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Message: 5677 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 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 >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.
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Message: 5678 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]
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Message: 5679 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]
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Message: 5680 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.
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Message: 5681 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 | -------------------------------
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Message: 5682 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.
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Message: 5683 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.
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Message: 5684 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.
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Message: 5685 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.
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Message: 5686 Date: Mon, 24 Dec 2001 03:43:11 Subject: Re: Keenan green Zometool struts From: paulerlich Dave, See Advanced Math Kit * -- this is the kit I was talking about.
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Message: 5687 Date: Mon, 24 Dec 2001 04:00:39 Subject: Re: Keenan green Zometool struts From: paulerlich More info: Green Line Kit *
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Message: 5689 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
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Message: 5692 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?
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Message: 5693 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 * 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 http://www.monz.org * "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
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Message: 5694 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.
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Message: 5696 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.
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Message: 5698 Date: Tue, 25 Dec 2001 08:48:46 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.) > > Presumably, however you define it, PB requires convexity, I just said "any shape that tiles the plane". Certainly, I've also tended to impose convexity on top of the PB property whenever a JI, untempered scale is meant. Do any of your examples still apply?
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