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Message: 2800 - Contents - Hide Contents Date: Fri, 28 Dec 2001 23:28:07 Subject: Re: Superparticular 5-limit scales From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> The three smallest 5-limit superparticulars are 81/80, 25/24, and16/15. Putting these into the form of a matrix and inverting gives us> [h3 h5 h7], and hence (81/80)^3 (25/24)^5 (16/15)^7 = 2. We canarrange these 15 scale steps in a number of ways given by the multinomial coefficient 15!/(3! 5! 7!) = 360360, which rotations and inversions would reduce further.> > All of these scales are epimorphic, with defining val h15 =h3+h5+h7, so singling out the interesting ones means putting on additional contraints; convexity and connectedness suggest themselves, of course.> > Anyone care to take a shot at it?Sure -- let's adopt convexity and connectedness (via consonances).
Message: 2801 - Contents - Hide Contents Date: Fri, 28 Dec 2001 03:14:44 Subject: Re: My top 5--for Paul From: clumma I wrote...>The listener will sort out notes one way or the other, and >far fewer of them 171 or 612. Dave's point, I think, is >that he or she wouldn't be able to tell the difference.For ets, it probably craps out somewhere around 282-tET, where the 19-limit is consistently represented to within a cent rms, and no interval has more than 2 cents absolute error. -Carl
Message: 2802 - Contents - Hide Contents Date: Fri, 28 Dec 2001 23:40:21 Subject: Re: Superparticular 5-limit scales From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Sure -- let's adopt convexity and connectedness (via consonances).It follows from Pick's theorem that the scale will be convex iff the area of the convex hull is 15, so if we had a quick way of finding the convex hull we could do that part of it. I'm open for other suggestions and ideas.
Message: 2803 - Contents - Hide Contents Date: Fri, 28 Dec 2001 08:26:41 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: dkeenanuqnetau I've modified my badness measure in ways that I hope take into account the fact (assuming it is one) that pelog is some kind of 5-limit temperament. I give the following possible ranking of 5-limit temperaments having a whole octave period. Gen Gens in RMS err Name (cents) 3 5 (cents) ------------------------------------ 503.8 [-1 -4] 4.2 meantone 498.3 [-1 8] 0.3 schismic 317.1 [ 6 5] 1.0 kleismic 380.0 [ 5 1] 4.6 163.0 [-3 -5] 8.0 387.8 [ 8 1] 1.1 271.6 [ 7 -3] 0.8 orwell 443.0 [ 7 9] 1.2 176.3 [ 4 9] 2.5 339.5 [-5 -13] 0.4 348.1 [ 2 8] 4.2 251.9 [-2 -8] 4.2 351.0 [ 2 1] 28.9 126.2 [-4 3] 6.0 522.9 [-1 3] 18.1 pelog?
Message: 2804 - Contents - Hide Contents Date: Fri, 28 Dec 2001 00:40:40 Subject: more 2-D periodicity-block math (was: new 1/6-comma meantone lattice) From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx>; <tuning@xxxxxxxxxxx.xxx> > Sent: Thursday, December 27, 2001 4:18 AM > Subject: [tuning-math] new 1/6-comma meantone lattice > > > I've added a new lattice to my "Lattice Diagrams comparing > rational implications of various meantone chains" webpage: > > Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) > > > It's about 2/3 of the way down the page: a new lattice > showing a definition of 1/6-comma meantone within a > 55-tone periodicity-block... just under the old 1/6-comma > lattice, below this text: > >>>> And here is a more accurate lattice of the above, >>> showing a closed 55-tone 1/6-comma meantone chain and >>> its implied pitches, all enclosed within a complete >>> periodicity-block defined by the two unison-vectors >>> 81:80 = [-4 4 -1] (the syntonic comma, the shorter >>> boundary extending from south-west to north-east on >>> this diagram) and [-51 19 9] (the long nearly vertical >>> boundary), portrayed here as the white area. >>> >>> For the bounding corners of the periodicity-block, I >>> arbitrarily chose the lattice coordinates [-7.5 -5] >>> for the north-west corner, [-11.5 -4] for north-east, >>> [11.5 4] for south-west, and [7.5 5] for south-east. >>> This produces a 55-tone system centered on n^0.Actually, these choices turn out not to be arbitrary. I found them intuitively, but now I've figured out how to formalize it. There's a very simple formula which finds the corners of a 2-dimensional periodicity-block from the unison-vectors. For matrix M: M = [a b] [c d] Each of the four corners NW, NE, SW, and SE are: NW = [ ( a-c)/2 , ( b-d)/2 ] NE = [ (-a-c)/2 , (-b-d)/2 ] SW = [ ( a+c)/2 , ( b+d)/2 ] SE = [ (-a+c)/2 , (-b+d)/2 ] (I invite canditates for better terms.) In simple generalized terms: [ (+/-a +/-c) (+/-b +/-d) ] --------------------------- 2 Plugging our example in, we get: M = [ 4 -1] [19 9] NW = [ -7.5 -5] NE = [-11.5 -4] SW = [ 11.5 4] SE = [ 7.5 5] Then, to get the corners of the tiling periodicity-blocks which fill the rest of the lattice, simply add or subtract either of the unison-vectors separately to any of these coordinates, and iterate the process as long as necessary to find as many tiles as desired. One of the unison-vectors will tile the plane along either of two parallel sides of the parallelogram, and the other unison-vector will tile the plane along either of the other two parallel sides. Has this ever been explained explicitly like this before? If so, please give references, tuning-math list URLs, etc. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2805 - Contents - Hide Contents Date: Fri, 28 Dec 2001 00:52:32 Subject: 2-D periodicity-block math From: monz (Please indulge my cross-posting. Thanks.) This is an expansion of a post I sent to tuning-math.> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx>; <tuning@xxxxxxxxxxx.xxx> > Sent: Thursday, December 27, 2001 4:18 AM > Subject: [tuning-math] new 1/6-comma meantone lattice > > > I've added a new lattice to my "Lattice Diagrams comparing > rational implications of various meantone chains" webpage: > > Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) > > > It's about 2/3 of the way down the page: a new lattice > showing a definition of 1/6-comma meantone within a > 55-tone periodicity-block... just under the old 1/6-comma > lattice, below this text: > >>> And here is a more accurate lattice of the above, >> showing a closed 55-tone 1/6-comma meantone chain and >> its implied pitches, all enclosed within a complete >> periodicity-block defined by the two unison-vectors >> 81:80 = [-4 4 -1] (the syntonic comma, the shorter >> boundary extending from south-west to north-east on >> this diagram) and [-51 19 9] (the long nearly vertical >> boundary), portrayed here as the white area. >> >> For the bounding corners of the periodicity-block, I >> arbitrarily chose the lattice coordinates [-7.5 -5] >> for the north-west corner, [-11.5 -4] for north-east, >> [11.5 4] for south-west, and [7.5 5] for south-east. >> This produces a 55-tone system centered on n^0. >> >> The grey area represents the part of the JI lattice >> outside the defined periodicity-block (and thus, with >> each of those pitch-classes in its own periodicity-block), >> and the lattice should be imagined as extending infinitely >> in all four directions. The other periodicity-blocks, >> all identical to this one, can be tiled against it to >> cover the entire space.Actually, these choices turn out not to be arbitrary. I found them intuitively, but now I've figured out how to formalize it. There's a very simple formula which finds the corners of a 2-dimensional periodicity-block from the unison-vectors. For matrix M: M = [a b] [c d] Each of the four corners NW, NE, SW, and SE are: NW = [ ( a-c)/2 , ( b-d)/2 ] NE = [ (-a-c)/2 , (-b-d)/2 ] SW = [ ( a+c)/2 , ( b+d)/2 ] SE = [ (-a+c)/2 , (-b+d)/2 ] (I invite canditates for better terms.) In simple generalized terms: [ (+/-a +/-c) (+/-b +/-d) ] --------------------------- 2 Plugging our example in, we get: M = [ 4 -1] [19 9] NW = [ -7.5 -5] NE = [-11.5 -4] SW = [ 11.5 4] SE = [ 7.5 5] Then, to get the corners of the tiling periodicity-blocks which fill the rest of the lattice, simply add or subtract either of the unison-vectors separately to any of these coordinates, and iterate the process as long as necessary to find as many tiles as desired. One of the unison-vectors will tile the plane along either of two parallel sides of the parallelogram, and the other unison-vector will tile the plane along either of the other two parallel sides. Here's something I'd like to be able to do to enhance this: Enable the user to find unison-vectors by: - click-and-drag on a pitch-height graph to specify an approximate size for a unison-vector - click on various lattice-points to create vectors directly - use mouse-click and the Ctrl or Shift button together, to drag a periodicity-block shape across the lattice JustMusic would then calculate appropriate candidates for the unison-vectors, displaying parameters of them in various ways. Then, the user could also explore the graphing of various linear or planar temperaments on the lattice, *within* the rational periodicity-block already defined. Of course, I'd also like to be able to work the other way around: specify parameters of a temperament first, then find the unison-vectors and periodicity-blocks from it. Manuel, at this point, if you'd like to incorporate *any* (or even all) of my JustMusic ideas into Scala, I'd be happy to collaborate. love / peace / harmony ... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2806 - Contents - Hide Contents Date: Fri, 28 Dec 2001 09:46:47 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> I've modified my badness measure in ways that I hope take into account > the fact (assuming it is one) that pelog is some kind of 5-limit > temperament. I give the following possible ranking of 5-limit > temperaments having a whole octave period.How are you ranking them, and how are you finding them? Here is your table, with my annotations:> Gen Gens in RMS err Name > (cents) 3 5 (cents) > ------------------------------------ > 503.8 [-1 -4] 4.2 meantone > 498.3 [-1 8] 0.3 schismic > 317.1 [ 6 5] 1.0 kleismic > 380.0 [ 5 1] 4.6 magic > 163.0 [-3 -5] 8.0 maxidiesic > 387.8 [ 8 1] 1.1 wuerschmidt > 271.6 [ 7 -3] 0.8 orwell > 443.0 [ 7 9] 1.2 minidiesic > 176.3 [ 4 9] 2.5 not on my list; badness=649 > 339.5 [-5 -13] 0.4 AMT > 348.1 [ 2 8] 4.2 meantone > 251.9 [-2 -8] 4.2 meantone > 351.0 [ 2 1] 28.9 neutral thirds > 126.2 [-4 3] 6.0 not on my list; badness=728 > 522.9 [-1 3] 18.1 pelogic
Message: 2807 - Contents - Hide Contents Date: Fri, 28 Dec 2001 03:24:55 Subject: observations on 12:7 (was: [tuning] u/otonality and major/minor) From: monz> From: jpehrson2 <jpehrson@xxx.xxx> > To: <tuning@xxxxxxxxxxx.xxx> > Sent: Thursday, December 27, 2001 11:14 AM > Subject: [tuning] Re: u/otonality and major/minor > > > --- In tuning@y..., "monz" <joemonz@y...> wrote: > > Yahoo groups: /tuning/message/31814 * [with cont.] > >>>> The dichotomy becomes even more clear when we progress from >> triads to tetrads. Traditional theory adds a "7th", continuing >> to stack the "minor chord" by ascending "3rds" the same as with >> the "major chord" -- so here the it's a "minor 7th" where with >> the "major chord" it's a "major 7th". But the dualistic theory >> continues the construction *downward* by adding a fourth >> chord-identity *below* the utonal triad given above, thus: >> >> C Ab F D >> 1/4 1/5 1/6 1/7 >> >> >> ****So, essentially, that would turn into a *sixth* rather than a > *seventh* yes??Yup. If you invert the chord so that the 1/7 is at the top, it forms an interval ~933.1290944 cents above the "root" 1/6. In my JustMusic harmonic analysis notation, I'd write that: 1 n^0 -- = ------- 7 2^0 * 7 8 2^3 * 1 10 2^1 * 5 12 2^2 * 3 Rewritten as ordinary ratios, from the top note down: 16/7 = 2/1 * 8/7 16/8 = 2/1 16/10 = 8/5 16/12 = 4/3 In other words, it's almost the same amount wider than the 12-EDO "major 6th", as the 7:4 is narrower than the 12-EDO "minor 7th". In fact, the difference between these two differences is exactly the same as the amount of tempering of the 3:2 in 12-EDO, a tiny unit of interval measurement known as a "grad". Definitions of tuning terms: grad, (c) 2001 by... * [with cont.] (Wayb.) I found this coincidence interesting, so I pursued it further for this post: Where "a" = excess of 12:7 "septimal major 6th" over 12-EDO "b" = deficit of 7:4 "septimal minor 7th" under 12-EDO (3/2) / ( 2^(7/12) ) = a - b [Note: "septimal minor 7th" is more frequently called "harmonic 7th".] PROOF: (3/2) / ( 2^(7/12) ) = [-12/12 1] 3:2 ratio = Pythagorean "perfect 5th" - [ 7/12 0] 12-EDO "perfect 5th" ------------- [ 19/12 1] = ~1.955000865 cents = 1 grad a = ( (16/7) / (4/3) ) / ( 2^(9/12) ) (16/7) / (4/3) = [ 4 0 0 -1] 8:7 ratio + "8ve" = "septimal 9th" - [ 2 -1 0 0] 4:3 ratio = Pythagorean "perfect 4th" --------------- [ 2 1 0 -1] = [ 24/12 1 0 -1] 12:7 = "septimal M6th" - [ 9/12 0 0 0] 12-EDO "major 6th" ------------------- [ 15/12 1 0 -1] = [ 5/4 1 0 -1] = ~33.1290944 cents b = ( 2^(10/12) ) / (7/4) = [ 10/12 0 0 0] 12-EDO "minor 7th" - [-24/12 0 0 1] 7:4 ratio = "septimal minor 7th" ------------------- [ 34/12 0 0 -1] [ 17/6 0 0 -1] = ~31.17409353 cents a - b = [ 15/12 1 0 -1] a - [ 34/12 0 0 -1] b ------------------- [ 19/12 1 0 0] = ~1.955000865 cents = 1 grad I would love to see what this looks like in elegant algebra. love / peace / harmony ... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2808 - Contents - Hide Contents Date: Fri, 28 Dec 2001 03:41:57 Subject: Re: Paul's lattice math and my diagrams From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, December 27, 2001 1:51 PM > Subject: [tuning-math] Re: Paul's lattice math and my diagrams > >>> Oh, OK Paul, I've got you now.Hope you didn't take that the wrong way... I meant that I understand (I think...)>> My description really is based on the planar representation, >> The wrong planar representation, in my opinion.Even after emphasizing the equivalence of tiled periodicity-blocks? I don't get it!>> while you were talking about the cylindrical representation. >> _Or_ a planar representation, like the ones in _The Forms Of > Tonality_.Yes, well... I no longer have that available for consultation, and am eagerly awaiting a new copy...>> Cool. But even tho it works, there still is something wrong >> with the mathematics in my spreadsheet. I'd appreciate some >> error correction. >> Since I think part 3 of the Gentle Introduction should answer the > mathematics part of your question, I'm not currently inclined to > decipher the meaning of the mathematical method you've come up with. > I'd be happy to work with you on understanding and implementing the > method in part 3 of the GI so that you may do what you're trying to > do.OK... when I have time, I'll have to sit down with my spreadsheet and create a section that works strictly according to your method. If I have trouble then, I'll ask for help.> P.S. How can you include W. A. Mozart under 55-EDO on your Equal > Temperament definition page? I could understand if you wanted to put > Mozart on a meantone page, but 55? Totally unjustified. Come on, > let's not just make things up.Well... his conception was clearly based on the "9 commas per whole-tone, 5 commas per diatonic semitone" idea. So his teaching of intonation definitely implied a *subset* of 55-EDO. I suppose amending my webpage to that effect would be best. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2809 - Contents - Hide Contents Date: Sat, 29 Dec 2001 11:20:34 Subject: Re: non-uniqueness of a^(b/c) type numbers From: monz> From: unidala <JGill99@xxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, December 29, 2001 10:23 AM > Subject: [tuning-math] Re: non-uniqueness of a^(b/c) type numbers > >> [me, monz]>> If b and c in a^(b/c) are always integers, the simple >> calculations needed for tuning math always gives results >> which are also integers, so there's never any error at all. >> JG: Monz, can you give an example of a prime number > taken to a rational power (where the power's numerator and > denominator are integer) which (directly, as numerically > evaluated)equals an integer? I couldn't. Of course, the > rational valued exponent must not be equal (itself) to > an integer value...No, I don't think I can give an example of that either. But does it matter? I'm simply having a hard time understanding how a^(b/c) can possibly equal d^(e/f) EXACTLY. As far as I can see, there are no six integers that will satisfy that equation. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2810 - Contents - Hide Contents Date: Sat, 29 Dec 2001 05:16:15 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote:>>> I've modified my badness measure in ways that I hope take into >>> account the fact (assuming it is one) that pelog is some kind >>> of 5-limit temperament. >>>> Was there evidence for this, or is this just an assumption for >> further exploration? It strikes me as extremely unlikely that >> any Indonesian tuning is a 5-limit temperament. >> Posted this before I saw the bit on six narrow and one wide > fifths. But: > > () The Scala scale archive is not a good source of actual pelogs, > or any other ethnic tunings for that matter.Why not? What are the pelog tunings Dave used? And Dave, what are the means and standard deviations of the two sizes of thirds?
Message: 2811 - Contents - Hide Contents Date: Sat, 29 Dec 2001 18:48:16 Subject: Re: the unison-vectordeterminant relationship From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, December 29, 2001 5:36 PM > Subject: Re: [tuning-math] Re: the unison-vector<-->determinant relationship > > >>> From: genewardsmith <genewardsmith@xxxx.xxx> >> To: <tuning-math@xxxxxxxxxxx.xxx> >> Sent: Saturday, December 29, 2001 4:10 PM >> Subject: [tuning-math] Re: the unison-vector<-->determinant relationship >> >> >> >>>>>> OK, fix what's wrong with this, if anything... >>> >>> One can find an infinity of closer and closer >>> fraction-of-a-comma meantone representations of that >>> 5-limit JI periodicity-block, which would be represented >>> on my lattice here as shiftings of the angle of the vector >>> representing the meantone. But one could never find one >>> that would go straight down the middle of the symmetrical >>> periodicity-block. >>> >>> Now, what does this mean? >>>> I'm not following you--why not spell it out, giving the block in question? > > >> OK. I'd much rather spell it out with 1/6-comma, since > I've been working more with that. > > But I already have a spreadsheet which shows how various > meantones "fit" within a particular periodicity-block. > I posted about it here last week: > > my original post: > Yahoo groups: /tuning-math/message/2102 * [with cont.] > > errata: > Yahoo groups: /tuning-math/message/2103 * [with cont.] > > Paul's criticism: > Yahoo groups: /tuning-math/message/2104 * [with cont.] > > > And guess what? Oops! It looks like the conclusion I > reached there is that the 7/25-comma meantone *does* indeed > go straight down the middle of the periodicity-block! > > Here's my Microsoft Excel spreadsheet latting these meantones: > > Yahoo groups: /tuning-math/files/monz/ * [with cont.] (6%20-14)%20(4%20-1)%20 > PB%20and%207-25cmt.xls > > I made all the graphs the same size and put them in > exactly the same place, so if you don't change anything, > you can click on the worksheet tabs at the bottom and > see how the different meantones lie within the periodicity-block. > > > Hmmm... *if* my lattice *was* wrapped as a cylinder or > torus, would there be any significance to this "middle > of the road" business? > > > -monzI should have specified... for the benefit of those who are not able to see my spreadsheet, it forms a 50-tone periodicity-block from the [3, 5] unison-vectors [6 -14],[-4 1]. Then each spreadsheet lattices a fraction-of-a-comma type meantone within it: 2/7-, 7/25-, 5/18-, and 3/11-comma. The 7/25-comma meantone is lattice right down the middle of the symmetrical periodicity-block. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2812 - Contents - Hide Contents Date: Sat, 29 Dec 2001 19:41:08 Subject: Re: Three and four tone scales From: clumma>These are the 5-limit 3 and 4 tone scales, up to isomorphim by mode >and inversion, with only superparticular scale steps and which are >connected. The connectivity number I give is the edge-connectivity, >which means the number of edges (consonant intervals) which would >need to be removed in order to make it disconnected. It is therefore >a measure of how connected by consonant intervals the scale in >question is.Right. Naturally, we're all looking forward to the 7-limit. Hopefully, there won't be too many superparticulars. -Carl
Message: 2813 - Contents - Hide Contents Date: Sat, 29 Dec 2001 05:24:53 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote:> I did not know that. They certainly get plenty of it. > But can't they minimize roughness and still have plenty of > beating? Roughness is pretty unpleasant. Beating can > be pleasant, though.OK, that may be part of it.>>> You'd have to plug in all the partials. The timbres are too >>> out there to just plug in the fundamentals as we do normally. >>> IOW, I'm not sure harmonic entropy is so significant for this >>> music. >>>> Try an experiment. Get three bells or gongs or whatever, as long >> as they each have a clear pitch (I guess you can use a synth for >> this). >> I don't have a synth that does inharmonic additive sythesis. > Besides, many gamelan instruments don't evoke a clear sense > of pitch to me at all.It's a matter of _how_ clear. Typically, according to Jacky, the 2nd and 3rd partials are about 50 cents from their harmonic series positions. That spells increased entropy (yes, timbres have entropy), but still within the "valley" of a particular pitch.>> Tune them to a Pelog major triad. You don't hear any sense of >> integrity? I sure do. >> What's your setup?A cheesy Ensoniq, or listen to real Gamelan music, or the Blackwood piece. Look, I'm not saying the tuning is _designed_ to approximate the major triad and its intervals, but statistically (pending further analysis) it sure seems to be playing a shaping role.
Message: 2814 - Contents - Hide Contents Date: Sat, 29 Dec 2001 20:50 +0 Subject: Re: non-uniqueness of a^(b/c) type numbers From: graham@xxxxxxxxxx.xx.xx monz wrote:> I'm simply having a hard time understanding how > a^(b/c) can possibly equal d^(e/f) EXACTLY. > > As far as I can see, there are no six integers > that will satisfy that equation.Um, how about 2^(2/3) = 2^(4/6) = 4^(1/3) Graham
Message: 2815 - Contents - Hide Contents Date: Sat, 29 Dec 2001 05:31:46 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> > Sure but ratios of 3 are part of the 5 limit and the ratios of 5 are > supposedly explaining why the ratio of 3 is so bad. It's supposedly to > get good enough ratios of 5 with only 3 and 4 gens (simpler than > meantone).I wouldn't put it past the Indonesians.>>>> Is there really any evidence that pelog is a 5-limit temperament? >>>> I think there's strong evidence it at least relates to the 3- limit, >> and that's the error you objected to. >> Yes but by claiming that the 523 c temperament isn't junk (as a > 5-limit temperament) because it corresponds closely enough to pelog, > you are claiming something more than that.That's not the only reason I was claiming that. Note that I referenced Margo Schulter for example. These 'errors' are well within the acceptable norm under quite a few interesting circumstances -- the only time they really get in the way is with sustained harmonic timbres in the absense of adaptive tuning. Try it!
Message: 2816 - Contents - Hide Contents Date: Sat, 29 Dec 2001 20:50 +0 Subject: Re: the unison-vectordeterminant relationship From: graham@xxxxxxxxxx.xx.xx monz wrote:>> Is there someplace online that explains these "wedge products"? >> Perhaps old tuning-math posts?See if you can find the post I made not long after I worked out the principles. All you need to know is ei^ej = -ej^ei and again:> I think I've answered my own question: > http://mathworld.wolfram.com/WedgeProduct.html * [with cont.] > > But it still looks like Chinese. :(Can you work out Python? It's related to Dutch. The code at <# Temperament finding library -- definitions * [with cont.] (Wayb.)> includes wedge products. If you can get it working in an interpreter, you can also try lots of examples. Graham
Message: 2817 - Contents - Hide Contents Date: Sat, 29 Dec 2001 05:34:43 Subject: Re: Paul's lattice math and my diagrams From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> Hmmm... what about my ears telling meI guess then I would change that reference to: 55: Joe Monzo interpretation of W. A. Mozart
Message: 2818 - Contents - Hide Contents Date: Sat, 29 Dec 2001 12:59:06 Subject: Re: non-uniqueness of a^(b/c) type numbers From: monz> From: <graham@xxxxxxxxxx.xx.xx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, December 29, 2001 12:50 PM > Subject: [tuning-math] Re: non-uniqueness of a^(b/c) type numbers > > > monz wrote: >>> I'm simply having a hard time understanding how >> a^(b/c) can possibly equal d^(e/f) EXACTLY. >> >> As far as I can see, there are no six integers >> that will satisfy that equation. >> Um, how about 2^(2/3) = 2^(4/6) = 4^(1/3)Oops! So did I really mean to write: "no six *coprime* integers that will satisfy that equation"? Help... I'm sinking... -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2819 - Contents - Hide Contents Date: Sat, 29 Dec 2001 06:14:02 Subject: Seven connected scales From: genewardsmith Here are the seven connected scales one gets from (9/8)^3 * (10/9)^2 * (16/15)^2 = 2, when modes and inversions are taken as equivalent: [9/8, 9/8, 10/9, 16/15, 9/8, 10/9, 16/15] [1, 9/8, 81/64, 45/32, 3/2, 27/16, 15/8] connectivity = 1 [9/8, 9/8, 10/9, 16/15, 9/8, 16/15, 10/9] [1, 9/8, 81/64, 45/32, 3/2, 27/16, 9/5] connectivity = 1 [9/8, 9/8, 16/15, 10/9, 9/8, 10/9, 16/15] [1, 9/8, 81/64, 27/20, 3/2, 27/16, 15/8] connectivity = 1 [9/8, 10/9, 9/8, 10/9, 9/8, 16/15, 16/15] [1, 9/8, 5/4, 45/32, 25/16, 225/128, 15/8] connectivity = 1 [9/8, 10/9, 9/8, 10/9, 16/15, 9/8, 16/15] [1, 9/8, 5/4, 45/32, 25/16, 5/3, 15/8] connectivity = 2 [9/8, 10/9, 9/8, 16/15, 9/8, 10/9, 16/15] [1, 9/8, 5/4, 45/32, 3/2, 27/16, 15/8] connectivity = 2 [9/8, 10/9, 10/9, 9/8, 16/15, 9/8, 16/15] [1, 9/8, 5/4, 25/18, 25/16, 5/3, 15/8] connectivity = 1
Message: 2820 - Contents - Hide Contents Date: Sat, 29 Dec 2001 21:37:03 Subject: Re: non-uniqueness of a^(b/c) type numbers From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> I understand that many different combinations of prime-factors > and fractional exponents can be found which *approach any > floating-point value arbitrarily closely*, but EXACT values > are STILL INCOMMENSURABLE!!I'm not sure what you mean by this, in any sense which would not also be true of ordinary rational numbers.> Why must numbers of the form a^(b/c) be understood in > terms of their floating-point decimal value? If we stick > to the a^(b/c) form we get exact values and can manipulate > them the same way as our regular rational numbers of the > form x^y.Since a>0, there is no problem interpreting these exponents. If you want to use prime factorization, you get a vector space over the rational numbers from the exponents, which in some ways is easier to treat than the abelian group you get from confining yourself to the rationals, with integer exponents.
Message: 2821 - Contents - Hide Contents Date: Sat, 29 Dec 2001 06:57:54 Subject: Classes of 5-limit superparticular scales From: genewardsmith Three tones: (6/5)(5/4)(4/3) = 2 (10/9)(6/5)(3/2) = 2 (16/15)(5/4)(3/2) = 2 Four tones: (10/9)(6/5)^2(5/4) = 2 (16/15)(6/5)(5/4)^2 = 2 (25/24)(6/5)^2(4/3) = 2 (81/80)(10/9)(4/3)^2 = 2 Five tones: (10/9)^2(9/8)(6/5)^2 = 2 (16/15)^2(9/8)(5/4)^2 = 2 (25/24)(10/9)(6/5)^3 = 2 Six tones: (25/24)^2(16/15)(6/5)^3 = 2 (81/80)(10/9)^3(6/5)^2 = 2 Seven tones: (16/15)^2(10/9)^2(9/8)^3 Nine tones: (25/24)^2(16/15)^4(9/8)^3 Ten tones: (81/80)^3(16/15)^2(10/9)^5 = 2 Fifteen tones: (81/80)^3(25/24)^5(16/15)^7 = 2 Interestingly, no twelve tones. Aside from these, we have 2/1 = 2 in the 2-limit, and (4/3)(3/2) = 2 and (9/8)^2(4/3) = 2 in the 3-limit.
Message: 2822 - Contents - Hide Contents Date: Sat, 29 Dec 2001 21:38:41 Subject: Re: the unison-vectordeterminant relationship From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> But it still looks like Chinese. :(There are some old posts--until recently I would not have tossed multilinear algebra at people so casually, but there's been a lot of discussion of the wedge product.
Message: 2823 - Contents - Hide Contents Date: Sat, 29 Dec 2001 07:24:30 Subject: Re: Seven connected scales From: clumma Gene, this is amazing! -C. --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> Here are the seven connected scales one gets from > (9/8)^3 * (10/9)^2 * (16/15)^2 = 2, when modes and inversions aretaken as equivalent:> > [9/8, 9/8, 10/9, 16/15, 9/8, 10/9, 16/15] > [1, 9/8, 81/64, 45/32, 3/2, 27/16, 15/8] > connectivity = 1 > > [9/8, 9/8, 10/9, 16/15, 9/8, 16/15, 10/9] > [1, 9/8, 81/64, 45/32, 3/2, 27/16, 9/5] > connectivity = 1 > > [9/8, 9/8, 16/15, 10/9, 9/8, 10/9, 16/15] > [1, 9/8, 81/64, 27/20, 3/2, 27/16, 15/8] > connectivity = 1 > > [9/8, 10/9, 9/8, 10/9, 9/8, 16/15, 16/15] > [1, 9/8, 5/4, 45/32, 25/16, 225/128, 15/8] > connectivity = 1 > > [9/8, 10/9, 9/8, 10/9, 16/15, 9/8, 16/15] > [1, 9/8, 5/4, 45/32, 25/16, 5/3, 15/8] > connectivity = 2 > > [9/8, 10/9, 9/8, 16/15, 9/8, 10/9, 16/15] > [1, 9/8, 5/4, 45/32, 3/2, 27/16, 15/8] > connectivity = 2 > > [9/8, 10/9, 10/9, 9/8, 16/15, 9/8, 16/15] > [1, 9/8, 5/4, 25/18, 25/16, 5/3, 15/8] > connectivity = 1
Message: 2824 - Contents - Hide Contents Date: Sat, 29 Dec 2001 19:24:52 Subject: Re: the unison-vectordeterminant relationship From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, December 29, 2001 6:52 PM > Subject: [tuning-math] Re: the unison-vector<-->determinant relationship > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> The 7/25-comma meantone is lattice right down the middle of >> the symmetrical periodicity-block. >> Monz, I can't view your .xls file right now, and I'm wondering what > it means. I've seen quite a few of your lattices for this topic, but > I have to [_sic_: no] clue as to how to picture what you're describing > above.OK, let's go back to 1/6-comma meantone, since that's what I've been mostly working with here. On my diagram here: Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) (you can click on the diagram to open a big version), you can see that 1/6-comma meantone does not run exactly down the center of the (19 9),(4 -1) periodicity-block. There are other fraction-of-a-comma meantones which come closer to the center, and it seems to me that the one which *does* run exactly down the middle is 8/49-comma. Is this derivable from the [19 9],[4 -1] matrix? Is there any kind of significance to it? It seems to me that a meantone chain that would run down the center of a periodicity-block would have the smallest overall deviation from the most closely implied JI ratios in the periodicity-block, assuming that the JI lattice is wrapped into a cylinder. Yes? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
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