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Message: 3850 - Contents - Hide Contents Date: Fri, 08 Feb 2002 09:21:04 Subject: A 58 tone epimorphic scale containing Genesis From: genewardsmith Genesis is not epimorphic, so we should find non-CS, inconsistent properties. We would also not be able to produce it as any kind of block. I took the steps of Genesis which were mapped to 2 by h58, which are 45/44, 49/48, 50/49 and 55/54, and split them in two as follows: 45/44 = 81/80 100/99 49/48 = 245/242 121/120 50/49 = 99/98 100/99 55/54 = 100/99 121/120 I then picked the one of the two possibilites presented in each case for filling the "gap" by choosing the option of least Tenney height, and obtained the following: 1, 81/80, 45/44, 33/32, 21/20, 16/15, 27/25, 12/11, 11/10, 10/9, 9/8, 8/7, 121/105, 7/6, 32/27, 6/5, 40/33, 11/9, 99/80, 5/4, 14/11, 9/7, 315/242, 21/16, 4/3, 27/20, 15/11, 11/8, 7/5, 99/70, 10/7, 16/11, 22/15, 40/27, 3/2, 32/21, 484/315, 14/9, 11/7, 8/5, 81/50, 18/11, 33/20, 5/3, 27/16, 12/7, 121/70, 7/4, 16/9, 9/5, 20/11, 11/6, 50/27, 15/8, 40/21, 64/33, 88/45, 160/81 It is readily verified that this *is* epimorphic, with map h58. The set of steps for this scale is 121/120, 100/99, 99/98, 245/242, 81/80, 64/63, and 56/55. Geometrically it is more complex than Genesis, with 30 verticies and 65 facets. However, a 4D parallepiped would have 16 verticies and 8 facets so there is a limit to how simple this can get, and I suppose Genesis is not so bad.

Message: 3851 - Contents - Hide Contents Date: Fri, 08 Feb 2002 18:48:53 Subject: Re: xenharmonic bridges in the 12edo comma pump (was: exactly what...) From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> so our matrix for these three bridges is: > > [2 3 5] [-4 4 -1] = 81:80 syntonic comma > [-19/6 2 0] = 12edo==9/8 bridge = ~3.910001731 cents > [ -5/6 2 -1] = 12edo==10/9 bridge = ~17.59628787 cents > > > and note that these three bridges are linearly dependent.The second bridge is just P^(1/6), where P is the Pythagorean comma. The Pythagorean and syntonic commas together define the 12-et in the 5-limit; one way to express that is 81/80 ^ P = 81/80 X P (where the wedge product in this case becomes the vector cross-product) = h12, the [12, 19, 28] val. Your other bridge is W^(1/6), where W = (81/80)^6 P^(-1). Is there anything gained by taking roots of commas? I don't see it.> so here's the entire list of bridges which i would say > are in effect for the comma pump in 12edo: > > 2 3 5 ~cents > > [ -4 4 -1 ] = 81:80 syntonic comma = 21.5062896 > [ -5/6 2 -1 ] = 12edo==10/9 bridge = 17.59628787 > [-36/11 36/11 -9/11] = 1/11cmt==10/9 bridge = 17.59605512 > [ -8/11 8/11 -2/11] = 1/11cmt==9/8 bridge = 3.910234472 > [-19/6 2 0 ] = 12edo==9/8 bridge = 3.910001731 > [161/66 -14/11 -2/11] = 12edo==1/11cmt bridge = 0.000232741It seems to me the only one on this list which is important for the comma pump is 81/80, and that defines it. The others define 12-et and 1/11 comma meantone, and have nothing to do with the pump so far as I can see.

Message: 3852 - Contents - Hide Contents Date: Fri, 8 Feb 2002 11:29:46 Subject: Re: xenharmonic bridges in the 12edo comma pump (was: exactly what...) From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, February 07, 2002 6:00 AM > Subject: [tuning-math] xenharmonic bridges in the 12edo comma pump (was: exactly what...) > > ... > > so here's the entire list of bridges which i would say > are in effect for the comma pump in 12edo: > > 2 3 5 ~cents > > [ -4 4 -1 ] = 81:80 syntonic comma = 21.5062896 > [ -5/6 2 -1 ] = 12edo==10/9 bridge = 17.59628787 > [-36/11 36/11 -9/11] = 1/11cmt==10/9 bridge = 17.59605512 > [ -8/11 8/11 -2/11] = 1/11cmt==9/8 bridge = 3.910234472 > [-19/6 2 0 ] = 12edo==9/8 bridge = 3.910001731 > [161/66 -14/11 -2/11] = 12edo==1/11cmt bridge = 0.000232741oops ... that really should say "are in effect for the note D for the comma pump in 12edo". i didn't examine the bridges which are in effect for any of the other notes. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3853 - Contents - Hide Contents Date: Fri, 8 Feb 2002 11:36:17 Subject: Re: xenharmonic bridges in the 12edo comma pump From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, February 08, 2002 10:48 AM > Subject: [tuning-math] Re: xenharmonic bridges in the 12edo comma pump > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> so our matrix for these three bridges is: >> >> [2 3 5] [-4 4 -1] = 81:80 syntonic comma >> [-19/6 2 0] = 12edo==9/8 bridge = ~3.910001731 cents >> [ -5/6 2 -1] = 12edo==10/9 bridge = ~17.59628787 cents >> >> >> and note that these three bridges are linearly dependent. >> The second bridge is just P^(1/6), where P is the Pythagorean comma. > The Pythagorean and syntonic commas together define the 12-et in > the 5-limit; one way to express that is 81/80 ^ P = 81/80 X P > (where the wedge product in this case becomes the vector cross-product) > = h12, the [12, 19, 28] val. Your other bridge is W^(1/6), where > W = (81/80)^6 P^(-1). Is there anything gained by taking roots > of commas? I don't see it. >>> so here's the entire list of bridges which i would say >> are in effect for the comma pump in 12edo: >> >> 2 3 5 ~cents >> >> [ -4 4 -1 ] = 81:80 syntonic comma = 21.5062896 >> [ -5/6 2 -1 ] = 12edo==10/9 bridge = 17.59628787 >> [-36/11 36/11 -9/11] = 1/11cmt==10/9 bridge = 17.59605512 >> [ -8/11 8/11 -2/11] = 1/11cmt==9/8 bridge = 3.910234472 >> [-19/6 2 0 ] = 12edo==9/8 bridge = 3.910001731 >> [161/66 -14/11 -2/11] = 12edo==1/11cmt bridge = 0.000232741 >> It seems to me the only one on this list which is important for > the comma pump is 81/80, and that defines it. The others define > 12-et and 1/11 comma meantone, and have nothing to do with the > pump so far as I can see.thanks for all of that, Gene. i'm not sure what any of this means, but my feeling is that all these small intervals (and many others which i did not list) have something to do with what we're hearing when composers intentionally mix the properties of different tuning systems like this, so they're worth looking at. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3854 - Contents - Hide Contents Date: Fri, 08 Feb 2002 22:31:27 Subject: Re: A 58 tone epimorphic scale containing Genesis From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>Genesis is not epimorphic, so we should find non-CS, inconsistent >properties. We would also not be able to produce it as any kind of >block.i thought my original question made it clear that we were to take 11/10 and 20/11 as auxillaries, not as part of the block.

Message: 3855 - Contents - Hide Contents Date: Fri, 08 Feb 2002 22:44:53 Subject: Re: A 58 tone epimorphic scale containing Genesis From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > i thought my original question made it clear that we were to take > 11/10 and 20/11 as auxillaries, not as part of the block.I don't know how to take something as an auxillary, but we can certainly look at the scale we get by leaving 11/10 and 20/11 out of Genesis. Is this what you mean? Perhaps you could tell us again what you are actually looking for.

Message: 3856 - Contents - Hide Contents Date: Fri, 08 Feb 2002 23:31:33 Subject: Re: A 58 tone epimorphic scale containing Genesis From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > >> i thought my original question made it clear that we were to take >> 11/10 and 20/11 as auxillaries, not as part of the block. >> I don't know how to take something as an auxillary, but we cancertainly look at the scale we get by leaving 11/10 and 20/11 out of> Genesis. Is this what you mean?yes, or whatever choice is best. there should be only 41 notes inside the parallelepiped and whatever other shape you may happen to investigate (what's the 4-dimensional analogue of a 3-d rhombic dodecahedron or a 2-d hexagon? i feel that should be the most general case, and certain vertices may coincide in specific instances, causing some edges/faces/cells to disappear).

Message: 3857 - Contents - Hide Contents Date: Sat, 09 Feb 2002 09:33:47 Subject: Genesis Minus as a block From: genewardsmith I undertook a proceedure to determine if "Genesis Minus", by which I mean Genesis less 11/10 and 20/11, is a block by my understanding of what a block is. That understanding is that a block is epimorphic and convex, where by "convex" I mean it is convexly closed: every lattice point contained in the convex hull of the octave equivalence classes of the scale are already in the scale. Equivalently, a block is epimorphic and has the property that there exists a norm such that no point can be added to the block without increasing its diameter. It seems Genesis Minus is a block by this definition (I hope Joe notices I said the word "definition".) I downloaded and ran the "qhull" program, which gave me a set of inequalities defining the convex hull, which I converted into a vector space norm. By this norm, Genesis Minus is within a radius of one of the unison, whereas the nearest comma seems to be 385/384 at a distance of three. The Voroni cells of the lattice of 41-et commas using this distance measure give the appropriate tiling of the 4D space. Here are the distances from unity of the tones of Genesis Minus according to this distance measure: 1 0 81/80 1. 33/32 1. 21/20 1. 16/15 1. 12/11 1. 10/9 1. 9/8 .5196155385 8/7 1. 7/6 1. 32/27 1. 6/5 1. 11/9 1. 5/4 1. 14/11 1. 9/7 1. 21/16 1. 4/3 .0392310770 27/20 1. 11/8 1. 7/5 1. 10/7 1. 16/11 1. 40/27 1. 3/2 .0392310770 32/21 1. 14/9 1. 11/7 1. 8/5 1. 18/11 1. 5/3 1. 27/16 1. 12/7 1. 7/4 1. 16/9 .5196155385 9/5 1. 11/6 1. 15/8 1. 40/21 1. 64/33 1. 160/81 1. I'll run some more computations; computing this distance measure is a little slow.

Message: 3859 - Contents - Hide Contents Date: Mon, 11 Feb 2002 00:35:55 Subject: h72 = h31 + h41 From: monz hi Gene, in an old post, you wrote: Yahoo groups: /tuning-math/message/1168 * [with cont.]> tuning-math Message 1168 > From: genewardsmith@j... > Date: Fri Oct 5, 2001 6:18 am > Subject: Re: 3rd-best 11-limit temperament > > > . . . I've been meaning to suggest that Manuel > consider putting into Scala a routine to calculate > Gen(m, n, p) and Mos(n,m,p) for two ets m and n and > a prime limit p; in case m and n are not relatively > prime this needs to be adjusted by working inside of > the interval of repetition. Of course one can also > think of this in terms of the ets generated by linear > combinations of hm and hn, as for instance > h53 = h22 + h31 and h72 = h31 + h41.by "h72 = h31 + h41", do you mean the following? 72-udo (unequal division of the octave), the combination of 31edo and 41edo ~cents (1200) 1170.731707 1161.290323 1141.463415 1122.580645 1112.195122 1083.870968 1082.926829 1053.658537 1045.16129 1024.390244 1006.451613 995.1219512 967.7419355 965.8536585 936.5853659 929.0322581 907.3170732 890.3225806 878.0487805 851.6129032 848.7804878 819.5121951 812.9032258 790.2439024 774.1935484 760.9756098 735.483871 731.7073171 702.4390244 696.7741935 673.1707317 658.0645161 643.902439 619.3548387 614.6341463 585.3658537 580.6451613 556.097561 541.9354839 526.8292683 503.2258065 497.5609756 468.2926829 464.516129 439.0243902 425.8064516 409.7560976 387.0967742 380.4878049 351.2195122 348.3870968 321.9512195 309.6774194 292.6829268 270.9677419 263.4146341 234.1463415 232.2580645 204.8780488 193.5483871 175.6097561 154.8387097 146.3414634 117.0731707 116.1290323 87.80487805 77.41935484 58.53658537 38.70967742 29.26829268 0 and can you explain a little more fully what you wrote there? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3860 - Contents - Hide Contents Date: Mon, 11 Feb 2002 01:22:06 Subject: secondary generator definition From: monz there was a need to supplement the definition of "generator", which my Dictionary had as a generator of a scale, with a secondary definition describing the generator of a kernel. Gene, if you already posted your definition of "generator", i'm sorry i can't recall it now. this is what i added: Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.)>> In periodicity-block theory, there are small intervals> called unison-vectors, a select few of which are able to > generate a kernel, which in JI is the periodicity-block > enclosing a finite set of ratios on the lattice.corrections, additions, etc. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3861 - Contents - Hide Contents Date: Mon, 11 Feb 2002 09:44 +0 Subject: Re: h72 = h31 + h41 From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <004101c1b2d7$1f67e5a0$af48620c@xxx.xxx.xxx> monz wrote:> by "h72 = h31 + h41", do you mean the following? > > > 72-udo (unequal division of the octave), > the combination of 31edo and 41edoNo, that's "the nearest-prime mapping of 72edo which is consistent with the nearest-prime mappings of 31edo and 41edo." If you know how many steps an interval approximates to in 31edo and 41edo, you can add them together to get the approximation to 72edo where this equation holds. Graham

Message: 3862 - Contents - Hide Contents Date: Mon, 11 Feb 2002 10:06:40 Subject: Re: h72 = h31 + h41 From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> by "h72 = h31 + h41", do you mean the following? > > > 72-udo (unequal division of the octave), > the combination of 31edo and 41edoNot at all. I start by assuming the temperaments I am looking at are both regular and consistent, and regard them as defined by mappings from JI to abstract "notes" consisting of generator steps, and the tuning of the temperament by a mapping in turn of the "notes" to "tones" which are real numbers. In the case of the 11-limit, "h31" is the mapping defined b sending 2 to 31, 3 to 49, 5 to 72, 7 to 87 and 11 to 107. BY definition h31(a*b) = h31(a) + h31(b), so this defines a mapping from any 11-limit interval to an integer. This one-dimensional mapping I call a "val"; it is in a sense the dual concept to interval. There is likewise a [41,65,95,115,142] mapping I call h41, and a [72,114,167,202,249] mapping I call h72. Denoting by "g+h" the mapping which sends a to g(a)+h(a), we have h72=h31+h41; in terms of the mappings above regarded as column vectors, this is vector addition.

Message: 3863 - Contents - Hide Contents Date: Mon, 11 Feb 2002 11:32:37 Subject: Re: h72 = h31 + h41 From: monz hi Graham and Gene,> From: <graham@xxxxxxxxxx.xx.xx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, February 11, 2002 1:44 AM > Subject: [tuning-math] Re: h72 = h31 + h41 > > > In-Reply-To: <004101c1b2d7$1f67e5a0$af48620c@xxx.xxx.xxx> > monz wrote: >>> by "h72 = h31 + h41", do you mean the following? >> >> >> 72-udo (unequal division of the octave), >> the combination of 31edo and 41edo >> No, that's "the nearest-prime mapping of 72edo which is consistent with > the nearest-prime mappings of 31edo and 41edo." If you know how many > steps an interval approximates to in 31edo and 41edo, you can add them > together to get the approximation to 72edo where this equation holds. > From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, February 11, 2002 2:06 AM > Subject: [tuning-math] Re: h72 = h31 + h41 > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> by "h72 = h31 + h41", do you mean the following? >> >> >> 72-udo (unequal division of the octave), >> the combination of 31edo and 41edo >> Not at all. I start by assuming the temperaments I am looking > at are both regular and consistent, and regard them as defined > by mappings from JI to abstract "notes" consisting of generator > steps, and the tuning of the temperament by a mapping in turn > of the "notes" to "tones" which are real numbers. In the > case of the 11-limit, "h31" is the mapping defined b sending > 2 to 31, 3 to 49, 5 to 72, 7 to 87 and 11 to 107. BY definition > h31(a*b) = h31(a) + h31(b), so this defines a mapping from any > 11-limit interval to an integer. This one-dimensional mapping > I call a "val"; it is in a sense the dual concept to interval. > There is likewise a [41,65,95,115,142] mapping I call h41, and > a [72,114,167,202,249] mapping I call h72. Denoting by "g+h" > the mapping which sends a to g(a)+h(a), we have h72=h31+h41; > in terms of the mappings above regarded as column vectors, > this is vector addition.so, then does this express what you're saying? : [ 31 41 72] [2 3 5 7 11] [ 49 65 114] [ 72 95 167] [ 87 115 202] [107 142 249] if it does, then i think i understand what you're doing. and can you explain what you mean by 'a "val" ... is in a sense the dual concept to interval' ? i don't get it. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3864 - Contents - Hide Contents Date: Mon, 11 Feb 2002 15:09:32 Subject: Clough / Engebretsen / Kochavi -- MTS article From: monz hi Gene, if you haven't yet read this, i strongly urge you to do so. i think you'll get a lot out of it ... and then, hopefully, you'll explain it to me! :) John Clough, Nora Engebretsen, and Jonathan Kochavi "Scales, Sets, and Interval Cycles: A Taxonomy" _Music Theory Spectrum_, vol 21 #1, Spring 1999, p 74-104 ABSTRACT>> Recent studies in the theory of scales by Agmon, Balzano, >> Carey and Clampitt, Clough and Douthett, Clough and Myerson, >> and Gamer have in common the central role of the interval >> cycle. Based on scale features defined in these studies, >> and an additional feature called *distributional evenness* >> defined here, a taxonomy is proposed for pitch-class sets >> (pcsets) the [_sic_: that] that correspond to interval cycles >> or to certain conjunctions thereof. Pairwise implicative >> relationships among the features are explored. Of 20 sets >> of features that are consistent with these relationships, >> 13 are found to be instantiated by actual pcsets and 7 others >> are shown to be incapable of instantiation. Most instantiated >> feature-sets correspond to infinite classes of pcsets which >> are shown to be enumerable; one such feature-set is found >> to be uniquely realized (up to transposition) in the usual >> diatonic pcset. -monz _________________________________________________________Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3865 - Contents - Hide Contents Date: Tue, 12 Feb 2002 12:49 +0 Subject: Re: h72 = h31 + h41 From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <00cb01c1b332$dcdd1e60$af48620c@xxx.xxx.xxx> monz wrote:> so, then does this express what you're saying? : > > [ 31 41 72] [2 3 5 7 11] > [ 49 65 114] > [ 72 95 167] > [ 87 115 202] > [107 142 249]Not really. The multiplication's the wrong way round. It should be [2 3 5 7 11][ 31 41 72] [ 49 65 114] [ 72 95 167] [ 87 115 202] [107 142 249] And then, you shouldn't be multiplying frequency integers like that, so change it to [log(2) log(3) log(5) log(7) log(11)][ 31 41 72] [ 49 65 114] [ 72 95 167] [ 87 115 202] [107 142 249] And you still aren't saying anything about nearest approximations or how they add up.> and can you explain what you mean by 'a "val" ... is in > a sense the dual concept to interval' ? i don't get it.It helps if you understand wedge products. But yes, it's something like that. I think the definition is that the wedge product of a val and an integer will always be a scalar. Graham

Message: 3866 - Contents - Hide Contents Date: Tue, 12 Feb 2002 10:29:31 Subject: Re: h72 = h31 + h41 From: monz> From: <graham@xxxxxxxxxx.xx.xx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, February 12, 2002 4:49 AM > Subject: [tuning-math] Re: h72 = h31 + h41 > > > In-Reply-To: <00cb01c1b332$dcdd1e60$af48620c@xxx.xxx.xxx> > monz wrote: >>> so, then does this express what you're saying? : >> >> [ 31 41 72] [2 3 5 7 11] >> [ 49 65 114] >> [ 72 95 167] >> [ 87 115 202] >> [107 142 249] >> Not really. The multiplication's the wrong way round. It should be > > [2 3 5 7 11][ 31 41 72] > [ 49 65 114] > [ 72 95 167] > [ 87 115 202] > [107 142 249]hmmm . . . i had a hunch that that was the case. i only wrote it that way so that the Yahoo interface would put the bigger matrix in proper columns. but why does it make a difference?> And then, you shouldn't be multiplying frequency integers like that, so > change it to > > [log(2) log(3) log(5) log(7) log(11)][ 31 41 72] > [ 49 65 114] > [ 72 95 167] > [ 87 115 202] > [107 142 249]i had a hunch about that too. can't that be written more easily as: log([2 3 5 7 11]) [ 31 41 72] [ 49 65 114] [ 72 95 167] [ 87 115 202] [107 142 249]> And you still aren't saying anything about nearest approximations or how > they add up.why not? please clarify, because that's exactly what i'm trying to understand here.>> and can you explain what you mean by 'a "val" ... is in >> a sense the dual concept to interval' ? i don't get it. >> It helps if you understand wedge products. But yes, it's something like > that. I think the definition is that the wedge product of a val and an > integer will always be a scalar.ok, thanks . . . i still haven't learned what a wedgie is. still studying. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3867 - Contents - Hide Contents Date: Tue, 12 Feb 2002 21:20 +0 Subject: Re: h72 = h31 + h41 From: graham@xxxxxxxxxx.xx.xx Me:>> Not really. The multiplication's the wrong way round. It should be >> >> [2 3 5 7 11][ 31 41 72] >> [ 49 65 114] >> [ 72 95 167] >> [ 87 115 202] >> [107 142 249] > Monz:> hmmm . . . i had a hunch that that was the case. > i only wrote it that way so that the Yahoo interface > would put the bigger matrix in proper columns. > but why does it make a difference?Try multiplying it out in Excel, using MMULT. You'll get different results.> i had a hunch about that too. can't that be written > more easily as: > > log([2 3 5 7 11]) [ 31 41 72] > [ 49 65 114] > [ 72 95 167] > [ 87 115 202] > [107 142 249]There is a definition of the logarithm of a matrix, but I don't think this is it. Me:>> And you still aren't saying anything about nearest approximations or >> how they add up. Monz:> why not? please clarify, because that's exactly what i'm > trying to understand here.There's nothing in the formula to say "take the nearest approximation" or "this column plus this other one makes that column". You can expend h31+h41=h72 to [ 31] [ 41] [ 72] [ 49] [ 65] [114] [ 72] + [ 95] = [167] [ 87] [115] [202] [107] [142] [249] If you want to understand it, try setting up a spreadsheet that calculates the last column from the first two. Then get it to generate those columns from their top entries, and the list of prime numbers.>> It helps if you understand wedge products. But yes, it's something >> like that. I think the definition is that the wedge product of a val >> and an integer will always be a scalar. > >> ok, thanks . . . i still haven't learned what a wedgie is. > still studying.The thing you have here as h72 is a val: [ 72] [114] [167] [202] [249] for an interval, take the comma 81:80. That's 2**-4 * 3**4 * 5**-1 or [-4 4 -1 0 0]. You get the representation of comma in h72 from the wedge product comma^h72. That expands to give the matrix product `[-4 4 1 0 0][ 72] = [1] ` [114] ` [167] ` [202] ` [249] If you want to play with wedge products, you'll have to get my Python library from <Automatically generated temperaments * [with cont.] (Wayb.)>. Then you can do things like>>> import temper >>> comma = temper.WedgableRatio(81,80) >>> h72 = temper.PrimeET(72, temper.primes[:4]) >>> comma^temper.Wedgable(h72).complement(){(0, 1, 2, 3, 4): 1} I think my complement() method is doing what Gene calls the "dual" so I'll rename it sometime. Graham

Message: 3868 - Contents - Hide Contents Date: Tue, 12 Feb 2002 00:13:29 Subject: Re: secondary generator definition From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> > there was a need to supplement the definition of "generator", > which my Dictionary had as a generator of a scale, with a > secondary definition describing the generator of a kernel. > > Gene, if you already posted your definition of "generator", > i'm sorry i can't recall it now. > > this is what i added: > > Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) >>>> In periodicity-block theory, there are small intervals>> called unison-vectors, a select few of which are able to >> generate a kernel, which in JI is the periodicity-block >> enclosing a finite set of ratios on the lattice.what does this have to do with the definition of 'generator'??

Message: 3869 - Contents - Hide Contents Date: Tue, 12 Feb 2002 00:37:58 Subject: Re: secondary generator definition From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> what does this have to do with the definition of 'generator'??I've been using "generator" in its meaning from the theory of finitely-generated abelian groups, in which sense both the generators of a kernel or the generators of a temperament are generators.

Message: 3870 - Contents - Hide Contents Date: Wed, 13 Feb 2002 14:13:56 Subject: Re: h72 = h31 + h41 From: monz hi Graham,> From: <graham@xxxxxxxxxx.xx.xx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, February 12, 2002 1:20 PM > Subject: [tuning-math] Re: h72 = h31 + h41 > > > The thing you have here as h72 is a val: > > [ 72] > [114] > [167] > [202] > [249] > > for an interval, take the comma 81:80. That's 2**-4 * 3**4 * 5**-1 or [-4 > 4 -1 0 0]. You get the representation of comma in h72 from the wedge > product comma^h72.ahh . . . so now we have to use ** instead of ^ for "raise to the power of", because now ^ is the wedge product. yes?> That expands to give the matrix product > > `[-4 4 1 0 0][ 72] = [1] > ` [114] > ` [167] > ` [202] > ` [249]there's a typo in the row matrix on the left. the exponent of 5 should be -1 not 1, so that matrix should read `[-4 4 -1 0 0]. so then if ^ is the wedge product, is (81/80)^h72 = 1 the proper way to notate this? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3871 - Contents - Hide Contents Date: Wed, 13 Feb 2002 22:46:12 Subject: Re: h72 = h31 + h41 From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> so then if ^ is the wedge product, is (81/80)^h72 = 1 > the proper way to notate this?It's certainly *possible* to interpret this in a way which makes sense, and notate it thusly, but I think it would be much preferable to write this as h72(81/80) = 1.

Message: 3872 - Contents - Hide Contents Date: Wed, 13 Feb 2002 21:20:59 Subject: Re: h72 = h31 + h41 From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, February 13, 2002 2:46 PM > Subject: [tuning-math] Re: h72 = h31 + h41 > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> so then if ^ is the wedge product, is (81/80)^h72 = 1 >> the proper way to notate this? >> It's certainly *possible* to interpret this in a way which > makes sense, and notate it thusly, but I think it would be > much preferable to write this as h72(81/80) = 1.well, ok . . . now t h a t notation looks familiar, and i understand it. but i've seen ^ used in connection with wedgies, so my question is still not really answered: do we have to use ** now to represent "raise to the power of"? apparently, whatever ^ is being used for, it's something else other than that. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3873 - Contents - Hide Contents Date: Wed, 13 Feb 2002 23:44:18 Subject: ^ and ** (was: h72 = h31 + h41) From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, February 13, 2002 11:03 PM > Subject: [tuning-math] Re: h72 = h31 + h41 > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> but i've seen ^ used in connection with wedgies, so my >> question is still not really answered: do we have to use ** >> now to represent "raise to the power of"? apparently, >> whatever ^ is being used for, it's something else other >> than that. >> The "^" symbol is well-established as a notation both > for exponentiationaha! i k n e w there was a one-word term for that, but i couldn't think of it. thanks.> and wedge product; I would use it for either myself so > long as there seemed no potential for confusion. Fortran > gave us "**" for exponentiation also, which is fine, > and not used for anything else to my knowledge.well, i've certainly used ^ a heck of a lot for exponentiation, and i sure don't want to change now. but in the interests of standardization and consistency (in the general sense, not the technical tuning sense), since ** is unique for exponentiation, then maybe we should adopt this separation. ^ for wedge product ** for exponentiation should we create a poll? any other opinions on it? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3874 - Contents - Hide Contents Date: Thu, 14 Feb 2002 07:57:41 Subject: Re: ^ and ** (was: h72 = h31 + h41) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> ^ for wedge product > ** for exponentiation > > > should we create a poll? any other opinions on it?let's keep ^ for exponentiation and use /\ for wedge product

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