Tuning-Math Digests messages 5653 - 5677

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Message: 5653

Date: Wed, 27 Nov 2002 05:02:52

Subject: Re: spiral lattices

From: Carl Lumma

>Spiral Lattices | Phyllotaxis *
>
>applicable to MOS scales . . .

Indeed; Erv Wilson once sent me home with a pine cone the
size of my head.

-Carl


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Message: 5654

Date: Wed, 27 Nov 2002 05:06:18

Subject: Re: Paul's new names

From: Carl Lumma

>What are the main formualtions,
> 
>we have graham's (unweighted minimax), unweighted rms,
>weighted rms, geometric (both unweighted and weighted??) . . .

For complexity?  What are they?

-Carl


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Message: 5655

Date: Thu, 28 Nov 2002 11:10:04

Subject: Re: Paul's new names

From: Carl Lumma

>>Minimax, rms, etc. of what?  The numbers in the map?
>
>the number of generators comprising each consonant interval.

Ok, thanks; as I thought.

Conceptually, if we're thinking in terms of Partchian limits,
I prefer simply the number of generators needed to span all
the identities (consonant intervals).  This can be 'weighted'
by simply dividing by the number of identities.  Reason being,
I view the choice of a complete Partchian limit as a statement
that all the identities in that limit will be treated as
consonances in the music, and thus are all equally important
in that sense.  If we're not talking about Partchian limits,
we can just omit the identities that make the range bad.  Why
we would want to smooth such bad approximations out with
something like rms I cannot guess.

I can see weighted error, but not weighted complexity, where
the weighting proporational to the identity.  Why should we
expect more generators to be required to render 7 than 5?

-Carl


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Message: 5659

Date: Thu, 28 Nov 2002 00:30:45

Subject: Re: Paul's new names

From: Carl Lumma

>>>we have graham's (unweighted minimax), unweighted rms,
>>>weighted rms, geometric (both unweighted and weighted??) . . .
>> 
>> For complexity?
>
>yup, doze R dem.
>
>> What are they?
>
>dem. doze.

Minimax, rms, etc. of what?  The numbers in the map?

-Carl


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Message: 5662

Date: Thu, 28 Nov 2002 18:40:46

Subject: Re: Paul's new names

From: Carl Lumma

>>Conceptually, if we're thinking in terms of Partchian limits,
>>I prefer simply the number of generators needed to span all
>>the identities (consonant intervals).
> 
>that's what i meant by "minimax" -- it's graham's way.

Thought so.  Isn't minimax a bad term for this?
 
>>This can be 'weighted' by simply dividing by the number of
>>identities.
> 
> that has no effect on the rankings.

But it allows you to compare different limits.  Actually I
haven't checked if something stronger than division would
be needed, but you get the idea.

>**but if modulation is usually accomplished by making a small
>number of "chromatic" changes to the basic "diatonic" scale,
>shouldn't extra points be awarded if the modulations more often
>move one by the *simpler* consonances, particularly the 3-limit
>ones?

Not in my view.  I'm thinking we should not even assume tonal
composition at this level.
 
-Carl


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Message: 5664

Date: Thu, 28 Nov 2002 18:55:25

Subject: Re: Paul's new names

From: Carl Lumma

>>smooth such bad approximations out? i'm not sure what you mean.
>>rms is similar to graham's method, but takes into account the
>>second-longest, third-longest, etc. chains of generators to a
>>small extent too. that seems like a good thing to me.
> 
>It really depends on whether or not you are interested in
>incomplete n-limit chords; normally, I think we would be.

If I'm interested in complete 7-limit tetrads on every beat,
have a temperament with really simple 3s and 7s but complex 5s,
rms will punish this less than I'd like.  At least, the history
of Western music seems to assert that for music employing
n-limit harmony, < n-limit harmony sounds too different to
fall back on for any length of time.

-Carl


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Message: 5666

Date: Thu, 28 Nov 2002 19:24:31

Subject: Re: Paul's new names

From: Carl Lumma

>>>the number of generators comprising each consonant interval.
>>
>>Ok, thanks; as I thought.
> 
>It doesn't cover geometric complexity, for which you should
>see my postings on this list.

Msg. 4533 is the one, I'm guessing.  Very cool.  But I don't
have the technique for choosing the defining commas.  Is
there any way this can be defined in terms of the map?

-Carl


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