Re: Mike S's "Dorian"-ish scale [discussion continued from MMM] - we

Every 12 tone temperament was conceived as being a singular variable chain.
This is the only example you will kind in history. look at Barbour's book
This scale shows the absurdity of calling things ranks at all as it makes complicated a simple thing into a convoluted process that tells us nothing no offers us anything we didn't have before.
It also alienates it from periodicity blocks and constant structures. These too are simply thought of as a single chain that varies which is also reflected in mapping scales to a keyboard.
also the recurrent sequence also fit into this category.

--

/^_,',',',_ //^/Kraig Grady_^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

On Sat, Nov 27, 2010 at 4:39 PM, Kraig Grady <kraiggrady@anaphoria.com> wrote:
>
> Every 12 tone temperament was conceived as being a singular
> variable chain.
> This is the only example you will kind in history. look at
> Barbour's book

So rank-2. But things like Werckmeister aren't like that, right?

> This scale shows the absurdity of calling things ranks at all
> as it makes complicated a simple thing into a convoluted process
> that tells us nothing no offers us anything we didn't have before.

I don't understand the protests here. The main point of my analysis
was simply to show that there are, generally speaking, three interval
classes per step size, within which there is a variation of no more
than 2 cents. And then if you analyze it that way, it ends up looking
like something that's halfway between 5-limit JI and meantone, where
the syntonic comma is shrunk but not eliminated.

This is basically the goal of any well-temperament, right? So what's
wrong with expressing this in the language of mathematics?

> It also alienates it from periodicity blocks and constant
> structures.

Wouldn't a 5-limit PB just be rank 1?

-Mike

>> It also alienates it from periodicity blocks and constant
>> structures.
>
>Wouldn't a 5-limit PB just be rank 1?

A 5-limit PB is rank 3. -Carl

On Sat, Nov 27, 2010 at 8:04 PM, Carl Lumma <carl@lumma.org> wrote:
>
> >> It also alienates it from periodicity blocks and constant
> >> structures.
> >
> >Wouldn't a 5-limit PB just be rank 1?
>
> A 5-limit PB is rank 3. -Carl

I thought it was that 5-limit JI was rank 3, and if you're creating a
PB that means you're denoting two unison vectors, which would bring it
down to rank 1.

-Mike

Mike wrote:

>> A 5-limit PB is rank 3. -Carl
>
>I thought it was that 5-limit JI was rank 3, and if you're creating a
>PB that means you're denoting two unison vectors, which would bring it
>down to rank 1.

Periodicity blocks are JI. If you temper you no longer
have a PB. You have a temperament that can be understood in
terms one or (usually many) periodicity blocks.

-Carl

I am still trying to see what this classification gets one.
As i mentioned all the temperaments were thought of in terms of a simgular 'variable' rank.
I see no reason to change that and pretend that they were thinking of a couple of different chains.
What happens with with 24Et . the rank depends on how you think of it.
if the fifth is 14 or you decide to use a coprime generator like 13.
even if i know that something is rank 3, it tells me nothing cause it doesn't tell me what or where they are.

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> Mike wrote:
>
> >> A 5-limit PB is rank 3. -Carl
> >
> >I thought it was that 5-limit JI was rank 3, and if you're creating a
> >PB that means you're denoting two unison vectors, which would bring it
> >down to rank 1.
>
> Periodicity blocks are JI. If you temper you no longer
> have a PB. You have a temperament that can be understood in
> terms one or (usually many) periodicity blocks.
>
> -Carl
>

Kraig wrote:

>What happens with with 24Et. the rank depends on how you think of it.
>if the fifth is 14 or you decide to use a coprime generator like 13.
>even if i know that something is rank 3, it tells me nothing cause it
>doesn't tell me what or where they are.

All ETs are rank 1. Or more precisely, any temperament defined
by a single val is rank 1.

The usual 5-limit val in 24-ET gives the same tunings as 12-ET.
<24 38 56| gives the same tuning of the 2:3:5 triad as <12 19 28|.
That doesn't make it rank 2.

The concept is very important in the understanding of various JI
systems and their related temperaments. In 5-limit JI everybody
knows all notes are made from some number of octaves, fifths, and
thirds. That's all the concept is saying (it is rank 3).

-Carl

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, Nov 27, 2010 at 4:39 PM, Kraig Grady <kraiggrady@...> wrote:
> >
> > Every 12 tone temperament was conceived as being a singular
> > variable chain.
> > This is the only example you will kind in history. look at
> > Barbour's book
>
> So rank-2. But things like Werckmeister aren't like that, right?
>
> > This scale shows the absurdity of calling things ranks at all
> > as it makes complicated a simple thing into a convoluted process
> > that tells us nothing no offers us anything we didn't have before.
>
> I don't understand the protests here. The main point of my analysis
> was simply to show that there are, generally speaking, three interval
> classes per step size, within which there is a variation of no more
> than 2 cents. And then if you analyze it that way, it ends up looking
> like something that's halfway between 5-limit JI and meantone, where
> the syntonic comma is shrunk but not eliminated.

What you're talking about here sounds similar to something we enable in the Dynamic Tonality synths TFS, Viking, 2032 (using the Tone Diamond interface). Essentially, we allow for a linear interpolation of pitches between a temperament (e.g., meantone) and two associated selections of notes from a 5-limit lattice (one mimimizes the number of major triads with wolf intervals, the other minimizes the number of minor triads with wolf intervals).

The process is described more fully on pp.79-80 of this paper

http://oro.open.ac.uk/21505/2/33.2.sethares.pdf

Andy

>
> This is basically the goal of any well-temperament, right? So what's
> wrong with expressing this in the language of mathematics?
>
> > It also alienates it from periodicity blocks and constant
> > structures.
>
> Wouldn't a 5-limit PB just be rank 1?
>
> -Mike
>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> > A 5-limit PB is rank 3. -Carl
>
> I thought it was that 5-limit JI was rank 3, and if you're creating a
> PB that means you're denoting two unison vectors, which would bring it
> down to rank 1.

Add to the two commas of a PB a chroma, and you have your three generators. The commas are not tempered out; if you do temper them out the chroma becomes an et step and you do get rank one, but that's the associated equal temperament, not the PB.

if the concept is important then one has to be able to say why it is useful.
If it is say you can resolve it into a single chain when you temper i would like to point out if you think of 5 limit how it has been thought of since the people in India as a single chain with slight variation you can do the same.

--

/^_,',',',_ //^/Kraig Grady_^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

Kraig wrote:

>if the concept is important then one has to be able to say why
>it is useful.

I thought I just did.

-Carl

--- In tuning-math@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> if the concept is important then one has to be able to say why
> it is useful.

Because there's more to regular tunings than just rank one tunings, or even rank one and rank two. Since five-limit JI is rank three, this should be obvious.

> If it is say you can resolve it into a single chain when you
> temper i would like to point out if you think of 5 limit how it
> has been thought of since the people in India as a single chain
> with slight variation you can do the same.

You seem to be talking about schismatic tuning. Again, there's more to approximating 5-limit JI than just this one rank two temperament, or even than both schismatic and meantone.

On Sun, Nov 28, 2010 at 4:33 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
> >
> > if the concept is important then one has to be able to say why
> > it is useful.
>
> Because there's more to regular tunings than just rank one tunings, or even rank one and rank two. Since five-limit JI is rank three, this should be obvious.
>
> > If it is say you can resolve it into a single chain when you
> > temper i would like to point out if you think of 5 limit how it
> > has been thought of since the people in India as a single chain
> > with slight variation you can do the same.
>
> You seem to be talking about schismatic tuning. Again, there's more to approximating 5-limit JI than just this one rank two temperament, or even than both schismatic and meantone.

And then there's that whole thing about puns and comma pumps, which is
where JI in general starts to become less meaningful.

-Mike

On 28 November 2010 01:39, Kraig Grady <kraiggrady@anaphoria.com> wrote:
> Every 12 tone temperament was conceived  as being a singular
> variable chain.
> This is the only example you will kind in history. look at
> Barbour's book

In that case, the rank is the number of variations in the chain, plus
one (for the octave).

They aren't the only examples we talk about -- or the ones the "rank"
terminology was invented for. We aren't constrained by history.

>  This scale shows the absurdity of calling things ranks at all
> as it makes complicated a simple thing into a convoluted process
> that tells us nothing no offers us anything we didn't have before.

What's complicated? What's convoluted?

>  It also alienates it from periodicity blocks and constant
> structures. These too are simply thought of as a single chain
> that varies which is also reflected in mapping scales to a
> keyboard.

It isn't alienated at all. Periodicity blocks and constant structures
are directly relevant to regular temperaments, which is where ranks
come from. The rank is the number of unison vectors you don't temper
out, plus one, assuming octave equivalence. You might think of them
as a single chain. Others don't.

>  also the recurrent sequence also fit into this category.

I don't think it's useful to talk about the ranks of them, which may
be why nobody has.

Graham

On 28 November 2010 02:45, Mike Battaglia <battaglia01@gmail.com> wrote:
> On Sat, Nov 27, 2010 at 4:39 PM, Kraig Grady <kraiggrady@anaphoria.com> wrote:
>>
>> Every 12 tone temperament was conceived as being a singular
>> variable chain.
>> This is the only example you will kind in history. look at
>> Barbour's book
>
> So rank-2. But things like Werckmeister aren't like that, right?

Werckmeister III is exactly like that, with two different sizes of
fifth, so it'll be rank 3. Unless there's some pair of generators
that makes it rank 2, I don't know.

> I don't understand the protests here. The main point of my analysis
> was simply to show that there are, generally speaking, three interval
> classes per step size, within which there is a variation of no more
> than 2 cents. And then if you analyze it that way, it ends up looking
> like something that's halfway between 5-limit JI and meantone, where
> the syntonic comma is shrunk but not eliminated.
>
> This is basically the goal of any well-temperament, right? So what's
> wrong with expressing this in the language of mathematics?

I don't think fractal dimensions are going to work for it. You need
to define how notes are added with increased resolution. For a rank 2
temperament, you can say the number of notes per octave is
proportional to the number of notes in a chain, so the dimension would
be 2.

>> It also alienates it from periodicity blocks and constant
>> structures.
>
> Wouldn't a 5-limit PB just be rank 1?

Its rank is 3 if it's JI. Its fractal dimension would be 1, because
it can only be extended by adding pitches in a linear series.

You can also define a rank 1 homomorphism for a periodicity block or a
well temperament. So for 12 tones, you say that 3:2 maps to 7 steps,
and so on. In this case there's no homomorphism from tempered
intervals to concrete pitch differences. You need to map pitches to a
fixed scale of higher rank.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> Werckmeister III is exactly like that, with two different sizes of
> fifth, so it'll be rank 3. Unless there's some pair of generators
> that makes it rank 2, I don't know.

It's generated by 2^(1/4) and 3, so rank 2.

On 30 November 2010 13:08, genewardsmith <genewardsmith@sbcglobal.net> wrote:
>
>
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
>> Werckmeister III is exactly like that, with two different sizes of
>> fifth, so it'll be rank 3.  Unless there's some pair of generators
>> that makes it rank 2, I don't know.
>
> It's generated by 2^(1/4) and 3, so rank 2.

Is it generally true that a circulating temperament has a rank equal
to the number of distinct intervals you use to circulate? This works
for meantone if you count the wolf.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> Is it generally true that a circulating temperament has a rank equal
> to the number of distinct intervals you use to circulate? This works
> for meantone if you count the wolf.

If there are r distinct intervals and P is the period, then there is a linear combination of the intervals equal to 0 mod P, and the r distinct intervals generate the scale mod P, so there can be no more than r+1 generators, including P. If one of the intervals appears only once, it can be written in terms of the others, bringing the rank down to r. However, this is not the only way that can happen: in the Werckmeister III example, we have 4f + 8j = 7P, where f is the flat fifth and j is the just fifth. Hence f + 2j = 7/4 P, so f = 7/4 P - 2j. Hence P/4 and j work as generators. An example where the rank was r+1 would be nice as then we would know that is possible, which I suspect is true.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

An example where the rank was r+1 would be nice as then we would know that is possible, which I suspect is true.
>

Come to think of it, we have

n1*f1 + n2*f2 + ... nr*fr = n*P

Dividing by n, this gives

P = n1*(f1/n) + n2*(f2/n) + ... + nr*(fr/n)

Hence f1/n ... fr/n are r generators, though it's quite likely that a smaller group would serve. In any case, we get rank r.

Gene wrote:

>If there are r distinct intervals and P is the period, then there is a
>linear combination of the intervals equal to 0 mod P, and the r
>distinct intervals generate the scale mod P, so there can be no more
>than r+1 generators, including P.

The key phrase being no more than. It's easy to construct
subsets of the 5-limit diamond with 4 or more sizes of 3rd,
for instance. So the question, it seems to me, is how to find
the minimal "distinct" generating intervals.

>If one of the intervals appears only
>once, it can be written in terms of the others, bringing the rank down
>to r.

What if three of the intervals appear once each?

>However, this is not the only way that can happen: in the
>Werckmeister III example, we have 4f + 8j = 7P, where f is the flat
>fifth and j is the just fifth. Hence f + 2j = 7/4 P,
>so f = 7/4 P - 2j. Hence P/4 and j work as generators.

Clever.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:

> Clever.

Not so much; see my next posting. For some reason I was thinking I couldn't divide the intervals, but the question was about rank. But you know by now you get the benefit of "off the top of my head" in dealing with me.