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Message: 6076

Date: Tue, 8 Jan 2002 12:09:47

Subject: Re: please simplify equation

From: monz

Hi Gene,


> From: genewardsmith <genewardsmith@xxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Tuesday, January 08, 2002 11:55 AM
> Subject: [tuning-math] Re: please simplify equation
>
>
> Hmmm...if it's in your dictionary, it might be well to
> be more precise and point out that for a quadratic number
> field, like the golden ratio field Q(r), that (a+br)/(c+dr)
> gives all of the elements, but for a field of degree greater
> than three that would not be so. In general, however, for
> a field of degee d, elements of the form
> a_0 + a_1 r + ... + a_{n-1} r^{d-1} with a_i a rational number
> and r an algebraic number define the number field Q(r) of
> degree d, where d is the degree of the irrreducible
> polynomial satisfied by r.
> 
> Maybe you should just skip the generalities and say that
> every number of the form (a+br)/(c+dr) with a,b,c,d rational
> can be reduced by the formula I gave to a unique form A + Br,
> with A and B rational, and that Q(r) defined by this is an
> algebraic number field, analogous to the ordinary rational
> numbers.


Thanks for the further elaboration!  But I hesitate to put all
of this into the "golden meantone" definition.  Wouldn't it be
better as part of the "algebraic number" definition?  If the
latter, then please suggest how the information specific to
each Dictionary entry should be placed and how they should
be linked.  I'm understanding you as saying that golden
meantone is an example of a specific type of algebriac number
field, and so the Dictionary entries should be written to
reflect that.



-monz


 




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top of page bottom of page up down Message: 6078 Date: Tue, 8 Jan 2002 13:13:32 Subject: observation on golden meantone formula From: monz Hmmm ... take a look at this: Yahoo groups: /tuning-math/files/monz/goldenMT-lattice.gif * Here I plotted the values for c = (8a + 11b) from the golden meantone formula (2^a)*(v^b) = 2^[ (c - b*PHI) / 11]. I found two things interesting about this graph. First, it exactly mirrors the placement of the PHI-related intervals in an interval matrix I made, ordered according to generator number. Secondly ... the plot of the "c" values creates a familiar-looking lattice-diagram-type pattern, where the southwest-to-northeast axis is the inverse of the generator (so it's the "4th") and the southeast-to-northwest axis is the "whole tone". What's going on here? Curious... -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
top of page bottom of page up down Message: 6086 Date: Wed, 9 Jan 2002 14:30:44 Subject: Re: background reading From: monz > From: mfeustl <mfeustl@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, January 09, 2002 9:55 AM > Subject: [tuning-math] background reading > > > Holy smoke! I did a search on Dowland tuning and I'm delighted to > have found this group. I'm double-majoring in applied math and > instrumental jazz, and I'm always looking for ways to put math and > music together. A fair amount of the discussion here is still a > little over my head, though-- is there any background reading someone > could suggest to get me up to speed? My online Dictionary of Tuning Terms should give you a lot to chew on. Definitions of tuning terms: index, (c) 1998 by Joe Monzo * And if you haven't already found it, I have a webpage devoted entirely to an exploration of Dowland's tuning. John Dowland's Lute Fretting (c)1998 by Joe Monzo * I just gave a presentation on this in Italy four months ago. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
top of page bottom of page up down Message: 6089 Date: Thu, 10 Jan 2002 23:34:57 Subject: Re: All in the spirit of friendship, Gene From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: >but I'll try to work more like this. I think this will be helpful for everyone, and again I apologize for my crankiness -- I got no sleep last night aside from a short piece of music that came to me in a dream -- and departed my mind almost as quickly. >In that spirit, can you explain >if the 27-et hyperpythagorean system is positive, or something else? 27-tET is considered a _triply positive_ system. See page 8 of http://www.anaphoria.com/xen2.PDF - Ok * . . .
top of page bottom of page up down Message: 6090 Date: Thu, 10 Jan 2002 05:38:47 Subject: Re: Some 8-tone 72-et scales From: clumma >I've decided to add the 9-limit numbers to my set of measures; that >gives us a better idea of the nature of these scales, in that we >can see how much of the 11-limit harmony, if any, actually involves >11. Good! > [0, 7, 14, 30, 37, 53, 60, 67] > [7, 7, 16, 7, 16, 7, 7, 5] > edges 11 17 21 22 connectivity 2 3 5 5 This scale looks better than it would have. 5 at the 9 limit is better than at the 11 limit. That's why I think we should be normalizing against limit and cardinality of the scale. -Carl
top of page bottom of page up down Message: 6091 Date: Thu, 10 Jan 2002 18:59:18 Subject: Re: Dictionary query From: monz > From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, January 10, 2002 1:56 PM > Subject: [tuning-math] Re: Dictionary query > > > Today, on these lists, we tend to call negative systems "meantone" > and positive systems "schismic". The reason 700 cents was chosen as > the dividing line between "negative" and "positive" is that when the > fifth is below 700 cents, the "meantone" (+4 fifths) approximation to > the 5/4 is better than the "schismic" (-8 fifths) approximation to > the 5/4. When the fifth is above 700 cents, the "schismic" > approximation to the 5/4 is better than the "meantone" approximation > to the 5/4. I might differ, saying that there is a "gray area", and > also factoring the 6/5 into consideration . . . but the definitions > are well-established and there is no reason to favor ones which could > breed potential contradictions. > > As for your definition pages, Monz, they definitely give the wrong > idea. Positive systems should be characterized by the fraction of a > _schisma_ that the fifths differ from just -- this is the relevant > measure of them. Knowing what fraction of a syntonic comma a positive > system's fifth might have been _increased_ by is irrelevant for > understanding the functioning of the system, and is potentially > misleading. Thanks very much for that, Paul. So how does it look now? Definitions of tuning terms: positive system, (c) 2001 by Joe Monzo * -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
top of page bottom of page up down Message: 6096 Date: Thu, 10 Jan 2002 19:13:58 Subject: Re: Dictionary query From: monz > From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, January 10, 2002 7:05 PM > Subject: [tuning-math] Re: Dictionary query > > > > Thanks very much for that, Paul. So how does it look now? > > Definitions of tuning terms: positive system, (c) 2001 by Joe Monzo * > > > > > > > > -monz > > Unfortunately, 22 is not a schismic temperament . . . this is my > fault, of course . . . I later alluded to the correct definition in > conversation with Gene, as you can see . . . I'm a bit too tired to > correct this now, but I'm sure Graham or John Chalmers can help you > if they're available before I can get back to you. OK, but if they don't post anything here, please do give me more info. > P.S. Monz, why do you like to keep incomplete/incorrect > definitions/descriptions at the top of your dictionary pages, or even > in there at all? Why not attempt for the more precise, univerally > agreed-on definitions/examples first, and then post > alternate/intermediary-stages-in-someone's-thinking stuff later, > preferably on entirely separate webpages? Umm... because I'm a decent author but a lousy editor? The Dictionary is always a work-in-progress, and I prefer to simply amend definitions that are not complete. But if they really are *incorrect*, then please, by all means, not only give me the correct information, but also tell me what to get rid of! I have no problem deleting something that really is wrong. (I may not get around to it as quickly as any of us would like... but that's another story...) If even one other person here agrees with you that the commatic description of positive systems is absolutely useless, then they're history. Let me know. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
top of page bottom of page up down Message: 6097 Date: Thu, 10 Jan 2002 19:44:44 Subject: Re: Optimal 5-Limit Generators For Dave From: paulerlich --- In tuning-math@y..., graham@m... wrote: > Paul: > > It should be quite straightforward to prove. How could you tell > > whether 50:49 produces torsion or not in an octave-invariant > > formulation? > > Do you care about it being [dis]proven, then? Sure, but this works pretty well as a "proof" for me. > I expect your algorithm for > generating periodicity blocks will solve everything. How so? It doesn't detect torsion . . . I needed Gene's fix, which takes the powers of 2 into account, to do that. > But I haven't looked > it up because people keep saying they aren't interested, while >asking more > and more questions. Looked what up? > It won't change anything musically. Not sure what you mean. > > > Paul: > > I thought Gene showed that the common-factor rule only works in the > > octave-specific case. > > I don't remember him considering the adjoint, rather than the wedge > product. But we may not need it anyway. Gene, any enlightenment? > > Me: > > > Pairs of ETs with > > > torsion don't work with wedge products either. It may be that the > > sign of the > > > mapping can be used to disambiguate them. Otherwise, give the > > range of generators > > > as part of the definition. > > Paul: > > You've lost me. Gene, any comments? > > Meaning contorsion here. The octave-specific wedge product can remove it, > but not use it. An octave-equivalent wedge product (the octave- equivalent > mapping) will treat such systems, wrongly, as requiring a division of the > octave. But starting from ETs it does make more sense to use > octave-specific vectors in the first place. Perhaps we should only ask if > unison vectors can work in an octave-equivalent system, in which case this > problem doesn't apply. > > Me: > > > Wouldn't it be nice to say whether or not Fokker's methods would > > have worked if he > > > had run into torsion? > > Paul: > > I'm pretty sure the answer is no. Gene? > > The main thing we've added to Fokker (after Wilson) is the mapping, > instead of merely counting the number of notes in the periodicity block. > The Monz-shruti example gives a periodicity block with more notes than you > need for the temperament, but the mappings still come out. What do you mean, "the mappings still come out"? The 3:2, for example, is not always represented by the same number of steps. > There are more > insidious examples of torsion where the mappings don't work either. The > problem being that octave-equivalent matrices don't differentiate commatic > torsion from systems requiring a period that isn't the octave. Exactly my point.
top of page bottom of page up down Message: 6098 Date: Thu, 10 Jan 2002 19:17:02 Subject: Re: Distinct p-limit intervals and ets From: monz > From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, January 10, 2002 6:34 PM > Subject: [tuning-math] Re: Distinct p-limit intervals and ets > > > Huh? I'm looking at Definitions of tuning terms: unique, (c) 1998 by Joe Monzo * , > and I see that the links are to > ftp://ella.mills.edu/ccm/tuning/papers/consist_limits.txt . . . You > should change that to > Consistency limits of equal temperaments * . . . OK, that's done, and it's been uploaded. I just wasn't sure if the other link should have been in the definition. Anyway, my request still stands: can you please explain these tables in more detail? I don't quite understand what's in them. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
top of page bottom of page up down Message: 6099 Date: Thu, 10 Jan 2002 19:45:16 Subject: Re: Distinct p-limit intervals and ets From: paulerlich --- In tuning-math@y..., <manuel.op.de.coul@e...> wrote: > > >Monz, the links to the tables are outdated. Manuel, could you provide > >the updated links? > > Consistency limits of equal temperaments * and > Equal temperament step size ranges for consistency limits * Monz, would you update your links in the "unique" definition, please?
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