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Message: 2875 - Contents - Hide Contents Date: Sun, 30 Dec 2001 20:07:18 Subject: Re: coordinates from unison-vectors From: monz A plea to all who understand matrix math: A week ago, I posted the pseudo-code for the formulas in my Excel spreadsheet which calculates the coordinates of a 2-dimensional periodicity-block from a given pair of unison-vectors. It's quoted below, with some additional comments inserted. The algorithm sometimes calculates the proper coordinates, but not always. I've tried to work from Paul's _Gentle Introduction to Fokker Periodicity Blocks, part 3_ <A gentle introduction to Fokker periodicity bl... * [with cont.] (Wayb.)>, but I'm just not getting it. I am *begging* someone who knows how to do this to have a look at my algorithm and correct it. The two big problems: 1) I find that sometimes in order to get the correct periodicity-block, I have to enter one or both pairs of the unison-vector exponents with the signs (=/-) reversed. 2) Sometimes the blocks are centered on 1/1, as I intend for them to be ... but sometimes they're not, with 1/1 being plotted at a corner. Also, it seems that sometimes the order in which I list the unison-vectors makes a difference. Is it supposed to? (I don't think so...) The code within the "LOOP" is the part that needs fixing. Thanks in advance for any help received! -monz> ----- Original Message ----- >>> From: monz <joemonz@xxxxx.xxx> >> To: <tuning-math@xxxxxxxxxxx.xxx> >> Sent: Sunday, December 23, 2001 12:28 AM >> Subject: Re: [tuning-math] coordinates from unison-vectors (was: 55-tET) >> >> >> >> Here's the pseudo-code for the formulas in my spreadsheet. >> Please feel free to correct any errors or to make the code >> more elegant. >> >> >> >> unison-vectors = >> >> (3^a) * (5^b) >> (3^c) * (5^d) >> >> >> >> unison-vector matrix = >> >> (a b) >> (c d) >> >> >> >> determinant n of the matrix : >> >> n = (a*d) - (c*b) >> >> >> >> inverse of the matrix = >> >> ( d -b) >> (-c a) >> ------- >> n >> >> >> >> inverse coordinates p, q : >> >> >> p = 0, q = 0 >> >> >> LOOP >> >> '-- This part finds the values p,q which fit into the unit-cube >> >> if ABS(p+d) > (ABS(n)/2) >> >> then p = MOD(p+d, ABS(n)) - ABS(n) >> >> else p = p + d >> >> end if >> >> >> >> if ABS(q-b) > (ABS(n)/2) >> >> then q = MOD(q-b, ABS(n)) - ABS(n) >> >> else q = q - b >> >> end if >> >> >> >> lattice coordinates x, y : >> >> '-- This part transforms back to the original space >> >> x = ( (q*c) + (p*a) ) / n >> >> y = ( (q*d) + (p*b) ) / n >> >> >> >> END LOOP >> >> > > _________________________________________________________Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2876 - Contents - Hide Contents Date: Sun, 30 Dec 2001 03:07:02 Subject: Re: the unison-vectordeterminant relationship From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> Ah, so this is related to the tensor product? >> A wedge product can be defined as a quotient of a tensor product;it is an antisymmetrized tensor product, in effect.> > Is there an ellipsoid>> associated with the wedge product?? >> There's a parallepiped associated to it, and you could associate anellipsoid to that if you wanted to--why do you ask? This parallelepiped -- does it exist in the lattice of notes? Perhaps after some transformation? (I really need to go through the GA Matlab tutorial you gave me -- haven't had time yet).
Message: 2877 - Contents - Hide Contents Date: Sun, 30 Dec 2001 20:38:34 Subject: Re: coordinates from unison-vectors From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 30, 2001 8:07 PM > Subject: Re: [tuning-math] coordinates from unison-vectors > <Yahoo groups: /tuning-math/message/2332 * [with cont.] > > > > A plea to all who understand matrix math: > > > A week ago, I posted the pseudo-code for the formulas > in my Excel spreadsheet which calculates the coordinates > of a 2-dimensional periodicity-block from a given pair > of unison-vectors. <etc.>I realized that if I elaborate more on what's going on in my spreadsheet, I'm more likely to get help. The first part of the "LOOP" treats the unison-vectors as boundaries of a unit-cube, and calculates values p,q for the coordinates within that unit-cube, on the transformed lattice. I think this is working OK... but if anyone wants to check... The second part of the "LOOP" transforms these coordinates back to the original lattice, as values x,y. This is the part that is definitely *not* working all the time. Paul has already mentioned that I should be dividing by the determinant n again at the end, but it doesn't work at all if I leave that out. I don't understand why he says that either, because the inverse matrix has the determinant as the denominator. I think Paul is confused because he uses the decimal fractional values of the adjoint, whereas I keep the exponents in integer form and require the determinant denominator. But of course, I'm way out of my league here and could definitely be very wrong... -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2878 - Contents - Hide Contents Date: Sun, 30 Dec 2001 22:16:08 Subject: Re: coordinates from unison-vectors From: monz Wow... big, *BIG* Oops!... my bad!> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 30, 2001 8:38 PM > Subject: Re: [tuning-math] coordinates from unison-vectors > > ... > Paul has already mentioned that I should be dividing by > the determinant n again at the end, ...That should be "Paul has already mentioned that I should *NOT* be dividing by the determinant n again ^^^^^ at the end,..." -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2879 - Contents - Hide Contents Date: Sun, 30 Dec 2001 03:30:15 Subject: Re: the unison-vectordeterminant relationship From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> This parallelepiped -- does it exist in the lattice of notes?It's simply the parallepiped defined by the vectors we are wedging together, so yes. The wedge of two vectors is the directed area of the associated parallogram, of three vectors the directed volume of the associated parallepiped, and so forth.
Message: 2880 - Contents - Hide Contents Date: Sun, 30 Dec 2001 03:48:29 Subject: Re: the unison-vectordeterminant relationship From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> 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.Right, and if you used a different unison vector besides (19 9) to define your JI block, you would also see this.> 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.> Is there > any kind of significance to it?In my opinion, no.> > 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?Assuming the lattice is not wrapped into a cylinder, it might make some infinitesimal difference. If the lattice is wrapped around a cylinder, as it should be if you're talking about an actual meantone tuning, then you're implying an infinite number of JI ratios no matter what meantone tuning you're using.
Message: 2881 - Contents - Hide Contents Date: Sun, 30 Dec 2001 03:49:29 Subject: Re: the unison-vectordeterminant relationship From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> This parallelepiped -- does it exist in the lattice of notes? >> It's simply the parallepiped defined by the vectors we are wedging >together, so yes. The wedge of two vectors is the directed area of >the associated parallogram, of three vectors the directed volume of >the associated parallepiped, and so forth.So the wegde product is simply the boundary of the Fokker periodicity block??
Message: 2882 - Contents - Hide Contents Date: Sun, 30 Dec 2001 04:07:07 Subject: Re: the unison-vectordeterminant relationship From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Saturday, December 29, 2001 7:48 PM >> Subject: [tuning-math] Re: the unison-vector<-->determinant relationship >> >> >> --- 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. > > Right, I understand that. But since any given meantone interval > can only be closest to *one* particular 5-limit JI ratio,While functioning as an infinite number.> which > should fall within the periodicity-block, the vector of the meantone > chain will *still* imply a unique periodicity-block, will it not?I don't get it. What's the vector of the meantone chain? Is (19 9) an example?> (assuming that the periodicity-block is replicated a comma away > as one travels around the cylinder)Assuming to the answer to the last question it "yes", then I'd say, "no, one would get a "strip" rather than a periodicity block", but then of course I'd be ignoring your "since any given meantone interval can only be closest to *one* particular 5-limit JI ratio". If I take that part seriously, I have two comments: (a) You are NOT, with your current method, mapping identical meantone intervals to identical JI ratios, and (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.
Message: 2883 - Contents - Hide Contents Date: Sun, 30 Dec 2001 04:40:54 Subject: Nonatonic scales From: genewardsmith I've put the ones with connectivity of two in front; the number is the edge connectivity. I'm afraid I was lazy and didn't try to find a good mode for these; it would be interesting to see what the lattice diagrams of these scales look like, and that would tell us a good choice for 1/1. [1, 25/24, 10/9, 32/27, 4/3, 25/18, 40/27, 5/3, 16/9] [25/24, 16/15, 16/15, 9/8, 25/24, 16/15, 9/8, 16/15, 9/8] 2 [1, 25/24, 10/9, 5/4, 125/96, 25/18, 25/16, 5/3, 15/8] [25/24, 16/15, 9/8, 25/24, 16/15, 9/8, 16/15, 9/8, 16/15] 2 [1, 25/24, 10/9, 5/4, 4/3, 25/18, 40/27, 5/3, 16/9] [25/24, 16/15, 9/8, 16/15, 25/24, 16/15, 9/8, 16/15, 9/8] 2 [1, 25/24, 10/9, 5/4, 4/3, 25/18, 25/16, 5/3, 16/9] [25/24, 16/15, 9/8, 16/15, 25/24, 9/8, 16/15, 16/15, 9/8] 2 [1, 25/24, 625/576, 125/108, 125/96, 25/18, 25/16, 5/3, 15/8] [25/24, 25/24, 16/15, 9/8, 16/15, 9/8, 16/15, 9/8, 16/15] 1 [1, 25/24, 10/9, 125/108, 100/81, 25/18, 40/27, 5/3, 16/9] [25/24, 16/15, 25/24, 16/15, 9/8, 16/15, 9/8, 16/15, 9/8] 1 [1, 25/24, 10/9, 125/108, 125/96, 25/18, 40/27, 5/3, 16/9] [25/24, 16/15, 25/24, 9/8, 16/15, 16/15, 9/8, 16/15, 9/8] 1 [1, 25/24, 10/9, 32/27, 100/81, 25/18, 40/27, 5/3, 16/9] [25/24, 16/15, 16/15, 25/24, 9/8, 16/15, 9/8, 16/15, 9/8] 1 [1, 25/24, 10/9, 32/27, 4/3, 25/18, 40/27, 128/81, 16/9] [25/24, 16/15, 16/15, 9/8, 25/24, 16/15, 16/15, 9/8, 9/8] 1 [1, 25/24, 10/9, 32/27, 4/3, 25/18, 25/16, 5/3, 16/9] [25/24, 16/15, 16/15, 9/8, 25/24, 9/8, 16/15, 16/15, 9/8] 1 [1, 25/24, 10/9, 32/27, 4/3, 64/45, 40/27, 5/3, 16/9] [25/24, 16/15, 16/15, 9/8, 16/15, 25/24, 9/8, 16/15, 9/8] 1 [1, 25/24, 10/9, 32/27, 4/3, 64/45, 8/5, 5/3, 16/9] [25/24, 16/15, 16/15, 9/8, 16/15, 9/8, 25/24, 16/15, 9/8] 1 [1, 25/24, 10/9, 32/27, 4/3, 3/2, 8/5, 5/3, 16/9] [25/24, 16/15, 16/15, 9/8, 9/8, 16/15, 25/24, 16/15, 9/8] 1 [1, 25/24, 10/9, 5/4, 125/96, 25/18, 25/16, 5/3, 16/9] [25/24, 16/15, 9/8, 25/24, 16/15, 9/8, 16/15, 16/15, 9/8] 1 [1, 25/24, 10/9, 5/4, 4/3, 25/18, 40/27, 5/3, 15/8] [25/24, 16/15, 9/8, 16/15, 25/24, 16/15, 9/8, 9/8, 16/15] 1 [1, 25/24, 10/9, 5/4, 4/3, 64/45, 8/5, 128/75, 16/9] [25/24, 16/15, 9/8, 16/15, 16/15, 9/8, 16/15, 25/24, 9/8] 1
Message: 2884 - Contents - Hide Contents Date: Sun, 30 Dec 2001 05:02:10 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: clumma>>> >t'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), >>>> Which is why I suggested that the plug-in-the-fundamentals-only >> shortcut shouldn't be applied. >>Why not? Afraid of a little assymetry?Only when it's spelled like that. ;) 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?>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. What do you have in mind? I'm not clear how one would go about testing anything that's been said here.>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. 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#. Scale is treated like a chord, not melodically -- tone cluster on harmonium for drone, melody is essentially a scale 'arpeggio' figure centered on notes of the scale (usually three notes are used as centers of this pattern, sometimes they form a 1st-inversion minor triad). "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). I still say there's nothing here that would turn up an optimized 5-limit temperament! This music bores the hell out of me when I'm not going for the glassy partial soup that I love so much. "The Gamelan of Cirebon" There's less triadic inuendo here. Their low 'phones have a more resolved timbre than the Balinese, above, ensemble had. Also, I haven't heard pelog on this disc. Seems to be sorog (one chinese, one pelog tetrachord). I guess it all depends if you consider these tonic changes or just points of symmetry in a melisma (sp?). If there's anything that would produce an optimum 5-limit temperament in there either way, I'll eat my shoe. -Carl
Message: 2885 - Contents - Hide Contents Date: Sun, 30 Dec 2001 00:00:09 Subject: Re: non-uniqueness of a^(b/c) type numbers From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> 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.You mean, if a and d are prime? Looks about right. Now let's look at a more relevant example, how about 7/26-comma meantone. How did you express the fifth and the major third of this tuning again?
Message: 2886 - Contents - Hide Contents Date: Sun, 30 Dec 2001 08:38:46 Subject: Decatonic scales From: genewardsmith Here they be: [1, 81/80, 27/25, 6/5, 243/200, 162/125, 36/25, 8/5, 81/50, 9/5] [81/80, 16/15, 10/9, 81/80, 16/15, 10/9, 10/9, 81/80, 10/9, 10/9] 2 [1, 81/80, 27/25, 6/5, 243/200, 27/20, 36/25, 8/5, 81/50, 9/5] [81/80, 16/15, 10/9, 81/80, 10/9, 16/15, 10/9, 81/80, 10/9, 10/9] 2 [1, 81/80, 27/25, 6/5, 243/200, 27/20, 3/2, 243/160, 81/50, 9/5] [81/80, 16/15, 10/9, 81/80, 10/9, 10/9, 81/80, 16/15, 10/9, 10/9] 2 [1, 81/80, 27/25, 6/5, 243/200, 27/20, 3/2, 243/160, 27/16, 9/5] [81/80, 16/15, 10/9, 81/80, 10/9, 10/9, 81/80, 10/9, 16/15, 10/9] 2 [1, 81/80, 27/25, 6/5, 243/200, 27/20, 3/2, 8/5, 81/50, 9/5] [81/80, 16/15, 10/9, 81/80, 10/9, 10/9, 16/15, 81/80, 10/9, 10/9] 2 [1, 81/80, 27/25, 6/5, 243/200, 162/125, 36/25, 729/500, 81/50, 9/5] [81/80, 16/15, 10/9, 81/80, 16/15, 10/9, 81/80, 10/9, 10/9, 10/9] 1 [1, 81/80, 27/25, 6/5, 243/200, 27/20, 36/25, 729/500, 81/50, 9/5] [81/80, 16/15, 10/9, 81/80, 10/9, 16/15, 81/80, 10/9, 10/9, 10/9] 1 [1, 81/80, 27/25, 6/5, 32/25, 162/125, 36/25, 8/5, 81/50, 9/5] [81/80, 16/15, 10/9, 16/15, 81/80, 10/9, 10/9, 81/80, 10/9, 10/9] 1 [1, 81/80, 27/25, 6/5, 4/3, 27/20, 36/25, 8/5, 16/9, 9/5] [81/80, 16/15, 10/9, 10/9, 81/80, 16/15, 10/9, 10/9, 81/80, 10/9] 1 [1, 81/80, 27/25, 6/5, 4/3, 27/20, 3/2, 243/160, 27/16, 9/5] [81/80, 16/15, 10/9, 10/9, 81/80, 10/9, 81/80, 10/9, 16/15, 10/9] 1 [1, 81/80, 27/25, 6/5, 4/3, 27/20, 3/2, 8/5, 16/9, 9/5] [81/80, 16/15, 10/9, 10/9, 81/80, 10/9, 16/15, 10/9, 81/80, 10/9] 1 [1, 81/80, 27/25, 6/5, 4/3, 27/20, 3/2, 5/3, 16/9, 9/5] [81/80, 16/15, 10/9, 10/9, 81/80, 10/9, 10/9, 16/15, 81/80, 10/9] 1 [1, 81/80, 9/8, 729/640, 81/64, 27/20, 3/2, 243/160, 27/16, 9/5] [81/80, 10/9, 81/80, 10/9, 16/15, 10/9, 81/80, 10/9, 16/15, 10/9] 1
Message: 2887 - Contents - Hide Contents Date: Sun, 30 Dec 2001 00:06:31 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: >>>>>> Simple neutral thirds? as opposed to the complex ones above? >>>> If 25/24 is a unison, then 6/5~5/4, and that is the basis of this > temperament. >> Sure. But if we call this temperament "neutral thirds temperament" > without qualification, this conflicts with the usage in Graham's > catalogReally? Why so? He also uses a generator of a neutral third in his "neutral thirds temperament".
Message: 2888 - Contents - Hide Contents Date: Sun, 30 Dec 2001 09:38:06 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:> Have you ever looked at it? It's basically everything that ever > entered anybody's fancy. There are scales in there named after > me I don't even remember making up.Not really--I don't think any of my scales are in there.
Message: 2889 - Contents - Hide Contents Date: Sun, 30 Dec 2001 00:10:09 Subject: Re: the unison-vectordeterminant relationship From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> ... you can see that the 1/6-comma meantone vector is has an > off-kilter relationship with 5-limit JI. Two different linear > algebras simultaneous at work, right? (I sure hope so...)If you look at u^v, then it is linear in u *and* v, hence bilinear. The same is true of a related doodad called the tensor product, and of the dot product, which is a bilinear form. So all this stuff is called "multilinear algebra", except in contexts where something called the Clifford algebra comes to the fore and we have "geometric algebra" instead.> > 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?
Message: 2890 - Contents - Hide Contents Date: Sun, 30 Dec 2001 10:58:20 Subject: Classes of 7-limit scales From: genewardsmith Here are the classes of 7-limit epimorphic superparticular scales. The larger ones are far beyond the limits of computation so far as classifying all of the connected scales. I think a subset of these, with some kind of propriety condition, and with the number of scale degrees not too large, is what needs to be considered. 4 [64/63, 21/20, 5/4, 3/2] [1, 1, 1, 1] [36/35, 28/27, 5/4, 3/2] [1, 1, 1, 1] [36/35, 10/9, 7/6, 3/2] [1, 1, 1, 1] [36/35, 7/6, 5/4, 4/3] [1, 1, 1, 1] [28/27, 15/14, 6/5, 3/2] [1, 1, 1, 1] [21/20, 10/9, 8/7, 3/2] [1, 1, 1, 1] [21/20, 8/7, 5/4, 4/3] [1, 1, 1, 1] [16/15, 15/14, 7/6, 3/2] [1, 1, 1, 1] [15/14, 7/6, 6/5, 4/3] [1, 1, 1, 1] [8/7, 7/6, 6/5, 5/4] [1, 1, 1, 1] 5 [225/224, 126/125, 10/9, 4/3] [1, 1, 1, 2] [225/224, 28/27, 6/5, 4/3] [1, 1, 2, 1] [225/224, 21/20, 16/15, 4/3] [1, 1, 1, 2] [225/224, 16/15, 7/6, 3/2] [1, 2, 1, 1] [126/125, 15/14, 10/9, 3/2] [1, 1, 2, 1] [81/80, 28/27, 15/14, 4/3] [1, 1, 1, 2] [64/63, 21/20, 6/5, 5/4] [1, 1, 1, 2] [49/48, 8/7, 6/5, 5/4] [1, 2, 1, 1] [36/35, 28/27, 6/5, 5/4] [1, 1, 1, 2] [36/35, 21/20, 10/9, 3/2] [1, 1, 2, 1] [36/35, 16/15, 7/6, 5/4] [1, 1, 1, 2] [36/35, 15/14, 7/6, 4/3] [1, 1, 2, 1] [36/35, 8/7, 7/6, 5/4] [1, 1, 2, 1] [28/27, 15/14, 6/5, 5/4] [1, 1, 2, 1] [25/24, 8/7, 7/6, 6/5] [1, 1, 1, 2] [21/20, 16/15, 8/7, 5/4] [1, 1, 1, 2] [21/20, 8/7, 7/6, 5/4] [1, 2, 1, 1] [15/14, 10/9, 7/6, 6/5] [1, 1, 1, 2] [15/14, 8/7, 7/6, 6/5] [1, 1, 2, 1] 6 [2401/2400, 15/14, 8/7, 5/4] [1, 1, 3, 1] [2401/2400, 15/14, 8/7, 4/3] [1, 2, 2, 1] [225/224, 64/63, 21/20, 4/3] [1, 1, 2, 2] [225/224, 28/27, 10/9, 6/5] [1, 1, 1, 3] [81/80, 10/9, 8/7, 7/6] [1, 1, 2, 2] [64/63, 21/20, 8/7, 5/4] [1, 2, 1, 2] [49/48, 36/35, 10/9, 3/2] [1, 2, 2, 1] [49/48, 25/24, 8/7, 6/5] [1, 1, 2, 2] [49/48, 21/20, 8/7, 5/4] [1, 1, 3, 1] [49/48, 16/15, 15/14, 3/2] [1, 2, 2, 1] [36/35, 28/27, 7/6, 5/4] [2, 1, 1, 2] [36/35, 25/24, 7/6, 4/3] [2, 1, 2, 1] [36/35, 15/14, 8/7, 7/6] [1, 1, 1, 3] [36/35, 10/9, 7/6, 5/4] [2, 1, 2, 1] [28/27, 25/24, 15/14, 6/5] [1, 1, 1, 3] [28/27, 15/14, 7/6, 6/5] [1, 2, 1, 2] [25/24, 21/20, 8/7, 4/3] [1, 2, 2, 1] [21/20, 16/15, 15/14, 5/4] [1, 2, 1, 2] [21/20, 15/14, 10/9, 6/5] [1, 1, 2, 2] [21/20, 15/14, 8/7, 7/6] [1, 1, 2, 2] [21/20, 10/9, 8/7, 5/4] [2, 1, 2, 1] [16/15, 15/14, 7/6, 6/5] [1, 2, 2, 1] 7 [4375/4374, 126/125, 15/14, 6/5] [1, 1, 2, 3] [2401/2400, 15/14, 8/7, 7/6] [1, 2, 3, 1] [225/224, 126/125, 10/9, 6/5] [1, 1, 3, 2] [225/224, 28/27, 15/14, 6/5] [1, 2, 1, 3] [225/224, 21/20, 16/15, 5/4] [1, 1, 3, 2] [126/125, 28/27, 15/14, 3/2] [1, 2, 3, 1] [81/80, 50/49, 28/27, 4/3] [2, 1, 2, 2] [64/63, 25/24, 21/20, 6/5] [1, 2, 1, 3] [64/63, 21/20, 9/8, 5/4] [2, 2, 1, 2] [50/49, 49/48, 8/7, 6/5] [1, 2, 2, 2] [50/49, 21/20, 16/15, 5/4] [1, 2, 2, 2] [50/49, 21/20, 10/9, 6/5] [1, 2, 2, 2] [50/49, 21/20, 8/7, 7/6] [1, 2, 2, 2] [49/48, 36/35, 8/7, 5/4] [2, 1, 3, 1] [49/48, 15/14, 8/7, 6/5] [2, 1, 3, 1] [36/35, 28/27, 25/24, 6/5] [1, 1, 2, 3] [36/35, 28/27, 9/8, 5/4] [2, 2, 1, 2] [36/35, 25/24, 8/7, 7/6] [2, 1, 1, 3] [36/35, 16/15, 15/14, 7/6] [1, 1, 2, 3] [36/35, 15/14, 10/9, 7/6] [2, 1, 1, 3] [36/35, 10/9, 9/8, 7/6] [2, 2, 1, 2] [28/27, 15/14, 9/8, 6/5] [2, 2, 1, 2] [25/24, 21/20, 8/7, 7/6] [1, 2, 3, 1] [21/20, 10/9, 9/8, 8/7] [2, 2, 1, 2] [16/15, 15/14, 9/8, 7/6] [2, 2, 1, 2] 8 [2401/2400, 49/48, 15/14, 8/7] [1, 1, 2, 4] [2401/2400, 16/15, 15/14, 4/3] [1, 2, 4, 1] [225/224, 49/48, 16/15, 3/2] [2, 1, 4, 1] [225/224, 28/27, 16/15, 6/5] [2, 2, 1, 3] [225/224, 16/15, 7/6, 6/5] [2, 3, 2, 1] [126/125, 64/63, 25/24, 6/5] [1, 1, 3, 3] [126/125, 25/24, 8/7, 4/3] [2, 3, 2, 1] [126/125, 15/14, 10/9, 5/4] [2, 2, 3, 1] [81/80, 49/48, 10/9, 8/7] [1, 2, 1, 4] [81/80, 36/35, 10/9, 7/6] [1, 2, 3, 2] [81/80, 21/20, 10/9, 8/7] [1, 2, 3, 2] [64/63, 21/20, 15/14, 5/4] [2, 3, 1, 2] [49/48, 25/24, 21/20, 8/7] [1, 1, 2, 4] [49/48, 21/20, 15/14, 8/7] [2, 1, 1, 4] [36/35, 28/27, 15/14, 7/6] [2, 1, 2, 3] [36/35, 25/24, 10/9, 7/6] [3, 1, 1, 3] [36/35, 21/20, 10/9, 5/4] [2, 2, 3, 1] [28/27, 21/20, 15/14, 6/5] [2, 1, 3, 2] [25/24, 21/20, 10/9, 8/7] [1, 3, 1, 3] [21/20, 16/15, 15/14, 7/6] [1, 2, 3, 2] [21/20, 15/14, 10/9, 8/7] [3, 1, 2, 2] 9 [2401/2400, 50/49, 15/14, 8/7] [2, 1, 2, 4] [2401/2400, 50/49, 8/7, 6/5] [2, 3, 2, 2] [2401/2400, 16/15, 15/14, 5/4] [1, 3, 4, 1] [225/224, 36/35, 28/27, 6/5] [2, 1, 3, 3] [225/224, 36/35, 16/15, 7/6] [2, 1, 3, 3] [225/224, 16/15, 9/8, 7/6] [2, 4, 1, 2] [126/125, 25/24, 8/7, 7/6] [2, 3, 3, 1] [126/125, 15/14, 10/9, 9/8] [2, 2, 4, 1] [126/125, 15/14, 10/9, 7/6] [2, 3, 3, 1] [81/80, 28/27, 15/14, 6/5] [1, 3, 3, 2] [64/63, 50/49, 21/20, 5/4] [2, 1, 4, 2] [64/63, 21/20, 10/9, 9/8] [2, 2, 2, 3] [50/49, 49/48, 21/20, 8/7] [1, 2, 2, 4] [50/49, 21/20, 10/9, 8/7] [1, 4, 2, 2] [49/48, 36/35, 15/14, 8/7] [3, 1, 1, 4] [36/35, 28/27, 10/9, 9/8] [2, 2, 2, 3] [36/35, 25/24, 16/15, 7/6] [3, 2, 1, 3] [36/35, 21/20, 10/9, 9/8] [2, 2, 4, 1] [28/27, 16/15, 15/14, 9/8] [2, 2, 2, 3] [25/24, 21/20, 16/15, 8/7] [2, 3, 1, 3] 10 [2401/2400, 64/63, 15/14, 8/7] [2, 1, 4, 3] [2401/2400, 16/15, 15/14, 7/6] [1, 3, 5, 1] [225/224, 64/63, 21/20, 5/4] [1, 3, 4, 2] [126/125, 81/80, 15/14, 10/9] [2, 1, 2, 5] [126/125, 49/48, 25/24, 8/7] [2, 1, 3, 4] [126/125, 21/20, 15/14, 10/9] [2, 1, 3, 4] [81/80, 64/63, 10/9, 7/6] [3, 2, 3, 2] [81/80, 36/35, 21/20, 10/9] [1, 2, 2, 5] [81/80, 28/27, 10/9, 8/7] [3, 2, 3, 2] [64/63, 25/24, 21/20, 8/7] [1, 2, 4, 3] [64/63, 21/20, 15/14, 7/6] [2, 3, 3, 2] [49/48, 36/35, 25/24, 8/7] [3, 2, 1, 4] [49/48, 36/35, 10/9, 5/4] [2, 4, 3, 1] [49/48, 16/15, 15/14, 6/5] [2, 3, 4, 1] [36/35, 28/27, 25/24, 7/6] [4, 1, 2, 3] [36/35, 21/20, 15/14, 10/9] [2, 3, 1, 4] [28/27, 21/20, 15/14, 8/7] [2, 3, 3, 2] [21/20, 16/15, 15/14, 10/9] [3, 2, 3, 2] 11 [4375/4374, 36/35, 21/20, 3/2] [2, 5, 3, 1] [2401/2400, 50/49, 21/20, 8/7] [2, 3, 2, 4] [225/224, 126/125, 15/14, 10/9] [1, 3, 2, 5] [225/224, 28/27, 16/15, 9/8] [2, 2, 4, 3] [225/224, 21/20, 16/15, 7/6] [3, 1, 5, 2] [126/125, 28/27, 15/14, 5/4] [2, 3, 5, 1] [126/125, 25/24, 15/14, 10/9] [3, 1, 3, 4] [126/125, 25/24, 10/9, 8/7] [3, 4, 1, 3] [50/49, 49/48, 36/35, 8/7] [1, 4, 2, 4] [50/49, 36/35, 21/20, 10/9] [1, 2, 4, 4] [50/49, 28/27, 21/20, 6/5] [3, 2, 4, 2] [50/49, 21/20, 16/15, 7/6] [3, 4, 2, 2] [49/48, 36/35, 10/9, 9/8] [2, 4, 4, 1] [49/48, 16/15, 15/14, 9/8] [2, 4, 4, 1] [36/35, 28/27, 15/14, 9/8] [2, 4, 2, 3] [36/35, 25/24, 21/20, 10/9] [3, 1, 3, 4] 12 [2401/2400, 225/224, 16/15, 4/3] [1, 4, 6, 1] [2401/2400, 49/48, 16/15, 15/14] [1, 1, 4, 6] [225/224, 36/35, 28/27, 7/6] [2, 4, 3, 3] [126/125, 28/27, 15/14, 7/6] [2, 3, 6, 1] [126/125, 25/24, 16/15, 8/7] [3, 5, 1, 3] [81/80, 64/63, 21/20, 10/9] [3, 2, 2, 5] [81/80, 49/48, 36/35, 10/9] [1, 2, 4, 5] [81/80, 36/35, 28/27, 10/9] [3, 2, 2, 5] [64/63, 21/20, 15/14, 10/9] [2, 5, 3, 2] [49/48, 21/20, 16/15, 15/14] [2, 1, 4, 5] [28/27, 21/20, 16/15, 15/14] [2, 3, 2, 5] [25/24, 21/20, 16/15, 15/14] [2, 3, 4, 3] 13 [4375/4374, 126/125, 15/14, 10/9] [1, 4, 5, 3] [2401/2400, 225/224, 16/15, 5/4] [1, 4, 7, 1] [2401/2400, 64/63, 16/15, 15/14] [2, 1, 3, 7] [2401/2400, 64/63, 15/14, 9/8] [2, 4, 4, 3] [2401/2400, 50/49, 16/15, 15/14] [2, 1, 4, 6] [225/224, 126/125, 16/15, 10/9] [3, 3, 2, 5] [225/224, 21/20, 16/15, 10/9] [3, 3, 5, 2] [126/125, 50/49, 49/48, 8/7] [2, 3, 4, 4] [126/125, 50/49, 21/20, 10/9] [2, 3, 4, 4] [126/125, 28/27, 15/14, 9/8] [2, 4, 6, 1] [81/80, 28/27, 15/14, 8/7] [3, 5, 3, 2] [64/63, 50/49, 21/20, 7/6] [2, 3, 6, 2] [64/63, 25/24, 21/20, 9/8] [4, 2, 4, 3] [50/49, 28/27, 21/20, 8/7] [3, 2, 6, 2] [50/49, 21/20, 16/15, 10/9] [3, 6, 2, 2] [49/48, 36/35, 16/15, 15/14] [3, 1, 4, 5] [49/48, 36/35, 15/14, 10/9] [3, 5, 1, 4] [36/35, 28/27, 25/24, 9/8] [4, 4, 2, 3] 14 [4375/4374, 49/48, 36/35, 3/2] [2, 3, 8, 1] [225/224, 49/48, 16/15, 6/5] [4, 2, 7, 1] [126/125, 64/63, 25/24, 8/7] [4, 1, 6, 3] [126/125, 36/35, 25/24, 10/9] [3, 3, 4, 4] [126/125, 28/27, 21/20, 15/14] [2, 4, 1, 7] [81/80, 64/63, 28/27, 10/9] [5, 2, 2, 5] [64/63, 28/27, 21/20, 15/14] [2, 2, 5, 5] [49/48, 36/35, 25/24, 10/9] [3, 6, 1, 4] [36/35, 28/27, 21/20, 15/14] [2, 4, 3, 5] 15 [2401/2400, 225/224, 16/15, 7/6] [1, 5, 8, 1] [2401/2400, 50/49, 36/35, 8/7] [4, 5, 2, 4] [225/224, 49/48, 16/15, 9/8] [4, 2, 8, 1] [225/224, 36/35, 28/27, 9/8] [2, 4, 6, 3] [225/224, 25/24, 21/20, 16/15] [3, 2, 3, 7] [126/125, 81/80, 28/27, 15/14] [2, 1, 5, 7] [126/125, 28/27, 25/24, 15/14] [3, 4, 1, 7] [126/125, 25/24, 16/15, 15/14] [3, 5, 4, 3] [81/80, 50/49, 28/27, 6/5] [4, 3, 6, 2] [81/80, 28/27, 16/15, 15/14] [3, 5, 2, 5] [64/63, 50/49, 21/20, 10/9] [2, 3, 8, 2] [50/49, 49/48, 36/35, 10/9] [1, 4, 6, 4] [50/49, 25/24, 21/20, 16/15] [3, 2, 6, 4] [36/35, 25/24, 21/20, 16/15] [3, 5, 3, 4] 16 [4375/4374, 126/125, 28/27, 15/14] [1, 4, 3, 8] [2401/2400, 64/63, 21/20, 15/14] [2, 4, 3, 7] [225/224, 126/125, 28/27, 15/14] [1, 3, 5, 7] [225/224, 64/63, 21/20, 7/6] [3, 5, 6, 2] [81/80, 64/63, 49/48, 10/9] [5, 4, 2, 5] [64/63, 49/48, 21/20, 15/14] [4, 2, 5, 5] [64/63, 25/24, 21/20, 15/14] [4, 2, 7, 3] 17 [4375/4374, 36/35, 21/20, 5/4] [3, 8, 5, 1] [2401/2400, 126/125, 50/49, 8/7] [4, 2, 7, 4] [225/224, 49/48, 21/20, 16/15] [5, 2, 1, 9] [225/224, 28/27, 21/20, 16/15] [5, 2, 3, 7] [126/125, 64/63, 25/24, 9/8] [4, 4, 6, 3] [126/125, 50/49, 25/24, 10/9] [6, 3, 4, 4] [81/80, 36/35, 28/27, 15/14] [3, 2, 7, 5] [50/49, 49/48, 21/20, 16/15] [5, 2, 6, 4] [50/49, 28/27, 21/20, 16/15] [5, 2, 8, 2] [49/48, 36/35, 28/27, 15/14] [3, 5, 4, 5] 18 [2401/2400, 225/224, 49/48, 16/15] [1, 6, 1, 10] [225/224, 126/125, 25/24, 16/15] [3, 3, 5, 7] [225/224, 64/63, 21/20, 10/9] [3, 5, 8, 2] [225/224, 49/48, 36/35, 16/15] [5, 3, 1, 9] [126/125, 36/35, 25/24, 16/15] [3, 3, 8, 4] [49/48, 36/35, 25/24, 16/15] [3, 6, 5, 4] 19 [2401/2400, 225/224, 50/49, 16/15] [2, 6, 1, 10] [2401/2400, 50/49, 36/35, 10/9] [4, 5, 6, 4] [2401/2400, 50/49, 21/20, 16/15] [2, 7, 6, 4] [81/80, 64/63, 28/27, 15/14] [5, 2, 7, 5] [81/80, 50/49, 28/27, 8/7] [6, 3, 8, 2] [64/63, 50/49, 28/27, 21/20] [2, 5, 2, 10] [64/63, 50/49, 25/24, 21/20] [4, 3, 2, 10] [64/63, 36/35, 25/24, 21/20] [4, 3, 5, 7] [50/49, 36/35, 28/27, 21/20] [5, 2, 4, 8] [36/35, 28/27, 25/24, 21/20] [7, 4, 5, 3] 20 [2401/2400, 225/224, 64/63, 16/15] [2, 7, 1, 10] [225/224, 81/80, 28/27, 16/15] [5, 3, 5, 7] 21 [4375/4374, 36/35, 21/20, 9/8] [4, 10, 6, 1] [126/125, 50/49, 49/48, 10/9] [6, 7, 4, 4] [126/125, 50/49, 28/27, 21/20] [2, 7, 4, 8] [126/125, 50/49, 25/24, 16/15] [6, 3, 8, 4] [64/63, 50/49, 49/48, 21/20] [4, 5, 2, 10] [64/63, 49/48, 36/35, 15/14] [4, 7, 5, 5] 22 [4375/4374, 126/125, 36/35, 15/14] [4, 7, 3, 8] [4375/4374, 49/48, 36/35, 5/4] [3, 5, 13, 1] [4375/4374, 36/35, 21/20, 15/14] [4, 10, 7, 1] [225/224, 64/63, 25/24, 21/20] [3, 7, 2, 10] [126/125, 36/35, 28/27, 25/24] [3, 7, 4, 8] [81/80, 64/63, 25/24, 21/20] [3, 7, 5, 7] [81/80, 36/35, 28/27, 25/24] [3, 7, 7, 5] [49/48, 36/35, 28/27, 25/24] [3, 10, 4, 5] 23 [4375/4374, 50/49, 36/35, 21/20] [4, 1, 10, 8] [4375/4374, 36/35, 25/24, 21/20] [4, 11, 1, 7] [2401/2400, 64/63, 50/49, 21/20] [2, 4, 7, 10] [225/224, 126/125, 28/27, 16/15] [8, 3, 5, 7] [126/125, 64/63, 25/24, 15/14] [7, 4, 9, 3] [50/49, 49/48, 36/35, 16/15] [5, 8, 6, 4] 24 [225/224, 64/63, 28/27, 21/20] [5, 7, 2, 10] [225/224, 36/35, 28/27, 21/20] [5, 7, 9, 3] 25 [4375/4374, 126/125, 50/49, 15/14] [4, 10, 3, 8] [4375/4374, 81/80, 36/35, 21/20] [5, 1, 12, 7] [4375/4374, 81/80, 36/35, 16/15] [5, 8, 5, 7] [2401/2400, 126/125, 50/49, 10/9] [4, 6, 11, 4] [81/80, 50/49, 28/27, 16/15] [8, 5, 10, 2] 26 [225/224, 64/63, 49/48, 21/20] [5, 9, 2, 10] [126/125, 64/63, 36/35, 25/24] [7, 4, 3, 12] [81/80, 64/63, 49/48, 15/14] [5, 9, 7, 5] [64/63, 49/48, 36/35, 25/24] [4, 7, 10, 5] 27 [4375/4374, 49/48, 36/35, 9/8] [4, 6, 16, 1] [225/224, 81/80, 36/35, 28/27] [5, 3, 7, 12] [225/224, 49/48, 36/35, 28/27] [5, 3, 10, 9] [81/80, 50/49, 36/35, 28/27] [8, 5, 2, 12] [50/49, 49/48, 36/35, 28/27] [5, 8, 10, 4] 29 [4375/4374, 49/48, 36/35, 15/14] [4, 7, 17, 1] [126/125, 81/80, 64/63, 25/24] [7, 3, 7, 12] [126/125, 81/80, 50/49, 28/27] [2, 8, 7, 12] [126/125, 64/63, 50/49, 25/24] [10, 4, 3, 12] [126/125, 50/49, 49/48, 16/15] [6, 11, 8, 4] [126/125, 50/49, 28/27, 25/24] [10, 7, 4, 8] [81/80, 64/63, 50/49, 28/27] [10, 2, 5, 12] [81/80, 64/63, 28/27, 25/24] [10, 7, 7, 5] 30 [4375/4374, 126/125, 36/35, 25/24] [4, 7, 11, 8] [4375/4374, 81/80, 64/63, 16/15] [5, 13, 5, 7] [4375/4374, 49/48, 36/35, 25/24] [4, 7, 18, 1] [2401/2400, 225/224, 64/63, 21/20] [2, 7, 11, 10] [225/224, 126/125, 36/35, 28/27] [8, 3, 7, 12] 31 [4375/4374, 50/49, 49/48, 36/35] [4, 1, 8, 18] [2401/2400, 50/49, 36/35, 16/15] [8, 13, 6, 4] [64/63, 50/49, 49/48, 36/35] [4, 5, 12, 10] 32 [4375/4374, 81/80, 49/48, 36/35] [5, 1, 7, 19] [4375/4374, 81/80, 36/35, 28/27] [5, 8, 12, 7] [225/224, 126/125, 64/63, 25/24] [3, 10, 7, 12] 33 [4375/4374, 126/125, 50/49, 21/20] [4, 10, 11, 8] 34 [225/224, 81/80, 64/63, 28/27] [5, 10, 7, 12] 35 [2401/2400, 50/49, 36/35, 28/27] [8, 13, 10, 4] 36 [225/224, 64/63, 49/48, 36/35] [5, 9, 12, 10] [81/80, 64/63, 49/48, 25/24] [10, 14, 7, 5] 37 [4375/4374, 81/80, 64/63, 21/20] [5, 13, 12, 7] [2401/2400, 126/125, 50/49, 16/15] [8, 6, 19, 4] [225/224, 126/125, 50/49, 28/27] [8, 10, 7, 12] [126/125, 50/49, 49/48, 28/27] [10, 15, 8, 4] 39 [4375/4374, 2401/2400, 50/49, 36/35] [4, 8, 9, 18] 41 [4375/4374, 126/125, 50/49, 25/24] [4, 18, 11, 8] [126/125, 64/63, 50/49, 49/48] [10, 4, 15, 12] [81/80, 64/63, 50/49, 49/48] [10, 14, 5, 12] 43 [2401/2400, 64/63, 50/49, 36/35] [12, 4, 17, 10] 44 [4375/4374, 81/80, 64/63, 28/27] [5, 20, 12, 7] [225/224, 126/125, 64/63, 28/27] [15, 10, 7, 12] 45 [2401/2400, 126/125, 50/49, 28/27] [8, 10, 23, 4] 46 [4375/4374, 126/125, 81/80, 36/35] [12, 7, 8, 19] [225/224, 81/80, 64/63, 49/48] [5, 10, 19, 12] 49 [4375/4374, 126/125, 50/49, 49/48] [4, 18, 19, 8] 51 [4375/4374, 81/80, 64/63, 49/48] [5, 20, 19, 7] 53 [2401/2400, 126/125, 64/63, 50/49] [12, 10, 4, 27] [2401/2400, 81/80, 64/63, 50/49] [12, 10, 14, 17] 54 [4375/4374, 225/224, 126/125, 36/35] [12, 8, 15, 19] 56 [225/224, 126/125, 64/63, 49/48] [15, 10, 19, 12] 57 [4375/4374, 2401/2400, 126/125, 50/49] [4, 8, 18, 27] 60 [2401/2400, 225/224, 64/63, 36/35] [12, 17, 21, 10] 65 [4375/4374, 126/125, 81/80, 64/63] [12, 7, 27, 19] [4375/4374, 126/125, 81/80, 50/49] [12, 26, 8, 19] 70 [2401/2400, 225/224, 81/80, 64/63] [12, 17, 10, 31] 73 [4375/4374, 225/224, 126/125, 50/49] [12, 8, 34, 19] 80 [2401/2400, 225/224, 126/125, 64/63] [12, 27, 10, 31] 92 [4375/4374, 225/224, 126/125, 64/63] [12, 27, 34, 19]
Message: 2891 - Contents - Hide Contents Date: Sun, 30 Dec 2001 17:22 +0 Subject: Re: Optimal 5-Limit Generators for Dave From: graham@xxxxxxxxxx.xx.xx Gene:>>> If 25/24 is a unison, then 6/5~5/4, and that is the basis of this >> temperament. Dave:>> Sure. But if we call this temperament "neutral thirds temperament" >> without qualification, this conflicts with the usage in Graham's >> catalog Paul:> Really? Why so? He also uses a generator of a neutral third in > his "neutral thirds temperament".Yes, it looks okay to me. To pick through my description "Two neutral thirds make up a perfect fifth." That's true for 6:5=~5:4 "The neutral thirds can be called 11:9." But don't have to be. "The `wolf third' to make up a 7-note scale can be identified with 6:5 or 7:6 or neither." And in this case neither. I'd call the temperament with the wolf third as 6:5 the "typical neutral third temperament" or the "meantone-like neutral third temperament" but that isn't the only way of doing it. Taking meantone and dividing the fifths in two, but not calling the result an 11:9, would also be a neutral third temperament. I'm not sure if this is what some of these examples are intending. I should really update that description to specify that the scale be an MOS generated by the neutral third, or some permutation (like rast). Graham
Message: 2892 - Contents - Hide Contents Date: Sun, 30 Dec 2001 23:59:37 Subject: Re: Optimal 5-Limit Generators for Dave From: dkeenanuqnetau --- In tuning-math@y..., graham@m... wrote:> Paul:>> Really? Why so? He also uses a generator of a neutral third in >> his "neutral thirds temperament". >> Yes, it looks okay to me. To pick through my description ...I was referring to your catalog page which I thought did not allow for 5 to map to 1 gen. In any case, when we're talking 5-limit temperaments I think we should have different names or qualifiers for the different mappings of the prime 5 when the prime 3 maps to 2 gens. I suggested "simple neutral thirds temperament" for the [2 1] mapping and "complex neutral thirds temperament" for the [2 8] mapping. What are the useful mappings when we go to 7-limit? It would be useful to have all these mappings listed on your catalog page, Graham.
Message: 2893 - Contents - Hide Contents Date: Sun, 30 Dec 2001 01:57:22 Subject: Hexatonic scales From: genewardsmith Here are the connected 5-limit hexatonic superparticular scales; "c" is the edge connectivity. 1--25/24--5/4--4/3--8/5--48/25 [25/24 6/5 16/15 6/5 6/5 25/24] c = 1 1--6/5--36/25--3/2--8/5--5/3 [6/5 6/5 25/24 16/15 25/24 6/5] c = 1 1--6/5--5/4--4/3--8/5--5/3 [6/5 25/24 16/15 6/5 25/24 6/5] c = 2 1--6/5--5/4--3/2--25/16--5/3 [6/5 25/24 6/5 25/24 16/15 6/5] c = 1 1--10/9--4/3--27/20--3/2--9/5 [10/9 6/5 81/80 10/9 6/5 10/9] c = 1 1--6/5--4/3--27/20--3/2--9/5 [6/5 10/9 81/80 10/9 6/5 10/9] c = 1
Message: 2894 - Contents - Hide Contents Date: Sun, 30 Dec 2001 02:38:06 Subject: Re: the unison-vectordeterminant relationship From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> ... you can see that the 1/6-comma meantone vector is has an >> off-kilter relationship with 5-limit JI. Two different linear >> algebras simultaneous at work, right? (I sure hope so...) >> If you look at u^v, then it is linear in u *and* v, hence bilinear. >The same is true of a related doodad called the tensor product,Ah, so this is related to the tensor product? Is there an ellipsoid associated with the wedge product??
Message: 2895 - Contents - Hide Contents Date: Mon, 31 Dec 2001 00:48:48 Subject: Re: Maple and graph theory From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: >>> Maple has a graph theory package, which allows me to take a scale >> and compute its connectivity fairly easily. >> Is this the same as Dave Keenan's connectivity?Almost certainly not. I assume you are referring to where I say e.g. a meantone diatonic scale is 86% connected at the 9-limit. Which means that of the 21 possible dyads (considered octave-equivalently) 18 of them are (reasonable approximations of) 9-limit ratios. I assume that Gene's measure gave the number of connected subgraphs that were disconnected from each other.
Message: 2896 - Contents - Hide Contents Date: Mon, 31 Dec 2001 20:33:09 Subject: Re: Some 7-limit superparticular pentatonics From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:> Gene, are you allowing 9-limit edges?No, but it would be easy enough to do so. I'm already buried under the varied possibilities as it is, though, and also want to explore tempered scales using graph theory methods.> Tomorrow, I'm going to try and tune these > up and see what they sound like.I was wondering also; I think I'll fiddle around on FTS with some of these.
Message: 2897 - Contents - Hide Contents Date: Mon, 31 Dec 2001 01:29:40 Subject: Re: Maple and graph theory From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> I assume that Gene's measure gave the number of connected subgraphs > that were disconnected from each other.It's edge-connectivity--the number of edges which would need to be removed to make the graph disconnected.
Message: 2898 - Contents - Hide Contents Date: Mon, 31 Dec 2001 12:52:38 Subject: Re: coordinates from unison-vectors From: monz Hi J!> From: unidala <JGill99@xxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, December 31, 2001 8:05 AM > Subject: [tuning-math] Re: coordinates from unison-vectors > > > ... > > J Gill: So your (dynamite presentation!) XLS Spreadsheet ...Thanks for the positive feedback!> ... will allow determination of mean-tone "coefficients" which > "best fit" any 5-limit periodicity block?Well... the spreadsheet *allows* determination of the meantone coordinates and axis... but it still can't *make* the determination! The user still has to do some of the work. The spreadsheet automatically figures out where to close the meantone chain, at (+/- 1/2 determinant) generators. But I haven't yet "coordinated" [groan pun intended] the mathematics to determine the "best-fit" meantone automatically. So far, you can only try out different values and see what they look like.> Not being very adept with this PB stuff you guys apply,Have patience, J... I've spent the night working on a terrific graphic which will go into a Dictionary entry webpage for "Transformation". That should clear up a lot of the fog for you.> what are the mean-tone "coefficients" which represent > a "best-fit" for Ellis' "Duodene", with 12 pitches at: > > 1/1--16/15--9/8--6/5--5/4--4/3--45/32--3/2--8/5--5/3--9/5--15/8 > and > 16/15--9/8--6/5--5/4--4/3--45/32--3/2--8/5--5/3--9/5--15/8--2/1 > > and for > 1/1--16/15--9/8--6/5--5/4--4/3--7/5--3/2--8/5--5/3--9/5--15/8 > and > 16/15--9/8--6/5--5/4--4/3--7/5--3/2--8/5--5/3--9/5--15/8--2/1 > > > Curiously, J GillWell, as I said, I don't have mathematics to do the job, so I just try a few by eye and intuition, and see what looks good. First of all, the first two of your Duodenes are equivalent to each other on my lattice, because it's 2-dimensional for prime-factors 3 and 5, with 2 ignored. Secondly, I could use only that first pair, because the second pair are 7-limit, making the lattice 3-dimensional, and my spreadsheet doesn't handle that ... yet ... So: Ellis's Duodene is a 12-tone set defined by the unison-vectors [0 3] and [4 -1], as Paul shows in his "Gentle Introduction to Fokker Periodicity Blocks, part 2": A gentle introduction to Fokker periodicity bl... * [with cont.] (Wayb.) Here's my spreadsheet comparing the Duodene with two different meantones: -2/9-comma and -1/4-comma: Yahoo groups: /tuning-math/files/monz/duodene.xls * [with cont.] And a graphic of the -1/4-comma meantone lattice: Yahoo groups: /tuning-math/files/monz/ * [with cont.] (0,3) (4,-1) PB and -1-4cmt.gif (or without dealing with the broken link, just go to Yahoo groups: /tuning-math/files/monz/ * [with cont.] and download the file named "(0,3) (4,-1) PB and -1-4cmt.gif".) Legend: * the pink outlines the parallelogram boundaries of the periodicity-block according to the shape defined by the pair of unison-vectors, * the blue plots the coordinates of the twelve 5-limit JI pitch-classes of Ellis's "Duodene" which lie within the boundaries of the periodicity-block, * the yellow plots the more-or-less arbitrarily chosen meantones, whose chains are closed according to the interval spanned by one of the unison-vectors, * the green connects each meantone pitch-class with its closest JI relative within the periodicity-block, closeness measured in pitch-height. (I drew those green lines in by hand... so if you change the meantone in the spreadsheet, the lines won't adjust.) Note that this periodicity-block has three pitch-classes which fall right on the eastern boundary: (2,-1), (2,0) and (2,1). All three of these thus have alternates a comma lower -- that's would be -[4,-1] unison-vector--, which would place them on the western boundary: (-2,0), (-2,1), and (-2,2), respectively. Also, since (-2,2) and (2,1) happen to fall right on northwest and northeast *corners* of the boundary (respectively), they also have lower alternates at the distance of the *other* unison-vector, which would be -[0 3], which would place the alternates at (-2,-1) and (2,-2), respectively. With a total of 17 possible pitch-classes which may be used to define this periodicity-block, the only meantone which goes right down the middle of all of them is -1/4-comma. That's why I chose to use that for the graphic. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2899 - Contents - Hide Contents Date: Mon, 31 Dec 2001 03:22:03 Subject: Some 10 note 22 et scales From: genewardsmith 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.
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