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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
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
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
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
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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