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Message: 5325 - Contents - Hide Contents Date: Wed, 16 Oct 2002 00:00:00 Subject: Re: mathematical model of torsion-block symmetry? From: wallyesterpaulrus hans, it looks like you're tripping up over this 2 thing, which may have confused graham in the past as well. 2 most certainly does make a difference, and watching it closely allows one to detect the (relatively few) torsional periodicity blocks from among the many non- torsional ones which are well-behaved. the example on monz's page has 24 notes but the group really only has 12 elements (per octave, anyway). if you temper out the unison vectors you get 12-equal, not 24-equal. hence it's "pathological" in terms of the old chalmers bit about justifying ETs in terms of fokker periodicity blocks with the same number of notes. --- In tuning-math@y..., "Hans Straub" <straub@d...> wrote:>> >> I've mentioned this before, but readers used to the "abeliangroup" terminology should keep in mind that abelian group and Z- module mean the same thing.>> >> I like the Z-module approach because it emphasizes the vector properties - > but if people are more used to abelian groups, I can use that, of course. >>>> Now, the quotient module being finite... >>>> Whups--you are sticking "2" into the mix when you conclude this. >> The math is more straightforward if you treat 2 as just another prime number. >> >> 2 is just another prime number, sure - but where exactly do you think I > confuse something? So far, I see no flaw in my reasoning... Itstill seems to> me that the "torsion" definition in the tuning dictionary describes a rather > trivial property that every finite periodicity block has (afterall, a "torsion> group", as decribed on mathworld.wolfram.com, is nothing but a finite group). > If this were so, I would suggest you remove this term from the dictionary. > > Regards, > > Hans Straub
Message: 5326 - Contents - Hide Contents Date: Wed, 16 Oct 2002 03:27:04 Subject: Re: mathematical model of torsion-block symmetry? From: Gene Ward Smith --- In tuning-math@y..., "Hans Straub" <straub@d...> wrote:>> >> I've mentioned this before, but readers used to the "abelian group" terminology should keep in mind that abelian group and Z-module mean the same thing. >> >> I like the Z-module approach because it emphasizes the vector properties - > but if people are more used to abelian groups, I can use that, of course.I suspect you like it because you are Swiss--Z-module is Continental, whereas an American mathematician will normally say abelian group.>>> Now, the quotient module being finite... >>>> Whups--you are sticking "2" into the mix when you conclude this. >> The math is more straightforward if you treat 2 as just another prime number. >> >> 2 is just another prime number, sure - but where exactly do you think I > confuse something?Monzo's example was the block defined by 2048/2025 and 648/625; if we mod out the free group on three generators {2,3,5} by the subgroup defined by the above, we produce a mapping onto Z x Z/2Z. This has a nontrivial torsion part, so the block is a torsion block. Using wedge products, which in the 5-limit we can identify with the cross-product, we have 2048/2025 ^ 648/625 = [11 -4 -2] ^ [3 4 -4] = [24 38 56] = 2 * [12 19 28], showing the 2-torsion. For this to work, the vectors need to be defined using the 2; Monzo unfortunately left this off and the page should be changed.
Message: 5327 - Contents - Hide Contents Date: Wed, 16 Oct 2002 04:58:40 Subject: Re: scales and periodicity blocks (from tuning-math2) From: Carl Lumma>> >A tuning system where each interval occurs always subtended >> by the same number of steps." >>I don't see any reference to JI in this definition, so I don't >think it means the same as epimorphic; certainly, however, any >epimorphic scale will have this property.While Kraig's statement is very clear about being the complete definition of CS (and thus epimorphic->CS but not CS->epimorphic), I believe Erv came up with CS for subsets of the JI lattice. If true, we're still finding new overlap between our steps and Erv's. -Carl
Message: 5329 - Contents - Hide Contents Date: Thu, 17 Oct 2002 10:56:52 Subject: Re: CS implies EPIMORPHISM From: Pierre Lamothe Gene wrote: << This raises a facinating possibility, but I can't see that it works. Taking the diatonic scale in 12-et as an example, the group generated by the notes of the scale is the 12-et; there is no morphism from here to 7-et. >> As I already said, 12-et lost the underlying 5-limit diatonic structure. The scale 0 2 4 5 7 9 11 (mod 12) is inconsistent in itself and worth only as a blurred image of the consistent underlying JI. It's precisely what reveals its non-epimorphic property ( 6 would belong to degree 3 and 4 ) or, in other words, its non-CS property ( 6 subtended by 3 or 4 steps ). On the other hand, the diatonic scale 0 9 17 22 31 39 48 (mod 53) is also an image of the Zarlino scale but consistent in itself, in other words, having the CS, or epimorphic, or congruity property (the concept I used in my theories). Look. In 53-et, class 1 = ( 5, 8, 9 ) == ( 16/15, 10/9, 9/8 ) and class 0 = ( 1, 3 ) == ( 81/80, 25/24 ). In comparaison, in 12-et it would be, class 1 = ( 1, 2 ) and class 0 = ( 1 ), what is inconsistent by definition. The vanishing of the comma 81/80, its splitting up in several little errors, is what is seeked normally by temperaments. In that case we have to understand that the result is necessarily a blurred image keeping not its underlying structure. When the stucture is keeped, as in 53-et, the same problems occurs like the comma drift. Pierre [This message contained attachments]
Message: 5330 - Contents - Hide Contents Date: Thu, 17 Oct 2002 00:21:12 Subject: also . . . From: wallyesterpaulrus Yahoo groups: /tuning/message/39703 * [with cont.]
Message: 5331 - Contents - Hide Contents Date: Thu, 17 Oct 2002 20:21:36 Subject: [tuning] Re: Everyone Concerned From: wallyesterpaulrus --- In tuning-math@y..., Bill Arnold <billarnoldfla@y...> wrote:> Also, if you play a C octave as "the most perfect harmonicstructure" as you say> "of just intonation," at least in that scale, would adding an Fnote in that octave,> as you play it, form for you, a consonant or dissonant chord?it would be consonant, but the root would now be F.> And what do you call the F > note relative to the C octave, as described? Would you call it a >harmonic chord?it could still be seen as a harmonic chord, with proportions 3:4:6 over a fundamental _F_ two octaves lower. it could also be seen as a subharmonic chord, with proportions (1/4):(1/3):(1/2) under a common overtone c' an octave above the higher c in the chord. since 4, 3, and 2 are simpler numbers than 3, 4, 6, the chord C-F-c has been considered as a "subharmonic" or "undertonal" trine, while C-G-c would the the "harmonic" or "overtonal" trine, with the proportions exactly reversed.
Message: 5332 - Contents - Hide Contents Date: Thu, 17 Oct 2002 02:37:58 Subject: CS implies EPIMORPHISM From: Pierre Lamothe Maybe I was not sufficiently clear the precedent times I wrote about that. Let S = (f0, f1, .., fn) be a scale, i.e. and ordered set of separate frequencies where fn is the octave of the tonic f0. Let (Tij) be the matrix of intervals Tij = fj / fi (modulo octave) whose content is the set T of all possible intervals x (of the first octave) within S or any derived scale S' (horizontal or vertical line in the matrix) with another tonic in S and/or possibly another direction (dual). The CS property means : if any interval x appears twice or more in the matrix, its positions belong to the same diagonal. That implies : the set of \ diagonals is a valid partition of the set T (the intersection of any two classes being empty). Consequently, let D : T --> diag be the surjective mapping of any interval Tij in its unique possible diagonal d, enumerated from 0 (the tonic diagonal) to n-1.The diagonal d represents the degree of the interval in these scales (or amount of steps from the tonic), each interval belonging to a distinct diagonal. The point : D is a congruence, i.e. an equivalence relation (since the partition is valid) which is also a morphism, since in a such ordered matrix D(xy) = D(x) + D(y) Now, since the name of a morphism as surjection is EPIMORPHISM, may I conclude, as many times before, that CS in a scale implies EPIMORPHISM Ok, it's not sufficient to insure it's a good scale : it's only epimorphic. However, it's epimorphic and CS means nothing else than it's epimorphic. (... and there is no need to restrict to rationals for that.) Finally, epimorphism don't imply periodicity block or convexity but only condition for that. For instance, the scale 1, 9/8, 5/4, 25/18, 40/27, 128/81, 16/9, 2 has the same steps (16/15, 10/9, 9/8) than the Zarlino scale and consequently the same epimorphism D (giving the diagonals) which, applied to coordinates (x,y,z) of its intervals, may be written D(x,y,z) = 7x + 11y + 16z (mod 7) so the same unison vectors, including 81/80 and 25/24, whose wedge product is [7,11,16], etc. But it's clearly not a convex periodicity block. The convex hull of that scale in the lattice <3 5>(y,z) has 5 holes. Pierre [This message contained attachments]
Message: 5333 - Contents - Hide Contents Date: Thu, 17 Oct 2002 20:50:39 Subject: Re: CS implies EPIMORPHISM From: wallyesterpaulrus --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:> Gene wrote: > > << This raises a facinating possibility, but I can't see that it >works. Taking the diatonic scale in 12-et as an example, the group >generated by the notes of the scale is the 12-et; there is no >morphism from here to 7-et. >> > > As I already said, 12-et lost the underlying 5-limit diatonic >structure. The scale 0 2 4 5 7 9 11 (mod 12) is inconsistent in >itself and worth only as a blurred image of the consistent >underlying JI.i can't think of a single point of view from which i would agree with this value judgment. rather, the underlying JI you give is defective, for example its ii chord is very out-of-tune.> It's precisely what reveals its non-epimorphic property ( 6 would >belong to degree 3 and 4 ) or, in other words, its non-CS property ( >6 subtended by 3 or 4 steps ).in 12-equal, you're right, it's not CS. but in 19-equal or 31-equal, it is.> Look. In 53-et, class 1 = ( 5, 8, 9 ) == ( 16/15, 10/9, 9/8 ) andclass 0 = ( 1, 3 ) == ( 81/80, 25/24 ).> > In comparaison, in 12-et it would be, class 1 = ( 1, 2 ) and class0 = ( 1 ), what is inconsistent by definition. "consistent" has got to be the most overloaded term on these forums . . . :)> > The vanishing of the comma 81/80, its splitting up in several >little errors, is what is seeked normally by temperaments. In that >case we have to understand that the result is necessarily a blurred >image keeping not its underlying structure. When the stucture is >keeped, as in 53-et, the same problems occurs like the comma drift.and the out-of-tune ii triad.
Message: 5334 - Contents - Hide Contents Date: Thu, 17 Oct 2002 08:17:39 Subject: Re: CS implies EPIMORPHISM From: Gene Ward Smith --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote: This raises a facinating possibility, but I can't see that it works. Taking the diatonic scale in 12-et as an example, the group generated by the notes of the scale is the 12-et; there is no morphism from here to 7-et.
Message: 5336 - Contents - Hide Contents Date: Fri, 18 Oct 2002 15:07:49 Subject: [tuning] Re: Everyone Concerned From: wallyesterpaulrus --- In tuning-math@y..., Bill Arnold <billarnoldfla@y...> wrote:>> it could still be seen as a harmonic chord, with proportions 3:4:6 >> over a fundamental _F_ two octaves lower. it could also be seen as a >> subharmonic chord, with proportions (1/4):(1/3):(1/2) under a common >> overtone c' an octave above the higher c in the chord. since 4, 3, >> and 2 are simpler numbers than 3, 4, 6, the chord C-F-c has been >> considered as a "subharmonic" or "undertonal" trine, while C-G-c >> would the the "harmonic" or "overtonal" trine, with the proportions >> exactly reversed. >> >>> Thank you for that response. > > Why is it not a C chord, with two C notes in C-F-c?i'm not sure what you're asking here.> Also, what if F # or Fb were substituted for the F note: > > in other words, C-F#-c? > > and: C-Fb-c?it would really depend on what tuning system you're using.> > Are the more harmonic chords? huh?
Message: 5337 - Contents - Hide Contents Date: Fri, 18 Oct 2002 17:40:22 Subject: Epimorphic From: manuel.op.de.coul@xxxxxxxxxxx.xxx Gene, I'm implementing the epimorphic property in Scala, but find the name a bit terse. Shall I call it prime-epimorphic or do you have a better name? Manuel
Message: 5338 - Contents - Hide Contents Date: Fri, 18 Oct 2002 08:41:04 Subject: Fw: Re: mathematical model of torsion-block symmetry? From: monz Yahoo has been bouncing my messages back to me lately. i'm trying this one again. -monz ----- Original Message ----- From: "monz" <monz@xxxxxxxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Friday, October 18, 2002 1:37 AM Subject: Re: [tuning-math] Re: mathematical model of torsion-block symmetry?> hi Gene, >>> From: "Gene Ward Smith" <genewardsmith@xxxx.xxx> >> To: <tuning-math@xxxxxxxxxxx.xxx> >> Sent: Tuesday, October 15, 2002 8:27 PM >> Subject: [tuning-math] Re: mathematical model of torsion-block symmetry? >> >> >> --- In tuning-math@y..., "Hans Straub" <straub@d...> wrote: >>>>>>> Now, the quotient module being finite... >>>>>>>> Whups--you are sticking "2" into the mix when you conclude this. >>>> The math is more straightforward if you treat 2 as just another prime > number. >>>> >>>>>> 2 is just another prime number, sure - but where exactly do you think I >>> confuse something? >>>> Monzo's example was the block defined by 2048/2025 and 648/625; >> if we mod out the free group on three generators {2,3,5} by the >> subgroup defined by the above, we produce a mapping onto Z x Z/2Z. >> This has a nontrivial torsion part, so the block is a torsion block. >> >> Using wedge products, which in the 5-limit we can identify with >> the cross-product, we have 2048/2025 ^ 648/625 = [11 -4 -2] ^ [3 4 -4] = >> [24 38 56] = 2 * [12 19 28], showing the 2-torsion. For this to work, >> the vectors need to be defined using the 2; Monzo unfortunately left >> this off and the page should be changed. > > >> hmmm ... somewhere on this list, about a month or two back, you > wrote a post explaining how to do the wedgie, and i had set up > a spreadsheet to do the calculation according to your formula. > the answer i just got for this one was: [0 0 24 56 -38 0]. > > > regarding the webpage: > Yahoo groups: /monz/files/dict/torsion.htm * [with cont.] > > ... not really knowing how to edit down what i already have > in the "torsion" definition, which is now quite confusing to me, > i simply added the above quote to the bottom of the "real" > definition (around the middle of the page). > > Gene, is there any way that you could edit this mess into > one good solid definition? perhaps with commentary after > it, but *useful* commentary? > > > > -monz > > > > > > > > >
Message: 5339 - Contents - Hide Contents Date: Fri, 18 Oct 2002 08:41:26 Subject: Re: help on diagrams for Gene's math (was : CS implies EPIMORPHISM) From: monz here's another one that never made it. -monz ----- Original Message ----- From: "monz" <monz@xxxxxxxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Friday, October 18, 2002 12:08 AM Subject: help on diagrams for Gene's math (was : CS implies EPIMORPHISM)> hi Gene (and the others as well), > > > i'd like to make some diagrams which help > explain your posts, which are in nearly > impetentrable mathematicalese to me. a good > start would be the discussion going on right > now about epimorphism vs. CS. > > how could i use a standard spreadsheet > (i use Microsoft Excel) to create graphs > which portray the concepts you describe with > equations or other math terminology? > > please help. > > > > -monz > > > > > > > > > > > >
Message: 5340 - Contents - Hide Contents Date: Fri, 18 Oct 2002 14:15:34 Subject: Re: CS implies EPIMORPHISM From: Pierre Lamothe I wrote: As I already said, 12-et lost the underlying 5-limit diatonic structure. The scale 0 2 4 5 7 9 11 (mod 12) is inconsistent in itself and worth only as a blurred image of the consistent underlying JI. Paul wrote: i can't think of a single point of view from which i would agree with this value judgment. rather, the underlying JI you give is defective, for example its ii chord is very out-of-tune. Probably the term worth leaved you to think it was a value judgment. I'm not musician and I leave to musicians the care to appreciate musical aspects. I wanted only to say that the diatonic scale in 12-et don't enclose (in thelist of its numbers) structural properties. You have to consider something else to reconstitute the structure. By underlying JI, I mean, here, what is enclosed, for instance, in the wedge product result (7,11,16). Beside, there exist also a macrotonal approach (using not the JI microtonalproperties) reconstituting the structure : the t-gammier ( 0 2 4 7 11). That structure is epimoph (CS)and naturally the interval 6, the tritone, don't exist within it. But once yet, the consistence is exterior to the isolated mode itself. The underlying JI refered is not the isolated Zarlino scale, so consonant ii chord (10/9 4/3 5/3) exists. What follows is not a judgment or a position but only a reference. It's represented in a portion of the Zarlino gammier (not the isolated Zarlino scale). If I well understood theexperience of Pierre-Yves Asselin ( Musique et tempérament ) the choice of intonation, a cappella, for i - vi - iv - ii - v - i was ooXo .oXXo oXXo .oXoo oXoo .XXoo XXoo .Xooo oooX .ooXX ooXo .oXXo without drift, inserting ( spontaneoulsly? ) the comma between ii and v. Paul wrote: "consistent" has got to be the most overloaded term on these forums . . . :) I imagine. I used it in macrotonal sense of structural consistence, qualifying so the imbrication of the elements rather than the individual (microtonal) properties. Is consistent an imbrication obeing to simple universal principles Pierre [This message contained attachments]
Message: 5341 - Contents - Hide Contents Date: Fri, 18 Oct 2002 18:42:27 Subject: Re: CS implies EPIMORPHISM From: wallyesterpaulrus --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:> If I well understood the experience of Pierre-Yves > Asselin ( Musique et tempérament ) the choice of intonation, acappella, for i - vi - iv - ii - v - i was> ooXo > .oXXo > > oXXo > .oXoo > > oXoo > .XXoo > > XXoo > .Xooo > > oooX > .ooXX > > ooXo > .oXXo > without drift, inserting ( spontaneoulsly? ) the comma between ii and v.i think a solution nearer to reality would use the vicentino's second tuning (adaptive just intonation), so that the simultaenous intervals are all just but the successive intervals are not. the comma will be distributed among the successive intervals. this way, instead of the disturbingly large full-comma shift in the intonation of the 2nd scale degree as in the solution you cite above, we have (ideally) four 1/4-comma shifts -- each just below the limen of melodic discriminability. what if the (rotated) progression occured in the dorian mode? would your source, or you, advocate shifting the *tonic* or *1/1* by a full comma in this way?> I used it in macrotonal sense of structural consistence, qualifyingso the imbrication of the> elements rather than the individual (microtonal) properties. Isconsistent an imbrication obeing to simple> universal principleswhat do the words "imbrication" and "obeing" mean?
Message: 5342 - Contents - Hide Contents Date: Fri, 18 Oct 2002 23:26:25 Subject: Re: mathematical model of torsion-block symmetry? From: wallyesterpaulrus --- In tuning-math@y..., "Hans Straub" <straub@d...> wrote:> > And another question concerning this: the periodicity blocks I have seen > displayed so far all seemed to be drawn without the 2 (one of the reasons for > my mistake above). Somehow you must use octave idenitifcation - or am I > missing something again? > > Regards, > Hans Straubit's true that you must use octave identification. most musicians think of pitch in "pitch-class" terms, which means "modulo" the octave. all of the BP periodicity blocks that have been displayed use 3, instead of 2, as the interval of equivalence. some musicians claim they can "hear" equivalence this way. as far as we know, though, octave-equivalence is universal among the world's musical cultures. projecting down to a 2-less subspace does tweak the various distance metrics, though. that is why i like to use a triangular lattice, instead of a rectangular one, when dealing with these 2-less subspaces. it represents the "average" or "effective" distance between pitch classes much better that way.
Message: 5344 - Contents - Hide Contents Date: Sat, 19 Oct 2002 16:16:26 Subject: Re: for monzoni: bloated list of 5-limit linear temperaments From: wallyesterpaulrus --- In tuning-math@y..., "monz" <monz@a...> wrote:> thanks, paul! i'll add it to my "linear temperaments" > definition when i get a chance. > > because of the tunings used in some of my favorites > of Herman Miller's _Pavane for a warped princess_, > there's a family of equal-temperaments which i've become > interested in lately, which all temper out the apotome, > {2,3}-vector [-11 7], ratio 2187:2048, ~114 cents: > 14-, 21-, and 28-edo. > > i noticed that these EDOs all have cardinalities which > are multiples of the exponent of 3 of the "vanishing comma". > > looking at the lattices on my "bingo-card-lattice" definition > Yahoo groups: /monz/files/dict/bingo.htm * [with cont.] > i can see it works the same way for 10-, 15-, and 20-edo, > which all temper out the _limma_, {2,3}-vector [8 -5] = ~90 cents. > > > so apparently, at least in these few cases (but my guess > is that it happens in many more), there is some relationship > between the logarithmic division of 2 which creates the > EDO and the exponent of 3 of a comma that's tempered out. > > has anyone noted this before? any further comments on it? > is it possible that for these two "commas" it's just > a coincidence? > > -monzexamine the table below -- you'll note that certain commas vanishing force the generator to be a fraction of an octave (600 cents, 400 cents, 300 cents, 240 cents) instead of a full octave . . . the reason i posted this is that i wanted to see you fill out the list on the eqtemp page . . . also lots of e-mails and post on the tuning list awaiting your attention . . .> > > ----- Original Message ----- > From: "wallyesterpaulrus" <wallyesterpaulrus@y...> > To: <tuning-math@y...> > Sent: Friday, October 18, 2002 6:40 PM > Subject: [tuning-math] for monzoni: bloated list of 5-limit linear > temperaments > > >> monzieurs, >>>> someone let me know if anything is wrong or missing . . . >> >> 25/24 ("neutral thirds"?) >> generators [1200., 350.9775007] >> ets 3 4 7 10 11 13 17 >> >> 81/80 (3)^4/(2)^4/(5) meantone >> generators [1200., 696.164845] >> ets 5 7 12 19 31 50 >> >> 128/125 (2)^7/(5)^3 augmented >> generators [400.0000000, 91.20185550] >> ets 3 9 12 15 27 39 66 >> >> 135/128 (3)^3*(5)/(2)^7 pelogic >> generators [1200., 677.137655] >> ets 7 9 16 23 >> >> 250/243 (2)*(5)^3/(3)^5 porcupine >> generators [1200., 162.9960265] >> ets 7 8 15 22 37 >> >> 256/243 (2)^8/(3)^5 quintal (blackwood?) >> generators [240.0000000, 84.66378778] >> ets 5 10 15 25 >> >> 648/625 (2)^3*(3)^4/(5)^4 diminished >> generators [300.0000000, 94.13435693] >> ets 4 8 12 16 20 28 32 40 52 64 >> >> 2048/2025 (2)^11/(3)^4/(5)^2 diaschismic >> generators [600.0000000, 105.4465315] >> ets 10 12 34 46 80 >> >> 3125/3072 (5)^5/(2)^10/(3) magic >> generators [1200., 379.9679493] >> ets 3 13 16 19 22 25 >> >> 15625/15552 (5)^6/(2)^6/(3)^5 kleismic >> generators [1200., 317.0796753] >> ets 4 11 15 19 34 53 87 >> >> 16875/16384 negri >> generators [1200., 126.2382718] >> ets 9 10 19 28 29 47 48 66 67 85 86 >> >> 20000/19683 (2)^5*(5)^4/(3)^9 quadrafifths >> generators [1200., 176.2822703] >> ets 7 13 20 27 34 41 48 61 75 95 >> >> 32805/32768 (3)^8*(5)/(2)^15 shismic >> generators [1200., 701.727514] >> ets 12 17 29 41 53 65 >> >> 78732/78125 (2)^2*(3)^9/(5)^7 hemisixths >> generators [1200., 442.9792975] >> ets 8 11 19 27 46 65 84 >> >> 393216/390625 (2)^17*(3)/(5)^8 wuerschmidt >> generators [1200., 387.8196733] >> ets 3 28 31 34 37 40 >> >> 531441/524288 (3)^12/(2)^19 pythagoric (NOT pythagorean)/aristoxenean? >> generators [100.0000000, 14.66378756] >> ets 12 48 60 72 84 96 >> >> 1600000/1594323 (2)^9*(5)^5/(3)^13 amt >> generators [1200., 339.5088256] >> ets 7 11 18 25 32 >> >> 2109375/2097152 (3)^3*(5)^7/(2)^21 orwell >> generators [1200., 271.5895996] >> ets 9 13 22 31 53 84 >> >> 4294967296/4271484375 (2)^32/(3)^7/(5)^9 septathirds >> generators [1200., 55.27549315] >> ets 22 43 65 87
Message: 5345 - Contents - Hide Contents Date: Sat, 19 Oct 2002 16:22:13 Subject: Re: for monzoni: bloated list of 5-limit linear temperaments From: wallyesterpaulrus --- In tuning-math@y..., "monz" <monz@a...> wrote:> oh, and of course, your list already shows that this > also happens with the "Pythagoric" temperaments, which > all temper out the Pythagorean comma, {2,3}-vector [-19 12], > and which all have cardinalities which are multiples of 12.i hope you'll update your eqtemp page -- it currently claims that 12- equal acts as a pythagorean tuning (with a link to 3-limit JI), but what you actually mean is "pythagoreic" or "aristoxenean" or whatever the vanishing of the pythagorean comma is called.> so it seems that any EDO which tempers out a 3-limit > "comma" has a cardinality (= logarithmic division of 2) > which is a multiple of the exponent of 3 in that "comma".it doesn't have to be 3 -- it can be any prime or composite (product and/or ratio) of primes. diesic, for example, tempers out 5^3, and so divides the octave into 3 equal parts. diminished tempers out (3/5) ^4, so 4 equal parts. blackwood tempers out 3^5, so 5 equal parts. the famous ennealimmal tempers out (3^3/5^2)^9, so 9 equal parts. once you get beyond the 5-limit, a linear temperament will have several vanishing commas, so things aren't as simple . . .
Message: 5346 - Contents - Hide Contents Date: Sat, 19 Oct 2002 11:06:27 Subject: Re: [tuning] Re: Everyone Concerned From: monz hi Bill,> From: "Bill Arnold" <billarnoldfla@xxxxx.xxx> > To: <tuning@xxxxxxxxxxx.xxx>; <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, October 19, 2002 7:41 AM > Subject: Re: [tuning-math] [tuning] Re: Everyone Concerned > > > I thank you for your suggestion that I read an > elementary book. However, I am not after an > elementary understanding. I look at Charles Lucy's > Charts, and they are not elementary nor do I find > any of this elementary, but complex. Not that that > scares me from investigating it.Jon is right. it's great that you're interested in more complex relationships, but you're asking questions from the Remedial Music Theory course. this is extremely basic stuff that you need to know in order to investigate the kinds of questions and speculations you have in mind. i suggest a splendid little book which, back in the day, taught me the things that really got me interested in music: Otto Karolyi. 1991. _Introducing Music_ Penguin, London ISBN 0-14-013520-0 (this must be a more recent revised edition)> I guess what I am after now: can you explain to me > if there IS such a thing as "Natural" scales? And > "Natural" chords? Or, are they, like words, the > creation of the mind? You know, as pointed out by > many authors such as Guy Murchie in Music of the Spheres, > that there are "Natural" shapes in Nature: six-sided > hexagonal gems like emeralds and eight-sided pyramidal > gems like diamonds. So: are their such "Natural" > shapes in scales and chords in Nature? Or, are they > creatings of minds?crystalline gems and tuning-theory lattice diagrams are expressions of the same mathematical concepts. see: Yahoo groups: /monz/lattices/lattices.htm * [with cont.] for my version of tonal-lattice theory. (please note that the lattice diagrams we use around here are usually somewhat different from the what mathematicians call lattices. i defer to others to explain if you need it.) seems to me that the question you're interested in is: is mathematics the creation of the human mind, or does it have some objective existence in the non-human world? -monz
Message: 5347 - Contents - Hide Contents Date: Sat, 19 Oct 2002 14:05:09 Subject: Re: for monzoni: bloated list of 5-limit linear temperaments From: monz hi paul,> From: "wallyesterpaulrus" <wallyesterpaulrus@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, October 19, 2002 9:22 AM > Subject: [tuning-math] Re: for monzoni: bloated list of 5-limit linear temperaments > > > i hope you'll update your eqtemp page -- it currently claims that 12- > equal acts as a pythagorean tuning (with a link to 3-limit JI), but > what you actually mean is "pythagoreic" or "aristoxenean" or whatever > the vanishing of the pythagorean comma is called. thanks.i decided to go with "aristoxenean" in honor of Aristoxenos. see the new Dictionary entry: Yahoo groups: /monz/files/dict/aristox.htm * [with cont.] -monz "all roads lead to n^0"
Message: 5348 - Contents - Hide Contents Date: Sat, 19 Oct 2002 17:45:03 Subject: Re: CS implies EPIMORPHISM From: Pierre Lamothe Paul wrote: i think a solution nearer to reality would use the vicentino's second tuning (adaptive just intonation), so that the simultaenous intervals are all just but the successive intervals are not. the comma will be distributed among the successive intervals. this way, instead of the disturbingly large full-comma shift in the intonation of the 2nd scale degree as in the solution you cite above, we have (ideally) four 1/4-comma shifts -- each just below the limen of melodic discriminability. I hoped your advice on the Asselin solution. I like such short and sweet answer. what if the (rotated) progression occured in the dorian mode? would your source, or you, advocate shifting the *tonic* or *1/1* by a full comma in this way I advocate nothing in the musical domain as such but perhaps a clear separation between what is a matter for musicians and what is a matter for scientists -- even if the same person may play often the two roles -- and then, for the scientific views and discourses, I would advocate, for sure, logic, coherence, rigourousness, etc. I don't believe M. Asselin had treated that question. I read that many years ago when I worked in his firm. Just like that, I ask me here what is the analog progression in dorian ? In the two exact "dorian" translation, the first has no triad on the tonic, and the progression in the second case seems rather to be i - iii - v ... Is it the case ? ...U UXXXoooU .XXXTooo .UooooooU .....U ...U UooooooU .oooTXXX .UoooXXXU .....U I wrote: I used it in macrotonal sense of structural consistence, qualifying so the imbrication of the elements rather than the individual (microtonal) properties. Is consistent an imbrication obeing to simple universal principles. Paul wrote: what do the words "imbrication" and "obeing" mean? Imbrication qualifie (macrotonally, i.e. independently of individual properties) how the elements are interwoven or interlinked or emmeshed. By obeing simple principles, I mean meet simple structural (math) conditions or axioms. Epimorphism and convexity are such topological conditions independant of microtonal metrics. For instance, one can easily enumerate all epimorphisms which are homotope in 3D for 5, 6, 7, 8... degrees, without considering harmonic possibilities. Pierre P.S. If I had'nt lost my computer and programs, some months ago, I could begin to talk about problems I resolved. For instance, the fundamental domain in 3D, (i.e. the convex hull of minimal unison vectors) varies with the microtonal metrics, but the shape is always an hexagon, as the figures above. In 2D, it's a segment. What is the polytope series, giving that shape in subsequent dimensions ? One can calculate easily (without computer) the amount of faces and cells, and the decomposition in cross polytopes. I found the corresponding name for 4D and 5D : cuboctahedron and prismatodecachoron. For the moment, I am in forced sabbatical. I have to borrow a computer for posting. [This message contained attachments]
Message: 5349 - Contents - Hide Contents Date: Sat, 19 Oct 2002 12:54:38 Subject: Re: A common notation for JI and ETs From: David C Keenan At 10:18 AM 4/10/2002 -0700, George Secor wrote:>I find 282 a little difficult, but still notatable. If we don't use >|\, then we can't have both matching symbols and ||\ as RC of /|. With >that constraint I would do 282 this way with rational complementation: > >282a: |( ~| ~)| |~ /| |) )|) (| (|( //| |~) (|~ /|\ (|) > ||( )||( ~|| ~||( )||~ )/|| ||) ||\ ~||) ~||\ //|| /||) >/||\ (RC) > >The )/|| symbol is the double-shaft version of the one that I am >proposing below for 306 and 494; here it is the proposed rational >complement of )|). > >But if we use |\ with matching symbols, then I get this: > >282b: |( ~| ~)| |~ /| |) )|) |\ (|( //| |~) (|~ /|\ (|) > ||( ~|| ~)|| ||~ /|| ||) )||) ||\ (||( //|| ||~) (||~ >/||\ (MS) > >But this shifts symbols such as ||) into the wrong positions and makes >them almost meaningless, besides not having ||\. So I prefer 282a.I agree with the single-shaft symbols of 282a but am unsure about that new )/|| symbol. I can't help feeling that we're drifting off into outer space here, and I have to admit that I'm losing interest in notating these big ones. Maybe |~) is silly too. Who cares about a 13:19 comma?>> Good point. Forget 318-ET, but 306-ET is of interest for being >strictly>> Pythagorean. The fifth is so close to 2:3 that even god can barely >tell the >> difference. ;-) >>What's making me hesitate about 306 is a 5 factor 49 percent of a >degree false. But I tried it anyway without looking at what you have >and came up with the following, which surprised me with how well it >works. It eliminates the shaky flag with a new symbol )/|, which I >will explain below when I discuss 494:OK. I'll wait 'til there to respond.>306: )| |( )|( ~|( /| )/| |) )|) |\ (|( |~) /|) (|~ /|\ >(|) > )|| (|\ )||( ~||( /|| )/| ||) )||) ||\ (||( ||~) /||) >(||~ /||\ (RC & MS) >>> If we can accept fuzzy arithmetic with the right wavy flag, and the >> addition of the 13:19 comma symbol |~) then the 31-limit-consistent >388-ET>> can be notated (but surprisingly, not 311-ET). >> >> 1 2 3 4 5 6 7 8 9 10 11 12 13 14 >> 388: )| |( ~| ~)| ~|( |~ /| ~|~ |) |\ (| ~|) ~|\ //| >> >> 15 16 17 18 19 20 21 22 >> |~) /|) /|\ (/| |\) (|) (|\ ||( ... (MS) >> >> The symbols (/| and |\) are of course the 31-comma symbols we agreed >on >> long ago. >>Yes, and they work quite well here, as well as in 494, below. Rational >complementation doesn't work very well when /| and |\ are 3 degrees >apart, so I will go along with the matching symbols, even if they don't >really mean much of anything; 388 is therefore agreed! > >I was wondering why you said that we can't do 311. Is it because (/| >is not the proper number of degrees for the 31 comma?That's the reason. But I would have put it this way: There is no symbol for 15deg311 because the only interpretation we have agreed for (/| and |\) are the relevant 31-commas which are respectively 14deg311 and 16deg311.>But neither is >|~ as 6deg388, the 23 comma, nor is )|~ as 8deg494 valid as the 19' >comma, but you have proposed these here. And I agree with your >decision, because there is no alternative.But I think there is at least one valid comma interpretation for each of these.>So I would do 311 thus: > >311: |( )|( ~)| ~|( |~ /| |) |\ (| (|( ~|\ //| /|) /|\ >(/| (|) (|\ > ~|| ~)|| ~|( )|~ /|| ||) ||\ ~||) (||( ~||\ ||~) >/||) /||\ (RC) > >I have selected the best single-shaft symbols and used their rational >complements. The symbols are not matched in the half-apotomes.OK. So I guess we are interpreting (/| as the comma resulting from combining the two flag commas. If that's so, that's fair enough.>> Here's another one I think should be on the list, 494-ET, if only >because>> of the fineness of the division, and because it shows all our >rational>> complements*. It is 17-limit consistent. Somewhat surprisingly, it is >fully>> notatable with the addition of the 13:19 comma symbol |~). It has the >same>> problem as 306 and 388, with right-wavy being fuzzy, taking on values >6, 7>> and 8 here.I now agree this is _too_ fuzzy, having _three_ different values.>> 1 2 3 4 5 6 7 8 9 10 11 12 13 14 >15>> 494: )| |( )|( ~| ~)| ~|( |~ )|~ /| ~|~ |) )|) |\ (| >~|) >>>> 16 17 18 19 20 21 22 23 24 25 26 27 28 >> (|( ~|\ //| |~) /|) (|~ /|\ (/| |\) (|) )|| (|\ )||( ... >> (RC* & MS) >> >> * It agrees with all our rational complements so far, except that >we'd need >> to accept >> ~|~ <---> )|) [where the |~ flag corresponds to 6 steps of >494] >> instead of >> ~|~ <---> /|( >> which might become an alternative complement. >> >> and we'd need to add >> >> )|( <---> |~) [where the |~ flag corresponds to 8 steps of >494] >>>> In all other symbols above, the |~ flag corresponds to 7 steps of >494. >>>> My interpretations are >> ~|~ 5:19 comma >> )|) 7:19 comma >> )|( 19 comma + 5:7 comma >> |~) 13:19 comma >> >> Obviously these symbols should be the last to be chosen for any >purpose. >>>> So we see that the addition of that one new symbol |~) for the 13:19 >comma>> and the acceptance of a fuzzy right wavy flag, lets the maximum >notatable>> ET leap from 217 to 494, more than double! >> >> So who cares about notating 282, 388 and 494? I dunno, but here's a >funny>> thing: The difference between them is 106. 176 is the next one down. >>And (surprise!) 600 is the next one up (but 7 and 17 are really bad). >All I can say about 106 is that it's twice 53.OK. That suggests that 229, 335 and 441 might be notatable.>I first found 494 in the 1970s when I was looking for a division with a >low-error 17 limit. I noticed that two excellent 7-limit divisions, 99 >and 171, have their 11 errors in opposite directions, so in their sum, >270, they cancel out (reckoned as fractions of a degree). For the 13 >limit both 224 and 270 are good, but their 17 errors are in opposite >directions, so in their sum, 494, they also cancel out. (Also note >their difference of 46, which is also quite good for the 17 limit.) >But I digress. Interesting. >I have a problem changing ~|~ to represent 10deg494 in that it must be >given a different complement to make this work. The proposed >complement, )||), has an offset of -2.64 cents, large enough that it >would be invalid in most other larger divisions.Yeah. Forget that.> This would also make >the complementation we previously had for ~|~ <--> /||( and /|( <--> >~||~ (offset of 0.49 cents) unavailable for other divisions such as 342 >and 388 (except as an alternate complement). Yeah. >Instead of ~|~ I propose )/| for 10deg494 (and 6deg306 above), which is >the correct number of degrees and has the actual flags for the 5:19 >comma (hence is easy to remember; besides, the symbol that I made for >this looks pretty good). Good. > This also makes a consistent complement to >)||) in 282, 306 and 494 (the three places where I have found a use for >it); the offset of -2.25 cents is still rather large, but not as much >as before.Oh dear. Large offset bad. But I accept.> (It makes a nice alternate complement with /|( with an >offset of 0.88 cents.)Don't really care.>It also restricts the fuzzy arithmetic to only >one symbol, |~), which has its two flags on the same side. This good. > This would >put the total number of single-shaft symbols at 30, and the only >symbols that would be left without rational complements are )|\ and >/|~. Don't care. >I don't object to the fuzzy |~) arithmetic for 19deg494, because this >makes it consistent with its proposed complement )||(, which has an >offset of only 0.09 cents (and would probably be valid a lot of other >places). The symbol does somewhat resemble |\), but I believe that the >two are sufficiently different in size that this shouldn't cause any >problem. Agree. >So I get: > >494: )| |( )|( ~| ~)| ~|( |~ )|~ /| )/| |) )|) |\ (| ~|) > (|( ~|\ //| |~) /|) (|~ /|\ (/| |\) (|) > )|| (|\ )||( ~|| ~)|| ~||( ||~ )||~ /|| )/|| ||) )||) ||\ > (|| ~||) (||( ~||\ //|| ||~) /||) (||~ /||\ (RC & MS) > >The only irregularities with this are the fuzzy symbol arithmetic with >|~) and ||~) and the fact that )|~ is not valid as the 19' comma. >Considering that 19 is not well represented in 494 and that the 19' >comma will be the much less used of the two 19 commas, I think that >this is inconsequential. >I tried messing around with some 3-flag symbols as alternatives to |~), >which would eliminate the remaining fuzzy symbol arithmetic. Since )/| >looked so good, I tried ~|\( for the 37 comma for 19deg494, which seems >pretty easy to distinguish from everything else. As a u-d complement >to )|( it has an offset of -2.60 cents, rather large, so it's not valid >in a lot of other places. I eventually decided that it wasn't worth >it, especially since the symbol would have 3 flags, so I would stick >with your proposal for |~). OK. >However, I am intrigued by the idea of )|)), the 19+7^2 diesis, as >being very close to half an apotome (and thus its own rational >complement); this would be very useful in a lot of places, e.g., 270, >311, and 400. We may have to explore this a bit more, or at least >leave open the possibility of future expansion, i.e., more flag >combinations. I figure that the more bells and whistles we have, the >less likely it is that anybody is ever going to use all of them.The only 3-flagger I can countenance at the moment is the Pythagorean comma symbol, probably only used in theoretical discussions. Is it ))|~ or ~~|( ? I suggest the first, to avoid any confusion caused by the 5:7 comma interpretation of |(. Hmm. You know a new (and very small) right flag type for the 5-schisma (32768:32805), (2^15:3^8*5), 1.95 cents, which I will symbolise for now as |`, would also give us a two-flag symbol for the Pythagorean comma, /|`, which is bit more theoretically meaningful. But I was thinking it would only be worth the trouble if it also gave us reasonable symbols for the diaschisma 2025:2048 and the 5-diesis 125:128. Unfortunately, what we need to get these, is a _negative_ 5-schisma flag, which I'll symbolise for now as |'. Then diaschisma is /|' and 5-diesis is //|'. Just a thort. But we should certainly leave open the possibility of 3-flaggers in future such as )|)). I think we should leave ETs above 217 out of the first article, except in so far as we may need to mention 494 in explaining why we made certain choices. Otherwise I'm afraid we'll scare people off.>> Here's another big one we can notate this way. Only 11-limit >consistent,>> but its relative accuracy at that limit is extremely good. 342 = >2*3*3*19. >> >> 342: >> )| |( )|( ~|( )|~ /| ~|~ |) |\ ~|) (|( //| |~) /|) /|\>(/| (|) (|\ > >Agreed! > >I spoke about 224 and 270 above, but we don't have a notation for them. > How about this: > >224: |( )|( ~|( /| |) |\ (|( //| /|) /|\ (|) (|\ > ~|| ~||( /|| ||) ||\ (||( ~||\ /||) /||\ (RC)I accept, but can't help wishing there was a better way to do 2 and 3 degrees.>270: |( ~| ~)| )|~ /| |) |\ (| (|( //| /|) /|\ (/| (|) >(|\ > ~|| ~||( )||~ /|| ||) ||\ (|| ~||\ //|| /||) /||\ (RC)Agreed. With this one it's 3 and 4 that are contentious. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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