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Message: 10875 - Contents - Hide Contents Date: Fri, 23 Apr 2004 15:10:48 Subject: Search for a JI-replacement temperament From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote: >>> If we loosened our 13-limit requirements a bit to allow approximately >> the same max error as Miracle in the 11 limit (~3.3c), then 130- ET >> would be our universal near-JI replacement. (BTW, I've found the >> semantics for a 130 notation to be quite elegant.) >> It would be one possibility.If I may elaborate, we all know how wonderful 72-ET is at the 11- limit. I would also like to point out that there is an elegance with 72-ET semantics in that there are separate accidentals for primes 5, 7, and 11: 1deg 5-comma (80:81) 2deg 7-comma (63:64) 3deg 11-diesis (32:33) With 130-ET there are also separate accidentals for primes 5, 7, 11, and 13: 1deg 7-comma less 5-comma (aka 5:7-kleisma, 5103:5120) 2deg 5-comma (80:81) 3deg 7-comma (63:64) 4deg 11-diesis less 5-comma (aka 55-comma, 54:55) 5deg 13-diesis (1024:1053); also 5-comma plus 7-comma (35:36) 6deg 11-diesis (32:33) It might be argued that the 17-comma (4096:4131) or 17-kleisma (2176:2187) might be used for 1deg130. In selecting sagittal symbols for ETs such as this, Dave and I have concluded that, since ratios of 17 are much less popular than 7/5 and 10/7, a 5:7 kleisma symbol would be the best choice for 1deg. (Besides, 130-ET is not 17-limit consistent.)> Others which have possibilities are 111, > 494 and the amazing 311.A problem that I have with 111 is that it's neither 1,5,25 nor 1,7,49- consistent, which can result in notational discrepancies between 111- ET and JI. Also, ratios of 5 and 7 are not easily distinguished by a 111 mapping, since the 5-comma and 7-comma are each 2 degrees of 111. On the other hand, 494 is excellent, and I believe that it would also satisfy Paul's personal requirements for a "universal tuning." Yes, 311 is amazing -- I only wish that 255879:256000 vanished in this one so that 25/16 and 13/10 could be notated using the same accidental. Dave and I failed to find a "perfect" ET into which JI could be mapped and thereby notated using that division's notation. We therefore decided to notate JI (at several levels of precision) without reference to any ETs. The foregoing is just to point out a few things we discovered when we actually went about deciding how to notate these tunings.>> Gene, I imagine that you might want to enlighten us as to which >> superparticular commas taken together define 224 and 130, >> respectively. >

Message: 10876 - Contents - Hide Contents Date: Fri, 23 Apr 2004 15:19:58 Subject: Re: 270 equal as the universal temperament From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> > In that case I propose 196608-equal as the universal temperament.Not a good choice: it's not 5-limit consistent, nor 1,3,9-consistent, plus 7th harmonic has error of >43% of a degree, etc., etc. But you weren't really serious about this, were you? And I can hear Dave telling me that I have better things to do, which reminds me that I'm not really here. :-| Please ignore this message. ;-) --George

Message: 10877 - Contents - Hide Contents Date: Fri, 23 Apr 2004 17:13:05 Subject: Re: 270 equal as the universal temperament From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: >>>> In that case I propose 196608-equal as the universal temperament. >> Not a good choice: it's not 5-limit consistent, nor 1,3,9- consistent, > plus 7th harmonic has error of >43% of a degree, etc., etc. > > But you weren't really serious about this, were you?My point was that this is the mu division--the "temperament" the midi tuning standard affords us.

Message: 10878 - Contents - Hide Contents 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/ * [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: 10879 - Contents - Hide Contents Date: Sat, 24 Apr 2004 23:26:37 Subject: What's with 14 From: Gene Ward Smith 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]; ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10880 - Contents - Hide Contents 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 * [with cont.]> 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 * [with cont.] 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: Sign In - * [with cont.] (Wayb.) tuning-math-2848-minkowski-reduced-schoenberg-12et.jpg ... delete the line-break in that URL, or use this one: Document Not Found * [with cont.] Search for http://tinyurl.com/yspc3 in Wayback Machine -monz ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10881 - Contents - Hide Contents 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 * [with cont.] 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: 10882 - Contents - Hide Contents Date: Sun, 25 Apr 2004 19:19:10 Subject: Re: What's with 14 From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Joseph Pehrson" <jpehrson@r...> wrote:> ***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... (??)You just chop the vals I list down. The first two give <14 22|, which gives the same fifth as 7-et ("contorsion".) The last is <14 23|; you could also try <14 25| I suppose. None of this is very good.

Message: 10883 - Contents - Hide Contents 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-temp... * [with cont.] (Wayb.) 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 * [with cont.] > > 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]; > >

Message: 10884 - Contents - Hide Contents Date: Sun, 25 Apr 2004 22:23:55 Subject: Re: What's with 14 From: Gene Ward Smith --- 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?

Message: 10885 - Contents - Hide Contents 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 * [with cont.] 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-temp... * [with cont.] (Wayb.) > > 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 - Contents - Hide Contents 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 * [with cont.] 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 - Contents - Hide Contents 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 - Contents - Hide Contents Date: Sun, 25 Apr 2004 23:45:58 Subject: another 'hanson' incidence From: Paul Erlich http://www.anaphoria.com/hrgm.PDF - Type Ok * [with cont.] (Wayb.) horagram 9 (p. 11)

Message: 10889 - Contents - Hide Contents Date: Mon, 26 Apr 2004 14:14:03 Subject: Re: More 270 From: George D. Secor --- 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 The ratios 11/7 and 14/11 may also be notated with: 2deg 891:896, the 7:11 kleisma, symbol )|( but both symbols are useful to provide alternate spellings. Should you question the advisability of using a ratio with such large numbers as 45056:45927 for a common symbol-element (Sagittal flag) in a notation, let me point out that when it is combined with other commonly used Sagittal flags, the resulting ratios are much simpler: |( + (| = (|(, 5103:5120 * 45056:45927 = 44:45 |) + (| = (|), 63:64 * 45056:45927 = 704:729 |\ + (| = (|\, 54:55 * 45056:45927 = 8192:8505, ~26:27 (which uses the same symbol) --George

Message: 10890 - Contents - Hide Contents 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 14/11Well, these have more than three digits in the numerator and denominator, so were explicitly excluded from Gene's list.

Message: 10891 - Contents - Hide Contents 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 * [with cont.] ? 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 - Contents - Hide Contents 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 * [with cont.] > ? > > 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: 10893 - Contents - Hide Contents Date: Mon, 26 Apr 2004 21:24:42 Subject: Re: More 270 From: George D. Secor --- 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

Message: 10894 - Contents - Hide Contents 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. > > --GeorgeI 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 - Contents - Hide Contents 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 * [with cont.] >> ? >> >> And, salivating, I ask, is there a straightforward calculation to > go>> from the vanishing tratio to the TOP error and/or complexity -- > like

Message: 10896 - Contents - Hide Contents 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 * [with cont.] >>> ? >>> >>> And, salivating, I ask, is there a straightforward calculation to >> go>>> from the vanishing tratio to the TOP error and/or complexity -- >> like

Message: 10897 - Contents - Hide Contents Date: Mon, 26 Apr 2004 02:34:00 Subject: Re: another 'hanson' incidence From: Carl Lumma What about it? -Carl ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10898 - Contents - Hide Contents 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?

Message: 10899 - Contents - Hide Contents Date: Tue, 27 Apr 2004 20:45:00 Subject: Re: Vanishing tratios From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: Speaking of vanishing, I was going to respond to a post of yours about tratios and yantras, and it seems to have vanished!

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