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Message: 10878 Date: Fri, 23 Apr 2004 17:17:46 Subject: Re: 270 equal as the universal temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > > > Then who came up with the subject line? > > I came up with the subject line; you came up with your own > interpretation of what you thought it should mean. So what did you mean, according to your own interpretation, when you said "I was not calling 270 a universal temperament"? ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * <*> 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 *
Message: 10880 Date: Sat, 24 Apr 2004 09:45:13 Subject: Re: lattices of Schoenberg's rational implications From: monz returning to an old subject ... during a big discussion i instigated concerning possible periodicity-blocks which might describe Schoenberg's 1911 12-tET theory as posited in his _Harmonielehre_, --- In tuning-math@xxxxxxxxxxx.xxxx "genewardsmith" <genewardsmith@j...> wrote: Yahoo groups: /tuning-math/message/2848 * > Message 2848 > From: "genewardsmith" <genewardsmith@j...> > Date: Sun Jan 20, 2002 7:20 pm > Subject: Re: lattices of Schoenberg's rational implications > > > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > > > However, I think the only reality for Schoenberg's > > system is a tuning where there is ambiguity, as defined > > by the kernel <33/32, 64/63, 81/80, 225/224>. BTW, > > is this Minkowski-reduced? > > Nope. The honor belongs to <22/21, 33/32, 36/35, 50/49>. a few messages after that, Yahoo groups: /tuning-math/message/2850 * Paul Erlich posted four different possible lattices based on those unison-vectors. i've made a rectangular-style lattice based on those same four unison-vectors, using the Tonalsoft software. this is what the software gave me: ~cents.. ratio 0000.000 1/1 0084.467 21/20 0231.174 8/7 0266.871 7/6 0386.314 5/4 0498.045 4/3 0582.512 7/5 0701.955 3/2 0813.686 8/5 0884.359 5/3 0968.826 7/4 1115.533 40/21 here is a screen-shot of the actual Tonalsoft lattice: Yahoo! - * tuning-math-2848-minkowski-reduced-schoenberg-12et.jpg ... delete the line-break in that URL, or use this one: Document Not Found * -monz ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * <*> 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 *
Message: 10881 Date: Sun, 25 Apr 2004 18:15:17 Subject: Re: What's with 14 From: Joseph Pehrson --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> Yahoo groups: /tuning-math/message/10323 * wrote: > > Joseph asked on MMM: > > "What's with 14, though... it scores pretty badly on the famed Paul > Erlich accuracy chart... :)" > > The tuning that the Zeta function likes for 14 has a flat octave, and > corresponds to <14 22 33 39 48|. It has the following TM bases: > > 5-limit: [27/25, 2048/1875] > 7-limit: [21/20, 27/25, 2048/1875] > 11-limit: [21/20, 27/25, 33/32, 242/225] > > 27/25 in the 5-limit, 21/20 and 27/25 together in the 7-limit, and > 21/20,27/25 and 33/32 in the 11-limit give the beep temperament, so > this 14-et val is closely associated to beep. The top tuning has > octaves around four cents flat. > > Another val regards 14 as a contorted version of 7 in the 5-limit; in > the 11-limit it is <14 22 32 39 48|. TM bases are > > 5-limit: [25/24, 81/80] > 7-limit: [25/24, 49/48, 81/80] > 11-limit: [25/24, 33/32, 45/44, 49/48] > > This involves decimal, meantone and jamesbond, and the TOP tuning of > the octave is now quite sharp, not flat; 1209.43 cents. > > Other vals are possible; for instance a father version is <14 23 33 40 > 49|. TM bases for this are > > 5-limit: [16/15, 15625/13122]; > 7-limit: [16/15, 50/49, 175/162]; > 11-limit: [16/15, 22/21, 50/49, 175/162]; ***Well, most of this is, admittedly, a bit over my head... but I believe I saw that the 3-limit is reflected in 14-tET, at least according to the Erlich chart, and I don't see it listed in the above... (??) Thanks! JP
Message: 10883 Date: Sun, 25 Apr 2004 19:56:50 Subject: Re: What's with 14 From: Paul Erlich Hi Joseph. As far as 5-limit goes, I have a suggestion. Remember the big ET chart on Monz's equal temperament page: Tonalsoft Encyclopaedia of Tuning - equal-temperament, (c) 2004 Tonalsoft Inc. * It's the first chart there . . . Now mouse over "zoom: 1" above the chart. If you can't see the yellow triangular grid, mouse over "zoom: 1" under "negatives". You'll see 14 occuring three times on that chart . . . once overlapping 7. These are three ways of "using" 14-equal in the 5-limit. Look at how large the errors are of the basic 5-limit consonances. Two of the instances of 14, it is true, are fairly close to the "just perfect fifths - just perfect fourths" line. But they both have thirds that are off by around 30-80 cents. The other instance of 14 (the one at the top) doesn't fare much better, and the "perfect fifth" is some 70 cents sharp!! Am I making any sense? -Paul --- In tuning-math@xxxxxxxxxxx.xxxx "Joseph Pehrson" <jpehrson@r...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > > Yahoo groups: /tuning-math/message/10323 * > > wrote: > > > > Joseph asked on MMM: > > > > "What's with 14, though... it scores pretty badly on the famed Paul > > Erlich accuracy chart... :)" > > > > The tuning that the Zeta function likes for 14 has a flat octave, > and > > corresponds to <14 22 33 39 48|. It has the following TM bases: > > > > 5-limit: [27/25, 2048/1875] > > 7-limit: [21/20, 27/25, 2048/1875] > > 11-limit: [21/20, 27/25, 33/32, 242/225] > > > > 27/25 in the 5-limit, 21/20 and 27/25 together in the 7-limit, and > > 21/20,27/25 and 33/32 in the 11-limit give the beep temperament, so > > this 14-et val is closely associated to beep. The top tuning has > > octaves around four cents flat. > > > > Another val regards 14 as a contorted version of 7 in the 5- limit; > in > > the 11-limit it is <14 22 32 39 48|. TM bases are > > > > 5-limit: [25/24, 81/80] > > 7-limit: [25/24, 49/48, 81/80] > > 11-limit: [25/24, 33/32, 45/44, 49/48] > > > > This involves decimal, meantone and jamesbond, and the TOP tuning of > > the octave is now quite sharp, not flat; 1209.43 cents. > > > > Other vals are possible; for instance a father version is <14 23 33 > 40 > > 49|. TM bases for this are > > > > 5-limit: [16/15, 15625/13122]; > > 7-limit: [16/15, 50/49, 175/162]; > > 11-limit: [16/15, 22/21, 50/49, 175/162]; > > > ***Well, most of this is, admittedly, a bit over my head... but I > believe I saw that the 3-limit is reflected in 14-tET, at least > according to the Erlich chart, and I don't see it listed in the > above... (??) > > Thanks! > > JP
Message: 10885 Date: Sun, 25 Apr 2004 22:33:29 Subject: Re: What's with 14 From: Joseph Pehrson --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> Yahoo groups: /tuning-math/message/10326 * wrote: > Hi Joseph. > > As far as 5-limit goes, I have a suggestion. > > Remember the big ET chart on Monz's equal temperament page: > > Tonalsoft Encyclopaedia of Tuning - equal-temperament, (c) 2004 Tonalsoft Inc. * > > It's the first chart there . . . > > Now mouse over "zoom: 1" above the chart. If you can't see the yellow > triangular grid, mouse over "zoom: 1" under "negatives". > > You'll see 14 occuring three times on that chart . . . once > overlapping 7. > > These are three ways of "using" 14-equal in the 5-limit. > > Look at how large the errors are of the basic 5-limit consonances. > > Two of the instances of 14, it is true, are fairly close to the "just > perfect fifths - just perfect fourths" line. > > But they both have thirds that are off by around 30-80 cents. > > The other instance of 14 (the one at the top) doesn't fare much > better, and the "perfect fifth" is some 70 cents sharp!! > > Am I making any sense? > > -Paul > ***Yes, I can see that a line drawn through the two 14s is practically parallel to the just 3:2 line, the thirds, however, being way off... I'd forgotten how nice these charts look in "negative" mode... I think certain features come out better that way, too... JP
Message: 10886 Date: Sun, 25 Apr 2004 22:34:42 Subject: Re: What's with 14 From: Joseph Pehrson --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> Yahoo groups: /tuning-math/message/10327 * wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: > > > Now mouse over "zoom: 1" above the chart. If you can't see the > yellow > > triangular grid, mouse over "zoom: 1" under "negatives". > > I'm getting a "not found" for these. Monz? ***Possibly you did what I first did, Gene, and actually *clicked on* the links, rather than just "mousing over" without clicking... I got that error message first, too... JP
Message: 10887 Date: Sun, 25 Apr 2004 23:11:54 Subject: Re: What's with 14 From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: > > > Now mouse over "zoom: 1" above the chart. If you can't see the > yellow > > triangular grid, mouse over "zoom: 1" under "negatives". > > I'm getting a "not found" for these. Monz? Are you clicking on them? You shouldn't. Just mouse over.
Message: 10888 Date: Sun, 25 Apr 2004 23:45:58 Subject: another 'hanson' incidence From: Paul Erlich http://www.anaphoria.com/hrgm.PDF - Ok * horagram 9 (p. 11)
Message: 10890 Date: Mon, 26 Apr 2004 17:32:56 Subject: Re: More 270 From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: > > In case anyone is inspired to consider 270 in the light of a 13- > limit > > notation system, this might help: the TM basis is 676/675, > 1001/1000, > > 1716/1715, 3025/3024 and 4096/4095. I used this to compute the first > > twenty notes of a Fokker block, and used that to compute all the > > 13-limit intervals with Tenney height less than a million up to > twenty > > 270-et steps, with the following result: > > > > 1 {625/624, 352/351, 351/350, 847/845, 385/384, 441/440, 540/539, > 364/363, 729/728, 325/324} > > ... > > 8 {49/48, 50/49, 729/715, 910/891, 875/858, 143/140, 864/847} > > This list misses a couple of intervals that Dave and I have found to > be valuable for notating 11-limit consonances, and we have chosen the > symbols for these two intervals to notate 1deg and 8deg of 311: > > 1deg 5103:5120 (3^6*7:2^10*5), the 5:7 kleisma, symbol |(, notates > 7/5 and 10/7 > 8deg 45056:45927 (2^12*11:3^8*7), the 7:11 comma, symbol (|, notates > 11/7 and 14/11 Well, these have more than three digits in the numerator and denominator, so were explicitly excluded from Gene's list.
Message: 10891 Date: Mon, 26 Apr 2004 19:41:19 Subject: Vanishing tratios From: Paul Erlich Before I finalize my paper, I'd like to explore the following idea. What if I take a 3-term ratio ("tratio"?) and have it vanish? Let's say 125:126:128. So 128:125 vanishes, 128:126 = 64:63 vanishes, and 126:125 vanishes. Any two of these three 'commas' of course would be enough to give you the result: in 7-limit multibreed/multival/wedgie form, <<3, 0, -6, -7, -18, -14]], the temperament formerly known as Tripletone. Other examples would seem to be: 243:252:256 for 7-limit Blackwood 245:252:256 for Dominant Sevenths 343:350:360 for 7-limit Diminished 441:448:450 for Pajara Does anyone know a way to find the simplest (lowest numbers) tratio for a given codimension-two temperament? How about for the 7- limit 'linear' temperaments listed here: Yahoo groups: /tuning-math/message/10266 * ? And, salivating, I ask, is there a straightforward calculation to go from the vanishing tratio to the TOP error and/or complexity -- like there is for vanishing ratios in the codimension-1 case? For a single vanishing ratio n:d, the TOP error is proportional to log(n/d)/log(n*d), and complexity [= 'L1 norm' of the wedgie] is proportional to log(n*d).
Message: 10892 Date: Mon, 26 Apr 2004 19:58:10 Subject: Re: Vanishing tratios From: Paul Erlich Since I just mentioned Negri on the MakeMicroMusic list, I think it's 672:675:686 --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > Before I finalize my paper, I'd like to explore the following idea. > > What if I take a 3-term ratio ("tratio"?) and have it vanish? > > Let's say 125:126:128. > > So 128:125 vanishes, 128:126 = 64:63 vanishes, and 126:125 vanishes. > > Any two of these three 'commas' of course would be enough to give you > the result: in 7-limit multibreed/multival/wedgie form, > > <<3, 0, -6, -7, -18, -14]], > > the temperament formerly known as Tripletone. > > Other examples would seem to be: > > 243:252:256 for 7-limit Blackwood > 245:252:256 for Dominant Sevenths > 343:350:360 for 7-limit Diminished > 441:448:450 for Pajara > > Does anyone know a way to find the simplest (lowest numbers) tratio > for a given codimension-two temperament? How about for the 7- > limit 'linear' temperaments listed here: > > Yahoo groups: /tuning-math/message/10266 * > ? > > And, salivating, I ask, is there a straightforward calculation to go > from the vanishing tratio to the TOP error and/or complexity -- like > there is for vanishing ratios in the codimension-1 case? > > For a single vanishing ratio n:d, the TOP error is proportional to > > log(n/d)/log(n*d), > > and complexity [= 'L1 norm' of the wedgie] is proportional to > > log(n*d).
Message: 10894 Date: Mon, 26 Apr 2004 21:33:26 Subject: Re: More 270 From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: > > --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" > <gdsecor@y...> > > wrote: > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" > <gwsmith@s...> wrote: > > > > In case anyone is inspired to consider 270 in the light of a 13- > limit > > > > notation system, this might help: the TM basis is 676/675, > 1001/1000, > > > > 1716/1715, 3025/3024 and 4096/4095. I used this to compute the > first > > > > twenty notes of a Fokker block, and used that to compute all the > > > > 13-limit intervals with Tenney height less than a million up to > twenty > > > > 270-et steps, with the following result: > > > > > > > > 1 {625/624, 352/351, 351/350, 847/845, 385/384, 441/440, > 540/539, 364/363, 729/728, 325/324} > > > > ... > > > > 8 {49/48, 50/49, 729/715, 910/891, 875/858, 143/140, 864/847} > > > > > > This list misses a couple of intervals that Dave and I have found > to > > > be valuable for notating 11-limit consonances, and we have chosen > the > > > symbols for these two intervals to notate 1deg and 8deg of 311: > > > > > > 1deg 5103:5120 (3^6*7:2^10*5), the 5:7 kleisma, symbol |(, > notates 7/5 and 10/7 > > > 8deg 45056:45927 (2^12*11:3^8*7), the 7:11 comma, symbol (|, > notates > > > 11/7 and 14/11 > > > > Well, these have more than three digits in the numerator and > > denominator, so were explicitly excluded from Gene's list. > > But I assume that you and Gene would both agree that it is essential > that symbols be provided to notate 7/5, 10/7, 11/7, and 14/11 in JI. > So if these are the principal ratios that will be mapped to 131, 139, > 176, and 94 degrees of 270, respectively, then the notational > semantics should take this into account. > > For example, taking C as 1/1, if a G-flat of 132deg (arrived at by a > chain of fifths) is lowered by 1deg to arrive at 7/5, then the > interval of 1deg alteration will be 5103:5120. > > The only other 15-limit consonances requiring a 1-deg alteration > (from tones in a chain of fifths) are 13/11 and 22/13. Raising the A > of 204deg by 1deg gives 22/13, with the interval of 1deg alteration > as 351:352. While the interval of alteration has smaller integers, > the ratios being notated are less simple. > > I submit that the semantics of a notation should be determined by the > simplicity of the ratios for the resulting pitches rather than those > for the intervals of alteration, so I question the imposition of a 3- > digit cutoff for the latter. > > --George I think Gene was just using it for purposes of illustration and certainly not considering the question of notation in the same way you and Dave have been.
Message: 10895 Date: Mon, 26 Apr 2004 22:39:35 Subject: Re: Vanishing tratios From: Paul Erlich 7-limit Miracle -- 7168:7200:7203? --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > Since I just mentioned Negri on the MakeMicroMusic list, I think it's > > 672:675:686 > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: > > Before I finalize my paper, I'd like to explore the following idea. > > > > What if I take a 3-term ratio ("tratio"?) and have it vanish? > > > > Let's say 125:126:128. > > > > So 128:125 vanishes, 128:126 = 64:63 vanishes, and 126:125 vanishes. > > > > Any two of these three 'commas' of course would be enough to give > you > > the result: in 7-limit multibreed/multival/wedgie form, > > > > <<3, 0, -6, -7, -18, -14]], > > > > the temperament formerly known as Tripletone. > > > > Other examples would seem to be: > > > > 243:252:256 for 7-limit Blackwood > > 245:252:256 for Dominant Sevenths > > 343:350:360 for 7-limit Diminished > > 441:448:450 for Pajara > > > > Does anyone know a way to find the simplest (lowest numbers) tratio > > for a given codimension-two temperament? How about for the 7- > > limit 'linear' temperaments listed here: > > > > Yahoo groups: /tuning-math/message/10266 * > > ? > > > > And, salivating, I ask, is there a straightforward calculation to > go > > from the vanishing tratio to the TOP error and/or complexity -- > like > > there is for vanishing ratios in the codimension-1 case? > > > > For a single vanishing ratio n:d, the TOP error is proportional to > > > > log(n/d)/log(n*d), > > > > and complexity [= 'L1 norm' of the wedgie] is proportional to > > > > log(n*d).
Message: 10896 Date: Mon, 26 Apr 2004 22:44:36 Subject: Re: Vanishing tratios From: Paul Erlich 5-limit 12-equal -- 625:640:648? I'm going tratio-wild, but I have to go! :( --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > 7-limit Miracle -- 7168:7200:7203? > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: > > Since I just mentioned Negri on the MakeMicroMusic list, I think > it's > > > > 672:675:686 > > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > > wrote: > > > Before I finalize my paper, I'd like to explore the following > idea. > > > > > > What if I take a 3-term ratio ("tratio"?) and have it vanish? > > > > > > Let's say 125:126:128. > > > > > > So 128:125 vanishes, 128:126 = 64:63 vanishes, and 126:125 > vanishes. > > > > > > Any two of these three 'commas' of course would be enough to give > > you > > > the result: in 7-limit multibreed/multival/wedgie form, > > > > > > <<3, 0, -6, -7, -18, -14]], > > > > > > the temperament formerly known as Tripletone. > > > > > > Other examples would seem to be: > > > > > > 243:252:256 for 7-limit Blackwood > > > 245:252:256 for Dominant Sevenths > > > 343:350:360 for 7-limit Diminished > > > 441:448:450 for Pajara > > > > > > Does anyone know a way to find the simplest (lowest numbers) > tratio > > > for a given codimension-two temperament? How about for the 7- > > > limit 'linear' temperaments listed here: > > > > > > Yahoo groups: /tuning-math/message/10266 * > > > ? > > > > > > And, salivating, I ask, is there a straightforward calculation to > > go > > > from the vanishing tratio to the TOP error and/or complexity -- > > like > > > there is for vanishing ratios in the codimension-1 case? > > > > > > For a single vanishing ratio n:d, the TOP error is proportional to > > > > > > log(n/d)/log(n*d), > > > > > > and complexity [= 'L1 norm' of the wedgie] is proportional to > > > > > > log(n*d).
Message: 10897 Date: Mon, 26 Apr 2004 02:34:00 Subject: Re: another 'hanson' incidence From: Carl Lumma > http://www.anaphoria.com/hrgm.PDF - Ok * > > horagram 9 (p. 11) What about it? -Carl ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * <*> 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 *
Message: 10898 Date: Tue, 27 Apr 2004 19:34:05 Subject: Re: Vanishing tratios From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > 5-limit 12-equal -- 625:640:648? > > I'm going tratio-wild, but I have to go! :( Let's define the function weird(a,b,c) = a*b*c/gcd(a,b)/gcd(a,c)/gcd(a,b) 5-limit ETs and lowest-weird tratios (by inspection) ET........tratio............weird 03-equal: 45:48:50......... 3600 04-equal: 24:25:27......... 5400 05-equal: 75:80:81......... 32400 07-equal: 384:400:405...... 259200 ("""""""""240:243:250...... 486000) 09-equal: 125:128:135...... 432000 10-equal: 729:768:800...... 4665600 12-equal: 625:640:648...... 6480000 15-equal: 243:250:256...... 7776000 16-equal: 3072:3125:3240... 259200000 19-equal: 15360:15552:15625 3888000000 22-equal: 6075:6144:6250... 1555200000 The monotonic pattern seems to break here. Did I miss any lower-weird and/or simpler tratios?
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