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Message: 10850 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 lattice basis, (c) 2004 Tonalsoft Inc. * > > > 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 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 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 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 Date: Wed, 21 Apr 2004 14:13:38 Subject: Re: notation of monzos From: Carl Lumma >> > > >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? -Carl
Message: 10855 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 Date: Wed, 21 Apr 2004 10:22:44 Subject: Re: notation of monzos From: Carl Lumma >However I don't use the word "monzo" Why not? -Carl
Message: 10857 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: > > > > Tonalsoft Encyclopaedia of Tuning - TM-reduced lattice basis, (c) 2004 Tonalsoft Inc. * > > 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) 2003 by Joe Monzo * 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 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? > > -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.
Message: 10859 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? > > -Carl XH can't do color. I don't know if it's "JI" enough for 1/1.
Message: 10860 Date: Wed, 21 Apr 2004 14:21:24 Subject: Re: notation of monzos From: Carl Lumma >> 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? >> >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 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/ * <*> 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 *
Message: 10862 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 lattice basis, (c) 2004 Tonalsoft Inc. * > > > > 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) 2003 by Joe Monzo * objectification? as a series of axes?
Message: 10863 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) 2003 by Joe Monzo * > > > > 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 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 * > > 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 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?
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