Tuning-Math Digests messages 7325 - 7349

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Message: 7325

Date: Thu, 14 Aug 2003 23:39:00

Subject: Re: Comments about Fokker's misfit metric

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > not lost; i just read three. i'm not sure i understand how 
> > consistency is enforced differently than just using the best 
> > approximations to the primes, though.
> 
> All it does is supply a different, and normally preferable, answer 
to 
> the question of how to define a standard val.

yes, i just wasn't seeing how it arrived at that val.

> Choosing any val 
> enforces consistency.

yup!


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Message: 7326

Date: Thu, 14 Aug 2003 17:55:38

Subject: Re: Comments about Fokker's misfit metric

From: Carl Lumma

>personally, i would also include "9/6", since a complete 11-limit 
>hexad contains both a 3:2 and a "9:6", so the perfect fifth should
>be weighted twice.

Here, here!

-Carl


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Message: 7331

Date: Thu, 14 Aug 2003 01:03:25

Subject: Re: tctmo!

From: Carl Lumma

>rather than being "what's been going on on the tuning-math list",

What rather than "what's been going on on the tuning-math list"?

>>		1.1.1-- A theme played in a different mode keeps
>>		generic intervals (3rds, etc.) the same while pitches
>>		change.
>
>>			1.1.1.1-- This is, in fact, only true for
>>			Rothenberg-proper scales, such as the familiar
>>			diatonic scale in 12-tone equal temperament.
>
>the pythagorean diatonic is improper but would seem to have the 
>property you're trying to describe. so would blackjack . . .

I suppose any scale would, if we define generic intervals as simply
being the intervals between consecutive scale degrees.  We need
the assumption that listeners keep track of the relative sizes of
intervals to build a map of generic intervals.  This is getting a
bit complicated, so I've axed the section.

-Carl


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Message: 7332

Date: Thu, 14 Aug 2003 12:36:26

Subject: Re: Clough & Myerson

From: Graham Breed

Gene Ward Smith wrote:

>Considering d note scales in c-equal, they define the genus as the 
>step sizes of the gaps between the notes of the chord, modulo 
>transposition, in terms of d, and the species as the same in terms of 
>c. In other words, in C major CEG is 0 2 4 in diatonic terms and 
>0 4 7 in chromatic terms, so the genus is 223 and the species is 435. 
>Theorem 5 from Variety helps explain why Eytan was hipped about 
>certain kinds of scales:
>
>Theorem 5
>For any reduced scale, if every chord species is unambiguous in its 
>generic membership, then c = 2d-1 or c = 2d-2. The converse is also 
>true.
>
The reason Eytan gives for using them is that they have his kind of 
efficiency (each chromatic interval is represented at least once), and 
are proper with no more than one ambiguous interval pair (the tritone).  
I proved this in

Maximal Evenness Proofs *

You can reduce scales without affecting propriety by ensuring 
Rothenberg's alpha matrix (showing the ordering of intervals, rather 
than their sizes) is preserved.  I believe that every strictly proper 
MOS has a maximally even counterpart with the same alpha matrix, but 
haven't proved it yet.

>The fine print here is "reduced scale"; they reduce a scale with 
>Myhill's Property by lowering the size of the et c so that the sizes 
>of the scale steps are 1 and 2. This turns out to be slick way of 
>proving things, but it doesn't seem to be more than that.
>
Do they insist on gcd(c,d)=1 as well?  The reduction is a necessary 
condition for Agmon's efficiency.  It does, however, mean that a lot of 
strictly proper scales lose their strictness.

>A nice fact which is proven as well is that if (c, d)=1 then there is 
>a unique scale with Myhill's Property and it can be obtained very 
>simply as floor(kc/d), 0 <= k < d. This can be generalized 
>immediately by considering periods rather than octaves, allowing the 
>elimination of the condition that (c, d)=1. This does all the "white-
>key black key" stuff I was doing a while back, (and Graham too, in 
>his own way?) in a slicker way.
>  
>
Yes, those are maximally even scales.  See the link above.

>That's not all but the papers are worth reading and done right, which 
>we cannot always count on to be the case.
>
Oh, then I might look them out one day.


                         Graham


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Message: 7335

Date: Fri, 15 Aug 2003 17:15:31

Subject: Re: Cartesian product of two ET scales

From: Graham Breed

Carlos wrote:

>Ok, I think I gave a wrong example. What you say is clear in the case we 
>take cyclic groups in which both generators are of the form g=2**(1/n) 
>where n is some number.  This is, in the case of assuming the octave 
>equivalence.
>
>I think what I was having in mind was something different and more general 
>like, gaving  cyclic groups with generators of the form
>
>g1= 3**(1/n)  (I am think of the Bohlen-Pierce scale, for example n=13), 
>
>g2= 2**(1/m) i.e a regular ET scale  (say m=2)
>  
>
If the equivalence interval isn the same, I think they should both be 
considered of infinite order.  In which case adding them won't give a 
cyclic group, but a linear temperament.  For which, see

How to find linear temperaments * 
<The Proxomitron Reveals... *>

>What would be the product group look like? It has to be cyclic of order nxm 
>but it does not have a single generator. It would seem that the generator 
>has to be the pair (g1,g2), or not?
>  
>
If it doesn't have a single generator then it isn't cyclic, by 
definition.  As they do have an interval in common, you could define 
12-equal to be 19 notes to a 3:1.  Then the resultant group will be 
cyclic of order 19*13=247.


                                              Graham


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Message: 7343

Date: Mon, 18 Aug 2003 20:13:33

Subject: Re: Cartesian product of two ET scales

From: Paul Erlich

balzano tried to make such application in his beautiful papers 
generalizing the 12-equal diatonic scale to any n*(n+1)-equal tuning, 
where the generalized diatonic scale has n+n+1 notes. i find his use 
of the direct product sneaky and his thesis invalid . . .

--- In tuning-math@xxxxxxxxxxx.xxxx Carlos <garciasuarez@y...> wrote:
> Graham and Gene,
> 
> Thanks. I think that I was trying to think of something which does 
not make 
> much sense, you need to have the same equivalence (octave or 
otherwise) in 
> both groups to have a meaninfull product.
> 
> What I was trying to do is to find an application to the concept 
of "direct 
> product" of groups.
> 
> Say you have the groups two cyclic groups G1= {0*,1**} and G2=
{0**,1**,2**} 
> the direct product would be G1 x G2 =
> 
> { (0*,0**),(0*,1**),(0*,2**), (1*,0**),(1*,1**),(1*,2**) }
> 
> which is cyclic of order 6 and which has as a generator (1*,1**)
> 
> I was trying to figure out if these could have some sort of 
application to 
> scales with more than one generator.
> 
> Thanks
> 
> Carlos
> 
> 
> 
> 
> On Friday 15 August 2003 22:19, Gene Ward Smith wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx Carlos <garciasuarez@y...> 
wrote:
> > > I think what I was having in mind was something different and 
more
> >
> > general
> >
> > > like, gaving  cyclic groups with generators of the form
> > >
> > > g1= 3**(1/n)  (I am think of the Bohlen-Pierce scale, for 
example
> >
> > n=13),
> >
> > > g2= 2**(1/m) i.e a regular ET scale  (say m=2)
> > >
> > > What would be the product group look like?
> >
> > It doesn't make much sense to reduce this to octave pitch 
classes; if
> > you don't you have a free group of rank two, generated by g1 and 
g2.
> >
> >
> >
> >
> > To unsubscribe from this group, send an email to:
> > tuning-math-unsubscribe@xxxxxxxxxxx.xxx
> >
> >
> >
> > Your use of Yahoo! Groups is subject to
> > Yahoo! Terms of Service *


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Message: 7345

Date: Mon, 18 Aug 2003 21:50:30

Subject: Re: Cartesian product of two ET scales

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > balzano tried to make such application in his beautiful papers 
> > generalizing the 12-equal diatonic scale to any n*(n+1)-equal 
> tuning, 
> > where the generalized diatonic scale has n+n+1 notes. i find his 
> use 
> > of the direct product sneaky and his thesis invalid . . .
> 
> Are the papers actually beautiful? The thesis seems frankly brain-
> damaged; should I read them anyway?

they're beautiful in their trickery. if you were a mathematician who 
knew little of music and tuning theory and history, i wouldn't blame 
you for being taken in by it.


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Message: 7347

Date: Tue, 19 Aug 2003 21:51:08

Subject: Re: New tuning group: microtuning

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Carlos <garciasuarez@y...> 
wrote:
> > Do you mean that the tuning@xxxxxxxxxxx.xxx
> > 
> > will not work anymore?
> > 
> > Please clarify
> 
> It will no longer accept new posts.

it's accepting them now, so i suggest setting the microtuning group 
not to accept any new posts.


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