Tuning-Math Digests messages 5426 - 5450

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Message: 5426

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


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Message: 5427

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]


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Message: 5436

Date: Wed, 23 Oct 2002 05:53:27

Subject: Re: NMOS

From: Carl Lumma

>>>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.
 
>>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


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Message: 5438

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


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Message: 5440

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


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Message: 5449

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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