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Message: 5475 - Contents - Hide Contents Date: Thu, 31 Oct 2002 22:29:38 Subject: Re: New file uploaded to tuning-math From: Carl Lumma>> >his .gif is golden. >> >> But I forget the coloring scheme... >> >> -Carl >>bluish = consistent > >reddish = inconsistent > >is that what you meant?Yes, but what's hot purple (14, 20), and real purple (24)? -Carl

Message: 5476 - Contents - Hide Contents Date: Fri, 01 Nov 2002 19:59:47 Subject: Re: New file uploaded to tuning-math From: wallyesterpaulrus --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>>> This .gif is golden. >>> >>> But I forget the coloring scheme... >>> >>> -Carl >>>> bluish = consistent >> >> reddish = inconsistent >> >> is that what you meant? >> Yes, but what's hot purple (14, 20), reddish > and real > purple (24)?bluish (isn't that violet, not purple?)

Message: 5477 - Contents - Hide Contents Date: Fri, 01 Nov 2002 21:52:23 Subject: Re: New file uploaded to tuning-math From: Carl Lumma>>> >luish = consistent >>> >>> reddish = inconsistent >>> >>> is that what you meant? >>>> Yes, but what's hot purple (14, 20), > >reddish > >> and real >> purple (24)? >>bluish (isn't that violet, not purple?)Got it. It's whatever you want to call it. Did you do this just for visibility?

Message: 5478 - Contents - Hide Contents Date: Fri, 01 Nov 2002 21:54:06 Subject: Re: New file uploaded to tuning-math From: wallyesterpaulrus --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:

Message: 5479 - Contents - Hide Contents Date: Fri, 01 Nov 2002 23:09:59 Subject: Re: New file uploaded to tuning-math From: Carl Lumma

Message: 5480 - Contents - Hide Contents Date: Fri, 01 Nov 2002 23:11:27 Subject: Re: New file uploaded to tuning-math From: wallyesterpaulrus --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>>> Did you do this just for visibility? >> >> yes. > > It worked! > > -C.notice how even 76 is visible in one of the zooms i posted. this wasn't easy to arrange, with all the numbers lying on top of one another!

Message: 5481 - Contents - Hide Contents Date: Sat, 02 Nov 2002 23:57:49 Subject: Re: Miracle and 72-et From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> The 72-et version of miracle really only dominates in the 11-limit.If we look at the convergents to the rms optimal secor, we find that> 175 is the next convergent beyond 72 for the 5 and 7 limits, and 113 > (and beyond that 298) are the next convergents in the primary 13-limit version of miracle. For the alternative 13-limit miracle, 72 is not even a convergent! We have however 113 and beyond that, 267. It would seem 175 and 113 have a place at the table. if i recall correctly, graham's program originally spat out the denominator 113 for the miracle generator.

Message: 5482 - Contents - Hide Contents Date: Sat, 02 Nov 2002 03:17:05 Subject: A look at 13-limit linear temperaments From: Gene Ward Smith I did this admissions-committee style, with two different ways to make the cut--either unweighted badness less than 250, or graham badness less than 660. Limits are graham complexity less than or equal to 50, and rms error below 20. Note the two different versions of miracle! [[1, 1, -5, -1, 2, 4], [0, 2, 25, 13, 5, -1]] [1200., 351.6171952] rms 4.164243901 comp 13.36413110 graham 26 bad 203.4451071 grabad 552.0725839 mos [10, 17, 24, 41, 58, 99] [[1, 2, 2, 3, 4, 4], [0, 4, -3, 2, 5, 3]] [1200., -126.4454893] rms 19.07231186 comp 5.059644256 graham 11 bad 217.0617566 grabad 695.8127255 mos [10, 19, 28, 47, 66, 85] [[1, 1, -1, 3, 6, 8], [0, 3, 17, -1, -13, -22]] [1200., 234.4801337] rms 2.768744646 comp 18.44857718 graham 39 bad 219.3952496 grabad 674.3413860 mos [11, 16, 21, 26, 31, 36, 41, 46, 87] [[1, 1, 3, 3, 2, 0], [0, 6, -7, -2, 15, 38]] [1200., 116.7795117] rms 2.215733090 comp 21.71865558 graham 45 bad 224.2677630 grabad 668.8615243 mos [10, 11, 21, 31, 41, 72] [[2, 5, 8, 5, 6, 8], [0, 6, 11, -2, -3, 2]] [600.0000000, -183.2252192] rms 3.167218102 comp 17.40114939 graham 30 bad 229.9031248 grabad 520.4270399 mos [14, 20, 26, 46, 72] [[3, 0, 3, 10, 8, 0], [0, 6, 5, -2, 3, 14]] [400.0000000, 316.9938762] rms 1.823490161 comp 25.26064132 graham 48 bad 231.5101214 grabad 606.4085004 mos [12, 15, 27, 42, 57, 72, 87] [[1, 0, -4, 17, -6, 10], [0, 1, 4, -9, 6, -4]] [1200., 1892.727763] rms 12.11708320 comp 7.338255924 graham 15 bad 240.8725835 grabad 703.9389215 mos [12, 19, 26, 45, 71, 97] [[1, 0, -31, -21, -14, -9], [0, 1, 21, 15, 11, 8]] [1200., 1904.391710] rms 6.209025908 comp 11.46080276 graham 21 bad 240.9051197 grabad 597.5199555 mos [12, 17, 29, 46, 63] [[1, 0, -6, 4, -12, -7], [0, 4, 21, -3, 39, 27]] [1200., 475.6946183] rms 2.245890676 comp 22.61415486 graham 42 bad 241.5233672 grabad 611.3114745 mos [13, 18, 23, 28, 33, 38, 43, 48, 53, 58] [[29, 46, 0, 14, 33, 40], [0, 0, 1, 1, 1, 1]] [41.37931034, 2788.239580] rms 2.277983567 comp 22.46330341 graham 29 bad 242.5275176 grabad 355.7521911 mos [58, 87] [[1, 1, 0, 2, 3, 3], [0, 5, 20, 7, 4, 6]] [1200., 139.4310901] rms 8.867498827 comp 9.091204541 graham 20 bad 243.0710758 grabad 793.1332067 mos [17, 26, 43, 69] [[3, 0, 7, 6, 8, 4], [0, 2, 0, 1, 1, 3]] [400.0000000, 950.4747331] rms 15.05561416 comp 6.434283174 graham 12 bad 245.7246539 grabad 625.8501279 mos [15, 24, 39, 63, 87] [[1, 0, 15, 25, -33, -28], [0, 1, -8, -14, 23, 20]] [1200., 1902.127698] rms 2.806870405 comp 19.72435043 graham 37 bad 245.8818240 grabad 631.7204666 mos [12, 17, 29, 41, 53, 94] [[1, 4, 5, 2, 4, 8], [0, 9, 10, -3, 2, 16]] [1200., -322.1404543] rms 6.393156943 comp 11.47388339 graham 21 bad 248.4740356 grabad 615.2396380 mos [11, 15, 26, 41, 67] [[2, 0, 11, 31, 45, 55], [0, 1, -2, -8, -12, -15]] [600.0000000, 1903.786589] rms 2.880707578 comp 19.68247952 graham 34 bad 251.5468524 grabad 571.1070885 mos [10, 12, 22, 34, 46, 58] [[1, 1, 3, 3, 2, 4], [0, 6, -7, -2, 15, -3]] [1200., 116.8457500] rms 6.194083836 comp 11.81524439 graham 22 bad 251.5597433 grabad 639.1622256 mos [10, 11, 21, 31, 41, 72] [[1, 0, -4, -3, 4, 0], [0, 3, 12, 11, -1, 7]] [1200., 633.5217226] rms 12.76012488 comp 7.304108433 graham 13 bad 251.8869934 grabad 598.0946990 mos [11, 13, 15, 17, 19, 36, 53, 89] [[2, 1, -12, 2, -9, -2], [0, 3, 23, 5, 22, 13]] [600.0000000, 434.1894677] rms 2.006172054 comp 26.72452058 graham 46 bad 277.1616007 grabad 625.8999601 mos [14, 22, 36, 58, 94] [[2, 1, 0, 5, 6, 4], [0, 7, 15, 2, 3, 11]] [600.0000000, 185.9945003] rms 4.007057140 comp 17.17556404 graham 30 bad 285.2279283 grabad 658.4266756 mos [14, 20, 26, 32, 58, 84] [[2, 2, 5, 6, 5, 7], [0, 3, -1, -1, 5, 1]] [600.0000000, 231.2498543] rms 12.34827552 comp 8.160882306 graham 14 bad 287.8803242 grabad 646.8422283 mos [10, 16, 26, 36, 62, 88]

Message: 5483 - Contents - Hide Contents Date: Sat, 02 Nov 2002 08:34:24 Subject: Miracle and 72-et From: Gene Ward Smith The 72-et version of miracle really only dominates in the 11-limit. If we look at the convergents to the rms optimal secor, we find that 175 is the next convergent beyond 72 for the 5 and 7 limits, and 113 (and beyond that 298) are the next convergents in the primary 13-limit version of miracle. For the alternative 13-limit miracle, 72 is not even a convergent! We have however 113 and beyond that, 267. It would seem 175 and 113 have a place at the table.

Message: 5484 - Contents - Hide Contents Date: Tue, 5 Nov 2002 01:36:12 Subject: 156edo, articulating septimal kleisma From: monz hmmm ... this is interesting: i was searching for an EDO which would articulate the "septimal kleisma" [2 2 -1] = 225:224 (~7.711522991 cents). 156edo looked very good, since the "augmented-6th" 225:128 and the "harmonic-minor-7th" 7:4 are approximated very closely by is 127 and 126 degrees, respectively. thus, the "septimal kleisma" is almost exactly 1 degree. but then, when i made a bingo-card-lattice of 156edo ("perfect-5th" ~3:2 = 91 degrees, "major-3rd" ~5:4 = 50), the representation of 225:128 turned out to be 126. so at least in this mapping (which to me is the one which makes the most sense ... i think ...), the "septimal kleisma" is not articulated after all! help. -monz

Message: 5485 - Contents - Hide Contents Date: Tue, 05 Nov 2002 19:09:25 Subject: Re: 156edo, articulating septimal kleisma From: Carl Lumma> i was searching for an EDO which would articulate the > "septimal kleisma" [2 2 -1] = 225:224 (~7.711522991 cents).You could really articulate it with ets like 27 and 37, which both represent it as one step, or 26, which shrinks it to -1 steps! If you really want to articulate it accurately, why not use JI? If Gene/Paul would follow through and make the tree zoom duals, you might be able to see this kind of thing at a glance! -Carl

Message: 5486 - Contents - Hide Contents Date: Tue, 5 Nov 2002 10:56:20 Subject: Re: 156edo, articulating septimal kleisma From: manuel.op.de.coul@xxxxxxxxxxx.xxx Use DIVIDE/CONSISTENT 225/224 You see the size is zero in 156-ET, 152 is better. Manuel

Message: 5487 - Contents - Hide Contents Date: Tue, 05 Nov 2002 19:58:14 Subject: Tree zoom duals From: Gene Ward Smith --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:> If Gene/Paul would follow through and make the tree zoom > duals, you might be able to see this kind of thing at a > glance!Here you go: We can consider 5-limit ets to be vals [a,b,c] with elements positive integers; they define a point in the projective plane. Since "a" is not zero, we can divide by "a" and get the point [b/a,c/a] in the affine plane as a coordinate patch. We now note that the points are all clustered about [log2(3),log2(5)], so we may move the origin to this point. We now have the "tree zoom" picture. Similarly, if q is a comma, then we may consider it to be [a,b,c] where 2^a 3^b 5^c = q. We no longer can always divide by "a", so we cannot take "a=0" to be the line at infinity as we did with ets. Instead, we can take a+log2(3)*b + log2(5)*c=0 to be the line at infinity, since the comma is not 1. This means we can divide through by log2(q) (or cents, etc--which log map we use is not important) and get [a/log2(q),b/log2(q),c/log2(q)]; taking the last two gives us an affine coordinate patch: [b/log2(q), c/log2(q)] which can be used to plot commas as points.

Message: 5488 - Contents - Hide Contents Date: Tue, 5 Nov 2002 02:02:37 Subject: Re: 156edo, articulating septimal kleisma From: monz thanks, Manuel.

Message: 5489 - Contents - Hide Contents Date: Tue, 05 Nov 2002 20:57:54 Subject: Re: Tree zoom duals From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:>taking the last two gives us an affine coordinate patch: [b/log2(q), >c/log2(q)] which can be used to plot commas as points.interesting . . . thanks so much gene!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Message: 5490 - Contents - Hide Contents Date: Tue, 05 Nov 2002 21:02:37 Subject: Re: 156edo, articulating septimal kleisma From: wallyesterpaulrus --- In tuning-math@y..., "monz" <monz@a...> wrote:> hmmm ... this is interesting: > > i was searching for an EDO which would articulate the > "septimal kleisma" [2 2 -1] = 225:224 (~7.711522991 cents). > > 156edo looked very good, since the "augmented-6th" 225:128 and > the "harmonic-minor-7th" 7:4 are approximated very closely > by is 127 and 126 degrees, respectively. thus, the > "septimal kleisma" is almost exactly 1 degree. > > but then, when i made a bingo-card-lattice of 156edo > ("perfect-5th" ~3:2 = 91 degrees, "major-3rd" ~5:4 = 50), > the representation of 225:128 turned out to be 126. > > so at least in this mapping (which to me is the one > which makes the most sense ... i think ...), the > "septimal kleisma" is not articulated after all! > > help.monz, this is really no different from 55-equal -- although the *just* 80:81 (syntonic comma) is very nearly one degree of 55-equal, the *native* 80:81 (syntonic comma) of 55-equal vanishes. in that case you just need a much better 5-limit approximation -- 53- equal of course does the trick. in the current case you just need a much better 7-limit approximation -- 152-equal, the universal tuning, will work. :) similarly, one of your webpages claims that the schisma is well- represented by one degree of 614-equal. of course, 612-equal would, functionally, work much better, since it offers a far better 5-limit approximation. maybe we should start using the symbol 81;80 for the native syntonic comma of any temperament?

Message: 5491 - Contents - Hide Contents Date: Tue, 05 Nov 2002 23:54:38 Subject: Re: 156edo, articulating septimal kleisma From: wallyesterpaulrus --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:> If Gene/Paul would follow through and make the tree zoom > duals, you might be able to see this kind of thing at a > glance! >unfortunately, 225:224 has components in all three directions (3, 5, and 7), so it might be hard without a nice VRML implementation of this . . .

Message: 5492 - Contents - Hide Contents Date: Tue, 5 Nov 2002 22:34:24 Subject: Re: Tree zoom duals From: monz> From: "Gene Ward Smith" <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, November 05, 2002 6:58 PM > Subject: [tuning-math] Re: Tree zoom duals > > > --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:>> --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote: >>>>> taking the last two gives us an affine coordinate patch: [b/log2(q), >>> c/log2(q)] which can be used to plot commas as points. >>>> interesting . . . thanks so much gene!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! >> Yer welcome--of course now you need to zoom out rather than > in if you want to fool with this. > > What would be really cool is a 3D applet, which let you look at > a 7-limit picture from various directions. We could have either > ets as points, linear temperaments as lines, and commas as planes, > or a dual picture with commas as points, linear temperaments > again as lines, and ets as planes.this is all stuff that i've always intended from the beginning to have available in my JustMusic software. JustMusic software, (c) 1999 by Joseph L. Monzo * [with cont.] (Wayb.) unfortunately, both the project and the Yahoo group have been slumbering for quite some time. any chance we can wake them up? -monz (still looking for a Microsoft Visual C++ programmer to help out)

Message: 5493 - Contents - Hide Contents Date: Wed, 06 Nov 2002 20:07:58 Subject: Re: Tree zoom duals From: wallyesterpaulrus --- In tuning-math@y..., "monz" <monz@a...> wrote:> > ----- Original Message ----- > From: "wallyesterpaulrus" <wallyesterpaulrus@y...> > To: <tuning-math@y...> > Sent: Tuesday, November 05, 2002 11:55 PM > Subject: [tuning-math] Re: Tree zoom duals > >>> --- In tuning-math@y..., "monz" <monz@a...> wrote: >> >>>>>>>> What would be really cool is a 3D applet, which let you look at >>>> a 7-limit picture from various directions. We could have either >>>> ets as points, linear temperaments as lines, and commas as planes, >>>> or a dual picture with commas as points, linear temperaments >>>> again as lines, and ets as planes. >>> >>> >>>>>> this is all stuff that i've always intended from the beginning >>> to have available in my JustMusic software. >>> >>> JustMusic software, (c) 1999 by Joseph L. Monzo * [with cont.] (Wayb.) >>>> is that really true? do you have any of the ideas above referenced >> anywhere? > > >> what's on the webpage and in the archives of the Yahoo JustMusic group > is all i've ever made public about it. the rest of my ideas are in > folders and folders of notes and old BASIC code ... stuff that i > haven't even looked at in over 5 years.monz, if you dig up notes in your folders that mention even one of these ideas, let alone all of them, i'll be absolutely stunned.

Message: 5494 - Contents - Hide Contents Date: Wed, 06 Nov 2002 20:09:45 Subject: Re: Tree zoom duals From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote: >>>> Yer welcome--of course now you need to zoom out rather than in if >>> you want to fool with this. >>>> what do you mean? >> The high-octane ets crowd in towards the origin, so you zoom in to >find them. The high-octane commas expand out to infinity, so you >would need to zoom out to include them.right -- i realized this as soon as i signed off last night. i also realized that each et line will intersect each axis at a distance from the origin inversely proportional to that consonance's error in that et. this will spare me the effort of having to construct a list of commas.

Message: 5495 - Contents - Hide Contents Date: Wed, 06 Nov 2002 21:51:43 Subject: Re: Tree zoom duals From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

Message: 5496 - Contents - Hide Contents Date: Wed, 06 Nov 2002 00:33:10 Subject: Re: 156edo, articulating septimal kleisma From: Carl Lumma>> >f Gene/Paul would follow through and make the tree zoom >> duals, you might be able to see this kind of thing at a >> glance! >> >>unfortunately, 225:224 has components in all three directions (3, >5, and 7), so it might be hard without a nice VRML implementation >of this . . .Maybe Robert Walker can help us there... For now, the 5-limit would still be cool. I suspect even cooler than the et-centric versions... -Carl

Message: 5497 - Contents - Hide Contents Date: Wed, 06 Nov 2002 02:58:54 Subject: Re: Tree zoom duals From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote: >>> taking the last two gives us an affine coordinate patch: [b/log2(q), >> c/log2(q)] which can be used to plot commas as points. >> interesting . . . thanks so much gene!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Yer welcome--of course now you need to zoom out rather than in if you want to fool with this. What would be really cool is a 3D applet, which let you look at a 7-limit picture from various directions. We could have either ets as points, linear temperaments as lines, and commas as planes, or a dual picture with commas as points, linear temperaments again as lines, and ets as planes.

Message: 5498 - Contents - Hide Contents Date: Wed, 06 Nov 2002 03:09:37 Subject: Re: 156edo, articulating septimal kleisma From: Gene Ward Smith --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:> Maybe Robert Walker can help us there... For now, the 5-limit > would still be cool. I suspect even cooler than the et-centric > versions...Plus you can always use inversive geometry, turn either version inside-out, and replace the lines with circles.

Message: 5499 - Contents - Hide Contents Date: Wed, 06 Nov 2002 07:53:52 Subject: Re: Tree zoom duals From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:>> --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote: >>>>> taking the last two gives us an affine coordinate patch: [b/log2 (q), >>> c/log2(q)] which can be used to plot commas as points. >>>> interesting . . . thanks so much gene!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! >> Yer welcome--of course now you need to zoom out rather than in if >you want to fool with this.what do you mean?> What would be really cool is a 3D applet, which let you look at a 7- >limit picture from various directions. We could have either ets as >points, linear temperaments as lines, and commas as planes, or a >dual picture with commas as points, linear temperaments again as >lines, and ets as planes.yup yup yup yup yup!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!

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