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Message: 10850 - Contents - Hide Contents Date: Wed, 21 Apr 2004 17:16:11 Subject: Re: TM-reduced bases for ETs From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> hi paul, > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: > >>> Tonalsoft Encyclopaedia of Tuning - TM-reduced... * [with cont.] (Wayb.) >>> Right off the bat: >> >> "A method for reducing the bases of a lattice." >> >> should read >> >> "A method for reducing the basis of a lattice." >> >> TM-reduction results in a *single* basis for the lattice, not >> multiple _bases_. > > >> thanks. i'd really like to add a little more "regular English" > to the opening part of that definition, describing exactly > what TM-reduction does ... before heading into Gene's > mathematical definition. can you or anyone else help? > > i'm thinking something like this: > > "A method for reducing the basis of a lattice to its > most compact representation, with all unison-vectors > as small as possible in prime-space."Well, you must mean the "a kernel lattice" rather than simply "a lattice", since it's a kernel lattice in general whose basis consists of commatic unison vectors. The TM-reduced basis of the ordinary 5- limit lattice, which is not normally a kernel lattice, is {2:1, 3:1, 5:1}, and these are normally not unison vectors. Otherwise, the above is roughly correct, as long as "prime space" means "the Tenney lattice with taxicab metric". If your usually- presented conception of "prime space" doesn't include an axis for 2 then the above will be misleading unless you clarify it. If the basis has more than two components, though, there is more than one possible interpretation of "as small as possible", so for a truly precise description one would have to peek into Gene's mathematical definition.

Message: 10851 - Contents - Hide Contents Date: Wed, 21 Apr 2004 21:01:36 Subject: Re: 270 equal as the universal temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Paul Erlich wrote: >>> 64:63 is famous for being hidden within chords . . . >> Yes, but it doesn't run to four digits, or not in each number. Exactly. > The > neutral third comma makes it to three -- 243:242.Where would you find this within a chord?

Message: 10852 - Contents - Hide Contents Date: Wed, 21 Apr 2004 17:18:45 Subject: Re: notation of monzos From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> hi paul and Gene, > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" > <gwsmith@s...> >> wrote: >>>>>> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >>>> hello all, >>>> >>>>>>>> did we ever reach a consensus on the use or >>>> non-use of angle-brakets and/or pipe symbols >>>> in expressing monzos? >>>>>> Not really. Angle-brakets seem to be well accepted, >>> but so far as "|" versus "]" goes, some people use one, >>> and some the other. >>>> The monzo would have a "[", not a "]". I'm using these, >> not pipe symbols or even vertical lines, in my paper. > >> it was my understanding that the monzo by itself uses [...> > and the val uses <...] , and that putting them together > one would use the pipe symbol instead of the two square > brackets thus: <...|...> . has this become established usage?That's the way my paper does it. However I don't use the word "monzo" (sorry joe), but I've suggested here in a post entitled "Who's Val?" that if they're called monzos, their counterparts should be called breeds.> also, what about the suggestion to use comma punctuation > after the exponents of 3, 11, 19, 31, etc.? is that > established at all?Some people liked that, but someone didn't -- was it Dave or George?

Message: 10853 - Contents - Hide Contents Date: Wed, 21 Apr 2004 21:10:31 Subject: Re: notation of monzos From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> hi paul, > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: >>>>>> However I don't use the word "monzo" >>> >>> Why not? >>> >>> -Carl >>>> I want the minimum of jargon, and the maximum sense >> of "I could have thought of this myself and I probably >> did at one point" in the reader. > >> i guess i'm just being selfish, but i am disappointed, > and wish you'd use "monzo" in your paper ...Don't worry . . . you (along with Carl Lumma and Joseph Pehrson) are credited in the paper with providing "crucial germinative interest". I don't refer to the general item enough to warrant giving it a name . . . i say "in vector form the syntonic comma is represented as [-4 4 -1>" in between an arithmetic and a geometrical explanation of this fact, and thereafter i simply give specific ones without referring to them by a general name (rather, i use them, as well as standard ratios, to represent commas, so i simply refer to them as commas).

Message: 10854 - Contents - Hide Contents Date: Wed, 21 Apr 2004 14:13:38 Subject: Re: notation of monzos From: Carl Lumma>>>>> >owever I don't use the word "monzo" >>>> >>>> Why not? >>>>>> I want the minimum of jargon, and the maximum sense >>> of "I could have thought of this myself and I probably >>> did at one point" in the reader. >>>> i guess i'm just being selfish, but i am disappointed, >> and wish you'd use "monzo" in your paper ... >>Don't worry . . . you (along with Carl Lumma and Joseph Pehrson) are >credited in the paper with providing "crucial germinative interest". > >I don't refer to the general item enough to warrant giving it a >name . . . i say "in vector form the syntonic comma is represented >as [-4 4 -1>" in between an arithmetic and a geometrical explanation >of this fact, and thereafter i simply give specific ones without >referring to them by a general name (rather, i use them, as well as >standard ratios, to represent commas, so i simply refer to them as >commas).I'm sure you're doing the right thing for your paper, Paul, as you did with The Forms of Tonality. Come to think of it, has Forms ever been submitted to something like Xenharmonikon or 1/1? -Carl

Message: 10855 - Contents - Hide Contents Date: Wed, 21 Apr 2004 17:21:46 Subject: Re: 270 equal as the universal temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Over on metatuning Graham had this to say: > > "It doesn't matter if anybody can tell the difference, because there's > no evidence that four digit frequency ratios have any audible meaning. > The 13-limit is borderline..." > > If "four digit frequency ratios" (which from context I take to mean > superparticular ones)I don't infer that meaning at all. Instead, I think Graham was talking about ratios that have meaning as harmonic simulteneities. The most complex interval I could tune by ear was 17:13, so I infer that in two-voice music, more complex frequency ratios have no audible meaning -- and this is exactly the sense in which Graham intended his statement above.

Message: 10856 - Contents - Hide Contents Date: Wed, 21 Apr 2004 10:22:44 Subject: Re: notation of monzos From: Carl Lumma

Message: 10857 - Contents - Hide Contents Date: Wed, 21 Apr 2004 21:13:15 Subject: Re: TM-reduced bases for ETs From: monz hi paul, --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> thanks. i'd really like to add a little more "regular English" >> to the opening part of that definition, describing exactly >> what TM-reduction does ... before heading into Gene's >> mathematical definition. can you or anyone else help? >> >> i'm thinking something like this: >> >> "A method for reducing the basis of a lattice to its >> most compact representation, with all unison-vectors >> as small as possible in prime-space." >> Well, you must mean the "a kernel lattice" rather than simply > "a lattice", since it's a kernel lattice in general whose > basis consists of commatic unison vectors. The TM-reduced basis > of the ordinary 5-limit lattice, which is not normally a > kernel lattice, is {2:1, 3:1, 5:1}, and these are normally > not unison vectors. > > Otherwise, the above is roughly correct, as long as > "prime space" means "the Tenney lattice with taxicab metric". > If your usually-presented conception of "prime space" doesn't > include an axis for 2 then the above will be misleading > unless you clarify it. > > If the basis has more than two components, though, there > is more than one possible interpretation of "as small as > possible", so for a truly precise description one would > have to peek into Gene's mathematical definition.ok, thanks. i need a defintiion of "kernel lattice" for the Encyclopaedia too, along with a good description of what makes a "kernel lattice" different from one that's not a kernel. here is my current definition of "prime-space": Definitions of tuning terms: prime-space, (c) ... * [with cont.] (Wayb.) please feel free to offer corrections, comments, etc. on that, and to provide me with a "kernel lattice" definition, and a good opening paragraph for the "TM-reduced basis" definition. i thank you in advance. my use of prime-space generally does not include 2 as a prime-factor, but it certainly can and sometimes must be included, depending on the precise nature of the tuning. -monz

Message: 10858 - Contents - Hide Contents Date: Wed, 21 Apr 2004 17:25:29 Subject: Re: notation of monzos From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> However I don't use the word "monzo" > > Why not? > > -CarlI want the minimum of jargon, and the maximum sense of "I could have thought of this myself and I probably did at one point" in the reader.

Message: 10859 - Contents - Hide Contents Date: Wed, 21 Apr 2004 21:16:01 Subject: Re: notation of monzos From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>>> However I don't use the word "monzo" >>>>> >>>>> Why not? >>>>>>>> I want the minimum of jargon, and the maximum sense >>>> of "I could have thought of this myself and I probably >>>> did at one point" in the reader. >>>>>> i guess i'm just being selfish, but i am disappointed, >>> and wish you'd use "monzo" in your paper ... >>>> Don't worry . . . you (along with Carl Lumma and Joseph Pehrson) are >> credited in the paper with providing "crucial germinative interest". >> >> I don't refer to the general item enough to warrant giving it a >> name . . . i say "in vector form the syntonic comma is represented >> as [-4 4 -1>" in between an arithmetic and a geometrical explanation >> of this fact, and thereafter i simply give specific ones without >> referring to them by a general name (rather, i use them, as well as >> standard ratios, to represent commas, so i simply refer to them as >> commas). >> I'm sure you're doing the right thing for your paper, Paul, as > you did with The Forms of Tonality. Come to think of it, has Forms > ever been submitted to something like Xenharmonikon or 1/1? > > -CarlXH can't do color. I don't know if it's "JI" enough for 1/1.

Message: 10860 - Contents - Hide Contents Date: Wed, 21 Apr 2004 14:21:24 Subject: Re: notation of monzos From: Carl Lumma>> >'m sure you're doing the right thing for your paper, Paul, as >> you did with The Forms of Tonality. Come to think of it, has Forms >> ever been submitted to something like Xenharmonikon or 1/1? >>>XH can't do color. I don't know if it's "JI" enough for 1/1.Drag. I suppose you could submit it to 1/1 and see, though I don't think they do color either. -Carl

Message: 10861 - Contents - Hide Contents Date: Wed, 21 Apr 2004 18:27:25 Subject: Re: 270 equal as the universal temperament From: Graham Breed Paul Erlich wrote:> I don't infer that meaning at all. Instead, I think Graham was > talking about ratios that have meaning as harmonic simulteneities. > The most complex interval I could tune by ear was 17:13, so I infer > that in two-voice music, more complex frequency ratios have no > audible meaning -- and this is exactly the sense in which Graham > intended his statement above.Yes, but Gene's interpretation is still an interesting one ;-) Graham ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10862 - Contents - Hide Contents Date: Wed, 21 Apr 2004 21:22:25 Subject: Re: TM-reduced bases for ETs From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> hi paul, > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: > >>>>> Tonalsoft Encyclopaedia of Tuning - TM-reduced... * [with cont.] (Wayb.) >>>> thanks. i'd really like to add a little more "regular English" >>> to the opening part of that definition, describing exactly >>> what TM-reduction does ... before heading into Gene's >>> mathematical definition. can you or anyone else help? >>> >>> i'm thinking something like this: >>> >>> "A method for reducing the basis of a lattice to its >>> most compact representation, with all unison-vectors >>> as small as possible in prime-space." >>>> Well, you must mean the "a kernel lattice" rather than simply >> "a lattice", since it's a kernel lattice in general whose >> basis consists of commatic unison vectors. The TM-reduced basis >> of the ordinary 5-limit lattice, which is not normally a >> kernel lattice, is {2:1, 3:1, 5:1}, and these are normally >> not unison vectors. >> >> Otherwise, the above is roughly correct, as long as >> "prime space" means "the Tenney lattice with taxicab metric". >> If your usually-presented conception of "prime space" doesn't >> include an axis for 2 then the above will be misleading >> unless you clarify it. >> >> If the basis has more than two components, though, there >> is more than one possible interpretation of "as small as >> possible", so for a truly precise description one would >> have to peek into Gene's mathematical definition. > >> ok, thanks. i need a defintiion of "kernel lattice" > for the Encyclopaedia too, along with a good description > of what makes a "kernel lattice" different from one > that's not a kernel.umm . . . it's not a lattice that's a kernel, it's the lattice *formed* by a kernel. The kernel consists of all the commatic unison vectors of a tuning -- in the case of an ET, it's all the occurences of "0" in the bingo-card. As you can see by examining any of my bingo- cards, the "0"s themselves form a lattice. This is true for the kernels of higher-dimensional temperaments as well.> here is my current definition of "prime-space": > Definitions of tuning terms: prime-space, (c) ... * [with cont.] (Wayb.)objectification? as a series of axes?

Message: 10863 - Contents - Hide Contents Date: Thu, 22 Apr 2004 20:22:12 Subject: definition of prime-space (was: TM-reduced bases for ETs) From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>>> here is my current definition of "prime-space": >>> Definitions of tuning terms: prime-space, (c) ... * [with cont.] (Wayb.) >>>> objectification? as a series of axes? >> My own preference would be for a more mathematically precise > definition, but I suppose normed real vector spaces is > nothing Monz wants to get into.you guys all know that i'm way out of my league here. i just want to get the best definitions i can get of all the terms i'm using. prime-space is perhaps the most important of all, so if any of you want to add to my definition of it, please do. -monz

Message: 10864 - Contents - Hide Contents Date: Thu, 22 Apr 2004 20:26:36 Subject: Re: notation of monzos From: monz hi George (and everyone else), --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >>>>>> also, what about the suggestion to use comma punctuation >>> after the exponents of 3, 11, 19, 31, etc.? is that >>> established at all? >>>> Some people liked that, but someone didn't -- was it >> Dave or George? >> I'm the one who suggested the idea of the comma > punctuation, because it remedies the difficulty in > reading simple ratios involving higher primes > (such as 23/16) -- something I perceived to be a > serious problem with Monz's notation. See: > Yahoo groups: /tuning-math/message/7435 * [with cont.] > > I think there was only one person who disliked the > idea (I'm pretty sure it wasn't Dave, because he's > a stickler for perceptual improvements in notation), > IIRC on the grounds that the commas are a convenience > rather than a necessity. I responded that commas are > widely accepted as serving a similar purpose as > place-markers in large decimal numbers. However, > not enough people expressed any further opinions one > way or another for my suggestion to be either > adopted or rejected. > > Now's the time for others to speak up so the issue > can be resolved.i did agree with George that comma punctuation is good, after the exponent of 3 and then after every third exponent after that. yes, let's please resolve it now so that i can put it into the Encyclopaedia and be done with it. -monz

Message: 10865 - Contents - Hide Contents Date: Thu, 22 Apr 2004 22:07:08 Subject: Re: 270 equal as the universal temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> Being just like JI (and ~2 cent errors don't bother me) doesn't >> strike me as a sufficient qualification for being a/the "universal >> temperament". In addition, multiple vals/breeds should be > supported,>> including some useful non-micro-temperaments. >> In that case I propose 196608-equal as the universal temperament. > However, I was not calling 270 a universal temperament, but an all- > purpose replacement for JI.Then who came up with the subject line?

Message: 10866 - Contents - Hide Contents Date: Thu, 22 Apr 2004 03:29:55 Subject: Re: notation of monzos From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> as Gene pointed out when he first named the "vector of > prime-factor exponents" after me, it's useful to have > one word to replace a whole phrase, if you're going to > be referring to it often.I actually named it after you initially in the descriptions of some of my Maple routines written for my own personal use, without any thought of making it public, but then I started thinking of them as monzos and didn't have another name for them. I *did* have a name for vals, but I suppose we could use a name for vals as explicitly written out bra vectors, if we want to become obsessional about it.

Message: 10867 - Contents - Hide Contents Date: Thu, 22 Apr 2004 03:34:16 Subject: Re: 270 equal as the universal temperament From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Being just like JI (and ~2 cent errors don't bother me) doesn't > strike me as a sufficient qualification for being a/the "universal > temperament". In addition, multiple vals/breeds should be supported, > including some useful non-micro-temperaments.In that case I propose 196608-equal as the universal temperament. However, I was not calling 270 a universal temperament, but an all- purpose replacement for JI.

Message: 10868 - Contents - Hide Contents Date: Thu, 22 Apr 2004 03:38:00 Subject: Re: TM-reduced bases for ETs From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> here is my current definition of "prime-space": >> Definitions of tuning terms: prime-space, (c) ... * [with cont.] (Wayb.) >> objectification? as a series of axes?My own preference would be for a more mathematically precise definition, but I suppose normed real vector spaces is nothing Monz wants to get into.

Message: 10869 - Contents - Hide Contents Date: Thu, 22 Apr 2004 15:35:40 Subject: Re: 270 equal as the universal temperament From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Over on metatuning Graham had this to say: > > "It doesn't matter if anybody can tell the difference, because there's > no evidence that four digit frequency ratios have any audible meaning. > The 13-limit is borderline..." > > If "four digit frequency ratios" (which from context I take to mean > superparticular ones) have no audible meaning, it seems like a nifty > idea to temper them out. In the 7-limit this gives ennealimmal, and in > the 11-limit hemiennealimmal. If we take Graham's view, which I think > has something to be said for it, we go up to the 13-limit but no > farther. In the 13-limit, there are twelve four digit superparticular > commas; the kernel of all of these taken together is 270-equal. This > sort of fact I've discussed before; it does seem there is some > justification for considering 270-et to be a sort of universal > replacement for just intonation. > > Incidentally, we have to stop here; if we try to add 729/728, the next > smallest comma, we find this is a step of 270-et.If you try to add 729/728, then I believe you'll get 224-ET (a schismic tuning, unlike 270), for a max. error <1.6c for 13 and 15- limit consonances -- still a microtemperament by Dave Keenan's standards. I think anyone would be hard pressed to distinguish 224 from 15-limit JI. (Dave and I have found that the notation for 224 is a bit simpler than 270.) If we loosened our 13-limit requirements a bit to allow approximately the same max error as Miracle in the 11 limit (~3.3c), then 130-ET would be our universal near-JI replacement. (BTW, I've found the semantics for a 130 notation to be quite elegant.) Gene, I imagine that you might want to enlighten us as to which superparticular commas taken together define 224 and 130, respectively. --George

Message: 10870 - Contents - Hide Contents Date: Thu, 22 Apr 2004 15:36:26 Subject: Re: notation of monzos From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >>>> it was my understanding that the monzo by itself uses [...> >> and the val uses <...] , and that putting them together >> one would use the pipe symbol instead of the two square >> brackets thus: <...|...> . has this become established usage? >> That's the way my paper does it. However I don't use the word "monzo" > (sorry joe), but I've suggested here in a post entitled "Who's Val?" > that if they're called monzos, their counterparts should be called > breeds. >>> also, what about the suggestion to use comma punctuation >> after the exponents of 3, 11, 19, 31, etc.? is that >> established at all? >> Some people liked that, but someone didn't -- was it Dave or George?I'm the one who suggested the idea of the comma punctuation, because it remedies the difficulty in reading simple ratios involving higher primes (such as 23/16) -- something I perceived to be a serious problem with Monz's notation. See: Yahoo groups: /tuning-math/message/7435 * [with cont.] I think there was only one person who disliked the idea (I'm pretty sure it wasn't Dave, because he's a stickler for perceptual improvements in notation), IIRC on the grounds that the commas are a convenience rather than a necessity. I responded that commas are widely accepted as serving a similar purpose as place-markers in large decimal numbers. However, not enough people expressed any further opinions one way or another for my suggestion to be either adopted or rejected. Now's the time for others to speak up so the issue can be resolved. --George ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 10871 - Contents - Hide Contents Date: Fri, 23 Apr 2004 23:25:12 Subject: More 270 From: Gene Ward Smith In case anyone is inspired to consider 270 in the light of a 13-limit notation system, this might help: the TM basis is 676/675, 1001/1000, 1716/1715, 3025/3024 and 4096/4095. I used this to compute the first twenty notes of a Fokker block, and used that to compute all the 13-limit intervals with Tenney height less than a million up to twenty 270-et steps, with the following result: 1 {625/624, 352/351, 351/350, 847/845, 385/384, 441/440, 540/539, 364/363, 729/728, 325/324} 2 {196/195, 225/224, 176/175, 896/891, 243/242, 169/168, 640/637} 3 {245/243, 126/125, 121/120, 975/968, 275/273, 144/143, 832/825, 572/567} 4 {99/98, 100/99, 105/104, 512/507, 91/90} 5 {78/77, 81/80, 875/864, 245/242, 507/500} 6 {65/64, 686/675, 64/63, 343/338, 66/65, 625/616} 7 {891/875, 648/637, 847/832, 55/54, 56/55, 637/625} 8 {49/48, 50/49, 729/715, 910/891, 875/858, 143/140, 864/847} 9 {864/845, 169/165, 351/343, 819/800, 968/945, 128/125, 45/44} 10 {352/343, 525/512, 77/75, 693/676, 416/405, 40/39} 11 {1001/972, 605/588, 250/243, 36/35, 147/143} 12 {375/364, 567/550, 624/605, 65/63, 33/32} 13 {121/117, 405/392, 91/88, 512/495, 507/490, 125/121, 336/325} 14 {28/27, 175/169, 648/625, 729/704, 660/637, 539/520, 363/350} 15 {26/25, 27/26, 80/77, 343/330} 16 {176/169, 729/700, 169/162, 126/121, 715/686, 1001/960, 704/675, 25/24} 17 {847/810, 392/375, 256/245, 735/704, 882/845, 117/112, 448/429} 18 {507/484, 245/234, 288/275, 22/21} 19 {150/143, 104/99, 360/343, 605/576, 1024/975, 21/20} 20 {625/594, 637/605, 455/432, 256/243, 825/784, 81/77, 616/585, 539/512, 875/832}

Message: 10872 - Contents - Hide Contents Date: Fri, 23 Apr 2004 00:34:55 Subject: Re: 270 equal as the universal temperament From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:> If you try to add 729/728, then I believe you'll get 224-ET (a > schismic tuning, unlike 270), for a max. error <1.6c for 13 and 15- > limit consonances -- still a microtemperament by Dave Keenan's > standards. I think anyone would be hard pressed to distinguish 224 > from 15-limit JI. (Dave and I have found that the notation for 224 > is a bit simpler than 270.)The list of 13-limit 4-digit superparticulars is consist with 270 and only 270, so we know without checking at least one such comma won't work with 224. In fact, there are two: 1001/1000, which we can readily dispense with, and 2401/2400, which is the most important comma on the list. Even so 224 is an interesting system.> If we loosened our 13-limit requirements a bit to allow approximately > the same max error as Miracle in the 11 limit (~3.3c), then 130-ET > would be our universal near-JI replacement. (BTW, I've found the > semantics for a 130 notation to be quite elegant.)It would be one possibility. Others which have possibilities are 111, 494 and the amazing 311.> Gene, I imagine that you might want to enlighten us as to which > superparticular commas taken together define 224 and 130, > respectively.13-limit superparticulars consistent with 224 are as listed here: 540/539, 625/624, 729/728, 1716/1715, 2080/2079, 3025/3024, 4096/4095, 4225/4224, 4375/4374, 6656/6655, 9801/9800, 10648/10647, 123201/123200 I've already pointed out 1001/1000 and 2401/2400 are missing. 13-limit superparticulars belonging to 130 are 243/242, 351/350, 364/363, 441/440, 540/539, 676/675, 729/728, 1001/1000, 1716/1715, 2080/2079, 2401/2400, 4096/4095, 4225/4224, 9801/9800 Four digit superparticulars missing from the list are 3025/3034, 4375/4374, 6656/6655, 10648/10647 and 123201/123200. In both cases, the comma list is consistent only with the corresponding equal temperament. However, the reduced list for the 11-limit in both cases corrrespond to a linear temperament, not an equal temperament. We have 130-et consistent <<12 34 20 30 26 -2 6 -49 -48 15|| [<2 4 7 7 9|, <0 -6 -17 -10 -15|] TOP generators 600.225 83.277 in terms of 130, 1/2 and 9/130; 9/130 splits the difference between 22/21 and 21/20 224-et consistent octoid <<24 32 40 24 -5 -4 -45 3 -55 -71|| [<8 13 19 23 28|, <0 -3 -4 -5 -3|] TOP generators of 150.034 16.238 In terms of 224, 1/8 and 3/224 I named the 224 temperament octoid some time back; the first turns up on a lot of my 7 and 11 limit lists, but I don't think it's been named.

Message: 10873 - Contents - Hide Contents Date: Fri, 23 Apr 2004 00:36:12 Subject: Re: 270 equal as the universal temperament From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Then who came up with the subject line?I came up with the subject line; you came up with your own interpretation of what you thought it should mean.

Message: 10874 - Contents - Hide Contents Date: Fri, 23 Apr 2004 08:15:39 Subject: 7-limit symmetric lattice Minkowski reductions From: Gene Ward Smith For some important 7-limit ets, I give below the TM basis, and then the Minkowski reduced basis using symmetric lattice distance instead of Tenney distance; to break any ties however I do use Tenney. The final number is a diameter measure; it is the square of the symmetric lattice distance from the origin of the most distant of the three commas making up the basis. It can be seen from this than some of these ets (eg 41, 68, 171) have a more compact and more nearly cubical shape than others (eg 72) which is an advantage. 12: [36/35, 50/49, 64/63]; [36/35, 50/49, 64/63]; 7 15: [28/27, 49/48, 126/125]; [49/48, 28/27, 126/125]; 7 19: [49/48, 81/80, 126/125]; [49/48, 126/125, 225/224]; 9 22: [50/49, 64/63, 245/243]; [50/49, 64/63, 875/864]; 10 27: [64/63, 126/125, 245/243]; [64/63, 126/125, 1728/1715]; 10 31: [81/80, 126/125, 1029/1024]; [126/125, 225/224, 1728/1715]; 10 41: [225/224, 245/243, 1029/1024]; [225/224, 875/864, 2401/2400]; 11 53: [225/224, 1728/1715, 3125/3087]; [225/224, 1728/1715, 3125/3087]; 19 68: [245/243, 2048/2025, 2401/2400]; [2401/2400, 6144/6125, 245/243]; 17 72: [225/224, 1029/1024, 4375/4374]; [225/224, 1029/1024, 4375/4374]; 35 99: [3136/3125, 2401/2400, 4375/4374]; [2401/2400, 6144/6125, 4375/4374]; 35 130: [2401/2400, 3136/3125, 19683/19600]; [2401/2400, 6144/6125, 19683/19600]; 57 140: [2401/2400, 5120/5103, 15625/15552]; [2401/2400, 15625/15552, 5120/5103]; 37 171: [2401/2400, 4375/4374, 32805/32768]; [2401/2400, 4375/4374, 65625/65536]; 38

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