Particular Step-sizes

On Dante Rosati's interesting webpage at:

http://users.rcn.com/dante.interport/justguitar2.html

He derives a set ("gamut") of rational JI scale pitches which are characterized by a (very aesthetically interesting) set of exclusively "superparticular" step-sizes (the ratios between a given set of rationally valued scale pitches, when those scale pitches are ordered in ascending pitch). There is a symmetrical pattern to the values of the "superparticular" step-sizes which he derives (or presents) as well, where a "mirror-image" symmetry exists in the numerical values of the presented step-sizes (centered around the step-size of 50/49, which is located in the center of the set of step-sizes). See the JPG which appears in the webpage itself in order to see what I am describing above. Pretty neat, eh?

If one (arbitrarily, of course) limits Dante's given set of step-sizes to those which do not contain integers (in their numerator or denominator) of a value greater than 25, a certain (basically familiar) 10-tone set of JI pitch ratios scale emerges [when one adds a 1/1 scale pitch, as well as adding a pitch which serves as a "suitable approximation" to the SQRT(2), which could be either 7/5, 10/7, or the "infamous" 45/32], which, then, contains the following pitch-values listed below:

1/1--16/15--10/9--6/5--5/4--4/3--Tritone--3/2--8/5--5/3--9/5--15/8

Having seen (but not comprehending) references on the ATL (and, perhaps, the tuning-math group as well) to the phrase "superparticular unison vectors", I [imagining if Dante's symmetrical set of "superparticular step-sizes" was interpreted, instead, as "superparticular unison vectors" (the ratios between a given set of rationally valued "step-sizes", when the set's distinct, non-repeating "step-sizes" are ordered in ascending pitch)], then (working "backwards" from the given set of "scale pitches", derived the set of ratios which (on the premise of the "superparticular" set being a set of "unison vectors") would (in this case) constitute a resultant set of *scale pitch ratios*.

The resultant (non-sorted for ascending numerical value) set of "scale pitch ratios" is as follows:

1/1--16/15--32/27--4/3--3/2--7/4--21/10--21/8--27/8--9/2

63/10

9/1--27/2--21/1--168/5--56/1--96/1--168/1--896/3--2688/59--5040/5

within which these (arbitrarily selected, and octave-reduced) scale pitch ratios are derived below:

16/15--4/3--3/2--7/4--27/16--9/8

Evidently (in this particular case of "superparticular" rational numbers, anyway) "superparticular step-size" appears to be more desirable than "superparticular unison vectors! The above seems like a rather small range of resultant rational "scale-pitch" choices... (to me, anyway).

Perhaps I have made a procedural error in my interpretation of the method by which "unison-vectors" are derived from "step-sizes"? Perhaps I chose a certain case where "superparticular unison vectors" do not work out so well (while, perhaps, yielding stellar results in other specific cases)? What do you think?

Curiously, J Gill

--- In tuning@y..., J Gill <JGill99@i...> wrote:

>
> 1/1--16/15--10/9--6/5--5/4--4/3--Tritone--3/2--8/5--5/3--9/5--15/8
>
> Having seen (but not comprehending) references on the ATL (and,
perhaps,
> the tuning-math group as well) to the phrase "superparticular
unison
> vectors", I [imagining if Dante's symmetrical set
of "superparticular
> step-sizes" was interpreted, instead, as "superparticular unison
vectors"
> (the ratios between a given set of rationally valued "step-sizes",
when the
> set's distinct, non-repeating "step-sizes" are ordered in ascending
> pitch)], then (working "backwards" from the given set of "scale
pitches",
> derived the set of ratios which (on the premise of
the "superparticular"
> set being a set of "unison vectors") would (in this case)
constitute a
> resultant set of *scale pitch ratios*.

Whoa -- can you chop that up into more digestible pieces?

> Evidently (in this particular case of "superparticular" rational
numbers,
> anyway) "superparticular step-size" appears to be more desirable
than
> "superparticular unison vectors! The above seems like a rather
small range
> of resultant rational "scale-pitch" choices... (to me, anyway).

I can't follow your purported demonstration.

> Perhaps I have made a procedural error in my interpretation of the
method
> by which "unison-vectors" are derived from "step-sizes"?

One can find the unison vectors OF A PERIODICITY BLOCK by taking the
quotients of _any_ pair of (not necessarily neighboring) step-size-
ratios. In the JI world, a good clue that something is a periodicity
block is if it has the CS property (all intervals of a given size
always subtend the same number of steps). For example, one such scale
is

1/1 9/8 5/4 4/3 3/2 5/3 15/8 (2/1)

You'll see this scale forms a major part of my paper, _The Forms Of
Tonality_, which you have a copy of.

The step sizes are

9:8, 10:9, and 16:15

Hence the unison vectors are

(9:8)/(10:9) = 81:80
(9:8)/(16:15) = 135:128
(10:9)/(25:24) = 25:24

In my paper, you'll see these three unison vectors occuring between
copies of the scale in the 5-limit lattice. You'll also see that
135:128 is relatively unimportant, separating copies of the scale
that are connected by very few consonant intervals. And, since this
is a 2-dimensional system, only two unison vectors are needed to
define it. Hence, it is usually considered that the unison vectors
defining the diatonic scale are 81:80 and 25:24 -- both
superparticular. See the "Gentle Introduction" for another
demonstration that the unison vectors defining the diatonic scale are
81:80 and 25:24.

In 31659

<<

The step sizes are

9:8, 10:9, and 16:15

Hence the unison vectors are

(9:8)/(10:9) = 81:80
(9:8)/(16:15) = 135:128
(10:9)/(25:24) = 25:24

>>

???

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., J Gill <JGill99@i...> wrote:

>PE: Whoa -- can you chop that up into more digestible pieces?

JG: I can empathize with your request (it's a run-on, allright!). Here is an edited/improved version of the same statements:
>
> >
> > 1/1--16/15--10/9--6/5--5/4--4/3--Tritone--3/2--8/5--5/3--9/5--15/8
> >
> > Having seen (but not comprehending) references on the ATL (and,
> perhaps,
> > the tuning-math group as well) to the phrase "superparticular
>> unison
> > vectors", I imagined a situation where Dante's symmetrical set
>> of "superparticular
>> step-sizes" might be interpreted, instead, as "superparticular >>unison
>> vectors", themselves.

> > I understand the term "unison vectors" to mean: the ratios >>between a given set of rationally valued "step-sizes",
> >when the
> > set's distinct, non-repeating "step-sizes" are ordered in >>ascending
> > pitch.

>> I then tried working "backwards" from the given set of "step->>sizes" (those shown in Dante's diagram in the left column)
>>AS IF that set of "step-sizes" was, *instead*, a set of "unison >>vectors".

The resultant (non-sorted for ascending numerical value) set of resulting "scale pitch ratios" which I derived from this operation
is as follows (from "bottom-to-top" in the diagram):

1/1--16/15--32/27--4/3--3/2--7/4--21/10--21/8--27/8--9/2

63/10---[this is the central value within the set]

9/1--27/2--21/1--168/5--56/1--96/1--168/1--896/3--2688/59--5040/5

within which these (arbitrarily selected, and octave-reduced) scale pitch ratios are derived below:

16/15--4/3--3/2--7/4--27/16--9/8

> > Evidently (in this particular case of "superparticular" rational
> >numbers,
> > anyway) "superparticular step-size" appears to be more desirable
> >than
> > "superparticular unison vectors! The above seems like a rather
>> small range
> > of resultant rational "scale-pitch" choices... (to me, anyway).

> PE: I can't follow your purported demonstration.
JG: Hopefully improved by edited restatement above.

JG: I corrected an error pointed out in your text below (see note).

J Gill

> > Perhaps I have made a procedural error in my interpretation of the
> method
> > by which "unison-vectors" are derived from "step-sizes"?
>
> One can find the unison vectors OF A PERIODICITY BLOCK by taking the
> quotients of _any_ pair of (not necessarily neighboring) step-size-
> ratios. In the JI world, a good clue that something is a periodicity
> block is if it has the CS property (all intervals of a given size
> always subtend the same number of steps). For example, one such scale
> is
>
> 1/1 9/8 5/4 4/3 3/2 5/3 15/8 (2/1)
>
> You'll see this scale forms a major part of my paper, _The Forms Of
> Tonality_, which you have a copy of.
>
>
> The step sizes are
>
> 9:8, 10:9, and 16:15
>
> Hence the unison vectors are
>
> (9:8)/(10:9) = 81:80
> (9:8)/(16:15) = 135:128
> (10:9)/(16:15) = 25:24--[NOTE:(16:15), CORRECTED. -J GILL]
>
> In my paper, you'll see these three unison vectors occuring between
> copies of the scale in the 5-limit lattice. You'll also see that
> 135:128 is relatively unimportant, separating copies of the scale
> that are connected by very few consonant intervals. And, since this
> is a 2-dimensional system, only two unison vectors are needed to
> define it. Hence, it is usually considered that the unison vectors
> defining the diatonic scale are 81:80 and 25:24 -- both
> superparticular. See the "Gentle Introduction" for another
> demonstration that the unison vectors defining the diatonic scale are
> 81:80 and 25:24.

--- In tuning@y..., "Pierre Lamothe" <plamothe@a...> wrote:
> In 31659
>
> <<
>
> The step sizes are
>
> 9:8, 10:9, and 16:15
>
> Hence the unison vectors are
>
> (9:8)/(10:9) = 81:80
> (9:8)/(16:15) = 135:128
> (10:9)/(25:24) = 25:24
>
> >>
>
> ???

Sorry, that last line should have read,

(10:9)/(16:15) = 25:24

Is that better, Pierre?

--- In tuning@y..., "unidala" <JGill99@i...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning@y..., J Gill <JGill99@i...> wrote:
>
> >PE: Whoa -- can you chop that up into more digestible pieces?
>
> JG: I can empathize with your request (it's a run-on, allright!).
Here is an edited/improved version of the same statements:
> >
> > >
> > > 1/1--16/15--10/9--6/5--5/4--4/3--Tritone--3/2--8/5--5/3--9/5--
15/8
> > >
> > > Having seen (but not comprehending) references on the ATL (and,
> > perhaps,
> > > the tuning-math group as well) to the phrase "superparticular
> >> unison
> > > vectors", I imagined a situation where Dante's symmetrical set
> >> of "superparticular
> >> step-sizes" might be interpreted, instead, as "superparticular
>>unison
> >> vectors", themselves.

Well that would be problematic, since a given system can only have a
set of unison vectors which span a subset of the ratio space in
question -- otherwise you're implicitly saying that _every interval_
is a unison -- and the whole structure collapses.
>
> > > I understand the term "unison vectors" to mean: the ratios
>>between a given set of rationally valued "step-sizes",
> > >when the
> > > set's distinct, non-repeating "step-sizes" are ordered in
>>ascending
> > > pitch.

As you've seen, it's not necessary to order them in pitch.

So you made two incorrect assumptions.

> JG: I corrected an error pointed out in your text below (see note).

Thank you! Hopefully that example, combined with a look at the paper
I sent you, clears up the nature of unison vectors in your mind.

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

/tuning/topicId_31636.html#31659

>
> One can find the unison vectors OF A PERIODICITY BLOCK by taking
the quotients of _any_ pair of (not necessarily neighboring) step-
size-ratios. In the JI world, a good clue that something is a
periodicity block is if it has the CS property (all intervals of a
given size always subtend the same number of steps). For example, one
such scale is
>
> 1/1 9/8 5/4 4/3 3/2 5/3 15/8 (2/1)
>
> You'll see this scale forms a major part of my paper, _The Forms Of
> Tonality_, which you have a copy of.
>
>
> The step sizes are
>
> 9:8, 10:9, and 16:15
>
> Hence the unison vectors are
>
> (9:8)/(10:9) = 81:80
> (9:8)/(16:15) = 135:128
> (10:9)/(25:24) = 25:24
>

Hi Paul!

You said there were three step sizes and then the 25:24 suddenly
appears. Where does that come from again? It's a "chromatic
semitone," yes, but where is it in this scale??

> In my paper, you'll see these three unison vectors occuring between
> copies of the scale in the 5-limit lattice. You'll also see that
> 135:128 is relatively unimportant, separating copies of the scale
> that are connected by very few consonant intervals. And, since this
> is a 2-dimensional system, only two unison vectors are needed to
> define it. Hence, it is usually considered that the unison vectors
> defining the diatonic scale are 81:80 and 25:24 -- both
> superparticular. See the "Gentle Introduction" for another
> demonstration that the unison vectors defining the diatonic scale
are 81:80 and 25:24.

By the way, the explanation above is *just fantastic* and seems to be
presented in a slightly different way than anything in _The Forms of
Tonality_ I think... Could this explanation be added to the
later "expanded" version of the paper?

Thanks again!

Joseph

Paul wrote:

<<

> Hence the unison vectors are
>
> (9:8)/(10:9) = 81:80
> (9:8)/(16:15) = 135:128
> (10:9)/(25:24) = 25:24
>
> >>
>
> ???

Sorry, that last line should have read,

(10:9)/(16:15) = 25:24

Is that better, Pierre?

>>

Yes. I would add only that unison vector may have two senses:
- element of class 0
- element of a basis subtenting the class 0.

Here the three elements are dependant since (81:80)(25:24) = 135:128.
So, only two elements could be simultaneaouly unison vectors in the second sense.

In the first sense, there not exist only three unison vectors but an infinite number.

I want only to avoid confusion with my description of the scale used.

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> /tuning/topicId_31636.html#31659
>
> >
> > One can find the unison vectors OF A PERIODICITY BLOCK by taking
> the quotients of _any_ pair of (not necessarily neighboring) step-
> size-ratios. In the JI world, a good clue that something is a
> periodicity block is if it has the CS property (all intervals of a
> given size always subtend the same number of steps). For example,
one
> such scale is
> >
> > 1/1 9/8 5/4 4/3 3/2 5/3 15/8 (2/1)
> >
> > You'll see this scale forms a major part of my paper, _The Forms
Of
> > Tonality_, which you have a copy of.
> >
> >
> > The step sizes are
> >
> > 9:8, 10:9, and 16:15
> >
> > Hence the unison vectors are
> >
> > (9:8)/(10:9) = 81:80
> > (9:8)/(16:15) = 135:128
> > (10:9)/(25:24) = 25:24
> >
>
> Hi Paul!
>
> You said there were three step sizes and then the 25:24 suddenly
> appears. Where does that come from again? It's a "chromatic
> semitone," yes, but where is it in this scale??

Sorry, Joseph, that last line should have read,

(10:9)/(16:15) = 25:24

Are things any clearer now?
>
>
> > In my paper, you'll see these three unison vectors occuring
between
> > copies of the scale in the 5-limit lattice. You'll also see that
> > 135:128 is relatively unimportant, separating copies of the scale
> > that are connected by very few consonant intervals. And, since
this
> > is a 2-dimensional system, only two unison vectors are needed to
> > define it. Hence, it is usually considered that the unison
vectors
> > defining the diatonic scale are 81:80 and 25:24 -- both
> > superparticular. See the "Gentle Introduction" for another
> > demonstration that the unison vectors defining the diatonic scale
> are 81:80 and 25:24.
>
> By the way, the explanation above is *just fantastic* and seems to
be
> presented in a slightly different way than anything in _The Forms
of
> Tonality_ I think... Could this explanation be added to the
> later "expanded" version of the paper?

You mean the part referring to the "Gentle Introduction"??

--- In tuning@y..., "Pierre Lamothe" <plamothe@a...> wrote:
> Paul wrote:
>
> <<
>
> > Hence the unison vectors are
> >
> > (9:8)/(10:9) = 81:80
> > (9:8)/(16:15) = 135:128
> > (10:9)/(25:24) = 25:24
> >
> > >>
> >
> > ???
>
> Sorry, that last line should have read,
>
> (10:9)/(16:15) = 25:24
>
> Is that better, Pierre?
>
> >>
>
>
> Yes. I would add only that unison vector may have two senses:
> - element of class 0
> - element of a basis subtenting the class 0.
>
> Here the three elements are dependant since (81:80)(25:24) =
135:128.
> So, only two elements could be simultaneaouly unison vectors in the
second sense.
>
> In the first sense, there not exist only three unison vectors but
an infinite number.

Everything you say is exactly correct, Pierre. But if you examine the
periodicity of the scale I mentioned in the 5-limit lattice, you'll
see that the scale is connected, BY CONSONANT INTERVALS, ONLY to
unisonious copies of itself in the lattice which are 80:81, 81:80,
24:25, 25:24, 128:135, and 135:128 away. In other words, if you start
with the scale I gave, and find a pitch outside the scale which is
_consonant_ with a pitch inside the scale, it will be either 80:81,
81:80, 24:25, 25:24, 128:135, or 135:128 away from some other pitch
inside the scale.

Hi Paul,

If you let the 16/15 10/9 9/8 be the three stepsizes in a Tribonacci
series, the expansion rule would be

a-->(b-a)
b-->(c-a)
c-->a

So the stepsizes for a 2,3,7,12,22,41,... would be

25/24 135/128 15/16

and the 2D UVs would be

2048/2025 81/80 (and 128/125)

The contraction rule would be

a-->c
b-->(a+c)
c-->(b+c)

How does this jibe with your superparticular PB ideas? Isn't the
second example here a 12-tone PB?

--Dan Stearns

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Paul,
>
> If you let the 16/15 10/9 9/8 be the three stepsizes in a Tribonacci
> series, the expansion rule would be
>
> a-->(b-a)
> b-->(c-a)
> c-->a
>
> So the stepsizes for a 2,3,7,12,22,41,... would be
>
> 25/24 135/128 15/16

Hmm? Those would be the stepsizes for the 12-tone scale in this
series, not the whole series, correct?

> and the 2D UVs would be
>
> 2048/2025 81/80 (and 128/125)

Any two of which define 12-tET if tempered out. If only the smallest,
2048/2025, is tempered out, you have diaschismic, which is consistent
with 22-tET.

> The contraction rule would be
>
> a-->c
> b-->(a+c)
> c-->(b+c)
>
> How does this jibe with your superparticular PB ideas?

Not sure what you're getting at.

> Isn't the
> second example here a 12-tone PB?

If you take 2048/2025 and 81/80, you get

http://www.ixpres.com/interval/td/erlich/ramospblock.htm

If you take 81/80 and 128/125, you get the two at the bottom of

http://www.ixpres.com/interval/td/erlich/intropblock2.htm

If you take 2048/2025 and 128/125, you get the 24-tone "periodicity
block" Monz derived at the bottom of

http://www.ixpres.com/interval/td/erlich/srutipblock.htm

This periodicity block is an example of what I once called "double
vision" or something -- 12 pairs of 81:80-separated notes. But
81:80*81:80 = 2048/2025*128/125, so clearly there is something
inconsistent about not considering 81:80 a unison vector here. It
turns out, according to Gene, that this corresponds to a mathematical
condition known as "torsion", and we consider it a pathological
condition especially if one or more of the defining unison vectors is
to be tempered out in a reasonable way.

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

/tuning/topicId_31636.html#31758

>> >
> > Hi Paul!
> >
> > You said there were three step sizes and then the 25:24 suddenly
> > appears. Where does that come from again? It's a "chromatic
> > semitone," yes, but where is it in this scale??
>
> Sorry, Joseph, that last line should have read,
>
> (10:9)/(16:15) = 25:24
>
> Are things any clearer now?
> >

Hi Paul!

Absolutely! I was having a hard time figuring out where that was
coming from... :)

Could this explanation be added to the
> > later "expanded" version of the paper?
>
> You mean the part referring to the "Gentle Introduction"??

I just mean that I don't remember a part in _The Forms of Tonality_
where unison vectors are found by dividing scale intervals... or,
maybe I just missed it, but this recent post made it *much* clearer
for me...

Thanks!

Joseph

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

> I just mean that I don't remember a part in _The Forms of Tonality_
> where unison vectors are found by dividing scale intervals... or,
> maybe I just missed it, but this recent post made it *much* clearer
> for me...
>
> Thanks!
>
> Joseph

You're right, there isn't such a part.

I think we just got a bit _lucky_ that, in the particular example of
this scale, the step sizes _gave_ the consonance-lattice-active
unison vectors straight away.

In some cases, like, I think, the partial detempering of the triple-
BP scale that Manuel calculated on tuning-math, we have to also look
at the quotients between "thirds" in the scale, and at
least "fourths" too I think, before we find the consonance-lattice-
active unison vectors of the system. Manuel, can you verify this?

Paul wrote:

<<
Everything you say is exactly correct, Pierre. But if you examine the
periodicity of the scale I mentioned in the 5-limit lattice, you'll
see that the scale is connected, BY CONSONANT INTERVALS, ONLY to
unisonious copies of itself in the lattice which are 80:81, 81:80,
24:25, 25:24, 128:135, and 135:128 away. In other words, if you start
with the scale I gave, and find a pitch outside the scale which is
_consonant_ with a pitch inside the scale, it will be either 80:81,
81:80, 24:25, 25:24, 128:135, or 135:128 away from some other pitch
inside the scale.
>>

So let us talk about the hexagone H = <81/80 135/128 25/24 80/81 128/135 2425>. It's what is
represented here in blue, where U = 81/80 and V = 25/24. Its contents in red is the complete space of
intervals between the elements of Zarlino scale, in other words, its transposition space (tonic rotation).

It contains 19 elements. It corresponds to the diamond generated by <1 3 5 9 15 27 45>. It's the reunion
of 3 periodicity block having 7 elements intersecting only at unison 0. The minimal diamond in which the
Zarlino mode is sui generis is <1 3 5 9 15> and represented in red in the third drawing.
. . . . . . 0 . . . . . . 0 . . . 0 .
. . . 0 . . . . . . 0 . . . . . . . .
0 . . . . . . 0 . . . . . . 0 . . . 0
. . . . 0 . . . . . . 0 . . . . . . .
. 0 . . . . . . V . . . . . . 0 . . .
. . . . . 0 . . . * * * 0 . . . . . .
. . 0 . . . . . . 0 * * * . . . 0 . .
. . . . . . 0 . . . . . . U . . . . .
. . . 0 . . . . . . 0 . . . . . . 0 .
0 . . . . . . 0 . . . . . . 0 . . . .
. . . . 0 . . . . . . 0 . . . . . . 0
. 0 . . . . . . 0 . . . . . . 0 . . .

. . . V . . . . .
0 * * 5 2 6 * 0 .
. * * 3 0 4 1 * .
. 0 * * * * * * U
. . . . . 0 . . .

. . . V . . . . .
0 4 1 5 2 6 3 0 .
. 2 6 3 0 4 1 5 .
. 0 4 1 5 2 6 3 U
. . . . . 0 . . .

I think it's important to understand that a system in which the modes have N distinct steps has to deal
with N distinct periodicity blocks, even if an isolated scale may be enclosed in one periodicity block ;
for each step is forcely in a distinct block.

Let P be the periodicity block <U V>. I add there exist an operator represented by the simple matrix
M = [(0,1) (-1,-1)], which is a third root of the identity matrix, transforming any of the three blocks
<P MP MMP> in the next block. (The next of MMP is P). Finally, <P MP MMP> is our hexagone H.

P = <U V> = <81/80 25/24>
MP = <V -U-V> = <25/24 128/135>
MMP = <-U-V U> = <128/135 81/80>

Pierre

> From: jpehrson2 <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, December 19, 2001 6:05 PM
> Subject: [tuning] great explanation [periodicity block]
>
>
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> /tuning/topicId_31636.html#31659
>
> >
> > One can find the unison vectors OF A PERIODICITY BLOCK by taking
> > the quotients of _any_ pair of (not necessarily neighboring) step-
> > size-ratios. <etc. -- snip>
>
> ...
>
> By the way, the explanation above is *just fantastic* and seems to be
> presented in a slightly different way than anything in _The Forms of
> Tonality_ I think... Could this explanation be added to the
> later "expanded" version of the paper?

I agree with you Joe. Paul, this post was a terrific explanation
of the unison-vector-definining-a-periodicity-block concept.

-monz

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>In some cases, like, I think, the partial detempering of the triple-
>BP scale that Manuel calculated on tuning-math, we have to also look
>at the quotients between "thirds" in the scale, and at
>least "fourths" too I think, before we find the consonance-lattice-
>active unison vectors of the system. Manuel, can you verify this?

Yes, the UVs were quotients of intervals of 1, 4 and 6 steps in that
scale.
There's a Scala command now to get a list of potential unison vectors
which is SHOW/DIFFERENCE INTERVALS.

Manuel

--- In tuning@y..., "Pierre Lamothe" <plamothe@a...> wrote:

> So let us talk about the hexagone H = <81/80 135/128 25/24 80/81
128/135 2425>. It's what is
> represented here in blue, where U = 81/80 and V = 25/24.

Those are the only blue things I saw in your message -- no blue
hexagon.

> Its contents in red is the complete space of
> intervals between the elements of Zarlino scale, in other words,
its transposition space (tonic rotation).
>
> It contains 19 elements.

With you so far.

>It corresponds to the diamond generated by <1 3 5 9 15 27 45>.

Well, yes, if the interval-ratios are re-intepreted as pitch-ratios.

> It's the reunion
> of 3 periodicity block having 7 elements intersecting only at >
unison 0.

I'm not seeing this as more than a mathematical curiosity at present,
because periodicity blocks are blocks of pitch-ratios, while this set
was derived as a set of interval-ratios.

Let me look at your message again and I'll continue.

--- In tuning@y..., "Pierre Lamothe" <plamothe@a...> wrote:

> I think it's important to understand that a system in which the
modes have N distinct steps has to deal
> with N distinct periodicity blocks, even if an isolated scale may
be enclosed in one periodicity block ;
> for each step is forcely in a distinct block.

Not sure what this means . . .

>
> Let P be the periodicity block <U V>. I add there exist an operator
represented by the simple matrix
> M = [(0,1) (-1,-1)], which is a third root of the identity matrix,
transforming any of the three blocks
> <P MP MMP> in the next block. (The next of MMP is P). Finally, <P
MP MMP> is our hexagone H.
>
>
> P = <U V> = <81/80 25/24>
> MP = <V -U-V> = <25/24 128/135>
> MMP = <-U-V U> = <128/135 81/80>

Well that's pretty neat stuff! Gene, any comment?

> > > One can find the unison vectors OF A PERIODICITY BLOCK by taking
> > > the quotients of _any_ pair of (not necessarily neighboring) step-
> > > size-ratios. <etc. -- snip>
> >
> > ...
> >
> > By the way, the explanation above is *just fantastic* and seems to be
> > presented in a slightly different way than anything in _The Forms of
> > Tonality_ I think... Could this explanation be added to the
> > later "expanded" version of the paper?
>
>
> I agree with you Joe. Paul, this post was a terrific explanation
of the unison-vector-definining-a-periodicity-block concept.
>
>
> -monz

Monz, do you remember the S-matrix used on my site, on this List and in private posts
with Paul and you ?

Let m be a valid mode. The S-matrix is Q(m)\Q(m) where Q(m) are the steps.

For instance :
m = <1 : 9/8 : 5/4 : 4/3 : 3/2 : 5/3 : 15/8> is the mode

t = Q(m) = <16/15 : 10/9 : 9/8> are the steps

'S = '(t\t) = <128/135 : 24/25 : 80/81 : 1 : 81/80 : 25/24 : 135/128> is the contents of the S-matrix.
-----

I add a commentary. Removing the unison in 'S, the other elements define an important polytope containing
more than one periodicity block. The fundamental periodicity block permitting to derive this polytope is
Q(S) = <81/80 : 25/24>.
And more generally Q(S) = Q(t), what means that in a "well-constructed" system, the unison vectors,
in sense of elements of a basis generating the class 0, are defined uniquely by the steps.
Q <16/15 : 10/9 : 9/8> = <81/80 : 25/24>
With a mode generating the Indian System, for instance, the contents of the S-matrix is
'S = <2048/2187 : 128/135 : 24/25 : 80/81 : 1 : 81/80 : 25/24 : 135/128 : 2187/2048>
but the unison vectors are strictly those of the precedent example for
Q <256/243 : 16/15 : 10/9 : 9/8> = <81/80 : 25/24>
Pierre

Paul wrote:

> I'm not seeing this as more than a mathematical curiosity at present,
> because periodicity blocks are blocks of pitch-ratios, while this set
> was derived as a set of interval-ratios.

I don't define a periodicity block with pitch-ratios but with interval-ratios. The pitch-ratios are
used only as an application in my views. What I study is always the operative structure on
a set of elements and in that sense the particularity "pitch" as nothing to do with that.

In mathematics the elements are defined by the structure and not inversely. It's important
in modelization to insure that the correspondance between real objects and symbols is
valid, but the proper of a good model is its capacity to derive things without interference, i.e.
unconscious subtle choices and corrections resulting from the objects comprehension. If the
model fail to derive by itself, we have to correct the model, not its results.

Pierre

Hello Pierre!

> From: Pierre Lamothe <plamothe@aei.ca>
> To: Tuning <tuning@yahoogroups.com>
> Sent: Thursday, December 20, 2001 12:51 PM
> Subject: [tuning] Re: great explanation [periodicity block]
>
>
> Monz, do you remember the S-matrix used on my site, on this
> List and in private posts with Paul and you ?

No, actually, I didn't remember... but that's because I had
such a hard time understanding what you were posting that got
into the habit of printing out all of your posts and saving
them for future comprehensive study, and I haven't yet had
the time to dig into that. Thanks for this explanation!

-monz

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

> > P = <U V> = <81/80 25/24>
> > MP = <V -U-V> = <25/24 128/135>
> > MMP = <-U-V U> = <128/135 81/80>
>
> Well that's pretty neat stuff! Gene, any comment?

A cycle of order three can act irreducibly on 81/80 and 25/24 as above, but I don't see it gets us anywhere. If for instance we write things in the notation <81/80,25/24,16/15>^(-1) = [h3,h5,h7] then we decompose the representation into irreducible factors, where 16/15 is left fixed by the second factor, the identity representation. We can then transform to the usual 2,3,5 notation, where the transformation becomes

[ 54 -25 -6]
[ 85 -40 -9]
[127 -59 -14]

This indeed has order three, but sending 2 to 2^54 3^-25 5^-6 and so forth makes no sense to me. I think the group of transformations which preserves triads, discussed previously, makes much more sense as a way to transform 5-limit scales or Fokker blocks.

--- In tuning@y..., "Pierre Lamothe" <plamothe@a...> wrote:

> 'S = '(t\t) = <128/135 : 24/25 : 80/81 : 1 : 81/80 : 25/24 : 135/128> is the contents of the S-matrix.

Why do you call this a matrix--it looks to me like a set? When I read S-matrix I think of quantum field theory, which is not so good, but if it must be S-matrix I want a matrix, at least.

> And more generally Q(S) = Q(t), what means that in a "well-constructed" system, the unison vectors,
> in sense of elements of a basis generating the class 0, are defined uniquely by the steps.
> Q <16/15 : 10/9 : 9/8> = <81/80 : 25/24>

This does generate the kernel of h7, (my name for the epimorphism given by [7,11,16]) but in what sense is it defined uniquely from the set of steps, given that <81/80,135/128> and <25/24,135/128> do likewise?

Gene wrote:
A cycle of order three can act irreducibly on 81/80 and 25/24 as above, but I
don't see it gets us anywhere. If for instance we write things in the notation
<81/80,25/24,16/15>^(-1) = [h3,h5,h7] [...]
I think you miss the point. It defines the pertinent region around the unison where the
transposition space of the simplest mode is enclosed.

I don't know which intervals are used in your seek of the best temperaments but I would
say that it would make no sense to use intervals far from a such region to calculate errors.

Why using 16/15 which is the minimal step (class 1) in the example used, while that
concerns only unison vectors (class 0)? What is in question here is the following:
Given two unison vectors U and V determining a block having periodicity
in octave, how to determine the minimal polytope centered around the
unison in which the transposition space of a possible simplest mode m
would be contained?
Let L = <2 p q >Z3 the appropriate lattice for the periodicity block B = <U V> defined by
U = <2 p q >(a,b,c)
V = <2 p q ><d,e,f)
The block B is represented in <2 p q >Z3 / <2>Z by the matrix
[b e]
[c f]
whose determinant D = bf - ec is the periodicity. The contents of this block (which is
eccentric) with the vertex unison is a mode (say m) since there exist only one interval
in each D classes.

How to determine the set of all intervals between the elements of m? In other words,
how to determine the region covered in the lattice if we use the D transpositions of this
mode m (which correspond to the D translations where the block contains unison)? In
other words, (supposing all gammier conditions respected), what are the elements of
the maximal gammier generated by m as total chordic generator?

It's only in that context I used this matrix M
[0 -1]
[1 -1]
(having the property M3 = I) permitting to determine the solution as the hexagone
MxB x = 1,2,3,...
which is the envelope of the three blocks (having the same D periodicity)
[e -b-e][-b-e e][b e]
[c -c-f][-b-e f][c f]
You may verify that, for instance with the three first simplest systems in <2 3 5>Z3
1 9/8 5/4 3/2 5/3 (2) - Chinese
1 16/15 4/3 3/2 8/5 (2) - Japanese
1 9/8 5/4 4/3 3/2 5/3 15/8 (2) - Zarlino
By the way, I would ask if the present state of your maths permits to show that these
systems are the simplest in <2 3 5>Z3 ?

Pierre

Gene wrote:
> 'S = '(t\t) = <128/135 : 24/25 : 80/81 : 1 : 81/80 : 25/24 : 135/128> is
the contents of the S-matrix.

Why do you call this a matrix--it looks to me like a set? When I read S-matrix
I think of quantum field theory, which is not so good, but if it must be
S-matrix I want a matrix, at least.
Effectively 'S, the contents of S, is an ordered set. My symbolic operator '( ) gives the
ordered set of elements contained in a math object. Il you want the matrix, that corresponds
to the first matrix here:

--------------------------------------------------------------------------------

N.B. The following arrays are visible only on the web or with html email.

S(z) 16/15 10/9 9/8
16/15 1 25/24 135/128
10/9 24/25 1 81/80
9/8 128/135 80/81 1
S(p) 256/243 9/8
256/243 1 2187/2048
9/8 2048/2187 1
S(s) 9/8 8/7 7/6
9/8 1 64/63 28/27
8/7 63/64 1 49/48
7/6 27/28 48/49 1

--------------------------------------------------------------------------------

Gene wrote:
This does generate the kernel of h7, (my name for the epimorphism given by
[7,11,16]) but in what sense is it defined uniquely from the set of steps,
given that <81/80,135/128> and <25/24,135/128> do likewise?
Maybe my remark was badly formulated. I didn't mean using the steps is the only way to
define the epimorphism but knowing the steps is sufficient to determine it. I wanted to react
at comments in other threads where it seemed say that it could be insufficient. It's not the
case with the gammier systems.

Pierre

--- In tuning@y..., "Pierre Lamothe" <plamothe@a...> wrote:

I've replied to this on tuning-math.