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Message: 5351 - Contents - Hide Contents Date: Sat, 19 Oct 2002 01:40:13 Subject: for monzoni: bloated list of 5-limit linear temperaments From: wallyesterpaulrus 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: 5352 - Contents - Hide Contents Date: Sat, 19 Oct 2002 04:38:10 Subject: Re: Epimorphic From: Gene Ward Smith --- In tuning-math@y..., manuel.op.de.coul@e... wrote:> 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?Great! It seems to me it would be better to say "JI-epimorphic" or "RI-epimorphic", leaving open the possibility of also implementing "meantone-epimorphic" or "starling-epimorphic" some fine day.

Message: 5353 - Contents - Hide Contents Date: Sat, 19 Oct 2002 06:54:01 Subject: Re: mathematical model of torsion-block symmetry? From: Gene Ward Smith --- In tuning-math@y..., "Hans Straub" <straub@d...> wrote:>BTW, are you sure the quotient is Z x Z/2Z? It appears to me > it should be Z x Z x Z/12Z. What would be the basis, then?Here's one way to explain it: Define the homomorphic mappings, or "vals", h12 = [12,19,28], h4 = [4,6,9] and g = [1,2,3]. Then we may define a mapping from the postive rationals into themselves by H(q) = (16/15)^h12(q) (2025/2048)^h4(q) (625/648)^g(q) You may verify that q/H(q) is always some power of 81/80; so that any positive rational number q may be written as q = H(q) (81/80)^n If we mod out the kernel generated by 2048/2025 and 648/625, H(q) is sent to h12(q), so that it maps to Z. Since (648/625)/(2048/2025) = (81/80)^2, even exponoents n are sent to the identity, and odd exponents tothe torsion part; this part of the mapping is the Z/2Z part. In other words, the mapping is q --> [h(12,q), n mod 2] If you like, you can think of the even n as being played on the blue piano,and the odd n as being played on the red piano, with any tuning differencebetween pianos being up to you.

Message: 5354 - Contents - Hide Contents Date: Sat, 19 Oct 2002 01:47:30 Subject: Re: for monzoni: bloated list of 5-limit linear temperaments From: monz 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? -monz ----- Original Message ----- From: "wallyesterpaulrus" <wallyesterpaulrus@xxxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> 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: 5355 - Contents - Hide Contents Date: Sat, 19 Oct 2002 02:00:42 Subject: Re: for monzoni: bloated list of 5-limit linear temperaments From: monz 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. 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". interesting. looks to me like there's some kind of "bridge between incommensurable primes" going on here. -monz ----- Original Message ----- From: "monz" <monz@xxxxxxxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Saturday, October 19, 2002 1:47 AM Subject: Re: [tuning-math] for monzoni: bloated list of 5-limit linear temperaments> 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? > > -monz

Message: 5356 - Contents - Hide Contents Date: Sun, 20 Oct 2002 09:09:07 Subject: [tuning] Re: Everyone Concerned From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:>> You should remember that many self-respecting mathematicians would >> not call something a lattice unless it inherited a group structure >> from R^n. >> any examples of one that doesn't?The hexagonal tiling of chords in the 5-limit, for one.

Message: 5358 - Contents - Hide Contents Date: Sun, 20 Oct 2002 16:07:06 Subject: Re: MUSIC OF THE SPHERES From: Gene Ward Smith --- In tuning-math@y..., Bill Arnold <billarnoldfla@y...> wrote:> So: I am trying explain myself to both > groups. Although some in both groups want me to send > to only ONE group, I have yet to figure out why.Most people subscribed to this list are also subscribed to tuning, so there is no need to cross-post here simply to reach the people who read this list. Things posted here should usually have something mathematical about them.> In conclusion: if someone KNOWS of a message board which WELCOMES a > discussion of MUSIC OF THE SPHERES and the MATH and PHYSICS thereof, > let me know. I will take my question THERE.I thought someone had made a list for that very topic.

Message: 5359 - Contents - Hide Contents Date: Sun, 20 Oct 2002 12:31:16 Subject: Re: A common notation for JI and ETs From: monz> From: "David C Keenan" <d.keenan@xx.xxx.xx> > To: "George Secor" <gdsecor@xxxxx.xxx> > Cc: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, September 18, 2002 6:12 PM > Subject: [tuning-math] Re: A common notation for JI and ETs > > > At 06:19 PM 17/09/2002 -0700, George Secor wrote:>> From: George Secor (9/17/02, #4626) >> >> Neither 306 nor 318 are 7-limit consistent, so I don't see much point >> in doing these, other than they may have presented an interesting >> challenge. >> 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 an interesting coincidence! i just noticed this bit because Dave quoted it in his latest post. just yesterday, i "discovered" for myself that 306edo is a great approximation of Pythagorean tuning, and that one degree of it designates "Mercator's comma" (2^84 * 3^53), which i think makes it particularly useful to those who are really interested in exploring Pythagorean tuning. see my latest additions to: Yahoo groups: /monz/files/dict/pythag.htm * [with cont.] -monz "all roads lead to n^0"

Message: 5360 - Contents - Hide Contents Date: Sun, 20 Oct 2002 12:41:48 Subject: Re: MUSIC OF THE SPHERES From: monz hello Bill,> From: "Gene Ward Smith" <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, October 20, 2002 9:07 AM > Subject: [tuning-math] Re: MUSIC OF THE SPHERES > --- In tuning-math@y..., Bill Arnold <billarnoldfla@y...> wrote: >>> In conclusion: if someone KNOWS of a message board which WELCOMES a >> discussion of MUSIC OF THE SPHERES and the MATH and PHYSICS thereof, >> let me know. I will take my question THERE. >> I thought someone had made a list for that very topic. Yahoo groups: /celestial-tuning/ * [with cont.]i know you already subscribe, and i also know that your questions have gone unanswered by members of the celestial-tuning list, but if you're going to lurk on tuning and tuning-math, i encourage you to keep posting there on celestial-tunings as your work is entirely relevant to the subject matter of that group. (and this time *i'll* aplogize for the cross-post) -monz

Message: 5362 - Contents - Hide Contents Date: Sun, 20 Oct 2002 01:00:05 Subject: [tuning] Re: Everyone Concerned From: wallyesterpaulrus --- In tuning-math@y..., "monz" <monz@a...> wrote:> (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.)this is a canard. lattices are lattices, even for mathematicians. it's just that if you're outside the field of geometry or related disciplines, you find other meanings for the term "lattice". no self- respecting mathematician would deny that our lattices are in fact "lattices".

Message: 5363 - Contents - Hide Contents Date: Sun, 20 Oct 2002 01:05:34 Subject: Re: for monzoni: bloated list of 5-limit linear temperaments From: wallyesterpaulrus hi monz, if you could add a footnote in carl lumma's (orphaned) table (on Yahoo groups: /monz/files/dict/eqtemp.htm * [with cont.] again) to the pythagorean comma entry, referencing and linking to aristoxenean temperament, i'd love you forever! (do anyway ;) ) -paul --- In tuning-math@y..., "monz" <monz@a...> wrote:> > hi paul, >>> From: "wallyesterpaulrus" <wallyesterpaulrus@y...> >> To: <tuning-math@y...> >> 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: 5364 - Contents - Hide Contents Date: Sun, 20 Oct 2002 01:12:07 Subject: Re: CS implies EPIMORPHISM From: wallyesterpaulrus --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:> 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 forscientists -- even> if the same person may play often the two roles -- and then, forthe 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 thatmany years ago when> I worked in his firm. > > Just like that, I ask me here what is the analog progression indorian ? In the two exact> "dorian" translation, the first has no triad on the tonic, and theprogression in the second> case seems rather to be i - iii - v ... Is it the case ? > ...U > UXXXoooU > .XXXTooo > .UooooooU > .....U > > ...U > UooooooU > .oooTXXX > .UoooXXXU > .....Ui'm confused as to what you mean. rotating the progression so as to begin and end on ii -- ii-V-I-vi-ii -- should tell you what i'm talking about (i hope). rewriting in terms of dorian functions, it's i-IV-VII-v-i, a progression one can find many examples of in pop and rock music. what's your "scientific" assessment of this progression?> 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 ofindividual properties) how the elements> are interwoven or interlinked or emmeshed. By obeing simpleprinciples, I mean meet simple> structural (math) conditions or axioms. Epimorphism and convexityare such topological> conditions independant of microtonal metrics. For instance, one caneasily 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 couldbegin 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 shapeis always an hexagon, as the> figures above. In 2D, it's a segment. > > What is the polytope series, giving that shape in subsequentdimensions ? One can calculate> easily (without computer) the amount of faces and cells, and thedecomposition 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 acomputer for posting. all sounds very interesting . . .

Message: 5365 - Contents - Hide Contents Date: Sun, 20 Oct 2002 01:33:55 Subject: Re: for monzoni: bloated list of 5-limit linear temperaments From: wallyesterpaulrus --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> --- 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? >> >> -monz >> examine 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 . . .in fact, your "limma" example is just the blackwood temperament below . . .> the reason i posted this is that i wanted to see you fill out the > list on the eqtemp page . . .specifically, pelogic (135/128 -- 7, 9, 16, 23 -tET) and blackwood (256/243 -- 5, 10, 15, 25 -tET) are entirely missing from the (carl's) list, and the names are missing for negri (16875/16834 -- 9, 10, 19, 28, 29, 47, 48, 66, 67, 85, 86 -tET) and hemisixths (78732/78125 -- 8, 11, 19, 27, 46, 65, 84 -tET). a few of the more complex 5-limit temperaments, such as ennealimmal, might be good to show on some of the "zooms" if you wish . . .

Message: 5367 - Contents - Hide Contents Date: Sun, 20 Oct 2002 06:00:07 Subject: [tuning] Re: Everyone Concerned From: wallyesterpaulrus --- In tuning-math@y..., "Jon Szanto" <jonszanto@y...> wrote:> --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:>> no self-respecting mathematician would deny that our lattices >> are in fact "lattices". > > Really?yes, there was a lot of confusion on this point a while back, when someone thought the algebraic definition of lattices was the only mathematical one. they missed the geometric one, which is ours. same as in crystallography, too.

Message: 5368 - Contents - Hide Contents Date: Sun, 20 Oct 2002 06:07:22 Subject: [tuning] Re: Everyone Concerned From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@y..., "Jon Szanto" <jonszanto@y...> wrote:>> --- In tuning-math@y..., "wallyesterpaulrus" > <wallyesterpaulrus@y...> wrote:>>> no self-respecting mathematician would deny that our lattices >>> are in fact "lattices". >> >> Really? >> yes, there was a lot of confusion on this point a while back, when > someone thought the algebraic definition of lattices was the only > mathematical one. they missed the geometric one, which is ours. same > as in crystallography, too.You should remember that many self-respecting mathematicians would not call something a lattice unless it inherited a group structure from R^n.

Message: 5369 - Contents - Hide Contents Date: Sun, 20 Oct 2002 06:13:29 Subject: [tuning] Re: Everyone Concerned From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:>> --- In tuning-math@y..., "Jon Szanto" <jonszanto@y...> wrote:>>> --- In tuning-math@y..., "wallyesterpaulrus" >> >>>> no self-respecting mathematician would deny that our lattices >>>> are in fact "lattices". >>> >>> Really? >>>> yes, there was a lot of confusion on this point a while back, when >> someone thought the algebraic definition of lattices was the only >> mathematical one. they missed the geometric one, which is ours. same >> as in crystallography, too. >> You should remember that many self-respecting mathematicians would >not call something a lattice unless it inherited a group structure >from R^n.any examples of one that doesn't?

Message: 5371 - Contents - Hide Contents Date: Mon, 21 Oct 2002 12:44:22 Subject: Re: Epimorphic From: manuel.op.de.coul@xxxxxxxxxxx.xxx Gene wrote:>Great! It seems to me it would be better to say "JI-epimorphic" or >"RI-epimorphic", leaving open the possibility of also implementing >"meantone-epimorphic" or "starling-epimorphic" some fine day.It turns out the question was moot since Pierre showed that it's equivalent to CS. Anyway I don't need to throw the new code straight away if I use it to print out the characterising val. I'll call that epimorphic prime-degree mapping. Isn't "meantone-epimorphic" covered by Myhill's property? Manuel

Message: 5372 - Contents - Hide Contents Date: Mon, 21 Oct 2002 10:48:55 Subject: Re: Epimorphic From: Gene Ward Smith --- In tuning-math@y..., manuel.op.de.coul@e... wrote:> It turns out the question was moot since Pierre showed that it's > equivalent to CS.Not so far as I can see.

Message: 5373 - Contents - Hide Contents Date: Mon, 21 Oct 2002 19:40:45 Subject: Re: Digest Number 497 From: John Chalmers Gene asked:>Has anyone paid attention to scales which have a number of steps a >multiple of a MOS? They inherit structure from the MOS, and using a 2MOS >or a 3MOS seems like a good way to fill in those annoying gaps.I think most of Messiaien's "Modes of Limited Transposition" in 12-tet are multiple MOS's of 3, 4 and 6-tet. I don't have a list handy on this computer to check, unfortunately. IIRC, William Lyman Young (in his "Report to the Swedish Royal Academy of Music" etc.) proposed a decatonic scale in 24-tet which was two 5-tone MOS's of 12 (2322323223) and a 14-tone scale of 2 sections of the 7-tone diatonic sequence as 22122212212221 in 24-tet. He considered these as generated from cycles of half-fourths or half-fiths. I suspect that some of Wyschnegradski's scales might be multiple MOS's too, but I don't have a list either. --John

Message: 5374 - Contents - Hide Contents Date: Mon, 21 Oct 2002 10:52:34 Subject: A {2,5,7,11} linear temperament list From: Gene Ward Smith These all have Graham complexity less than 35 and badness computed using that and rms error less than 20. It is not guaranteed to be complete, but for a list like this I'm not worried about that. The theory here is exactly like the theory of 7-limit linear temperaments. [[3, 7, 0, 2], [0, 0, 1, 1]] [0, 3, 3, -2, -7, 7] generators [400.0000000, 3366.915067] rms 12.83021490 comp 3 bad 115.4719341 ets [3] [[1, 9, 10, 5], [0, 13, 14, 3]] [13, 14, 3, -40, 38, -4] generators [1200., -616.5022610] rms .7436768490 comp 14 bad 145.7606624 ets [2, 35, 37, 72, 109, 146, 183, 255] [[1, 0, -3, -7], [0, 2, 5, 9]] [2, 5, 9, 8, -14, 6] generators [1200., 1394.625830] rms 2.941218562 comp 9 bad 238.2387035 ets [6, 25, 31, 37, 43, 68, 105] [[1, 17, -6, 13], [0, 20, -12, 13]] [20, -12, 13, 78, 39, -84] generators [1200., -880.6991383] rms .2817232452 comp 32 bad 288.4846031 ets [15, 94, 109, 124, 139, 233, 342] [[1, 22, 4, 13], [0, 33, 2, 16]] [33, 2, 16, 38, 77, -88] generators [1200., -715.5638524] rms .2735926046 comp 33 bad 297.9423464 ets [52, 57, 109, 161, 270, 379, 649] [[1, 3, 2, 4], [0, 5, -6, 4]] [5, -6, 4, 32, 8, -28] generators [1200., -161.9526191] rms 2.548055120 comp 11 bad 308.3146695 ets [7, 8, 15, 22, 37, 52, 59] [[1, 8, 12, 4], [0, 21, 34, 2]] [21, 34, 2, -112, 68, 20] generators [1200., -324.4522543] rms .2735960964 comp 34 bad 316.2770874 ets [37, 196, 233, 270, 307, 503, 773] [[1, 6, 12, 3], [0, 8, 20, -1]] [8, 20, -1, -72, 30, 24] generators [1200., -551.5574843] rms .8192559146 comp 21 bad 361.2918583 ets [13, 37, 50, 87, 124, 161] [[1, 5, 5, 2], [0, 11, 9, -6]] [11, 9, -6, -48, 52, -10] generators [1200., -292.1250007] rms 1.354209854 comp 17 bad 391.3666478 ets [4, 33, 37, 41, 78, 115] [[2, 0, 1, -7], [0, 1, 1, 3]] [2, 2, 6, 10, -7, -1] generators [600.0000000, 2784.425579] rms 11.23571334 comp 6 bad 404.4856802 ets [6, 16, 22, 28] [[1, 0, 4, 7], [0, 2, -1, -3]] [2, -1, -3, -5, 14, -8] generators [1200., 1407.928958] rms 18.69483318 comp 5 bad 467.3708295 ets [5, 6, 11] [[1, 3, 4, 5], [0, 4, 7, 9]] [4, 7, 9, 1, -7, 5] generators [1200., -205.2910898] rms 5.985277878 comp 9 bad 484.8075081 ets [6, 29, 35, 41, 47] [[1, 0, 0, 3], [0, 5, 6, 1]] [5, 6, 1, -18, 15, 0] generators [1200., 560.6334332] rms 13.76729468 comp 6 bad 495.6226085 ets [2, 13, 15]

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