I've finally written a PDF about the badness measure I thought up the other year. This is part of a series of articles that was supposed to be finished by...
Very good to see you carrying on. Congratulations! I love the first-person style and professional typesetting. Have you considered submitting one of these...
... http://tonalsoft.com/enc/e/equal-temperament.aspx has a list of 5-limit temperaments. 53&270 is vulture. I've used "vulture" in the 7-limit for a 53&58...
... Bill Sethares suggested the complete search one might be publishable. But I haven't done any work towards it -- it's extra hassle. The priorities getting...
... Thanks! There's a kind of vulture in the 19-limit as well. ... I put it in the 11-limit. It also figures in the 13 and 17 limits. (The computer sorts all...
... I don't see the optimization there. TOP-max should be (max(E)-min(E))/(2+max(E)+min(E)) in algebraic form. ... For the optimized calculation, it's...
... In the recent doc, I saw "(Smith sevlat)" and "Smith (sevlat)" I think, though it wasn't sevlat both times. Neither one was linked, while most of the...
... I think you're talking about the usual trick for catching errors of the interior intervals, e.g. 5:3. But I thought it wasn't needed for TOP, because in...
... One's for referring to the article directly, like "Smith said this". The other where you add the citation for something else, like "lattices are really...
... It isn't needed for TOP because the optimization takes care of it. That formula takes you directly to the optimal error without needing the optimal scale...
... Understood. ... The above is an argument that, e.g. in the 5-limit, the RMS of the Tenney-weighted errors of 3 & 5 would be *greater than* or equal to the...
... As shown in the code, I simply take the max weighted error among the primes to generate the below table. Not max-min/2. ... I get 12.49595001136964 for...
... I was finding the best mapping for TOP-RMS. Here's the table using the best TOP-max mappings: 2: 33.0 77.3 77.3 77.3 77.6 77.6 3: 30.2 30.2 39.8...
... I get 2 a b^2 c d^2 <= a^2 d^4 + 2 a^2 b d^3 + b^4 c^2 + 2 b^3 c^2 d. That may still be wrong but I don't see how you can get a^2 b^2 c^2. ... Why? ... I...
... Any Tenney limit, or the intersection of any Tenney and prime limits, is a particular set of prime weights for an RMS calculation. So the optimal RMS over...
... Yes, that's what I remember from your paper. Except the other day when I was looking at it again, the tables seemed to show Farey limits intsead. ... ...
... There are tables for both. ... If you're looking for temperament classes you naturally compare them with their optimal tunings. The computation is the...
... That it's easier to work with the optimizations algebraically. I think the badness in the title is a very interesting function and should be studied as...
... My numbers agree, though this no longer shows all the successive improvements in 17-limit damage. Here's the complete list: (2 (33.0 77.3 77.3 77.3 77.6...
... I was playing with logflat badness the other night, generating this list, where I look at ETs up to 1000 and down to the number of primes (e.g. 4-ET is...
... Here are the best 3 ETs up to 100 ET for several different limits using this scheme. These are (val comma badness), where val is the non-torsional val with...
... Oh, and I'm assuming for rank 1 temperaments, removing torsion is as simple as ignoring vals with GCD > 1. Somebody please correct me if that's not right....
Have been studying Myhill's Property and also Rothenberg Scales. Interesting that the diatonic and the pentatonic scales have Myhill's property. So I thought I...
I have analyzed the 5 Rothenberg Proprietary Scales. Diatonic - CDEFGAB Mel Minor - CDEbFGAB Harm Major - CDEFGAbB Harm Minor - CDEbFGBbB Locrian Major...
... produces ... Now, of course the combinations for white and black keys are based on 7 * 1, 21 * 5, 35 * 10, 35 * 10, 21 * 5, 7 * 1 which is simple ...
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