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Message: 5550 - Contents - Hide Contents Date: Fri, 08 Nov 2002 23:19:12 Subject: Re: from the realms of private correspondence From: Carl Lumma>>> >ts 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: 5551 - Contents - Hide Contents Date: Fri, 08 Nov 2002 23:31:13 Subject: Re: from the realms of private correspondence From: wallyesterpaulrus --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>>> I don't understand how a pitch can have concordance. >>>> They have it with respect to "1". >> Dyad. So what you have is a function that assigns a value > to pitches, and then you subtract them.not if you think of it as a metric. then there is no value assigned to pitches at all, but rather there is a way of measuring the "harmonic distance" between any two pitches in the JI lattice.

Message: 5552 - Contents - Hide Contents Date: Fri, 08 Nov 2002 23:39:07 Subject: Re: from the realms of private correspondence From: wallyesterpaulrus --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>> 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?right. actually, i remember my suggestion better now. it's the total length of the edges of the smallest hyperrectangle that can enclose the points (a direct generalization of the tenney metric for dyads). and this ends up being proportional to the *LCM*, rather than to the product. so 4:5:6 and 10:12:15 actually end up having the same generalized tenney complexity, since they both have LCM of 60. this accords better with the lattice itself being symmetric along utonal/otonal lines. 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.>>> 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.Tenney HD is most certainly a metric, not a pseudometric!

Message: 5553 - Contents - Hide Contents Date: Fri, 08 Nov 2002 23:42:15 Subject: Re: from the realms of private correspondence From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> probably it was benedetti. in the renaissance. he had very > interesting reasons for choosing it -- margo has posted on this on > the tuning list . . .Wow--it goes back a ways. I was screwing my thinking cap on, and decided I got it from a book--which probably means indirectly from Tenney. In any case I quickly decided I liked p+q better than pq as a consonance measure anyway.

Message: 5554 - Contents - Hide Contents 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: 5555 - Contents - Hide Contents Date: Sat, 09 Nov 2002 10:47:41 Subject: Re: otonally-weighted lattices (was: from the realms of private correspondence) From: Gene Ward Smith --- In tuning-math@y..., "monz" <monz@a...> wrote:>> From: "wallyesterpaulrus" <wallyesterpaulrus@y...> >>> 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!The only method which occurs to me is to take the lattice of otonal *chords*, and ignore utonal anything.

Message: 5557 - Contents - Hide Contents Date: Sat, 09 Nov 2002 20:34:28 Subject: Fwd: Re: from the realms of private correspondence From: wallyesterpaulrus i'm not sure what you're asking. if it's about harmonic entropy, perhaps it's better asked on that list. --- In tuning-math@y..., <Josh@o...> wrote:> Always? > > ---- Original message ---->> Date: Fri, 08 Nov 2002 22:42:21 -0000 >> From: "wallyesterpaulrus" <wallyesterpaulrus@y...> >> Subject: Fwd: [tuning-math] Re: from the realms of private > correspondence >> To: tuning-math@y... >> >> 2:1 >> >> --- In tuning-math@y..., <Josh@o...> wrote:>>> How are we defining "octave"? >>> >>> ---- Original message ---->>>> Date: Fri, 08 Nov 2002 21:50:29 -0000 >>>> From: "wallyesterpaulrus" <wallyesterpaulrus@y...> >>>> Subject: [tuning-math] Re: from the realms of private >>> correspondence >>>> To: tuning-math@y... >>>> >>>> --- In tuning-math@y..., "Carl Lumma" <clumma@y...> > wrote:>>>>>> I don't understand how the term "limit" got into your >>> question,>>>>>> or is this what others have called it? >>>>>>>>>> It comes from that it can be used as an alternative to >>> odd limit.>>>>> Interestingly, IIRC Paul showed odd-limit is as close > to >>> the>>>>> product thing as we can get in an octave-equivalent >>> measure. Is>>>>> that right, Paul? >>>>>>>> in a certain sense dealing with harmonic entropy, yes. >>>> >>>> >>>> To unsubscribe from this group, send an email to: >>>> tuning-math-unsubscribe@y... >>>> >>>> >>>> >>>> Your use of Yahoo! Groups is subject to >>> Yahoo! Terms of Service * [with cont.] (Wayb.) >>>> >>>> >> >>>> To unsubscribe from this group, send an email to: >> tuning-math-unsubscribe@y... >> >> >> >> Your use of Yahoo! Groups is subject to > Yahoo! Terms of Service * [with cont.] (Wayb.) >> >>

Message: 5558 - Contents - Hide Contents Date: Sat, 09 Nov 2002 00:01:28 Subject: Re: from the realms of private correspondence From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> p+q is the mann measure.It's also the height function number theorists use most often, for what that is worth.

Message: 5559 - Contents - Hide Contents Date: Sat, 09 Nov 2002 20:36:56 Subject: Fwd: Re: from the realms of private correspondence From: wallyesterpaulrus --- In tuning-math@y..., <Josh@o...> wrote:> Neither of the specific inharmonicity articles I'm > thinking of is dogmatic enough to require much hole-poking. > Sorry I can't remember the authors' names right now. > One article basically shows that inharmonic partials > produced by putting threads under the ends of strings on > Indian string instruments (standard technique) were > well-correlated with intonation variances of some scale > degrees; that at least some good performers intone > with at least implicit reference to inharmonics.this is sensible.> The piano tuning article, if I remember correctly, shows > that the extent to which piano tuners stretch the octave > on a piano correlates with the degree of stretching of > the harmonic series on piano strings.this is well-known (see Hall, _Musical Acoustics_).

Message: 5560 - Contents - Hide Contents Date: Sat, 09 Nov 2002 00:07:45 Subject: Re: from the realms of private correspondence From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote: >>> p+q is the mann measure. >> It's also the height function number theorists use most often, for >what that is worth.there an important difference in that number theorists tend to view the rationals in linear space, while music theorists tend to view the rationals in logarithmic space . . .

Message: 5561 - Contents - Hide Contents Date: Sat, 09 Nov 2002 20:39:22 Subject: Re: otonally-weighted lattices (was: from the realms of private correspondence) From: wallyesterpaulrus --- In tuning-math@y..., "monz" <monz@a...> wrote:> > hi paul, > >>> From: "wallyesterpaulrus" <wallyesterpaulrus@y...> >> To: <tuning-math@y...> >> Sent: Friday, November 08, 2002 3:39 PM >> Subject: [tuning-math] Re: from the realms of private correspondence >> >> >>> 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?as far as i can tell, you're already doing the angles that way; as for the lengths, i'm not sure if erv used a specific formula, but it looks like a logarithmic spiral, so log(p) would be a good choice for the lengths.

Message: 5562 - Contents - Hide Contents Date: Sat, 09 Nov 2002 20:47:23 Subject: Re: Fwd: otonally-weighted lattices (was: from the realms of private correspondence) From: wallyesterpaulrus --- In tuning-math@y..., <Josh@o...> wrote:> IMHO, the question of whether you want to weight things > otonally or utonally might entail issues of what timbres > and textures you're intending to use.i don't know what it means to weight things otonally or utonally.> One unstated criterion of good counterpoint, for example > has come to be that more of the consonances should imply > utonal shapes than otonal shapes (except, generally, at > points of "resolution").can you give an example? i could imagine some serious controversy on this point -- george secor seems to feel that utonalities, even mere 7-limit ones, sound like "parodies" of true just intonation chords . . . personally, i feel that this conclusion may come from an over-reliance on synthesized chords -- with acoustic instruments, in particular the guitar, the overtones of utonal chords induce sympathetic vibrations among the strings which make the chord very much an "attractor" when attempting to tune a nearby chord "justly".> In looking at numerous ensembles styles from various > cultures, I've developed the opinion that otonality > is more likely to predominate in denser, more spectrally > arithmetic and less "controlled" textures, idioms, whereas > the utonal is likely to predominate either in thinner > textures with more fixed parts and less > arithmetic harmonic spectra.what do you mean by "less arithmetic harmonic spectra"? do you mean inharmonic spectra? personally, i see no way for utonalities to arise without the presence of harmonic partials to "match up"; meanwhile, otonalities are powerful "attractors" even for inharmonic timbres or sine waves, due to the confluence of the combinational tones, and of course virtual pitch . . .> Not that this is absolutely true; my point is that > you might want to ask yourself how the intervals > are going to be used before trying to decide which > of them are "better".true enough . . . admittedly, we're in some pretty abstract territory here, which may be why this discussion is not going on on the tuning list but rather here on tuning-math.

Message: 5563 - Contents - Hide Contents Date: Sun, 10 Nov 2002 01:41:31 Subject: Re: from the realms of private correspondence From: Carl Lumma>> >f 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. 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: 5564 - Contents - Hide Contents Date: Mon, 11 Nov 2002 03:54:04 Subject: Re: from the realms of private correspondence From: wallyesterpaulrus --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>>> 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.if you sit down and verify for yourself all the properties of "metric" for euclidean distance and taxicab distance, i think you'll begin to have a better idea of what a metric is, in general. note that my geometric generalization of tenney's HD function so that it amounts to the LCM of the N integers when the N-ad is expressed in lowest harmonic terms (the LCM of great importance to measures like Euler's and, more recently, Marion's) allows us to forget that it's a metric altogether . . . and to forget about taxicab, since the construction of the rectangle for a dyad already contains the definition of taxicab inside it. of course, i have no idea what such a construction is called, but who cares?

Message: 5565 - Contents - Hide Contents Date: Mon, 11 Nov 2002 13:26:57 Subject: recurrent sequence of ETs from george secor From: wally paulrus george secor wrote me off-list, and i replied: ------------------------------------------------------------------------------------------------------------------------->By the way, what would you call a series where the nth member is the >sum of the (n-3) plus the (n-2) members?it's a type of "plastic" sequence, because all sequences obeying this recurrence, such as Padovan Sequence -- from MathWorld * [with cont.] and Perrin Sequence -- from MathWorld * [with cont.] have the limiting ratio of successive terms equal to the plastic constant: Plastic Constant -- from MathWorld * [with cont.]>I found that such a series >gave me most of my "top-rated" divisions under 100 if I started with >12, 19, 22: >12 19 22 31 41 53 72 94cool! if you like 3, 9, and 10, you could start off with them . . . [i just realized you could start with 1, 2, 7 -- that might be easier to remember] we've discussed a lot of patterns like these on tuning and tuning-math. gene has brought a lot of insight into them: why they work, etc. --------------------------------- Do you Yahoo!? U2 on LAUNCH - Exclusive medley & videos from Greatest Hits CD [This message contained attachments]

Message: 5566 - Contents - Hide Contents Date: Tue, 12 Nov 2002 19:21:55 Subject: Re: Osmium-Orwell-Secor From: gdsecor --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., genewardsmith@j... wrote: >>> ... and s is the Osmium secor, (43-11z-3z^2)/241, of >> 115.3367774 cents."Osmium secor" sounds like a good record album title for an alternative tuning heavy metal band performing in the Miracle temperament. :) --George

Message: 5567 - Contents - Hide Contents Date: Tue, 12 Nov 2002 01:39:47 Subject: Re: recurrent sequence of ETs from george secor From: Gene Ward Smith --- In tuning-math@y..., wally paulrus <wallyesterpaulrus@y...> wrote: The constant here is particularly interesting as the smallest PV number: Pisot-Vijayaraghavan Constant -- from MathWorld * [with cont.] Also of possible musical interest are Salem numbers: Salem Constants -- from MathWorld * [with cont.] I've mentioned some of this stuff in connection with "metallic" tunings (gold, platinum, etc.) and if I gave the name "Osmium" to a sequence coming from x^3-x-1, since Osmium is (probably) the densest metal, and the "plastic constant" is the smallest PV number.>> I found that such a series >> gave me most of my "top-rated" divisions under 100 if I started with >> 12, 19, 22: > >> 12 19 22 31 41 53 72 94 >> cool! if you like 3, 9, and 10, you could start off with them . . .I mentioned this sequence in Yahoo groups: /tuning-math/message/1297 * [with cont.] as being reasonable, but not leading to an Osmium generator. George might also like Yahoo groups: /tuning-math/message/1375 * [with cont.] if only because of the name.

Message: 5568 - Contents - Hide Contents Date: Tue, 12 Nov 2002 23:39:20 Subject: question about et density on graphs From: wallyesterpaulrus in each of the graphs below, i zoom in on the previous graph by a factor of 10, and raise the highest ET plotted by a factor of sqrt (10): Yahoo groups: /tuning-math/files/Paul/monzoome... * [with cont.] Yahoo groups: /tuning-math/files/Paul/monzoome... * [with cont.] Yahoo groups: /tuning-math/files/Paul/monzoome... * [with cont.] Yahoo groups: /tuning-math/files/Paul/monzoome... * [with cont.] Yahoo groups: /tuning-math/files/Paul/monzoome... * [with cont.] the points seem to be getting more and more sparse. what do i do to keep a constant density?

Message: 5570 - Contents - Hide Contents 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: 5571 - Contents - Hide Contents Date: Wed, 13 Nov 2002 15:25:57 Subject: Re: Osmium-Orwell-Secor From: gdsecor --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> --- In tuning-math@y..., genewardsmith@j... wrote:>> --- In tuning-math@y..., genewardsmith@j... wrote: >>>>> ... and s is the Osmium secor, (43-11z-3z^2)/241, of >>> 115.3367774 cents. >> "Osmium secor" sounds like a good record album title for an > alternative tuning heavy metal band performing in the Miracle > temperament. :) > > --GeorgeDid anybody get this joke? Osmium is one of the heavy metals in the platinum group. --GS

Message: 5572 - Contents - Hide Contents Date: Wed, 13 Nov 2002 20:24:13 Subject: Re: Osmium-Orwell-Secor From: wallyesterpaulrus --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:>> --- In tuning-math@y..., genewardsmith@j... wrote:>>> --- In tuning-math@y..., genewardsmith@j... wrote: >>>>>>> ... and s is the Osmium secor, (43-11z-3z^2)/241, of >>>> 115.3367774 cents. >>>> "Osmium secor" sounds like a good record album title for an >> alternative tuning heavy metal band performing in the Miracle >> temperament. :) >> >> --George >> Did anybody get this joke? Osmium is one of the heavy metals in the > platinum group.yup, gene mentioned that osmium is the densest metal, so i got it. now, if gene would help me with the graph density problem . . .

Message: 5573 - Contents - Hide Contents Date: Wed, 13 Nov 2002 01:34:46 Subject: Re: sorry, gotta go From: Gene Ward Smith --- In tuning-math@y..., <Josh@o...> wrote:> Hopefully, I'll be back.I hope so.

Message: 5574 - Contents - Hide Contents 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 ... * [with cont.] (Wayb.) (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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