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Message: 2650 - Contents - Hide Contents Date: Sun, 23 Dec 2001 07:37:58 Subject: Temperament names From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> 2109375/2097152 = 2^-21 3^3 5^7 OrwellI like it, but the comma is called the semicomma on Manuel's list.> 27/25"Large limmic"? Margo seemed dubious about calling it any kind of a limma.> 16/15 Semitonic? > 135/128Major Limmic? Major Chromic? Map: [ 0 1] [-1 2] [ 3 1] Generators: a = 10.0215 / 23; b = 1 badness: 46.1 rms: 18.1 g: 2.94 errors: [-24.8, -17.7, 7.1] 25/24 I called it "Neutral Thirds". "Minor chromic" is another possibility.> 648/625Too many things called a diesis, I fear--Major Diesic?> 250/243 Maximal diesic? > 128/125Minor diesic? I think Paul named this; his names sounded good and maybe I should dig them out and keep them handy!> 3125/3072Magic--or Small Diesic for anyone who really digs this nomenclature. 78732/78125 = 2^2 3^9 5^-7 You'd think this would be some sort of diesis, but I can't find it. "The Quite Small Diesis" won't do, I suppose.> 393216/390625 = 2^17 3 5^-8This is Wuerschmidt's comma, so obviously the temperament is the Wuerschmidt. Who the heck is Wuerschmidt?
Message: 2651 - Contents - Hide Contents Date: Sun, 23 Dec 2001 13:24:04 Subject: Re: 55-tET & 1/6-comma meantone From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 1:13 PM > Subject: [tuning-math] Re: 55-tET & 1/6-comma meantone > > > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>> I'm having a hard time following Gene's comments because >>> I don't understand why (2^62 * 3^-23 * 5^-11) "really doesn't >>> work very well for anything *but* 65-et" when in fact it >>> *is* also a 67-EDO comma. >>>> I thought when you asked me to run it through my program you meant >> to analyze the linear temperament it defines; >> That's not what Monz meant. Plus Monz has an additional conceptual > idiosyncracy -- he sees the JI scale implied _in toto_, not just via > consonant intervals, and thus the smaller (in cents) unison vectors > have a special meaning to him which to us, they do not.Paul, I'm having a hard time understanding the difference between these two conceptions, but I think I'm beginning to get it. The implied ratios on my lattices follow the general trend of the meantone axis itself, which implies a handful of intervals which can be stacked to build the entire scale. But some of these JI intervals are emphatically *not consonant*, and are the "wolf intervals" which cause the displacement of the trend-line of the periodicity-block to align it with the meantone axis. Am I on the right track? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2652 - Contents - Hide Contents Date: Sun, 23 Dec 2001 23:57:24 Subject: Re: a different example From: paulerlich> It's hard for me to visualize what happens on a cylinder > or torus since I'm dealing with planar graphs.Well then, the Hall article I'm sending you tomorrow may help.
Message: 2653 - Contents - Hide Contents Date: Sun, 23 Dec 2001 07:48:13 Subject: Re: Temperament names From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>> 135/128 >> Major Limmic? Major Chromic?Paul said it was associated with Pelog. Pelogic?>> 648/625 >> Too many things called a diesis, I fear--Major Diesic?Paul suggested "Octo-diminished", since it can be done very well by the 64-et. Sounds fine to me.>> 128/125"It's the augmented system, since the 6-tone MOS is commonly known as the augmented scale" according to Paul.
Message: 2654 - Contents - Hide Contents Date: Sun, 23 Dec 2001 21:22:39 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> lattice coordinates x, y : > > > x = ( (q*c) + (p*a) ) / n > > y = ( (q*d) + (p*b) ) / nThere shouldn't be an "/ n" at the end of that. Maybe that's what's causing the weirdness, because sometimes n is negative.
Message: 2655 - Contents - Hide Contents Date: Sun, 23 Dec 2001 00:28:19 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx>; <jhchalmers@xxxx.xxx>; <paul@xxxxxxxxxxxxx.xxx> > Sent: Saturday, December 22, 2001 7:35 PM > Subject: Re: [tuning-math] coordinates from unison-vectors (was: 55-tET) > > > By brute force, a bit of research into matrix transformations, > and a whole lot of luck, I figured out how to do it.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 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 : x = ( (q*c) + (p*a) ) / n y = ( (q*d) + (p*b) ) / n END LOOP love / peace / harmony ... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2656 - Contents - Hide Contents
Date: Sun, 23 Dec 2001 19:40:42
Subject: Re: For Pierre, from tuning
From: Pierre Lamothe
Gene wrote:
Judging by these examples, I would propose the following definitions:
(1) We may write any positive rational number r in the form
2^n p/q, where p and q are odd integers and p/q is reduced to lowest form. The
fraction p/q we call the *odd part* of r.
(2) For any set S of rational numbers, we may take the odd parts of each
element, and the least common multiple D of their denominators. The set of
integers gammier(S) is defined as the set of the odd parts of the elements of S
times D; gammier(S) is the *gammier of S*.
Is there anything about this you want to accept, or to modify and then accept?
There is no problem with (1).
I would add only the terms I use for that.
r = 2n p/q is the rational number
r mod 2 = {2x p/q | all x in Z} is the corresponding octave class
Ton ( r ) = 2k p/q | k such that r in [1,2[ is the tone representing the class
Pivot ( r ) = p/q is the pivot representing the class : your odd part.
There are few problems with (2).
Gammier and generator
First, the expression "gammier <a b c ...>" means the structure generated by
<a b c ...>. The term <a b c ...> is refered as the generator g of the structure,
while the elements of that structure g\g are
a\a a\b a\c ...
b\a b\b b\c ,,,
c\a c\b c\c ...
,,,
where u\v means here the interval between u and v. The symbol \ is independant
of the composition law type, multiplicative or additive. So
a.. 4\5 = 5/4
b.. (log4)\(log5) = log5 - log4
Gammier conditions
More important, the gammier structure implies the existence of four conditions,
the last axioms of the gammier structure, which are
a.. regularity
b.. contiguity
c.. congruity
d.. fertility
so you refer simply to the harmoid structure if these conditions being unknow,
you use only
a.. this type of generation derived from the chordoid theory
b.. giving a finite set of rational numbers (mod 2)
implying implicitely
a.. the appropriate restriction of the multiplication as the law
b.. and the standard rational ordering
all that being necessary to formulate the axioms.
Minimal odd generator
Any finite set of rational numbers may be considered as a chordic generator of an
harmoid. Any line and any column of the matrix g\g, generated by a such chordic
generator g, may generate the same harmoid. There exist also an infinity of odd
set <a b c ...> generating the same harmoid.
It is important to find the minimal odd generator of a given harmoid. The canonical
order on the space of harmoids corresponds to the order of their minimal generator.
Your definition may permit to find it but also may fail to find that minimal generator.
For instance,
<1 10/9 5/4 5/3 20/11>
is a chordic generator of the gammier number 4 which contains the rast scale. Your
definition gives the lcm D = 99 and then the following odd generator
<99 110 165 180 495>
while the minimal generator is
<1 3 5 9 11>
I imagine you have already understood the problem linked to the duality in chordoid
structures: lines and columns of the chordic matrices being equigenerative.
There exist two ways to reduce a set of "odd parts" and we have to compare them
to find the minimal odd set.
Pierre.
[This message contained attachments]
Message: 2657 - Contents - Hide Contents Date: Sun, 23 Dec 2001 21:28:54 Subject: Re: 55-tET & 1/6-comma meantone From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> Paul, I'm having a hard time understanding the difference between > these two conceptions, but I think I'm beginning to get it. > > The implied ratios on my lattices follow the general trend of > the meantone axis itself, which implies a handful of intervals > which can be stacked to build the entire scale. > > But some of these JI intervals are emphatically *not consonant*, > and are the "wolf intervals" which cause the displacement of > the trend-line of the periodicity-block to align it with the > meantone axis. > > Am I on the right track?No, not really. This doesn't have quite that much to do with the wolf intervals, though it's related to the fact that once 81/80 is tempered out, Gene (if I may speak for him) and I would view the operative lattice as a cylindrical one -- the planar 2-d JI lattice no longer applies to the tuning, musically, psychologically, or spiritually.
Message: 2658 - Contents - Hide Contents
Date: Sun, 23 Dec 2001 03:47:55
Subject: Re: For Pierre, from tuning
From: Pierre Lamothe
Gene wrote:
The scales we get in this way are
scale1: [1, 10/9, 6/5, 4/3, 3/2, 5/3, 9/5]
scale2: [1, 9/8, 32/27, 4/3, 3/2, 27/16, 16/9]
scale3: [1, 16/15, 5/4, 4/3, 3/2, 8/5, 15/8]
There is overlap among these; their union is
[1, 16/15, 10/9, 9/8, 32/27, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 27/16, 16/9, 9/5, 15/8];
Does this have anything to do with what you are saying?
These scales and their union have certainly to do with what I am saying since there
is confinement in a region near unison and your scales are true modes in the sense of
the gammier theory. Moreover these modes are pretty well chosen to illustrate what
I wrote in my recent posts about unison vectors, hexagonal region, S-matrix, etc.
About the union I will say only that it misses only 27/20 and 40/27 to obtain the
gammier generated by <1 3 5 9 15 27>. But none of the three scales is a suigeneris
mode in that gammier, which is the Zarlino gammier <1 3 5 9 15> completed with the
odd 27 and whose steps are <16/15 10/9 9/8>.
I recall here what is a sui generis mode, that I would call "proper" modeif the term
was not already used. On a finite set of intervals modulo 2 with a partial order
defined by the partial composition law in the set, all totally ordered maximal
subset are modes in that structure. Among the sui generis modes of a structure,
those remarkable have a minimal transposition space, what is strongly correlated
to the harmonic properties of such modes.
I recall also that a gammier is obtained like a diamond with the differences between
the elements of an appropriate odd generator which is necessarily non convex.
I will add scale4: <1 9/8 5/4 4/3 3/2 5/3 15/8>, the Zarlino scale, for comparison.
What are the steps of the scales?
scale1: <27/25 10/9 9/8>
scale2: <256/243 9/8>
scale3: <16/15 9/8 75/64>
scale4: <16/15 10/9 9/8>
What are the transposition spaces of these scales, in other words, the intervals spaces
spanned by the elements of these scales, in other words, the gammiers generated by
these scales considered as chordic generators?
scale1: gammier <15 25 27 45 75 81 135>
scale2: gammier <1 3 9 27 81 243>
scale3: gammier <1 3 5 15 45 75 225>
scale4: gammier <1 3 5 9 15 27 45>
What are the S-matrix associated with your scales?
--------------------------------------------------------------------------------
HTML arrays
scale1 27/25 10/9 9/8
27/25 1 250/243 25/24
10/9 243/250 1 81/80
9/8 24/25 80/81 1
scale2 256/243 9/8
256/243 1 2187/2048
9/8 2048/2187 1
scale3 16/15 9/8 75/64
16/15 1 135/128 1125/1024
9/8 128/135 1 25/24
75/64 1024/1125 24/25 1
scale4 16/15 10/9 9/8
16/15 1 25/24 135/128
10/9 24/25 1 81/80
9/8 128/135 80/81 1
--------------------------------------------------------------------------------
What are the hexagones associated with these scales?
scale1: 81/80-25/24-250/243-80/81-24/25-243/250
scale2: no hexagone
scale3: 25/24-1125/1024-135/128-24/25-1024/1125-128/35
scale4: 81/80-135/128-25/24-80/81-128/135-24/25
That corresponds to elements in the matrices in order of a "8" shape.
We can look now at lattice representation in <2 3>Z2/ <2>Z for scale2 and
<2 3 5>Z3 / <2>Z for the others scales and see that the contents of the
segment or the hexagone is precisely the maximal gammier corresponding to
the transposition space. (In blue: class 0 - in red: the scale).
scale1
V . . . . . . 0 . . .
. 2 6 3 0 . . . . . .
. 0 4 1 5 2 . . . . .
. . . 6 3 0 4 1 . . .
. . . . . 5 2 6 3 U .
. . . . . . 0 4 1 5 .
. . . 0 . . . . . . 0
scale2
0 4 1 5 2 6 3 0 4 1 5 2 6 3 U
scale3
. . . . . 0 .
. . V 4 1 5 .
. . 5 2 6 3 U
. 6 3 0 4 1 .
0 4 1 5 2 . .
. 2 6 3 0 . .
. 0 . . . . .
scale4
. . . V . . . . .
0 4 1 5 2 6 3 0 .
. 2 6 3 0 4 1 5 .
. 0 4 1 5 2 6 3 U
. . . . . 0 . . .
What are the periodicity blocks associated with these gammiers in segment or hexagone?
scale1: <81/80 250/243> + <250/243 24/25> + <24/25 81/80>
scale2: <2187/2048> + <2048/2187>
scale3: <135/128 25/24> + <25/24 1024/1125> + <1024/1125 135/128>
scale4: <81/80 25/24> + <25/24 128/135> + <128/135 81/80>
As you may verified, the segment or the hexagone is obtained applying to the first block B
a matrix M being a third root of the identity matrix
[0 -1]
[1 -1]
in 3D and in the 2D case, at the reduced
[-1]
which is the square root of the unity.
--------------------------------------------------------------------------------
Gene wrote:
I need a definition before I get a statement or a proof — what do you mean by "simplest"?
There exist for any "well-structured scale" a minimal gammier in which thatscale corresponds
to a sui generis mode. And there exist a canonical order on the gammier space : the order of
their minimal generator.
What are the minimal generators of the minimal gammiers corresponding to our scales?
scale1: <15 25 27 45 135>
scale2: <1 3 27 243>
scale3: <1 5 45 75>
scale4: <1 5 9 15>
For comparison, the three simplest generators in <2 3 5>Z3 are
<1 3 5 9>
<1 3 9 15>
<1 5 9 15>
Pierre
[This message contained attachments]
Message: 2659 - Contents - Hide Contents Date: Sun, 23 Dec 2001 13:38:40 Subject: Re: For Pierre, from tuning From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 3:25 AM > Subject: [tuning-math] Re: For Pierre, from tuning > > > --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote: >>> What are the transposition spaces of these scales, in other words, the intervals spaces >> spanned by the elements of these scales, in other words, the gammiers generated by >> these scales considered as chordic generators? >> scale1: gammier <15 25 27 45 75 81 135> >> scale2: gammier <1 3 9 27 81 243> >> scale3: gammier <1 3 5 15 45 75 225> >> scale4: gammier <1 3 5 9 15 27 45> >> Judging by these examples, I would propose the following definitions: > > (1) We may write any positive rational number r in the form > 2^n p/q, where p and q are odd integers and p/q is reduced > to lowest form. The fraction p/q we call the *odd part* of r.I like this. What's interesting to me is to ponder the difference between thinking of a quantity in this form as opposed to the one I've preferred, which is simply 2^x 3^y 5^z... P^n, where P is the limiting prime-factor and x,y,z,n are integers (or often lately, fractions of integer terms). So is the general consensus that the former [2^n p/q] is best for describing dyads/intervals, and the latter [2^x 3^y 5^z... P^n] is best for describing larger entities such as entire tuning systems? What about those cases falling between, such as tri-, tetr-, pent-, hex-ads etc.? Most of you feel that the 2^n p/q notation is best for these, yes? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2660 - Contents - Hide Contents Date: Sun, 23 Dec 2001 11:25:18 Subject: Re: For Pierre, from tuning From: genewardsmith --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:> What are the transposition spaces of these scales, in other words, the intervals spaces > spanned by the elements of these scales, in other words, the gammiers generated by > these scales considered as chordic generators? > scale1: gammier <15 25 27 45 75 81 135> > scale2: gammier <1 3 9 27 81 243> > scale3: gammier <1 3 5 15 45 75 225> > scale4: gammier <1 3 5 9 15 27 45>Judging by these examples, I would propose the following definitions: (1) We may write any positive rational number r in the form 2^n p/q, where p and q are odd integers and p/q is reduced to lowest form. The fraction p/q we call the *odd part* of r. (2) For any set S of rational numbers, we may take the odd parts of each element, and the least common multiple D of their denominators. The set of integers gammier(S) is defined as the set of the odd parts of the elements ofS times D; gammier(S) is the *gammier of S*. Is there anything about this you want to accept, or to modify and then accept?
Message: 2661 - Contents - Hide Contents Date: Sun, 23 Dec 2001 13:41:40 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 1:18 PM > Subject: [tuning-math] Re: coordinates from unison-vectors (was: 55-tET) > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> I know that if the sign of the 3-exponent is changed, the sign >> for the 5-exponent must be reversed accordingly. But I find >> sometimes that using, for example, (4 -1) for the syntonic comma >> doesn't always give me the PB I expected, whereas making it >> (-4 1) does. >> As long as it's IN THE SAME FORM when you apply the inverse of the > matrix as well as when you apply the matrix itself, it won't matter -- > if you're centering around (0,0).My code does it automatically, so I guess it works correctly. All the user has to enter are the exponents of 3 and 5 for the two unison-vectors. Everything else is calculated from that. I've posted the Excel spreadsheet to the Files section. Yahoo groups: /tuning-math/files/monz/matrix m... * [with cont.] Also, I believe I've noticed that it makes a difference whether the larger or the smaller comma is listed first. Can someone check that and explain it? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2662 - Contents - Hide Contents Date: Sun, 23 Dec 2001 13:47:38 Subject: Re: 55-tET & 1/6-comma meantone From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 1:28 PM > Subject: [tuning-math] Re: 55-tET & 1/6-comma meantone > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> Am I on the right track? >> No, not really. This doesn't have quite that much to do with the wolf > intervals, though it's related to the fact that once 81/80 is > tempered out, Gene (if I may speak for him) and I would view the > operative lattice as a cylindrical one -- the planar 2-d JI lattice > no longer applies to the tuning, musically, psychologically, or > spiritually.And I've agreed with both of you many times in the past, and wish to emphasize again that the only reason I'm using a planar lattice is because it's beyond my abilities to draw cylindrical ones. In fact, I'd very much appreciate someone posting the mathematics for converting my Excel lattices into cylindrical ones. What would emerge from my meantone/JI-implication lattices if they were to be "cylindrified", is that each flavor of meantone would slice the cylinder diagonally at a different angle, and would also impart a unique diameter to each cylinder. Right? C'mon, guys... I'm itching to draw this stuff now... -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2663 - Contents - Hide Contents Date: Sun, 23 Dec 2001 13:50:16 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 1:22 PM > Subject: [tuning-math] Re: coordinates from unison-vectors (was: 55-tET) > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> lattice coordinates x, y : >> >> >> x = ( (q*c) + (p*a) ) / n >> >> y = ( (q*d) + (p*b) ) / n >> There shouldn't be an "/ n" at the end of that.Why not? "n" is the determinant of the matrix, and plays a crucial role in the transformation from one perspective to another.> Maybe that's what's causing the weirdness, because > sometimes n is negative.Hmmm... so perhaps I need to keep "/ abs(n)"? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2664 - Contents - Hide Contents Date: Sun, 23 Dec 2001 14:00:01 Subject: Re: Temperament names From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 1:20 PM > Subject: [tuning-math] Re: Temperament names > > > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: >>>>> 393216/390625 = 2^17 3 5^-8 >>>> This is Wuerschmidt's comma, so obviously the temperament is the >> Wuerschmidt. Who the heck is Wuerschmidt? >> Do you read German? If so, you should seek out his research. He found > some very interesting stuff, essentially looking into periodicity > blocks before Fokker.This piqued my interest, so I took a look at the Tuning and Temperament Bibliography Tuning & temperament bibliography * [with cont.] (Wayb.) and take the liberty of quoting the Joseph Würschmidt listings: "Logarithmische und graphische Darstellung der musikalischen Intervalle", Zeitschrift für Physik vol. 3, 1920, p. 89. "Viertel- und Sechsteltonmusik, eine kritische Studie", Neue Musikzeitung vol. 42, 1921, p. 183. "Über die neunzehnstufige Temperatur", Neue Musikzeitung vol. 42, 1921, p. 215. "Buchstabentonschrift und Von Oettingensches Tongewebe", Zeitschrift für Physik vol. 5, 1922, p. 111. "Die rationellen Tonsysteme in Quinten-Terzengewebe", Zeitschrift für Physik vol. 46, January 1928, p. 527. "Tonleitern, Tonarten und Tonsysteme. Eine historisch-theoretische Untersuchung", Sitzungsberichte der Physikalisch-medizinischen Sozietät zu Erlangen, Band 63, 1931, pp. 133-238. "Die neunzehn-stufige Skala; eine natürliche Erweiterung unseres Tonsystems" (The 19-tone scale; a natural expansion of our tonal system), Neue Musikzeitung vol. 14 no. 4, 1921, pp. 215-216. Which of these have you read, Paul? Can you summarize? I wonder if Tanaka wrote about periodicity-blocks before Würschmidt? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2665 - Contents - Hide Contents Date: Sun, 23 Dec 2001 14:04:33 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 1:41 PM > Subject: Re: [tuning-math] Re: coordinates from unison-vectors (was: 55-tET) > > > I've posted the Excel spreadsheet to the Files section. > Yahoo groups: /tuning-math/files/monz/matrix m... * [with cont.]I should have specified: the only worksheet in the file which draws the 5-limit periodicity-blocks is the one named "5-L PBs from UVs". The other worksheets illustrate 3-d examples from Graham's matrix tutorial webpage, and one of them was copied and currently has the unison-vectors for 22-EDO which were in my posts of a few days ago. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2666 - Contents - Hide Contents Date: 23 Dec 2001 19:15:55 -080 Subject: Keenan green Zometool struts From: paul@xxxxxxxxxxxxx.xxx Hey Dave, From Dave Keenan's Home Page * [with cont.] (Wayb.) one might get the idea that the Zome folks haven't implemented your green strut idea yet. But I recently saw a kit called "Advanced Mathematics" which did contain green struts. Did your ideas in fact help this product to be developed? Should I buy the kit? It's between 100 and 200 US$.
Message: 2667 - Contents - Hide Contents Date: 23 Dec 2001 19:16:24 -080 Subject: Keenan green Zometool struts From: paul@xxxxxxxxxxxxx.xxx Hey Dave, From Dave Keenan's Home Page * [with cont.] (Wayb.) one might get the idea that the Zome folks haven't implemented your green strut idea yet. But I recently saw a kit called "Advanced Mathematics" which did contain green struts. Did your ideas in fact help this product to be developed? Should I buy the kit? It's between 100 and 200 US$.
Message: 2668 - Contents - Hide Contents Date: Sun, 23 Dec 2001 22:11:56 Subject: Re: For Pierre, from tuning From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> So is the general consensus that the former [2^n p/q] is > best for describing dyads/intervals,Yes, if octave-equivalence is assumed.> and the latter > [2^x 3^y 5^z... P^n] is best for describing larger entities > such as entire tuning systems?This doesn't make sense, as you'll only be describing a _single ratio_ here, and that single ratio can not correspond to more notes than a dyad. However, I would agree that if one is looking at JI tuning systems, the highest prime number P is one of the most important pieces of information you could want about a system . . . as well as which, if any, primes are not used.
Message: 2669 - Contents - Hide Contents Date: Sun, 23 Dec 2001 22:16:13 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> I'm interested in seeing the differences between his formula and > the method I jury-rigged. :) > > ... always searching for greater elegance ...Why don't you give an example and I'll work it out in a different way?
Message: 2670 - Contents - Hide Contents Date: Sun, 23 Dec 2001 22:18:06 Subject: Re: 55-tET & 1/6-comma meantone From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> And I've agreed with both of you many times in the past, and wish > to emphasize again that the only reason I'm using a planar lattice > is because it's beyond my abilities to draw cylindrical ones.It shouldn't be -- you can simply "ink" the cylinder and then "roll" it a bunch of times over a flat sheet.> In > fact, I'd very much appreciate someone posting the mathematics for > converting my Excel lattices into cylindrical ones.Your Excel lattices, though, are currently referring to JI ratios, including some rather complex ones -- we have to get rid of this feature first. Gene, any clever ideas?> > What would emerge from my meantone/JI-implication lattices if > they were to be "cylindrified", is that each flavor of meantone > would slice the cylinder diagonally at a different angle, and > would also impart a unique diameter to each cylinder. Right?Hmm . . . not really. It seems to me that the angle and the diameter would be fixed, and the _second_ unison vector tells you how _long_ the cylinder is before it meets itself, when bent into a torus representing the ET you're approximating. If there is no second unison vector, than the different flavors of meantone are functionally identical, their only salient difference being the level of beating in the consonant intervals. It *would* be nice, I admit, to see some actual cylindrical arrangements of notes, particularly in a VRML implementation or something. So far, I've simply printed out flat, repeating lattices and then rolled them up by hand.
Message: 2671 - Contents - Hide Contents Date: Sun, 23 Dec 2001 22:19:39 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Sunday, December 23, 2001 1:22 PM >> Subject: [tuning-math] Re: coordinates from unison-vectors (was: 55-tET) >> >> >> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>> lattice coordinates x, y : >>> >>> >>> x = ( (q*c) + (p*a) ) / n >>> >>> y = ( (q*d) + (p*b) ) / n >>>> There shouldn't be an "/ n" at the end of that. > >> Why not? "n" is the determinant of the matrix, and plays > a crucial role in the transformation from one perspective > to another.All you need for the transformation in one direction is the matrix itself; and in the other direction, its inverse. You don't _additionally_ apply the determinant.>> Maybe that's what's causing the weirdness, because >> sometimes n is negative. >> Hmmm... so perhaps I need to keep "/ abs(n)"? Nope.
Message: 2672 - Contents - Hide Contents Date: Sun, 23 Dec 2001 22:22:11 Subject: Re: Temperament names From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> Which of these have you read, Paul?None, but Mandelbaum touched on his work. A lot of it involves a conception of "rational tone-systems", which are essentially 5-limit periodicity blocks, expressed with only three step sizes, but then equally tempered anyway. W.'s "rational tone systems" included 12-, 19-, 22-, 31-, 34-, 41-, 53-, 65-, and 118-tET, and there weren't many more given the constrains he imposed.
Message: 2673 - Contents - Hide Contents Date: Sun, 23 Dec 2001 14:26:29 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 2:19 PM > Subject: [tuning-math] Re: coordinates from unison-vectors (was: 55-tET) > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>> From: paulerlich <paul@s...> >>> To: <tuning-math@y...> >>> Sent: Sunday, December 23, 2001 1:22 PM >>> Subject: [tuning-math] Re: coordinates from unison-vectors >>> >>> >>> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>>>> lattice coordinates x, y : >>>> >>>> >>>> x = ( (q*c) + (p*a) ) / n >>>> >>>> y = ( (q*d) + (p*b) ) / n >>>>>> There shouldn't be an "/ n" at the end of that. >> >>>> Why not? "n" is the determinant of the matrix, and plays >> a crucial role in the transformation from one perspective >> to another. >> All you need for the transformation in one direction is the matrix > itself; and in the other direction, its inverse. You don't > _additionally_ apply the determinant.But when the inverse is described in integer terms, the determinant is part of it! Did you try my spreadsheet yet? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2674 - Contents - Hide Contents Date: Sun, 23 Dec 2001 22:25:04 Subject: Re: 55-tET & 1/6-comma meantone From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Your Excel lattices, though, are currently referring to JI ratios, > including some rather complex ones -- we have to get rid of this > feature first. Gene, any clever ideas?I thought you were the visual aids wizard; but if his lattices have numbersall over them why not take them off and see if you can get that to work for starters?
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