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Message: 7951 Date: Mon, 03 Nov 2003 16:54:14 Subject: Re: Some 11-limit TM reduced et bases From: Carl Lumma >This gives the basis for the corresponding standard val, and then the >characteristic temperament (the linear temperament obtained from the >first three of the four basis commas.) It seems you order these largest-to-smallest. Why do we want to leave out the smallest comma in the basis -- my guess was we'd want to omit the largest. -Carl
Message: 7952 Date: Mon, 03 Nov 2003 19:25:58 Subject: Re: Some 11-limit TM reduced et bases From: Carl Lumma >> It seems you order these largest-to-smallest. Why do we want to >>leave out the smallest comma in the basis -- my guess was we'd want >>to omit the largest. > >No, I ordered them by Tenney height, but why in the world would we >ditch the largest comma? Why in the world would we ditch the highest comma? Once again, isn't it a combination of these two factors that decides a comma's 'goodness'? Then again, I'm not sure of the relative benefit of ditching vs. keeping the best commas. -Carl
Message: 7953 Date: Tue, 04 Nov 2003 15:35:00 Subject: Re: Some 11-limit TM reduced et bases From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: > >This gives the basis for the corresponding standard val, and then the > >characteristic temperament (the linear temperament obtained from the > >first three of the four basis commas.) > > It seems you order these largest-to-smallest. no, simplest to most complex. > Why do we want to leave > out the smallest comma in the basis -- my guess was we'd want to omit > the largest. he leaves out the most complex, which is intuitive. the simplest will have the most effect on harmonic progressions in the tuning.
Message: 7954 Date: Tue, 04 Nov 2003 15:55:14 Subject: Re: reduced basis for 24-ET?? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: > hi Gene, > > > i tried deriving a periodicity-block for > 24-ET from a <3,5,11>-prime-space by using > the following unison-vectors: > > > [2, 3, 5, 11]-monzo ratio > > > [ -4, 4, -1, 0] 81:80 > > [ 7, 0, -3, 0] 128:125 > > [-17, 2, 0, 4] 131769:131072 > > > > but instead of getting a 24-tone periodicity-block, > i got a 48-tone torsional-block. > > > 24-ET represents ratios-of-11 so well that there > has to be a periodicity-block hiding in here somewhere. > can you help? > > > > -monz i'm not sure what you're asking gene. you'd like to remove the torsion? simple -- note that the sum of the three rows in the matrix above is [-14 6 -4 4] which is the square of [-7 3 -2 2] 3267:3200 using this for the third row of the matrix, you get [ -4, 4, -1, 0] 81:80 [ 7, 0, -3, 0] 128:125 [-7, 3, -2, 2] 3267:3200 and the torsion is gone.
Message: 7955 Date: Tue, 04 Nov 2003 15:59:17 Subject: Re: hey Paul From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: > I'm interested in these scales... > > >> [4, -3, 2, 13, 8, -14] [[1, 2, 2, 3], [0, 4, -3, 2]] > >> complexity 14.729697 rms 12.188571 badness 2644.480844 > >> generators [1200., -125.4687958] > > > >25:24 chroma = -6 - 4 = -10 generators -> 10 note scale > >graham complexity = 7 -> 6 tetrads > > Not sure of the significance of the - in -125. I realize > that might have been Gene. yup. anyway, this is a negri scale, yes? > >> [4, 2, 2, -1, 8, -6] [[2, 0, 3, 4], [0, 2, 1, 1]] > >> complexity 10.574200 rms 23.945252 badness 2677.407574 > >> generators [600.0000000, 950.9775006] > > > >// > > > >> [2, 6, 6, -3, -4, 5] [[2, 0, -5, -4], [0, 1, 3, 3]] > >> complexity 11.925109 rms 18.863889 badness 2682.600333 > >> generators [600.0000000, 1928.512337] > > > >25:24 chroma = 6 - 1 = 5 generators -> 10 note scale > >graham complexity = 3*2 = 6 -> 8 tetrads > > I've never noticed "generators" being expressed as larger > than "periods". Why? Can't we just reduce by the periods > here, getting > > 350.9775006 > > and > > 600., 128.512337 > > resp.? > > Again, sorry if this is more of a question for the poster > of the >>'d text (Gene?). it is. but the answer is yes, you can so reduce by the periods. gene was just trying to give a "hermite-reduced basis" or some such abstractly interesting form for the generators. > >> [6, -2, -2, 1, 20, -17] [[2, 2, 5, 6], [0, 3, -1, -1]] > >> complexity 19.126831 rms 11.798337 badness 4316.252447 > >> generators [600.0000000, 231.2978354] > > > >25:24 chroma = -2 - 3 = 5 generators -> 10 note scale > >graham complexity = 8 -> 4 tetrads > > By the way, do these temperaments have names? many of them do, thanks to gene . . .
Message: 7956 Date: Tue, 04 Nov 2003 16:51:45 Subject: Re: More TM base postings From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: > ... of course, noting that the chromatic semitone disappears > because 2^(7/24) is 350 cents, and is thus a neutral-3rd > which represents both the major-3rd and minor-3rd on this > particular bingo-card, which in turn means that it's not > really doing much in the way of representing 5-limit JI > in the first place. well, it's still doing so at least in theory, just like all the ETs on the 'dicot' line on the zoom-1 and zoom-10 charts: Definitions of tuning terms: equal temperament, (c) 1998 by Joe Monzo * (this mapping of 24-equal would appear at the intersection of the aristoxenean and dicot lines; i didn't label it because 24-equal is consistent in the 5-limit with the same approximations as 12-equal, so you'll see 24 on the same point as 12 instead.) and like those with a bingo card where the chromatic semitone vanishes, for example Yahoo groups: /tuning/files/perlich/10.gif * Yahoo groups: /tuning-math/files/Paul/7p.gif *
Message: 7957 Date: Tue, 04 Nov 2003 08:52:57 Subject: Re: Some 11-limit TM reduced et bases From: Carl Lumma >he leaves out the most complex, which is intuitive. the simplest will >have the most effect on harmonic progressions in the tuning. But isn't this also true for chromatic vectors? -Carl
Message: 7958 Date: Tue, 04 Nov 2003 09:02:43 Subject: Re: hey Paul From: Carl Lumma >> >> [4, -3, 2, 13, 8, -14] [[1, 2, 2, 3], [0, 4, -3, 2]] >> >> complexity 14.729697 rms 12.188571 badness 2644.480844 >> >> generators [1200., -125.4687958] >> > >> >25:24 chroma = -6 - 4 = -10 generators -> 10 note scale >> >graham complexity = 7 -> 6 tetrads >> >> Not sure of the significance of the - in -125. I realize >> that might have been Gene. > >yup. anyway, this is a negri scale, yes? Yep. >> >> [4, 2, 2, -1, 8, -6] [[2, 0, 3, 4], [0, 2, 1, 1]] >> >> complexity 10.574200 rms 23.945252 badness 2677.407574 >> >> generators [600.0000000, 950.9775006] >> > >> >// >> > >> >> [2, 6, 6, -3, -4, 5] [[2, 0, -5, -4], [0, 1, 3, 3]] >> >> complexity 11.925109 rms 18.863889 badness 2682.600333 >> >> generators [600.0000000, 1928.512337] >> > >> >25:24 chroma = 6 - 1 = 5 generators -> 10 note scale >> >graham complexity = 3*2 = 6 -> 8 tetrads >> >> I've never noticed "generators" being expressed as larger >> than "periods". Why? Can't we just reduce by the periods >> here, getting >> >> 350.9775006 >> >> and >> >> 600., 128.512337 >> >> resp.? >> >> Again, sorry if this is more of a question for the poster >> of the >>'d text (Gene?). > >it is. but the answer is yes, you can so reduce by the periods. gene >was just trying to give a "hermite-reduced basis" or some such >abstractly interesting form for the generators. Tx. >> >> [6, -2, -2, 1, 20, -17] [[2, 2, 5, 6], [0, 3, -1, -1]] >> >> complexity 19.126831 rms 11.798337 badness 4316.252447 >> >> generators [600.0000000, 231.2978354] >> > >> >25:24 chroma = -2 - 3 = 5 generators -> 10 note scale >> >graham complexity = 8 -> 4 tetrads >> >> By the way, do these temperaments have names? > >many of them do, thanks to gene . . . Is there a way for people to look them up? -Carl
Message: 7959 Date: Tue, 04 Nov 2003 17:07:58 Subject: Re: Some 11-limit TM reduced et bases From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: > >he leaves out the most complex, which is intuitive. the simplest will > >have the most effect on harmonic progressions in the tuning. > > But isn't this also true for chromatic vectors? > > -Carl not really, chromatic vectors only determine how far the temperament is carried out to form a scale, and can look the same for unrelated temperaments and scales, but the temperament itself is characterized by the commatic vectors. if you take one of the simplest commas which vanishes in the equal temperament and re-interpret it as a chromatic vector, you'll end up with a system that differs more strongly from the 'native harmony' of the equal temperament than when you do this with a more complex comma. for example, 31 in the 5-limit is [81/80, 393216/390625], and making the 393216/390625 chromatic maintains the meantone character that dominates 31-equal's 5-limit behavior, while making 81/80 chromatic yields the more tenuous würschmidt system . . .
Message: 7960 Date: Tue, 04 Nov 2003 17:18:58 Subject: Re: Some 11-limit TM reduced et bases From: Paul Erlich moreover, when there are three or more commatic vectors, the reduction definition is more arbitrary -- the simplest (or shortest in the lattice) comma is uniquely and unambiguously defined, but the rest depend on the precise reduction definition -- for example minkowski reduction may lead to a very simple second comma and a more complex third comma, while another basis may sacrifice the simplicity of the second comma so that the third comma comes out less complex . . . --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: > > >he leaves out the most complex, which is intuitive. the simplest > will > > >have the most effect on harmonic progressions in the tuning. > > > > But isn't this also true for chromatic vectors? > > > > -Carl > > not really, chromatic vectors only determine how far the temperament > is carried out to form a scale, and can look the same for unrelated > temperaments and scales, but the temperament itself is characterized > by the commatic vectors. if you take one of the simplest commas which > vanishes in the equal temperament and re-interpret it as a chromatic > vector, you'll end up with a system that differs more strongly from > the 'native harmony' of the equal temperament than when you do this > with a more complex comma. for example, 31 in the 5-limit is [81/80, > 393216/390625], and making the 393216/390625 chromatic maintains the > meantone character that dominates 31-equal's 5-limit behavior, while > making 81/80 chromatic yields the more tenuous würschmidt system . . .
Message: 7961 Date: Tue, 04 Nov 2003 09:19:20 Subject: Re: Some 11-limit TM reduced et bases From: Carl Lumma >>>he leaves out the most complex, which is intuitive. the simplest >>>will have the most effect on harmonic progressions in the tuning. >> >> But isn't this also true for chromatic vectors? >> >> -Carl > >not really, chromatic vectors only determine how far the temperament >is carried out to form a scale, and can look the same for unrelated >temperaments and scales, But still seems important, in light of the current "hey paul" thread, and Gene's T[n] thread. >but the temperament itself is characterized >by the commatic vectors. if you take one of the simplest commas which >vanishes in the equal temperament and re-interpret it as a chromatic >vector, you'll end up with a system that differs more strongly from >the 'native harmony' of the equal temperament than when you do this >with a more complex comma. for example, 31 in the 5-limit is [81/80, >393216/390625], and making the 393216/390625 chromatic maintains the >meantone character that dominates 31-equal's 5-limit behavior, while >making 81/80 chromatic yields the more tenuous würschmidt system . . . Ok, I'll buy that. -Carl
Message: 7970 Date: Tue, 04 Nov 2003 06:46:54 Subject: Re: More TM base postings From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: > > > > > the first one assumes the standard val (or something) > > > for inconsistent ets; i'd at least show the results > > > for other vals. > > > > The 5-limit needs to be what it is in order to include > > 81/80, and for some of these, the 7-limit needs to be > > what it is to avoid torsion. > > yes, that process should be made explicit though -- i'd > rather explain that 24-equal sometimes doesn't have a > reasonable basis due to torsion, and in those cases is > best understood as an equal halving of each 12-equal step, > instead of using hidden rules to provide a nice-looking > answer. yes, i agree totally with paul. in fact, the precise thing that prompted me to write my original post requesting these TM-reduced bases was that i tried to create 41-ET with our software, and while it did give 41edo as the temperament, it gave an 82-tone periodicity-block for the JI scale. as soon as i saw that, i suspected that it was due to torsion, and sure enough, that turned out to be the case. in the specific case of 24-ET in a [3,5]-prime-space, it's nice to be able to see how choosing a val of h(5)=8 results in a "double 12-ET", whereas h(5)=7 results in a true 24-tone periodicity-block ... ... of course, noting that the chromatic semitone disappears because 2^(7/24) is 350 cents, and is thus a neutral-3rd which represents both the major-3rd and minor-3rd on this particular bingo-card, which in turn means that it's not really doing much in the way of representing 5-limit JI in the first place. -monz
Message: 7971 Date: Tue, 04 Nov 2003 13:16:53 Subject: Re: hey Paul From: Carl Lumma >> Is there a way for people to look them up? > >Should I put up a web page? Dave, do you have an objection? Graham's catalog is neither complete or up to date, last I checked. The existence of a single resource is a lot to ask, I know... -Carl
Message: 7972 Date: Tue, 04 Nov 2003 07:44:29 Subject: reduced basis for 24-ET? From: monz hi Gene, i tried deriving a periodicity-block for 24-ET from a <3,5,11>-prime-space by using the following unison-vectors: [2, 3, 5, 11]-monzo ratio [ -4, 4, -1, 0] 81:80 [ 7, 0, -3, 0] 128:125 [-17, 2, 0, 4] 131769:131072 but instead of getting a 24-tone periodicity-block, i got a 48-tone torsional-block. 24-ET represents ratios-of-11 so well that there has to be a periodicity-block hiding in here somewhere. can you help? -monz
Message: 7974 Date: Tue, 04 Nov 2003 07:53:43 Subject: Re: reduced basis for 24-ET?? From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: > hi Gene, > > > i tried deriving a periodicity-block for > 24-ET from a <3,5,11>-prime-space by using > the following unison-vectors: > > > [2, 3, 5, 11]-monzo ratio > > > [ -4, 4, -1, 0] 81:80 > > [ 7, 0, -3, 0] 128:125 > > [-17, 2, 0, 4] 131769:131072 > > > > but instead of getting a 24-tone periodicity-block, > i got a 48-tone torsional-block. > > > 24-ET represents ratios-of-11 so well that there > has to be a periodicity-block hiding in here somewhere. > can you help? i tried using [14 -3 1 2] = 16384:16335 for the third unison-vector, along with 81:80 and 125:128, and it worked beautifully. i got one 12-tone PB in the 11^0 [3,5]-plane, and another very much like it in the 11^1 plane. -monz
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