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Message: 2675 - Contents - Hide Contents Date: Mon, 24 Dec 2001 04:19:44 Subject: Re: For Pierre, from tuning From: Pierre Lamothe Gene wote:> r mod 2 = {2^x p/q | all x in Z} is the corresponding octave classThere's a problem with this--number theorists already mean something quite specific by r mod 2, and this isn't it. r mod 2 is 0 if x>0, is 1 if x=0, and is 'infinity' or undefined if x<0. How about <2^n r> for this? I made an inattention error here. Forget r mod 2 = in the line and read simply {2^x p/q | all x in Z} is the corresponding octave class I use normally r mod <2> as equivalent to Ton (r) or Pivot (r) implying only a multiplicative modulo. -------------------------------------------------------------------------------- Gene wrote: So if {a, b, c} is a set of odd integers, the structure it generates are the ratios greater than one between them? The structure has two levels. I don't have time to explain that in details. I will use simply an example. Let {a, b, c} = {1, 3, 5}. The ordered odd generator g is noted <1 5 3> and the corresponding harmoid structure is represented by the chordic matrix g\g .1. 5/4 3/2 8/5 .1. 6/5 4/3 5/3 .1. The ordered contents '(g\g) {1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3} is the class level of the corresponding harmoid structure. I use most often the tones of the first octave to represent the classes. In that case the ratios are greater than one. But there exist equivalent manners to represent the classes: the pivots and also the tones of the centered octave and then the ratios are not all greater than one. {1/5, 1/3, 3/5, 1, 5/3, 3/1, 5/1} {2/3, 4/5, 5/6, 1, 6/5, 5/4, 3/2} The harmoid structure is not only a class structure, it's first an interval structure: {... 3/10, 3/5, 6/5, 12/5, ...} for instance, belong to the structure at interval level. I add here that the three first axioms are respected: that harmoid is a gammoid. --------------------------------------------------------------------------------> <99 110 165 180 495>Opps! I forgot to reduce. So <99 55 165 45 495>. -------------------------------------------------------------------------------- Gene wrote: However, taking 2 out of the picture gives me <1, 5/9, 5, 5/3, 5/11> which leads to <45,55,99,165,495> and finally to 1,3,5,9,5/3,9/5,11/3,11/5,11/9 which is not at all the same.> while the minimal generator is > <1 3 5 9 11>From this I get 1,3,5,9,5/3,9/5,11/3,11/5,11/9, with an extra 11 in there. Surely an error here. 495/45 = 11/1 Ton(11/1) = 11/8 -------------------------------------------------------------------------------- Gene wrote:> I imagine you have already understood the problem linked to the duality in chordoid > structures: lines and columns of the chordic matrices being equigenerative.I don't understand the above sentence; to follow you, I need clear, mathematical definitions. Ok, forget that for the moment. I thought you would have seen that there exist two ways to reduce a set of rational numbers to a set of odds. You have used only one. Using the precedent example, reduced first to pivots, I hope it will be now clear. These sets are strictly identical: {1, 5/9, 5/1, 5/3, 5/11} = {99/99, 55/99, 495/99, 165/99, 45/99} = {5/5, 5/9, 5/1, 5/3, 5/11} -------------------------------------------------------------------------------- I close here that session. I could only reply very shortly in these days. Merry Christmas! Pierre [This message contained attachments]
Message: 2676 - Contents - Hide Contents Date: Mon, 24 Dec 2001 13:59:01 Subject: Re: For Pierre, from tuning From: Pierre Lamothe Gene wrote: This is Partch's "Tonality Diamond", but what does the order do for you? Partch called it arbitary, and it seems to me that he got that right. In any case, "tonality diamond" is the recognized name here. Shortly. May I conclude you didn't know that the Zarlino gammier corresponds to a matrix like the "Tonality Diamond"? I repeated often that the chordic matrix is like the "Tonality Diamond", but this matrix is justified by the chordoid theory, which reconstruct the abelian group ending, rather than starting, with the closure axiom. So, without the closure, that structure has well-defined properties like being generated by a chordic matrix. It's not only limited to convex generators of N-limit type. More, none of the N-limit diamond is a gammier, the non-convexity is essential to respect the fertility condition. Many are even not gammoid, like the 11-limit structure, having not the CS property. The order has no importance. The choice of an order is only to permit direct reading, fo instance, of a.. the treillis which exhibit modes, like <15 1 9 5> b.. the chords structure, like <1 5 3 15 9> I will give later the axioms definition with their sense. Now it's holidays. Pierre [This message contained attachments]
Message: 2677 - Contents - Hide Contents Date: Mon, 24 Dec 2001 09:52:20 Subject: Re: For Pierre, from tuning From: genewardsmith --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:> Let {a, b, c} = {1, 3, 5}. The ordered odd generator g is noted <1 5 3> and the > corresponding harmoid structure is represented by the chordic matrix g\g > .1. 5/4 3/2 > 8/5 .1. 6/5 > 4/3 5/3 .1.This is Partch's "Tonality Diamond", but what does the order do for you? Partch called it arbitary, and it seems to me that he got that right. In any case, "tonality diamond" is the recognized name here.> The ordered contents '(g\g) > {1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3} > is the class level of the corresponding harmoid structure.I use most often the> tones of the first octave to represent the classes. In that case the ratios are > greater than one. But there exist equivalent manners to represent the classes:Let's see then--the "structure" in your nomenclature of a set of odd integers is a set of octave equivalence class representatives of the elements of the corresponding tonality diamond, ordered by size?> The harmoid structure is not only a class structure, it's first an interval structure: > {... 3/10, 3/5, 6/5, 12/5, ...} > for instance, belong to the structure at interval level.You've got to give different names to different things.> I add here that the three first axioms are respected: that harmoid is a gammoid.This does not convey much without precise definitions.
Message: 2678 - Contents - Hide Contents Date: Mon, 24 Dec 2001 21:07:03 Subject: Re: For Pierre, from tuning From: genewardsmith --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:> Gene wrote: > This is Partch's "Tonality Diamond", but what does the order do for you? Partch > called it arbitary, and it seems to me that he got that right. > > In any case, "tonality diamond" is the recognized name here. > Shortly. > > May I conclude you didn't know that the Zarlino gammier corresponds to a > matrix like the "Tonality Diamond"?Since the Zarlino gammier was not mentioned, I don't think you can concludeanything. However, I am the last person to suspect of having deep knowledge of the history of tuning theory, and to start with I need to ask if by the Zarlino gammier you mean the JI diatonic scale?> I repeated often that the chordic matrix is like the "Tonality Diamond", but > this matrix is justified by the chordoid theory, which reconstruct the abelian > group ending, rather than starting, with the closure axiom.It seems to me that the chordic matrix *is* the tonality diamond. Wherein do they differ? I can't see any abelian group structure in either, myself, because of the *lack* of closure.> It's not only limited to convex generators of N-limit type. More, none ofthe > N-limit diamond is a gammier, the non-convexity is essential to respect the > fertility condition. Many are even not gammoid, like the 11-limit structure, > having not the CS property.Again, without precise definitions mathematics can't operate. You need to clue me in.
Message: 2679 - Contents - Hide Contents Date: Mon, 24 Dec 2001 21:14:06 Subject: Re: For Pierre, from tuning From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote: > >> Gene wrote:>> This is Partch's "Tonality Diamond", but what does the order dofor you? Partch>> called it arbitary, and it seems to me that he got that right. >> >> In any case, "tonality diamond" is the recognized name here. >> Shortly. >> >> May I conclude you didn't know that the Zarlino gammiercorresponds to a>> matrix like the "Tonality Diamond"? >> Since the Zarlino gammier was not mentioned, I don't think you can >conclude anything. However, I am the last person to suspect of >having deep knowledge of the history of tuning theory,Don't worry -- the history of tuning theory will provide no clue as to what "Zarlino gammier" means.>and to start with I need to ask if by the Zarlino gammier you mean >the JI diatonic scale?He means the set of 19 ratios "mod 2" representing all the intervals that are found between pitches in the JI diatonic scale.
Message: 2680 - Contents - Hide Contents
Date: Mon, 24 Dec 2001 18:12:03
Subject: Re: For Pierre, from tuning
From: Pierre Lamothe
Gene,
I have really no time to define all with precision on that forum. If you want more
precision, you could find someone to translate that: the first definitions permitting
to explicit my chord theorem which suggests the chordicity as next axiom, just
before the closure giving then the abelian group.
--------------------------------------------------------------------------------
Structure de simploïde
Soit E un ensemble. Une partie non-vide S de E x E x E est un simploïde sur E, si elle admet les relations ternaires
(x,y,z) = (x,y,z')
(x,y,z) = (x,y',z)
La première relation détermine une loi de composition partielle dans E
pr12<S> --> pr3<S>
qui fait de S un groupoïde large et, réciproquement, la seconde relation détermine à son tour une loi d'accordance partielle dans E
\ : pr13<S> --> pr2<S>
Comme ces lois se déterminent mutuellement par la bijection canonique
((x,y),z) <--> ((x,z),y)
la donnée (parfois possible), pour un simploïde <E,S>, d'un graphe valué (N,F,V), appelé graphe d'accordance, où
les noeuds N = E
les flèches F = pr13<S>
les valeurs V = \(F)
équivaut
strictement à la donnée de la table de composition du simploïde, dont
les entrées, lorsqu'elles ne sont pas vides, correspondent, pour
chaque flèche, à T(source, valeur) = but.
On peut montrer aisément que l'ajout de la seconde relation correspond
à celui de l'axiome de simplicité à droite défini comme
ak = ak' implique k = k'
Commutativité et associativité
La commutativité et l'associativité sont généralement définies, en
dépendance de la fermeture algébrique,
ab = ba
a(bc) = (ab)c
De façon autonome, sans requérir la fermeture, on peut redéfinir ainsi
les axiomes de commutativité et d'associativité, où cette dernière est
réduite à une associativité à droite
k = ab implique k = ba
ak = (ab)c implique k = bc
Ajouter la commutativité à un simploïde entraîne la simplicité à
gauche, ou encore, pour compléter les deux relations ternaires
initiales, la relation
(x,y,z) = (x',y,z)
De même, en ajoutant la commutativité, l'associativité à droite
entraîne l'associativité à gauche et on peut ainsi parler simplement
de simploïde commutatif et associatif.
Lemme de composition transitive
L'existence des intervalles a\b, b\c et a\c dans un simploïde muni de
l'associativité à droite implique la composition transitive
(a\b)(b\c) = a\c
Démonstration.
Dans l'axiome d'associativité à droite
ak = (ab)c implique k = bc
la partie gauche de l'implication
ak = (ab)c
s'explicite
(a,b,x) appartient à S // où x = ab
(x,c,y) appartient à S // où y = xc = (ab)c
(a,k,z) appartient à S // où z = ak = (ab)c = y
et la partie droite k = bc s'explicite
(b,c,k) appartient à S // où k = bc.
En utilisant les trois intervalles de la partie gauche, et l'égalité z
= y, la partie droite peut être réécrite
(a\x,x\y,a\y) appartient à S
ce qui se réécrit encore, en assumant l'existence des intervalles
(a\x)(x\y) = a\y
Intervalles, accords, faisceaux
Dans un simploïde, les équations linéaires de la forme ax = b ont,
tout au plus, une solution. Cette solution, lorqu'elle existe, est
notée a\b et appelée intervalle entre a et b. On peut dire autrement
que a est accordé à b.
La loi d'accordance \ peut s'étendre à l'ensemble des parties p(E).
Soit A et B deux éléments de p(E). Si pour tout élément (a,b) de A x
Bil existe un élément k dans E tel que ak = b, on peut dès lors dire
que A est accordé à B, et que l'équation AX = B a une solution K,notée
A\B, qui désigne l'élément de p(E) correspondant aux intervalles
sous-tendus par (A,B).
Définition : Une partie A de E accordée à elle-même, est appelée un
accord dans E.
L'élément A\A de p(E), la solution de AX = A, formé des
intervallessous-tendus par l'accordance interne de A, est dit le
domaine générépar A, lequel est dit un générateur chordique de ce
domaine.
Deux accords A et B dans un simploïde sont
a.. égaux s'ils ont les mêmes éléments
b.. équigénératifs s'ils sous-tendent le même domaine A\A = B\B
c.. équipollents s'il existe une bijection f : A --> B tel que pour
tout x et y dans A
f(x)\f(y) = x\y
Soit k un élément de l'accord A. Les équations {k}X = A et AX ={k}
ont forcément une solution dans p(E). Ce sont des parties du
domaineA\A appelées respectivement faisceau divergent k\A et faisceau
convergent A\k du domaine A\A.
Par convention, l'absence de parenthèses est réservée à la notation
des faisceaux. Pour pouvoir écrire k\A et A\k au lieu de {k}\A et
A\{k}, qui sont des éléments de p(E), et où {k} est un ensemble à un
élément, il faut que A soit un accord et que k soit un élément de cet
accord.
Théorème des accords
Dans un simploïde,
a.. les faisceaux divergents k\A sont des accords équipollents à
A,s'il est muni de l'associativité à droite ;
b.. les faisceaux convergents A\k sont des accords équigénératifsà
A, et équipollents entre eux, s'il est muni en plus de la
commutativité.
Démonstration.
Soit <E,S> un simploïde sur E muni de l'associativité à droite, et k
un élément de l'accord A dans ce simploïde.
Deux éléments génériques x et y de A et l'élément k étant toujours
accordés entre eux, le lemme de composition transitive, qui découle de
l'associativité à droite, permet d'écrire la relation toujours avérée
(k\x)(x\y) = k\y
impliquant que k\x est toujours accordé à k\y
(k\x)\(k\y) = (x\y)
et, par extension à A tout entier, que le faisceau k\A est bien
accordé à lui-même, et constitue, de ce fait, un accord dans <E,S>.
Soit f une application de l'accord A sur l'accord k\A telle que x
--> k\x. Puisque la relation précédente peut se réécrire
f(x)\f(y) = x\y
il ne reste à montrer, pour assurer que k\A est bien équipollent à
A, que f est bijective, autrement dit que f(x) = f(y) entraîne x = y.
De la relation toujours avérée k(k\x) = x on tire
k f(x) = x
qui permet d'expliciter ce lien
f(x) = f(y)
k f(x) = k f(y)
x = y
(...)
--------------------------------------------------------------------------------
I cut here, after the demonstration of the first part of the chord
theorem.
Pierre
[This message contained attachments]
Message: 2681 - Contents - Hide Contents
Date: Mon, 24 Dec 2001 21:02:00
Subject: Re: For Pierre, from tuning
From: Pierre Lamothe
G. Why not simply go to the abelian group, and stay there?
P. I don't go to the abelian group for closure is not possible (without temperament)
in finite JI group. How determine a pertinent region in an infinite JI group?
G. You present some definitions, but they seem unmotivated, in other words.
P. It's your point of view. It's really funny. When I see the orgy of numbers and
technical talk here without any justification, I could ask me if all that is motivated.
Peace on earth at men of good will!
[This message contained attachments]
Message: 2682 - Contents - Hide Contents Date: Mon, 24 Dec 2001 00:34:27 Subject: Re: Flat 7 limit ET badness? (was: Badness with gentle rolloff) From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>>> I understand this to be equivalent to putting a sharp cutoff at > 600 >> on>>> steps*cents. The usual objection to sharp cutoffs applies, namely >>> people don't usually apply sharp cutoffs when making decisions > about>>> the usefulness of tunings. >>>> Oops! Only when cents is max-absolute error, not rms. >> Maybe it should be minimax.That's a tough choice. I'd rather see two separate rankings. One based on minimum-rms and another based on minimum maximum-absolute (minimax).> Maybe we should give a _range_ of optimal > generators, rather than just one, when the same minimax is achieved > for all within the range.Nah! Wouldn't you still want to know what value of generator minimises the max-absolute error of all those intervals that actually _depend_ on the generator.> Maybe we should also give the points at > which the minimax is doubled. This would give an idea of the > sensitivity of the tuning.The error-sensitivity of the tuning is already given by the maximum over the diamond, of the absolute value of the number of generators required for an interval. Which is the same as the maximum over the relevant primes minus the minimum over those primes, of the (signed) number of generators required for a prime.
Message: 2683 - Contents - Hide Contents Date: Mon, 24 Dec 2001 00:47:50 Subject: Re: a different example From: dkeenanuqnetau --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> It's hard for me to visualize what happens on a cylinder > or torus since I'm dealing with planar graphs.Here is 31-tET mapped onto the surface of a toroid as a 5-limit lattice. If you print out the lattice below (in a monospaced font), cut out the rectangle (cutting a half character width or height inside the lines), loop and tape it first side to side and then top to bottom, and you'll have it. Unfortunately you have to flatten it after the first looping to get it to loop in the other dimension, unless you printed it on rubber. [If you're viewing this from Yahoo's web interface, you will need to choose Message Index then Expand Messages to see it correctly formatted.] ------------------------------- | Gx | | Cx | | Fx | | B# | | E# | | A# | | D# | | G# | | C# | | F# | | B | | E | | A | |D | | G | | C | | F | |b B| | Eb | | Ab | | Db | | Gb| | Cb | | Fb | | BB | | EB | | AB | | DB | | GB | | Ax | | Ex | -------------------------------
Message: 2684 - Contents - Hide Contents Date: Mon, 24 Dec 2001 01:12:57 Subject: Re: Flat 7 limit ET badness? (was: Badness with gentle rolloff) From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> Maybe we should give a _range_ of > optimal>> generators, rather than just one, when the same minimax is achieved >> for all within the range. >> Nah! Wouldn't you still want to know what value of generator minimises > the max-absolute error of all those intervals that actually _depend_ > on the generator.Sorry, I was actually thinking MAD (mean absolute deviation), not minimax.
Message: 2685 - Contents - Hide Contents Date: Mon, 24 Dec 2001 01:16:11 Subject: Re: a different example From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> It's hard for me to visualize what happens on a > cylinder>> or torus since I'm dealing with planar graphs. >> Here is 31-tET mapped onto the surface of a toroid as a 5-limit > lattice. If you print out the lattice below (in a monospaced font), > cut out the rectangle (cutting a half character width or height inside > the lines), loop and tape it first side to side and then top to > bottom, and you'll have it. Unfortunately you have to flatten it after > the first looping to get it to loop in the other dimension, unless you > printed it on rubber.I'd suggest just taping the right edge to the left edge, as the resulting cylinder represents the 31 central tones of _any_ meantone, not just 31-tET.
Message: 2686 - Contents - Hide Contents Date: Mon, 24 Dec 2001 01:47:29 Subject: Re: My top 5--for Paul From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> I don't know if it deserves a name;You're right. It doesn't.> I tried to give it a name > because of its very low badness, but it's kind of absurd.Yes. It is a fine example of the musical irrelevance of a flat badness measure. I think musicians would rate it somewhere between 5 and infinity times as bad as the other four you listed. 50 notes for one triad? The problem, as usual is that an error of 0.5 c is imperceptible and so an error of 0.0002 c is no better, and does not compensate for a huge number of generators. Sorry if I'm sounding like a stuck record.
Message: 2687 - Contents - Hide Contents Date: Mon, 24 Dec 2001 01:51:25 Subject: Re: My top 5--for Paul From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>> I don't know if it deserves a name; >> You're right. It doesn't. >>> I tried to give it a name >> because of its very low badness, but it's kind of absurd. >> Yes. It is a fine example of the musical irrelevance of a flat badness > measure. I think musicians would rate it somewhere between 5 and > infinity times as bad as the other four you listed. 50 notes for one > triad? The problem, as usual is that an error of 0.5 c is > imperceptible and so an error of 0.0002 c is no better, and does not > compensate for a huge number of generators. Sorry if I'm sounding like > a stuck record.Let's not make decisions for musicians. Many theorists have delved into systems such as 118, 171, and 612. We would be doing no harm to have something to say about this range, even if we don't personally feel that it would be musically useful.
Message: 2688 - Contents - Hide Contents Date: Mon, 24 Dec 2001 03:43:11 Subject: Re: Keenan green Zometool struts From: paulerlich Dave, See Advanced Math Kit * [with cont.] (Wayb.) -- this is the kit I was talking about.
Message: 2689 - Contents - Hide Contents Date: Mon, 24 Dec 2001 04:00:39 Subject: Re: Keenan green Zometool struts From: paulerlich More info: Green Line Kit * [with cont.] (Wayb.)
Message: 2690 - Contents - Hide Contents Date: Mon, 24 Dec 2001 04:04:21 Subject: Re: a different example (was: coordinates from unison-vectors) From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> So anyway, I put in the matrix: > > ( 6 -14) > (-4 1) > > and could see that the resulting periodicity-block had a strong > correlation (in the sense of my meantone-JI implied lattices) > with the neighborhood of 2/7- to 3/11-comma meantone. > > The determinant is 50, so this agrees with my observation.I'll work this out in another way; however, I'm afraid that while in some ways it is more elegant, it only becomes computationally easier when the number of scale steps is high, and I don't know that 50 is high enough for that to be the case. First I put the 2 back into the above commas, and get q1 = 2^23 3^6 5^-15 and q2 = 80/81. I have one et, h50, such that h50(q1) = h50(q2) = 0. Now I search for something where h(q1)=0 and h(q2)=1, obtainining h19, h69, -h31 and -h81. The simplest of these is h19, and I choose it. Next, I look for something such that h(q1)=1 and h(q2)=0, and I get h16, -h34, -h84; I choose h16. Now I form a 3x3 matrix from these, and invert it: [ 50 19 16] [ 79 30 25]^(-1) = [116 44 37] [-10 -1 5] [ 23 6 -14] [ 4 -4 1] The rows of the inverted matrix correspond to the commas q0 = 2^-10 3^-1 5 = 3125/3072 (small diesis), q1 = 2^23 3^6 5^14, and q2=80/80. Now I calculate the scale; the nth step is scale[n] = q0^n * q1^round(19n/50) * q2^round(16n/50), where "round" rounds to the nearest integer. It doesn't matter which paticular hn I selected when I do this, or where I start and end; though my definition of "nearest integer" does matter. I got in this way: 1, 3125/3072, 128/125, 25/24, 82944/78125, 16/15, 625/576, 3456/3125, 10/9, 15625/13824, 144/125, 125/108, 18432/15625, 6/5, 625/512, 768/625, 5/4, 15625/12288, 32/25, 125/96, 20736/15625, 4/3, 3125/2304, 864/625, 25/18, 110592/78125, 36/25, 625/432, 4608/3125, 3/2, 15625/10368, 192/125, 25/16, 24576/15625, 8/5, 625/384, 1024/625, 5/3, 15625/9216, 216/125, 125/72, 27648/15625, 9/5, 3125/1728, 1152/625, 15/8, 78125/41472, 48/25, 125/64, 6144/3125 How does this compare with your results?
Message: 2691 - Contents - Hide Contents
Date: Mon, 24 Dec 2001 04:33:41
Subject: Re: a different example (was: coordinates from unison-vectors)
From: paulerlich
--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
With 50 notes, some arbitrary decision has to be made -- no note can
be exactly in the center, since 50 is an even number. But you should
be getting the following block or its reflection through the origin:
p5's M3's
---- -----
3 -7
4 -7
1 -6
2 -6
3 -6
4 -6
1 -5
2 -5
3 -5
0 -4
1 -4
2 -4
3 -4
0 -3
1 -3
2 -3
3 -3
0 -2
1 -2
2 -2
-1 -1
0 -1
1 -1
2 -1
-1 0
0 0
1 0
-2 1
-1 1
0 1
1 1
-2 2
-1 2
0 2
-3 3
-2 3
-1 3
0 3
-3 4
-2 4
-1 4
0 4
-3 5
-2 5
-1 5
-4 6
-3 6
-2 6
-1 6
-4 7
Message: 2692 - Contents - Hide Contents Date: Mon, 24 Dec 2001 04:46:09 Subject: Re: For Pierre, from tuning From: genewardsmith --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:> r mod 2 = {2^x p/q | all x in Z} is the corresponding octave classThere's a problem with this--number theorists already mean something quite specific by r mod 2, and this isn't it. r mod 2 is 0 if x>0, is 1 if x=0, and is 'infinity' or undefined if x<0. How about <2^n r> for this? 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 - log4So if {a, b, c} is a set of odd integers, the structure it generates are the ratios greater than one between them?> 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>I get <396,440,495,660,720>, with the ratios they generate being 1, 12/11, 10/9, 9/8, 5/4, 4/3, 16/11, 3/2, 18/11, 5/3, 20/11 However, taking 2 out of the picture gives me <1, 5/9, 5, 5/3, 5/11> which leads to <45,55,99,165,495> and finally to 1,3,5,9,5/3,9/5,11/3,11/5,11/9 which is not at all the same.> while the minimal generator is > <1 3 5 9 11>From this I get 1,3,5,9,5/3,9/5,11/3,11/5,11/9, with an extra 11 in there. From this I get> I imagine you have already understood the problem linked to the duality in chordoid > structures: lines and columns of the chordic matrices being equigenerative.I don't understand the above sentence; to follow you, I need clear, mathematical definitions.
Message: 2693 - Contents - Hide Contents Date: Mon, 24 Dec 2001 09:15:47 Subject: Shismic & co From: genewardsmith 6115295232/6103515625 = 2^23 3^6 5^-15 Semisuper Map: [ 0 2] [ 7 -3] [ 3 2] Generators: a = 52.00397043/118 (~3125/2304); b = 1/2 I gave the generator the silly name of "semisuper fourth" so as to get the silly name "Semisuper" for this system. I'm open for suggestions. badness: 190 rms: .1940 g: 9.933 errors: [.0226, .2081, .2255] 2^-15 3^-19 5^19 [(5/3)^19 ~ 2^14] Enneadecal Map: [ 0 19] [-1 38] [-1 52] Generators: a = 204.99802229/494 (~4/3); b = 1/19 badness: 391 rms: .1048 g: 15.513 errors: [.0741, -.0741, -.1482] 152, 171 and 323 work also, but 494 is the only et which really hits it. 32805/32768 Shismic Map: [ 0 1] [-1 2] [ 8 1] Generators: a = 120.000624/289 (~4/3); b = 1 badness: 55 rms: .1617 g: 6.976 errors: [-.2275, -.1338, .0937] 2^-52 3^-17 5^34 [(25/24)^17 ~ 2] Heptadecal Map: [ 0 17] [ 2 16] [ 1 34] a = 196.998525/612 (~5/4); b = 1/17 612 does this, Ennealimmal, *and* the similar 12 system whose proposed name I've forgotten! badness: 478 rms: .03437 g: 24.042 errors: [0, -.0421, .0421]
Message: 2694 - Contents - Hide Contents Date: Tue, 25 Dec 2001 08:42:56 Subject: Re: The epimorphic property From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> OK, but CS ==> PB in all "reasonable" cases where the unison vectors >> are not "ridiculously large" relative to the step sizes -- right? >> (Clearly a definition of "ridiculously large" is needed.) >> How does CS allow you to conclude you even have unison vectors?I've never seen a counterexample.>Having enough unison vectors to define the map is equivalent to >being epimorphic, by the way.So can you come up with a counterexample?
Message: 2695 - Contents - Hide Contents Date: Tue, 25 Dec 2001 15:44:28 Subject: lattices of Schoenberg's rational implications From: monz I was looking again at the post I sent here regarding the math of Schoenberg's tuning ideas: Yahoo groups: /tuning-math/messages/516?expand=1 * [with cont.] I constructed matrices of the unison-vectors mentioned by Schoenberg. The matrix of the later 13-limit system (explained by Schoenberg in 1934) expectedly has a determinant of 12. But interestingly, the 11-limit matrix (as described in 1911 in _Harmonielehre_) has a determinant of 7. I found this interesting first of all because Schoenberg, in _Harmonielehre_, defines the 7-tone diatonic scale in familiar 5-limit terms, then introduces the 11-limit ratios in an attempt to explain the origin of the chromatic notes. But his explanation is somewhat vague and incomplete, and introduces a notational inconsistency about which I say more below, so it's not really a surprise that the determinant of this periodicity-block is not 12. For the unison-vectors, I specified the relationships between pairs of notes which Schoenberg described as equivalent. Unison-vector matrices: 1911 _Harmonielehre_ 11-limit system ( 1 0 0 1 ) = 33:32 (-2 0 -1 0 ) = 64:63 ( 4 -1 0 0 ) = 81:80 ( 2 1 0 -1 ) = 45:44 Determinant = 7 1934 _Problems of Harmony_ 13-limit system (-2 0 0 -1 1 ) = 104:99 ( 2 0 -1 0 1 ) = 117:112 (-2 0 -1 0 0 ) = 64:63 ( 4 -1 0 0 0 ) = 81:80 ( 2 1 0 -1 0 ) = 45:44 Determinant = 12 The 13-limit system gives me no surprises. But the 11-limit system is intriguing. I have noted many times (as in my book and in that post) that Schoenberg was inconsistent in his naming of the pitches of this system. From my tuning-math post:>> (Note also that Schoenberg was unsystematic in his naming >> of the nearly-1/4-tone 11th partials, calling 11th/F by the >> higher of its nearest 12-EDO relatives, "b", while calling >> 11th/C and 11th/G by the lower, "f" and "c" respectively. >> This, ironically, is the reverse of the actual proximity >> of these overtones to 12-EDO: ~10.49362941, ~5.513179424, >> and ~0.532729432 Semitones, respectively).What I found is that eliminating this inconsistency, i.e., calling 11th/F a "Bb" instead of "B", also destroys the periodicity-block aspect of this system. The 45:44 unison-vector which results from calling that note "B" is *necessary* in order to define the 7-tone periodicity-block. Calling it a "Bb" removes the 45:44 UV and replaces it with the 22:21 UV already found as a result of combining the existing 33:32 and 64:63 UVs for the various "F"s. Thus, the matrix has only 3 UVs and lacks the remaining one which is necessary to define a PB. But why do I get a determinant of 7 for the 11-limit system? Schoenberg includes Bb and Eb as 7th harmonics in his description, which gives a set of 9 distinct pitches. But even when I include the 15:14 unison-vector, I still get a determinant of -7. And if I use 16:15 instead, then the determinant is only 5. Can someone explain what's going on here, and what candidates may be found for unison-vectors by extending the 11-limit system, in order to define a 12-tone periodicity-block? Thanks. (and Merry Christmas to all) 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: 2696 - Contents - Hide Contents Date: Tue, 25 Dec 2001 08:43:39 Subject: Re: The epimorphic property From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: >>> How does CS allow you to conclude you even have unison vectors?>Having enough unison vectors to define the map is equivalent to >being epimorphic, by the way. > > Plus, the map has to correctly order the scale, so we do need a >little more. Example, please.
Message: 2697 - Contents - Hide Contents Date: Tue, 25 Dec 2001 08:45:07 Subject: Re: The epimorphic property From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> OK, but CS ==> PB in all "reasonable" cases where the unison vectors > are not "ridiculously large" relative to the step sizes -- right? > (Clearly a definition of "ridiculously large" is needed.)Presumably, however you define it, PB requires convexity, in which case 1--9/8--5/4--4/3--40/27--5/3--15/8 would be an example of a scale which was epimorphic but not PB. Similarly, assuming (always a possibility) I understand Joe's definition correctly, 1--9/8--5/4--4/3--1024/675--5/3--15/8 would be an example of a scale which is CS, but neither epimorphic nor PB. How am I doing?
Message: 2698 - Contents - Hide Contents Date: Tue, 25 Dec 2001 08:46:38 Subject: Re: For Pierre, from tuning From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: genewardsmith <genewardsmith@j...> >> To: <tuning-math@y...> >> Sent: Monday, December 24, 2001 11:01 PM >> Subject: [tuning-math] Re: For Pierre, from tuning >> >> >> We've also devoted quite a lot of time to JI scales, >> mostly of those (Fokker blocks and the like) which are >> preimages of an equal temperament mapping. >> Hmmm... this description sounds very much like what I'm > trying to portray with my "acoustical rational implications > of meantones" lattices. The JI periodicity-blocks I derive > could be called "preimages of a meantone mapping", which in > turn in many cases equate to an equal-temperament mapping.Right . . . but you seem too concerned with certain qualities of the preimage which end up having no relevance once the meantone and/or ET tempering happens. For example all the 55-tone periodicity blocks I gave are exactly equivalent to one another once the 81:80 is tempered out -- so there's no point in preferring one to another, unless you really intend to use the JI tuning, wolves and all, rather than the meantone temperament.
Message: 2699 - Contents - Hide Contents Date: Tue, 25 Dec 2001 00:52:27 Subject: Re: For Pierre, from tuning From: genewardsmith --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:> I have really no time to define all with precision on that forum. If you want more > > precision, you could find someone to translate that: the first definitions permitting > > to explicit my chord theorem which suggests the chordicity as next axiom,just > > before the closure giving then the abelian group.After struggling through this, I still don't know why you want to mess around with groupoids. Why not simply go to the abelian group, and stay there? You present some definitions, but they seem unmotivated, in other words.
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