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Message: 6606 Date: Sun, 9 Mar 2003 15:07:32 Subject: good 11-limit meantones From: monz hey all, i've found some interesting meantones which give the lowest error-from-JI values thru the 11-limit: 11/48-, 8/35-, and 3/13-comma are three representative examples. can anyone (Gene?) give some data on these? if it's been done before, please supply links. thanks. -monz
Message: 6607 Date: Sun, 9 Mar 2003 15:10:59 Subject: Re: good 11-limit meantones From: monz > From: "monz" <monz@xxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, March 09, 2003 3:07 PM > Subject: [tuning-math] good 11-limit meantones > > > i've found some interesting meantones which give > the lowest error-from-JI values thru the 11-limit: > 11/48-, 8/35-, and 3/13-comma are three representative > examples. > > can anyone (Gene?) give some data on these? > if it's been done before, please supply links. > thanks. 2/9-comma is also quite good in 11-limit, and in fact is the one that got me started on this. -monz
Message: 6608 Date: Sun, 9 Mar 2003 16:07:11 Subject: Re: good 11-limit meantones From: monz ----- Original Message ----- From: "monz" <monz@xxxxxxxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Sunday, March 09, 2003 3:07 PM Subject: [tuning-math] good 11-limit meantones > hey all, > > > i've found some interesting meantones which give > the lowest error-from-JI values thru the 11-limit: > 11/48-, 8/35-, and 3/13-comma are three representative > examples. as for equal-temperaments which fall in this range, 31edo is pretty darn good for a low-cardinality EDO, generator 2^(18/31). 2^(79/136) is very good, 2^(97/167) better still, and 2^(176/303) really fantastic. feedback appreciated. -monz
Message: 6609 Date: Mon, 10 Mar 2003 08:56:18 Subject: Re: good 11-limit meantones From: monz hi Graham, > From: "Graham Breed" <graham@xxxxxxxxxx.xx.xx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, March 10, 2003 2:19 AM > Subject: [tuning-math] Re: good 11-limit meantones > > > monz wrote: > > > 2^(79/136) is very good, 2^(97/167) better still, > > and 2^(176/303) really fantastic. > > What mapping are you using for 136? i made the generator 2^(79/136), and the mapping of ratios to generators follows table i put at the bottom of this webpage: Definitions of tuning terms: meantone-from-JI error, (c) 2003 by Joe Monzo * gen. ratio -18 16/11 -17 12/11 -16 18/11 ... -10 8/7 -9 12/7 -8 14/11 ... -6 10/7 ... -4 8/5 -3 6/5 ... -1 4/3 ... +1 3/2 ... +3 5/3 +4 5/4 ... +6 7/5 ... +8 11/7 +9 7/6 +10 7/4 ... +16 11/9 +17 11/6 +18 11/8 > The red line shows 3, so remember 9 is twice as far out. Why aren't > ratios of 9 included in your applet? I can see that 3/13-comma meantone > would be close to the minimax for the simpler mapping if you ignore 9:8, > 10:9 and 9:7. oops ... i'm always thinking in terms of prime-numbers, and including ratios according to prime-limit rather than odd-limit. i guess i should add ratios of 9, but that will be a lot of work as i'll have to redo every graph. -monz
Message: 6610 Date: Mon, 10 Mar 2003 10:19:39 Subject: Re: good 11-limit meantones From: Graham Breed monz wrote: > 2^(79/136) is very good, 2^(97/167) better still, > and 2^(176/303) really fantastic. What mapping are you using for 136? I can find two fairly good meantones [136, 215, 316, 382, 470] [136, 215, 316, 381, 470] Both have a fifth of 79 steps, which matches your generator. And both have a worst 11-limit error of 11.7 cents. For 31-equal, this is only 11.1 cents. There are two relatively simple 11-limt meantone mappings. This one, consistent with 31- and 43-equal: 1 0 2 -1 4 -4 7 -10 11 -18 and this one 1 0 2 -1 4 -4 7 -10 -2 13 consistent with 31 and 50. The first one optimizes with a fourth of 503.3 cents and a worst 11-limit error of 11.0 cents. That's 0.2437-comma meantone, you can find rational approximations I'm sure. The other one optimizes at 502.4 cents (exactly 1/4-comma by the looks of it) and a worst error of 10.8 cents. My approximation graphs are here: Meantone temperaments * The red line shows 3, so remember 9 is twice as far out. Why aren't ratios of 9 included in your applet? I can see that 3/13-comma meantone would be close to the minimax for the simpler mapping if you ignore 9:8, 10:9 and 9:7. Graham
Message: 6613 Date: Wed, 12 Mar 2003 09:23:03 Subject: Rothenberg giveaway From: Carl Lumma All; I've got a spare copy of Rothenberg's three seminal, and very mathy papers on scale theory. The first person to write off-list with a valid snail address and legible sentence saying why you're interested, gets it. -Carl
Message: 6620 Date: Sat, 22 Mar 2003 18:43:19 Subject: Re: More on MOS/temperaments From: Carl Lumma Hmm, something's amiss. Anybody else get this list sent to them by e-mail? I got msg. 6056 but not 6057-6067. Just got 6068 & 9. >we already started on this in a series of posts, in particular we >were looking at scales where the major tetrad and the minor tetrad >arise from the same pattern of scale steps. ! Just when I was thinking, "when would I ever be interested in this incomplete-chain extra-intervals stuff"... Good looking out, Paul! More later. -C.
Message: 6621 Date: Sat, 22 Mar 2003 22:24:22 Subject: Re: this T[n] business From: Carl Lumma >>>So it seems my assertion is wrong; simple ratios don't tend >>>to be bigger. >> >>that's the great thing about tenney complexity (as opposed to >>farey, mann, etc.)! > >Ah, you've said that before, I think! I've just verified this for the Farey series. -Carl
Message: 6622 Date: Sat, 22 Mar 2003 19:12:26 Subject: Re: More on MOS/temperaments From: Carl Lumma >>And recall, this is only one comma! > >i'm not sure what you mean to imply by that -- and of course this >chroma is only defined plus or minus any arbitrary number of >commatic unison vectors. Maybe Gene's alluding to the planar and higher cases. -Carl
Message: 6623 Date: Sat, 22 Mar 2003 22:27:54 Subject: T[n] where n is small From: Carl Lumma Gene, What about turning this on scales, n < 11? I don't know how many lines of maple you do this with, but if they're few you can post them here and I can either translate to Scheme or run them in maple myself. -Carl
Message: 6624 Date: Sat, 22 Mar 2003 19:14:57 Subject: Re: 12 notes with 36/35 a chromatic comma From: Carl Lumma >Below I give the name, the wedgie, a TM reduced scale, major and >minor tetrads going around this scale. Bingo! -C.
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