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Message: 503 - Contents - Hide Contents Date: Fri, 13 Jul 2001 20:16:49 Subject: Re: Lamothe SD and "Octave Modularity" From: Pierre Lamothe In post #496, #497, #499 ... J Gill wrote << So here's my perplexment. Pierre seems to include the contribution of the prime number 2 ... -- SNIP -- I am very interested in anyone's (included Mr Lamothe's, if he is able to find the time) feedback on this.>>It's like a BANG BANG BANG ... at my door. I imagine I don't have the choice to make an exception and answer. However, I would avoid to quote more over all. :-) About the post #18625 on the Tuning List, the 'reduced modular complexity' was a temporary notion having utility only to show the effect of the representation in Z-module using octave modularity. That modularity is expressed by a rotation, as I wrote to Paul Erlich in post #18656 << As you may know, my definition of the sonance (or log complexity) C = C(X) = |x|*log 2 + |y|*log p + |z|*log q corresponds to the Tenney's Harmonic Distance. What I wanted to show is the effect of the representation of this sonance by the distance MC = MC(X) = |x| + |y| + |z| in <2,p,q> as Z-module and the distance RMC = RMC(X) = |y| + |z| in <p,q> where octave modularity is used. I underline that it is not question of complexity metric but topological invariance using MC and RMC to represent sonance (or log complexity). The first reduction corresponds to a variation in axis elongation ratios and the second to a counterclockwise rotation. The invariance concerns the spatial location of the intervals between them and the convexity property is not changed by these two transformations. >> While MC and RMC had only a temporary use, the _Sonance degree_ is rather a significant concept having sense in all congruent system (using unison vectors) having a Degree function (a surjective morphism on Z) preserving both the 'width order' and the 'sonance order' of the intervals. Since I did'nt precise that in the definition, I will give in APPENDICE the condition to preserve the sonance order. Comparing here the _Sonance degree_ (SD) using the operator D = [k2 kp kq] with the precedent definitions of sonance (S) and temporary MC and RMC SD * k2 = |x| * k2 + |y| * kp + |z| * kq S * log(2) = |x| * log(2) + |y| * log(p) + |z| * log(q) MC = |x| + |y| + |z| RMC = |y| + |z| we can see that only the _Sonance degree_ is an approximation of the _Sonance_ and as long as log(2):log(p):log(q) is well approximated by k2:kp:kq. I would E M P H A S I S here these remarks : (1) Like the Sonance (S), the _Sonance degree_ (SD) concerns first the intervals and it is applied to octave classes only in a derived manner. The value SD for an interval class modulo 2 is determined (by convention) as the value SD for the interval in the first octave of this class. (2) The variation of SD with octave changing is not proper to SD. It is strictly the same as the variation of S. So if that variability is a source of perplexity, this has got nothing to do with the specificity of SD as approximant of S but with the Sonance (Tenny's distance) as such. ----- The variation of the sonance with the octave changing is independant of the approximation of S by SD, and the sonance of an irreducible ratio n/d between any rational pair (kn:kd) is both independant of any musical context and any acoustical spectrum. It is simply (log n + log d) / log 2 which is the log in base 2 of the complexity n*d. ----- The following table shows for the classes of 9/8, 6/5, 5/4 and 4/3 the distinct values in 7 octaves for ratio, complexity, and the variation of the sonance S (or sonance degree SD since the variation is the same) compared to a = S(9/8), b = S(6/5), c = S(5/4) and d = S(4/3) in the first octave (i.e. the values used for the classes). -3 -2 -1 0 1 2 3 9/64 9/32 9/16 9/8 9/4 9/2 ( 9/1 ) 3/20 3/10 ( 3/5 ) 6/5 12/5 24/5 48/5 5/32 5/16 5/8 5/4 5/2 ( 5/1 ) 10/1 1/6 ( 1/3 ) 2/3 4/3 8/3 16/3 32/2 596 288 144 72 36 18 ( 9 ) 60 30 ( 15 ) 30 60 120 240 160 80 40 20 10 ( 5 ) 10 6 ( 3 ) 6 12 24 48 64 a+3 a+2 a+1 a a-1 a-2 ( a-3 ) b+1 b ( b-1 ) b b+1 b+2 b+3 c+3 c+2 c+1 c c-1 ( c-2 ) c-1 d-1 ( d-2 ) d-1 d d+1 d+2 d+3 The minimal sonance inside a class occurs for the "pivot" of the class which are here ( 9/1 ), ( 3/5 ), ( 5/1 ) and ( 1/3 ). Increasing or decreasing by N octaves relatively to the pivot results to add N to the sonance value of the pivot. It's there a microtonal property (i.e. independant of the organisation of the intervals). Besides, for a strutural understanding of musical modes it is sufficient to use the conventional sonance value attributed to the interval in the first octave which represents the class. The microtonal property is not neglected here since it is well reflected in the classification +/- major or minor of the modes (inside the structure). An interval like 9/8 is said "major" for its pivot 9/1 is higher while an interval like 8/5 is said "minor" for its pivot 1/5 is lower. So a major interval is more consonant an octave above while a minor interval is more consonant an octave under. A mode like 1 9/8 5/4 4/3 3/2 5/3 15/8 is major relatively to its dual 1 16/15 6/5 4/3 3/2 16/9 for "major" intervals predominate in the first mode while (forcely) "minor" in its dual. ----- The conventional use of the intervals in the first ascending octave to represent the class modulo 2 introduce an asymmetry for the complexity between an interval like 5/4 and its reverse 8/5. A symmetric complexity would result if pivots were used or if the conventional octave of reference was centered 1/sqrt(2) < octave < sqrt(2) When I need a symmetric sonance for the classes in particular context, I use simply the sum SS(5/4) = SS(8/5) = S(5/4) + S(8/5) which would be the same with the first descending octave representing the class SS(4/5) = SS(5/8) = S(4/5) + S(5/8). ----- Any acoustical consideration here seems to me at hundred miles of the question. There exist, for sure, acoustical conditions permitting to perceive an interval and contextual conditions permitting to perceive it as a category. Inside the limits of these good conditions, we need to change the semiotic level to understand relational properties like sonance (microtonal) and sonance degree (macrotonal). Since this post is an exception, I would add, without translation, what I wrote to Yves Hellegouarch, almost a year ago : << Aussi justes soient-elles, les considérations acoustiques, en regard de la question des gammes, m'apparaissent piégeantes. Elles introduisent subrepticement une erreur de perspective. La place privilégiée qu'occupe la notion d'harmonique dans le son musical laisse croire à une semblable primauté au niveau des structures tonales. Et on confond généralement trois aspects de l'harmonicité, selon qu'elle a trait à la paramétrisation du son musical, à l'explicitation de la sonance des accords, ou à la génération des modes musicaux. Or l'harmonique n'occupe une place privilégiée que dans le son musical. Mais alors pour cause. Pour qu'apparaisse l'intervalle de hauteur, à titre d'atome d'intelligibilité dans des processus d'évaluation de similitude formelle, il faut que les sons soient musicaux, autrement dit, qu'ils possèdent une périodicité principale marquée. Et c'est en raison seulement de cette périodicité qu'un tel son est analysable en terme d'harmoniques (et non de sous-harmoniques) de la fréquence principale. Au plan de la sonance, les théories acoustiques de la dissonance, de l'alignement des harmoniques d'Helmholtz aux bandes critiques de Zwicker, privilégient aussi l'harmonique. Mais j'estime qu'elles sont foncièrement erronées, au sens où elles peuvent espérer rendre compte de la rugosité mais nullement de la dissonance. Enfin, au niveau macrotonal, que j'estime avoir adéquatement théorisé, autour des problématiques de la liaison de la cohérence à la consonance puis de l'émergence de la simplicité, rien ne peut plus permettre de privilégier le générateur harmonique en regard de son dual. Mieux, sans la dualité, la musique ne serait probablement que rythmique et peinture sonore.>> -----About the << mother nature >> I would quote also a text on my website << Il faut savoir apprécier les contorsions mentales auxquelles on devait se livrer pour justifier la triade mineure à partir des idées de Rameau, puis cette assurance d’Hindemith qui se voyait écraser le dualisme de Riemann d’un simple diktat à croire en béton : « les sous-harmoniques n’existent pas dans la nature ». On commence à peine à comprendre que la linéarité (autrement dit le fait qu’on dispose de composantes qui n’interfèrent pas entre elles) n’est pas imposée par la nature — bien qu’elle ait à voir avec le phénomène physique de la résonance — mais par l’esprit. C’est l’exigence de périodicité qui détermine et non une quelconque tendance de la nature à créer des sons harmonieux, dont on se demande bien d’ailleurs, mis-à-part l’exemple des oiseaux, où ils seraient présents. Depuis la découverte du chaos déterministe on prend conscience, surtout, que la nature ne se contente pas des scénarios qui paraissent les plus simples, et que cette omniprésence de la linéarité dans les sciences physiques ne reflète en rien une prééminence naturelle séparée de notre esprit.>> Pierre Lamothe APPENDICEI did'nt precise in my _Sonance degree_ definition the conditions about the unison vectors giving a true coherent system. I added here that the resulting Degree function X --> D(X) has to be a surjective morphism on Z preserving both the 'width order' and the 'sonance order' of the intervals. It's not the good place to develop my theory of the coherent unison vectors. But since SD is in question I would like to precise that the condition which preserves the 'sonance order' is simply that the canonical basis have integer values. I wrote in post #272 about the definition << in the basis B = <B0 B1 .. BN> where B0 = 2 (octaviant system) and the other independant components are normally simple primes (primal basis).>>Normally, in simple systems, the canonical basis is a primal basis. But there exist Z-modules in which links exist between primes. The two types of such links are caracterized by these examples <2 3*5 7> Z^3 is a submodule of <2 3 5 7> Z^4 <2 3 7/11> Z^3 is a submodule of <2 3 7 11> Z^4 There is no problem in the first case whose canonical basis <2 15 7> has integer values while a D function associated with a periodicity block in the second case don't preserve the 'sonance order' of intervals. Scales of Margo Schulter using third 14/11 rather than 5/4 are concerned with that but it is sufficient to show here the pentatonic system generatedby <7 11 33 99> == <1 3 9 7/11> which is an approximant or the 5-limit chinese system <1 3 5 9> like the Ling Lun <1 3 9 81> is a such approximant. With a canonical basis such <2 3 7/11> a Sonance degree (SD) has no sense since the sonance S of (x,y,z) is |x|log(2) + |y|log(3) + |z|log(7) + |z|log(11) and not |x|log(2) + |y|log(3) + |z|log(7/11) which is equal to |x|log(2) + |y|log(3) + |z|log(7) - |z|log(11) The systems where we find scales near to Margo Schulter's ones don't preserve the 'sonance order', but it's not incoherent with the Gothic style of music using fifth and fourth as "stable sonorities". It would be different trying to consider the approximant chord 22:28:33 as a "stable sonority" like its chord image 4:5:6 in <1 3 5 9>.
Message: 504 - Contents - Hide Contents
Date: Sat, 14 Jul 2001 17:51 +0
Subject: Temperament catalog
From: graham@m...
I'm uploading a list of notable linear temperaments to
<Catalogue of linear temperaments * [with cont.] (Wayb.)>. Please report any errors or
omissions. The idea is that next time somebody thinks they have come up
with something new, they can see if it's on the list. Also if they have a
previous reference for anything, they can submit it.
Those 13-limit mappings of Erv Wilson should be added, if anybody can
remember them.
Monz already has a list of equal temperaments. Push the link my way and
I'll add it!
Graham
Message: 505 - Contents - Hide Contents Date: Sat, 14 Jul 2001 11:41:36 Subject: Re: Temperament catalog From: monz> From: <graham@m...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, July 14, 2001 9:51 AM > Subject: [tuning-math] Temperament catalog > > > I'm uploading a list of notable linear temperaments to > <Catalogue of linear temperaments * [with cont.] (Wayb.)>. Please report any errors or > omissions. The idea is that next time somebody thinks they have come up > with something new, they can see if it's on the list. Also if they have a > previous reference for anything, they can submit it.Good show, Graham! The first sentence under "Miracle" says "Has its own page [a link to Graham's], better covered by Monz, I'll link to that sometime." This should make it easier: Definitions of tuning terms: MIRACLE scale, (c... * [with cont.] (Wayb.) Note that I updated that page so that the mapping is given the way Graham does it: (period, generator). Near the end, under "Golden Section Tuning", there's a typo: "confussed" should be "confused".> Monz already has a list of equal temperaments. Push the link my way and > I'll add it!Sure thing... and I've just updated it again. Definitions of tuning terms: equal temperament... * [with cont.] (Wayb.) And, I've added a link to your catalog in my "linear temperament" definition (which, BTW, could probably be expanded... hint...) Definitions of tuning terms: linear temperamen... * [with cont.] (Wayb.) -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: 506 - Contents - Hide Contents Date: Sat, 14 Jul 2001 21:53:39 Subject: interesting observation on sruti proportions From: monz> From: monz <joemonz@y...> > To: <tuning@xxxxxxxxxxx.xxx> > Sent: Saturday, July 14, 2001 2:25 PM > Subject: [tuning] "typical" sruti ratios [was: what a dope (user)] > > > You can see here the three Pythagorean equivalents for > the sruti names given by Lentz. The three types of > srutis have the following Pythagorean ratios and > interval sizes in approximate cents: > > > Ratio Lentz's 5-limit Indian > 2^x * 3^y ~cents equivalent sruti name > > | -19 12 | 23.46001038 81/80 pramana > | 27 -17 | 66.76498529 25/24 nyuna > | 8 - 5 | 90.22499567 -- puranaI was playing around with these Pythagorean srutis on an Excel graph, trying to see what kind of proportions they exhibit among themselves, and found something very interesting. First, I noticed just by changing the scale of the y-axis to a major unit of 22.5 cents that the approximate proportions of the cents-values of these three srutis is 1:3:4. I vaguely remember reading something about this somewhere, in that the intervals between pitches of the Indian scale were given in that book as 1, 3, or 4 srutis. The specific type of sruti would thus have been the purana, tho not so indicated in that book. Then I went on and did the math, and look what I found: 90.22499567 / 23.46001038 = 3.845906042 = 3 & 3151/3725 66.76498529 / 23.46001038 = 2.845906042 = 2 & 3151/3725 Isn't it odd that these proportions both have *exactly* the same fractional part? Or is it a property of Pythagorean tuning that makes this so? The denominator, 3725, is prime-factored into (5^2) * 149, in case there's any significance in that. Very curious, -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: 507 - Contents - Hide Contents Date: Sun, 15 Jul 2001 09:56 +0 Subject: Re: interesting observation on sruti proportions From: graham@m... monz wrote:> First, I noticed just by changing the scale of the y-axis to > a major unit of 22.5 cents that the approximate proportions > of the cents-values of these three srutis is 1:3:4.I think you've discovered 53 tone equal temperament.> Then I went on and did the math, and look what I found: > > 90.22499567 / 23.46001038 = 3.845906042 = 3 & 3151/3725 > 66.76498529 / 23.46001038 = 2.845906042 = 2 & 3151/3725 > > Isn't it odd that these proportions both have *exactly* > the same fractional part? Or is it a property of Pythagorean > tuning that makes this so?The largest sruti is the sum of the other two. 90.22499567 / 23.46001038 = (66.76498529 + 23.46001038) / 23.46001038 = 66.76498529 / 23.46001038 + 1 Graham
Message: 508 - Contents - Hide Contents Date: Sun, 15 Jul 2001 08:01:48 Subject: Re: interesting observation on sruti proportions From: monz> From: <graham@m...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, July 15, 2001 1:56 AM > Subject: [tuning-math] Re: interesting observation on sruti proportions > > > monz wrote: >>> First, I noticed just by changing the scale of the y-axis to >> a major unit of 22.5 cents that the approximate proportions >> of the cents-values of these three srutis is 1:3:4. >> I think you've discovered 53 tone equal temperament.Yup, I noticed that too! Thanks.> >>> Then I went on and did the math, and look what I found: >> >> 90.22499567 / 23.46001038 = 3.845906042 = 3 & 3151/3725 >> 66.76498529 / 23.46001038 = 2.845906042 = 2 & 3151/3725 >> >> Isn't it odd that these proportions both have *exactly* >> the same fractional part? Or is it a property of Pythagorean >> tuning that makes this so? >> The largest sruti is the sum of the other two. > > 90.22499567 / 23.46001038 > > = (66.76498529 + 23.46001038) / 23.46001038 > > = 66.76498529 / 23.46001038 + 1Hmmm... thanks for explaining this, Graham. Could you go into a little more detail as to why the fractions work out the way they do? It's interesting to me, with my numerology fetish. -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: 509 - Contents - Hide Contents Date: Sun, 15 Jul 2001 20:39 +0 Subject: Re: interesting observation on sruti proportions From: graham@m... "The" "Monz" wrote:>> The largest sruti is the sum of the other two. >> >> 90.22499567 / 23.46001038 >> >> = (66.76498529 + 23.46001038) / 23.46001038 >> >> = 66.76498529 / 23.46001038 + 1 > >> Hmmm... thanks for explaining this, Graham. Could you go > into a little more detail as to why the fractions work > out the way they do? It's interesting to me, with my > numerology fetish.Pythagorean tuning can be defined using two intervals. So you can never have more than two Pythagorean intervals that are all independent. That's the only mathematically interesting thing I see here. Graham
Message: 510 - Contents - Hide Contents Date: Mon, 16 Jul 2001 22:43:28 Subject: Re: Temperament catalog From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> I'm uploading a list of notable linear temperaments to > <Catalogue of linear temperaments * [with cont.] (Wayb.)>. Please report any errors or > omissions.The Kleisma is a small interval generated from a series of just minor thirds. You must be thinking of 225:224 which is the "_septimal_ kleisma". As for omissions -- How about BP, for one?
Message: 511 - Contents - Hide Contents Date: Tue, 17 Jul 2001 11:02 +0 Subject: Re: Temperament catalog From: graham@m... In-Reply-To: <9ivqmg+507h@e...> Paul wrote:> The Kleisma is a small interval generated from a series of just minor > thirds. You must be thinking of 225:224 which is the "_septimal_ > kleisma".You mean the difference between four 6:5s and an octave?> As for omissions -- How about BP, for one?Either a JI scale or an ET, but not a linear temperament, unless you know better. Graham
Message: 512 - Contents - Hide Contents Date: Tue, 17 Jul 2001 15:10: Subject: Re: Naming intervals using Miracle From: manuel.op.de.coul@e... Dave Keenan wrote:>Does anyone feel that any of these names are somehow wrong? >Does this conflict with any existing use of "wide" and "narrow"? e.g. Scala.No direct conflicts, but I haven't seen the combinations "narrow neutral" and "wide neutral" before, and find it a bit counterintuitive. I'd prefer "large minor" and "small major". Manuel
Message: 514 - Contents - Hide Contents Date: Wed, 18 Jul 2001 05:29:21 Subject: Re: Temperament catalog From: Dave Keenan --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <9ivqmg+507h@e...> > Paul wrote: >>> The Kleisma is a small interval generated from a series of just minor >> thirds. You must be thinking of 225:224 which is the "_septimal_ >> kleisma". >> You mean the difference between four 6:5s and an octave?No. See Scala's intnam.par: 15552:15625 (8.11 c) or my 11 note chain-of-minor-thirds scale * [with cont.] (Wayb.) "A kleisma is the difference between a just fifth (2:3) and an octave reduced chain of 6 just minor thirds (5:6)" Presumably the septimal kleisma was so named because it is close in size to the kleisma. The catalog is a brilliant effort Graham. Thanks. You haven't told us what sort of interval the generator is, for some of them, and in some you've given a "generator" which is in fact the period. You could add to the Miracle entry, the fact that Secor proposed it as a way of making sense of Partch's 43 note gamut (i.e. not out of a musical vacuum.) Some others that have been found interesting by more than one person (one of those people being me) are: (-4 3 -2), a wide minor second generator (about 125c), octave period, 19 and 29-EDO. (-3, -5, 6), a wide neutral second generator (about 163c), octave period, 15 and 22-EDO. (7, -3, 8), a subminor third generator (about 272c), octave period, 22 and 31-EDO. (5, 1), a major third generator (about 380c), octave period, 19 and 60-EDO. Wilson's 11-limit mapping is: (-1, 8, 14, 18), perfect fourth generator (about 497 c), octave period, 41-EDO. Wilson's 13-limit mapping is: (-1, 8, 14, -23, -20), perfect fourth generator (about 498 c), octave period, 41 and 53-EDO. Your own 13-limit mapping that gives a more compact diamond (68 notes instead of 75) but with slightly higher errors is: (1, -2, -8, -12, -15), minor second generator (about 104 c) or equivalently a perfect fourth generator (about 496c), half-octave period, 46 and 104-EDO. -- Dave Keenan
Message: 515 - Contents - Hide Contents Date: Wed, 18 Jul 2001 05:40:56 Subject: Re: Naming intervals using Miracle From: Dave Keenan --- In tuning-math@y..., manuel.op.de.coul@e... wrote:> > Dave Keenan wrote:>> Does anyone feel that any of these names are somehow wrong? >> Does this conflict with any existing use of "wide" and "narrow"? e.g. > Scala. >> No direct conflicts, but I haven't seen the combinations "narrow neutral" > and "wide neutral" before, and find it a bit counterintuitive. I'd > prefer "large minor" and "small major".Yes, wide minor and narrow major are certainly valid alternatives in this system for the case of thirds and sixths. But don't you agree that both 10:11 and 11:12 are neutral seconds and so one of them must be a narrow or a wide neutral? And similarly for sevenths. -- Dave Keenan
Message: 516 - Contents - Hide Contents Date: Wed, 18 Jul 2001 02:16:42 Subject: lattices of Schoenberg's rational implications From: monz> <Yahoo groups: /tuning-math/message/44 * [with cont.] > > From: "monz" <joemonz@y...> > Date: Sun May 27, 2001 5:55 pm > Subject: Re: Fwd: optimizing octaves in MIRACLE scale. > > (My quotes of Schoenberg are from the English translation > of _Harmonielehre_ by Roy Carter, and the page numbers > refer to that edition.) > > > Schoenberg [p 23] posits the existences of two "forces", one > pulling downward and one pulling upward around the tonic, > which he illustrates as: F <- C -> G and likens to resistance > against gravity. In mathematical terms, he is referring to > the harmonic relationships of 3^-1 and 3^1, respectively. >>> [Schoenberg, p 24:] >> >> ...thus it is explained how the scale that finally emerged >> is put together from the most important components of a >> fundamental tone and its nearest relatives. These nearest >> relatives are just what gives the fundamental tone stability; >> for it represents the point of balance between their opposing >> tendencies. This scale appears as the residue of the properties >> of the three factors, as a vertical projection, as addition: > >> Schoenberg then presents a diagram of the overtones and the > resulting scale, which I have adaptated, adding the partial-numbers > which relate all the overtones together as a single set: > > b-45 > g-36 > e-30 > d-27 > c-24 > a-20 > g-18 g-18 > f-16 > c-12 c-12 > f-8 > > > f c g a d e b > 8 12 18 20 27 30 45 >I will now lattice these pitches, using as nomenclature for the notes my ASCII 72-EDO notation; legend: - + ~cents alteration from 12-EDO b # 100 [i.e., 12-EDO] v ^ 50 < > 33&1/3 - + 16&2/3 no accidental 0 [12-EDO] Here is a "standard triangular" 5-limit lattice of this diatonic scale: 20--- 30--- 45 A- E- B- / \ / \ / \ / \ / \ / \ 8 ---12--- 18--- 27 F C G D Look familiar? It should. An ASCII representation of a Monzo lattice of this scale looks like this: D /27 B- / /45'-._ / / G / /18 E- / 30'-._ / / ' C / /12 A- / 20'-._ / ' F 8> >> [Schoenberg:] >>>> Adding up the overtones (omitting repetitions) we get the seven >> tones of our scale. Here they are not yet arranged consecutively. >> But even the scalar order can be obtained if we assume that the >> further overtones are also in effect. And that assumption is >> in fact not optional; we must assume the presence of the other >> overtones. The ear could also have defined the relative pitch >> of the tones discovered by comparing them with taut strings, >> which of course become longer or shorter as the tone is lowered >> or raised. But the more distant overtones were also a >> dependable guide. Adding these we get the following: > > >> Schoenberg then extends the diagram to include the > following overtones: > > fundamental partials > > F 2...12, 16 > C 2...11 > G 2...12 > > (Note, therefore, that he is not systematic in his employment > of the various partials.) > > > Again, I adapt the diagram by adding partial-numbers: > > d-108 > c-99 > b-90 > a-81 > g-72 > f-66 > f-64 > (f-63) > e-60 > d-54 d-54 > c-48 c-48 > b-45 > b-44 > (bb-42) > a-40 > g-36 g-36 g-36 > f-32 > e-30 > (eb-28) > d-27 > c-24 c-24 > a-20 > g-18 g-18 > f-16 > c-12 c-12 > f-8 > > > (eb) (bb) > c d e f g a b c d e f g a b c d > [44] [64] > (28) (42) [66] > 24 27 30 32 36 40 45 48 54 60 63 72 81 90 99 108 > > > (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). > > > The partial-numbers are also given for the resulting scale > at the bottom of the diagram, showing that 7th/F (= eb-28) > is weaker than 5th/C (= e-30), and 7th/C (= bb-42) is weaker > than 5th/G (= b-45). > > Also note that 11th/F (= b-44), 16th/F (= f-64) and 11th/C > (= f-66) are all weaker still, thus I have included them in > square brackets. These overtones are not even mentioned by > Schoenberg.Here is a triangular 5-limit lattice of this expanded scale, showing 7- and 11-limit ratios in () and [] which are near in pitch to the 5-limit ones as physically close to them on the diagram. * indicates notes which are not exact 8ve-equivalents even tho Schoenberg implied that they are. *81*----- ------90 40------60--(42)[44]45 20--(28)30------ A- E- B- / \ / \ / \ / \ / \ / \ / \ / \ / \ 63[64][66]-*99*------72-----108 32------48-------36------54 16------24-------18------27 8------12------- ------ F C G D And here is an ASCII-fied 11-limit Monzo lattice showing where all of these pitches actually fall according to my lattice formula, when prime-factored: C^ / \99 / \ / \ A F^ \ /81 / \[66] \ / / \ \ / / \ \ D Bb^ \ \ /27 54 108 \[44] \ B- \ / \ \/90-._ \ / \ /\45 ' G ---------------F< \ / \ /18 36 72 /63 \ E- \ / / \/60-._ \ / / /\30 ' C ---------------Bb< / \ /12 24 48 /(42) A- \ / / 40'-._ \ / / 20 ' F ---------------Eb< 8 16 32 [64] (28) Of course, this description of the scale is only valid where "C" is the "tonic". Given Schoenberg's ideas about "pantonality", it will be difficult if not impossible to determine what the "tonic" is. Perhaps an interval analysis of the composition can reveal something. Of course, the tonic will be dynamically shifting all the time. Also, this explanation was intended to describe the rational implications of the *diatonic* scale, allowing only a few of the chromatic pitches. For the full chromatic scale, Schoenberg's later 13-limit system presented in "Problems of Harmony" must be consulted. Could anyone out there do some periodicity-block calculations on this theory and say something about that? Altho I analyze this tuning exactly according to Schoenberg's analysis (i.e., 8ve-specific), he himself considered it to represent 8ve-invariant pitches. His unison-vectors are 33:32, 45:44 64:63, and 81:80. His unusual explanation of Eb< (28) and Bb< (42) also makes 16:15 some type of special interval, if not a unison-vector. -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: 518 - Contents - Hide Contents Date: Wed, 18 Jul 2001 10:51 +0 Subject: Re: Temperament catalog From: graham@m... In-Reply-To: <9j36rh+ipou@e...> Dave Keenan wrote:> No. See Scala's intnam.par: 15552:15625 (8.11 c) > or my 11 note chain-of-minor-thirds scale * [with cont.] (Wayb.) > "A kleisma is the difference between a just fifth (2:3) and an octave > reduced chain of 6 just minor thirds (5:6)" Thanks! > Presumably the septimal kleisma was so named because it is close in > size to the kleisma. > > The catalog is a brilliant effort Graham. Thanks. > > You haven't told us what sort of interval the generator is, for some > of them, and in some you've given a "generator" which is in fact the > period.The mapping's enough to define the temperament. I say the period where it isn't the octave as well, and that should all be correct now.> You could add to the Miracle entry, the fact that Secor proposed it as > a way of making sense of Partch's 43 note gamut (i.e. not out of a > musical vacuum.)Too much information for a brief list. <snipped> temperaments added.> Wilson's 11-limit mapping is: > (-1, 8, 14, 18), perfect fourth generator (about 497 c), octave > period, 41-EDO. > > Wilson's 13-limit mapping is: > (-1, 8, 14, -23, -20), perfect fourth generator (about 498 c), octave > period, 41 and 53-EDO.I can't see this one. Does his layout contradict his numbers?> Your own 13-limit mapping that gives a more compact diamond (68 notes > instead of 75) but with slightly higher errors is: > (1, -2, -8, -12, -15), minor second generator (about 104 c) or > equivalently a perfect fourth generator (about 496c), half-octave > period, 46 and 104-EDO.I don't want to add *everything* from my automatically generated lists, because it'd mean duplicating those lists! I'm not sure it is more compact. I think it should be 70 notes for the diamond, what with it having a half-octave period. Cassandra(?) is more efficient, but not unique. Graham
Message: 519 - Contents - Hide Contents Date: Wed, 18 Jul 2001 12:03 +0 Subject: Re: Temperament catalog From: graham@m... In-Reply-To: <memo.301982@c...> I wrote:> I don't want to add *everything* from my automatically generated lists, > because it'd mean duplicating those lists! I'm not sure it is more > compact. I think it should be 70 notes for the diamond, what with it > having a half-octave period. Cassandra(?) is more efficient, but not > unique.Oops! The "I'm not sure" comment is a leftover from before I worked it out. Graham
Message: 524 - Contents - Hide Contents Date: Wed, 18 Jul 2001 15:08:47 Subject: Re: Lamothe Web Translation Update From: monz> rom: Pierre Lamothe <plamothe@a...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, July 18, 2001 2:13 PM > Subject: [tuning-math] Re: Lamothe Web Translation Update > > > > Hi J Gill and Monz, > > > I planned to work on my website (on which I did'nt written since Aug 2000) > in few weeks (or months). I will consider the 35k limit for Google > translation if I don't find another way to produce an English version. I > think that fine shades of thought cannot be translated by any machine and > when I read a such translation I ask to me most often how someone could > understand what I mean with a so poor result. > > Besides, I am unsatisfied with the French version itself. Almost all the > texts are unachieved and/or have to be updated and simplified. I would like > to ask help for translation but I want to produce also more substantial > texts before. > > Yet besides, there is a delay before Google add new pages, so the > translation would be available. However I divided, as suggested, and added > the resulting pages on my site. It is temporary and it will probably > disapear when I will revise in few months. So the following index of these > pages will not maintain in future. > > <division temporaire * [with cont.] (Wayb.)> > > > Any suggestion for help in translation or simply revision of my translation > in future ? Bonjour Pierre,I have enough fluency in French to make a decent English translation. For an example, see my translation of Patrice Bailhache's "Music translated into Mathematics: Leonhard Euler" at <Euler and music, by Patrice Bailhache, transla... * [with cont.] (Wayb.)>. The original French version is here <Euler et la musique * [with cont.] (Wayb.)>. The problem I would have translating your webpages is that I have trouble following the mathematical terminology. If someone else can provide a rudimentary English translation from a machine translator, and I can work out any questions via email with you, I can revise it into an acceptable English version. -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.)
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