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Message: 2825 - Contents - Hide Contents Date: Sat, 29 Dec 2001 21:45:34 Subject: Re: non-uniqueness of a^(b/c) type numbers From: genewardsmith --- In tuning-math@y..., "unidala" <JGill99@i...> wrote:> JG: Monz, can you give an example of a prime number > taken to a rational power (where the power's numerator and > denominator are integer) which (directly, as numerically > evaluated)equals an integer? I couldn't. Of course, the > rational valued exponent must not be equal (itself) to > an integer value...If x = p^(a/b), where a/b is in its lowest terms, then x^b = p^a, and x^b - p^a = 0 is an irreducible polynomial of degree b. Any solution will therefore be an algebraic integer (since p is an integer) but not an ordinary "rational integer". In other words, it can't happen. :)
Message: 2826 - Contents - Hide Contents Date: Sat, 29 Dec 2001 07:28:43 Subject: Re: Classes of 5-limit superparticular scales From: clumma --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> Three tones: > > (6/5)(5/4)(4/3) = 2 > (10/9)(6/5)(3/2) = 2 > (16/15)(5/4)(3/2) = 2 > /.../ > Six tones: > > (25/24)^2(16/15)(6/5)^3 = 2 > (81/80)(10/9)^3(6/5)^2 = 2Not quite following this post, Gene...> Seven tones: > > (16/15)^2(10/9)^2(9/8)^3 =? > Nine tones:Eight tones was your last message? So the =x is the number of connected permutations for each set of superparticulars? More with eight tones than with 1-7 or 9, 10, 12, 15!? -Carl
Message: 2827 - Contents - Hide Contents Date: Sat, 29 Dec 2001 21:47:25 Subject: Re: Three and four tone scales From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:> Right. Naturally, we're all looking forward to the 7-limit. > Hopefully, there won't be too many superparticulars.I imagine it's too much for me; however I'll try to do some of the smaller scales.
Message: 2828 - Contents - Hide Contents Date: Sat, 29 Dec 2001 07:34:33 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: clumma>>> >ry an experiment. Get three bells or gongs or whatever, as long >>> as they each have a clear pitch (I guess you can use a synth for >>> this). >>>> I don't have a synth that does inharmonic additive sythesis. >> Besides, many gamelan instruments don't evoke a clear sense >> of pitch to me at all. >>It's a matter of _how_ clear. Typically, according to Jacky, the >2nd and 3rd partials are about 50 cents from their harmonic >series positions. That spells increased entropy (yes, timbres >have entropy),Which is why I suggested that the plug-in-the-fundamentals-only shortcut shouldn't be applied. The variety of instruments in the gamelan is huge. I find that the main melodic insts. have a fairly clear sense of timbre at the attack (which outlines the melody and rhythm of the music), but often three or four distinct pitches over the rest of the envelope (which constitutes the harmony of the music, and thus my view of Sethares' treatment).>but still within the "valley" of a particular pitch.Have you ever tried a bell timbre? Wonder what the curve looks like...>A cheesy Ensoniq, or listen to real Gamelan music, or the Blackwood >piece. Look, I'm not saying the tuning is _designed_ to approximate >the major triad and its intervals, but statistically (pending >further analysis) it sure seems to be playing a shaping role.I've got a share of gamelan music, thanks to Kraig Grady's suggestions. I'm listening to the Blackwood now. The Blackwood sounds more triadic than the gamelan music. -Carl
Message: 2829 - Contents - Hide Contents Date: Sat, 29 Dec 2001 21:56:47 Subject: Re: non-uniqueness of a^(b/c) type numbers From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> Oops! So did I really mean to write: > "no six *coprime* integers that will satisfy that equation"?The first leads to the polynomial x^c - a^b = 0, and the second to x^f - d^e = 0. Since gcd(a,d)=1 they have no common factor.
Message: 2830 - Contents - Hide Contents Date: Sat, 29 Dec 2001 07:38:18 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: clumma>> >) The Scala scale archive is not a good source of actual pelogs, >> or any other ethnic tunings for that matter. > >Why not?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. I admit I've never looked at the pelogs. But I'll bet eggs benedict there are some of Wilson's in there! And I have looked very closely indeed at the bagpipe and mbira tunings. Just whatever anybody was dreaming, in ratios -- exactly the kind of thing you've pined against so often. -Carl
Message: 2831 - Contents - Hide Contents Date: Sat, 29 Dec 2001 14:48:30 Subject: Re: the unison-vectordeterminant relationship From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, December 29, 2001 1:38 PM > Subject: [tuning-math] Re: the unison-vector<-->determinant relationship > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> But it still looks like Chinese. :( >> There are some old posts--until recently I would not have > tossed multilinear algebra at people so casually, but there's > been a lot of discussion of the wedge product.Ah... I think I'm getting it. "Multilinear algebra"... So, for example, on my "1/6-comma meantone within (4 -1) (19 9) periodicity-block" lattice... [plug alert] (which I've just added in miniature with some new text on my JustMusic software webpage: <Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.)>,) ... 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...) 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? (please use as much English as possible along with the algebra) I have similiar questions for the octave-specific versions of my lattices, which utilize prime-factor 2, and on which I can also lattice EDOs. -monz (so sorrowfully deficient in math...) _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2832 - Contents - Hide Contents Date: Sat, 29 Dec 2001 07:45:03 Subject: Blackwood comment and call for review From: clumma Listening now to the 15-tone etude for the first time in a year or two. I still think it's my single favorite explicitly microtonal piece. It's off the hook! Would anybody with Sethares' book care to contribute a review of his gamelan tuning derivation (to the main list, if you like)? It's been some time since I've read it, and I don't have my copy handy. -Carl
Message: 2833 - Contents - Hide Contents Date: Sat, 29 Dec 2001 19:57:26 Subject: Re: the unison-vectordeterminant relationship From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > 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?>> 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.Right, I understand that. But since any given meantone interval can only be closest to *one* particular 5-limit JI ratio, which should fall within the periodicity-block, the vector of the meantone chain will *still* imply a unique periodicity-block, will it not? (assuming that the periodicity-block is replicated a comma away as one travels around the cylinder) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2834 - Contents - Hide Contents Date: Sat, 29 Dec 2001 22:54:51 Subject: Re: Three and four tone scales From: clumma>> >ight. Naturally, we're all looking forward to the 7-limit. >> Hopefully, there won't be too many superparticulars. >>I imagine it's too much for me; however I'll try to do some of >the smaller scales.You mean too much for your computer, or too much to post, or is there busy work involved? As I'm sure you know, scales of 5-10 notes are most interesting from a compositional point of view. Perhaps you could restrict yourself to superparticulars which have factors of 7 and scales with connectivity >= 2. -C.
Message: 2835 - Contents - Hide Contents Date: Sat, 29 Dec 2001 07:55:07 Subject: Re: Classes of 5-limit superparticular scales From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:>> (25/24)^2(16/15)(6/5)^3 = 2 >> (81/80)(10/9)^3(6/5)^2 = 2 >> Not quite following this post, Gene...The only 5-limit superparticulars are 2/1,3/2,4/3,5/4,6/5,9/8,10/9, 16/15,25/24 and 81/80. I took these three at a time, and calculated whether or not they lead to a partition of the octave, and if so, how many of each tone were required. So, for intance, 2 25/24's one 16/15 and three 6/5 make up an octave, and by arraging them in various ways give us six-tone scales.>> Seven tones: >> >> (16/15)^2(10/9)^2(9/8)^3> Eight tones was your last message?Seven tones--the same steps as above, which it seems is the only seven-tone possibility.
Message: 2836 - Contents - Hide Contents Date: Sat, 29 Dec 2001 23:14:47 Subject: more tetrachordality results From: clumma I've rev'd my tetraindex function a bit. It does this: Returns the minimum mean deviation (in cents, and as a percentage of the smallest interval in the scale), of the pitches in any order, of a pitch set representing the given scale, and its transposition at 702 cents, for all modes of the given scale. Thus: (tetraindex3 12 ls 'any) Pentatonic Scale (0 2 5 7 9) -> ((21 $) (10 %)) Diatonic Scale (0 2 4 5 7 9 11) -> ((16 $) (16 %)) Diminished chord (0 3 6 9) -> ((102 $) (34 %)) Wholetone scale (0 2 4 6 8 10) -> ((102 $) (51 %)) Diminished scale (0 2 3 5 6 8 9 11) -> ((50 $) (50 %)) Minor scales w/'gypsy' tetrachord (0 1 4 5 7 8 11) -> ((44 $) (44 %)) (0 1 4 5 7 8 10) -> ((44 $) (44 %)) (0 1 4 5 7 9 10) -> ((44 $) (44 %)) And here are the three connectivity=2 pentatonics from Gene's recent post, in 34-tET: (tetraindex3 34 ls 'any) 1--6/5--4/3--3/2--5/3 [6/5, 10/9, 9/8, 10/9, 6/5] (0 9 14 20 25) -> ((73 $) (41 %)) 1--5/4--4/3--3/2--8/5 [5/4, 16/15, 9/8, 16/15, 5/4] (0 11 14 20 23) -> ((129 $) (122 %)) 1--6/5--5/4--3/2--5/3 [6/5, 25/24, 6/5, 10/9, 6/5] (0 9 11 20 25) -> ((73 $) (103 %)) And one of the connectivity=1 scales: 1--25/24--5/4--3/2--9/5 [25/24, 6/5, 6/5, 6/5, 10/9] (0 2 11 20 29) -> ((128 $) (181 %)) -Carl
Message: 2837 - Contents - Hide Contents Date: Sat, 29 Dec 2001 08:29:17 Subject: Re: Classes of 5-limit superparticular scales From: clumma>>> >even tones: >>> >>> (16/15)^2(10/9)^2(9/8)^3 >>>> Eight tones was your last message? >> Seven tones--the same steps as above, which it seems is the only > seven-tone possibility.Right, they all =2, means they sum to an octave, duh. So you only checked for connectednes in the 7-tone case. Got it, thanks. -Carl
Message: 2839 - Contents - Hide Contents Date: Sat, 29 Dec 2001 20:25:51 Subject: Re: the unison-vectordeterminant relationship From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, December 29, 2001 8:07 PM > Subject: [tuning-math] Re: the unison-vector<-->determinant relationship > > > --- 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.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. I suppose that if some of the ratios in the periodicity-block were transposed, so that the block had a different shape, then 10/57 *would* run down the center ... but not for this one.>> >> 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?Well... the only way I can explain it is that on my lattice, the meantone chain which runs right down the center of this periodicity-block *does* follow the same angle on the lattice as (19 9).>>> (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, andNot really sure what you mean by this.> (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. So I don't understand why you're pointing out cases where the consonances that wrap are not used. 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? If I assume that my flat lattice is supposed to wrap cylindrically, then why does it matter whether I'm considering pitches or intervals? Isn't it the same? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2840 - Contents - Hide Contents Date: Sat, 29 Dec 2001 00:17:21 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: clumma>> >) There may be many explanations for this pattern of fifths, >> including something like Sethares' treatment... have you seen >> his derivation of gamelan tunings in his book? >>Looks totally contrived,As opposed to what we've seen here recently?>and what about harmonic entropy?You'd have to plug in all the partials. The timbres are too out there to just plug in the fundamentals as we do normally. IOW, I'm not sure harmonic entropy is so significant for this music.>> () In any case, because Indonesian music doesn't use 5-limit >> consonances -- let alone modulate them >>It modulates plenty, as we've recently discussed on the tuning >list. And, listen to some Pelog-scale Indonesian music. It >doesn't evoke 5-limit harmony to your ears?It has in the past, but I attributed that to my cultural conditioning.>At least, we can call it a "creative interpretation" of Pelog,Def. Just don't want that important qualifier to be left out. -Carl
Message: 2841 - Contents - Hide Contents Date: Sat, 29 Dec 2001 09:05:59 Subject: Three and four tone scales From: genewardsmith Three tones: 1--5/4--3/2 [5/4, 6/5, 4/3] connectivity = 2 1--3/2--5/3 [3/2, 10/9, 6/5] connectivity = 1 1--3/2--8/5 [3/2, 16/15, 5/4] connectivity = 1 Four tones: 1--5/4--3/2--5/3 [5/4, 6/5, 10/9, 6/5] connectivity = 2 1--6/5--3/2--5/3 [6/5, 5/4, 10/9, 6/5] connectivity = 1 1--6/5--3/2--8/5 [6/5, 5/4, 16/15, 5/4] connectivity = 2 1--6/5--3/2--15/8 [6/5, 5/4, 5/4, 16/15] connectivity = 1 1--6/5--5/4--3/2 [6/5, 25/24, 6/5, 4/3] connectivity = 2 1--25/24--5/4--3/2 [25/24, 6/5, 6/5, 4/3] connectivity = 1
Message: 2842 - Contents - Hide Contents Date: Sat, 29 Dec 2001 20:34:08 Subject: Re: the unison-vectordeterminant relationship From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, December 29, 2001 8:25 PM > Subject: Re: [tuning-math] Re: the unison-vector<-->determinant relationship > > >>> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> >> To: <tuning-math@xxxxxxxxxxx.xxx> >> Sent: Saturday, December 29, 2001 8:07 PM >> Subject: [tuning-math] Re: the unison-vector<-->determinant relationship >> >> >> --- 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. > >> 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. > > I suppose that if some of the ratios in the periodicity-block were > transposed, so that the block had a different shape, then 10/57 *would* > run down the center ... but not for this one. > > >>>>>> 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? > >> Well... the only way I can explain it is that on my lattice, > the meantone chain which runs right down the center of this > periodicity-block *does* follow the same angle on the lattice > as (19 9).And I felt that I should point out that since many periodicity-blocks could be found which have the same determinant, there are thus many different meantones which "go with" a particular determinant, depending on the actual unison-vectors employed in the definition of the periodicity-block. For this particular (19 9),(4 -1) block, 8/49-comma meantone seems *by eye* to be the meantone which splits the periodicity-block exactly in half, and thus has the lowest average deviation from the entire set of JI ratios within the block. Again I ask: how can this be derived mathematically? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2843 - Contents - Hide Contents Date: Sat, 29 Dec 2001 23:34:28 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>>>> 348.1 [ 2 8] 4.2 meantone >>>> 251.9 [-2 -8] 4.2 meantone >> >> Not really. >> You have two meantone systems, and you can't pass from one to the > other using a consonant interval.I don't understand what you mean here, or the point you are making.> I don't want to count these, since Ithink they are pointless, but other people do. I'd like to hear what the point is.>They are different melodically from meantone chains and they are better than some other temperaments that you are including. They form different MOS scales. Here are denominators of convergents and (semiconvergents). 503.8 c 5 7 12 19 31 50 81 348.1 c 7 (10 17 24) 31 (38 69 100) 251.9 c 5 (9 14) 19 (24 43 62) 81 (100>>>> 351.0 [ 2 1] 28.9 neutral thirds >>>> 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 catalog and the other neutral thirds temperament above.
Message: 2844 - Contents - Hide Contents Date: Sat, 29 Dec 2001 00:54:55 Subject: Maple and graph theory From: genewardsmith Maple has a graph theory package, which allows me to take a scale and compute its connectivity fairly easily. What does Matlab, etc. have?
Message: 2845 - Contents - Hide Contents Date: Sat, 29 Dec 2001 09:31:46 Subject: Re: Three and four tone scales From: clumma I assume these are all the 3- and 4-tone permutations with connectivity greater than 1? This is great... sorry to reply so much, I can't wait for the 6, 8, 9, and 10 tone results! -C. --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> Three tones: > 1--5/4--3/2 [5/4, 6/5, 4/3] connectivity = 2 > 1--3/2--5/3 [3/2, 10/9, 6/5] connectivity = 1 > 1--3/2--8/5 [3/2, 16/15, 5/4] connectivity = 1 > > Four tones: > 1--5/4--3/2--5/3 [5/4, 6/5, 10/9, 6/5] connectivity = 2 > 1--6/5--3/2--5/3 [6/5, 5/4, 10/9, 6/5] connectivity = 1 > 1--6/5--3/2--8/5 [6/5, 5/4, 16/15, 5/4] connectivity = 2 > 1--6/5--3/2--15/8 [6/5, 5/4, 5/4, 16/15] connectivity = 1 > 1--6/5--5/4--3/2 [6/5, 25/24, 6/5, 4/3] connectivity = 2 > 1--25/24--5/4--3/2 [25/24, 6/5, 6/5, 4/3] connectivity = 1
Message: 2846 - Contents - Hide Contents Date: Sat, 29 Dec 2001 23:49:58 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote:>>> () The Scala scale archive is not a good source of actual pelogs, >>> or any other ethnic tunings for that matter. >> >> Why not? >> 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. > > I admit I've never looked at the pelogs. But I'll bet eggs > benedict there are some of Wilson's in there!Did you use any of those, Dave? I would assume those would be specified by ratios. I assumed Dave used some "observed" examples.
Message: 2847 - Contents - Hide Contents Date: Sat, 29 Dec 2001 01:01:21 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote:>>> () There may be many explanations for this pattern of fifths, >>> including something like Sethares' treatment... have you seen >>> his derivation of gamelan tunings in his book? >>>> Looks totally contrived, >> As opposed to what we've seen here recently?Huh? You have something to say? Please, this was a strange pairing of instruments Sethares used, and all the evidence is that Indonesian music _cultivates_ beating, rather than trying to minimize it.>> and what about harmonic entropy? >> You'd have to plug in all the partials. The timbres are too > out there to just plug in the fundamentals as we do normally. > IOW, I'm not sure harmonic entropy is so significant for this > music.Try an experiment. Get three bells or gongs or whatever, as long as they each have a clear pitch (I guess you can use a synth for this). Tune them to a Pelog major triad. You don't hear any sense of integrity? I sure do.>>> At least, we can call it a "creative interpretation" of Pelog, >> Def. Just don't want that important qualifier to be left out. >Fine. If you look at what Wilson did with Pelog, I don't think we're crossing any lines. The creative potential of this is not to be trifled. Listen to Blackwood's 23-tET etude, where he emulates Indonesian music. You don't hear 5-limit harmony there? Isn't it beautiful?
Message: 2848 - Contents - Hide Contents Date: Sat, 29 Dec 2001 02:23:17 Subject: the unison-vectordeterminant relationship From: monz I just noticed something about the relationship between unison-vectors and determinants. Let me know if I've discovered something. Here goes... I found that for matrix M = [a b] [c d] matrix M' = [(a +/- c) (b +/- d)] [ c d ] results in the same determinant. Has anyone ever noticed this before? Is it simply a logical result of periodicity-block math? Is it common knowledge that I missed? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2849 - Contents - Hide Contents Date: Sat, 29 Dec 2001 23:55:46 Subject: Re: non-uniqueness of a^(b/c) type numbers From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> > I've been having a discussion with a friend in private email > about the business of numbers with fractional exponents > not following the Fundamental Theorem of Arithmetic. He > was generous enough to send me a very long and detailed > explanation, and Paul has gone over this with me more concisely > in the past, but in spite of all of the explanation, I still > don't get it. > > I understand that many different combinations of prime-factors > and fractional exponents can be found which *approach any > floating-point value arbitrarily closely*, but EXACT values > are STILL INCOMMENSURABLE!! > > Why must numbers of the form a^(b/c) be understood in > terms of their floating-point decimal value? If we stick > to the a^(b/c) form we get exact values and can manipulate > them the same way as our regular rational numbers of the > form x^y. > > If b and c in a^(b/c) are always integers, the simple > calculations needed for tuning math always gives results > which are also integers, so there's never any error at all. > > I've been trying to understand this for three years... > someone please help.I'm not sure what you're trying to understand here, so please clarify.
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