Tuning-Math Digests messages 10403 - 10427

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Message: 10403

Date: Sat, 28 Feb 2004 15:49:12

Subject: Re: DE scales with the stepwise harmonization property

From: Carl Lumma

>Augmented[9]
>[28/25, 35/32, 15/14, 16/15] [1, 2, 3, 3]
>
>(28/25)/(35/32) = 128/125
>(15/14)/(16/15) = 225/224

Augmented[9], eh?  How far is the 7-limit TOP version
from...

!
 TOP 5-limit Augmented[9].
 9
!
 93.15
 306.77
 399.92
 493.07
 706.69
 799.84
 892.99
 1106.61
 1199.76
!

...?

-Carl


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Message: 10404

Date: Sat, 28 Feb 2004 15:51:21

Subject: Re: Harmonized melody in the 7-limit

From: Carl Lumma

>> >> Rad.  No 6- or 7-toners, eh?  I wonder about the "9-limit".
>> >> The 9-limit has the further advantage that you can hit more
>> >> fifths, and thus improve omnitetrachordality.
>> >
>> >The 9-limit would be different, for sure. The simple symmetrical 
>> >lattice criterion wouldn't work, but it would be easy enough to
>> >find what does.
>> 
>> Nobody ever answered me if symmetrical is synonymous with unweighted. 
>
>Probably no one was sure what the question meant. It means 3, 5, 7,
>5/3, 7/3 and 7/5 are all the same size, however.

As I thought then.

-Carl



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Message: 10407

Date: Sun, 29 Feb 2004 12:01:00

Subject: Re: Stepwise harmonizing property

From: Carl Lumma

>I was implicitly assuming that one of the chords harmonized to the
>root, which doesn't make a lot of sense to assume. Dropping that makes
>the analysis far easier--steps have this property iff they are
>products (or ratios, but that adds nothing) of consonant intervals
>(including 1 as a consonant interval.)

How is this any different than a symmetric lattice distance of 2,
which is what I thought you used in the first place.

>This is obvious enough if you
>think about it; the situation no longer depends on fine distinctions.
>You get one consonant interval from the unison to the common note in
>one chord, and from the common note to another interval in the next
>chord; that interval, by definition one reachable by a chord with a
>common note, is therefore a product of consonances of the system.

I can't parse this.

>In the 7-limit, this has the effect of adding 50/49 (= (10/7)(5/7)) to
>the list of harmonizable intervals. I did, and also extended the size
>up to 8/7, getting a considerably larger list this time, and including
>some six and seven note scales for Carl.

Well this is cool, but since many classic scales contain steps up to a
minor third apart, perhaps 6/5 should be the cutoff.

-Carl


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Message: 10408

Date: Sun, 29 Feb 2004 12:07:59

Subject: Re: Harmonized melody in the 7-limit

From: Carl Lumma

>>> >> Rad.  No 6- or 7-toners, eh?  I wonder about the "9-limit".
>>> >> The 9-limit has the further advantage that you can hit more
>>> >> fifths, and thus improve omnitetrachordality.
>>> >
>>> >The 9-limit would be different, for sure. The simple symmetrical 
>>> >lattice criterion wouldn't work, but it would be easy enough to
>>> >find what does.

And why, pray tell, does symmetrical lattice distance not work in
the 9-limit?

-Carl


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Message: 10412

Date: Sun, 29 Feb 2004 13:19:30

Subject: Re: Harmonized melody in the 7-limit

From: Carl Lumma

>> >>> >> Rad.  No 6- or 7-toners, eh?  I wonder about the "9-limit".
>> >>> >> The 9-limit has the further advantage that you can hit more
>> >>> >> fifths, and thus improve omnitetrachordality.
>> >>> >
>> >>> >The 9-limit would be different, for sure. The simple symmetrical 
>> >>> >lattice criterion wouldn't work, but it would be easy enough to
>> >>> >find what does.
>> 
>> And why, pray tell, does symmetrical lattice distance not work in
>> the 9-limit?
>
>If you call something which makes 3 half as large as 5 or 7
>"symmetrical", it does.

One of us is still misunderstanding Paul Hahn's 9-limit approach.
In the unweighted version 3, 5, 7 and 9 are all the same length.
If you prefer I think you can just use your product-of-two-consonances
rule where the ratios of 9 have been included.

-Carl


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Message: 10415

Date: Mon, 01 Mar 2004 04:58:54

Subject: Re: Harmonized melody in the 7-limit

From: Carl Lumma

>> What you said was that symmetrical lattice distance won't work.
>
>It doesn't.
>
>> I asked why, and said Paul Hahn's version works.
>
>That's a symmetrical lattice, but it isn't a lattice of note-classes.

If I didn't know better I'd say you were trying to BS me.  What
is a lattice of note classes?

-Carl


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Message: 10416

Date: Mon, 01 Mar 2004 23:42:32

Subject: Re: DE scales with the stepwise harmonization property

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> Augmented[9], eh?  How far is the 7-limit TOP version
> >> from...
> >> 
> >> !
> >>  TOP 5-limit Augmented[9].
> >>  9
> >> !
> >>  93.15
> >>  306.77
> >>  399.92
> >>  493.07
> >>  706.69
> >>  799.84
> >>  892.99
> >>  1106.61
> >>  1199.76
> >> !
> >> 
> >> ...?
> >> 
> >> -Carl
> >
> >Just look at the horagram, Carl!
> >
> >107.31
> >292.68
> >399.99
> >507.3
> >692.67
> >799.98
> >907.29
> >1092.66
> >1199.97
> 
> Oh!  Where are the 7-limit horagrams?
> 
> -C.

Some of them are in

Yahoo groups: /tuning/files/perlich/ *

some of them are in

Yahoo groups: /tuning_files/files/Erlich/sevenlimit.zip *

and this one, aug7.gif, is in both.


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Message: 10417

Date: Mon, 01 Mar 2004 04:02:32

Subject: Re: Hanzos

From: Carl Lumma

>> My recollection is that Paul H.'s algorithm assigns a unique
>> lattice route (and therefore hanzo) to each 9-limit interval.
>
>So what? You still get an infinite number representing each interval,
>since you can multiply by arbitary powers of the dummy comma 9/3^2.

An infinite number from where?  If you look at the algorithm, that
dummy comma has zero length.

>> Certainly it can be used to find the set of lattice points
>> within distance <= 2 of a given point.
>
>Hahn's alogorithm can, or hahnzos can, or what?

The algorithm can.

-Carl


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Message: 10418

Date: Mon, 01 Mar 2004 23:44:21

Subject: Re: 9-limit stepwise

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
> > wrote:
> > > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...>
> > > wrote:
> > > 
> > > On the other end of the size scale we have these. Paul, have 
you 
> > ever
> > > considered Pajara[6] as a possible melody scale?
> > 
> > Seems awfully improper, but descending it resembles a famous 
> > Stravisky theme.
> 
> What I should have asked was if you've tried 443443 as a melody 
scale
> in 22-et.

Right; I thought you were talking about 2 2 7 2 2 7, but then you 
clarified. See Yahoo groups: /tuning-math/message/9886 *.


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Message: 10420

Date: Mon, 01 Mar 2004 23:47:10

Subject: Re: 9-limit stepwise

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
> <gwsmith@s...> 
> > > wrote:
> > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
> <gwsmith@s...>
> > > > wrote:
> > > > 
> > > > On the other end of the size scale we have these. Paul, have 
> you 
> > > ever
> > > > considered Pajara[6] as a possible melody scale?
> > > 
> > > Seems awfully improper, but descending it resembles a famous 
> > > Stravisky theme.
> > 
> > What I should have asked was if you've tried 443443 as a melody 
> scale
> > in 22-et.
> 
> Right; I thought you were talking about 2 2 7 2 2 7, but then you 
> clarified. See Yahoo groups: /tuning- *
math/message/9886.

Oh yeah, the answer is yes.


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Message: 10424

Date: Mon, 01 Mar 2004 18:45:24

Subject: Re: Hanzos

From: Carl Lumma

>>>>My recollection is that Paul H.'s algorithm assigns a unique
>>>>lattice route (and therefore hanzo) to each 9-limit interval.
>>>
>>>So what? You still get an infinite number representing each 
>>>interval, since you can multiply by arbitary powers of the dummy
>>>comma 9/3^2.
>> 
>>An infinite number from where?  If you look at the algorithm, that
>>dummy comma has zero length.
>
>If it has length zero then we are not talking about a lattice at all, 
>though a quotient of it (modding out the dummy comma) might be. In a 
>symmetrical lattice it necessarily has the same length as, for 
>example, 11/3^2, which is of length sqrt(1^2+2^2-1*2)=sqrt(3). It 
>does *not* have the same length as 11/9, which is of length one, of 
>course.

Yes, you're onto something here.  In the unweighted lattice there is
a point for 9/3^2, which lies on the diameter-1 hull.

>> >> Certainly it can be used to find the set of lattice points
>> >> within distance <= 2 of a given point.
>> >
>> >Hahn's alogorithm can, or hahnzos can, or what?
>> 
>> The algorithm can.
>
>Why do you think Hahn's definition of "distance" would work for this 
>problem? If you tell me what it is, we could check and see. However, 
>you've just told me it does not have the basic properties of a 
>metric, so I'm inclined to object to calling it a "distance".

I thought you acknowledged the receipt of the algorithm...

>Given a Fokker-style interval vector (I1, I2, . . . In):
>
>1.  Go to the rightmost nonzero exponent; add its absolute value
>to the total.
>
>2.  Use that exponent to cancel out as many exponents of the opposite
>sign as possible, starting to its immediate left and working right;
>discard anything remaining of that exponent.
>
>3.  If any nonzero exponents remain, go back to step one, otherwise
>stop.

As for the problem, let's start over.  Call the position occupied by
1/1 in 1/1,8/7,4/3,8/5 the root of utonal tetrads.  Now 7-limit tetrads
sharing a common dyad (pair of pitches) with an otonal tetrad rooted
on 1/1 will have roots...

5/4, 3/2, 15/8, 7/4, 35/32, 21/16

...and those sharing a common tone (single pitch) will have roots...
ack, this is visually exhausting... am I correct that they are the
1- and 2-combinations of:

1, 3, 5, 7, 1/3, 1/5, 1/7

?

If so, can you say why adding 9 and 1/9 to this list will not produce
an equivalent 9-limit result?

-Carl


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