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Message: 9225 - Contents - Hide Contents Date: Sat, 17 Jan 2004 16:07:51 Subject: Re: summary -- are these right? From: Carl Lumma>> >he Thing I was referring to here was most certainly rectangular. >> >> -Carl >>Well then it's no Thing that I've ever thought about or talked about >or heard of before!This was a different thing from our thread. -Carl

Message: 9226 - Contents - Hide Contents Date: Sat, 17 Jan 2004 05:44:16 Subject: Re: 11-limit Starling From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> 176;175 is |4 0 -2 -1 1> : this is one of the commas of my 11-limit > extension of porcupine.I would hope it would be a comma of anyone's version of porcupine, but I could run the naming convention check on it; I've been calling this simply "procupine" or 11-limit porcupine if we need to be precise. The top tunings for 5-limit and 7-limit porcupine are the same; the 11-limit tuning involves less sharpening of the octave, and moves it close to 22-equal--less than a cent away in val space. Compatible starling ET's include 12, 15, 16, 31,> 43, 46, 58, 73, 89.I get 31, 58, 65, and 89 for 11-limit starling, and your list for thrush.> As it turns out, DNA experiments have revealed that starlings are more > closely related to thrushes and mockingbirds than previously suspected. If > we accept this classification, it's a good argument for keeping the name > "thrush" for a temperament that has a close relationship with "starling".Thanks, Herman. I guess that settles it. :)

Message: 9227 - Contents - Hide Contents Date: Sat, 17 Jan 2004 19:24:31 Subject: Re: summary -- are these right? From: Carl Lumma>>>>>> >an you demonstrate how to get length log(9) out >>>>>> of 9/5? >>>>>>>>>> 9/5 is a ratio of 9. >>>>>>>> I meant on the lattice. >>>>>> Yes, that's how this 'lattice' is defined, isn't it? >>>> I was asking for any way it could be defined to make it >> equal odd-limit, but this seems like cheating because >> you require odd-limit infinity, and thus you're never >> taking any multi-stop routes. >>OK -- but without 'cheating', how can one do in the octave-equivalent >case what Tenney does in the octave-specific case?My question exactly.>> My point, if any, is that I think this will be impossible >> with odd-limit < inf. on a triangular lattice. >>Well, that's exactly what this: > >lattice orientation * [with cont.] (Wayb.) > >was attempting to address, at least for a prime limit of 5. Hmm... -Carl

Message: 9228 - Contents - Hide Contents Date: Sat, 17 Jan 2004 05:49:51 Subject: Re: summary -- are these right? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> On a unit-length odd-limit lattice both 9 and 11 have length 1.Doesn't 9 have length 2?

Message: 9229 - Contents - Hide Contents Date: Sat, 17 Jan 2004 19:27:30 Subject: Re: summary -- are these right? From: Carl Lumma>>>> >he two obvious variations are rectangular odd-limit >>>>>> How can odd-limit be rectangular? Makes no sense to me. >>>> One can certainly have a rectangular lattice with a 9-axis. >>A 'lattice'-like thing, yes.If we're going to be going over to the mathematical definition of lattice, we should come up with a term that means "anything with rungs".>But then it has nothing to do with odd-limit. And is there a >2-axis too?What would happen either way?>>>> and triangular octave-specific. >>>>>> Then the metric is not log(n*d) anymore. >>>> We actually haven't specified how to find the lengths of >> rungs like 9:5... >>True, but if you use something different from what Tenney gives, >you'll be hard pressed to get all the consonant intervals within a >given range (say, 260-500 cents) in the correct order of consonance.So summing up, can we say that we're happy with our octave-specific concordance heuristic and associated lattice/metric, and that we have an octave-equivalent concordance heuristic but *no* associated lattice/metric? On the other hand, given Gene's recent post, "we" might not include him... -Carl

Message: 9230 - Contents - Hide Contents Date: Sat, 17 Jan 2004 08:04:39 Subject: 12-note 5-limit Fokker blocks From: Gene Ward Smith Anyone who wants the zipped collection will find it here: Yahoo groups: /tuning-math/files/Gene/fokker12... * [with cont.]

Message: 9231 - Contents - Hide Contents Date: Sat, 17 Jan 2004 17:18:10 Subject: Re: summary -- are these right? From: Carl Lumma>>>>>> >ut the basic insight is that a triangular lattice, with >>>>>> Tenney-like lengths, a city-block metric, and odd axes or >>>>>> wormholes, agrees with the odd limit perfectly, and so is >>>>>> the best octave-invariant lattice representation (with >>>>>> associated metric) for anyone as Partchian as me. >>>>>>>>>> Right -- you need those odd axes, which screws up uniqueness, >>>>> and thus most of how we've been approaching temperament. >>>>>>>> But does the metric agree with log(odd-limit) or not? >>>> For 9:5, log(oddlimit) is log(9). If you run it through >>>> the "norm" you get... 2log(3) + log(5). >>>>>> No, because 9 has its own axis. >>>> It's still different than log(odd-limit), and in fact >> log(5) + log(9) = 2log(3) + log(5). >>You're forgetting that 5:3 has its own rung in this lattice, with >length log(5), since the 'odd-limit' of 5:3 is 5 (more correctly, 5:3 >is a ratio of 5).I guess so. Can you demonstrate how to get length log(9) out of 9/5?>>>> Not the same, >>>> it seems. However if you followed the >>>> lumma.org/stuff/latice1999.txt link, >>>>>> The page cannot be found. >>>> Typo here; "lattice". >>The page cannot be found.Geez, I'm so sorry, it's [Paul Hahn] * [with cont.] (Wayb.) -Carl

Message: 9232 - Contents - Hide Contents Date: Sat, 17 Jan 2004 10:23:24 Subject: Re: TOP history From: Graham Breed Paul Erlich wrote:> It's exactly what I've been pleading to you guys to help me figure > out last year and probably even earlier, except without octave- > equivalence. The idea was to temper out commas uniformly over their > length in the lattice, to see what error function this was optimal > with respect to, and to then apply this same error function to > optimize temperaments with more than one comma. The posts asking > about this can be found in the archives here.Was it? Oh. Well, I found this thread: Yahoo groups: /tuning-math/message/2857 * [with cont.] The new thing is the concept of weighted minimax. Graham

Message: 9233 - Contents - Hide Contents Date: Sat, 17 Jan 2004 17:25:14 Subject: Re: summary -- are these right? From: Carl Lumma>> >his was a different thing from our thread. >>You were talking about odd-limit thing: > >Yahoo groups: /tuning-math/message/8662 * [with cont.] > >When and where did you switch to a rectangular thing?Let's start over. I'm fishing for something we can use to weed down the number of "lattices" we're interested in. Am I correct that you think log(odd-limit) is the best octave-equivalent concordance heuristic, and that it constitutes a norm on the triangular odd-limit lattice with log weighting? Am I correct that you believe log(n*d) is the best octave-specific concordance heuristic and that it constitutes a norm on the Tenney lattice? The two obvious variations are rectangular odd-limit and triangular octave-specific. What say you about those? Finally, for each of these four lattice types, we can inquire about what happens when we use no weighting, ("unit lengths"). -Carl

Message: 9234 - Contents - Hide Contents Date: Sat, 17 Jan 2004 03:33:00 Subject: Re: summary -- are these right? From: Carl Lumma>> >n a unit-length odd-limit lattice both 9 and 11 have length 1. >>Doesn't 9 have length 2?9 occurs in two places on an odd-limit lattice of odd-limit >= 9. With unit lengths, it has length 1 on the 9 axis or length 2 on the 3 axis. You can allow this, or impose log weighting which makes them the same length on a rectangular lattice. -Carl

Message: 9235 - Contents - Hide Contents Date: Sat, 17 Jan 2004 12:22:20 Subject: Dual space example From: Gene Ward Smith I had thought of this business of subspaces and the JIP as a way of formulating what we were already doing, but didn't realize it led to anything new. I wish I'd seen the possible link to Paul's heuristic, but another interesting aspect I didn't consider is this bounded relative error business. Here's a basic example of what I mean. As I've pointed out from time to time, the norm which gives the equilateral triangular lattice for 5-limit octave classes is ||3^a 5^b|| = sqrt(a^2 + ab + b^2) where the class is represented by 3^a 5^b. The norm on the dual space is then ||(x3, x5)|| = sqrt(x3^2 - x3 x5 + x5^2) where x2 is the tuning of 3 (in log2, cents or whatever your favorite log is terms) and x5 is the tuning of 5. The nearest point to [log(3), log(5)] on a subspace corresponding to a temperament is the 5-limit rms tuning. If we look on the line x5=4x3 - 4, for instance, we get the Woolhouse tuning. If we now take the error in cents for the Woolhouse tuning of any 5-limit interval q and divide it by the norm of q, it will be bounded by the worst case, namely 81/80, for which we get 5.965. Other values are 5.790 for 3/2, 1.654 for 5/4, 4.136 for 5/3, 5.940 for 27/20 and so forth--all bounded by 5.965.

Message: 9236 - Contents - Hide Contents Date: Sat, 17 Jan 2004 19:59:46 Subject: Re: summary -- are these right? From: Carl Lumma>> >f we're going to be going over to the mathematical definition >> of lattice, we should come up with a term that means "anything >> with rungs". >>A graph (as in graph-theory) but with lengths for each rung?That could be a "directed graph" I think. But all the flavors of graph I'm aware of lack orientation, fixed dimensionality, and so forth. Maybe "space" would work here?>>> But then it has nothing to do with odd-limit. And is there a >>> 2-axis too? >>>> What would happen either way? >>If there is a 2-axis, a 9-axis in rectangular lattice seems >superfluous (it doesn't change anything in terms of the taxicab >distances you get, but adds an infinite number of copies of each >pitch), unless you have a reason for treating '9' as different >from '3*3' (and therefore '9/3' different than '3'), etc., such >as a constraint to 768-equal partials. > >If there is no 2-axis, you get bad consonance evaluation, for the >usual reasons.So rectangular must have 2, and triangular probably shouldn't...>> So summing up, can we say that we're happy with our >> octave-specific concordance heuristic and associated >> lattice/metric, and that we have an octave-equivalent >> concordance heuristic but *no* associated lattice/metric? >>I'd prefer not to say 'concordance heuristic', but yes.What would you say? -Carl

Message: 9237 - Contents - Hide Contents Date: Sat, 17 Jan 2004 12:28:16 Subject: Re: summary -- are these right? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> On a unit-length odd-limit lattice both 9 and 11 have length 1. >>>> Doesn't 9 have length 2? >> 9 occurs in two places on an odd-limit lattice of odd-limit >= 9.Is this "lattice" one of those goofy things people insist on calling lattices even though they are not? What in the world do you mean? 9 is always 3^2, so it necessarily is twice as far from the origin as 3, and in the same direction. If 9 and 11 are the same length, 3 is half that length. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 9238 - Contents - Hide Contents Date: Sun, 18 Jan 2004 01:56:35 Subject: Re: Duals to ems optimization From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Gene, note that I've always counted 3/1 and 9/3, etc., separately in > these optimizations. If you use that "weighting", do things look less > dubious? (The weight is proportional to the number of ways the > interval class can be represented by a ratio of odd numbers within > the limit.)It weighs 3 more, but it still seems weird. The norm on vals is sqrt(-6x3x11-2x5x11-2x5x7-6x3x7-6x3x5-2x7x11+21x3^2+5x5^2+5x7^2+5x11^2) and the corresponding norm on pcs is sqrt(3e3^2+6e3e5+6e3e7+6e3e11+11e5^2+10e5e7+10e5e11+11e7^2+10e11e7+11e11^2) This gives us ||3|| = ||3/2|| = ||4/3|| = sqrt(3), ||9/8||=2sqrt(3), ||5/4|| = ||7/4|| = ||11/8|| = sqrt(11). What seems more dubious is ||11/6|| = ||7/6|| = ||6/5|| = 2sqrt(2). I think my idea of using the dual norm to my "geometric" norm makes more sense.

Message: 9239 - Contents - Hide Contents Date: Sun, 18 Jan 2004 04:20:41 Subject: Re: A new graph for Paul? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> For one thing, the generator is not unique, and its multiplicity is >> proportional to periods per octave. >> Let's stick with octave periods for now. >>> If you're not transforming continuously in terms of the generator in >> cents, what are you transforming continuously in terms of? >> It might be something you can do by using 7-equal as a way to pass > from one temperament to another. The lines in question are 7-et lines; > if you take the generator val <0 a b| and wedge/cross-product it with > <1 11/7 16/7| you get the monzos for the various temperaments.Let me get back to this after we're done talking about error functions and the metrics of their duals.

Message: 9240 - Contents - Hide Contents Date: Sun, 18 Jan 2004 21:46:32 Subject: Strange9 From: Gene Ward Smith On Dave's web page is a discussion of "Strange9" KeenanTuning * [with cont.] (Wayb.) This is either Pajara, Pajara[10], or both. Dave wanted to beat Paul's Pajara[10] tuned to 22-equal, and ended up detempering it, though not on purpose. Paul came up with "Pajara" as a name for the temperament; I'm wondering if Dave was proposing "Strange" or "Strange9" for it.

Message: 9241 - Contents - Hide Contents Date: Sun, 18 Jan 2004 02:14:00 Subject: Re: Question for Dave Keenan From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> >> Thanks for that. Sounds to come. >> Thanks. I've listened to them. Definitely single pitches. Can you tell > us the relative amplitides of all the partials.sqrt(1), sqrt(2), . . . sqrt(6).> And can we hear a > sustained note around middle C. Yahoo groups: /tuning_files/files/Erlich/dave1... * [with cont.]>>>> The fact is that, when using inharmonic timbres of the sort I >>>> described, Western music seems to retain all it meaning: certain >>>> (dissonant) chords resolving to other (consonant) chords, etc., >> all>>>> sounds quite logical. My sense (and the opinion expressed in >>>> Parncutt's book, for example) is that *harmony* is in fact very >>>> closely related to the virtual pitch phenomenon. We already know, >>>> from our listening tests on the harmonic entropy list, that the >>>> sensory dissonance of a chord isn't a function of the sensory >>>> dissonances of its constituent dyads. Furthermore, you seem to be >>>> defining "something special" in a local sense as a function of >>>> interval size, but in real music you don't get to evaluate each >>>> sonority by detuning various intervals various amounts, which >>>> this "specialness" would seem to require for its detection. >>>> >>>> The question I'm asking is, with what other tonal systems, >> besides>>>> the Western one, is this going to be possible in. >>>>>> If by "Western tonal systems", you mean any based on approximating >>> small whole number ratios of frequency, >>>> No, I meant diatonic/meantone. >> OK. So is your question, "In what tonal systems other than > diatonic/meantone is it going to be possible to have dissonant chords > resolving to consonant chords?"?Yes, in a sense, and with timbres that are primarily sine-wave but have spectral and envelopular aids to being individually heard out.> The obvious answer would seem to be systems in which there are > consonant chords, i.e which approximate (or are) JI at least >partially.Right -- now is Top pelogic such a system?>>> What's your point? >>>> Did the above really not say anything to you? >> Certainly not until you clarified the above. And it might still be a > good idea for you to spell out the conclusion you intend.That the phenomenon responsible for central (aka 'virtual') pitch allows x amount of Tenney-weighted error in the partials of a single pitch, and should the phenomenon be at least partially responsible also for the consonance of triads, temperaments with x amount of Tenney-weighted error stand a chance of exhibiting this triadic consonance, especially if roughness-inducing harmonic overtones are absent from each of the pitches. So it's more than matching tuning and timbre to acheive 'sensory consonance' -- defined as a local minimum of roughness -- but it's less than the multiply-caused (central pitch, combinational tones, and low roughness) consonance that occurs with very-low-error temperaments or JI when using harmonic timbres.

Message: 9242 - Contents - Hide Contents Date: Sun, 18 Jan 2004 04:24:02 Subject: Re: summary -- are these right? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> If we're going to be going over to the mathematical definition >>> of lattice, we should come up with a term that means "anything >>> with rungs". >>>> A graph (as in graph-theory) but with lengths for each rung? >> That could be a "directed graph" I think.A directed graph can have both ways or one way streets between the nodes. A multigraph allows for multiplicity in the connection, which is a little like a shorter length. Nodes connected by lines at various distances sounds most like a polytope.

Message: 9243 - Contents - Hide Contents Date: Sun, 18 Jan 2004 16:19:15 Subject: A potentially informative property of tunings From: Herman Miller Take a generator of 260.76 cents and a period of 1206.55 cents. This defines a linear tuning which belongs to a family of related linear temperaments. The simplest mapping is the "beep" mapping, which distributes the 27;25 interval: [(1, 0), (2, -2), (3, -3)] but after 6 iterations of the generator, there's a better 5:1 at (1, 6), about 15 cents flat (compared with the 51 cent sharp "beep" version of the interval). That means this particular tuning is consistent with "beep" temperament only up to a range of 5 generators -- or to coin a phrase, its "consistency range" with respect to "beep" is 5. In comparison, top meantone has a "consistency range" of 34: its (17, -35) version of 5:1 is only 2 cents flat, compared with the 4-cent sharp (4, -4). Quarter-comma meantone has a "consistency range" of 29, since it has a better 3:1 at (-11, 30). First of all, I don't like the term "consistency range", but I couldn't think of anything better. I'd appreciate ideas for what to call this property. Secondly, the fact that this property varies from one particular tuning of a temperament to another implies that there's a particular tuning with the maximum "consistency range" for any given temperament. This seems like it would be a useful property to maximize. Once a tuning exceeds its "consistency range" limit for a specific temperament, it gives way to (mutates into?) a related temperament that shares part of the mapping. So "beep" mutates into "superpelog" at 6 iterations, and "superpelog" in turn mutates into a new temperament at 31 iterations: [(1, 0), (2, -2), (3, -3)] beep [(1, 0), (2, -2), (1, 6)] superpelog [(1, 0), (2, -2), (9, -31)] unnamed temperament The same thing happens with 7-limit versions of top meantone: [(1, 0), (2, -1), (4, -4), (2, 2)] [(1, 0), (2, -1), (4, -4), (-1, 9)] [(1, 0), (2, -1), (4, -4), (7, -10)] The mutation sequence for quarter-comma meantone skips over (-1, 9) and goes straight to (7, -10), followed by (-6, 21). Obviously, the mutation sequence varies from one tuning of the generator and period to another; a 442.2 cent tuning of "father" mutates to semisixths, but "top father" mutates to a bizarre temperament with mapping [(-2, 8), (5, -9), (5, -7)]. On the other hand, knowing the mutation sequence and the consistency range limit could be important when trying to get a feel for the usefulness of a new tuning. -- see my music page ---> ---<The Music Page * [with cont.] (Wayb.)>-- hmiller (Herman Miller) "If all Printers were determin'd not to print any @io.com email password: thing till they were sure it would offend no body, \ "Subject: teamouse" / there would be very little printed." -Ben Franklin

Message: 9244 - Contents - Hide Contents Date: Sun, 18 Jan 2004 02:19:50 Subject: Re: A new graph for Paul? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> Dual to the 5-limit symmetrical lattice of intervals is a 5-limit >> symmetrical lattice of vals whose first component is zero--which >> includes the generators in the period-generator of linear >> temperaments. >> Don't get it.What's the hang-up? Do you understand the part about pairs of integers representing generators for 5-limit linear temperaments? Do you understand the norm I've placed on these? Do you understand this makes it a lattice, and what that lattice looks like?>> The 3 axis and the 5 axis for intervals is 60 degrees apart; for a >> graph of the lattice of generators, |0 1> and |1 0> should be 120 >> degrees apart. There are interesting lines to draw on such a graph; >> the |-3 -5> of porcupine, |1 4> of meantone and |5 13> of amity lie >> along a line, for instance. >> What does that line mean?That's to be explored. We have, along a line, porcupine-->meantone-->tetracot-->amity In terms of generator mapping, but not the period part of the map, and therefore not in terms of the generators in cents, we can transform one to the next continuously. That's the sort of thing I think is worth thiking about from a compositional point of view, for starters. I suggest we might find out more about what it might mean with a graph in front of us.>> Each generator can be graphed twice, by >> graphing +-|1 4>, etc. This would give us additional lines; the line >> between |-1 -4> and |-3 -5> includes pelogic at |1 -3>. > > Meaning?Let's find out.

Message: 9245 - Contents - Hide Contents Date: Sun, 18 Jan 2004 04:32:55 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> If we're going to be going over to the mathematical definition >>>> of lattice, we should come up with a term that means "anything >>>> with rungs". >>>>>> A graph (as in graph-theory) but with lengths for each rung? >>>> That could be a "directed graph" I think. >> A directed graph can have both ways or one way streets between the > nodes. A multigraph allows for multiplicity in the connection, which > is a little like a shorter length. Nodes connected by lines at various > distances sounds most like a polytope. ? Polytope -- from MathWorld * [with cont.]

Message: 9246 - Contents - Hide Contents Date: Sun, 18 Jan 2004 22:33:36 Subject: Re: Question for Dave Keenan From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...>>> And can we hear a >> sustained note around middle C. > > Yahoo groups: /tuning_files/files/Erlich/dave1... * [with cont.]I didn't find a sustained note there.>> OK. So is your question, "In what tonal systems other than >> diatonic/meantone is it going to be possible to have dissonant > chords>> resolving to consonant chords?"? >> Yes, in a sense, and with timbres that are primarily sine-wave but > have spectral and envelopular aids to being individually heard out. >>> The obvious answer would seem to be systems in which there are >> consonant chords, i.e which approximate (or are) JI at least >> partially. >> Right -- now is Top pelogic such a system?Possibly. But if so, it's marginal. This is of course what I've been saying for some time. I don't see that you've given me any reason to change this position.>> And it might still be a >> good idea for you to spell out the conclusion you intend. >> That the phenomenon responsible for central (aka 'virtual') pitch > allows x amount of Tenney-weighted error in the partials of a single > pitch, and should the phenomenon be at least partially responsible > also for the consonance of triads, temperaments with x amount of > Tenney-weighted error stand a chance of exhibiting this triadic > consonance, especially if roughness-inducing harmonic overtones are > absent from each of the pitches. > > So it's more than matching tuning and timbre to acheive 'sensory > consonance' -- defined as a local minimum of roughness -- but it's > less than the multiply-caused (central pitch, combinational tones, > and low roughness) consonance that occurs with very-low-error > temperaments or JI when using harmonic timbres.But remember that the disagreement on things like "beep" and "father" is not whether they contain consonances but whether those consonances have anything to do with their supposed 5-limit mappings.

Message: 9247 - Contents - Hide Contents Date: Sun, 18 Jan 2004 23:25:10 Subject: Annotated Dave Keenan file From: Gene Ward Smith I've taken this file Good 7-limit generators * [with cont.] (Wayb.) and annotated it by adding names when available, and descriptions when not, of the 7-limit linear temperaments it discusses. While I think the value judgments are a little eccentric, it surely is worth looking at the temperaments here which don't have names. Maybe Dave wants to suggest something? The nameless ones are these: <3 7 -1 4 -10 -22| {36/35, 1029/1024} TOP: [1205.820043, 1890.417958, 2803.215176, 3389.260823] rms gens: [1200., 227.4263791] <2 -6 -6 -14 -15 3| {50/49, 135/128} TOP: [1206.548264, 1891.576247, 2771.109113, 3374.383246] rms gens: [600.0000000, 79.05132740] <6 -2 -2 -17 -20 1| {50/49, 525/512} TOP: [1203.400986, 1896.025764, 2777.627538, 3379.328030] rms gens: [600.0000000, 231.2978354] Dave's article: This appeared in Tuning Digest 245, 10 Jul 1999. [A typo has been corrected where the MA err for 227c was given as 22.4c] Message: 10 Date: Sat, 10 Jul 1999 10:39:56 +1000 From: David C Keenan [address removed] Subject: Good 7-limit generators I've been doing some serious number crunching and can now announce that there are only 3 single-chain generators of good (octave-based) 7-limit scales, and only 6 double-chain (with a half-octave) generators. To qualify, all they had to do was to have _more_ complete tetrads in a chain of 10 notes, than meantone using augmented sixths (2), and have lower errors (either RMS or max-absolute) than the best chain of fourths/fifths where the dominant 7th chord is the 4:5:6:7 approximation. i.e. around a 702.5c fifth (= 497.5c fourth). Here's the info on these not-quite-good-enough generators for comparison: Not good enough: No. generators in Min Min interval Generator No. tetrads 7-limit 7-limit 2 4 5 4 5 6 (+-0.5c) in 10 notes RMS error MA err. 3 5 6 7 7 7 ------------------------------------------------------------- 503.4c 2 3.6c 5.4c -1 -4 3-10 -6 -9 meantone 497.5c 8 20.2c 25.4c -1 -4 3 2 6 3 dominant7 Single chain: No. generators in Min Min interval Generator No. tetrads 7-limit 7-limit 2 4 5 4 5 6 (+-0.5c) in 10 notes RMS error MA err. 3 5 6 7 7 7 ------------------------------------------------------------- 125c 6 12.2c 17.9c -4 3 -7 -2 -5 2 tertiathirds 227c 4 16.5c 24.4c 3 7 -4 -1 -8 -4 <3 7 -1 4 -10 -22| {36/35, 1029/1024} 317c 8 12.3c 17.9c 6 5 1 3 -2 -3 kleismic The generator sizes are only given to +-0.5c because the exact value will depend on whether RMS error or Max-Absolute error or Max-otonal-beat-rate (not shown) is the measure to be optimised. Note that the minor-third generator, that we've been discussing recently, is the best possible for a single chain. Note also that 227c is equivalent to 1200-227 = 973c, a 4:7 generator. Double chain: No. generators in Min Min interval Generator No. tetrads 7-limit 7-limit 2 4 5 4 5 6 (+-0.5c) in 10 notes RMS error MA err. 3 5 6 7 7 7 ------------------------------------------------------------- 71c 4 12.5c 18.2c 1 -3 -4 -3 0 4 <2 -6 -6 -14 -15 3| {50/49, 135/128} 230c 4 11.8c 17.5c 3 -1 4 -1 0 -4 <6 -2 -2 -17 -20 1| {50/49, 525/512} 380.5c 4 10.3c 17.5c -3 1 4 1 0 -4 <6 -2 -2 -17 -20 1| {50/49, 525/512} 491c 8 10.9c 17.5c -1 2 -3 2 0 3 pajara 506.5c 4 11.2c 17.5c -1 -4 3 -4 0 -3 injera 521c 4 16.9c 23.1c -1 3 -4 3 0 4 <2 -6 -6 -14 -15 3| {50/49, 135/128} In the table above, I haven't shown whether the half octave is included in an interval or not, but that's not difficult to figure out. Note that 230c is equivalent to 970c, an approximate 4:7. 380.5c is a major third. The last three are fourths that correspond to fifths of 709c, 693.5c and 679c. The first two of these last three were recognised by Paul Erlich as being in the vicinity of the 22-tET and 26-tET fifths, and the last one is not really good enough, with a 23.1c error in its fifths. So far, 491c with a half-octave (22-tET) is the winner, with the (recently discovered?) 317c minor-third a close second, and 125c in third place (based on treating number-of-tetrads as more important than accuracy). I wonder which of these have been discovered before? I wonder if I've missed any? I'm 99% sure I haven't, but I'll be rechecking. Any journals likely to be interested in this? Does somone want to work out what ET's these embed in with sufficient accuracy? Or how many notes in the largest 125c MOS with 12 notes or less? I could look at higher multiple chains with the appropriate fraction of an octave, if anyone cares. Does anyone want me to change my criteria in any way? Regards, -- Dave Keenan Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 9248 - Contents - Hide Contents Date: Sun, 18 Jan 2004 02:25:08 Subject: Re: Duals to ems optimization From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> Gene, note that I've always counted 3/1 and 9/3, etc., separately in >> these optimizations. If you use that "weighting", do things look less >> dubious? (The weight is proportional to the number of ways the >> interval class can be represented by a ratio of odd numbers within >> the limit.) >> It weighs 3 more, but it still seems weird. The norm on vals is > > sqrt(-6x3x11-2x5x11-2x5x7-6x3x7-6x3x5- 2x7x11+21x3^2+5x5^2+5x7^2+5x11^2) > > and the corresponding norm on pcs is > > sqrt (3e3^2+6e3e5+6e3e7+6e3e11+11e5^2+10e5e7+10e5e11+11e7^2+10e11e7+11e11^2 ) > > This gives us ||3|| = ||3/2|| = ||4/3|| = sqrt(3),Perfect . . . (1.7321)> ||9/8||=2sqrt(3),Excellent . . . (3.4641)> ||5/4|| = ||7/4|| = ||11/8|| = sqrt(11).OK . . . (3.3166)> What seems more dubious > is ||11/6|| = ||7/6|| = ||6/5|| = 2sqrt(2). (2.8284)Doesn't seem unduly dubious, though, given that all the lengths are about equal here, except 3 is shorter. Pretty much in accordance with what we put in. Now what if we apply 'odd-limit-weighting' to each of the intervals, including 9:3 which is treated as having an odd-limit of 9? Try using 'odd-limit' plus-or-minus 1 or 1/2 too.> I think my idea of using > the dual norm to my "geometric" norm makes more sense.Why is that?

Message: 9249 - Contents - Hide Contents Date: Sun, 18 Jan 2004 05:19:25 Subject: Re: Question for Dave Keenan From: Carl Lumma>> >nd can we hear a >> sustained note around middle C. > > Yahoo groups: /tuning_files/files/Erlich/dave1... * [with cont.]Wrong file, Paul? -Carl

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