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Message: 2975 - Contents - Hide Contents Date: Fri, 04 Jan 2002 05:59:18 Subject: Re: Optimal 5-Limit Generators For Dave From: paulerlich --- In tuning-math@y..., graham@m... wrote:> genewardsmith@j... (genewardsmith) wrote: >>> You have an even and odd set of pitches, meaning an even or odd number >> of generators to the pitch. You can't get from even to odd by way of >> consonant 7-limit intervals, so basically we have two unrelated >> meantone systems a half-fifth or half-fourth apart. You can always glue >> together two unrelated systems and call it a temperament, and this >> differs only because it does have a single generator. >> These are the [2 8] systems.Not really. It's similar to torsion, but not quite the same.> There is some ambiguity, but if you mean the > half-fifth system, isn't that Vicentino's enharmonic? That's 31&24 or > [(1, 0), (1, 2), (0, 8)]. Two meantone scales, only 5-limit consonances > recognize, but neutral intervals used in melody. It may not be a > temperament, but does have a history of both theory and music, so don't > write it off so lightly.I doubt this reflects Vicentino's practice well at all. For instance, he didn't base any consonant harmonies on the second meantone scale, did he?> The half-fifth system is 24&19 or [(1, 0), (2, -2), (4, -8)].You mean half-fourth system?> There's > also a half-octave system, [(2, 0), (3, 1), (4, 4)]. That's the one my > program would deduce from the octave-equivalent mapping [2 8].From that unison vector? If so, I think you're confusion torsion with "contorsion".> If I had > such a program. If anybody cares, is it possible to write one? Where > torsion's present, we'll have to assume it means divisions of the octave > for uniqueness.Huh? Clearly this doesn't work in the Monz sruti 24 case.> Gene said it isn't possible, but I'm not convinced. How > could [1 4] be anything sensible but meantone?Not sure what the connection is.> Perhaps the first step is to find an interval that's only one generator > step, take the just value, period-reduce it and work everything else out > from that.If the half-fifth is the generator, what's the just value?> But there may be some cases where the optimal value should > cross a period boundary. ?? > If you think it can't be done, show a counter-example: an > octave-equivalent mapping without torsion that can lead to two different > but equally good temperaments.Equally good? Under what criteria? Look, why do we care about the octave-equivalent mapping? Certainly we can't object to asking the mapping to be octave-specific, can we?
Message: 2976 - Contents - Hide Contents Date: Fri, 4 Jan 2002 12:10 +00 Subject: Re: Some 9-tone 72-et scales From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a13hji+jr57@xxxxxxx.xxx> Me:>> 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. Paul:> The 22-tET "Pythagorean diatonic" works exceptionally well.You mean 4 4 1 4 4 4 1 ? Isn't it proper, and only one interval away from being strictly proper? Graham
Message: 2977 - Contents - Hide Contents 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: 2978 - Contents - Hide Contents Date: Fri, 04 Jan 2002 21:04:32 Subject: Quasi-diatonic 46-et scales From: genewardsmith These are from my list of 46-et scale classes derived from 126/125~1. The various goodness measures seem a little less correlated than they did in the case of the 72-et, and while the 46-et diatonic (and Indian diatonic)do well, they do not emerge as clear winners; the first scale listed in particular providing stiff competition. Two semitones in succession would certainly make for a different shape melodically, and the shift in empasis to the 11-limit means something quite different harmonically also. I thought of adding the 13 and 17 limits, but it almost seems as if everything connects to everything else by the time we go there. [0, 8, 15, 23, 30, 38, 42] [8, 7, 8, 7, 8, 4, 4] edges 8 9 17 connectivity 1 1 4 [0, 8, 15, 23, 27, 35, 42] [8, 7, 8, 4, 8, 7, 4] edges 11 11 16 connectivity 2 2 3 [0, 8, 15, 23, 30, 34, 42] [8, 7, 8, 7, 4, 8, 4] edges 9 10 16 connectivity 2 2 3 [0, 8, 15, 22, 30, 34, 42] [8, 7, 7, 8, 4, 8, 4] edges 8 10 16 connectivity 1 2 4 [0, 8, 16, 23, 30, 38, 42] [8, 8, 7, 7, 8, 4, 4] edges 5 7 16 connectivity 0 0 3 [0, 8, 16, 23, 27, 35, 42] [8, 8, 7, 4, 8, 7, 4] edges 9 9 15 connectivity 1 1 3 [0, 8, 16, 20, 27, 35, 42] [8, 8, 4, 7, 8, 7, 4] edges 8 9 15 connectivity 1 1 3 [0, 8, 16, 23, 27, 35, 39] [8, 8, 7, 4, 8, 4, 7] edges 8 8 15 connectivity 1 1 4 [0, 8, 16, 23, 31, 35, 39] [8, 8, 7, 8, 4, 4, 7] edges 7 7 15 connectivity 1 1 3 [0, 8, 16, 24, 31, 35, 39] [8, 8, 8, 7, 4, 4, 7] edges 6 7 15 connectivity 1 2 2 [0, 8, 16, 23, 31, 38, 42] [8, 8, 7, 8, 7, 4, 4] edges 6 7 15 connectivity 0 1 2 [0, 8, 16, 23, 31, 35, 42] [8, 8, 7, 8, 4, 7, 4] edges 8 8 14 connectivity 1 1 3 [0, 8, 16, 24, 31, 38, 42] [8, 8, 8, 7, 7, 4, 4] edges 3 5 14 connectivity 0 0 1 [0, 8, 16, 23, 27, 34, 42] [8, 8, 7, 4, 7, 8, 4] edges 7 7 13 connectivity 0 0 3 [0, 8, 16, 20, 28, 35, 42] [8, 8, 4, 8, 7, 7, 4] edges 6 7 13 connectivity 0 0 2 [0, 8, 16, 24, 31, 35, 42] [8, 8, 8, 7, 4, 7, 4] edges 5 6 13 connectivity 0 1 2 [0, 8, 16, 23, 30, 34, 42] [8, 8, 7, 7, 4, 8, 4] edges 5 6 13 connectivity 0 0 2 [0, 8, 16, 24, 28, 35, 42] [8, 8, 8, 4, 7, 7, 4] edges 4 5 12 connectivity 0 0 2
Message: 2979 - Contents - Hide Contents 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: 2980 - Contents - Hide Contents Date: Fri, 04 Jan 2002 21:53:05 Subject: Some 46-et decatonics From: genewardsmith We've talked about this system before--it still looks lovely. [0, 4, 8, 12, 16, 23, 27, 31, 35, 39] [4, 4, 4, 4, 7, 4, 4, 4, 4, 7] edges 18 18 28 connectivity 3 3 5 [0, 4, 8, 12, 16, 20, 27, 31, 35, 39] [4, 4, 4, 4, 4, 7, 4, 4, 4, 7] edges 18 18 28 connectivity 3 3 5 [0, 4, 8, 12, 16, 20, 24, 31, 35, 39] [4, 4, 4, 4, 4, 4, 7, 4, 4, 7] edges 16 17 28 connectivity 2 3 4 [0, 4, 8, 12, 16, 20, 24, 28, 35, 39] [4, 4, 4, 4, 4, 4, 4, 7, 4, 7] edges 13 15 27 connectivity 1 2 5 [0, 4, 8, 12, 16, 20, 24, 28, 32, 39] [4, 4, 4, 4, 4, 4, 4, 4, 7, 7] edges 10 13 27 connectivity 0 0 4
Message: 2981 - Contents - Hide Contents 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 consonantand they may>> even be sung in contemporaneous music, he doesn't use themhimself 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: 2982 - Contents - Hide Contents 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 planartemperament 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: 2983 - Contents - Hide Contents 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: 2984 - Contents - Hide Contents 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: 2985 - Contents - Hide Contents Date: Fri, 04 Jan 2002 01:44:10 Subject: Pentatonic 72-et scales From: genewardsmith [0, 7, 19, 42, 49] [7, 12, 23, 7, 23] edges 6 7 8 connectivity 1 2 3 [0, 7, 19, 26, 49] [7, 12, 7, 23, 23] edges 6 6 8 connectivity 2 2 3 [0, 7, 14, 37, 49] [7, 7, 23, 12, 23] edges 4 7 8 connectivity 1 2 2 [0, 7, 14, 26, 49] [7, 7, 12, 23, 23] edges 4 6 8 connectivity 0 2 2
Message: 2987 - Contents - Hide Contents Date: Fri, 04 Jan 2002 04:06:39 Subject: Re: "Connectedness" ? [was: The epimorph From: genewardsmith --- In tuning-math@y..., "unidala" <JGill99@i...> wrote:> Any chance that you could explain how you have > (in posting the various scales) characterized > (and subsequently quantized) what you refer to > as "connectedness"?I took the standard set of consonant intervals in the 5, 7 and 11 limits, and defined a graph (in the sense of a simple graph from the mathematical theory of graphs) on octave equivalence classes by taking the verticies of the graph to be the tones, and the edges of the graph (which you can think of as a line drawn between two verticies) as the consonant intervals connecting the tones. "Connectivity" is simply the standard edge-connectivity of graph theory, meaning the number of edges which would need to be removed from the graph in order to render it disconnected.
Message: 2988 - Contents - Hide Contents 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: 2989 - Contents - Hide Contents 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: 2990 - Contents - Hide Contents 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: 2991 - Contents - Hide Contents 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 thanRI 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: 2992 - Contents - Hide Contents 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: 2993 - Contents - Hide Contents 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: 2994 - Contents - Hide Contents Date: Sat, 05 Jan 2002 02:11:42 Subject: Some 46-et nonatonics From: genewardsmith These all, for some reason, have an 11-connectivity of 5. [0, 3, 6, 15, 18, 27, 30, 39, 42] [3, 3, 9, 3, 9, 3, 9, 3, 4] edges 13 22 29 connectvity 1 2 5 [0, 3, 6, 15, 18, 27, 30, 34, 37] [3, 3, 9, 3, 9, 3, 4, 3, 9] edges 15 22 28 connectvity 2 3 5 [0, 3, 6, 15, 18, 27, 30, 33, 42] [3, 3, 9, 3, 9, 3, 3, 9, 4] edges 14 21 28 connectvity 2 2 5 [0, 3, 6, 15, 18, 22, 25, 28, 37] [3, 3, 9, 3, 4, 3, 3, 9, 9] edges 11 20 28 connectvity 0 3 5 [0, 3, 6, 9, 18, 21, 30, 33, 42] [3, 3, 3, 9, 3, 9, 3, 9, 4] edges 10 18 28 connectvity 0 1 5 [0, 3, 6, 15, 18, 27, 36, 39, 42] [3, 3, 9, 3, 9, 9, 3, 3, 4] edges 9 18 28 connectvity 0 2 5 [0, 3, 6, 9, 18, 27, 30, 33, 42] [3, 3, 3, 9, 9, 3, 3, 9, 4] edges 9 18 28 connectvity 0 2 5 [0, 3, 6, 9, 18, 27, 30, 39, 42] [3, 3, 3, 9, 9, 3, 9, 3, 4] edges 8 17 28 connectvity 0 2 5 [0, 3, 6, 9, 12, 21, 30, 33, 42] [3, 3, 3, 3, 9, 9, 3, 9, 4] edges 7 15 28 connectvity 0 2 5 [0, 3, 6, 15, 18, 21, 30, 33, 37] [3, 3, 9, 3, 3, 9, 3, 4, 9] edges 14 19 27 connectvity 1 2 5 [0, 3, 6, 15, 18, 21, 30, 33, 42] [3, 3, 9, 3, 3, 9, 3, 9, 4] edges 13 19 27 connectvity 1 1 5 [0, 3, 6, 9, 18, 21, 30, 33, 37] [3, 3, 3, 9, 3, 9, 3, 4, 9] edges 12 18 27 connectvity 0 2 5 [0, 3, 6, 9, 18, 27, 30, 33, 37] [3, 3, 3, 9, 9, 3, 3, 4, 9] edges 10 18 27 connectvity 0 3 5 [0, 3, 6, 15, 18, 21, 30, 39, 42] [3, 3, 9, 3, 3, 9, 9, 3, 4] edges 10 18 27 connectvity 0 1 5 [0, 3, 6, 9, 12, 15, 24, 33, 37] [3, 3, 3, 3, 3, 9, 9, 4, 9] edges 8 17 27 connectvity 1 3 5 [0, 3, 6, 9, 12, 21, 24, 33, 37] [3, 3, 3, 3, 9, 3, 9, 4, 9] edges 9 16 27 connectvity 1 2 5 [0, 3, 6, 9, 12, 15, 24, 33, 42] [3, 3, 3, 3, 3, 9, 9, 9, 4] edges 7 16 27 connectvity 0 2 5 [0, 3, 6, 9, 12, 21, 30, 33, 37] [3, 3, 3, 3, 9, 9, 3, 4, 9] edges 8 15 27 connectvity 0 3 5 [0, 3, 6, 9, 18, 21, 30, 39, 42] [3, 3, 3, 9, 3, 9, 9, 3, 4] edges 7 15 27 connectvity 0 1 5 [0, 3, 6, 9, 12, 21, 24, 33, 42] [3, 3, 3, 3, 9, 3, 9, 9, 4] edges 7 15 27 connectvity 0 1 5 [0, 3, 6, 9, 18, 27, 36, 39, 42] [3, 3, 3, 9, 9, 9, 3, 3, 4] edges 6 15 27 connectvity 0 2 5 [0, 3, 6, 9, 12, 21, 30, 39, 42] [3, 3, 3, 3, 9, 9, 9, 3, 4] edges 6 14 27 connectvity 0 2 5 [0, 3, 6, 9, 18, 21, 24, 33, 37] [3, 3, 3, 9, 3, 3, 9, 4, 9] edges 11 17 26 connectvity 0 2 5 [0, 3, 6, 9, 18, 21, 25, 28, 37] [3, 3, 3, 9, 3, 4, 3, 9, 9] edges 10 17 26 connectvity 0 2 5 [0, 3, 6, 9, 18, 21, 30, 34, 37] [3, 3, 3, 9, 3, 9, 4, 3, 9] edges 11 16 26 connectvity 1 2 5 [0, 3, 6, 9, 18, 21, 24, 33, 42] [3, 3, 3, 9, 3, 3, 9, 9, 4] edges 8 16 26 connectvity 0 1 5 [0, 3, 6, 9, 12, 21, 24, 28, 37] [3, 3, 3, 3, 9, 3, 4, 9, 9] edges 8 15 26 connectvity 1 2 5 [0, 3, 6, 9, 18, 21, 24, 28, 37] [3, 3, 3, 9, 3, 3, 4, 9, 9] edges 9 16 25 connectvity 0 2 5
Message: 2995 - Contents - Hide Contents Date: Sat, 05 Jan 2002 23:58:21 Subject: More 12-tone 46-et scales From: genewardsmith There were 85 of these, so I culled out eight of the best. Regularity does not seem to win the day, as the fifth on this list does not stand out. [0, 3, 6, 9, 12, 15, 21, 27, 30, 36, 39, 42] [3, 3, 3, 3, 3, 6, 6, 3, 6, 3, 3, 4] edges 17 31 53 connectivity 0 3 8 [0, 3, 6, 9, 12, 15, 21, 27, 30, 33, 36, 42] [3, 3, 3, 3, 3, 6, 6, 3, 3, 3, 6, 4] edges 17 30 53 connectivity 0 4 8 [0, 3, 6, 9, 12, 15, 21, 27, 33, 36, 39, 42] [3, 3, 3, 3, 3, 6, 6, 6, 3, 3, 3, 4] edges 16 29 53 connectivity 0 4 8 [0, 3, 6, 9, 15, 18, 21, 27, 30, 33, 37, 40] [3, 3, 3, 6, 3, 3, 6, 3, 3, 4, 3, 6] edges 20 34 52 connectivity 2 4 8 [0, 3, 6, 12, 15, 18, 24, 27, 30, 36, 39, 42] [3, 3, 6, 3, 3, 6, 3, 3, 6, 3, 3, 4] edges 19 34 52 connectivity 1 4 8 [0, 3, 6, 9, 15, 18, 21, 27, 30, 34, 37, 40] [3, 3, 3, 6, 3, 3, 6, 3, 4, 3, 3, 6] edges 20 33 52 connectivity 2 3 8 [0, 3, 6, 9, 12, 15, 18, 21, 27, 30, 36, 40] [3, 3, 3, 3, 3, 3, 3, 6, 3, 6, 4, 6] edges 18 33 51 connectivity 1 5 7 [0, 3, 6, 9, 12, 15, 18, 21, 24, 30, 36, 40] [3, 3, 3, 3, 3, 3, 3, 3, 6, 6, 4, 6] edges 18 32 51 connectivity 2 5 7
Message: 2996 - Contents - Hide Contents Date: Sat, 05 Jan 2002 03:28:42 Subject: Some 12-tone, 2-step 46-et scales From: genewardsmith I was facinated to discover that the 7,5 system did a little better than the completely symmetrical 6,6 system. [0, 4, 8, 12, 16, 20, 23, 27, 31, 35, 39, 43] [4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 3] edges 24 24 40 connectivity 3 3 6 [0, 4, 8, 12, 16, 20, 24, 27, 31, 35, 39, 43] [4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 3] edges 24 25 41 connectivity 3 3 6 [0, 4, 8, 12, 16, 20, 24, 28, 31, 35, 39, 43] [4, 4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 3] edges 22 24 40 connectivity 2 3 6 [0, 4, 8, 12, 16, 20, 24, 28, 32, 35, 39, 43] [4, 4, 4, 4, 4, 4, 4, 4, 3, 4, 4, 3] edges 18 21 38 connectivity 1 2 5 [0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 39, 43] [4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 4, 3] edges 15 20 38 connectivity 1 2 5 [0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 43] [4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3] edges 12 19 38 connectivity 0 0 5
Message: 2997 - Contents - Hide Contents Date: Sat, 05 Jan 2002 04:40:31 Subject: Some hexatonic 46-et scales From: genewardsmith The one that had the best scores seems a little irregular, with a repeated large step, a triple repeat works pretty well also. The large step in question is a minor third. [0, 12, 24, 27, 39, 43] [12, 12, 3, 12, 4, 3] edges 9 10 12 connectivity 2 2 3 [0, 12, 24, 36, 39, 43] [12, 12, 12, 3, 4, 3] edges 7 10 12 connectivity 1 2 3 [0, 12, 16, 28, 31, 43] [12, 4, 12, 3, 12, 3] edges 9 9 11 connectivity 2 2 3 [0, 12, 24, 28, 31, 43] [12, 12, 4, 3, 12, 3] edges 8 9 11 connectivity 1 1 2 [0, 12, 24, 36, 40, 43] [12, 12, 12, 4, 3, 3] edges 5 8 11 connectivity 0 0 2 [0, 12, 24, 28, 40, 43] [12, 12, 4, 12, 3, 3] edges 6 7 10 connectivity 1 1 3
Message: 2998 - Contents - Hide Contents Date: Sat, 05 Jan 2002 09:35:52 Subject: More types of 72-et scales From: genewardsmith These were derived from 1029/1024, so it is not surprising many of them are really 36-et scales. I would not expect too much in the way of 5-harmony from these, but perhaps they will make up for it. 3 tones [44, 14] [1, 2] 4 tones [14, 30] [3, 1] 5 tones [14, 16] [4, 1] 6 tones [14, 2] [5, 1] 8 tones [11, 3] [6, 2] [7, 23] [7, 1] 9 tones [4, 10] [3, 6] 10 tones [7, 9] [9, 1] [6, 8] [4, 6] 11 tones [8, 6] [3, 8] [12, 2] [5, 6] 12 tones [13, 1] [5, 7] [4, 10] [8, 4] [5, 9] [9, 3] 15 tones [4, 6] [9, 6] 18 tones [2, 6] [9, 9]
Message: 2999 - Contents - Hide Contents 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 - The best web-based email! * [with cont.] (Wayb.)
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