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Message: 2950 - Contents - Hide Contents Date: Thu, 03 Jan 2002 11:16:19 Subject: Some 8-tone 72-et scales From: genewardsmith [0, 5, 12, 19, 35, 42, 58, 65] [5, 7, 7, 16, 7, 16, 7, 7] edges 11 17 22 connectivity 2 3 5 [0, 5, 12, 19, 35, 42, 49, 65] [5, 7, 7, 16, 7, 7, 16, 7] edges 11 18 21 connectivity 1 3 4 [0, 5, 12, 19, 26, 42, 49, 65] [5, 7, 7, 7, 16, 7, 16, 7] edges 9 15 21 connectivity 0 2 4 [0, 5, 12, 19, 26, 42, 58, 65] [5, 7, 7, 7, 16, 16, 7, 7] edges 7 13 21 connectivity 0 2 5 [0, 5, 12, 28, 35, 42, 49, 65] [5, 7, 16, 7, 7, 7, 16, 7] edges 9 17 20 connectivity 1 3 4 [0, 5, 12, 19, 35, 42, 49, 56] [5, 7, 7, 16, 7, 7, 7, 16] edges 8 17 20 connectivity 0 3 4 [0, 5, 12, 19, 26, 42, 49, 56] [5, 7, 7, 7, 16, 7, 7, 16] edges 8 16 20 connectivity 0 2 4 [0, 5, 12, 19, 26, 33, 49, 56] [5, 7, 7, 7, 7, 16, 7, 16] edges 6 13 19 connectivity 0 1 3 [0, 5, 12, 19, 26, 33, 49, 65] [5, 7, 7, 7, 7, 16, 16, 7] edges 5 11 19 connectivity 0 1 3 [0, 5, 12, 28, 35, 42, 49, 56] [5, 7, 16, 7, 7, 7, 7, 16] edges 6 14 18 connectivity 0 2 4 [0, 5, 21, 28, 35, 42, 49, 56] [5, 16, 7, 7, 7, 7, 7, 16] edges 4 13 18 connectivity 0 2 3 [0, 5, 12, 19, 26, 33, 40, 56] [5, 7, 7, 7, 7, 7, 16, 16] edges 3 11 18 connectivity 0 1 3
Message: 2951 - Contents - Hide Contents Date: Thu, 3 Jan 2002 13:38 +00 Subject: Re: Some 9-tone 72-et scales From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a1173r+ob05@xxxxxxx.xxx> gene wrote:> I'd need to write the code for it, and it isn't a graph property so I'm > not going to start with any advantage from the Maple graph theory > package. Paul did not think propriety was very important--what's your > take on it?I don't have a definitive answer. It, or something like it, may be important for modality. Especially for subsets of "comprehensible" ETs. The Pythagorean diatonic works fine despite being slightly improper, so you shouldn't be over-strict. For a scale with three step sizes to be proper shows that it has a certain level of cohesion. The extremely improper Magic subsets you gave composers, performers and listeners are likely to expect the large gaps to be filled in by more notes. This is a general problem with Magic scales of between 3 and 19 notes. The Decimal MOS has the opposite problem -- its step sizes are so closely equal that it doesn't have any shape. So it's great as a basis for notation, but doesn't have any sense of tonal center. Your top 10 note scale might solve this problem, because it's largest interval is almost twice the size of its smallest. And that single 5/72 step could be extremely important for leading to the tonic. 10 notes still seems like a lot for a mode. Perhaps the 6 note [12, 14, 9, 14, 9, 14] would work. How is it in terms of connectedness? Graham>> It's interesting that so many scales came out proper >> when that wasn't a criterion in the search. All the 10-note 72= >> scales are strictly proper. >> It's also interesting that the best scores were all proper. > > > > 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: 2952 - Contents - Hide Contents Date: Thu, 03 Jan 2002 20:19:01 Subject: Re: Some 9-tone 72-et scales From: clumma>> >ou might like to read Rothenberg's original papers on the subject. >> There's graph stuff in there that none of us have touched (propriety >> was just a starting point for Rothenberg), plus a fancy algorithm >> generating all the proper subsets of a scale. >>Where might Rothenberg's papers be found?Tuning bibliography at: Tuning & temperament bibliography * [with cont.] (Wayb.) The papers: Rothenberg, David. "A Model for Pattern Perception with Musical Applications. Part I: Pitch Structures as Order-Preserving Maps", Mathematical Systems Theory vol. 11, 1978, pp. 199-234. Rothenberg, David. "A Model for Pattern Perception with Musical Applications Part II: The Information Content of Pitch structures", Mathematical Systems Theory vol. 11, 1978, pp. 353-372. Rothenberg, David. "A Model for Pattern Perception with Musical Applications Part III: The Graph Embedding of Pitch Structures", Mathematical Systems Theory vol. 12, 1978, pp. 73-101. I'd send you copies, but my copies are locked away in Montana. -Carl
Message: 2953 - Contents - Hide Contents Date: Thu, 03 Jan 2002 20:27:13 Subject: Re: Some 9-tone 72-et scales From: clumma>I don't have a definitive answer. It, or something like it, may be >important for modality. Especially for subsets of "comprehensible" >ETs.Needless to say, I think it's very important. There's nothing against improper scales -- it isn't that kind of criterion. But R. shows that certain musical devices rely on proper scales. These effects are still available on proper subsets of improper scales, but if you want some of the effects of traditional diatonic music, where the entire pitch set is involved with these effects, then you need propriety over the whole scale.>The Pythagorean diatonic works fine despite being slightly >improper, so you shouldn't be over-strict.Right, which is why R. never uses propriety -- he uses stability.>For a scale with three step sizes to be proper shows that it has a >certain level of cohesion.I'll take this opportunity to stamp out the myth of the importance of 2nds. The variety of all the classes of scale intervals are equally important. So you need the variety of 2nds, and the ordering. To speak of only the variety is scale voodoo. IMO, the best measure is simply R.'s mean variety.>The Decimal MOS has the opposite problem -- its step sizes are so >closely equal that it doesn't have any shape.That's an issue of efficiency, not propriety. -Carl
Message: 2954 - Contents - Hide Contents Date: Thu, 03 Jan 2002 20:35:53 Subject: connectivity index? From: clumma Gene, have you considered making a general measure out of connectedness that takes into account the number of notes and the limit? Not sure how c looks over the saturated otonal chords, but certainly things get tougher directly with the number of notes and inversely with the limit. -Carl
Message: 2955 - Contents - Hide Contents Date: Thu, 03 Jan 2002 21:46:21 Subject: Re: Some 9-tone 72-et scales From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:>> The Decimal MOS has the opposite problem -- its step sizes are so >> closely equal that it doesn't have any shape. >> That's an issue of efficiency, not propriety.I don't know what "efficiency" means in this connection, but it sounds to me like what you were talking about--variety. I haven't jumped on the Miracle bandwagon precisely because the pudding-like sameness of the Decimal put me off; but the 72-et scales I've been cooking up are just the sort of thing I like.
Message: 2956 - Contents - Hide Contents Date: Thu, 03 Jan 2002 22:03:53 Subject: Re: Some 9-tone 72-et scales From: clumma>>> >he Decimal MOS has the opposite problem -- its step sizes are so >>> closely equal that it doesn't have any shape. >>>> That's an issue of efficiency, not propriety. >>I don't know what "efficiency" means in this connection, but it >sounds to me like what you were talking about--variety. I haven't >jumped on the Miracle bandwagon precisely because the pudding-like >sameness of the Decimal put me off; but the 72-et scales I've been >cooking up are just the sort of thing I like.Try message number 4044 on the main list for a general overview of and quotes from the Rothenberg papers, including the formula for efficiency. -Carl
Message: 2957 - Contents - Hide Contents Date: Thu, 03 Jan 2002 22:09:37 Subject: Re: Some 9-tone 72-et scales From: clumma I wrote...>Try message number 4044 on the main list for a general overview >of and quotes from the Rothenberg papers, including the formula >for efficiency.Yahoo has removed the tabs from my message, and least on the web, so you can tell where the citations start and stop.. Egregious. -C.
Message: 2958 - Contents - Hide Contents Date: Thu, 03 Jan 2002 22:17:31 Subject: Re: Some 9-tone 72-et scales From: clumma>> >ry message number 4044 on the main list for a general overview >> of and quotes from the Rothenberg papers, including the formula >> for efficiency. >>Yahoo has removed the tabs from my message, and least on >the web, so you can tell where the citations start and stop.. >Egregious.That's supposed to be _can't_ tell. -Carl
Message: 2959 - Contents - Hide Contents Date: Thu, 3 Jan 2002 22:22:30 Subject: Re: coordinates from unison-vectors From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, January 03, 2002 9:37 PM > Subject: [tuning-math] Re: coordinates from unison-vectors > > > Here's my Matlab code for doing this. I arbitrarily start with a 101- > by-101 square of lattice points. Rye is the 2-by-2 matrix of unison > vectors. t is the set of points inside the PB. > > > > for a=-50:50; > l(a+51,:)=a*ones(1,101); > m(:,a+51)=a*ones(101,1); > end; > p=[l(:) m(:)]; > s=p*inv(rye); > s(find(s(:,1)>.50000001),:)=[]; > s(find(s(:,2)>.50000001),:)=[]; > s(find(s(:,2)<-.49999999),:)=[]; > s(find(s(:,1)<-.49999999),:)=[]; > t=s*ryeThanks, Paul! This should help me a bit. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2960 - Contents - Hide Contents Date: Thu, 03 Jan 2002 03:14:43 Subject: Some 8-note 7-limit scales From: genewardsmith These are based on (21/20)^3 (15/14) (10/9)^2 (8/7)^2 = 2 [1, 21/20, 9/8, 5/4, 21/16, 3/2, 63/40, 7/4] [21/20, 15/14, 10/9, 21/20, 8/7, 21/20, 10/9, 8/7] edges 16 connectivity 3 [1, 21/20, 7/6, 49/40, 7/5, 147/100, 49/30, 7/4] [21/20, 10/9, 21/20, 8/7, 21/20, 10/9, 15/14, 8/7] edges 16 connectivity 2 [1, 21/20, 9/8, 5/4, 21/16, 3/2, 63/40, 9/5] [21/20, 15/14, 10/9, 21/20, 8/7, 21/20, 8/7, 10/9] edges 15 connectivity 2 [1, 21/20, 9/8, 5/4, 21/16, 3/2, 5/3, 7/4] [21/20, 15/14, 10/9, 21/20, 8/7, 10/9, 21/20, 8/7] edges 15 connectivity 2 [1, 21/20, 9/8, 9/7, 27/20, 3/2, 63/40, 9/5] [21/20, 15/14, 8/7, 21/20, 10/9, 21/20, 8/7, 10/9] edges 15 connectivity 1 [1, 21/20, 9/8, 9/7, 27/20, 3/2, 12/7, 9/5] [21/20, 15/14, 8/7, 21/20, 10/9, 8/7, 21/20, 10/9] edges 14 connectivity 2 [1, 21/20, 9/8, 9/7, 10/7, 3/2, 12/7, 9/5] [21/20, 15/14, 8/7, 10/9, 21/20, 8/7, 21/20, 10/9] edges 14 connectivity 2 [1, 21/20, 9/8, 189/160, 21/16, 3/2, 63/40, 7/4] [21/20, 15/14, 21/20, 10/9, 8/7, 21/20, 10/9, 8/7] edges 14 connectivity 1 [1, 21/20, 9/8, 189/160, 21/16, 3/2, 63/40, 9/5] [21/20, 15/14, 21/20, 10/9, 8/7, 21/20, 8/7, 10/9] edges 14 connectivity 1 [1, 21/20, 9/8, 5/4, 10/7, 3/2, 5/3, 7/4] [21/20, 15/14, 10/9, 8/7, 21/20, 10/9, 21/20, 8/7] edges 14 connectivity 1 [1, 21/20, 7/6, 5/4, 25/18, 35/24, 5/3, 7/4] [21/20, 10/9, 15/14, 10/9, 21/20, 8/7, 21/20, 8/7] edges 14 connectivity 1 [1, 21/20, 441/400, 63/50, 7/5, 147/100, 63/40, 7/4] [21/20, 21/20, 8/7, 10/9, 21/20, 15/14, 10/9, 8/7] edges 13 connectivity 2 [1, 21/20, 9/8, 5/4, 21/16, 3/2, 12/7, 9/5] [21/20, 15/14, 10/9, 21/20, 8/7, 8/7, 21/20, 10/9] edges 13 connectivity 2 [1, 21/20, 9/8, 5/4, 10/7, 3/2, 12/7, 9/5] [21/20, 15/14, 10/9, 8/7, 21/20, 8/7, 21/20, 10/9] edges 13 connectivity 2 [1, 21/20, 441/400, 49/40, 7/5, 147/100, 63/40, 7/4] [21/20, 21/20, 10/9, 8/7, 21/20, 15/14, 10/9, 8/7] edges 13 connectivity 2 [1, 21/20, 441/400, 49/40, 21/16, 3/2, 63/40, 7/4] [21/20, 21/20, 10/9, 15/14, 8/7, 21/20, 10/9, 8/7] edges 13 connectivity 1 [1, 21/20, 441/400, 49/40, 7/5, 147/100, 49/30, 7/4] [21/20, 21/20, 10/9, 8/7, 21/20, 10/9, 15/14, 8/7] edges 13 connectivity 1 [1, 21/20, 441/400, 49/40, 7/5, 3/2, 63/40, 7/4] [21/20, 21/20, 10/9, 8/7, 15/14, 21/20, 10/9, 8/7] edges 13 connectivity 1 [1, 21/20, 7/6, 49/40, 49/36, 343/240, 49/30, 7/4] [21/20, 10/9, 21/20, 10/9, 21/20, 8/7, 15/14, 8/7] edges 12 connectivity 2 [1, 21/20, 9/8, 189/160, 27/20, 3/2, 63/40, 7/4] [21/20, 15/14, 21/20, 8/7, 10/9, 21/20, 10/9, 8/7] edges 12 connectivity 2 [1, 21/20, 9/8, 9/7, 27/20, 3/2, 63/40, 7/4] [21/20, 15/14, 8/7, 21/20, 10/9, 21/20, 10/9, 8/7] edges 12 connectivity 2 [1, 21/20, 9/8, 5/4, 21/16, 35/24, 5/3, 7/4] [21/20, 15/14, 10/9, 21/20, 10/9, 8/7, 21/20, 8/7] edges 12 connectivity 1 [1, 21/20, 441/400, 189/160, 21/16, 3/2, 63/40, 7/4] [21/20, 21/20, 15/14, 10/9, 8/7, 21/20, 10/9, 8/7] edges 12 connectivity 1 [1, 21/20, 7/6, 49/40, 49/36, 35/24, 5/3, 7/4] [21/20, 10/9, 21/20, 10/9, 15/14, 8/7, 21/20, 8/7] edges 12 connectivity 1 [1, 21/20, 9/8, 9/7, 10/7, 3/2, 5/3, 7/4] [21/20, 15/14, 8/7, 10/9, 21/20, 10/9, 21/20, 8/7] edges 11 connectivity 2 [1, 21/20, 441/400, 189/160, 21/16, 3/2, 63/40, 9/5] [21/20, 21/20, 15/14, 10/9, 8/7, 21/20, 8/7, 10/9] edges 11 connectivity 1 [1, 21/20, 441/400, 189/160, 21/16, 441/320, 63/40, 7/4] [21/20, 21/20, 15/14, 10/9, 21/20, 8/7, 10/9, 8/7] edges 11 connectivity 1 [1, 21/20, 441/400, 49/40, 21/16, 3/2, 63/40, 9/5] [21/20, 21/20, 10/9, 15/14, 8/7, 21/20, 8/7, 10/9] edges 11 connectivity 1 [1, 21/20, 441/400, 49/40, 1029/800, 147/100, 63/40, 7/4] [21/20, 21/20, 10/9, 21/20, 8/7, 15/14, 10/9, 8/7] edges 11 connectivity 1 [1, 21/20, 9/8, 5/4, 21/16, 35/24, 49/32, 7/4] [21/20, 15/14, 10/9, 21/20, 10/9, 21/20, 8/7, 8/7] edges 11 connectivity 1 [1, 21/20, 441/400, 49/40, 21/16, 441/320, 63/40, 7/4] [21/20, 21/20, 10/9, 15/14, 21/20, 8/7, 10/9, 8/7] edges 11 connectivity 1 [1, 21/20, 9/8, 189/160, 21/16, 441/320, 63/40, 7/4] [21/20, 15/14, 21/20, 10/9, 21/20, 8/7, 10/9, 8/7] edges 11 connectivity 1 [1, 21/20, 441/400, 189/160, 27/20, 3/2, 63/40, 9/5] [21/20, 21/20, 15/14, 8/7, 10/9, 21/20, 8/7, 10/9] edges 11 connectivity 1 [1, 21/20, 441/400, 49/40, 7/5, 147/100, 63/40, 9/5] [21/20, 21/20, 10/9, 8/7, 21/20, 15/14, 8/7, 10/9] edges 11 connectivity 1 [1, 21/20, 441/400, 63/50, 1323/1000, 147/100, 63/40, 7/4] [21/20, 21/20, 8/7, 21/20, 10/9, 15/14, 10/9, 8/7] edges 11 connectivity 1 [1, 21/20, 441/400, 63/50, 27/20, 3/2, 63/40, 7/4] [21/20, 21/20, 8/7, 15/14, 10/9, 21/20, 10/9, 8/7] edges 11 connectivity 1 [1, 21/20, 9/8, 9/7, 27/20, 54/35, 12/7, 9/5] [21/20, 15/14, 8/7, 21/20, 8/7, 10/9, 21/20, 10/9] edges 11 connectivity 1 [1, 21/20, 441/400, 49/40, 1029/800, 147/100, 49/30, 7/4] [21/20, 21/20, 10/9, 21/20, 8/7, 10/9, 15/14, 8/7] edges 10 connectivity 1 [1, 21/20, 9/8, 9/7, 27/20, 3/2, 5/3, 7/4] [21/20, 15/14, 8/7, 21/20, 10/9, 10/9, 21/20, 8/7] edges 10 connectivity 1 [1, 21/20, 9/8, 9/7, 72/49, 54/35, 12/7, 9/5] [21/20, 15/14, 8/7, 8/7, 21/20, 10/9, 21/20, 10/9] edges 10 connectivity 1 [1, 21/20, 441/400, 189/160, 27/20, 3/2, 63/40, 7/4] [21/20, 21/20, 15/14, 8/7, 10/9, 21/20, 10/9, 8/7] edges 10 connectivity 1 [1, 21/20, 441/400, 49/40, 1029/800, 441/320, 63/40, 7/4] [21/20, 21/20, 10/9, 21/20, 15/14, 8/7, 10/9, 8/7] edges 9 connectivity 1 [1, 21/20, 9/8, 189/160, 27/20, 567/400, 63/40, 7/4] [21/20, 15/14, 21/20, 8/7, 21/20, 10/9, 10/9, 8/7] edges 9 connectivity 1 [1, 21/20, 441/400, 189/160, 27/20, 567/400, 63/40, 7/4] [21/20, 21/20, 15/14, 8/7, 21/20, 10/9, 10/9, 8/7] edges 8 connectivity 1 [1, 21/20, 441/400, 9261/8000, 1323/1000, 147/100, 63/40, 7/4] [21/20, 21/20, 21/20, 8/7, 10/9, 15/14, 10/9, 8/7] edges 8 connectivity 1 [1, 21/20, 441/400, 189/160, 3969/3200, 567/400, 63/40, 7/4] [21/20, 21/20, 15/14, 21/20, 8/7, 10/9, 10/9, 8/7] edges 7 connectivity 1
Message: 2961 - Contents - Hide Contents Date: Thu, 03 Jan 2002 23:10:12 Subject: Some 11-tone 72-et scales From: genewardsmith [0, 5, 14, 19, 28, 33, 42, 47, 56, 61, 70] [5, 9, 5, 9, 5, 9, 5, 9, 5, 9, 2] edges 13 28 41 connectivity 0 4 7 [0, 5, 14, 19, 28, 37, 42, 47, 56, 61, 70] [5, 9, 5, 9, 9, 5, 5, 9, 5, 9, 2] edges 13 26 40 connectivity 0 3 6 [0, 5, 14, 19, 28, 37, 42, 51, 56, 61, 70] [5, 9, 5, 9, 9, 5, 9, 5, 5, 9, 2] edges 12 26 40 connectivity 0 3 6 [0, 9, 14, 23, 28, 37, 46, 51, 56, 65, 70] [9, 5, 9, 5, 9, 9, 5, 5, 9, 5, 2] edges 12 26 40 connectivity 0 3 6 [0, 5, 14, 19, 28, 33, 42, 51, 56, 61, 70] [5, 9, 5, 9, 5, 9, 9, 5, 5, 9, 2] edges 12 26 40 connectivity 0 3 6 [0, 5, 10, 19, 28, 33, 42, 47, 56, 61, 70] [5, 5, 9, 9, 5, 9, 5, 9, 5, 9, 2] edges 12 25 40 connectivity 0 2 6 [0, 9, 14, 23, 28, 37, 46, 51, 60, 65, 70] [9, 5, 9, 5, 9, 9, 5, 9, 5, 5, 2] edges 11 25 40 connectivity 0 3 6 [0, 9, 14, 19, 28, 37, 42, 51, 56, 61, 70] [9, 5, 5, 9, 9, 5, 9, 5, 5, 9, 2] edges 13 24 40 connectivity 0 2 6 [0, 9, 14, 19, 28, 33, 42, 47, 56, 61, 70] [9, 5, 5, 9, 5, 9, 5, 9, 5, 9, 2] edges 12 24 40 connectivity 0 1 6 [0, 5, 14, 19, 28, 33, 42, 51, 56, 65, 70] [5, 9, 5, 9, 5, 9, 9, 5, 9, 5, 2] edges 11 26 39 connectivity 0 3 6 [0, 5, 14, 19, 28, 33, 42, 47, 56, 65, 70] [5, 9, 5, 9, 5, 9, 5, 9, 9, 5, 2] edges 12 26 39 connectivity 0 2 6 [0, 5, 14, 23, 28, 37, 42, 47, 56, 61, 70] [5, 9, 9, 5, 9, 5, 5, 9, 5, 9, 2] edges 12 24 39 connectivity 0 3 6 [0, 5, 14, 23, 28, 33, 42, 47, 56, 61, 70] [5, 9, 9, 5, 5, 9, 5, 9, 5, 9, 2] edges 12 24 39 connectivity 0 2 6 [0, 9, 14, 23, 32, 37, 42, 51, 56, 65, 70] [9, 5, 9, 9, 5, 5, 9, 5, 9, 5, 2] edges 12 24 39 connectivity 0 2 5 [0, 5, 14, 23, 28, 37, 42, 51, 56, 61, 70] [5, 9, 9, 5, 9, 5, 9, 5, 5, 9, 2] edges 11 24 39 connectivity 0 2 6 [0, 5, 14, 19, 24, 33, 38, 47, 56, 61, 70] [5, 9, 5, 5, 9, 5, 9, 9, 5, 9, 2] edges 11 24 39 connectivity 0 3 5 [0, 9, 14, 23, 32, 37, 46, 51, 60, 65, 70] [9, 5, 9, 9, 5, 9, 5, 9, 5, 5, 2] edges 10 23 39 connectivity 0 2 5 [0, 9, 14, 19, 28, 37, 42, 47, 56, 61, 70] [9, 5, 5, 9, 9, 5, 5, 9, 5, 9, 2] edges 12 22 39 connectivity 0 1 6 [0, 9, 14, 23, 28, 37, 42, 47, 56, 61, 70] [9, 5, 9, 5, 9, 5, 5, 9, 5, 9, 2] edges 11 22 39 connectivity 0 2 6 [0, 5, 14, 23, 28, 33, 42, 47, 56, 65, 70] [5, 9, 9, 5, 5, 9, 5, 9, 9, 5, 2] edges 13 24 38 connectivity 0 3 6 [0, 5, 14, 19, 28, 37, 42, 47, 56, 65, 70] [5, 9, 5, 9, 9, 5, 5, 9, 9, 5, 2] edges 12 24 38 connectivity 0 2 6 [0, 9, 18, 23, 28, 37, 42, 51, 56, 65, 70] [9, 9, 5, 5, 9, 5, 9, 5, 9, 5, 2] edges 12 23 38 connectivity 0 1 5 [0, 9, 14, 19, 28, 37, 46, 51, 56, 65, 70] [9, 5, 5, 9, 9, 9, 5, 5, 9, 5, 2] edges 10 23 38 connectivity 0 2 5 [0, 5, 10, 19, 28, 33, 42, 47, 52, 61, 70] [5, 5, 9, 9, 5, 9, 5, 5, 9, 9, 2] edges 13 22 38 connectivity 0 2 6 [0, 9, 18, 23, 28, 37, 46, 51, 60, 65, 70] [9, 9, 5, 5, 9, 9, 5, 9, 5, 5, 2] edges 12 22 38 connectivity 0 2 6 [0, 5, 14, 23, 28, 33, 42, 51, 56, 61, 70] [5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 2] edges 11 22 38 connectivity 0 2 6 [0, 9, 18, 23, 32, 37, 46, 51, 60, 65, 70] [9, 9, 5, 9, 5, 9, 5, 9, 5, 5, 2] edges 11 22 38 connectivity 0 1 5 [0, 5, 10, 19, 28, 37, 42, 47, 56, 61, 70] [5, 5, 9, 9, 9, 5, 5, 9, 5, 9, 2] edges 11 22 38 connectivity 0 1 5 [0, 5, 10, 19, 28, 33, 42, 51, 56, 61, 70] [5, 5, 9, 9, 5, 9, 9, 5, 5, 9, 2] edges 11 21 38 connectivity 0 1 6 [0, 5, 10, 19, 28, 33, 38, 47, 56, 61, 70] [5, 5, 9, 9, 5, 5, 9, 9, 5, 9, 2] edges 11 21 38 connectivity 0 2 5 [0, 5, 10, 19, 28, 33, 42, 47, 56, 65, 70] [5, 5, 9, 9, 5, 9, 5, 9, 9, 5, 2] edges 11 23 37 connectivity 0 2 5 [0, 5, 14, 23, 28, 37, 46, 51, 56, 65, 70] [5, 9, 9, 5, 9, 9, 5, 5, 9, 5, 2] edges 11 22 37 connectivity 0 1 5 [0, 9, 18, 23, 32, 37, 42, 51, 56, 65, 70] [9, 9, 5, 9, 5, 5, 9, 5, 9, 5, 2] edges 11 21 37 connectivity 0 2 5 [0, 9, 18, 23, 28, 37, 46, 51, 56, 65, 70] [9, 9, 5, 5, 9, 9, 5, 5, 9, 5, 2] edges 11 21 37 connectivity 0 1 5 [0, 9, 14, 23, 32, 37, 42, 51, 56, 61, 70] [9, 5, 9, 9, 5, 5, 9, 5, 5, 9, 2] edges 11 21 37 connectivity 0 1 4 [0, 5, 14, 19, 24, 33, 38, 47, 52, 61, 70] [5, 9, 5, 5, 9, 5, 9, 5, 9, 9, 2] edges 10 21 37 connectivity 0 2 5 [0, 5, 10, 19, 28, 33, 38, 47, 52, 61, 70] [5, 5, 9, 9, 5, 5, 9, 5, 9, 9, 2] edges 12 20 37 connectivity 0 1 5 [0, 9, 14, 19, 28, 37, 46, 51, 56, 61, 70] [9, 5, 5, 9, 9, 9, 5, 5, 5, 9, 2] edges 10 20 37 connectivity 0 1 4 [0, 9, 14, 19, 28, 37, 46, 51, 60, 65, 70] [9, 5, 5, 9, 9, 9, 5, 9, 5, 5, 2] edges 9 20 37 connectivity 0 1 5 [0, 5, 14, 19, 28, 37, 42, 51, 60, 65, 70] [5, 9, 5, 9, 9, 5, 9, 9, 5, 5, 2] edges 10 21 36 connectivity 0 1 5 [0, 5, 14, 23, 28, 37, 46, 51, 60, 65, 70] [5, 9, 9, 5, 9, 9, 5, 9, 5, 5, 2] edges 10 21 36 connectivity 0 1 4 [0, 5, 14, 23, 32, 37, 42, 51, 56, 65, 70] [5, 9, 9, 9, 5, 5, 9, 5, 9, 5, 2] edges 10 21 36 connectivity 0 1 4 [0, 5, 14, 19, 28, 37, 46, 51, 56, 65, 70] [5, 9, 5, 9, 9, 9, 5, 5, 9, 5, 2] edges 9 21 36 connectivity 0 1 4 [0, 9, 18, 23, 28, 37, 42, 51, 56, 61, 70] [9, 9, 5, 5, 9, 5, 9, 5, 5, 9, 2] edges 11 19 36 connectivity 0 1 4 [0, 5, 14, 23, 28, 33, 38, 47, 56, 61, 70] [5, 9, 9, 5, 5, 5, 9, 9, 5, 9, 2] edges 10 19 36 connectivity 0 1 4 [0, 5, 14, 23, 28, 37, 46, 51, 56, 61, 70] [5, 9, 9, 5, 9, 9, 5, 5, 5, 9, 2] edges 9 19 36 connectivity 0 1 4 [0, 5, 14, 23, 28, 33, 42, 47, 52, 61, 70] [5, 9, 9, 5, 5, 9, 5, 5, 9, 9, 2] edges 11 20 35 connectivity 0 1 4 [0, 5, 10, 19, 28, 37, 42, 47, 56, 65, 70] [5, 5, 9, 9, 9, 5, 5, 9, 9, 5, 2] edges 10 20 35 connectivity 0 1 5 [0, 5, 14, 23, 32, 37, 42, 51, 56, 61, 70] [5, 9, 9, 9, 5, 5, 9, 5, 5, 9, 2] edges 9 20 35 connectivity 0 1 3 [0, 9, 18, 23, 32, 41, 46, 51, 60, 65, 70] [9, 9, 5, 9, 9, 5, 5, 9, 5, 5, 2] edges 11 19 35 connectivity 0 1 4 [0, 9, 14, 23, 32, 41, 46, 51, 60, 65, 70] [9, 5, 9, 9, 9, 5, 5, 9, 5, 5, 2] edges 9 19 35 connectivity 0 1 3 [0, 5, 10, 19, 28, 37, 42, 47, 52, 61, 70] [5, 5, 9, 9, 9, 5, 5, 5, 9, 9, 2] edges 11 18 35 connectivity 0 1 4 [0, 9, 14, 23, 32, 37, 42, 47, 56, 61, 70] [9, 5, 9, 9, 5, 5, 5, 9, 5, 9, 2] edges 9 18 35 connectivity 0 1 3 [0, 9, 14, 23, 32, 37, 46, 55, 60, 65, 70] [9, 5, 9, 9, 5, 9, 9, 5, 5, 5, 2] edges 8 18 35 connectivity 0 1 4 [0, 9, 18, 23, 32, 37, 42, 51, 56, 61, 70] [9, 9, 5, 9, 5, 5, 9, 5, 5, 9, 2] edges 10 18 34 connectivity 0 1 4 [0, 9, 18, 23, 28, 37, 42, 47, 56, 61, 70] [9, 9, 5, 5, 9, 5, 5, 9, 5, 9, 2] edges 10 18 34 connectivity 0 1 3 [0, 5, 10, 19, 28, 33, 38, 47, 56, 65, 70] [5, 5, 9, 9, 5, 5, 9, 9, 9, 5, 2] edges 9 18 34 connectivity 0 1 4 [0, 9, 18, 23, 28, 37, 46, 55, 60, 65, 70] [9, 9, 5, 5, 9, 9, 9, 5, 5, 5, 2] edges 9 18 34 connectivity 0 1 4 [0, 5, 10, 19, 24, 33, 42, 51, 56, 65, 70] [5, 5, 9, 5, 9, 9, 9, 5, 9, 5, 2] edges 8 18 34 connectivity 0 2 4 [0, 9, 18, 27, 32, 37, 46, 51, 60, 65, 70] [9, 9, 9, 5, 5, 9, 5, 9, 5, 5, 2] edges 9 17 34 connectivity 0 1 4 [0, 5, 14, 23, 28, 33, 38, 47, 52, 61, 70] [5, 9, 9, 5, 5, 5, 9, 5, 9, 9, 2] edges 9 17 33 connectivity 0 1 4 [0, 5, 14, 19, 24, 29, 38, 47, 52, 61, 70] [5, 9, 5, 5, 5, 9, 9, 5, 9, 9, 2] edges 9 17 33 connectivity 0 1 3 [0, 5, 14, 19, 24, 33, 38, 43, 52, 61, 70] [5, 9, 5, 5, 9, 5, 5, 9, 9, 9, 2] edges 8 17 32 connectivity 0 1 3 [0, 9, 14, 23, 32, 41, 46, 55, 60, 65, 70] [9, 5, 9, 9, 9, 5, 9, 5, 5, 5, 2] edges 7 16 32 connectivity 0 1 3 [0, 9, 18, 23, 32, 37, 42, 47, 56, 61, 70] [9, 9, 5, 9, 5, 5, 5, 9, 5, 9, 2] edges 8 16 31 connectivity 0 2 3 [0, 5, 14, 19, 24, 29, 38, 43, 52, 61, 70] [5, 9, 5, 5, 5, 9, 5, 9, 9, 9, 2] edges 7 15 29 connectivity 0 1 3
Message: 2962 - Contents - Hide Contents Date: Thu, 3 Jan 2002 20:05:08 Subject: updated definition: transformation From: monz I've corrected and updated my Dictionary entry for "transformation": Definitions of tuning terms: transformation, (... * [with cont.] (Wayb.) I'd appreciate errata, criticism, etc., before I post an announcement on the big list. In particular: - is is correct to use the word "translation"? (at the beginning of the definition) - When I look at the final (bottom) diagram on the "transformation" webpage (the one that has the d,q unit-vectors divided into 1/7ths), my spatial imagination tells me that the correct algebraic solution is: x = (di + cj) / n y = (bi + aj) / n I can see by looking at the diagram that this is the way the coordinates transpose according to the matrix determinant, in a sort of criss-cross pattern. But the formula I derive (about 1/5 of the way down the page) is x = (di - bj) / n y = (aj - ci) / n They're not the same. ... ? Is the first set correct generally, or only for this particular example? Or is it the second set that's only correct in this particular example? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2963 - Contents - Hide Contents Date: Fri, 04 Jan 2002 05:37:11 Subject: Re: coordinates from unison-vectors From: paulerlich Here's my Matlab code for doing this. I arbitrarily start with a 101- by-101 square of lattice points. Rye is the 2-by-2 matrix of unison vectors. t is the set of points inside the PB. for a=-50:50; l(a+51,:)=a*ones(1,101); m(:,a+51)=a*ones(101,1); end; p=[l(:) m(:)]; s=p*inv(rye); s(find(s(:,1)>.50000001),:)=[]; s(find(s(:,2)>.50000001),:)=[]; s(find(s(:,2)<-.49999999),:)=[]; s(find(s(:,1)<-.49999999),:)=[]; t=s*rye
Message: 2964 - Contents - Hide Contents Date: Fri, 04 Jan 2002 08:38:40 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: clumma>No. It just assumes that the overtones are pretty close to harmonic, >because they will then lead to the same ratio-intepretations for the >fundamentals as the fundamentals by themselves. If they're 50 cents >from harmonic, they will lead to a larger s value for the resulting >harmonic entropy curve, but that's about it.s represents the blur of the spectral components coming in. How could an inharmonic timbre change that?>You can synthesize inharmonic sounds, yes?No, that's the problem.>You can use a high-limit JI scale that sounds like a pelog scale, >yes?The scale isn't stuck in JI, the timbre is.>>> The gamelan scales sound like they contain a rough major >>> triad and a rough minor triad, forming a very rough major >>> seventh chord together, plus one extra note -- don't they? >>>> Yes, to me, pelog sounds like a I and a III with a 4th in the >> middle. But the music seems to use a fixed tonic, with not >> much in the way of triadic structure. >> How about 5-limit intervals?Not sure what you're asking.>> Okay, let's take a >> journey... >> >> "Instrumental music of Northeast Thailand" >> >> Characteristic stop rhythm. Harmonium and marimba-sounding >> things play major pentatonic on C# (A=440) or relative minor >> on A#. >>This is clearly not a pelog tuning!Right, it's the chinese pentatonic. I threw it in for completeness.>> I suppose there is some argument for triadic structure here >> too, but if I hadn't heard the last disc beforehand, I'd >> say they were just doing the 'start the figure on different >> scale members' thing, as in the first disc. I don't know >> Paul, this is not life as we know it (or hear it). >>What on earth does that mean?It's easy for me to hear triadic structure. I'll bend over backwards to do it. It's easy for me to hear pelog as a subset of the diatonic scale, too. Indonesians might hear it differently.>> I still say there's nothing here that would turn up an optimized >> 5-limit temperament! >>Forget the optimization. All you need is the mapping -- that >chains of three fifths make a major third and that chains of >four fifths make a minor third. This seems to be a definite >characteristic of pelog! Just as much as the "opposite" is a >characteristic of Western music, regardless of whether strict >JI, optimized meantone, 12-tET, or whatever is used.Western music uses progressions of four fifths and expects to wind up on a major third. I didn't notice anything like this for the [1 -3] map (right?) on the cited discs.>> I guess it all depends if you consider these tonic changes >> or just points of symmetry in a melisma (sp?). >>Why does that matter?One's a harmonic device, the other melodic. All other things being equal, it wouldn't matter. But I think a lot of the other stuff that goes along with harmonic music is missing from this music. Western music requires meantone. The pelog 5-limit map is far more extreme, but what suffers in this music as we change the tuning from 5-of- 7, to 23, to 16, all the way to strict JI? I think the tuning on these discs is closer to JI than 23-tET, and I don't hear them avoiding a disjoint interval. Do you? Incidentally, I think Wilson agrees with your point of view here. While he does caution against eager interps. of his ethno music theory, I think he thinks that harmonic mapping is inevitable, and atomic in music. I'm not sure I agree. Not sure I disagree. -Carl
Message: 2965 - Contents - Hide Contents Date: Fri, 04 Jan 2002 05:41:54 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > Ask a musician, e.g Paul. I don't think I've ever seen them before. >I > wouldn't miss them. But I do think they look better than pelogic.As a musician, I have to say pelogic is one of my favorites. Perhaps I'm really advocating a gens^4 (weighted, of course) times mistuning badness measure here . . . but I'm happy as long as pelogic is in there. Try it with a marimba or other appropriate, inharmonic sound -- it instantly transports you to Bali -- big time! My keyboardist friend was improvising on a 12-tone mapping of this generator -- now that was some awesome music!
Message: 2966 - Contents - Hide Contents Date: Fri, 04 Jan 2002 08:49:36 Subject: Re: tetrachordality From: clumma>> >o obviously, these two scales will come out >> the same. But you've view -- and I remember >> doing some listening experiments that back you >> up (the low efficiency of the symmetrical >> version was the other theory there) -- is that >> the symmetrical version is not tetrachordal. >> >> So what's going on here? Where's the error >> in tetrachordality = similarity at transposition >> by a 3:2? >>An octave species is homotetrachordal if it has identical melodic >structure within two 4:3 spans, separated by either a 4:3 or a 3:2. >In the pentachordal scale, _all_ of the octave species are >homotetrachordal (some in more than one way). In the symmetrical >scale, _none_ of the octave species are homotetrachordal.That's the def. in your paper. But: () I never understood how it reflects symmetry at the 3:2. () "homotetrachordal" is a new term on me. Are there precise defs. of homo- vs. omni- around? How did you choose these prefixes? () We agreed a bit ago that 'the number of notes that change when a scale is transposed by 3:2 index its omnitetrachordality', right? My current approach is just a re-scaling of this. So do we want to revise this agreement? -Carl
Message: 2967 - Contents - Hide Contents Date: Fri, 04 Jan 2002 05:43:31 Subject: Re: Some 12-tone meantone scales/temperaments From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> Here are up to isomorphism by mode and inversion all of the >meantone scales of twelve tones which have a 7-limit edge- >connectivity greater than two. While the usual meantone scale (with >a connectivity of six) wins, it does not dominate, and the other >scales/temperaments are worth considering.One of them ought to be the Keenan scale, which is the meantonized Lumma/Fokker scale. Which one? I'm suprised no one said anything.>While the results are given in terms of the 31-et, they do not >depend on the precise tuning, and are generic meantone results. > > I am not aware if this sort of thing has ever been investigated, >but it certainly seems worth pursuing. Oh yes. > > [0, 2, 5, 8, 10, 13, 15, 18, 20, 23, 26, 28] > [2, 3, 3, 2, 3, 2, 3, 2, 3, 3, 2, 3] 6 > > [0, 3, 6, 8, 11, 13, 16, 19, 21, 23, 26, 29] > [3, 3, 2, 3, 2, 3, 3, 2, 2, 3, 3, 2] 5 > > [0, 3, 6, 8, 11, 13, 16, 18, 21, 23, 26, 29] > [3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 3, 2] 5 > > [0, 3, 6, 8, 10, 13, 16, 19, 21, 23, 26, 29] > [3, 3, 2, 2, 3, 3, 3, 2, 2, 3, 3, 2] 5 > > [0, 3, 6, 8, 11, 13, 16, 18, 21, 23, 26, 28] > [3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3] 4 > > [0, 3, 6, 8, 11, 13, 16, 18, 21, 24, 26, 28] > [3, 3, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3] 4 > > [0, 2, 5, 8, 10, 13, 15, 17, 20, 23, 25, 28] > [2, 3, 3, 2, 3, 2, 2, 3, 3, 2, 3, 3] 4 > > [0, 3, 6, 8, 11, 13, 15, 18, 21, 23, 26, 28] > [3, 3, 2, 3, 2, 2, 3, 3, 2, 3, 2, 3] 3 > > [0, 3, 6, 8, 11, 13, 16, 18, 20, 23, 26, 28] > [3, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 3] 3 > > [0, 3, 6, 9, 11, 13, 15, 18, 21, 24, 26, 28] > [3, 3, 3, 2, 2, 2, 3, 3, 3, 2, 2, 3] 3 > > [0, 3, 6, 9, 11, 13, 15, 17, 19, 22, 25, 28] > [3, 3, 3, 2, 2, 2, 2, 2, 3, 3, 3, 3] 3 > > [0, 2, 5, 8, 10, 12, 15, 18, 20, 22, 25, 28] > [2, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 3] 3 > > [0, 2, 5, 8, 11, 13, 15, 17, 20, 23, 26, 28] > [2, 3, 3, 3, 2, 2, 2, 3, 3, 3, 2, 3] 3 > > [0, 3, 5, 7, 10, 13, 15, 18, 20, 22, 25, 28] > [3, 2, 2, 3, 3, 2, 3, 2, 2, 3, 3, 3] 3
Message: 2968 - Contents - Hide Contents Date: Fri, 04 Jan 2002 09:07:52 Subject: Some types of 46-et scales From: genewardsmith These were derived from the 126/125 planar temperament as interpreted by 46-et. 6 tones [12, 4, 3] [3, 1, 2] 7 tones [8, 4, 7] [3, 2, 2] 9 tones [3, 9, 4] [5, 3, 1] 10 tones [7, 1, 4] [5, 3, 2] [4, 4, 7] [5, 3, 2] 12 tones [1, 7, 3] [5, 5, 2] [4, 4, 3] [5, 5, 2] [4, 4, 3] [7, 3, 2] [3, 6, 4] [8, 3, 1] 13 tones [3, 6, 1] [9, 3, 1] 15 tones [3, 4, 1] [5, 7, 3] [6, 1, 4] [5, 8, 2] 17 tones [1, 4, 3] [5, 5, 7] 19 tones [1, 3, 4] [7, 9, 3] 20 tones [2, 4, 1] [5, 7, 8]
Message: 2969 - Contents - Hide Contents Date: Fri, 4 Jan 2002 04:07:19 Subject: Some definitions From: Pierre Lamothe Reduced set of short definitions about chordoid and gammier structures permitting to see their relations -------------------------------------------------------------------------------- Gammier structure Gammier structure is Gammoid structure with Fertility axiom Gammoid structure is Harmoid structure with Regularity axiom Contiguity axiom Congruity axiom Harmoid structure is Chordoid structure on rational numbers with standard multiplication standard order finite chordoid congruence modulo 2 Chordoid structure See Chordoid structure It is sufficient to know at this level that any finite set of odd numbers A = <k1 k2 ... kn> generates a finite chordoid of classes modulo 2 with the matrix A\A = [aij] where the generic element is aij = kj/ki and a corresponding harmoid with the set {2xaij} where the x are relative integers. Inversely, for any harmoid there exist a such set of minimal odd values generating it and so called its minimal harmonic generator. The minimal genericity is the rank of that minimal generator. Atom definition in an harmoid a is an atom if a > u (where u is the unison) and xy = a has no solution where both (u < x < a) and (u < y < a) Regularity axiom is a < 2/a for any atom a Contiguity axiom is any interval k is divisible by an atom or there exist an atom a such that ax = k has a solution Congruity axiom is for any interval k there exist a stable number D of atoms in any variant of a complete atomic decomposition of k Degree function definition in gammoid number D(X) of atoms in an interval X Octave periodicity definition in gammoid number D(X) where X is the octave Fertility axiom is octave periodicity > minimal genericity -------------------------------------------------------------------------------- Chordoid structure Chordoid structure is Simploid structure with Right associativity axiom Commutativity axiom Chordicity axiom Simploid structure is set of elements with partial binary law Right simplicity axiom Right simplicity axiom is ak = ak' Þ k = k' Lemme 1 in simploid ab = c Þ b = a\c behind the reverse law \ the interval a\b the interval domain A\B which is all x\y where x in A and y in B Right associativity axiom is ak = (ab)c Þ k = bc Commutativity axiom is k = ab Þ k = ba Chordicity axiom is There exist a subset A in E such that E = A\A -------------------------------------------------------------------------------- [This message contained attachments]
Message: 2970 - Contents - Hide Contents Date: Fri, 04 Jan 2002 05:44:58 Subject: Re: The 7-limit connectivity of 7-tone meantone scales From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> I might as well give this: > > [0 5 10 15 18 23 28] > 5553553 c = 4 > > [0 5 10 15 18 23 26] > 5553535 c = 3 > > [0 5 10 15 18 21 26] > 5553355 c = 3Yes, that augmented sixth in the last scale gives you a nice 7-limit consonance that doesn't occur in the other scales.
Message: 2971 - Contents - Hide Contents Date: Fri, 04 Jan 2002 05:48:03 Subject: Re: 7 and 10 note Magic scales From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> These are 7-connected scales in the Magic temperament > [5 1 12 25 -5 -10]. They are presented in terms of the 41-et, but >are generic 7-limit Magic scales. It is striking that in both the 7 >and 10 note cases, there is a non-MOS scale with the same >connectivity as the MOS scale.This would be very striking if the 7- and 10-tone scales are really intended to be used for 7-limit harmony. I thought we really had to go to 13 notes. Anyway, I'd like to see lattices -- repeating ones to show the cylidricality of Magic. Maybe I should make some myself -- someone please remind me sometime.
Message: 2972 - Contents - Hide Contents Date: Fri, 04 Jan 2002 09:51:54 Subject: Re: Some 9-tone 72-et scales From: clumma>> >t's interesting that so many scales came out proper >> when that wasn't a criterion in the search. All the >> 10-note 72= scales are strictly proper. >>It's also interesting that the best scores were all proper.The max connectedness c of a scale of cardinality k is k-1, right? To actually get in octave-equivalent scales at odd-limit n, you have to have k <= (n+1)/2, right? For scales with a high c for k > n, some n-limit interval(s) will have to appear between more than one pair of scale members. Can we get propriety from this? It seems that any scale with one interval class always consonant will be connected. Given that the sizes of the consonances are fairly well distributed across the octave. . . Oh well. Something interesting is going on here, that's for sure. -Carl
Message: 2973 - Contents - Hide Contents Date: Fri, 04 Jan 2002 05:49:12 Subject: Re: Nine tone Orwell scales From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> Here are the 7-limit connectivities of the nine note Orwells--I>wish I had known about these excellent non-MOS scales when I was>doing my Orwell piece!Any of these show greater tetrachordality than the original MOS?
Message: 2974 - Contents - Hide Contents Date: Fri, 4 Jan 2002 12:10 +00 Subject: Re: Optimal 5-Limit Generators For Dave From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a13gbn+gl7s@xxxxxxx.xxx> Me:>> There is some ambiguity, but if you mean the >> half-fifth system, isn't that Vicentino's enharmonic? That's 31&24 > or>> [(1, 0), (1, 2), (0, 8)]. Two meantone scales, only 5-limit > consonances>> recognize, but neutral intervals used in melody. It may not be a >> temperament, but does have a history of both theory and music, so > don't>> write it off so lightly. Paul:> I doubt this reflects Vicentino's practice well at all. For instance, > he didn't base any consonant harmonies on the second meantone scale, > did he?How do you mean? The two meantones fit snugly on the two different keyboards, and chords in the enharmonic genus typically alternate between them. As most chords are consonances, there's no other way of getting the enharmonic melodies right. For you to ask this question suggests either I didn't understand you, or you don't have a copy of Vicentino's book. It is worth reading. I thought you had it because you recommended it to somebody else.>> The half-fifth system is 24&19 or [(1, 0), (2, -2), (4, -8)]. >> You mean half-fourth system?Looks like it. Me:>> There's >> also a half-octave system, [(2, 0), (3, 1), (4, 4)]. That's the > one my>> program would deduce from the octave-equivalent mapping [2 8]. Paul: >> From that unison vector? If so, I think you're confusion torsion > with "contorsion".This has nothing directly to do with unison vectors. Me:>> If I had >> such a program. If anybody cares, is it possible to write one? > Where>> torsion's present, we'll have to assume it means divisions of the > octave >> for uniqueness. Paul:> Huh? Clearly this doesn't work in the Monz sruti 24 case.No, that can't be expressed in this particular octave equivalent system. It may be possible to include it later, but let's deal with the simple cases first.>> Gene said it isn't possible, but I'm not convinced. How >> could [1 4] be anything sensible but meantone? >> Not sure what the connection is.[1 4] is a definition of meantone: 4 fifths are equivalent to a major third. Is that a unique definition, or do we have to add "plus two octaves"?>> Perhaps the first step is to find an interval that's only one > generator>> step, take the just value, period-reduce it and work everything > else out >> from that. >> If the half-fifth is the generator, what's the just value?Well, it could be either 5:4 or 6:5. Or 11:9 or 27:22. Or 49:40 or 60:49. But if you mean the case where all consonances are specified in terms of fifths, but the generator is a half-fifth, I thought I defined those out of existence above. If not, you can take the square root. Me:>> But there may be some cases where the optimal value should >> cross a period boundary. Paul: > ??Say you have a system that divides the octave into two equal parts, and 7:5 is a single generator steps. It may happen that 7:5 approximates best to be larger than a half octave, so taking its just value for calculating the mapping will get the wrong results. This may be a real problem when the octave is divided into 41 equal parts, like one of the higher-limit temperaments I came up with, and the generator is a fairly complex interval. Me:>> If you think it can't be done, show a counter-example: an >> octave-equivalent mapping without torsion that can lead to two > different>> but equally good temperaments. Paul:> Equally good? Under what criteria? Look, why do we care about the > octave-equivalent mapping? Certainly we can't object to asking the > mapping to be octave-specific, can we?It should be fairly obvious if you get the mapping right because the errors will be small. If you can find an example that depends on the choice of reasonable criteria, that'll do as a counterexample. You were the one originally pushing for octave-equivalent calculations. If you aren't bothered any more, I'm not; I was only trying to answer your questions. But it would be elegant to describe systems in the simplest possible way, and one consistent with Fokker. It's up to you if you don't want the paper to cover that. Graham
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