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Message: 11300 - Contents - Hide Contents Date: Tue, 06 Jul 2004 20:18:30 Subject: Beethoven's Appassionata comma From: Paul Erlich On page 509 of _The Harmonic Experience_, W. A. Mathieu provides a harmonic map of Beethoven's "Appassionata" sonata, which begins and ends in F minor. The first thing to notice is that Beethoven invokes enharmonic equivalence at two points in the piece. If the notation is kept "consistent" and enharmonic equivalence is not used, then the piece begins in F minor and ends in Abbbb minor! But clearly Beethoven wanted to return to the home key at the end. This makes it clear that Beethoven, unlike Mozart and most other composers since c.1480, was not assuming meantone temperament, but was instead assuming a closed, cyclic system of 12 pitches per octave. According to Mathieu, the music (if analysed in JI), through its exploratory harmonic path, moves up a Great Diesis (128:125) to land in the key of Ab major (which would actually be Gbbb major had enharmonic equivalence not been employed). Then it moves up a Diaschisma (2048:2025) to return to F minor (Abbbb minor without enharmonic equivalence). (The second half of the piece spends a lot of time in a harmonically static mode.) If these commas both vanish, the tuning system must be 12-equal or some other closed, cyclic 12- tone system. If Mathieu's analysis is correct and the Didymic comma (syntonic comma or 81:80, which vanishes in meantone) doesn't actually come into play in this piece, a JI rendition of the piece would end up (128:125)*(2048:2025) = 262144:253125 higher than it began. My new paper (for those who have been looking at the draft) calls the temperament where 262144:253125 vanishes "Subchrome". But I'm changing this name to "Passion". Correspondingly, I'd like to change the name of "Superchrome". The first half of the alphabet is off-limits, since I've already done that part of the paper. Any ideas? "Papaya"?
Message: 11301 - Contents - Hide Contents Date: Tue, 06 Jul 2004 20:26:45 Subject: Re: bimonzos, and naming tunings (was: Gene's mail server)) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: > > I'm a little late replying, because I've got a new computer now. I'm > taking a break from things like config files. >>>> It's really, really, really easy. Simply replace the bimonzos you >> list>>> with the corresponding bivals, and you are done. >>>> You've got to be kidding me. >> Why? You said you were not introducing any multilinar algebra, I > thought. Certainly you cannot do so and stay suitably nonmathematical.Cannot do so? Cannot do what?>>> You need explain >>> nothing, nor define anything. >>>> I'd like to do better by my readers. >> How does simply tossing a bimonzo in their face do better by them?It won't be tossed in their face, it'll be clearly illustrated.>> Show me how a bimonzo gets so bad in a higher prime limit. >> The bimonzo isn't even what you would use in a higher prime limit, > for starters. To get a linear temperament,Stop right there. You're changing the subject. We were only talking about bimonzos. And we weren't only talking about linear temperaments. You keep telling me this stuff I know, but you're not looking at the context of the paper.> (1) In any limit, the first pi(p)-1 entries of the bival give us the > period, and the generator part of the period-generator map. For any > limit above 5, the advantage goes on this point to bivals.You're setting up an unfair comparison, bivals vs. multimonzos. What if I set it up as bimonzos vs. multivals? That's still a middle path, just one that rides closer to the JI edge than to the ET edge.> All of the commas have two exponents of one sign and one of the > opposite sign in terms of the components of the bival. If we take the > complement of this, and so use the bimonzo signs, two of the commas > (the odd comma, {3,5,7}, and the 5-limit comma, {2,3,5}) have all the > exponents the same sign,Oh. I was only testing the 5-limit comma, {2,3,5), and I liked the fact that you didn't have to selectively mess with the exponents, the way you do with the bival.
Message: 11302 - Contents - Hide Contents Date: Tue, 06 Jul 2004 21:21:40 Subject: Re: Beethoven's Appassionata comma From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> If Mathieu's analysis is correct and the Didymic comma (syntonic > comma or 81:80, which vanishes in meantone) doesn't actually come > into play in this piece, a JI rendition of the piece would end up > (128:125)*(2048:2025) = 262144:253125 higher than it began. My new > paper (for those who have been looking at the draft) calls the > temperament where 262144:253125 vanishes "Subchrome". But I'm > changing this name to "Passion".Sounds like an overrated movie by Mel Gibson, or a perfume. Why not call it "appassionata"?> Correspondingly, I'd like to change the name of "Superchrome". The > first half of the alphabet is off-limits, since I've already done > that part of the paper. Any ideas? "Papaya"?The single most striking thing about the comma remains how close it is to 21/20; another factor to bear in mind is that it is a 12-equal comma, but also a 23 and 35 et comma, so you might think it is in a natural partnership with 36/35 for a 7-limit version ("Number 58".)
Message: 11303 - Contents - Hide Contents Date: Tue, 06 Jul 2004 21:30:21 Subject: Re: from linear to equal From: monz hi Paul (and Gene), --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:>> hi Gene and Paul, >> >> >> huygens or fokker are indeed the two most appropriate names >> for 31edo. >> >> but if your main criteria in naming is to honor someone >> who advocated 11-limit, >> It would have to be a particular mapping of the 11-limit > along a particular chain of fifths that would, for example, > correspond to a particular path in the 31-equal circle of > fifths. Huygens never went beyond 7-limit, and Fokker didn't > prescribe one method of generating (by fifths) 31-equal's > approximation of 11 over another.well, of course neither Ptolemy nor Partch advocated equal-temperament ... but if it's any help, these are three xenharmonic-bridges that i've posited for Ptolemy (in my book): monzo ratio ~cents 3 5 7 11 23 [-4, 0, 1,-1, 0 > 896 / 891 9.687960643 [-6, 1,-1, 0, 0 > 5120 / 5103 5.757802203 [ 2,-1, 2,-1, 0 > 441 / 440 3.930158439 in a sense, one could say that all the differences between Ptolemy's genera for notes of the same name (i.e., all the different _lichanoi_ etc.) are unison-vectors, since the notes all do carry the same name. but within Ptolemy's diatonic genera alone, the difference between two of the _parhypatai_ (the "even diatonic" and "tonic diatonic") is as large as 81/77 = ~88 cents. the entire range of _parhypatai_ and _tritai_ in Ptolemy's genera is covered by the difference in size between his "even diatonic" and his enharmonic, and is 3;5,7,11,23-monzo [3, 1, 0,-1,-1> ratio 270 / 253 = ~112.5864268 cents. the difference between Ptolemy's _lichanoi_ and _paranetai_ is even greater: the usual 9/8 = ~203.9100017 cents. the three xenharmonic-bridges in my table above are the ones that are smallest in size. for more on Ptolemy's genera, see: Tonalsoft Encyclopaedia of Tuning - diatonic, ... * [with cont.] (Wayb.) Tonalsoft Encyclopaedia of Tuning - chromatic,... * [with cont.] (Wayb.) Definitions of tuning terms: enharmonic, (c) 1... * [with cont.] (Wayb.) -monz
Message: 11304 - Contents - Hide Contents Date: Tue, 06 Jul 2004 21:41:04 Subject: Re: Beethoven's Appassionata comma From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> If Mathieu's analysis is correct and the Didymic comma (syntonic >> comma or 81:80, which vanishes in meantone) doesn't actually come >> into play in this piece, a JI rendition of the piece would end up >> (128:125)*(2048:2025) = 262144:253125 higher than it began. My new >> paper (for those who have been looking at the draft) calls the >> temperament where 262144:253125 vanishes "Subchrome". But I'm >> changing this name to "Passion". >> Sounds like an overrated movie by Mel Gibson, or a perfume. Why not > call it "appassionata"?I've already cut out, pasted, and all that stuff that Carl couldn't believe I was doing, for the first half of the alphabet. Moreover, I want short names.
Message: 11305 - Contents - Hide Contents Date: Tue, 06 Jul 2004 22:30:54 Subject: Ptolemy's genera (was: from linear to equal) From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> well, of course neither Ptolemy nor Partch advocated > equal-temperament ... but if it's any help, these are > three xenharmonic-bridges that i've posited for Ptolemy > (in my book): > > monzo ratio ~cents > 3 5 7 11 23 > > [-4, 0, 1,-1, 0 > 896 / 891 9.687960643 > [-6, 1,-1, 0, 0 > 5120 / 5103 5.757802203 > [ 2,-1, 2,-1, 0 > 441 / 440 3.930158439those interested in pursuing this further will find helpful a brand new webpage i just wrote. as i have time, i'll add much more to this one. Tonalsoft Encyclopaedia of Tuning - Ptolemy, (... * [with cont.] (Wayb.) -monz
Message: 11306 - Contents - Hide Contents Date: Tue, 06 Jul 2004 22:50:13 Subject: Re: Ptolemy's genera (was: from linear to equal) From: Paul Erlich Hi Monz, Why is it that you're always creating new webpages and ignoring corrections to your old ones? This seems to be a pattern with you. The latest correction you ignored: Yahoo! - * [with cont.] (Wayb.) -Paul --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >>> well, of course neither Ptolemy nor Partch advocated >> equal-temperament ... but if it's any help, these are >> three xenharmonic-bridges that i've posited for Ptolemy >> (in my book): >> >> monzo ratio ~cents >> 3 5 7 11 23 >> >> [-4, 0, 1,-1, 0 > 896 / 891 9.687960643 >> [-6, 1,-1, 0, 0 > 5120 / 5103 5.757802203 >> [ 2,-1, 2,-1, 0 > 441 / 440 3.930158439 > > >> those interested in pursuing this further will find > helpful a brand new webpage i just wrote. as i have > time, i'll add much more to this one. > > Tonalsoft Encyclopaedia of Tuning - Ptolemy, (... * [with cont.] (Wayb.) > > > > -monz
Message: 11307 - Contents - Hide Contents Date: Tue, 06 Jul 2004 03:01:12 Subject: Re: NOT tuning From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>>> except where TOP already had pure octaves, in >>>>>> which case it would actually change! >>>>>>>>>> That's impossible given the criterion of NOT. >>>>> >>>>> Maybe I don't comprehend you. >>>>>>>> I didn't say NOT, I said "Graham" and "pure-octaves TOP". >>>>>> Ok, it would seem to violate the definition of >>> "pure-octaves TOP". >>>> It doesn't. It still has pure octaves. >> Oh, I thought you were saying the octaves changed. So in > fact I have no clue what you were trying to say.Graham's "pure-octaves TOP" is just a uniform stretching or compression of normal TOP -- except in those cases where normal TOP's already got pure octaves, in which case the change is not a mere uniform stretching or compression.
Message: 11308 - Contents - Hide Contents Date: Tue, 06 Jul 2004 23:13:55 Subject: Ptolemy and leapday From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> well, of course neither Ptolemy nor Partch advocated > equal-temperament ... but if it's any help, these are > three xenharmonic-bridges that i've posited for Ptolemy > (in my book): > > monzo ratio ~cents > 3 5 7 11 23 > > [-4, 0, 1,-1, 0 > 896 / 891 9.687960643 > [-6, 1,-1, 0, 0 > 5120 / 5103 5.757802203 > [ 2,-1, 2,-1, 0 > 441 / 440 3.930158439If these three are related, so they define an 11-limit planar temperament. If you add 3388/3375 to this, you get Graham's mystery, with a 1/29 period; if you add 385/384, rodan; if 243/242 an 11-limit hemififths; and if 100/99, an 11-limit version of garibaldi (schismic family) which I have listed as "garybald". If we add 121/120 we get the 11-limit reduction of what Herman dubbed "leapday" in the 13 limit. The TOP tunings are not identical but they are close, and I suggest giving them the same name, and perhaps "ptolemy" could be that name. The 11 and 13 limit temperaments also have a common poptimal generator of 19/46, which again supports giving them the same name, and could be another naming idea along the lines of 19/84 I suppose. The corresponding fifth is 27/46; 2.39 cents sharp. I figured a Hellenistic Greek might like a fifth as a generator, if introduced to temperament; but more importantly, this (and "garybald"), is a "brigable" temperament since it has a 1 as the first wedgie element. The xenharmonic bridge to 5 is |31 -21 1>.
Message: 11309 - Contents - Hide Contents Date: Tue, 06 Jul 2004 23:30:16 Subject: Re: bimonzos, and naming tunings (was: Gene's mail server)) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> Why? You said you were not introducing any multilinar algebra, I >> thought. Certainly you cannot do so and stay suitably > nonmathematical. >> Cannot do so? Cannot do what?You can be not very mathematical, or you can introduce wedge products, but you can't do both.>> How does simply tossing a bimonzo in their face do better by them? >> It won't be tossed in their face, it'll be clearly illustrated.Is this why you want bimonzos--to show a wedge product of two commas pictorially?>>> Show me how a bimonzo gets so bad in a higher prime limit. >>>> The bimonzo isn't even what you would use in a higher prime limit, >> for starters. To get a linear temperament, >> Stop right there. You're changing the subject. We were only talking > about bimonzos.Your paper does not introduce bimonzos for 11-limit planar temperaments. The only time they get used, they are used for linear temperaments. Hence, this is what we are discussing. And we weren't only talking about linear> temperaments.I saw nothing about planar temperaments.> Oh. I was only testing the 5-limit comma, {2,3,5), and I liked the > fact that you didn't have to selectively mess with the exponents, the > way you do with the bival.Commas are exellent for 5-limit temperaments, or for codimension one in general. However, most of the time that isn't what we are looking at. In any case, the bival for a 5-limit linear temperament does, in fact, give the mapping; <1 4 4| gets to the mapping for meantone more directly than |-4 4 -1> does.
Message: 11310 - Contents - Hide Contents Date: Tue, 06 Jul 2004 03:14:26 Subject: Re: NOT tuning From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> Graham's "pure-octaves TOP" is just a uniform stretching or >> compression of normal TOP -- except in those cases where normal TOP's >> already got pure octaves, in which case the change is not a mere >> uniform stretching or compression. > > Aha! Got'cha. >> That *is* interesting, and a bit unsettling I suppose.Yet true nonetheless. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
Message: 11311 - Contents - Hide Contents Date: Tue, 06 Jul 2004 23:41:26 Subject: Re: Beethoven's Appassionata comma From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> I've already cut out, pasted, and all that stuff that Carl couldn't > believe I was doing, for the first half of the alphabet. Moreover, I > want short names."Murky" would place it next to "misty"
Message: 11312 - Contents - Hide Contents Date: Tue, 06 Jul 2004 23:11:04 Subject: Re: Beethoven's Appassionata comma From: Herman Miller Paul Erlich wrote:> Correspondingly, I'd like to change the name of "Superchrome". The > first half of the alphabet is off-limits, since I've already done > that part of the paper. Any ideas? "Papaya"?It's been called "diaschizoid" and "ragitonic".
Message: 11313 - Contents - Hide Contents Date: Wed, 07 Jul 2004 20:48:40 Subject: Re: Ptolemy and leapday From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> Did you see my question about this definition of yours? > > No. Yahoo groups: /tuning/message/54026 * [with cont.]
Message: 11314 - Contents - Hide Contents Date: Wed, 07 Jul 2004 20:50:24 Subject: Re: Beethoven's Appassionata comma From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> Paul Erlich wrote: >>> Correspondingly, I'd like to change the name of "Superchrome". The >> first half of the alphabet is off-limits, since I've already done >> that part of the paper. Any ideas? "Papaya"? >> It's been called "diaschizoid" and "ragitonic".The first name is in the first half of the alphabet, and the second doesn't seem appropriate to me (as I've mentioned). But thanks.
Message: 11315 - Contents - Hide Contents Date: Wed, 07 Jul 2004 20:56:38 Subject: Re: Beethoven's Appassionata comma From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> jjensen142000 wrote: >>> I *almost* got that book today from the music library... I had >> the call number on a slip of paper in my pocket and everything, >> but I was just too busy :( >> >> How am I ever going to make it through 500+ pages though? >> My copy covers Appassionata on p.349, and not in the detail Paul > describes. Could he have a different edition? Amazon doesn't mention it.This looks like the one I've seen two copies of, and was referring to: Amazon.com: Books: Harmonic Experience: Tonal ... * [with cont.] (Wayb.) 0174200?v=glance 563 pages. Yup.
Message: 11317 - Contents - Hide Contents Date: Wed, 07 Jul 2004 21:48:51 Subject: Re: Beethoven's Appassionata comma From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "jjensen142000" <jjensen14@h...> wrote:> I still haven't actually seen this book, but upon thinking about > this posting a little, it seems that the relevant issue is the > particular path of modulations: Fm --> Ab --> Fm, not whether > they occurred in the Appassionata sonata or somewhere else. > Therefore, maybe you would want names that describe this function? > In other words, name (or classify) the temperments by what > modulations they enable...?That would probably be the comma method of naming; base the name on sucessive commas for the 5, 7, 11 etc limits, where at each limit they form a basis for the kernel, or alternatively if they determine the temperament after reduction of the wedgie, and we choose the smallest Tenney height which works. Each method leads to a unique name for each linear temperament which makes the family relationships clear. I suppose along with the TM basis for a temperament I could give the one or both of the comma names sometimes: 7-limit pajara TM basis: {50/49, 64/63} 7-limit comma name, first method: (2048/2025, 64/63) 7-limit comma name, second method: (2048/2025, 50/49)
Message: 11318 - Contents - Hide Contents Date: Wed, 07 Jul 2004 22:38:10 Subject: Re: Beethoven's Appassionata comma From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "jjensen142000" <jjensen14@h...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:>> On page 509 of _The Harmonic Experience_, W. A. Mathieu provides a >> harmonic map of Beethoven's "Appassionata" sonata, which begins and >> ends in F minor. >> >> The first thing to notice is that Beethoven invokes enharmonic >> equivalence at two points in the piece. If the notation is >> kept "consistent" and enharmonic equivalence is not used, then the >> piece begins in F minor and ends in Abbbb minor! But clearly >> Beethoven wanted to return to the home key at the end. This makes > it>> clear that Beethoven, unlike Mozart and most other composers since >> c.1480, was not assuming meantone temperament, but was instead >> assuming a closed, cyclic system of 12 pitches per octave. >> >> According to Mathieu, the music (if analysed in JI), through its >> exploratory harmonic path, moves up a Great Diesis (128:125) to > land>> in the key of Ab major (which would actually be Gbbb major had >> enharmonic equivalence not been employed). Then it moves up a >> Diaschisma (2048:2025) to return to F minor (Abbbb minor without >> enharmonic equivalence). (The second half of the piece spends a lot >> of time in a harmonically static mode.) If these commas both > vanish,>> the tuning system must be 12-equal or some other closed, cyclic 12- >> tone system. >> >> If Mathieu's analysis is correct and the Didymic comma (syntonic >> comma or 81:80, which vanishes in meantone) doesn't actually come >> into play in this piece, a JI rendition of the piece would end up >> (128:125)*(2048:2025) = 262144:253125 higher than it began. My new >> paper (for those who have been looking at the draft) calls the >> temperament where 262144:253125 vanishes "Subchrome". But I'm >> changing this name to "Passion". >> >> Correspondingly, I'd like to change the name of "Superchrome". The >> first half of the alphabet is off-limits, since I've already done >> that part of the paper. Any ideas? "Papaya"? > > >> I still haven't actually seen this book, but upon thinking about > this posting a little, it seems that the relevant issue is the > particular path of modulations: Fm --> Ab --> Fm, not whether > they occurred in the Appassionata sonata or somewhere else. > Therefore, maybe you would want names that describe this function? > In other words, name (or classify) the temperments by what > modulations they enable...? > > just a thought... > JeffSeems a bit far-fetched. There are a whole heck of a lot of different paths one can follow to traverse a 262144:253125, and of course the starting point is completely irrelevant. Meanwhile, "Fm --> Ab --> Fm" alone doesn't tell you anything about temperament at all -- this could just as well be a JI chord progression.
Message: 11319 - Contents - Hide Contents Date: Wed, 07 Jul 2004 23:16:21 Subject: 3-d ("planar") temperaments request From: Paul Erlich Hi Gene, I'd like to include a table in my paper which summarizes a bunch of 7- limit, codimension-1 temperaments. As you'd guess, for each, I want a set of three generators, one of which generates 2:1 all by itself. And then the mappings from these generators to primes. The criteria for choosing the generators should be something I can explain, like, "the second generator is constrained to be narrower than the first generator, and given that constraint, is chosen so as to map to the simplest ratio possible. The third generator is constrained to be smaller than the second generator, and given that constraint, is chosen so as to map to the simplest ratio possible." This is the list of commas: 28 27 36 35 49 48 50 49 64 63 81 80 126 125 128 125 225 224 245 243 250 243 256 245 405 392 525 512 648 625 686 675 729 700 875 864 1029 1000 1029 1024 1323 1280 1728 1715 2048 2025 2240 2187 2401 2400 2430 2401 3125 3024 3125 3072 3125 3087 3136 3125 3645 3584 4000 3969 4375 4374 5103 5000 5120 5103 5625 5488 6144 6125 8748 8575 10976 10935 15625 15552 16875 16807 19683 19600 32805 32768 Thanks a bunch, Paul
Message: 11320 - Contents - Hide Contents Date: Wed, 07 Jul 2004 00:15:03 Subject: Re: bimonzos, and naming tunings (was: Gene's mail server)) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:>>> How does simply tossing a bimonzo in their face do better by them? >>>> It won't be tossed in their face, it'll be clearly illustrated. >> Is this why you want bimonzos--to show a wedge product of two commas > pictorially?Something like that. Did you not see the post I wrote to you yesterday in which I mentioned that the 12-equal case will be presented, similarly to what's in the gentle introduciton to periodicity blocks?>>>> Show me how a bimonzo gets so bad in a higher prime limit. >>>>>> The bimonzo isn't even what you would use in a higher prime > limit,>>> for starters. To get a linear temperament, >>>> Stop right there. You're changing the subject. We were only talking >> about bimonzos. >> Your paper does not introduce bimonzos for 11-limit planar > temperaments. The only time they get used, they are used for linear > temperaments.Once again, the initial exposition of a "bimonzo" (not what I call it) will be in the context of 12-equal, not a linear temperament.>> And we weren't only talking about linear >> temperaments. >> I saw nothing about planar temperaments.I'm going to give 7-limit "planar" temperament very brief coverage, but they'll be mentioned, certainly.>> Oh. I was only testing the 5-limit comma, {2,3,5), and I liked the >> fact that you didn't have to selectively mess with the exponents, > the>> way you do with the bival. >> Commas are exellent for 5-limit temperaments,I'm talking about 7-limit here.> or for codimension one > in general. However, most of the time that isn't what we are looking > at. In any case, the bival for a 5-limit linear temperament does, in > fact, give the mapping;I know that . . . :)
Message: 11321 - Contents - Hide Contents Date: Wed, 07 Jul 2004 00:17:29 Subject: Re: Ptolemy and leapday From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> The TOP tunings are not identical > but they are close, and I suggest giving them the same name, and > perhaps "ptolemy" could be that name.Note that Monz is only listing a small percentage of the "ptolemy commas" that he found. You guys are playing real fast and loose! (Don't take that as a complaint.)> is a "brigable" temperament sinceDid you see my question about this definition of yours? You didn't reply. (Deja vu, anyone?)
Message: 11322 - Contents - Hide Contents Date: Wed, 07 Jul 2004 00:18:04 Subject: Re: Beethoven's Appassionata comma From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> I've already cut out, pasted, and all that stuff that Carl couldn't >> believe I was doing, for the first half of the alphabet. Moreover, > I>> want short names. >> "Murky" would place it next to "misty" ?
Message: 11323 - Contents - Hide Contents Date: Wed, 07 Jul 2004 04:08:47 Subject: Re: Ptolemy and leapday From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: