Tuning-Math Digests messages 10426 - 10450

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Message: 10426

Date: Mon, 01 Mar 2004 18:52:18

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.
>
>How do you get that????

>Given a Fokker-style interval vector (I1, I2, . . . In):

[-2 0 0 1]

>1.  Go to the rightmost nonzero exponent; add its absolute value
>to the total.

T=1

>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.

[-1 0 0 0]
T=1

>3.  If any nonzero exponents remain, go back to step one, otherwise
>stop.

T=... whoops, I forgot an absolute value here.  The correct
value is 2.

-Carl


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Message: 10428

Date: Mon, 01 Mar 2004 12:25:59

Subject: Re: Harmonized melody in the 7-limit

From: Carl Lumma

>> If I didn't know better I'd say you were trying to BS me.  What
>> is a lattice of note classes?
>
>It's the kind of lattice I was talking about--for each octave 
>ewquivalence class, we have a lattice point. Hence there is a lattice 
>point representing 9,9/4,9/8... etc but only one.

If you take the 2s out of the hanzos, we have that.

-Carl


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Message: 10429

Date: Mon, 01 Mar 2004 19:01:22

Subject: Re: Hanzos

From: Carl Lumma

>>>>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).

I don't know where sqrt would be coming from.  I thought everything
would have to have whole number lengths.

>>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.

That was based on the rather simplistic idea that one steps out and
two back leaves you one away from where you started.

-Carl



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Message: 10433

Date: Mon, 01 Mar 2004 01:26:20

Subject: Re: Hanzos

From: Carl Lumma

>> 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.
>
>In this system you don't exactly, have 7-limit notes and intervals.
>You do have "hanzos", with basis 2,3,5,7,9.

The usual point of odd-limit is to get octave equivalence, and
therefore I'd say the 2s should be dropped from the basis.

>The hahnzo |0 -2 0 0 1> is a comma, 9/3^2, which obviously would
>play a special role. Hahnzos map onto 7-limit intervals, but not
>1-1. Are you happy with the idea that two scales could be
>different, since they have steps and notes which are distinct as
>hahnzos, even though they have exactly the same steps and notes
>in the 7-limit?

I think the answer here is yes, though I'm at a loss for why
you're mapping hanzos to the 7-limit.

>We've got three hahnzos corresponding to 81/80;

My recollection is that Paul H.'s algorithm assigns a unique
lattice route (and therefore hanzo) to each 9-limit interval.
Certainly it can be used to find the set of lattice points
within distance <= 2 of a given point.

-Carl


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Message: 10434

Date: Mon, 01 Mar 2004 23:04:55

Subject: Re: TOP and Tenney space webpage

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > I have no idea how to reconcile this with
> > 
> > Yahoo groups: /tuning-math/message/9797 *
> > 
> > and with the fact that it's obvious that this linear operator, 
when 
> > acting on a monzo, is the only one returns its pitch (or interval 
> > size) in cents.
> 
> I'm being contrary; it seems to me "measure" isn't really how we 
want 
> to look at it, since just intonation is now being viewed as one of 
an 
> infinite set of possible tuning maps.

True. It only measures pitch (or interval size) in the untempered 
case.


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Message: 10435

Date: Mon, 01 Mar 2004 01:18:50

Subject: Re: Harmonized melody in the 7-limit

From: Carl Lumma

>> One of us is still misunderstanding Paul Hahn's 9-limit approach.
>
>What in the world makes you think this has anything to do with me or
>anything I've said?

""
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.

If you call something which makes 3 half as large as 5 or 7
"symmetrical", it does.
""

-Carl


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Message: 10436

Date: Mon, 01 Mar 2004 23:05:58

Subject: Re: non-1200: Tenney/heuristic meantone temperament

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > Amazingly, Gene's page would have you believe you need to search 
> even 
> > in the codimension 1 case!
> 
> You want I should derive the codimension 1 formula instead?

Why not?


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Message: 10438

Date: Mon, 01 Mar 2004 23:11:57

Subject: Re: Harmonized melody in the 7-limit

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> 
> > Also, could we screen based on which of the above
> > combinations, and which orderings of those, produce the
> > most low-numbered ratios in the scale?  Or does such
> > an approach fail on the grounds that it ignores temperament
> > (aka TOLERANCE)?
> 
> One approach would be to use tempering to simplify the problem. If 
we 
> pick linear temperaments which reduce the four sizes of scale step 
to 
> two, we also automatically enforce Myhill's property.

That would seem to depend on the ordering, and if the period isn't an 
octave would seem to be impossible.


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Message: 10440

Date: Mon, 01 Mar 2004 23:18:08

Subject: Re: DE scales with the stepwise harmonization property

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >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

Just look at the horagram, Carl!

107.31
292.68
399.99
507.3
692.67
799.98
907.29
1092.66
1199.97


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Message: 10443

Date: Mon, 01 Mar 2004 04:07:13

Subject: Re: Harmonized melody in the 7-limit

From: Carl Lumma

>> >> One of us is still misunderstanding Paul Hahn's 9-limit approach.
>> >
>> >What in the world makes you think this has anything to do with me or
>> >anything I've said?
>> 
>> ""
>> 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.
>> 
>> If you call something which makes 3 half as large as 5 or 7
>> "symmetrical", it does.
>> ""
>
>Can you point out where in the above quote you found the words "Paul
>Hahn?"

What you said was that symmetrical lattice distance won't work.
I asked why, and said Paul Hahn's version works.

-Carl


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Message: 10444

Date: Mon, 01 Mar 2004 23:20:44

Subject: Re: 9-limit stepwise

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> Here are some 9-limit stepwise harmonizable scales, with the same
> bound on size of steps--8/7 is the largest. In order to keep the
> numbers down, I also enforced that the size of the largest step (in
> cents) is less than four times that of the smallest step--the
> logarithmic ratio is the fourth number listed.
> 
> In order I give scale type number on the list, scale steps,
> multiplicities, largest/smallest, and number of steps in the scale.
> 
> As you can see, the largest scale listed has 41 steps, which is
> getting up there. (64/63)/(81/80)=5120/5103 and (49/48)/(50/49) =
> 2401/2400; putting these together gives us hemififths, and hence
> Hemififths[41] as a DE for this. Hemififths is into the
> microtemperament range by most standards; it has octave-generator 
with
> TOP values [1199.700, 351.365] and a mapping of
> [<1 1 -5 -1|, <0 2 25 13|]. I've never tried to use it, and so far 
as
> I know neither has anyone else, but this certainly gives a 
motivation.
> Ets for hemififths are 41, 58, 99 and 140. We also get Schismic[29]
> out of scale 4, Diaschismic[22] out of scale 5,

I thought Diaschismic had no unique definition in the 7-limit.


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Message: 10446

Date: Mon, 01 Mar 2004 23:22:48

Subject: Re: 9-limit stepwise

From: Paul Erlich

--- 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.


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Message: 10447

Date: Mon, 01 Mar 2004 15:23:55

Subject: Re: DE scales with the stepwise harmonization property

From: Carl Lumma

>> 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.


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Message: 10448

Date: Mon, 01 Mar 2004 23:23:55

Subject: Re: 9-limit stepwise

From: Paul Erlich

--- 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?

I'm confused -- I thought the largest step was supposed to be less 
than 200 cents?


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