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Message: 2850 - Contents - Hide Contents Date: Sat, 29 Dec 2001 01:02:59 Subject: Re: Maple and graph theory From: paulerlich --- 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?>What does Matlab, etc. have?I'm not sure if Matlab has a graph theory toolbox, but I probably wouldn't be willing to pay for it anyway. Look at the Matlab website - - I have the basic package, plus the optimization and statistics toolboxes.
Message: 2851 - Contents - Hide Contents Date: Sat, 29 Dec 2001 02:31:07 Subject: Re: the unison-vectordeterminant relationship From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, December 29, 2001 2:23 AM > Subject: [tuning-math] the unison-vector<-->determinant relationship > > > 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?Oops... my bad. That's not quite right. Change the second part to: for x = any integer, matrix M' = [(a+cx) (b+dx)] and [(a-cx) (b-dx)] [ c d ] [ c d ] results in the same determinant. Is there a better way to write that? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2852 - Contents - Hide Contents
Date: Sat, 29 Dec 2001 21:17:54
Subject: flexible mapping of meantones to PBs
From: monz
Paul, I finally understand your objections to what I've been
saying!
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?
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.
-monz
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Message: 2853 - Contents - Hide Contents Date: Sat, 29 Dec 2001 23:57:43 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- 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? What are the pelog tunings Dave used?All the 7-note ones with names pelog*.scl. There are 10 or so. Only a few didn't fit the pattern and they looked like "theoretical" ones.> And Dave, what are the > means and standard deviations of the two sizes of thirds?No time to investigate this now, sorry. But it certainly should be looked at.
Message: 2854 - Contents - Hide Contents Date: Sat, 29 Dec 2001 01:17:28 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: clumma>> >ooks totally contrived, >> >> As opposed to what we've seen here recently? >>Huh? You have something to say?No way. Just asking what you thought the differences were. Scanning the scala archive, and reporting, "they do"?>Please, this was a strange pairing of instruments Sethares >used,Really? My memory is that he based the stuff on DATs he made while visiting. I'd look it up, but I've left his book in Montana.>and all the evidence is that Indonesian music _cultivates_ >beating, rather than trying to minimize it.I did not know that. They certainly get plenty of it. But can't they minimize roughness and still have plenty of beating? Roughness is pretty unpleasant. Beating can be pleasant, though.>>> 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).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. At least, I've usually done analytical listening (I forget Sethares' term -- where you try to listen to the partials) when I've enjoyed Balinese/Javanese music.>Tune them to a Pelog major triad. You don't hear any sense of >integrity? I sure do.What's your setup?>Fine. If you look at what Wilson did with Pelog, I don't think >we're crossing any lines.Wilson strongly disclaims making any conclusions about what is actually going on, says he's doing a creative interp, though his papers do leave out this important disclaimer...>The creative potential of this is not to be trifled.No argument here!>Listen to Blackwood's 23-tET etude, where he emulates Indonesian >music. You don't hear 5-limit harmony there? Isn't it beautiful?I seem to remember thinking it was triadic, and quite beautiful. I'll listen again tonight. -Carl
Message: 2855 - Contents - Hide Contents Date: Sat, 29 Dec 2001 10:33:54 Subject: Pentatonic scales From: genewardsmith Here are the connected superparticular 5-limit pentatonics: 1--10/9--4/3--3/2--9/5 [10/9, 6/5, 9/8, 6/5, 10/9] connectivity = 1 1--6/5--4/3--3/2--5/3 [6/5, 10/9, 9/8, 10/9, 6/5] connectivity = 2 1--6/5--4/3--3/2--9/5 [6/5, 10/9, 9/8, 6/5, 10/9] connectivity = 1 1--16/15--4/3--3/2--15/8 [16/15, 5/4, 9/8, 5/4, 16/15] connectivity = 1 1--5/4--4/3--3/2--8/5 [5/4, 16/15, 9/8, 16/15, 5/4] connectivity = 2 1--5/4--4/3--3/2--15/8 [5/4, 16/15, 9/8, 5/4, 16/15] connectivity = 1 1--6/5--36/25--3/2--5/3 [6/5, 6/5, 25/24, 10/9, 6/5] connectivity = 1 1--6/5--5/4--3/2--5/3 [6/5, 25/24, 6/5, 10/9, 6/5] connectivity = 2 1--25/24--5/4--3/2--9/5 [25/24, 6/5, 6/5, 6/5, 10/9] connectivity = 1
Message: 2856 - Contents - Hide Contents Date: Sat, 29 Dec 2001 01:18:09 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich I wrote,> 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?To say nothing of the potential of such systems were a combination of adaptive tempering and adaptive timbring to be used.
Message: 2857 - Contents - Hide Contents Date: Sat, 29 Dec 2001 10:41:51 Subject: Re: Three and four tone scales From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:> I assume these are all the 3- and 4-tone permutations with > connectivity greater than 1?These are the 5-limit 3 and 4 tone scales, up to isomorphim by mode and inversion, with only superparticular scale steps and which are connected. The connectivity number I give is the edge-connectivity, which means the number of edges (consonant intervals) which would need to be removed in order to make it disconnected. It is therefore a measure of how connected by consonant intervals the scale in question is.
Message: 2858 - Contents - Hide Contents Date: Sat, 29 Dec 2001 01:38:38 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:>> I've modified my badness measure in ways that I hope take into account >> the fact (assuming it is one) that pelog is some kind of 5-limit >> temperament. I give the following possible ranking of 5-limit >> temperaments having a whole octave period. >> How are you ranking them,gens * exp((err/7.4c)^0.5) where gens = sqrt((gens(1:3)/ln(3))^2 + (gens(1:5)/ln(5))^2 + (gens(3:5)/ln(5))^2) /(1/ln(3) + 2/ln(5)) and err = sqrt(err(1:3)^2 + err(1:5)^2 + err(3:5)^2) /3 and how are you finding them? Brute force search of all generators from 0 to 600 c in increments of 0.1 c. Three passes, limiting the max absolute number of gens for any prime to 4, then 10, then 36 gens.> > Here is your table, with my annotations: >>> Gen Gens in RMS err Name >> (cents) 3 5 (cents) >> ------------------------------------ >> 503.8 [-1 -4] 4.2 meantone >> 498.3 [-1 8] 0.3 schismic >> 317.1 [ 6 5] 1.0 kleismic >> 380.0 [ 5 1] 4.6 magic >> 163.0 [-3 -5] 8.0 maxidiesic >> 387.8 [ 8 1] 1.1 wuerschmidt >> 271.6 [ 7 -3] 0.8 orwell >> 443.0 [ 7 9] 1.2 minidiesic >> 176.3 [ 4 9] 2.5 not on my list; badness=649 >> 339.5 [-5 -13] 0.4 AMTWhat does that stand for?>> 348.1 [ 2 8] 4.2 meantone >> 251.9 [-2 -8] 4.2 meantone Not really. >> 351.0 [ 2 1] 28.9 neutral thirdsSimple neutral thirds? as opposed to the complex ones above?>> 126.2 [-4 3] 6.0 not on my list; badness=728 >> 522.9 [-1 3] 18.1 pelogic
Message: 2859 - Contents - Hide Contents Date: Sat, 29 Dec 2001 10:45:57 Subject: Re: the unison-vectordeterminant relationship From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote: This is a basic property of determinants; one proof uses the definition of an nxn determinant in terms of an n-fold wedge product. If D = v1^v2^v3 ... ^vn, then (v1+x*v2)^v2^v3...^vn = v1^v2^v3 ... ^vn + x*v2^v2^v3...^vn (distributive law for wedge products). Since v2^v2=0, the second term cancels, and the determinant equals D.
Message: 2860 - Contents - Hide Contents Date: Sat, 29 Dec 2001 02:16:06 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: genewardsmith --- 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 want to count these, since I think they are pointless, but other people do. I'd like to hear what the point is.>>> 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.
Message: 2861 - Contents - Hide Contents Date: Sat, 29 Dec 2001 08:46:09 Subject: Re: the unison-vectordeterminant relationship From: monz Hi Gene!> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, December 29, 2001 2:45 AM > Subject: [tuning-math] Re: the unison-vector<-->determinant relationship > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > > This is a basic property of determinants; one proof uses > the definition of an nxn determinant in terms of an n-fold > wedge product. > > If D = v1^v2^v3 ... ^vn, then (v1+x*v2)^v2^v3...^vn = > v1^v2^v3 ... ^vn + x*v2^v2^v3...^vn (distributive law > for wedge products). Since v2^v2=0, the second term cancels, > and the determinant equals D.Thanks for that explanation... unfortunately, it looks like Chinese to me. If you can at least make it look like Sumerian, maybe I'll get it... :) Now that I finally understand a bit about how matrices work, can you use them to rewrite what you wrote above? I simply don't comprehend your algebra. Is there someplace online that explains these "wedge products"? Perhaps old tuning-math posts? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2862 - Contents - Hide Contents Date: Sat, 29 Dec 2001 17:02:47 Subject: Re: non-uniqueness of a^(b/c) type numbers From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, December 29, 2001 4:00 PM > Subject: [tuning-math] Re: non-uniqueness of a^(b/c) type numbers > > > --- 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?In each vector [a b c] below, a = exponent of prime-factor 2 b = exponent of prime-factor 3 c = exponent of prime-factor 5 7/26-comma meantone ------------------- "5th" = [-1*(26/26) 1*(26/26) 0*(26/26)] = 3/2 = Pythagorean "perfect 5th" - [-4*(7/26) 4*(7/26) -1*(7/26) ] = (81/80)^(7/26) = 7/26-comma ----------------------------------- [ 2/26 -2/26 7/26] = (3/2)/ ((81/80)^(7/26)) = [ 1/13 -1/13 7/26] = 7/26-comma meantone "5th" "major 3rd" = [ 1/13 -1/13 7/26] = 7/26-comma meantone "5th" * [ 4/1 4/1 4/1 ] = 4 times (= 4 generators) ----------------------- [ 4/13 -4/13 28/26] = ~5/1 = 7/26-meantone "5th harmonic" = [ 4/13 -4/13 14/13] (= reduced) = [ 4/13 -4/13 14/13] = 7/26-meantone "5th harmonic" - [ 2*(13/13) 0 0 ] = subtract 2 "8ves" ------------------------ [ -22/13 -4/13 14/13] = 7/26-comma meantone "major 3rd" And how about throwing in the 55-tone 1/6-comma meantone as well, since I'm actually exploring that a lot lately? 1/6-comma meantone ------------------- "5th" = [-1*(6/6) 1*(6/6) 0*(6/6)] = 3/2 = Pythagorean "perfect 5th" - [-4*(1/6) 4*(1/6) -1*(1/6)] = (81/80)^(7/26) = 1/6-comma ------------------------------- [ -2/6 2/6 1/6 ] = (3/2)/ ((81/80)^(1/6)) = [ -1/3 1/3 1/6 ] = 1/6-comma meantone "5th" "major 3rd" = [-1/3 1/3 1/6] = 1/6-comma meantone "5th" * [ 4/1 4/1 4/1] = 4 times (= 4 generators) ----------------------- [-4/3 4/3 4/6] = ~5/1 = 1/6-meantone 5th harmonic = [-4/3 4/3 2/3] (= reduced) = [ -4/3 4/3 2/3] = 1/6-meantone 5th harmonic - [2*(3/3) 0 0 ] = subtract 2 "8ves" ------------------------ [ -10/3 4/3 2/3] = 1/6-comma meantone "major 3rd _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2863 - Contents - Hide Contents Date: Sat, 29 Dec 2001 02:22:03 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>> Paul, I think you're severely distorting what I wrote. I didn't say >> pelog is junk. I said "as a 5-limit approximation ..." >> But it's really the ratio of *3* that had the error you objected to.Sure but ratios of 3 are part of the 5 limit and the ratios of 5 are supposedly explaining why the ratio of 3 is so bad. It's supposedly to get good enough ratios of 5 with only 3 and 4 gens (simpler than meantone).>> Is there really any evidence that pelog is a 5-limit temperament? >> I think there's strong evidence it at least relates to the 3-limit, > and that's the error you objected to.Yes but by claiming that the 523 c temperament isn't junk (as a 5-limit temperament) because it corresponds closely enough to pelog, you are claiming something more than that.> As far as 5-limit, it's > definitely a matter of opinion, but I'm referring to Herman Miller's > use of the scale, not necessarily the traditional one. I'm also > referring to Margo and Bill's use of consonant sonorities where the > departures from 5-limit JI are even larger than this. >>>>> So there are >>>> plenty of other temperaments as good as this. >>>>>> By _as good as_, I mean having an equal or lower RMS error ANS > and>>> equal or lower 'gens' measure. >>>> Why can't I use my own criteria for "as good as"? >> Well, we're trying to find out if Gene is missing anything with his > methods. But if you'd like to suggest a different measure of cents > error and/or a different complexity measure, I'd hope Gene could be > accomodating . . .He might be missing something because of his badness measure as well as because of his method. I'll just give my list and let others decide whether the temperaments in it are worth examining.
Message: 2864 - Contents - Hide Contents Date: Sat, 29 Dec 2001 08:57:42 Subject: non-uniqueness of a^(b/c) type numbers From: monz 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. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2865 - Contents - Hide Contents Date: Sat, 29 Dec 2001 17:36:24 Subject: Re: the unison-vectordeterminant relationship From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, December 29, 2001 4:10 PM > Subject: [tuning-math] Re: the unison-vector<-->determinant relationship > > > >>>> OK, fix what's wrong with this, if anything... >> >> One can find an infinity of closer and closer >> fraction-of-a-comma meantone representations of that >> 5-limit JI periodicity-block, which would be represented >> on my lattice here as shiftings of the angle of the vector >> representing the meantone. But one could never find one >> that would go straight down the middle of the symmetrical >> periodicity-block. >> >> Now, what does this mean? >> I'm not following you--why not spell it out, giving the block in question?OK. I'd much rather spell it out with 1/6-comma, since I've been working more with that. But I already have a spreadsheet which shows how various meantones "fit" within a particular periodicity-block. I posted about it here last week: my original post: Yahoo groups: /tuning-math/message/2102 * [with cont.] errata: Yahoo groups: /tuning-math/message/2103 * [with cont.] Paul's criticism: Yahoo groups: /tuning-math/message/2104 * [with cont.] And guess what? Oops! It looks like the conclusion I reached there is that the 7/25-comma meantone *does* indeed go straight down the middle of the periodicity-block! Here's my Microsoft Excel spreadsheet latting these meantones: Yahoo groups: /tuning-math/files/monz/ * [with cont.] (6%20-14)%20(4%20-1)%20 PB%20and%207-25cmt.xls I made all the graphs the same size and put them in exactly the same place, so if you don't change anything, you can click on the worksheet tabs at the bottom and see how the different meantones lie within the periodicity-block. Hmmm... *if* my lattice *was* wrapped as a cylinder or torus, would there be any significance to this "middle of the road" business? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2866 - Contents - Hide Contents Date: Sat, 29 Dec 2001 02:26:54 Subject: Re: Keenan green Zometool struts From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >> Are you getting the bundle of the Adv Math kit with George Hart and >> Henri Picitto's Zome Geometry book? >> That's this one, yes? > > Advanced Math Creator Kit Bundle * [with cont.] (Wayb.) Yes.
Message: 2867 - Contents - Hide Contents Date: Sat, 29 Dec 2001 09:30:49 Subject: Re: the unison-vectordeterminant relationship From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, December 29, 2001 8:46 AM > Subject: Re: [tuning-math] Re: the unison-vector<-->determinant relationship > > > Is there someplace online that explains these "wedge products"? > Perhaps old tuning-math posts?I think I've answered my own question: http://mathworld.wolfram.com/WedgeProduct.html * [with cont.] But it still looks like Chinese. :( -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2868 - Contents - Hide Contents Date: Sat, 29 Dec 2001 02:29:46 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Dave, I was hoping that, instead of doing this, you would think in > terms of two separate badness factors, an 'error' factor and > a 'complexity' factor -- and let us know if you could find anything > that was better on _both_ factors than _any_ of the temperaments I > listed, but was not in the list anywhere . . . see?Ok. When I get time.
Message: 2870 - Contents - Hide Contents Date: Sun, 30 Dec 2001 02:40:34 Subject: Re: non-uniqueness of a^(b/c) type numbers From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Saturday, December 29, 2001 4:00 PM >> Subject: [tuning-math] Re: non-uniqueness of a^(b/c) type numbers >> >> >> --- 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? > >> In each vector [a b c] below, > > a = exponent of prime-factor 2 > b = exponent of prime-factor 3 > c = exponent of prime-factor 5 > > > > 7/26-comma meantone > ------------------- > > "5th" = > > [-1*(26/26) 1*(26/26) 0*(26/26)] = 3/2 =Pythagorean "perfect 5th"> - [-4*(7/26) 4*(7/26) -1*(7/26) ] = (81/80)^(7/26) = 7/26- comma > ----------------------------------- > [ 2/26 -2/26 7/26] = (3/2)/ ((81/80)^(7/26)) > = [ 1/13 -1/13 7/26] = 7/26-comma meantone "5th" > > > > > "major 3rd" = > > > [ 1/13 -1/13 7/26] = 7/26-comma meantone "5th" > * [ 4/1 4/1 4/1 ] = 4 times (= 4 generators) > ----------------------- > [ 4/13 -4/13 28/26] = ~5/1 = 7/26-meantone "5th harmonic" > = [ 4/13 -4/13 14/13] (= reduced) > > > = [ 4/13 -4/13 14/13] = 7/26-meantone "5th harmonic" > - [ 2*(13/13) 0 0 ] = subtract 2 "8ves" > ------------------------ > [ -22/13 -4/13 14/13] = 7/26-comma meantone "major 3rd"Well, now I'm confused, because I thought that when you were latticing 7/26-comma meantone using this method (and ignoring octaves), you were somehow getting a two-dimensional arrangement of points. Is that not correct?
Message: 2871 - Contents - Hide Contents Date: Sun, 30 Dec 2001 02:47:48 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote:>>>> Try an experiment. Get three bells or gongs or whatever, as long >>>> as they each have a clear pitch (I guess you can use a synth for >>>> this). >>>>>> I don't have a synth that does inharmonic additive sythesis. >>> Besides, many gamelan instruments don't evoke a clear sense >>> of pitch to me at all. >>>> It's a matter of _how_ clear. Typically, according to Jacky, the >> 2nd and 3rd partials are about 50 cents from their harmonic >> series positions. That spells increased entropy (yes, timbres >> have entropy), >> Which is why I suggested that the plug-in-the-fundamentals-only > shortcut shouldn't be applied. >Why not? Afraid of a little assymetry?>>> but still within the "valley" of a particular pitch. >> Have you ever tried a bell timbre? Wonder what the curve > looks like...That's an instrument where the hear fundamental is not even in the spectrum . . . the Gamelan instruments are a bit different.
Message: 2872 - Contents - Hide Contents Date: Sun, 30 Dec 2001 02:49:42 Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE From: paulerlich By the way Carl, have you tried any actual _listening experiments_ yet?>> 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. > > -CarlThe 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?
Message: 2873 - Contents - Hide Contents Date: Sun, 30 Dec 2001 02:52:25 Subject: Re: the unison-vectordeterminant relationship From: paulerlich --- 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 clue as to how to picture what you're describing above. Can you help? Have you received my package yet?
Message: 2874 - Contents - Hide Contents Date: Sun, 30 Dec 2001 03:01:47 Subject: Re: the unison-vectordeterminant relationship From: genewardsmith --- 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 an ellipsoid to that if you wanted to--why do you ask?
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