Wedge Product

Hi there,

I thought I'd add a 3d calculation to the newbie
wedge product definition.

In the end it turned into more of an encyclopedia entry,
so I will have to trim this back down again, but I
thought that as it is, it could be of interest
to tuning group members who are learning about wedge
products - if you want to use any of this material in your
encyclopedia Monz then you are welcome :-)

So anyway here it is:

=term=
wedge product:
=definition=
One accessible example of its use is to find out the
area or volume of a periodicity block, and so its
number of notes.

The two dimensional case is the simplest - the
magnitude of the wedge product of [a b> and [c d>
is a*c - b*d

For example, the wedge product of 81/80 = [4,-1> and
128/125 = [ 0,-3> has magnitude:
abs( 4*-3 - -1*0 ) = abs(-12) = 12
which shows that there are twelve notes in the
scales constructed using these as unison vectors.

The idea can also be extended to 3D and higher
dimensions. It turns out that in 3D, the magnitude
of a/\b/\c is the volume of the parallelopiped
spanned by a, b and c. which will give us the number
of notes in the periodicity block.
When you wedge two vectors you get a bivector
which is a directed area - the parallelogram
area swept out from one to the next. When you
wedge three of them you get a trivector or
a directed volume.

To try this out, write out the unison vectors
using unit vectors for the directions of travel
of interest

So for instance:
Syntonic comma [* 4, -1> = 4u1 - u2
Diesis [* 0, -3> = -3u2
Septimal diesis [* 2, 0 1> = 2u1 + u3
(4u1 - u2)/\(-3u2)/\(2u1 + u3)
Now any vector wedged with itself is 0, so
you can ignore terms like u1/\u1 so multiplying out you get:
= (-12u1u2) (2u1 + u3)
Here I've written u1u2 for u1/\u2 to make it easier to read
= -12u1u2u3. So any scale built up using 3/2s, 5/4s
and 7/4s and with no pairs of notes in it at any
of these intervals will have twelve notes in it.
So septimal scales have 12 notes just like the
five limit ones.

Now Let's try out a more complex case, with the
lattice generated by 3/2, 5/4 and 13/8.
Using *s for the ignored prime factors, lets
uses these as our unison vectors:

tridecimal diesis 26/25 = [* 0, -2 * *, 1> = (-2u2 + u3)
syntonic comma 81/80 = [* 4, -1 * *, 0> = (4u1 - u2)
schisma [* 8, 1> = (8u1 + u2)
(8u1 + u2)/\(4u1 - u2)/\(-2u2 + u3)
=(-8u1u2 + 4u2u1)/\(-2u2 + u3)
= -8u1u2u3
So the periodicity block has 8 notes. To make it
just try and construct an 8 note scale with none
of those dieses or commas, here is an example:
1/1 9/8 5/4 4/3 3/2 8/5 13/8 16/9 2/1
There are many ways to do it, anyway
the use of the calculation is just that it shows
that any such scale has 8 notes so when you
get that far you are done.

For another ex:
Syntonic comma 81/80 = [* 4, -1 0> = -4u1 - u2
Septimal diesis 64/63 = [* 4, -1 0> = 4u1 - u2
tritonic diesis 50/49 = [* 0, 2 -2> = 2u2 - 2u3
(-4u1 - u2)/\(4u1 - u2)/\(2u2 - 2u3)
= (-4u1u2 -4u2u1)/\(2u2 - 2u3)
= 8u1u2u3 - 8u2u1u3
where I'm writing u1u2 for u1/\u2 etc. to make it easier to read
Now, since the bivector u1u2 is a unit of directed area,
if you reverse the order you change the direction so change
sign, so u1u2 = -u2u1. So u2u1u3 = -u1u2u3
So the answer is 8u1u2u3 + 8u1u2u3
= 16 u1u2u3
So this time all the scales have 16 notes.

I was surprised to find in my calculations that
nearly all the 3D periodicity blocks I made using
the 3, 5 and 7 , 11 or 13 limit dieses
and commas came out as 12 note scales.
I had to search for a while to find interesting
examples with other numbers of notes for
the calculations.

For instance if you use the syntonic comma
and the schisma instead of the syntonic comma
and the diesis as your unison vectors
you still get a twelve tone scale.
If you use the pythagorean comma and the
diesis though you get a 36 note scale
which makes sense as the pythagorean
comma and diesis are indpenedent
- one is 3 notes in the 5/4 direction
and the other is 12 notes in the 3/2
direction so you wuold expect the answer
to be 36 indeed.

Robert

hi Robert,

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

> Hi there,
>
> I thought I'd add a 3d calculation to the newbie
> wedge product definition.
>
> In the end it turned into more of an encyclopedia entry,
> so I will have to trim this back down again, but I
> thought that as it is, it could be of interest
> to tuning group members who are learning about wedge
> products - if you want to use any of this material in your
> encyclopedia Monz then you are welcome :-)

thanks, Robert. i'm sure that i will want to put
some or all of it into my Encyclopaedia entry.

i'll probably have more comments after i've read
it more carefully ... but i have to point out one
thing to you, as it's fairly important (at the
bottom of this post).

> I was surprised to find in my calculations that
> nearly all the 3D periodicity blocks I made using
> the 3, 5 and 7 , 11 or 13 limit dieses
> and commas came out as 12 note scales.
> I had to search for a while to find interesting
> examples with other numbers of notes for
> the calculations.

you could have gotten some 5-limit examples
right off of my bingo-card lattices:

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

> For instance if you use the syntonic comma
> and the schisma instead of the syntonic comma
> and the diesis as your unison vectors
> you still get a twelve tone scale.
> If you use the pythagorean comma and the
> diesis though you get a 36 note scale
> which makes sense as the pythagorean
> comma and diesis are indpenedent
> - one is 3 notes in the 5/4 direction
> and the other is 12 notes in the 3/2
> direction so you wuold expect the answer
> to be 36 indeed.

here, you've disclosed a tuning which has
"torsion".

the 2,3,5-monzos for the pythagorean-comma
and diesis are:

[-19 12, 0 > = 531441 / 524288 pythagorean-comma
[ 7 0, -3 > = 128 / 125 diesis

the resulting bimonzo is [[ 84 -57 36 >> .

converting that to a val does indeed give < 36 57 84 ],
which is a prime-mapping valid for 36-ET.

however, note that all three of those numbers have
a common divisor, which means that this is not a
periodicity-block by rather a torsional-block.

this means that the third unison-vector, which you
get by vector-adding the monzos (or multiplying the
ratios) of the pythagorean-comma and the diesis, is
(81/80)^3 :

[-19 12, 0 >
+ [ 7 0, -3 >
----------------
[-12 12, -3 >

obviously all three elements of this monzo can be divided
by 3, which gives you [ -4 4, 1 > ... that should look
familiar as the syntonic-comma.

so guess what you get if you divide all three elements
of the val by 3?

yep, < 12 19 28 ] ... good old 12-ET!

so in other words, the wedge-product is giving 36-ET
as 3 bike-chains of 12-ET, each a syntonic-comma apart.

this is precisely a description of Groven's tuning,
which, however, i derived a different way:

http://tonalsoft.com/monzo/groven/groven.htm

-monz

Hi Monz,

thanks that's a great help.

Now I understand what a bimonzo is,
bike-chain, torsion, and torsional
block - so many terms inter-related.

Maybe I can begin to understand val better too.

When you say:

> the 2,3,5-monzos for the pythagorean-comma
> and diesis are:

> [-19 12, 0 > = 531441 / 524288 pythagorean-comma
> [ 7 0, -3 > = 128 / 125 diesis

> the resulting bimonzo is [[ 84 -57 36 >> .

okay I see that, just the wedge product
(-19e1 + 12 e2)/\(7 e1 - -3 e3)
= 84 e21 +57 e13 -36 e23
= 84 e21 -57 e31 + 36 e32

where I now remember the convention is to
do them with the highest number first in
each of e21 etc, and to use e for the unit
vectors (I'm used to u) and e12 for what I
did before as u1u2.

> converting that to a val does indeed give < 36 57 84 ],
> which is a prime-mapping valid for 36-ET.

I found the 36 more simply by just leaving
out the exponent of 2 in the calculation.
But can see - if you leave out the
exponent of 2 (our e1 direction) then you just need the
e32 term. similarly leave out 3 assuming
tritave equivalence and you get the
e31 and leaving out 5 you get the
e21 term so the bimonzo does
give the vals as Gene described in
his post.

I'll have another go at understanding
val now

I can understand the notion
as explained in your encyclopedia
of an addition preserving
homomorphism from the lattice
exponents to the integers

- one example of a val is
the valuation that just extracts
the exponent of e.g. 3 in a ratio.

Another example could be the sum of
the exponents of 2 and 3 - because
if you multiply together two
ratios you add the exponents.
so multiplication of ratios corresponds
to addition of vectors in the lattice.

Indeed a sum or linear combination
of any number of the exponents
will also be a val.

E.g. 10*exponent of 2
+ 3 * exponent of 3
+ 11 * exponent of 5
would also be a val.

You would then have in this particular case
val(15/8) = val [-3 1, 1>
= -30 + 3 + 11 = -16

But then, why is the list of periodicity areas
referred to as a val?

Why do you call
< 36 57 84 ],
a val - what is the homomorphism
from the lattice to the integers
that it represents?

One would think on the face of it that
it should be the homomorphism that
takes e.g. 15/8 to
< 36 57 84 ], [-3 1, 1>
= -3*36 + 57 + 84 = 33
but I can't yet make any sense of that.

I feel that I'm just missing some
little bit of the puzzle to understand
the concept and then it will probably click
as is so often the case in maths
:-).

Robert

hi Robert (and Gene, Herman, Graham),

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

> Hi Monz,
>
> thanks that's a great help.
>
> Now I understand what a bimonzo is,
> bike-chain, torsion, and torsional
> block - so many terms inter-related.
>
> Maybe I can begin to understand val better too.

please be aware that i'm still struggling to understand
all of this too! i'm learning it right along with you
... you understand more of the algebra, and i have a
good intuitive grasp of the geometry.

> When you say:
>
> > the 2,3,5-monzos for the pythagorean-comma
> > and diesis are:
>
> > [-19 12, 0 > = 531441 / 524288 pythagorean-comma
> > [ 7 0, -3 > = 128 / 125 diesis
>
> > the resulting bimonzo is [[ 84 -57 36 >> .
>
> okay I see that, just the wedge product
> (-19e1 + 12 e2)/\(7 e1 - -3 e3)
> = 84 e21 +57 e13 -36 e23
> = 84 e21 -57 e31 + 36 e32
>
> where I now remember the convention is to
> do them with the highest number first in
> each of e21 etc, and to use e for the unit
> vectors (I'm used to u) and e12 for what I
> did before as u1u2.
>
> > converting that to a val does indeed give < 36 57 84 ],
> > which is a prime-mapping valid for 36-ET.
>
> I found the 36 more simply by just leaving
> out the exponent of 2 in the calculation.
> But can see - if you leave out the
> exponent of 2 (our e1 direction) then you just need the
> e32 term. similarly leave out 3 assuming
> tritave equivalence and you get the
> e31 and leaving out 5 you get the
> e21 term so the bimonzo does
> give the vals as Gene described in
> his post.

the val < 36 57 84 ] gives the "conventional"
prime-mapping for 36edo.

2:1 is 36 degrees of 36edo
3:1 is 57 degrees of 36edo
5:1 is 84 degrees of 36edo

36edo has no other likely candidates for 2:1 or 3:1,
but it is possible to map 5:1 to 83 degrees, which
would give a rather narrow "major-3rd" of ~366.67 cents.

> I'll have another go at understanding
> val now
>
> I can understand the notion
> as explained in your encyclopedia
> of an addition preserving
> homomorphism from the lattice
> exponents to the integers
>
> - one example of a val is
> the valuation that just extracts
> the exponent of e.g. 3 in a ratio.
>
> Another example could be the sum of
> the exponents of 2 and 3 - because
> if you multiply together two
> ratios you add the exponents.
> so multiplication of ratios corresponds
> to addition of vectors in the lattice.
>
> Indeed a sum or linear combination
> of any number of the exponents
> will also be a val.
>
> E.g. 10*exponent of 2
> + 3 * exponent of 3
> + 11 * exponent of 5
> would also be a val.
>
> You would then have in this particular case
> val(15/8) = val [-3 1, 1>
> = -30 + 3 + 11 = -16

you'll have to look to Gene to confirm that
what you're doing there is correct and/or
makes and sense ... i don't know.

> But then, why is the list of periodicity areas
> referred to as a val?

again, Gene is the one to explain that.

> Why do you call
> < 36 57 84 ],
> a val - what is the homomorphism
> from the lattice to the integers
> that it represents?
>
> One would think on the face of it that
> it should be the homomorphism that
> takes e.g. 15/8 to
> < 36 57 84 ], [-3 1, 1>
> = -3*36 + 57 + 84 = 33
> but I can't yet make any sense of that.

see my explanation above of the prime-mapping
to 36edo.

> I feel that I'm just missing some
> little bit of the puzzle to understand
> the concept and then it will probably click
> as is so often the case in maths
> :-).
>
> Robert

indeed, by asking each other questions about this
stuff we both learn.

-monz

HI Monz,

> the val < 36 57 84 ] gives the "conventional"
> prime-mapping for 36edo.

> 2:1 is 36 degrees of 36edo
> 3:1 is 57 degrees of 36edo
> 5:1 is 84 degrees of 36edo

> 36edo has no other likely candidates for 2:1 or 3:1,
> but it is possible to map 5:1 to 83 degrees, which
> would give a rather narrow "major-3rd" of ~366.67 cents.

...

> > One would think on the face of it that
> > it should be the homomorphism that
> > takes e.g. 15/8 to
> > < 36 57 84 ], [-3 1, 1>
> > = -3*36 + 57 + 84 = 33
> > but I can't yet make any sense of that.

see my explanation above of the prime-mapping
to 36edo.

Oh right I see.

So if the 2/1 is 36 degrees, and the 3/1
is 57 degrees and 5/1 is 84 then
the 33 I got there is just the
number of degrees for 15/8 in the scale.
Divide by 3 and you get the 12-et
position which is 1 as expected.

I.e. if you map 3/1 to 57 degrees of
36-et and 5/1 to 84 degrees and then
stack those intervals consistently
then you will get to degree 33
as the interval for 15/8.
The val then tells you how to
do that mapping.

Trying with a few other ratios
e.g. where is 5/4 in a twelve tone scale:
<12 19 28] [-2 0, 1>
= -24 + 28 = 4
so 5/4 is the fourth degree.

Or 16/15

<12 19 28] [4 -1, -1>
= 48 - 19 - 28 = 1
:-).

Or, 25/16 which we know is G#
<12, 19 28] [-4 0, 2>
= -48 + 56 = 8
and indeed G# is the 8th degree
C C# D D# E F F# G G#
0 1 2 3 4 5 6 7 8

So, I'm convinced - but how does
it work?

Surely it also applies to the original
pitches as well as the et approximation
- if you actually have
a 15/8 in your scale and arrange
the 16 notes in ascending order,
and you have your periodicity
block as a rectangle rather than
some irregular shape,
presumably this shows that
the 15/8 is the 33rd note inthe
scale, so the val applied
to the original periodicity block pitches
again gives their scale degrees
which is quite impressive given
that it is a 2D region in the lattice
and you wouldn't easily be able
to see what the scale degrees are.

You wouldn't expect that to work if
your block was irregular in shape
of course - if you replaced e.g.
5/4 by a 5/4 moved up by a dozen
diesis shifts or whatever it will
surely put the notes of the scale
out of order, so it is something
to do with the original
periodicity block representations
of the equivalence classes
under the unison vectors.

That's all making sense except
that I don't see why it works.

If I understood that then all the
rest would become perfectly clear.
Is there some theorem that
gets invoked to prove this?

Let's see, look at the lattice:

16/15.....8/5.....6/5.....9/5...(27/20)
\ / \ / \ / \ / \
4/3.....1/1.....3/2.....9/8...(27/16)
\ / \ / \ / \ / \
5/3.....5/4.....15/8....45/32..(135/64)
\ / \ / \ / \ /
(25/24...25/16...75/64...225/128)

Yes of course, I see it now.
The periodicity block for octave equivalence
has 12 values, and spans 1 to 2/1 so of course
2/1 is 12 degrees.
The periodicity block for tritave equivalence
has 19 notes as we calculated, and if you
just use the parallelogram in its original
shape again all the notes are between 1/1
and 3/1 so 3/1 is 19 degrees under tritave
equivalence.

The 3/1 block we know is
1/1 2/1 4/3 8/3 16/9 32/27 64/27 128/81 ... 2^19/3^5

There - one could also go in the 5 direction alternatively
and repeat the block with a 19 unit repeat vector
vertically instead of horizonatally.

Then some of those are already in the lattice
1/1 9/8 4/3 3/2 2/1 9/4 16/3
Others will be close to lattice points.
We can move any of them up or down by a syntonic comma
and I suppose the one thing left to be shown
is that each one is within a syntonic comma
(in this case) of one of the numbers in the
octave equivalence periodicity block.

Given that, then moving pitches by a syntonic
comma won't change the number of notes
between the 1/1 and the 3/1,
and we will just end up with the five
limit scale all the way up to 3/1.

Similarly for 5/1, moving notes by a syntonic
comma we should get to the original.

So - I can understand at least intuitively
how it works and can see how this could
be worked up into a proof.

You might think that the result depends
on the size of the commas and dieses
used. If you have a 12 tone periodicity
block for instance generated using
a unison vector of more than 100 cents
then it wouldn't be expected to work.
But normally if the unison vector
gets larger, the periodicity block
gets smaller.

E.g. with

E.g. try 25/24 as one unison vector and
45/32 as the other one.

Monzos:
[-3 -1, 2>
[-5 2, 1>

Bimonzo
(-3 e1 - e2 + 2 e3) /\ (-5 e1 + 2 e2 + e3)
=
-6 e12 - 3 e13 + 5 e21 - e23 - 10 e31 + 4 e32
= 11 e21 - 7 e31 - 5 e32
Bimonzo:
[11 -7, -5>
So that generates our pentatonic scale surely.

val (reverse the order of the bimonzo numerals)
<-5 -7, 11]
and obviously you take the absolute values
for the number of notes:
<5 7, 11]

The 5/1 is then 11 notes in the pentatonic scale,
the 3/1 is 7 notes and the 2/1 of cours 5

So applying it to e.g.
8/5 then
<5 7, 11] [3 0, -1>
= 15 - 11 = 4
as expected.

It's all making sense now.
Not what I'd call a proof
but can see how one could
prove this, at least, seems
reasonably feasible that one
could do it - the big gap
is to show that if you have a comma
defining a periodicity block then
the comma is smaller than
the smallest step in the
resulting scale, so that when
you shift the notes by the commas
then they don't get out of order.

If that could be proved I think
most of the rest would be
details to fill in. I can't
see how to prove it right
now but if one could prove
that, it would be done.

Gene must have done that, or maybe
there is some nice result
one can use.

:-)

I'd be interested in the mathematical
details of how it gets proved
- perhaps on tuning-math if it is
rather technical.

> indeed, by asking each other questions about this
> stuff we both learn.

Yes that's how it goes indeed :-).
thanks,

Robert

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

> =term=
> wedge product:
> =definition=
> One accessible example of its use is to find out the
> area or volume of a periodicity block, and so its
> number of notes.

I like this, but why not say "basic example" or "simple example"?

> The two dimensional case is the simplest - the
> magnitude of the wedge product of [a b> and [c d>
> is a*c - b*d

You need your wild cards in there, don't you? In fact, you could do it
all with wild cards: det([|* 4 -1>, |* 0 -3>]) = -12,
det([|-4 * -1>, |7 * -3>]) = 19, det([|-4 4 *>, |7 0 *>]) =-28; which
are of course minors. This seems to be what you do below, you may as
well start out with it.

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

> so in other words, the wedge-product is giving 36-ET
> as 3 bike-chains of 12-ET, each a syntonic-comma apart.

It's also saying (81/80)^3 ~ 1, which is where the torsion comes in,
so calling it Groven or any other precise tuning doesn't seem really
legit.

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

> One would think on the face of it that
> it should be the homomorphism that
> takes e.g. 15/8 to
> < 36 57 84 ], [-3 1, 1>
> = -3*36 + 57 + 84 = 33
> but I can't yet make any sense of that.

That's exactly what it is; the homomorphism need not be onto. Why is
that not sensible?

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

> Surely it also applies to the original
> pitches as well as the et approximation
> - if you actually have
> a 15/8 in your scale and arrange
> the 16 notes in ascending order,
> and you have your periodicity
> block as a rectangle rather than
> some irregular shape,
> presumably this shows that
> the 15/8 is the 33rd note inthe
> scale, so the val applied
> to the original periodicity block pitches
> again gives their scale degrees
> which is quite impressive given
> that it is a 2D region in the lattice
> and you wouldn't easily be able
> to see what the scale degrees are.

This is where the business of "epimorphic" scales comes in; when Scala
says something is "JI epimorphic" and gives a val it means the val
mapping order corresponds to the scale order.

> If that could be proved I think
> most of the rest would be
> details to fill in. I can't
> see how to prove it right
> now but if one could prove
> that, it would be done.
>
> Gene must have done that, or maybe
> there is some nice result
> one can use.

This goes back to the discussion of Paul's "Hypothesis", which was
indeed a major topic at one point.

> > If that could be proved I think
> > most of the rest would be
> > details to fill in. I can't
> > see how to prove it right
> > now but if one could prove
> > that, it would be done.
> >
> > Gene must have done that, or maybe
> > there is some nice result
> > one can use.
>
> This goes back to the discussion of Paul's "Hypothesis",
> which was indeed a major topic at one point.

How does epimorphism relate to the Hypothesis?

-Carl

hi Robert,

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

> HI Monz,
>
> > the val < 36 57 84 ] gives the "conventional"
> > prime-mapping for 36edo.
>
> > 2:1 is 36 degrees of 36edo
> > 3:1 is 57 degrees of 36edo
> > 5:1 is 84 degrees of 36edo
>
> > > One would think on the face of it that
> > > it should be the homomorphism that
> > > takes e.g. 15/8 to
> > > < 36 57 84 ], [-3 1, 1>
> > > = -3*36 + 57 + 84 = 33
> > > but I can't yet make any sense of that.
>
> see my explanation above of the prime-mapping
> to 36edo.
>
> Oh right I see.
>
> So if the 2/1 is 36 degrees, and the 3/1
> is 57 degrees and 5/1 is 84 then
> the 33 I got there is just the
> number of degrees for 15/8 in the scale.
> Divide by 3 and you get the 12-et
> position which is 1 as expected.

looks like you made a typo there ... of course
the 15/8 ratio maps to 11 degrees of 12edo, not 1.

> I.e. if you map 3/1 to 57 degrees of
> 36-et and 5/1 to 84 degrees and then
> stack those intervals consistently
> then you will get to degree 33
> as the interval for 15/8.
> The val then tells you how to
> do that mapping.
>
> Trying with a few other ratios
> e.g. where is 5/4 in a twelve tone scale:
> <12 19 28] [-2 0, 1>
> = -24 + 28 = 4
> so 5/4 is the fourth degree.
>
> Or 16/15
>
> <12 19 28] [4 -1, -1>
> = 48 - 19 - 28 = 1
> :-).
>
> Or, 25/16 which we know is G#
> <12, 19 28] [-4 0, 2>
> = -48 + 56 = 8
> and indeed G# is the 8th degree
> C C# D D# E F F# G G#
> 0 1 2 3 4 5 6 7 8

yes, that's all correct, as you could see.

> So, I'm convinced - but how does
> it work?

ah ... i think that's the $64000 question.

i haven't asked him about it, but my guess
is that Gene saw that this stuff worked out
algebraically, and *now* he and the rest of
us are all trying to figure out *why*.

... or more likely, Gene already understands
all of it and the rest of us don't.

anyway, i think now you can see why vals and
wedgies are useful. the purpose of temperament
is to emulate JI in some way, but trading off
accuracy of intonation for lower cardinality,
more harmonic senses, etc.

the vals and wedgies show us how the temperaments
and JI intersect, and they also provide compact
sets of numbers that uniquely identify temperaments
and even whole temperament families, and help us
to see the relationships between different tunings.

> Surely it also applies to the original
> pitches as well as the et approximation
> - if you actually have
> a 15/8 in your scale and arrange
> the 16 notes in ascending order,
> and you have your periodicity
> block as a rectangle rather than
> some irregular shape,
> presumably this shows that
> the 15/8 is the 33rd note inthe
> scale, so the val applied
> to the original periodicity block pitches
> again gives their scale degrees
> which is quite impressive given
> that it is a 2D region in the lattice
> and you wouldn't easily be able
> to see what the scale degrees are.

yes, absolutely. the first periodicity-block
theory was formulated by Fokker, and it was
all in connection with JI ... even tho he too
mapped 7-limit JI to 31-ET, he explained everything
in terms of JI.

the area of any 2-D segment of the lattice
(usually a paralellogram) encloses all of
the ratios formed by any two prime-factors
within the boundaries defined by the unison-vectors.
i think you grasp that.

> You wouldn't expect that to work if
> your block was irregular in shape
> of course - if you replaced e.g.
> 5/4 by a 5/4 moved up by a dozen
> diesis shifts or whatever it will
> surely put the notes of the scale
> out of order, so it is something
> to do with the original
> periodicity block representations
> of the equivalence classes
> under the unison vectors.

transposing any JI ratio by one instance
of a unison-vector will change the shape
of the periodicity-block, but only slightly,
and may be desirable if one wants to have,
for example, a 5-limit hexagon rather than
a 5-limit parallelogram.

but i think it's not likely that one would
transpose a note by more one instance of a
unison-vector. so what you say is true,
but i don't think it's very meaningful.

anyway ... there's the business about the
TM-reduced lattice basis, which puts all the
JI pitches as compactly as possible near 1/1.

(BTW, the lattices on my webpage about this
are wrong, and should only be showing the
unison-vectors and not all the other pitches too.
i'll have to redo them when i have time.)

> <big snip>
>
> It's all making sense now.
> Not what I'd call a proof
> but can see how one could
> prove this, at least, seems
> reasonably feasible that one
> could do it - the big gap
> is to show that if you have a comma
> defining a periodicity block then
> the comma is smaller than
> the smallest step in the
> resulting scale, so that when
> you shift the notes by the commas
> then they don't get out of order.
>
> If that could be proved I think
> most of the rest would be
> details to fill in. I can't
> see how to prove it right
> now but if one could prove
> that, it would be done.
>
> Gene must have done that, or maybe
> there is some nice result
> one can use.
>
> :-)
>
> I'd be interested in the mathematical
> details of how it gets proved
> - perhaps on tuning-math if it is
> rather technical.

i'm hoping to see it there too.

it may even already be in the archives!
Gene wrote a lot about periodicity-blocks
when he first found out about them 2 years ago,
because he had apparently already formulated
similar concepts in his tuning theories
without being aware of Fokker's work.

-monz

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

> > This goes back to the discussion of Paul's "Hypothesis",
> > which was indeed a major topic at one point.
>
> How does epimorphism relate to the Hypothesis?

You can prove it under the condition that the block is epimorphic.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > so in other words, the wedge-product is giving 36-ET
> > as 3 bike-chains of 12-ET, each a syntonic-comma apart.
>
> It's also saying (81/80)^3 ~ 1,

what exactly does that mean?

(if the answer gets technical, we can go to tuning-math)

> which is where the torsion comes in,
> so calling it Groven or any other precise tuning
> doesn't seem really legit.

Groven created a 36-tone tuning which had three chains
of tempered 3:2s a syntonic comma apart from each other.

http://tonalsoft.com/monzo/groven/groven.htm

as far as the pitch-height of the notes in Groven's
tuning, it's not 36edo ... the smallest cardinality
which represents it well is 53edo, so it's like a
36-tone subset of 53edo.

re: "legit" ... what *is* the criteria for "legitimacy"?

i never understood the "illigetimacy" of torsion ...
if a composer likes to play with comma shifts, and
so therefore wants a temperament which gives a nice
approximation to the syntonic-comma, what's illegitimate
about that?

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > > so in other words, the wedge-product is giving 36-ET
> > > as 3 bike-chains of 12-ET, each a syntonic-comma apart.
> >
> > It's also saying (81/80)^3 ~ 1,
>
>
> what exactly does that mean?

If I temper out 128/125 and the Pythagorean comma, (81/80)^3 =
531441/512000 is a comma of the system, and is tempered out also, but
81/80 is not a product of 128/125 and 531441/524288 unless you allow
fractional exponents. Hence, has a cube which is equivalent to a
unison, and hence is a comma, without itself being equivalent to a
unison; this is torsion.

It seems to make more sense to always include the commas you get with
fractional exponents, which is one of the things working with wedgies
accomplishes.

> i never understood the "illigetimacy" of torsion ...
> if a composer likes to play with comma shifts, and
> so therefore wants a temperament which gives a nice
> approximation to the syntonic-comma, what's illegitimate
> about that?

It's fine, but if he wants the cube of a number greater than one to
equal one, he's in trouble; you can't very well have a comma equal to
(sqrt(-3)-1)/2, after all.

Hi Gene

> > One would think on the face of it that
> > it should be the homomorphism that
> > takes e.g. 15/8 to
> > < 36 57 84 ], [-3 1, 1>
> > = -3*36 + 57 + 84 = 33
> but I can't yet make any sense of that.

> That's exactly what it is; the homomorphism need not be onto. Why is
> that not sensible?

Yes I understood that. All that was clear long ago.

It's just what the 33 is that I couldn't
make sense of, what the number was the number
of.

When Monz explained that
it is a scale degree then everything began to
fall into place. Probably that got explained
somewhere too - now that I know what everything
means then the encyclopedia entries on this
will all be a much easier read. The thing
is when you don't understand the basic
idea behind it all then you can't see
the connections between scale degrees,
and numbers of notes in periodicity blocks
assuming octave tritave, and pentave
equivalence. It's just that basic
idea that the number of notes in the
periodicity block gives the number of notes
spanned by the interval of equivalence
used. Once you see that then all the
various interconnected concepts come
together and you understand it all.
If you haven't seen that connection,
then it is just like a whole lot of
pieces of a jigsaw puzzle that don't
seem to fit any of them with any of the
others, and it doesn't make sense.

I expect you've had the same
experience, well every mathematician
loves a puzzle, and this is just
like trying a puzzle and having
the pleasure of solving it (and
I've enjoyed this one too :-) )

So anyway, somehow one needs
a newbie style explanation of that connection
which shouldn't be too hard to do - and
that needs to go first, rather than last.
Then I think visitors to the encyclopedia
will find it all a much easier read.
Everyone likes a puzzle true, but
this is too hard a puzzle because
the concepts are so many and so abstract
that everyone except a mathematician
will prohably give up.

(whether by nature or trained - I'm not
suggesting you have to be a trained mathematician
to practice maths any more than you need to
be a trained composer to compose, but if
trained you may have rather more kind of
familiarity with the length of time
needed to come to terms with abstract
concepts and see what they mean)

So anyway to make this more user friendly,
one needs to say first of all in the entry
on vals that the
val <12 19 28] gives the number of
scale degrees spanned by 2/1, 3/1 and
5/1 in the periodicity block under
octave equivalence. Before you
even explain how it is calculated.

They can see that too, no need to be
technical yet, just
do it empirically first, that
3/1 is the 19th degree and
5/1 the 28th which isn't hard to see.

Then, maybe immediately, still before you
explain how they get calculated,
you then give some examples
to show how you can find the scale
degree of e.g. 16/15 etc.

as 12 * exponent of 2
+ 19 * exponent of 3
+ 28 * exponent of 5

That's kind of intriguing and
leads the newbie reader on - this
topic is worth exploring...

Then after that, can explain how
to find the degrees of 3 and 5
from the periodicity blocks
by analogy with the degree of 2
which should be clear by then.

Then they need the tool of wedge
products to find the numbers
for themselves, so you bring that in or
send them off to that part of the
encyclopedia to brush up on it.

Then one can introduce consistency.
You want the number of scale
degrees spanned by the product of two
ratios to be the sum of the number of degrees
spanned by each one - so that
then gives the motivation for
the val to be a homomorphism.

Maybe one doesn't need to give
the rules in detail algebraically.
Just say that you want the
degree numbers of stacked intervals
to add up consistently - it is clear that
the number of degrees spanned by 1/1
has to be 0 and that if r spans n degrees
then 1/r spans -n degrees. You
could even say that now that it is
understood what a val is - it
is the matter of introducing the algebraic
properties of the val before you know yet
what it is used for that makes it a
particularly hard read for a non mathematician.

So you know it is a homomorphism, but don't
have to use that word (maybe in a techy
footnote at the end if you want)
any more than you need to say that
the rationals form a field
when teaching someone about addition
or multiplication (unless they
are maths students at university
of course).

I'm not saying this to you personally
btw - you've done a maths of tuning
for mathematicans that's fine
(would probably be fine for me
too if I wasn't quite so rusty
and slow at reading maths these
days - give me three months
solid maths research and I'd
be able to read stuff like that
again much more easily).

Anyway it all falls into place now.

I understand your point of view well
since after all I trained for years
as a mathematician too, and I nearly
went on to do postgraduate maths
immediately after my first degree
- if I'd done that, I'd have found this
back to front approach really hard
to follow too.

I think it was studying philosophy
in my case that first made
clear the poetential for explaining
things in natural language.
Also philosophy is kind
of back to front too - once
you understand it it has
clarity too, great clarity
at times, but no axiomatics,
well at least no axiomatics
that all philosophers agree on
(except perhaps the use of pure
logic itself, which is rarely challenged in
philosophy).

Robert

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

> So anyway to make this more user friendly,
> one needs to say first of all in the entry
> on vals that the
> val <12 19 28] gives the number of
> scale degrees spanned by 2/1, 3/1 and
> 5/1 in the periodicity block under
> octave equivalence. Before you
> even explain how it is calculated.

You should also say <12 19 28| gives the number of steps of 12-equal
to get to the approximations of 2, 3, and 5; and point out that the
two statements are closely related.

> Also philosophy is kind
> of back to front too - once
> you understand it it has
> clarity too, great clarity
> at times, but no axiomatics,
> well at least no axiomatics
> that all philosophers agree on
> (except perhaps the use of pure
> logic itself, which is rarely challenged in
> philosophy).

What about Quine?

>> > so in other words, the wedge-product is giving 36-ET
>> > as 3 bike-chains of 12-ET, each a syntonic-comma apart.
>>
>> It's also saying (81/80)^3 ~ 1,
>
>what exactly does that mean?

It looks like this condition would also be met by a non-
torsional temperament.

>Groven created a 36-tone tuning which had three chains
>of tempered 3:2s a syntonic comma apart from each other.
>
>http://tonalsoft.com/monzo/groven/groven.htm
>
>as far as the pitch-height of the notes in Groven's
>tuning, it's not 36edo ... the smallest cardinality
>which represents it well is 53edo, so it's like a
>36-tone subset of 53edo.

It's schismic. At least I seem to remember reading
something about the -8 mapping for the major 3rd.

-Carl

>> > > so in other words, the wedge-product is giving 36-ET
>> > > as 3 bike-chains of 12-ET, each a syntonic-comma apart.
>> >
>> > It's also saying (81/80)^3 ~ 1,
>>
>>
>> what exactly does that mean?
>
>If I temper out 128/125 and the Pythagorean comma, (81/80)^3 =
>531441/512000 is a comma of the system, and is tempered out also, but
>81/80 is not a product of 128/125 and 531441/524288 unless you allow
>fractional exponents. Hence, has a cube which is equivalent to a
>unison, and hence is a comma, without itself being equivalent to a
>unison; this is torsion.

Sorry, this is too terse to parse. "Hence, has a cube"?
What's a cube?

It almost sounds like you're saying: 81/80 is represented by a
nonzero number of steps in the temperament, but some power of
it is represented by the unison.

I thought the problem of torsional tunings was that no modulation
path in "JI" covered them.

>> i never understood the "illigetimacy" of torsion ...
>> if a composer likes to play with comma shifts, and
>> so therefore wants a temperament which gives a nice
>> approximation to the syntonic-comma, what's illegitimate
>> about that?
>
>It's fine, but if he wants the cube of a number greater than one to
>equal one, he's in trouble; you can't very well have a comma equal to
>(sqrt(-3)-1)/2, after all.

Here are concepts I'm not familiar with again, seemingly new with
this post: "cube" and "comma" (usually comma means nothing more than
"small interval").

-Carl

Carl Lumma schrieb:
>>>>>so in other words, the wedge-product is giving 36-ET
>>>>>as 3 bike-chains of 12-ET, each a syntonic-comma apart.
>>>>
>>>>It's also saying (81/80)^3 ~ 1,
>>>
>>>
>>>what exactly does that mean?
>>
>>If I temper out 128/125 and the Pythagorean comma, (81/80)^3 =
>>531441/512000 is a comma of the system, and is tempered out also, but
>>81/80 is not a product of 128/125 and 531441/524288 unless you allow
>>fractional exponents. Hence, has a cube which is equivalent to a
>>unison, and hence is a comma, without itself being equivalent to a
>>unison; this is torsion. > > > Sorry, this is too terse to parse. "Hence, has a cube"?
> What's a cube?

glad to be able to help:

power of 2: "square"
power of 3: "cube"

but what the heck does it mean (in a different post) that something is "onto"? JARGON!!!!!!!!!!!!

klaus

p.s.
sorry for not participating in the anti-jargon discussion i set off the other week. something blew up in my system (the computer, i mean) when i tried to respond to that and monz's robert-johnson-mail (hyphens for monz :O) ). but if this series of explanations is a consequence of this discussion, i feel justified and satisfied.

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

> >If I temper out 128/125 and the Pythagorean comma, (81/80)^3 =
> >531441/512000 is a comma of the system, and is tempered out also, but
> >81/80 is not a product of 128/125 and 531441/524288 unless you allow
> >fractional exponents. Hence, has a cube which is equivalent to a
> >unison, and hence is a comma, without itself being equivalent to a
> >unison; this is torsion.

> Sorry, this is too terse to parse. "Hence, has a cube"?
> What's a cube?

(81/80)^3 is a cube.

> It almost sounds like you're saying: 81/80 is represented by a
> nonzero number of steps in the temperament, but some power of
> it is represented by the unison.

That's one way of describing what is going on, which is why I think
getting rid of torsion is a good idea.

> I thought the problem of torsional tunings was that no modulation
> path in "JI" covered them.

The problem with them is that we don't know what they are, because we
can't interpret the idea that, for instance, 81/80 is not a unison but
its cube is in a way which makes much sense.

> >> i never understood the "illigetimacy" of torsion ...
> >> if a composer likes to play with comma shifts, and
> >> so therefore wants a temperament which gives a nice
> >> approximation to the syntonic-comma, what's illegitimate
> >> about that?
> >
> >It's fine, but if he wants the cube of a number greater than one to
> >equal one, he's in trouble; you can't very well have a comma equal to
> >(sqrt(-3)-1)/2, after all.
>
> Here are concepts I'm not familiar with again, seemingly new with
> this post: "cube" and "comma" (usually comma means nothing more than
> "small interval").

Generally it means a small interval *tempered out by a temperament*.
Surely cube is familiar, so what are you saying?

--- In tuning@yahoogroups.com, klaus schmirler <KSchmir@z...> wrote:

> but what the heck does it mean (in a different post) that
> something is "onto"? JARGON!!!!!!!!!!!!

I rejected "epimorphism" and "surjective homomorphism" precisely to
avoid complaints about jargon. After a certain point, the complaint
isn't that it's jargon, it's that its math.

Gene Ward Smith wrote:

> I rejected "epimorphism" and "surjective homomorphism" precisely to
> avoid complaints about jargon. After a certain point, the complaint
> isn't that it's jargon, it's that its math.

Yes, which is why there's a "tuning-math" list to take the math away from this one. But the point's hardly been reached this time, has it? "Onto" is as much a jargon term as "surjective". The only difference is that a Google search for "onto" is utterly useless in this context.

Graham

Hi Gene,

> > One accessible example of its use is to find out the
> > area or volume of a periodicity block, and so its
> > number of notes.

> I like this, but why not say "basic example" or "simple example"?

Because though mathematically it is maybe simple because
it uses concepts that aren't so very abstract, and
the calculations are often a few lines, the newbie
idea of a simple calculation is one that is easy to do
and calculating the volume of even a 3D periodicity
block by hand isn't particularly easy - I know
nothing compared with proofs that run to many
hundreds of pages or whatever, but it is complex
enough so that one can easily make slips when doing
the calculations by hand - that is newbie complicated.

But anyway, probably a better idea to just say "one example"
and not qualify it in any way at all.

> > The two dimensional case is the simplest - the
> > magnitude of the wedge product of [a b> and [c d>
> > is a*c - b*d

> You need your wild cards in there, don't you? In fact, you could do it
> all with wild cards: det([|* 4 -1>, |* 0 -3>]) = -12,
> det([|-4 * -1>, |7 * -3>]) = 19, det([|-4 4 *>, |7 0 *>]) =-28; which
> are of course minors. This seems to be what you do below, you may as
> well start out with it.

Yes, I need the wild cards - will put them in. Don't want to use Det though
as too techy. I try to avoid using the word determinant here - every use of a
techy math term will stop a newbie in their tracks if they haven't
learnt it yet.

Here is my latest definition, and I've changed it
around again, somewhat back the way it was originally,
and start by explaining what a wedge
product is mathematically but simply.
Then one carefully chosen example from first principles.

=term=
wedge product

=definition=
The wedge product x/\y of two vectors x and y is the area
swept out from one to the other. So it is an
area with a sense of direction and x /\y = - y/\x.
A vector is made up of unit vectors for two or more
directions of travel - similarly a wedge product
of vectors is made up of wedge products of unit vectors.

One example of its use in tuning theory is to find out the
area or volume of a periodicity block. To do that you
write out the monzos in the form of a sum of unit vectors
then multiply them out using two rules: ei/\ei = 0 (area 0)
and because it is a directed area, ei/\ej = - ej/\ei
You can only change the order of the unit vectors using the
second rule.

Example:
tridecimal diesis 26/25 = [* 0, -2 * *, 1> = (-2e2 + e3)
syntonic comma 81/80 = [* 4, -1 * *, 0> = (4e1 - e2)
schisma [* 8, 1> = (8e1 + e2)
(-2e2 + e3) /\ (4e1 - e2) /\ (8e1 + e2)
= (-2e2 + e3) /\ (4e12 -8e21)
= (-2e2 + e3) /\ (-12e21)
= -12 e321
An example would be David Canright's
13 limit twelve tone scale.
You can simplify this calculation
by using the diesis in place of the schisma.

Well the best I can do at present. Don't know how many
hours it took me :-).

Be sure to say if anything there is unclearly
expressed or of course if there is a
typo or something not correct.

Robert

Hi Klaus:

> but what the heck does it mean (in a different post) that
> something is "onto"? JARGON!!!!!!!!!!!!

Gene just meant that a val need not take
all possible integers as values.

A homomorphism is an operation preserving
mapping.

The vals are mappings from the lattice to
the integers because to every interval they assign
an integer - the number of degrees spanned by the interval.

They are operation preserving because if you add two intervals in the
lattice, you add the number of degrees that they span in the scale.

An onto homomorphism, or surjection as it is also
called is one that can take all the possible
values. That's important to know sometimes if you
want to be able to go both ways - if you
want for instance to find all the intervals
that span to a particular number of scale
degrees.

The val will be onto of course in this case
because every possible number of scale
degrees must be reached in some way
or another. But a valid val is also
something such as
[0 19, 0>
which just gives you the number of
scale degrees spanned by the
power of 3 in the fraction.
That wouldn't be onto because
it can only take values that are
a multiple of 19.

I think jargon is particularly confusing
for a newbie when it uses a familiar
word like "onto" in a totally unfamiliar
way - and here, there is no way one could guess
what it means by the context.

Try replacing it by some other word
e.g. crumble or something - the
vals are crumble. Does it make sense
from the context?

If not then a newbie can't be expected
to know what it means unless they know
the definition of the word.

Robert

Robert Walker schrieb:

> .... >
> I think jargon is particularly confusing
> for a newbie when it uses a familiar
> word like "onto" in a totally unfamiliar
> way - and here, there is no way one could guess
> what it means by the context. My Big Dictionary actually has "into" and "onto" as
adjectives, and it explains "surjection" as an "onto
mapping". Projektion and Superjektion do indeed sound like
something I could have learned at school. Still, I wonder
whether terms like "identical" (self-identical, if
necessary) or an explicit "mapping on the whole/a subset"
won't do.

> > Try replacing it by some other word
> e.g. crumble or something - the
> vals are crumble. Does it make sense
> from the context?

Just as much. In fact, I always (up to this series of
explanations, thanks again) took "val" as an abbreviation
for value. Very satisfying explanation. Btw, I can see the
similarity between a vector and a wedge. but where does the
term "val" come from (isn't etymology the very essence of
definition)?

> > If not then a newbie can't be expected
> to know what it means unless they know
> the definition of the word.

Also, waste a thought on the poor people who may have the
solidest grasp of their high school math (that's not me!),
but were taught in another language than English.

klaus

--- In tuning@yahoogroups.com, klaus schmirler <KSchmir@z...> wrote:

> Robert Walker schrieb:
> >
> >
> > Try replacing it by some other word
> > e.g. crumble or something - the
> > vals are crumble. Does it make sense
> > from the context?
>
> Just as much. In fact, I always (up to this series of
> explanations, thanks again) took "val" as an abbreviation
> for value. Very satisfying explanation. Btw, I can see the
> similarity between a vector and a wedge. but where does the
> term "val" come from (isn't etymology the very essence of
> definition)?

Gene made it up.

Paul Erlich advocates calling it a "breed" in honor
of Graham Breed, similar to the way Gene has named
the interval vector "monzo" after me.

-monz

>> Sorry, this is too terse to parse. "Hence, has a cube"?
>> What's a cube?
>
>(81/80)^3 is a cube.

Yes; I was thinking you were introducing some new
lattice-geometrical term.

>> It almost sounds like you're saying: 81/80 is represented by a
>> nonzero number of steps in the temperament, but some power of
>> it is represented by the unison.
>
>That's one way of describing what is going on, which is why I
>think getting rid of torsion is a good idea.

But in, say, augmented with a generator of 400 cents, 5/4 is
represented by a nonzero number of steps but (5/4)^3 is
represented by the unison. How does this fail the description
above?

>> >> i never understood the "illigetimacy" of torsion ...
>> >> if a composer likes to play with comma shifts, and
>> >> so therefore wants a temperament which gives a nice
>> >> approximation to the syntonic-comma, what's illegitimate
>> >> about that?
>> >
>> >It's fine, but if he wants the cube of a number greater
>> >than one to equal one, he's in trouble; you can't very
>> >well have a comma equal to (sqrt(-3)-1)/2, after all.
>>
>> Here are concepts I'm not familiar with again, seemingly
>> new with this post: "cube" and "comma" (usually comma
>> means nothing more than "small interval").
>
>Generally it means a small interval *tempered out by a
>temperament*. Surely cube is familiar, so what are you saying?

Maybe you can explain where the (sqrt(-3)-1)/2 term came from.

-Carl

> "Onto" is as much a jargon term as "surjective". The only
> difference is that a Google search for "onto" is utterly
> useless in this context.

It shouldn't be. One could search for "homomorphism onto",
or even use proximity searching...

http://www.staggernation.com/cgi-bin/gaps.cgi

-Carl

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

> I think jargon is particularly confusing
> for a newbie when it uses a familiar
> word like "onto" in a totally unfamiliar
> way - and here, there is no way one could guess
> what it means by the context.

Maybe I worry about the wrong things--I knew that a google search on
"epimorphism" would likely bring up the category theory definition,
which would be both confusing and for our purposes here downright
incorrect.

--- In tuning@yahoogroups.com, klaus schmirler <KSchmir@z...> wrote:

> Just as much. In fact, I always (up to this series of
> explanations, thanks again) took "val" as an abbreviation
> for value. Very satisfying explanation. Btw, I can see the
> similarity between a vector and a wedge. but where does the
> term "val" come from (isn't etymology the very essence of
> definition)?

It comes from "valuation", since vals are linear combinations with
integer coefficients of the additive form of p-adic valuations.
Readers interesting in reading technical math definitions can chew on
this:

http://en.wikipedia.org/wiki/Valuation_%28mathematics%29

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

> But in, say, augmented with a generator of 400 cents, 5/4 is
> represented by a nonzero number of steps but (5/4)^3 is
> represented by the unison. How does this fail the description
> above?

It fails because (5/4)^3 most certainly does not represent a unison,
it is an octave.

This is a very good example of why people should remember that

*** 2 is a prime number ***

I wish I could cure people of the tendency to assume 2 is actually 1.

> Maybe you can explain where the (sqrt(-3)-1)/2 term came from.

A number which is not 1 but whose cube is 1 must be +-(sqrt(-3)-1)/2.
Hence you could say 81/80 is being tempered to (sqrt(-3)-1)/2 except
for the fact that this makes no sense whatever.

--- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> > "Onto" is as much a jargon term as "surjective". The only
> > difference is that a Google search for "onto" is utterly
> > useless in this context.
>
> It shouldn't be. One could search for "homomorphism onto",
> or even use proximity searching...
>
> http://www.staggernation.com/cgi-bin/gaps.cgi

Doing that brought up, for instance, this:

http://mathworld.wolfram.com/UniversalAlgebra.html

which would have worked.

>> But in, say, augmented with a generator of 400 cents, 5/4 is
>> represented by a nonzero number of steps but (5/4)^3 is
>> represented by the unison. How does this fail the description
>> above?
>
>It fails because (5/4)^3 most certainly does not represent a unison,
>it is an octave.
>
>This is a very good example of why people should remember that
>
>*** 2 is a prime number ***
>
>I wish I could cure people of the tendency to assume 2 is actually 1.

Aha.

>> Maybe you can explain where the (sqrt(-3)-1)/2 term came from.
>
>A number which is not 1 but whose cube is 1 must be +-(sqrt(-3)-1)/2.
>Hence you could say 81/80 is being tempered to (sqrt(-3)-1)/2 except
>for the fact that this makes no sense whatever.

Ah.

-Carl

Hi Gene,

> Would this be clearer as [* 8, 1 * *, 0>?

Yes, done.

> > An example would be David Canright's
> 13 limit twelve tone scale.

> Is Canright's scale a Fokker block using these commas? I'd make that
> explicit, and also point out that the "-12 e321" means that it is a 12
> note scale

Sorry no its not at all. Has a 7 and an 11. Nonsense there,
sorry about that.

> > You can simplify this calculation
> > by using the diesis in place of the schisma.

> And get different Fokker blocks too. Another try would be with 65/64.

Oh right, same number of notes but different scale. I was
assuming that it would be the same scale for not good
reasons at all.

Perhaps 65/64 might be an interesting one to use indeed
as the example. Scala calls it the "Thirteenth partial
chroma".

(-2e2 + e3) /\ (4e1 - e2) /\ (e2 + e3)
= - 8e213 + e312

I make that a nine note scale:

Thanks for the corrections. Also indeed yes I
forgot with all the trimming to say that it
is a 12 note scale :-).

Here is the example now:

Example:
tridecimal diesis 26/25 = [* 0, -2 * *, 1> = (-2e2 + e3)
syntonic comma 81/80 = [* 4, -1 * *, 0> = (4e1 - e2)
schisma [* 8, 1 * *, 0> = (8e1 + e2)
(-2e2 + e3) /\ (4e1 - e2) /\ (8e1 + e2)
= (-2e2 + e3) /\ (4e12 -8e21)
= (-2e2 + e3) /\ (-12e21)
= -12 e321
That's the directed volume, and we can ignore the sign
to get the volume as 12, so it is a 12 note scale.
You could try the diesis in place of the schisma
- or the thirteenth partial chroma 65/64.

Kind of nice to leave them with a couple of
examples to try like that :-).

Though - I suppose one kind of quite
large gap there is - how do you set
about making these scales once
you have figured out the number
of notes. Not so easy by hand,
in 3D or higher especially.

So I think maybe should add at the end:

You can make periodicity blocks
like this in SCALA using File | New
| Periodicity Block -
then enter the list of unison
intervals.

Robert

HI Gene

sorry forgot to delete my first attempt in that
e-mail which came out as a nine note scale until I found the
mistakea nd corrected it.

Robert

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

> Though - I suppose one kind of quite
> large gap there is - how do you set
> about making these scales once
> you have figured out the number
> of notes. Not so easy by hand,
> in 3D or higher especially.

I can do it easily enough, and in fact can survey for every possible
block; I'm not sure what system Manuel uses. In this case, the result
isn't too interesting; if you start from 128/125, 81/80, 26/25 and
65/64, it turns out that only the first two matter, and you end up
with things like the Ellis duodene.