Tuning-Math Digests messages 5550 - 5574

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Message: 5550

Date: Fri, 08 Nov 2002 23:19:12

Subject: Re: from the realms of private correspondence

From: Carl Lumma

>>>its vertices are
>>>
>>>1    4
>>>
>>>5    20
>>>
>>>25   100
>>>
>>>125  500
>>>
>>>and it also intersects 5 (again), 20 (again), 25 (again), 100 
>>>(again), 2, and 10. 
>> 
>> My chord is:
>> 
>> 25
>> |
>> 5
>> |
>> 1
> 
> you said 4:5:25!

That's true, but I didn't mean it.  :)
 
>> Here are your verticies on the lattice:
>> 
>> 125 - x - 500
>>  |    |    |
>> 25 -  x - 100
>>  |    |    |
>>  5 -  x - 20
>>  |    |    |
>>  1 -  x -  4
> 
>i don't know if you read my diagram right. it was meant to
>represent the eight vertices of (in concept) a cube.

Ah, it was a diagram!

>> How did you figure
>> out that the perimeter of these structures
>> would be a consistent taxicab distance for three
>> points?
> 
> it's easy. there are 12 edges. the three representing
> each of the pitches' distances from 1/1 (when they are
> expressed as simply as possible as harmonics thereof)
> are each present four times.

?

> so you can divide through by four, and you simply have
> log(a) + log(b) + log (c), which equals log(a*b*c). get it?

Oh, dear, I certainly don't... You're in favor of my
suggestion after all, just that you don't consider it
a metric?

>>I don't understand how a pitch can have concordance.
> 
>it doesn't! that's why we have a concordance *metric*!

I understand what you're getting at now on this point, but I
still would call both Tenney HD and my suggestion pseudometrics
based on the notation at mathworld.

-Carl


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Message: 5554

Date: Sat, 9 Nov 2002 02:10:19

Subject: otonally-weighted lattices (was: from the realms of private correspondence)

From: monz

hi paul,


> From: "wallyesterpaulrus" <wallyesterpaulrus@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Friday, November 08, 2002 3:39 PM
> Subject: [tuning-math] Re: from the realms of private correspondence
> 
>
> <snip> ... i don't see how one could ever hope to
> embody favoritism for otonal over utonal in a lattice,
> as much as i believe in such favoritism myself.


hmmm ... wow, you really "struck a chord" here
with me!

several years ago, when i had first moved to San Diego
and was setting up the Sonic Arts website, i was 
pondering how one might favor otonality in a lattice.

i haven't thought about it since then, and don't really
remember what ideas i had come up with, but i do recall
that i was trying to incorporate Erv Wilson's famous
"harmonic spiral" diagram into my own lattice formula,
whereby the angles and lengths of each prime-axis would
radiate outward from each lattice-point according to
the measurements in Erv's diagram.

any thoughts on that?



-monz


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Message: 5563

Date: Sun, 10 Nov 2002 01:41:31

Subject: Re: from the realms of private correspondence

From: Carl Lumma

>>If you know what norms are and how to work with them.  I'm
>>still struggling with metrics.  But do tell.  Maybe Paul
>>will follow.
> 
>Did you see my mathworld citation? Here is another:

I did.  Unfortunately, most of it is straight over my head.
Why this is so is a matter of some interest to me... I can't
tell if it's really hard, just some notational hurdle, or both.

>Normed vector space - Wikipedia, the free encyclopedia *

That's better, thanks.

>There is an error on this page--the field need not be either
>C or R, but can be any local field of characteristic 0. In
>particular, it can be the rational numbers.

You should fix the page... it's a Wiki, after all.  Just click
"Edit this page" near the top.

-Carl


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Message: 5570

Date: Wed, 13 Nov 2002 12:12:58

Subject: Re: sorry, gotta go

From: Carl Lumma

> Hopefully, I'll be back.

Hey, thanks for the kind words.  I worry sometimes about
the vibe around here, starting with my own posts.  I'll
look forward to hearing your side of the story again, and
to getting over the initial terminology gap that happens
often around here (the good news is that so far, every
time new terminology has come around, we end up learning
something important about tuning!).

-Carl


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Message: 5574

Date: Wed, 13 Nov 2002 13:55:52

Subject: 43edo 7-limit periodicity-block

From: monz

i've just added some 7-limit lattices to my
Tuning Dictionary "meride" entry, showing the
"closest to 1/1" 7-limit periodicity-block
for 43edo.

Definitions of tuning terms: meride, (c) 1998 by Joe Monzo *

(at the bottom of the page)

just above the lattice, i refer to Gene's
"7-limit MT reduced bases for 43edo".  but
i find that on these lattices, 225:224 is closer
than 126:125.  is that because i'm using the
rectangular rather than triangular/hexagonal 
taxicab metric?

so anyway, the bases i see are 81:80 and 225:224.
what's the third one?

here's a list of [3,5,7] vectors for the ratios
in my periodicity-block; asterisks indicate pitches
which occur twice (**) or 3 times (***) equally far
away from 1/1, with the 43edo-degree number -- they're
shown in darker shades of grey on the 5-limit "sheets"
lattices:

[ 0  5  0] ***27
[ 0  4  0] **13
[ 0  3  0] **42
[-1  2  0] **3
[ 0  2  0]
[-1  1  0]
[ 0  1  0]
[ 1  1  0]
[-2  0  0]
[-1  0  0]
[ 0  0  0]
[ 1  0  0]
[ 2  0  0]
[-1 -1  0]
[ 0 -1  0]
[ 1 -1  0]
[ 0 -2  0]
[ 1 -2  0] **40
[ 0 -3  0] **1
[ 0 -4  0] **30
[ 0 -5  0] **16
[ 0  3  1]
[-1  2  1]
[ 0  2  1]
[ 1  2  1]
[ 2  2  1] ***27
[-1  1  1]
[ 0  1  1]
[ 1  1  1]
[ 2  1  1] **13
[-1  0  1]
[ 0  0  1]
[ 1  0  1]
[ 2  0  1] **42
[ 0 -1  1]
[ 1 -1  1] **3
[-1  1 -1] **40
[ 0  1 -1]
[-2  0 -1] **1
[-1  0 -1]
[ 0  0 -1]
[ 1  0 -1]
[-2 -1 -1] **30
[-1 -1 -1]
[ 0 -1 -1]
[ 1 -1 -1]
[-2 -2 -1] **16
[-1 -2 -1]
[ 0 -2 -1]
[ 1 -2 -1]
[-1 -3 -1] ***27
[ 0 -3 -1]




-monz
"all roads lead to n^0"


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