Tuning-Math Digests messages 10630 - 10654

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Message: 10630

Date: Mon, 15 Mar 2004 07:55:21

Subject: Re: Gene's private reserve -- 7-limit

From: Graham Breed

Gene Ward Smith wrote:

> Rats--two temperaments with identical TOP errors *and* Graham
> complexities. I was aware this could happen, but guessed, apparently
> incorrectly, that it was quite unlikely and then forgot to check. The
> missing one gets 64th place since it is better in the 9-limit.

I've verified that, contorsion aside, each temperament I look at does 
have a unique badness.  I need this to test my temperament finder (now 
in three languages!).  I can't prove it but it's always worked so far.

> 64 [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]]
> bad 238.136875 top err 2.939961 graham 9
> 
> 
>>64 [9, 0, 9, -21, -11, 21] [[9, 14, 21, 25], [0, 1, 0, 1]]
>>bad 238.136875 top err 2.939961 graham 9

Oh yes, that'll be because it's a kind of minimax.  The 7 mapping 
presumable doesn't contribute to the TOP error, and the complexities 
happen to be the same.  They should be distinct if you switch to some 
kind of RMS.


                        Graham


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Message: 10638

Date: Wed, 17 Mar 2004 23:53:40

Subject: Re: Minimal filled scale

From: Carl Lumma

Greetings from beautiful Portland!

>Suppose we have a linear temperament with octave period. A
>chord-type in this temperament is a set of generators. A
>question we might ask is what is the what is the cardinality
>of the smallest set of contiguous generators which arise from
>contiguous generator translates of the chord--the minimal
>filled scale for the chord.

"Continuous generator translates"??

>This doesn't depend on the temperament, but only on the chord,
>considered as a set or list.

Huh?  It must depend on the mapping.

>The minimal filled scale for septimal miracle is Miracle[19],
>and for 11-limit miracle is Canasta (Miracle[31].) For 5-limit
>meantone we get the diatonic scale (Meantone[7]), and in the
>7-limit, Meantone[16]. And so on and so forth...

How is this different from Graham complexity?

-Carl


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Message: 10641

Date: Thu, 18 Mar 2004 22:03:33

Subject: Re: Symmetric 7-limit comma badness and 2401/2400

From: Carl Lumma

>> >If you take, for a 7-limit interval q and a symmetric
>> >lattice distance dist, the function cents(q) dist(q)^4,
>> 
>> Why 4?  It used to be pi(lim)-1, which would be 3 in the
>> 7-limit.
>
>The exponent should be rank(Group)/rank(Kernel); the rank of
>the group for p-limit will be pi(p), and the rank of the
>kernel for codimension one temperaments is of course one.

You were among the people to review my code, which used
pi(lim)-1.  Is the above due to that we're no longer
assuming octave equivalence or something?

I also asked:

>...I'm only returned the 10 best results, and I only
>search q with d <= 3000 and cents(q) <= 600.  What bounds
>does your method require?

-Carl


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Message: 10642

Date: Thu, 18 Mar 2004 22:01:06

Subject: Re: Minimal filled scale

From: Carl Lumma

>> Greetings from beautiful Portland!
>> 
>> >Suppose we have a linear temperament with octave period. A
>> >chord-type in this temperament is a set of generators. A
>> >question we might ask is what is the what is the cardinality
>> >of the smallest set of contiguous generators which arise from
>> >contiguous generator translates of the chord--the minimal
>> >filled scale for the chord.
>> 
>> "Continuous generator translates"??
>
>Contiguous.

Weird; high-level typo; I read it correctly.

>I mean if we have for example a chord [0 1 4 10], we take
>[1 2 5 11] [1 3 6 12] etc.

You mean [2 3 6 12]?

>until we've filled all the holes,

I still don't get it.  You're harmonizing every note of the
original chord?

>and every
>note is harmonizable by at least one such chord.

The original chord has this property...

>> >The minimal filled scale for septimal miracle is Miracle[19],
>> >and for 11-limit miracle is Canasta (Miracle[31].) For 5-limit
>> >meantone we get the diatonic scale (Meantone[7]), and in the
>> >7-limit, Meantone[16]. And so on and so forth...
>> 
>> How is this different from Graham complexity?
>
>How is it the same?

I was hoping an explanation of the difference would help us
understand it.

-Carl


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Message: 10644

Date: Thu, 18 Mar 2004 00:36:36

Subject: Re: Symmetric 7-limit comma badness and 2401/2400

From: Carl Lumma

>If you take, for a 7-limit interval q and a symmetric
>lattice distance dist, the function cents(q) dist(q)^4,

Why 4?  It used to be pi(lim)-1, which would be 3 in the
7-limit.

>you get a log-flat symmetric
>badness measure for 7-limit commas. Euclidean and Hahn are
>not very different; below I use Hahn distance, and order
>some commas with badness less than 3000 from best to worst.
>The list is completely dominated by superparticulars, and
>it looks to me as if 2401/2400 is likely to be an absolute
>minimum in badness. At any rate looking at this makes the
>Erlich phenomenon--the great importance of 2401/2400
>for 7-limit micro ets--more understandable.
>
>2401/2400 184.626652
>8/7 231.174094
>7/6 266.870906
>6/5 315.641287
>5/4 386.313714
>4/3 498.044999
>50/49 559.609830
>49/48 571.148984
>7/5 582.512193
>10/7 617.487808

If q = n/d, then using (n-d)/d instead of cents(q) and
log(d)^3 instead of dist(q)^4, I get...

((0.19645692845300844 7 2401/2400)
 (0.4419896533813025 3 4/3)
 (0.6660493039778589 5 5/4)
 (0.7075223009389495 7 225/224)
 (0.8337823128571303 5 6/5)
 (0.9004848978857011 7 126/125)
 (0.9587113625980545 7 7/6)
 (1.0518045661034596 5 81/80)
 (1.0526168298843461 7 8/7)
 (1.1239582004626365 3 9/8))

...I'm only returned the 10 best results, and I only
search q with d <= 3000 and cents(q) <= 600.  What bounds
does your method require?

-Carl



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Message: 10645

Date: Thu, 18 Mar 2004 17:44:12

Subject: Re: Minimal filled scale

From: Carl Lumma

>> >I mean if we have for example a chord [0 1 4 10], we take
>> >[1 2 5 11] [1 3 6 12] etc.
>> 
>> You mean [2 3 6 12]?
>
>Right.
>
>> >until we've filled all the holes,
>> 
>> I still don't get it.  You're harmonizing every note of the
>> original chord?
>
>No, I'm harmonizing everything with translates of the chord in a
>minimal contiguous-generator scale containing the chord.
>
>> >and every
>> >note is harmonizable by at least one such chord.
>> 
>> The original chord has this property...
>
>No, the numbers from 0 to 10 only find harmonies for 0, 1, 4 and 10.
>2, 3, 5, 6, 7, 8 and 9 have no major tetrad. If, however, I take the
>numbers from 0 to 15, every one of them has a major tetrad to
>harmonize it. The union of the sets {i,i+1,i+4,i+10} as i ranges from
>0 to 5 is {0..15}; no smaller value than 5 will work.

Got it.  Cool.

-Carl


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