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Message: 9275 - Contents - Hide Contents Date: Sun, 18 Jan 2004 01:49:24 Subject: Re: A new graph for Paul? From: Paul Erlich --- 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.> 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? The dual graph I recently produced puts 'linear temperaments' that share an ET on straight lines . . .

Message: 9276 - Contents - Hide Contents Date: Sun, 18 Jan 2004 01:50:38 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>>>> But 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?9/5 is a ratio of 9.>>>>> 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. >OK, which part were we talking about?

Message: 9277 - Contents - Hide Contents Date: Sun, 18 Jan 2004 04:08:56 Subject: Re: summary -- are these right? From: Paul Erlich --- 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.Directed means each rung has a specific beginning point and ending point.> But all the flavors > of graph I'm aware of lack orientation, fixed dimensionality, > and so forth. Maybe "space" would work here?A space has an infinite number of points between any two points.>>> 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. >

Message: 9278 - Contents - Hide Contents Date: Sun, 18 Jan 2004 01:53:30 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> This 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,That or something very similar to it, like perhaps log(2*odd-limit - 1) or log(2*odd-limit + 1) etc.> and that it constitutes a norm > on the triangular odd-limit lattice with log weighting?Technically, it can't, because you don't have uniqueness, etc.> 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? Yes. > The two obvious variations are rectangular odd-limitHow can odd-limit be rectangular? Makes no sense to me.> and triangular octave-specific.Then the metric is not log(n*d) anymore. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9279 - Contents - Hide Contents Date: Sun, 18 Jan 2004 04:11:22 Subject: Re: A new graph for Paul? From: Gene Ward Smith --- 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. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9280 - Contents - Hide Contents Date: Mon, 19 Jan 2004 07:55:34 Subject: Re: Question for Dave Keenan From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> After all, Gene and others would have us > believe that meantone dominant seventh chords were "experienced as" > 4:5:6:7 chords, even though the 6:7 interval would typically be tuned > far closer to 5:6.Would harmonic entropy suggest something different?

Message: 9281 - Contents - Hide Contents Date: Mon, 19 Jan 2004 11:42:54 Subject: Re: Question for Dave Keenan From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> It hasn't been done yet, but I promise to do it post haste, if you > first give me a sincere effort to improve the computational > efficiency of the calculation (starting, of course, with the dyadic > case). I'd prefer to discuss this on the harmonic entropy list, for > the convenience of those who wish to follow its development.It's your definition; I've thought about trying to compute it, but it looks nasty. Are you suggesting trying to find something similar, but easier to compute?

Message: 9282 - Contents - Hide Contents Date: Mon, 19 Jan 2004 08:06:46 Subject: Re: Question for Dave Keenan From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> e.g. with TOP Beep, you were asking us to believe that the 260 c > generator could be "experienced as" an approximate 5:6 even though it > is only 7 cents away from 6:7, and 55 cents away from 5:6.A feeble objection. The comma for equating 7/6 and 6/5 is 36/35, and the wedgie we get with that and 27/25 is <<2 3 1 0 -4 -6||. Now compare TOP tunings: TOP 27/25: <1200 1879.486 2819.230| TOP <<2 3 1 0 -4 -6||: <1200 1879.486 2819.230 3329.029| In other words, 7-limit beep equates 7/6 and 6/5 anyway.

Message: 9283 - Contents - Hide Contents Date: Mon, 19 Jan 2004 12:08:03 Subject: Harmonic Entropy (was: Re: Question for Dave Keenan) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> It hasn't been done yet, but I promise to do it post haste, if you >> first give me a sincere effort to improve the computational >> efficiency of the calculation (starting, of course, with the dyadic >> case). I'd prefer to discuss this on the harmonic entropy list, for >> the convenience of those who wish to follow its development. >> It's your definition; I've thought about trying to compute it, but it > looks nasty.The idea is to find an easier way to express it or compute it, but the direct computation is pretty easy for dyads. Let's use means instead of mediants, which will allow us to use voronoi cells in the generalization to higher dimensions. The harmonic entropy of a dyad i is simply -SUM (p(j,i) log(p(j,i))) j where j runs in order of cents size over all lowest-term ratios where n*d < 10000 or 65536 or some large number, and p(j,i) is proportional to (cents(j+1)-cents(j-1))*exp( -(cents(j)-cents(i))^2 / (2s^2) ) and the constant of proportionality is such that -SUM (p(j,i)) = 1 j or, if you prefer, you could define p(j,i) as 1/(s*sqrt(2*pi)) times the integral from (cents(j-1) + cents(j))/2 to (cents(j+1) + cents(j))/2 of exp( -(cents(t)-cents(i))^2 / (2s^2) ) dt> Are you suggesting trying to find something similar, but > easier to compute?Yes -- similar or even equal.

Message: 9284 - Contents - Hide Contents Date: Mon, 19 Jan 2004 08:21:29 Subject: Re: Annotated Dave Keenan file From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> However, <4 -3 -17 -14 -38 -31| is closer, <4 -3 21 -14 22 57| much >> closer yet, and <4 -3 40 -14 52 101| has the identical TOP tuning. >> The TOP tuning of what? Negri.>> What to do? >> I don't get it. Paul's temperament database doesn't list tertiathirds > so I don't know what comma(s) tertiathirds was based on. If it's > previously been a 5-limit linear temperament I don't see how it could > have <-4 3] the same as negri.Tertiathirds is a name for the 7-limit temperament with TM comma base {49/48, 225/224}, wedgie <<4 -3 2 -14 -8 13|| and mapping [<1 2 2 3|, <0 -4 3 -2|]. There's been a question all along as to whether it should be called "negri", and I suppose it should be; it's pretty closely tied to 2/19 as a generator either way.

Message: 9285 - Contents - Hide Contents Date: Mon, 19 Jan 2004 18:41:47 Subject: Re: A potentially informative property of tunings From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> 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.Since you're describing a relationship or comparison between two temperaments, I would suggest "compatibility range". The term "consistency" is usually used to describe only relationships within a single temperament. --George

Message: 9286 - Contents - Hide Contents Date: Mon, 19 Jan 2004 08:35:22 Subject: Re: A potentially informative property of tunings From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> The mutation sequence for quarter-comma meantone skips over (-1, 9) and > goes straight to (7, -10), followed by (-6, 21).If you want to go out this far to map 7, it probably makes more sense to use a sharper fifth and (12, -22) for the map. This is a temperament for meantones in the neighborhood of 67-et, and since 55-et falls in there we could be evil and call it "moztone", but given the nature of Mozart's use of meantone that hardly makes sense. In any case, its badness is so high because of the high complexity to get to 7s harmony it hardly makes sense to use this for an alternative to septimal meantone. Jon might be happy with its TOP tuning, since octaves are only a smidgen flat.

Message: 9287 - Contents - Hide Contents Date: Mon, 19 Jan 2004 21:48:40 Subject: Re: Question for Dave Keenan From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> After all, Gene and others would have us >> believe that meantone dominant seventh chords were "experienced as" >> 4:5:6:7 chords, even though the 6:7 interval would typically be tuned >> far closer to 5:6. >> Would harmonic entropy suggest something different?I assume you mean here the _minimisation_ of harmonic entropy. There's also the posssibility that the dominant seventh chord functions best when its harmonic entropy (or maybe only the HE of one of its dyads) is locally _maximised_. George Secor alluded to this recently (on the tuning list I think).

Message: 9288 - Contents - Hide Contents Date: Mon, 19 Jan 2004 08:37:10 Subject: Re: Question for Dave Keenan From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> After all, Gene and others would have us >> believe that meantone dominant seventh chords were "experienced as" >> 4:5:6:7 chords, even though the 6:7 interval would typically be tuned >> far closer to 5:6. >> Would harmonic entropy suggest something different?It might, depending on the value of 's' or hearing resolution assumed (this is essentially the only free parameter in harmonic entropy, which subsumes considerations of timbre, register, etc.). The meantone dominant seventh could land outside the low-entropy region surrounding 4:5:6:7 in the 3-dimensional space of tetrads (where the axes corresponding to these harmonics lie at 60-degree angles).

Message: 9289 - Contents - Hide Contents Date: Mon, 19 Jan 2004 22:06:43 Subject: Re: Question for Dave Keenan From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote: >>> e.g. with TOP Beep, you were asking us to believe that the 260 c >> generator could be "experienced as" an approximate 5:6 even though it >> is only 7 cents away from 6:7, and 55 cents away from 5:6. >> A feeble objection. ... > In other words, 7-limit beep equates 7/6 and 6/5 anyway.Mad if it didn't. However I don't see that that affects anything I said about 5-limit TOP Beep. Paul's objection was better, namely that if you provide a large enough context of other pitches in an approximate harmonic-series segment, then maybe even a 55 cent error can be pulled into line, as it were. But even if that were true, there is a very significant difference between a normal 5-limit temperament where all the 5-limit harmonies work (including bare dyads and utonalities), and one in which only otonal tetrads and larger otonalities "work".

Message: 9290 - Contents - Hide Contents Date: Mon, 19 Jan 2004 00:07:04 Subject: Re: TOP on the web From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> I can plug the formulas into a program, and they seem to produce accurate > results, but I have no idea how to generalize them to higher limits.I've added something on how to set it up as a linear programming problem.> Also, how would this work for more than one comma? I'd like to eventually > be able to figure out things like the TOP for a linear tuning with a > mapping of [(1, 0), (2, -2), (1, 6), (3, -1), (3, 2)] and commas of [-7 3 > 1], [-4 -1 0 2], and [-5 1 0 0 1] (what I'm calling superpelog), or any > other interesting and unusual things that come up.Here's some data on superpelog. You can see the top tunings for it correspond in the 5, 7 and 11 limits, so would you object to simply calling this "pelog", and saying "11-limit pelog" or "7-limit pelog" if you want to be more specific? They have the same generators. TOP 5: [1206.548265, 1891.576247, 2771.109113] TOP 7: [1206.548264, 1891.576247, 2771.109113, 3358.884653] TOP 11: [1206.548264, 1891.576247, 2771.109113, 3358.884653, 4141.165078] Wedgies pelog7: <<2 -6 1 -14, -4 19|| pelog11: <<2 -6 1 -2 -14 -4 -10 19 16 -9}|| TM comma basis 5 limit: {135/128} 7 limit: {135/128, 49/48} 11 limit: {33/32, 45/44, 49/48, 33/32} I'd like to know if the> 260.76 cent generator and 1206.55 cent period is still optimal, or whether > the extension to 11 limit changes anything. Apparently not. > (I've noticed there's a new notation for monzos that involves different > symbols like angle brackets and vertical bars, but I'm continuing to use > the old notation with square brackets since I don't know the appropriate > usage of the new notation.) Monzos: |...> Vals: <...|Linear temperament wedgies: <<...|| Planar temperament wedgies: <<<...||| It makes sense to use <...| also for tuning maps, but I didn't above, as I'm not sure if doing so would sow confusion.

Message: 9291 - Contents - Hide Contents Date: Mon, 19 Jan 2004 08:41:12 Subject: Re: Question for Dave Keenan From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> It might, depending on the value of 's' or hearing resolution assumed > (this is essentially the only free parameter in harmonic entropy, > which subsumes considerations of timbre, register, etc.). The > meantone dominant seventh could land outside the low-entropy region > surrounding 4:5:6:7 in the 3-dimensional space of tetrads (where the > axes corresponding to these harmonics lie at 60-degree angles).Can you calculate this for various s?

Message: 9292 - Contents - Hide Contents Date: Mon, 19 Jan 2004 22:12:50 Subject: Re: Annotated Dave Keenan file From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Tertiathirds is a name for the 7-limit temperament with TM comma base > {49/48, 225/224}, wedgie <<4 -3 2 -14 -8 13|| and mapping > [<1 2 2 3|, <0 -4 3 -2|]. There's been a question all along as to > whether it should be called "negri", and I suppose it should be; it's > pretty closely tied to 2/19 as a generator either way.I assumed that tertiathirds and negri were alternative names for the same thing, in my own 7-limit temperament spreadsheet from long ago. One being a systematic name, the other a common name, like sodium chloride and salt.

Message: 9293 - Contents - Hide Contents Date: Mon, 19 Jan 2004 00:52:19 Subject: Re: A potentially informative property of tunings From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> [(1, 0), (2, -2), (1, 6)] superpelogWhy are you calling this "superpelog"? It's a Dave Keenan style duplicated pelog, isn't it?> [(1, 0), (2, -2), (9, -31)] unnamed temperamentThis can't be what you want; it gives a complex "comma" of 588 cents.> 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).The latter isn't going to win any awards, as the error is almost identical. They differ by 31 generators; going up by 205 generators to (-79, 195) or down 174 generators to (80, -184) cuts the error almost in half, but at the cost of absurd complexity. If you are going to do that you may as well go whole-hog to (-164, 400) or (166,-389) giving you two microtemperaments (error less than 1/5 cent) for {2,5,7}-JI bizarrely attached to a meantone fifth. I don't think this really works as a way of analyzing meantone.

Message: 9294 - Contents - Hide Contents Date: Mon, 19 Jan 2004 08:49:16 Subject: Re: Question for Dave Keenan From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> It might, depending on the value of 's' or hearing resolution assumed > (this is essentially the only free parameter in harmonic entropy, > which subsumes considerations of timbre, register, etc.). The > meantone dominant seventh could land outside the low-entropy region > surrounding 4:5:6:7 in the 3-dimensional space of tetrads (where the > axes corresponding to these harmonics lie at 60-degree angles).If we analyze 1--5/4--3/2--9/5 as a chord of septimal meantone, using the approximations native to that, we get that 9/5 is a 9-limit consonance and 5/4--9/5 is equivalent to 10/7 by 126/125, so this would be a 9-limit magic chord for septimal meantone.

Message: 9295 - Contents - Hide Contents Date: Mon, 19 Jan 2004 15:50:21 Subject: Re: Question for Dave Keenan From: Carl Lumma>There's also the posssibility that the dominant seventh chord >functions best when its harmonic entropy (or maybe only the HE of one >of its dyads) is locally _maximised_.What would this look like? The dominant seventh chord is defined as a local minimum of entropy. -Carl

Message: 9296 - Contents - Hide Contents Date: Mon, 19 Jan 2004 08:49:41 Subject: Re: Question for Dave Keenan From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> It might, depending on the value of 's' or hearing resolution assumed >> (this is essentially the only free parameter in harmonic entropy, >> which subsumes considerations of timbre, register, etc.). The >> meantone dominant seventh could land outside the low-entropy region >> surrounding 4:5:6:7 in the 3-dimensional space of tetrads (where the >> axes corresponding to these harmonics lie at 60-degree angles). >> Can you calculate this for various s?It hasn't been done yet, but I promise to do it post haste, if you first give me a sincere effort to improve the computational efficiency of the calculation (starting, of course, with the dyadic case). I'd prefer to discuss this on the harmonic entropy list, for the convenience of those who wish to follow its development.

Message: 9297 - Contents - Hide Contents Date: Mon, 19 Jan 2004 15:51:39 Subject: Re: Annotated Dave Keenan file From: Carl Lumma>> >ertiathirds is a name for the 7-limit temperament with TM comma base >> {49/48, 225/224}, wedgie <<4 -3 2 -14 -8 13|| and mapping >> [<1 2 2 3|, <0 -4 3 -2|]. There's been a question all along as to >> whether it should be called "negri", and I suppose it should be; it's >> pretty closely tied to 2/19 as a generator either way. >>I assumed that tertiathirds and negri were alternative names for the >same thing, in my own 7-limit temperament spreadsheet from long ago. >One being a systematic name, the other a common name, like sodium >chloride and salt.Works for me. -Carl

Message: 9298 - Contents - Hide Contents Date: Mon, 19 Jan 2004 01:13:21 Subject: Re: Annotated Dave Keenan file From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> 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 >> Why isn't this negri?Interesting question. This is the only 7-limit version of negri with a badness score which is much good, so using the "reasonable tuning" criterion perhaps it should be. However, <4 -3 -17 -14 -38 -31| is closer, <4 -3 21 -14 22 57| much closer yet, and <4 -3 40 -14 52 101| has the identical TOP tuning. What to do?

Message: 9299 - Contents - Hide Contents Date: Mon, 19 Jan 2004 08:50:39 Subject: Re: Question for Dave Keenan From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> It might, depending on the value of 's' or hearing resolution assumed >> (this is essentially the only free parameter in harmonic entropy, >> which subsumes considerations of timbre, register, etc.). The >> meantone dominant seventh could land outside the low-entropy region >> surrounding 4:5:6:7 in the 3-dimensional space of tetrads (where the >> axes corresponding to these harmonics lie at 60-degree angles). >> If we analyze 1--5/4--3/2--9/5 as a chord of septimal meantone, using > the approximations native to that, we get that 9/5 is a 9-limit > consonance and 5/4--9/5 is equivalent to 10/7 by 126/125, so this > would be a 9-limit magic chord for septimal meantone. Yup.

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