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Message: 5950 - Contents - Hide Contents Date: Mon, 13 Jan 2003 22:50:59 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma>> > thought of that, but I thought also that as long as one always >> uses the same set of targets across temperaments, one is ok. >> Whaddya think? //>why would keeping the same set of targets help? complexity >shouldn't be this arbitrary!Because complexity is comparitive. The idea of a complete otonal chord is not less arbitrary.>>> multiply the generator span of the otonal (or utonal) n-ad by the >>> number of periods per octave. >>>> That's what I thought. How does this compare to the taxicab >> approach? Say, for Pajara. >>i'm unclear on what taxicab approach you mean. be patient with me, >i know this would be easier in person. but i have to go now.Oh, sure, dude. You still abroad? I've got to go now, too. Just count the number of generators it takes, on the shortest route in a rect. lattice, to get to the approx. of the target interval.>i don't think anyone's been doing weighted error on this list.Oh, crap. What is it you've been pushing, then? Weighted complexity?>but if you did, you'd minimize > >f(w3*error(3),w5*error(5),w5*error(5:3)) > >where f is either RMS or MAD or MAX or whatever, and w3 is your >weight on ratios of 3, and w5 is your weight on ratios of 5.Thanks. So my f is +, where you tend to use RMS. And I've got complexity in the w's, which is a mistake, as I'll post about shortly... -Carl

Message: 5951 - Contents - Hide Contents Date: Mon, 13 Jan 2003 23:04:16 Subject: Re: Nonoctave scales and linear temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:>> Which I cooked up >> as a generalization of Graham complexity for temperaments that >> don't necessarily have octaves. >> there seems to be a problem, in that by defining the generators as an > octave and a fifth, you get different numbers than by defining them > as an octave and a twelfth, say. > > plus, graham complexity doesn't operate on a per-identity basis.All of which being why I came up with geometric complexity, which is invariant with respect to choice of generators and does not have this problem.

Message: 5952 - Contents - Hide Contents Date: Mon, 13 Jan 2003 23:09:14 Subject: Re: Notating Pajara From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith > <genewardsmith@j...>" <genewardsmith@j...> wrote: >>> This is the system with wedgie [2, -4, -4, 2, 12, -11] which we >> used to call Paultone. It has [1/2, 5/56] as poptimal in both the 7- >> limit and the 9-limit, and my recommendation is that the 56-et be >> used to notate it. The alternative is 22, but with all due respect >> for Paul's favorite division, 56 is in much better tune. >> in my paper, i found that the fifth should be in the range 708.8143 > to 710.0927 cents, with a preference for the former (equal-weighted > RMS). 22-equal is 709.0909 cents, so it's obviously quite fine, while > 56-equal is 707.1428, which is not even in the range.Hmmm, on what basis? I'd better go look. The advantage of 56 is that it brings us nearer to diaschismic; it gives us better 5-limit harmony without much disadvantage in the 7-limit, which already has the tritone as a 7/5 anyway.

Message: 5953 - Contents - Hide Contents Date: Tue, 14 Jan 2003 16:28:37 Subject: Re: Notating Pajara From: Kalle Aho --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Gene: >>> minimax 706.8431431 > > Manuel:>> Could this be wrong? I have 709.363 in the scale archive. >> I agree with Manuel > > > Graham Me too.I get 709.363 for 7-limit and 708.128 for 9-limit. Kalle

Message: 5954 - Contents - Hide Contents Date: Tue, 14 Jan 2003 16:33:43 Subject: Re: Notating Pajara From: Graham Breed Kalle Aho wrote:> I get 709.363 for 7-limit and 708.128 for 9-limit.But I agree with Gene's 706.843 for the 9-limit pajara minimax. It looks like he accidentally copied that for the 7-limit. Graham

Message: 5955 - Contents - Hide Contents Date: Tue, 14 Jan 2003 18:38:47 Subject: Re: Notating Hemififths From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" <gdsecor@y...> wrote: >>>> The vals corresponding to 58, 99, 140 and 239 which cover it are as >> follows: >>>>>> l58 := [58, 92, 135, 163, 201, 215] >>> l99 := [99, 157, 230, 278, 343, 367] >>> l140 := [140, 222, 325, 393, 485, 519] >>> l239 := [239, 379, 555, 671, 828, 886]>> For 41, and 58: 3 = +2G, 11 = +5G, 13 = -1G > >> 99, 140, 157 are not 1,3,11,13-consistent; they take 11 as one degree >> lower than +5G and 13 as one degree higher than -1G. >> That's why I gave the vals; you are supposed to use them, notthe "standard" val. I don't know what those numbers are supposed to be indictating, so we have a language barrier. Until now there was no pressing need for me to learn your mathematical notation for some of these things, but I did jot down a reference that I saw in a recent message that some of these things are explained beginning in message #682, so I have some reading to do. Are there any other things that might be helpful? Anyway, in looking at the problem, I'm just trying to arrive at symbols that will notate whatever is in your range of generators, including as many of the ET's as possible.>> 181 is 1,3,11,13-consistent, but takes 11 as one degree lower than >> +5G and 13 as one degree lower than -1G. >> >> To put it another way, none of those that are larger than 58-ET take >> either 9:11 or 13:16 as half of 2:3. It looks like we need something >> below the 11 limit. >> Your notation system assumes you are using the mapping which roundsn*log2(p) to the nearest integer for primes p and division n? That's correct.>> To digress for a moment, if you consider only 41 and 58, you might as >> well include all of these: >> >> 9deg31 < 7deg24 < 12deg41 < 17deg58 < 5deg17 < 3deg10 >> You are wandering out of the good range for the temperament we aretrying to notate.>>> which lets you notate using the 11-diesis: >> >> 1/1 27/22 3/2 18/11 9/8 11/8 27/16 >> C E\!/ G B\!/ D F/|\ A >> Can you give me what the 11-diesis is?It's 32:33. For 27/22 and 18/11 rational sagittal notation would use the symbol (!), which represents the 11' diesis, 704:729. But since 726:729 vanishes in these divisions, the 11 diesis symbol suffices. Since we're still lacking a formal presentation of the sagittal notation, we have another language barrier. I made a file that provides a quick reference to ascii simulations of single-shaft sagittal symbols for you (and anyone else interested): Yahoo groups: /tuning- * [with cont.] math/files/secor/notation/quickref.txt This includes our latest notation for the down and up 5' comma (or "schisma", 32768:32805), which will *not* be appearing in my XH18 paper. So maybe now these symbols won't seem so cryptic.>> But you're considering a narrower range that includes higher >> divisions, so you want a 7-limit just ratio that's approximately half >> of 2:3. The simplest one is 40:49, which is good for all the ET's >> you gave except for 215 (which is not 1,7,49-consistent). >> (49/40)^2/(3/2) = 2401/2400 is a comma of Hemififths, so it worksfor any of the mappings I gave including the 239 val. Yes.>> Fortunately, Dave has just proposed some symbols involving 7^2. I >> was also looking (previously) for a decent symbol that would notate >> half an apotome as nearly as possible (for such divisions as 99 and >> 311-ET), so it looks as if Dave's proposal of the symbol '|)) for the >> 5:49' diesis, 392:405, is going to be the answer for both. >> 405/392 is seven generator steps of Hemififths, down two octaves.This kind of description is independent of the exact tuning, so perhaps it could be used to figure notations out. Yes, I would like to see a notation that will work for the whole range of generators. After taking a closer look at this I see that 5 is +25G (generator steps) and 7 is +13G. I said that 215 isn't 1,7,49-consistent, but now I see that the real reason that this notation won't apply to it is that ~4:5 (taken as +25G) is so large that it isn't the best 4:5 of 215. --George

Message: 5956 - Contents - Hide Contents Date: Tue, 14 Jan 2003 18:43:34 Subject: Re: Notating Pajara From: Kalle Aho --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Kalle Aho wrote: >>> I get 709.363 for 7-limit and 708.128 for 9-limit. >> But I agree with Gene's 706.843 for the 9-limit pajara minimax. It > looks like he accidentally copied that for the 7-limit. > > > GrahamWith a generator of 706.843 cents the worst error (7:4) is 17.488 cents. With a generator of 708.128 cents it's 14.918. Kalle

Message: 5957 - Contents - Hide Contents Date: Tue, 14 Jan 2003 19:09:14 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> I thought of that, but I thought also that as long as one always >>> uses the same set of targets across temperaments, one is ok. >>> Whaddya think? > //>> why would keeping the same set of targets help? complexity >> shouldn't be this arbitrary! >> Because complexity is comparitive. The idea of a complete > otonal chord is not less arbitrary.right, but whether a particular mapping is more complex than another shouldn't be this arbitrary!>> i don't think anyone's been doing weighted error on this list. >> Oh, crap. What is it you've been pushing, then? > Weighted complexity?that's been much more common, yes.>> but if you did, you'd minimize >> >> f(w3*error(3),w5*error(5),w5*error(5:3)) >> >> where f is either RMS or MAD or MAX or whatever, and w3 is your >> weight on ratios of 3, and w5 is your weight on ratios of 5. >> Thanks. So my f is +,are you sure? aren't there absolute values, in which case it's equivalent to MAD? (or p=1, which gene doesn't want to consider)

Message: 5958 - Contents - Hide Contents Date: Tue, 14 Jan 2003 19:10:14 Subject: Re: Nonoctave scales and linear temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:>>> Which I cooked up >>> as a generalization of Graham complexity for temperaments that >>> don't necessarily have octaves. >>>> there seems to be a problem, in that by defining the generators as an >> octave and a fifth, you get different numbers than by defining them >> as an octave and a twelfth, say. >> >> plus, graham complexity doesn't operate on a per-identity basis. >> All of which being why I came up with geometric complexity, which >is invariant with respect to choice of generators and does not have >this problem.can you demonstate the "problem" for other complexity measures, say for meantone?

Message: 5959 - Contents - Hide Contents Date: Tue, 14 Jan 2003 19:11:19 Subject: Re: Notating Pajara From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith >>> >>> This is the system with wedgie [2, -4, -4, 2, 12, -11] which we >>> used to call Paultone. It has [1/2, 5/56] as poptimal in both the 7- >>> limit and the 9-limit, and my recommendation is that the 56-et be >>> used to notate it. The alternative is 22, but with all due respect >>> for Paul's favorite division, 56 is in much better tune. >>>> in my paper, i found that the fifth should be in the range 708.8143 >> to 710.0927 cents, with a preference for the former (equal- weighted >> RMS). 22-equal is 709.0909 cents, so it's obviously quite fine, while >> 56-equal is 707.1428, which is not even in the range. >> Hmmm, on what basis?i said, equal-weighted RMS. 7-limit.

Message: 5960 - Contents - Hide Contents Date: Tue, 14 Jan 2003 19:27:08 Subject: Re: Notating Pajara From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:>>> in my paper, i found that the fifth should be in the range 708.8143 >> to 710.0927 cents, with a preference for the former (equal- weighted >> RMS). 22-equal is 709.0909 cents, so it's obviously quite fine, while >> 56-equal is 707.1428, which is not even in the range. >> Opinions on that might differ. For 7-limit, I get > > least squares 708.8143306i get 708.814329511048; close enough!> least cubes 708.9699583i get 708.969922809224; close enough!> least fourth powers 709.0364438i get 709.051244958997 -- we have some discrepancy!! i get a p=4 norm of 0.0162470021271903 for your value, and 0.0162469162102062 for mine, so clearly something's wrong with your algorithm!> minimax 706.8431431i'm not sure, but i do get this same figure for MAD.> From this we can't prove that 22-equal is poptimal, though it at >least comes close and might be.according to my calculations, for p=5 we get 709.112411004975, so 22- equal is in there! I can't decypher your figures one >and two, so I don't know where 710.0927 comes from--this wouldn't me >the notorious MAD value, by any chance? no, it's least squares with limit weighting (instead of the more common inverse-limit weighting or equal weighting) -- the corresponding meantone is 175/634-comma, as you can see in my paper or here: Definitions of tuning terms: meantone, (c) 199... * [with cont.] (Wayb.)> > For the 9 limit, we have > > least squares 707.2464833 > least cubes 707.4361060 > least fourth powers 707.5632154 > minimax 706.8431431 > > The 56-et fifth of 707.1429 clearly works better here. > > If we intersect the range of known 7 and 9 limit poptimals, we get >values between 706.8431 and 707.5632 cents, with 33/56 right in the >midst of it. The best choice might still be 22, but it isn't a walk.on my keyboard it is!

Message: 5961 - Contents - Hide Contents Date: Tue, 14 Jan 2003 19:31:02 Subject: Re: Notating Hemififths From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" <gdsecor@y...> wrote:>> Your notation system assumes you are using the mapping which rounds> n*log2(p) to the nearest integer for primes p and division n? > > That's correct.that's terrible -- for example, for 64-equal in the 5-limit, this is a notably poor choice.

Message: 5962 - Contents - Hide Contents Date: Tue, 14 Jan 2003 19:32:24 Subject: Re: Notating Pajara From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho <kalleaho@m...>" <kalleaho@m...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:>> Kalle Aho wrote: >>>>> I get 709.363 for 7-limit and 708.128 for 9-limit. >>>> But I agree with Gene's 706.843 for the 9-limit pajara minimax. It >> looks like he accidentally copied that for the 7-limit. >> >> >> Graham >> With a generator of 706.843 cents the worst error (7:4) is 17.488 > cents. > With a generator of 708.128 cents it's 14.918. > > Kalleactually there is no single minimax value, since 7:5 is the worst error, since it's stuck at 600 cents, across a range of generator values.

Message: 5963 - Contents - Hide Contents Date: Tue, 14 Jan 2003 19:35:33 Subject: Re: Notating Pajara From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho <kalleaho@m...>" > <kalleaho@m...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> > wrote:>>> Kalle Aho wrote: >>>>>>> I get 709.363 for 7-limit and 708.128 for 9-limit. >>>>>> But I agree with Gene's 706.843 for the 9-limit pajara minimax. > It>>> looks like he accidentally copied that for the 7-limit. >>> >>> >>> Graham >>>> With a generator of 706.843 cents the worst error (7:4) is 17.488 >> cents. >> With a generator of 708.128 cents it's 14.918. >> >> Kalle >> actually there is no single minimax value, since 7:5 is the worst > error, since it's stuck at 600 cents, across a range of generator > values.though i suppose if you were taking the limit as p approached infinity, you'd essentially take the minimax without considering 7:5. so gene, it looks like you're not getting the right answer for p>3.

Message: 5964 - Contents - Hide Contents Date: Tue, 14 Jan 2003 21:09:30 Subject: Re: Notating Hemififths From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" > <gdsecor@y...> wrote: >>>> Your notation system assumes you are using the mapping which > rounds>> n*log2(p) to the nearest integer for primes p and division n? >> >> That's correct. >> that's terrible -- for example, for 64-equal in the 5-limit, this is > a notably poor choice.But this isn't the whole story. We have two ways of notating 64-ET. The first uses its native fifth in conjunction with the 13 diesis (1024:1053) and 13' diesis (26:27) symbols: 64: /|) (|\ /||\ But our preferred method notates it as a subset of 128-ET: 128: )| ~|( /| (|( (|~ /|\ (|) )|| ~||( ||\ (||( (||~ /||\ This notation uses the 5 and 7 commas, the 11 and 11' dieses, and the (|( symbol (valid in all three of its comma roles), in addition to the 19 and 17' commas and the 11:19 diesis. This was not a particularly easy one for us, and it was only after spending a lot of time on a lot of different ETs and constantly fine-tuning our symbol definitions and symbol selection procedure that Dave and I came to an agreement on this as the 128-ET standard symbol set. --George

Message: 5965 - Contents - Hide Contents Date: Tue, 14 Jan 2003 01:46:51 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma>All of which being why I came up with geometric complexity, >which is invariant with respect to choice of generators and >does not have this problem.Unfortunately, you may be the only one on this list that understands geometric complexity. :( -Carl

Message: 5966 - Contents - Hide Contents Date: Tue, 14 Jan 2003 21:27:14 Subject: Re: Notating Hemififths From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" <gdsecor@y...> wrote:> ... We have two ways of notating 64-ET. > > The first uses its native fifth in conjunction with the 13 diesis > (1024:1053) and 13' diesis (26:27) symbols: > > 64: /|) (|\ /||\ > > But our preferred method notates it as a subset of 128-ET: > > 128: )| ~|( /| (|( (|~ /|\ (|) )|| ~||( ||\ (||( > (||~ /||\ > > This notation uses the 5 and 7 commas,Oops, sorry! There's no 7 comma there.> the 11 and 11' dieses, and the > (|( symbol (valid in all three of its comma roles),as the 5:11, 7:13, and 11:17 commas; it's highly desirable for symbols that have multiple roles to be valid in all of them, and that, in conjunction with a low product complexity for its primary comma role, nailed this symbol for 4 degrees.> in addition to > the 19 and 17' commas and the 11:19 diesis.These higher primes were unavoidable. We pick the ones that are least unusual, and sometimes there's only a single possiblity. --George

Message: 5967 - Contents - Hide Contents Date: Tue, 14 Jan 2003 04:40:30 Subject: Re: Notating Pajara From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> in my paper, i found that the fifth should be in the range 708.8143 > to 710.0927 cents, with a preference for the former (equal-weighted > RMS). 22-equal is 709.0909 cents, so it's obviously quite fine, while > 56-equal is 707.1428, which is not even in the range.Opinions on that might differ. For 7-limit, I get least squares 708.8143306 least cubes 708.9699583 least fourth powers 709.0364438 minimax 706.8431431 From this we can't prove that 22-equal is poptimal, though it at least comes close and might be. I can't decypher your figures one and two, so I don't know where 710.0927 comes from--this wouldn't me the notorious MAD value, by any chance? For the 9 limit, we have least squares 707.2464833 least cubes 707.4361060 least fourth powers 707.5632154 minimax 706.8431431 The 56-et fifth of 707.1429 clearly works better here. If we intersect the range of known 7 and 9 limit poptimals, we get values between 706.8431 and 707.5632 cents, with 33/56 right in the midst of it. The best choice might still be 22, but it isn't a walk.

Message: 5968 - Contents - Hide Contents Date: Tue, 14 Jan 2003 21:49:59 Subject: Re: Notating Pajara From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Kalle Aho wrote: >>> I get 709.363 for 7-limit and 708.128 for 9-limit. >> But I agree with Gene's 706.843 for the 9-limit pajara minimax. It > looks like he accidentally copied that for the 7-limit.I'm afraid minimax can sometimes involve a range of values, and that's what's happened here. I was simply feeding it to Maple's simplex routine and turning the crank, but I need to redo the code so that the range is returned when there is one. My minimax value has an error on major thirds of zero, and hence 7 and 7/5 with the same error. Your milage may vary, and it seems it does.

Message: 5969 - Contents - Hide Contents Date: Tue, 14 Jan 2003 08:13:47 Subject: Notating Unidec From: Gene Ward Smith This is another important system, particularly interesting in the 11-limit. It is a half-octave system; in the 13-limit it has wedgie [12, 22, -4, -6, 4, 7, -40, -51, -38, -71, -90, -72, -3, 26, 36] and mapping [[2, 5, 8, 5, 6, 8], [0, -6, -11, 2, 3, -2]] The range of generators is 18/118 < 29/190 < 40/262 < 11/72 The 5-limit poptimal range is a cat's whisker above 18/118, with 281/1842 the best; it's possible 118 is 5-poptimal. In the other limits, we have 40/262 in the 7, 9, 11 and 13 limits, but 29/190 is an improvement in the 13 limit. The corresponding vals are l118 := [118, 187, 274, 331, 408, 436] l1842 := [1842, 2919, 4277, 5167, 6369, 6806] l190 := [190, 301, 441, 533, 657, 702] l262 := [262, 415, 608, 735, 906, 968] l72 := [72, 114, 167, 202, 249, 266] with errors err118 := [-.2601, .1270, -2.7242, -2.1655, -6.6295] err1842 := [-.3263, .0055, -2.7021, -2.1323, -6.6516] err190 := [-.9024, -1.0505, -2.5101, -1.8443, -6.8436] err262 := [-1.1916, -1.5809, -2.4137, -1.6997, -6.9400] err72 := [-1.9550, -2.9804, -2.1592, -1.3180, -7.1945] The choice appears to be between 72 and 118; I'd be inclined to give it to 118.

Message: 5970 - Contents - Hide Contents Date: Tue, 14 Jan 2003 22:08:21 Subject: Re: Notating Hemififths From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" <gdsecor@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus > <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" >>> >>>> Your notation system assumes you are using the mapping which >> rounds>>> n*log2(p) to the nearest integer for primes p and division n? >>> >>> That's correct. >>>> that's terrible -- for example, for 64-equal in the 5-limit, this > is>> a notably poor choice. >> But this isn't the whole story. We have two ways of notating 64-ET. > > The first uses its native fifth in conjunction with the 13 diesis > (1024:1053) and 13' diesis (26:27) symbols: > > 64: /|) (|\ /||\ > > But our preferred method notates it as a subset of 128-ET: > > 128: )| ~|( /| (|( (|~ /|\ (|) )|| ~||( ||\ (||( > (||~ /||\ > > This notation uses the 5 and 7 commas, the 11 and 11' dieses, and the > (|( symbol (valid in all three of its comma roles), in addition to > the 19 and 17' commas and the 11:19 diesis. This was not a > particularly easy one for us, and it was only after spending a lot of > time on a lot of different ETs and constantly fine-tuning our symbol > definitions and symbol selection procedure that Dave and I came to an > agreement on this as the 128-ET standard symbol set. > > --Georgewell, 64 is an interesting case. depending on how you map the 5-limit consonances, 64 can either be: (a) meantone (80:81 vanishing) -- narrow perfect fifth, narrow major third, ; (b) both diminished (648:625 vanishing) and escapade (4294967296:4271484375 vanishing) -- narrow perfect fifth, wide major third; (c) kleismic (15625:15552 vanishing) -- wide perfect fifth, wide major third. i imagine the consequences for notation could be frightening . . . and this is without even proceeding to the 7-limit . . .

Message: 5971 - Contents - Hide Contents Date: Tue, 14 Jan 2003 08:28:55 Subject: Re: Notating Hemififths From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" <gdsecor@y...> wrote:>> The vals corresponding to 58, 99, 140 and 239 which cover it are as > follows: >>>> l58 := [58, 92, 135, 163, 201, 215] >> l99 := [99, 157, 230, 278, 343, 367] >> l140 := [140, 222, 325, 393, 485, 519] >> l239 := [239, 379, 555, 671, 828, 886]> For 41, and 58: 3 = +2G, 11 = +5G, 13 = -1G > 99, 140, 157 are not 1,3,11,13-consistent; they take 11 as one degree > lower than +5G and 13 as one degree higher than -1G.That's why I gave the vals; you are supposed to use them, not the "standard" val.> 181 is 1,3,11,13-consistent, but takes 11 as one degree lower than > +5G and 13 as one degree lower than -1G. > > To put it another way, none of those that are larger than 58-ET take > either 9:11 or 13:16 as half of 2:3. It looks like we need something > below the 11 limit.Your notation system assumes you are using the mapping which rounds n*log2(p) to the nearest integer for primes p and division n?> To digress for a moment, if you consider only 41 and 58, you might as > well include all of these: > > 9deg31 < 7deg24 < 12deg41 < 17deg58 < 5deg17 < 3deg10You are wandering out of the good range for the temperament we are trying to notate.> which lets you notate using the 11-diesis: > > 1/1 27/22 3/2 18/11 9/8 11/8 27/16 > C E\!/ G B\!/ D F/|\ ACan you give me what the 11-diesis is?> But you're considering a narrower range that includes higher > divisions, so you want a 7-limit just ratio that's approximately half > of 2:3. The simplest one is 40:49, which is good for all the ET's > you gave except for 215 (which is not 1,7,49-consistent).(49/40)^2/(3/2) = 2401/2400 is a comma of Hemififths, so it works for any of the mappings I gave including the 239 val.> Fortunately, Dave has just proposed some symbols involving 7^2. I > was also looking (previously) for a decent symbol that would notate > half an apotome as nearly as possible (for such divisions as 99 and > 311-ET), so it looks as if Dave's proposal of the symbol '|)) for the > 5:49' diesis, 392:405, is going to be the answer for both.405/392 is seven generator steps of Hemififths, down two octaves. This kind of description is independent of the exact tuning, so perhaps it could be used to figure notations out.

Message: 5972 - Contents - Hide Contents Date: Tue, 14 Jan 2003 22:09:44 Subject: Re: Notating Pajara From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:>> Kalle Aho wrote: >>>>> I get 709.363 for 7-limit and 708.128 for 9-limit. >>>> But I agree with Gene's 706.843 for the 9-limit pajara minimax. It >> looks like he accidentally copied that for the 7-limit. >> I'm afraid minimax can sometimes involve a range of values, and >that's what's happened here. I was simply feeding it to Maple's >simplex routine and turning the crank, but I need to redo the code >so that the range is returned when there is one. My minimax value >has an error on major thirds of zero, and hence 7 and 7/5 with the >same error. Your milage may vary, and it seems it does.so what's going wrong with p>3?

Message: 5973 - Contents - Hide Contents Date: Tue, 14 Jan 2003 11:51:41 Subject: Re: Notating Pajara From: manuel.op.de.coul@xxxxxxxxxxx.xxx Gene wrote:>minimax 706.8431431Could this be wrong? I have 709.363 in the scale archive. Manuel

Message: 5974 - Contents - Hide Contents Date: Tue, 14 Jan 2003 22:38:49 Subject: Re: Nonoctave scales and linear temperaments From: Carl Lumma>> >ecause complexity is comparitive. The idea of a complete >> otonal chord is not less arbitrary. >>right, but whether a particular mapping is more complex >than another shouldn't be this arbitrary!I'm lost. If you agree with that, then what's arbitrary?>> Oh, crap. What is it you've been pushing, then? >> Weighted complexity? >>that's been much more common, yes.Ok. Here's my latest thinking, as promised. Ideally we'd base everything on complete n-ads, with harmonic entropy. Since that's not available, we'll look at dyadic breakdowns. If you use the concept of odd limits, and your best way of measuring the error of an n-ad is to break it down into dyads, you're basically saying that ratio containing n is much different than any ratio containing at most n-2. Thus, I suspect that my sum of abs-errors for each odd identity up to the limit would make sense despite the fact that for dyads like 5:3 the errors may cancel. If we throw out odd-limit, however, we might be better off. If there were a weighting that followed Tenney limit but was steep enough to make near-perfect 2:1s a fact of life and anything much beyond the 17-limit go away, we could have individually-weighted errors and 'limit infinity'. We should be able to search map space and assign generator values from scratch. Pure 2:1 generators should definitely not be assumed. Instead, we might use the appearence of many near-octave generators as evidence the weighting is right. As far as my combining error and complexity before optimizing generators, that was wrong. Moreover, combining them at all is not for me. I'm not bound to ask, "What's the 'best' temp. in size range x?". Rather, I might ask, "What's the most accurate temperament in complexity range x?". Which is just a sort on all possible temperaments, first by complexity, then by accuracy. Which is how I set up Dave's 5-limit spreadsheet after endlessly trying exponents in the badness calc. without being able to get a sensicle ranking. As for which complexity to use, we have the question of how to define a map at 'limit infinity'. . . In the meantime, what about standard n-limit complexity? () Gene's geometric complexity sounds interesting (assuming it's limit-specific...). () The number of notes of the temperament needed to get all of the n-limit dyads. () The taxicab complexity of the n-limit commas on the harmonic lattice. Or something that measured how much smaller the average harmonic structure would be in the temperament than in JI. This sort of formulation is probably best, and may in fact be what Gene's geometric complexity does...>>> f(w3*error(3),w5*error(5),w5*error(5:3)) >>> >>> where f is either RMS or MAD or MAX or whatever, and w3 is >>> your weight on ratios of 3, and w5 is your weight on ratios >>> of 5. >>>> Thanks. So my f is +, >>are you sure? aren't there absolute values, in which case it's >equivalent to MAD? (or p=1, which gene doesn't want to consider)Yep, you're right. Though its choice was just an expedient here, and it would be the MAD of just the identities, not of all the dyads in the limit. Last time we tested these things for all-the-dyads-in-a-chord, I believe I preferred RMS. Which is not to say that MAD shouldn't be included in the poptimal series. -Carl

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