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Message: 5450 Date: Thu, 24 Oct 2002 01:46:00 Subject: Re: NMOS From: Carl Lumma >>I don't see how the symmetrical decatonics can be MOS, since >>they don't have Myhill's property. > > They look awfully Myhill to me. Look again, or use Scala. -C.
Message: 5451 Date: Thu, 24 Oct 2002 03:44:53 Subject: Re: NMOS From: Carl Lumma >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! So my only point/question there was why should continuing the chain be any better than superposing at some other interval. IOW, why would NMOS be special scales of this type? >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'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. -Carl
Message: 5454 Date: Thu, 24 Oct 2002 07:22:22 Subject: Re: NMOS From: Carl Lumma >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. -Carl
Message: 5457 Date: Fri, 25 Oct 2002 05:49:14 Subject: Re: Helmholtz's schismic temperament (was: NMOS) From: monz hi Gene, > From: "monz" <monz@xxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, October 24, 2002 11:15 PM > Subject: Re: [tuning-math] Re: NMOS > > ... > > 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]. re-reading this, i realized that including prime-factor 2 in the vectors would be a good idea, since you've pointed out that it's necessary in order to see the torsion. so ... Helmholtz's tuning, as {2,3,5}-prime-vectors with C=n^0, viewed as: - a Pythagorean chain, Fbb [26 -16 0] ... C# [-11 7 0] - a 5-limit Euler genus, Ab [7 -4 0] ... C# [-11 7 0] + Ab [11 -8 1] ... C# [-7 3 1] generating unision-vectors: [-19 12 0] Pythagorean comma [-4 4 -1] syntonic comma from that data, can you explain how the torsion works in this tuning? -monz
Message: 5463 Date: Mon, 28 Oct 2002 10:36:04 Subject: Re: A common notation for JI and ETs From: David C Keenan At 07:35 PM 26/10/2002 -0700, you wrote: >From: George Secor, 10/26/2002 (#4897) >Subject: A common notation for JI and ETs > >--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote >[#4662]: > > --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: > > > --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote: > > > > --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: > > > > > > > > > >159a: |( ~|( /| |\ ~|\ /|) /|\ (|) (|\ ~||( /|| ||\ >~||\ /||) /||\ > > > > > > > > I prefer > > > > 159b: ~| ~|( /| |\ (|( /|) /|\ (|) (|\ ~||( /|| ||\ >(||( /||) /||\ (RC & MS) > > > > > > The (| flag is not the same number of degrees in (|( and (|\, so >(|( is > > > not valid. > > > > > > I prefer |( because it is valid as the 5:7, 11:13, and 17'-17 >commas, > > > hence is more desirable for its lower-prime applications than a > > > 17-comma symbol. In addition, it is consistent as the rational > > > complement of /||). Neither of our options has rational > > > complementation throughout. > > > OK. I'll go with yours. 159a. > >I've been reviewing a lot of these divisions over the past few days, >and I now think that ~|\ is not a very good choice for 5deg159. If we >used the 7 comma instead of the 11-5 comma for 4deg (thereby foregoing >matching symbols, as you proposed for 125) and the 5+5 comma instead of >the 13 diesis for 6deg, then a combination of our two proposals would >work very nicely: > >159c: |( ~|( /| |) (|( //| /|\ (|) ~|| ~||( ||) ||\ (||( >/||) /||\ (RC) > >This has the advantage of having the lowest prime-number choices for >all of the single-shaft symbols, which are (in addition) valid in all >of their roles. Besides, 7 is slightly more accurate than 13 in 159. OK. Yes. This looks fine. >--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote [#4656]: > > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > > > --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: > > > > > > > > 171: |( ~|( /| |) |\ ~|\ /|) /|\ (|\ ~||( /|| ||) >||\ ~||\ /||) /||\ > > > > > > I think I prefer > > > 171b: |( ~|( /| |) |\ //| /|) /|\ (|\ ~||( /|| ||) ||\ > //|| /||) /||\ > > > > Okay! > >After reviewing 171, I am totally in agreement with your choice of >single-shaft symbols, but I would like to see meaningful double-shaft >symbols. May we use their rational complements? > >171c: |( ~|( /| |) |\ //| /|) /|\ (|\ ~|| /|| ||) ||\ >(||( /||) /||\ (RC) > >Since abandonment of matching sequences in 125 and 159 would require >that rational complementation be the determining principle for the >double-shaft symbols, and since it is RC rather than MS that are the >usual determining principle for the smaller divisions, I believe that >RC will be as least as important as MS as an organizing principle for >aiding in the memorization of symbols in ETs. If a division has both, >then that's great, but if it can have just one, shouldn't it be RC, >especially if there are no unusual single-shaft symbols? After all, >the more a principle is used, the better it is remembered. OK. I'll go along with this. >Change of subject: Regarding 152, our discussion about this has been >extensive, but I will quote only the end of my last message (#4682) >about this: > >[GS:] > > You followed up that thought in a subsequent message (#4673), which I >will > > include here: > >[DK:] > > > Now that I've looked at this myself, I definitely agree that >2deg152 should > > > be ~|(. There's more of an argument for 3deg217 being |~. I can >accept > > > either /|~ or (|( for 5deg152. I also realised that we should not >be using > > > 13-comma symbols in 152. It has inconsistent 13s. The symbol //| is >quite > > > valid (in all its roles) for 6deg152. > > > > > > 152j: )| ~|( /| |\ (|( //| /|\ (|) )|| ~||( /|| ||\ >(||( //|| /||\ (MS) > > > 152k: )| ~|( /| |\ /|~ //| /|\ (|) )|| ~||( /|| ||\ >/||~ //|| /||\ (MS) > > > You have a couple of good points there. Using ~|( agrees with its >use in 494 > > as 2/3 of the 5 comma. > > > If those who attach harmonic meaning to the symbols recognize that >ratios of > > 13 are compromised in 152, then they would readily accept the fact >that (|( is > > not valid as the 7:13 comma. So I have no objections to using the >single-shaft > > symbols of 152j. I have some reservations about the meaning of //|| >being > > misleading, since it's not valid as the complement of the 17 comma, >but at this > > point I can provide neither a good alternative proposal nor a good >rationale for > > using something else, so I can't disagree with what you have. > > > I will have to see if there are any other divisions for which //| >might be more > > appropriate than /|) and if there is any advantage in changing those >from what > > we already have. > >Relevant to my last remark is my proposal for 159, above. > >After doing the above divisions and making my latest comments, I would >now like to use your single-shaft symbols from 152j and have rational >complements instead of matching symbols: > >152m: )| ~|( /| |\ (|( //| /|\ (|) ~|| ~||( /|| ||\ (||( >(||~ /||\ (RC) > >This for the same reasons I gave for 171, above. Yes. This makes sense. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page *
Message: 5466 Date: Wed, 30 Oct 2002 15:20:24 Subject: Re: NMOS From: monz hey guys, i missed something at the beginning of this thread. what the heck is "NMOS" and "2MOS"? (sheesh ... i had a hard enough time understanding "MOS" at first ...) -monz
Message: 5467 Date: Wed, 30 Oct 2002 22:05:01 Subject: Re: NMOS From: monz > From: "Gene Ward Smith" <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, October 30, 2002 5:35 PM > Subject: [tuning-math] Re: NMOS > > > --- In tuning-math@y..., "monz" <monz@a...> wrote: > > hey guys, > > > > > > i missed something at the beginning of this thread. > > what the heck is "NMOS" and "2MOS"? > > A 2MOS would be a scale of 2M notes to each period (eg, octaves) with generator g, where M notes to a period with generator g is a MOS. thanks, Gene, but ... hmmm -- i think i'd understand this a whole lot better with an example. and i still don't have any idea what "NMOS" is. -monz
Message: 5470 Date: Thu, 31 Oct 2002 19:06:08 Subject: Re: New file uploaded to tuning-math From: Carl Lumma > Yahoo groups: /tuning-math/files/Paul/xoomcont.gif * This .gif is golden. But I forget the coloring scheme... -Carl
Message: 5473 Date: Thu, 31 Oct 2002 11:33:57 Subject: Dictionary of Tuning Terms URL (was: NMOS) From: monz ----- Original Message ----- From: <paul.hjelmstad@xx.xxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Thursday, October 31, 2002 11:28 AM Subject: Re: [tuning-math] Re: NMOS > > Message for Joe Monzo: > > I can no longer get to your Definition of Tuning Terms > page. Please provide a new link. THANKS! Definitions of tuning terms: index, (c) 1998 by Joe Monzo * -monz
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