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Message: 5354 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 * 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 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 * > 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: 5359 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 * -monz "all roads lead to n^0"
Message: 5360 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/ * 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: 5371 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: 5373 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
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