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Message: 5425 - Contents - Hide Contents Date: Wed, 23 Oct 2002 12:24:24 Subject: Re: Epimorphic From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote: >>> --- In tuning-math@y..., manuel.op.de.coul@e... wrote:>>> So I may conclude that the simplest example of a JI, >>> CS and non-epimorphic scale is this one: 1/1--4/1 >>>> If you want to call that a scale. >> i'm confused. are we assuming a 2/1 interval of equivalence? then > wouldn't be a scale at all, would it?I made no assumptions about intervals of equivalence, but I did assume any mapping would map every positive rational number to an integer, which v2(q)/2, where v2 is the 2-adic valuation, does not. Other definitions are possible.

Message: 5426 - Contents - Hide Contents Date: Wed, 23 Oct 2002 16:25:45 Subject: Re: Epimorphic From: manuel.op.de.coul@xxxxxxxxxxx.xxx I've uploaded the new Scala version so the epimorphic code can be tried now by others (show data). Also the Edit->Sound settings dialog has been improved a bit. Perhaps I should once mention another new feature which is the automatic keyboard mapping available in the MIDI relay dialog. You need to set "Automatic note name mapping" to use it, and select an appropriate key for the music. It doesn't respond to key changes, but it's not a trivial feature either. So if you have a midi loopback or keyboard, you can use bigger scales like 31-tET and the results will still make sense. For ETs it's easy to make the notation system follow the tuning if you set the "Set corresponding notation system" tick box in File->New->Equal temperament. Manuel

Message: 5427 - Contents - Hide Contents Date: Wed, 23 Oct 2002 11:02:42 Subject: Do you sleep Gene? :)) From: Pierre Lamothe Gene wrote: Try 1/1--2700/2401--5/4--4/3--3/2--5/3--2401/2400--2/1 Paul wrote: 2401/2400 is not between 5/3 and 2/1, gene! Gene wrote: OK, so be picky about it. :) 2401/1280 is, though. I intended to put up a slightly modified version of 1--9/8--5/4--4/3--3/2--5/3--15/8--2, with the 9/8 adjusted down by 2400/2401 and the 15/8 adjusted up by the same amount. Since h7(2401/2400) = 2, this throws a spanner in the works. You conclude from there, your presumed counterexample is not epimorphic. I hope I don't need to show that that CS implies EPIMORPHISM as I shown that. However you have doubts and believe it's not epimorphic with your definition. You're wong! Using your words I say there is a val h such that if qn is the nth scale degree, then h(qn) = n. I don't say h(qn) = 4n which would be true even with the false way you represent it. No, there is a true val such that h(qn) = n. It was not bad to try to show off with your spanner, but it would have been better to try to understand what you were doing. You simply forgotten to reduce the basis before wedging. You used the basis <2 3 5 7> while there exist a dependance about 7. The minimal basis is <2 3 5 2401>. Using the false one, you introduce inappropriate lattice points, so the corresponding val would be [28 44 64 78]. It's the type of error where modifying unwittingly the representration you attribute the thing to the represented object. There is no problem with that CS and epimorph scale, but with your way to described it with a spanned primal basis, where the octave periodicity 7 appears 28, for each block is filled with supplement lattice points never used. In the basis <2 3 5 49>, the val would be [14 22 32 78] while in the minimal basis, the val is [7 11 16 78] and your example is perfectly epimorph. Pierre [This message contained attachments]

Message: 5428 - Contents - Hide Contents Date: Wed, 23 Oct 2002 18:43:14 Subject: Re: NMOS From: wallyesterpaulrus --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>>>> I meant a chain of generators where the number of generators is >>>> a multiple of a number giving a MOS--or in other words, is a>>>>> multiple of something arising from a semiconvergent. >>>>>> How would the multiple property justify itself againt scales >>> that were two MOSs superposed at some other interval (besides >>> the comma)? >>>> huh? did you mean the generator? >> Nope, I meant the chromatic unison vector.then i have no idea what this has to do with what gene was talking about.>>> In the case of Messiaien, the octatonic scale is an NMOS. >>>> it's also an MOS, plain and simple. >> Of what generator and ie?the generator is the semitone (or fifth, etc.), the interval of repetition is 1/4 octave. the ie is irrelevant, but is usually taken to be an octave.>>> For the interlaced diatonic scales in >>> 24-tET, Paul has pointed out that this has excellent 7-limit >>> harmony in 26. I forget at what interval this is, but I >>> don't think it's the comma. >> >> the comma? > > 2187/2048 ??????>>> But Paul's excellent decatonics in 22 are two pentatonic MOSs >>> apart by a non-comma (the half-octave). >>>> these are the symmetrical decatonics, and they _are_ MOSs. the >> pentachordal decatonics aren't. the same goes for the 14-note >> scales in 26 -- the symmetrical ones are MOS, the "tetrachordal" >> ones aren't. >> I don't see how the symmetrical decatonics can be MOS, since > they don't have Myhill's property. Or were you the person > who was saying the fractional-period temperaments were MOSs > without Myhill's property?the problem is that most theorists speak of the interval of equivalence and the interval of repetition as the same thing. this is poor because (a) when you derive these beasts from unison vectors, the two can differ; and (b) it leads to fumbles like omitting the octatonic scale from the famous self-similar paper (though it clearly qualifies). if you regress to this way of thinking, then the ie for the symmetrical decatonic would have to be the half-octave, and then yes, they are Myhill.> Be nice to get a FAQ on torsion v. fractional period v. > MOS v. NMOS v. Myhill.sure . . .

Message: 5429 - Contents - Hide Contents Date: Wed, 23 Oct 2002 18:49:54 Subject: Re: NMOS From: wallyesterpaulrus --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>>>>> I meant a chain of generators where the number of generators is >>>>> a multiple of a number giving a MOS--or in other words, is a>>>>>> multiple of something arising from a semiconvergent. >>>>>>>> How would the multiple property justify itself againt scales >>>> that were two MOSs superposed at some other interval (besides >>>> the comma)? >>>>>> huh? did you mean the generator? >>>> Nope, I meant the chromatic unison vector. >> then i have no idea what this has to do with what gene was talking > about.scratch that -- i see now! they *are* superposed at the chromatic unison vector -- i was thinking of the end of one being attached to the beginning of the other via the generator, but this is equivalent! thanks carl! the reason torsion comes in is that we are treating the chromatic unison vector as a *step* in the NMOS, while an *integer multiple* (or exponent, in frequency-ratio space) of the chromatic unison vector is a 0-step interval. thus the paradox -- if you temper out the latter, you end up tempering out the former, and you end up with the "wrong" number of notes -- a fraction of what the determinant of the fokker matrix would tell you.>> 2187/2048 > > ??????yes, i see now that this is one possible expression for the chromatic unison vector in the diatonic scale. which is what you were thinking of. sorry.

Message: 5430 - Contents - Hide Contents Date: Wed, 23 Oct 2002 18:53:16 Subject: Fwd: [tuning] Re: Everyone Concerned From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote: >>>> If you take a lattice (p-limit, let's say) and then take a >>> sublattice (defined by commas, lets say) of the same rank n, if you >>> mod out R^n by the sublattice you get a discrete group on the >>> compact quotient. As I remarked, you should not assume every >>> mathematican would call this a lattice! >>>> what's the musical analogue? i'm lost. >> A 5-limit octave-reduced Fokker block painted on a donut, hopefullyusing something edible. but if you unroll this onto an infinite bingo card, which hopefully you've seen examples of lately, then it *is* a lattice. and what if you consider the donut to be merely a flawed representation of a non- euclidean closed universe with constant curvature everywhere? wouldn't the block then qualify as a lattice, even though it's finite?

Message: 5431 - Contents - Hide Contents Date: Wed, 23 Oct 2002 18:57:31 Subject: Fwd: [tuning] Re: Everyone Concerned From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> but if you unroll this onto an infinite bingo card, which hopefully > you've seen examples of lately, then it *is* a lattice.It's a quotient of a lattice. and what if

Message: 5432 - Contents - Hide Contents Date: Wed, 23 Oct 2002 19:01:50 Subject: Fwd: [tuning] Re: Everyone Concerned From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote: >>> but if you unroll this onto an infinite bingo card, which hopefully >> you've seen examples of lately, then it *is* a lattice. >> It's a quotient of a lattice.why? it's just R^2, isn't it?>> and what if >> you consider the donut to be merely a flawed representation of a non- >> euclidean closed universe with constant curvature everywhere? >> wouldn't the block then qualify as a lattice, even though it's finite? > > Topologically impossible.i don't see the big deal -- a lot of video games work this way.

Message: 5433 - Contents - Hide Contents Date: Wed, 23 Oct 2002 21:04:28 Subject: Fwd: [tuning] Re: Everyone Concerned From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> i don't see the big deal -- a lot of video games work this way.Video games work on manifolds?? Genus 0 corresponds to Riemann sphere, which is to say the complex projective line; positive curvature. Genus 1, the donut or algebraically an elliptic curve, zero curvature. Genus > 1, negative curvature. Relevance to tuning theory--to be determined.

Message: 5434 - Contents - Hide Contents Date: Wed, 23 Oct 2002 21:09:08 Subject: Fwd: [tuning] Re: Everyone Concerned From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote: >>> i don't see the big deal -- a lot of video games work this way. >> Video games work on manifolds??the edges of the screen wrap around and meet one another, hence a donut topology.> Genus 0 corresponds to Riemann sphere, which is to say the complexprojective line; positive curvature.> > Genus 1, the donut or algebraically an elliptic curve, zero curvature.i did say constant curvature, didn't i? last time i checked, zero was a constant . . . :)

Message: 5436 - Contents - Hide Contents Date: Wed, 23 Oct 2002 05:53:27 Subject: Re: NMOS From: Carl Lumma>>> > meant a chain of generators where the number of generators is >>> a multiple of a number giving a MOS--or in other words, is a>>>> multiple of something arising from a semiconvergent. >>>> How would the multiple property justify itself againt scales >> that were two MOSs superposed at some other interval (besides >> the comma)? >>huh? did you mean the generator?Nope, I meant the chromatic unison vector.>> In the case of Messiaien, the octatonic scale is an NMOS. >>it's also an MOS, plain and simple.Of what generator and ie?>> For the interlaced diatonic scales in >> 24-tET, Paul has pointed out that this has excellent 7-limit >> harmony in 26. I forget at what interval this is, but I >> don't think it's the comma. > >the comma? 2187/2048>> But Paul's excellent decatonics in 22 are two pentatonic MOSs >> apart by a non-comma (the half-octave). >>these are the symmetrical decatonics, and they _are_ MOSs. the >pentachordal decatonics aren't. the same goes for the 14-note >scales in 26 -- the symmetrical ones are MOS, the "tetrachordal" >ones aren't.I don't see how the symmetrical decatonics can be MOS, since they don't have Myhill's property. Or were you the person who was saying the fractional-period temperaments were MOSs without Myhill's property? Be nice to get a FAQ on torsion v. fractional period v. MOS v. NMOS v. Myhill. -Carl

Message: 5437 - Contents - Hide Contents Date: Thu, 24 Oct 2002 10:16:53 Subject: Re: NMOS From: Gene Ward Smith --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>> scratch that -- i see now! they *are* superposed at the chromatic >> the reason torsion comes in is that we are treating the chromatic >> unison vector as a *step* in the NMOS, while an *integer multiple* >> (or exponent, in frequency-ratio space) of the chromatic unison >> vector is a 0-step interval. thus the paradox -- if you temper out >> the latter, you end up tempering out the former, and you end up >> with the "wrong" number of notes -- a fraction of what the >> determinant of the fokker matrix would tell you. >> Okay, I've cut and pasted that, and look forward to when I have > some pegs to hang it on.I don't see why you two are talking about torsion.> I'm afraid I also don't understand the difference between the ie > (in the MOS sense) and the interval of repetition. It looks to > me like the symmetrical decatonic is MOS at the half-octave.Which is why it is MOS--which you just were denying!

Message: 5438 - Contents - Hide Contents Date: Thu, 24 Oct 2002 12:39:31 Subject: Re: NMOS From: manuel.op.de.coul@xxxxxxxxxxx.xxx> It looks to > me like the symmetrical decatonic is MOS at the half-octave. >Which is why it is MOS--which you just were denying!I have to disagree with this, if the octave is the IE but the scale repeats at the half octave it doesn't have Myhill's property because there is only one size of tritone. If you cut the scale in half then it does have this property, but it would be a different scale. I don't like the MOS term much because of this kind of confusion. Manuel

Message: 5439 - Contents - Hide Contents Date: Thu, 24 Oct 2002 11:13:52 Subject: Re: NMOS From: Gene Ward Smith --- In tuning-math@y..., manuel.op.de.coul@e... wrote:> I don't like > the MOS term much because of this kind of confusion.I've assumed MOS referred to the period; obviously you are asking for trouble if you assume otherwise. Since Joe's dictionary supports my usage, can we just take it as definitive here?

Message: 5440 - Contents - Hide Contents Date: Thu, 24 Oct 2002 13:41:57 Subject: Re: NMOS From: manuel.op.de.coul@xxxxxxxxxxx.xxx>I've assumed MOS referred to the period;Hey, I've always assumed it referred to the IE.>obviously you are asking for trouble if you assume otherwise. What trouble? >Since Joe's dictionary supports my usage, can we just take it >as definitive here?Just took a look. It says "only two different size intervals". This might be confusing, it doesn't exclude the possibility of less than two different size intervals. For Myhill's property it's exactly two different size intervals, prime and octave excluded. The interval of equivalence is musically more important than the period, that's why I assumed it is based on that. Manuel

Message: 5441 - Contents - Hide Contents Date: Thu, 24 Oct 2002 18:58:43 Subject: Re: NMOS From: wallyesterpaulrus --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>> sounds fine to me! the ie (in the MOS sense) *is* the interval of >> repetition. it's just that i prefer not to call it the ie, and >> reserve ie to mean the ratio by which pitches are reduced to pitch >> classes (usually 2 -- the octave). we had a discussion here where >> many names were proposed for the former thing -- i think maybe >> "period" actually won out. >> I like "period" for the MOS sense and "interval of equivalence" > or "equivalence interval" for the psychoacoustic sense. > > -Carlgreat! then you're on board with some of the developments that happened in your absence . . .

Message: 5442 - Contents - Hide Contents Date: Thu, 24 Oct 2002 19:00:07 Subject: Fwd: [tuning] Re: Everyone Concerned From: wallyesterpaulrus --- In tuning-math@y..., <Josh@o...> wrote:> Oh, you're talking about the game content, not > the mechanics of the CRT! right. > Your use of the term "screen" confused me. sorry. > The property you're describing varies from game > to game.didn't mean to suggest otherwise.> While it defies basic spatial properties > in some unobvious ways that are probably more > important than the obvious ones, I think it may > be a useful analogy for the quasi-spatial modeling > of non-strictly-spatial mathematical properties. > > My question about "lattices" would be whether it's relevant > if they're physically impossible, as long as the spatial > aspect is modeling non-spatial mathematical relationships. > > Anyone?i don't understand any of the above.

Message: 5443 - Contents - Hide Contents Date: Thu, 24 Oct 2002 19:00:59 Subject: Re: NMOS From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>>> scratch that -- i see now! they *are* superposed at the chromatic > >>> the reason torsion comes in is that we are treating the chromatic >>> unison vector as a *step* in the NMOS, while an *integer multiple* >>> (or exponent, in frequency-ratio space) of the chromatic unison >>> vector is a 0-step interval. thus the paradox -- if you temper out >>> the latter, you end up tempering out the former, and you end up >>> with the "wrong" number of notes -- a fraction of what the >>> determinant of the fokker matrix would tell you. >>>> Okay, I've cut and pasted that, and look forward to when I have >> some pegs to hang it on. >> I don't see why you two are talking about torsion.doesn't my statement above make sense to you??>> I'm afraid I also don't understand the difference between the ie >> (in the MOS sense) and the interval of repetition. It looks to >> me like the symmetrical decatonic is MOS at the half-octave. >> Which is why it is MOS--which you just were denying!clearly carl was correcting himself here!

Message: 5444 - Contents - Hide Contents Date: Thu, 24 Oct 2002 19:04:18 Subject: Re: NMOS From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., manuel.op.de.coul@e... wrote: >>> I don't like >> the MOS term much because of this kind of confusion. >> I've assumed MOS referred to the period; obviously you are asking >for trouble if you assume otherwise. Since Joe's dictionary supports >my usage, can we just take it as definitive here?i don't see any problem or confusion with the MOS term at this point - - however it's the academic terms, like myhill, which can be problematic as manuel pointed out.

Message: 5445 - Contents - Hide Contents Date: Thu, 24 Oct 2002 19:05:46 Subject: Re: NMOS From: wallyesterpaulrus --- In tuning-math@y..., manuel.op.de.coul@e... wrote:>> I've assumed MOS referred to the period; >> Hey, I've always assumed it referred to the IE. >>> obviously you are asking for trouble if you assume otherwise. > > What trouble?we have it on kraig's word that erv considers 3 2 2 2 2 3 2 2 2 2 to be MOS.

Message: 5446 - Contents - Hide Contents Date: Thu, 24 Oct 2002 19:44:25 Subject: Re: NMOS From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> doesn't my statement above make sense to you??Not really--what does NMOS have to do with torsion?

Message: 5447 - Contents - Hide Contents Date: Thu, 24 Oct 2002 20:01:30 Subject: Re: NMOS From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote: >>> doesn't my statement above make sense to you?? >> Not really--what does NMOS have to do with torsion?think about the Hypothesis. the Hypothesis says that if you temper out all but one of the unison vectors of a fokker periodicity block, you get an MOS. well, if the fokker periodicity block has torsion, you (may?) end up with an NMOS instead! cases in point: helmoltz 24 and groven 36.

Message: 5448 - Contents - Hide Contents Date: Thu, 24 Oct 2002 23:07:53 Subject: Re: NMOS From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> think about the Hypothesis. the Hypothesis says that if you temper > out all but one of the unison vectors of a fokker periodicity block, > you get an MOS. well, if the fokker periodicity block has torsion, > you (may?) end up with an NMOS instead! cases in point: helmoltz 24 > and groven 36.Helmholtz 24 is simply 23 consecutive fifths; it can be pretty well equated with the 24 out of 53 2MOS I gave the notes for. Torsion is not a consideration.

Message: 5449 - Contents - Hide Contents Date: Thu, 24 Oct 2002 23:15:10 Subject: Re: NMOS From: monz hi Gene,> From: "Gene Ward Smith" <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, October 24, 2002 4:07 PM > Subject: [tuning-math] Re: NMOS > > > --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote: >>> think about the Hypothesis. the Hypothesis says that if you temper >> out all but one of the unison vectors of a fokker periodicity block, >> you get an MOS. well, if the fokker periodicity block has torsion, >> you (may?) end up with an NMOS instead! cases in point: helmoltz 24 >> and groven 36. >> Helmholtz 24 is simply 23 consecutive fifths; it can be pretty > well equated with the 24 out of 53 2MOS I gave the notes for. > Torsion is not a consideration.torsion *is* a consideration in Helmholtz's tuning if one considers the skhisma (~2 cents) to be under the margin of error of pitch perception (usually considered to be around ~5 cents). Helmholtz's tuning can be viewed as the Pythagorean chain 3^(-16...+7). but Helmholtz himself viewed it as a skhismic temperament described by the Euler genus 3^(-8...+7) * 5^(0...+1). with C as n^0 (= 1/1), this gives a 12-tone Pythagorean chain from Ab 3^-4 to C# 3^7 which has a counterpart one syntonic comma lower at (using {3,5}-prime-vector notation) Ab [-8 1] to C# [3 1]. this is a 24-tone torsional periodicity-block defined by the Pythagorean and syntonic commas, [12 0] and [4 -1].

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