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Message: 11350 Date: Fri, 09 Jul 2004 23:36:19 Subject: Re: 43 7-limit planar temperaments From: Paul Erlich That's not quite what I was asking for, but thanks! That's awesome. What I was asking for was the simplest possible criterion for determining that it's *this* set of generators and not any of the other equivalent sets. *That's* what I feel I owe the readers, even more than a method for the mathematical ones to be able to do it themselves. Of course, I could just give the readers the results without explaining the criterion, but that's a last resort.
Message: 11351 Date: Fri, 09 Jul 2004 00:01:14 Subject: Re: Joining Post From: Carl Lumma >some of you know me from tuning list and Make Micro Music. >Just to say, I'll be listening on this list for while. Hiya Mark! For some resaon, I thought you were already onboard. Anyway, welcome (or welcome back)! -Carl
Message: 11354 Date: Sat, 10 Jul 2004 00:50:30 Subject: Re: 43 7-limit planar temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: > > > What I was asking for was the simplest possible criterion for > > determining that it's *this* set of generators and not any of the > > other equivalent sets. *That's* what I feel I owe the readers, even > > more than a method for the mathematical ones to be able to do it > > themselves. > > Don't the rules I give do that job? They force the result. Yes, but it requires a lot more math than I've introduced in the paper. If you were able to implement a criterion like the one I sketched out, then I could provide a complete explanation for the reader. But don't worry if it's not easy . . . I'll just use what you did, with your rules in a footnote.
Message: 11359 Date: Sun, 11 Jul 2004 22:08:23 Subject: Vals (was: Please stop the jargon explosion From: Graham Breed Moved to tuning-math because it belongs here. Gene Ward Smith wrote: > A {2,3,7}-val would be fine, so long as you called it that; otherwise > you would have no way of knowing <118 187 331| means the mapping > applies to 2, 3 and 7 except by guessing. If "val", without > qualification, is what you say then you know what the prime limit is > by counting. What about inharmonic timbres? If the "prime intervals" were empirically determined, can we still talk about vals? All your definitions require rationals. I work with prime intervals whose magnitudes happen to be the logarithms of prime numbers by default. Graham
Message: 11365 Date: Tue, 13 Jul 2004 02:57:10 Subject: Re: 50 From: monz hi Gene, have you ever seen this? http://tonalsoft.com/monzo/woolhouse/essay.htm#temp * Woolhouse and Paul Erlich both (independently, about 160 years apart) discovered that 7/26-comma meantone is an optimal meantone by one type of measure. Woolhouse then advocates 50-et as a very good approximation to that tuning, and then 19-et as a more practical (but not as close) alternative. good luck trying to find Woolhouse's book itself if you want it. the only copy i've ever discovered is at the U. of Pennsylvania library in Philadelphia. -monz --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > Here are the TM bases for 50-et up to the 19-limit. It > might be noted that in terms of the 50-equal version of > meantone, 20/13 maps to -11, 11/8 to -13, and 16/13 to -15, > leading to the 12, 14, and 16 note wolves of the same size. > > > 5-limit > [81/80, 1207959552/1220703125] > > 7-limit > [16807/16384, 81/80, 126/125] > > 11-limit > [81/80, 245/242, 126/125, 385/384] > > 13-limit > [81/80, 245/242, 105/104, 126/125, 144/143] > > 17-limit > [81/80, 105/104, 126/125, 144/143, 170/169, 221/220] > > 19-limit > [81/80, 105/104, 126/125, 133/132, 144/143, 153/152, 170/169]
Message: 11369 Date: Tue, 13 Jul 2004 06:05:33 Subject: Re: 50 From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > but you quote > Woolhouse as saying: > > [Woolhouse 1835, p 46:] > > This system is precisely the same as that which Dr. Smith, in his > Treatise on harmonics [Smith 1759], calls the scale of equal harmony. > It is decidedly the most perfect of any systems in which the tones are > all alike. > > Is Smith's tuning 50-equal? It's close, but it's much closer to 5/18-comma meantone than to 50- equal. Search the tuning list for more info ;) Also see Jorgenson.
Message: 11370 Date: Tue, 13 Jul 2004 06:10:29 Subject: Re: A 14-note modmos of meantone From: Paul Erlich Is this distinct from Injera in some way? The two scales I originally proposed for Injera both had 14 notes: the DE one and the omnitetrachordal variant. They're both "double diatonic". --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > This is interesting as one way to construct these, though it's > cheating in a way because from another point of view it is mos, or at > least ce. > > If you take my comma list for 50 and dispense with one of the TM basis > commas, one of the temperaments you get (above the 7-limit) is 12&50, > which is actually pretty good if you want higher limit consonances. > The 50-et generators are 1/2 and 2/25, and if you take the 14-note > (MOS? DE?) you get, when the result is translated into meantone, > > -24, -23, -22, -3, -2, -1, 0, 1, 2, 3, 22, 23, 24, 25 > > This is a 14-note modmos; it has two 50-et diatonic scales hence the name. > > ! bidiatonic.scl > 14 note modmos of meantone, mos of 12&50 > 14 > ! > 96.000000 > 192.000000 > 288.000000 > 312.000000 > 408.000000 > 504.000000 > 600.000000 > 696.000000 > 792.000000 > 888.000000 > 912.000000 > 1008.000000 > 1104.000000 > 1200.000000
Message: 11372 Date: Tue, 13 Jul 2004 09:31:02 Subject: Re: monz back to math school From: Graham Breed jjensen142000 wrote: > Trigonometry and calculus are really irrelevant to linear algebra, > and Grassmann algebra is (i think) abstract algebra (groups, rings, > homomorphisms, etc) applied to linear algebra. In other words, > it is *much* harder, as in you get a bachelors degree in math and > then you take it in graduate school. Of course, you wouldn't need > the full force of it to follow most of the tuning-math discussions. Trig and calculus are both handy to know, and do come up in tuning-math theory. So if they're pre-requisites, why not give them a try? It depends on how advanced the courses get. Easy trig and calculus are all you need. Grassman algebra is based on wedge products. It's not that difficult. But it isn't that well known, so they'll only expect hard core mathematicians to learn it. Get basic algebra, and you should be able to learn it from the online materials, and questions here. Also, you can avoid a lot of linear algebra if you know Grassman algebra. So choose your poison ... Graham
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