Tuning-Math Digests messages 5450 - 5474

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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.


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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


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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


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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


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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 *


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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


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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


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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


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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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