4000 4050 4100 4150 4200 4250 4300 4350 4400 4450 4500 4550 4600 4650 4700 4750 4800 4850 4900 4950 5000 5050 5100 5150 5200 5250 5300 5350 5400 5450 5500 5550 5600 5650 5700 5750 5800 5850 5900 5950 6000 6050 6100 6150 6200 6250 6300 6350 6400 6450 6500 6550
5950 - 5975 -
![]()
![]()
Message: 5975 Date: Fri, 04 Jan 2002 06:00:20 Subject: Re: 12 note, 225/224 planar temperament scales From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > These are, of course, expressed in terms of the 72-et. They have >three step sizes, which are 2, 5, and 7 in the 72-et, so this is >closely related to Miracle, as one would expect. They are sorted in >terms of number of edges (consonant intervals) and then connectivity. I think Dave Keenan gave a "tetrachordal" example of one of these. Dave? > > [0, 5, 12, 19, 21, 28, 35, 42, 49, 51, 58, 65] > [5, 7, 7, 2, 7, 7, 7, 7, 2, 7, 7, 7] > edges 34 connectivity 4 > > [0, 5, 12, 19, 21, 28, 35, 42, 49, 56, 58, 65] > [5, 7, 7, 2, 7, 7, 7, 7, 7, 2, 7, 7] > edges 34 connectivity 4 > > [0, 5, 12, 14, 21, 28, 35, 42, 49, 51, 58, 65] > [5, 7, 2, 7, 7, 7, 7, 7, 2, 7, 7, 7] > edges 33 connectivity 4 > > [0, 5, 12, 19, 26, 28, 35, 42, 49, 51, 58, 65] > [5, 7, 7, 7, 2, 7, 7, 7, 2, 7, 7, 7] > edges 33 connectivity 3 > > [0, 5, 7, 14, 21, 28, 35, 42, 49, 56, 63, 65] > [5, 2, 7, 7, 7, 7, 7, 7, 7, 7, 2, 7] > edges 32 connectivity 5 > > [0, 5, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70] > [5, 2, 7, 7, 7, 7, 7, 7, 7, 7, 7, 2] > edges 32 connectivity 5 > > [0, 5, 7, 14, 21, 28, 35, 42, 44, 51, 58, 65] > [5, 2, 7, 7, 7, 7, 7, 2, 7, 7, 7, 7] > edges 32 connectivity 4 > > [0, 5, 7, 14, 21, 28, 35, 42, 49, 51, 58, 65] > [5, 2, 7, 7, 7, 7, 7, 7, 2, 7, 7, 7] > edges 32 connectivity 4 > > [0, 5, 7, 14, 21, 28, 35, 42, 49, 56, 58, 65] > [5, 2, 7, 7, 7, 7, 7, 7, 7, 2, 7, 7] > edges 32 connectivity 4 > > [0, 5, 12, 14, 21, 28, 35, 42, 44, 51, 58, 65] > [5, 7, 2, 7, 7, 7, 7, 2, 7, 7, 7, 7] > edges 32 connectivity 4 > > [0, 5, 12, 14, 21, 28, 35, 42, 49, 56, 58, 65] > [5, 7, 2, 7, 7, 7, 7, 7, 7, 2, 7, 7] > edges 32 connectivity 4 > > [0, 5, 12, 19, 21, 28, 35, 42, 44, 51, 58, 65] > [5, 7, 7, 2, 7, 7, 7, 2, 7, 7, 7, 7] > edges 32 connectivity 3 > > [0, 5, 12, 14, 21, 28, 35, 42, 49, 56, 63, 65] > [5, 7, 2, 7, 7, 7, 7, 7, 7, 7, 2, 7] > edges 31 connectivity 4 > > [0, 5, 7, 14, 21, 28, 35, 37, 44, 51, 58, 65] > [5, 2, 7, 7, 7, 7, 2, 7, 7, 7, 7, 7] > edges 31 connectivity 4 > > [0, 5, 12, 14, 21, 28, 35, 37, 44, 51, 58, 65] > [5, 7, 2, 7, 7, 7, 2, 7, 7, 7, 7, 7] > edges 30 connectivity 3 > > [0, 5, 7, 14, 21, 28, 30, 37, 44, 51, 58, 65] > [5, 2, 7, 7, 7, 2, 7, 7, 7, 7, 7, 7] > edges 30 connectivity 3 > > [0, 5, 12, 19, 26, 28, 35, 42, 44, 51, 58, 65] > [5, 7, 7, 7, 2, 7, 7, 2, 7, 7, 7, 7] > edges 29 connectivity 2 > > [0, 5, 7, 14, 21, 23, 30, 37, 44, 51, 58, 65] > [5, 2, 7, 7, 2, 7, 7, 7, 7, 7, 7, 7] > edges 29 connectivity 2 > > [0, 5, 12, 19, 21, 28, 35, 37, 44, 51, 58, 65] > [5, 7, 7, 2, 7, 7, 2, 7, 7, 7, 7, 7] > edges 28 connectivity 3 > > [0, 5, 7, 9, 16, 23, 30, 37, 44, 51, 58, 65] > [5, 2, 2, 7, 7, 7, 7, 7, 7, 7, 7, 7] > edges 28 connectivity 1 > > [0, 5, 7, 14, 16, 23, 30, 37, 44, 51, 58, 65] > [5, 2, 7, 2, 7, 7, 7, 7, 7, 7, 7, 7] > edges 28 connectivity 1 > > [0, 5, 12, 14, 21, 28, 30, 37, 44, 51, 58, 65] > [5, 7, 2, 7, 7, 2, 7, 7, 7, 7, 7, 7] > edges 27 connectivity 2 > > [0, 5, 12, 14, 16, 23, 30, 37, 44, 51, 58, 65] > [5, 7, 2, 2, 7, 7, 7, 7, 7, 7, 7, 7] > edges 24 connectivity 1 > > [0, 5, 12, 14, 21, 23, 30, 37, 44, 51, 58, 65] > [5, 7, 2, 7, 2, 7, 7, 7, 7, 7, 7, 7] > edges 24 connectivity 1 > > [0, 5, 12, 19, 21, 28, 30, 37, 44, 51, 58, 65] > [5, 7, 7, 2, 7, 2, 7, 7, 7, 7, 7, 7] > edges 23 connectivity 2 > > [0, 5, 12, 19, 26, 28, 35, 37, 44, 51, 58, 65] > [5, 7, 7, 7, 2, 7, 2, 7, 7, 7, 7, 7] > edges 23 connectivity 1 > > [0, 5, 12, 19, 26, 33, 35, 42, 44, 51, 58, 65] > [5, 7, 7, 7, 7, 2, 7, 2, 7, 7, 7, 7] > edges 23 connectivity 1 > > [0, 5, 12, 19, 21, 23, 30, 37, 44, 51, 58, 65] > [5, 7, 7, 2, 2, 7, 7, 7, 7, 7, 7, 7] > edges 21 connectivity 1 > > [0, 5, 12, 19, 26, 28, 30, 37, 44, 51, 58, 65] > [5, 7, 7, 7, 2, 2, 7, 7, 7, 7, 7, 7] > edges 19 connectivity 1 > > [0, 5, 12, 19, 26, 33, 35, 37, 44, 51, 58, 65] > [5, 7, 7, 7, 7, 2, 2, 7, 7, 7, 7, 7] > edges 18 connectivity 1
![]()
![]()
![]()
Message: 5977 Date: Fri, 04 Jan 2002 06:01:11 Subject: Re: Optimal 5-Limit Generators For Dave From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > --- In tuning-math@y..., graham@m... wrote: > > > There is some ambiguity, but if you mean the > > half-fifth system, isn't that Vicentino's enharmonic? > > I thought Vicentino was 31-et. Vicentino did a lot of things both inside and outside 31-tET.
![]()
![]()
![]()
Message: 5979 Date: Fri, 04 Jan 2002 06:04:00 Subject: Re: Optimal 5-Limit Generators For Dave From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > --- In tuning-math@y..., Graham Breed <graham@m...> wrote: > > > Although he > > mentions, briefly, that he considers neutral thirds as consonant and they may > > even be sung in contemporaneous music, he doesn't use them himself in chords. > > And he doesn't quite give the 11-limit interpretation. > > If neutral thirds are consonant we are not talking about the 5-> >limit and the entire argument is moot. They're not considered consonant on the level of the 5-limit consonances. And anyway, the music makes the argument not moot, if Graham's interpretation is reasonable. How far out in the chain of generators do you have to go to account for V's simplest example of "enharmonic genus" music?
![]()
![]()
![]()
Message: 5980 Date: Fri, 04 Jan 2002 06:07:57 Subject: Re: Some 10-tone, 72-et scales From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > I started out looking at these as 7-limit 225/224 planar temperament scales, but decided it made more sense to check the 5 and 11 limits also, and to take them as 72-et scales; if they are ever used that is probably how they will be used. I think anyone interested in the > 72-et should take a look at the top three, which are all 5- connected, and the top scale in particular, which is a clear winner. The "edges" number counts edges (consonant intervals) in the 5, 7, and 11 limits, and the connectivity is the edge-connectivity in the 5, 7 and 11 limits. > > [0, 5, 12, 19, 28, 35, 42, 49, 58, 65] > [5, 7, 7, 9, 7, 7, 7, 9, 7, 7] Was this Dave Keenan's 72-tET version of my Pentachordal Decatonic?
![]()
![]()
![]()
Message: 5981 Date: Fri, 04 Jan 2002 06:14:53 Subject: Re: tetrachordality From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote: > Paul, > > My current model works like this: > > pentachordal > (0 109 218 382 491 600 709 873 982 1091) > (1193 102 211 375 484 593 702 811 920 1084) > 7 7 7 7 7 7 7 62 62 7 > > symmetrical > (0 109 218 382 491 600 709 818 982 1091) > (1193 102 211 320 484 593 702 811 920 1084) > 7 7 7 62 7 7 7 7 62 7 > > So obviously, these two scales will come out > the same. But you've view -- and I remember > doing some listening experiments that back you > up (the low efficiency of the symmetrical > version was the other theory there) -- is that > the symmetrical version is not tetrachordal. > > So what's going on here? Where's the error > in tetrachordality = similarity at transposition > by a 3:2? An octave species is homotetrachordal if it has identical melodic structure within two 4:3 spans, separated by either a 4:3 or a 3:2. In the pentachordal scale, _all_ of the octave species are homotetrachordal (some in more than one way). In the symmetrical scale, _none_ of the octave species are homotetrachordal.
![]()
![]()
![]()
Message: 5982 Date: Fri, 04 Jan 2002 06:20:34 Subject: Re: Some 9-tone 72-et scales From: paulerlich --- In tuning-math@y..., graham@m... wrote: > In-Reply-To: <a1173r+ob05@e...> > gene wrote: > > > I'd need to write the code for it, and it isn't a graph property so I'm > > not going to start with any advantage from the Maple graph theory > > package. Paul did not think propriety was very important--what's your > > take on it? > > I don't have a definitive answer. It, or something like it, may be > important for modality. Especially for subsets of "comprehensible" ETs. > The Pythagorean diatonic works fine despite being slightly improper, so > you shouldn't be over-strict. The 22-tET "Pythagorean diatonic" works exceptionally well.
![]()
![]()
![]()
Message: 5986 Date: Fri, 04 Jan 2002 05:04:26 Subject: Re: the unison-vectordeterminant relationship From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > > > > > 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? > > > > > > > > You should find that the interval corresponding to (19 9), AS IT > > > > APPEARS in 8/49-comma meantone, is a very tiny interval. > > > > > > > > > Ah... so then 8/49-comma meantone does *not* run *exactly* > > > down the middle. How could one calculate the meantone which > > > *does* run exactly down the middle? > > > > It's 55-tET. > > > Not if the periodicity-block is a parallelogram. 10/57-comma meantone > is much closer to 55-EDO than 1/6-comma meantone, yet it is further > away from the center of this periodicity-block. Hmm . . . the line you want is the vector (19 9). So any generator 3^a/b * 5^c/d that is a solution to the equation a/b * 19 + c/d * 9 = 0 would work. This gives a/b*19 = -c/d*9 Does this help? > > (a) You are NOT, with your current method, mapping identical meantone > > intervals to identical JI ratios, and > > > Not really sure what you mean by this. For example, if you look at the fifth D-A, in some of your diagrams this is mapped to 40:27, while C-G is mapped to 3:2, and yet the two intervals are tuned _identically_ in any of the meantones in question. > > (b) if you really meant "pitches" rather than "intervals", I'd argue > > that the mappings you are producing involve a rather arbitrary rule, > > and don't reflect the musical properties of the meantone tunings. The > > only case in which they would is if you specifically knew you were > > not going to use any of the consonances that "wrap" around the block, > > AND you were interested in using a simultaneous JI tuning with the > > meantone that would minimize the _pitch_ differences between the two - > > - a very contrived scenario. > > > OK, now I'm getting more confused again. > > Again -- the reader is supposed to *imagine* that my mappings > wrap cylindrically. If they are so wrapped, all the diagrams for the different meantones will look _exactly the same_. So what's the point of all the different shapes? > So I don't understand why you're pointing > out cases where the consonances that wrap are not used. Because in those cases, you're mapping a JI tuning which will function similarly to the meantone in question. > Also, I'm not thinking specifically of pitches. I asked this > before but didn't get an answer that was clear -- what's the > real difference? See my remark about D-A and C-G above. You did the same thing when you mapped Partch's 43 to 72-tET. You concerned yourself with approximating the pitches, while approximating the consonant intervals leads to a more musically relevant result. > If I assume that my flat lattice is supposed > to wrap cylindrically, then why does it matter whether I'm > considering pitches or intervals? Well, again, in that case all your diagrams would look identical, but they don't, so it looks like you're implying something different.
![]()
![]()
![]()
Message: 5987 Date: Fri, 04 Jan 2002 05:13:29 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote: > Not sure what you mean. The reason I suggested the shortcut not > be applied for inharmonic timbres is because... it is a shortcut. > Which assumes you have clearly resolved fundamentals. No? No. It just assumes that the overtones are pretty close to harmonic, because they will then lead to the same ratio-intepretations for the fundamentals as the fundamentals by themselves. If they're 50 cents from harmonic, they will lead to a larger s value for the resulting harmonic entropy curve, but that's about it. > >By the way Carl, have you tried any actual _listening > >experiments_ yet? > > You mean with a synthesizer? As I explained, I don't have the > right gear -- I've got an additive synth that's stuck in JI. You can synthesize inharmonic sounds, yes? You can use a high-limit JI scale that sounds like a pelog scale, yes? > What do you have in mind? I'm not clear how one would go about > testing anything that's been said here. Well, what I'm saying seems most clear and powerful to me as a musician actually playing this stuff. > > >The gamelan scales sound like they contain a rough major > >triad and a rough minor triad, forming a very rough major > >seventh chord together, plus one extra note -- don't they? > > Yes, to me, pelog sounds like a I and a III with a 4th in the > middle. But the music seems to use a fixed tonic, with not > much in the way of triadic structure. How about 5-limit intervals? > Okay, let's take a > journey... > > "Instrumental music of Northeast Thailand" > > Characteristic stop rhythm. Harmonium and marimba-sounding > things play major pentatonic on C# (A=440) or relative minor > on A#. This is clearly not a pelog tuning! > "JAVA Tembang Sunda" (Inedit) > > This is unlike the gamelan music I've heard (it's a plucked > string ensemble with vocalists and flute). Jeez, I forgot I > had this CD! There _is_ I -> III, and even I -> IV motion > here. > > "Gamelan Semar Pagulingan from Besang-Ababi/Karangasem > Music from Bali" > > I suppose there is some argument for triadic structure here > too, but if I hadn't heard the last disc beforehand, I'd > say they were just doing the 'start the figure on different > scale members' thing, as in the first disc. I don't know > Paul, this is not life as we know it (or hear it). What on earth does that mean? > I still > say there's nothing here that would turn up an optimized > 5-limit temperament! Forget the optimization. All you need is the mapping -- that chains of three fifths make a major third and that chains of four fifths make a minor third. This seems to be a definite characteristic of pelog! Just as much as the "opposite" is a characteristic of Western music, regardless of whether strict JI, optimized meantone, 12-tET, or whatever is used. > > I guess it all depends if you consider these tonic changes > or just points of symmetry in a melisma (sp?). Why does that matter?
![]()
![]()
![]()
Message: 5988 Date: Fri, 04 Jan 2002 05:17:49 Subject: Re: flexible mapping of meantones to PBs From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > Paul, I finally understand your objections to what I've been > saying! I hope so . . . we've been "stuck" on the very same set of issues for several years now. > > > Here's another example I just found: > > (This is one of the examples in Fokker 1968, "Selections from the > Harmonic Lattice of Perfect Fifths and Major Thirds Containing > 12, 19, 22, 31, 41 or 53 Notes".) > > > > unison-vector matrix = > > [ 4 -1] > [-1 5] > > > determinant = | 19 | > > > periodicity-block coordinates: > > > 3^x 5^y ~cents > > -1 -2 925.4175714 > 0 -2 427.3725723 > 1 -2 1129.327573 > 2 -2 631.282574 > -1 -1 111.7312853 > 0 -1 813.6862861 > 1 -1 315.641287 > 2 -1 1017.596288 > -1 0 498.0449991 > 0 0 0 > 1 0 701.9550009 > -2 1 182.4037121 > -1 1 884.358713 > 0 1 386.3137139 > 1 1 1088.268715 > -2 2 568.717426 > -1 2 70.67242686 > 0 2 772.6274277 > 1 2 274.5824286 > > > Ah!... actually now I see what's happening. > > The meantones most commonly associated with this periodicity-block > would be 1/3-comma and 19-EDO. I'm not latticing EDOs on this > spreadsheet, so we'll just stick with the fraction-of-a-comma type. > > 1/3-comma does indeed split the periodicity-block exactly in half, > just not along an axis I expected, as it doesn't follow the same > angle as either of the unison-vectors. > > The meantone I found by eye to split it according to the same angle > as the unison-vector [-1 5] is 16/61-comma. > > > But I think now I understand what you've been getting at, Paul. > > In the 1/3-comma chain, > > closest JI > generator coordinate > > +1 ( 1 0) - 1/3-comma > +2 (-2 1) + 1/3-comma > +3 (-1 1) exactly > +4 ( 0 1) - 1/3-comma > +5 ( 1 1) - 2/3-comma > +6 (-2 2) exactly > +6 (-1 2) - 1/3-comma > +6 ( 0 2) - 2/3-comma > +6 ( 1 2) - 1 comma > etc. > > In my mapping done by eye, everything would be the same up > to +4 generator. Then I'd set +5 generator equal to > (-3 2) - 1/3-comma, rather than (1 1) - 2/3-comma, since > it's closer. And so on. > > But then we end up with +6 generator mapped to (1 2) - 1 comma > instead of to exactly (-3 3), which is what I would get. > > But *it doesn't matter which periodicity-block contains the > closest-approach ratio, because they're all equivalent!* Right? Sure, but not sure what you did above to decide that. > > Got it now. Whew! > > > It doesn't matter which fraction-of-a-comma meantone I lattice > within a periodicity-block -- they'll *all* split the block > exactly symmetrically in half. Only the angles and resulting > areas differ. Well, this has nothing to do with any of my objections to what you've been saying. You could perfectly well be interested, for some strange musical contrivance or pure mathematical curiosity, in the meantone that "splits" the "periodicity block" along an "axis" parallel to one of the unison vectors, in the way you've been diagramming things, and I'd have no problem helping you do so.
![]()
![]()
![]()
Message: 5989 Date: Fri, 04 Jan 2002 05:20:58 Subject: Re: Some 10 note 22 et scales From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > We can also use the assoicated graph to analyze scales other than RI scales; here is the connectivity of the scales having eight steps of size 2 and two steps of size 3 in the 22-et: > > c = 6 > > 2222322223 > > c = 5 > > 2222232223 > > c = 4 > > 2222223223 > > c = 3 > > 2222222323 and 2222222233 > > No surprises here, but there might be other things people think > would be worth analyzing. Is an MOS (in an ET or linear temperament) always more connected than any of its permutations? Probably not -- what conditions can we place on the situations in which it is?
![]()
![]()
![]()
Message: 5990 Date: Fri, 04 Jan 2002 05:23:27 Subject: Re: Some 7-limit superparticular pentatonics From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > These are the ones which employ the two most proper possibilities, > (15/14)(8/7)(7/6)^2(6/5) and (15/14)(10/9)(7/6)(6/5)^2; both with a > Blackwood index of 2.64 (largest over smallest scale step ratio.) This term "Blackwood index" should only be applied when there are only two step sizes.
![]()
![]()
![]()
Message: 5991 Date: Fri, 04 Jan 2002 05:34:23 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > > > > There were the usual repetions (meantone, 1/2 fifth meantone, 1/2 > fourth meantone, etc) > > To a mathematician focussing on approximation of ratios for harmony > these may be repetitions, but to a musician they are quite distinct > and it is quite wrong to call them "meantones". But it is important to > point out their relationship to meantone. Hello folks. These systems can be derived from a combined-ET viewpoint but cannot be derived from a unison vector viewpoint. This is very reminiscent of torsional blocks, which can be derived from a unison vector viewpoint but not from a combined-ET viewpoint. So is this, mathematically, the "dual of torsion"? Can we deal with torsion, as well as "contorsion" or whatever we call this, in the beginning of our paper, and leave out the specific examples, as they follow a fairly obvious pattern??
![]()
![]()
![]()
Message: 5997 Date: Sat, 5 Jan 2002 22:07:54 Subject: please simplify equation From: monz Can this equation be simplified? (I've added brackets above the section whose log is taken, and above the entire power of 10, to make them easier to see.) |----------------------------------------------------| |-----------------------| v = 10 ^ ( LOG( 1 / (2 ^ (9r - 1/r) ) ) / ( -15r + 2/r - 1) ) Thanks. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
4000 4050 4100 4150 4200 4250 4300 4350 4400 4450 4500 4550 4600 4650 4700 4750 4800 4850 4900 4950 5000 5050 5100 5150 5200 5250 5300 5350 5400 5450 5500 5550 5600 5650 5700 5750 5800 5850 5900 5950 6000 6050 6100 6150 6200 6250 6300 6350 6400 6450 6500 6550
5950 - 5975 -