This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).
Contents Hide Contents S 1110000 10050 10100 10150 10200 10250 10300 10350 10400 10450 10500 10550 10600 10650 10700 10750 10800 10850 10900 10950
10600 - 10625 -
Message: 10630 Date: Mon, 15 Mar 2004 07:55:21 Subject: Re: Gene's private reserve -- 7-limit From: Graham Breed Gene Ward Smith wrote: > Rats--two temperaments with identical TOP errors *and* Graham > complexities. I was aware this could happen, but guessed, apparently > incorrectly, that it was quite unlikely and then forgot to check. The > missing one gets 64th place since it is better in the 9-limit. I've verified that, contorsion aside, each temperament I look at does have a unique badness. I need this to test my temperament finder (now in three languages!). I can't prove it but it's always worked so far. > 64 [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]] > bad 238.136875 top err 2.939961 graham 9 > > >>64 [9, 0, 9, -21, -11, 21] [[9, 14, 21, 25], [0, 1, 0, 1]] >>bad 238.136875 top err 2.939961 graham 9 Oh yes, that'll be because it's a kind of minimax. The 7 mapping presumable doesn't contribute to the TOP error, and the complexities happen to be the same. They should be distinct if you switch to some kind of RMS. Graham ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10638 Date: Wed, 17 Mar 2004 23:53:40 Subject: Re: Minimal filled scale From: Carl Lumma Greetings from beautiful Portland! >Suppose we have a linear temperament with octave period. A >chord-type in this temperament is a set of generators. A >question we might ask is what is the what is the cardinality >of the smallest set of contiguous generators which arise from >contiguous generator translates of the chord--the minimal >filled scale for the chord. "Continuous generator translates"?? >This doesn't depend on the temperament, but only on the chord, >considered as a set or list. Huh? It must depend on the mapping. >The minimal filled scale for septimal miracle is Miracle[19], >and for 11-limit miracle is Canasta (Miracle[31].) For 5-limit >meantone we get the diatonic scale (Meantone[7]), and in the >7-limit, Meantone[16]. And so on and so forth... How is this different from Graham complexity? -Carl
Message: 10641 Date: Thu, 18 Mar 2004 22:03:33 Subject: Re: Symmetric 7-limit comma badness and 2401/2400 From: Carl Lumma >> >If you take, for a 7-limit interval q and a symmetric >> >lattice distance dist, the function cents(q) dist(q)^4, >> >> Why 4? It used to be pi(lim)-1, which would be 3 in the >> 7-limit. > >The exponent should be rank(Group)/rank(Kernel); the rank of >the group for p-limit will be pi(p), and the rank of the >kernel for codimension one temperaments is of course one. You were among the people to review my code, which used pi(lim)-1. Is the above due to that we're no longer assuming octave equivalence or something? I also asked: >...I'm only returned the 10 best results, and I only >search q with d <= 3000 and cents(q) <= 600. What bounds >does your method require? -Carl
Message: 10642 Date: Thu, 18 Mar 2004 22:01:06 Subject: Re: Minimal filled scale From: Carl Lumma >> Greetings from beautiful Portland! >> >> >Suppose we have a linear temperament with octave period. A >> >chord-type in this temperament is a set of generators. A >> >question we might ask is what is the what is the cardinality >> >of the smallest set of contiguous generators which arise from >> >contiguous generator translates of the chord--the minimal >> >filled scale for the chord. >> >> "Continuous generator translates"?? > >Contiguous. Weird; high-level typo; I read it correctly. >I mean if we have for example a chord [0 1 4 10], we take >[1 2 5 11] [1 3 6 12] etc. You mean [2 3 6 12]? >until we've filled all the holes, I still don't get it. You're harmonizing every note of the original chord? >and every >note is harmonizable by at least one such chord. The original chord has this property... >> >The minimal filled scale for septimal miracle is Miracle[19], >> >and for 11-limit miracle is Canasta (Miracle[31].) For 5-limit >> >meantone we get the diatonic scale (Meantone[7]), and in the >> >7-limit, Meantone[16]. And so on and so forth... >> >> How is this different from Graham complexity? > >How is it the same? I was hoping an explanation of the difference would help us understand it. -Carl
Message: 10644 Date: Thu, 18 Mar 2004 00:36:36 Subject: Re: Symmetric 7-limit comma badness and 2401/2400 From: Carl Lumma >If you take, for a 7-limit interval q and a symmetric >lattice distance dist, the function cents(q) dist(q)^4, Why 4? It used to be pi(lim)-1, which would be 3 in the 7-limit. >you get a log-flat symmetric >badness measure for 7-limit commas. Euclidean and Hahn are >not very different; below I use Hahn distance, and order >some commas with badness less than 3000 from best to worst. >The list is completely dominated by superparticulars, and >it looks to me as if 2401/2400 is likely to be an absolute >minimum in badness. At any rate looking at this makes the >Erlich phenomenon--the great importance of 2401/2400 >for 7-limit micro ets--more understandable. > >2401/2400 184.626652 >8/7 231.174094 >7/6 266.870906 >6/5 315.641287 >5/4 386.313714 >4/3 498.044999 >50/49 559.609830 >49/48 571.148984 >7/5 582.512193 >10/7 617.487808 If q = n/d, then using (n-d)/d instead of cents(q) and log(d)^3 instead of dist(q)^4, I get... ((0.19645692845300844 7 2401/2400) (0.4419896533813025 3 4/3) (0.6660493039778589 5 5/4) (0.7075223009389495 7 225/224) (0.8337823128571303 5 6/5) (0.9004848978857011 7 126/125) (0.9587113625980545 7 7/6) (1.0518045661034596 5 81/80) (1.0526168298843461 7 8/7) (1.1239582004626365 3 9/8)) ...I'm only returned the 10 best results, and I only search q with d <= 3000 and cents(q) <= 600. What bounds does your method require? -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: 10645 Date: Thu, 18 Mar 2004 17:44:12 Subject: Re: Minimal filled scale From: Carl Lumma >> >I mean if we have for example a chord [0 1 4 10], we take >> >[1 2 5 11] [1 3 6 12] etc. >> >> You mean [2 3 6 12]? > >Right. > >> >until we've filled all the holes, >> >> I still don't get it. You're harmonizing every note of the >> original chord? > >No, I'm harmonizing everything with translates of the chord in a >minimal contiguous-generator scale containing the chord. > >> >and every >> >note is harmonizable by at least one such chord. >> >> The original chord has this property... > >No, the numbers from 0 to 10 only find harmonies for 0, 1, 4 and 10. >2, 3, 5, 6, 7, 8 and 9 have no major tetrad. If, however, I take the >numbers from 0 to 15, every one of them has a major tetrad to >harmonize it. The union of the sets {i,i+1,i+4,i+10} as i ranges from >0 to 5 is {0..15}; no smaller value than 5 will work. Got it. Cool. -Carl
10000 10050 10100 10150 10200 10250 10300 10350 10400 10450 10500 10550 10600 10650 10700 10750 10800 10850 10900 10950
10600 - 10625 -