us that 7:1 is also -2 generators. As the
generator is still a semitone and octaves are still unisons, that means
7:4 is an octave less two semitones.

This is all an accurate description of paultone/twintone/pajara in a
tritone-equivalent system. So what's the problem? In general, making an
octave a unison vector is like imposing octave equivalence. That's
actually quite similar to something Fokker said.

But hey, octave equivalence is a strange concept to pull out of the hat,
so perhaps you're still not convinced. Well, remember the unison vector
Gene gave describes the equal temperament h22, and the octave as unison
vector describes something I called g0. Well, I will now show that g0 is
indeed an equal temperament. It's a consistent mapping to a temperament
with no equal steps to the octave.

Another unison vector that isn't a unison could be 3:1. Tempering this
out gives two tone equal temperament:

>>> g2 = (wedgie^{(1,):1}).equalTemperament()
>>> g2.basis
(2, 0, 11, 12)

g0 and g2 together define paultone/twintone/pajara

>>> (g0^g2).complement() == wedgie
1

It's also possible to combine g0 and g2 to get h22.

>>> g22 = g2 + g0
>>> for each in range(10):
... g22 = g22 + g2
...
>>> for each in range(34):
... g22 = g22 + g0
...
>>> h22.basis
(22, 35, 51, 62)
>>> g22.basis
[22, 35, 51, 62]

Unfortunately, I didn't set my library up to allow equal temperaments to
be multiplied by integers. Also, addition causes the basis to be a list
rather than a tuple. But if it weren't for these things, that could have
been written

>>> (g0*35 + g2*11) == h22
1

I'm assuming we all accept that 12-equal is a temperament. It's also
consistent with paultone/twintone/pajara.

>>> h12 = temper.PrimeET(12, temper.primes[:3])
>>> h12.basis
[12, 19, 28, 34]
>>> wedgie.complement()^h12
{}

If h12 and h22 are pajara-consistent ETs, so must h10 be

>>> h10 = h22-h12
>>> h10.basis
[10, 16, 23, 28]

and h2 (not the prime mapping, but still distinct from g2)

>>> h2 = h12-h10
>>> h2.basis
[2, 3, 5, 6]

We can also get another 2 note equal temperament from h12 and h22, which
I'll have to call i2 because I'm running out of letters.

>>> h8 = h10-h2
>>> h6 = h8-h2
>>> h4 = h6-h2
>>> i2 = h4-h2
>>> i2.basis
[2, 4, 3, 4]

As we have two 2 note equal temperaments, the difference between them must
be an equal temperament with no notes.

>>> i0 = i2-h2
>>> i0.basis
[0, 1, -2, -2]

Hey, that's the same as g0 above!

>>> g0.basis
(0, 1, -2, -2)

So the pajara equal temperament you get with the octave as a commatic
unison vector certainly looks like an equal temperament as I understand
the concept. Perhaps, specifically, it's a regular equal temperament as
not any no notes are allowed.

Unless somebody wants to stipulate that an equal temperament has to have
more than no notes, but I don't remember that ever being mentioned before.

Graham

In-Reply-To: <a3cadl+f19b@eGroups.com>
Me:
> > If the octave is a chromatic unison vector in twintone,

Paul:
> How odd to call an octave a chromatic unison vector. When a note is
> altered by a chromatic unison vector, it is supposed to undergo a
> small but nonzero change in pitch. In neither the octave-invariant
> nor the octave-specific case is this true for the octave!

We've been through this before. Although an octave isn't a unison vector
because it isn't small, we still don't have a word for something that's
like a unison vector but small. So I have to keep calling them "things
like unison vectors". Do you have a mathematical definition for "like a
unison"?

Me:
> > it also gets
> > divided into two equal parts because of torsion.

Paul:
> Warning -- this does not agree with the definition of torsion that
> Gene was talking about.

How so?

Is 25:24 a unison vector?

Does it have torsion when used with the Miracle wedgie?

Does it define a decimal scale?

Is it always divided into two equal parts in other Miracle tunings?

Where am I going wrong?

Graham

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

> Oops, I meant a non-octave linear temperament, approximately 11
> periods per octave?

As long as 2 is represented, it seems to me any temperament is an octave temperament. The basis I gave was for a fifth and a tritone below a fifth, and I could if I wanted make the fifth a pure fifth,
but I could do that, and temper octaves, in the octave basis also. There are three considerations: interval of equivalence of a scale using a given temperament, a basis of generators for the temperament, and the tuning of the temperament. This are independent.

--- In tuning-math@y..., graham@m... wrote:

> The octave is acting as a unison, but it's more complicated than
that. As
> it has torsion, it's actually half an octave that's acting as a
commatic
> unison vector.

No offense, Graham, but could you at least invent some terminology
that makes sense for what you're talking about, instead of
misappropriating terminology that makes no sense the way you're using
it? Half an octave does not act a commatic unison vector here -- this
is very frustrating because I thought I had spent dozens of posts
explaining to you what a commatic unison vector is, and convincing
you that an octave isn't one and a fifth isn't one . . . did all that
arguing make no impression on you?

A chromatic unison vector is a generalized "augmented unison".
Nothing else.

> (BTW, in an octave-equivalent system, half a unison is a half-
octave as
> well as a unison. This is obvious if you think of octave-
equivalent
> frequency space as a Hilbert space, and remember that half the
pitch is
> the same as the square root of the frequency.)

Huh? So if the frequency is 6400 Hz, the square root of that is 80,
and that's half the pitch??

> A tritone is a
> unison.

Right . . .

> This is all an accurate description of paultone/twintone/pajara in
a
> tritone-equivalent system. So what's the problem? In general,
making an
> octave a unison vector is like imposing octave equivalence. That's
> actually quite similar to something Fokker said.

Please fill us in!

--- In tuning-math@y..., graham@m... wrote:

> Where am I going wrong?

I'm not saying you're wrong, only that your methods are different
from Gene's -- most recently exemplified with the case that he
considered "not a temperament" and you considered "22-tET".

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > Oops, I meant a non-octave linear temperament, approximately 11
> > periods per octave?
>
> As long as 2 is represented, it seems to me any temperament is an
>octave temperament. The basis I gave was for a fifth and a tritone
>below a fifth, and I could if I wanted make the fifth a pure fifth,
>> but I could do that, and temper octaves, in the octave basis also.
>There are three considerations: interval of equivalence of a scale
>using a given temperament, a basis of generators for the
>temperament, and the tuning of the temperament. This are independent.

So why did you say "this was not a temperament"? And isn't it true
that, if you took it out to, say, 10 notes per approximate octave,
and tuned the octaves pure, it would _not_ be an octave-repeating
scale? This seems to be the point Graham is missing.

Me:
> > The octave is acting as a unison, but it's more complicated that that. As
> > it has torsion, it's actually half an octave that's acting as a commatic
> > unison vector.

Paul:
> No offense, Graham, but could you at least invent some terminology
> that makes sense for what you're talking about, instead of
> misappropriating terminology that makes no sense the way you're using
> it? Half an octave does not act a commatic unison vector here -- this
> is very frustrating because I thought I had spent dozens of posts
> explaining to you what a commatic unison vector is, and convincing
> you that an octave isn't one and a fifth isn't one . . . did all that
> arguing make no impression on you?

I'm fully aware that an octave is not a unison vector. I've said so before
and I didn't say otherwise in that quote. All I said is that it (or the
tritone) acts as a unison vector. Which it does. As far as the algebra's
concerned, it's exactly like a unison vector. I could invent terminology,
but don't need to here because the tritone is already what we're calling the
"period". The thread actually started because Monz was confused about the
difference between "unison vector", "equivalence interval" and "period". He
was right to be because they are very similar.

Paul:
> A chromatic unison vector is a generalized "augmented unison".
> Nothing else.

Well, can you think of a word for something that acts like a unison vector
but isn't? To cover the meanings of "unison vector", "generator", "period"
and "equivalence interval"?

Me:
> > (BTW, in an octave-equivalent system, half a unison is a half- octave as
> > well as a unison. This is obvious if you think of octave-equivalent
> > frequency space as a Hilbert space, and remember that half the pitch is
> > the same as the square root of the frequency.)

Paul:
> Huh? So if the frequency is 6400 Hz, the square root of that is 80,
> and that's half the pitch??

Hmm, something wrong there. I meant the square root of a frequency *ratio*.
Yes? That seems to make sense.

Me:
> > A tritone is a unison.

Paul:
> Right . . .

That's what the algebra says if you assume an octave is a unison vector
(which it isn't, but I was assuming it in that passage).

Me:
> > octave a unison vector is like imposing octave equivalence. That's
> > actually quite similar to something Fokker said.

Paul:
> Please fill us in!

In <http://www.xs4all.nl/~huygensf/doc/fokkerpb.html>, "By common general
agreement all notes differing by an arbitrary number of octaves only, are
considered as unison, and as one and the same note." I mentioned this last
time round as well.

Graham

Me:
> > Where am I going wrong?

Paul:
> I'm not saying you're wrong, only that your methods are different
> from Gene's -- most recently exemplified with the case that he
> considered "not a temperament" and you considered "22-tET".

I meant there where I was wrong with Gene's terminology. I didn't call the
thing he called "not a temperament" 22-tET. I called it
paultone/twintone/pajara. Because that's what it is. It's actually a
mapping of 0-tET. You could call it a paradox that something with no notes
counts as a temperament. You could then analyse the assumptions that led to
it instead of shouting back "you're wrong" at the person who pointed it out.

I haven't yet seen that my methods are different from Gene's at all. I've
actually adopted wholesale his stuff about wedgies, so far as I understand
it. All we disagree on is interpretation. What he's saying is pretty much
where I started at <http://x31eq.com/lintemp.htm> anyway. Two
generators (I didn't know the word then) which can be any size, and a number
of commas which approximate to unisons.

Graham

Gene:
> > As long as 2 is represented, it seems to me any temperament is an
> >octave temperament. The basis I gave was for a fifth and a tritone
> >below a fifth, and I could if I wanted make the fifth a pure fifth,
> > but I could do that, and temper octaves, in the octave basis also.
> >
> >There are three considerations: interval of equivalence of a scale
> >using a given temperament, a basis of generators for the
> >temperament, and the tuning of the temperament. This are independent.

Paul:
> So why did you say "this was not a temperament"? And isn't it true
> that, if you took it out to, say, 10 notes per approximate octave,
> and tuned the octaves pure, it would _not_ be an octave-repeating
> scale? This seems to be the point Graham is missing.

The thing he said wasn't a temperament has no notes to an octave, so you
could say 2 isn't represented and so it isn't an octave temperament by that
definition.

What *has* octave repetition got to do with anything?

Graham

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <a3d5gb+ldpj@e...>
> genewardsmith wrote:

All elements are
> > torsion elements, and we have a finite group, so this is rather
> > different than a block with torsion elements.
>
> But it still involves torsion?

Certainly, but not in the sense of a torsion block with torsion, since it isn't a block.

> > This is *not* a temperament, or at least not one
> > I'm interested in hearing, so 2 is not acting as a unison, which is
> > hardly a surprise.
>
> Of course it's a temperament. It's twintone/paultone/pajara.

Is pajara the new official name? I'd like to get this settled. As for this val, which defines only one of two required generator mappings being a temperament, that's only if you layer on some interpretation and perform the extra calculations to find a good choice for the second generator; taken by itself, it isn't one. It's telling us to send the octave to a unison, and 5 and 7 both to 1/9; it's only after you stick in half-octaves and send 7 to some tuning of 64/9 and 5 to a half-octave below that that pajara emerges. Read literally as a temperament, it sends 2 to 1 and 5 and 7 to 1/9, and I don't think that qualifies.

> The octave is acting as a unison, but it's more complicated than that. As
> it has torsion, it's actually half an octave that's acting as a commatic
> unison vector.

I would say it's acting as a generator, but if you make 2 a unison it becomes a torsion element, since its square is an octave.

> (BTW, in an octave-equivalent system, half a unison is a half-octave as
> well as a unison. This is obvious if you think of octave-equivalent
> frequency space as a Hilbert space, and remember that half the pitch is
> the same as the square root of the frequency.)

You get a real Hilbert space if you allow anything of the form
3^e3 5^e5 ... which can have an infinite number of prime exponents so
long as e3^2 + e5^2 + ... converges. Is this what you mean? The result isn't even guaranteed to be a real number, and I don't know what it would be good for.

> >>> i0 = i2-h2
> >>> i0.basis
> [0, 1, -2, -2]
>
> Hey, that's the same as g0 above!

And which I think hardly counts as a temperament. As I said, it's not one I want to listen to.

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

> So why did you say "this was not a temperament"?

Because a "temperament" which sends

1-9/8--5/4--4/3--3/2--5/3--15/8 to

1--9--1/9--1/3--3--1/27--1/3

hardly seems worthy of the name. In any case, 2 isn't represented!

And isn't it true
> that, if you took it out to, say, 10 notes per approximate octave,
> and tuned the octaves pure, it would _not_ be an octave-repeating
> scale? This seems to be the point Graham is missing.

We seem to be talking about different things--what is "it"? If you mean pajara, it's a temperament, not a scale.

--- In tuning-math@y..., Graham Breed <graham@m...> wrote:
> Well, can you think of a word for something that acts like a unison vector
> but isn't? To cover the meanings of "unison vector", "generator", "period"
> and "equivalence interval"?

What about kernel element? Of course, a period is a kernel element only if you make it one, by having a corresponding mapping, but that is the case here. The same would be true of an equivalence interval--if we send the half-octave to 1, it is a kernel element, but if we send 2 to 1 but not sqrt(2), then sqrt(2) is an element of order 2. One way we get a cyclic group of order 11, the other way of order 22.

--- In tuning-math@y..., Graham Breed <graham@m...> wrote:
> Me:
> > > The octave is acting as a unison, but it's more complicated
that that. As
> > > it has torsion, it's actually half an octave that's acting as a
commatic
> > > unison vector.
>
> Paul:
> > No offense, Graham, but could you at least invent some terminology
> > that makes sense for what you're talking about, instead of
> > misappropriating terminology that makes no sense the way you're
using
> > it? Half an octave does not act a commatic unison vector here --
this
> > is very frustrating because I thought I had spent dozens of posts
> > explaining to you what a commatic unison vector is, and convincing
> > you that an octave isn't one and a fifth isn't one . . . did all
that
> > arguing make no impression on you?
>
> I'm fully aware that an octave is not a unison vector. I've said
so before
> and I didn't say otherwise in that quote. All I said is that it
(or the
> tritone) acts as a unison vector. Which it does. As far as the
algebra's
> concerned, it's exactly like a unison vector.

But why a "chromatic" unison vector? A chromatic unison vector
indicates something that is actually tuned differently from
an "equivalence".
>
> Paul:
> > A chromatic unison vector is a generalized "augmented unison".
> > Nothing else.
>
> Well, can you think of a word for something that acts like a unison
vector
> but isn't?

It does in your mechanics, but not in Gene's.

> Me:
> > > (BTW, in an octave-equivalent system, half a unison is a half-
octave as
> > > well as a unison. This is obvious if you think of octave-
equivalent
> > > frequency space as a Hilbert space, and remember that half the
pitch is
> > > the same as the square root of the frequency.)
>
> Paul:
> > Huh? So if the frequency is 6400 Hz, the square root of that is
80,
> > and that's half the pitch??
>
> Hmm, something wrong there. I meant the square root of a frequency
*ratio*.
> Yes? That seems to make sense.

But half the pitch? I think you mean half the interval, or half the
pitch _difference_.
>
> Me:
> > > octave a unison vector is like imposing octave equivalence.
That's
> > > actually quite similar to something Fokker said.
>
> Paul:
> > Please fill us in!
>
> In <http://www.xs4all.nl/~huygensf/doc/fokkerpb.html>, "By common
general
> agreement all notes differing by an arbitrary number of octaves
only, are
> considered as unison, and as one and the same note." I mentioned
this last
> time round as well.

Well, there are apparantly different ways of implementing this
observation mathematically.

--- In tuning-math@y..., Graham Breed <graham@m...> wrote:
> Me:
> > > Where am I going wrong?
>
> Paul:
> > I'm not saying you're wrong, only that your methods are different
> > from Gene's -- most recently exemplified with the case that he
> > considered "not a temperament" and you considered "22-tET".
>
> I meant there where I was wrong with Gene's terminology. I didn't
call the
> thing he called "not a temperament" 22-tET. I called it
> paultone/twintone/pajara. Because that's what it is.

Not really, because paultone/twintone/pajara repeat themselves every
octave, while I don't think Gene's construction does -- that's why he
said it's "not a temperament", I believe.

> It's actually a mapping of 0-tET. You could call it a paradox that
something with no notes
> counts as a temperament. You could then analyse the assumptions
that led to
> it instead of shouting back "you're wrong" at the person who
pointed it out.

Good point. But you're not going to help anyone understand this stuff
by using misleading terminology. That's all I'm trying to say.

--- In tuning-math@y..., Graham Breed <graham@m...> wrote:
> Gene:
> > > As long as 2 is represented, it seems to me any temperament is
an
> > >octave temperament. The basis I gave was for a fifth and a
tritone
> > >below a fifth, and I could if I wanted make the fifth a pure
fifth,
> > > but I could do that, and temper octaves, in the octave basis
also.
> > >
> > >There are three considerations: interval of equivalence of a
scale
> > >using a given temperament, a basis of generators for the
> > >temperament, and the tuning of the temperament. This are
independent.
>
> Paul:
> > So why did you say "this was not a temperament"? And isn't it true
> > that, if you took it out to, say, 10 notes per approximate octave,
> > and tuned the octaves pure, it would _not_ be an octave-repeating
> > scale? This seems to be the point Graham is missing.
>
> The thing he said wasn't a temperament has no notes to an octave,

No notes? He said it was generated by a fifth and a fifth-tritone --
so it seems like it could have plenty of notes, up to an infinite
number, in fact.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., graham@m... wrote:
> > In-Reply-To: <a3d5gb+ldpj@e...>
> > genewardsmith wrote:
>
> All elements are
> > > torsion elements, and we have a finite group, so this is rather
> > > different than a block with torsion elements.
> >
> > But it still involves torsion?
>
> Certainly, but not in the sense of a torsion block with torsion,
since it isn't a block.
>
> > > This is *not* a temperament, or at least not one
> > > I'm interested in hearing, so 2 is not acting as a unison,
which is
> > > hardly a surprise.
> >
> > Of course it's a temperament. It's twintone/paultone/pajara.
>
> Is pajara the new official name? I'd like to get this settled.

OK, no more "paultone" or "twintone".

>As for this val, which defines only one of two required generator
>mappings being a temperament, that's only if you layer on some
>interpretation and perform the extra calculations to find a good
>choice for the second generator; taken by itself, it isn't one. It's
>telling us to send the octave to a unison, and 5 and 7 both to 1/9;
>it's only after you stick in half-octaves and send 7 to some tuning
>of 64/9 and 5 to a half-octave below that that pajara emerges. Read
>literally as a temperament, it sends 2 to 1 and 5 and 7 to 1/9, and
>I don't think that qualifies.
>
> > The octave is acting as a unison, but it's more complicated than
that. As
> > it has torsion, it's actually half an octave that's acting as a
commatic
> > unison vector.
>
> I would say it's acting as a generator, but if you make 2 a unison
it becomes a torsion element, since its square is an octave.

This, along with my message to Monzo this morning, seems to show the
very real problems with considering 2 a unison!

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > So why did you say "this was not a temperament"?
>
> Because a "temperament" which sends
>
> 1-9/8--5/4--4/3--3/2--5/3--15/8 to
>
> 1--9--1/9--1/3--3--1/27--1/3
>
> hardly seems worthy of the name.

You'd have to invoke "tritone-equivalence", which is clearly not a
recognized psychoacoustical phenomenon!

> > And isn't it true
> > that, if you took it out to, say, 10 notes per approximate
octave,
> > and tuned the octaves pure, it would _not_ be an octave-repeating
> > scale? This seems to be the point Graham is missing.
>
> We seem to be talking about different things--what is "it"? If you
> mean pajara, it's a temperament, not a scale.

What I thought we were talking about, and I thought you agreed, was
the fact that you derived temperaments originally in terms of two
generators, neither of which was guaranteed to be an octave, and then
came up with a different basis for the temperament such that the
octave was either a member of the basis or a power of a member of the
basis. For example, didn't you originally state that a form of
Blackjack which has a period of ~5/3 came out of your mechanism? I
took that to mean that if there were no such phenomenon as octave
equivalence, a Blackjack scale with a period of ~5/3 would better
exploit the consonances than the standard Blackjack scale.

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

> > The thing he said wasn't a temperament has no notes to an octave,
>
> No notes? He said it was generated by a fifth and a fifth-tritone --
> so it seems like it could have plenty of notes, up to an infinite
> number, in fact.

One is a basis for pajara/twintone. We have octaves in it, since
(15/14)^(-2) (3/2)^2 = 49/25 ~ 2. The other thing, which I declined to call a temperament, doesn't even represent octaves, so it depends on which thing you are talking about.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > > The thing he said wasn't a temperament has no notes to an
octave,
> >
> > No notes? He said it was generated by a fifth and a fifth-
tritone --
> > so it seems like it could have plenty of notes, up to an infinite
> > number, in fact.
>
> One is a basis for pajara/twintone. We have octaves in it, since
> (15/14)^(-2) (3/2)^2 = 49/25 ~ 2.

This is the thing I was trying to call Graham's attention to earlier.

>The other thing, which I declined >to call a temperament, doesn't
>even represent octaves, so it depends >on which thing you are
>talking about.

Well I confused the two things, which is completely my fault, but was
not helped by Graham's opinion that the thing you declined to call a
temperament was in fact pajara.

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

> What I thought we were talking about, and I thought you agreed, was
> the fact that you derived temperaments originally in terms of two
> generators, neither of which was guaranteed to be an octave, and then
> came up with a different basis for the temperament such that the
> octave was either a member of the basis or a power of a member of the
> basis.

That's a trick you can do with linear temperaments, and since this form is both useful and commonly used by Graham, it seems like a good one. This thread in my mind is partly about the point that the octave does not have a special status, in that you can do exactly the same for other intervals, such as a fifth.

For example, didn't you originally state that a form of
> Blackjack which has a period of ~5/3 came out of your mechanism?

That didn't come out of my mechanism, it came out of my misunderstanding of a comment you made.

I
> took that to mean that if there were no such phenomenon as octave
> equivalence, a Blackjack scale with a period of ~5/3 would better
> exploit the consonances than the standard Blackjack scale.

It seems like an interesting plan, at any rate. As I say, I can take any linear temperament such as miracle, and use anything I like (and 5/3 would seem to be a particularly good choice) as one of the generators, or at least as a power of one of the generators.

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

> Well I confused the two things, which is completely my fault, but was
> not helped by Graham's opinion that the thing you declined to call a
> temperament was in fact pajara.

If you ever get around to trying 222223 with a period of 3/2 in the 22-et, tell us about it.

> From: Graham Breed <graham@microtonal.co.uk>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, February 01, 2002 2:23 PM
> Subject: [tuning-math] Re: interval of equivalence, unison-vector, period
>
>
> "period". The thread actually started because Monz was
> confused about the difference between "unison vector",
> "equivalence interval" and "period". He was right to be
> because they are very similar.

Hi Graham,

No need to use the past tense here ... halfway thru today's
spate of messages on this subject, I'm more confused now
than ever.

:) and :( at the same time.

-monz

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> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, February 01, 2002 3:56 PM
> Subject: [tuning-math] Re: interval of equivalence, unison-vector, period
>
>
> > I would say it's acting as a generator, but if you make 2 a unison
> > it becomes a torsion element, since its square is an octave.
>
> This, along with my message to Monzo this morning, seems to show the
> very real problems with considering 2 a unison!

How is 2^2 an octave? By definition, it's simply 2.
Now you guys have really lost me.

(I'm glad I understand at least most of what was posted here
up to last September!)

-monz

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--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > Well I confused the two things, which is completely my fault, but
was
> > not helped by Graham's opinion that the thing you declined to
call a
> > temperament was in fact pajara.
>
> If you ever get around to trying 222223 with a period of 3/2 in the
>22-et, tell us about it.

I think I will!

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Friday, February 01, 2002 3:56 PM
> > Subject: [tuning-math] Re: interval of equivalence, unison-
vector, period
> >
> >
> > > I would say it's acting as a generator, but if you make 2 a
unison
> > > it becomes a torsion element, since its square is an octave.
> >
> > This, along with my message to Monzo this morning, seems to show
the
> > very real problems with considering 2 a unison!
>
>
> How is 2^2 an octave? By definition, it's simply 2.
> Now you guys have really lost me.

Dude, what exactly are you referring to? I thought this was amazingly
clear, but I guess I'm wrong!

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, February 01, 2002 11:27 PM
> Subject: [tuning-math] Re: interval of equivalence, unison-vector, period
>
>
> > > > I would say it's acting as a generator, but if you
> > > > make 2 a unison it becomes a torsion element, since
> > > > its square is an octave.
> > >
> > > This, along with my message to Monzo this morning,
> > > seems to show the very real problems with considering
> > > 2 a unison!
> >
> >
> > How is 2^2 an octave? By definition, it's simply 2.
> > Now you guys have really lost me.
>
> Dude, what exactly are you referring to? I thought this
> was amazingly clear, but I guess I'm wrong!

Oh, OK ... I think I get it.

If 2 = a unison, then 2^2 = an octave. Yes?

But I'm still confused, because if 2 is a unison, then
essentially for purposes of tuning math 2=1. So how does
squaring that get you to the octave?

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Friday, February 01, 2002 11:27 PM
> > Subject: [tuning-math] Re: interval of equivalence, unison-
vector, period
> >
> >
> > > > > I would say it's acting as a generator, but if you
> > > > > make 2 a unison it becomes a torsion element, since
> > > > > its square is an octave.
> > > >
> > > > This, along with my message to Monzo this morning,
> > > > seems to show the very real problems with considering
> > > > 2 a unison!
> > >
> > >
> > > How is 2^2 an octave? By definition, it's simply 2.
> > > Now you guys have really lost me.
> >
> > Dude, what exactly are you referring to? I thought this
> > was amazingly clear, but I guess I'm wrong!
>
>
> Oh, OK ... I think I get it.
>
> If 2 = a unison, then 2^2 = an octave. Yes?

Umm . . . not exactly. 2 is an octave, and 2^2 is a double octave.

monz wrote:

> Oh, OK ... I think I get it.
>
> If 2 = a unison, then 2^2 = an octave. Yes?

No, if 2 were a unison (which it isn't) 2^2 would also be a unison.

> But I'm still confused, because if 2 is a unison, then
> essentially for purposes of tuning math 2=1. So how does
> squaring that get you to the octave?

It doesn't. Except that if an octave is a unison, squaring gives you a
unison which is also an octave because an octave is a unison.

Graham

Me:
> > Well, can you think of a word for something that acts like a unison
> > vector but isn't? To cover the meanings of "unison vector",
> > "generator", "period" and "equivalence interval"?

Gene:
> What about kernel element? Of course, a period is a kernel element only
> if you make it one, by having a corresponding mapping, but that is the
> case here. The same would be true of an equivalence interval--if we
> send the half-octave to 1, it is a kernel element, but if we send 2 to
> 1 but not sqrt(2), then sqrt(2) is an element of order 2. One way we
> get a cyclic group of order 11, the other way of order 22.

Yes, that'll do. Although can the period and interval of equivalence both
be kernel elements?

From group theory, I think "identity" will do for things like unisons.
Adding or subtracting unisons gives you what you started, which is like
identities. So I'll go with "identity vector" for things like commatic
unison vectors. I'm not sure we need a word for things like chromatic
unison vectors. Will "generator" do instead?

Graham

genewardsmith wrote:

> Because a "temperament" which sends
>
> 1-9/8--5/4--4/3--3/2--5/3--15/8 to
>
> 1--9--1/9--1/3--3--1/27--1/3
>
> hardly seems worthy of the name. In any case, 2 isn't represented!

Why not? Why is 2 special?

Graham

paulerlich wrote:

> You'd have to invoke "tritone-equivalence", which is clearly not a
> recognized psychoacoustical phenomenon!

You certainly would. Psychoacoustics will have to make its own mind up.

Graham

genewardsmith wrote:

> Is pajara the new official name? I'd like to get this settled. As for
> this val, which defines only one of two required generator mappings
> being a temperament, that's only if you layer on some interpretation
> and perform the extra calculations to find a good choice for the second
> generator; taken by itself, it isn't one. It's telling us to send the
> octave to a unison, and 5 and 7 both to 1/9; it's only after you stick
> in half-octaves and send 7 to some tuning of 64/9 and 5 to a
> half-octave below that that pajara emerges. Read literally as a
> temperament, it sends 2 to 1 and 5 and 7 to 1/9, and I don't think that
> qualifies.

It defines one of the required mappings for an octave-specific linear
temperament. On it's own it is an equal temperament, but a somewhat
strange one. It could also be an octave-equivalent mapping for pajara.
An equal temperament with no steps to the octave is an octave-equivalent
linear temperament. I don't see what other sense making an octave an
identity vector could make.

Yes, it's telling us to map the octave to a unison which we could call
"imposing octave equivalence" or making the octave an identity vector. As
it has torsion, that makes a tritone the real identity vector.

Yes, it sends 2 to 1. They only differ by identity vectors. That's what
octave equivalent temperaments are like. Also 5, 7 and 1/9 differ only by
identity vectors. So it's a tritone equivalent linear temperament.

> > The octave is acting as a unison, but it's more complicated than
> > that. As it has torsion, it's actually half an octave that's acting
> > as a commatic unison vector.
>
> I would say it's acting as a generator, but if you make 2 a unison it
> becomes a torsion element, since its square is an octave.

It only acts as a generator if it isn't tempered out. In this case it is
being tempered out, so it must be an identity vector, which means it's
like a unison vector. Torsion is certainly involved, but I don't
understand what you're saying there.

> > (BTW, in an octave-equivalent system, half a unison is a half-octave
> > as well as a unison. This is obvious if you think of
> > octave-equivalent frequency space as a Hilbert space, and remember
> > that half the pitch is the same as the square root of the frequency.)
>
> You get a real Hilbert space if you allow anything of the form
> 3^e3 5^e5 ... which can have an infinite number of prime exponents so
> long as e3^2 + e5^2 + ... converges. Is this what you mean? The result
> isn't even guaranteed to be a real number, and I don't know what it
> would be good for.

3 and 5 don't enter into it. I mean octave equivalent frequency ratios
behave like the complex numbers with modulus 1. Partly because it's a
circular system, and also because nth roots have n values.

> > >>> i0 = i2-h2
> > >>> i0.basis
> > [0, 1, -2, -2]
> >
> > Hey, that's the same as g0 above!
>
> And which I think hardly counts as a temperament. As I said, it's not
> one I want to listen to.

There are all kinds of temperaments I wouldn't want to listen to. Why
single this one out for opprobrium?

Graham