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Message: 8853 - Contents - Hide Contents Date: Tue, 23 Dec 2003 04:24:13 Subject: Re: Chord mapping From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> Just a note, chord lookup table with common-tone matching is the > basis of my adaptive tuning algorithm... > > MIDI-based adaptive tuning by common-tone matc... * [with cont.] (Wayb.)So when do you code it?

Message: 8854 - Contents - Hide Contents Date: Tue, 23 Dec 2003 22:38:33 Subject: plot of 5-limit commas From: Paul Erlich I plotted all 5-limit commas for which tenney complexity [TC] < 30 and tenney-based heuristic error [(n-d)/TC/log(TC)] < 0.001 and the comma is not a higher power of some other comma Yahoo groups: /tuning-math/files/Paul/waterfal... * [with cont.] who can explain the "woof" and "warp", or curved 'grid', apparent in this plot? here is the plot with the ratios of the commas written in: Yahoo groups: /tuning-math/files/Paul/waterfal... * [with cont.] the size of the font used for each comma was 16^(1-epimericity) where the epimerity is tenney based and equal to log(n-d)/log(n*d). one can easily imagine the epimericity 'contours' running across this plot, though i wasn't able to show them as easily as i thought i would . . . anyway, i hope herman miller, at least, takes a look at this!

Message: 8856 - Contents - Hide Contents Date: Tue, 23 Dec 2003 23:39:44 Subject: Re: epimorphism From: Manuel Op de Coul>Manuel, you are wrong. This is indeed a torsional block. The four >determinants are 20, 32, 46, and 56 -- obviously these are all >multiples of 2, so we have torsion!Drag, you're right. Why is it that when you know there's a bug in the code you can spot it immediately, when otherwise it remains unnoticed. Manuel

Message: 8857 - Contents - Hide Contents Date: Tue, 23 Dec 2003 22:42:14 Subject: Re: epimorphism From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:>>> Manuel, you are wrong. This is indeed a torsional block. The four >> determinants are 20, 32, 46, and 56 -- obviously these are all >> multiples of 2, so we have torsion! >> Drag, you're right. Why is it that when you know there's a bug in > the code you can spot it immediately, when otherwise it remains unnoticed. > > Manuelso now can you find a *real* counterexample?

Message: 8858 - Contents - Hide Contents Date: Tue, 23 Dec 2003 10:21:32 Subject: Magical mystery meanpop wreckage From: Gene Ward Smith If we take the duodene, which is the genus 3^i 5^j where i runs from -1 to 2 and j from -1 to 1, we get something which mapped by meantone gives us i+4j, which runs from -1-4=-5 to 2+4=6, the meantone 12-note MOS. If we take instead the major/minor thirds parallogram, which is the same as the duodene but with 5/3 in place of 3, we do not get a MOS. Instead, we get 3i+4j, which gives us the values -7,-4,-3,-1,0,1,2,3,4,6,7,10. In septimal meantone, this gives us the magic chords of the augmented triad, based on (5/4)^2(9/7)/2 =225/224, and the diminished seventh, based on (6/5)^3(7/6)/2 = 126/125. We don't need 81/80 to find this temperament; the existence of the above two magic chords determines it. Beyond the 7-limit, septimal meantone divides into what I call "meantone" and "meanpop", depending on how 11 maps, with the dividing line being 31-et for which the two are the same. Meanpop territory is where the fifths are just a tad flatter, as for example 50-et or Wilson meantone. Further magic awaits us in meanpop land. We have (9/7)/(6/5)/(20/13) = 351/350 (the ratwolf triad) and (5/4)(7/6)/(16/11) = 385/384 (no name known to me.) If we ask for all four of these magic chords at once, we get 13-limit meanpop, for which 50-et is a good choice. Consider what happens to the "thirds" scale discussed above when the temperament is meanpop. The 5-limit JI scale is ! thirds.scl ! Major and minor thirds paralleogram 12 ! 25/24 10/9 6/5 5/4 4/3 25/18 3/2 8/5 5/3 125/72 48/25 2 If we take the 50-et version of this, we get ! wreckpop.scl ! "Wreckmeister" 13-limit meanpop (50 et) tempered thirds.scl 12 ! 72.000000 192.000000 312.000000 384.000000 504.000000 576.000000 696.000000 816.000000 888.000000 960.000000 1128.000000 1200.000000 The 0-4-7 triads for this goes [432, 312, 744] [384, 312, 696] [384, 264, 648] [432, 312, 744] [384, 312, 696] [384, 312, 696] [432, 264, 696] [384, 312, 696] [384, 312, 696] [432, 312, 744] [384, 264, 648] [384, 312, 696] We have six meantone major triads, [384, 312, 696], but we also have three ratwolf triads [432, 312, 744], and two as yet unnamed magical [384, 264, 648] triads. Finally, we have a nice, old-fashioned supermajor triad, [432, 264, 696]. Each scale step has a triad, magic or mundane, to go with it (plus the minor versions of course.) This is, of course, related to my Wreckmeister A scale, which however I only tempered via 126/125 planar. Looking at the matter from the point of view of meanpop seems like a much better way to go about Wreckmeistering. ________________________________________________________________________ ________________________________________________________________________ To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx ------------------------------------------------------------------------ Yahoo! Groups Links To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] 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: 8860 - Contents - Hide Contents Date: Wed, 24 Dec 2003 21:21:32 Subject: Re: epimorphism From: Manuel Op de Coul Paul wrote:>so now can you find a *real* counterexample?I guess not... I've uploaded a new release with the torsion and epimorphism bugs fixed, updated Sagittal symbols, improved periodicity block dialog, and smaller improvements. http://www.huygens-fokker.org/software/Scala_Setup.exe - Type Ok * [with cont.] (Wayb.) Manuel

Message: 8861 - Contents - Hide Contents Date: Wed, 24 Dec 2003 21:19:23 Subject: Re: plot of 5-limit commas From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> who can explain the "woof" and "warp", or curved 'grid', apparent > in >> this plot? >> I'd start by taking ratios of commas along lines, and see if there is > a single high-quality comma behind the line.No; the lines are essentially single numerators in one direction, and single denominators in the other.

Message: 8862 - Contents - Hide Contents Date: Wed, 24 Dec 2003 14:42:57 Subject: definitions From: Carl Lumma Gene, If you could collect all your definitions, "scale", "tuning", "notation", etc. and put them on xenharmony.org or Wikipedia I would be enthralled. -Carl ________________________________________________________________________ ________________________________________________________________________ To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx ------------------------------------------------------------------------ Yahoo! Groups Links To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] 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: 8863 - Contents - Hide Contents Date: Wed, 24 Dec 2003 08:52:13 Subject: Re: plot of 5-limit commas From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> who can explain the "woof" and "warp", or curved 'grid', apparent in > this plot?I'd start by taking ratios of commas along lines, and see if there is a single high-quality comma behind the line. ________________________________________________________________________ ________________________________________________________________________ To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx ------------------------------------------------------------------------ Yahoo! Groups Links To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] 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: 8864 - Contents - Hide Contents Date: Thu, 25 Dec 2003 23:59:12 Subject: Re: Transitive groups of degree 12 and low order containing a 12-cycle From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> Gene, > > If it's not too much trouble, could you run S(4) X C(4) and also > S(4) X S(3)? And give the generators and polynomials, if its not too > much troubleS(4)XC(3) order 72 Generators a := [[12, 4, 8], [1, 5, 9], [2, 6, 10], [3, 7, 11]] e := [[12, 3, 6, 9], [1, 4, 7, 10], [2, 5, 8, 11]] B := [[1, 10], [2, 5], [6, 9]] Polynomial x^12+x^11+3*x^10+8*x^9+14*x^8+19*x^7+24*x^6+19*x^5+14*x^4+8*x^3+3*x^2+x+1 S(4)XS(3) order 144 Generators a := [[12, 4, 8], [1, 5, 9], [2, 6, 10], [3, 7, 11]] b := [[1, 5], [2, 10], [4, 8], [7, 11]] e := [[12, 3, 6, 9], [1, 4, 7, 10], [2, 5, 8, 11]] B := [[1, 10], [2, 5], [6, 9]] Polynomial x^12+x^11+3*x^10+6*x^9+11*x^8+13*x^7+17*x^6+13*x^5+11*x^4+6*x^3+3*x^2+x+1 Merry Christmas!

Message: 8866 - Contents - Hide Contents Date: Fri, 26 Dec 2003 00:44:15 Subject: Re: Attention Gene From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Yahoo groups: /tuning-math/message/8269 * [with cont.] > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" > <gwsmith@s...> >> wrote:>>> After fixing my program, here is what I am getting for Prooijen > and>>> geometric 11-limit reductions: >>> >>> ! red72_11pro.scl >>> Prooijen 11-limit reduced scale >>> 72 >>> ! >>> 81/80 >>> 64/63 >>>> Gene -- why isn't this 45/44?I guess I'm using the wrong definition of Prooijen complexity.

Message: 8868 - Contents - Hide Contents Date: Fri, 26 Dec 2003 06:16:03 Subject: Re: Attention Gene From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >> Yahoo groups: /tuning-math/message/8269 * [with cont.] >>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> >> wrote:>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" >>>> wrote: >>>> After fixing my program, here is what I am getting for Prooijen >> and>>>> geometric 11-limit reductions: >>>> >>>> ! red72_11pro.scl >>>> Prooijen 11-limit reduced scale >>>> 72 >>>> ! >>>> 81/80 >>>> 64/63 >>>>>> Gene -- why isn't this 45/44? >> I guess I'm using the wrong definition of Prooijen complexity.It's called 'expressibility', and it's simply (the log of) the "ratio of" (or, imprecisely speaking, "odd-limit") measure of the ratio. Definitions of tuning terms: ratio of, (c) 200... * [with cont.] (Wayb.) Searching Small Intervals * [with cont.] (Wayb.) Since log(45)<log(63), you must indeed have the wrong definition.

Message: 8871 - Contents - Hide Contents Date: Fri, 26 Dec 2003 06:23:06 Subject: Re: Attention Gene From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Since log(45)<log(63), you must indeed have the wrong definition.log(11*45)>log(63), which is what I think I used.

Message: 8872 - Contents - Hide Contents Date: Fri, 26 Dec 2003 08:32:43 Subject: Re: Attention Gene From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> Since log(45)<log(63), you must indeed have the wrong definition. >> log(11*45)>log(63), which is what I think I used.I thought I had already straightened you out on that particular misunderstanding. ________________________________________________________________________ ________________________________________________________________________ To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx ------------------------------------------------------------------------ Yahoo! Groups Links To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] 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: 8873 - Contents - Hide Contents Date: Sat, 27 Dec 2003 20:52:20 Subject: Re: 5-limit, 12-note Fokker blocks From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> The Duodene is well-known as a 5-limit, 12-note Fokker block; > I > decided to check on whether Scala knows of four other examples, and > got this.Scala has many others, including Marpurg's Monochord #1, Ramos/Ramis, etc . . . see the Gentle Introduction to Fokker Periodicity Blocks.

Message: 8874 - Contents - Hide Contents Date: Sat, 27 Dec 2003 00:21:53 Subject: 5-limit, 12-note Fokker blocks From: Gene Ward Smith The Duodene is well-known as a 5-limit, 12-note Fokker block; I decided to check on whether Scala knows of four other examples, and got this. (The scales can be obtained by letting i range from 1 to 12.) (16/15)^i (2048/2025)^(-5i/12) (81/80)^(-2i/12) No Scala name (16/15)^i (81/80)^(8i/12) (648/625)^(-5i/12) No Scala name (16/15)^i (128/125)^(-8i/12) (648/625)^(3i/12) No Scala name, a mode of what I've called the thirds scale (16/15)^i (2048/2025)^(-3i/12) (128/125)^(-2i/12) Scala identifies it as lumma5r.scl ! lumma5r.scl ! Carl Lumma's scale, 5-limit just version, TL 19-2- 99 12 ! 16/15 9/8 75/64 5/4 4/3 45/32 3/2 8/5 5/3 225/128 15/8 2/1

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