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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!
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
Message: 7327 Date: Thu, 14 Aug 2003 00:50:08 Subject: Re: Comments about Fokker's misfit metric From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carlos <garciasuarez@y...> wrote:
> The intervals I have considered then are > > Just fifth 3/2 > Just mayor third 5/4 > Just minor third 6/5 > Harmonic seventh 7/4 > Subminor fifth 7/5 > Subminor third 7/6 > Supersecond 8/7 > Major tone 9/8 > Super major third 9/7 > Acute minor seventh 9/5 > Trumpet interval 11/10 > The 11th harmonic 11/8 > Meshaqah quartertones 11/6 > Unamed_1 11/9 > Unamed_2 11/7
Having both 7/4 and 8/7 is redundent. If you eliminate that, you have a table of representatives for the 14 11-limit consonances, in the terminology of this group.
Message: 7328 Date: Thu, 14 Aug 2003 00:52:18 Subject: Re: interval vector From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx <monz@a...> wrote:
> > From: Gene Ward Smith [mailto:gwsmith@s...] > > Sent: Wednesday, August 13, 2003 5:42 PM
> i don't think other scholars would be too accepting > of *me* being the first to promote the use of my own > name for a term, so ... can *you* publish something > with it? that would be a start in gaining acceptance.
I've decided that this is the 21st century and paper is obsolecent. It's up on www.xenharmony.org.
Message: 7330 Date: Thu, 14 Aug 2003 05:16:27 Subject: Clough & Myerson From: Gene Ward Smith Reprints of two papers by Clough and Myerson appeared in my mailbox today from Down Under, courtesy of Gerry Myerson. One is "Variety and Multiplicity of Diatonic System", the other (from the Math Monthly) "Musical Scales and the Generalized Circle of Fifths". 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 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. 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. That's not all but the papers are worth reading and done right, which we cannot always count on to be the case.
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
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 * [with cont.] (Wayb.) 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
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 * [with cont.] (Wayb.) <http://microtonal.co.uk/temper/method.html * [with cont.] (Wayb.)>
>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
Message: 7336 Date: Fri, 15 Aug 2003 20:19:05 Subject: Re: Cartesian product of two ET scales From: Gene Ward Smith --- 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.
Message: 7339 Date: Fri, 15 Aug 2003 07:20:32 Subject: Re: Clough & Myerson From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> >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?
Yes, always. Nonoctave periods do not seem to have been considered.
Message: 7341 Date: Fri, 15 Aug 2003 11:09:43 Subject: Re: Cartesian product of two ET scales From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carlos <garciasuarez@y...> wrote:
> I am wondering which, if any of the historical relevant tunnings
produces
> scales which are cyclic groups of this kind, i.e. with two
independent
> generators.
Cyclic groups by definition can be expressed in terms of a single generator. They can also be factored into cyclic groups of prime power order, and if N and M are relatively prime, we are simply combining the two factorizations. If you take N, M (not necessarily coprime) and combine the corresponding equal temperaments then you will get the cyclic group of order LCM(N, M).
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 * [with cont.] (Wayb.)
Message: 7344 Date: Mon, 18 Aug 2003 21:44:24 Subject: Re: Cartesian product of two ET scales From: Gene Ward Smith --- 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?
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.
Message: 7346 Date: Tue, 19 Aug 2003 20:19:46 Subject: Re: New tuning group: microtuning From: Gene Ward Smith --- 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.
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.
Message: 7348 Date: Tue, 19 Aug 2003 01:45:44 Subject: Re: Cartesian product of two ET scales From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> 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.
I'm a mathematician who got disgusted and quit doing music theory when Balzono turned down his attempt to get published, so I'm prejudiced against.
Message: 7349 Date: Tue, 19 Aug 2003 07:24:24 Subject: New tuning group: microtuning From: Gene Ward Smith It seems we need a replacement for the tuning group; the Yahoo version of that is now at microtuning@xxxxx.xxxxxx.
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